<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="#org1f375bb">1</a>: the change of its dynamics due to the spindle rotation speed is acceptable</li>
<li>Section <ahref="#orgcf5798c">2</a>: the support compliance dynamics is not much present in the nano-hexapod dynamics</li>
<li>Section <ahref="#org1a9d09c">3</a>: the change of payload impedance has acceptable effect on the plant dynamics</li>
<li>Figure <ahref="#org2b01fbe">2</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <ahref="#org9bfe588">3</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <ahref="#org9f887c8">4</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
<li>Figure <ahref="#org5926aca">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>
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.
<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>
<li>Figure <ahref="#org6257db9">7</a>: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)</li>
<li>Figure <ahref="#orgcea6354">8</a>: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)</li>
<li>Figure <ahref="#org5c6f89c">9</a>: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)</li>
<li>Figure <ahref="#org7f2360c">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>
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.
<li>Figure <ahref="#orgcce8740">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="#org87ecfc8">12</a>: same, but for all tested nano-hexapod stiffnesses</li>
<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>
<li>Figure <ahref="#org4eb6bfc">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="#org6e013a7">14</a>: same, but for all tested nano-hexapod stiffnesses</li>
<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>
And finally, in Figures <ahref="#org0a4e875">21</a> and <ahref="#orge890242">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>
