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Q = INVERSEKINEMATIC_KUKA_KR5_SIXX_R850(robot, T)
Solves the inverse kinematic problem for the ABB IRB 140 robot
where:
robot stores the robot parameters.
T is an homogeneous transform that specifies the position/orientation
of the end effector.
A call to Q=INVERSEKINEMATIC_KUKA_KR5_SIXX_R850 returns 8 possible solutions, thus,
Q is a 6x8 matrix where each column stores 6 feasible joint values.
Example code:
robot=load_robot('kuka', 'KR5_sixx_R850');
q = [0 0 0 0 0 0];
T = directkinematic(robot, q);
%Call the inversekinematic for this robot
qinv = inversekinematic(robot, T);
check that all of them are feasible solutions!
and every Ti equals T
for i=1:8,
Ti = directkinematic(robot, qinv(:,i))
end
See also DIRECTKINEMATIC.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%0001 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0002 % Q = INVERSEKINEMATIC_KUKA_KR5_SIXX_R850(robot, T) 0003 % Solves the inverse kinematic problem for the ABB IRB 140 robot 0004 % where: 0005 % robot stores the robot parameters. 0006 % T is an homogeneous transform that specifies the position/orientation 0007 % of the end effector. 0008 % 0009 % A call to Q=INVERSEKINEMATIC_KUKA_KR5_SIXX_R850 returns 8 possible solutions, thus, 0010 % Q is a 6x8 matrix where each column stores 6 feasible joint values. 0011 % 0012 % 0013 % Example code: 0014 % 0015 % robot=load_robot('kuka', 'KR5_sixx_R850'); 0016 % q = [0 0 0 0 0 0]; 0017 % T = directkinematic(robot, q); 0018 % %Call the inversekinematic for this robot 0019 % qinv = inversekinematic(robot, T); 0020 % check that all of them are feasible solutions! 0021 % and every Ti equals T 0022 % for i=1:8, 0023 % Ti = directkinematic(robot, qinv(:,i)) 0024 % end 0025 % See also DIRECTKINEMATIC. 0026 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0027 0028 0029 % Copyright (C) 2012, by Arturo Gil Aparicio 0030 % 0031 % This file is part of ARTE (A Robotics Toolbox for Education). 0032 % 0033 % ARTE is free software: you can redistribute it and/or modify 0034 % it under the terms of the GNU Lesser General Public License as published by 0035 % the Free Software Foundation, either version 3 of the License, or 0036 % (at your option) any later version. 0037 % 0038 % ARTE is distributed in the hope that it will be useful, 0039 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0040 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0041 % GNU Lesser General Public License for more details. 0042 % 0043 % You should have received a copy of the GNU Leser General Public License 0044 % along with ARTE. If not, see <http://www.gnu.org/licenses/>. 0045 function q = inversekinematic_kuka_kr5_sixx_r850(robot, T) 0046 0047 %initialize q, 0048 %eight possible solutions are generally feasible 0049 q=zeros(6,8); 0050 0051 % %Evaluate the parameters 0052 % theta = eval(robot.DH.theta); 0053 d = eval(robot.DH.d); 0054 L6=abs(d(6)); 0055 0056 0057 %T= [ nx ox ax Px; 0058 % ny oy ay Py; 0059 % nz oz az Pz]; 0060 Px=T(1,4); 0061 Py=T(2,4); 0062 Pz=T(3,4); 0063 0064 %Compute the position of the wrist, being W the Z component of the end effector's system 0065 W = T(1:3,3); 0066 0067 % Pm: wrist position 0068 Pm = [Px Py Pz]' - L6*W; 0069 0070 %first joint, two possible solutions admited: 0071 % if q(1) is a solution, then q(1) + pi is also a solution 0072 q1=atan2(Pm(2), Pm(1)); 0073 0074 0075 %solve for q2 0076 q2_1=solve_for_theta2(robot, [q1 0 0 0 0 0 0], Pm); 0077 %the other possible solution is q1 + pi 0078 q2_2=solve_for_theta2(robot, [q1+pi 0 0 0 0 0 0], Pm); 0079 0080 %solve for q3 0081 q3_1=solve_for_theta3(robot, [q1 0 0 0 0 0 0], Pm); 0082 %solver for q3 for both cases 0083 q3_2=solve_for_theta3(robot, [q1+pi 0 0 0 0 0 0], Pm); 0084 0085 0086 %the next matrix doubles each column. For each two