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Q = INVERSEKINEMATIC_IRB7600_400_255_m2000(robot, T)
Solves the inverse kinematic problem for the ABB IRB7600_400_255_m2000 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_IRB7600_400_255_m2000 returns 8 possible solutions, thus,
Q is a 6x8 matrix where each column stores 6 feasible joint values.
Example code:
abb=load_robot('ABB', 'IRB7600_400_255_m2000');
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_IRB7600_400_255_m2000(robot, T) 0003 % Solves the inverse kinematic problem for the ABB IRB7600_400_255_m2000 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_IRB7600_400_255_m2000 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', 'IRB7600_400_255_m2000'); 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_irb7600_400_255_m2000(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 %leave only the real part of the solutions 0100 q=real(q); 0101 0102 %Note that in this robot, the joint q3 has a non-simmetrical range. In this 0103 %case, the joint ranges from 60 deg to -219 deg, thus, the typical normalizing 0104 %step is avoided in this angle (the next line is commented). When solving 0105 %for the orientation, the solutions are normalized to the [-pi, pi] range 0106 %only for the theta4, theta5 and theta6 joints. 0107 0108 %normalize q to [-pi, pi] 0109 q(1,:) = normalize(q(1,:)); 0110 q(2,:) = normalize(q(2,:)); 0111 0112 % solve for the last three joints 0113 % for any of the possible combinations (theta1, theta2, theta3) 0114 for i=1:2:size(q,2), 0115 % use solve_spherical_wrist2 for the particular orientation 0116 % of the systems in this ABB robot 0117 % use either the geometric or algebraic method. 0118 % the function solve_spherical_wrist2 is used due to the relative 0119 % orientation of the last three DH reference systems. 0120 0121 %use either one algebraic method or the geometric 0122 %qtemp = solve_spherical_wrist2(robot, q(:,i), T, 1, 'geometric'); %wrist up 0123 qtemp = solve_spherical_wrist2(robot, q(:,i), T, 1,'algebraic'); %wrist up 0124 qtemp(4:6)=normalize(qtemp(4:6)); 0125 q(:,i)=qtemp; 0126 0127 %qtemp = solve_spherical_wrist2(robot, q(:,i), T, -1, 'geometric'); %wrist down 0128 qtemp = solve_spherical_wrist2(robot, q(:,i), T, -1, 'algebraic'); %wrist down 0129 qtemp(4:6)=normalize(qtemp(4:6)); 0130 q(:,i+1)=qtemp; 0131 end 0132 0133 0134 0135 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0136 % solve for second joint theta2, two different 0137 % solutions are returned, corresponding 0138 % to elbow up and down solution 0139 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0140 function q2 = solve_for_theta2(robot, q, Pm) 0141 0142 %Evaluate the parameters 0143 theta = eval(robot.DH.theta); 0144 d = eval(robot.DH.d); 0145 a = eval(robot.DH.a); 0146 alpha = eval(robot.DH.alpha); 0147 0148 %See geometry 0149 L2=a(2); 0150 L3=d(4); 0151 A2=a(3); 0152 0153 %See geometry of the robot 0154 %compute L4 0155 L4 = sqrt(A2^2 + L3^2); 0156 0157 %The inverse kinematic problem can be solved as in the IRB 140 (for example) 0158 0159 %given q1 is known, compute first DH transformation 0160 T01=dh(robot, q, 1); 0161 0162 %Express Pm in the reference system 1, for convenience 0163 p1 = inv(T01)*[Pm; 1]; 0164 0165 r = sqrt(p1(1)^2 + p1(2)^2); 0166 0167 beta = atan2(-p1(2), p1(1)); 0168 gamma = real(acos((L2^2+r^2-L4^2)/(2*r*L2))); 0169 0170 %return two possible solutions 0171 %elbow up and elbow down 0172 %the order here is important and is coordinated with the function 0173 %solve_for_theta3 0174 q2(1) = pi/2 - beta - gamma; %elbow up 0175 q2(2) = pi/2 - beta + gamma; %elbow down 0176 0177 0178 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0179 % solve for third joint theta3, two different 0180 % solutions are returned, corresponding 0181 % to elbow up and down solution 0182 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0183 function q3 = solve_for_theta3(robot, q, Pm) 0184 0185 %Evaluate the parameters 0186 theta = eval(robot.DH.theta); 0187 d = eval(robot.DH.d); 0188 a = eval(robot.DH.a); 0189 alpha = eval(robot.DH.alpha); 0190 0191 %See geometry 0192 L2=a(2); 0193 L3=d(4); 0194 A2=a(3); 0195 0196 %See geometry of the robot 0197 %compute L4 0198 L4 = sqrt(A2^2 + L3^2); 0199 0200 %the angle phi is fixed 0201 phi=acos((A2^2+L4^2-L3^2)/(2*A2*L4)); 0202 0203 %given q1 is known, compute first DH transformation 0204 T01=dh(robot, q, 1); 0205 0206 %Express Pm in the reference system 1, for convenience 0207 p1 = inv(T01)*[Pm; 1]; 0208 0209 r = sqrt(p1(1)^2 + p1(2)^2); 0210 0211 eta = real(acos((L2^2 + L4^2 - r^2)/(2*L2*L4))); 0212 0213 %return two possible solutions 0214 %elbow up and elbow down solutions 0215 %the order here is important 0216 q3(1) = pi - phi- eta; 0217 q3(2) = pi - phi + eta; 0218