MECH572
Introduction To Robotics
Lecture 11
Dept,Of Mechanical Engineering
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
θ,θθ,??? tt?,
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)
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
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 O
i
O
Ci-1
Ci
ci-1
c
?i-1
?i
Oi+1
Natural Orthogonal Complement
? Constraint Equations & Twist Shape Matrix – R Joint
Equations (6.63) and (6.64) pertaining to the first link,
Natural Orthogonal Complement
? Constraint Equations & Twist Shape Matrix – R Joint
6n? 6n matrix
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
Natural Orthogonal Complement
? Constraint Equations & Twist Shape Matrix – R Joint
Natural Orthogonal Complement
? Constraint Equation and Twist Shape Matrix – R Joint
Easy to verify
Recall
Natural Orthogonal Complement
? Constraint equation and Twist Shape Matrix – P Joint
Oi-1
Oi
O'i
Oi+1
Ci-1
Ci
Oi
?i-1
ci-1
?i
ci
bi
ai
ai+1
Natural Orthogonal Complement
? Constraint equation and Twist Shape Matrix – P Joint
Regroup (6.74a) and (6.77),
Natural Orthogonal Complement
? Constraint equation and Twist Shape Matrix – P Joint
If the first joint is prismatic,then
where
Define partial Jacobian
Natural Orthogonal Complement
? Constraint equation and Twist Shape Matrix
Compute
If kth joint is prismatic,then
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)
Forward Dynamics
? Overview
Purpose of forward dynamics – Simulation,Model-based control
Method – Solving Ordinary Differential equation (System E.O.M),
Generalized
Inertia
Matrix
Inertial Force Active
working
wrench
Dissipative
force
Gravitational
force
EE Static wrench
acting on the joint
Forward Dynamics
? Problem Description
Known,at
To find,at
Solution,Integration to compute at
Need to compute I,?,and
Forward Dynamics
? Computation Procedure
(1) Compute I
Using T,the Natural Orthogonal Complement
Recall
M – Positive Semi-Definite
Factoring,
Forward Dynamics
? Computation Procedure
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
Forward Dynamics
? Computation Procedure
(3) Solving Equations
Cholesky decomposition of the generalized inertia matrix
Solving two linear systems of equations
Alternative solution
Planar Manipulator
? Fundamentals
Basic definitions in 2-D
Newton-Euler Equation in 2-D
Matrix forms,
Element level,
System level,
Scalar
2-D Vector
Scalar
2-D Vector
Planar Manipulators
? Fundamentals
Constraint equations/Natural Orthogonal Compliment
K – 3n?3n matrix
T – 3n?n matrix
Equation of Motion
Planar Manipulators
? Example
Planar Manipulators
? Example
Solution,
Angular velocities,
Twist-Shape matrix
Planar Manipulators
? Example
Planar Manipulators
? Example
The inertial matrix
Elements
Generalized Inertial Matrix
Planar Manipulators
? Example
Twist Shape Matrix Rate
Let represent (i,j) entry of
Planar Manipulators
? Example
Now define
Planar Manipulators
? Example
Gravity wrench
Planar Manipulators
? Example
Final form
Dynamic Model Review
? Summary
Dynamic Model of a system
Euler-Lagarange Equation
(System Level Model)
Newton-Euler Equation
(Element Level Model –
Uncoupled)
Apply Kinematic constraint
conditions in terms of K and T
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,
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
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
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
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,
Course Review
? Overview of Robotics
Mechanics
Control
Computer
Vision
Artificial
Intelligence
Kinematics
(Static)
Dynamics
Analysis for robot structure design,operation
procedure planning,singularity/work space analysis
Math model for control and simulation,
dynamical analysis
Force/torque control
Trajectory/position control Robotics
(Chapter 1)
Course Review
? Robotics Topics
Kinematics
(Chapter 4)
Static
(Chapter 4)
Dynamics
(Chapter 6)
Basic definition & Notation – DH Notation,Coord,Trans,.,
General Analysis
Specific problems
Geometrical – Closed form solution
Differential – Jacobian analysis,vel,acceleration
Direct Kinematic Problem
Inverse Kinematic Problem
Mapping between EE wrench and joint torque
Basic definition & Notation – Mass/inertia matrices,twist,wrench
General Analysis – Multi-body dynamics(N-E,E-L),constraints (K,T)
Specific problems
Forward Dynamics – Solve ODE
Inverse Dynamics – Recursive algorithm
Course Review
? Analysis & Modelling Tools
Math Tools
(Chapter 2)
Rigid-body
Mechanics
(Chapter 3)
Linear Transformation – Q,Coordinate Trans,[p]1 = a + Q[p]2
Invariant Concept – vect(Q),tr(Q)
Definition &
Concept
Analysis – Screw motion (screw axis,pitch,properties)
E.O.M
Motion - twist
Force/Torque - wrench
Mass property – m,I,c
Newton-Euler – body level
Euler-Lagarange – System level
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