diff --git a/docs/figs/comp_relative_motion_sensor_act_leg.pdf b/docs/figs/comp_relative_motion_sensor_act_leg.pdf new file mode 100644 index 0000000..2e34657 Binary files /dev/null and b/docs/figs/comp_relative_motion_sensor_act_leg.pdf differ diff --git a/docs/figs/comp_relative_motion_sensor_act_leg.png b/docs/figs/comp_relative_motion_sensor_act_leg.png new file mode 100644 index 0000000..b3aa6fb Binary files /dev/null and b/docs/figs/comp_relative_motion_sensor_act_leg.png differ diff --git a/docs/figs/comp_relative_motion_sensor_act_leg_encoder.pdf b/docs/figs/comp_relative_motion_sensor_act_leg_encoder.pdf new file mode 100644 index 0000000..53ecffb Binary files /dev/null and b/docs/figs/comp_relative_motion_sensor_act_leg_encoder.pdf differ diff --git a/docs/figs/comp_relative_motion_sensor_act_leg_encoder.png b/docs/figs/comp_relative_motion_sensor_act_leg_encoder.png new file mode 100644 index 0000000..f66b59c Binary files /dev/null and b/docs/figs/comp_relative_motion_sensor_act_leg_encoder.png differ diff --git a/docs/flexible-stewart-platform.html b/docs/flexible-stewart-platform.html index 585377f..f224367 100644 --- a/docs/flexible-stewart-platform.html +++ b/docs/flexible-stewart-platform.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Stewart Platform with Flexible Elements @@ -28,48 +28,48 @@ @@ -80,8 +80,8 @@

1 Simscape Model

-
-

1.1 Flexible APA

+
+

1.1 Flexible APA

apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
@@ -323,8 +323,8 @@
 
-
-

1.3 Identification

+
+

1.3 Identification

And we identify the dynamics from force actuators to force sensors. @@ -433,8 +433,8 @@ Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3',

-
-

1.6 Flexible APA

+
+

1.6 Flexible APA

@@ -564,8 +564,8 @@ In order to model a flexible element with only few mass-spring-damper elements:
-
-

1.12 Conclusion

+
+

1.12 Conclusion

@@ -621,8 +621,8 @@ references = initializeReferences(stewart);

-
-

2.2 Identification

+
+

2.2 Identification

And we identify the dynamics from force actuators to force sensors. @@ -718,8 +718,8 @@ Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);

3 Flexible Joint Specifications

-
-

3.1 Stewart Platform Initialization

+
+

3.1 Stewart Platform Initialization

stewart = initializeStewartPlatform();
@@ -796,11 +796,11 @@ controller = initializeController('type', 'open-loop');
 
-
-

3.6 Comparison of perfect joint and worst specified joint

+
+

3.6 Comparison of perfect joint and worst specified joint

-
-

3.7 Conclusion

+
+

3.7 Conclusion

Qualitatively: @@ -889,8 +889,8 @@ Quantitatively:

4 Flexible Joint Specifications with the APA300ML

-
-

4.1 Stewart Platform Initialization

+
+

4.1 Stewart Platform Initialization

apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
@@ -932,22 +932,17 @@ controller = initializeController('type', 'open-loop');
 
-
-

4.2 Comparison of perfect joint and worst specified joint

+
+

4.2 Comparison of perfect joint and worst specified joint

5 Relative Motion Sensors

-
-

5.1 Stewart Platform Initialization

+
+

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);
@@ -957,33 +952,24 @@ 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));
+
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');
 
-
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');
+
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));
 
@@ -994,7 +980,6 @@ stewart = computeJointsPose(stewart);
stewart = initializeCylindricalStruts(stewart);
-% stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
 
@@ -1018,18 +1003,13 @@ controller = initializeController('type', 'open-loop');
disturbances = initializeDisturbances();
 
- -
-
open('stewart_platform_model.slx')
-
-

Author: Dehaeze Thomas

-

Created: 2020-09-01 mar. 13:18

+

Created: 2020-09-07 lun. 23:16

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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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CURVE() GEOMETRIC_REPRESENTATION_ITEM() +RATIONAL_B_SPLINE_CURVE((1.,0.5,1.,0.5,1.,0.5,1.)) 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+#11715 = ORIENTED_EDGE ( 'NONE', *, *, #486, .T. ) ; +#11716 = CYLINDRICAL_SURFACE ( 'NONE', #1377, 2.750000000000002665 ) ; +ENDSEC; +END-ISO-10303-21; diff --git a/matlab/stewart_platform_model.slx b/matlab/stewart_platform_model.slx index f82b86b..1dce7a0 100644 Binary files a/matlab/stewart_platform_model.slx and b/matlab/stewart_platform_model.slx differ diff --git a/org/flexible-stewart-platform.org b/org/flexible-stewart-platform.org index 798667c..0dd8222 100644 --- a/org/flexible-stewart-platform.org +++ b/org/flexible-stewart-platform.org @@ -1450,11 +1450,11 @@ Quantitatively: simulinkproject('../'); #+end_src -** Stewart Platform Initialization #+begin_src matlab - apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K'); + open('stewart_platform_model.slx') #+end_src +** Stewart Platform Initialization #+begin_src matlab stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart); @@ -1463,32 +1463,23 @@ Quantitatively: #+end_src #+begin_src matlab - 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)); + 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'); #+end_src #+begin_src matlab - 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'); + 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)); #+end_src #+begin_src matlab @@ -1497,7 +1488,6 @@ Quantitatively: #+begin_src matlab stewart = initializeCylindricalStruts(stewart); - % stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none'); #+end_src #+begin_src matlab @@ -1518,10 +1508,6 @@ Quantitatively: disturbances = initializeDisturbances(); #+end_src -#+begin_src matlab - open('stewart_platform_model.slx') -#+end_src - ** Initialization :noexport: #+begin_src matlab %% Options for Linearized @@ -1533,9 +1519,9 @@ Quantitatively: %% 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, '/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 Motion - Actuators [m] - io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Lm'); io_i = io_i + 1; % Relative Motion - Legs [m] + io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Lm'); io_i = io_i + 1; % Relative Motion - Legs [m] #+end_src #+begin_src matlab :exports none @@ -1581,3 +1567,156 @@ Quantitatively: linkaxes([ax1,ax2],'x'); #+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/comp_relative_motion_sensor_act_leg.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:comp_relative_motion_sensor_act_leg +#+caption: Comparison of the dynamique from actuator to relative motion sensor located in parallel with the actuator or with the leg (flexible joints included) +#+RESULTS: +[[file:figs/comp_relative_motion_sensor_act_leg.png]] + +* Struts with Encoders +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab :results none :exports none + simulinkproject('../'); +#+end_src + +** Flexible Strut +#+begin_src matlab + apa = load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K'); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f '); +#+end_src + +#+RESULTS: +| Total number of Nodes | 8 | +| Number of interface Nodes | 8 | +| Number of Modes | 6 | +| Size of M and K matrices | 54 | + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f '); +#+end_src + +#+caption: Coordinates of the interface nodes +#+RESULTS: +| 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 | + +** Stewart Platform +#+begin_src matlab + 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); +#+end_src + +#+begin_src matlab + disturbances = initializeDisturbances(); + ground = initializeGround('type', 'none'); + payload = initializePayload('type', 'rigid', 'm', 1); + controller = initializeController('type', 'open-loop'); + references = initializeReferences(stewart); +#+end_src + + +#+begin_src matlab + %% 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] +#+end_src + +#+begin_src matlab + %% 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'}; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:6 + plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]); + end + for i = 1:6 + plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]); + end + for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_relative_motion_sensor_act_leg_encoder.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:comp_relative_motion_sensor_act_leg_encoder +#+caption: +#+RESULTS: +[[file:figs/comp_relative_motion_sensor_act_leg_encoder.png]] diff --git a/org/stewart-architecture.html b/org/stewart-architecture.html new file mode 100644 index 0000000..55564f0 --- /dev/null +++ b/org/stewart-architecture.html @@ -0,0 +1,2927 @@ + + + + + + +Stewart Platform - Definition of the Architecture + + + + + + + + + + + + +
+ UP + | + HOME +
+

Stewart Platform - Definition of the Architecture

+
+

Table of Contents

+
+ +
+
+ +

+In this document is explained how the Stewart Platform architecture is defined. +

+ +

+Some efforts has been made such that the procedure for the definition of the Stewart Platform architecture is as logical and clear as possible. +

+ +

+When possible, the notations are compatible with the one used in (Taghirad 2013). +

+ +

+The definition of the Stewart platform is done in three main parts: +

+
    +
  • First, the geometry if defined (Section 1)
  • +
  • Then, the inertia of the mechanical elements are defined (Section 2)
  • +
  • Finally, the Stiffness and Damping characteristics of the elements are defined (Section 3)
  • +
+ +

+In section 4, the procedure the initialize the Stewart platform is summarize and the associated Matlab code is shown. +

+ +

+Finally, all the Matlab function used to initialize the Stewart platform are described in section 5. +

+ +
+

1 Definition of the Stewart Platform Geometry

+
+

+ +

+

+Stewart platforms are generated in multiple steps: +

+
    +
  • Definition of the frames
  • +
  • Definition of the location of the joints
  • +
  • Computation of the length and orientation of the struts
  • +
  • Choice of the rest position of the mobile platform
  • +
+ +

