126 lines
4.5 KiB
Matlab
126 lines
4.5 KiB
Matlab
function [pi_lgr_base, baseQR] = base_params_qr(includeMotorDynamics)
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% ----------------------------------------------------------------------
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% In this function QR decomposition is applied to regressor in closed
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% form obtained from Lagrange formulation of dynamics.
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%
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% You should already have a function to compute the regressor matrix of the
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% robot full_regressor_UR10E.m
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%
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% In the beginning you need to choose if you want to include motor dynamics
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% (reflected inertia of the motor). A rule of thumb is to include it.
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%
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% After finding base parametrs the test is performed which compares the
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% output torques with base and full parameters for randomly generated data.
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%
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% Finally, a structure is generated and saved with parameters necessary to
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% find base parameters from standard ones, and base regressor from a
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% strandard one.
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% ----------------------------------------------------------------------
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% Seed the random number generator based on the current time
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rng('shuffle');
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% ------------------------------------------------------------------------
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% Set limits on posistion and velocities
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% ------------------------------------------------------------------------
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q_min = -pi*ones(6,1);
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q_max = pi*ones(6,1);
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qd_max = 3*pi*ones(6,1);
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q2d_max = 6*pi*ones(6,1);
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% -----------------------------------------------------------------------
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% Standard dynamics paramters of the robot in symbolic form
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% -----------------------------------------------------------------------
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m = sym('m%d',[6,1],'real');
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hx = sym('h%d_x',[6,1],'real');
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hy = sym('h%d_y',[6,1],'real');
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hz = sym('h%d_z',[6,1],'real');
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ixx = sym('i%d_xx',[6,1],'real');
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ixy = sym('i%d_xy',[6,1],'real');
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ixz = sym('i%d_xz',[6,1],'real');
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iyy = sym('i%d_yy',[6,1],'real');
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iyz = sym('i%d_yz',[6,1],'real');
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izz = sym('i%d_zz',[6,1],'real');
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im = sym('im%d',[6,1],'real');
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% Load parameters attached to the end-effector
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syms ml hl_x hl_y hl_z il_xx il_xy il_xz il_yy il_yz il_zz real
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% Vector of symbolic parameters
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for i = 1:6
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if includeMotorDynamics
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pi_lgr_sym(:,i) = [ixx(i),ixy(i),ixz(i),iyy(i),iyz(i),izz(i),...
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hx(i),hy(i),hz(i),m(i),im(i)]';
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else
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pi_lgr_sym(:,i) = [ixx(i),ixy(i),ixz(i),iyy(i),iyz(i),izz(i),...
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hx(i),hy(i),hz(i),m(i)]';
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end
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end
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[nLnkPrms, nLnks] = size(pi_lgr_sym);
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pi_lgr_sym = reshape(pi_lgr_sym, [nLnkPrms*nLnks, 1]);
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% -----------------------------------------------------------------------
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% Find relation between independent columns and dependent columns
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% -----------------------------------------------------------------------
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% Get observation matrix of identifiable paramters
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W = [];
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for i = 1:25
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q_rnd = q_min + (q_max - q_min).*rand(6,1);
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qd_rnd = -qd_max + 2*qd_max.*rand(6,1);
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q2d_rnd = -q2d_max + 2*q2d_max.*rand(6,1);
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if includeMotorDynamics
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Y = regressorWithMotorDynamics(q_rnd,qd_rnd,q2d_rnd);
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else
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Y = full_regressor_UR10E(q_rnd,qd_rnd,q2d_rnd);
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end
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W = vertcat(W,Y);
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end
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% QR decomposition with pivoting: W*E = Q*R
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% R is upper triangular matrix
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% Q is unitary matrix
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% E is permutation matrix
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[Q, R, E] = qr(W);
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% matrix W has rank bb which is number number of base parameters
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bb = rank(W);
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% R = [R1 R2;
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% 0 0]
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% R1 is bbxbb upper triangular and reguar matrix
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% R2 is bbx(c-bb) matrix where c is number of standard parameters
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R1 = R(1:bb,1:bb);
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R2 = R(1:bb,bb+1:end);
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beta = R1\R2; % the zero rows of K correspond to independent columns of WP
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beta(abs(beta)<sqrt(eps)) = 0; % get rid of numerical errors
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% W2 = W1*beta
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% Make sure that the relation holds
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W1 = W*E(:,1:bb);
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W2 = W*E(:,bb+1:end);
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assert(norm(W2 - W1*beta) < 1e-6,...
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'Found realationship between W1 and W2 is not correct\n');
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% -----------------------------------------------------------------------
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% Find base parmaters
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% -----------------------------------------------------------------------
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pi1 = E(:,1:bb)'*pi_lgr_sym; % independent paramters
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pi2 = E(:,bb+1:end)'*pi_lgr_sym; % dependent paramteres
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% all of the expressions below are equivalent
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pi_lgr_base = pi1 + beta*pi2;
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% pi_lgr_base = [eye(bb) beta]*[pi1;pi2];
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% pi_lgr_base = [eye(bb) beta]*E'*pi_lgr_sym;
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% ---------------------------------------------------------------------
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% Create structure with the result of QR decompositon a
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% ---------------------------------------------------------------------
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baseQR = struct;
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baseQR.numberOfBaseParameters = bb;
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baseQR.permutationMatrix = E;
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baseQR.beta = beta;
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baseQR.motorDynamicsIncluded = includeMotorDynamics;
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