nass-simscape/simscape_subsystems/index.org

Subsystems used for the Simscape Models

• Introduction
• Ground
  • Simscape Model
  • Function description
  • Function content
• Granite
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Translation Stage
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Tilt Stage
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Spindle
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Micro Hexapod
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Center of gravity compensation
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Mirror
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Nano Hexapod
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Sample
  • Simscape Model
  • Function description
  • Optional Parameters
  • Function content
• Generate Reference Signals
  • Function Declaration and Documentation
  • Optional Parameters
  • Initialize Parameters
  • Translation Stage
  • Tilt Stage
  • Spindle
  • Micro Hexapod
  • Axis Compensation
  • Nano Hexapod
  • Save
• Initialize Disturbances
  • Function Declaration and Documentation
  • Optional Parameters
  • Load Data
  • Parameters
  • Ground Motion
  • Translation Stage - X direction
  • Translation Stage - Z direction
  • Spindle - Z direction
  • Direct Forces
  • Set initial value to zero
  • Save
• Z-Axis Geophone
• Z-Axis Accelerometer

Introduction   ignore

The full Simscape Model is represented in Figure fig:simscape_picture.

Screenshot of the Multi-Body Model representation

This model is divided into multiple subsystems that are independent. These subsystems are saved in separate files and imported in the main file using a block balled "subsystem reference".

Each stage is configured (geometry, mass properties, dynamic properties …) using one function.

These functions are defined below.

Ground

<<sec:initializeGround>>

Simscape Model

The model of the Ground is composed of:

• A Cartesian joint that is used to simulation the ground motion
• A solid that represents the ground on which the granite is located

Simscape model for the Ground

Simscape picture for the Ground

Function description

  function [ground] = initializeGround()

Function content

First, we initialize the granite structure.

  ground = struct();

We set the shape and density of the ground solid element.

  ground.shape = [2, 2, 0.5]; % [m]
  ground.density = 2800; % [kg/m3]

The ground structure is saved.

  save('./mat/stages.mat', 'ground', '-append');

Granite

<<sec:initializeGranite>>

Simscape Model

The Simscape model of the granite is composed of:

• A cartesian joint such that the granite can vibrations along x, y and z axis
• A solid

The output sample_pos corresponds to the impact point of the X-ray.

Simscape model for the Granite

Simscape picture for the Granite

Function description

  function [granite] = initializeGranite(args)

Optional Parameters

   arguments
       args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3]
       args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m]
       args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m]
       args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m]
   end

Function content

First, we initialize the granite structure.

  granite = struct();

Properties of the Material and link to the geometry of the granite.

  granite.density = args.density; % [kg/m3]
  granite.STEP    = './STEPS/granite/granite.STEP';

Stiffness of the connection with Ground.

  granite.k.x = 4e9; % [N/m]
  granite.k.y = 3e8; % [N/m]
  granite.k.z = 8e8; % [N/m]

Damping of the connection with Ground.

  granite.c.x  = 4.0e5; % [N/(m/s)]
  granite.c.y  = 1.1e5; % [N/(m/s)]
  granite.c.z  = 9.0e5; % [N/(m/s)]

Equilibrium position of the Cartesian joint.

  granite.x0 = args.x0;
  granite.y0 = args.y0;
  granite.z0 = args.z0;

Z-offset for the initial position of the sample with respect to the granite top surface.

  granite.sample_pos = 0.8; % [m]

The granite structure is saved.

  save('./mat/stages.mat', 'granite', '-append');

Translation Stage

<<sec:initializeTy>>

Simscape Model

The Simscape model of the Translation stage consist of:

• One rigid body for the fixed part of the translation stage
• One rigid body for the moving part of the translation stage
• Four 6-DOF Joints that only have some rigidity in the X and Z directions. The rigidity in rotation comes from the fact that we use multiple joints that are located at different points
• One 6-DOF joint that represent the Actuator. It is used to impose the motion in the Y direction
• One 6-DOF joint to inject force disturbance in the X and Z directions

Simscape model for the Translation Stage

Simscape picture for the Translation Stage

Function description

  function [ty] = initializeTy(args)

Optional Parameters

  arguments
      args.x11 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z11 (1,1) double {mustBeNumeric} = 0 % [m]
      args.x21 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z21 (1,1) double {mustBeNumeric} = 0 % [m]
      args.x12 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z12 (1,1) double {mustBeNumeric} = 0 % [m]
      args.x22 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z22 (1,1) double {mustBeNumeric} = 0 % [m]
  end

Function content

First, we initialize the ty structure.

  ty = struct();

Define the density of the materials as well as the geometry (STEP files).

