Stéphane on Locomotion - Comments: ZMP Support Areas for Multi-contact Mobility Under Frictional Constraintshttps://scaron.info/publications/tro-2016.html/2016-12-14T16:42:00+01:00Posted by: Stéphane2016-12-14T16:42:00+01:002016-12-14T16:42:00+01:00Stéphanetag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-reply-static-equilibrium-polygon<p>In the case of the CoM polygon, it is convenient but not necessary to choose a vertical <span class="math">$$\bfn$$</span>. You can take a non-vertical one: it will give you the same polygon but in a tilted plane.</p> <p>In general, <span class="math">$$\bfn$$</span> represents the plane in which you are controlling your CoM. Because …</p><p>In the case of the CoM polygon, it is convenient but not necessary to choose a vertical <span class="math">$$\bfn$$</span>. You can take a non-vertical one: it will give you the same polygon but in a tilted plane.</p> <p>In general, <span class="math">$$\bfn$$</span> represents the plane in which you are controlling your CoM. Because the CoM is constrained to lie in this plane, you should choose it according to the motions to be realized: for instance, if the robot is climbing stairs, the plane should have the same inclination as the staircase.</p> Posted by: Stéphane2016-12-14T16:42:00+01:002016-12-14T16:42:00+01:00Stéphanetag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-reply-cross-product-by-n<p>A core assumption of this paper is that dynamics are decoupled into 2D plane dynamics (involving horizontal CoM and ZMP motions) and 4D complementary dynamics. Taking the cross product by <span class="math">$$\bfn$$</span> is a convenient way to project onto the plane, as the projection of any vector <span class="math">$$\bfv$$</span> onto a plane …</p><p>A core assumption of this paper is that dynamics are decoupled into 2D plane dynamics (involving horizontal CoM and ZMP motions) and 4D complementary dynamics. Taking the cross product by <span class="math">$$\bfn$$</span> is a convenient way to project onto the plane, as the projection of any vector <span class="math">$$\bfv$$</span> onto a plane of normal <span class="math">$$\bfn$$</span> can be written <span class="math">$$\bfn \times \bfv \times \bfn$$</span> (the cross product is not associative in general but in this particular case it is).</p> Posted by: Stéphane2016-12-14T16:42:00+01:002016-12-14T16:42:00+01:00Stéphanetag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-reply-equation-51<p>Rather than being added, equation (51) is already present in Newton and Euler equations (<em>i.e.</em> the dynamics of the floating base) and we are not ignoring it. Specifically, we decompose the 6D floating base dynamics into:</p> <ul class="simple"> <li>2D plane dynamics, <em>e.g.</em> equation (50)</li> <li>complementary 3D linear dynamics, <em>e.g …</em></li></ul><p>Rather than being added, equation (51) is already present in Newton and Euler equations (<em>i.e.</em> the dynamics of the floating base) and we are not ignoring it. Specifically, we decompose the 6D floating base dynamics into:</p> <ul class="simple"> <li>2D plane dynamics, <em>e.g.</em> equation (50)</li> <li>complementary 3D linear dynamics, <em>e.g.</em> equation (49)</li> <li>complementary 1D angular dynamics along the vertical axis, <em>e.g.</em> equation (51)</li> </ul> <p>In total we model the full dynamics, while leaving out equation (51) would be an approximation of these dynamics.</p> Posted by: Stéphane2016-12-14T16:42:00+01:002016-12-14T16:42:00+01:00Stéphanetag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-reply-pendular-area-on-flat-terrain<p>There are actually two assumptions where the ZMP support area becomes the convex hull of ground contact points: coplanar contact points, and infinite friction. Without the latter, we can run into cases such as the one depicted in Figure 6 of the paper, and the area is again CoM-dependent (even …</p><p>There are actually two assumptions where the ZMP support area becomes the convex hull of ground contact points: coplanar contact points, and infinite friction. Without the latter, we can run into cases such as the one depicted in Figure 6 of the paper, and the area is again CoM-dependent (even on a horizontal floor, imagine for instance standing on ice). With these two assumptions, we can indeed show that the algorithm to compute the pendular support area <a class="reference external" href="https://scaron.info/publications/tro-2016.html/robot-locomotion/zmp-support-area.html#simplification-on-flat-floors">reduces to a simple convex hull</a>.