Stéphane on Locomotion - Comments: Leveraging Cone Double Description for Multi-contact Stability of Humanoids with Applications to Statics and Dynamicshttps://scaron.info/publications/rss-2015.html/2015-07-13T16:42:00+02:00Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-dirac-fields<p>Yes, in this case we refer to the <a class="reference external" href="https://en.wikipedia.org/wiki/Dirac_delta_function">Dirac delta function</a>, that is to say, a
distribution that concentrates forces at the vertices of the contact area. You
will find a bit more discussion on this point, along with a proof of the
corresponding proposition, at the end of Section …</p><p>Yes, in this case we refer to the <a class="reference external" href="https://en.wikipedia.org/wiki/Dirac_delta_function">Dirac delta function</a>, that is to say, a
distribution that concentrates forces at the vertices of the contact area. You
will find a bit more discussion on this point, along with a proof of the
corresponding proposition, at the end of Section III of <a class="reference external" href="https://scaron.info/publications/rss-2015.html/publications/icra-2015.html">this study of surface
contacts</a>.</p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-need-double-description<p>It is not, as we later found out: local span matrices are straightforward to
compute, as detailed for instance in the introduction of <a class="reference external" href="https://scaron.info/publications/tro-2016.html">this paper</a>. The only point where the double
description remains necessary so far is for computing the gravito-inertial face
matrix <span class="math">\(U_\mathit{stance}.\)</span></p>
<p>It is not, as we later found out: local span matrices are straightforward to
compute, as detailed for instance in the introduction of <a class="reference external" href="https://scaron.info/publications/rss-2015.html/publications/tro-2016.html">this paper</a>. The only point where the double
description remains necessary so far is for computing the gravito-inertial face
matrix <span class="math">\(U_\mathit{stance}.\)</span></p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-numerical-instabilities-more-contacts<p>Unfortunately I have only an intuition here: making more contacts implies a
resultant polyhedron with more faces, i.e. with more rays in the V-description
and more hyperplanes in the H-one. This increases the risk of "tricky" faces
(of very small area) in the output polyhedron, especially if some of …</p><p>Unfortunately I have only an intuition here: making more contacts implies a
resultant polyhedron with more faces, i.e. with more rays in the V-description
and more hyperplanes in the H-one. This increases the risk of "tricky" faces
(of very small area) in the output polyhedron, especially if some of your input
rays are nearly aligned. My guess is that these make the double-description
fail when it tries to cut (c.f. Fukuda's paper on the algorithm) by an "almost
meaningless" hyperplane.</p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-cdd-canonicalize<p>In my experience this extra step only make matters worse (with regards to
numerical-inconsistency errors), so I would advise against doing it. Also, note
that, even if your input V-description is irreducible, cdd may output you a
non-minimal H-description.</p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-cdd-numerical-inconsistency<p>It may happen that your problem is "doable" but you ended up hitting some
