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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');
@@ -76,12 +109,12 @@
 
 
 Number of Modes
-6
+120
 
 
 
 Size of M and K matrices
-48
+162
 
 
 
@@ -112,23 +145,23 @@
 
 
 1.0
-53917.0
+697783.0
+0.0
 0.0
 -0.015
-0.0
 
 
 
 2.0
-53918.0
+697784.0
+0.0
 0.0
 0.015
-0.0
 
 
 
 3.0
-53919.0
+697785.0
 -0.0325
 0.0
 0.0
@@ -136,7 +169,7 @@
 
 
 4.0
-53920.0
+697786.0
 -0.0125
 0.0
 0.0
@@ -144,7 +177,7 @@
 
 
 5.0
-53921.0
+697787.0
 -0.0075
 0.0
 0.0
@@ -152,7 +185,7 @@
 
 
 6.0
-53922.0
+697788.0
 0.0125
 0.0
 0.0
@@ -160,7 +193,7 @@
 
 
 7.0
-53923.0
+697789.0
 0.0325
 0.0
 0.0
@@ -174,7 +207,7 @@
 
1.2 Flexible Joint

 

 

-
flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
 

 

 
@@ -235,7 +268,7 @@
 
 
 1.0
-181278.0
+528875.0
 0.0
 0.0
 0.0
@@ -243,7 +276,7 @@
 
 
 2.0
-181279.0
+528876.0
 0.0
 0.0
 -0.0
@@ -268,22 +301,22 @@
 
 
 Axial Stiffness [N/um]
-119
+94
 
 
 
 Bending Stiffness [Nm/rad]
-33
+5
 
 
 
 Bending Stiffness [Nm/rad]
-33
+5
 
 
 
 Torsion Stiffness [Nm/rad]
-236
+260
 
 
 
@@ -334,9 +367,17 @@ controller = initializeController('type', 'open-loop');
 stewart = initializeFramesPositions(stewart);
 stewart = generateGeneralConfiguration(stewart);
 stewart = computeJointsPose(stewart);
-% stewart = initializeStrutDynamics(stewart, 'K', 1.8e6*ones(6,1));
-stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
-stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235*ones(6,1), 'Kf_F', 33*ones(6,1), 'Kt_F', 235*ones(6,1));
+stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
+stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
+                                           'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
+                                           'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
+                                           'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
+                                           'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
+                                           'type_F', 'universal_3dof', ...
+                                           'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
+                                           'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
+                                           'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
+                                           'Kt_F', flex_joint.K(6,6)*ones(6,1));
 stewart = initializeCylindricalPlatforms(stewart);
 stewart = initializeCylindricalStruts(stewart);
 stewart = computeJacobian(stewart);
@@ -372,7 +413,7 @@ G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', '
 stewart = initializeFramesPositions(stewart);
 stewart = generateGeneralConfiguration(stewart);
 stewart = computeJointsPose(stewart);
-stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
+stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
 stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
 stewart = initializeCylindricalPlatforms(stewart);
 stewart = initializeCylindricalStruts(stewart);
@@ -392,8 +433,8 @@ Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3',
-
-

1.6 Flexible APA

+
+

1.6 Flexible APA

@@ -407,8 +448,17 @@ Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', stewart = initializeFramesPositions(stewart); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); -stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP'); -stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235, 'Kf_F', 33*ones(6,1), 'Kt_F', 235); +stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP'); +stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ... + 'Kr_M', flex_joint.K(1,1)*ones(6,1), ... + 'Ka_M', flex_joint.K(3,3)*ones(6,1), ... + 'Kf_M', flex_joint.K(4,4)*ones(6,1), ... + 'Kt_M', flex_joint.K(6,6)*ones(6,1), ... + 'type_F', 'universal_3dof', ... + 'Kr_F', flex_joint.K(1,1)*ones(6,1), ... + 'Ka_F', flex_joint.K(3,3)*ones(6,1), ... + 'Kf_F', flex_joint.K(4,4)*ones(6,1), ... + 'Kt_F', flex_joint.K(6,6)*ones(6,1)); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none'); stewart = computeJacobian(stewart); @@ -442,7 +492,7 @@ Ga.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', stewart = initializeFramesPositions(stewart); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); -stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP'); +stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP'); stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP'); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none'); @@ -461,9 +511,19 @@ Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3',
-
-

1.8 Direct Velocity Feedback

+
+

1.8 Save

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

1.9 Direct Velocity Feedback

+

flexible_elements_effect_dvf.png @@ -474,8 +534,8 @@ Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3',

-

1.9 Integral Force Feedback

-
+

1.10 Integral Force Feedback

+

flexible_elements_effect_iff.png @@ -484,11 +544,492 @@ Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3',

+ +
+

1.11 Procedure to include flexible elements into Simscape

+
+

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

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

1.12 Conclusion

+
+
+

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

+ +
+
+
+
+ +
+

2 Control with flexible elements

+
+
+
+

2.1 Flexible APA and Flexible Joint

+
+
+
apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
+
+ +
+
stewart = initializeStewartPlatform();
+stewart = initializeFramesPositions(stewart);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
+stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+stewart = initializeInertialSensor(stewart);
+
+
+ +
+
ground = initializeGround('type', 'none');
+payload = initializePayload('type', 'rigid', 'm', 50);
+controller = initializeController('type', 'open-loop');
+
+
+ +
+
disturbances = initializeDisturbances();
+references = initializeReferences(stewart);
+
+
+
+
+ +
+

2.2 Identification

+
+

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

+
+
%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'stewart_platform_model';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
+io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
+
+
+ +
+
G = -linearize(mdl, io, options);
+G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
+
+
+
+
+ +
+

2.3 Decentralized Direct Velocity Feedback

+
+

+Controller Design +

+
+
Kl = -1e5*s/(1 + s/2/pi/2e2)/(1 + s/2/pi/5e2) * eye(6);
+
+
+ +
+
isstable(feedback(G(1:6,1:6)*Kl, eye(6), -1))
+
+
+ +
+1
+
+
+
+ +
+

2.4 HAC

+
+
+
Kx = tf(zeros(6));
+
+controller = initializeController('type', 'ref-track-hac-dvf');
+
+
+ +
+
%% Name of the Simulink File
+mdl = 'stewart_platform_model';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Controller'],              1, 'input');  io_i = io_i + 1; % Actuator Force Inputs [N]
+io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]
+
+%% Run the linearization
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
+
+
+ +
+
Gl = -stewart.kinematics.J*G;
+Gl.OutputName  = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
+
+
+ +
+
wc = 2*pi*300;
+Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);
+
+
+
+
+
+ +
+

3 Flexible Joint Specifications

+
+
+
+

3.1 Stewart Platform Initialization

+
+
+
stewart = initializeStewartPlatform();
+stewart = initializeFramesPositions(stewart);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 43e6*ones(6,1), 'K1', 0.4e6*ones(6,1), 'C1', 10*ones(6,1));
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart);
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+stewart = initializeInertialSensor(stewart);
+
+references = initializeReferences(stewart);
+
+
+ +
+
ground = initializeGround('type', 'none');
+payload = initializePayload('type', 'rigid', 'm', 50);
+controller = initializeController('type', 'open-loop');
+
+
+ +
+
disturbances = initializeDisturbances();
+
+
+ +
+
open('stewart_platform_model.slx')
+
+
+
+
+ +
+

