Inverse kinematics

Inverse kinematics (IK) is the problem of computing motions (velocities, accelerations) that achieve a given set of tasks, such as putting a foot on a surface, moving the center of mass (CoM) to a target location, etc. These tasks can be defined by a set x=(x1,,xN)\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfx = (x_1, \ldots, x_N) of points or frames attached to the robot, with for instance x1\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} x_1 the CoM, x2\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} x_2 the center of the right foot sole, etc. The problems of forward and inverse kinematics are:

However, the mapping FK\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} FK is not always invertible due to kinematic redundancy: there may be infinitely many ways to achieve a given set of tasks. For example, if there are only two tasks x=(xlf,xrf)\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfx = (x_{lf}, x_{rf}) to keep left and right foot frames at a constant location on the ground, the humanoid can move its upper body while keeping its legs in the same configuration. In this post, we will see a common solution for inverse kinematics where redundancy is used achieve multiple tasks at once.

Kinematic task

Let us consider the task of bringing a point p\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfp, located on one of the robot’s links, to a goal position p\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfp^*, both point coordinates being expressed in the world frame. When the robot is in configuration q\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfq, the (position) residual of this task is:

r(q)=pp(q)\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfr(\bfq) = \bfp^* - \bfp(\bfq)

The goal of the task is to bring this residual to zero. Next, from forward kinematics we know how to compute the Jacobian matrix of p\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfp:

J(q)=pq(q),\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ(\bfq) = \frac{\partial \bfp}{\partial \bfq}(\bfq),

which maps joint velocities q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd to end-point velocities p˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfpd via J(q)q˙=p˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ(\bfq) \qd = \bfpd. Suppose that we apply a velocity q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd over a small duration δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \delta t. The new residual after δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \delta t is r=rp˙δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfr' = \bfr - \bfpd \delta t. Our goal is to cancel it, that is r=0p˙δt=r\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfr' = \boldsymbol{0} \Leftrightarrow \bfpd \delta t = \bfr, which leads us to define the velocity residual:

v(q,δt)=defr(q)δt=pp(q)δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfv(\bfq, \delta t) \defeq \frac{\bfr(\bfq)}{\delta t} = \frac{\bfp^* - \bfp(\bfq)}{\delta t}

The best option is then to select q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd such that:

J(q)q˙=p˙=v(q,δt)\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ(\bfq) \qd = \bfpd = \bfv(\bfq, \delta t)

If the Jacobian were invertible, we could take q˙=J1v\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd = \bfJ^{-1} \bfv. However, that’s usually not the case (think of a point task where J\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ has three rows and one column per DOF). The best solution that we can get in the least-square sense is the solution to:

minimizeq˙ Jq˙v2,\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \underset{\qd}{\textrm{minimize}}\ \| \bfJ \qd - \bfv \|^2,

and is given by the pseudo-inverse J\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ^\dagger of J\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ: q˙=Jv\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd = \bfJ^\dagger \bfv. By writing this equivalently as (JJ)q=Jv\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} (\bfJ^\top \bfJ) \bfq = \bfJ^\top \bfv, we see that this approach is exactly the Gauss-Newton algorithm. (There is a sign difference compared with the Gauss-Newton update rule, which comes from our use of the end-effector Jacobian p/q\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \partial \bfp / \partial \bfq rather than the residual Jacobian r/q\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \partial \bfr / \partial \bfq.)

Task gains

For this solution to work, the time step δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \delta t should be sufficiently small, so that the variations of the Jacobian term between q\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfq and q+q˙δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfq + \qd \delta t can be neglected. The total variation is

J(q+q˙δt)q˙J(q)q˙=δtq˙H(q)q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ(\bfq + \qd \delta t) \qd - \bfJ(\bfq) \qd = \delta t\,\qd^\top \bfH(\bfq) \qd

where H(q)\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfH(\bfq) is the Hessian matrix of the task. This matrix is more expensive to compute than J\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ. Rather than checking that the variation above is small enough, a common practice is to multiply the velocity residual by a proportional gain Kp[0,1]\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} K_p \in [0, 1]:

J(q)q˙=Kpv\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfJ(\bfq) \qd = K_p\, \bfv

For example, Kp=0.5\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} K_p = 0.5 means that the system will (try at best to) cut the residual by half at each time step δt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \delta t. Adding this gain does not change the exponential convergence to the solution r=0\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfr = \boldsymbol{0}, and helps avoid overshooting of the real solution. When you observe instabilities in your IK tracking, reducing task gains is usually a good idea.