near-singular configuration. In such cases, a simple hack is to perturb the
input (contact locations, or coordinates where the CWC is taken) by some
epsilon.</p>
<p>Another cause of numerical instabilities are vectors that are "almost"
rational …</p><p>It may happen that your problem is "doable" but you ended up hitting some
near-singular configuration. In such cases, a simple hack is to perturb the
input (contact locations, or coordinates where the CWC is taken) by some
epsilon.</p>
<p>Another cause of numerical instabilities are vectors that are "almost"
rational, for instance with coordinates such as <tt class="docutils literal">0.4 + <span class="pre">1e-12</span></tt>. Rounding up
coordinates to a sound number of decimals can filter out those.</p>
<p>If none of the above apply, you can try with rational rather than floating
point numbers. In that case, I'd advise using PPL or its Python wrapper
<a class="reference external" href="https://github.com/haudren/pyparma">pyparma</a> rather than cdd.</p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-analytical-wrench-cone<p>Unfortunately the <a class="reference external" href="https://scaron.info/publications/icra-2015.html">analytical formula of the contact wrench cone</a> is only known (yet) for a <em>single</em>
contact. In multi-contact, the best call so far is to use numerical algorithms
such as the double description method.</p>
<p>Unfortunately the <a class="reference external" href="https://scaron.info/publications/rss-2015.html/publications/icra-2015.html">analytical formula of the contact wrench cone</a> is only known (yet) for a <em>single</em>
contact. In multi-contact, the best call so far is to use numerical algorithms
such as the double description method.</p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-dual-matrix-shape<p>It is not always the case, as you can see in Table I of the paper. In single
support, redundant inequalities simplify and the <span class="math">\(6 \times 16\)</span> span
matrix of the wrench cone has a <span class="math">\(16 \times 6\)</span> face dual. However, in
double-support the dual of the <span class="math">\(6 \times 32 …</span></p><p>It is not always the case, as you can see in Table I of the paper. In single
support, redundant inequalities simplify and the <span class="math">\(6 \times 16\)</span> span
matrix of the wrench cone has a <span class="math">\(16 \times 6\)</span> face dual. However, in
double-support the dual of the <span class="math">\(6 \times 32\)</span> span matrix typically has
more than 100 rows.</p>
Posted by: Stéphane2015-07-13T16:42:00+02:002015-07-13T16:42:00+02:00Stéphanetag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-reply-link-with-3d-com-pattern-generator<p>In this paper, the gravito-inertial wrench coordinates <span class="math">\(w_{GI} = (f_{GI},
\tau_{GI})\)</span> are taken at the origin <span class="math">\(O\)</span> of the world frame. Using proper
notations, we should have written <span class="math">\(w^{gi}_O\)</span> to acknowledge this
reference point. (We apologize for the non-standard notations.) Our <a class="reference external" href="https://scaron.info/publications/humanoids-2016.html">later
works</a> uses the <em>net …</em></p><p>In this paper, the gravito-inertial wrench coordinates <span class="math">\(w_{GI} = (f_{GI},