3.2 Effect of the Bending Stiffness

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

3.3 Effect of the Torsion Stiffness

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

3.4 Effect of the Axial Stiffness

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

3.5 Effect of the Radial (Shear) Stiffness

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

3.6 Comparison of perfect joint and worst specified joint

+
+
+

3.7 Conclusion

+
+

+Qualitatively: +

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

+Quantitatively: +

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

4 Flexible Joint Specifications with the APA300ML

+
+
+
+

4.1 Stewart Platform Initialization

+
+
+
apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
+
+ +
+
stewart = initializeStewartPlatform();
+stewart = initializeFramesPositions(stewart);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
+stewart = initializeCylindricalPlatforms(stewart);
+stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
+stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+stewart = initializeInertialSensor(stewart);
+
+references = initializeReferences(stewart);
+
+
+ +
+
ground = initializeGround('type', 'none');
+payload = initializePayload('type', 'rigid', 'm', 50);
+controller = initializeController('type', 'open-loop');
+
+
+ +
+
disturbances = initializeDisturbances();
+
+
+ +
+
open('stewart_platform_model.slx')
+
+
+
+
+ +
+

4.2 Comparison of perfect joint and worst specified joint

+
+
+
+

5 Relative Motion Sensors

+
+
+
+

5.1 Stewart Platform Initialization

+
+
+
apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
+
+ +
+
stewart = initializeStewartPlatform();
+stewart = initializeFramesPositions(stewart);
+stewart = generateGeneralConfiguration(stewart);
+stewart = computeJointsPose(stewart);
+
+
+ +
+
stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 43e6*ones(6,1), 'K1', 0.4e6*ones(6,1), 'C1', 10*ones(6,1));
+% stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
+
+
+ +
+
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
+                                           'type_M', 'spherical_3dof', ...
+                                           'Kf_M', 100*ones(6,1), ...
+                                           'Kt_M', 500*ones(6,1), ...
+                                           'Kf_F', 100*ones(6,1), ...
+                                           'Kt_F', 500*ones(6,1), ...
+                                           'Ka_M', 200e6*ones(6,1), ...
+                                           'Kr_M', 1e6*ones(6,1), ...
+                                           'Ka_F', 200e6*ones(6,1), ...
+                                           'Kr_F', 1e6*ones(6,1));
+% stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
+%                                            'type_M', 'spherical_3dof', ...
+%                                            'Kf_M', 100*ones(6,1), ...
+%                                            'Kt_M', 500*ones(6,1), ...
+%                                            'Kf_F', 100*ones(6,1), ...
+%                                            'Kt_F', 500*ones(6,1), ...
+%                                            'Ka_M', 1e6*ones(6,1), ...
+%                                            'Kr_M', 1e6*ones(6,1), ...
+%                                            'Ka_F', 1e6*ones(6,1), ...
+%                                            'Kr_F', 1e6*ones(6,1));
+% stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
+
+
+ +
+
stewart = initializeCylindricalPlatforms(stewart);
+
+
+ +
+
stewart = initializeCylindricalStruts(stewart);
+% stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
+
+
+ +
+
stewart = computeJacobian(stewart);
+stewart = initializeStewartPose(stewart);
+stewart = initializeInertialSensor(stewart);
+
+references = initializeReferences(stewart);
+
+
+ +
+
ground = initializeGround('type', 'none');
+payload = initializePayload('type', 'rigid', 'm', 50);
+controller = initializeController('type', 'open-loop');
+
+
+ +
+
disturbances = initializeDisturbances();
+
+
+ +
+
open('stewart_platform_model.slx')
+
+
+
+

Author: Dehaeze Thomas

-

Created: 2020-08-07 ven. 09:55

+

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

diff --git a/docs/stewart-architecture.html b/docs/stewart-architecture.html index 6274689..810d785 100644 --- a/docs/stewart-architecture.html +++ b/docs/stewart-architecture.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Stewart Platform - Definition of the Architecture @@ -63,94 +63,94 @@
-
-

Documentation

-
+
+

Documentation

+
- -
-

Function description

-
+
+

Function description

+
function [stewart] = initializeStewartPlatform()
 % initializeStewartPlatform - Initialize the stewart structure
@@ -747,11 +747,11 @@ This Matlab function is accessible
-
-

Documentation

-
+
+

Documentation

+
- -
-

Function description

-
+
+

Function description

+
function [stewart] = initializeFramesPositions(stewart, args)
 % initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
@@ -784,9 +784,9 @@ This Matlab function is accessible
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     stewart
@@ -814,9 +814,9 @@ FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]
-
-

Populate the stewart structure

-
+
+

Populate the stewart structure

+
stewart.geometry.H      = H;
 stewart.geometry.FO_M   = FO_M;
@@ -840,9 +840,9 @@ This Matlab function is accessible 
-
Documentation

-

+
 
-

-
Function description

-

+

+
Function description

+

 

 
function [stewart] = generateGeneralConfiguration(stewart, args)
 % generateGeneralConfiguration - Generate a Very General Configuration
@@ -884,9 +884,9 @@ The radius of the circles can be chosen as well as the angles where the joints a
 

 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -921,9 +921,9 @@ end
 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
stewart.platform_F.Fa = Fa;
 stewart.platform_M.Mb = Mb;
@@ -945,9 +945,9 @@ This Matlab function is accessible here
 

 

 
-

-
Documentation

-

+

+
Documentation

+

 
 

 
stewart-struts.png
@@ -957,9 +957,9 @@ This Matlab function is accessible here
 

 

 
-

-
Function description

-

+

+
Function description

+

 

 
function [stewart] = computeJointsPose(stewart, args)
 % computeJointsPose -
@@ -995,9 +995,9 @@ This Matlab function is accessible here
 

 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -1009,9 +1009,9 @@ end
 

 

 
-

-
Check the stewart structure elements

-

+

+
Check the stewart structure elements

+

 

 
assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
 Fa = stewart.platform_F.Fa;
@@ -1097,9 +1097,9 @@ end
 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
stewart.geometry.Aa = Aa;
 stewart.geometry.Ab = Ab;
@@ -1132,9 +1132,9 @@ This Matlab function is accessible here
 

 

 
-

-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeStewartPose(stewart, args)
 % initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
@@ -1158,9 +1158,9 @@ This Matlab function is accessible here
 

 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -1182,9 +1182,9 @@ end
 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
stewart.actuators.Leq = dLi;
 

@@ -1205,9 +1205,9 @@ This Matlab function is accessible 
-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeCylindricalPlatforms(stewart, args)
 % initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
@@ -1241,9 +1241,9 @@ This Matlab function is accessible 
-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -1278,9 +1278,9 @@ end
 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
stewart.platform_F.type = 1;
 
@@ -1316,9 +1316,9 @@ This Matlab function is accessible 
-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeCylindricalStruts(stewart, args)
 % initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
@@ -1351,9 +1351,9 @@ This Matlab function is accessible 
-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -1403,9 +1403,9 @@ end
 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
switch args.type_M
   case 'cylindrical'
@@ -1451,9 +1451,9 @@ This Matlab function is accessible he
 

 

 
-

-
Documentation

-

+

+
Documentation

+

 
 

 
piezoelectric_stack.jpg
@@ -1482,9 +1482,9 @@ A simplistic model of such amplified actuator is shown in Figure 
-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeStrutDynamics(stewart, args)
 % initializeStrutDynamics - Add Stiffness and Damping properties of each strut
@@ -1506,9 +1506,9 @@ A simplistic model of such amplified actuator is shown in Figure 
-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -1546,9 +1546,9 @@ This Matlab function is accessible 
-
Documentation

-

+

+
Documentation

+

 

 An amplified piezoelectric actuator is shown in Figure 13.
 