+This steps are detailed below. +

+
+
+

1.1 Frames Definition

+
+

+We define 4 important frames (see Figure 1): +

+
    +
  • \(\{F\}\): Frame fixed to the Fixed base and located at the center of its bottom surface. +This is used to fix the Stewart platform to some support.
  • +
  • \(\{M\}\): Frame fixed to the Moving platform and located at the center of its top surface. +This is used to place things on top of the Stewart platform.
  • +
  • \(\{A\}\): Frame fixed to the fixed base.
  • +
  • \(\{B\}\): Frame fixed to the moving platform.
  • +
+ +

+Even though frames \(\{A\}\) and \(\{B\}\) don’t usually correspond to physical elements, they are of primary importance. +Firstly, they are used for the definition of the motion of the Mobile platform with respect to the fixed frame: +

+
    +
  • In position: \({}^A\bm{P}_{B}\) (read: Position of frame \(\{B\}\) expressed in frame \(\{A\}\))
  • +
  • In rotation: \({}^A\bm{R}_{B}\) (read: The rotation matrix that express the orientation of frame \(\{B\}\) expressed in frame \(\{A\}\))
  • +
+

+The frames \(\{A\}\) and \(\{B\}\) are used for all the kinematic analysis (Jacobian, Stiffness matrix, …). +

+ +

+Typical choice of \(\{A\}\) and \(\{B\}\) are: +

+
    +
  • Center of mass of the payload
  • +
  • Location where external forces are applied to the mobile platform (for instance when the mobile platform is in contact with a stiff environment)
  • +
  • Center of the cube for the cubic configuration
  • +
+ +

+The definition of the frames is done with the initializeFramesPositions function (link); +

+ + +
+

frame_definition.png +

+

Figure 1: Definition of the Frames for the Stewart Platform

+
+
+
+ +
+

1.2 Location of the Spherical Joints

+
+

+Then, we define the location of the spherical joints (see Figure 2): +

+
    +
  • \(\bm{a}_{i}\) are the position of the spherical joints fixed to the fixed base
  • +
  • \(\bm{b}_{i}\) are the position of the spherical joints fixed to the moving platform
  • +
+ +

+The location of the joints will define the Geometry of the Stewart platform. +Many characteristics of the platform depend on the location of the joints. +

+ +

+The location of the joints can be set to arbitrary positions or it can be computed to obtain specific configurations such as: +

+
    +
  • A cubic configuration: function generateCubicConfiguration (described in this file)
  • +
  • A symmetrical configuration
  • +
+ +

+A function (generateGeneralConfiguration) to set the position of the joints on a circle is described here. +

+ +

+The location of the spherical joints are then given by \({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\). +

+ + +
+

joint_location.png +

+

Figure 2: Position of the Spherical/Universal joints for the Stewart Platform

+
+
+
+ +
+

1.3 Length and orientation of the struts

+
+

+From the location of the joints (\({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\)), we compute the length \(l_i\) and orientation of each strut \(\hat{\bm{s}}_i\) (unit vector aligned with the strut). +The length and orientation of each strut is represented in figure 3. +

+ +

+This is done with the computeJointsPose function (link). +

+ + +
+

length_orientation_struts.png +

+

Figure 3: Length \(l_i\) and orientation \(\hat{\bm{s}}_i\) of the Stewart platform struts

+
+
+
+ +
+

1.4 Rest Position of the Stewart platform

+
+

+We may want to initialize the Stewart platform in some position and orientation that corresponds to its rest position. +

+ +

+To do so, we choose: +

+
    +
  • the position of \(\bm{O}_B\) expressed in \(\{A\}\) using \({}^A\bm{P}\)
  • +
  • the orientation of \(\{B\}\) expressed in \(\{A\}\) using a rotation matrix \({}^{A}\bm{R}_{B}\)
  • +
+ +

+Then, the function initializeStewartPose (link) compute the corresponding initial and rest position of each of the strut. +

+
+
+
+ +
+

2 Definition of the Inertia and geometry of the Fixed base, Mobile platform and Struts

+
+

+ +

+

+Now that the geometry of the Stewart platform has been defined, we have to choose the inertia of: +

+
    +
  • The Fixed base
  • +
  • The Mobile platform
  • +
  • The two parts of the struts
  • +
+ +

+The inertia of these elements will modify the dynamics of the systems. +It is thus important to set them properly. +

+
+
+

2.1 Inertia and Geometry of the Fixed and Mobile platforms

+
+

+In order to set the inertia of the fixed and mobile platforms, we can use the following function that assume that both platforms are cylindrical: +

+
    +
  • initializeCylindricalPlatforms (link): by choosing the height, radius and mass of the platforms, it computes the inertia matrix that will be used for simulation
  • +
+
+
+ +
+

2.2 Inertia and Geometry of the struts

+
+

+Similarly for the struts, we suppose here that they have a cylindrical shape. +They are initialize with the following function: +

+
    +
  • initializeCylindricalStruts (link): the two parts of each strut are supposed to by cylindrical. We can set the mass and geometry of both strut parts.
  • +
+
+
+
+ +
+

3 Definition of the stiffness and damping of the joints

+
+

+ +

+

+The global stiffness and damping of the Stewart platform depends on its geometry but also on the stiffness and damping of: +

+
    +
  • the actuator because of the finite stiffness of the actuator / linear guide
  • +
  • the spherical joints
  • +
+
+ +
+

3.1 Stiffness and Damping of the Actuator

+
+

+Each Actuator is modeled by 3 elements in parallel (Figure 4): +

+
    +
  • A spring with a stiffness \(k_{i}\)
  • +
  • A dashpot with a damping \(c_{i}\)
  • +
  • An ideal force actuator generating a force \(\tau_i\)
  • +
+ + +
+

stewart_platform_actuator.png +

+

Figure 4: Model of the Stewart platform actuator

+
+ +

+The initialization of the stiffness and damping properties of the actuators is done with the initializeStrutDynamics (link). +

+
+
+ +
+

3.2 Stiffness and Damping of the Spherical Joints

+
+

+Even though we often suppose that the spherical joint are perfect in the sense that we neglect its stiffness and damping, we can set some rotation stiffness and damping of each of the spherical/universal joints. +

+ +

+This is done with the initializeJointDynamics function (link). +

+
+
+
+ +
+

4 Summary of the Initialization Procedure and Matlab Example

+
+

+ +

+

+The procedure to define the Stewart platform is the following: +

+
    +
  1. Define the initial position of frames \(\{A\}\), \(\{B\}\), \(\{F\}\) and \(\{M\}\). +We do that using the initializeFramesPositions function. +We have to specify the total height of the Stewart platform \(H\) and the position \({}^{M}\bm{O}_{B}\) of \(\{B\}\) with respect to \(\{M\}\).
  2. +
  3. Compute the positions of joints \({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\). +We can do that using various methods depending on the wanted architecture: +
      +
    • generateCubicConfiguration permits to generate a cubic configuration
    • +
  4. +
  5. Compute the position and orientation of the joints with respect to the fixed base and the moving platform. +This is done with the computeJointsPose function. +If wanted, compute the rest position of each strut to have the wanted pose of the mobile platform with the function initializeStewartPose.
  6. +
  7. Define the mass and inertia of each element of the Stewart platform with the initializeCylindricalPlatforms and initializeCylindricalStruts
  8. +
  9. Define the dynamical properties of the Stewart platform by setting the stiffness and damping of the actuators and joints.
  10. +
+ +

+By following this procedure, we obtain a Matlab structure stewart that contains all the information for the Simscape model and for further analysis. +

+
+
+

4.1 Example of the initialization of a Stewart Platform

+
+

+Let’s first define the Stewart Platform Geometry. +

+
+
stewart = initializeStewartPlatform();
+stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeStewartPose(stewart, 'AP', [0;0;0], 'ARB', eye(3));
+
+
+ +

+Then, define the inertia and geometry of the fixed base, mobile platform and struts. +

+
+
stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+
+
+ +

+We initialize the strut stiffness and damping properties. +

+
+
stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e2*ones(6,1));
+stewart = initializeAmplifiedStrutDynamics(stewart);
+stewart = initializeJointDynamics(stewart);
+
+
+ +

+And finally the inertial sensors included in each strut. +

+
+
stewart = initializeInertialSensor(stewart, 'type', 'none');
+
+
+ +

+The obtained stewart Matlab structure contains all the information for analysis of the Stewart platform and for simulations using Simscape. +

+ +

+The function displayArchitecture can be used to display the current Stewart configuration: +

+
+
displayArchitecture(stewart, 'views', 'all');
+
+
+ + +
+

stewart_architecture_example.png +

+

Figure 5: Display of the current Stewart platform architecture (png, pdf)

+
+ +

+There are many options to show or hides elements such as labels and frames. +The documentation of the function is available here. +

+ +

+Let’s now move a little bit the top platform and re-display the configuration: +

