98 lines
4.0 KiB
Matlab
98 lines
4.0 KiB
Matlab
function generate_rb_dynamics(path_to_urdf)
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% ----------------------------------------------------------------------
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% Function generates rigid body dynamics equations, namely M, C matrices
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% and g vector: M(q) ddq + C(q, dq) dq + g(q) = tau
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% ----------------------------------------------------------------------
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% Parse urdf to get robot description
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ur10 = parse_urdf(path_to_urdf);
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% Create symbolic generilized coordiates, their first and second deriatives
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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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% ------------------------------------------------------------------------
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% Getting gradient of energy functions, to derive dynamics
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% ------------------------------------------------------------------------
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T_pk = sym(zeros(4,4,6)); % transformation between links
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w_kk(:,1) = sym(zeros(3,1)); % angular velocity k in frame k
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v_kk(:,1) = sym(zeros(3,1)); % linear velocity of the origin of frame k in frame k
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g_kk(:,1) = sym([0,0,9.81])'; % vector of graviatational accelerations in frame k
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p_kk(:,1) = sym(zeros(3,1)); % origin of frame k in frame k
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for i = 1:6
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jnt_axs_k = str2num(ur10.robot.joint{i}.axis.Attributes.xyz)';
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% Transformation from parent link frame p to current joint frame
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rpy_k = sym(str2num(ur10.robot.joint{i}.origin.Attributes.rpy));
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R_pj = RPY(rpy_k);
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R_pj(abs(R_pj)<sqrt(eps)) = sym(0); % to avoid numerical errors
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p_pj = str2num(ur10.robot.joint{i}.origin.Attributes.xyz)';
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T_pj = sym([R_pj, p_pj; zeros(1,3), 1]); % to avoid numerical errors
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% Tranformation from joint frame of the joint that rotaties body k to
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% link frame. The transformation is pure rotation
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R_jk = Rot(q_sym(i),sym(jnt_axs_k));
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p_jk = sym(zeros(3,1));
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T_jk = [R_jk, p_jk; sym(zeros(1,3)),sym(1)];
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% Transformation from parent link frame p to current link frame k
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T_pk(:,:,i) = T_pj*T_jk;
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z_kk(:,i) = sym(jnt_axs_k);
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w_kk(:,i+1) = T_pk(1:3,1:3,i)'*w_kk(:,i) + sym(jnt_axs_k)*qd_sym(i);
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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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beta_K(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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beta_P(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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end
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beta_Lf = [beta_K(1,:) - beta_P(1,:), beta_K(2,:) - beta_P(2,:),...
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beta_K(3,:) - beta_P(3,:), beta_K(4,:) - beta_P(4,:),...
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beta_K(5,:) - beta_P(5,:), beta_K(6,:) - beta_P(6,:)];
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% Lagrangian dynamics
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pi_sndrd_sym = sym('pi%d%d', [60,1], 'real'); % standard parameters
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Lagr = beta_Lf*pi_sndrd_sym; % Lagrangian of the system
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P = [beta_P(1,:), beta_P(2,:), beta_P(3,:), beta_P(4,:),...
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beta_P(5,:), beta_P(6,:)]*pi_sndrd_sym; % Potential energy
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dLagr_dqd = jacobian(Lagr, qd_sym)';
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% Get inertia matrix M and gravity vector G
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M_mtrx_sym = jacobian(dLagr_dqd, qd_sym);
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G_vctr_sym = jacobian(P, q_sym)';
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% Get velocity matrix C using Christoffel symbols of the first kind
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cs1 = sym(zeros(6,6,6));
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for i = 1:1:6
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for j = 1:1:6
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for k = 1:1:6
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cs1(i,j,k) = 0.5*(diff(M_mtrx_sym(i,j), q_sym(k)) + ...
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diff(M_mtrx_sym(i,k), q_sym(j)) - ...
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diff(M_mtrx_sym(j,k), q_sym(i)));
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end
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end
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end
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C_mtrx_sym = sym(zeros(6, 6));
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for i = 1:1:6
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for j = 1:1:6
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for k = 1:1:6
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C_mtrx_sym(i,j) = C_mtrx_sym(i,j)+cs1(i,j,k)*qd_sym(k);
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end
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end
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end
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% Generate functions
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matlabFunction(M_mtrx_sym, 'File','autogen/M_mtrx_fcn',...
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'Vars',{q_sym, pi_sndrd_sym}, 'Optimize', true);
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matlabFunction(C_mtrx_sym, 'File','autogen/C_mtrx_fcn',...
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'Vars',{q_sym, qd_sym, pi_sndrd_sym}, 'Optimize', false);
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matlabFunction(G_vctr_sym, 'File','autogen/G_vctr_fcn',...
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'Vars',{q_sym, pi_sndrd_sym}, 'Optimize', true);
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