When dealing with a planar surface contact, such as the foot of a humanoid on a flat ground or the palm of its hand against a wall, friction fights against the two surfaces rotating agaisnt each other, that is, rotation along the contact normal. This rotation is often called twisting or yaw rotation, and so there should be some twisting friction balancing moments along the contact normal to prevent it. Coulomb friction gives us a model for linear forces:
∥ f t ∥ ≤ μ f 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}}
\| \bff_{t} \| \leq \mu f_z ∥ f t ∥ ≤ μ f z where we align the 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}}
z z -axis of our frame with the contact normal, f 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}}
f_z f z is the normal force, f 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}}
\bff_t f t is the orthogonal tangential force, and μ \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}}
\mu μ is the adimensional Coulomb friction coefficient.
How can we find a similar model for twisting friction?
Coulomb-like model
A straightforward idea is to mimic Coulomb friction, but with the moment τ 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}}
\tau_z τ z of contact forces along the contact normal instead of the tangential force f 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}}
\bff_t f t . The formula would look like this:
∣ τ z ∣ ≤ μ t w i s t f 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}}
| \tau_z | \leq \mu_\mathit{twist} f_z ∣ τ z ∣ ≤ μ twist f z where τ 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}}
\tau_z τ z is the moment along the contact normal, f 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}}
f_z f z is the net normal force, and μ t w i s 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}}
\mu_\mathit{twist} μ twist looks like a friction coefficient. Yet, if we look at its physical unit, it is not a friction coefficient at all: the Coulomb friction μ \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}}
\mu μ is adimensional, but this μ t w i s 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}}
\mu_\mathit{twist} μ twist has the unit of a distance . We should thus rather write it as:
∣ τ z ∣ ≤ d t w i s t f 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}}
| \tau_z | \leq d_\mathit{twist} f_z ∣ τ z ∣ ≤ d twist f z where the SI unit of d t w i s 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}}
d_\mathit{twist} d twist is the meter. But then, how do we identify the value of d t w i s 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}}
d_\mathit{twist} d twist on an actual system? Does it depend on the sizes of the two surfaces in contact?
Twisting friction model
As it turns out, yes. This distance not only depends on the sizes of the two surfaces in contact, but also on the location of the center of pressure within the contact area. For a rectangular 2 X × 2 Y \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}}
2X \times 2Y 2 X × 2 Y contact, we can actually compute the inequality exactly by integrating linearized friction cones over the contact area (short version in this paper , full version around p. 82 of this thesis ). Considering the contact wrench w = ( f , 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}}
\bfw = (\bff, \bftau) w = ( f , τ ) at the center of the area, the resulting formula is:
∣ τ z − τ z ∗ ∣ ≤ μ d C o P f 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}}
| \tau_z - \tau_z^* | \leq \mu d_\mathit{CoP} f_z ∣ τ z − τ z ∗ ∣ ≤ μ d CoP f z where d C o P \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}}
d_\mathit{CoP} d CoP is the distance from the center of pressure (CoP) to the nearest edge of the contact area, μ \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}}
\mu μ is the usual coefficient of friction, and a "nominal" moment τ 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}}
\tau_z^* τ z ∗ appears. This moment is defined as:
τ z ∗ : = ∣ X f y − μ x C o P f z ∣ + ∣ Y f x − μ y C o P f 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}}
\tau_z^* := |X f_y - \mu x_\mathit{CoP} f_z| + |Y f_x - \mu y_\mathit{CoP} f_z| τ z ∗ := ∣ X f y − μ x CoP f z ∣ + ∣ Y f x − μ y CoP f z ∣ It is a CoP-dependent "safest" moment to avoid twisting slippage. Expressed like this, the twisting-friction inequality constraint looks nonlinear, but it is fortunately linear when we formulate it as a linear wrench friction cone :
[ − Y − X − ( X + Y ) μ + μ + μ − 1 − Y + X − ( X + Y ) μ + μ − μ − 1 + Y − X − ( X + Y ) μ − μ + μ − 1 + Y + X − ( X + Y ) μ − μ − μ − 1 + Y + X − ( X + Y ) μ + μ + μ + 1 + Y − X − ( X + Y ) μ + μ − μ + 1 − Y + X − ( X + Y ) μ − μ + μ + 1 − Y − X − ( X + Y ) μ − μ − μ + 1 ] [ f x f y f z τ x τ x τ z ] ≤ [ 0 0 0 0 0 0 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}}
\begin{bmatrix}
-Y & -X & -(X + Y) \mu & +\mu & +\mu & -1 \\
-Y & +X & -(X + Y) \mu & +\mu & -\mu & -1 \\
+Y & -X & -(X + Y) \mu & -\mu & +\mu & -1 \\
+Y & +X & -(X + Y) \mu & -\mu & -\mu & -1 \\
+Y & +X & -(X + Y) \mu & +\mu & +\mu & +1 \\
+Y & -X & -(X + Y) \mu & +\mu & -\mu & +1 \\
-Y & +X & -(X + Y) \mu & -\mu & +\mu & +1 \\
-Y & -X & -(X + Y) \mu & -\mu & -\mu & +1
\end{bmatrix}
\begin{bmatrix}
f_x \\
f_y \\
f_z \\
\tau_x \\
\tau_x \\
\tau_z
\end{bmatrix}
\leq
\begin{bmatrix}
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0
\end{bmatrix} − Y − Y + Y + Y + Y + Y − Y − Y − X + X − X + X + X − X + X − X − ( X + Y ) μ − ( X + Y ) μ − ( X + Y ) μ − ( X + Y ) μ − ( X + Y ) μ − ( X + Y ) μ − ( X + Y ) μ − ( X + Y ) μ + μ + μ − μ − μ + μ + μ − μ − μ + μ − μ + μ − μ + μ − μ + μ − μ − 1 − 1 − 1 − 1 + 1 + 1 + 1 + 1 f x f y f z τ x τ x τ z ≤ 0 0 0 0 0 0 0 0 The nonlinear expression is obtained when we divide these linear inequalities by f 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}}
f_z f z to reformulate them in terms of the CoP.
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