+
+
tx = 0.1; % [rad]
+ty = 0.2; % [rad]
+tz = 0.05; % [rad]
+
+Rx = [1 0        0;
+      0 cos(tx) -sin(tx);
+      0 sin(tx)  cos(tx)];
+
+Ry = [ cos(ty) 0 sin(ty);
+      0        1 0;
+      -sin(ty) 0 cos(ty)];
+
+Rz = [cos(tz) -sin(tz) 0;
+      sin(tz)  cos(tz) 0;
+      0        0       1];
+
+ARB = Rz*Ry*Rx;
+AP = [0.08; 0; 0]; % [m]
+
+displayArchitecture(stewart, 'AP', AP, 'ARB', ARB);
+view([0 -1 0]);
+
+
+ + +
+

stewart_architecture_example_pose.png +

+

Figure 6: Display of the Stewart platform architecture at some defined pose (png, pdf)

+
+ +

+One can also use the describeStewartPlatform function to have a description of the current Stewart platform’s state. +

+ +
+describeStewartPlatform(stewart)
+GEOMETRY:
+- The height between the fixed based and the top platform is 90 [mm].
+- Frame {A} is located 45 [mm] above the top platform.
+- The initial length of the struts are:
+	 95.2, 95.2, 95.2, 95.2, 95.2, 95.2 [mm]
+
+ACTUATORS:
+- The actuators are mechanicaly amplified.
+- The vertical stiffness and damping contribution of the piezoelectric stack is:
+	 ka = 2e+07 [N/m] 	 ca = 1e+01 [N/(m/s)]
+- Vertical stiffness when the piezoelectric stack is removed is:
+	 kr = 5e+06 [N/m] 	 cr = 1e+01 [N/(m/s)]
+
+JOINTS:
+- The joints on the fixed based are universal joints
+- The joints on the mobile based are spherical joints
+- The position of the joints on the fixed based with respect to {F} are (in [mm]):
+	  113 	 -20 	  15
+	  113 	  20 	  15
+	 -39.3 	  108 	  15
+	 -73.9 	  88.1 	  15
+	 -73.9 	 -88.1 	  15
+	 -39.3 	 -108 	  15
+- The position of the joints on the mobile based with respect to {M} are (in [mm]):
+	  57.9 	 -68.9 	 -15
+	  57.9 	  68.9 	 -15
+	  30.8 	  84.6 	 -15
+	 -88.6 	  15.6 	 -15
+	 -88.6 	 -15.6 	 -15
+	  30.8 	 -84.6 	 -15
+
+KINEMATICS:
+'org_babel_eoe'
+ans =
+    'org_babel_eoe'
+
+
+
+
+ +
+

5 Functions

+
+

+ +

+
+ +
+

5.1 initializeStewartPlatform: Initialize the Stewart Platform structure

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Documentation

+
+ +
+

stewart-frames-position.png +

+

Figure 7: Definition of the position of the frames

+
+
+
+ +
+

Function description

+
+
+
function [stewart] = initializeStewartPlatform()
+% initializeStewartPlatform - Initialize the stewart structure
+%
+% Syntax: [stewart] = initializeStewartPlatform(args)
+%
+% Outputs:
+%    - stewart - A structure with the following sub-structures:
+%      - platform_F -
+%      - platform_M -
+%      - joints_F   -
+%      - joints_M   -
+%      - struts_F   -
+%      - struts_M   -
+%      - actuators  -
+%      - geometry   -
+%      - properties -
+
+
+
+
+ +
+

Initialize the Stewart structure

+
+
+
stewart = struct();
+stewart.platform_F = struct();
+stewart.platform_M = struct();
+stewart.joints_F   = struct();
+stewart.joints_M   = struct();
+stewart.struts_F   = struct();
+stewart.struts_M   = struct();
+stewart.actuators  = struct();
+stewart.sensors    = struct();
+stewart.sensors.inertial = struct();
+stewart.sensors.force    = struct();
+stewart.sensors.relative = struct();
+stewart.geometry   = struct();
+stewart.kinematics = struct();
+
+
+
+
+
+ +
+

5.2 initializeFramesPositions: Initialize the positions of frames {A}, {B}, {F} and {M}

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Documentation

+
+ +
+

stewart-frames-position.png +

+

Figure 8: Definition of the position of the frames

+
+
+
+ +
+

Function description

+
+
+
function [stewart] = initializeFramesPositions(stewart, args)
+% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
+%
+% Syntax: [stewart] = initializeFramesPositions(stewart, args)
+%
+% Inputs:
+%    - args - Can have the following fields:
+%        - H    [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
+%        - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
+%
+% Outputs:
+%    - stewart - A structure with the following fields:
+%        - geometry.H      [1x1] - Total Height of the Stewart Platform [m]
+%        - geometry.FO_M   [3x1] - Position of {M} with respect to {F} [m]
+%        - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]
+%        - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.H    (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
+    args.MO_B (1,1) double {mustBeNumeric} = 50e-3
+end
+
+
+
+
+ +
+

Compute the position of each frame

+
+
+
H = args.H; % Total Height of the Stewart Platform [m]
+
+FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]
+
+MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]
+
+FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.geometry.H      = H;
+stewart.geometry.FO_M   = FO_M;
+stewart.platform_M.MO_B = MO_B;
+stewart.platform_F.FO_A = FO_A;
+
+
+
+
+
+ +
+

5.3 generateGeneralConfiguration: Generate a Very General Configuration

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Documentation

+
+

+Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}. +The radius of the circles can be chosen as well as the angles where the joints are located (see Figure 9). +

+ + +
+

stewart_bottom_plate.png +

+

Figure 9: Position of the joints

+
+
+
+ +
+

Function description

+
+
+
function [stewart] = generateGeneralConfiguration(stewart, args)
+% generateGeneralConfiguration - Generate a Very General Configuration
+%
+% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)
+%
+% Inputs:
+%    - args - Can have the following fields:
+%        - FH  [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]
+%        - FR  [1x1] - Radius of the position of the fixed joints in the X-Y [m]
+%        - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]
+%        - MH  [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]
+%        - FR  [1x1] - Radius of the position of the mobile joints in the X-Y [m]
+%        - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%        - platform_F.Fa  [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
+%        - platform_M.Mb  [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.FH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
+    args.FR  (1,1) double {mustBeNumeric, mustBePositive} = 115e-3;
+    args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180);
+    args.MH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
+    args.MR  (1,1) double {mustBeNumeric, mustBePositive} = 90e-3;
+    args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180);
+end
+
+
+
+
+ +
+

Compute the pose

+
+
+
Fa = zeros(3,6);
+Mb = zeros(3,6);
+
+
+ +
+
for i = 1:6
+  Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i));  args.FH];
+  Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
+end
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.platform_F.Fa = Fa;
+stewart.platform_M.Mb = Mb;
+
+
+
+
+
+ +
+

5.4 computeJointsPose: Compute the Pose of the Joints

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Documentation

+
+ +
+

stewart-struts.png +

+

Figure 10: Position and orientation of the struts

+
+
+
+ +
+

Function description

+
+
+
function [stewart] = computeJointsPose(stewart, args)
+% computeJointsPose -
+%
+% Syntax: [stewart] = computeJointsPose(stewart, args)
+%
+% Inputs:
+%    - stewart - A structure with the following fields
+%        - platform_F.Fa   [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
+%        - platform_M.Mb   [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
+%        - platform_F.FO_A [3x1] - Position of {A} with respect to {F}
+%        - platform_M.MO_B [3x1] - Position of {B} with respect to {M}
+%        - geometry.FO_M   [3x1] - Position of {M} with respect to {F}
+%    - args - Can have the following fields:
+%        - AP   [3x1] - The wanted position of {B} with respect to {A}
+%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
+%
+% Outputs:
+%    - stewart - A structure with the following added fields
+%        - geometry.Aa    [3x6]   - The i'th column is the position of ai with respect to {A}
+%        - geometry.Ab    [3x6]   - The i'th column is the position of bi with respect to {A}
+%        - geometry.Ba    [3x6]   - The i'th column is the position of ai with respect to {B}
+%        - geometry.Bb    [3x6]   - The i'th column is the position of bi with respect to {B}
+%        - geometry.l     [6x1]   - The i'th element is the initial length of strut i
+%        - geometry.As    [3x6]   - The i'th column is the unit vector of strut i expressed in {A}
+%        - geometry.Bs    [3x6]   - The i'th column is the unit vector of strut i expressed in {B}
+%        - struts_F.l     [6x1]   - Length of the Fixed part of the i'th strut
+%        - struts_M.l     [6x1]   - Length of the Mobile part of the i'th strut
+%        - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
+%        - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
+    args.ARB (3,3) double {mustBeNumeric} = eye(3)
+end
+
+
+
+
+ +
+

Check the stewart structure elements

+
+
+
assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
+Fa = stewart.platform_F.Fa;
+
+assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
+Mb = stewart.platform_M.Mb;
+
+assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
+FO_A = stewart.platform_F.FO_A;
+
+assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
+MO_B = stewart.platform_M.MO_B;
+
+assert(isfield(stewart.geometry,   'FO_M'), 'stewart.geometry should have attribute FO_M')
+FO_M = stewart.geometry.FO_M;
+
+
+
+
+ +
+

Compute the position of the Joints

+
+
+
Aa = Fa - repmat(FO_A, [1, 6]);
+Bb = Mb - repmat(MO_B, [1, 6]);
+
+
+ +

+Original: +

+
+
Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]);
+Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]);
+
+
+ +

+Translation & Rotation: (Rotation and then translation) +

+
+
Ab = args.ARB *Bb - repmat(-args.AP, [1, 6]);
+Ba = args.ARB'*Aa - repmat( args.AP, [1, 6]);
+
+
+
+
+ +
+