  % Ty Granite frame
  ty.granite_frame.density = 7800; % [kg/m3] => 43kg
  ty.granite_frame.STEP    = './STEPS/Ty/Ty_Granite_Frame.STEP';

  % Guide Translation Ty
  ty.guide.density         = 7800; % [kg/m3] => 76kg
  ty.guide.STEP            = './STEPS/ty/Ty_Guide.STEP';

  % Ty - Guide_Translation12
  ty.guide12.density       = 7800; % [kg/m3]
  ty.guide12.STEP          = './STEPS/Ty/Ty_Guide_12.STEP';

  % Ty - Guide_Translation11
  ty.guide11.density       = 7800; % [kg/m3]
  ty.guide11.STEP          = './STEPS/ty/Ty_Guide_11.STEP';

  % Ty - Guide_Translation22
  ty.guide22.density       = 7800; % [kg/m3]
  ty.guide22.STEP          = './STEPS/ty/Ty_Guide_22.STEP';

  % Ty - Guide_Translation21
  ty.guide21.density       = 7800; % [kg/m3]
  ty.guide21.STEP          = './STEPS/Ty/Ty_Guide_21.STEP';

  % Ty - Plateau translation
  ty.frame.density         = 7800; % [kg/m3]
  ty.frame.STEP            = './STEPS/ty/Ty_Stage.STEP';

  % Ty Stator Part
  ty.stator.density        = 5400; % [kg/m3]
  ty.stator.STEP           = './STEPS/ty/Ty_Motor_Stator.STEP';

  % Ty Rotor Part
  ty.rotor.density         = 5400; % [kg/m3]
  ty.rotor.STEP            = './STEPS/ty/Ty_Motor_Rotor.STEP';

Stiffness of the stage.

  ty.k.ax  = 5e8; % Axial Stiffness for each of the 4 guidance (y) [N/m]
  ty.k.rad = 5e7; % Radial Stiffness for each of the 4 guidance (x-z) [N/m]

Damping of the stage.

  ty.c.ax  = 70710; % [N/(m/s)]
  ty.c.rad = 22360; % [N/(m/s)]

Equilibrium position of the joints.

  ty.x0_11 = args.x11;
  ty.z0_11 = args.z11;
  ty.x0_12 = args.x12;
  ty.z0_12 = args.z12;
  ty.x0_21 = args.x21;
  ty.z0_21 = args.z21;
  ty.x0_22 = args.x22;
  ty.z0_22 = args.z22;

The ty structure is saved.

  save('./mat/stages.mat', 'ty', '-append');

Tilt Stage

<<sec:initializeRy>>

Simscape Model

The Simscape model of the Tilt stage is composed of:

• Two solid bodies for the two part of the stage
• Four 6-DOF joints to model the flexibility of the stage. These joints are virtually located along the rotation axis and are connecting the two solid bodies. These joints have some translation stiffness in the u-v-w directions aligned with the joint. The stiffness in rotation between the two solids is due to the fact that the 4 joints are connecting the two solids are different locations
• A Bushing Joint used for the Actuator. The Ry motion is imposed by the input.

Simscape model for the Tilt Stage

Simscape picture for the Tilt Stage

Function description

  function [ry] = initializeRy(args)

Optional Parameters

  arguments
      args.x11 (1,1) double {mustBeNumeric} = 0 % [m]
      args.y11 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z11 (1,1) double {mustBeNumeric} = 0 % [m]
      args.x12 (1,1) double {mustBeNumeric} = 0 % [m]
      args.y12 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z12 (1,1) double {mustBeNumeric} = 0 % [m]
      args.x21 (1,1) double {mustBeNumeric} = 0 % [m]
      args.y21 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z21 (1,1) double {mustBeNumeric} = 0 % [m]
      args.x22 (1,1) double {mustBeNumeric} = 0 % [m]
      args.y22 (1,1) double {mustBeNumeric} = 0 % [m]
      args.z22 (1,1) double {mustBeNumeric} = 0 % [m]
  end

Function content

First, we initialize the ry structure.

  ry = struct();

Properties of the Material and link to the geometry of the Tilt stage.

  % Ry - Guide for the tilt stage
  ry.guide.density = 7800; % [kg/m3]
  ry.guide.STEP    = './STEPS/ry/Tilt_Guide.STEP';

  % Ry - Rotor of the motor
  ry.rotor.density = 2400; % [kg/m3]
  ry.rotor.STEP    = './STEPS/ry/Tilt_Motor_Axis.STEP';

  % Ry - Motor
  ry.motor.density = 3200; % [kg/m3]
  ry.motor.STEP    = './STEPS/ry/Tilt_Motor.STEP';

  % Ry - Plateau Tilt
  ry.stage.density = 7800; % [kg/m3]
  ry.stage.STEP    = './STEPS/ry/Tilt_Stage.STEP';

Stiffness of the stage.

  ry.k.tilt = 1e4; % Rotation stiffness around y [N*m/deg]
  ry.k.h    = 1e8; % Stiffness in the direction of the guidance [N/m]
  ry.k.rad  = 1e8; % Stiffness in the top direction [N/m]
  ry.k.rrad = 1e8; % Stiffness in the side direction [N/m]

Damping of the stage.

  ry.c.tilt = 2.8e2;
  ry.c.h    = 2.8e4;
  ry.c.rad  = 2.8e4;
  ry.c.rrad = 2.8e4;

Equilibrium position of the joints.

