Stewart Platform with Flexible Elements

1. Simscape Model
2. Control with flexible elements
3. Flexible Joint Specifications
4. Flexible Joint Specifications with the APA300ML
- 4.1. Stewart Platform Initialization
- 4.2. Comparison of perfect joint and worst specified joint
5. Relative Motion Sensors
- 5.1. Stewart Platform Initialization

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]	z [m]
1.0	697783.0	0.0	-0.015
2.0	697784.0	0.0	0.015
3.0	697785.0	-0.0325	0.0
4.0	697786.0	-0.0125	0.0
5.0	697787.0	-0.0075	0.0
6.0	697788.0	0.0125	0.0
7.0	697789.0	0.0325	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

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

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

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

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

1.10 Integral Force Feedback

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))

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

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 = 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 = 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', 'universal_3dof', ...
                                           'type_M', 'spherical_3dof', ...
                                           'Kf_M', 100*ones(6,1), ...
                                           'Kt_M', 500*ones(6,1), ...
                                           'Kf_F', 100*ones(6,1), ...
                                           'Kt_F', 500*ones(6,1), ...
                                           'Ka_M', 200e6*ones(6,1), ...
                                           'Kr_M', 1e6*ones(6,1), ...
                                           'Ka_F', 200e6*ones(6,1), ...
                                           'Kr_F', 1e6*ones(6,1));
% stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
%                                            'type_M', 'spherical_3dof', ...
%                                            'Kf_M', 100*ones(6,1), ...
%                                            'Kt_M', 500*ones(6,1), ...
%                                            'Kf_F', 100*ones(6,1), ...
%                                            'Kt_F', 500*ones(6,1), ...
%                                            'Ka_M', 1e6*ones(6,1), ...
%                                            'Kr_M', 1e6*ones(6,1), ...
%                                            'Ka_F', 1e6*ones(6,1), ...
%                                            'Kr_F', 1e6*ones(6,1));
% stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');

stewart = initializeCylindricalPlatforms(stewart);

stewart = initializeCylindricalStruts(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')