ECE 450 Introduction to Robotics

Section: 50883

Instructor: Linda A. Gee

10/07/99

Lecture 11

Lecture 11 2

Joint 2 Solution Method

• Project p onto the x1y1 plane

• Use ARM and ELBOW indicators

• Solve for 2 in terms of sin 2 and cos 2

2 = tan-1 (sin 2/cos 2) where - 2

Lecture 11 3

Arm Configurations

*Fu, Page 63

Lecture 11 4

Joint 2 Solution Setup

*Fu, Page 67

Lecture 11 5

Joint 3 Solution Method

• Project p onto the x2y2 plane

• Define (2p4)y

• Define 3 = -

• Solve for 3 in terms of sin 3 and cos 3

3 = tan-1 (sin 3/cos 3) where - 3

Lecture 11 6

Joint 3 Solution Setup

*Fu, Page 68

Lecture 11 7

PUMA Robot

*Fu, page 37

Lecture 11 8

Arm Solution for the Last Three Joints (i = 4,5,6)

• Having solved for the first three joint angles (1, 2, 3), defines the transformation matrix 0T3

• Set joint 4 so rotation about joint 5 aligns with axis of motion of joint 6 with the approach vector, a

» z4 = (z3 x a)/ z3 x a

• Set joint 5 to align axis of motion of joint 6 with a» a = z5

• Set joint 6 to align orientation vector, s = y6 and n

» s = y6

Lecture 11 9

Joint 4 Solution Setup

x3

4

4

x4

y3

z4

*wrist orientation affects sign

sin4 = - (z4 • x3)

cos4 = z4 • y3

Lecture 11 10

Hand Coordinate System

*Fu, page 43

Lecture 11 11

Joint 5 Solution Setup

5

5

x4

cos5 = - (a • y4)sin5 = a • x4

y4

an

Lecture 11 12

Joint 6 Solution Setup

x5

6

6

n

y5

s

cos6 = s • y5

sin6 = n • y5

*align orientation of gripper to easepicking up object: s = y6

Lecture 11 13

Decision Equations for Arm Configuration Indicators

• For the PUMA robot arm, there are 8 solutions to the inverse kinematic problem– First three joint solutions (1,2,3) positions the arm

– Last three joint solutions (4,5,6) provides hand orientation

• Four solutions to the first three joints: – RIGHT shoulder

– LEFT shoulder

• FLIP toggle yields balance of solutions

Lecture 11 14

Decision Equations for Arm Indicators

• ARM = sign(-d4 S23 -a3 C23 -a2 C2)

+1 RIGHT arm

-1 LEFT arm

• ELBOW = ARM•sign(d4 C3 - a3 S3)

+1 ELBOW above wrist

-1 ELBOW below wrist

• WRIST = sign(s•z4)0 = +1 WRIST DOWN

sign(n•z4)=0 = -1 WRIST UP

Lecture 11 15

Decision Equations for Arm Indicators cont’d

• Arm configuration indicators can be determined from joint angles

• Useful for providing verification of the arm solution

Lecture 11 16

Computer Simulation

• Useful to verify and validate the inverse kinematics solution

• Software generates the allowable locations in the workspace given the joint angle limitations

• Joint angles are used to produce T (arm matrix) and used to compute the decision equations which yield three arm indicators

Lecture 11 17

System Flow for Joint Solution

Direct Kinematics

Decision Equations

Inverse Kinematics

+

-Error

Joint anglesPosition and orientationof end-effector

ARM, ELBOW,WRIST

T

n s a p0 0 0 1

For T =

Lecture 11 18

Summary

• General Robotic Control• Object description• Direct Kinematics

• Robot arm parameters: links, joints• 4x4 Transformation Matrices

• Inverse Kinematics• Euler angle solution• Inverse transformation• Geometric approach• Arm indicators