  ry.x0_11 = args.x11;
  ry.y0_11 = args.y11;
  ry.z0_11 = args.z11;
  ry.x0_12 = args.x12;
  ry.y0_12 = args.y12;
  ry.z0_12 = args.z12;
  ry.x0_21 = args.x21;
  ry.y0_21 = args.y21;
  ry.z0_21 = args.z21;
  ry.x0_22 = args.x22;
  ry.y0_22 = args.y22;
  ry.z0_22 = args.z22;

Z-Offset so that the center of rotation matches the sample center;

  ry.z_offset = 0.58178; % [m]

The ty structure is saved.

  save('./mat/stages.mat', 'ry', '-append');

Spindle

<<sec:initializeRz>>

Simscape Model

The Simscape model of the Spindle is composed of:

• Two rigid bodies: the stator and the rotor
• A Bushing Joint that is used both as the actuator (the Rz motion is imposed by the input) and as the force perturbation in the Z direction.
• The Bushing joint has some flexibility in the X-Y-Z directions as well as in Rx and Ry rotations

Simscape model for the Spindle

Simscape picture for the Spindle

Function description

  function [rz] = initializeRz(args)

Optional Parameters

  arguments
      args.x0  (1,1) double {mustBeNumeric} = 0 % Equilibrium position of the Joint [m]
      args.y0  (1,1) double {mustBeNumeric} = 0 % Equilibrium position of the Joint [m]
      args.z0  (1,1) double {mustBeNumeric} = 0 % Equilibrium position of the Joint [m]
      args.rx0 (1,1) double {mustBeNumeric} = 0 % Equilibrium position of the Joint [rad]
      args.ry0 (1,1) double {mustBeNumeric} = 0 % Equilibrium position of the Joint [rad]
  end

Function content

First, we initialize the rz structure.

  rz = struct();

Properties of the Material and link to the geometry of the spindle.

  % Spindle - Slip Ring
  rz.slipring.density = 7800; % [kg/m3]
  rz.slipring.STEP    = './STEPS/rz/Spindle_Slip_Ring.STEP';

  % Spindle - Rotor
  rz.rotor.density    = 7800; % [kg/m3]
  rz.rotor.STEP       = './STEPS/rz/Spindle_Rotor.STEP';

  % Spindle - Stator
  rz.stator.density   = 7800; % [kg/m3]
  rz.stator.STEP      = './STEPS/rz/Spindle_Stator.STEP';

Stiffness of the stage.

  rz.k.rot  = 1e6; % TODO - Rotational Stiffness (Rz) [N*m/deg]
  rz.k.tilt = 1e6; % Rotational Stiffness (Rx, Ry) [N*m/deg]
  rz.k.ax   = 2e9; % Axial Stiffness (Z) [N/m]
  rz.k.rad  = 7e8; % Radial Stiffness (X, Y) [N/m]

Damping of the stage.

  rz.c.rot  = 1.6e3;
  rz.c.tilt = 1.6e3;
  rz.c.ax   = 7.1e4;
  rz.c.rad  = 4.2e4;

Equilibrium position of the joints.

  rz.x0 = args.x0;
  rz.y0 = args.y0;
  rz.z0 = args.z0;
  rz.rx0 = args.rx0;
  rz.ry0 = args.ry0;

The rz structure is saved.

  save('./mat/stages.mat', 'rz', '-append');

Micro Hexapod

<<sec:initializeMicroHexapod>>

Simscape Model

Simscape model for the Micro-Hexapod

Simscape picture for the Micro-Hexapod

Function description

  function [micro_hexapod] = initializeMicroHexapod(args)

Optional Parameters

  arguments
      % initializeFramesPositions
      args.H    (1,1) double {mustBeNumeric, mustBePositive} = 350e-3
      args.MO_B (1,1) double {mustBeNumeric} = 270e-3
      % generateGeneralConfiguration
      args.FH  (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
      args.FR  (1,1) double {mustBeNumeric, mustBePositive} = 175.5e-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} = 45e-3
      args.MR  (1,1) double {mustBeNumeric, mustBePositive} = 118e-3
      args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180)
      % initializeStrutDynamics
      args.Ki (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e7*ones(6,1)
      args.Ci (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.4e3*ones(6,1)
      % initializeCylindricalPlatforms
      args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 10
      args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
      args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 207.5e-3
      args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 10
      args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
      args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3
      % initializeCylindricalStruts
      args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1
      args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
      args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
      args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1
      args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
      args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
      % inverseKinematics
      args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
      args.ARB (3,3) double {mustBeNumeric} = eye(3)
      % Equilibrium position of each leg
      args.dLeq (6,1) double {mustBeNumeric} = zeros(6,1)
  end