@@ -1561,29 +1561,58 @@ An amplified piezoelectric actuator is shown in Figure 13<
 

 
 

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

-
    
-
  • \(K_{r}\) represent the vertical stiffness when the piezoelectric stack is removed
    • 
-
  • \(K_{a}\) is the vertical stiffness contribution of the piezoelectric stack
    • 
-
  • \(F_{i}\) represents the part of the piezoelectric stack that is used as a force actuator
    • 
-
  • \(F_{m,i}\) represents the remaining part of the piezoelectric stack that is used as a force sensor
    • 
-
  • \(v_{m,i}\) represents the absolute velocity of the top part of the actuator
    • 
-
  • \(d_{m,i}\) represents the total relative displacement of the actuator
    • 
-

+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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

 
 
-

-
iff_1dof.png
+

+
souleille18_model_piezo.png
 

-
Figure 14: Model of an amplified actuator

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

 

 

 

 
-

-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeAmplifiedStrutDynamics(stewart, args)
 % initializeAmplifiedStrutDynamics - Add Stiffness and Damping properties of each strut
@@ -1592,35 +1621,35 @@ A simplistic model of such amplified actuator is shown in Figure 
-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
-    args.Kr (6,1) double {mustBeNumeric, mustBeNonnegative} = 5e6*ones(6,1)
-    args.Cr (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
-    args.Ka (6,1) double {mustBeNumeric, mustBeNonnegative} = 15e6*ones(6,1)
-    args.Ca (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+    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
 

 

@@ -1631,24 +1660,23 @@ end
 
Compute the total stiffness and damping

 

 

-
K = args.Ka + args.Kr;
-C = args.Ca + args.Cr;
+
K = args.K1 + args.Ka.*args.Ke./(args.Ka + args.Ke);
+C = args.C1;
 

 

 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
stewart.actuators.type = 2;
 
 stewart.actuators.Ka = args.Ka;
-stewart.actuators.Ca = args.Ca;
-
-stewart.actuators.Kr = args.Kr;
-stewart.actuators.Cr = args.Cr;
+stewart.actuators.Ke = args.Ke;
+stewart.actuators.K1 = args.K1;
+stewart.actuators.C1 = args.C1;
 
 stewart.actuators.K = K;
 stewart.actuators.C = C;
@@ -1670,9 +1698,9 @@ This Matlab function is accessible 
-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeFlexibleStrutDynamics(stewart, args)
 % initializeFlexibleStrutDynamics - Add Stiffness and Damping properties of each strut
@@ -1685,6 +1713,7 @@ This Matlab function is accessible 
 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -1704,6 +1733,7 @@ This Matlab function is accessible 
-
Populate the stewart structure

-

+
 
-

-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeJointDynamics(stewart, args)
 % initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
@@ -1791,22 +1823,30 @@ This Matlab function is accessible he
 

 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
-    args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'universal'
-    args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'spherical'
-    args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
+    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} = 20*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} = 15*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} = 20*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)
@@ -1838,6 +1878,10 @@ end
     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
@@ -1851,6 +1895,10 @@ switch args.type_M
     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
 

 

@@ -1861,16 +1909,14 @@ end
 
Add Stiffness and Damping in Translation of each strut

 

 

-Translation Stiffness
+Axial and Radial (shear) Stiffness
 

 

-
stewart.joints_M.Kx = zeros(6,1);
-stewart.joints_M.Ky = zeros(6,1);
-stewart.joints_M.Kz = zeros(6,1);
+
stewart.joints_M.Ka = args.Ka_M;
+stewart.joints_M.Kr = args.Kr_M;
 
-stewart.joints_F.Kx = zeros(6,1);
-stewart.joints_F.Ky = zeros(6,1);
-stewart.joints_F.Kz = zeros(6,1);
+stewart.joints_F.Ka = args.Ka_F;
+stewart.joints_F.Kr = args.Kr_F;
 

 

 
@@ -1878,13 +1924,11 @@ stewart.joints_F.Kz = zeros(6,1);
 Translation Damping
 

 

-
stewart.joints_M.Cx = zeros(6,1);
-stewart.joints_M.Cy = zeros(6,1);
-stewart.joints_M.Cz = zeros(6,1);
+
stewart.joints_M.Ca = args.Ca_M;
+stewart.joints_M.Cr = args.Cr_M;
 
-stewart.joints_F.Cx = zeros(6,1);
-stewart.joints_F.Cy = zeros(6,1);
-stewart.joints_F.Cz = zeros(6,1);
+stewart.joints_F.Ca = args.Ca_F;
+stewart.joints_F.Cr = args.Cr_F;
 

 

 

@@ -1898,10 +1942,10 @@ Rotational Stiffness
 

 

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

 

 
@@ -1910,10 +1954,10 @@ Rotational Damping
 

 

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

 

 

@@ -2027,9 +2071,9 @@ Note that there is trade-off between:
 

 

 
-

-
Function description

-

+

+
Function description

+

 

 
function [stewart] = initializeInertialSensor(stewart, args)
 % initializeInertialSensor - Initialize the inertial sensor in each strut
@@ -2055,9 +2099,9 @@ Note that there is trade-off between:
 

 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -2098,9 +2142,9 @@ end
 

 

 
-

-
Populate the stewart structure

-

+

+
Populate the stewart structure

+

 

 
stewart.sensors.inertial = sensor;
 

@@ -2121,9 +2165,9 @@ This Matlab function is accessible here
 

 
-

-
Function description

-

+

+
Function description

+

 

 
function [] = displayArchitecture(stewart, args)
 % displayArchitecture - 3D plot of the Stewart platform architecture
@@ -2152,9 +2196,9 @@ This Matlab function is accessible here
 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -2175,9 +2219,9 @@ end
 

 

 
-

-
Check the stewart structure elements

-

+

+
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;
@@ -2515,9 +2559,9 @@ This Matlab function is accessible he
 

 

 
-

-
Function description

-

+

+
Function description

+

 

 
function [] = describeStewartPlatform(stewart)
 % describeStewartPlatform - Display some text describing the current defined Stewart Platform
@@ -2533,9 +2577,9 @@ This Matlab function is accessible he
 

 

 
-

-
Optional Parameters

-

+

+
Optional Parameters

+

 

 
arguments
     stewart
@@ -2684,7 +2728,7 @@ end
 

 

 
Author: Dehaeze Thomas

-
Created: 2020-08-05 mer. 16:23

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

 

 
 
diff --git a/matlab/stewart_platform_model.slx b/matlab/stewart_platform_model.slx
index 1abc349..f82b86b 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 5b1c17e..798667c 100644
--- a/org/flexible-stewart-platform.org
+++ b/org/flexible-stewart-platform.org
@@ -62,10 +62,10 @@
 #+end_src
 
 #+RESULTS:
-| Total number of Nodes     |  7 |
-| Number of interface Nodes |  7 |
-| Number of Modes           |  6 |
-| Size of M and K matrices  | 48 |
+| Total number of Nodes     |   7 |
+| Number of interface Nodes |   7 |
+| Number of Modes           | 120 |
+| Size of M and K matrices  | 162 |
 
 #+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 ');
@@ -73,19 +73,19 @@
 
 #+caption: Coordinates of the interface nodes
 #+RESULTS:
-| Node i | Node Number |   x [m] |  y [m] | z [m] |
-|--------+-------------+---------+--------+-------|
-|    1.0 |     53917.0 |     0.0 | -0.015 |   0.0 |
-|    2.0 |     53918.0 |     0.0 |  0.015 |   0.0 |
-|    3.0 |     53919.0 | -0.0325 |    0.0 |   0.0 |
-|    4.0 |     53920.0 | -0.0125 |    0.0 |   0.0 |
-|    5.0 |     53921.0 | -0.0075 |    0.0 |   0.0 |
-|    6.0 |     53922.0 |  0.0125 |    0.0 |   0.0 |
-|    7.0 |     53923.0 |  0.0325 |    0.0 |   0.0 |
+| Node i | Node Number |   x [m] | y [m] |  z [m] |
+|--------+-------------+---------+-------+--------|
+|    1.0 |    697783.0 |     0.0 |   0.0 | -0.015 |
+|    2.0 |    697784.0 |     0.0 |   0.0 |  0.015 |
+|    3.0 |    697785.0 | -0.0325 |   0.0 |    0.0 |
+|    4.0 |    697786.0 | -0.0125 |   0.0 |    0.0 |
+|    5.0 |    697787.0 | -0.0075 |   0.0 |    0.0 |
+|    6.0 |    697788.0 |  0.0125 |   0.0 |    0.0 |
+|    7.0 |    697789.0 |  0.0325 |   0.0 |    0.0 |
 