\tau_{GI})\)</span> are taken at the origin <span class="math">\(O\)</span> of the world frame. Using proper
notations, we should have written <span class="math">\(w^{gi}_O\)</span> to acknowledge this
reference point. (We apologize for the non-standard notations.) Our <a class="reference external" href="https://scaron.info/publications/rss-2015.html/publications/humanoids-2016.html">later
works</a> uses the <em>net contact
wrench</em>, <em>i.e.</em> the sum of all contact wrenches applied to the robot, which
yields the opposite of the gravito-inertial one: <span class="math">\(w^{c}_O = -w^{gi}_O.\)</span>
Therefore, the face matrices are opposed as well: <span class="math">\(U_\mathit{stance} =
-A_O.\)</span></p>
Posted by: Attendee #82015-07-13T16:18:00+02:002015-07-13T16:18:00+02:00Attendee #8tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-dirac-fields<p>One footnote of the paper mentions Dirac fields. Are you referring here to the
Dirac delta function and its higher dimensional versions?</p>Posted by: Attendee #72015-07-13T16:17:00+02:002015-07-13T16:17:00+02:00Attendee #7tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-numerical-instabilities-more-contacts<p>Numerical instabilities seem to happen more often in stances with more contacts. Why?</p>Posted by: Attendee #52015-07-13T16:06:00+02:002015-07-13T16:06:00+02:00Attendee #5tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-cdd-canonicalize<p>What about calling the <code>Canonicalize()</code> function in cdd to remove redundant
rays from the input V-description?</p>Posted by: Attendee #52015-07-13T16:05:00+02:002015-07-13T16:05:00+02:00Attendee #5tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-cdd-numerical-inconsistency<p>The cdd library gives me the error "Numerical inconsistency is found. Use the
GMP exact arithmetic." What can I do?</p>Posted by: Attendee #42015-07-13T16:04:00+02:002015-07-13T16:04:00+02:00Attendee #4tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-dual-matrix-shape<p>Given an <span class="math">\(n \times m\)</span> matrix <span class="math">\(V\)</span>, is it the case that its dual (face or
H-representation) matrix <span class="math">\(U = V^F\)</span> has shape <span class="math">\(m \times n\)</span>?</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em …</script><p>Given an <span class="math">\(n \times m\)</span> matrix <span class="math">\(V\)</span>, is it the case that its dual (face or
H-representation) matrix <span class="math">\(U = V^F\)</span> has shape <span class="math">\(m \times n\)</span>?</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em" : indent;
linebreak = (screen.width < 768) ? 'true' : linebreak;
}
var mathjaxscript = document.createElement('script');
mathjaxscript.id = 'mathjaxscript_pelican_#%@#$@#';
mathjaxscript.type = 'text/javascript';
mathjaxscript.src = '/MathJax/MathJax.js?config=TeX-AMS-MML_HTMLorMML';
var configscript = document.createElement('script');
configscript.type = 'text/x-mathjax-config';
configscript[(window.opera ? "innerHTML" : "text")] =
"MathJax.Hub.Config({" +
" config: ['MMLorHTML.js']," +
" TeX: { extensions: ['AMSmath.js','AMSsymbols.js','noErrors.js','noUndefined.js'], equationNumbers: { autoNumber: 'none' } }," +
" jax: ['input/TeX','input/MathML','output/HTML-CSS']," +
" extensions: ['tex2jax.js','mml2jax.js','MathMenu.js','MathZoom.js']," +
" displayAlign: '"+ align +"'," +
" displayIndent: '"+ indent +"'," +
" showMathMenu: true," +
