experiment design is changed as to have two options for optimization algorithms - pattern search and genetic algorithm. Generation of full regressor and well as load regressor are put into seperate file
This commit is contained in:
Shamil Mamedov
parent
856c868a51
commit
7e7f3512c4
Binary file not shown.
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@ -129,9 +129,10 @@ wr = invG*we;
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% -----------------------------------------------------------------------
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fprintf('Validation of mapping from standard parameters to base ones\n')
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if includeMotorDynamics
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ur10.pi = rand(nLnkPrms*nLnks, 1);
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ur10.pi(end+1,:) = rand(1,nLnks);
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ur10.pi = reshape(ur10.pi,[nLnkPrms*nLnks, 1]);
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else
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ur10.pi = reshape(ur10.pi,[60,1]);
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ur10.pi = reshape(ur10.pi,[nLnkPrms*nLnks, 1]);
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end
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% On random positions, velocities, aceeleations
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@ -1,175 +0,0 @@
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% ----------------------------------------------------------------------
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% In this script 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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% Get robot description
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run('main_ur.m')
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% Seed the random number generator based on the current time
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rng('shuffle');
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includeMotorDynamics = 1;
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% ------------------------------------------------------------------------
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% Getting limits on posistion and velocities
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% ------------------------------------------------------------------------
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q_min = zeros(6,1);
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q_max = zeros(6,1);
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qd_max = zeros(6,1);
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q2d_max = 2*ones(6,1); % it is chosen by us as it is not given in URDF
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for i = 1:6
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q_min(i) = str2double(ur10.robot.joint{i}.limit.Attributes.lower);
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q_max(i) = str2double(ur10.robot.joint{i}.limit.Attributes.upper);
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qd_max(i) = str2double(ur10.robot.joint{i}.limit.Attributes.velocity);
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end
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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 = 1:20
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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 identifiable 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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if norm(W2 - W1*beta) > 1e-6
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fprintf('Found realationship between W1 and W2 is not correct\n');
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return
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end
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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_base2 = [eye(bb) beta]*[pi1;pi2];
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pi_lgr_base3 = [eye(bb) beta]*E'*pi_lgr_sym;
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% Relationship needed for identifcation using physical feasibility
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%{
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KG = [eye(bb) beta; zeros(size(W,2)-bb,bb) eye(size(W,2)-bb)];
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G = KG*E';
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invG = E*[eye(bb) -beta; zeros(size(W,2)-bb,bb) eye(size(W,2)-bb)];
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we = G*pi_lgr_sym;
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vpa(we,3)
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wr = invG*we;
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%}
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return
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% -----------------------------------------------------------------------
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% Validation of obtained mappings
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% -----------------------------------------------------------------------
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fprintf('Validation of mapping from standard parameters to base ones\n')
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ur10.pi = reshape(ur10.pi,[60,1]);
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% On random positions, velocities, aceeleations
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for i = 1:100
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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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Yi = regressorWithMotorDynamics(q_rnd,qd_rnd,q2d_rnd);
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else
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Yi = full_regressor_UR10E(q_rnd,qd_rnd,q2d_rnd);
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end
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Yi = full_regressor_UR10E(q_rnd, qd_rnd, q2d_rnd);
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tau_full = Yi*ur10.pi;
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pi_lgr_base = [eye(bb) beta]*E'*ur10.pi;
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Y_base = Yi*E(:,1:bb);
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tau_base = Y_base*pi_lgr_base;
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nrm_err1(i) = norm(tau_full - tau_base);
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end
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figure
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plot(nrm_err1)
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ylabel('||\tau - \tau_b||')
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grid on
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% -----------------------------------------------------------------------
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% Additional functions
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% -----------------------------------------------------------------------
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function Y = regressorWithMotorDynamics(q,qd,q2d)
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% ----------------------------------------------------------------------
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% This function adds motor dynamics to rigid body regressor.
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% It is simplified model of motor dynamics, it adds only reflected
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% inertia i.e. I_rflctd = Im*N^2 where N is reduction ratio - I_rflctd*q_2d
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% parameter is added to existing vector of each link [pi_i I_rflctd_i]
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% so that each link has 11 parameters
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% ----------------------------------------------------------------------
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Y_rgd_bdy = full_regressor_UR10E(q,qd,q2d);
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Y_mtrs = diag(q2d);
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Y = [Y_rgd_bdy(:,1:10), Y_mtrs(:,1), Y_rgd_bdy(:,11:20), Y_mtrs(:,2),...
