nass-simscape/org/control_virtual_mass.org

Decentralize control to add virtual mass

• Initialization
• Identification
• Adding Virtual Mass in the Leg's Space
  • Plant
  • Controller Design
  • Identification of the Primary Plant
• Adding Virtual Mass in the Task Space
  • Plant
  • Controller Design
  • Identification of the Primary Plant

Initialization

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

  initializeSimscapeConfiguration();
  initializeDisturbances('enable', false);
  initializeLoggingConfiguration('log', 'none');

  initializeController('type', 'hac-dvf');

The nano-hexapod has the following leg's stiffness and damping.

  initializeNanoHexapod('k', 1e5, 'c', 2e2);

We set the stiffness of the payload fixation:

  Kp = 1e8; % [N/m]

Identification

We identify the system for the following payload masses:

  Ms = [1, 10, 50];

Identification of the transfer function from $\tau$ to $d\mathcal{L}$.

Identification of the Primary plant without virtual add of mass

Adding Virtual Mass in the Leg's Space

Plant

Transfer function from $\tau_i$ to $d\mathcal{L}_i$ for three payload masses

Controller Design

  Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6);

Loop Gain for the addition of virtual mass in the leg's space

Identification of the Primary Plant

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

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

Adding Virtual Mass in the Task Space

Plant

Let's look at the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$: \[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \]

Dynamics from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ used for virtual mass addition in the task space

Controller Design

  KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0]));

Loop gain for virtual mass addition in the task space

  Kdvf = inv(nano_hexapod.J')*KmX*inv(nano_hexapod.J);

Identification of the Primary Plant

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

Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the task space