Simscape Model
+
+
+
+Table of Contents
+ ++This file is used to explain how this Simscape Model works. +
+-
+
- In section 1, the simulink project with the associated scripts are presented +
- In section 2, an introduction to Simscape Multibody is done +
- In section 3, each simscape files are presented with the associated signal names and joint architectures +
- In section 4, the list of the Simulink library elements are described +
- In section 5, the scripts used for the simulations (initialization for instance) are described +
- In section 6, a list of Matlab function that will be used are defined here +
- In section 7, all the functions that are used to initialize the Simscape Multibody elements are defined here. This includes the mass of all solids for instance. +
+
+
+1 Simulink Project - Startup and Shutdown scripts
+
+
+
+
++From the Simulink project mathworks page: +
+++ ++Simulink® and Simulink Projects provide a collaborative, scalable environment that enables teams to manage their files and data in one place. +
+ ++With Simulink Projects, you can: +
++
+- Collaborate: Enforce companywide standards such as company tools, libraries, and standard startup and shutdown scripts. Share your work with rich sharing options including MATLAB® toolboxes, email, and archives.
+- Automate: Set up your project environment correctly every time by automating steps such as loading the data, managing the path, and opening the models.
+- Integrate with source control: Enable easy integration with source control and configuration management tools.
+
+The project can be opened using the simulinkproject function:
+
+
+
+simulinkproject('./'); ++
+When the project opens, a startup script is ran.
+The startup script is defined below and is exported to the project_startup.m script.
+
+
+
+freqs = logspace(-1, 3, 1000); +save_fig = false; +save('./mat/config.mat', 'freqs', 'save_fig'); + +project = simulinkproject; +projectRoot = project.RootFolder; + +myCacheFolder = fullfile(projectRoot, '.SimulinkCache'); +myCodeFolder = fullfile(projectRoot, '.SimulinkCode'); + +Simulink.fileGenControl('set',... + 'CacheFolder', myCacheFolder,... + 'CodeGenFolder', myCodeFolder,... + 'createDir', true); ++
+When the project closes, it runs the project_shutdown.m script defined below.
+
+
+
+Simulink.fileGenControl('reset'); ++
+The project also permits to automatically add defined folder to the path when the project is opened. +
+
+
+
+2 Simscape Multibody - Presentation
+
+
+
+
+
++A simscape model permits to model multi-physics systems. +
+ ++Simscape Multibody permits to model multibody systems using blocks representing bodies, joints, constraints, force elements, and sensors. +
+
+
+
+2.1 Solid bodies
+
+
+
+Each solid body is represented by a solid block.
+The geometry of the solid body can be imported using a step file. The properties of the solid body such as its mass, center of mass and moment of inertia can be derived from its density and its geometry as defined by the step file. They also can be set by hand.
+
+
+
+2.2 Frames
+
+
++Frames are very important in simscape multibody, they defined where the forces are applied, where the joints are located and where the measurements are made. +
+ ++They can be defined from the solid body geometry, or using the rigid transform block. +
+
+
+
+2.3 Joints
+
+
+
+
+
+
+
+
++Solid Bodies are connected with joints (between frames of the two solid bodies). +
+ ++There are various types of joints that are all described here. +
+ +| Joint Block | +Translational DOFs | +Rotational DOFs | +
|---|---|---|
| 6-DOF | +3 | +3 | +
| Bearing | +1 | +3 | +
| Bushing | +3 | +3 | +
| Cartesian | +3 | +0 | +
| Constant Velocity | +0 | +2 | +
| Cylindrical | +1 | +1 | +
| Gimbal | +0 | +3 | +
| Leadscrew | +1 (coupled) | +1 (coupled) | +
| Pin Slot | +1 | +1 | +
| Planar | +2 | +1 | +
| Prismatic | +1 | +0 | +
| Rectangular | +2 | +0 | +
| Revolute | +0 | +1 | +
| Spherical | +1 | +3 | +
| Telescoping | +1 | +3 | +
| Universal | +0 | +2 | +
| Weld | +0 | +0 | +
+Joint blocks are assortments of joint primitives: +
+-
+
- Prismatic: allows translation along a single standard axis:
Px,Py,Pz
+ - Revolute: allows rotation about a single standard axis:
Rx,Ry,Rz
+ - Spherical: allow rotation about any 3D axis:
S
+ - Lead Screw: allows coupled rotation and translation on a standard axis:
LSz
+ - Constant Velocity: Allows rotation at constant velocity between intersection through arbitrarily aligned shafts:
CV
+
| Joint Block | +Px | +Py | +Pz | +Rx | +Ry | +Rz | +S | +CV | +LSz | +
|---|---|---|---|---|---|---|---|---|---|
| 6-DOF | +x | +x | +x | ++ | + | + | x | ++ | + |
| Bearing | ++ | + | x | +x | +x | +x | ++ | + | + |
| Bushing | +x | +x | +x | +x | +x | +x | ++ | + | + |
| Cartesian | +x | +x | +x | ++ | + | + | + | + | + |
| Constant Velocity | ++ | + | + | + | + | + | + | x | ++ |
| Cylindrical | ++ | + | x | ++ | + | x | ++ | + | + |
| Gimbal | ++ | + | + | x | +x | +x | ++ | + | + |
| Leadscrew | ++ | + | + | + | + | + | + | + | x | +
| Pin Slot | +x | ++ | + | + | + | x | ++ | + | + |
| Planar | +x | +x | ++ | + | + | x | ++ | + | + |
| Prismatic | ++ | + | x | ++ | + | + | + | + | + |
| Rectangular | +x | +x | ++ | + | + | + | + | + | + |
| Revolute | ++ | + | + | + | + | x | ++ | + | + |
| Spherical | ++ | + | + | + | + | + | x | ++ | + |
| Telescoping | ++ | + | x | ++ | + | + | x | ++ | + |
| Universal | ++ | + | + | x | +x | ++ | + | + | + |
| Weld | ++ | + | + | + | + | + | + | + | + |
+For each of the joint primitive, we can specify the dynamical properties: +
+-
+
- The spring stiffness: either linear or rotational one +
- The damping coefficient +
+For the actuation, we can either specify the motion or the force: +
+-
+
- the force applied in the corresponding DOF is provided by the input +
- the motion is provided by the input +
+A sensor can be added to measure either the position, velocity or acceleration of the joint DOF. +
+ ++Composite Force/Torque sensing: +
+-
+
- Constraint force +
- Total force: gives the sum across all joint primitives over all sources: actuation inputs, internal springs and dampers. +
+
+
+2.4 Measurements
+
+
+
+A transform sensor block measures the spatial relationship between two frames: the base B and the follower F.
+
+It can give the rotational and translational position, velocity and acceleration. +
+ +
+The measurement frame should be specified: it corresponds to the frame in which to resolve the selected spatial measurements. The default is world.
+
+If we want to simulate an inertial sensor, we just have to choose B to be the world frame.