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Y_rgd_bdy(:,21:30), Y_mtrs(:,3), Y_rgd_bdy(:,31:40), Y_mtrs(:,4),...
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Y_rgd_bdy(:,41:50), Y_mtrs(:,5), Y_rgd_bdy(:,51:60), Y_mtrs(:,6)];
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end
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@ -1,5 +1,3 @@
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% Get robot description
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run('main_ur.m')
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% -----------------------------------------------------------------------
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%
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% Getting base parameters of the UR10 manipulator based on the
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@ -11,10 +9,13 @@ run('main_ur.m')
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% Here we tried to generilize what was given there to more general
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% parametrization compared to their modified DH parameters.
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% ------------------------------------------------------------------------
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% Get robot description
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run('main_ur.m')
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generateBaseRegressorFunction = 0;
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generateBaseDynamicsFunctions = 0;
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% -----------------------------------------------------------------------
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% Create symbolic parameters
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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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@ -36,9 +37,7 @@ for i = 1:6
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hx(i),hy(i),hz(i),m(i)]';
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end
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% ------------------------------------------------------------------------
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% Symbolic generilized coordiates, their first and second deriatives
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% -----------------------------------------------------------------------
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q_sym = sym('q%d',[6,1],'real');
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qd_sym = sym('qd%d',[6,1],'real');
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q2d_sym = sym('q2d%d',[6,1],'real');
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@ -75,72 +74,14 @@ for i = 1:6
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v_kk(:,i+1) = T_pk(1:3,1:3,i)'*(v_kk(:,i) + cross(w_kk(:,i),sym(p_pj)));
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g_kk(:,i+1) = T_pk(1:3,1:3,i)'*g_kk(:,i);
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p_kk(:,i+1) = T_pk(1:3,1:3,i)'*(p_kk(:,i) + sym(p_pj));
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K_reg(i,:) = [sym(0.5)*w2wtlda(w_kk(:,i+1)),...
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v_kk(:,i+1)'*vec2skewSymMat(w_kk(:,i+1)),...
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sym(0.5)*v_kk(:,i+1)'*v_kk(:,i+1)];
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P_reg(i,:) = [sym(zeros(1,6)), g_kk(:,i+1)',...
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g_kk(:,i+1)'*p_kk(:,i+1)];
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lmda(:,:,i) = getLambda(T_pk(1:3,1:3,i),sym(p_pj));
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f(:,i) = getF(v_kk(:,i+1),w_kk(:,i+1),sym(jnt_axs_k),qd_sym(i));
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DK(:,i+1) = lmda(:,:,i)*DK(:,i) + qd_sym(i)*f(:,i);
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DP(:,i+1) = lmda(:,:,i)*DP(:,i);
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end
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% ---------------------------------------------------------
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% Kinetic and potential energy of the load
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% ---------------------------------------------------------
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% Transformation from link 6 frame to end-effector frame
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rpy_ee = sym(str2num(ur10.robot.joint{7}.origin.Attributes.rpy));
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R_6ee = RPY(rpy_ee);
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R_6ee(abs(R_6ee)<sqrt(eps)) = sym(0); % to avoid numerical errors
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p_6ee = str2num(ur10.robot.joint{7}.origin.Attributes.xyz)';
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T_6ee = sym([R_6ee, p_6ee; zeros(1,3), 1]); % to avoid numerical errors
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w_eeee = T_6ee(1:3,1:3)'*w_kk(:,7);
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v_eeee = T_6ee(1:3,1:3)'*(v_kk(:,7) + cross(w_kk(:,i+1),sym(p_6ee)));
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g_eeee = T_6ee(1:3,1:3)'*g_kk(:,7);
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p_eeee = T_6ee(1:3,1:3)'*(p_kk(:,7) + sym(p_6ee));
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K_l = [sym(0.5)*w2wtlda(w_eeee), v_eeee'*vec2skewSymMat(w_eeee),...
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sym(0.5)*(v_eeee'*v_eeee)];
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P_l = [sym(zeros(1,6)), g_eeee', g_eeee'*p_eeee];
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% -----------------------------------------------------
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% Dynamic regressor of the load
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% -----------------------------------------------------
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Lagrl = K_l - P_l;
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dLagrl_dq = jacobian(Lagrl,q_sym)';
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dLagrl_dqd = jacobian(Lagrl,qd_sym)';
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tl = sym(zeros(6,10));
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for i = 1:6
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tl = tl + diff(dLagrl_dqd,q_sym(i))*qd_sym(i)+...
