The capture point is a characteristic quantity of the linear inverted
pendulum model used in legged
robot locomotion. It was originally coined by Pratt et al. (2006) to address a question of push
recovery: where should the robot step right now to eliminate linear momentum
and come asymptotically to a stop?
Derivation
Let us start from the equation of motion of the linear inverted pendulum, where
all the mass is concentrated at the center of mass p G \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_G p G :
p ¨ G = ω 2 ( p G − p Z ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfpdd_G = \omega^2 (\bfp_G - \bfp_Z) p ¨ G = ω 2 ( p G − p Z ) We assume that the robot steps instantly at time t = 0 \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
t=0 t = 0 and maintains its
ZMP p Z \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z p Z at a constant
location in its new foothold, so that p Z \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z p Z is stationary. Since the
natural frequency ω \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\omega ω of the pendulum is also a model constant, we
can solve this second-order linear differential equation as:
p G ( t ) = p Z + e ω t 2 [ p G ( 0 ) + p ˙ G ( 0 ) ω − p Z ] + e − ω t 2 [ p G ( 0 ) − p ˙ G ( 0 ) ω − p Z ] \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_G(t) = \displaystyle \bfp_Z + \frac{e^{\omega t}}{2} \left[\bfp_G(0) +
\frac{\bfpd_G(0)}{\omega} - \bfp_Z\right] + \frac{e^{-\omega t}}{2}
\left[\bfp_G(0) - \frac{\bfpd_G(0)}{\omega} - \bfp_Z\right] p G ( t ) = p Z + 2 e ω t [ p G ( 0 ) + ω p ˙ G ( 0 ) − p Z ] + 2 e − ω t [ p G ( 0 ) − ω p ˙ G ( 0 ) − p Z ] This function is the sum of a stationary term p Z \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z p Z , a convergent term
factored by e − ω t \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
e^{-\omega t} e − ω t that vanishes as t → ∞ \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
t \to \infty t → ∞ , and a
term factored by e ω t \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
e^{\omega t} e ω t that diverges as t → ∞ \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
t \to \infty t → ∞ .
Let us define the capture point as:
p C = d e f p G + p ˙ G ω \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_C \defeq \bfp_G + \frac{\bfpd_G}{\omega} p C = def p G + ω p ˙ G The divergent term in p G ( t ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_G(t) p G ( t ) is then e ω t / 2 ( p C ( 0 ) − p Z ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
e^{\omega t}/2
(\bfp_C(0) - \bfp_Z) e ω t /2 ( p C ( 0 ) − p Z ) . In particular, the only way for the center of mass
trajectory to be bounded is for the stationary ZMP to be equal to the
instantaneous capture point:
p Z = p C ( 0 ) ⟹ p G ( t ) ⟶ t → ∞ p C ( 0 ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z = \bfp_C(0) \ \Longrightarrow \ \bfp_G(t) \underset{t \to
\infty}{\longrightarrow} \bfp_C(0) p Z = p C ( 0 ) ⟹ p G ( t ) t → ∞ ⟶ p C ( 0 ) We can thus interpret the capture point as a point where the robot should step
(shift its ZMP) in order to come (asymptotically) to a stop.
Connection to balance
The capture point is a divergent component of motion of the linear inverted pendulum. Shifting the
ZMP to the capture point prevents divergence from the unstable dynamics of the
model, but does not control the other (stable) component. In effect, the system
comes to a stop following its natural dynamics:
p ˙ G = ω ( p C − p G ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfpd_G = \omega (\bfp_C - \bfp_G) p ˙ G = ω ( p C − p G ) This phenomenon is noticable in balance controllers based on
capture point feedback such as Englsberger et al. (2011) and Morisawa et al. (2012) . Take the robot standing,
push it in a given direction and sustain your push, then suddenly release it:
the robot will come back to its reference standing position following its
natural dynamics (which only depend on ω \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\omega ω , i.e. gravity and the
height of the center of mass), regardless of the values of the various feedback
gains used in the balance controller. You can for instance test this behavior
in dynamic simulations with the lipm_walking_controller .
