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< h1 class = "title" > Study of the Flexible Joints< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orge032d30" > 1. Bending and Torsional Stiffness< / a >
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< ul >
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< li > < a href = "#org4af6fbb" > 1.1. Initialization< / a > < / li >
< li > < a href = "#orgde60939" > 1.2. Realistic Bending Stiffness Values< / a >
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< ul >
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< li > < a href = "#org1f64e69" > 1.2.1. Direct Velocity Feedback< / a > < / li >
< li > < a href = "#org7eb4054" > 1.2.2. Primary Plant< / a > < / li >
< li > < a href = "#org81a1a77" > 1.2.3. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org8ad3f34" > 1.3. Parametric Study< / a >
< ul >
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< li > < a href = "#org1575b3d" > 1.3.1. Direct Velocity Feedback< / a > < / li >
< li > < a href = "#orgb35fa00" > 1.3.2. Primary Control< / a > < / li >
< li > < a href = "#org4a1264f" > 1.3.3. Conclusion< / a > < / li >
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< / ul >
< / li >
< / ul >
< / li >
< li > < a href = "#org81f1d95" > 2. Translation Stiffness< / a >
< ul >
< li > < a href = "#org969d9e7" > 2.1. Realistic Translation Stiffness Values< / a >
< ul >
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< li > < a href = "#org14d57c4" > 2.1.1. Initialization< / a > < / li >
< li > < a href = "#org790d5e4" > 2.1.2. Direct Velocity Feedback< / a > < / li >
< li > < a href = "#orgddae25e" > 2.1.3. Primary Plant< / a > < / li >
< li > < a href = "#org7ebf071" > 2.1.4. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org0275632" > 2.2. Parametric study< / a >
< ul >
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< li > < a href = "#org5ed48b8" > 2.2.1. Direct Velocity Feedback< / a > < / li >
< li > < a href = "#org5d9965b" > 2.2.2. Primary Control< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org8ee81cd" > 2.3. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb8a9692" > 3. Conclusion< / a > < / li >
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< / ul >
< / div >
< / div >
< p >
In this document is studied the effect of the mechanical behavior of the flexible joints that are located the extremities of each nano-hexapod’ s legs.
< / p >
< p >
Ideally, we want the x and y rotations to be free and all the translations to be blocked.
However, this is never the case and be have to consider:
< / p >
< ul class = "org-ul" >
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< li > Finite bending stiffnesses (Section < a href = "#org3eb4121" > 1< / a > )< / li >
< li > Axial stiffness in the direction of the legs (Section < a href = "#org8f4d83b" > 2< / a > )< / li >
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< / ul >
< p >
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This may impose some limitations, also, the goal is to specify the required joints stiffnesses (summarized in Section < a href = "#org6614f42" > 3< / a > ).
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< / p >
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< div id = "outline-container-orge032d30" class = "outline-2" >
< h2 id = "orge032d30" > < span class = "section-number-2" > 1< / span > Bending and Torsional Stiffness< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
< a id = "org3eb4121" > < / a >
< / p >
< p >
In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.
< / p >
< / div >
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< div id = "outline-container-org4af6fbb" class = "outline-3" >
< h3 id = "org4af6fbb" > < span class = "section-number-3" > 1.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< p >
Let’ s initialize all the stages with default parameters.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
< / pre >
< / div >
< p >
Let’ s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeSample('mass', 50, 'freq', 200*ones(6,1));
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgde60939" class = "outline-3" >
< h3 id = "orgde60939" > < span class = "section-number-3" > 1.2< / span > Realistic Bending Stiffness Values< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
< p >
Let’ s compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:
< / p >
< ul class = "org-ul" >
< li > \(K_{\theta, \phi} = 15\,[Nm/rad]\) stiffness in flexion< / li >
< li > \(K_{\psi} = 20\,[Nm/rad]\) stiffness in torsion< / li >
< / ul >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kf_M = 15*ones(6,1);
Kf_F = 15*ones(6,1);
Kt_M = 20*ones(6,1);
Kt_F = 20*ones(6,1);
< / pre >
< / div >
< p >
The stiffness and damping of the nano-hexapod’ s legs are:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > k_opt = 1e5; % [N/m]
c_opt = 2e2; % [N/(m/s)]
< / pre >
< / div >
< p >
This corresponds to the optimal identified stiffness.