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diff(dLagrl_dqd,qd_sym(i))*q2d_sym(i);
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end
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Y_l = tl - dLagrl_dq;
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% matlabFunction(Y_l,'File','idntfcn/load_regressor_UR10E',...
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% 'Vars',{q_sym,qd_sym,q2d_sym});
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% ----------------------------------------------------
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% Dynamic regressor of the full paramters
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% ----------------------------------------------------
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Lagrf = [K_reg(1,:) - P_reg(1,:), K_reg(2,:) - P_reg(2,:),...
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K_reg(3,:) - P_reg(3,:), K_reg(4,:) - P_reg(4,:),...
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K_reg(5,:) - P_reg(5,:), K_reg(6,:) - P_reg(6,:)];
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dLagrf_dq = jacobian(Lagrf,q_sym)';
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dLagrf_dqd = jacobian(Lagrf,qd_sym)';
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tf = sym(zeros(6,60));
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for i = 1:6
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tf = tf + diff(dLagrf_dqd,q_sym(i))*qd_sym(i)+...
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diff(dLagrf_dqd,qd_sym(i))*q2d_sym(i);
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end
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Y_f = tf - dLagrf_dq;
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matlabFunction(Y_f,'File','autogen/full_regressor_UR10E',...
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'Vars',{q_sym,qd_sym,q2d_sym});
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return
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% -----------------------------------------------------------------------
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% Regrouping parameters of the links
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% -----------------------------------------------------------------------
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@ -160,6 +101,7 @@ for i = 6:-1:2
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end
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pir_ur10_num = double(pir_ur10_num);
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% -----------------------------------------------------------------------
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% Getting base parameters
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% -----------------------------------------------------------------------
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@ -209,20 +151,23 @@ for i = 1:6
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pir_base_vctr = vertcat(pir_base_vctr,pir_base_num{i});
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end
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% -----------------------------------------------------------------------
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% Computing Regressor
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% -----------------------------------------------------------------------
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dLagr_dq = jacobian(grad_Lagr,q_sym)';
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dLagr_dqd = jacobian(grad_Lagr,qd_sym)';
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t1 = sym(zeros(6,length(pir_base_vctr)));
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% t1 = sym(zeros(6,42));
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for i = 1:6
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t1 = t1 + diff(dLagr_dqd,q_sym(i))*qd_sym(i)+...
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diff(dLagr_dqd,qd_sym(i))*q2d_sym(i);
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end
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Y_hat = t1 - dLagr_dq;
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% matlabFunction(Y_hat,'File','idntfcn/base_regressor_UR10E',...
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% 'Vars',{q_sym,qd_sym,q2d_sym});
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if generateBaseRegressorFunction
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matlabFunction(Y_hat,'File','autogen/base_regressor_UR10E',...
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'Vars',{q_sym,qd_sym,q2d_sym});
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end
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% ------------------------------------------------------------
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% Finding mapping from standard parameters to base parameters
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@ -234,6 +179,7 @@ pi_ur10_base = [pir_base_sym{1};pir_base_sym{2};pir_base_sym{3};...
