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docs/flexible_joints_study.html
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@ -4,7 +4,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-05-05 mar. 11:26 --> <!-- 2020-05-05 mar. 11:50 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Study of the Flexible Joints</title> <title>Study of the Flexible Joints</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -37,43 +37,43 @@
<ul> <ul>
<li><a href="#orge032d30">1. Bending and Torsional Stiffness</a> <li><a href="#orge032d30">1. Bending and Torsional Stiffness</a>
<ul> <ul>
<li><a href="#org14d57c4">1.1. Initialization</a></li> <li><a href="#org8fdef7f">1.1. Initialization</a></li>
<li><a href="#orgde60939">1.2. Realistic Bending Stiffness Values</a> <li><a href="#orgde60939">1.2. Realistic Bending Stiffness Values</a>
<ul> <ul>
<li><a href="#org5ed48b8">1.2.1. Direct Velocity Feedback</a></li> <li><a href="#orgdb214f9">1.2.1. Direct Velocity Feedback</a></li>
<li><a href="#orgddae25e">1.2.2. Primary Plant</a></li> <li><a href="#org4069e58">1.2.2. Primary Plant</a></li>
<li><a href="#orgb8a9692">1.2.3. Conclusion</a></li> <li><a href="#orga32adf0">1.2.3. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8ad3f34">1.3. Parametric Study</a> <li><a href="#org8ad3f34">1.3. Parametric Study</a>
<ul> <ul>
<li><a href="#org44ccdbe">1.3.1. Direct Velocity Feedback</a></li> <li><a href="#org4adf147">1.3.1. Direct Velocity Feedback</a></li>
<li><a href="#org5d9965b">1.3.2. Primary Control</a></li> <li><a href="#org53e5f08">1.3.2. Primary Control</a></li>
<li><a href="#org0f9f990">1.3.3. Conclusion</a></li> <li><a href="#orgc45ccb0">1.3.3. Conclusion</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</li> </li>
<li><a href="#org81f1d95">2. Translation Stiffness</a> <li><a href="#orgdaf7b6c">2. Axial Stiffness</a>
<ul> <ul>
<li><a href="#org969d9e7">2.1. Realistic Translation Stiffness Values</a> <li><a href="#org969d9e7">2.1. Realistic Translation Stiffness Values</a>
<ul> <ul>
<li><a href="#org8fdef7f">2.1.1. Initialization</a></li> <li><a href="#orge82a7c2">2.1.1. Initialization</a></li>
<li><a href="#orgc087bb9">2.1.2. Direct Velocity Feedback</a></li> <li><a href="#org44f67b8">2.1.2. Direct Velocity Feedback</a></li>
<li><a href="#org4069e58">2.1.3. Primary Plant</a></li> <li><a href="#orgd5fd59b">2.1.3. Primary Plant</a></li>
<li><a href="#org3d8a1a7">2.1.4. Conclusion</a></li> <li><a href="#org552093a">2.1.4. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org0275632">2.2. Parametric study</a> <li><a href="#org0275632">2.2. Parametric study</a>
<ul> <ul>
<li><a href="#orgdb214f9">2.2.1. Direct Velocity Feedback</a></li> <li><a href="#orge13b41c">2.2.1. Direct Velocity Feedback</a></li>
<li><a href="#org53e5f08">2.2.2. Primary Control</a></li> <li><a href="#org15c2c08">2.2.2. Primary Control</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org1ddd8bf">2.3. Conclusion</a></li> <li><a href="#orgce1052e">2.3. Conclusion</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orga32adf0">3. Conclusion</a></li> <li><a href="#org865157e">3. Conclusion</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -106,8 +106,8 @@ In this section, we wish to study the effect of the rotation flexibility of the
</p> </p>
</div> </div>
<div id="outline-container-org14d57c4" class="outline-3"> <div id="outline-container-org8fdef7f" class="outline-3">
<h3 id="org14d57c4"><span class="section-number-3">1.1</span> Initialization</h3> <h3 id="org8fdef7f"><span class="section-number-3">1.1</span> Initialization</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
Let&rsquo;s initialize all the stages with default parameters. Let&rsquo;s initialize all the stages with default parameters.
