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<h1 class="title">Control of the Nano-Active-Stabilization-System</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org25c471e">1. Control Configuration - Introduction</a></li>
<li><a href="#orgcd0731f">2. Tracking Control in the Frame of the Nano-Hexapod - Basic Architectures</a>
<ul>
<li><a href="#orgbc69eea">2.1. Control in the frame of the Legs</a></li>
<li><a href="#org65bc213">2.2. Control in the Cartesian frame</a></li>
</ul>
</li>
<li><a href="#org9ef6b25">3. Active Damping Architecture - Collocated Control (link)</a>
<ul>
<li><a href="#orge25231e">3.1. Integral Force Feedback</a></li>
<li><a href="#org66ad123">3.2. Direct Relative Velocity Feedback</a></li>
</ul>
</li>
<li><a href="#orgf83465a">4. HAC-LAC Architectures (link)</a>
<ul>
<li><a href="#org3a2dfa2">4.1. HAC-LAC using IFF and Tracking control in the frame of the Legs</a></li>
<li><a href="#org27fd54b">4.2. HAC-LAC using IFF and Tracking control in the Cartesian frame</a></li>
<li><a href="#org8454531">4.3. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the task space</a></li>
<li><a href="#org89a2695">4.4. HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</a></li>
<li><a href="#orgac21cc9">4.5. HAC-LAC using DVF - the HAC controller is positioning the sample w.r.t. the granite in the task space</a></li>
<li><a href="#org6676bde">4.6. HAC-LAC using DVF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</a></li>
</ul>
</li>
<li><a href="#orge5dd5fd">5. Cascade Architectures (link)</a>
<ul>
<li><a href="#org9b331a4">5.1. Cascade Control with HAC-LAC Inner Loop and Primary Controller in the task space</a></li>
<li><a href="#org2f8d9f9">5.2. Cascade Control with HAC-LAC Inner Loop and Primary Controller in the joint space</a></li>
</ul>
</li>
<li><a href="#org3a980c5">6. Force Control (link)</a></li>
<li><a href="#org73661f1">7. Other Control Architectures</a>
<ul>
<li><a href="#org26f61ba">7.1. Control to force the nano-hexapod to not do any vertical rotation</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
The system consist of the following inputs and outputs (Figure <a href="#org37cd4b0">1</a>):
</p>
<ul class="org-ul">
<li>\(\bm{\tau}\): Forces applied in each leg</li>
<li>\(\bm{\tau}_m\): Force sensor located in each leg</li>
<li>\(\bm{\mathcal{X}}\): Measurement of the payload position with respect to the granite</li>
<li>\(d\bm{\mathcal{L}}\): Measurement of the (small) relative motion of each leg</li>
</ul>
<div id="org37cd4b0" class="figure">
<p><img src="figs/control_architecture_plant.png" alt="control_architecture_plant.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Block diagram with the inputs and outputs of the system</p>
</div>
<p>
In order to position the Sample with respect to the granite, we must use the measurement \(\bm{\mathcal{X}}\) in the control loop.
The wanted position of the sample with respect to the granite is represented by \(\bm{r}_\mathcal{X}\).
From \(\bm{r}_\mathcal{X}\) and \(\bm{\mathcal{X}}\), we can compute the required small change of pose of the nano-hexapod’s top platform expressed in the frame of the nano-hexapod’s base as shown in Figure <a href="#orgb843e60">2</a>.
</p>
<p>
This can we considered as:
</p>
<ul class="org-ul">
<li>the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) expressed in a frame attach to the base of the nano-hexapod</li>
<li>the wanted (small) pose displacement \(\bm{r}_{d\mathcal{X}_n}\) of the nano-hexapod mobile platform with respect to its base</li>
</ul>
<div id="orgb843e60" class="figure">
<p><img src="figs/control_architecture_pos_error.png" alt="control_architecture_pos_error.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Block diagram corresponding to the computation of the position error in the frame of the nano-hexapod</p>
</div>
<p>
In this document, we see how the different outputs of the system can be used to control of position \(\bm{\mathcal{X}}\).
</p>
<div id="outline-container-org25c471e" class="outline-2">
<h2 id="org25c471e"><span class="section-number-2">1</span> Control Configuration - Introduction</h2>
<div class="outline-text-2" id="text-1">
<p>
In this section, we discuss the control configuration for the NASS.