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dbase_dfull = jacobian(pi_ur10_base,pi_ur10_full);
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Kd = jacobian(pi_ur10_base - Pb*pi_ur10_full, Pd*pi_ur10_full);
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% --------------------------------------------------------------------
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% Computing matrices of dynamic equations of motion
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% --------------------------------------------------------------------
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@ -277,18 +223,19 @@ for i = 1:1:6
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end
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end
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% Generating matrices
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fprintf('Generating dynamic equatio elements:\n')
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%{
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fprintf('\t Mass matrix\n')
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matlabFunction(M_mtrx_sym,'File','autogen/M_mtrx_fcn','Vars',{q_sym,xi});
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if generateBaseDynamicsFunctions
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fprintf('Generating dynamic equation elements:\n');
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fprintf('\t Matrix of Coriolis and Centrifugal Forces\n')
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matlabFunction(C_mtrx_sym,'File','autogen/C_mtrx_fcn','Vars',{q_sym,qd_sym,xi});
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fprintf('\t Mass matrix\n');
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matlabFunction(M_mtrx_sym,'File','autogen/M_mtrx_fcn','Vars',{q_sym,xi});
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fprintf('\t Matrix of Coriolis and Centrifugal Forces\n');
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matlabFunction(C_mtrx_sym,'File','autogen/C_mtrx_fcn','Vars',{q_sym,qd_sym,xi});
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fprintf('\t Vector of gravitational forces\n');
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matlabFunction(G_vctr_sym,'File','autogen/G_vctr_fcn','Vars',{q_sym,xi});
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end
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fprintf('\t Vector of gravitational forces\n')
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matlabFunction(G_vctr_sym,'File','autogen/G_vctr_fcn','Vars',{q_sym,xi});
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%}
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return
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% --------------------------------------------------------------------
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% Tests
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@ -4,6 +4,9 @@
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% ---------------------------------------------------------------------
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run('main_ur.m'); % get robot description
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% Choose optimization algorithm: 'patternsearch', 'ga'
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optmznAlgorithm = 'patternsearch';
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% Getting limits on posistion and velocities. Moreover we get some
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% constant parmeters of the robot that allow us to accelerate computation
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% of the robot regressor.
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@ -38,52 +41,73 @@ traj_par.t = 0:traj_par.t_smp:traj_par.T; % time
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traj_par.N = 5; % number of harmonics
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||||
|
||||
|
||||
% ------------------------------------------------------------------------
|
||||
% ----------------------------------------------------------------------
|
||||
% Otimization
|
||||
% ------------------------------------------------------------------------
|
||||
% -----------------------------------------------------------------------
|
||||
A = []; b = [];
|
||||
Aeq = []; beq = [];
|
||||
lb = []; ub = [];
|
||||
x0 = rand(6*2*traj_par.N,1);
|
||||
|
||||
optns_pttrnSrch = optimoptions('patternsearch');
|
||||
optns_pttrnSrch.Display = 'iter';
|
||||
if strcmp(optmznAlgorithm, 'patternsearch')
|
||||
x0 = rand(6*2*traj_par.N,1);
|
||||
optns_pttrnSrch = optimoptions('patternsearch');
|
||||
optns_pttrnSrch.Display = 'iter';
|
||||
optns_pttrnSrch.StepTolerance = 1e-3;
|
||||
|
||||
x = patternsearch(@(x)traj_cost_lgr(x,traj_par,ur10),x0,A,b,Aeq,beq,lb,ub,...
|
||||
@(x)traj_cnstr(x,traj_par,ur10), optns_pttrnSrch);
|
||||
|
||||
|
||||
return
|
||||
|
||||
options = optimoptions('ga');
|
||||
options.Display = 'iter';
|
||||
options.PlotFcn = 'gaplotbestf'; % {'gaplotbestf', 'gaplotscores'}
|
||||
options.MaxGenerations = 50;
|
||||
options.PopulationSize = 1e+3; % in each generation.
|
||||
options.InitialPopulationRange = [-100; 100];
|
||||
options.SelectionFcn = 'selectionroulette';
|
||||
|
||||
[x,fval,exitflag,output,population,scores] = ga(@(x)traj_cost(x,traj_par,ur10),...
|
||||
6*2*traj_par.N,A,b,Aeq,beq,lb,ub,@(x)traj_cnstr(x,traj_par,ur10),options);
|
||||
[x,fval] = patternsearch(@(x)traj_cost_lgr(x,traj_par,ur10), x0, ...
|
||||
A, b, Aeq, beq, lb, ub, ...
|
||||
@(x)traj_cnstr(x,traj_par,ur10), optns_pttrnSrch);
|
||||
|
||||
elseif strcmp(optmznAlgorithm, 'ga')
|
||||
optns_ga = optimoptions('ga');
|
||||
optns_ga.Display = 'iter';
|
||||
optns_ga.PlotFcn = 'gaplotbestf'; % {'gaplotbestf', 'gaplotscores'}
|
||||
optns_ga.MaxGenerations = 50;
|
||||
optns_ga.PopulationSize = 1e+3; % in each generation.
|
||||
optns_ga.InitialPopulationRange = [-100; 100];
|
||||
optns_ga.SelectionFcn = 'selectionroulette';
|
||||
|
||||
[x,fval] = ga(@(x)traj_cost_lgr(x,traj_par,ur10), 6*2*traj_par.N,...
