MECH572A
Introduction To Robotics
Lecture 7
Dept,Of Mechanical Engineering
Review
? Basic Robotic Kinematic Problems
Direct (forward) Kinematics
Inverse Kinematics
? DH Notation
Oi-
1
Oi
Oi+
1
Zi-
1
Zi
Zi+1
i-1
i
i+1
Xi-
1
Xi X
i+1
Revolute joints
bi-1
bi
?i
ai-1
ai
?i ?i-
1
?i-1
Review
? Transformation Between Neighboring Links
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,
Linear transformation of vector in ?3 to vector in ?1
Norm of the vector is invariant,i.e.,|| VLHS|| = || VRHS||
eliminate ?2
?2 ?3 ?1
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),12121 ?? sc
?1 is eliminated
Inverse Kinematics
? 6-R Decoupled Manipulator
– Position Problem (cont'd)
– Use trigonometric identities to treat (4.22)
Tan-half-angle-
identities
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)
C Z1 Z2
Z3
X1 Y1
a1
a2 a3
? ? ? ?
? ?
? ? 1T1T
1
11
bcQacQc
accQ
cccQa
????
??
???
)(2
21
12
The first two components of
[c]2 vanish
C lies on Z2 axis
[c]1
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
The quadric equation in ?3
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
1 root – if radical = 0
No root – if radical < 0
Solution for ?4
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]
Solve for ?6
where
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