</p>
<p>
From (<a href="#citeproc_bib_item_2">Skogestad and Postlethwaite 2007</a>):
</p>
<blockquote>
<p>
We define the <b>control configuration</b> to be the restrictions imposed on the overall controller \(K\) by decomposing it into a set of <b>local controllers</b> with predetermined links and with a possibly predetermined design sequence where subcontrollers are designed locally.
</p>
<p>
Some elements used to build up a specific control configuration are:
</p>
<ul class="org-ul">
<li><b>Cascade controllers</b>. The output from one controller is the input to another</li>
<li><b>Decentralized controllers</b>. The control system consists of independent feedback controllers which interconnect a subset of the output measurements with a subset of the manipulated inputs.
These subsets should not be used by any other controller</li>
<li><b>Feedforward elements</b>. Link measured disturbances and manipulated inputs</li>
<li><b>Decoupling elements</b>. Link one set of manipulated inputs with another set of manipulated inputs.
They are used to improve the performance of decentralized control systems.</li>
</ul>
</blockquote>
<p>
Decoupling elements will be used to convert quantities from the joint space to the task space and vice-versa.
</p>
<p>
Decentralized controllers will be largely used both for Tracking control (Section <a href="#orga1c5122">2</a>) and for Active Damping techniques (Section <a href="#orgaf5a850">3</a>)
</p>
<p>
Combining both can be done in an HAC-LAC topology presented in Section <a href="#org4b1b4af">4</a>.
</p>
<p>
The use of decentralized controllers is proposed in Section <a href="#org697801a">5</a>.
</p>
</div>
</div>
<div id="outline-container-orgcd0731f" class="outline-2">
<h2 id="orgcd0731f"><span class="section-number-2">2</span> Tracking Control in the Frame of the Nano-Hexapod - Basic Architectures</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orga1c5122"></a>
</p>
<p>
In this section, we suppose that we want to track some reference position \(\bm{r}_{\mathcal{X}_n}\) corresponding to the pose of the nano-hexapod’s mobile platform with respect to its fixed base.
</p>
<p>
To do so, we have to the use the leg’s length measurement \(d\bm{\mathcal{L}}\).
</p>
<p>
However, thanks to the forward and inverse kinematics, the controller can either be designed in the task space or in the joint space.
</p>
<p>
These to configuration are described in the next two sections.
</p>
</div>
<div id="outline-container-orgbc69eea" class="outline-3">
<h3 id="orgbc69eea"><span class="section-number-3">2.1</span> Control in the frame of the Legs</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<a id="org92ab294"></a>
</p>
<p>
From the wanted small change in pose of the nano-hexapod’s mobile platform \(\bm{r}_{d\mathcal{X}_n}\), we can use the Inverse Kinematics of the nano-hexapod to compute the corresponding small change of the leg length of the nano-hexapod \(\bm{r}_{d\mathcal{L}}\).
Then, this is subtracted by the measurement of the leg relative displacement \(d\bm{\mathcal{L}}\) to obtain to displacement error of each leg \(\bm{\epsilon}_{d\mathcal{L}}\).
Finally, a diagonal (Decentralized) controller \(\bm{K}_\mathcal{L}\) can be used.
</p>
<div id="org6c88afe" class="figure">
<p><img src="figs/control_architecture_leg_frame.png" alt="control_architecture_leg_frame.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Control in the frame of the legs</p>
</div>
</div>
</div>
<div id="outline-container-org65bc213" class="outline-3">
<h3 id="org65bc213"><span class="section-number-3">2.2</span> Control in the Cartesian frame</h3>
<div class="outline-text-3" id="text-2-2">
<p>
<a id="orgd4d12e5"></a>
</p>
<p>
From the relative displacement of each leg \(d\bm{\mathcal{L}}\), the pose of the nano-hexapod’s mobile platform \(\bm{\mathcal{X}_n}\) is estimated.
It is then subtracted from reference pose of the nano-hexapod \(\bm{r}_{\mathcal{X}_n}\) to obtain the pose error \(\bm{\epsilon}_{\mathcal{X}_n}\).