Compute the strut length and orientation

+
+
+
As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As
+
+l = vecnorm(Ab - Aa)';
+
+
+ +
+
Bs = (Bb - Ba)./vecnorm(Bb - Ba);
+
+
+
+
+ +
+

Compute the orientation of the Joints

+
+
+
FRa = zeros(3,3,6);
+MRb = zeros(3,3,6);
+
+for i = 1:6
+  FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
+  FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));
+
+  MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
+  MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
+end
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.geometry.Aa = Aa;
+stewart.geometry.Ab = Ab;
+stewart.geometry.Ba = Ba;
+stewart.geometry.Bb = Bb;
+stewart.geometry.As = As;
+stewart.geometry.Bs = Bs;
+stewart.geometry.l  = l;
+
+stewart.struts_F.l  = l/2;
+stewart.struts_M.l  = l/2;
+
+stewart.platform_F.FRa = FRa;
+stewart.platform_M.MRb = MRb;
+
+
+
+
+
+ +
+

5.5 initializeStewartPose: Determine the initial stroke in each leg to have the wanted pose

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeStewartPose(stewart, args)
+% initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
+%                         It uses the inverse kinematic
+%
+% Syntax: [stewart] = initializeStewartPose(stewart, args)
+%
+% Inputs:
+%    - stewart - A structure with the following fields
+%        - Aa   [3x6] - The positions ai expressed in {A}
+%        - Bb   [3x6] - The positions bi expressed in {B}
+%    - args - Can have the following fields:
+%        - AP   [3x1] - The wanted position of {B} with respect to {A}
+%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - actuators.Leq [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
+    args.ARB (3,3) double {mustBeNumeric} = eye(3)
+end
+
+
+
+
+ +
+

Use the Inverse Kinematic function

+
+
+
[Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.actuators.Leq = dLi;
+
+
+
+
+
+ +
+

5.6 initializeCylindricalPlatforms: Initialize the geometry of the Fixed and Mobile Platforms

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeCylindricalPlatforms(stewart, args)
+% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
+%
+% Syntax: [stewart] = initializeCylindricalPlatforms(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - Fpm [1x1] - Fixed Platform Mass [kg]
+%        - Fph [1x1] - Fixed Platform Height [m]
+%        - Fpr [1x1] - Fixed Platform Radius [m]
+%        - Mpm [1x1] - Mobile Platform Mass [kg]
+%        - Mph [1x1] - Mobile Platform Height [m]
+%        - Mpr [1x1] - Mobile Platform Radius [m]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - platform_F [struct] - structure with the following fields:
+%        - type = 1
+%        - M [1x1] - Fixed Platform Mass [kg]
+%        - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]
+%        - H [1x1] - Fixed Platform Height [m]
+%        - R [1x1] - Fixed Platform Radius [m]
+%      - platform_M [struct] - structure with the following fields:
+%        - M [1x1] - Mobile Platform Mass [kg]
+%        - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]
+%        - H [1x1] - Mobile Platform Height [m]
+%        - R [1x1] - Mobile Platform Radius [m]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
+    args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
+    args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3
+    args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
+    args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
+    args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
+end
+
+
+
+
+ +
+

Compute the Inertia matrices of platforms

+
+
+
I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
+            1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
+            1/2 *args.Fpm * args.Fpr^2]);
+
+
+ +
+
I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
+            1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
+            1/2 *args.Mpm * args.Mpr^2]);
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.platform_F.type = 1;
+
+stewart.platform_F.I = I_F;
+stewart.platform_F.M = args.Fpm;
+stewart.platform_F.R = args.Fpr;
+stewart.platform_F.H = args.Fph;
+
+
+ +
+
stewart.platform_M.type = 1;
+
+stewart.platform_M.I = I_M;
+stewart.platform_M.M = args.Mpm;
+stewart.platform_M.R = args.Mpr;
+stewart.platform_M.H = args.Mph;
+
+
+
+
+
+ +
+

5.7 initializeSolidPlatforms: Initialize the geometry of the Fixed and Mobile Platforms

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeSolidPlatforms(stewart, args)
+% initializeSolidPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
+%
+% Syntax: [stewart] = initializeSolidPlatforms(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - density [1x1] - Density of the platforms [kg]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - platform_F [struct] - structure with the following fields:
+%        - type = 2
+%        - M [1x1] - Fixed Platform Density [kg/m^3]
+%      - platform_M [struct] - structure with the following fields:
+%        - type = 2
+%        - M [1x1] - Mobile Platform Density [kg/m^3]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.density (1,1) double {mustBeNumeric, mustBePositive} = 7800
+end
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.platform_F.type = 2;
+
+stewart.platform_F.density = args.density;
+
+
+ +
+
stewart.platform_M.type = 2;
+
+stewart.platform_M.density = args.density;
+
+
+
+
+
+ +
+

5.8 initializeCylindricalStruts: Define the inertia of cylindrical struts

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeCylindricalStruts(stewart, args)
+% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
+%
+% Syntax: [stewart] = initializeCylindricalStruts(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - Fsm [1x1] - Mass of the Fixed part of the struts [kg]
+%        - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]
+%        - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]
+%        - Msm [1x1] - Mass of the Mobile part of the struts [kg]
+%        - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]
+%        - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - struts_F [struct] - structure with the following fields:
+%        - M [6x1]   - Mass of the Fixed part of the struts [kg]
+%        - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
+%        - H [6x1]   - Height of cylinder for the Fixed part of the struts [m]
+%        - R [6x1]   - Radius of cylinder for the Fixed part of the struts [m]
+%      - struts_M [struct] - structure with the following fields:
+%        - M [6x1]   - Mass of the Mobile part of the struts [kg]
+%        - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
+%        - H [6x1]   - Height of cylinder for the Mobile part of the struts [m]
+%        - R [6x1]   - Radius of cylinder for the Mobile part of the struts [m]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.type_F    char   {mustBeMember(args.type_F,{'cylindrical', 'none'})} = 'cylindrical'
+    args.type_M    char   {mustBeMember(args.type_M,{'cylindrical', 'none'})} = 'cylindrical'
+    args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+    args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
+    args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
+    args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+    args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
+    args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
+end
+
+
+
+
+ +
+

Compute the properties of the cylindrical struts

+
+
+
Fsm = ones(6,1).*args.Fsm;
+Fsh = ones(6,1).*args.Fsh;
+Fsr = ones(6,1).*args.Fsr;
+
+Msm = ones(6,1).*args.Msm;
+Msh = ones(6,1).*args.Msh;
+Msr = ones(6,1).*args.Msr;
+
+
+ +
+
I_F = zeros(3, 3, 6); % Inertia of the "fixed" part of the strut
+I_M = zeros(3, 3, 6); % Inertia of the "mobile" part of the strut
+
+for i = 1:6
+  I_F(:,:,i) = diag([1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
+                     1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
+                     1/2  * Fsm(i) * Fsr(i)^2]);
+
+  I_M(:,:,i) = diag([1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
+                     1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
+                     1/2  * Msm(i) * Msr(i)^2]);
+end
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
switch args.type_M
+  case 'cylindrical'
+    stewart.struts_M.type = 1;
+  case 'none'
+    stewart.struts_M.type = 2;
+end
+
+stewart.struts_M.I = I_M;
+stewart.struts_M.M = Msm;
+stewart.struts_M.R = Msr;
+stewart.struts_M.H = Msh;
+
+
+ +
+
switch args.type_F
+  case 'cylindrical'
+    stewart.struts_F.type = 1;
+  case 'none'
+    stewart.struts_F.type = 2;
+end
+
+stewart.struts_F.I = I_F;
+stewart.struts_F.M = Fsm;
+stewart.struts_F.R = Fsr;
+stewart.struts_F.H = Fsh;
+
+
+
+
+
+ +
+

5.9 initializeStrutDynamics: Add Stiffness and Damping properties of each strut

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Documentation

+
+ +
+

piezoelectric_stack.jpg +

+

Figure 11: Example of a piezoelectric stach actuator (PI)

+
+ +

+A simplistic model of such amplified actuator is shown in Figure 12 where: +

+
    +
  • \(K\) represent the vertical stiffness of the actuator
  • +
  • \(C\) represent the vertical damping of the actuator
  • +
  • \(F\) represents the force applied by the actuator
  • +
  • \(F_{m}\) represents the total measured force
  • +
  • \(v_{m}\) represents the absolute velocity of the top part of the actuator
  • +
  • \(d_{m}\) represents the total relative displacement of the actuator
  • +
+ + +
+

actuator_model_simple.png +

+

Figure 12: Simple model of an Actuator

+
+
+
+ +
+

Function description

+
+
+
function [stewart] = initializeStrutDynamics(stewart, args)
+% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
+%
+% Syntax: [stewart] = initializeStrutDynamics(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - K [6x1] - Stiffness of each strut [N/m]
+%        - C [6x1] - Damping of each strut [N/(m/s)]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - actuators.type = 1
+%      - actuators.K [6x1] - Stiffness of each strut [N/m]
+%      - actuators.C [6x1] - Damping of each strut [N/(m/s)]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = 20e6*ones(6,1)
+    args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e1*ones(6,1)
+end
+
+
+
+
+ +
+