" messageStyle: 'normal'," +
" tex2jax: { " +
" inlineMath: [ ['\\\\(','\\\\)'] ], " +
" displayMath: [ ['$$','$$'] ]," +
" processEscapes: true," +
" preview: 'TeX'," +
" }, " +
" 'HTML-CSS': { " +
" availableFonts: ['STIX', 'TeX']," +
" preferredFont: 'STIX'," +
" styles: { '.MathJax_Display, .MathJax .mo, .MathJax .mi, .MathJax .mn': {color: '#333 ! important'} }," +
" linebreaks: { automatic: "+ linebreak +", width: '90% container' }," +
" }, " +
" 'TeX': { " +
" Macros: { " +
" defeq: \"{\\\\stackrel{\\\\mathrm{def}}{=}}\"," +
" Ld: \"{\\\\dot{L}}\"," +
" LdG: \"{\\\\dot{L}_G}\"," +
" bfA: \"{\\\\boldsymbol{A}}\"," +
" bfB: \"{\\\\boldsymbol{B}}\"," +
" bfC: \"{\\\\boldsymbol{C}}\"," +
" bfD: \"{\\\\boldsymbol{D}}\"," +
" bfE: \"{\\\\boldsymbol{E}}\"," +
" bfF: \"{\\\\boldsymbol{F}}\"," +
" bfG: \"{\\\\boldsymbol{G}}\"," +
" bfH: \"{\\\\boldsymbol{H}}\"," +
" bfI: \"{\\\\boldsymbol{I}}\"," +
" bfJ: \"{\\\\boldsymbol{J}}\"," +
" bfK: \"{\\\\boldsymbol{K}}\"," +
" bfL: \"{\\\\boldsymbol{L}}\"," +
" bfM: \"{\\\\boldsymbol{M}}\"," +
" bfN: \"{\\\\boldsymbol{N}}\"," +
" bfO: \"{\\\\boldsymbol{O}}\"," +
" bfP: \"{\\\\boldsymbol{P}}\"," +
" bfQ: \"{\\\\boldsymbol{Q}}\"," +
" bfR: \"{\\\\boldsymbol{R}}\"," +
" bfS: \"{\\\\boldsymbol{S}}\"," +
" bfT: \"{\\\\boldsymbol{T}}\"," +
" bfU: \"{\\\\boldsymbol{U}}\"," +
" bfV: \"{\\\\boldsymbol{V}}\"," +
" bfW: \"{\\\\boldsymbol{W}}\"," +
" bfX: \"{\\\\boldsymbol{X}}\"," +
" bfY: \"{\\\\boldsymbol{Y}}\"," +
" bfZ: \"{\\\\boldsymbol{Z}}\"," +
" bfa: \"{\\\\boldsymbol{a}}\"," +
" bfb: \"{\\\\boldsymbol{b}}\"," +
" bfc: \"{\\\\boldsymbol{c}}\"," +
" bfd: \"{\\\\boldsymbol{d}}\"," +
" bfe: \"{\\\\boldsymbol{e}}\"," +
" bff: \"{\\\\boldsymbol{f}}\"," +
" bfg: \"{\\\\boldsymbol{g}}\"," +
" bfh: \"{\\\\boldsymbol{h}}\"," +
" bfi: \"{\\\\boldsymbol{i}}\"," +
" bfj: \"{\\\\boldsymbol{j}}\"," +
" bfk: \"{\\\\boldsymbol{k}}\"," +
" bfl: \"{\\\\boldsymbol{l}}\"," +
" bfm: \"{\\\\boldsymbol{m}}\"," +
" bfn: \"{\\\\boldsymbol{n}}\"," +
" bfo: \"{\\\\boldsymbol{o}}\"," +
" bfp: \"{\\\\boldsymbol{p}}\"," +
" bfq: \"{\\\\boldsymbol{q}}\"," +
" bfr: \"{\\\\boldsymbol{r}}\"," +
" bfs: \"{\\\\boldsymbol{s}}\"," +
" bft: \"{\\\\boldsymbol{t}}\"," +
" bfu: \"{\\\\boldsymbol{u}}\"," +
" bfv: \"{\\\\boldsymbol{v}}\"," +
" bfw: \"{\\\\boldsymbol{w}}\"," +
" bfx: \"{\\\\boldsymbol{x}}\"," +
" bfy: \"{\\\\boldsymbol{y}}\"," +
" bfz: \"{\\\\boldsymbol{z}}\"," +
" bfalpha: \"{\\\\boldsymbol{\\\\alpha}}\"," +
" bfbeta: \"{\\\\boldsymbol{\\\\beta}}\"," +
" bfchi: \"{\\\\boldsymbol{\\\\chi}}\"," +
" bfgamma: \"{\\\\boldsymbol{\\\\gamma}}\"," +
" bflambda: \"{\\\\boldsymbol{\\\\lambda}}\"," +
" bfomega: \"{\\\\boldsymbol{\\\\omega}}\"," +
" bfone: \"{\\\\boldsymbol{1}}\"," +
" bfsigma: \"{\\\\boldsymbol{\\\\sigma}}\"," +
" bftau: \"{\\\\boldsymbol{\\\\tau}}\"," +
" bfxi: \"{\\\\boldsymbol{\\\\xi}}\"," +
" bfzero: \"{\\\\boldsymbol{0}}\"," +
" calA: \"{\\\\cal A}\"," +
" calB: \"{\\\\cal B}\"," +
" calC: \"{\\\\cal C}\"," +
" calD: \"{\\\\cal D}\"," +
" calE: \"{\\\\cal E}\"," +
" calF: \"{\\\\cal F}\"," +
" calG: \"{\\\\cal G}\"," +
" calH: \"{\\\\cal H}\"," +
" calI: \"{\\\\cal I}\"," +
" calJ: \"{\\\\cal J}\"," +
" calK: \"{\\\\cal K}\"," +
" calL: \"{\\\\cal L}\"," +
" calM: \"{\\\\cal M}\"," +
" calN: \"{\\\\cal N}\"," +
" calO: \"{\\\\cal O}\"," +
" calP: \"{\\\\cal P}\"," +
" calQ: \"{\\\\cal Q}\"," +
" calR: \"{\\\\cal R}\"," +
" calS: \"{\\\\cal S}\"," +
