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Stewart Platform with Flexible Elements

Table of Contents

1 Simscape Model

1.1 Flexible APA

  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
Total number of Nodes 7
Number of interface Nodes 7
Number of Modes 120
Size of M and K matrices 162
Table 1: Coordinates of the interface nodes
Node i Node Number x [m] y [m] z [m]
1.0 697783.0 0.0 0.0 -0.015
2.0 697784.0 0.0 0.0 0.015
3.0 697785.0 -0.0325 0.0 0.0
4.0 697786.0 -0.0125 0.0 0.0
5.0 697787.0 -0.0075 0.0 0.0
6.0 697788.0 0.0125 0.0 0.0
7.0 697789.0 0.0325 0.0 0.0

1.2 Flexible Joint

  flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
Total number of Nodes 2
Number of interface Nodes 2
Number of Modes 6
Size of M and K matrices 18
Table 2: Coordinates of the interface nodes
Node i Node Number x [m] y [m] z [m]
1.0 528875.0 0.0 0.0 0.0
2.0 528876.0 0.0 0.0 -0.0
Caracteristic Value
Axial Stiffness [N/um] 94
Bending Stiffness [Nm/rad] 5
Bending Stiffness [Nm/rad] 5
Torsion Stiffness [Nm/rad] 260

1.3 Identification

And we identify the dynamics from force actuators to force sensors. 

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

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');

  disturbances = initializeDisturbances();

1.4 No Flexible Elements

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
                                             'type_F', 'universal_3dof', ...
                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

1.5 Flexible joints

simscape_flex_joints.png

Figure 1: Figure caption

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  %% Run the linearization
  Gj = linearize(mdl, io, options);
  Gj.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

1.6 Flexible APA

simscape_flex_apa.png

Figure 2: Figure caption

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
                                             'type_F', 'universal_3dof', ...
                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  %% Run the linearization
  Ga = -linearize(mdl, io, options);
  Ga.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Ga.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

1.7 Flexible Joints and APA

simscape_flexible.png

Figure 3: Figure caption

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  Gf = -linearize(mdl, io, options);
  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

1.8 Save

  save('./mat/flexible_stewart_platform.mat', 'G', 'Gj', 'Ga', 'Gf');

1.9 Direct Velocity Feedback

flexible_elements_effect_dvf.png

Figure 4: Change of the DVF plant dynamics with the added flexible elements

1.10 Integral Force Feedback

flexible_elements_effect_iff.png

Figure 5: Change of the IFF plant dynamics with the added flexible elements

1.11 Procedure to include flexible elements into Simscape

In order to model a flexible element with only few mass-spring-damper elements:

  • FEM of the flexible element
  • Super-Element extraction
  • Parameters to extract
    • For the flexible joint: axial, shear, bending and torsion stiffnesses
    • For the APA: k1, ka, ke, c1
  • This can be done directly from the Stiffness matrix or using identification from a simple Simscape model

1.12 Conclusion

The results seems to indicate that the model is well represented with only few degrees of freedom. This permit to have a relatively sane number of states for the model.

2 Control with flexible elements

2.1 Flexible APA and Flexible Joint

  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
  flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');

  disturbances = initializeDisturbances();
  references = initializeReferences(stewart);

2.2 Identification

And we identify the dynamics from force actuators to force sensors. 

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

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]

  G = -linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};

2.3 Decentralized Direct Velocity Feedback

Controller Design 

  Kl = -1e5*s/(1 + s/2/pi/2e2)/(1 + s/2/pi/5e2) * eye(6);

  isstable(feedback(G(1:6,1:6)*Kl, eye(6), -1))

1

2.4 HAC

  Kx = tf(zeros(6));

  controller = initializeController('type', 'ref-track-hac-dvf');

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],              1, 'input');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]

  %% Run the linearization
  G = linearize(mdl, io);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

  Gl = -stewart.kinematics.J*G;
  Gl.OutputName  = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};

  wc = 2*pi*300;
  Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);