+
+Force sensors are included in the joints blocks. +
+
+
+2.5 Excitation
+
+
++We can apply external forces to the model by using an external force and torque block. +
+ ++Internal force, acting reciprocally between base and following origins is implemented using the internal force block even though it is usually included in one joint block. +
+
+
+
+3 Simulink files and signal names
+ + +
+
+
+3.1 List of Simscape files
+
+
+
+
++Few different Simulink files are used: +
+-
+
- kinematics +
- identification - micro station +
- identification - nano station +
- control +
| Simscape Name | +Ty | +Ry | +Rz | +Hexa | +NASS | +
|---|---|---|---|---|---|
| id micro station | +F | +F | +F | +F | ++ |
| id nano station stages | +F | +F | +F | +F | +F | +
| id nano station config | +D | +D | +D | +D | +F | +
| control nano station | +D | +D | +D | +D | +F | +
+
+
+3.2 List of Inputs
+
+
+
+
+
+3.2.1 Perturbations
+
+
+
+
+| Variable | +Meaning | +Size | +Unit | +
|---|---|---|---|
Dw |
+Ground motion | +3 | +[m] | +
Fg |
+External force applied on granite | +3 | +[N] | +
Fs |
+External force applied on the Sample | +3 | +[N] | +
+
+
+3.2.2 Measurement Noise
+
+
+
+
+| Variable | +Meaning | +Size | +Unit | +
|---|---|---|---|
| + | + | + | + |
+
+3.2.3 Control Inputs
+
+
+
+
+| Variable | +Meaning | +Size | +Unit | +
|---|---|---|---|
Fy |
+Actuation force for Ty | +1 | +[N] | +
Dy |
+Imposed displacement for Ty | +1 | +[m] | +
My |
+Actuation torque for Ry | +1 | +[N.m] | +
Ry |
+Imposed rotation for Ry | +1 | +[rad] | +
Mz |
+Actuation torque for Rz | +1 | +[N.m] | +
Rz |
+Imposed rotation for Rz | +1 | +[rad] | +
Fh |
+Actuation force/torque for hexapod (cart) | +6 | +[N, N.m] | +
Fhl |
+Actuation force/torque for hexapod (legs) | +6 | +[N] | +
Dh |
+Imposed position for hexapod (cart) | +6 | +[m, rad] | +
Rm |
+Position of the two masses | +2 | +[rad] | +
Fn |
+Actuation force for the NASS (cart) | +6 | +[N, N.m] | +
Fnl |
+Actuation force for the NASS's legs | +6 | +[N] | +
Dn |
+Imposed position for the NASS (cart) | +6 | +[m, rad] | +
+
+3.3 List of Outputs
+
+
+
+
+| Variable | +Meaning | +Size | +Unit | +
|---|---|---|---|
Dgm |
+Absolute displacement of the granite | +3 | +[m] | +
Vgm |
+Absolute Velocity of the granite | +3 | +[m/s] | +
Dym |
+Measured displacement of Ty | +1 | +[m] | +
Rym |
+Measured rotation of Ry | +1 | +[rad] | +
Rzm |
+Measured rotation of Rz | +1 | +[rad] | +
Dhm |
+Measured position of hexapod (cart) | +6 | +[m, rad] | +
Fnlm |
+Measured force of NASS's legs | +6 | +[N] | +
Dnlm |
+Measured elongation of NASS's legs | +6 | +[m] | +
Dnm |
+Measured position of NASS w.r.t NASS's base | +6 | +[m, rad] | +
Vnm |
+Measured absolute velocity of NASS platform | +6 | +[m/s, rad/s] | +
Vnlm |
+Measured absolute velocity of NASS's legs | +6 | +[m/s] | +
Dsm |
+Position of Sample w.r.t. granite frame | +6 | +[m, rad] | +
+
+
+4 Simulink Library
+
+
+
+
+
++A simulink library is developed in order to share elements between the different simulink files. +
+
+4.1
+
+
+4.1 inputs
+
+
++inputs.slx +
+
+4.2
+
+
+4.2 nass_library
+
+
+
+
+4.3
+
+
+4.3 pos_error_wrt_nass_base
+
+
+
+
+4.4
+
+
+4.4 QuaternionToAngles
+
+
+
+
+4.5
+
+4.5 RotationMatrixToAngle
+
+
+
+
+
+
+5 Scripts
+ + +
+
+5.1 Simulation Initialization
+
+
+
+
++This Matlab script is accessible here. +
+ ++This script runs just before the simulation is started. +It is used to load the simulation configuration and the controllers used for the simulation. +
+ +
+
+%% Load all the data used for the simulation +load('sim_conf.mat'); + +%% Load Controller +load('controllers.mat'); ++
+
+6 Functions
+ + +
+
+
+6.1 computePsdDispl
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [psd_object] = computePsdDispl(sys_data, t_init, n_av) + i_init = find(sys_data.time > t_init, 1); + + han_win = hanning(ceil(length(sys_data.Dx(i_init:end, :))/n_av)); + Fs = 1/sys_data.time(2); + + [pdx, f] = pwelch(sys_data.Dx(i_init:end, :), han_win, [], [], Fs); + [pdy, ~] = pwelch(sys_data.Dy(i_init:end, :), han_win, [], [], Fs); + [pdz, ~] = pwelch(sys_data.Dz(i_init:end, :), han_win, [], [], Fs); + + [prx, ~] = pwelch(sys_data.Rx(i_init:end, :), han_win, [], [], Fs); + [pry, ~] = pwelch(sys_data.Ry(i_init:end, :), han_win, [], [], Fs); + [prz, ~] = pwelch(sys_data.Rz(i_init:end, :), han_win, [], [], Fs); + + psd_object = struct(... + 'f', f, ... + 'dx', pdx, ... + 'dy', pdy, ... + 'dz', pdz, ... + 'rx', prx, ... + 'ry', pry, ... + 'rz', prz); +end ++
+
+
+6.2 computeSetpoint
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function setpoint = computeSetpoint(ty, ry, rz) +%% +setpoint = zeros(6, 1); + +%% Ty +Ty = [1 0 0 0 ; + 0 1 0 ty ; + 0 0 1 0 ; + 0 0 0 1 ]; + +% Tyinv = [1 0 0 0 ; +% 0 1 0 -ty ; +% 0 0 1 0 ; +% 0 0 0 1 ]; + +%% Ry +Ry = [ cos(ry) 0 sin(ry) 0 ; + 0 1 0 0 ; + -sin(ry) 0 cos(ry) 0 ; + 0 0 0 1 ]; + +% TMry = Ty*Ry*Tyinv; + +%% Rz +Rz = [cos(rz) -sin(rz) 0 0 ; + sin(rz) cos(rz) 0 0 ; + 0 0 1 0 ; + 0 0 0 1 ]; + +% TMrz = Ty*TMry*Rz*TMry'*Tyinv; + +%% All stages +% TM = TMrz*TMry*Ty; + +TM = Ty*Ry*Rz; + +[thetax, thetay, thetaz] = RM2angle(TM(1:3, 1:3)); + +setpoint(1:3) = TM(1:3, 4); +setpoint(4:6) = [thetax, thetay, thetaz]; + +%% Custom Functions +function [thetax, thetay, thetaz] = RM2angle(R) + if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1 + thetay = -asin(R(3, 1)); + thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay)); + thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay)); + else + thetaz = 0; + if abs(R(3, 1)+1) < 1e-6 % R31 = -1 + thetay = pi/2; + thetax = thetaz + atan2(R(1, 2), R(1, 3)); + else + thetay = -pi/2; + thetax = -thetaz + atan2(-R(1, 2), -R(1, 3)); + end + end +end +end ++
+
+
+6.3 converErrorBasis