Function content

  micro_hexapod = initializeFramesPositions('H', args.H, 'MO_B', args.MO_B);
  micro_hexapod = generateGeneralConfiguration(micro_hexapod, 'FH', args.FH, 'FR', args.FR, 'FTh', args.FTh, 'MH', args.MH, 'MR', args.MR, 'MTh', args.MTh);
  micro_hexapod = computeJointsPose(micro_hexapod);
  micro_hexapod = initializeStrutDynamics(micro_hexapod, 'Ki', args.Ki, 'Ci', args.Ci);
  micro_hexapod = initializeCylindricalPlatforms(micro_hexapod, 'Fpm', args.Fpm, 'Fph', args.Fph, 'Fpr', args.Fpr, 'Mpm', args.Mpm, 'Mph', args.Mph, 'Mpr', args.Mpr);
  micro_hexapod = initializeCylindricalStruts(micro_hexapod, 'Fsm', args.Fsm, 'Fsh', args.Fsh, 'Fsr', args.Fsr, 'Msm', args.Msm, 'Msh', args.Msh, 'Msr', args.Msr);
  micro_hexapod = computeJacobian(micro_hexapod);
  [Li, dLi] = inverseKinematics(micro_hexapod, 'AP', args.AP, 'ARB', args.ARB);
  micro_hexapod.Li = Li;
  micro_hexapod.dLi = dLi;

Equilibrium position of the each joint.

  micro_hexapod.dLeq = args.dLeq;

The micro_hexapod structure is saved.

  save('./mat/stages.mat', 'micro_hexapod', '-append');

Center of gravity compensation

<<sec:initializeAxisc>>

Simscape Model

The Simscape model of the Center of gravity compensator is composed of:

• One main solid that is connected to two other solids (the masses to position of center of mass) through two revolute joints
• The angle of both revolute joints is set by the input

Simscape model for the Center of Mass compensation system

Simscape picture for the Center of Mass compensation system

Function description

  function [axisc] = initializeAxisc()

Optional Parameters

Function content

First, we initialize the axisc structure.

  axisc = struct();

Properties of the Material and link to the geometry files.

  % Structure
  axisc.structure.density = 3400; % [kg/m3]
  axisc.structure.STEP    = './STEPS/axisc/axisc_structure.STEP';

  % Wheel
  axisc.wheel.density     = 2700; % [kg/m3]
  axisc.wheel.STEP        = './STEPS/axisc/axisc_wheel.STEP';

  % Mass
  axisc.mass.density      = 7800; % [kg/m3]
  axisc.mass.STEP         = './STEPS/axisc/axisc_mass.STEP';

  % Gear
  axisc.gear.density      = 7800; % [kg/m3]
  axisc.gear.STEP         = './STEPS/axisc/axisc_gear.STEP';

The axisc structure is saved.

  save('./mat/stages.mat', 'axisc', '-append');

Mirror

<<sec:initializeMirror>>

Simscape Model

The Simscape Model of the mirror is just a solid body. The output mirror_center corresponds to the center of the Sphere and is the point of measurement for the metrology

Simscape model for the Mirror

Simscape picture for the Mirror

Function description

  function [] = initializeMirror(args)

Optional Parameters

  arguments
      args.shape       char   {mustBeMember(args.shape,{'spherical', 'conical'})} = 'spherical'
      args.angle (1,1) double {mustBeNumeric, mustBePositive} = 45 % [deg]
  end

Function content

First, we initialize the mirror structure.

  mirror = struct();

We define the geometrical values.

  mirror.h = 50; % Height of the mirror [mm]
  mirror.thickness = 25; % Thickness of the plate supporting the sample [mm]
  mirror.hole_rad = 120; % radius of the hole in the mirror [mm]
  mirror.support_rad = 100; % radius of the support plate [mm]
  mirror.jacobian = 150; % point of interest offset in z (above the top surfave) [mm]
  mirror.rad = 180; % radius of the mirror (at the bottom surface) [mm]

  mirror.density = 2400; % Density of the material [kg/m3]

  mirror.cone_length = mirror.rad*tand(args.angle)+mirror.h+mirror.jacobian; % Distance from Apex point of the cone to jacobian point

Now we define the Shape of the mirror. We first start with the internal part.

  mirror.shape = [...
      0 mirror.h-mirror.thickness
      mirror.hole_rad mirror.h-mirror.thickness; ...
      mirror.hole_rad 0; ...
      mirror.rad 0 ...
  ];

Then, we define the reflective used part of the mirror.

  if strcmp(args.shape, 'spherical')
      mirror.sphere_radius = sqrt((mirror.jacobian+mirror.h)^2+mirror.rad^2); % Radius of the sphere [mm]

      for z = linspace(0, mirror.h, 101)
          mirror.shape = [mirror.shape; sqrt(mirror.sphere_radius^2-(z-mirror.jacobian-mirror.h)^2) z];
      end
  elseif strcmp(args.shape, 'conical')
      mirror.shape = [mirror.shape; mirror.rad+mirror.h/tand(args.angle) mirror.h];
  else
      error('Shape should be either conical or spherical');
  end

Finally, we close the shape.

  mirror.shape = [mirror.shape; 0 mirror.h];

The mirror structure is saved.

  save('./mat/stages.mat', 'mirror', '-append');

Nano Hexapod

<<sec:initializeNanoHexapod>>

Simscape Model

Simscape model for the Nano Hexapod

Simscape picture for the Nano Hexapod

Function description

  function [nano_hexapod] = initializeNanoHexapod(args)

