MECH572-lecture8.ppt

文档介绍：MECH572AIntroduction To Robotics
Lecture 8
Dept. Of Mechanical Engineering
Review
Robot Kinematics
Geometric Analysis

Differential analysis
Forward (direct) vs. Inverse Kinematics problem
Inverse Kinematics Problem (IKP)
- Problem description: Known QEE and pEE, Seek 1, …n
- Possibility of Analytical (closed form) solution depends on the architecture of the manipulator
- 6-R Decoupled manipulator (., PUMA)
Position problem – position of C (wrist centre)
Orientation problem – EE orientation
Review
IKP – 6-R Decoupled Manipulator
Solution process overview:
Arm (position)
Wrist (orientation)
1, 2, 3
Equ's in 1, 3
Quartic equ in 3 (3)
Eliminate 2
Eliminate 1
1  0
3
1
2
2  0
Max Number of Solution: 4
Elimination
Solution
4, 5, 6
Quadratic equ in 4 (4)
4
5
6
Radical  0
Max Number of Solution: 2
Special geometry in wrist axis
Manipulator Kinematics
Velocity (Differential) Analysis
Manipulator Kinematics
Velocity Analysis
Angular velocity of EE
Position of EE
Manipulator Kinematics
Velocity Analysis (cont'd)
Define
Let
Position vector from Oi to P
Recall twist
Manipulator Kinematics
Velocity Analysis (cont'd)
Jacobian matrix:
ith column (revolute joints)
Linear transformation between joint rates and Cartesian rates (EE)
The Plücker array of ith axis point P of EE
Manipulator Kinematics
Velocity Analysis (cont'd)
Prismatic joint:
The ith column of Jacobian matrix
Manipulator Kinematics
Velocity Analysis (cont'd)
For 6 joint manipulator, J is a 66 square matrix

Solve equations using Gauss-elimination (LU position) algorithm
Compute y
(Forward substitution)
Compute
(Backward substitution)
Manipulator Kinematics
Velocity Analysis (cont'd)
Transformation of Jacobian matrix
In general, Jacobian can be defined wrt different points. For decoupled manipulators:

Recall twist transformation
Property:
Wrist Centre
Point P at EE
Two point A and B on EE