Decentralize control to add virtual mass
+Table of Contents
+ +1 Initialization
+initializeGround(); +initializeGranite(); +initializeTy(); +initializeRy(); +initializeRz(); +initializeMicroHexapod(); +initializeAxisc(); +initializeMirror(); + +initializeSimscapeConfiguration(); +initializeDisturbances('enable', false); +initializeLoggingConfiguration('log', 'none'); + +initializeController('type', 'hac-dvf'); ++
+We set the stiffness of the payload fixation: +
+Kp = 1e8; % [N/m]
+
+2 Identification
+2.1 Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\)
+K = tf(zeros(6)); +Kdvf = tf(zeros(6)); ++
+We identify the system for the following payload masses: +
+Ms = [1, 10, 50]; ++
+The nano-hexapod has the following leg’s stiffness and damping. +
+initializeNanoHexapod('k', 1e5, 'c', 2e2); ++
2.2 Identification of the Primary plant without virtual add of mass
+3 Adding Virtual Mass in the Leg’s Space
+3.1 Plant
++
+Figure 1: Transfer function from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses
+3.2 Controller Design
+Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6); ++
+
+Figure 2: Loop Gain for the addition of virtual mass in the leg’s space
+3.3 Identification of the Primary Plant
++
+Figure 3: Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the leg’s space
++
+Figure 4: Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the leg’s space
+4 Adding Virtual Mass in the Task Space
+4.1 Plant
++Let’s look at the transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\): +\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \] +
+ + ++
+Figure 5: Dynamics from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) used for virtual mass addition in the task space
+4.2 Controller Design
+KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0])); ++
+
+Figure 6: Loop gain for virtual mass addition in the task space
+Kdvf = inv(nano_hexapod.J')*KmX*inv(nano_hexapod.J); ++
4.3 Identification of the Primary Plant
++
+Figure 7: Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the task space
++
+Figure 8: Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the task space
+