18 KiB
18 KiB
Matlab Functions used for the NASS Project
Functions
<<sec:functions>>
computePsdDispl
<<sec: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);
endcomputeSetpoint
<<sec: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
endconverErrorBasis
<<sec: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
endgenerateDiagPidControl
<<sec: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;
endidentifyPlant
<<sec: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, '/Fd'], 1, 'input'); % Disturbance Forces
io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
io(9) = linio([mdl, '/Es'], 1, 'output'); % Position Error w.r.t. NASS base
io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the legs
%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz', ...
'Dgx', 'Dgy', 'Dgz', ...
'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz', ...
'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
'Frzz', 'Ftyx', 'Ftyz'};
G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6', ...
'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz', ...
'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
%% Create the sub transfer functions
minreal_tol = sqrt(eps);
% 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'}), minreal_tol, false);
% From ground motion to Sample displacement
sys.G_gm = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Dgx', 'Dgy', 'Dgz'}), minreal_tol, false);
% 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'}), minreal_tol, false);
% 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'}), minreal_tol, false);
% 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'}), minreal_tol, false);
% From forces/torques applied by the NASS to position error
sys.G_plant = minreal(G({'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}), minreal_tol, false);
% From forces/torques applied by the NASS to velocity of the actuator
sys.G_geoph = minreal(G({'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), minreal_tol, false);
% From various disturbance forces to position error
sys.G_dist = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Frzz', 'Ftyx', 'Ftyz'}), minreal_tol, false);
%% We remove low frequency and high frequency dynamics that are usually unstable
% using =freqsep= is risky as it may change the shape of the transfer functions
% f_min = 0.1; % [Hz]
% f_max = 1e4; % [Hz]
% [~, sys.G_cart] = freqsep(freqsep(sys.G_cart, 2*pi*f_max), 2*pi*f_min);
% [~, sys.G_gm] = freqsep(freqsep(sys.G_gm, 2*pi*f_max), 2*pi*f_min);
% [~, sys.G_fs] = freqsep(freqsep(sys.G_fs, 2*pi*f_max), 2*pi*f_min);
% [~, sys.G_iff] = freqsep(freqsep(sys.G_iff, 2*pi*f_max), 2*pi*f_min);
% [~, sys.G_dleg] = freqsep(freqsep(sys.G_dleg, 2*pi*f_max), 2*pi*f_min);
% [~, sys.G_plant] = freqsep(freqsep(sys.G_plant, 2*pi*f_max), 2*pi*f_min);
%% We finally verify that the system is stable
if ~isstable(sys.G_cart) || ~isstable(sys.G_gm) || ~isstable(sys.G_fs) || ~isstable(sys.G_iff) || ~isstable(sys.G_dleg) || ~isstable(sys.G_plant)
warning('One of the identified system is unstable');
end
endInverse Kinematics of the Hexapod
<<sec:inverseKinematicsHexapod>>
This Matlab function is accessible here.
function [L] = inverseKinematicsHexapod(hexapod, AP, ARB)
% inverseKinematicsHexapod - Compute the initial position of each leg to have the wanted Hexapod's position
%
% Syntax: inverseKinematicsHexapod(hexapod, AP, ARB)
%
% Inputs:
% - hexapod - Hexapod object containing the geometry of the hexapod
% - AP - Position vector of point OB expressed in frame {A} in [m]
% - ARB - Rotation Matrix expressed in frame {A}
% Wanted Length of the hexapod's legs [m]
L = zeros(6, 1);
for i = 1:length(L)
Bbi = hexapod.pos_top_tranform(i, :)' - 1e-3*[0 ; 0 ; hexapod.TP.thickness+hexapod.Leg.sphere.top+hexapod.SP.thickness.top+hexapod.jacobian]; % [m]
Aai = hexapod.pos_base(i, :)' + 1e-3*[0 ; 0 ; hexapod.BP.thickness+hexapod.Leg.sphere.bottom+hexapod.SP.thickness.bottom-hexapod.h-hexapod.jacobian]; % [m]
L(i) = sqrt(AP'*AP + Bbi'*Bbi + Aai'*Aai - 2*AP'*Aai + 2*AP'*(ARB*Bbi) - 2*(ARB*Bbi)'*Aai);
end
endcomputeReferencePose
<<sec:computeReferencePose>>
This Matlab function is accessible here.
function [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn)
% computeReferencePose - Compute the homogeneous transformation matrix corresponding to the wanted pose of the sample
%
% Syntax: [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn)
%
% Inputs:
% - Dy - Reference of the Translation Stage [m]
% - Ry - Reference of the Tilt Stage [rad]
% - Rz - Reference of the Spindle [rad]
% - Dh - Reference of the Micro Hexapod (Pitch, Roll, Yaw angles) [m, m, m, rad, rad, rad]
% - Dn - Reference of the Nano Hexapod [m, m, m, rad, rad, rad]
%
% Outputs:
% - WTr -
%% Translation Stage
Rty = [1 0 0 0;
0 1 0 Dy;
0 0 1 0;
0 0 0 1];
%% Tilt Stage - Pure rotating aligned with Ob
Rry = [ cos(Ry) 0 sin(Ry) 0;
0 1 0 0;
-sin(Ry) 0 cos(Ry) 0;
0 0 0 1];
%% Spindle - Rotation along the Z axis
Rrz = [cos(Rz) -sin(Rz) 0 0 ;
sin(Rz) cos(Rz) 0 0 ;
0 0 1 0 ;
0 0 0 1 ];
%% Micro-Hexapod
Rhx = [1 0 0;
0 cos(Dh(4)) -sin(Dh(4));
0 sin(Dh(4)) cos(Dh(4))];
Rhy = [ cos(Dh(5)) 0 sin(Dh(5));
0 1 0;
-sin(Dh(5)) 0 cos(Dh(5))];
Rhz = [cos(Dh(6)) -sin(Dh(6)) 0;
sin(Dh(6)) cos(Dh(6)) 0;
0 0 1];
Rh = [1 0 0 Dh(1) ;
0 1 0 Dh(2) ;
0 0 1 Dh(3) ;
0 0 0 1 ];
Rh(1:3, 1:3) = Rhz*Rhy*Rhx;
%% Nano-Hexapod
Rnx = [1 0 0;
0 cos(Dn(4)) -sin(Dn(4));
0 sin(Dn(4)) cos(Dn(4))];
Rny = [ cos(Dn(5)) 0 sin(Dn(5));
0 1 0;
-sin(Dn(5)) 0 cos(Dn(5))];
Rnz = [cos(Dn(6)) -sin(Dn(6)) 0;
sin(Dn(6)) cos(Dn(6)) 0;
0 0 1];
Rn = [1 0 0 Dn(1) ;
0 1 0 Dn(2) ;
0 0 1 Dn(3) ;
0 0 0 1 ];
Rn(1:3, 1:3) = Rnx*Rny*Rnz;
%% Total Homogeneous transformation
WTr = Rty*Rry*Rrz*Rh*Rn;
end