|
||||
A, b, Aeq, beq, lb, ub, ...
|
||||
@(x)traj_cnstr(x,traj_par,ur10),optns_ga);
|
||||
else
|
||||
error('Chosen algorithm is not found among implemented ones');
|
||||
end
|
||||
|
||||
%% ------------------------------------------------------------------------
|
||||
% Verifying obtained trajectory
|
||||
% ------------------------------------------------------------------------
|
||||
ab = reshape(x,[12,traj_par.N]);
|
||||
a = ab(1:6,:); % sin coeffs
|
||||
b = ab(7:12,:); % cos coeffs
|
||||
c_pol = pol_coeffs(traj_par.T,a,b,traj_par.wf,traj_par.N); % polynomial coeffs
|
||||
[q,qd,q2d] = mixed_traj(traj_par.t,c_pol,a,b,traj_par.wf,traj_par.N);
|
||||
|
||||
traj_cnstr(x,traj_par,ur10)
|
||||
c_pol = getPolCoeffs(traj_par.T, a, b, traj_par.wf, traj_par.N, ur10.q0);
|
||||
[q,qd,q2d] = mixed_traj(traj_par.t, c_pol, a, b, traj_par.wf, traj_par.N);
|
||||
|
||||
|
||||
figure
|
||||
subplot(3,1,1)
|
||||
plot(traj_par.t,q)
|
||||
ylabel('$q$','interpreter','latex')
|
||||
grid on
|
||||
legend('q1','q2','q3','q4','q5','q6')
|
||||
subplot(3,1,2)
|
||||
plot(traj_par.t,qd)
|
||||
ylabel('$\dot{q}$','interpreter','latex')
|
||||
grid on
|
||||
legend('qd1','qd2','qd3','qd4','qd5','qd6')
|
||||
subplot(3,1,3)
|
||||
plot(traj_par.t,q2d)
|
||||
legend('q2d1','q2d2','q2d3','q2d4','q2d5','q2d6')
|
||||
ylabel('$\ddot{q}$','interpreter','latex')
|
||||
grid on
|
||||
legend('q2d1','q2d2','q2d3','q2d4','q2d5','q2d6')
|
||||
|
||||
|
||||
pathToFolder = 'trajectory_optmzn/';
|
||||
if strcmp(optmznAlgorithm, 'patternsearch')
|
||||
t1 = strcat('N',num2str(traj_par.N),'T',num2str(traj_par.T));
|
||||
filename = strcat(pathToFolder,'ptrnSrch_',t1,'.mat');
|
||||
elseif strcmp(optmznAlgorithm, 'ga')
|
||||
t1 = strcat('N',num2str(traj_par.N),'T',num2str(traj_par.T));
|
||||
filename = strcat(pathToFolder,'ga_',t1,'.mat');
|
||||
end
|
||||
save(filename,'a','b','c_pol','traj_par')
|
||||
|
|
|
|||
|
|
@ -0,0 +1,108 @@
|
|||
% ---------------------------------------------------------------------
|
||||
% In this script trajectory optimization otherwise called experiment
|
||||
% design for dynamic paramters identification is carried out.
|
||||
% ---------------------------------------------------------------------
|
||||
run('main_ur.m'); % get robot description
|
||||
|
||||
% Choose optimization algorithm: 'patternsearch', 'ga'
|
||||
optmznAlgorithm = 'patternsearch';
|
||||
|
||||
% Getting limits on posistion and velocities. Moreover we get some
|
||||
% constant parmeters of the robot that allow us to accelerate computation
|
||||
% of the robot regressor.
