stewart-simscape/org/dynamics-study.org

Stewart Platform - Dynamics Study

• Some tests
  • Simscape Model
  • test
  • Compare external forces and forces applied by the actuators
  • Comparison of the static transfer function and the Compliance matrix
  • Transfer function from forces applied in the legs to the displacement of the legs

Some tests

Simscape Model

  open('stewart_platform_dynamics.slx')

test

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);

Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

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

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/J-T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/X'],   1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

  G_cart = minreal(G*inv(stewart.J'));
  G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};

  figure; bode(G, G_cart)

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

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

  %% Run the linearization
  Gd = linearize(mdl, io, options);
  Gd.InputName  = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
  Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

  freqs = logspace(0, 3, 1000);

  figure;
  bode(Gd, G)

Compare external forces and forces applied by the actuators

Initialization of the Stewart platform.

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);

Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

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

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

Estimation of the transfer function from $\mathcal{\bm{F}}_{d}$ to $\mathcal{\bm{X}}$:

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

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

  %% Run the linearization
  Gd = linearize(mdl, io, options);
  Gd.InputName  = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
  Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

Comparison of the two transfer function matrices.

  freqs = logspace(0, 4, 1000);

  figure;
  bode(Gd, G, freqs)

Seems quite similar.

Comparison of the static transfer function and the Compliance matrix

Initialization of the Stewart platform.

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);

Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$:

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

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

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};

Let's first look at the low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.

2.0e-06	-9.1e-19	-5.3e-12	7.3e-18	1.7e-05	1.3e-18
-1.7e-18	2.0e-06	8.6e-19	-1.7e-05	-1.5e-17	6.7e-12
3.6e-13	3.2e-19	5.0e-07	-2.5e-18	8.1e-12	-1.5e-19
1.0e-17	-1.7e-05	-5.0e-18	1.9e-04	9.1e-17	-3.5e-11
1.7e-05	-6.9e-19	-5.3e-11	6.9e-18	1.9e-04	4.8e-18
-3.5e-18	-4.5e-12	1.5e-18	7.1e-11	-3.4e-17	4.6e-05

And now at the Compliance matrix.

2.0e-06	2.9e-22	2.8e-22	-3.2e-21	1.7e-05	1.5e-37
-2.1e-22	2.0e-06	-1.8e-23	-1.7e-05	-2.3e-21	1.1e-22
3.1e-22	-1.6e-23	5.0e-07	1.7e-22	2.2e-21	-8.1e-39
2.1e-21	-1.7e-05	2.0e-22	1.9e-04	2.3e-20	-8.7e-21
1.7e-05	2.5e-21	2.0e-21	-2.8e-20	1.9e-04	1.3e-36
3.7e-23	3.1e-22	-6.0e-39	-1.0e-20	3.1e-22	4.6e-05

The low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$ corresponds to the compliance matrix of the Stewart platform.

Transfer function from forces applied in the legs to the displacement of the legs

Initialization of the Stewart platform.

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);

Estimation of the transfer function from $\bm{\tau}$ to $\bm{L}$:

  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/J-T'], 1, 'openinput');  io_i = io_i + 1;
  io(io_i) = linio([mdl, '/L'], 1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};

  freqs = logspace(1, 3, 1000);
  figure; bode(G, 2*pi*freqs)

  bodeFig({G(1,1), G(1,2)}, freqs, struct('phase', true));