Subsystems used for the Simscape Models
Table of Contents
The full Simscape Model is represented in Figure 1.
Figure 1: 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.
1 Ground
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
Figure 2: Simscape model for the Ground
Figure 3: Simscape picture for the Ground
Function description
function [ground] = initializeGround()
1.1 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');
2 Granite
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.
Figure 4: Simscape model for the Granite
Figure 5: 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');
3 Translation Stage
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
Figure 6: Simscape model for the Translation Stage
Figure 7: 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');
4 Tilt Stage
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.
Figure 8: Simscape model for the Tilt Stage
Figure 9: 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');
5 Spindle
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
Figure 10: Simscape model for the Spindle
Figure 11: Simscape picture for the Spindle
Function description
function [rz] = initializeRz(args)
Optional Parameters
arguments args.rigid logical {mustBeNumericOrLogical} = false args.x0 (1,1) double {mustBeNumeric} = 0 % [m] args.y0 (1,1) double {mustBeNumeric} = 0 % [m] args.z0 (1,1) double {mustBeNumeric} = 0 % [m] args.rx0 (1,1) double {mustBeNumeric} = 0 % [rad] args.ry0 (1,1) double {mustBeNumeric} = 0 % [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');
6 Micro Hexapod
Simscape Model
Figure 12: Simscape model for the Micro-Hexapod
Figure 13: 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');
7 Center of gravity compensation
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
Figure 14: Simscape model for the Center of Mass compensation system
Figure 15: 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');
8 Mirror
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
Figure 16: Simscape model for the Mirror
Figure 17: 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');
9 Nano Hexapod
Simscape Model
Figure 18: Simscape model for the Nano Hexapod
Figure 19: 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');
10 Sample
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.
Figure 20: Simscape model for the Sample
Figure 21: 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');
11 Generate Reference Signals
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(:) = args.Rz_amplitude+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); 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
12 Initialize Disturbances
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');
13 Z-Axis Geophone
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
14 Z-Axis Accelerometer
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