Multiple tasks

So far, we have seen what happens for a single task, but redundant systems like humanoid robots need to achieve multiple tasks at once (moving feet from contact to contact, following with the center of mass, regulating the angular momentum, etc.) A common practice to combine tasks is weighted combination. We saw that a task i\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} i can be seen as minimizing Jiq˙Kivi2\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \| \bfJ_i \qd - K_i \bfv_i\|^2. Then, by associating a weight wi\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} w_i to each task, the IK problem becomes:

minimizeq˙ task iwiJiq˙Kivi2\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \underset{\qd}{\textrm{minimize}}\ \sum_{\textrm{task }i} w_i \| \bfJ_i \qd - K_i \bfv_i \|^2

The solution to this problem can again be computed by pseudo-inverse or using a quadratic programming (QP) solver. This formulation has some convergence properties, but its solutions are always a compromise between tasks, which can be a problem in practice. For example, when a humanoid tries to make three contacts while it is only possible to make at most two, the solution found by a weighted combination will be somewhere “in the middle” and achieve no contact at all:

Weighted solution to an unfeasible three-contact problem

In practice, we can set implicit priorities between tasks by having e.g. wi=104\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} w_i = 10^4 for the most important task, wi=102\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} w_i = 10^2 for the next one, etc. This is for instance how the pymanoid IK is used in this paper. This solution emulates the behavior of a hierarchical quadratic program (HQP) (unfortunately, there has been no open source HQP solver available as of 2016–2021). See this talk by Wieber (2015) for an deeper overview of the question.

Inequality constraints

Last but not least, we need to enforce a number of inequality constraints to avoid solutions that violate e.g. joint limits. The overall IK problem becomes:

minimize q˙task iwiJiq˙Kivi2subject to q˙q˙q˙+\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \begin{align*} \underset{\qd}{\textrm{minimize }} & \sum_{\textrm{task }i} w_i \| \bfJ_i \qd - K_i \bfv_i \|^2 \\ \textrm{subject to } & \qd^- \leq \qd \leq \qd^+ \end{align*}

Inequalities between vectors being taken componentwise. This time, the pseudo-inverse solution cannot be applied (as it doesn’t handle inequalities), but it is still possible to use a QP solver, such as CVXOPT or quadprog in Python. These solvers work on problems with a quadratic cost function and linear inequalities:

minimize x(1/2)xPx+rxsubject to Gxh\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \begin{align*} \underset{\bfx}{\textrm{minimize }} & (1/2)\,\bfx^\top \bfP \bfx + \bfr^\top \bfx \\ \textrm{subject to } & \bfG \bfx \leq \bfh \end{align*}

The weighted IK problem falls under this framework.

Cost function

Consider one of the squared norms in the task summation:

Jiq˙Kivi2=(Jiq˙Kivi)(Jiq˙Kivi)=q˙JiJiq˙KiviJiq˙Kiq˙Jivi+Ki2vivi=q˙(JiJi)q˙2(KiviJi)q˙+Ki2vivi\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \begin{align*} \| \bfJ_i \qd - K_i \bfv_i \|^2 & = (\bfJ_i \qd - K_i \bfv_i)^\top (\bfJ_i \qd - K_i \bfv_i) \\ & = \qd^\top \bfJ_i^\top \bfJ_i \qd - K_i \bfv_i^\top \bfJ_i \qd - K_i \qd^\top \bfJ_i^\top \bfv_i + K_i^2 \bfv_i^\top \bfv_i \\ & = \qd^\top (\bfJ_i^\top \bfJ_i) \qd - 2 (K_i \bfv_i^\top \bfJ_i) \qd + K_i^2 \bfv_i^\top \bfv_i \end{align*}

