MECH572
Introduction To Robotics
Lecture 4
Dept,Of Mechanical Engineering
Review
? Concept of Screw Motion of
a rigid body
O
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Q
P
P'
A
A'
?
P0
dp
e
dA
L
Review
? Use Plücker array to represent a line (screw axis)
? Instantaneous Motion
Velocity Analysis,
Angular velocity tensor (Skew-symmetric)
Rigid-Body Mechanics
? Instant Screw of Rigid-Body Motion (cont'd)
Define ISA
Consider 2 points A and P
Recall Theorem 2.3.4
If
Then
Ref,Eq,
(2.6a&b)
Rigid-Body Mechanics
? Instant Screw of Rigid-Body Motion (cont'd)
Points on ISA
O
P'0
Rigid-Body Mechanics
? Instant Screw of Rigid-Body Motion (cont'd)
Rigid-Body Mechanics
? Instant Screw of Rigid-Body Motion - Example
e'1 = e3
e'2 = -e1 Q =
e'3 = -e2
z
x
y A
B
C
D
E
F G
H
A'
B'
C'
D'
E'
F'
G'
H'
e1
e2
e3
e'1
e'3
e'2 1
Unit Cube
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001
100
010
Seek,L
Rigid-Body Mechanics
? Instant Screw of Rigid-Body Motion – Example
tr(Q) = 1 + 2 cos? = 0 cos? = -
vect(Q) = sin?e = sin? =
e =
To get equation for L,we still need a,
2
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Rigid-Body Mechanics
? Instant Screw of Rigid-Body Motion – Example (cont'd)
Use point A as reference point
a = a' = Q – I =
(Q – I) (- a') = p0 =
L,x + 1 = y – 3 = z + 2
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101
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Rigid-Body Mechanics
? Twist of Rigid-Body
Concept - decompose velocity into two parts
Define
describes the velocity field of a rigid body
Velocity perpendicular
to L'
Velocity along L'
L''
O
P
p
Angular velocity of the rigid
body
Linear velocity of a point on the
rigid body
(3-D figure)
Rigid-Body Mechanics
? Twist of Rigid-Body (cont'd)
t depends on the point from which the linear velocity is measured,
Assume the twist at point A and P are,
A and P denote the cross-product matrices of a and p,respectively
Recall
Rigid-Body Mechanics
? Twist of Rigid-Body – Alternative interpretation
Modified Plücker array,
Multiply the amplitude A = ||?||
eωωωωω ??
Add a pitch element
Rigid-Body Mechanics
? Acceleration Analysis
Velocity expression,
Differentiate wrt time
Angular Acceleration tensor
Define twist rate which describes acceleration field
Skew-symmetric
Symmetric
or
Rigid-Body Mechanics
? Rigid-Body Motion w.r.t Moving Frames
From geometry,
Velocity,
Rigid-Body Mechanics
? Rigid-Body Motion w.r.t Moving Frames (cont'd)
Acceleration
Recall concept – Coriolis force
Acceleration when M is fixed Acceleration
within M frame Coriolis Acceleration
x
y
z
O
vb
m Coriolis force = 2m(v
b ×?)
?
Rigid-Body Mechanics
? Static Analysis
Equivalent force
and moment acting
on a rigid-body
Rigid-Body Mechanics
? Static Analysis (cont'd)
Similarities between velocity and force
Force f - Angular velocity ? Same for the whole rigid-body
Moment n – Linear velocity v Depends on location
Recall the velocity analysis,
and the concept of velocity screw
Apply the similarity to force and moment analysis
Rigid-Body Mechanics
? Static Analysis (cont'd)
O
P0''
f n0
e'' L''
p0''
a
nA
A
Rigid-Body Mechanics
? Static Analysis (cont'd)
Concept of Wrench
Define
representing a system of forces & moments acting on a rigid body
Power of a rigid-body
Wrench acting at A and P
Wrench transform
A and P denote the cross-product matrices of a and p,respectively
Rigid-Body Mechanics
? Static Analysis (cont'd) - Example
Unit Cube
Unit forces f1 and f2
Resultant force
Z
X
Y
f1
f2
A
B
C
D
E
F G
H
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Rigid-Body Mechanics
? Static Analysis – Example (cont'd)
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=
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2
1
Rigid-Body Mechanics
? Dynamics of Rigid-Body
Rigid body Mass density
Mass
Mass 1st moment
Mass 2nd moment
Mass centre
Positive definite
and symmetric
X
Y
Z
O
YC
ZC
C
p
c
r
XC
Rigid-Body Mechanics
? Dynamics of Rigid-Body (cont'd)
Kinetic energy
Definition
Recall,Skew-symmetric A
Rigid-Body Mechanics
? Dynamics of Rigid-Body (cont'd)
Parallel Axes Theorem,
Principle Axes – Eigenvalue/vector
det(I0 - ?1) = 0
Eigenvalues ?1,?2,?3 --- Principle moments
Eigenvectors u1,u2,u3 --- representing principle axes
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zzzyzx
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xzxyxx
III
III
III
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General form Principle Moments
Rigid-Body Mechanics
? Dynamics of Rigid-Body (cont'd)
So far we have defined,
m,IC Rigid-Body Mass/Inertia Property
w Wrench – force and moment applied to rigid body
t Twist – angular and linear velocities (motion)
Equation of motion – describes the relationship between force/moment
applied to a rigid body and the resultant motion
w t m,I
C
Rigid-Body Mechanics
? Dynamics of Rigid-Body (cont'd)
Newton-Euler equations
In form of momentum and angular momentum,
Integrated form
Momentum Screw
Newton's equation - Translational
Euler's equation - Rotational
Assignment #2
? 3.1,3.2,3.5,3.6,3.14
? Due in two weeks,