Optional Parameters

  arguments
      % initializeFramesPositions
      args.H    (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
      args.MO_B (1,1) double {mustBeNumeric} = 175e-3
      % generateGeneralConfiguration
      args.FH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
      args.FR  (1,1) double {mustBeNumeric, mustBePositive} = 100e-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)
      % initializeStrutDynamics
      args.actuator  char   {mustBeMember(args.actuator,{'piezo', 'lorentz'})} = 'piezo'
      % initializeCylindricalPlatforms
      args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
      args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
      args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-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
      % initializeCylindricalStruts
      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
      % inverseKinematics
      args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
      args.ARB (3,3) double {mustBeNumeric} = eye(3)
      % Equilibrium position of each leg
      args.dLeq (6,1) double {mustBeNumeric} = zeros(6,1)
  end

Function content

  nano_hexapod = initializeFramesPositions('H', args.H, 'MO_B', args.MO_B);
  nano_hexapod = generateGeneralConfiguration(nano_hexapod, 'FH', args.FH, 'FR', args.FR, 'FTh', args.FTh, 'MH', args.MH, 'MR', args.MR, 'MTh', args.MTh);
  nano_hexapod = computeJointsPose(nano_hexapod);
  if strcmp(args.actuator, 'piezo')
      nano_hexapod = initializeStrutDynamics(nano_hexapod, 'Ki', 1e7*ones(6,1), 'Ci', 1e2*ones(6,1));
  elseif strcmp(args.actuator, 'lorentz')
      nano_hexapod = initializeStrutDynamics(nano_hexapod, 'Ki', 1e4*ones(6,1), 'Ci', 1e2*ones(6,1));
  else
      error('args.actuator should be piezo or lorentz');
  end
  nano_hexapod = initializeCylindricalPlatforms(nano_hexapod, 'Fpm', args.Fpm, 'Fph', args.Fph, 'Fpr', args.Fpr, 'Mpm', args.Mpm, 'Mph', args.Mph, 'Mpr', args.Mpr);
  nano_hexapod = initializeCylindricalStruts(nano_hexapod, 'Fsm', args.Fsm, 'Fsh', args.Fsh, 'Fsr', args.Fsr, 'Msm', args.Msm, 'Msh', args.Msh, 'Msr', args.Msr);
  nano_hexapod = computeJacobian(nano_hexapod);
  [Li, dLi] = inverseKinematics(nano_hexapod, 'AP', args.AP, 'ARB', args.ARB);
  nano_hexapod.Li = Li;
  nano_hexapod.dLi = dLi;

  nano_hexapod.dLeq = args.dLeq;

  save('./mat/stages.mat', 'nano_hexapod', '-append');

Sample

<<sec:initializeSample>>

Simscape Model

The Simscape model of the sample environment is composed of:

• A rigid transform that can be used to translate the sample (position offset)
• A cartesian joint to add some flexibility to the sample environment mount
• A solid that represent the sample
• An input is added to apply some external forces and torques at the center of the sample environment. This could be the case for cable forces for instance.

Simscape model for the Sample

Simscape picture for the Sample

Function description

  function [sample] = initializeSample(args)

Optional Parameters

  arguments
      args.radius (1,1) double {mustBeNumeric, mustBePositive} = 0.1 % [m]
      args.height (1,1) double {mustBeNumeric, mustBePositive} = 0.3 % [m]
      args.mass   (1,1) double {mustBeNumeric, mustBePositive} = 50 % [kg]
      args.freq   (1,1) double {mustBeNumeric, mustBePositive} = 100 % [Hz]
      args.offset (1,1) double {mustBeNumeric} = 0 % [m]
      args.x0     (1,1) double {mustBeNumeric} = 0 % [m]
      args.y0     (1,1) double {mustBeNumeric} = 0 % [m]
      args.z0     (1,1) double {mustBeNumeric} = 0 % [m]
  end

Function content

First, we initialize the sample structure.

  sample = struct();

We define the geometrical parameters of the sample as well as its mass and position.

  sample.radius = args.radius; % [m]
  sample.height = args.height; % [m]
  sample.mass = args.mass; % [kg]
  sample.offset = args.offset; % [m]

Stiffness of the sample fixation.

  sample.k.x = sample.mass * (2*pi * args.freq)^2; % [N/m]
  sample.k.y = sample.mass * (2*pi * args.freq)^2; % [N/m]
  sample.k.z = sample.mass * (2*pi * args.freq)^2; % [N/m]

Damping of the sample fixation.

  sample.c.x = 0.1*sqrt(sample.k.x*sample.mass); % [N/(m/s)]
  sample.c.y = 0.1*sqrt(sample.k.y*sample.mass); % [N/(m/s)]
  sample.c.z = 0.1*sqrt(sample.k.z*sample.mass); % [N/(m/s)]

Equilibrium position of the Cartesian joint corresponding to the sample fixation.

  sample.x0 = args.x0; % [m]
  sample.y0 = args.y0; % [m]
  sample.z0 = args.z0; % [m]

The sample structure is saved.

  save('./mat/stages.mat', 'sample', '-append');

Generate Reference Signals

<<sec:initializeReferences>>

Function Declaration and Documentation

  function [ref] = initializeReferences(args)

