nass-simscape/org/stewart_platform.org

Stewart Platform - Simscape Model

• Introduction
• initializeStewartPlatform: Initialize the Stewart Platform structure
  • Documentation
  • Function description
  • Initialize the Stewart structure
• initializeFramesPositions: Initialize the positions of frames {A}, {B}, {F} and {M}
  • Documentation
  • Function description
  • Optional Parameters
  • Compute the position of each frame
  • Populate the stewart structure
• generateGeneralConfiguration: Generate a Very General Configuration
  • Documentation
  • Function description
  • Optional Parameters
  • Compute the pose
  • Populate the stewart structure
• computeJointsPose: Compute the Pose of the Joints
  • Documentation
  • Function description
  • Check the stewart structure elements
  • Compute the position of the Joints
  • Compute the strut length and orientation
  • Compute the orientation of the Joints
  • Populate the stewart structure
• initializeStewartPose: Determine the initial stroke in each leg to have the wanted pose
  • Function description
  • Optional Parameters
  • Use the Inverse Kinematic function
  • Populate the stewart structure
• initializeCylindricalPlatforms: Initialize the geometry of the Fixed and Mobile Platforms
  • Function description
  • Optional Parameters
  • Compute the Inertia matrices of platforms
  • Populate the stewart structure
• initializeCylindricalStruts: Define the inertia of cylindrical struts
  • Function description
  • Optional Parameters
  • Compute the properties of the cylindrical struts
  • Populate the stewart structure
• initializeStrutDynamics: Add Stiffness and Damping properties of each strut
  • Documentation
  • Function description
  • Optional Parameters
  • Add Stiffness and Damping properties of each strut
• initializeAmplifiedStrutDynamics: Add Stiffness and Damping properties of each strut for an amplified piezoelectric actuator
  • Documentation
  • Function description
  • Optional Parameters
  • Compute the total stiffness and damping
  • Populate the stewart structure
• initializeJointDynamics: Add Stiffness and Damping properties for spherical joints
  • Function description
  • Optional Parameters
  • Add Actuator Type
  • Add Stiffness and Damping in Translation of each strut
  • Add Stiffness and Damping in Rotation of each strut
• initializeInertialSensor: Initialize the inertial sensor in each strut
  • Geophone - Working Principle
  • Accelerometer - Working Principle
  • Function description
  • Optional Parameters
  • Compute the properties of the sensor
  • Populate the stewart structure
• displayArchitecture: 3D plot of the Stewart platform architecture
  • Function description
  • Optional Parameters
  • Check the stewart structure elements
  • Figure Creation, Frames and Homogeneous transformations
  • Fixed Base elements
  • Mobile Platform elements
  • Legs
  • Figure parameters
  • Subplots
• describeStewartPlatform: Display some text describing the current defined Stewart Platform
  • Function description
  • Optional Parameters
  • Geometry
  • Actuators
  • Joints
  • Kinematics
• generateCubicConfiguration: Generate a Cubic Configuration
  • Function description
  • Documentation
  • Optional Parameters
  • Check the stewart structure elements
  • Position of the Cube
  • Compute the pose
  • Populate the stewart structure
• computeJacobian: Compute the Jacobian Matrix
  • Function description
  • Check the stewart structure elements
  • Compute Jacobian Matrix
  • Compute Stiffness Matrix
  • Compute Compliance Matrix
  • Populate the stewart structure
• inverseKinematics: Compute Inverse Kinematics
  • Theory
  • Function description
  • Optional Parameters
  • Check the stewart structure elements
  • Compute
• forwardKinematicsApprox: Compute the Approximate Forward Kinematics
  • Function description
  • Optional Parameters
  • Check the stewart structure elements
  • Computation

Introduction   ignore

Stewart platforms are generated in multiple steps.

We define 4 important frames:

• $\{F\}$: Frame fixed to the Fixed base and located at the center of its bottom surface. This is used to fix the Stewart platform to some support.
• $\{M\}$: Frame fixed to the Moving platform and located at the center of its top surface. This is used to place things on top of the Stewart platform.
• $\{A\}$: Frame fixed to the fixed base. It defined the center of rotation of the moving platform.
• $\{B\}$: Frame fixed to the moving platform. The motion of the moving platforms and forces applied to it are defined with respect to this frame $\{B\}$.