|
||||
q_min = zeros(6,1); q_max = zeros(6,1); qd_max = zeros(6,1);
|
||||
for i = 1:6
|
||||
rot_axes(:,i) = str2num(ur10.robot.joint{i}.axis.Attributes.xyz)';
|
||||
R_pj = RPY(str2num(ur10.robot.joint{i}.origin.Attributes.rpy));
|
||||
p_pj = str2num(ur10.robot.joint{i}.origin.Attributes.xyz)';
|
||||
T_pj(:,:,i) = [R_pj, p_pj; zeros(1,3), 1];
|
||||
r_com(:,i) = str2num(ur10.robot.link{i+1}.inertial.origin.Attributes.xyz)';
|
||||
|
||||
q_min(i) = str2double(ur10.robot.joint{i}.limit.Attributes.lower);
|
||||
q_max(i) = str2double(ur10.robot.joint{i}.limit.Attributes.upper);
|
||||
qd_max(i) = str2double(ur10.robot.joint{i}.limit.Attributes.velocity);
|
||||
end
|
||||
|
||||
ur10.rot_axes = rot_axes; % axis of rotation of each joint
|
||||
ur10.T_pj = T_pj;
|
||||
ur10.r_com = r_com;
|
||||
ur10.qd_max = qd_max;
|
||||
ur10.q2d_max = 2*ones(6,1);
|
||||
% Use different limit for positions for safety
|
||||
ur10.q_min = deg2rad([-180 -180 -90 -180 -90 -90]');
|
||||
ur10.q_max = deg2rad([180 0 90 0 90 90]');
|
||||
ur10.q0 = deg2rad([0 -90 0 -90 0 0 ]');
|
||||
|
||||
% Trajectory parameters
|
||||
traj_par.T = 20; % period of signal
|
||||
traj_par.wf = 2*pi/traj_par.T; % fundamental frequency
|
||||
traj_par.t_smp = 1e-1; % sampling time
|
||||
traj_par.t = 0:traj_par.t_smp:traj_par.T; % time
|
||||
traj_par.N = 5; % number of harmonics
|
||||
|
||||
|
||||
% ----------------------------------------------------------------------
|
||||
% Otimization
|
||||
% -----------------------------------------------------------------------
|
||||
A = []; b = [];
|
||||
Aeq = []; beq = [];
|
||||
lb = []; ub = [];
|
||||
|
||||
if strcmp(optmznAlgorithm, 'patternsearch')
|
||||
x0 = rand(6*2*traj_par.N,1);
|
||||
optns_pttrnSrch = optimoptions('patternsearch');
|
||||
optns_pttrnSrch.Display = 'iter';
|
||||
optns_pttrnSrch.StepTolerance = 1e-3;
|
||||
|
||||
[x,fval] = patternsearch(@(x)traj_cost_lgr(x,traj_par,ur10), x0, ...
|
||||
A, b, Aeq, beq, lb, ub, ...
|
||||
@(x)traj_cnstr(x,traj_par,ur10), optns_pttrnSrch);
|
||||
|
||||
elseif strcmp(optmznAlgorithm, 'ga')
|
||||
optns_ga = optimoptions('ga');
|
||||
optns_ga.Display = 'iter';
|
||||
optns_ga.PlotFcn = 'gaplotbestf'; % {'gaplotbestf', 'gaplotscores'}
|
||||
optns_ga.MaxGenerations = 50;
|
||||
optns_ga.PopulationSize = 1e+3; % in each generation.
|
||||
optns_ga.InitialPopulationRange = [-100; 100];
|
||||
optns_ga.SelectionFcn = 'selectionroulette';
|
||||
|
||||
[x,fval] = ga(@(x)traj_cost_lgr(x,traj_par,ur10), 6*2*traj_par.N,...
|
||||
A, b, Aeq, beq, lb, ub, ...
|
||||
@(x)traj_cnstr(x,traj_par,ur10),optns_ga);
|
||||
else
|
||||
error('Chosen algorithm is not found among implemented ones');
|
||||
end
|
||||
|
||||
%% ------------------------------------------------------------------------
|
||||
% Verifying obtained trajectory
|
||||
% ------------------------------------------------------------------------
|
||||
ab = reshape(x,[12,traj_par.N]);
|
||||
a = ab(1:6,:); % sin coeffs
|
||||
b = ab(7:12,:); % cos coeffs
|
||||
c_pol = getPolCoeffs(traj_par.T, a, b, traj_par.wf, traj_par.N, ur10.q0);
|
||||
[q,qd,q2d] = mixed_traj(traj_par.t, c_pol, a, b, traj_par.wf, traj_par.N);
|
||||
|
||||
|
||||
if strcmp(optmznAlgorithm, 'patternsearch')
|
||||
strcat('N',num2str(traj_par.N),'T',num2str(traj_par.T))
|
||||
elseif strcmp(optmznAlgorithm, 'ga')
|
||||
|
||||
end
|
||||
|
||||
figure
|
||||
subplot(3,1,1)
|
||||
plot(traj_par.t,q)
|
||||
ylabel('$q$','interpreter','latex')
|
||||
grid on
|
||||
legend('q1','q2','q3','q4','q5','q6')
|
||||
subplot(3,1,2)
|
||||
plot(traj_par.t,qd)
|
||||
ylabel('$\dot{q}$','interpreter','latex')
|
||||
grid on
|
||||
legend('qd1','qd2','qd3','qd4','qd5','qd6')
|
||||
subplot(3,1,3)
|
||||
plot(traj_par.t,q2d)
|
||||
ylabel('$\ddot{q}$','interpreter','latex')
|
||||
grid on
|
||||
legend('q2d1','q2d2','q2d3','q2d4','q2d5','q2d6')
|
||||
|
|
@ -0,0 +1,115 @@
|
|||
% ----------------------------------------------------------------------
|
||||
% In this file full regressor of the robot as well as load regressor
|
||||
% are obtained using Lagrange formulation of dynamics.