Optional Parameters

  arguments
      % Sampling Frequency [s]
      args.Ts           (1,1) double {mustBeNumeric, mustBePositive} = 1e-3
      % Maximum simulation time [s]
      args.Tmax         (1,1) double {mustBeNumeric, mustBePositive} = 100
      % Either "constant" / "triangular" / "sinusoidal"
      args.Dy_type      char {mustBeMember(args.Dy_type,{'constant', 'triangular', 'sinusoidal'})} = 'constant'
      % Amplitude of the displacement [m]
      args.Dy_amplitude (1,1) double {mustBeNumeric} = 0
      % Period of the displacement [s]
      args.Dy_period    (1,1) double {mustBeNumeric, mustBePositive} = 1
      % Either "constant" / "triangular" / "sinusoidal"
      args.Ry_type      char {mustBeMember(args.Ry_type,{'constant', 'triangular', 'sinusoidal'})} = 'constant'
      % Amplitude [rad]
      args.Ry_amplitude (1,1) double {mustBeNumeric} = 0
      % Period of the displacement [s]
      args.Ry_period    (1,1) double {mustBeNumeric, mustBePositive} = 1
      % Either "constant" / "rotating"
      args.Rz_type      char {mustBeMember(args.Rz_type,{'constant', 'rotating'})} = 'constant'
      % Initial angle [rad]
      args.Rz_amplitude (1,1) double {mustBeNumeric} = 0
      % Period of the rotating [s]
      args.Rz_period    (1,1) double {mustBeNumeric, mustBePositive} = 1
      % For now, only constant is implemented
      args.Dh_type      char {mustBeMember(args.Dh_type,{'constant'})} = 'constant'
      % Initial position [m,m,m,rad,rad,rad] of the top platform (Pitch-Roll-Yaw Euler angles)
      args.Dh_pos       (6,1) double {mustBeNumeric} = zeros(6, 1), ...
      % For now, only constant is implemented
      args.Rm_type      char {mustBeMember(args.Rm_type,{'constant'})} = 'constant'
      % Initial position of the two masses
      args.Rm_pos       (2,1) double {mustBeNumeric} = [0; pi]
      % For now, only constant is implemented
      args.Dn_type      char {mustBeMember(args.Dn_type,{'constant'})} = 'constant'
      % Initial position [m,m,m,rad,rad,rad] of the top platform
      args.Dn_pos       (6,1) double {mustBeNumeric} = zeros(6,1)
  end

Initialize Parameters

  %% Set Sampling Time
  Ts = args.Ts;
  Tmax = args.Tmax;

  %% Low Pass Filter to filter out the references
  s = zpk('s');
  w0 = 2*pi*10;
  xi = 1;
  H_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2);

Translation Stage

      %% Translation stage - Dy
      t = 0:Ts:Tmax; % Time Vector [s]
      Dy   = zeros(length(t), 1);
      Dyd  = zeros(length(t), 1);
      Dydd = zeros(length(t), 1);
      switch args.Dy_type
        case 'constant'
          Dy(:) = args.Dy_amplitude;
          Dyd(:)   = 0;
          Dydd(:)  = 0;
        case 'triangular'
          % This is done to unsure that we start with no displacement
          Dy_raw = args.Dy_amplitude*sawtooth(2*pi*t/args.Dy_period,1/2);
          i0 = find(t>=args.Dy_period/4,1);
          Dy(1:end-i0+1) = Dy_raw(i0:end);
          Dy(end-i0+2:end) = Dy_raw(end); % we fix the last value

          % The signal is filtered out
          Dy   = lsim(H_lpf,     Dy, t);
          Dyd  = lsim(H_lpf*s,   Dy, t);
          Dydd = lsim(H_lpf*s^2, Dy, t);
        case 'sinusoidal'
          Dy(:) = args.Dy_amplitude*sin(2*pi/args.Dy_period*t);
          Dyd   = args.Dy_amplitude*2*pi/args.Dy_period*cos(2*pi/args.Dy_period*t);
          Dydd  = -args.Dy_amplitude*(2*pi/args.Dy_period)^2*sin(2*pi/args.Dy_period*t);
        otherwise
          warning('Dy_type is not set correctly');
      end

      Dy = struct('time', t, 'signals', struct('values', Dy), 'deriv', Dyd, 'dderiv', Dydd);

Tilt Stage

      %% Tilt Stage - Ry
      t = 0:Ts:Tmax; % Time Vector [s]
      Ry   = zeros(length(t), 1);
      Ryd  = zeros(length(t), 1);
      Rydd = zeros(length(t), 1);

      switch args.Ry_type
        case 'constant'
          Ry(:) = args.Ry_amplitude;
          Ryd(:)   = 0;
          Rydd(:)  = 0;
        case 'triangular'
          Ry_raw = args.Ry_amplitude*sawtooth(2*pi*t/args.Ry_period,1/2);
          i0 = find(t>=args.Ry_period/4,1);
          Ry(1:end-i0+1) = Ry_raw(i0:end);
          Ry(end-i0+2:end) = Ry_raw(end); % we fix the last value