 ** Flexible Joint
 #+begin_src matlab
-  flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+  flex_joint = load('./mat/flexor_025.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
@@ -106,8 +106,8 @@
 #+RESULTS:
 | Node i | Node Number | x [m] | y [m] | z [m] |
 |--------+-------------+-------+-------+-------|
-|    1.0 |    181278.0 |   0.0 |   0.0 |   0.0 |
-|    2.0 |    181279.0 |   0.0 |   0.0 |  -0.0 |
+|    1.0 |    528875.0 |   0.0 |   0.0 |   0.0 |
+|    2.0 |    528876.0 |   0.0 |   0.0 |  -0.0 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
   data2orgtable([1e-6*flex_joint.K(3,3), flex_joint.K(4,4), flex_joint.K(5,5), flex_joint.K(6,6)]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %0.f ');
@@ -116,10 +116,10 @@
 #+RESULTS:
 | *Caracteristic*            | *Value* |
 |----------------------------+---------|
-| Axial Stiffness [N/um]     |     119 |
-| Bending Stiffness [Nm/rad] |      33 |
-| Bending Stiffness [Nm/rad] |      33 |
-| Torsion Stiffness [Nm/rad] |     236 |
+| Axial Stiffness [N/um]     |      94 |
+| Bending Stiffness [Nm/rad] |       5 |
+| Bending Stiffness [Nm/rad] |       5 |
+| Torsion Stiffness [Nm/rad] |     260 |
 
 ** Identification
 And we identify the dynamics from force actuators to force sensors.
@@ -154,15 +154,23 @@ And we identify the dynamics from force actuators to force sensors.
   stewart = initializeFramesPositions(stewart);
   stewart = generateGeneralConfiguration(stewart);
   stewart = computeJointsPose(stewart);
-  % stewart = initializeStrutDynamics(stewart, 'K', 1.8e6*ones(6,1));
-  stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
-  stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235*ones(6,1), 'Kf_F', 33*ones(6,1), 'Kt_F', 235*ones(6,1));
+  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
+  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
+                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
+                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
+                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
+                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
+                                             'type_F', 'universal_3dof', ...
+                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
+                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
+                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
+                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
   stewart = initializeCylindricalPlatforms(stewart);
   stewart = initializeCylindricalStruts(stewart);
   stewart = computeJacobian(stewart);
   stewart = initializeStewartPose(stewart);
   stewart = initializeInertialSensor(stewart);
- 
+
   references = initializeReferences(stewart);
 #+end_src
 
@@ -184,7 +192,7 @@ And we identify the dynamics from force actuators to force sensors.
   stewart = initializeFramesPositions(stewart);
   stewart = generateGeneralConfiguration(stewart);
   stewart = computeJointsPose(stewart);
-  stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
+  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
   stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
   stewart = initializeCylindricalPlatforms(stewart);
   stewart = initializeCylindricalStruts(stewart);
@@ -211,8 +219,17 @@ And we identify the dynamics from force actuators to force sensors.
   stewart = initializeFramesPositions(stewart);
   stewart = generateGeneralConfiguration(stewart);
   stewart = computeJointsPose(stewart);
-  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
-  stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235, 'Kf_F', 33*ones(6,1), 'Kt_F', 235);
+  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
+  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
+                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
+                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
+                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
+                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
+                                             'type_F', 'universal_3dof', ...
+                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
+                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
+                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
+                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
   stewart = initializeCylindricalPlatforms(stewart);
   stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
   stewart = computeJacobian(stewart);
@@ -238,7 +255,7 @@ And we identify the dynamics from force actuators to force sensors.
   stewart = initializeFramesPositions(stewart);
   stewart = generateGeneralConfiguration(stewart);
   stewart = computeJointsPose(stewart);
-  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
+  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
   stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
   stewart = initializeCylindricalPlatforms(stewart);
   stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
@@ -253,6 +270,11 @@ And we identify the dynamics from force actuators to force sensors.
   Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
 #+end_src
 
+** Save
+#+begin_src matlab
+  save('./mat/flexible_stewart_platform.mat', 'G', 'Gj', 'Ga', 'Gf');
+#+end_src
+
 ** Direct Velocity Feedback
 #+begin_src matlab :exports none
   freqs = logspace(0, 3, 1000);
@@ -268,10 +290,10 @@ And we identify the dynamics from force actuators to force sensors.
       plot(freqs, abs(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
   end
   for i = 1:6
-      plot(freqs, abs(squeeze(freqresp(44*Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
+      plot(freqs, abs(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
   end
   for i = 1:6
-      plot(freqs, abs(squeeze(freqresp(44*Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
+      plot(freqs, abs(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
   end
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -330,14 +352,14 @@ And we identify the dynamics from force actuators to force sensors.
       plot(freqs, abs(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
   end
   for i = 1:6
-      plot(freqs, abs(squeeze(freqresp(1e9*Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
+      plot(freqs, abs(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
   end
   for i = 1:6
-      plot(freqs, abs(squeeze(freqresp(1e9*Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
+      plot(freqs, abs(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
   end
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
 