columns, two different 0087 %configurations for theta4, theta5 and theta6 will be computed. These 0088 %configurations are generally referred as wrist up and wrist down solution 0089 q = [q1 q1 q1 q1 q1+pi q1+pi q1+pi q1+pi; 0090 q2_1(1) q2_1(1) q2_1(2) q2_1(2) q2_2(1) q2_2(1) q2_2(2) q2_2(2); 0091 q3_1(1) q3_1(1) q3_1(2) q3_1(2) q3_2(1) q3_2(1) q3_2(2) q3_2(2); 0092 0 0 0 0 0 0 0 0; 0093 0 0 0 0 0 0 0 0; 0094 0 0 0 0 0 0 0 0]; 0095 %leave only the real part of the solutions 0096 q=real(q); 0097 0098 %Note that in this robot, the joint q3 has a non-simmetrical range. In this 0099 %case, the joint ranges from 60 deg to -219 deg, thus, the typical normalizing 0100 %step is avoided in this angle (the next line is commented). When solving 0101 %for the orientation, the solutions are normalized to the [-pi, pi] range 0102 %only for the theta4, theta5 and theta6 joints. 0103 0104 %normalize q to [-pi, pi] 0105 q(1,:) = normalize(q(1,:)); 0106 q(2,:) = normalize(q(2,:)); 0107 0108 0109 0110 % for any of the possible combinations (theta1, theta2, theta3) 0111 for i=1:2:size(q,2), 0112 qtemp = solve_spherical_wrist(robot, q(:,i), T, 1,'geometric'); %wrist up 0113 qtemp(4:6)=normalize(qtemp(4:6)); 0114 q(:,i)=qtemp; 0115 0116 qtemp = solve_spherical_wrist(robot, q(:,i), T, -1, 'geometric'); %wrist up 0117 qtemp(4:6)=normalize(qtemp(4:6)); 0118 q(:,i+1)=qtemp; 0119 end 0120 0121 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0122 % solve for second joint theta2, two different 0123 % solutions are returned, corresponding 0124 % to elbow up and down solution 0125 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0126 function q2 = solve_for_theta2(robot, q, Pm) 0127 0128 %Evaluate the parameters 0129 theta = eval(robot.DH.theta); 0130 d = eval(robot.DH.d); 0131 a = eval(robot.DH.a); 0132 alpha = eval(robot.DH.alpha); 0133 0134 %See geometry 0135 L2=abs(a(2)); 0136 L3=abs(d(4)); 0137 A2 = abs(a(3)); 0138 0139 %See geometry of the robot 0140 %compute L4 0141 L4 = sqrt(A2^2 + L3^2); 0142 0143 %The inverse kinematic problem can be solved as in the IRB 140 (for example) 0144 0145 %given q1 is known, compute first DH transformation 0146 T01=dh(robot, q, 1); 0147 0148 %Express Pm in the reference system 1, for convenience 0149 p1 = inv(T01)*[Pm; 1]; 0150 0151 r = sqrt(p1(1)^2 + p1(2)^2); 0152 0153 beta = atan2(-p1(2), p1(1)); 0154 gamma = real(acos((L2^2+r^2-L4^2)/(2*r*L2))); 0155 0156 %return two possible solutions 0157 %elbow up and elbow down 0158 %the order here is important and is coordinated with the function 0159 %solve_for_theta3 0160 q2(1) = pi/2 - beta - gamma; %elbow up 0161 q2(2) = pi/2 - beta + gamma; %elbow down 0162 0163 0164 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0165 % solve for third joint theta3, two different 0166 % solutions are returned, corresponding 0167 % to elbow up and down solution 0168 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0169 function q3 = solve_for_theta3(robot, q, Pm) 0170 0171 %Evaluate the parameters 0172 theta = eval(robot.DH.theta); 0173 d = eval(robot.DH.d); 0174 a = eval(robot.DH.a); 0175 alpha = eval(robot.DH.alpha); 0176 0177 %See geometry 0178 L2=abs(a(2)); 0179 L3=abs(d(4)); 0180 0181 A2 = abs(a(3)); 0182 0183 %See geometry of the robot 0184 %compute L4 0185 L4 = sqrt(A2^2 + L3^2); 0186 0187 %the angle phi is fixed 0188 phi=acos((A2^2+L4^2-L3^2)/(2*A2*L4)); 0189 0190 %given q1 is known, compute first DH transformation 0191 T01=dh(robot, q, 1); 0192 0193 %Express Pm in the reference system 1, for convenience 0194 p1 = inv(T01)*[Pm; 1]; 0195 0196 r = sqrt(p1(1)^2 + p1(2)^2); 0197 0198 eta = real(acos((L2^2 + L4^2 - r^2)/(2*L2*L4))); 0199 0200 %return two possible solutions 0201 %elbow up and elbow down solutions 0202 %the order here is important 0203 q3(1) = pi - phi - eta; 0204 q3(2) = pi - phi + eta; 0205