</p> Posted by: Stéphane2016-12-14T16:42:00+01:002016-12-14T16:42:00+01:00Stéphanetag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-reply-wrench-coordinates-at-com<p>Wrench coordinates are indeed taken at the origin of the world frame (or any other fixed point, for what matters). Taking screw coordinates at the origin of the world frame is typical of <a class="reference external" href="http://royfeatherstone.org/spatial/">spatial vector algebra</a>, which was formalized by Roy Featherstone.</p> <p>The main reason for working with the CWC …</p><p>Wrench coordinates are indeed taken at the origin of the world frame (or any other fixed point, for what matters). Taking screw coordinates at the origin of the world frame is typical of <a class="reference external" href="http://royfeatherstone.org/spatial/">spatial vector algebra</a>, which was formalized by Roy Featherstone.</p> <p>The main reason for working with the CWC at the origin is that it only depends on the stance (set of contacts), while the CWC taken at the CoM would also depend on CoM coordinates. Actually, once you have the former, it is straightforward to compute the latter using a simple dual transformation formula, as we described in Section III of the <a class="reference external" href="https://hal.archives-ouvertes.fr/hal-01349880/document">following paper</a>.</p> Posted by: Reader #52016-12-14T16:05:00+01:002016-12-14T16:05:00+01:00Reader #5tag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-question-pendular-area-on-flat-terrain<p>The definition of the pendular support area depends on the instantaneous CoM position, but it is known that the support area on flat terrains is simply the convex hull of the feet. Then, shouldn't the algorithm for the pendular support area be independent on the CoM position at least for …</p><p>The definition of the pendular support area depends on the instantaneous CoM position, but it is known that the support area on flat terrains is simply the convex hull of the feet. Then, shouldn't the algorithm for the pendular support area be independent on the CoM position at least for the flat terrain case?</p>Posted by: Reader #42016-12-14T16:04:00+01:002016-12-14T16:04:00+01:00Reader #4tag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-question-static-equilibrium-polygon<p>When computing the static-equilibrium polygon, why is <span class="math">$$\bfn$$</span> chosen to be vertical? You mention earlier in the paper that <span class="math">$$\bfn$$</span> can be "a fixed unit space vector, not necessarily aligned with the gravity vector". How is it determined or what does it represent in general?</p> <script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#@#')) { var …</script><p>When computing the static-equilibrium polygon, why is <span class="math">$$\bfn$$</span> chosen to be vertical? 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" + "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')).appendChild(configscript); (document.body || document.getElementsByTagName('head')).appendChild(mathjaxscript); } </script>Posted by: Reader #32016-12-14T16:03:00+01:002016-12-14T16:03:00+01:00Reader #3tag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-question-equation-51<p>What was the need for adding equation (51)? Aren't equations (49) and (50) sufficient to characterize static equilibrium?</p>Posted by: Reader #22016-12-14T16:02:00+01:002016-12-14T16:02:00+01:00Reader #2tag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-question-cross-product-by-n<p>What was the reason for introducing the cross product by <span class="math">$$\bfn$$</span> on both sides of equation (50)? Is it just here to expand the cross product and isolate <span class="math">$$\bfp_G$$</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 …</script><p>What was the reason for introducing the cross product by <span class="math">$$\bfn$$</span> on both sides of equation (50)? 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" + "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')).appendChild(configscript); (document.body || document.getElementsByTagName('head')).appendChild(mathjaxscript); } </script>Posted by: Reader #12016-12-14T16:01:00+01:002016-12-14T16:01:00+01:00Reader #1tag:scaron.info,2016-12-14:/publications/tro-2016.html//comment-question-wrench-coordinates-at-com<p>In Equation (5), the wrench coordinates <span class="math">$$\bfw_O^c$$</span> are taken with respect to the origin of the world frame. Why don't you rather take this wrench at the CoM, as is usually done?</p> <script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#@#')) { var align = "center", indent = "0em", linebreak = "false"; if (false) { align = (screen.width < 768 …</script><p>In Equation (5), the wrench coordinates <span class="math">$$\bfw_O^c$$</span> are taken with respect to the origin of the world frame. Why don't you rather take this wrench at the CoM, as is usually done?</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) ? 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