MECH572A
Introduction To Robotics
Lecture 9
Dept,Of Mechanical Engineering
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)
Joint Rate Cartesian rate EE Wrench Joint force/torque
Rate analysis Static analysis
Review
? Velocity,Acceleration and Static Analysis
6R Decoupled Manipulator
-> Solve two linear systems of equations (3 equations 3 unknowns)
Rate problem
Static problem
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
Planar Manipulator
? 3-Revolute Planar Manipulator
Properties,
1) e1 // e2 // e3
2) ?1 = ?2 = ?3
3) X1,X2 and X3
coplanar
4) b1 = b2 = b3=0
5) a1,a2,and a3
none zero
(Link length)
e1
e2
e3
X1
X2
X3
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
Planar Manipulator
? Displacement Analysis
The case of two real roots,
Planar Manipulator
? Velocity Analysis
2-D cross-
product
matrix
Planar Manipulator
? Velocity Analysis
Mapping rates between joint and Cartesian space
Planar Manipulator
? Acceleration Analysis
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
2? 2?
Scalar
2-D vector
Planar Manipulator
? Example (cont'd)
Solution,
3 equations,3 unknowns
Planar Manipulator
? Example (cont'd)
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
Manipulator Dynamics
? Overview
– Robotic Dynamic Problems
Forward Dynamics
Inverse Dynamics
Joint
torque
Joint
Coord,
?i ?i
?1(t),… ?n(t)
?1(t),… ?n(t)
?1(t),… ?n(t)
?1(t),… ?n(t)
t t
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
(6×6)
Angular Velocity
Dyad (6×6)
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)
Manipulator Dynamics
? Newton-Euler Formulation
Momentum,
Newton-Euler equ,
Compact form,
c
C
fw
nw
fc
nc
?i
Bi
(E)
(N)
Manipulator Dynamics
? Euler-Lagarange Formulation
T – System kinetic energy
? - Joint displacement vector (generalized coordinates)
- Generalized force
Alternative form
Non conservative force
Conservative force
Lagrangian
Power supplied
to the system
Dissipation
function
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Active wrench at ith joint
Overall system,
System level definitions,
Twist Momentum Constraint Wrench Active Wrench Dissipative Wrench
Working wrench
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 )
(6n×6n)
(6n×6n)
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Differentiate w.r.t time,
Write the E-L equation as
Generalized Inertia matrix
Generalized momentum
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Alternative form of E-L equation,
Example
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
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,
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
3) Square the magnitude of mass-centre velocity
4) Compute the kinetic energy
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Components of the generalized inertia matrix,
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,
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
8) Compute the partial derivative
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Compute
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
10) Compute partial derivative of potential energy,
11) The final form of E.O.M
Manipulator Dynamics
? Euler-Lagrange Formulation (cont'd)
Conclusion,
o Straightforward differentiation approach to derive Euler-Lagrange
equation is not practical
o Alternative approach is desirable – Natural Orthogonal Decomposition
method will be introduced later on