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function error_nass = convertErrorBasis(pos, setpoint, ty, ry, rz) +% convertErrorBasis - +% +% Syntax: convertErrorBasis(p_error, ty, ry, rz) +% +% Inputs: +% - p_error - Position error of the sample w.r.t. the granite [m, rad] +% - ty - Measured translation of the Ty stage [m] +% - ry - Measured rotation of the Ry stage [rad] +% - rz - Measured rotation of the Rz stage [rad] +% +% Outputs: +% - P_nass - Position error of the sample w.r.t. the NASS base [m] +% - R_nass - Rotation error of the sample w.r.t. the NASS base [rad] +% +% Example: +% + +%% If line vector => column vector +if size(pos, 2) == 6 + pos = pos'; +end + +if size(setpoint, 2) == 6 + setpoint = setpoint'; +end + +%% Position of the sample in the frame fixed to the Granite +P_granite = [pos(1:3); 1]; % Position [m] +R_granite = [setpoint(1:3); 1]; % Reference [m] + +%% Transformation matrices of the stages +% T-y +TMty = [1 0 0 0 ; + 0 1 0 ty ; + 0 0 1 0 ; + 0 0 0 1 ]; + +% R-y +TMry = [ cos(ry) 0 sin(ry) 0 ; + 0 1 0 0 ; + -sin(ry) 0 cos(ry) 0 ; + 0 0 0 1 ]; + +% R-z +TMrz = [cos(rz) -sin(rz) 0 0 ; + sin(rz) cos(rz) 0 0 ; + 0 0 1 0 ; + 0 0 0 1 ]; + +%% Compute Point coordinates in the new reference fixed to the NASS base +% P_nass = TMrz*TMry*TMty*P_granite; +P_nass = TMrz\TMry\TMty\P_granite; +R_nass = TMrz\TMry\TMty\R_granite; + +dx = R_nass(1)-P_nass(1); +dy = R_nass(2)-P_nass(2); +dz = R_nass(3)-P_nass(3); + +%% Compute new basis vectors linked to the NASS base +% ux_nass = TMrz*TMry*TMty*[1; 0; 0; 0]; +% ux_nass = ux_nass(1:3); +% uy_nass = TMrz*TMry*TMty*[0; 1; 0; 0]; +% uy_nass = uy_nass(1:3); +% uz_nass = TMrz*TMry*TMty*[0; 0; 1; 0]; +% uz_nass = uz_nass(1:3); + +ux_nass = TMrz\TMry\TMty\[1; 0; 0; 0]; +ux_nass = ux_nass(1:3); +uy_nass = TMrz\TMry\TMty\[0; 1; 0; 0]; +uy_nass = uy_nass(1:3); +uz_nass = TMrz\TMry\TMty\[0; 0; 1; 0]; +uz_nass = uz_nass(1:3); + +%% Rotations error w.r.t. granite Frame +% Rotations error w.r.t. granite Frame +rx_nass = pos(4); +ry_nass = pos(5); +rz_nass = pos(6); + +% Rotation matrices of the Sample w.r.t. the Granite +Mrx_error = [1 0 0 ; + 0 cos(-rx_nass) -sin(-rx_nass) ; + 0 sin(-rx_nass) cos(-rx_nass)]; + +Mry_error = [ cos(-ry_nass) 0 sin(-ry_nass) ; + 0 1 0 ; + -sin(-ry_nass) 0 cos(-ry_nass)]; + +Mrz_error = [cos(-rz_nass) -sin(-rz_nass) 0 ; + sin(-rz_nass) cos(-rz_nass) 0 ; + 0 0 1]; + +% Rotation matrix of the Sample w.r.t. the Granite +Mr_error = Mrz_error*Mry_error*Mrx_error; + +%% Use matrix to solve +R = Mr_error/[ux_nass, uy_nass, uz_nass]; % Rotation matrix from NASS base to Sample + +[thetax, thetay, thetaz] = RM2angle(R); + +error_nass = [dx; dy; dz; thetax; thetay; thetaz]; + +%% Custom Functions +function [thetax, thetay, thetaz] = RM2angle(R) + if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1 + thetay = -asin(R(3, 1)); + % thetaybis = pi-thetay; + thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay)); + % thetaxbis = atan2(R(3, 2)/cos(thetaybis), R(3, 3)/cos(thetaybis)); + thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay)); + % thetazbis = atan2(R(2, 1)/cos(thetaybis), R(1, 1)/cos(thetaybis)); + else + thetaz = 0; + if abs(R(3, 1)+1) < 1e-6 % R31 = -1 + thetay = pi/2; + thetax = thetaz + atan2(R(1, 2), R(1, 3)); + else + thetay = -pi/2; + thetax = -thetaz + atan2(-R(1, 2), -R(1, 3)); + end + end +end + +end ++
+
+6.4 generateDiagPidControl
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [K] = generateDiagPidControl(G, fs) + %% + pid_opts = pidtuneOptions(... + 'PhaseMargin', 50, ... + 'DesignFocus', 'disturbance-rejection'); + + %% + K = tf(zeros(6)); + + for i = 1:6 + input_name = G.InputName(i); + output_name = G.OutputName(i); + K(i, i) = tf(pidtune(minreal(G(output_name, input_name)), 'PIDF', 2*pi*fs, pid_opts)); + end + + K.InputName = G.OutputName; + K.OutputName = G.InputName; +end ++
+
+6.5 identifyPlant
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [sys] = identifyPlant(opts_param) + %% Default values for opts + opts = struct(); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% Options for Linearized + options = linearizeOptions; + options.SampleTime = 0; + + %% Name of the Simulink File + mdl = 'sim_nano_station_id'; + + %% Input/Output definition + io(1) = linio([mdl, '/Fn'], 1, 'input'); % Cartesian forces applied by NASS + io(2) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion + io(3) = linio([mdl, '/Fs'], 1, 'input'); % External forces on the sample + io(4) = linio([mdl, '/Fnl'], 1, 'input'); % Forces applied on the NASS's legs + io(5) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample + io(6) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs + io(7) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs + io(8) = linio([mdl, '/Es'], 1, 'output'); % Position Error w.r.t. NASS base + + %% Run the linearization + G = linearize(mdl, io, 0); + G.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz', ... + 'Dgx', 'Dgy', 'Dgz', ... + 'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz', ... + 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', ... + 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6', ... + 'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ... + 'Edx', 'Rdy', 'Edz', 'Erx', 'Ery', 'Erz'}; + + %% Create the sub transfer functions + % From forces applied in the cartesian frame to displacement of the sample in the cartesian frame + sys.G_cart = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'})); + % From ground motion to Sample displacement + sys.G_gm = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Dgx', 'Dgy', 'Dgz'})); + % From direct forces applied on the sample to displacement of the sample + sys.G_fs = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz'})); + % From forces applied on NASS's legs to force sensor in each leg + sys.G_iff = minreal(G({'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})); + % From forces applied on NASS's legs to displacement of each leg + sys.G_dleg = minreal(G({'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})); + % From forces applied on NASS's legs to displacement of each leg + sys.G_plant = minreal(G({'Edx', 'Rdy', 'Edz', 'Erx', 'Ery', 'Erz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'})); +end ++
+
+6.6 runSimulation
