<p><spanclass="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 <ahref="#orgc4acef7">2</a>.
</p>
<p>
This can we considered as:
</p>
<ulclass="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>
<p><spanclass="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 \(\bm{\mathcal{X}}\).
<h2id="org15699e9"><spanclass="section-number-2">1</span> Control Configuration - Introduction</h2>
<divclass="outline-text-2"id="text-1">
<p>
In this section, we discuss the control configuration for the NASS.
</p>
<p>
From <aclass='org-ref-reference'href="#skogestad07_multiv_feedb_contr">skogestad07_multiv_feedb_contr</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>
<ulclass="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 <ahref="#org251e3c9">2</a>) and for Active Damping techniques (Section <ahref="#org1b3cc21">3</a>)
</p>
<p>
Combining both can be done in an HAC-LAC topology presented in Section <ahref="#org31fa800">4</a>.
</p>
<p>
The use of decentralized controllers is proposed in Section <ahref="#orga038762">5</a>.
<h2id="orgfb6ef96"><spanclass="section-number-2">2</span> Tracking Control - Basic Architectures</h2>
<divclass="outline-text-2"id="text-2">
<p>
<aid="org251e3c9"></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.
<h3id="org970ab39"><spanclass="section-number-3">2.1</span> Control in the frame of the Legs</h3>
<divclass="outline-text-3"id="text-2-1">
<p>
<aid="org8583193"></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.
<h3id="org82193fb"><spanclass="section-number-3">2.2</span> Control in the Cartesian frame</h3>
<divclass="outline-text-3"id="text-2-2">
<p>
<aid="orgbd7e263"></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 \(\bm{J}^{-T}\).
<h2id="orgbfd345b"><spanclass="section-number-2">3</span> Active Damping Architecture - Collocated Control</h2>
<divclass="outline-text-2"id="text-3">
<p>
<aid="org1b3cc21"></a>
</p>
<p>
From <aclass='org-ref-reference'href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</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.
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 <aclass='org-ref-reference'href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</a>:
</p>
<blockquote>
<p>
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <ahref="#org1b2c5c7">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>
<ulclass="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>
The principle of Cascade control is shown in Figure <ahref="#org03ef231">11</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 <ahref="#org03ef231">11</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. –<aclass='org-ref-reference'href="#taghirad13_paral">taghirad13_paral</a>
<p><spanclass="figure-number">Figure 11: </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>
<ulclass="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><spanclass="figure-number">Figure 12: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
<p><spanclass="figure-number">Figure 13: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the Cartesian Frame</p>
<ulclass='org-ref-bib'><li><aid="skogestad07_multiv_feedb_contr">[skogestad07_multiv_feedb_contr]</a><aname="skogestad07_multiv_feedb_contr"></a>Skogestad & Postlethwaite, Multivariable Feedback Control: Analysis and Design, John Wiley (2007).</li>
<li><aid="preumont18_vibrat_contr_activ_struc_fourt_edition">[preumont18_vibrat_contr_activ_struc_fourt_edition]</a><aname="preumont18_vibrat_contr_activ_struc_fourt_edition"></a>Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).</li>
<li><aid="taghirad13_paral">[taghirad13_paral]</a><aname="taghirad13_paral"></a>Taghirad, Parallel robots : mechanics and control, CRC Press (2013).</li>