Then, we define the location of the spherical joints:

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

We define the rest position of the Stewart platform:

• For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.
• Thus, to define the rest position of the Stewart platform, we just have to defined its total height $H$. $H$ corresponds to the distance from the bottom of the fixed base to the top of the moving platform.

From $\bm{a}_{i}$ and $\bm{b}_{i}$, we can determine the length and orientation of each strut:

• $l_{i}$ is the length of the strut
• ${}^{A}\hat{\bm{s}}_{i}$ is the unit vector align with the strut

The position of the Spherical joints can be computed using various methods:

• Cubic configuration
• Circular configuration
• Arbitrary position
• These methods should be easily scriptable and corresponds to specific functions that returns ${}^{F}\bm{a}_{i}$ and ${}^{M}\bm{b}_{i}$. The input of these functions are the parameters corresponding to the wanted geometry.

For Simscape, we need:

• The position and orientation of each spherical joint fixed to the fixed base: ${}^{F}\bm{a}_{i}$ and ${}^{F}\bm{R}_{a_{i}}$
• The position and orientation of each spherical joint fixed to the moving platform: ${}^{M}\bm{b}_{i}$ and ${}^{M}\bm{R}_{b_{i}}$
• The rest length of each strut: $l_{i}$
• The stiffness and damping of each actuator: $k_{i}$ and $c_{i}$
• The position of the frame $\{A\}$ with respect to the frame $\{F\}$: ${}^{F}\bm{O}_{A}$
• The position of the frame $\{B\}$ with respect to the frame $\{M\}$: ${}^{M}\bm{O}_{B}$

initializeStewartPlatform: Initialize the Stewart Platform structure

<<sec:initializeStewartPlatform>>

This Matlab function is accessible here.

Documentation

Definition of the position of the frames

Function description

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

Initialize the Stewart structure

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

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

<<sec:initializeFramesPositions>>

This Matlab function is accessible here.

Documentation

Definition of the position of the frames

Function description

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

Optional Parameters

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

Compute the position of each frame

  H = args.H; % Total Height of the Stewart Platform [m]

  FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]

  MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]

  FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]

Populate the stewart structure

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

generateGeneralConfiguration: Generate a Very General Configuration

<<sec:generateGeneralConfiguration>>

This Matlab function is accessible here.

Documentation

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

  \begin{tikzpicture}
    % Internal and external limit
    \draw[fill=white!80!black] (0, 0) circle [radius=3];
    % Circle where the joints are located
    \draw[dashed] (0, 0) circle [radius=2.5];

    % Bullets for the positions of the joints
    \node[] (J1) at ( 80:2.5){$\bullet$};
    \node[] (J2) at (100:2.5){$\bullet$};
    \node[] (J3) at (200:2.5){$\bullet$};
    \node[] (J4) at (220:2.5){$\bullet$};
    \node[] (J5) at (320:2.5){$\bullet$};
    \node[] (J6) at (340:2.5){$\bullet$};

    % Name of the points
    \node[above right] at (J1) {$a_{1}$};
    \node[above left]  at (J2) {$a_{2}$};
    \node[above left]  at (J3) {$a_{3}$};
    \node[right     ]  at (J4) {$a_{4}$};
    \node[left      ]  at (J5) {$a_{5}$};
    \node[above right] at (J6) {$a_{6}$};

    % First 2 angles
    \draw[dashed, ->] (0:1)   arc [start angle=0, end angle=80, radius=1]    node[below right]{$\theta_{1}$};
    \draw[dashed, ->] (0:1.5) arc [start angle=0, end angle=100, radius=1.5] node[left       ]{$\theta_{2}$};

    % Division of 360 degrees by 3
    \draw[dashed] (0, 0) -- ( 80:3.2);
    \draw[dashed] (0, 0) -- (100:3.2);
    \draw[dashed] (0, 0) -- (200:3.2);
    \draw[dashed] (0, 0) -- (220:3.2);
    \draw[dashed] (0, 0) -- (320:3.2);
    \draw[dashed] (0, 0) -- (340:3.2);

    % Radius for the position of the joints
    \draw[<->] (0, 0) --node[near end, above]{$R$} (180:2.5);

    \draw[->] (0, 0) -- ++(3.4, 0) node[above]{$x$};
    \draw[->] (0, 0) -- ++(0, 3.4) node[left]{$y$};
  \end{tikzpicture}

Position of the joints

Function description

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

Optional Parameters

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

Compute the pose

  Fa = zeros(3,6);
  Mb = zeros(3,6);

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

Populate the stewart structure

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

computeJointsPose: Compute the Pose of the Joints

<<sec:computeJointsPose>>

This Matlab function is accessible here.