   ax2 = subplot(2, 1, 2);
   hold on;
@@ -376,3 +398,1186 @@ And we identify the dynamics from force actuators to force sensors.
 #+caption: Change of the IFF plant dynamics with the added flexible elements
 #+RESULTS:
 [[file:figs/flexible_elements_effect_iff.png]]
+
+** Procedure to include flexible elements into Simscape
+In order to model a flexible element with only few mass-spring-damper elements:
+- FEM of the flexible element
+- Super-Element extraction
+- Parameters to extract
+  - For the flexible joint: axial, shear, bending and torsion stiffnesses
+  - For the APA: k1, ka, ke, c1
+- This can be done directly from the Stiffness matrix or using identification from a simple Simscape model
+
+** Conclusion
+#+begin_important
+  The results seems to indicate that the model is well represented with only few degrees of freedom.
+  This permit to have a relatively sane number of states for the model.
+#+end_important
+
+* Control with flexible elements
+** 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 APA and Flexible Joint
+#+begin_src matlab
+  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+  flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+#+begin_src matlab
+  stewart = initializeStewartPlatform();
+  stewart = initializeFramesPositions(stewart);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
+  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
+  stewart = initializeCylindricalPlatforms(stewart);
+  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
+  stewart = computeJacobian(stewart);
+  stewart = initializeStewartPose(stewart);
+  stewart = initializeInertialSensor(stewart);
+#+end_src
+
+#+begin_src matlab
+  ground = initializeGround('type', 'none');
+  payload = initializePayload('type', 'rigid', 'm', 50);
+  controller = initializeController('type', 'open-loop');
+#+end_src
+
+#+begin_src matlab
+  disturbances = initializeDisturbances();
+  references = initializeReferences(stewart);
+#+end_src
+
+** Identification
+And we identify the dynamics from force actuators to force sensors.
+#+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]
+#+end_src
+
+#+begin_src matlab
+  G = -linearize(mdl, io, options);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
+#+end_src
+
+** Decentralized Direct Velocity Feedback
+#+begin_src matlab :exports none
+  freqs = logspace(1, 4, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 2, 1);
+  hold on;
+  for i = 1:6
+    plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  title('Diagonal elements of the Plant');
+
+  ax2 = subplot(2, 2, 3);
+  hold on;
+  for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\epsilon_{\\mathcal{L}_%i}/\\tau_%i$', i, i));
+  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]);
+  legend('location', 'northwest');
+
+  ax3 = subplot(2, 2, 2);
+  hold on;
+  for i = 1:5
+    for j = i+1:6
+      plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
+    end
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  title('Off-Diagonal elements of the Plant');
+
+  ax4 = subplot(2, 2, 4);
+  hold on;
+  for i = 1:5
+    for j = i+1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
+    end
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
+  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,ax3,ax4],'x');
+#+end_src
+
+Controller Design
+#+begin_src matlab
+  Kl = -1e5*s/(1 + s/2/pi/2e2)/(1 + s/2/pi/5e2) * eye(6);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(1, 4, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G(1, 1)*Kl(1,1), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1)*Kl(1,1), freqs, 'Hz'))));
+  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 :results replace value
+  isstable(feedback(G(1:6,1:6)*Kl, eye(6), -1))
+#+end_src
+
+#+RESULTS:
+: 1
+
+** HAC
+#+begin_src matlab
+  Kx = tf(zeros(6));
+
+  controller = initializeController('type', 'ref-track-hac-dvf');
+#+end_src
+
+#+begin_src matlab
+  %% Name of the Simulink File
+  mdl = 'stewart_platform_model';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Controller'],              1, 'input');  io_i = io_i + 1; % Actuator Force Inputs [N]
+  io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]
+
+  %% Run the linearization
+  G = linearize(mdl, io);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
+#+end_src
+
+#+begin_src matlab
+  Gl = -stewart.kinematics.J*G;
+  Gl.OutputName  = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(1, 4, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 2, 1);
+  hold on;
+  for i = 1:6
+    plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  title('Diagonal elements of the Plant');
+
+  ax2 = subplot(2, 2, 3);
+  hold on;
+  for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\epsilon_{\\mathcal{L}_%i}/\\tau_%i$', i, i));
+  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]);
+  legend('location', 'northwest');
+
+  ax3 = subplot(2, 2, 2);
+  hold on;
+  for i = 1:5
+    for j = i+1:6
+      plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
+    end
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  title('Off-Diagonal elements of the Plant');
+
+  ax4 = subplot(2, 2, 4);
+  hold on;
+  for i = 1:5
+    for j = i+1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
+    end
+  end
+  set(gca,'ColorOrderIndex',1);
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
+  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,ax3,ax4],'x');
+#+end_src
+
+#+begin_src matlab
+  wc = 2*pi*300;
+  Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(1, 3, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  for i = 1:6
+    plot(freqs, abs(squeeze(freqresp(Kl(i, i)*Gl(i, i), freqs, 'Hz'))));
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 1:6
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*Gl(i, i), freqs, 'Hz'))));
+  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
+
+* Flexible Joint Specifications
+** 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
+
+** Stewart Platform Initialization
+#+begin_src matlab
+  stewart = initializeStewartPlatform();
+  stewart = initializeFramesPositions(stewart);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 43e6*ones(6,1), 'K1', 0.4e6*ones(6,1), 'C1', 10*ones(6,1));
+  stewart = initializeCylindricalPlatforms(stewart);
+  stewart = initializeCylindricalStruts(stewart);
+  stewart = computeJacobian(stewart);
+  stewart = initializeStewartPose(stewart);
+  stewart = initializeInertialSensor(stewart);
+
+  references = initializeReferences(stewart);
+#+end_src
+
+#+begin_src matlab
+  ground = initializeGround('type', 'none');
+  payload = initializePayload('type', 'rigid', 'm', 50);
+  controller = initializeController('type', 'open-loop');
+#+end_src
+
+#+begin_src matlab
+  disturbances = initializeDisturbances();
+#+end_src
+
+#+begin_src matlab
+  open('stewart_platform_model.slx')
+#+end_src
+
+** Initialization                                                  :noexport:
+#+begin_src matlab
+  freqs = logspace(0, 3, 1000);
+
+  c1 = [     0 0.4470 0.7410 0.2]; % Blue
+  c2 = [0.8500 0.3250 0.0980 0.2]; % Orange
+  c3 = [0.9290 0.6940 0.1250 0.2]; % Yellow
+  c4 = [0.4940 0.1840 0.5560 0.2]; % Purple
+  c5 = [0.4660 0.6740 0.1880 0.2]; % Green
+  c6 = [0.3010 0.7450 0.9330 0.2]; % Light Blue
+  c7 = [0.6350 0.0780 0.1840 0.2]; % Red
+  colors = [c1; c2; c3; c4; c5; c6; c7];
+#+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 Motion - Legs [m]
+  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
+#+end_src
+
+** Effect of the Bending Stiffness
+#+begin_src matlab
+  Kfs = [1, 10, 100, 1000]; % [Nm/rad]
+#+end_src
+
+#+begin_src matlab :exports none
+  Gs = {zeros(length(Kfs))};
+
+  for Kfs_i = 1:length(Kfs)
+      stewart = initializeJointDynamics(stewart, 'type_F', 'universal', ...
+                                                 'type_M', 'spherical', ...
+                                                 'Kf_M', Kfs(Kfs_i)*ones(6,1), ...
+                                                 'Kt_M', 0*ones(6,1), ...
+                                                 'Kf_F', Kfs(Kfs_i)*ones(6,1), ...
+                                                 'Kt_F', 0*ones(6,1), ...
+                                                 'Ka_M', 0*ones(6,1), ...
+                                                 'Kr_M', 0*ones(6,1), ...
+                                                 'Ka_F', 0*ones(6,1), ...
+                                                 'Kr_F', 0*ones(6,1));
+
+      G = linearize(mdl, io, options);
+      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+      Gs(Kfs_i) = {G};
+  end
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Kfs_i = 1:length(Kfs)
+      plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kfs(Kfs_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Kfs_i = 1:length(Kfs)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kfs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :));
+      end
+  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 :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Kfs_i = 1:length(Kfs)
+      plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kfs(Kfs_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Kfs_i = 1:length(Kfs)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kfs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :));
+      end
+  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
+
+** Effect of the Torsion Stiffness
+#+begin_src matlab
+  Kts = [10, 100, 500, 1000]; % [Nm/rad]
+#+end_src
+
+#+begin_src matlab :exports none
+  Gt = {zeros(length(Kts))};
+
+  for Kts_i = 1:length(Kts)
+      stewart = initializeJointDynamics(stewart, 'type_F', 'universal', ...
+                                                 'type_M', 'spherical', ...
+                                                 'Kf_M', 0*ones(6,1), ...
+                                                 'Kt_M', Kts(Kts_i)*ones(6,1), ...
+                                                 'Kf_F', 0*ones(6,1), ...
+                                                 'Kt_F', Kts(Kts_i)*ones(6,1), ...
+                                                 'Ka_M', 0*ones(6,1), ...
+                                                 'Kr_M', 0*ones(6,1), ...
+                                                 'Ka_F', 0*ones(6,1), ...
+                                                 'Kr_F', 0*ones(6,1));
+
+      G = linearize(mdl, io, options);
+      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+      Gt(Kts_i) = {G};
+  end
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Kts_i = 1:length(Kts)
+      plot(freqs, abs(squeeze(freqresp(Gt{Kts_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'DisplayName', sprintf('$k_t = %.0f [Nm/rad]$', Kts(Kts_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Gt{Kts_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Kts_i = 1:length(Kts)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Gt{Kts_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :));
+      end
+  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 :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Kts_i = 1:length(Kts)
+      plot(freqs, abs(squeeze(freqresp(Gs{Kts_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kts(Kts_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Gs{Kts_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Kts_i = 1:length(Kts)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kts_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :));
+      end
+  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
+
+** Effect of the Axial Stiffness
+#+begin_src matlab
+  Kas = [1e6, 1e7, 1e8, 5e8, 1e9]; % [N/m]
+#+end_src
+
