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Q = INVERSEKINEMATIC_IRB120(robot, T)
Solves the inverse kinematic problem for the ABB IRB 120 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_IRB120 returns 8 possible solutions, thus,
Q is a 6x8 matrix where each column stores 6 feasible joint values.
Example code:
abb=load_robot('abb', 'IRB120');
q = [0 0 0 0 0 0];
T = directkinematic(abb, q);
%Call the inversekinematic for this robot
qinv = inversekinematic(abb, T);
check that all of them are feasible solutions!
and every Ti equals T
for i=1:8,
Ti = directkinematic(abb, qinv(:,i))
end
See also DIRECTKINEMATIC.
Author: Arturo Gil Aparicio
Universitas Miguel Hernandez, SPAIN.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%0001 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0002 % Q = INVERSEKINEMATIC_IRB120(robot, T) 0003 % Solves the inverse kinematic problem for the ABB IRB 120 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_IRB120 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 % abb=load_robot('abb', 'IRB120'); 0016 % q = [0 0 0 0 0 0]; 0017 % T = directkinematic(abb, q); 0018 % %Call the inversekinematic for this robot 0019 % qinv = inversekinematic(abb, T); 0020 % check that all of them are feasible solutions! 0021 % and every Ti equals T 0022 % for i=1:8, 0023 % Ti = directkinematic(abb, qinv(:,i)) 0024 % end 0025 % See also DIRECTKINEMATIC. 0026 % 0027 % Author: Arturo Gil Aparicio 0028 % Universitas Miguel Hernandez, SPAIN. 0029 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0030 0031 % Copyright (C) 2012, by Arturo Gil Aparicio 0032 % 0033 % This file is part of ARTE (A Robotics Toolbox for Education). 0034 % 0035 % ARTE is free software: you can redistribute it and/or modify 0036 % it under the terms of the GNU Lesser General Public License as published by 0037 % the Free Software Foundation, either version 3 of the License, or 0038 % (at your option) any later version. 0039 % 0040 % ARTE is distributed in the hope that it will be useful, 0041 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0042 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0043 % GNU Lesser General Public License for more details. 0044 % 0045 % You should have received a copy of the GNU Leser General Public License 0046 % along with ARTE. If not, see <http://www.gnu.org/licenses/>. 0047 function q = inversekinematic_irb120(robot, T) 0048 0049 %initialize q, 0050 %eight possible solutions are generally feasible 0051 q=zeros(6,8); 0052 0053 % %Evaluate the parameters 0054 % theta = eval(robot.DH.theta); 0055 d = eval(robot.DH.d); 0056 L6=d(6); 0057 0058 0059 %T= [ nx ox ax Px; 0060 % ny oy ay Py; 0061 % nz oz az Pz]; 0062 Px=T(1,4); 0063 Py=T(2,4); 0064 Pz=T(3,4); 0065 0066 %Compute the position of the wrist, being W the Z component of the end effector's system 0067 W = T(1:3,3); 0068 0069 % Pm: wrist position 0070 Pm = [Px Py Pz]' - L6*W; 0071 0072 %first joint, two possible solutions admited: 0073 % if q(1) is a solution, then q(1) + pi is also a solution 0074 q1=atan2(Pm(2), Pm(1)); 0075 0076 0077 %solve for q2 0078 q2_1=solve_for_theta2(robot, [q1 0 0 0 0 0 0], Pm); 0079 %the other possible solution is q1 + pi 0080 q2_2=solve_for_theta2(robot, [q1+pi 0 0 0 0 0 0], Pm); 0081 0082 %solve for q3 0083 q3_1=solve_for_theta3(robot, [q1 0 0 0 0 0 0], Pm); 0084 %solver for q3 for both cases 0085 q3_2=solve_for_theta3(robot, [q1+pi 0 0 0 0 0 0], Pm); 0086 0087 0088 0089 %the next matrix doubles each column. For each two columns, two different 0090 %configurations for theta4, theta5 and theta6 will be computed. These 0091 %configurations are generally referred as wrist up and wrist down