@ -168,8 +168,8 @@ This corresponds to the optimal identified stiffness.
</p> </p>
</div> </div>
<div id="outline-container-org5ed48b8" class="outline-4"> <div id="outline-container-orgdb214f9" class="outline-4">
<h4 id="org5ed48b8"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4> <h4 id="orgdb214f9"><span class="section-number-4">1.2.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-1-2-1"> <div class="outline-text-4" id="text-1-2-1">
<p> <p>
We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness. We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness.
@ -189,8 +189,8 @@ It is shown that the adding of stiffness for the flexible joints does increase a
</div> </div>
</div> </div>
<div id="outline-container-orgddae25e" class="outline-4"> <div id="outline-container-org4069e58" class="outline-4">
<h4 id="orgddae25e"><span class="section-number-4">1.2.2</span> Primary Plant</h4> <h4 id="org4069e58"><span class="section-number-4">1.2.2</span> Primary Plant</h4>
<div class="outline-text-4" id="text-1-2-2"> <div class="outline-text-4" id="text-1-2-2">
<p> <p>
Let&rsquo;s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs). Let&rsquo;s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs).
@ -210,8 +210,8 @@ The plant dynamics is not found to be changing significantly.
</div> </div>
</div> </div>
<div id="outline-container-orgb8a9692" class="outline-4"> <div id="outline-container-orga32adf0" class="outline-4">
<h4 id="orgb8a9692"><span class="section-number-4">1.2.3</span> Conclusion</h4> <h4 id="orga32adf0"><span class="section-number-4">1.2.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-1-2-3"> <div class="outline-text-4" id="text-1-2-3">
<div class="important"> <div class="important">
<p> <p>
@ -248,8 +248,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p> </p>
</div> </div>
<div id="outline-container-org44ccdbe" class="outline-4"> <div id="outline-container-org4adf147" class="outline-4">
<h4 id="org44ccdbe"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4> <h4 id="org4adf147"><span class="section-number-4">1.3.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-1-3-1"> <div class="outline-text-4" id="text-1-3-1">
<p> <p>
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#org8fbbf9d">3</a>. The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#org8fbbf9d">3</a>.
@ -279,8 +279,8 @@ It is shown that the bending stiffness of the flexible joints does indeed change
</div> </div>
</div> </div>
<div id="outline-container-org5d9965b" class="outline-4"> <div id="outline-container-org53e5f08" class="outline-4">
<h4 id="org5d9965b"><span class="section-number-4">1.3.2</span> Primary Control</h4> <h4 id="org53e5f08"><span class="section-number-4">1.3.2</span> Primary Control</h4>
<div class="outline-text-4" id="text-1-3-2"> <div class="outline-text-4" id="text-1-3-2">
<p> <p>
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#orgb739560">5</a>. The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#orgb739560">5</a>.
@ -299,8 +299,8 @@ It is shown that the bending stiffness of the flexible joints have very little i
</div> </div>
</div> </div>
<div id="outline-container-org0f9f990" class="outline-4"> <div id="outline-container-orgc45ccb0" class="outline-4">
<h4 id="org0f9f990"><span class="section-number-4">1.3.3</span> Conclusion</h4> <h4 id="orgc45ccb0"><span class="section-number-4">1.3.3</span> Conclusion</h4>
<div class="outline-text-4" id="text-1-3-3"> <div class="outline-text-4" id="text-1-3-3">
<div class="important"> <div class="important">
<p> <p>
@ -313,8 +313,8 @@ The bending stiffness of the flexible joint does not significantly change the dy
</div> </div>
</div> </div>
<div id="outline-container-org81f1d95" class="outline-2"> <div id="outline-container-orgdaf7b6c" class="outline-2">
<h2 id="org81f1d95"><span class="section-number-2">2</span> Translation Stiffness</h2> <h2 id="orgdaf7b6c"><span class="section-number-2">2</span> Axial Stiffness</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="org8f4d83b"></a> <a id="org8f4d83b"></a>
@ -341,8 +341,8 @@ Cz_M = 1*ones(6,1); % [N/(m/s)]
</div> </div>
</div> </div>
<div id="outline-container-org8fdef7f" class="outline-4"> <div id="outline-container-orge82a7c2" class="outline-4">
<h4 id="org8fdef7f"><span class="section-number-4">2.1.1</span> Initialization</h4> <h4 id="orge82a7c2"><span class="section-number-4">2.1.1</span> Initialization</h4>
<div class="outline-text-4" id="text-2-1-1"> <div class="outline-text-4" id="text-2-1-1">
<p> <p>
Let&rsquo;s initialize all the stages with default parameters. Let&rsquo;s initialize all the stages with default parameters.
