MECH572-lecture4.ppt

MECH572Introduction To Robotics
Lecture 4
Dept. Of Mechanical Engineering
Review
Concept of Screw Motion of
a rigid body
O

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
If
Then
Ref. Eq. (&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
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:
= 120º
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
Rigid-Body Mechanics
Twist of Rigid-Body
Concept - pose 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)