solution 0092 q = [q1 q1 q1 q1 q1+pi q1+pi q1+pi q1+pi; 0093 q2_1(1) q2_1(1) q2_1(2) q2_1(2) q2_2(1) q2_2(1) q2_2(2) q2_2(2); 0094 q3_1(1) q3_1(1) q3_1(2) q3_1(2) q3_2(1) q3_2(1) q3_2(2) q3_2(2); 0095 0 0 0 0 0 0 0 0; 0096 0 0 0 0 0 0 0 0; 0097 0 0 0 0 0 0 0 0]; 0098 0099 0100 %leave only the real part of the solutions 0101 q=real(q); 0102 0103 %Note that in this robot, the joint q3 has a non-simmetrical range. In this 0104 %case, the joint ranges from 60� to -219�, thus, the typical normalizing 0105 %step is avoided in this angle (the next line is commented). When solving 0106 %for the orientation, the solutions are normalized to the [-pi, pi] range 0107 %only for the theta4, theta5 and theta6 joints. 0108 0109 %normalize q to [-pi, pi] 0110 q(1,:) = normalize(q(1,:)); 0111 q(2,:) = normalize(q(2,:)); 0112 0113 % solve for the last three joints 0114 % for any of the possible combinations (theta1, theta2, theta3) 0115 for i=1:2:size(q,2), 0116 qtemp = solve_spherical_wrist2(robot, q(:,i), T, 1,'algebraic'); %wrist up 0117 qtemp(4:6)=normalize(qtemp(4:6)); 0118 q(:,i)=qtemp; 0119 0120 qtemp = solve_spherical_wrist2(robot, q(:,i), T, -1, 'algebraic'); %wrist up 0121 qtemp(4:6)=normalize(qtemp(4:6)); 0122 q(:,i+1)=qtemp; 0123 end 0124 0125 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0126 % solve for second joint theta2, two different 0127 % solutions are returned, corresponding 0128 % to elbow up and down solution 0129 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0130 function q2 = solve_for_theta2(robot, q, Pm) 0131 0132 %Evaluate the parameters 0133 %theta = eval(robot.DH.theta); 0134 d = eval(robot.DH.d); 0135 a = eval(robot.DH.a); 0136 %alpha = eval(robot.DH.alpha); 0137 0138 %See geometry 0139 L2=a(2); 0140 L3=d(4); 0141 A2=a(3); 0142 0143 %See geometry of the robot 0144 %compute L4 0145 L4 = sqrt(A2^2 + L3^2); 0146 0147 %The inverse kinematic problem can be solved as in the IRB 140 (for example) 0148 0149 %given q1 is known, compute first DH transformation 0150 T01=dh(robot, q, 1); 0151 0152 %Express Pm in the reference system 1, for convenience 0153 p1 = inv(T01)*[Pm; 1]; 0154 0155 r = sqrt(p1(1)^2 + p1(2)^2); 0156 0157 beta = atan2(-p1(2), p1(1)); 0158 gamma = real(acos((L2^2+r^2-L4^2)/(2*r*L2))); 0159 0160 %return two possible solutions 0161 %elbow up and elbow down 0162 %the order here is important and is coordinated with the function 0163 %solve_for_theta3 0164 q2(1) = pi/2 - beta - gamma; %elbow up 0165 q2(2) = pi/2 - beta + gamma; %elbow down 0166 0167 0168 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0169 % solve for third joint theta3, two different 0170 % solutions are returned, corresponding 0171 % to elbow up and down solution 0172 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0173 function q3 = solve_for_theta3(robot, q, Pm) 0174 0175 %Evaluate the parameters 0176 %theta = eval(robot.DH.theta); 0177 d = eval(robot.DH.d); 0178 a = eval(robot.DH.a); 0179 %alpha = eval(robot.DH.alpha); 0180 0181 %See geometry 0182 L2=a(2); 0183 L3=d(4); 0184 A2=a(3); 0185 0186 %See geometry of the robot 0187 %compute L4 0188 L4 = sqrt(A2^2 + L3^2); 0189 0190 %the angle phi is fixed 0191 phi=acos((A2^2+L4^2-L3^2)/(2*A2*L4)); 0192 0193 %given q1 is known, compute first DH transformation 0194 T01=dh(robot, q, 1); 0195 0196 %Express Pm in the reference system 1, for convenience 0197 p1 = inv(T01)*[Pm; 1]; 0198 0199 r = sqrt(p1(1)^2 + p1(2)^2); 0200 0201 eta = real(acos((L2^2 + L4^2 - r^2)/(2*L2*L4))); 0202 0203 %return two possible solutions 0204 %elbow up and elbow down solutions 0205 %the order here is important 0206 q3(1) = pi - phi- eta; 0207 q3(2) = pi - phi + eta; 0208 0209 0210