This behavior highlights how balance controllers based on capture-point
feedback are not trying to come to a stop as fast as possible. Rather, they
focus on preventing divergence, and leverage passive dynamics to absorb
undesired linear momentum. When using linear feedback, Sugihara (2009) showed that this approach
maximizes the basin of attraction of the resulting controller.
Boundedness condition
The derivation above can be generalized to the case where p Z ( t ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z(t) p Z ( t ) is
time-varying rather than time-invariant. Consider the equation of motion split
as follows into divergent and convergent components:
p ˙ C = ω ( p C − p Z ) p ˙ G = ω ( p C − p G ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\begin{align*}
\bfpd_C & = \omega (\bfp_C - \bfp_Z) \\
\bfpd_G & = \omega (\bfp_C - \bfp_G)
\end{align*} p ˙ C p ˙ G = ω ( p C − p Z ) = ω ( p C − p G ) The capture point diverges away from the ZMP while the center of mass is
attracted to the capture point:
As the center-of-mass dynamics are convergent, the system diverges if and only
if its capture point diverges. We can therefore focus on the
capture point dynamics alone.
The solution to a first-order linear time-varying differential equation is:
y ˙ ( t ) − a ( t ) y ( t ) = b ( t ) ⟹ y ( t ) = e A ( t ) ( y ( 0 ) + ∫ τ = 0 t b ( τ ) e − A ( τ ) d τ ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\dot{\bfy}(t) - a(t) \bfy(t) = \bfb(t) \ \Longrightarrow
\ \bfy(t) = e^{A(t)} \left(\bfy(0) + \int_{\tau=0}^t \bfb(\tau) e^{-A(\tau)}
{\rm d} \tau \right) y ˙ ( t ) − a ( t ) y ( t ) = b ( t ) ⟹ y ( t ) = e A ( t ) ( y ( 0 ) + ∫ τ = 0 t b ( τ ) e − A ( τ ) d τ ) where A \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
A A is the antiderivative of a \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
a a such that A ( 0 ) = 0 \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
A(0)=0 A ( 0 ) = 0 .
Applied to capture point dynamics, this formula becomes:
p C ( t ) = e ω t ( p C ( 0 ) − ω ∫ τ = 0 t p Z ( t ) e − ω τ d τ ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_C(t) = e^{\omega t} \left(\bfp_C(0) - \omega \int_{\tau=0}^t \bfp_Z(t)
e^{-\omega \tau} {\rm d}\tau\right) p C ( t ) = e ω t ( p C ( 0 ) − ω ∫ τ = 0 t p Z ( t ) e − ω τ d τ ) We can check how, in the previous case where p Z \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z p Z is stationary, this
formula becomes:
p C ( t ) = p Z + e ω t ( p C ( 0 ) − p Z ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_C(t) = \bfp_Z + e^{\omega t} (\bfp_C(0) - \bfp_Z) p C ( t ) = p Z + e ω t ( p C ( 0 ) − p Z ) The capture point trajectory is then bounded if and only if p Z = p C ( 0 ) \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_Z =
\bfp_C(0) p Z = p C ( 0 ) , which is indeed the result we obtained above. In the general case,
the capture point stays bounded if and only if:
p C ( 0 ) = ω ∫ τ = 0 t p Z ( t ) e − ω τ d τ \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\bftheta{\boldsymbol{\theta}}
\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\defeq{\stackrel{\mathrm{def}}{=}}
\def\p{\boldsymbol{p}}
\def\qdd{\ddot{\bfq}}
\def\qd{\dot{\bfq}}
\def\q{\boldsymbol{q}}
\def\xd{\dot{x}}
\def\yd{\dot{y}}
\def\zd{\dot{z}}
\bfp_C(0) = \omega \int_{\tau=0}^t \bfp_Z(t) e^{-\omega \tau} {\rm d}\tau p C ( 0 ) = ω ∫ τ = 0 t p Z ( t ) e − ω τ d τ This condition was coined boundedness condition by Lanari et al. (2014) . It relates future system
inputs to the present state, and characterizes the subset of these inputs that
will actually stabilize the system in the long run. The boundedness condition
is, for instance, a core component of the walking trajectory generator from
Scianca et al. (2019) . It can also be
applied to more general reduced models such as the variable-height inverted
pendulum .
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© Stéphane Caron — Text and figures licensed under the CC BY 4.0 by the author.
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