# 305-572A Mechanics of Robotics Systems I by 81jnAQ

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```									      Dept. Of Mechanical Engineering

MECH572A
Introduction To Robotics

Lecture 7
Review
• Basic Robotic Kinematic Problems
Direct (forward) Kinematics
Inverse Kinematics
• DH Notation
Zi-
1

i-                 i
bi-1                   1          Zi
Oi-                   ai-1
1

Xi-                       i
i-1 1 Oi bi
ai
i
i-1                                               Oi+
Xi                        1
Xi+1
i+1

Revolute joints                                               Zi+1
Review
Fi to Fi+1

Orientation:

Position:
Review
• Forward Kinematics

Known joint angles            End Effector Position + Orientation
Inverse Kinematics

• Overview
- Problem description:
Known EE position and orientation, find joint angles (inverse process)
Direct Kinematics Problem (DKP) -> Solution unique
Inverse Kinematics Problem (IKP) -> May have multiple solutions,
not always solvable (Kinematic Invertibility)
- Equations in IKP are usually highly nonlinear, analytically solvable
(closed form solution available) only for certain types of
manipulators, examples:
PUMA (6R decoupled)
Stanford Arm (5R-1P)
Canadarm 2 (7R with 3 parallel pitch joint axes)
other types of manipulator rely on numerical methods for solution
Inverse Kinematics

• Overview (cont'd)
- PUMA – 6R decoupled (Arm + Wrist)
Inverse Kinematics

• Overview (cont'd)
- Canada Arm 2 – 7R (Off-pitch Joints + Pitch joints)

3 parallel pitch joints
4 off-pitch joints
Inverse Kinematics

• Overview (cont'd)
Scope of this course – Decoupled manipulators
- Have Special architecture that allows the decoupling of
position problem from orientation problem. e.g. PUMA
- Analytical IKP solution available
Inverse Kinematics

• 6-R Decoupled Manipulator

Arm (Position)
Wrist (Orientation)

C: wrist centre
Inverse Kinematics
• 6-R Decoupled Manipulator
– Position Problem

Recall ai =Qibi -
eq(4.3d)
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
EE position vector p, wrist centre position vector c

c expressed in
terms of p and Q
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)

From eq. (4.18c)
position problem can be
decoupled from
orientation problem.
Three equations in
(4.17) for three
unknowns 1, 2, and 3
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
Solve the equation:

2              3                   1
Linear transformation of vector in 3 to vector in 1
Norm of the vector is invariant, i.e., || VLHS|| = || VRHS||
eliminate 2
Inverse Kinematics
• 6-R Decoupled Manipulator
– Position Problem (cont'd) - coefficients of eq. (4.19a)

The 3rd scalar equation of (4.17) does not contain 2 and thus leads to
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
Two equations (4.19a) and (4.20a) linear in c1, s1, c3, s3, solve c1, s1 in
terms of c3, s3 :

If       0, then we have a singularity. To be discussed later

(4.21a) & (4.21b):   c12  s12  1

1 is eliminated
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)

Tan-half-angle-
identities
– Use trigonometric identities to treat (4.22)
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)

Solving (4.22):

Substitute c3 and s3 to (4.21a) & (4.21b) to determinefour different
solutions of 1
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
To get 2, use the first two equations of (4.17), which yield,
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)

If          there is a Singularity, which is to be discussed next
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
Discussion on solutions

e1 // e2   1 = 0
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
e1 intersects with e2
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
PUMA Robot – A special case

Four solutions for same
given wrist centre
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd)

Z2
Z1                      C
The first two components of
[c]2 vanish

a 1  Q 1 c2  c1  c                                 Z3
Q1 c2  c  a 1
a2    a3
c2  Q   T
(c  a 1 )  Q c  b 1
T
a1
[c]1

C lies on Z2 axis                    X1               Y1
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd) – Example

From figure

Compute the coefficients
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd) –Example

solve the equation
Inverse Kinematics

• 6-R Decoupled Manipulator
– Position Problem (cont'd) –Example
Compute 1

Compute 2

The remaining roots are computed likewise:
Inverse Kinematics

• 6-R Decoupled Manipulator
– Orientation Problem
The EE orientation in terms
of Q along with 1, 2, and
3 are known data. 4, 5
and 6 are to be computed
Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
– Orientation problem (cont'd)
Geometric relationship yields

2 roots – if radical > 0
Solution for 4     1 root – if radical = 0
No root – if radical < 0
Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
– Orientation problem (cont'd)
Workspace of spherical wrist:

The workspace description
Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
– Orientation problem
Recall
Assume

Equating the first two elements of the 3rd column (independent of
6)

Solve for 5
Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
– Orientation problem

Take the first column of both side
Recall
Q6= [p6, q6, u6]

where

Solve for 6
Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
– Orientation problem summary
2 sets of solution:  {4, 5}1, 6
{4, 5}2, 6
Inverse Kinematics
• 6-R Decoupled Manipulator (Cont’d)
– Overall Solution of IKP

Arm (Position)          Wrist (Orientation)   Total
4                          2               8

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