<li>The payload mass and dynamical properties (studied <ahref="uncertainty_payload.html">here</a> and <ahref="uncertainty_experiment.html">here</a>)</li>
<li>The experimental conditions, mainly the spindle rotation speed (studied <ahref="uncertainty_experiment.html">here</a>)</li>
</ul>
<p>
As seen before, the stiffness of the nano-hexapod greatly influence the effect of such parameters.
</p>
<p>
We wish here to see if we can determine an optimal stiffness of the nano-hexapod such that:
</p>
<ulclass="org-ul">
<li>Section <ahref="#org902923f">1</a>: the change of its dynamics due to the spindle rotation speed is acceptable</li>
<li>Section <ahref="#orgabe2ab2">2</a>: the support compliance dynamics is not much present in the nano-hexapod dynamics</li>
<li>Section <ahref="#org2bd8390">3</a>: the change of payload impedance has acceptable effect on the plant dynamics</li>
</ul>
<p>
The overall goal is to design a nano-hexapod that will allow the highest possible control bandwidth.
<h3id="org7dcfddb"><spanclass="section-number-3">1.3</span> Change of dynamics</h3>
<divclass="outline-text-3"id="text-1-3">
<p>
We plot the change of dynamics due to the change of the spindle rotation speed (from 0rpm to 60rpm):
</p>
<ulclass="org-ul">
<li>Figure <ahref="#orgfd21b56">2</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <ahref="#org2a4cc54">3</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <ahref="#orgbf48d68">4</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
<li>Figure <ahref="#org16be775">5</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_y\) (coupling of the centralized positioning plant)</li>
<p><spanclass="figure-number">Figure 1: </span>Root Locus plot for IFF control when not rotating (in red) and when rotating at 60rpm (in blue) for 4 different nano-hexapod stiffnesses (<ahref="./figs/opti_stiffness_iff_root_locus.png">png</a>, <ahref="./figs/opti_stiffness_iff_root_locus.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 2: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (<ahref="./figs/opt_stiffness_wz_iff.png">png</a>, <ahref="./figs/opt_stiffness_wz_iff.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 3: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (<ahref="./figs/opt_stiffness_wz_dvf.png">png</a>, <ahref="./figs/opt_stiffness_wz_dvf.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 4: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (<ahref="./figs/opt_stiffness_wz_fx_dx.png">png</a>, <ahref="./figs/opt_stiffness_wz_fx_dx.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 5: </span>Change of Coupling from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_y\) for a spindle rotation speed from 0rpm to 60rpm (<ahref="./figs/opt_stiffness_wz_coupling.png">png</a>, <ahref="./figs/opt_stiffness_wz_coupling.pdf">pdf</a>)</p>
</div>
</div>
</div>
<divclass="outline-text-2"id="text-1">
<divclass="important">
<p>
The leg stiffness should be at higher than \(k_i = 10^4\ [N/m]\) such that the main resonance frequency does not shift too much when rotating.
For the coupling, it is more difficult to conclude about the minimum required leg stiffness.
</p>
</div>
<divclass="notes">
<p>
Note that we can use very soft nano-hexapod if we limit the spindle rotating speed.
And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.
The diagonal element of the identified Micro-Station compliance matrix are shown in Figure <ahref="#org6cfb14b">6</a>.
<p><spanclass="figure-number">Figure 6: </span>Identified Compliance of the Micro-Station (<ahref="./figs/opt_stiff_micro_station_compliance.png">png</a>, <ahref="./figs/opt_stiff_micro_station_compliance.pdf">pdf</a>)</p>
We plot the change of dynamics due to the compliance of the Micro-Station.
The solid curves are corresponding to the nano-hexapod without the micro-station, and the dashed curves with the micro-station:
</p>
<ulclass="org-ul">
<li>Figure <ahref="#org71f5400">7</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <ahref="#org32aef29">8</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <ahref="#org8a33fed">9</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
<li>Figure <ahref="#orge9bd08b">10</a>: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)</li>
<p><spanclass="figure-number">Figure 7: </span>Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) due to the micro-station compliance (<ahref="./figs/opt_stiffness_micro_station_iff.png">png</a>, <ahref="./figs/opt_stiffness_micro_station_iff.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 8: </span>Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) due to the micro-station compliance (<ahref="./figs/opt_stiffness_micro_station_dvf.png">png</a>, <ahref="./figs/opt_stiffness_micro_station_dvf.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 9: </span>Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance (<ahref="./figs/opt_stiffness_micro_station_fx_dx.png">png</a>, <ahref="./figs/opt_stiffness_micro_station_fx_dx.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 10: </span>Change of dynamics from force \(\mathcal{F}_z\) to displacement \(\mathcal{X}_z\) due to the micro-station compliance (<ahref="./figs/opt_stiffness_micro_station_fz_dz.png">png</a>, <ahref="./figs/opt_stiffness_micro_station_fz_dz.pdf">pdf</a>)</p>
</div>
</div>
</div>
<divclass="outline-text-2"id="text-2">
<divclass="important">
<p>
The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when the stiffness of the legs is less than \(10^6\ [N/m]\).
When the nano-hexapod is stiff (\(k>10^7\ [N/m]\)), the compliance of the micro-station appears in the primary plant.