Add Stiffness and Damping properties of each strut

+
+
+
stewart.actuators.type = 1;
+
+stewart.actuators.K = args.K;
+stewart.actuators.C = args.C;
+
+
+
+
+
+ +
+

5.10 initializeAmplifiedStrutDynamics: Add Stiffness and Damping properties of each strut for an amplified piezoelectric actuator

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Documentation

+
+

+An amplified piezoelectric actuator is shown in Figure 13. +

+ + +
+

amplified_piezo_with_displacement_sensor.jpg +

+

Figure 13: Example of an Amplified piezoelectric actuator with an integrated displacement sensor (Cedrat Technologies)

+
+ +

+A simplistic model of such amplified actuator is shown in Figure 14 where the parameters are described in Table 1. +

+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: Parameters used for the model of the APA 100M
 Meaning
\(k_e\)Stiffness used to adjust the pole of the isolator
\(k_1\)Stiffness of the metallic suspension when the stack is removed
\(k_a\)Stiffness of the actuator
\(c_1\)Added viscous damping
+ + +
+

souleille18_model_piezo.png +

+

Figure 14: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator

+
+
+
+ +
+

Function description

+
+
+
function [stewart] = initializeAmplifiedStrutDynamics(stewart, args)
+% initializeAmplifiedStrutDynamics - Add Stiffness and Damping properties of each strut
+%
+% Syntax: [stewart] = initializeAmplifiedStrutDynamics(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - Ka [6x1] - Stiffness of the actuator [N/m]
+%        - Ke [6x1] - Stiffness used to adjust the pole of the isolator [N/m]
+%        - K1 [6x1] - Stiffness of the metallic suspension when the stack is removed [N/m]
+%        - C1 [6x1] - Added viscous damping [N/(m/s)]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - actuators.type = 2
+%      - actuators.K   [6x1] - Total Stiffness of each strut [N/m]
+%      - actuators.C   [6x1] - Total Damping of each strut [N/(m/s)]
+%      - actuators.Ka [6x1] - Stiffness of the actuator [N/m]
+%      - actuators.Ke [6x1] - Stiffness used to adjust the pole of the isolator [N/m]
+%      - actuators.K1 [6x1] - Stiffness of the metallic suspension when the stack is removed [N/m]
+%      - actuators.C1 [6x1] - Added viscous damping [N/(m/s)]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.Ke (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.5e6*ones(6,1)
+    args.Ka (6,1) double {mustBeNumeric, mustBeNonnegative} = 43e6*ones(6,1)
+    args.K1 (6,1) double {mustBeNumeric, mustBeNonnegative} = 0.4e6*ones(6,1)
+    args.C1 (6,1) double {mustBeNumeric, mustBeNonnegative} = 10*ones(6,1)
+end
+
+
+
+
+ +
+

Compute the total stiffness and damping

+
+
+
K = args.K1 + args.Ka.*args.Ke./(args.Ka + args.Ke);
+C = args.C1;
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.actuators.type = 2;
+
+stewart.actuators.Ka = args.Ka;
+stewart.actuators.Ke = args.Ke;
+stewart.actuators.K1 = args.K1;
+stewart.actuators.C1 = args.C1;
+
+stewart.actuators.K = K;
+stewart.actuators.C = C;
+
+
+
+
+
+ +
+

5.11 initializeFlexibleStrutDynamics: Model each strut with a flexible element

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeFlexibleStrutDynamics(stewart, args)
+% initializeFlexibleStrutDynamics - Add Stiffness and Damping properties of each strut
+%
+% Syntax: [stewart] = initializeFlexibleStrutDynamics(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - K [nxn] - Vertical stiffness contribution of the piezoelectric stack [N/m]
+%        - M [nxn] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
+%        - xi        [1x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
+%        - step_file [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
+%        - Gf [6x1] - Gain from strain in [m] to measured [N] such that it matches
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.K        double {mustBeNumeric} = zeros(6,6)
+    args.M        double {mustBeNumeric} = zeros(6,6)
+    args.H        double {mustBeNumeric} = 0
+    args.n_xyz    double {mustBeNumeric} = zeros(2,3)
+    args.xi       double {mustBeNumeric} = 0.1
+    args.Gf       double {mustBeNumeric} = 1
+    args.step_file char {} = ''
+end
+
+
+
+
+ +
+

Compute the axial offset

+
+
+
stewart.actuators.ax_off = (stewart.geometry.l(1) - args.H)/2; % Axial Offset at the ends of the actuator
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.actuators.type = 3;
+
+stewart.actuators.Km = args.K;
+stewart.actuators.Mm = args.M;
+
+stewart.actuators.n_xyz = args.n_xyz;
+stewart.actuators.xi = args.xi;
+
+stewart.actuators.step_file = args.step_file;
+
+stewart.actuators.K = args.K(3,3); % Axial Stiffness
+
+stewart.actuators.Gf = args.Gf;
+
+
+
+
+
+ +
+

5.12 initializeJointDynamics: Add Stiffness and Damping properties for spherical joints

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeJointDynamics(stewart, args)
+% initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
+%
+% Syntax: [stewart] = initializeJointDynamics(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'
+%        - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'
+%        - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]
+%        - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
+%        - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
+%        - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
+%        - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
+%        - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
+%        - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
+%        - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - stewart.joints_F and stewart.joints_M:
+%        - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)
+%        - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]
+%        - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]
+%        - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]
+%        - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]
+%        - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]
+%        - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof', 'spherical_3dof', 'flexible'})} = 'universal'
+    args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof', 'spherical_3dof', 'flexible'})} = 'spherical'
+    args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 33*ones(6,1)
+    args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
+    args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 236*ones(6,1)
+    args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
+    args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 33*ones(6,1)
+    args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
+    args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 236*ones(6,1)
+    args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
+    args.Ka_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8*ones(6,1)
+    args.Ca_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+    args.Kr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7*ones(6,1)
+    args.Cr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+    args.Ka_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8*ones(6,1)
+    args.Ca_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+    args.Kr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7*ones(6,1)
+    args.Cr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+    args.K_M        double {mustBeNumeric} = zeros(6,6)
+    args.M_M        double {mustBeNumeric} = zeros(6,6)
+    args.n_xyz_M    double {mustBeNumeric} = zeros(2,3)
+    args.xi_M       double {mustBeNumeric} = 0.1
+    args.step_file_M char {} = ''
+    args.K_F        double {mustBeNumeric} = zeros(6,6)
+    args.M_F        double {mustBeNumeric} = zeros(6,6)
+    args.n_xyz_F    double {mustBeNumeric} = zeros(2,3)
+    args.xi_F       double {mustBeNumeric} = 0.1
+    args.step_file_F char {} = ''
+end
+
+
+
+
+ +
+

Add Actuator Type

+
+
+
switch args.type_F
+  case 'universal'
+    stewart.joints_F.type = 1;
+  case 'spherical'
+    stewart.joints_F.type = 2;
+  case 'universal_p'
+    stewart.joints_F.type = 3;
+  case 'spherical_p'
+    stewart.joints_F.type = 4;
+  case 'flexible'
+    stewart.joints_F.type = 5;
+  case 'universal_3dof'
+    stewart.joints_F.type = 6;
+  case 'spherical_3dof'
+    stewart.joints_F.type = 7;
+end
+
+switch args.type_M
+  case 'universal'
+    stewart.joints_M.type = 1;
+  case 'spherical'
+    stewart.joints_M.type = 2;
+  case 'universal_p'
+    stewart.joints_M.type = 3;
+  case 'spherical_p'
+    stewart.joints_M.type = 4;
+  case 'flexible'
+    stewart.joints_M.type = 5;
+  case 'universal_3dof'
+    stewart.joints_M.type = 6;
+  case 'spherical_3dof'
+    stewart.joints_M.type = 7;
+end
+
+
+
+
+ +
+

Add Stiffness and Damping in Translation of each strut

+
+

+Axial and Radial (shear) Stiffness +

+
+
stewart.joints_M.Ka = args.Ka_M;
+stewart.joints_M.Kr = args.Kr_M;
+
+stewart.joints_F.Ka = args.Ka_F;
+stewart.joints_F.Kr = args.Kr_F;
+
+
+ +

+Translation Damping +

+
+
stewart.joints_M.Ca = args.Ca_M;
+stewart.joints_M.Cr = args.Cr_M;
+
+stewart.joints_F.Ca = args.Ca_F;
+stewart.joints_F.Cr = args.Cr_F;
+
+
+
+
+ +
+

Add Stiffness and Damping in Rotation of each strut

+
+

+Rotational Stiffness +

+
+
stewart.joints_M.Kf = args.Kf_M;
+stewart.joints_M.Kt = args.Kt_M;
+
+stewart.joints_F.Kf = args.Kf_F;
+stewart.joints_F.Kt = args.Kt_F;
+
+
+ +

+Rotational Damping +

+
+
stewart.joints_M.Cf = args.Cf_M;
+stewart.joints_M.Ct = args.Ct_M;
+
+stewart.joints_F.Cf = args.Cf_F;
+stewart.joints_F.Ct = args.Ct_F;
+
+
+
+
+ +
+

