MECH572A
Introduction 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 (e.g.,PUMA)
Position problem – position of C (wrist centre)
Orientation problem – EE orientation
θθfx ?? ???
)(θfx?
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 w.r.t
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 decomposition) 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
Manipulator Kinematics
? Velocity Analysis (cont'd)
The Jacobian matrix of decoupled Manipulator has special form
Partition arm and wrist rates,
Arm rate Wrist rate
Manipulator Kinematics
? Velocity Analysis (cont'd)
Decoupled Manipulator – solve 2 systems of three equations and three
unknowns
Manipulator Kinematics
? Application Example – MSS/Canadarm2
Operating and control overview
Robotic Work Station (RWS) MSS
Commands
Telemetry
Hand Controllers Display & Control Panel
Manipulator Kinematics
? Application Example (cont'd)
Kinematic aspects of Canadarm2 control modes
1,Human-in-the-loop modes (commanding the arm via hand
controllers)
a) Manual Augmented Mode (MAM)
Description,Control the manipulator by commanding EE rate
Kinematics,IKP rate problem
b) Single Joint Rate Mode (SJRM)
Description,Control the manipulator by commanding a single joint
Kinematics,DKP rate problem
Manipulator Kinematics
? Application Example (cont'd)
Kinematic aspects of Canadarm2 control modes (Cont'd)
2,Automatic Modes
a) Joint Modes
Operator Commanded/Pre-Stored Joint Auto Modes (OJAM & PJAM)
Description,Execute joints movements to a pre-set joint positions
Kinematics,DKP position/orientation problem
b) EE Modes (POR <Point Of Resolution> Mode)
Operator Commanded/Pre-Stored POR Auto Modes (OPAM & PPAM)
Description,Execute manipulator movements to a pre-set EE position/orientation
Kinematics,IKP position/orientation problem
)(θfx?
)(xgθ?
Manipulator Kinematics
? Singularity Analysis – Decoupled Manipulators
Observe the Jacobian Matrix
- If neither J12 nor J21 is singular,IKP problem is solvable
- Singularities of sub-Jacobian can be analyzed separately for decoupled
manipulators
a) Singularity of J21
- Singularity of J21 depends on the relative orientation of the first three
column vectors
- ?1 does not change relative orientation (viewpoint only)
Manipulator Kinematics
? Singularity Analysis (cont'd)
General concept
L L3
L2
L1 Locus of L
One sheet hyperboloid
surface
Manipulator Kinematics
? Singularity Analysis (cont'd)
In summary,Let L1,L2 and L3 represent e1,e2 and e3,respectively,if wrist
centre C fall on the surface of hyperboloid,singularity occurs,
Example – PUMA Robot
Case 1,
C lies in the plane determined by
intersecting e1 and e2
e1? r1 and e2 ? r2 are coplanar
Velocity of C along the direction
perpendicular to e3? r3 and n12
(L direction) can not be produced
L intersects with e1at I,with
e2 and e3 at ?
Manipulator Kinematics
? Singularity Analysis (cont'd)
Case 2,
e2 and e3 are parallel
r2 and r3 lie in the same
plane
e2? r2 and e3? r3 are
coplanar
Velocity of C in the plane
determined by e2 and e3
normal to e1? r1 (L direction)
can not be produced
Manipulator Kinematics
? Singularity Analysis (cont'd)
Geometric representation of singularity
Singularity, lies in the line that represents nullspace of
no mapping between and
The range of J21 is perpendicular to the nullspace of
Wrist singularity – J12 is singular
e4,e5 and e6 are coplanar
e.g.,
Manipulator Kinematics
? Acceleration Analysis
Rate relationship
Differentiate wrt time
Solving equation using LU decomposition
Compute z (forward substitution)
Compute (Backward substitution)
Manipulator Kinematics
? Acceleration Analysis (cont'd)
Computing the Jacobian rate
Recall
Differentiate
Manipulator Kinematics
? Acceleration Analysis (cont;d)
Computing the Jacobian rate (cont'd)
where
Static Analysis
? Mapping between joint torques and EE wrench
Joint torques
Wrench acting at EE
Power at EE
Power at joints
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X
Z
Y ?1
?n ?
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Static Analysis
? Mapping between joint torques and EE wrench (cont'd)
Power conservation condition,
Mapping EE wrench in Cartesian space to joint torques in joint space,
Recall
Static Analysis
? Mapping between joint torques and EE wrench (cont'd)
6-R Decoupled Manipulator
Solve the static problem for decoupled arm,
Arm torques
Wrist torques
Manipulator Kinematics
? Interpretation Jacobian Matrix
–Mapping from n-D joint space to 6-D Cartesian space
–The range of J (Column space) represents all possible EE twist that
can be produced by the manipulator
–If t lies in the range of J,then there exist a that produces t at EE
–The nullspace of J transpose represent all singularities
Assignment #3
? Problems 4.4,4.7,4.19
? Due in two weeks