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [] = runSimulation(sys_name, sys_mass, ctrl_type, act_damp) + %% Load the controller and save it for the simulation + if strcmp(ctrl_type, 'cl') && strcmp(act_damp, 'none') + K_obj = load('./mat/K_fb.mat'); + K = K_obj.(sprintf('K_%s_%s', sys_mass, sys_name)); %#ok + save('./mat/controllers.mat', 'K'); + elseif strcmp(ctrl_type, 'cl') && strcmp(act_damp, 'iff') + K_obj = load('./mat/K_fb_iff.mat'); + K = K_obj.(sprintf('K_%s_%s_iff', sys_mass, sys_name)); %#ok + save('./mat/controllers.mat', 'K'); + elseif strcmp(ctrl_type, 'ol') + K = tf(zeros(6)); %#ok + save('./mat/controllers.mat', 'K'); + else + error('ctrl_type should be cl or ol'); + end + + %% Active Damping + if strcmp(act_damp, 'iff') + K_iff_crit = load('./mat/K_iff_crit.mat'); + K_iff = K_iff_crit.(sprintf('K_iff_%s_%s', sys_mass, sys_name)); %#ok + save('./mat/controllers.mat', 'K_iff', '-append'); + elseif strcmp(act_damp, 'none') + K_iff = tf(zeros(6)); %#ok + save('./mat/controllers.mat', 'K_iff', '-append'); + end + + %% + if strcmp(sys_name, 'pz') + initializeNanoHexapod(struct('actuator', 'piezo')); + elseif strcmp(sys_name, 'vc') + initializeNanoHexapod(struct('actuator', 'lorentz')); + else + error('sys_name should be pz or vc'); + end + + if strcmp(sys_mass, 'light') + initializeSample(struct('mass', 1)); + elseif strcmp(sys_mass, 'heavy') + initializeSample(struct('mass', 50)); + else + error('sys_mass should be light or heavy'); + end + + %% Run the simulation + sim('sim_nano_station_ctrl.slx'); + + %% Split the Dsample matrix into vectors + [Dx, Dy, Dz, Rx, Ry, Rz] = matSplit(Es.Data, 1); %#ok + time = Dsample.Time; %#ok + + %% Save the result + filename = sprintf('sim_%s_%s_%s_%s', sys_mass, sys_name, ctrl_type, act_damp); + save(sprintf('./mat/%s.mat', filename), ... + 'time', 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', 'K'); +end ++
+
+7 Initialize Elements
+ +
+
+
+7.1 Simulation Configuration
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [] = initializeSimConf(opts_param) + %% Default values for opts + opts = struct('Ts', 1e-4, ... % Sampling time [s] + 'Tsim', 10, ... % Simulation time [s] + 'cl_time', 0, ... % Close Loop time [s] + 'gravity', false ... % Gravity along the z axis + ); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% + sim_conf = struct(); + + %% + sim_conf.Ts = opts.Ts; + sim_conf.Tsim = opts.Tsim; + sim_conf.cl_time = opts.cl_time; + + %% Gravity + if opts.gravity + sim_conf.g = -9.8; %#ok + else + sim_conf.g = 0; %#ok + end + + %% Save + save('./mat/sim_conf.mat', 'sim_conf'); +end ++
+
+
+7.2 Experiment
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [] = initializeExperiment(exp_name, sys_mass) + if strcmp(exp_name, 'tomography') + opts_sim = struct(... + 'Tsim', 5, ... + 'cl_time', 5 ... + ); + initializeSimConf(opts_sim); + + if strcmp(sys_mass, 'light') + opts_inputs = struct(... + 'Dw', true, ... + 'Rz', 60 ... % rpm + ); + elseif strcpm(sys_mass, 'heavy') + opts_inputs = struct(... + 'Dw', true, ... + 'Rz', 1 ... % rpm + ); + else + error('sys_mass should be light or heavy'); + end + + initializeInputs(opts_inputs); + else + error('exp_name is only configured for tomography'); + end +end ++
+
+
+7.3 Inputs
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [inputs] = initializeInputs(opts_param) + %% Default values for opts + opts = struct( ... + 'Dw', false, ... + 'Dy', false, ... + 'Ry', false, ... + 'Rz', false, ... + 'Dh', false, ... + 'Rm', false, ... + 'Dn', false ... + ); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% Load Sampling Time and Simulation Time + load('./mat/sim_conf.mat', 'sim_conf'); + + %% Define the time vector + t = 0:sim_conf.Ts:sim_conf.Tsim; + + %% Ground motion - Dw + if islogical(opts.Dw) && opts.Dw == true + load('./mat/perturbations.mat', 'Wxg'); + Dw = 1/sqrt(2)*100*random('norm', 0, 1, length(t), 3); + Dw(:, 1) = lsim(Wxg, Dw(:, 1), t); + Dw(:, 2) = lsim(Wxg, Dw(:, 2), t); + Dw(:, 3) = lsim(Wxg, Dw(:, 3), t); + elseif islogical(opts.Dw) && opts.Dw == false + Dw = zeros(length(t), 3); + else + Dw = opts.Dw; + end + + %% Translation stage - Dy + if islogical(opts.Dy) && opts.Dy == true + Dy = zeros(length(t), 1); + elseif islogical(opts.Dy) && opts.Dy == false + Dy = zeros(length(t), 1); + else + Dy = opts.Dy; + end + + %% Tilt Stage - Ry + if islogical(opts.Ry) && opts.Ry == true + Ry = 3*(2*pi/360)*sin(2*pi*0.2*t); + elseif islogical(opts.Ry) && opts.Ry == false + Ry = zeros(length(t), 1); + elseif isnumeric(opts.Ry) && length(opts.Ry) == 1 + Ry = opts.Ry*(2*pi/360)*ones(length(t), 1); + else + Ry = opts.Ry; + end + + %% Spindle - Rz + if islogical(opts.Rz) && opts.Rz == true + Rz = 2*pi*0.5*t; + elseif islogical(opts.Rz) && opts.Rz == false + Rz = zeros(length(t), 1); + elseif isnumeric(opts.Rz) && length(opts.Rz) == 1 + Rz = opts.Rz*(2*pi/60)*t; + else + Rz = opts.Rz; + end + + %% Micro Hexapod - Dh + if islogical(opts.Dh) && opts.Dh == true + Dh = zeros(length(t), 6); + elseif islogical(opts.Dh) && opts.Dh == false + Dh = zeros(length(t), 6); + else + Dh = opts.Dh; + end + + %% Axis Compensation - Rm + if islogical(opts.Rm) + Rm = zeros(length(t), 2); + Rm(:, 2) = pi*ones(length(t), 1); + else + Rm = opts.Rm; + end + + %% Nano Hexapod - Dn + if islogical(opts.Dn) && opts.Dn == true + Dn = zeros(length(t), 6); + elseif islogical(opts.Dn) && opts.Dn == false + Dn = zeros(length(t), 6); + else + Dn = opts.Dn; + end + + %% Setpoint - Ds + Ds = zeros(length(t), 6); + for i = 1:length(t) + Ds(i, :) = computeSetpoint(Dy(i), Ry(i), Rz(i)); + end + + %% External Forces applied on the Granite + Fg = zeros(length(t), 3); + + %% External Forces applied on the Sample + Fs = zeros(length(t), 6); + + %% Create the input Structure that will contain all the inputs + inputs = struct( ... + 'Ts', sim_conf.Ts, ... + 'Dw', timeseries(Dw, t), ... + 'Dy', timeseries(Dy, t), ... + 'Ry', timeseries(Ry, t), ... + 'Rz', timeseries(Rz, t), ... + 'Dh', timeseries(Dh, t), ... + 'Rm', timeseries(Rm, t), ... + 'Dn', timeseries(Dn, t), ... + 'Ds', timeseries(Ds, t), ... + 'Fg', timeseries(Fg, t), ... + 'Fs', timeseries(Fs, t) ... + ); + + %% Save + save('./mat/inputs.mat', 'inputs'); + + %% Custom Functions + function setpoint = computeSetpoint(ty, ry, rz) + %% + setpoint = zeros(6, 1); + + %% Ty + TMTy = [1 0 0 0 ; + 0 1 0 ty ; + 0 0 1 0 ; + 0 0 0 1 ]; + + %% Ry + TMRy = [ cos(ry) 0 sin(ry) 0 ; + 0 1 0 0 ; + -sin(ry) 0 cos(ry) 0 ; + 0 0 0 1 ]; + + %% Rz + TMRz = [cos(rz) -sin(rz) 0 0 ; + sin(rz) cos(rz) 0 0 ; + 0 0 1 0 ; + 0 0 0 1 ]; + + %% All stages + TM = TMTy*TMRy*TMRz; + + [thetax, thetay, thetaz] = RM2angle(TM(1:3, 1:3)); + + setpoint(1:3) = TM(1:3, 4); + setpoint(4:6) = [thetax, thetay, thetaz]; + + %% Custom Functions + function [thetax, thetay, thetaz] = RM2angle(R) + if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1 + thetay = -asin(R(3, 1)); + thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay)); + thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay)); + else + thetaz = 0; + if abs(R(3, 1)+1) < 1e-6 % R31 = -1 + thetay = pi/2; + thetax = thetaz + atan2(R(1, 2), R(1, 3)); + else + thetay = -pi/2; + thetax = -thetaz + atan2(-R(1, 2), -R(1, 3)); + end + end + end + end +end ++
+
+
+7.4 Ground
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [ground] = initializeGround() + %% + ground = struct(); + + ground.shape = [2, 2, 0.5]; % [m] + ground.density = 2800; % [kg/m3] + ground.color = [0.5, 0.5, 0.5]; + + %% Save + save('./mat/stages.mat', 'ground', '-append'); +end ++
+
+
+7.5 Granite
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [granite] = initializeGranite() + %% + granite = struct(); + + %% Static Properties + granite.density = 2800; % [kg/m3] + granite.volume = 0.72; % [m3] TODO - should + granite.mass = granite.density*granite.volume; % [kg] + granite.color = [1 1 1]; + granite.STEP = './STEPS/granite/granite.STEP'; + + %% Dynamical Properties + granite.k.x = 1e8; % [N/m] + granite.c.x = 1e4; % [N/(m/s)] + + granite.k.y = 1e8; % [N/m] + granite.c.y = 1e4; % [N/(m/s)] + + granite.k.z = 1e8; % [N/m] + granite.c.z = 1e4; % [N/(m/s)] + + %% Positioning parameters + granite.sample_pos = 0.8; % Z-offset for the initial position of the sample [m] + + %% Save + save('./mat/stages.mat', 'granite', '-append'); +end ++
+
+
+7.6 Translation Stage
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [ty] = initializeTy(opts_param) + %% Default values for opts + opts = struct('rigid', false); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% + ty = struct(); + + %% Y-Translation - Static Properties + % Ty Granite frame + ty.granite_frame.density = 7800; % [kg/m3] + ty.granite_frame.color = [0.753 1 0.753]; + ty.granite_frame.STEP = './STEPS/Ty/Ty_Granite_Frame.STEP'; + % Guide Translation Ty + ty.guide.density = 7800; % [kg/m3] + ty.guide.color = [0.792 0.820 0.933]; + ty.guide.STEP = './STEPS/ty/Ty_Guide.STEP'; + % Ty - Guide_Translation12 + ty.guide12.density = 7800; % [kg/m3] + ty.guide12.color = [0.792 0.820 0.933]; + ty.guide12.STEP = './STEPS/Ty/Ty_Guide_12.STEP'; + % Ty - Guide_Translation11 + ty.guide11.density = 7800; % [kg/m3] + ty.guide11.color = [0.792 0.820 0.933]; + ty.guide11.STEP = './STEPS/ty/Ty_Guide_11.STEP'; + % Ty - Guide_Translation22 + ty.guide22.density = 7800; % [kg/m3] + ty.guide22.color = [0.792 0.820 0.933]; + ty.guide22.STEP = './STEPS/ty/Ty_Guide_22.STEP'; + % Ty - Guide_Translation21 + ty.guide21.density = 7800; % [kg/m3] + ty.guide21.color = [0.792 0.820 0.933]; + ty.guide21.STEP = './STEPS/Ty/Ty_Guide_21.STEP'; + % Ty - Plateau translation + ty.frame.density = 7800; % [kg/m3] + ty.frame.color = [0.792 0.820 0.933]; + ty.frame.STEP = './STEPS/ty/Ty_Stage.STEP'; + % Ty Stator Part + ty.stator.density = 5400; % [kg/m3] + ty.stator.color = [0.792 0.820 0.933]; + ty.stator.STEP = './STEPS/ty/Ty_Motor_Stator.STEP'; + % Ty Rotor Part + ty.rotor.density = 5400; % [kg/m3] + ty.rotor.color = [0.792 0.820 0.933]; + ty.rotor.STEP = './STEPS/ty/Ty_Motor_Rotor.STEP'; + + ty.m = 250; % TODO [kg] + + %% Y-Translation - Dynamicals Properties + if opts.rigid + ty.k.ax = 1e10; % Axial Stiffness for each of the 4 guidance (y) [N/m] + else + ty.k.ax = 1e7/4; % Axial Stiffness for each of the 4 guidance (y) [N/m] + end + ty.k.rad = 9e9/4; % Radial Stiffness for each of the 4 guidance (x-z) [N/m] + + ty.c.ax = 100*(1/5)*sqrt(ty.k.ax/ty.m); + ty.c.rad = 100*(1/5)*sqrt(ty.k.rad/ty.m); + + %% Save + save('./mat/stages.mat', 'ty', '-append'); +end ++
+
+
+7.7 Tilt Stage
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [ry] = initializeRy(opts_param) + %% Default values for opts + opts = struct('rigid', false); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% + ry = struct(); + + %% Tilt Stage - Static Properties + % Ry - Guide for the tilt stage + ry.guide.density = 7800; % [kg/m3] + ry.guide.color = [0.792 0.820 0.933]; + ry.guide.STEP = './STEPS/ry/Tilt_Guide.STEP'; + % Ry - Rotor of the motor + ry.rotor.density = 2400; % [kg/m3] + ry.rotor.color = [0.792 0.820 0.933]; + ry.rotor.STEP = './STEPS/ry/Tilt_Motor_Axis.STEP'; + % Ry - Motor + ry.motor.density = 3200; % [kg/m3] + ry.motor.color = [0.792 0.820 0.933]; + ry.motor.STEP = './STEPS/ry/Tilt_Motor.STEP'; + % Ry - Plateau Tilt + ry.stage.density = 7800; % [kg/m3] + ry.stage.color = [0.792 0.820 0.933]; + ry.stage.STEP = './STEPS/ry/Tilt_Stage.STEP'; + + ry.m = 200; % TODO [kg] + + %% Tilt Stage - Dynamical Properties + if opts.rigid + ry.k.tilt = 1e10; % Rotation stiffness around y [N*m/deg] + else + ry.k.tilt = 1e4; % Rotation stiffness around y [N*m/deg] + end + + ry.k.h = 357e6/4; % Stiffness in the direction of the guidance [N/m] + ry.k.rad = 555e6/4; % Stiffness in the top direction [N/m] + ry.k.rrad = 238e6/4; % Stiffness in the side direction [N/m] + + ry.c.h = 10*(1/5)*sqrt(ry.k.h/ry.m); + ry.c.rad = 10*(1/5)*sqrt(ry.k.rad/ry.m); + ry.c.rrad = 10*(1/5)*sqrt(ry.k.rrad/ry.m); + ry.c.tilt = 10*(1/1)*sqrt(ry.k.tilt/ry.m); + + %% Positioning parameters + ry.z_offset = 0.58178; % Z-Offset so that the center of rotation matches the sample center [m] + + %% Save + save('./mat/stages.mat', 'ry', '-append'); +end ++
+
+
+7.8 Spindle
