-
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
diff --git a/mat/APA300ML.mat b/mat/APA300ML.mat
index ef64ba7..560064a 100644
Binary files a/mat/APA300ML.mat and b/mat/APA300ML.mat differ
diff --git a/mat/base_plate.STEP b/mat/base_plate.STEP
new file mode 100644
index 0000000..6539b61
--- /dev/null
+++ b/mat/base_plate.STEP
@@ -0,0 +1,12918 @@
+ISO-10303-21;
+HEADER;
+FILE_DESCRIPTION (( 'STEP AP214' ),
+ '1' );
+FILE_NAME ('base-plate-2.STEP',
+ '2020-10-06T11:52:22',
+ ( '' ),
+ ( '' ),
+ 'SwSTEP 2.0',
+ 'SolidWorks 2019',
+ '' );
+FILE_SCHEMA (( 'AUTOMOTIVE_DESIGN' ));
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diff --git a/mat/motion_error_ol.mat b/mat/motion_error_ol.mat
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index 0000000..4ee593f
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diff --git a/mat/strut_encoder.mat b/mat/strut_encoder.mat
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index 0000000..500cb25
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diff --git a/mat/strut_encoder.step b/mat/strut_encoder.step
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+#11695 = DIRECTION ( 'NONE', ( 0.0000000000000000000, 0.0000000000000000000, 1.000000000000000000 ) ) ;
+#11696 = FACE_OUTER_BOUND ( 'NONE', #5906, .T. ) ;
+#11697 = VERTEX_POINT ( 'NONE', #8964 ) ;
+#11698 = DIRECTION ( 'NONE', ( -0.0000000000000000000, 1.000000000000000000, -0.0000000000000000000 ) ) ;
+#11699 = EDGE_LOOP ( 'NONE', ( #11238, #10377, #2271, #7166 ) ) ;
+#11700 = DIRECTION ( 'NONE', ( 0.0000000000000000000, 0.0000000000000000000, 1.000000000000000000 ) ) ;
+#11701 = ORIENTED_EDGE ( 'NONE', *, *, #9962, .F. ) ;
+#11702 = EDGE_LOOP ( 'NONE', ( #1084, #918 ) ) ;
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+#11704 = ADVANCED_FACE ( 'NONE', ( #10036 ), #7233, .T. ) ;
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+#11709 = VERTEX_POINT ( 'NONE', #10924 ) ;
+#11710 = FACE_OUTER_BOUND ( 'NONE', #4193, .T. ) ;
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+#11712 = PRESENTATION_STYLE_ASSIGNMENT (( #876 ) ) ;
+#11713 = ORIENTED_EDGE ( 'NONE', *, *, #4562, .F. ) ;
+#11714 = UNCERTAINTY_MEASURE_WITH_UNIT (LENGTH_MEASURE( 1.000000000000000082E-05 ), #10245, 'distance_accuracy_value', 'NONE');
+#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
+
+
Stewart Platform - Definition of the Architecture
+
+
+
+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 );
+
+
+
+
+
+
+
+
+
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 \({}^{M}\bm{b}_{i}\).
+
+
+
+
+
+
+
+
+
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 ).
+
+
+
+
+
+
+
+
+
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 \(\tau_i\)
+
+
+
+
+
+
+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:
+
+
+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 \(\{M\}\).
+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
+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.
+Define the mass and inertia of each element of the Stewart platform with the initializeCylindricalPlatforms and initializeCylindricalStruts
+Define the dynamical properties of the Stewart platform by setting the stiffness and damping of the actuators and joints.
+
+
+
+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' );
+
+
+
+
+
+
+
+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;
+ty = 0.2;
+tz = 0.05;
+
+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];
+
+displayArchitecture(stewart, 'AP' , AP, 'ARB' , ARB);
+view([0 - 1 0]);
+
+
+
+
+
+
+
+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 .
+
+
+
+
+
+
+
Function description
+
+
+
function [stewart] = initializeStewartPlatform ()
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 .
+
+
+
+
+
+
+
Function description
+
+
+
function [stewart] = initializeFramesPositions (stewart , args )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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;
+
+FO_M = [0; 0; H];
+
+MO_B = [0; 0; args.MO_B];
+
+FO_A = MO_B + FO_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 ).
+
+
+
+
+
+
+
+
+
Function description
+
+
+
function [stewart] = generateGeneralConfiguration (stewart , args )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 .
+
+
+
+
+
+
+
Function description
+
+
+
function [stewart] = computeJointsPose (stewart , args )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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);
+
+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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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);
+I_M = zeros(3, 3, 6);
+
+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
+
+
+
+
+
+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
+
+
+
+
+
+
+
+
+
Function description
+
+
+
function [stewart] = initializeStrutDynamics (stewart , args )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 .
+
+
+
+
+
+
+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
+
+
+
+
+
+
+
+
+
+
+
Function description
+
+
+
function [stewart] = initializeAmplifiedStrutDynamics (stewart , args )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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;
+
+
+
+
+
+
+
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);
+
+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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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;
+
+
+
+
+
+
+
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);
+
+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:
+
+
+\(\omega_0 = \sqrt{\frac{k}{m}}\)
+\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)
+
+
+
+
+
+
+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:
+
+
+\(\omega_0 = \sqrt{\frac{k}{m}}\)
+\(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\)
+
+
+
+
+
+
+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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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.
+
+
+
+
+
+
+
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];
+ v = null([0; 0; 1]' );
+ center = [0; 0; 0];
+ radius = stewart.platform_F.R;
+
+ 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];
+
+if args.frames
+ FM_uv = FTm* [d_unit_vector* eye(3); zeros(1,3)];
+ 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)];
+ 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];
+ v = null((FTm(1: 3,1: 3)* [0;0;1])' );
+ center = Fm(1: 3);
+ radius = stewart.platform_M.R;
+
+ 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 )
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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
+
>
+
+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;
+
+
+
+
+
+
+
+
+
+
+
+
+
+