< / p >
< / div >
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< div id = "outline-container-org1f64e69" class = "outline-4" >
< h4 id = "org1f64e69" > < span class = "section-number-4" > 1.2.1< / span > Direct Velocity Feedback< / h4 >
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< div class = "outline-text-4" id = "text-1-2-1" >
< 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.
< / p >
< p >
The obtained dynamics are shown in Figure < a href = "#org656fd1c" > 1< / a > .
It is shown that the adding of stiffness for the flexible joints does increase a little bit the frequencies of the mass suspension modes. It stiffen the structure.
< / p >
< div id = "org656fd1c" class = "figure" >
< p > < img src = "figs/flex_joint_rot_dvf.png" alt = "flex_joint_rot_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint stiffness< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7eb4054" class = "outline-4" >
< h4 id = "org7eb4054" > < span class = "section-number-4" > 1.2.2< / span > Primary Plant< / h4 >
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< div class = "outline-text-4" id = "text-1-2-2" >
< p >
Let’ 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).
< / p >
< p >
The dynamics is compare with and without the joint flexibility in Figure < a href = "#org4322feb" > 2< / a > .
The plant dynamics is not found to be changing significantly.
< / p >
< div id = "org4322feb" class = "figure" >
< p > < img src = "figs/flex_joints_rot_primary_plant_L.png" alt = "flex_joints_rot_primary_plant_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints (dashed) and with flexible joints (solid)< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org81a1a77" class = "outline-4" >
< h4 id = "org81a1a77" > < span class = "section-number-4" > 1.2.3< / span > Conclusion< / h4 >
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< div class = "outline-text-4" id = "text-1-2-3" >
< div class = "important" >
< p >
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Considering realistic flexible joint bending stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.
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< / p >
< p >
It only increases a little bit the suspension modes of the sample on top of the nano-hexapod.
< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org8ad3f34" class = "outline-3" >
< h3 id = "org8ad3f34" > < span class = "section-number-3" > 1.3< / span > Parametric Study< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
< p >
We wish now to see what is the impact of the rotation stiffness of the flexible joints on the dynamics.
This will help to determine the requirements on the joint’ s stiffness.
< / p >
< p >
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Let’ s consider the following bending stiffness of the flexible joints:
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ks = [1, 5, 10, 50, 100]; % [Nm/rad]
< / pre >
< / div >
< p >
We also consider here a nano-hexapod with the identified optimal actuator stiffness.
< / p >
< / div >
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< div id = "outline-container-org1575b3d" class = "outline-4" >
< h4 id = "org1575b3d" > < span class = "section-number-4" > 1.3.1< / span > Direct Velocity Feedback< / h4 >
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< div class = "outline-text-4" id = "text-1-3-1" >
< 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 > .
< / p >
< p >
The corresponding Root Locus plot is shown in Figure < a href = "#orgb9f3389" > 4< / a > .
< / p >
< p >
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It is shown that the bending stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback.
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< / p >
< div id = "org8fbbf9d" class = "figure" >
< p > < img src = "figs/flex_joints_rot_study_dvf.png" alt = "flex_joints_rot_study_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered Rotation Stiffnesses< / p >
< / div >
< div id = "orgb9f3389" class = "figure" >
< p > < img src = "figs/flex_joints_rot_study_dvf_root_locus.png" alt = "flex_joints_rot_study_dvf_root_locus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Root Locus for all the considered Rotation Stiffnesses< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb35fa00" class = "outline-4" >
< h4 id = "orgb35fa00" > < span class = "section-number-4" > 1.3.2< / span > Primary Control< / h4 >
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< div class = "outline-text-4" id = "text-1-3-2" >
< 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 > .
< / p >
< p >
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It is shown that the bending stiffness of the flexible joints have very little impact on the dynamics.