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [rz] = initializeRz(opts_param) + %% Default values for opts + opts = struct('rigid', false); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% + rz = struct(); + + %% Spindle - Static Properties + % Spindle - Slip Ring + rz.slipring.density = 7800; % [kg/m3] + rz.slipring.color = [0.792 0.820 0.933]; + rz.slipring.STEP = './STEPS/rz/Spindle_Slip_Ring.STEP'; + % Spindle - Rotor + rz.rotor.density = 7800; % [kg/m3] + rz.rotor.color = [0.792 0.820 0.933]; + rz.rotor.STEP = './STEPS/rz/Spindle_Rotor.STEP'; + % Spindle - Stator + rz.stator.density = 7800; % [kg/m3] + rz.stator.color = [0.792 0.820 0.933]; + rz.stator.STEP = './STEPS/rz/Spindle_Stator.STEP'; + + % Estimated mass of the mooving part + rz.m = 250; % [kg] + + %% Spindle - Dynamical Properties + % Estimated stiffnesses + rz.k.ax = 2e9; % Axial Stiffness [N/m] + rz.k.rad = 7e8; % Radial Stiffness [N/m] + rz.k.rot = 100e6*2*pi/360; % Rotational Stiffness [N*m/deg] + + if opts.rigid + rz.k.tilt = 1e10; % Vertical Rotational Stiffness [N*m/deg] + else + rz.k.tilt = 1e2; % TODO what value should I put? [N*m/deg] + end + + % TODO + rz.c.ax = 2*sqrt(rz.k.ax/rz.m); + rz.c.rad = 2*sqrt(rz.k.rad/rz.m); + rz.c.tilt = 100*sqrt(rz.k.tilt/rz.m); + rz.c.rot = 100*sqrt(rz.k.rot/rz.m); + + %% Save + save('./mat/stages.mat', 'rz', '-append'); +end ++
+
+
+7.9 Micro Hexapod
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [micro_hexapod] = initializeMicroHexapod(opts_param) + %% Default values for opts + opts = struct(); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% Stewart Object + micro_hexapod = struct(); + micro_hexapod.h = 350; % Total height of the platform [mm] +% micro_hexapod.jacobian = 269.26; % Distance from the top platform to the Jacobian point [mm] + micro_hexapod.jacobian = 270; % Distance from the top platform to the Jacobian point [mm] + + %% Bottom Plate - Mechanical Design + BP = struct(); + + BP.rad.int = 110; % Internal Radius [mm] + BP.rad.ext = 207.5; % External Radius [mm] + BP.thickness = 26; % Thickness [mm] + BP.leg.rad = 175.5; % Radius where the legs articulations are positionned [mm] + BP.leg.ang = 9.5; % Angle Offset [deg] + BP.density = 8000; % Density of the material [kg/m^3] + BP.color = [0.6 0.6 0.6]; % Color [rgb] + BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness]; + + %% Top Plate - Mechanical Design + TP = struct(); + + TP.rad.int = 82; % Internal Radius [mm] + TP.rad.ext = 150; % Internal Radius [mm] + TP.thickness = 26; % Thickness [mm] + TP.leg.rad = 118; % Radius where the legs articulations are positionned [mm] + TP.leg.ang = 12.1; % Angle Offset [deg] + TP.density = 8000; % Density of the material [kg/m^3] + TP.color = [0.6 0.6 0.6]; % Color [rgb] + TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness]; + + %% Struts + Leg = struct(); + + Leg.stroke = 10e-3; % Maximum Stroke of each leg [m] + Leg.k.ax = 5e7; % Stiffness of each leg [N/m] + Leg.ksi.ax = 3; % Maximum amplification at resonance [] + Leg.rad.bottom = 25; % Radius of the cylinder of the bottom part [mm] + Leg.rad.top = 17; % Radius of the cylinder of the top part [mm] + Leg.density = 8000; % Density of the material [kg/m^3] + Leg.color.bottom = [0.5 0.5 0.5]; % Color [rgb] + Leg.color.top = [0.5 0.5 0.5]; % Color [rgb] + + Leg.sphere.bottom = Leg.rad.bottom; % Size of the sphere at the end of the leg [mm] + Leg.sphere.top = Leg.rad.top; % Size of the sphere at the end of the leg [mm] + Leg.m = TP.density*((pi*(TP.rad.ext/1000)^2)*(TP.thickness/1000)-(pi*(TP.rad.int/1000^2))*(TP.thickness/1000))/6; % TODO [kg] + Leg = updateDamping(Leg); + + + %% Sphere + SP = struct(); + + SP.height.bottom = 27; % [mm] + SP.height.top = 27; % [mm] + SP.density.bottom = 8000; % [kg/m^3] + SP.density.top = 8000; % [kg/m^3] + SP.color.bottom = [0.6 0.6 0.6]; % [rgb] + SP.color.top = [0.6 0.6 0.6]; % [rgb] + SP.k.ax = 0; % [N*m/deg] + SP.ksi.ax = 10; + + SP.thickness.bottom = SP.height.bottom-Leg.sphere.bottom; % [mm] + SP.thickness.top = SP.height.top-Leg.sphere.top; % [mm] + SP.rad.bottom = Leg.sphere.bottom; % [mm] + SP.rad.top = Leg.sphere.top; % [mm] + SP.m = SP.density.bottom*2*pi*((SP.rad.bottom*1e-3)^2)*(SP.height.bottom*1e-3); % TODO [kg] + + SP = updateDamping(SP); + + %% + Leg.support.bottom = [0 SP.thickness.bottom; 0 0; SP.rad.bottom 0; SP.rad.bottom SP.height.bottom]; + Leg.support.top = [0 SP.thickness.top; 0 0; SP.rad.top 0; SP.rad.top SP.height.top]; + + %% + micro_hexapod.BP = BP; + micro_hexapod.TP = TP; + micro_hexapod.Leg = Leg; + micro_hexapod.SP = SP; + + %% + micro_hexapod = initializeParameters(micro_hexapod); + + %% Save + save('./mat/stages.mat', 'micro_hexapod', '-append'); + + %% + function [element] = updateDamping(element) + field = fieldnames(element.k); + for i = 1:length(field) + element.c.(field{i}) = 1/element.ksi.(field{i})*sqrt(element.k.(field{i})/element.m); + end + end + + %% + function [stewart] = initializeParameters(stewart) + %% Connection points on base and top plate w.r.t. World frame at the center of the base plate + stewart.pos_base = zeros(6, 3); + stewart.pos_top = zeros(6, 3); + + alpha_b = stewart.BP.leg.ang*pi/180; % angle de décalage par rapport à 120 deg (pour positionner les supports bases) + alpha_t = stewart.TP.leg.ang*pi/180; % +- offset angle from 120 degree spacing on top + + height = (stewart.h-stewart.BP.thickness-stewart.TP.thickness-stewart.Leg.sphere.bottom-stewart.Leg.sphere.top-stewart.SP.thickness.bottom-stewart.SP.thickness.top)*0.001; % TODO + + radius_b = stewart.BP.leg.rad*0.001; % rayon emplacement support base + radius_t = stewart.TP.leg.rad*0.001; % top radius in meters + + for i = 1:3 + % base points + angle_m_b = (2*pi/3)* (i-1) - alpha_b; + angle_p_b = (2*pi/3)* (i-1) + alpha_b; + stewart.pos_base(2*i-1,:) = [radius_b*cos(angle_m_b), radius_b*sin(angle_m_b), 0.0]; + stewart.pos_base(2*i,:) = [radius_b*cos(angle_p_b), radius_b*sin(angle_p_b), 0.0]; + + % top points + % Top points are 60 degrees offset + angle_m_t = (2*pi/3)* (i-1) - alpha_t + 2*pi/6; + angle_p_t = (2*pi/3)* (i-1) + alpha_t + 2*pi/6; + stewart.pos_top(2*i-1,:) = [radius_t*cos(angle_m_t), radius_t*sin(angle_m_t), height]; + stewart.pos_top(2*i,:) = [radius_t*cos(angle_p_t), radius_t*sin(angle_p_t), height]; + end + + % permute pos_top points so that legs are end points of base and top points + stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom + stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)]; + + %% leg vectors + legs = stewart.pos_top - stewart.pos_base; + leg_length = zeros(6, 1); + leg_vectors = zeros(6, 3); + for i = 1:6 + leg_length(i) = norm(legs(i,:)); + leg_vectors(i,:) = legs(i,:) / leg_length(i); + end + + stewart.Leg.lenght = 1000*leg_length(1)/1.5; + stewart.Leg.shape.bot = [0 0; ... + stewart.Leg.rad.bottom 0; ... + stewart.Leg.rad.bottom stewart.Leg.lenght; ... + stewart.Leg.rad.top stewart.Leg.lenght; ... + stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ... + 0 0.2*stewart.Leg.lenght]; + + %% Calculate revolute and cylindrical axes + rev1 = zeros(6, 3); + rev2 = zeros(6, 3); + cyl1 = zeros(6, 3); + for i = 1:6 + rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]); + rev1(i,:) = rev1(i,:) / norm(rev1(i,:)); + + rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:)); + rev2(i,:) = rev2(i,:) / norm(rev2(i,:)); + + cyl1(i,:) = leg_vectors(i,:); + end + + + %% Coordinate systems + stewart.lower_leg = struct('rotation', eye(3)); + stewart.upper_leg = struct('rotation', eye(3)); + + for i = 1:6 + stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)']; + stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)']; + end + + %% Position Matrix + stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.thickness+stewart.Leg.sphere.top+stewart.SP.thickness.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)]; + + %% Compute Jacobian Matrix + aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)]; + stewart.J = getJacobianMatrix(leg_vectors', aa'); + end + + function J = getJacobianMatrix(RM,M_pos_base) + % RM: [3x6] unit vector of each leg in the fixed frame + % M_pos_base: [3x6] vector of the leg connection at the top platform location in the fixed frame + J = zeros(6); + J(:, 1:3) = RM'; + J(:, 4:6) = cross(M_pos_base, RM)'; + end +end ++
+
+7.10 Center of gravity compensation
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [axisc] = initializeAxisc() + %% + axisc = struct(); + + %% Axis Compensator - Static Properties + % Structure + axisc.structure.density = 3400; % [kg/m3] + axisc.structure.color = [0.792 0.820 0.933]; + axisc.structure.STEP = './STEPS/axisc/axisc_structure.STEP'; + % Wheel + axisc.wheel.density = 2700; % [kg/m3] + axisc.wheel.color = [0.753 0.753 0.753]; + axisc.wheel.STEP = './STEPS/axisc/axisc_wheel.STEP'; + % Mass + axisc.mass.density = 7800; % [kg/m3] + axisc.mass.color = [0.792 0.820 0.933]; + axisc.mass.STEP = './STEPS/axisc/axisc_mass.STEP'; + % Gear + axisc.gear.density = 7800; % [kg/m3] + axisc.gear.color = [0.792 0.820 0.933]; + axisc.gear.STEP = './STEPS/axisc/axisc_gear.STEP'; + + axisc.m = 40; % TODO [kg] + + %% Axis Compensator - Dynamical Properties + axisc.k.ax = 1; % TODO [N*m/deg)] + axisc.c.ax = (1/1)*sqrt(axisc.k.ax/axisc.m); + + %% Save + save('./mat/stages.mat', 'axisc', '-append'); +end ++
+
+
+7.11 Mirror
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [] = initializeMirror(opts_param) + %% Default values for opts + opts = struct(... + 'shape', 'spherical', ... % spherical or conical + 'angle', 45 ... + ); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% + mirror = struct(); + 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 mirror [kg/m3] + mirror.color = [0.4 1.0 1.0]; % Color of the mirror + + mirror.cone_length = mirror.rad*tand(opts.angle)+mirror.h+mirror.jacobian; % Distance from Apex point of the cone to jacobian point + + %% Shape + mirror.shape = [... + 0 mirror.h-mirror.thickness + mirror.hole_rad mirror.h-mirror.thickness; ... + mirror.hole_rad 0; ... + mirror.rad 0 ... + ]; + + if strcmp(opts.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(opts.shape, 'conical') + mirror.shape = [mirror.shape; mirror.rad+mirror.h/tand(opts.angle) mirror.h]; + else + error('Shape should be either conical or spherical'); + end + + mirror.shape = [mirror.shape; 0 mirror.h]; + + %% Save + save('./mat/stages.mat', 'mirror', '-append'); +end ++
+
+
+7.12 Nano Hexapod
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [nano_hexapod] = initializeNanoHexapod(opts_param) + %% Default values for opts + opts = struct('actuator', 'piezo'); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + opts.(opt{1}) = opts_param.(opt{1}); + end + end + + %% Stewart Object + nano_hexapod = struct(); + nano_hexapod.h = 90; % Total height of the platform [mm] + nano_hexapod.jacobian = 175; % Point where the Jacobian is computed => Center of rotation [mm] +% nano_hexapod.jacobian = 174.26; % Point where the Jacobian is computed => Center of rotation [mm] + + %% Bottom Plate + BP = struct(); + + BP.rad.int = 0; % Internal Radius [mm] + BP.rad.ext = 150; % External Radius [mm] + BP.thickness = 10; % Thickness [mm] + BP.leg.rad = 100; % Radius where the legs articulations are positionned [mm] + BP.leg.ang = 5; % Angle Offset [deg] + BP.density = 8000;% Density of the material [kg/m^3] + BP.color = [0.7 0.7 0.7]; % Color [rgb] + BP.shape = [BP.rad.int BP.thickness; BP.rad.int 0; BP.rad.ext 0; BP.rad.ext BP.thickness]; + + %% Top Plate + TP = struct(); + + TP.rad.int = 0; % Internal Radius [mm] + TP.rad.ext = 100; % Internal Radius [mm] + TP.thickness = 10; % Thickness [mm] + TP.leg.rad = 90; % Radius where the legs articulations are positionned [mm] + TP.leg.ang = 5; % Angle Offset [deg] + TP.density = 8000;% Density of the material [kg/m^3] + TP.color = [0.7 0.7 0.7]; % Color [rgb] + TP.shape = [TP.rad.int TP.thickness; TP.rad.int 0; TP.rad.ext 0; TP.rad.ext TP.thickness]; + + %% Leg + Leg = struct(); + + Leg.stroke = 80e-6; % Maximum Stroke of each leg [m] + if strcmp(opts.actuator, 'piezo') + Leg.k.ax = 1e7; % Stiffness of each leg [N/m] + elseif strcmp(opts.actuator, 'lorentz') + Leg.k.ax = 1e4; % Stiffness of each leg [N/m] + else + error('opts.actuator should be piezo or lorentz'); + end + Leg.ksi.ax = 10; % Maximum amplification at resonance [] + Leg.rad.bottom = 12; % Radius of the cylinder