Stiffness and Mass matrices for flexible joint

+
+
+
stewart.joints_F.M = args.M_F;
+stewart.joints_F.K = args.K_F;
+stewart.joints_F.n_xyz = args.n_xyz_F;
+stewart.joints_F.xi = args.xi_F;
+stewart.joints_F.xi = args.xi_F;
+stewart.joints_F.step_file = args.step_file_F;
+
+stewart.joints_M.M = args.M_M;
+stewart.joints_M.K = args.K_M;
+stewart.joints_M.n_xyz = args.n_xyz_M;
+stewart.joints_M.xi = args.xi_M;
+stewart.joints_M.step_file = args.step_file_M;
+
+
+
+
+
+ +
+

5.13 initializeFlexibleStrutAndJointDynamics: Model each strut with a flexible element

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [stewart] = initializeFlexibleStrutAndJointDynamics(stewart, args)
+% initializeFlexibleStrutAndJointDynamics - Add Stiffness and Damping properties of each strut
+%
+% Syntax: [stewart] = initializeFlexibleStrutAndJointDynamics(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - K [nxn] - Vertical stiffness contribution of the piezoelectric stack [N/m]
+%        - M [nxn] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
+%        - xi        [1x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
+%        - step_file [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
+%        - Gf [6x1] - Gain from strain in [m] to measured [N] such that it matches
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.K        double {mustBeNumeric} = zeros(6,6)
+    args.M        double {mustBeNumeric} = zeros(6,6)
+    args.H        double {mustBeNumeric} = 0
+    args.n_xyz    double {mustBeNumeric} = zeros(2,3)
+    args.xi       double {mustBeNumeric} = 0.1
+    args.Gf       double {mustBeNumeric} = 1
+    args.step_file char {} = ''
+end
+
+
+
+
+ +
+

Compute the axial offset

+
+
+
stewart.actuators.ax_off = (stewart.geometry.l(1) - args.H)/2; % Axial Offset at the ends of the actuator
+
+
+
+
+ +
+

Populate the stewart structure

+
+

+No discrete joints: +

+
+
stewart.joints_F.type = 10;
+stewart.joints_M.type = 10;
+
+
+
+ +

+No discrete struts: +

+
+
stewart.struts_F.type = 3;
+stewart.struts_M.type = 3;
+
+
+ +
+
stewart.actuators.type = 4;
+
+stewart.actuators.Km = args.K;
+stewart.actuators.Mm = args.M;
+
+stewart.actuators.n_xyz = args.n_xyz;
+stewart.actuators.xi = args.xi;
+
+stewart.actuators.step_file = args.step_file;
+
+stewart.actuators.K = args.K(3,3); % Axial Stiffness
+
+stewart.actuators.Gf = args.Gf;
+
+
+
+
+
+ +
+

5.14 initializeInertialSensor: Initialize the inertial sensor in each strut

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Geophone - Working Principle

+
+

+From the schematic of the Z-axis geophone shown in Figure 15, we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\): +\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \] +with: +

+
    +
  • \(\omega_0 = \sqrt{\frac{k}{m}}\)
  • +
  • \(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)
  • +
+ + +
+

inertial_sensor.png +

+

Figure 15: Schematic of a Z-Axis geophone

+
+ +

+We see that at frequencies above \(\omega_0\): +\[ \frac{\dot{d}}{\dot{w}} \approx -1 \] +

+ +

+And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support. +

+ +

+We generally want to have the smallest resonant frequency \(\omega_0\) to measure low frequency absolute velocity, however there is a trade-off between \(\omega_0\) and the mass of the inertial mass. +

+
+
+ +
+

Accelerometer - Working Principle

+
+

+From the schematic of the Z-axis accelerometer shown in Figure 16, we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\): +\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \] +with: +

+
    +
  • \(\omega_0 = \sqrt{\frac{k}{m}}\)
  • +
  • \(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)
  • +
+ + +
+

inertial_sensor.png +

+

Figure 16: Schematic of a Z-Axis geophone

+
+ +

+We see that at frequencies below \(\omega_0\): +\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \] +

+ +

+And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support. +

+ +

+Note that there is trade-off between: +

+
    +
  • the highest measurable acceleration \(\omega_0\)
  • +
  • the sensitivity of the accelerometer which is equal to \(-\frac{1}{{\omega_0}^2}\)
  • +
+
+
+ +
+

Function description

+
+
+
function [stewart] = initializeInertialSensor(stewart, args)
+% initializeInertialSensor - Initialize the inertial sensor in each strut
+%
+% Syntax: [stewart] = initializeInertialSensor(args)
+%
+% Inputs:
+%    - args - Structure with the following fields:
+%        - type       - 'geophone', 'accelerometer', 'none'
+%        - mass [1x1] - Weight of the inertial mass [kg]
+%        - freq [1x1] - Cutoff frequency [Hz]
+%
+% Outputs:
+%    - stewart - updated Stewart structure with the added fields:
+%      - stewart.sensors.inertial
+%        - type    - 1 (geophone), 2 (accelerometer), 3 (none)
+%        - K [1x1] - Stiffness [N/m]
+%        - C [1x1] - Damping [N/(m/s)]
+%        - M [1x1] - Inertial Mass [kg]
+%        - G [1x1] - Gain
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.type       char   {mustBeMember(args.type,{'geophone', 'accelerometer', 'none'})} = 'none'
+    args.mass (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e-2
+    args.freq (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e3
+end
+
+
+
+
+ +
+

Compute the properties of the sensor

+
+
+
sensor = struct();
+
+switch args.type
+  case 'geophone'
+    sensor.type = 1;
+
+    sensor.M = args.mass;
+    sensor.K = sensor.M * (2*pi*args.freq)^2;
+    sensor.C = 2*sqrt(sensor.M * sensor.K);
+  case 'accelerometer'
+    sensor.type = 2;
+
+    sensor.M = args.mass;
+    sensor.K = sensor.M * (2*pi*args.freq)^2;
+    sensor.C = 2*sqrt(sensor.M * sensor.K);
+    sensor.G = -sensor.K/sensor.M;
+  case 'none'
+    sensor.type = 3;
+end
+
+
+
+
+ +
+

Populate the stewart structure

+
+
+
stewart.sensors.inertial = sensor;
+
+
+
+
+
+ +
+

5.15 displayArchitecture: 3D plot of the Stewart platform architecture

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [] = displayArchitecture(stewart, args)
+% displayArchitecture - 3D plot of the Stewart platform architecture
+%
+% Syntax: [] = displayArchitecture(args)
+%
+% Inputs:
+%    - stewart
+%    - args - Structure with the following fields:
+%        - AP   [3x1] - The wanted position of {B} with respect to {A}
+%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
+%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
+%        - F_color [color] - Color used for the Fixed elements
+%        - M_color [color] - Color used for the Mobile elements
+%        - L_color [color] - Color used for the Legs elements
+%        - frames    [true/false] - Display the Frames
+%        - legs      [true/false] - Display the Legs
+%        - joints    [true/false] - Display the Joints
+%        - labels    [true/false] - Display the Labels
+%        - platforms [true/false] - Display the Platforms
+%        - views     ['all', 'xy', 'yz', 'xz', 'default'] -
+%
+% Outputs:
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+    args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
+    args.ARB (3,3) double {mustBeNumeric} = eye(3)
+    args.F_color = [0 0.4470 0.7410]
+    args.M_color = [0.8500 0.3250 0.0980]
+    args.L_color = [0 0 0]
+    args.frames    logical {mustBeNumericOrLogical} = true
+    args.legs      logical {mustBeNumericOrLogical} = true
+    args.joints    logical {mustBeNumericOrLogical} = true
+    args.labels    logical {mustBeNumericOrLogical} = true
+    args.platforms logical {mustBeNumericOrLogical} = true
+    args.views     char    {mustBeMember(args.views,{'all', 'xy', 'xz', 'yz', 'default'})} = 'default'
+end
+
+
+
+
+ +
+

Check the stewart structure elements

+
+
+
assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
+FO_A = stewart.platform_F.FO_A;
+
+assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
+MO_B = stewart.platform_M.MO_B;
+
+assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
+H = stewart.geometry.H;
+
+assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
+Fa = stewart.platform_F.Fa;
+
+assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
+Mb = stewart.platform_M.Mb;
+
+
+
+
+ + +
+

Figure Creation, Frames and Homogeneous transformations

+
+

+The reference frame of the 3d plot corresponds to the frame \(\{F\}\). +

+
+
if ~strcmp(args.views, 'all')
+  figure;
+else
+  f = figure('visible', 'off');
+end
+
+hold on;
+
+
+ +

+We first compute homogeneous matrices that will be useful to position elements on the figure where the reference frame is \(\{F\}\). +

+
+
FTa = [eye(3), FO_A; ...
+       zeros(1,3), 1];
+ATb = [args.ARB, args.AP; ...
+       zeros(1,3), 1];
+BTm = [eye(3), -MO_B; ...
+       zeros(1,3), 1];
+
+FTm = FTa*ATb*BTm;
+
+
+ +

+Let’s define a parameter that define the length of the unit vectors used to display the frames. +

+
+
d_unit_vector = H/4;
+
+
+ +

+Let’s define a parameter used to position the labels with respect to the center of the element. +

+
+
d_label = H/20;
+
+
+
+
+ +
+

Fixed Base elements

+
+

+Let’s first plot the frame \(\{F\}\). +

+
+
Ff = [0, 0, 0];
+if args.frames
+  quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
+          [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
+
+  if args.labels
+    text(Ff(1) + d_label, ...
+        Ff(2) + d_label, ...
+        Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
+  end
+end
+
+
+ +