@ -370,8 +370,8 @@ initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
</div> </div>
</div> </div>
<div id="outline-container-orgc087bb9" class="outline-4"> <div id="outline-container-org44f67b8" class="outline-4">
<h4 id="orgc087bb9"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4> <h4 id="org44f67b8"><span class="section-number-4">2.1.2</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-2-1-2"> <div class="outline-text-4" id="text-2-1-2">
<p> <p>
The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified. The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified.
@ -390,8 +390,8 @@ The obtained dynamics are shown in Figure <a href="#org78dd87a">6</a>.
</div> </div>
</div> </div>
<div id="outline-container-org4069e58" class="outline-4"> <div id="outline-container-orgd5fd59b" class="outline-4">
<h4 id="org4069e58"><span class="section-number-4">2.1.3</span> Primary Plant</h4> <h4 id="orgd5fd59b"><span class="section-number-4">2.1.3</span> Primary Plant</h4>
<div class="outline-text-4" id="text-2-1-3"> <div class="outline-text-4" id="text-2-1-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6); <pre class="src src-matlab">Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
@ -415,8 +415,8 @@ The dynamics is compare with and without the joint flexibility in Figure <a href
</div> </div>
</div> </div>
<div id="outline-container-org3d8a1a7" class="outline-4"> <div id="outline-container-org552093a" class="outline-4">
<h4 id="org3d8a1a7"><span class="section-number-4">2.1.4</span> Conclusion</h4> <h4 id="org552093a"><span class="section-number-4">2.1.4</span> Conclusion</h4>
<div class="outline-text-4" id="text-2-1-4"> <div class="outline-text-4" id="text-2-1-4">
<div class="important"> <div class="important">
<p> <p>
@ -448,8 +448,8 @@ We also consider here a nano-hexapod with the identified optimal actuator stiffn
</p> </p>
</div> </div>
<div id="outline-container-orgdb214f9" class="outline-4"> <div id="outline-container-orge13b41c" class="outline-4">
<h4 id="orgdb214f9"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4> <h4 id="orge13b41c"><span class="section-number-4">2.2.1</span> Direct Velocity Feedback</h4>
<div class="outline-text-4" id="text-2-2-1"> <div class="outline-text-4" id="text-2-2-1">
<p> <p>
The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#orgab9ab86">8</a>. The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure <a href="#orgab9ab86">8</a>.
@ -491,8 +491,8 @@ It can be seen that very little active damping can be achieve for axial stiffnes
</div> </div>
</div> </div>
<div id="outline-container-org53e5f08" class="outline-4"> <div id="outline-container-org15c2c08" class="outline-4">
<h4 id="org53e5f08"><span class="section-number-4">2.2.2</span> Primary Control</h4> <h4 id="org15c2c08"><span class="section-number-4">2.2.2</span> Primary Control</h4>
<div class="outline-text-4" id="text-2-2-2"> <div class="outline-text-4" id="text-2-2-2">
<p> <p>
The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#org6070692">11</a>. The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure <a href="#org6070692">11</a>.