of the bottom part [mm] + Leg.rad.top = 10; % Radius of the cylinder of the top part [mm] + Leg.density = 8000; % Density of the material [kg/m^3] + Leg.color.bottom = [0.5 0.5 0.5]; % Color [rgb] + Leg.color.top = [0.5 0.5 0.5]; % Color [rgb] + + Leg.sphere.bottom = Leg.rad.bottom; % Size of the sphere at the end of the leg [mm] + Leg.sphere.top = Leg.rad.top; % Size of the sphere at the end of the leg [mm] + Leg.m = TP.density*((pi*(TP.rad.ext/1000)^2)*(TP.thickness/1000)-(pi*(TP.rad.int/1000^2))*(TP.thickness/1000))/6; % TODO [kg] + + Leg = updateDamping(Leg); + + + %% Sphere + SP = struct(); + + SP.height.bottom = 15; % [mm] + SP.height.top = 15; % [mm] + SP.density.bottom = 8000; % [kg/m^3] + SP.density.top = 8000; % [kg/m^3] + SP.color.bottom = [0.7 0.7 0.7]; % [rgb] + SP.color.top = [0.7 0.7 0.7]; % [rgb] + SP.k.ax = 0; % [N*m/deg] + SP.ksi.ax = 3; + + SP.thickness.bottom = SP.height.bottom-Leg.sphere.bottom; % [mm] + SP.thickness.top = SP.height.top-Leg.sphere.top; % [mm] + SP.rad.bottom = Leg.sphere.bottom; % [mm] + SP.rad.top = Leg.sphere.top; % [mm] + SP.m = SP.density.bottom*2*pi*((SP.rad.bottom*1e-3)^2)*(SP.height.bottom*1e-3); % TODO [kg] + + SP = updateDamping(SP); + + %% + Leg.support.bottom = [0 SP.thickness.bottom; 0 0; SP.rad.bottom 0; SP.rad.bottom SP.height.bottom]; + Leg.support.top = [0 SP.thickness.top; 0 0; SP.rad.top 0; SP.rad.top SP.height.top]; + + %% + nano_hexapod.BP = BP; + nano_hexapod.TP = TP; + nano_hexapod.Leg = Leg; + nano_hexapod.SP = SP; + + %% + nano_hexapod = initializeParameters(nano_hexapod); + + %% Save + save('./mat/stages.mat', 'nano_hexapod', '-append'); + + %% + function [element] = updateDamping(element) + field = fieldnames(element.k); + for i = 1:length(field) + element.c.(field{i}) = 1/element.ksi.(field{i})*sqrt(element.k.(field{i})/element.m); + end + end + + %% + function [stewart] = initializeParameters(stewart) + %% Connection points on base and top plate w.r.t. World frame at the center of the base plate + stewart.pos_base = zeros(6, 3); + stewart.pos_top = zeros(6, 3); + + alpha_b = stewart.BP.leg.ang*pi/180; % angle de décalage par rapport à 120 deg (pour positionner les supports bases) + alpha_t = stewart.TP.leg.ang*pi/180; % +- offset angle from 120 degree spacing on top + + height = (stewart.h-stewart.BP.thickness-stewart.TP.thickness-stewart.Leg.sphere.bottom-stewart.Leg.sphere.top-stewart.SP.thickness.bottom-stewart.SP.thickness.top)*0.001; % TODO + + radius_b = stewart.BP.leg.rad*0.001; % rayon emplacement support base + radius_t = stewart.TP.leg.rad*0.001; % top radius in meters + + for i = 1:3 + % base points + angle_m_b = (2*pi/3)* (i-1) - alpha_b; + angle_p_b = (2*pi/3)* (i-1) + alpha_b; + stewart.pos_base(2*i-1,:) = [radius_b*cos(angle_m_b), radius_b*sin(angle_m_b), 0.0]; + stewart.pos_base(2*i,:) = [radius_b*cos(angle_p_b), radius_b*sin(angle_p_b), 0.0]; + + % top points + % Top points are 60 degrees offset + angle_m_t = (2*pi/3)* (i-1) - alpha_t + 2*pi/6; + angle_p_t = (2*pi/3)* (i-1) + alpha_t + 2*pi/6; + stewart.pos_top(2*i-1,:) = [radius_t*cos(angle_m_t), radius_t*sin(angle_m_t), height]; + stewart.pos_top(2*i,:) = [radius_t*cos(angle_p_t), radius_t*sin(angle_p_t), height]; + end + + % permute pos_top points so that legs are end points of base and top points + stewart.pos_top = [stewart.pos_top(6,:); stewart.pos_top(1:5,:)]; %6th point on top connects to 1st on bottom + stewart.pos_top_tranform = stewart.pos_top - height*[zeros(6, 2),ones(6, 1)]; + + %% leg vectors + legs = stewart.pos_top - stewart.pos_base; + leg_length = zeros(6, 1); + leg_vectors = zeros(6, 3); + for i = 1:6 + leg_length(i) = norm(legs(i,:)); + leg_vectors(i,:) = legs(i,:) / leg_length(i); + end + + stewart.Leg.lenght = 1000*leg_length(1)/1.5; + stewart.Leg.shape.bot = [0 0; ... + stewart.Leg.rad.bottom 0; ... + stewart.Leg.rad.bottom stewart.Leg.lenght; ... + stewart.Leg.rad.top stewart.Leg.lenght; ... + stewart.Leg.rad.top 0.2*stewart.Leg.lenght; ... + 0 0.2*stewart.Leg.lenght]; + + %% Calculate revolute and cylindrical axes + rev1 = zeros(6, 3); + rev2 = zeros(6, 3); + cyl1 = zeros(6, 3); + for i = 1:6 + rev1(i,:) = cross(leg_vectors(i,:), [0 0 1]); + rev1(i,:) = rev1(i,:) / norm(rev1(i,:)); + + rev2(i,:) = - cross(rev1(i,:), leg_vectors(i,:)); + rev2(i,:) = rev2(i,:) / norm(rev2(i,:)); + + cyl1(i,:) = leg_vectors(i,:); + end + + + %% Coordinate systems + stewart.lower_leg = struct('rotation', eye(3)); + stewart.upper_leg = struct('rotation', eye(3)); + + for i = 1:6 + stewart.lower_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)']; + stewart.upper_leg(i).rotation = [rev1(i,:)', rev2(i,:)', cyl1(i,:)']; + end + + %% Position Matrix + stewart.M_pos_base = stewart.pos_base + (height+(stewart.TP.thickness+stewart.Leg.sphere.top+stewart.SP.thickness.top+stewart.jacobian)*1e-3)*[zeros(6, 2),ones(6, 1)]; + + %% Compute Jacobian Matrix + aa = stewart.pos_top_tranform + (stewart.jacobian - stewart.TP.thickness - stewart.SP.height.top)*1e-3*[zeros(6, 2),ones(6, 1)]; + stewart.J = getJacobianMatrix(leg_vectors', aa'); + end + + function J = getJacobianMatrix(RM,M_pos_base) + % RM: [3x6] unit vector of each leg in the fixed frame + % M_pos_base: [3x6] vector of the leg connection at the top platform location in the fixed frame + J = zeros(6); + J(:, 1:3) = RM'; + J(:, 4:6) = cross(M_pos_base, RM)'; + end +end ++
+
+7.13 Sample
+
+
+
+
++This Matlab function is accessible here. +
+ +
+
+function [sample] = initializeSample(opts_param) + %% Default values for opts + sample = struct('radius', 100, ... + 'height', 300, ... + 'mass', 50, ... + 'offset', 0, ... + 'color', [0.45, 0.45, 0.45] ... + ); + + %% Populate opts with input parameters + if exist('opts_param','var') + for opt = fieldnames(opts_param)' + sample.(opt{1}) = opts_param.(opt{1}); + end + end + + %% + sample.k.x = 1e8; + sample.c.x = sqrt(sample.k.x*sample.mass)/10; + + sample.k.y = 1e8; + sample.c.y = sqrt(sample.k.y*sample.mass)/10; + + sample.k.z = 1e8; + sample.c.z = sqrt(sample.k.y*sample.mass)/10; + + %% Save + save('./mat/stages.mat', 'sample', '-append'); +end ++