A diagonal controller \(\bm{K}_\mathcal{X}\) is used to generate forces and torques applied on the payload in a frame attached to the nano-hexapod’s base.
These forces are then converted to forces applied in each of the nano-hexapod’s actuators by the use of the Jacobian \(\bm{J}^{-T}\).
</p>
<div id="orga34a56c" class="figure">
<p><img src="figs/control_architecture_cartesian_frame.png" alt="control_architecture_cartesian_frame.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Control in the cartesian Frame (rotating frame attached to the nano-hexapod’s base)</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org9ef6b25" class="outline-2">
<h2 id="org9ef6b25"><span class="section-number-2">3</span> Active Damping Architecture - Collocated Control (<a href="control_active_damping.html">link</a>)</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="orgaf5a850"></a>
</p>
<p>
From (<a href="#citeproc_bib_item_1">Preumont 2018</a>):
</p>
<blockquote>
<p>
Active damping is very effective in reducing the settling time of transient disturbances and the effect of steady state disturbances near the resonance frequencies of the system; however, away from the resonances, the active damping is completely ineffective and leaves the closed-loop response essentially unchanged.
Such low-gain controllers are often called Low Authority Controllers (LAC), because they modify the poles of the system only slightly.
</p>
</blockquote>
<p>
Two very well known active damping techniques are <b>Integral Force Feedback</b> and <b>Direct Velocity Feedback</b>.
</p>
<p>
These two active damping techniques are collocated control techniques.
</p>
<p>
The active damping techniques are studied in <a href="control_active_damping.html">this</a> document.
</p>
</div>
<div id="outline-container-orge25231e" class="outline-3">
<h3 id="orge25231e"><span class="section-number-3">3.1</span> Integral Force Feedback</h3>
<div class="outline-text-3" id="text-3-1">
<p>
<a id="org71c8197"></a>
</p>
<p>
In this active damping technique, the force sensors in each leg is used.
</p>
<p>
The controller \(\bm{K}_\text{IFF}\) is a diagonal matrix, each of its diagonal element consists of:
</p>
<ul class="org-ul">
<li>an pure integrator</li>
<li>a gain \(g\) that can be tuned to achieve a maximum damping</li>
</ul>
\begin{equation}
\bm{K}_\text{IFF}(s) = \frac{g}{s} \bm{I}_{6}
\end{equation}
<p>
A lead-lag can also be used instead of a pure integrator.
</p>
<div id="orga842725" class="figure">
<p><img src="figs/control_architecture_iff.png" alt="control_architecture_iff.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Integral Force Feedback</p>
</div>
</div>
</div>
<div id="outline-container-org66ad123" class="outline-3">
<h3 id="org66ad123"><span class="section-number-3">3.2</span> Direct Relative Velocity Feedback</h3>
<div class="outline-text-3" id="text-3-2">
<p>
<a id="org4acc137"></a>
</p>
<p>
The controller \(\bm{K}_\text{DVF}\) is a diagonal matrix.
Each diagonal element consists of:
</p>
<ul class="org-ul">
<li>a derivative action up to some frequency \(\omega_0\)</li>
<li>a gain \(g\) that can be tuned to achieve a maximum damping</li>
</ul>
\begin{equation}
\bm{K}_\text{DVF}(s) = \frac{g s}{\omega_0 + s} \bm{I}_{6}
\end{equation}
<div id="org76615cc" class="figure">
<p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Direct Velocity Feedback</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgf83465a" class="outline-2">
<h2 id="orgf83465a"><span class="section-number-2">4</span> HAC-LAC Architectures (<a href="control_hac_lac.html">link</a>)</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org4b1b4af"></a>
</p>
<p>
Here we can combine Active Damping Techniques (Low authority control) with a tracking controller (high authority control).
Usually, the low authority controller is designed first, and the high authority controller is designed based on the damped plant.
</p>
<p>
From (<a href="#citeproc_bib_item_1">Preumont 2018</a>):
</p>
<blockquote>
<p>
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <a href="#orgfe203dc">7</a>.
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:
</p>
<ul class="org-ul">
<li>The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth</li>
<li>The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)</li>
<li>The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)</li>
</ul>
</blockquote>
<div id="orgfe203dc" class="figure">
<p><img src="figs/control_architecture_hac_lac.png" alt="control_architecture_hac_lac.png" />
</p>
<p><span class="figure-number">Figure 7: </span>HAC-LAC Control Architecture</p>
</div>
<p>
If there is only one input to the system, the HAC-LAC topology can be represented as depicted in Figure <a href="#org8e5c9da">8</a>.