Documentation

Position and orientation of the struts

Function description

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

Check the stewart structure elements

  assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
  Fa = stewart.platform_F.Fa;

  assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
  Mb = stewart.platform_M.Mb;

  assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
  FO_A = stewart.platform_F.FO_A;

  assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
  MO_B = stewart.platform_M.MO_B;

  assert(isfield(stewart.geometry,   'FO_M'), 'stewart.geometry should have attribute FO_M')
  FO_M = stewart.geometry.FO_M;

Compute the position of the Joints

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

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

Compute the strut length and orientation

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

  l = vecnorm(Ab - Aa)';

  Bs = (Bb - Ba)./vecnorm(Bb - Ba);

Compute the orientation of the Joints

  FRa = zeros(3,3,6);
  MRb = zeros(3,3,6);

  for i = 1:6
    FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
    FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));

    MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
    MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
  end

Populate the stewart structure

  stewart.geometry.Aa = Aa;
  stewart.geometry.Ab = Ab;
  stewart.geometry.Ba = Ba;
  stewart.geometry.Bb = Bb;
  stewart.geometry.As = As;
  stewart.geometry.Bs = Bs;
  stewart.geometry.l  = l;

  stewart.struts_F.l  = l/2;
  stewart.struts_M.l  = l/2;

  stewart.platform_F.FRa = FRa;
  stewart.platform_M.MRb = MRb;

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

<<sec:initializeStewartPose>>

This Matlab function is accessible here.

Function description

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

Optional Parameters

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

Use the Inverse Kinematic function

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

Populate the stewart structure

  stewart.actuators.Leq = dLi;

initializeCylindricalPlatforms: Initialize the geometry of the Fixed and Mobile Platforms

<<sec:initializeCylindricalPlatforms>>

This Matlab function is accessible here.

Function description

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

Optional Parameters

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

Compute the Inertia matrices of platforms

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

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

Populate the stewart structure

  stewart.platform_F.type = 1;

  stewart.platform_F.I = I_F;
  stewart.platform_F.M = args.Fpm;
  stewart.platform_F.R = args.Fpr;
  stewart.platform_F.H = args.Fph;

  stewart.platform_M.type = 1;

  stewart.platform_M.I = I_M;
  stewart.platform_M.M = args.Mpm;
  stewart.platform_M.R = args.Mpr;
  stewart.platform_M.H = args.Mph;

initializeCylindricalStruts: Define the inertia of cylindrical struts

<<sec:initializeCylindricalStruts>>

This Matlab function is accessible here.

Function description

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

Optional Parameters

  arguments
      stewart
      args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
      args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
      args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
      args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
      args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
      args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
  end

Compute the properties of the cylindrical struts

  Fsm = ones(6,1).*args.Fsm;
  Fsh = ones(6,1).*args.Fsh;
  Fsr = ones(6,1).*args.Fsr;

  Msm = ones(6,1).*args.Msm;
  Msh = ones(6,1).*args.Msh;
  Msr = ones(6,1).*args.Msr;

  I_F = zeros(3, 3, 6); % Inertia of the "fixed" part of the strut
  I_M = zeros(3, 3, 6); % Inertia of the "mobile" part of the strut

  for i = 1:6
    I_F(:,:,i) = diag([1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
                       1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
                       1/2  * Fsm(i) * Fsr(i)^2]);

    I_M(:,:,i) = diag([1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
                       1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
                       1/2  * Msm(i) * Msr(i)^2]);
  end

Populate the stewart structure

  stewart.struts_M.type = 1;

  stewart.struts_M.I = I_M;
  stewart.struts_M.M = Msm;
  stewart.struts_M.R = Msr;
  stewart.struts_M.H = Msh;

  stewart.struts_F.type = 1;

  stewart.struts_F.I = I_F;
  stewart.struts_F.M = Fsm;
  stewart.struts_F.R = Fsr;
  stewart.struts_F.H = Fsh;

initializeStrutDynamics: Add Stiffness and Damping properties of each strut

<<sec:initializeStrutDynamics>>

This Matlab function is accessible here.