" calT: \"{\\\\cal T}\"," +
" calU: \"{\\\\cal U}\"," +
" calV: \"{\\\\cal V}\"," +
" calW: \"{\\\\cal W}\"," +
" calX: \"{\\\\cal X}\"," +
" calY: \"{\\\\cal Y}\"," +
" calZ: \"{\\\\cal Z}\"," +
" d: [\"{\\\\rm d}{#1}\", 1]," +
" bfcd: \"{\\\\dot{\\\\bfc}}\"," +
" bfpd: \"{\\\\dot{\\\\bfp}}\"," +
" bfpdd: \"{\\\\ddot{\\\\bfp}}\"," +
" dim: \"{\\\\rm dim}\"," +
" p: \"{\\\\boldsymbol{p}}\"," +
" q: \"{\\\\boldsymbol{q}}\"," +
" qd: \"{\\\\dot{\\\\bfq}}\"," +
" qdd: \"{\\\\ddot{\\\\bfq}}\"," +
" xd: \"{\\\\dot{x}}\"," +
" xdd: \"{\\\\ddot{x}}\"," +
" yd: \"{\\\\dot{y}}\"," +
" ydd: \"{\\\\ddot{y}}\"," +
" zd: \"{\\\\dot{z}}\"," +
" zdd: \"{\\\\ddot{z}}\"," +
" defeq: \"{\\\\stackrel{\\\\mathrm{def}}{\\\\ =\\\\ }}\"," +
" } " +
" } " +
"}); " +
"if ('default' !== 'default') {" +
"MathJax.Hub.Register.StartupHook('HTML-CSS Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax['HTML-CSS'].FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"}";
(document.body || document.getElementsByTagName('head')[0]).appendChild(configscript);
(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
}
</script>Posted by: Attendee #32015-07-13T16:03:00+02:002015-07-13T16:03:00+02:00Attendee #3tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-link-with-3d-com-pattern-generator<p>What is the link between the matrix <span class="math">\(U_\mathit{stance}\)</span> in this paper and the
matrices <span class="math">\(A_O\)</span> or <span class="math">\(A_G\)</span> used in your follow-up 3D-COM pattern generator?</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em …</script><p>What is the link between the matrix <span class="math">\(U_\mathit{stance}\)</span> in this paper and the
matrices <span class="math">\(A_O\)</span> or <span class="math">\(A_G\)</span> used in your follow-up 3D-COM pattern generator?</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em" : indent;
linebreak = (screen.width < 768) ? 'true' : linebreak;
}
var mathjaxscript = document.createElement('script');
mathjaxscript.id = 'mathjaxscript_pelican_#%@#$@#';
mathjaxscript.type = 'text/javascript';
mathjaxscript.src = '/MathJax/MathJax.js?config=TeX-AMS-MML_HTMLorMML';
var configscript = document.createElement('script');
configscript.type = 'text/x-mathjax-config';
configscript[(window.opera ? "innerHTML" : "text")] =
"MathJax.Hub.Config({" +
" config: ['MMLorHTML.js']," +
" TeX: { extensions: ['AMSmath.js','AMSsymbols.js','noErrors.js','noUndefined.js'], equationNumbers: { autoNumber: 'none' } }," +
" jax: ['input/TeX','input/MathML','output/HTML-CSS']," +
" extensions: ['tex2jax.js','mml2jax.js','MathMenu.js','MathZoom.js']," +
" displayAlign: '"+ align +"'," +
" displayIndent: '"+ indent +"'," +
" showMathMenu: true," +
" messageStyle: 'normal'," +
" tex2jax: { " +
" inlineMath: [ ['\\\\(','\\\\)'] ], " +
" displayMath: [ ['$$','$$'] ]," +
" processEscapes: true," +
" preview: 'TeX'," +
" }, " +
" 'HTML-CSS': { " +
" availableFonts: ['STIX', 'TeX']," +
" preferredFont: 'STIX'," +
" styles: { '.MathJax_Display, .MathJax .mo, .MathJax .mi, .MathJax .mn': {color: '#333 ! important'} }," +
" linebreaks: { automatic: "+ linebreak +", width: '90% container' }," +
" }, " +
" 'TeX': { " +
" Macros: { " +
" defeq: \"{\\\\stackrel{\\\\mathrm{def}}{=}}\"," +
" Ld: \"{\\\\dot{L}}\"," +
" LdG: \"{\\\\dot{L}_G}\"," +
" bfA: \"{\\\\boldsymbol{A}}\"," +
" bfB: \"{\\\\boldsymbol{B}}\"," +
" bfC: \"{\\\\boldsymbol{C}}\"," +
