nass-simscape/org/flexible_joints_study.org

Study of the Flexible Joints

• Introduction
• Rotational Stiffness
  • Introduction
  • Initialization
  • Realistic Rotational Stiffness Values
    • Introduction
    • Direct Velocity Feedback
    • Primary Plant
    • Conclusion
  • Parametric Study
    • Introduction
    • Direct Velocity Feedback
    • Primary Control
    • Conclusion
• Translation Stiffness
  • Introduction
  • Initialization
  • Direct Velocity Feedback
  • Primary Plant
  • Parametric study
    • Introduction
    • Direct Velocity Feedback
    • Primary Control
  • Conclusion

Introduction   ignore

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.

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:

• Finite x and y rotational stiffnesses (Section sec:rot_stiffness)
• Translation stiffness in the direction of the legs (Section sec:trans_stiffness)

This may impose some limitations, also, the goal is to specify the required joints stiffnesses.

Rotational Stiffness

<<sec:rot_stiffness>>

Introduction   ignore

In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints.

Initialization

Let's initialize all the stages with default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

Let's consider the heaviest mass which should we the most problematic with it comes to the flexible joints.

  initializeSample('mass', 50, 'freq', 200*ones(6,1));
  initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);

Realistic Rotational Stiffness Values

Introduction   ignore

Let's compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case:

• $K_{\theta, \phi} = 15\,[Nm/rad]$ stiffness in flexion
• $K_{\psi} = 20\,[Nm/rad]$ stiffness in torsion

  Kf_M = 15*ones(6,1);
  Kt_M = 20*ones(6,1);
  Kf_F = 15*ones(6,1);
  Kt_F = 20*ones(6,1);

The stiffness and damping of the nano-hexapod's legs are:

  k = 1e5; % [N/m]
  c = 2e2; % [N/(m/s)]

Direct Velocity Feedback

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.

The obtained dynamics are shown in Figure fig:flex_joint_rot_dvf. 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.

Dynamics from actuators force $\tau_i$ to relative motion sensors $d\mathcal{L}_i$ with (blue) and without (red) considering the flexible joint stiffness

Primary Plant

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).

The dynamics is compare with and without the joint flexibility in Figure fig:flex_joints_rot_primary_plant_L. The plant dynamics is not found to be changing significantly.

Dynamics from $\bm{\tau}^\prime_i$ to $\bm{\epsilon}_{\mathcal{X}_n,i}$ with perfect joints (dashed) and with flexible joints (solid)

Conclusion

Considering realistic flexible joint rotational stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control.

It only increases a little bit the suspension modes of the sample on top of the nano-hexapod.

Parametric Study

Introduction   ignore

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.

Let's consider the following rotational stiffness of the flexible joints:

  Ks = [1, 10, 100]; % [Nm/rad]

We also consider here a nano-hexapod with the identified optimal actuator stiffness.

Direct Velocity Feedback

The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure fig:flex_joints_rot_study_dvf.

The corresponding Root Locus plot is shown in Figure fig:flex_joints_rot_study_dvf_root_locus.

It is shown that the rotational stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback.

Dynamics from $\tau_i$ to $d\mathcal{L}_i$ for all the considered Rotation Stiffnesses

Root Locus for all the considered Rotation Stiffnesses

Primary Control

  Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);

Diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the considered rotational stiffnesses

Conclusion

Translation Stiffness

<<sec:trans_stiffness>>

Introduction   ignore

Let's know consider a flexibility in translation of the flexible joint.

Initialization

Let's initialize all the stages with default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

Let's consider the heaviest mass which should we the most problematic with it comes to the flexible joints.

  initializeSample('mass', 50, 'freq', 200*ones(6,1));
  initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);

Direct Velocity Feedback

  Kz_F = 60e6*ones(6,1); % [N/m]
  Kz_M = 60e6*ones(6,1); % [N/m]
  Cz_F = 1e2*ones(6,1); % [N/m]
  Cz_M = 1e2*ones(6,1); % [N/m]

Primary Plant

  Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);

Parametric study

Introduction   ignore

  Kzs = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; % [N/m]

Direct Velocity Feedback

Primary Control

  Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);

Conclusion