Documentation

Example of a piezoelectric stach actuator (PI)

A simplistic model of such amplified actuator is shown in Figure fig:actuator_model_simple where:

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

  \begin{tikzpicture}
    \draw (-1, 0) -- (1, 0);

    % Spring, Damper, and Actuator
    \draw[spring]   (-1, 0) -- (-1, 1.5) node[midway, left=0.1]{$K$};
    \draw[damper]   ( 0, 0) -- ( 0, 1.5) node[midway, left=0.2]{$C$};
    \draw[actuator] ( 1, 0) -- ( 1, 1.5) node[midway, left=0.1](F){$F$};

    \node[forcesensor={2}{0.2}] (fsens) at (0, 1.5){};

    \node[left] at (fsens.west) {$F_{m}$};

    \draw[dashed] (1, 0) -- ++(0.4, 0);
    \draw[dashed] (1, 1.7) -- ++(0.4, 0);

    \draw[->] (0, 1.7)node[]{$\bullet$} -- ++(0, 0.5) node[right]{$v_{m}$};

    \draw[<->] (1.4, 0) -- ++(0, 1.7) node[midway, right]{$d_{m}$};
  \end{tikzpicture}

Simple model of an Actuator

Function description

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

Optional Parameters

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

Add Stiffness and Damping properties of each strut

  stewart.actuators.type = 1;

  stewart.actuators.K = args.K;
  stewart.actuators.C = args.C;

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

<<sec:initializeAmplifiedStrutDynamics>>

This Matlab function is accessible here.

Documentation

An amplified piezoelectric actuator is shown in Figure fig:cedrat_apa95ml.

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

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

  \begin{tikzpicture}
    % Ground
    \draw (-1.2, 0) -- (1, 0);

    % 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}

Model of an amplified actuator

Function description

  function [stewart] = initializeAmplifiedStrutDynamics(stewart, args)
  % initializeAmplifiedStrutDynamics - Add Stiffness and Damping properties of each strut
  %
  % Syntax: [stewart] = initializeAmplifiedStrutDynamics(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - Ka [6x1] - 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)]
  %
  % 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)]

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)
  end

Compute the total stiffness and damping

  K = args.Ka + args.Kr;
  C = args.Ca + args.Cr;

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.K = K;
  stewart.actuators.C = K;

initializeJointDynamics: Add Stiffness and Damping properties for spherical joints

<<sec:initializeJointDynamics>>

This Matlab function is accessible here.

Function description

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

Optional Parameters

  arguments
      stewart
      args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof'})} = 'universal'
      args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'spherical_3dof'})} = 'spherical'
      args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*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.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
      args.Kz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
      args.Cz_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
      args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 15*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.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
      args.Kz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 60e6*ones(6,1)
      args.Cz_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e2*ones(6,1)
  end

Add Actuator Type

  switch args.type_F
    case 'universal'
      stewart.joints_F.type = 1;
    case 'spherical'
      stewart.joints_F.type = 2;
    case 'universal_p'
      stewart.joints_F.type = 3;
    case 'spherical_p'
      stewart.joints_F.type = 4;
    case 'universal_3dof'
      stewart.joints_F.type = 5;
  end

  switch args.type_M
    case 'universal'
      stewart.joints_M.type = 1;
    case 'spherical'
      stewart.joints_M.type = 2;
    case 'universal_p'
      stewart.joints_M.type = 3;
    case 'spherical_p'
      stewart.joints_M.type = 4;
    case 'spherical_3dof'
      stewart.joints_M.type = 6;
  end

Add Stiffness and Damping in Translation of each strut

Translation Stiffness

  stewart.joints_M.Kx = zeros(6,1);
  stewart.joints_M.Ky = zeros(6,1);
  stewart.joints_M.Kz = args.Kz_M;

  stewart.joints_F.Kx = zeros(6,1);
  stewart.joints_F.Ky = zeros(6,1);
  stewart.joints_F.Kz = args.Kz_F;

Translation Damping

  stewart.joints_M.Cx = zeros(6,1);
  stewart.joints_M.Cy = zeros(6,1);
  stewart.joints_M.Cz = args.Cz_M;

  stewart.joints_F.Cx = zeros(6,1);
  stewart.joints_F.Cy = zeros(6,1);
  stewart.joints_F.Cz = args.Cz_F;

Add Stiffness and Damping in Rotation of each strut

Rotational Stiffness

  stewart.joints_M.Kf = args.Kf_M;
  stewart.joints_M.Kt = args.Kf_M;

  stewart.joints_F.Kf = args.Kf_F;
  stewart.joints_F.Kt = args.Kf_F;

Rotational Damping

  stewart.joints_M.Cf = args.Cf_M;
  stewart.joints_M.Ct = args.Cf_M;

  stewart.joints_F.Cf = args.Cf_F;
  stewart.joints_F.Ct = args.Cf_F;

initializeInertialSensor: Initialize the inertial sensor in each strut

<<sec:initializeInertialSensor>>

This Matlab function is accessible here.