+Now plot the frame \(\{A\}\) fixed to the Base. +

+
+
if args.frames
+  quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
+          [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
+
+  if args.labels
+    text(FO_A(1) + d_label, ...
+         FO_A(2) + d_label, ...
+         FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
+  end
+end
+
+
+ +

+Let’s then plot the circle corresponding to the shape of the Fixed base. +

+
+
if args.platforms && stewart.platform_F.type == 1
+  theta = [0:0.01:2*pi+0.01]; % Angles [rad]
+  v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
+  center = [0; 0; 0]; % Center of the circle
+  radius = stewart.platform_F.R; % Radius of the circle [m]
+
+  points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
+
+  plot3(points(1,:), ...
+        points(2,:), ...
+        points(3,:), '-', 'Color', args.F_color);
+end
+
+
+ +

+Let’s now plot the position and labels of the Fixed Joints +

+
+
if args.joints
+  scatter3(Fa(1,:), ...
+           Fa(2,:), ...
+           Fa(3,:), 'MarkerEdgeColor', args.F_color);
+  if args.labels
+    for i = 1:size(Fa,2)
+      text(Fa(1,i) + d_label, ...
+           Fa(2,i), ...
+           Fa(3,i), sprintf('$a_{%i}$', i), 'Color', args.F_color);
+    end
+  end
+end
+
+
+
+
+ +
+

Mobile Platform elements

+
+

+Plot the frame \(\{M\}\). +

+
+
Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}
+
+if args.frames
+  FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
+  quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
+          FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)
+
+  if args.labels
+    text(Fm(1) + d_label, ...
+         Fm(2) + d_label, ...
+         Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
+  end
+end
+
+
+ +

+Plot the frame \(\{B\}\). +

+
+
FB = FO_A + args.AP;
+
+if args.frames
+  FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
+  quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
+          FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)
+
+  if args.labels
+    text(FB(1) - d_label, ...
+         FB(2) + d_label, ...
+         FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
+  end
+end
+
+
+ +

+Let’s then plot the circle corresponding to the shape of the Mobile platform. +

+
+
if args.platforms && stewart.platform_M.type == 1
+  theta = [0:0.01:2*pi+0.01]; % Angles [rad]
+  v = null((FTm(1:3,1:3)*[0;0;1])'); % Two vectors that are perpendicular to the circle normal
+  center = Fm(1:3); % Center of the circle
+  radius = stewart.platform_M.R; % Radius of the circle [m]
+
+  points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
+
+  plot3(points(1,:), ...
+        points(2,:), ...
+        points(3,:), '-', 'Color', args.M_color);
+end
+
+
+ +

+Plot the position and labels of the rotation joints fixed to the mobile platform. +

+
+
if args.joints
+  Fb = FTm*[Mb;ones(1,6)];
+
+  scatter3(Fb(1,:), ...
+           Fb(2,:), ...
+           Fb(3,:), 'MarkerEdgeColor', args.M_color);
+
+  if args.labels
+    for i = 1:size(Fb,2)
+      text(Fb(1,i) + d_label, ...
+           Fb(2,i), ...
+           Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
+    end
+  end
+end
+
+
+
+
+ +
+

Legs

+
+

+Plot the legs connecting the joints of the fixed base to the joints of the mobile platform. +

+
+
if args.legs
+  for i = 1:6
+    plot3([Fa(1,i), Fb(1,i)], ...
+          [Fa(2,i), Fb(2,i)], ...
+          [Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);
+
+    if args.labels
+      text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
+           (Fa(2,i)+Fb(2,i))/2, ...
+           (Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
+    end
+  end
+end
+
+
+
+
+ +
+

5.15.1 Figure parameters

+
+
+
switch args.views
+  case 'default'
+      view([1 -0.6 0.4]);
+  case 'xy'
+      view([0 0 1]);
+  case 'xz'
+      view([0 -1 0]);
+  case 'yz'
+      view([1 0 0]);
+end
+axis equal;
+axis off;
+
+
+
+
+ +
+

5.15.2 Subplots

+
+
+
if strcmp(args.views, 'all')
+  hAx = findobj('type', 'axes');
+
+  figure;
+  s1 = subplot(2,2,1);
+  copyobj(get(hAx(1), 'Children'), s1);
+  view([0 0 1]);
+  axis equal;
+  axis off;
+  title('Top')
+
+  s2 = subplot(2,2,2);
+  copyobj(get(hAx(1), 'Children'), s2);
+  view([1 -0.6 0.4]);
+  axis equal;
+  axis off;
+
+  s3 = subplot(2,2,3);
+  copyobj(get(hAx(1), 'Children'), s3);
+  view([1 0 0]);
+  axis equal;
+  axis off;
+  title('Front')
+
+  s4 = subplot(2,2,4);
+  copyobj(get(hAx(1), 'Children'), s4);
+  view([0 -1 0]);
+  axis equal;
+  axis off;
+  title('Side')
+
+  close(f);
+end
+
+
+
+
+
+ + +
+

5.16 describeStewartPlatform: Display some text describing the current defined Stewart Platform

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [] = describeStewartPlatform(stewart)
+% describeStewartPlatform - Display some text describing the current defined Stewart Platform
+%
+% Syntax: [] = describeStewartPlatform(args)
+%
+% Inputs:
+%    - stewart
+%
+% Outputs:
+
+
+
+
+ +
+

Optional Parameters

+
+
+
arguments
+    stewart
+end
+
+
+
+
+ +
+

5.16.1 Geometry

+
+
+
fprintf('GEOMETRY:\n')
+fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)
+
+if stewart.platform_M.MO_B(3) > 0
+  fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n',  1e3*stewart.platform_M.MO_B(3))
+else
+  fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
+end
+
+fprintf('- The initial length of the struts are:\n')
+fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
+fprintf('\n')
+
+
+
+
+ +
+

5.16.2 Actuators

+
+
+
fprintf('ACTUATORS:\n')
+if stewart.actuators.type == 1
+    fprintf('- The actuators are classical.\n')
+    fprintf('- The Stiffness and Damping of each actuators is:\n')
+    fprintf('\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.K(1), stewart.actuators.C(1))
+elseif stewart.actuators.type == 2
+    fprintf('- The actuators are mechanicaly amplified.\n')
+    fprintf('- The vertical stiffness and damping contribution of the piezoelectric stack is:\n')
+    fprintf('\t ka = %.0e [N/m] \t ca = %.0e [N/(m/s)]\n', stewart.actuators.Ka(1), stewart.actuators.Ca(1))
+    fprintf('- Vertical stiffness when the piezoelectric stack is removed is:\n')
+    fprintf('\t kr = %.0e [N/m] \t cr = %.0e [N/(m/s)]\n', stewart.actuators.Kr(1), stewart.actuators.Cr(1))
+end
+fprintf('\n')
+
+
+
+
+ +
+

5.16.3 Joints

+
+
+
fprintf('JOINTS:\n')
+
+
+ +

+Type of the joints on the fixed base. +

+
+
switch stewart.joints_F.type
+  case 1
+    fprintf('- The joints on the fixed based are universal joints\n')
+  case 2
+    fprintf('- The joints on the fixed based are spherical joints\n')
+  case 3
+    fprintf('- The joints on the fixed based are perfect universal joints\n')
+  case 4
+    fprintf('- The joints on the fixed based are perfect spherical joints\n')
+end
+
+
+ +

+Type of the joints on the mobile platform. +

+
+
switch stewart.joints_M.type
+  case 1
+    fprintf('- The joints on the mobile based are universal joints\n')
+  case 2
+    fprintf('- The joints on the mobile based are spherical joints\n')
+  case 3
+    fprintf('- The joints on the mobile based are perfect universal joints\n')
+  case 4
+    fprintf('- The joints on the mobile based are perfect spherical joints\n')
+end
+
+
+ +

+Position of the fixed joints +

+
+
fprintf('- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n')
+fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_F.Fa)
+
+
+ +

+Position of the mobile joints +

+
+
fprintf('- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n')
+fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_M.Mb)
+fprintf('\n')
+
+
+
+
+ +
+

5.16.4 Kinematics

+
+
+
fprintf('KINEMATICS:\n')
+
+if isfield(stewart.kinematics, 'K')
+  fprintf('- The Stiffness matrix K is (in [N/m]):\n')
+  fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.K)
+end
+
+if isfield(stewart.kinematics, 'C')
+  fprintf('- The Damping matrix C is (in [m/N]):\n')
+  fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.C)
+end
+
+
+
+
+
+
+ +

+ +

+ +

Bibliography

+
+
Taghirad, Hamid. 2013. Parallel Robots : Mechanics and Control. Boca Raton, FL: CRC Press.
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-11-03 mar. 08:50