          % The signal is filtered out
          Ry   = lsim(H_lpf,     Ry, t);
          Ryd  = lsim(H_lpf*s,   Ry, t);
          Rydd = lsim(H_lpf*s^2, Ry, t);
        case 'sinusoidal'
          Ry(:) = args.Ry_amplitude*sin(2*pi/args.Ry_period*t);

          Ryd  = args.Ry_amplitude*2*pi/args.Ry_period*cos(2*pi/args.Ry_period*t);
          Rydd = -args.Ry_amplitude*(2*pi/args.Ry_period)^2*sin(2*pi/args.Ry_period*t);
        otherwise
          warning('Ry_type is not set correctly');
      end

      Ry = struct('time', t, 'signals', struct('values', Ry), 'deriv', Ryd, 'dderiv', Rydd);

Spindle

  %% Spindle - Rz
  t = 0:Ts:Tmax; % Time Vector [s]
  Rz   = zeros(length(t), 1);
  Rzd  = zeros(length(t), 1);
  Rzdd = zeros(length(t), 1);

  switch args.Rz_type
    case 'constant'
      Rz(:)   = args.Rz_amplitude;
      Rzd(:)  = 0;
      Rzdd(:) = 0;
    case 'rotating'
      Rz(:) = 2*pi/args.Rz_period*t;

      % The signal is filtered out
      Rz   = lsim(H_lpf,     Rz, t);
      Rzd  = lsim(H_lpf*s,   Rz, t);
      Rzdd = lsim(H_lpf*s^2, Rz, t);

      % We add the angle offset
      Rz = Rz + args.Rz_amplitude;
    otherwise
      warning('Rz_type is not set correctly');
  end

  Rz = struct('time', t, 'signals', struct('values', Rz), 'deriv', Rzd, 'dderiv', Rzdd);

Micro Hexapod

  %% Micro-Hexapod
  t = [0, Ts];
  Dh = zeros(length(t), 6);
  Dhl = zeros(length(t), 6);

  switch args.Dh_type
    case 'constant'
      Dh = [args.Dh_pos, args.Dh_pos];

      load('mat/stages.mat', 'micro_hexapod');

      AP = [args.Dh_pos(1) ; args.Dh_pos(2) ; args.Dh_pos(3)];

      tx = args.Dh_pos(4);
      ty = args.Dh_pos(5);
      tz = args.Dh_pos(6);

      ARB = [cos(tz) -sin(tz) 0;
             sin(tz)  cos(tz) 0;
             0        0       1]*...
            [ cos(ty) 0 sin(ty);
              0       1 0;
             -sin(ty) 0 cos(ty)]*...
            [1 0        0;
             0 cos(tx) -sin(tx);
             0 sin(tx)  cos(tx)];

      [~, Dhl] = inverseKinematics(micro_hexapod, 'AP', AP, 'ARB', ARB);
      Dhl = [Dhl, Dhl];
    otherwise
      warning('Dh_type is not set correctly');
  end

  Dh = struct('time', t, 'signals', struct('values', Dh));
  Dhl = struct('time', t, 'signals', struct('values', Dhl));

Axis Compensation

      %% Axis Compensation - Rm
      t = [0, Ts];

      Rm = [args.Rm_pos, args.Rm_pos];
      Rm = struct('time', t, 'signals', struct('values', Rm));

Nano Hexapod

      %% Nano-Hexapod
      t = [0, Ts];
      Dn = zeros(length(t), 6);

      switch args.Dn_type
        case 'constant'
          Dn = [args.Dn_pos, args.Dn_pos];
        otherwise
          warning('Dn_type is not set correctly');
      end

      Dn = struct('time', t, 'signals', struct('values', Dn));

Save

      %% Save
      save('mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'Rm', 'Dn', 'Ts');
  end

Initialize Disturbances

<<sec:initializeDisturbances>>

Function Declaration and Documentation

  function [] = initializeDisturbances(args)
  % initializeDisturbances - Initialize the disturbances
  %
  % Syntax: [] = initializeDisturbances(args)
  %
  % Inputs:
  %    - args -

Optional Parameters

  arguments
      % Global parameter to enable or disable the disturbances
      args.enable logical {mustBeNumericOrLogical} = true
      % Ground Motion - X direction
      args.Dwx logical {mustBeNumericOrLogical} = true
      % Ground Motion - Y direction
      args.Dwy logical {mustBeNumericOrLogical} = true
      % Ground Motion - Z direction
      args.Dwz logical {mustBeNumericOrLogical} = true
      % Translation Stage - X direction
      args.Fty_x logical {mustBeNumericOrLogical} = true
      % Translation Stage - Z direction
      args.Fty_z logical {mustBeNumericOrLogical} = true
      % Spindle - Z direction
      args.Frz_z logical {mustBeNumericOrLogical} = true
  end

Load Data

  load('./disturbances/mat/dist_psd.mat', 'dist_f');

We remove the first frequency point that usually is very large.

Parameters

We define some parameters that will be used in the algorithm.