+#+begin_src matlab :exports none
+  Ga = {zeros(length(Kas))};
+
+  for Kas_i = 1:length(Kas)
+      stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
+                                                 'type_M', 'spherical_3dof', ...
+                                                 'Kf_M', 0*ones(6,1), ...
+                                                 'Kt_M', 0*ones(6,1), ...
+                                                 'Kf_F', 0*ones(6,1), ...
+                                                 'Kt_F', 0*ones(6,1), ...
+                                                 'Ka_M', Kas(Kas_i)*ones(6,1), ...
+                                                 'Kr_M', 0*ones(6,1), ...
+                                                 'Ka_F', Kas(Kas_i)*ones(6,1), ...
+                                                 'Kr_F', 0*ones(6,1));
+
+      G = linearize(mdl, io, options);
+      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+      Ga(Kas_i) = {G};
+  end
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Kas_i = 1:length(Kas)
+      plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'DisplayName', sprintf('$k_a = %.0e [N/m]$', Kas(Kas_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Kas_i = 1:length(Kas)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Ga{Kas_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :));
+      end
+  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 :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Kas_i = 1:length(Kas)
+      plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'DisplayName', sprintf('$k_a = %.0e [N/m]$', Kas(Kas_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Kas_i = 1:length(Kas)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Ga{Kas_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :));
+      end
+  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
+
+** Effect of the Radial (Shear) Stiffness
+#+begin_src matlab
+  Krs = [1e4, 1e5, 1e6, 1e7]; % [N/m]
+#+end_src
+
+#+begin_src matlab :exports none
+  Gr = {zeros(length(Kas))};
+
+  for Krs_i = 1:length(Krs)
+      stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
+                                                 'type_M', 'spherical_3dof', ...
+                                                 'Kf_M', 0*ones(6,1), ...
+                                                 'Kt_M', 0*ones(6,1), ...
+                                                 'Kf_F', 0*ones(6,1), ...
+                                                 'Kt_F', 0*ones(6,1), ...
+                                                 'Ka_M', 1e15*ones(6,1), ...
+                                                 'Kr_M', Krs(Krs_i)*ones(6,1), ...
+                                                 'Ka_F', 1e15*ones(6,1), ...
+                                                 'Kr_F', Krs(Krs_i)*ones(6,1));
+
+      G = linearize(mdl, io, options);
+      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+      Gr(Krs_i) = {G};
+  end
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Krs_i = 1:length(Krs)
+      plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'DisplayName', sprintf('$k_r = %.0e [N/m]$', Krs(Krs_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Krs_i = 1:length(Krs)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Gr{Krs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :));
+      end
+  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 :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+
+  for Krs_i = 1:length(Krs)
+      plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'DisplayName', sprintf('$k_r = %.0e [N/m]$', Krs(Krs_i)));
+      for i = 2:6
+          plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'HandleVisibility', 'off');
+      end
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for Krs_i = 1:length(Krs)
+      for i = 1:6
+          plot(freqs, 180/pi*angle(squeeze(freqresp(Gr{Krs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :));
+      end
+  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
+
+** Comparison of perfect joint and worst specified joint
+#+begin_src matlab :exports none
+  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', ...
+                                             'type_M', 'spherical_p');
+
+  Gp = linearize(mdl, io, options);
+  Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  Gp.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+  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));
+
+  G = linearize(mdl, io, options);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
+  end
+  plot(freqs, abs(squeeze(freqresp(Gp('L1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
+  end
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
+  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 :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G('Fm1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
+  end
+  plot(freqs, abs(squeeze(freqresp(Gp('Fm1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
+  end
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
+  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
+
+** Conclusion
+Qualitatively:
+|                   | *Specification*                                  |
+|-------------------+--------------------------------------------------|
+| Axial Stiffness   | Much larger than the Actuator axial stiffness    |
+| Shear Stiffness   |                                                  |
+| Bending Stiffness | Much smaller than the Actuator bending stiffness |
+| Torsion Stiffness |                                                  |
+
+Quantitatively:
+|                   | *Specification* |
+|-------------------+-----------------|
+| Axial Stiffness   | > 200  [N/um]   |
+| Shear Stiffness   | > 1 [N/um]      |
+| Bending Stiffness | < 100  [Nm/rad] |
+| Torsion Stiffness | < 500  [Nm/rad] |
+
+* Flexible Joint Specifications with the APA300ML
+** 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
+
+** Stewart Platform Initialization
+#+begin_src matlab
+  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+#+begin_src matlab
+  stewart = initializeStewartPlatform();
+  stewart = initializeFramesPositions(stewart);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
+  stewart = initializeCylindricalPlatforms(stewart);
+  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
+  stewart = computeJacobian(stewart);
+  stewart = initializeStewartPose(stewart);
+  stewart = initializeInertialSensor(stewart);
+
+  references = initializeReferences(stewart);
+#+end_src
+
+#+begin_src matlab
+  ground = initializeGround('type', 'none');
+  payload = initializePayload('type', 'rigid', 'm', 50);
+  controller = initializeController('type', 'open-loop');
+#+end_src
+
+#+begin_src matlab
+  disturbances = initializeDisturbances();
+#+end_src
+
+#+begin_src matlab
+  open('stewart_platform_model.slx')
+#+end_src
+
+** Initialization                                                  :noexport:
+#+begin_src matlab
+  freqs = logspace(0, 3, 1000);
+
+  c1 = [     0 0.4470 0.7410 0.2]; % Blue
+  c2 = [0.8500 0.3250 0.0980 0.2]; % Orange
+  c3 = [0.9290 0.6940 0.1250 0.2]; % Yellow
+  c4 = [0.4940 0.1840 0.5560 0.2]; % Purple
+  c5 = [0.4660 0.6740 0.1880 0.2]; % Green
+  c6 = [0.3010 0.7450 0.9330 0.2]; % Light Blue
+  c7 = [0.6350 0.0780 0.1840 0.2]; % Red
+  colors = [c1; c2; c3; c4; c5; c6; c7];
+#+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 Motion - Legs [m]
+  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
+#+end_src
+
+** Comparison of perfect joint and worst specified joint
+#+begin_src matlab :exports none
+  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', ...
+                                             'type_M', 'spherical_p');
+
+  Gp = linearize(mdl, io, options);
+  Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  Gp.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+
+  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));
+
+  G = linearize(mdl, io, options);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
+#+end_src
+
+#+begin_src matlab :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
+  end
+  plot(freqs, abs(squeeze(freqresp(Gp('L1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
+  end
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
+  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 :exports none
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G('Fm1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
+  end
+  plot(freqs, abs(squeeze(freqresp(Gp('Fm1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
+  end
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
+  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
+
+* Relative Motion Sensors
+** 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
+
+** Stewart Platform Initialization
+#+begin_src matlab
+  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+#+begin_src matlab
+  stewart = initializeStewartPlatform();
+  stewart = initializeFramesPositions(stewart);
+  stewart = generateGeneralConfiguration(stewart);
+  stewart = computeJointsPose(stewart);
+#+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));
+  % 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');
+#+end_src
+
+#+begin_src matlab
+  stewart = initializeCylindricalPlatforms(stewart);
+#+end_src
+
+#+begin_src matlab
+  stewart = initializeCylindricalStruts(stewart);
+  % stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
+#+end_src
+
+#+begin_src matlab
+  stewart = computeJacobian(stewart);
+  stewart = initializeStewartPose(stewart);
+  stewart = initializeInertialSensor(stewart);
+
+  references = initializeReferences(stewart);
+#+end_src
+
+#+begin_src matlab
+  ground = initializeGround('type', 'none');
+  payload = initializePayload('type', 'rigid', 'm', 50);
+  controller = initializeController('type', 'open-loop');
+#+end_src
+
+#+begin_src matlab
+  disturbances = initializeDisturbances();
+#+end_src
+
+#+begin_src matlab
+  open('stewart_platform_model.slx')
+#+end_src
+
+** Initialization                                                  :noexport:
+#+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 Motion - Actuators [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
+  G = linearize(mdl, io, options);
+  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
+                  'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(1, 4, 5000);
+
+  figure;
+
+  ax1 = subplot(2, 1, 1);
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Actuator Stroke');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
+  end
+  plot(freqs, abs(squeeze(freqresp(G('D1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Leg Stroke');
+  for i = 2:6
+      plot(freqs, abs(squeeze(freqresp(G(sprintf('D%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
+  end
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+  legend('location', 'northeast');
+
+  ax2 = subplot(2, 1, 2);
+  hold on;
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
+  end
+  for i = 1:6
+      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('D%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
+  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
diff --git a/org/stewart-architecture.org b/org/stewart-architecture.org
index caedb4a..e41c392 100644
--- a/org/stewart-architecture.org
+++ b/org/stewart-architecture.org
@@ -1168,45 +1168,20 @@ An amplified piezoelectric actuator is shown in Figure [[fig:cedrat_apa95ml]].
 #+caption: Example of an Amplified piezoelectric actuator with an integrated displacement sensor (Cedrat Technologies)
 [[file:figs/amplified_piezo_with_displacement_sensor.jpg]]
 
