1. Introduction To Robotics

2. 1) Outward Recursion – Kinematic Computation
Review
Recursive Inverse Dynamics
Inverse Dynamics – Known joint angles compute joint torques
1) Outward Recursion – Kinematic Computation
Known Compute
From 0 to n, recursively based on geometrical and differential relationship associated with each link.
2) Inward Recursion – Dynamics Computation
Compute wrench wi based on wi+1 and kinematic quantities obtained from 1)
From n+1 to 0, recursively using Newton-Euler equation

3. Each link – 6-DOF; Within the system – 1-DOF
Review
The Natural Orthogonal Compliment
Each link – 6-DOF; Within the system – 1-DOF
5-DOF constrained
Kinematic Constraint equation
T : Natural Orthogonal Complement (Twist Shape Function)

4. Review Natural Orthogonal Complement (cont'd)
Use T in the Newton-Euler Equation, the system equation of motion becomes:
where
Consistent with the result obtained from Euler-Lagrange equation
Generalized inertia matrix
Active force
Dissipative force
Gravitational force
Vector of Coriolis and centrifugal force

5. Natural Orthogonal Complement
Constraint Equations & Twist-Shape Matrix
1) Angular velocity Constraint
Ei : Cross-product matrix of ei
2) Linear Velocity Constraints
ci = ci-1+ i-1 + i
Differentiate: 
Oi+1
Oi-1 Oi O
Ci-1 Ci
ci-1 c
i-1 i

6. Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint
Equations (6.63) and (6.64) pertaining to the first link:

7. Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint
6n 6n matrix

8. Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint
Define partial Jacobian
6 n matrix with its element defined as
Mapping the first i joint rates to ti of the ith link

9. Natural Orthogonal Complement
Constraint Equations & Twist Shape Matrix – R Joint

10. Natural Orthogonal Complement
Constraint Equation and Twist Shape Matrix – R Joint
Easy to verify
Recall

11. Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix – P Joint
Oi-1 Oi
O'i
Oi+1
Ci-1 Ci
i-1
ci-1 i ci bi ai
ai+1

12. Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix – P Joint
Regroup (6.74a) and (6.77):

13. Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix – P Joint
If the first joint is prismatic, then
where
Define partial Jacobian

14. Natural Orthogonal Complement
Constraint equation and Twist Shape Matrix
Compute
If kth joint is prismatic, then

15. Natural Orthogonal Complement
Noninertial Base Link
Include it in the joint rate vector - 6(n+1)
The generalized velocity:
Twist of base link (6-DOF)

16. Forward Dynamics Overview
Purpose of forward dynamics – Simulation, Model-based control
Method – Solving Ordinary Differential equation (System E.O.M):
Dissipative force
Gravitational force
EE Static wrench acting on the joint
Generalized Inertia Matrix
Inertial Force
Active working wrench

17. Forward Dynamics Problem Description Known: at To find: at
Solution: Integration to compute at
Need to compute I, , and

18. Forward Dynamics Computation Procedure (1) Compute I
Using T, the Natural Orthogonal Complement
Recall
M – Positive Semi-Definite
Factoring:

19. Forward Dynamics
Computation Procedure

20. Forward Dynamics Computation Procedure (2) Compute
Rewrite system equation as
the problem can be solved as an inverse dynamics problem using the recursive algorithm.
Know current compute
Torque required to produce the current joint angles and rates when joint acceleration and dissipative force vanish

21. Forward Dynamics Computation Procedure (3) Solving Equations
Cholesky decomposition of the generalized inertia matrix
Solving two linear systems of equations
Alternative solution

22. Planar Manipulator Fundamentals Basic definitions in 2-D System level:
Newton-Euler Equation in 2-D
Matrix forms:
Element level:
System level:
Scalar
Scalar
2-D Vector
2-D Vector

23. Planar Manipulators Fundamentals
Constraint equations/Natural Orthogonal Compliment
K – 3n3n matrix
T – 3nn matrix
Equation of Motion

24. Planar Manipulators
Example

25. Planar Manipulators Example Solution: Angular velocities:
Twist-Shape matrix

26. Planar Manipulators
Example

27. Planar Manipulators Example The inertial matrix Elements
Generalized Inertial Matrix

28. Planar Manipulators Example Twist Shape Matrix Rate
Let represent (i,j) entry of

29. Planar Manipulators
Example
Now define

30. Planar Manipulators
Example
Gravity wrench

31. Planar Manipulators
Example
Final form

32. Dynamic Model Review Summary Dynamic Model of a system
Euler-Lagarange Equation (System Level Model)
Apply Kinematic constraint conditions in terms of K and T
Newton-Euler Equation (Element Level Model – Uncoupled)

33. Gravity Term in E.O.M Model Gravitational Force
Incorporate gravity into recursive inverse dynamics algorithm
Using the natural orthogonal complement T
No change in the algorithm.

34. Dissipative Term in E.O.M
Model Friction Forces
Viscous Friction – Solid vs viscous fluids
Coulomb Friction – Solid vs Solid (Dry friction)
(1) Viscous Friction
Velocity field v = v(r, t)
v vanishes at the interface surface v
Symmetric
Skew-Symmetric

35. Dissipative Term in E.O.M
Model Friction Forces
Only the symmetric part of the gradient is responsible for power dissipation
 - Viscosity coefficient of the fluids
For revolute joint pair velocity field can be
modeled as pure tangential
The dissipative function
At each joint
System level

36. Dissipative Term in E.O.M
Model Friction Forces
The dissipative force
(2) Coulomb Friction
Simplified model
Constant determined experimentally (R – Force; P – Torque)
Dissipative function:
At joint i
Overall

37. Dissipative Term in E.O.M
Model Friction Forces
Property:
Lower relative speed -> Coulomb friction is high
High relative speed -> Coulomb friction is low
Enhanced model:

38. Course Review Overview of Robotics
Analysis for robot structure design, operation procedure planning, singularity/work space analysis
Kinematics (Static)
Mechanics
Math model for control and simulation, dynamical analysis
Dynamics
Force/torque control
Control
Robotics
(Chapter 1)
Trajectory/position control
Computer Vision
Artificial Intelligence

39. Course Review Robotics Topics
Basic definition & Notation – DH Notation, Coord. Trans, ..
Geometrical – Closed form solution
Kinematics(Chapter 4)
General Analysis
Differential – Jacobian analysis, vel, acceleration
Direct Kinematic Problem
Specific problems
Inverse Kinematic Problem
Static (Chapter 4)
Mapping between EE wrench and joint torque
Basic definition & Notation – Mass/inertia matrices, twist, wrench
Dynamics (Chapter 6)
General Analysis – Multi-body dynamics(N-E, E-L), constraints (K, T)
Forward Dynamics – Solve ODE
Specific problems
Inverse Dynamics – Recursive algorithm

40. Course Review Analysis & Modelling Tools
Linear Transformation – Q, Coordinate Trans: [p]1 = a + Q[p]2
Math Tools (Chapter 2)
Invariant Concept – vect(Q), tr(Q)
Motion - twist
Definition & Concept
Force/Torque - wrench
Mass property – m, I, c
Rigid-body Mechanics (Chapter 3)
Analysis – Screw motion (screw axis, pitch, properties)
Newton-Euler – body level
E.O.M
Euler-Lagarange – System level

41. Office Hour Next Week Mon/Tues (Dec 6, 7) 17:00-18:00 MD 457
Assignment #4 Due on Dec 6. Submit your assignment during the office hour and get the solution.
Final Exam (Open Book): 14:00 – 17:00 Dec 8, 2004