  Fs = 2*dist_f.f(end);      % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
  N  = 2*length(dist_f.f);   % Number of Samples match the one of the wanted PSD
  T0 = N/Fs;                 % Signal Duration [s]
  df = 1/T0;                 % Frequency resolution of the DFT [Hz]
                             % Also equal to (dist_f.f(2)-dist_f.f(1))
  t = linspace(0, T0, N+1)'; % Time Vector [s]
  Ts = 1/Fs;                 % Sampling Time [s]

Ground Motion

  phi = dist_f.psd_gm;
  C = zeros(N/2,1);
  for i = 1:N/2
    C(i) = sqrt(phi(i)*df);
  end

  if args.Dwx && args.enable
    theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
    Cx = [0 ; C.*complex(cos(theta),sin(theta))];
    Cx = [Cx; flipud(conj(Cx(2:end)))];;
    Dwx = N/sqrt(2)*ifft(Cx); % Ground Motion - x direction [m]
  else
    Dwx = zeros(length(t), 1);
  end

  if args.Dwy && args.enable
    theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
    Cx = [0 ; C.*complex(cos(theta),sin(theta))];
    Cx = [Cx; flipud(conj(Cx(2:end)))];;
    Dwy = N/sqrt(2)*ifft(Cx); % Ground Motion - y direction [m]
  else
    Dwy = zeros(length(t), 1);
  end

  if args.Dwy && args.enable
    theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
    Cx = [0 ; C.*complex(cos(theta),sin(theta))];
    Cx = [Cx; flipud(conj(Cx(2:end)))];;
    Dwz = N/sqrt(2)*ifft(Cx); % Ground Motion - z direction [m]
  else
    Dwz = zeros(length(t), 1);
  end

Translation Stage - X direction

  if args.Fty_x && args.enable
    phi = dist_f.psd_ty; % TODO - we take here the vertical direction which is wrong but approximate
    C = zeros(N/2,1);
    for i = 1:N/2
      C(i) = sqrt(phi(i)*df);
    end
    theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
    Cx = [0 ; C.*complex(cos(theta),sin(theta))];
    Cx = [Cx; flipud(conj(Cx(2:end)))];;
    u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty x [N]
    Fty_x = u;
  else
    Fty_x = zeros(length(t), 1);
  end

Translation Stage - Z direction

  if args.Fty_z && args.enable
    phi = dist_f.psd_ty;
    C = zeros(N/2,1);
    for i = 1:N/2
      C(i) = sqrt(phi(i)*df);
    end
    theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
    Cx = [0 ; C.*complex(cos(theta),sin(theta))];
    Cx = [Cx; flipud(conj(Cx(2:end)))];;
    u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty z [N]
    Fty_z = u;
  else
    Fty_z = zeros(length(t), 1);
  end

Spindle - Z direction

  if args.Frz_z && args.enable
    phi = dist_f.psd_rz;
    C = zeros(N/2,1);
    for i = 1:N/2
      C(i) = sqrt(phi(i)*df);
    end
    theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
    Cx = [0 ; C.*complex(cos(theta),sin(theta))];
    Cx = [Cx; flipud(conj(Cx(2:end)))];;
    u = N/sqrt(2)*ifft(Cx); % Disturbance Force Rz z [N]
    Frz_z = u;
  else
    Frz_z = zeros(length(t), 1);
  end

Direct Forces

  u = zeros(length(t), 6);
  Fd = u;

Set initial value to zero

  Dwx    = Dwx   - Dwx(1);
  Dwy    = Dwy   - Dwy(1);
  Dwz    = Dwz   - Dwz(1);
  Fty_x  = Fty_x - Fty_x(1);
  Fty_z  = Fty_z - Fty_z(1);
  Frz_z  = Frz_z - Frz_z(1);

Save

  save('mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't');

Z-Axis Geophone

<<sec:initializeZAxisGeophone>>

  function [geophone] = initializeZAxisGeophone(args)
      arguments
          args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % [kg]
          args.freq (1,1) double {mustBeNumeric, mustBePositive} = 1    % [Hz]
      end

      %%
      geophone.m = args.mass;

      %% The Stiffness is set to have the damping resonance frequency
      geophone.k = geophone.m * (2*pi*args.freq)^2;

      %% We set the damping value to have critical damping
      geophone.c = 2*sqrt(geophone.m * geophone.k);

      %% Save
      save('./mat/geophone_z_axis.mat', 'geophone');
  end

Z-Axis Accelerometer

<<sec:initializeZAxisAccelerometer>>

  function [accelerometer] = initializeZAxisAccelerometer(args)
      arguments
          args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % [kg]
          args.freq (1,1) double {mustBeNumeric, mustBePositive} = 5e3  % [Hz]
      end

      %%
      accelerometer.m = args.mass;

      %% The Stiffness is set to have the damping resonance frequency
      accelerometer.k = accelerometer.m * (2*pi*args.freq)^2;

      %% We set the damping value to have critical damping
      accelerometer.c = 2*sqrt(accelerometer.m * accelerometer.k);

      %% Gain correction of the accelerometer to have a unity gain until the resonance
      accelerometer.gain = -accelerometer.k/accelerometer.m;

      %% Save
      save('./mat/accelerometer_z_axis.mat', 'accelerometer');
  end