KINEMATICS ANALYSIS OF ROBOTS (Part 4)

This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. After this lecture, the student should be able to: Solve problems of robot kinematics analysis using transformation matrices Kinematics Analysis of Robots IV

Example: A 3 DOF RRR Robot Inverse Kinematics if no restriction on the orientation of point “P”. This is useful if we are concerned only on the position of “P” Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A=3 B=2C=1 P

Example: A 3 DOF RRR Robot In this case we are not given the following matrix for the description of the orientation and position of point “P”: Instead, we need to find  1,  2, and  3 given only the location of “P” w.r.t. to the base frame:

Example: A 3 DOF RRR Robot We know that

Example: A 3 DOF RRR Robot Equate elements: Be Careful!

Example: A 3 DOF RRR Robot Be careful when using your calculator to find inverse tangent!  y x 1 st quadrant (x=+ve, y=+ve) 2 nd quadrant (x=-ve, y=+ve) 4 th quadrant (x=+ve, y=-ve) 3 rd quadrant (x=-ve, y=-ve) Your calculator can only give the angle in the 1 st quadrant. You have to adjust the answer from the calculator

Example: A 3 DOF RRR Robot If cos(  1 )  0, let Otherwise, let

Example: A 3 DOF RRR Robot We now have From the comparison of the 3 rd row elements: Combining these equations, we get

Example: A 3 DOF RRR Robot Using Let

Example: A 3 DOF RRR Robot We can now solve the above equations to get If a  0, then If b  0, then

Example: A 3 DOF RRR Robot Y 0, Y 1 X 0, X 1 Z 0, Z 1 Z2Z2 X2X2 Y2Y2 Z3Z3 X3X3 Y3Y3 A=3 B=2C=1 P Now find  1,  2, and  3 to move point “P” to

Example: A 3 DOF RRR Robot Now cos(  1 )  0, let I.e. Assume elbow down

Example: A 3 DOF RRR Robot Now a  0, then Compare this with the results form the previous lecture! The difference is that now we are not concerned with the orientation of point “P”.

Example: A 3 DOF RPR Robot Link and Joint Assignment Link (0)Link (1)Link (2)Link (3) Revolute joint Prismatic joint Revolute joint

Example: A 3 DOF RPR Robot Frame Assignment Z1Z1 Z1Z1 X1X1 X1X1 Y1Y1

Example: A 3 DOF RPR Robot Frame Assignment Z 0, Z 1 Y 0, Y 1 X 0, X 1 Z2Z2 Z2Z2 X2X2 Y2Y2 Y2Y2

Example: A 3 DOF RPR Robot Frame Assignment Z 0, Z 1 Y 0, Y 1 X 0, X 1 Z2Z2 Y2Y2 X2X2 Z3Z3 X3X3 Y3Y3 Y3Y3

Example: A 3 DOF RPR Robot Frame Assignment Z 0, Z 1 Y 0, Y 1 X 0, X 1 Z2Z2 Y2Y2 X2X2 Z3Z3 Y3Y3 X3X3 11 d2d2 33

Example: A 3 DOF RPR Robot Tabulation of D-H parameters Z 0, Z 1 Y 0, Y 1 X 0, X 1 Z2Z2 Y2Y2 X2X2 Z3Z3 Y3Y3 X3X3  0 = (angle from Z 0 to Z 1 measured along X 0 ) = 0° a 0 = (distance from Z 0 to Z 1 measured along X 0 ) = 0 d 1 = (distance from X 0 to X 1 measured along Z 1 )= 0  1 = variable (angle from X 0 to X 1 measured along Z 1 )  1 = 0° (at home position) but  1 can change as the arm moves

Example: A 3 DOF RPR Robot Tabulation of D-H parameters Z 0, Z 1 Y 0, Y 1 X 0, X 1 Z2Z2 Y2Y2 X2X2 Z3Z3 Y3Y3 X3X3  1 = (angle from Z 1 to Z 2 measured along X 1 ) = -90° a 1 = (distance from Z 1 to Z 2 measured along X 1 ) = 0 d 2 = variable (distance from X 1 to X 2 measured along Z 2 )  2 = variable (angle from X 1 to X 2 measured along Z 2 ) = 180°

Example: A 3 DOF RPR Robot Tabulation of D-H parameters Z 0, Z 1 Y 0, Y 1 X 0, X 1 Z2Z2 Y2Y2 X2X2 Z3Z3 Y3Y3 X3X3  2 = (angle from Z 2 to Z 3 measured along X 2 ) = 0° a 2 = (distance from Z 2 to Z 3 measured along X 2 ) = 0 d 3 = (distance from X 2 to X 3 measured along Z 3 ) = A  3 = variable (angle from X 2 to X 3 measured along Z 3 )  3 = 0° (at home position) but  3 can change as the arm moves A

Link i Twist  i Link length a i Link offset d i Joint angle  i i=000…… i=1-90°00  1 (  1 =0° at home position) i=200d2d2  2 =180° i=3……A  3 (  3 =-0° at home position) Summary of D-H parameters

Example: A 3 DOF RPR Robot Tabulation of Transformation Matrices from the D-H table

Example: A 3 DOF RPR Robot Tabulation of Transformation Matrices from the D-H table

Example: A 3 DOF RPR Robot Once all the transformation matrices are obtained, you can then proceed to get the overall transformation matrix for the forward kinematics. After that, given the position and orientation of the point “P” on the gripper, you can proceed to compare the terms of the matrices to get the inverse kinematics. For the inverse kinematics, you will be solving for  1, d 2, and  3 Try it out as Homework.

Summary This lecture continues the discussion on the analysis of the forward and inverse kinematics of robots. The following were covered: Problems of robot kinematics analysis using transformation matrices