where we used the fact that, for any real number x\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} x, x=x\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} x^\top = x. As the term in vivi\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfv_i^\top \bfv_i does not depend on q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd, the minimum of the squared norm is the same as the minimum of (1/2)q˙Pq˙+rq˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} (1/2)\, \qd^\top \bfP \qd + \bfr^\top \qd with Pi=JiJi\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfP_i = \bfJ_i^\top \bfJ_i and ri=KiviJi\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfr_i = -K_i \bfv_i^\top \bfJ_i. The pair (P,r)\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} (\bfP, \bfr) for the weighted IK problem is finally P=iwiPi\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfP = \sum_i w_i \bfP_i and r=iwiri\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfr = \sum_i w_i \bfr_i.

Joint limits

Since we solve a problem in q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \qd, velocity limits can be directly written as Gq˙h\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfG \qd \leq \bfh where G\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfG stacks the matrices +E3\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} +\bfE_3 and E3\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} -\bfE_3, while h\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfh stacks the corresponding vectors +q˙+\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} +\qd^+ and q˙\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} -\qd^-. Next, we want to implement limits on joint angles qqq+\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfq^- \leq \bfq \leq \bfq^+ so that the robot stays within its mechanical range of motion. A solution for this is to add velocity bounds:

q˙=Klimqqδtq˙+=Klimq+qδt\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \begin{align*} \qd^- & = K_{\textrm{lim}} \frac{\q^- - \q}{\delta t} \\ \qd^+ & = K_{\textrm{lim}} \frac{\q^+ - \q}{\delta t} \\ \end{align*}

where Klim[0,1]\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} K_{\textrm{lim}} \in [0, 1] is a proportional gain. For example, a value of Klim=0.5\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} K_{\textrm{lim}} = 0.5 means that a joint angle update will not exceed half the gap separating its current value from its bounds (Kanoun, 2011).

To go further

The formulation that we have seen here is implemented in Python in Pink, an inverse kinematics library based on Pinocchio. Pink comes with a number of examples for manipulators, wheeled biped or humanoid robots. It also runs in real-time motion control loops on the Upkie wheeled biped.

Levenberg-Marquardt damping

One efficient technique to make the IK more numerically stable is to add Levenberg-Marquardt damping, an extension of Thikonov regularization where the damping matrix is chosen proportional to the task error. This strategy is implemented e.g. by the BodyTask of Pink. See (Sugihara, 2011) for a deeper dive.

Research software

Whole-body control at the acceleration level is currently more common than the velocity level presented here. Implementation details have been reported for instance for ladder-climbing with the HRP-2 humanoid which uses the Tasks C++ library, or for whole-body control of a torque-controlled humanoid.

History of the expression “inverse kinematics”

