nass-simscape/org/amplified_piezoelectric_stack.org

Amplified Piezoelectric Stack Actuator

• Introduction
• Simplified Model
  • Parameters
  • Identification
  • Root Locus
  • Analytical Model
  • Analytical Analysis
• Rotating X-Y platform
  • Parameters
  • Identification
  • Root Locus
  • Analysis
• Stewart Platform with Amplified Actuators
  • Initialization
  • Identification
  • Controller Design
  • Effect of the Low Authority Control on the Primary Plant
    • Introduction
    • Identification of the undamped plant
    • Identification of the damped plant
    • Effect of the Damping on the plant diagonal dynamics
    • Effect of the Damping on the coupling dynamics
  • Effect of the Low Authority Control on the Sensibility to Disturbances
    • Introduction
    • Identification
    • Results
    • Conclusion
  • Optimal Stiffnesses

Introduction   ignore

The presented model is based on cite:souleille18_concep_activ_mount_space_applic.

The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure fig:souleille18_model_piezo). The parameters are shown in the table below.

Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator

	Value	Meaning
$m$	$1\,[kg]$	Payload mass
$k_e$	$4.8\,[N/\mu m]$	Stiffness used to adjust the pole of the isolator
$k_1$	$0.96\,[N/\mu m]$	Stiffness of the metallic suspension when the stack is removed
$k_a$	$65\,[N/\mu m]$	Stiffness of the actuator
$c_1$	$10\,[N/(m/s)]$	Added viscous damping

Parameters used for the model of the APA 100M

Simplified Model

Parameters

  m = 1; % [kg]

  ke = 4.8e6; % [N/m]
  ce = 5; % [N/(m/s)]
  me = 0.001; % [kg]

  k1 = 0.96e6; % [N/m]
  c1 = 10; % [N/(m/s)]

  ka = 65e6; % [N/m]
  ca = 5; % [N/(m/s)]
  ma = 0.001; % [kg]

  h = 0.2; % [m]

IFF Controller:

  Kiff = -8000/s;

Identification

Identification in open-loop.

  %% Name of the Simulink File
  mdl = 'amplified_piezo_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion
  io(io_i) = linio([mdl, '/f'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % External Force

  io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
  io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement

  G = linearize(mdl, io, 0);
  G.InputName  = {'w', 'f', 'F'};
  G.OutputName = {'Fs', 'x1'};

Identification in closed-loop.

  %% Name of the Simulink File
  mdl = 'amplified_piezo_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/w'],  1, 'input');  io_i = io_i + 1; % Base Motion
  io(io_i) = linio([mdl, '/f'],  1, 'input');  io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/F'],  1, 'input');  io_i = io_i + 1; % External Force

  io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
  io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement

  Giff = linearize(mdl, io, 0);
  Giff.InputName  = {'w', 'f', 'F'};
  Giff.OutputName = {'Fs', 'x1'};

Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)

Root Locus

Root Locus

Analytical Model

If we apply the Newton's second law of motion on the top mass, we obtain: \[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]

Then, we can write that the measured force $F_s$ is equal to: \[ F_s = k_a(w - x_e) + f = -k_e (x_1 - x_e) \] which gives: \[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \]

Re-injecting that into the previous equations gives: \[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \] \[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]

  Ga = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * ...
       [ 1 ,              k1 + ke*ka/(ke + ka)  , ke/(ke + ka) ;
        -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke+ka)*(m*s^2 + k1)];
  Ga.InputName = {'F', 'w', 'f'};
  Ga.OutputName = {'x1', 'Fs'};

Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)

Analytical Analysis

For Integral Force Feedback Control, the plant is: \[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]

As high frequency, this converge to: \[ \frac{F_s}{f} \underset{\omega\to\infty}{\longrightarrow} \frac{k_e}{k_e + k_a} \] And at low frequency: \[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]

It has two complex conjugate zeros at: \[ z = \pm j \sqrt{\frac{k_1}{m}} \] And two complex conjugate poles at: \[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \]

If maximal damping is to be attained with IFF, the distance between the zero and the pole is to be maximized. Thus, we wish to maximize $p/z$, which is equivalent as to minimize $k_1$ and have $k_e \approx k_a$ (supposing $k_e + k_a \approx \text{cst}$).

Rotating X-Y platform

Parameters

  m = 1; % [kg]

  ke = 4.8e6; % [N/m]
  ce = 5; % [N/(m/s)]
  me = 0.001; % [kg]

  k1 = 0.96e6; % [N/m]
  c1 = 10; % [N/(m/s)]

  ka = 65e6; % [N/m]
  ca = 5; % [N/(m/s)]
  ma = 0.001; % [kg]

  h = 0.2; % [m]

  Kiff = tf(0);

Identification

Rotating speed in rad/s:

  Ws = 2*pi*[0, 1, 10, 100];

  Gs = {zeros(length(Ws), 1)};

Identification in open-loop.

  %% Name of the Simulink File
  mdl = 'amplified_piezo_xy_rotating_stage';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/fx'],  1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/fy'],  1, 'openinput');  io_i = io_i + 1;

  io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;

  for i = 1:length(Ws)
      ws = Ws(i);
      G = linearize(mdl, io, 0);
      G.InputName  = {'fx', 'fy'};
      G.OutputName = {'Fsx', 'Fsy'};
      Gs(i) = {G};
  end

Transfer function matrix from forces to force sensors for multiple rotation speed

Root Locus

Root locus for 3 rotating speed

Analysis

The negative stiffness induced by the rotation is equal to $m \omega_0^2$. Thus, the maximum rotation speed where IFF can be applied is: \[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]

Let's verify that.

  Ws = 2*pi*[140, 160];

Identification

  %% Name of the Simulink File
  mdl = 'amplified_piezo_xy_rotating_stage';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/fx'],  1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/fy'],  1, 'openinput');  io_i = io_i + 1;

  io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;

  for i = 1:length(Ws)
      ws = Ws(i);
      G = linearize(mdl, io, 0);
      G.InputName  = {'fx', 'fy'};
      G.OutputName = {'Fsx', 'Fsy'};
      Gs(i) = {G};
  end

Root Locus for the two considered rotation speed. For the red curve, the system is unstable.

Stewart Platform with Amplified Actuators

Initialization

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

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

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

We set the stiffness of the payload fixation:

  Kp = 1e8; % [N/m]

Identification

  K = tf(zeros(6));
  Kiff = 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('actuator', 'amplified');

Controller Design

Dynamics for the Integral Force Feedback for three payload masses

Root Locus for the IFF control for three payload masses

Damping as function of the gain

Damping ratio of the poles as a function of the IFF gain

Finally, we use the following controller for the Decentralized Direct Velocity Feedback:

  Kiff = -1e4/s*eye(6);

Effect of the Low Authority Control on the Primary Plant

Introduction   ignore

Identification of the undamped plant   ignore

Identification of the damped plant   ignore

Effect of the Damping on the plant diagonal dynamics   ignore

Primary plant in the task space with (dashed) and without (solid) IFF

Primary plant in the space of the legs with (dashed) and without (solid) IFF

Effect of the Damping on the coupling dynamics   ignore

Coupling in the primary plant in the task with (dashed) and without (solid) IFF

Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF

Effect of the Low Authority Control on the Sensibility to Disturbances

Introduction   ignore

Identification   ignore

Results   ignore

Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied

Conclusion   ignore

Optimal Stiffnesses