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< / p >
< div id = "orgb739560" class = "figure" >
< p > < img src = "figs/flex_joints_rot_study_primary_plant.png" alt = "flex_joints_rot_study_primary_plant.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered bending stiffnesses< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org4a1264f" class = "outline-4" >
< h4 id = "org4a1264f" > < span class = "section-number-4" > 1.3.3< / span > Conclusion< / h4 >
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< div class = "outline-text-4" id = "text-1-3-3" >
< div class = "important" >
< p >
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The bending stiffness of the flexible joint does not significantly change the dynamics.
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< / p >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org81f1d95" class = "outline-2" >
< h2 id = "org81f1d95" > < span class = "section-number-2" > 2< / span > Translation Stiffness< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
< a id = "org8f4d83b" > < / a >
< / p >
< p >
Let’ s know consider a flexibility in translation of the flexible joint, in the axis of the legs.
< / p >
< / div >
< div id = "outline-container-org969d9e7" class = "outline-3" >
< h3 id = "org969d9e7" > < span class = "section-number-3" > 2.1< / span > Realistic Translation Stiffness Values< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< p >
We choose realistic values of the axial stiffness of the joints:
\[ K_a = 60\,[N/\mu m] \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kz_F = 60e6*ones(6,1); % [N/m]
Kz_M = 60e6*ones(6,1); % [N/m]
Cz_F = 1*ones(6,1); % [N/(m/s)]
Cz_M = 1*ones(6,1); % [N/(m/s)]
< / pre >
< / div >
< / div >
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< div id = "outline-container-org14d57c4" class = "outline-4" >
< h4 id = "org14d57c4" > < span class = "section-number-4" > 2.1.1< / span > Initialization< / h4 >
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< div class = "outline-text-4" id = "text-2-1-1" >
< p >
Let’ s initialize all the stages with default parameters.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
< / pre >
< / div >
< p >
Let’ s consider the heaviest mass which should we the most problematic with it comes to the flexible joints.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeSample('mass', 50, 'freq', 200*ones(6,1));
initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org790d5e4" class = "outline-4" >
< h4 id = "org790d5e4" > < span class = "section-number-4" > 2.1.2< / span > Direct Velocity Feedback< / h4 >
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< div class = "outline-text-4" id = "text-2-1-2" >
< 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.
< / p >
< p >
The obtained dynamics are shown in Figure < a href = "#org78dd87a" > 6< / a > .
< / p >
< div id = "org78dd87a" class = "figure" >
< p > < img src = "figs/flex_joint_trans_dvf.png" alt = "flex_joint_trans_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint axis stiffness< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgddae25e" class = "outline-4" >
< h4 id = "orgddae25e" > < span class = "section-number-4" > 2.1.3< / span > Primary Plant< / h4 >
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< div class = "outline-text-4" id = "text-2-1-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
< / pre >
< / div >
< p >
Let’ 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).
< / p >
< p >
The dynamics is compare with and without the joint flexibility in Figure < a href = "#org9bd0791" > 7< / a > .
< / p >
< div id = "org9bd0791" class = "figure" >
< p > < img src = "figs/flex_joints_trans_primary_plant_L.png" alt = "flex_joints_trans_primary_plant_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed)< / p >
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< / div >
< / div >
< / div >
< div id = "outline-container-org7ebf071" class = "outline-4" >
< h4 id = "org7ebf071" > < span class = "section-number-4" > 2.1.4< / span > Conclusion< / h4 >
< div class = "outline-text-4" id = "text-2-1-4" >
< div class = "important" >
< p >
For the realistic value of the flexible joint axial stiffness, the dynamics is not much impact, and this should not be a problem for control.
< / p >
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< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org0275632" class = "outline-3" >
< h3 id = "org0275632" > < span class = "section-number-3" > 2.2< / span > Parametric study< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< p >
We wish now to see what is the impact of the < b > axial< / b > stiffness of the flexible joints on the dynamics.