" bfD: \"{\\\\boldsymbol{D}}\"," +
" bfE: \"{\\\\boldsymbol{E}}\"," +
" bfF: \"{\\\\boldsymbol{F}}\"," +
" bfG: \"{\\\\boldsymbol{G}}\"," +
" bfH: \"{\\\\boldsymbol{H}}\"," +
" bfI: \"{\\\\boldsymbol{I}}\"," +
" bfJ: \"{\\\\boldsymbol{J}}\"," +
" bfK: \"{\\\\boldsymbol{K}}\"," +
" bfL: \"{\\\\boldsymbol{L}}\"," +
" bfM: \"{\\\\boldsymbol{M}}\"," +
" bfN: \"{\\\\boldsymbol{N}}\"," +
" bfO: \"{\\\\boldsymbol{O}}\"," +
" bfP: \"{\\\\boldsymbol{P}}\"," +
" bfQ: \"{\\\\boldsymbol{Q}}\"," +
" bfR: \"{\\\\boldsymbol{R}}\"," +
" bfS: \"{\\\\boldsymbol{S}}\"," +
" bfT: \"{\\\\boldsymbol{T}}\"," +
" bfU: \"{\\\\boldsymbol{U}}\"," +
" bfV: \"{\\\\boldsymbol{V}}\"," +
" bfW: \"{\\\\boldsymbol{W}}\"," +
" bfX: \"{\\\\boldsymbol{X}}\"," +
" bfY: \"{\\\\boldsymbol{Y}}\"," +
" bfZ: \"{\\\\boldsymbol{Z}}\"," +
" bfa: \"{\\\\boldsymbol{a}}\"," +
" bfb: \"{\\\\boldsymbol{b}}\"," +
" bfc: \"{\\\\boldsymbol{c}}\"," +
" bfd: \"{\\\\boldsymbol{d}}\"," +
" bfe: \"{\\\\boldsymbol{e}}\"," +
" bff: \"{\\\\boldsymbol{f}}\"," +
" bfg: \"{\\\\boldsymbol{g}}\"," +
" bfh: \"{\\\\boldsymbol{h}}\"," +
" bfi: \"{\\\\boldsymbol{i}}\"," +
" bfj: \"{\\\\boldsymbol{j}}\"," +
" bfk: \"{\\\\boldsymbol{k}}\"," +
" bfl: \"{\\\\boldsymbol{l}}\"," +
" bfm: \"{\\\\boldsymbol{m}}\"," +
" bfn: \"{\\\\boldsymbol{n}}\"," +
" bfo: \"{\\\\boldsymbol{o}}\"," +
" bfp: \"{\\\\boldsymbol{p}}\"," +
" bfq: \"{\\\\boldsymbol{q}}\"," +
" bfr: \"{\\\\boldsymbol{r}}\"," +
" bfs: \"{\\\\boldsymbol{s}}\"," +
" bft: \"{\\\\boldsymbol{t}}\"," +
" bfu: \"{\\\\boldsymbol{u}}\"," +
" bfv: \"{\\\\boldsymbol{v}}\"," +
" bfw: \"{\\\\boldsymbol{w}}\"," +
" bfx: \"{\\\\boldsymbol{x}}\"," +
" bfy: \"{\\\\boldsymbol{y}}\"," +
" bfz: \"{\\\\boldsymbol{z}}\"," +
" bfalpha: \"{\\\\boldsymbol{\\\\alpha}}\"," +
" bfbeta: \"{\\\\boldsymbol{\\\\beta}}\"," +
" bfchi: \"{\\\\boldsymbol{\\\\chi}}\"," +
" bfgamma: \"{\\\\boldsymbol{\\\\gamma}}\"," +
" bflambda: \"{\\\\boldsymbol{\\\\lambda}}\"," +
" bfomega: \"{\\\\boldsymbol{\\\\omega}}\"," +
" bfone: \"{\\\\boldsymbol{1}}\"," +
" bfsigma: \"{\\\\boldsymbol{\\\\sigma}}\"," +
" bftau: \"{\\\\boldsymbol{\\\\tau}}\"," +
" bfxi: \"{\\\\boldsymbol{\\\\xi}}\"," +
" bfzero: \"{\\\\boldsymbol{0}}\"," +
" calA: \"{\\\\cal A}\"," +
" calB: \"{\\\\cal B}\"," +
" calC: \"{\\\\cal C}\"," +
" calD: \"{\\\\cal D}\"," +
" calE: \"{\\\\cal E}\"," +
" calF: \"{\\\\cal F}\"," +
" calG: \"{\\\\cal G}\"," +
" calH: \"{\\\\cal H}\"," +
" calI: \"{\\\\cal I}\"," +
" calJ: \"{\\\\cal J}\"," +
" calK: \"{\\\\cal K}\"," +
" calL: \"{\\\\cal L}\"," +
" calM: \"{\\\\cal M}\"," +
" calN: \"{\\\\cal N}\"," +
" calO: \"{\\\\cal O}\"," +
" calP: \"{\\\\cal P}\"," +
" calQ: \"{\\\\cal Q}\"," +
" calR: \"{\\\\cal R}\"," +
" calS: \"{\\\\cal S}\"," +
" calT: \"{\\\\cal T}\"," +
" calU: \"{\\\\cal U}\"," +
" calV: \"{\\\\cal V}\"," +
" calW: \"{\\\\cal W}\"," +
" calX: \"{\\\\cal X}\"," +
" calY: \"{\\\\cal Y}\"," +
" calZ: \"{\\\\cal Z}\"," +
" d: [\"{\\\\rm d}{#1}\", 1]," +
" bfcd: \"{\\\\dot{\\\\bfc}}\"," +
" bfpd: \"{\\\\dot{\\\\bfp}}\"," +
" bfpdd: \"{\\\\ddot{\\\\bfp}}\"," +
" dim: \"{\\\\rm dim}\"," +
" p: \"{\\\\boldsymbol{p}}\"," +
" q: \"{\\\\boldsymbol{q}}\"," +
" qd: \"{\\\\dot{\\\\bfq}}\"," +
" qdd: \"{\\\\ddot{\\\\bfq}}\"," +
" xd: \"{\\\\dot{x}}\"," +
" xdd: \"{\\\\ddot{x}}\"," +
" yd: \"{\\\\dot{y}}\"," +
" ydd: \"{\\\\ddot{y}}\"," +
" zd: \"{\\\\dot{z}}\"," +
" zdd: \"{\\\\ddot{z}}\"," +
" defeq: \"{\\\\stackrel{\\\\mathrm{def}}{\\\\ =\\\\ }}\"," +