+
+ + diff --git a/org/stewart-architecture.org b/org/stewart-architecture.org index e41c392..ff2e217 100644 --- a/org/stewart-architecture.org +++ b/org/stewart-architecture.org @@ -926,6 +926,66 @@ This Matlab function is accessible [[file:../src/initializeCylindricalPlatforms. stewart.platform_M.H = args.Mph; #+end_src +** =initializeSolidPlatforms=: Initialize the geometry of the Fixed and Mobile Platforms +:PROPERTIES: +:header-args:matlab+: :tangle ../src/initializeSolidPlatforms.m +:header-args:matlab+: :comments none :mkdirp yes :eval no +:END: +<> + +This Matlab function is accessible [[file:../src/initializeSolidPlatforms.m][here]]. + +*** Function description +:PROPERTIES: +:UNNUMBERED: t +:END: +#+begin_src matlab + function [stewart] = initializeSolidPlatforms(stewart, args) + % initializeSolidPlatforms - Initialize the geometry of the Fixed and Mobile Platforms + % + % Syntax: [stewart] = initializeSolidPlatforms(args) + % + % Inputs: + % - args - Structure with the following fields: + % - density [1x1] - Density of the platforms [kg] + % + % Outputs: + % - stewart - updated Stewart structure with the added fields: + % - platform_F [struct] - structure with the following fields: + % - type = 2 + % - M [1x1] - Fixed Platform Density [kg/m^3] + % - platform_M [struct] - structure with the following fields: + % - type = 2 + % - M [1x1] - Mobile Platform Density [kg/m^3] +#+end_src + +*** Optional Parameters +:PROPERTIES: +:UNNUMBERED: t +:END: +#+begin_src matlab + arguments + stewart + args.density (1,1) double {mustBeNumeric, mustBePositive} = 7800 + end +#+end_src + +*** Populate the =stewart= structure +:PROPERTIES: +:UNNUMBERED: t +:END: +#+begin_src matlab + stewart.platform_F.type = 2; + + stewart.platform_F.density = args.density; +#+end_src + +#+begin_src matlab + stewart.platform_M.type = 2; + + stewart.platform_M.density = args.density; +#+end_src + ** =initializeCylindricalStruts=: Define the inertia of cylindrical struts :PROPERTIES: :header-args:matlab+: :tangle ../src/initializeCylindricalStruts.m @@ -1513,6 +1573,96 @@ Rotational Damping stewart.joints_M.step_file = args.step_file_M; #+end_src +** TODO =initializeFlexibleStrutAndJointDynamics=: Model each strut with a flexible element +:PROPERTIES: +:header-args:matlab+: :tangle ../src/initializeFlexibleStrutAndJointDynamics.m +:header-args:matlab+: :comments none :mkdirp yes :eval no +:END: +<> + +This Matlab function is accessible [[file:../src/initializeFlexibleStrutAndJointDynamics.m][here]]. + +*** Function description +:PROPERTIES: +:UNNUMBERED: t +:END: +#+begin_src matlab + function [stewart] = initializeFlexibleStrutAndJointDynamics(stewart, args) + % initializeFlexibleStrutAndJointDynamics - Add Stiffness and Damping properties of each strut + % + % Syntax: [stewart] = initializeFlexibleStrutAndJointDynamics(args) + % + % Inputs: + % - args - Structure with the following fields: + % - K [nxn] - Vertical stiffness contribution of the piezoelectric stack [N/m] + % - M [nxn] - Vertical damping contribution of the piezoelectric stack [N/(m/s)] + % - xi [1x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m] + % - step_file [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)] + % - Gf [6x1] - Gain from strain in [m] to measured [N] such that it matches + % + % Outputs: + % - stewart - updated Stewart structure with the added fields: +#+end_src + +*** Optional Parameters +:PROPERTIES: +:UNNUMBERED: t +:END: +#+begin_src matlab + arguments + stewart + args.K double {mustBeNumeric} = zeros(6,6) + args.M double {mustBeNumeric} = zeros(6,6) + args.H double {mustBeNumeric} = 0 + args.n_xyz double {mustBeNumeric} = zeros(2,3) + args.xi double {mustBeNumeric} = 0.1 + args.Gf double {mustBeNumeric} = 1 + args.step_file char {} = '' + end +#+end_src + +*** Compute the axial offset +:PROPERTIES: +:UNNUMBERED: t +:END: + +#+begin_src matlab + stewart.actuators.ax_off = (stewart.geometry.l(1) - args.H)/2; % Axial Offset at the ends of the actuator +#+end_src + +*** Populate the =stewart= structure +:PROPERTIES: +:UNNUMBERED: t +:END: +No discrete joints: +#+begin_src matlab + stewart.joints_F.type = 10; + stewart.joints_M.type = 10; + +#+end_src + +No discrete struts: +#+begin_src matlab + stewart.struts_F.type = 3; + stewart.struts_M.type = 3; +#+end_src + +#+begin_src matlab + stewart.actuators.type = 4; + + stewart.actuators.Km = args.K; + stewart.actuators.Mm = args.M; + + stewart.actuators.n_xyz = args.n_xyz; + stewart.actuators.xi = args.xi; + + stewart.actuators.step_file = args.step_file; + + stewart.actuators.K = args.K(3,3); % Axial Stiffness + + stewart.actuators.Gf = args.Gf; +#+end_src + ** =initializeInertialSensor=: Initialize the inertial sensor in each strut :PROPERTIES: :header-args:matlab+: :tangle ../src/initializeInertialSensor.m diff --git a/simscape_subsystems/Fixed_Based.slx b/simscape_subsystems/Fixed_Based.slx index d1bf72d..0ba8592 100644 Binary files a/simscape_subsystems/Fixed_Based.slx and b/simscape_subsystems/Fixed_Based.slx differ diff --git a/simscape_subsystems/Mobile_Platform.slx b/simscape_subsystems/Mobile_Platform.slx index 54e2092..0606808 100644 Binary files a/simscape_subsystems/Mobile_Platform.slx and b/simscape_subsystems/Mobile_Platform.slx differ diff --git a/simscape_subsystems/Stewart_Platform.slx b/simscape_subsystems/Stewart_Platform.slx index b0176ed..67eb303 100644 Binary files a/simscape_subsystems/Stewart_Platform.slx and b/simscape_subsystems/Stewart_Platform.slx differ diff --git a/simscape_subsystems/stewart_strut.slx b/simscape_subsystems/stewart_strut.slx index c20cb87..a2c8bb9 100644 Binary files a/simscape_subsystems/stewart_strut.slx and b/simscape_subsystems/stewart_strut.slx differ diff --git a/src/initializeFlexibleStrutAndJointDynamics.m b/src/initializeFlexibleStrutAndJointDynamics.m new file mode 100644 index 0000000..fed70d2 --- /dev/null +++ b/src/initializeFlexibleStrutAndJointDynamics.m @@ -0,0 +1,48 @@ +function [stewart] = initializeFlexibleStrutAndJointDynamics(stewart, args) +% initializeFlexibleStrutAndJointDynamics - Add Stiffness and Damping properties of each strut +% +% Syntax: [stewart] = initializeFlexibleStrutAndJointDynamics(args) +% +% Inputs: +% - args - Structure with the following fields: +% - K [nxn] - Vertical stiffness contribution of the piezoelectric stack [N/m] +% - M [nxn] - Vertical damping contribution of the piezoelectric stack [N/(m/s)] +% - xi [1x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m] +% - step_file [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)] +% - Gf [6x1] - Gain from strain in [m] to measured [N] such that it matches +% +% Outputs: +% - stewart - updated Stewart structure with the added fields: + +arguments + stewart + args.K double {mustBeNumeric} = zeros(6,6) + args.M double {mustBeNumeric} = zeros(6,6) + args.H double {mustBeNumeric} = 0 + args.n_xyz double {mustBeNumeric} = zeros(2,3) + args.xi double {mustBeNumeric} = 0.1 + args.Gf double {mustBeNumeric} = 1 + args.step_file char {} = '' +end + +stewart.actuators.ax_off = (stewart.geometry.l(1) - args.H)/2; % Axial Offset at the ends of the actuator + +stewart.joints_F.type = 10; +stewart.joints_M.type = 10; + +stewart.struts_F.type = 3; +stewart.struts_M.type = 3; + +stewart.actuators.type = 4; + +stewart.actuators.Km = args.K; +stewart.actuators.Mm = args.M; + +stewart.actuators.n_xyz = args.n_xyz; +stewart.actuators.xi = args.xi; + +stewart.actuators.step_file = args.step_file; + +stewart.actuators.K = args.K(3,3); % Axial Stiffness + +stewart.actuators.Gf = args.Gf; diff --git a/src/initializeSolidPlatforms.m b/src/initializeSolidPlatforms.m new file mode 100644 index 0000000..6d92100 --- /dev/null +++ b/src/initializeSolidPlatforms.m @@ -0,0 +1,30 @@ +function [stewart] = initializeSolidPlatforms(stewart, args) +% initializeSolidPlatforms - Initialize the geometry of the Fixed and Mobile Platforms +% +% Syntax: [stewart] = initializeSolidPlatforms(args) +% +% Inputs: +% - args - Structure with the following fields: +% - density [1x1] - Density of the platforms [kg] +% +% Outputs: +% - stewart - updated Stewart structure with the added fields: +% - platform_F [struct] - structure with the following fields: +% - type = 2 +% - M [1x1] - Fixed Platform Density [kg/m^3] +% - platform_M [struct] - structure with the following fields: +% - type = 2 +% - M [1x1] - Mobile Platform Density [kg/m^3] + +arguments + stewart + args.density (1,1) double {mustBeNumeric, mustBePositive} = 7800 +end + +stewart.platform_F.type = 2; + +stewart.platform_F.density = args.density; + +stewart.platform_M.type = 2; + +stewart.platform_M.density = args.density;