@ -508,8 +508,8 @@ The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for
</div> </div>
</div> </div>
<div id="outline-container-org1ddd8bf" class="outline-3"> <div id="outline-container-orgce1052e" class="outline-3">
<h3 id="org1ddd8bf"><span class="section-number-3">2.3</span> Conclusion</h3> <h3 id="orgce1052e"><span class="section-number-3">2.3</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<div class="important"> <div class="important">
<p> <p>
@ -533,30 +533,52 @@ We may interpolate the results and say that the axial joint stiffness should be
</div> </div>
</div> </div>
<div id="outline-container-orga32adf0" class="outline-2"> <div id="outline-container-org865157e" class="outline-2">
<h2 id="orga32adf0"><span class="section-number-2">3</span> Conclusion</h2> <h2 id="org865157e"><span class="section-number-2">3</span> Conclusion</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
<a id="org6614f42"></a> <a id="org6614f42"></a>
</p> </p>
<div class="important"> <div class="important">
<p>
In this study we considered the bending, torsional and axial stiffnesses of the flexible joints used for the nano-hexapod.
</p>
<p>
The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the legs.
It is not found to be much problematic for the considered control architecture (it is however, if Integral Force Feedback is to be used).
As a consequence of the added stiffness, it could increase a little bit the required actuator force.
</p>
<p>
The axial stiffness of the flexible joints can be very problematic for control.
Small values of the axial stiffness are shown to limit the achievable damping with Direct Velocity Feedback.
The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod.
</p>
<p> <p>
For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties: For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>Axial Stiffness: \(K_a > 10^7\,[N/m]\)</li>
<li>Bending Stiffness: \(K_b < 50\,[Nm/rad]\)</li> <li>Bending Stiffness: \(K_b < 50\,[Nm/rad]\)</li>
<li>Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)</li> <li>Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)</li>
<li>Axial Stiffness: \(K_a > 10^7\,[N/m]\)</li>
</ul> </ul>
<p>
As there is generally a trade-off between bending stiffness and axial stiffness, it should be highlighted that the <b>axial</b> stiffness is the most important property of the flexible joints.
</p>
</div> </div>
</div> </div>
</div> </div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-05-05 mar. 11:26</p> <p class="date">Created: 2020-05-05 mar. 11:50</p>
</div> </div>
</body> </body>
</html> </html>

24
org/flexible_joints_study.org
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@ -504,7 +504,7 @@ It is shown that the bending stiffness of the flexible joints have very little i
The bending stiffness of the flexible joint does not significantly change the dynamics. The bending stiffness of the flexible joint does not significantly change the dynamics.
#+end_important #+end_important
* Translation Stiffness * Axial Stiffness
<<sec:trans_stiffness>> <<sec:trans_stiffness>>
** Introduction :ignore: ** Introduction :ignore:
@ -993,8 +993,22 @@ The dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the
<<sec:conclusion>> <<sec:conclusion>>
#+begin_important #+begin_important
For the identified optimal actuator stiffness $k = 10^5\,[N/m]$, the flexible joint should have the following stiffness properties: In this study we considered the bending, torsional and axial stiffnesses of the flexible joints used for the nano-hexapod.
- Bending Stiffness: $K_b < 50\,[Nm/rad]$
- Torsion Stiffness: $K_t < 50\,[Nm/rad]$ The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the legs.
- Axial Stiffness: $K_a > 10^7\,[N/m]$ It is not found to be much problematic for the considered control architecture (it is however, if Integral Force Feedback is to be used).
As a consequence of the added stiffness, it could increase a little bit the required actuator force.
The axial stiffness of the flexible joints can be very problematic for control.
Small values of the axial stiffness are shown to limit the achievable damping with Direct Velocity Feedback.
The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod.
For the identified optimal actuator stiffness $k = 10^5\,[N/m]$, the flexible joint should have the following stiffness properties:
- Axial Stiffness: $K_a > 10^7\,[N/m]$
- Bending Stiffness: $K_b < 50\,[Nm/rad]$
- Torsion Stiffness: $K_t < 50\,[Nm/rad]$
As there is generally a trade-off between bending stiffness and axial stiffness, it should be highlighted that the *axial* stiffness is the most important property of the flexible joints.
#+end_important #+end_important