< / p >
< p >
Let’ s consider the following values for the axial stiffness:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kzs = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; % [N/m]
< / pre >
< / div >
< p >
We also consider here a nano-hexapod with the identified optimal actuator stiffness (\(k = 10^5\,[N/m]\)).
< / p >
< / div >
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< div id = "outline-container-org5ed48b8" class = "outline-4" >
< h4 id = "org5ed48b8" > < span class = "section-number-4" > 2.2.1< / span > Direct Velocity Feedback< / h4 >
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< div class = "outline-text-4" id = "text-2-2-1" >
< 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 > .
< / p >
< p >
It is shown that the axial stiffness of the flexible joints does have a huge impact on the dynamics.
< / p >
< p >
If the axial stiffness of the flexible joints is \(K_a > 10^7\,[N/m]\) (here \(100\) times higher than the actuator stiffness), then the change of dynamics stays reasonably small.
< / p >
< p >
This is more clear by looking at the root locus (Figures < a href = "#org9d43966" > 9< / a > and < a href = "#org987d98e" > 10< / a > ).
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It can be seen that very little active damping can be achieve for axial stiffnesses below \(10^7\,[N/m]\).
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< div id = "orgab9ab86" class = "figure" >
< p > < img src = "figs/flex_joints_trans_study_dvf.png" alt = "flex_joints_trans_study_dvf.png" / >
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< p > < span class = "figure-number" > Figure 8: < / span > Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered axis Stiffnesses< / p >
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< div id = "org9d43966" class = "figure" >
< p > < img src = "figs/flex_joints_trans_study_dvf_root_locus.png" alt = "flex_joints_trans_study_dvf_root_locus.png" / >
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< p > < span class = "figure-number" > Figure 9: < / span > Root Locus for all the considered axial Stiffnesses< / p >
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< div id = "org987d98e" class = "figure" >
< p > < img src = "figs/flex_joints_trans_study_root_locus_unzoom.png" alt = "flex_joints_trans_study_root_locus_unzoom.png" / >
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< p > < span class = "figure-number" > Figure 10: < / span > Root Locus (unzoom) for all the considered axial Stiffnesses< / p >
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< div id = "outline-container-org5d9965b" class = "outline-4" >
< h4 id = "org5d9965b" > < span class = "section-number-4" > 2.2.2< / span > Primary Control< / h4 >
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< div class = "outline-text-4" id = "text-2-2-2" >
< 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 > .
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< div id = "org6070692" class = "figure" >
< p > < img src = "figs/flex_joints_trans_study_primary_plant.png" alt = "flex_joints_trans_study_primary_plant.png" / >
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< p > < span class = "figure-number" > Figure 11: < / span > Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered axial stiffnesses< / p >
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< div id = "outline-container-org8ee81cd" class = "outline-3" >
< h3 id = "org8ee81cd" > < span class = "section-number-3" > 2.3< / span > Conclusion< / h3 >
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< div class = "important" >
< p >
The axial stiffness of the flexible joints should be maximized.
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< p >
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For the considered actuator stiffness \(k = 10^5\,[N/m]\), the axial stiffness of the flexible joints should ideally be above \(10^7\,[N/m]\).
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< p >
This is a reasonable stiffness value for such joints.
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< p >
We may interpolate the results and say that the axial joint stiffness should be 100 times higher than the actuator stiffness, but this should be confirmed with further analysis.
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< div id = "outline-container-orgb8a9692" class = "outline-2" >
< h2 id = "orgb8a9692" > < span class = "section-number-2" > 3< / span > Conclusion< / h2 >
< div class = "outline-text-2" id = "text-3" >
< p >
< a id = "org6614f42" > < / a >
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< p >
For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:
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< ul class = "org-ul" >
< li > Bending Stiffness: \(K_b < 50 \ , [ Nm / rad ] \ ) < / li >
< li > Torsion Stiffness: \(K_t < 50 \ , [ Nm / rad ] \ ) < / li >
< li > Axial Stiffness: \(K_a > 10^7\,[N/m]\)< / li >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-05-05 mar. 10:44< / p >
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