12 KiB
12 KiB
Stewart Platform - Dynamics Study
- Compare external forces and forces applied by the actuators
- Comparison of the static transfer function and the Compliance matrix
Compare external forces and forces applied by the actuators
Introduction ignore
Comparison with fixed support
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none'); ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
%% 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, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'}; Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$:
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
%% 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 with a flexible support
We redo the identification for when the Stewart platform is on a flexible support.
ground = initializeGround('type', 'flexible');Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
%% 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, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'}; Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$:
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
%% Run the linearization
Gd = linearize(mdl, io, options);
Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};Conclusion
The transfer function from forces/torques applied by the actuators on the payload $\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}$ to the pose of the mobile platform $\bm{\mathcal{X}}$ is the same as the transfer function from external forces/torques to $\bm{\mathcal{X}}$ as long as the Stewart platform's base is fixed.
Comparison of the static transfer function and the Compliance matrix
Introduction ignore
Analysis
Initialization of the Stewart platform.
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'none'); ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');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_model';
%% 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;
%% 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, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'}; Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};Let's first look at the low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.
| 4.7e-08 | -7.2e-19 | 5.0e-18 | -8.9e-18 | 3.2e-07 | 9.9e-18 |
| 4.7e-18 | 4.7e-08 | -5.7e-18 | -3.2e-07 | -1.6e-17 | -1.7e-17 |
| 3.3e-18 | -6.3e-18 | 2.1e-08 | 4.4e-17 | 6.6e-18 | 7.4e-18 |
| -3.2e-17 | -3.2e-07 | 6.2e-18 | 5.2e-06 | -3.5e-16 | 6.3e-17 |
| 3.2e-07 | 2.7e-17 | 4.8e-17 | -4.5e-16 | 5.2e-06 | -1.2e-19 |
| 4.0e-17 | -9.5e-17 | 8.4e-18 | 4.3e-16 | 5.8e-16 | 1.7e-06 |
And now at the Compliance matrix.
| 4.7e-08 | -2.0e-24 | 7.4e-25 | 5.9e-23 | 3.2e-07 | 5.9e-24 |
| -7.1e-25 | 4.7e-08 | 2.9e-25 | -3.2e-07 | -5.4e-24 | -3.3e-23 |
| 7.9e-26 | -6.4e-25 | 2.1e-08 | 1.9e-23 | 5.3e-25 | -6.5e-40 |
| 1.4e-23 | -3.2e-07 | 1.3e-23 | 5.2e-06 | 4.9e-22 | -3.8e-24 |
| 3.2e-07 | 7.6e-24 | 1.2e-23 | 6.9e-22 | 5.2e-06 | -2.6e-22 |
| 7.3e-24 | -3.2e-23 | -1.6e-39 | 9.9e-23 | -3.3e-22 | 1.7e-06 |
Conclusion
The low frequency transfer function matrix from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$ corresponds to the compliance matrix of the Stewart platform.