" } " +
" } " +
"}); " +
"if ('default' !== 'default') {" +
"MathJax.Hub.Register.StartupHook('HTML-CSS Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax['HTML-CSS'].FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"}";
(document.body || document.getElementsByTagName('head')[0]).appendChild(configscript);
(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
}
</script>Posted by: Attendee #22015-07-13T16:02:00+02:002015-07-13T16:02:00+02:00Attendee #2tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-need-double-description<p>According to Figure 2, you use the double-description method to compute the
span matrix <span class="math">\(V_\mathit{surf}\)</span> at each contact. Is this necessary?</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em" : indent;
linebreak = (screen …</script><p>According to Figure 2, you use the double-description method to compute the
span matrix <span class="math">\(V_\mathit{surf}\)</span> at each contact. Is this necessary?</p>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "false";
if (false) {
align = (screen.width < 768) ? "left" : align;
indent = (screen.width < 768) ? "0em" : indent;
linebreak = (screen.width < 768) ? 'true' : linebreak;
}
var mathjaxscript = document.createElement('script');
mathjaxscript.id = 'mathjaxscript_pelican_#%@#$@#';
mathjaxscript.type = 'text/javascript';
mathjaxscript.src = '/MathJax/MathJax.js?config=TeX-AMS-MML_HTMLorMML';
var configscript = document.createElement('script');
configscript.type = 'text/x-mathjax-config';
configscript[(window.opera ? "innerHTML" : "text")] =
"MathJax.Hub.Config({" +
" config: ['MMLorHTML.js']," +
" TeX: { extensions: ['AMSmath.js','AMSsymbols.js','noErrors.js','noUndefined.js'], equationNumbers: { autoNumber: 'none' } }," +
" jax: ['input/TeX','input/MathML','output/HTML-CSS']," +
" extensions: ['tex2jax.js','mml2jax.js','MathMenu.js','MathZoom.js']," +
" displayAlign: '"+ align +"'," +
" displayIndent: '"+ indent +"'," +
" showMathMenu: true," +
" messageStyle: 'normal'," +
" tex2jax: { " +
" inlineMath: [ ['\\\\(','\\\\)'] ], " +
" displayMath: [ ['$$','$$'] ]," +
" processEscapes: true," +
" preview: 'TeX'," +
" }, " +
" 'HTML-CSS': { " +
" availableFonts: ['STIX', 'TeX']," +
" preferredFont: 'STIX'," +
" styles: { '.MathJax_Display, .MathJax .mo, .MathJax .mi, .MathJax .mn': {color: '#333 ! important'} }," +
" linebreaks: { automatic: "+ linebreak +", width: '90% container' }," +
" }, " +
" 'TeX': { " +
" Macros: { " +
" defeq: \"{\\\\stackrel{\\\\mathrm{def}}{=}}\"," +
" Ld: \"{\\\\dot{L}}\"," +
" LdG: \"{\\\\dot{L}_G}\"," +
" bfA: \"{\\\\boldsymbol{A}}\"," +
" bfB: \"{\\\\boldsymbol{B}}\"," +
" bfC: \"{\\\\boldsymbol{C}}\"," +
" bfD: \"{\\\\boldsymbol{D}}\"," +
" bfE: \"{\\\\boldsymbol{E}}\"," +
" bfF: \"{\\\\boldsymbol{F}}\"," +
" bfG: \"{\\\\boldsymbol{G}}\"," +
" bfH: \"{\\\\boldsymbol{H}}\"," +
" bfI: \"{\\\\boldsymbol{I}}\"," +
" bfJ: \"{\\\\boldsymbol{J}}\"," +
" bfK: \"{\\\\boldsymbol{K}}\"," +
" bfL: \"{\\\\boldsymbol{L}}\"," +
" bfM: \"{\\\\boldsymbol{M}}\"," +
" bfN: \"{\\\\boldsymbol{N}}\"," +
" bfO: \"{\\\\boldsymbol{O}}\"," +
" bfP: \"{\\\\boldsymbol{P}}\"," +
" bfQ: \"{\\\\boldsymbol{Q}}\"," +
" bfR: \"{\\\\boldsymbol{R}}\"," +
" bfS: \"{\\\\boldsymbol{S}}\"," +
" bfT: \"{\\\\boldsymbol{T}}\"," +
" bfU: \"{\\\\boldsymbol{U}}\"," +
" bfV: \"{\\\\boldsymbol{V}}\"," +
" bfW: \"{\\\\boldsymbol{W}}\"," +