Usually, the Low Authority Controller is first design, and then the High Authority Controller is designed based on the damped plant.
</p>
<div id="org8e5c9da" class="figure">
<p><img src="figs/control_architecture_hac_lac_one_input.png" alt="control_architecture_hac_lac_one_input.png" />
</p>
<p><span class="figure-number">Figure 8: </span>HAC-LAC Architecture with a system having only one input</p>
</div>
</div>
<div id="outline-container-org3a2dfa2" class="outline-3">
<h3 id="org3a2dfa2"><span class="section-number-3">4.1</span> HAC-LAC using IFF and Tracking control in the frame of the Legs</h3>
<div class="outline-text-3" id="text-4-1">
<div id="org259b2b4" class="figure">
<p><img src="figs/control_architecture_hac_iff_L.png" alt="control_architecture_hac_iff_L.png" />
</p>
<p><span class="figure-number">Figure 9: </span>IFF + Control in the frame of the legs</p>
</div>
</div>
</div>
<div id="outline-container-org27fd54b" class="outline-3">
<h3 id="org27fd54b"><span class="section-number-3">4.2</span> HAC-LAC using IFF and Tracking control in the Cartesian frame</h3>
<div class="outline-text-3" id="text-4-2">
<div id="org0bde593" class="figure">
<p><img src="figs/control_architecture_hac_iff_X.png" alt="control_architecture_hac_iff_X.png" />
</p>
<p><span class="figure-number">Figure 10: </span>IFF + Control in the cartesian frame</p>
</div>
</div>
</div>
<div id="outline-container-org8454531" class="outline-3">
<h3 id="org8454531"><span class="section-number-3">4.3</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the task space</h3>
<div class="outline-text-3" id="text-4-3">
<div id="orgd88bdcb" class="figure">
<p><img src="figs/control_architecture_hac_iff_pos_X.png" alt="control_architecture_hac_iff_pos_X.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org89a2695" class="outline-3">
<h3 id="org89a2695"><span class="section-number-3">4.4</span> HAC-LAC using IFF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</h3>
<div class="outline-text-3" id="text-4-4">
<div id="orgc4e63e2" class="figure">
<p><img src="figs/control_architecture_hac_iff_pos_L.png" alt="control_architecture_hac_iff_pos_L.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-orgac21cc9" class="outline-3">
<h3 id="orgac21cc9"><span class="section-number-3">4.5</span> HAC-LAC using DVF - the HAC controller is positioning the sample w.r.t. the granite in the task space</h3>
<div class="outline-text-3" id="text-4-5">
<div id="org4e63dc4" class="figure">
<p><img src="figs/control_architecture_hac_dvf_pos_X.png" alt="control_architecture_hac_dvf_pos_X.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org6676bde" class="outline-3">
<h3 id="org6676bde"><span class="section-number-3">4.6</span> HAC-LAC using DVF - the HAC controller is positioning the sample w.r.t. the granite in the space of the legs</h3>
<div class="outline-text-3" id="text-4-6">
<div id="org2cc76e1" class="figure">
<p><img src="figs/control_architecture_hac_dvf_pos_L.png" alt="control_architecture_hac_dvf_pos_L.png" />
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orge5dd5fd" class="outline-2">
<h2 id="orge5dd5fd"><span class="section-number-2">5</span> Cascade Architectures (<a href="control_cascade.html">link</a>)</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org697801a"></a>
</p>
<p>
The principle of Cascade control is shown in Figure <a href="#org8e45511">15</a> and explained as follow:
</p>
<blockquote>
<p>
To follow <b>two objectives</b> with different properties in one control system, usually a <b>hierarchy</b> of two feedback loops is used in practice.
This kind of control topology is called <b>cascade control</b>, which is used when there are <b>several measurements and one prime control variable</b>.
Cascade control is implemented by <b>nesting</b> the control loops, as shown in Figure <a href="#org8e45511">15</a>.