Geophone - Working Principle

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

• $\omega_0 = \sqrt{\frac{k}{m}}$
• $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$

Schematic of a Z-Axis geophone

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

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

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

Accelerometer - Working Principle

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

• $\omega_0 = \sqrt{\frac{k}{m}}$
• $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$

Schematic of a Z-Axis geophone

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

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

Note that there is trade-off between:

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

Function description

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

Optional Parameters

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

Compute the properties of the sensor

  sensor = struct();

  switch args.type
    case 'geophone'
      sensor.type = 1;

      sensor.M = args.mass;
      sensor.K = sensor.M * (2*pi*args.freq)^2;
      sensor.C = 2*sqrt(sensor.M * sensor.K);
    case 'accelerometer'
      sensor.type = 2;

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

Populate the stewart structure

  stewart.sensors.inertial = sensor;

displayArchitecture: 3D plot of the Stewart platform architecture

<<sec:displayArchitecture>>

This Matlab function is accessible here.

Function description

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

Optional Parameters

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

Check the stewart structure elements

  assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
  FO_A = stewart.platform_F.FO_A;

  assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
  MO_B = stewart.platform_M.MO_B;

  assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
  H = stewart.geometry.H;

  assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
  Fa = stewart.platform_F.Fa;

  assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
  Mb = stewart.platform_M.Mb;

Figure Creation, Frames and Homogeneous transformations

The reference frame of the 3d plot corresponds to the frame $\{F\}$.

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

  hold on;

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

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

  FTm = FTa*ATb*BTm;

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

  d_unit_vector = H/4;

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

  d_label = H/20;

Fixed Base elements

Let's first plot the frame $\{F\}$.

  Ff = [0, 0, 0];
  if args.frames
    quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
            [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)

    if args.labels
      text(Ff(1) + d_label, ...
          Ff(2) + d_label, ...
          Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
    end
  end

Now plot the frame $\{A\}$ fixed to the Base.

  if args.frames
    quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
            [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)

    if args.labels
      text(FO_A(1) + d_label, ...
           FO_A(2) + d_label, ...
           FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
    end
  end

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

  if args.platforms && stewart.platform_F.type == 1
    theta = [0:0.01:2*pi+0.01]; % Angles [rad]
    v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
    center = [0; 0; 0]; % Center of the circle
    radius = stewart.platform_F.R; % Radius of the circle [m]

    points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));

    plot3(points(1,:), ...
          points(2,:), ...
          points(3,:), '-', 'Color', args.F_color);
  end

Let's now plot the position and labels of the Fixed Joints

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

Mobile Platform elements

Plot the frame $\{M\}$.

  Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}

  if args.frames
    FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
    quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
            FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)

    if args.labels
      text(Fm(1) + d_label, ...
           Fm(2) + d_label, ...
           Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
    end
  end

Plot the frame $\{B\}$.

  FB = FO_A + args.AP;

  if args.frames
    FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
    quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
            FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)

    if args.labels
      text(FB(1) - d_label, ...
           FB(2) + d_label, ...
           FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
    end
  end

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

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

    points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));

    plot3(points(1,:), ...
          points(2,:), ...
          points(3,:), '-', 'Color', args.M_color);
  end

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

  if args.joints
    Fb = FTm*[Mb;ones(1,6)];

    scatter3(Fb(1,:), ...
             Fb(2,:), ...
             Fb(3,:), 'MarkerEdgeColor', args.M_color);

    if args.labels
      for i = 1:size(Fb,2)
        text(Fb(1,i) + d_label, ...
             Fb(2,i), ...
             Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
      end
    end
  end

Legs

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

  if args.legs
    for i = 1:6
      plot3([Fa(1,i), Fb(1,i)], ...
            [Fa(2,i), Fb(2,i)], ...
            [Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);

      if args.labels
        text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
             (Fa(2,i)+Fb(2,i))/2, ...
             (Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
      end
    end
  end