-A simplistic model of such amplified actuator is shown in Figure [[fig:amplified_piezo_model]] where:
-- $K_{r}$ represent the vertical stiffness when the piezoelectric stack is removed
-- $K_{a}$ is the vertical stiffness contribution of the piezoelectric stack
-- $F_{i}$ represents the part of the piezoelectric stack that is used as a force actuator
-- $F_{m,i}$ represents the remaining part of the piezoelectric stack that is used as a force sensor
-- $v_{m,i}$ represents the absolute velocity of the top part of the actuator
-- $d_{m,i}$ represents the total relative displacement of the actuator
+A simplistic model of such amplified actuator is shown in Figure [[fig:souleille18_model_piezo]] where the parameters are described in Table [[tab:parameters_amplified_piezo]].
 
-#+begin_src latex :file iff_1dof.pdf
-  \begin{tikzpicture}
-    % Ground
-    \draw (-1.2, 0) -- (1, 0);
+#+name: tab:parameters_amplified_piezo
+#+caption: 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                                          |
 
-    % Mass
-    \draw (-1.2, 1.4) -- ++(2.2, 0);
-    \node[forcesensor={0.4}{0.4}] (fsensn) at (0, 1){};
-    \draw[] (-0.4, 1) -- (0.4, 1);
-    \node[right] at (fsensn.east) {$F_{m}$};
-
-    % Spring, Damper, and Actuator
-    \draw[spring] (-0.4, 0) -- (-0.4, 1) node[midway, right=0.1]{$K_{a}$};
-    \draw[actuator={0.4}{0.2}] (0.4, 0) -- (0.4, 1) node[midway, right=0.1]{$F$};
-
-    \draw[spring] (-1, 0) -- (-1, 1.4) node[midway, left=0.1]{$K_{r}$};
-
-    \draw[dashed] (1, 0) -- ++(0.4, 0);
-
-    \draw[dashed] (1, 1.4) -- ++(0.4, 0);
-
-    \draw[->] (0, 1.4)node[]{$\bullet$} -- ++(0, 0.5) node[right]{$v_{m}$};
-
-    \draw[<->] (1.4, 0) -- ++(0, 1.4) node[midway, right]{$d_{m}$};
-  \end{tikzpicture}
-#+end_src
-
-#+name: fig:amplified_piezo_model
-#+caption: Model of an amplified actuator
-#+RESULTS:
-[[file:figs/iff_1dof.png]]
+#+name: fig:souleille18_model_piezo
+#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
+[[file:./figs/souleille18_model_piezo.png]]
 
 *** Function description
 :PROPERTIES:
@@ -1220,20 +1195,20 @@ A simplistic model of such amplified actuator is shown in Figure [[fig:amplified
   %
   % Inputs:
   %    - args - Structure with the following fields:
-  %        - Ka [6x1] - Vertical stiffness contribution of the piezoelectric stack [N/m]
-  %        - Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
-  %        - Kr [6x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
-  %        - Cr [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
+  %        - 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] - Vertical stiffness contribution of the piezoelectric stack [N/m]
-  %      - actuators.Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
-  %      - actuators.Kr [6x1] - Vertical stiffness when the piezoelectric stack is removed [N/m]
-  %      - actuators.Cr [6x1] - Vertical damping when the piezoelectric stack is removed [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)]
 #+end_src
 
 *** Optional Parameters
@@ -1243,10 +1218,10 @@ A simplistic model of such amplified actuator is shown in Figure [[fig:amplified
 #+begin_src matlab
   arguments
       stewart
-      args.Kr (6,1) double {mustBeNumeric, mustBeNonnegative} = 5e6*ones(6,1)
-      args.Cr (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
-      args.Ka (6,1) double {mustBeNumeric, mustBeNonnegative} = 15e6*ones(6,1)
-      args.Ca (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+      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
 #+end_src
 
@@ -1255,8 +1230,8 @@ A simplistic model of such amplified actuator is shown in Figure [[fig:amplified
 :UNNUMBERED: t
 :END:
 #+begin_src matlab
-  K = args.Ka + args.Kr;
-  C = args.Ca + args.Cr;
+  K = args.K1 + args.Ka.*args.Ke./(args.Ka + args.Ke);
+  C = args.C1;
 #+end_src
 
 *** Populate the =stewart= structure
@@ -1267,10 +1242,9 @@ A simplistic model of such amplified actuator is shown in Figure [[fig:amplified
   stewart.actuators.type = 2;
 
   stewart.actuators.Ka = args.Ka;
-  stewart.actuators.Ca = args.Ca;
-
-  stewart.actuators.Kr = args.Kr;
-  stewart.actuators.Cr = args.Cr;
+  stewart.actuators.Ke = args.Ke;
+  stewart.actuators.K1 = args.K1;
+  stewart.actuators.C1 = args.C1;
 
   stewart.actuators.K = K;
   stewart.actuators.C = C;
@@ -1301,6 +1275,7 @@ This Matlab function is accessible [[file:../src/initializeFlexibleStrutDynamics
   %        - 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:
@@ -1318,6 +1293,7 @@ This Matlab function is accessible [[file:../src/initializeFlexibleStrutDynamics
       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
@@ -1347,6 +1323,8 @@ This Matlab function is accessible [[file:../src/initializeFlexibleStrutDynamics
   stewart.actuators.step_file = args.step_file;
 
   stewart.actuators.K = args.K(3,3); % Axial Stiffness
+
+  stewart.actuators.Gf = args.Gf;
 #+end_src
 
 ** TODO =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints
@@ -1400,16 +1378,24 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
 #+begin_src matlab
   arguments
       stewart
-      args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'universal'
-      args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'spherical'
-      args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
+      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} = 20*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} = 15*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} = 20*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)
@@ -1439,6 +1425,10 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
       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
@@ -1452,6 +1442,10 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
       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
 #+end_src
 