" bfX: \"{\\\\boldsymbol{X}}\"," +
" bfY: \"{\\\\boldsymbol{Y}}\"," +
" bfZ: \"{\\\\boldsymbol{Z}}\"," +
" bfa: \"{\\\\boldsymbol{a}}\"," +
" bfb: \"{\\\\boldsymbol{b}}\"," +
" bfc: \"{\\\\boldsymbol{c}}\"," +
" bfd: \"{\\\\boldsymbol{d}}\"," +
" bfe: \"{\\\\boldsymbol{e}}\"," +
" bff: \"{\\\\boldsymbol{f}}\"," +
" bfg: \"{\\\\boldsymbol{g}}\"," +
" bfh: \"{\\\\boldsymbol{h}}\"," +
" bfi: \"{\\\\boldsymbol{i}}\"," +
" bfj: \"{\\\\boldsymbol{j}}\"," +
" bfk: \"{\\\\boldsymbol{k}}\"," +
" bfl: \"{\\\\boldsymbol{l}}\"," +
" bfm: \"{\\\\boldsymbol{m}}\"," +
" bfn: \"{\\\\boldsymbol{n}}\"," +
" bfo: \"{\\\\boldsymbol{o}}\"," +
" bfp: \"{\\\\boldsymbol{p}}\"," +
" bfq: \"{\\\\boldsymbol{q}}\"," +
" bfr: \"{\\\\boldsymbol{r}}\"," +
" bfs: \"{\\\\boldsymbol{s}}\"," +
" bft: \"{\\\\boldsymbol{t}}\"," +
" bfu: \"{\\\\boldsymbol{u}}\"," +
" bfv: \"{\\\\boldsymbol{v}}\"," +
" bfw: \"{\\\\boldsymbol{w}}\"," +
" bfx: \"{\\\\boldsymbol{x}}\"," +
" bfy: \"{\\\\boldsymbol{y}}\"," +
" bfz: \"{\\\\boldsymbol{z}}\"," +
" bfalpha: \"{\\\\boldsymbol{\\\\alpha}}\"," +
" bfbeta: \"{\\\\boldsymbol{\\\\beta}}\"," +
" bfchi: \"{\\\\boldsymbol{\\\\chi}}\"," +
" bfgamma: \"{\\\\boldsymbol{\\\\gamma}}\"," +
" bflambda: \"{\\\\boldsymbol{\\\\lambda}}\"," +
" bfomega: \"{\\\\boldsymbol{\\\\omega}}\"," +
" bfone: \"{\\\\boldsymbol{1}}\"," +
" bfsigma: \"{\\\\boldsymbol{\\\\sigma}}\"," +
" bftau: \"{\\\\boldsymbol{\\\\tau}}\"," +
" bfxi: \"{\\\\boldsymbol{\\\\xi}}\"," +
" bfzero: \"{\\\\boldsymbol{0}}\"," +
" calA: \"{\\\\cal A}\"," +
" calB: \"{\\\\cal B}\"," +
" calC: \"{\\\\cal C}\"," +
" calD: \"{\\\\cal D}\"," +
" calE: \"{\\\\cal E}\"," +
" calF: \"{\\\\cal F}\"," +
" calG: \"{\\\\cal G}\"," +
" calH: \"{\\\\cal H}\"," +
" calI: \"{\\\\cal I}\"," +
" calJ: \"{\\\\cal J}\"," +
" calK: \"{\\\\cal K}\"," +
" calL: \"{\\\\cal L}\"," +
" calM: \"{\\\\cal M}\"," +
" calN: \"{\\\\cal N}\"," +
" calO: \"{\\\\cal O}\"," +
" calP: \"{\\\\cal P}\"," +
" calQ: \"{\\\\cal Q}\"," +
" calR: \"{\\\\cal R}\"," +
" calS: \"{\\\\cal S}\"," +
" calT: \"{\\\\cal T}\"," +
" calU: \"{\\\\cal U}\"," +
" calV: \"{\\\\cal V}\"," +
" calW: \"{\\\\cal W}\"," +
" calX: \"{\\\\cal X}\"," +
" calY: \"{\\\\cal Y}\"," +
" calZ: \"{\\\\cal Z}\"," +
" d: [\"{\\\\rm d}{#1}\", 1]," +
" bfcd: \"{\\\\dot{\\\\bfc}}\"," +
" bfpd: \"{\\\\dot{\\\\bfp}}\"," +
" bfpdd: \"{\\\\ddot{\\\\bfp}}\"," +
" dim: \"{\\\\rm dim}\"," +
" p: \"{\\\\boldsymbol{p}}\"," +
" q: \"{\\\\boldsymbol{q}}\"," +
" qd: \"{\\\\dot{\\\\bfq}}\"," +
" qdd: \"{\\\\ddot{\\\\bfq}}\"," +
" xd: \"{\\\\dot{x}}\"," +
" xdd: \"{\\\\ddot{x}}\"," +
" yd: \"{\\\\dot{y}}\"," +
" ydd: \"{\\\\ddot{y}}\"," +
" zd: \"{\\\\dot{z}}\"," +
" zdd: \"{\\\\ddot{z}}\"," +
" defeq: \"{\\\\stackrel{\\\\mathrm{def}}{\\\\ =\\\\ }}\"," +
" } " +
" } " +
"}); " +
"if ('default' !== 'default') {" +
"MathJax.Hub.Register.StartupHook('HTML-CSS Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax['HTML-CSS'].FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"}";
(document.body || document.getElementsByTagName('head')[0]).appendChild(configscript);
(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
}
</script>Posted by: Attendee #12015-07-13T16:01:00+02:002015-07-13T16:01:00+02:00Attendee #1tag:scaron.info,2015-07-13:/publications/rss-2015.html//comment-question-analytical-wrench-cone<p>In your previous work, you provided an analytical formula for the contact wrench cone. Why don't you use it in this paper?</p>