MECH572
Introduction To Robotics
Lecture 10
Dept,Of Mechanical Engineering
Review
? Fundamentals of Multibody Dynamics
Newton-Euler Equation
Rely on free-body diagram,Constraint force involved,
Compact form,
Working
Wrench
Constraint
Wrench
Review
? Fundamentals of Multibody Dynamics
Euler-Lagarange Equation
Kinetic energy,
Alternative form of E-L equation
or
In terms of twist
In terms of generalized coordinates and inertia
Review
? Fundamentals of Multibody Dynamics
Summary,
Newton-Euler Equation
- Element (body) level formulation
- All forces/moments involved (active,constraints,…)
- Reference point – mass centre
Euler-Lagarange Equation
- System level formulation
- System kinetic/potential energy involved
- Generalized coordinates
Y
Z
111,,??? ???
222,,??? ???
nnn ??? ???,,
…
11,cc ??
22,cc ??? 33,cc ???
11,ωω ?
22,ωω ? 33
,ωω ?
Kinematics Computation
Dynamics Computation
f
Inverse Dynamics
? Overview of Recursive Algorithm
Inverse Dynamics,Known time history of joint position,rate and
acceleration,compute joint torque
Recursive Algorithm,The problem is formulated as a recursive process
in such a way that the computation can be proceed from one link to the
next,
X
n
Recursive Inverse Dynamics
? Procedure Summary
(1) Kinematic Computation (Outward)
Known,Compute θ,θθ,??? tt?,
111,,??? ??? 222,,??? ???11,cc ?? 22,cc ???11,ωω ? 22,ωω ?
),( 11 tt ? ),( 22 tt ?
…
Link 1 Link 2
Recursive Inverse Dynamics
? Procedure Summary
(2) Dynamic Computation (Inward)
Known,Kinematic quantities of each link (from outward recursion)
Compute,Joint wrench and external wrench
WEE (fEE,nEE) Wn (fn,nn) Wn-1 (fn-1,nn-1)
… N.E
Recursive Inverse Dynamics
? Outward Recursions - Kinematics Computation
(i) Angular velocity and acceleration
Expressed in (i+1) frame
Initial conditions
Recursive Inverse Dynamics
? Outward Recursions - Kinematics Computation
Computational complexity for angular velocity and acceleration
Coordinate Transformation
The extra term in computation,
Also
Recursive Inverse Dynamics
? Outward Recursions - Kinematics Computation
Complexity,
8M 5A (R)
8M 4A (P)
10M 7A (R)
8M 4A (P)
Recursive Inverse Dynamics
? Outward Recursions - Kinematics Computation
(ii) Linear velocity and acceleration
Revolute Joint (R)
Differentiate,
Recursive Inverse Dynamics
? Outward Recursions - Kinematics Computation
(ii) Linear velocity and acceleration (cont'd)
Prismatic Joint (P)
Intermediate
variables
Recursive Inverse Dynamics
? Outward Recursions - Kinematics Computation
(ii) Linear velocity and acceleration (cont'd)
Initial Conditions,
Recursive Inverse Dynamics
? Outward Recursions - Algorithm
Recursive Inverse Dynamics
? Inward Recursions - Dynamics Computation
Wrench exerted by (i-1)st link
to ith link
Revolute joint
Active torque produced by actuator for rotation (driving torque)
Nonworking constraint forces
Recursive Inverse Dynamics
? Inward Recursions - Dynamics Computation (cont'd)
Prismatic joint
The driving force/torque =
Active force applied by motor (driving force)
(R)
(P)
Recursive Inverse Dynamics
? Inward Recursions - Dynamics Computation
Using Newton-Euler Equation
starting with the EE
Recursive Inverse Dynamics
? Inward Recursions - Dynamics Computation
Using Newton-Euler Equation
Compute ith link based on results from (i+1)st link
Inward recursion – express variables in ith frame (multiply by Qi)
Recursive Inverse Dynamics
? Inward Recursions - Dynamics Computation
Recursive Inverse Dynamics
? Inward Recursions - Dynamics Computation
Computation complexity
For 6-R manipulator
Incorporation of Gravity
Base link
Gravitational effect will be propagated forward to EE
Natural Orthogonal Complement
? The Concept
– The mathematical model of a system is usually represented by
ordinary differential equations (ODE)
– Dynamic analysis and simulation use system math model to predict
the next state of the system based on the current state,
– System state equation
x – State vector (minimum number of variables needed to describe
dynamic behaviour of a system)
u – Control input
w - Disturbance
Initial condition
x(t0) = x0
Linear System
),,( wuxfx ??
??
?
???
???
FwEuDxC
CwBuAxx
)( t
?
State equation
Output function
Natural Orthogonal Complement
? The Concept (cont'd)
– Robot Manipulator state vector
– To derive the state equation,we first analyze the Newton-Euler
equation,
ith element
System level
a) Uncoupled equations derived based on free-body diagram
b) Each link has 6-DOF as a free body; couple together –> 1-DOF,
5-DOF constrained,There exists a set of constraint equations
governing the coupling of consecutive links,
???
?
???
??
?
?
?x
Natural Orthogonal Complement
? Kinemtic constraint equations
Linear homogeneous system of algebraic equation on link twist
t,6n vector
K,6n?6n matrix
Since t can attain non-zero values,K must be singular,
Take transpose of (6.50) and multiply a 6n vector
Since constraint wrench does not do work,we have
Comparing (6.51) and (6.52),
Natural Orthogonal Complement
? Kinemtic constraint equations
Mathematical interpretation,
Recall Linear Algebra Rule – the following two subspaces of A are
perpendicular to each other,
a) Null space of A
b) Range of A Transpose
Examine the kinemtic constraints,
(6.50),t lies in the null space of K
(6.53),lies in the range of
(6.51) t and are perpendicular to each other
Verified – obey the linear algebra rule
Natural Orthogonal Complement
? Kinemtic constraint equations
From velocity analysis,t is linear transformation of
Since can be non zero
Differentiate (6.54) wrt time
Substitute (6.56) to N-E equ,
left multiply and rearrange the equation,
Generalized Inertia matrix
Natural Orthogonal Complement
? Kinemtic constraint equations
System equation of motion becomes,
where
If considering the static wrench acting at EE and propagated to each joint
Vector of Corilios and
centrifugal force
Assignment #4
? 6.3,6.11
? Due in two weeks