3 Flexible Joint Specifications

3.1 Stewart Platform Initialization

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 43e6*ones(6,1), 'K1', 0.4e6*ones(6,1), 'C1', 10*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');

  disturbances = initializeDisturbances();

  open('stewart_platform_model.slx')

3.2 Effect of the Bending Stiffness

  Kfs = [1, 10, 100, 1000]; % [Nm/rad]

3.3 Effect of the Torsion Stiffness

  Kts = [10, 100, 500, 1000]; % [Nm/rad]

3.4 Effect of the Axial Stiffness

  Kas = [1e6, 1e7, 1e8, 5e8, 1e9]; % [N/m]

3.5 Effect of the Radial (Shear) Stiffness

  Krs = [1e4, 1e5, 1e6, 1e7]; % [N/m]

3.6 Comparison of perfect joint and worst specified joint

3.7 Conclusion

Qualitatively:

  Specification
Axial Stiffness Much larger than the Actuator axial stiffness
Shear Stiffness  
Bending Stiffness Much smaller than the Actuator bending stiffness
Torsion Stiffness  

Quantitatively:

  Specification
Axial Stiffness > 200 [N/um]
Shear Stiffness > 1 [N/um]
Bending Stiffness < 100 [Nm/rad]
Torsion Stiffness < 500 [Nm/rad]

4 Flexible Joint Specifications with the APA300ML

4.1 Stewart Platform Initialization

  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');

  disturbances = initializeDisturbances();

  open('stewart_platform_model.slx')

4.2 Comparison of perfect joint and worst specified joint

5 Relative Motion Sensors

5.1 Stewart Platform Initialization

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);

  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
  % stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');

  flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
                                             'type_F', 'universal_3dof', ...
                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));

  stewart = initializeCylindricalPlatforms(stewart);

  stewart = initializeCylindricalStruts(stewart);

  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');

  disturbances = initializeDisturbances();

6 Struts with Encoders

6.1 Flexible Strut

  apa = load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
Total number of Nodes 8
Number of interface Nodes 8
Number of Modes 6
Size of M and K matrices 54
Table 3: Coordinates of the interface nodes
Node i Node Number x [m] y [m] z [m]
1.0 504411.0 0.0 0.0 0.0405
2.0 504412.0 0.0 0.0 -0.0405
3.0 504413.0 -0.0325 0.0 0.0
4.0 504414.0 -0.0125 0.0 0.0
5.0 504415.0 -0.0075 0.0 0.0
6.0 504416.0 0.0325 0.0 0.0
7.0 504417.0 0.004 0.0145 -0.00175
8.0 504418.0 0.004 0.0166 -0.00175

6.2 Stewart Platform

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', 95e-3, 'MO_B', 220e-3);
  stewart = generateGeneralConfiguration(stewart, 'FH', 22.5e-3, 'FR', 114e-3, 'FTh', [   -11,    11, 120-11, 120+11, 240-11, 240+11]*(pi/180), ...
                                                  'MH', 22.5e-3, 'MR', 110e-3, 'MTh', [-60+15, 60-15,  60+15, 180-15, 180+15, -60-15]*(pi/180));
  stewart = computeJointsPose(stewart);

  stewart = initializeFlexibleStrutAndJointDynamics(stewart, 'H', (apa.int_xyz(1,3) - apa.int_xyz(2,3)), ...
                                                             'K', apa.K, ...
                                                             'M', apa.M, ...
                                                             'n_xyz', apa.n_xyz, ...
                                                             'xi', 0.1, ...
                                                             'Gf', -2.65e7, ...
                                                             'step_file', 'mat/APA300ML.STEP');

  stewart = initializeSolidPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  disturbances = initializeDisturbances();
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 1);
  controller = initializeController('type', 'open-loop');
  references = initializeReferences(stewart);

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

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Lm'); io_i = io_i + 1; % Force Sensors [N]

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
                  'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};

comp_relative_motion_sensor_act_leg_encoder.png

Author: Dehaeze Thomas

Created: 2021-01-08 ven. 15:29