Figure parameters

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

Subplots

  if strcmp(args.views, 'all')
    hAx = findobj('type', 'axes');

    figure;
    s1 = subplot(2,2,1);
    copyobj(get(hAx(1), 'Children'), s1);
    view([0 0 1]);
    axis equal;
    axis off;
    title('Top')

    s2 = subplot(2,2,2);
    copyobj(get(hAx(1), 'Children'), s2);
    view([1 -0.6 0.4]);
    axis equal;
    axis off;

    s3 = subplot(2,2,3);
    copyobj(get(hAx(1), 'Children'), s3);
    view([1 0 0]);
    axis equal;
    axis off;
    title('Front')

    s4 = subplot(2,2,4);
    copyobj(get(hAx(1), 'Children'), s4);
    view([0 -1 0]);
    axis equal;
    axis off;
    title('Side')

    close(f);
  end

describeStewartPlatform: Display some text describing the current defined Stewart Platform

<<sec:describeStewartPlatform>>

This Matlab function is accessible here.

Function description

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

Optional Parameters

  arguments
      stewart
  end

Geometry

  fprintf('GEOMETRY:\n')
  fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)

  if stewart.platform_M.MO_B(3) > 0
    fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n',  1e3*stewart.platform_M.MO_B(3))
  else
    fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
  end

  fprintf('- The initial length of the struts are:\n')
  fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
  fprintf('\n')

Actuators

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

Joints

  fprintf('JOINTS:\n')

Type of the joints on the fixed base.

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

Type of the joints on the mobile platform.

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

Position of the fixed joints

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

Position of the mobile joints

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

Kinematics

  fprintf('KINEMATICS:\n')

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

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

generateCubicConfiguration: Generate a Cubic Configuration

<<sec:generateCubicConfiguration>>

This Matlab function is accessible here.

Function description

  function [stewart] = generateCubicConfiguration(stewart, args)
  % generateCubicConfiguration - Generate a Cubic Configuration
  %
  % Syntax: [stewart] = generateCubicConfiguration(stewart, args)
  %
  % Inputs:
  %    - stewart - A structure with the following fields
  %        - geometry.H [1x1] - Total height of the platform [m]
  %    - args - Can have the following fields:
  %        - Hc  [1x1] - Height of the "useful" part of the cube [m]
  %        - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
  %        - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
  %        - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %        - platform_F.Fa  [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
  %        - platform_M.Mb  [3x6] - Its i'th column is the position vector of joint bi with respect to {M}

Documentation

Cubic Configuration

Optional Parameters

  arguments
      stewart
      args.Hc  (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
      args.FOc (1,1) double {mustBeNumeric} = 50e-3
      args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
      args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
  end

Check the stewart structure elements

  assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
  H = stewart.geometry.H;

Position of the Cube

We define the useful points of the cube with respect to the Cube's center. ${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}.

  sx = [ 2; -1; -1];
  sy = [ 0;  1; -1];
  sz = [ 1;  1;  1];

  R = [sx, sy, sz]./vecnorm([sx, sy, sz]);

  L = args.Hc*sqrt(3);

  Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];

  CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
  CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg

Compute the pose

We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).

  CSi = (CCm - CCf)./vecnorm(CCm - CCf);

We now which to compute the position of the joints $a_{i}$ and $b_{i}$.

  Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
  Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;

Populate the stewart structure

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

computeJacobian: Compute the Jacobian Matrix

<<sec:computeJacobian>>

This Matlab function is accessible here.

Function description

  function [stewart] = computeJacobian(stewart)
  % computeJacobian -
  %
  % Syntax: [stewart] = computeJacobian(stewart)
  %
  % Inputs:
  %    - stewart - With at least the following fields:
  %      - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
  %      - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
  %      - actuators.K [6x1] - Total stiffness of the actuators
  %
  % Outputs:
  %    - stewart - With the 3 added field:
  %        - kinematics.J [6x6] - The Jacobian Matrix
  %        - kinematics.K [6x6] - The Stiffness Matrix
  %        - kinematics.C [6x6] - The Compliance Matrix

Check the stewart structure elements

  assert(isfield(stewart.geometry, 'As'),   'stewart.geometry should have attribute As')
  As = stewart.geometry.As;

  assert(isfield(stewart.geometry, 'Ab'),   'stewart.geometry should have attribute Ab')
  Ab = stewart.geometry.Ab;

  assert(isfield(stewart.actuators, 'K'),   'stewart.actuators should have attribute K')
  Ki = stewart.actuators.K;

Compute Jacobian Matrix

  J = [As' , cross(Ab, As)'];

Compute Stiffness Matrix

  K = J'*diag(Ki)*J;

Compute Compliance Matrix

  C = inv(K);

Populate the stewart structure

  stewart.kinematics.J = J;
  stewart.kinematics.K = K;
  stewart.kinematics.C = C;

inverseKinematics: Compute Inverse Kinematics

<<sec:inverseKinematics>>

This Matlab function is accessible here.