|
||||
% ----------------------------------------------------------------------
|
||||
% Get robot description
|
||||
run('main_ur.m')
|
||||
|
||||
generateLoadRegressorFunction = 0;
|
||||
generateFullRegressorFunction = 0;
|
||||
|
||||
% Symbolic generilized coordiates, their first and second deriatives
|
||||
q_sym = sym('q%d',[6,1],'real');
|
||||
qd_sym = sym('qd%d',[6,1],'real');
|
||||
q2d_sym = sym('q2d%d',[6,1],'real');
|
||||
|
||||
% ------------------------------------------------------------------------
|
||||
% Getting gradient of energy functions, to derive dynamics
|
||||
% ------------------------------------------------------------------------
|
||||
T_pk = sym(zeros(4,4,6)); % transformation between links
|
||||
w_kk(:,1) = sym(zeros(3,1)); % angular velocity k in frame k
|
||||
v_kk(:,1) = sym(zeros(3,1)); % linear velocity of the origin of frame k in frame k
|
||||
g_kk(:,1) = sym([0,0,9.81])'; % vector of graviatational accelerations in frame k
|
||||
p_kk(:,1) = sym(zeros(3,1)); % origin of frame k in frame k
|
||||
DK(:,1) = sym(zeros(10,1)); % gradient of kinetic energy
|
||||
DP(:,1) = sym([zeros(1,6),[0,0,9.81],0]'); % gradient of gravitational energy
|
||||
|
||||
for i = 1:6
|
||||
jnt_axs_k = str2num(ur10.robot.joint{i}.axis.Attributes.xyz)';
|
||||
% Transformation from parent link frame p to current joint frame
|
||||
rpy_k = sym(str2num(ur10.robot.joint{i}.origin.Attributes.rpy));
|
||||
R_pj = RPY(rpy_k);
|
||||
R_pj(abs(R_pj)<sqrt(eps)) = sym(0); % to avoid numerical errors
|
||||
p_pj = str2num(ur10.robot.joint{i}.origin.Attributes.xyz)';
|
||||
T_pj = sym([R_pj, p_pj; zeros(1,3), 1]); % to avoid numerical errors
|
||||
% Tranformation from joint frame of the joint that rotaties body k to
|
||||
% link frame. The transformation is pure rotation
|
||||
R_jk = Rot(q_sym(i),sym(jnt_axs_k));
|
||||
p_jk = sym(zeros(3,1));
|
||||
T_jk = [R_jk, p_jk; sym(zeros(1,3)),sym(1)];
|
||||
% Transformation from parent link frame p to current link frame k
|
||||
T_pk(:,:,i) = T_pj*T_jk;
|
||||
z_kk(:,i) = sym(jnt_axs_k);
|
||||
|
||||
w_kk(:,i+1) = T_pk(1:3,1:3,i)'*w_kk(:,i) + sym(jnt_axs_k)*qd_sym(i);
|
||||
v_kk(:,i+1) = T_pk(1:3,1:3,i)'*(v_kk(:,i) + cross(w_kk(:,i),sym(p_pj)));
|
||||
g_kk(:,i+1) = T_pk(1:3,1:3,i)'*g_kk(:,i);
|
||||
p_kk(:,i+1) = T_pk(1:3,1:3,i)'*(p_kk(:,i) + sym(p_pj));
|
||||
|
||||
beta_K(i,:) = [sym(0.5)*w2wtlda(w_kk(:,i+1)),...