The output control loop is called the <b>primary loop</b>, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system. – (<a href="#citeproc_bib_item_3">Taghirad 2013</a>)
</p>
</blockquote>
<div id="org8e45511" class="figure">
<p><img src="figs/control_architecture_cascade_control.png" alt="control_architecture_cascade_control.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Cascade Control Architecture</p>
</div>
<p>
This control topology seems adapted for the NASS, as indeed we have more inputs than outputs
</p>
<p>
In the NASS’s case:
</p>
<ul class="org-ul">
<li>The primary objective is to position the sample with respect to the granite, thus the outer loop (and primary controller) should corresponds to a motion control loop</li>
</ul>
<p>
The inner loop can be composed of the system controlled with the HAC-LAC topology.
</p>
</div>
<div id="outline-container-org9b331a4" class="outline-3">
<h3 id="org9b331a4"><span class="section-number-3">5.1</span> Cascade Control with HAC-LAC Inner Loop and Primary Controller in the task space</h3>
<div class="outline-text-3" id="text-5-1">
<div id="orge54ab8a" class="figure">
<p><img src="figs/control_architecture_cascade_L.png" alt="control_architecture_cascade_L.png" />
</p>
<p><span class="figure-number">Figure 16: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
</div>
</div>
</div>
<div id="outline-container-org2f8d9f9" class="outline-3">
<h3 id="org2f8d9f9"><span class="section-number-3">5.2</span> Cascade Control with HAC-LAC Inner Loop and Primary Controller in the joint space</h3>
<div class="outline-text-3" id="text-5-2">
<div id="orgdb3211a" class="figure">
<p><img src="figs/control_architecture_cascade_X.png" alt="control_architecture_cascade_X.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the Cartesian Frame</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org3a980c5" class="outline-2">
<h2 id="org3a980c5"><span class="section-number-2">6</span> Force Control (<a href="control_force.html">link</a>)</h2>
<div class="outline-text-2" id="text-6">
<p>
Signals:
</p>
<ul class="org-ul">
<li>\(\bm{r}_\mathcal{F}\) is the wanted total force/torque to be applied to the payload</li>
<li>\(\bm{\epsilon}_\mathcal{F}\) is the force/torque errors that should be applied to the payload</li>
<li>\(\bm{\tau}\) is the force applied in each actuator</li>
</ul>
<div id="org17f57fd" class="figure">
<p><img src="figs/control_architecture_force.png" alt="control_architecture_force.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org73661f1" class="outline-2">
<h2 id="org73661f1"><span class="section-number-2">7</span> Other Control Architectures</h2>
<div class="outline-text-2" id="text-7">
</div>
<div id="outline-container-org26f61ba" class="outline-3">
<h3 id="org26f61ba"><span class="section-number-3">7.1</span> Control to force the nano-hexapod to not do any vertical rotation</h3>
<div class="outline-text-3" id="text-7-1">
<p>
As the sample rotation around the vertical axis is not measure, the best we can do with the nano-hexapod is to not rotate around this same axis.
</p>
<p>
One way to do it is shown in Figure <a href="#org6559cc5">19</a>.
</p>
<p>
The controller \(\bm{K}_{R_z}\) is decomposed as shown in Figure <a href="#org1d551e2">20</a>.
</p>
<div id="org6559cc5" class="figure">
<p><img src="figs/control_architecture_fixed_rz.png" alt="control_architecture_fixed_rz.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Figure caption</p>
</div>
<div id="org1d551e2" class="figure">
<p><img src="figs/control_architecture_fixed_Krz.png" alt="control_architecture_fixed_Krz.png" />
</p>
<p><span class="figure-number">Figure 20: </span>Figure caption</p>
</div>
</div>
</div>
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<p>
</p>
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
<div class="csl-bib-body">
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. <a href="https://doi.org/10.1007/978-3-319-72296-2">https://doi.org/10.1007/978-3-319-72296-2</a>.</div>
<div class="csl-entry"><a name="citeproc_bib_item_2"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. <i>Multivariable Feedback Control: Analysis and Design</i>. John Wiley.</div>
<div class="csl-entry"><a name="citeproc_bib_item_3"></a>Taghirad, Hamid. 2013. <i>Parallel Robots : Mechanics and Control</i>. Boca Raton, FL: CRC Press.</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-02-20 sam. 23:08</p>
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