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MECH572A Introduction To Robotics

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Mapping between Cartesian and joint space (Linear transformation) ... Inertia Dyad (6 6) Angular Velocity Dyad (6 6) Manipulator Dynamics ... – PowerPoint PPT presentation

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Title: MECH572A Introduction To Robotics


1
MECH572A Introduction To Robotics
  • Lecture 9

2
Review
  • Velocity, Acceleration and Static Analysis
  • Mapping between Cartesian and joint space
    (Linear transformation)
  • Jacobian Matrix General form for n-R
    manipulator
  • Special case 6R Decoupled Manipulator
    (PUMA)

EE Wrench
Joint force/torque
Joint Rate
Cartesian rate
Rate analysis
Static analysis
3
Review
  • Velocity, Acceleration and Static Analysis
  • 6R Decoupled Manipulator
  • -gt Solve two linear systems of equations
    (3 equations 3 unknowns)
  • Rate problem
  • Static problem

4
Review
  • Singularity Analysis
  • - Based on the analysis of Jacobian Matrix
  • - Singularity analysis of 6R decoupled
    manipulator
  • - General concept
  • Conditioning analysis of J12
    J21
  • Acceleration Analysis


5
Planar Manipulator
  • 3-Revolute Planar Manipulator
  • Properties
  • 1) e1 // e2 // e3
  • 2) ?1 ?2 ?3
  • 3) X1, X2 and X3
  • coplanar
  • 4) b1 b2 b30
  • 5) a1, a2, and a3
  • none zero
  • (Link length)

6
Planar Manipulator
  • Displacement Analysis
  • From geometry
  • (4.103a)² (4.103b)²
  • Also
  • Solution depends on the relative position
    between line L and circle C
  • a) L intersects with C
    2 roots
  • b) L tangent to C
    1 root
  • c) L does not intersect with C No
    root

7
Planar Manipulator
  • Displacement Analysis
  • The case of two real roots

8
Planar Manipulator
  • Velocity Analysis

2-D cross-product matrix
9
Planar Manipulator
  • Velocity Analysis

Mapping rates between joint and Cartesian space
10
Planar Manipulator
  • Acceleration Analysis

11
Planar Manipulator
  • Static Analysis
  • Example
  • 3-R planar manipulator. Known
  • Link length a1 a2 a3 1m
  • Joint angles ?1 ?2 ?3 45º
  • Joint Torques ?1 ?2 Nm, ?3 1
    Nm
  • Seek Wrench acting at EE

Scalar
2-D vector
12
Planar Manipulator
  • Example (cont'd)
  • Solution

3 equations, 3 unknowns
13
Planar Manipulator
  • Example (cont'd)

14
Manipulator Dynamics
  • Overview
  • Dynamics Study the relationship between
    force/torque and the manipulator motion
  • This relationship can be expressed mathematically
    by a set of differential equations Equation of
    Motion (E.O.M)
  • Establish E.O.M
  • Newton-Euler Formulation
  • Direct interpretation of Newton's 2nd law.
    Constraint forces appear in the EOM
  • Lagarangian Formulation
  • Described using work/energy of the system.
    Constraint/workless forces eliminated from EOM

15
Manipulator Dynamics
  • Overview
  • Robotic Dynamic Problems

?1(t), ?n(t)
?1(t), ?n(t)
Forward Dynamics
?i
?i
Joint torque
Joint Coord.
t
t
Inverse Dynamics
?1(t), ?n(t)
?1(t), ?n(t)
16
Manipulator Dynamics
  • Basic Definitions in Multibody Dynamics
  • A system of rigid-body Bi, i 1, 2 r
  • Mass and angular velocity representations of
    each body in matrix form
  • Ii - Inertia matrix
  • ?i - Angular velocity matrix
  • mi - mass

Inertia Dyad (66)
Angular Velocity Dyad (66)
17
Manipulator Dynamics
  • Basic Definitions in Multibody Dynamics (cont'd)
  • Twist, Momentum and wrench

Twist
Momentum screw
Working wrench (Actuator forces, environmental
forces, dissipation forces)
Constraint wrench (non-working force)
18
Manipulator Dynamics
  • Newton-Euler Formulation
  • Momentum
  • Newton-Euler equ.
  • Compact form

(E)
(N)
19
Manipulator Dynamics
  • Euler-Lagarange Formulation
  • T System kinetic energy
  • ? - Joint displacement vector (generalized
    coordinates)
  • - Generalized force
  • Alternative form

Power supplied to the system
Non conservative force
Dissipation function
Conservative force
Lagrangian
20
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Active wrench at ith joint
  • Overall system
  • System level definitions

Working wrench
Constraint Wrench
Active Wrench
Dissipative Wrench
Twist
Momentum
21
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Manipulator Mass
  • Manipulator Angular Velocity
  • System Momentum
  • System Kinetic Energy
  • Recall for series manipulators
  • Alternative form of Kinetic Energy (in terms
    of generalized coordinates)

  • (Homogeneous
    in )

(6n6n)
(6n6n)
22
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Differentiate w.r.t time
  • Write the E-L equation as

Generalized Inertia matrix
Generalized momentum
23
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Alternative form of E-L equation
  • Example

24
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)

25
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Solution
  • 1) Compute the angular velocity of each link
  • 2) Compute the linear velocity of mass centre
    of each link

26
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • 3) Square the magnitude of mass-centre
    velocity
  • 4) Compute the kinetic energy

27
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Components of the generalized inertia matrix

28
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • 5) Potential Energy
  • 6) Compute the power delivered to the
    manipulator
  • 7) Compute time-derivative of generalized
    inertia matrix

29
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)

30
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • 8) Compute the partial derivative

31
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Compute

32
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • 10) Compute partial derivative of potential
    energy
  • 11) The final form of E.O.M

33
Manipulator Dynamics
  • Euler-Lagrange Formulation (cont'd)
  • Conclusion
  • Straightforward differentiation approach to
    derive Euler-Lagrange equation is not practical
  • Alternative approach is desirable Natural
    Orthogonal Decomposition method will be
    introduced later on
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