|
||||
v_kk(:,i+1)'*vec2skewSymMat(w_kk(:,i+1)),...
|
||||
sym(0.5)*v_kk(:,i+1)'*v_kk(:,i+1)];
|
||||
beta_P(i,:) = [sym(zeros(1,6)), g_kk(:,i+1)',...
|
||||
g_kk(:,i+1)'*p_kk(:,i+1)];
|
||||
end
|
||||
|
||||
|
||||
% --------------------------------------------------------------------
|
||||
% Gradient of the kinetic and potential energy of the load
|
||||
% --------------------------------------------------------------------
|
||||
% Transformation from link 6 frame to end-effector frame
|
||||
rpy_ee = sym(str2num(ur10.robot.joint{7}.origin.Attributes.rpy));
|
||||
R_6ee = RPY(rpy_ee);
|
||||
R_6ee(abs(R_6ee)<sqrt(eps)) = sym(0); % to avoid numerical errors
|
||||
p_6ee = str2num(ur10.robot.joint{7}.origin.Attributes.xyz)';
|
||||
T_6ee = sym([R_6ee, p_6ee; zeros(1,3), 1]); % to avoid numerical errors
|
||||
|
||||
w_eeee = T_6ee(1:3,1:3)'*w_kk(:,7);
|
||||
v_eeee = T_6ee(1:3,1:3)'*(v_kk(:,7) + cross(w_kk(:,i+1),sym(p_6ee)));
|
||||
g_eeee = T_6ee(1:3,1:3)'*g_kk(:,7);
|
||||
p_eeee = T_6ee(1:3,1:3)'*(p_kk(:,7) + sym(p_6ee));
|
||||
|
||||
beta_Kl = [sym(0.5)*w2wtlda(w_eeee), v_eeee'*vec2skewSymMat(w_eeee),...
|
||||
sym(0.5)*(v_eeee'*v_eeee)];
|
||||
|
||||
beta_Pl = [sym(zeros(1,6)), g_eeee', g_eeee'*p_eeee];
|
||||
|
||||
|
||||
% ---------------------------------------------------------------------
|
||||
% Dynamic regressor of the load
|
||||
% ---------------------------------------------------------------------
|
||||
Lagrl = beta_Kl - beta_Pl;
|
||||
dLagrl_dq = jacobian(Lagrl,q_sym)';
|
||||
dLagrl_dqd = jacobian(Lagrl,qd_sym)';
|
||||
tl = sym(zeros(6,10));
|
||||
for i = 1:6
|
||||
tl = tl + diff(dLagrl_dqd,q_sym(i))*qd_sym(i)+...
|
||||
diff(dLagrl_dqd,qd_sym(i))*q2d_sym(i);
|
||||
end
|
||||
Y_l = tl - dLagrl_dq;
|
||||
|
||||
if generateLoadRegressorFunction
|
||||
matlabFunction(Y_l,'File','autogen/load_regressor_UR10E',...
|
||||
'Vars',{q_sym,qd_sym,q2d_sym});
|
||||
end
|
||||
|
||||
|
||||
% ---------------------------------------------------------------------
|
||||
% Dynamic regressor of the full paramters
|
||||
% ---------------------------------------------------------------------
|
||||
Lagrf = [beta_K(1,:) - beta_P(1,:), beta_K(2,:) - beta_P(2,:),...
|
||||
beta_K(3,:) - beta_P(3,:), beta_K(4,:) - beta_P(4,:),...
|
||||
beta_K(5,:) - beta_P(5,:), beta_K(6,:) - beta_P(6,:)];
|
||||
dLagrf_dq = jacobian(Lagrf,q_sym)';
|
||||
dLagrf_dqd = jacobian(Lagrf,qd_sym)';
|
||||
tf = sym(zeros(6,60));
|
||||
for i = 1:6
|
||||
tf = tf + diff(dLagrf_dqd,q_sym(i))*qd_sym(i)+...
|
||||
diff(dLagrf_dqd,qd_sym(i))*q2d_sym(i);
|
||||
end
|
||||
Y_f = tf - dLagrf_dq;
|
||||
|
||||
if generateFullRegressorFunction
|
||||
matlabFunction(Y_f,'File','autogen/full_regressor_UR10E',...
|
||||
'Vars',{q_sym,qd_sym,q2d_sym});
|
||||
end
|
||||
Loading…
Reference in New Issue