MECH572A Introduction To Robotics

1
MECH572AIntroduction To Robotics

• Lecture 8

2
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

3
Review

• IKP 6-R Decoupled Manipulator
• Solution process
  overview

Arm (position)
Wrist (orientation)
?1, ?2, ?3
?4, ?5, ?6
Eliminate ?2
Special geometry in wrist axis
Elimination
Equ's in ?1, ?3
Eliminate ?1
?1 ? 0
Quadratic equ in ?4 (?4)
Quartic equ in ?3 (?3)
Radical ? 0
?4
?3
Solution
?5
?2 ? 0
?1
?6
?2
Max Number of Solution 4
Max Number of Solution 2
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Manipulator Kinematics

• Velocity (Differential) Analysis

5
Manipulator Kinematics

• Velocity Analysis
• Angular velocity of EE
• Position of EE

6
Manipulator Kinematics

• Velocity Analysis (cont'd)
• Define
• Let

Position vector from Oi to P
Recall twist
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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
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Manipulator Kinematics

• Velocity Analysis (cont'd)
• Prismatic joint
• The ith column of Jacobian matrix

9
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)
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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

Wrist Centre
Point P at EE
Two point A and B on EE
Property
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Manipulator Kinematics

• Velocity Analysis (cont'd)
• The Jacobian matrix of decoupled Manipulator
  has special form
• Partition arm and wrist rates

Arm rate
Wrist rate
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Manipulator Kinematics

• Velocity Analysis (cont'd)
• Decoupled Manipulator solve 2 systems of
  three equations and three unknowns

13
Manipulator Kinematics

• Application Example MSS/Canadarm2
• Operating and control overview

Display Control Panel
Hand Controllers
Commands
Telemetry
Robotic Work Station (RWS)
MSS
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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

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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 ltPoint Of Resolutiongt 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

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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)

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Manipulator Kinematics

• Singularity Analysis (cont'd)
• General concept

L1
Locus of L One sheet hyperboloid surface
L2
L3
L
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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 ?

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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

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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.,

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Manipulator Kinematics

• Acceleration Analysis
• Rate relationship
• Differentiate wrt time
• Solving equation using LU decomposition

Compute z (forward substitution)
Compute (Backward substitution)
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Manipulator Kinematics

• Acceleration Analysis (cont'd)
• Computing the Jacobian rate
• Recall
• Differentiate

23
Manipulator Kinematics

• Acceleration Analysis (contd)
• Computing the Jacobian rate (cont'd)
• where

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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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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
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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
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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

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Assignment 3

• Problems 4.4, 4.7, 4.19
• Due in two weeks