<h4id="orgb44d421"><spanclass="section-number-4">3.3.1</span> Frequency variation</h4>
<divclass="outline-text-4"id="text-3-3-1">
<p>
We here compare the dynamics for the same payload mass, but different stiffness resulting in different resonance frequency of the payload:
</p>
<ulclass="org-ul">
<li>Figure <ahref="#org00db693">11</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
<li>Figure <ahref="#org76716ad">12</a>: same, but for all tested nano-hexapod stiffnesses</li>
</ul>
<p>
We can see two mass lines for the soft nano-hexapod (Figure <ahref="#org00db693">11</a>):
</p>
<ulclass="org-ul">
<li>The first mass line corresponds to \(\frac{1}{(m_n + m_p)s^2}\) where \(m_p = 10\ [kg]\) is the mass of the payload and \(m_n = 15\ [Kg]\) is the mass of the nano-hexapod top platform and attached mirror</li>
<li>The second mass line corresponds to \(\frac{1}{m_n s^2}\)</li>
<li>The zero corresponds to the resonance of the payload alone (fixed nano-hexapod’s top platform)</li>
<p><spanclass="figure-number">Figure 11: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod (<ahref="./figs/opt_stiffness_payload_freq_fz_dz.png">png</a>, <ahref="./figs/opt_stiffness_payload_freq_fz_dz.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 12: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency (<ahref="./figs/opt_stiffness_payload_freq_all.png">png</a>, <ahref="./figs/opt_stiffness_payload_freq_all.pdf">pdf</a>)</p>
<h4id="orgfc270b0"><spanclass="section-number-4">3.3.2</span> Mass variation</h4>
<divclass="outline-text-4"id="text-3-3-2">
<p>
We here compare the dynamics for different payload mass with the same resonance frequency (100Hz):
</p>
<ulclass="org-ul">
<li>Figure <ahref="#orga1343a7">13</a>: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.</li>
<li>Figure <ahref="#org35aebae">14</a>: same, but for all tested nano-hexapod stiffnesses</li>
</ul>
<p>
We can see here that for the soft nano-hexapod:
</p>
<ulclass="org-ul">
<li>the first resonance \(\omega_n\) is changing with the mass of the payload as \(\omega_n = \sqrt{\frac{k_n}{m_p + m_n}}\) with \(k_p\) the stiffness of the nano-hexapod, \(m_p\) the payload’s mass and \(m_n\) the mass of the nano-hexapod top platform</li>
<li>the first mass line corresponding to \(\frac{1}{(m_p + m_n)s^2}\) is changing with the payload mass</li>
<li>the zero at 100Hz is not changing as it corresponds to the resonance of the payload itself</li>
<p><spanclass="figure-number">Figure 13: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod (<ahref="./figs/opt_stiffness_payload_mass_fz_dz.png">png</a>, <ahref="./figs/opt_stiffness_payload_mass_fz_dz.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 14: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass (<ahref="./figs/opt_stiffness_payload_mass_all.png">png</a>, <ahref="./figs/opt_stiffness_payload_mass_all.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 15: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<ahref="./figs/opt_stiffness_payload_impedance_all_fz_dz.png">png</a>, <ahref="./figs/opt_stiffness_payload_impedance_all_fz_dz.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 16: </span>Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (<ahref="./figs/opt_stiffness_payload_impedance_fz_dz.png">png</a>, <ahref="./figs/opt_stiffness_payload_impedance_fz_dz.pdf">pdf</a>)</p>
<h2id="org973d2e3"><spanclass="section-number-2">4</span> Total Change of dynamics</h2>
<divclass="outline-text-2"id="text-4">
<p>
We now consider the total change of nano-hexapod dynamics due to:
</p>
<ulclass="org-ul">
<li><code>Gk_wz_err</code> - Change of spindle rotation speed</li>
<li><code>Gf_err</code> and <code>Gm_err</code> - Change of payload resonance</li>
<li><code>Gmf_err</code> and <code>Gmr_err</code> - Micro-Station compliance</li>
</ul>
<p>
The obtained dynamics are shown:
</p>
<ulclass="org-ul">
<li>Figure <ahref="#orgcf64eb6">17</a> for a stiffness \(k = 10^3\ [N/m]\)</li>
<li>Figure <ahref="#org175cc57">18</a> for a stiffness \(k = 10^5\ [N/m]\)</li>
<li>Figure <ahref="#org998cf87">19</a> for a stiffness \(k = 10^7\ [N/m]\)</li>
<li>Figure <ahref="#orgd3db91c">20</a> for a stiffness \(k = 10^9\ [N/m]\)</li>
</ul>
<p>
And finally, in Figures <ahref="#orge05feb5">21</a> and <ahref="#org17c5c95">22</a> are shown an animation of the change of dynamics with the nano-hexapod’s stiffness.
<p><spanclass="figure-number">Figure 17: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^3\ [N/m]\) (<ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.png">png</a>, <ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e3.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 18: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^5\ [N/m]\) (<ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.png">png</a>, <ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e5.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 19: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^7\ [N/m]\) (<ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.png">png</a>, <ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e7.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 20: </span>Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^9\ [N/m]\) (<ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.png">png</a>, <ahref="./figs/opt_stiffness_plant_dynamics_fx_dx_k_1e9.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 21: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
<p><spanclass="figure-number">Figure 22: </span>Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness</p>