Theory

For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.

From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as

\begin{align*} l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\ &= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i \end{align*}

To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself:

\begin{equation} l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right] \end{equation}

Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by:

\begin{equation} l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i} \end{equation}

If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation. Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.

Function description

  function [Li, dLi] = inverseKinematics(stewart, args)
  % inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
  %
  % Syntax: [stewart] = inverseKinematics(stewart)
  %
  % Inputs:
  %    - stewart - A structure with the following fields
  %        - geometry.Aa   [3x6] - The positions ai expressed in {A}
  %        - geometry.Bb   [3x6] - The positions bi expressed in {B}
  %        - geometry.l    [6x1] - Length of each strut
  %    - args - Can have the following fields:
  %        - AP   [3x1] - The wanted position of {B} with respect to {A}
  %        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
  %
  % Outputs:
  %    - Li   [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
  %    - dLi  [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}

Optional Parameters

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

Check the stewart structure elements

  assert(isfield(stewart.geometry, 'Aa'),   'stewart.geometry should have attribute Aa')
  Aa = stewart.geometry.Aa;

  assert(isfield(stewart.geometry, 'Bb'),   'stewart.geometry should have attribute Bb')
  Bb = stewart.geometry.Bb;

  assert(isfield(stewart.geometry, 'l'),   'stewart.geometry should have attribute l')
  l = stewart.geometry.l;

Compute

  Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));

  dLi = Li-l;

forwardKinematicsApprox: Compute the Approximate Forward Kinematics

<<sec:forwardKinematicsApprox>>

This Matlab function is accessible here.

Function description

  function [P, R] = forwardKinematicsApprox(stewart, args)
  % forwardKinematicsApprox - Computed the approximate pose of {B} with respect to {A} from the length of each strut and using
  %                           the Jacobian Matrix
  %
  % Syntax: [P, R] = forwardKinematicsApprox(stewart, args)
  %
  % Inputs:
  %    - stewart - A structure with the following fields
  %        - kinematics.J  [6x6] - The Jacobian Matrix
  %    - args - Can have the following fields:
  %        - dL [6x1] - Displacement of each strut [m]
  %
  % Outputs:
  %    - P  [3x1] - The estimated position of {B} with respect to {A}
  %    - R  [3x3] - The estimated rotation matrix that gives the orientation of {B} with respect to {A}

Optional Parameters

  arguments
      stewart
      args.dL (6,1) double {mustBeNumeric} = zeros(6,1)
  end

Check the stewart structure elements

  assert(isfield(stewart.kinematics, 'J'),   'stewart.kinematics should have attribute J')
  J = stewart.kinematics.J;

Computation

From a small displacement of each strut $d\bm{\mathcal{L}}$, we can compute the position and orientation of {B} with respect to {A} using the following formula: \[ d \bm{\mathcal{X}} = \bm{J}^{-1} d\bm{\mathcal{L}} \]

  X = J\args.dL;

The position vector corresponds to the first 3 elements.

  P = X(1:3);

The next 3 elements are the orientation of {B} with respect to {A} expressed using the screw axis.

  theta = norm(X(4:6));
  s = X(4:6)/theta;

We then compute the corresponding rotation matrix.

  R = [s(1)^2*(1-cos(theta)) + cos(theta) ,        s(1)*s(2)*(1-cos(theta)) - s(3)*sin(theta), s(1)*s(3)*(1-cos(theta)) + s(2)*sin(theta);
       s(2)*s(1)*(1-cos(theta)) + s(3)*sin(theta), s(2)^2*(1-cos(theta)) + cos(theta),         s(2)*s(3)*(1-cos(theta)) - s(1)*sin(theta);
       s(3)*s(1)*(1-cos(theta)) - s(2)*sin(theta), s(3)*s(2)*(1-cos(theta)) + s(1)*sin(theta), s(3)^2*(1-cos(theta)) + cos(theta)];