Thirty-five years ago (Richard, 1981), “inverse kinematics” was defined as the problem of finding joint angles q\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} \bfq fulfilling a set of tasks g(q)=0\def\LdG{\dot{L}_G} \def\Ld{\dot{L}} \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ{\boldsymbol{J}} \def\bfK{\boldsymbol{K}} \def\bfL{\boldsymbol{L}} \def\bfM{\boldsymbol{M}} \def\bfN{\boldsymbol{N}} \def\bfO{\boldsymbol{O}} \def\bfP{\boldsymbol{P}} \def\bfQ{\boldsymbol{Q}} \def\bfR{\boldsymbol{R}} \def\bfS{\boldsymbol{S}} \def\bfT{\boldsymbol{T}} \def\bfU{\boldsymbol{U}} \def\bfV{\boldsymbol{V}} \def\bfW{\boldsymbol{W}} \def\bfX{\boldsymbol{X}} \def\bfY{\boldsymbol{Y}} \def\bfZ{\boldsymbol{Z}} \def\bfalpha{\boldsymbol{\alpha}} \def\bfa{\boldsymbol{a}} \def\bfbeta{\boldsymbol{\beta}} \def\bfb{\boldsymbol{b}} \def\bfcd{\dot{\bfc}} \def\bfchi{\boldsymbol{\chi}} \def\bfc{\boldsymbol{c}} \def\bfd{\boldsymbol{d}} \def\bfe{\boldsymbol{e}} \def\bff{\boldsymbol{f}} \def\bfgamma{\boldsymbol{\gamma}} \def\bfg{\boldsymbol{g}} \def\bfh{\boldsymbol{h}} \def\bfi{\boldsymbol{i}} \def\bfj{\boldsymbol{j}} \def\bfk{\boldsymbol{k}} \def\bflambda{\boldsymbol{\lambda}} \def\bfl{\boldsymbol{l}} \def\bfm{\boldsymbol{m}} \def\bfn{\boldsymbol{n}} \def\bfomega{\boldsymbol{\omega}} \def\bfone{\boldsymbol{1}} \def\bfo{\boldsymbol{o}} \def\bfpdd{\ddot{\bfp}} \def\bfpd{\dot{\bfp}} \def\bfphi{\boldsymbol{\phi}} \def\bfp{\boldsymbol{p}} \def\bfq{\boldsymbol{q}} \def\bfr{\boldsymbol{r}} \def\bfsigma{\boldsymbol{\sigma}} \def\bfs{\boldsymbol{s}} \def\bftau{\boldsymbol{\tau}} \def\bft{\boldsymbol{t}} \def\bfu{\boldsymbol{u}} \def\bfv{\boldsymbol{v}} \def\bfw{\boldsymbol{w}} \def\bfxi{\boldsymbol{\xi}} \def\bfx{\boldsymbol{x}} \def\bfy{\boldsymbol{y}} \def\bfzero{\boldsymbol{0}} \def\bfz{\boldsymbol{z}} \def\calA{\mathcal{A}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \def\calD{\mathcal{D}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calG{\mathcal{G}} \def\calH{\mathcal{H}} \def\calI{\mathcal{I}} \def\calJ{\mathcal{J}} \def\calK{\mathcal{K}} \def\calL{\mathcal{L}} \def\calM{\mathcal{M}} \def\calN{\mathcal{N}} \def\calO{\mathcal{O}} \def\calP{\mathcal{P}} \def\calQ{\mathcal{Q}} \def\calR{\mathcal{R}} \def\calS{\mathcal{S}} \def\calT{\mathcal{T}} \def\calU{\mathcal{U}} \def\calV{\mathcal{V}} \def\calW{\mathcal{W}} \def\calX{\mathcal{X}} \def\calY{\mathcal{Y}} \def\calZ{\mathcal{Z}} \def\d#1{{\rm d}{#1}} \def\defeq{\stackrel{\mathrm{def}}{=}} \def\dim{\rm dim} \def\p{\boldsymbol{p}} \def\qdd{\ddot{\bfq}} \def\qd{\dot{\bfq}} \def\q{\boldsymbol{q}} \def\xdd{\ddot{x}} \def\xd{\dot{x}} \def\ydd{\ddot{y}} \def\yd{\dot{y}} \def\zdd{\ddot{z}} \def\zd{\dot{z}} g(\bfq) = 0, which is a fully geometric problem. It was solved by computing successive velocities bringing the robot to this desired configuration, which was called “differential inverse kinematics” (Nakamura, 1990) and corresponds to what we call “inverse kinematics” here. Considering that “kinematic” means motion, a number of recent papers choose to call “inverse geometry” (or “posture generation” for humanoids) the former problem, and reserve “kinematic” for computations including velocities or accelerations.

Discussion

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    sjauhri

    Posted on

    Thanks for this post! It's super useful.

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    David

    Posted on

    I'm having a hard time bridging the gap between the theory here and the inputs to the solver (for example, the solve_lp function you link to elsewhere on your site). Could someone help me understand what variables here lead to the inputs c, G, and h to the function? Thanks for your help.

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      Stéphane

      Posted on

      Inverse kinematics will yield a quadratic program (QP) rather than a linear program (LP), so we end up calling a solve_qp rather than solve_lp function. Rather than c, we thus get a matrix P and a vector q for the cost function. The derivation of P, q, G and h is detailed in the inequality constraints section above, which assumes some familiarity with kinematics, linear algebra and quadratic programming. Feel free to ask more specific questions on that part. You can also check out the code of Pink to see how these matrices are built in practice.

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