@@ -1459,26 +1453,22 @@ This Matlab function is accessible [[file:../src/initializeJointDynamics.m][here
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
-Translation Stiffness
+Axial and Radial (shear) Stiffness
 #+begin_src matlab
-  stewart.joints_M.Kx = zeros(6,1);
-  stewart.joints_M.Ky = zeros(6,1);
-  stewart.joints_M.Kz = zeros(6,1);
+  stewart.joints_M.Ka = args.Ka_M;
+  stewart.joints_M.Kr = args.Kr_M;
 
-  stewart.joints_F.Kx = zeros(6,1);
-  stewart.joints_F.Ky = zeros(6,1);
-  stewart.joints_F.Kz = zeros(6,1);
+  stewart.joints_F.Ka = args.Ka_F;
+  stewart.joints_F.Kr = args.Kr_F;
 #+end_src
 
 Translation Damping
 #+begin_src matlab
-  stewart.joints_M.Cx = zeros(6,1);
-  stewart.joints_M.Cy = zeros(6,1);
-  stewart.joints_M.Cz = zeros(6,1);
+  stewart.joints_M.Ca = args.Ca_M;
+  stewart.joints_M.Cr = args.Cr_M;
 
-  stewart.joints_F.Cx = zeros(6,1);
-  stewart.joints_F.Cy = zeros(6,1);
-  stewart.joints_F.Cz = zeros(6,1);
+  stewart.joints_F.Ca = args.Ca_F;
+  stewart.joints_F.Cr = args.Cr_F;
 #+end_src
 
 *** Add Stiffness and Damping in Rotation of each strut
@@ -1488,19 +1478,19 @@ Translation Damping
 Rotational Stiffness
 #+begin_src matlab
   stewart.joints_M.Kf = args.Kf_M;
-  stewart.joints_M.Kt = args.Kf_M;
+  stewart.joints_M.Kt = args.Kt_M;
 
   stewart.joints_F.Kf = args.Kf_F;
-  stewart.joints_F.Kt = args.Kf_F;
+  stewart.joints_F.Kt = args.Kt_F;
 #+end_src
 
 Rotational Damping
 #+begin_src matlab
   stewart.joints_M.Cf = args.Cf_M;
-  stewart.joints_M.Ct = args.Cf_M;
+  stewart.joints_M.Ct = args.Ct_M;
 
   stewart.joints_F.Cf = args.Cf_F;
-  stewart.joints_F.Ct = args.Cf_F;
+  stewart.joints_F.Ct = args.Ct_F;
 #+end_src
 
 *** Stiffness and Mass matrices for flexible joint
diff --git a/simscape_subsystems/Stewart_Platform.slx b/simscape_subsystems/Stewart_Platform.slx
index e2e1ac9..b0176ed 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 4bc7797..c20cb87 100644
Binary files a/simscape_subsystems/stewart_strut.slx and b/simscape_subsystems/stewart_strut.slx differ
diff --git a/src/initializeAmplifiedStrutDynamics.m b/src/initializeAmplifiedStrutDynamics.m
index 950bd6d..f3665bc 100644
--- a/src/initializeAmplifiedStrutDynamics.m
+++ b/src/initializeAmplifiedStrutDynamics.m
@@ -5,39 +5,38 @@ function [stewart] = initializeAmplifiedStrutDynamics(stewart, args)
 %
 % Inputs:
 %    - args - Structure with the following fields:
-%        - Ka [6x1] - Vertical stiffness contribution of the piezoelectric stack [N/m]
-%        - Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
-%        - Kr [6x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
-%        - Cr [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
+%        - 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] - Vertical stiffness contribution of the piezoelectric stack [N/m]
-%      - actuators.Ca [6x1] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
-%      - actuators.Kr [6x1] - Vertical stiffness when the piezoelectric stack is removed [N/m]
-%      - actuators.Cr [6x1] - Vertical damping when the piezoelectric stack is removed [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)]
 
 arguments
     stewart
-    args.Kr (6,1) double {mustBeNumeric, mustBeNonnegative} = 5e6*ones(6,1)
-    args.Cr (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
-    args.Ka (6,1) double {mustBeNumeric, mustBeNonnegative} = 15e6*ones(6,1)
-    args.Ca (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
+    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
 
-K = args.Ka + args.Kr;
-C = args.Ca + args.Cr;
+K = args.K1 + args.Ka.*args.Ke./(args.Ka + args.Ke);
+C = args.C1;
 
 stewart.actuators.type = 2;
 
 stewart.actuators.Ka = args.Ka;
-stewart.actuators.Ca = args.Ca;
-
-stewart.actuators.Kr = args.Kr;
-stewart.actuators.Cr = args.Cr;
+stewart.actuators.Ke = args.Ke;
+stewart.actuators.K1 = args.K1;
+stewart.actuators.C1 = args.C1;
 
 stewart.actuators.K = K;
 stewart.actuators.C = C;
diff --git a/src/initializeFlexibleStrutDynamics.m b/src/initializeFlexibleStrutDynamics.m
index 5030f68..b338da0 100644
--- a/src/initializeFlexibleStrutDynamics.m
+++ b/src/initializeFlexibleStrutDynamics.m
@@ -9,6 +9,7 @@ function [stewart] = initializeFlexibleStrutDynamics(stewart, args)
 %        - 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:
@@ -20,6 +21,7 @@ arguments
     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
 
@@ -36,3 +38,5 @@ 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/initializeJointDynamics.m b/src/initializeJointDynamics.m
index d974981..cc78b3a 100644
--- a/src/initializeJointDynamics.m
+++ b/src/initializeJointDynamics.m
@@ -29,16 +29,24 @@ function [stewart] = initializeJointDynamics(stewart, args)
 
 arguments
     stewart
-    args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'universal'
-    args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'flexible'})} = 'spherical'
-    args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*ones(6,1)
+    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} = 20*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} = 15*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} = 20*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)
@@ -62,6 +70,10 @@ switch args.type_F
     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
@@ -75,35 +87,35 @@ switch args.type_M
     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
 
-stewart.joints_M.Kx = zeros(6,1);
-stewart.joints_M.Ky = zeros(6,1);
-stewart.joints_M.Kz = zeros(6,1);
+stewart.joints_M.Ka = args.Ka_M;
+stewart.joints_M.Kr = args.Kr_M;
 
-stewart.joints_F.Kx = zeros(6,1);
-stewart.joints_F.Ky = zeros(6,1);
-stewart.joints_F.Kz = zeros(6,1);
+stewart.joints_F.Ka = args.Ka_F;
+stewart.joints_F.Kr = args.Kr_F;
 
-stewart.joints_M.Cx = zeros(6,1);
-stewart.joints_M.Cy = zeros(6,1);
-stewart.joints_M.Cz = zeros(6,1);
+stewart.joints_M.Ca = args.Ca_M;
+stewart.joints_M.Cr = args.Cr_M;
 
-stewart.joints_F.Cx = zeros(6,1);
-stewart.joints_F.Cy = zeros(6,1);
-stewart.joints_F.Cz = zeros(6,1);
+stewart.joints_F.Ca = args.Ca_F;
+stewart.joints_F.Cr = args.Cr_F;
 
 stewart.joints_M.Kf = args.Kf_M;
-stewart.joints_M.Kt = args.Kf_M;
+stewart.joints_M.Kt = args.Kt_M;
 
 stewart.joints_F.Kf = args.Kf_F;
-stewart.joints_F.Kt = args.Kf_F;
+stewart.joints_F.Kt = args.Kt_F;
 
 stewart.joints_M.Cf = args.Cf_M;
-stewart.joints_M.Ct = args.Cf_M;
+stewart.joints_M.Ct = args.Ct_M;
 
 stewart.joints_F.Cf = args.Cf_F;
-stewart.joints_F.Ct = args.Cf_F;
+stewart.joints_F.Ct = args.Ct_F;
 
 stewart.joints_F.M = args.M_F;
 stewart.joints_F.K = args.K_F;

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