Study of the Flexible Joints

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 1. 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.

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 2. The plant dynamics is not found to be changing significantly.

The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure 3.

The corresponding Root Locus plot is shown in Figure 4.

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.

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 5.

It is shown that the bending stiffness of the flexible joints have very little impact on the dynamics.

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);

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.

The obtained dynamics are shown in Figure 6.

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

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 7.

The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure 8.

It is shown that the axial stiffness of the flexible joints does have a huge impact on the dynamics.

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.

This is more clear by looking at the root locus (Figures 9 and 10). It can be seen that very little active damping can be achieve for axial stiffnesses below \(10^7\,[N/m]\).

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 11.