MECH572
Introduction To Robotics
Lecture 3
Dept,Of Mechanical Engineering
Review
? Rigid-body Rotation - Representations
Representat
ion
Matrix Linear Invariant Quadratic Invariant
(Euler Parameters)
Natural Invariant
Definition
Number of
Elements 9 4 4 4
Constraints ||e1|| = 1,||e2|| = 1,
||e3|| = 1
e1?e2= 0,e2?e3= 0,
e3?e1= 0
||e|| = 1
Independent
Elements 9 - 6 = 3 4 – 1 = 3 4 – 1 = 3 4 – 1 = 3
Review
? Alternative form to represent a rotation – Euler Angles
A sequence of rotation,
Q = Q(?)Q(?)Q(?)
?,?,? rotation angles about certain axes,
? Coordinate Transformation
General form
Homogeneous form
Origin offset
Review
? Similarity Transformations
- Transformation of matrix entries (compare with vector entries
which uses linear transformation)
? The concept of invariance
After transformation between frames,certain quantities are
unchanged or frame invariant (inner product,trace,moments,
etc)
Overview of Rigid-Body Mechanics
? Purpose – Lay down foundations of kinetostatics (kinematics
+ statics) and dynamics of rigid bodies using matrix method
? Scope
– Linear and angular displacement,velocity,and acceleration analysis
– Static analysis
– Mass & Inertial properties
– Equation of motion for single rigid body
? Useful tools/concepts to be introduced
– Screw theory
– Twist
– Wrench
Q
O
P
P' A
p a p'
a'
P – Arbitrary
A - Reference
Rigid-Body Mechanics
? Description of a Rigid-Body Motion
Rigid-Body motion preserves distance
Define
Left multiply Take transpose = 0
A'
Displacement of any point projected onto the
rotation axis are same
Rigid-Body Mechanics
? General Rigid-Body Motion
e
Rigid-Body Mechanics
? General Rigid-Body Motion
Geometric Interpretation
A
B A'
B'
O
2-D case 3-D case
e
? Pitch p0
Any rigid body motion in 2-D can be
regarded as a pure rotation around
one point O
Any rigid body motion in 3-D can be
regarded as a Screw-like motion along
an axis,e
Rigid-Body Mechanics
? Screw Motion of a Rigid-Body
A rigid body can attain any configuration from its original to arbitrary
position following a screw-like motion defined by
– Screw Axis L
– Pitch p
L
?
Pitch p
d0 – minimum displacement
? - Magnitude
Rigid-Body Mechanics
? Screw of Rigid-Body Motion
Question to answer – how to define the screw axis of rigid-body motion?
Known Q,a and a' (direction+reference point)
Seek p0 (a point to define the screw axis)
Rearrange (3.10b) & (3.9)
e
O
A
P0
a
p0
Linear systems of equation
(overdetermined)
Rigid-Body Mechanics
? Screw of Rigid-Body Motion (cont'd)
Left multiply
Assume
O
P2
P
p1
p2
p
Rigid-Body Mechanics
? Plücker Coordinates of a Line
Co-linear
P1
Plücker array fully defining a line
Moment of L (unit force acting in the
dir,of the line about the ref,Point)
P
e
A
B
b a
p
Rigid-Body Mechanics
? Plücker coordinate of a line (cont'd)
Four independent variables to define a line
Plücker coordinate w.r.t different points A & B
O
Constraints
Rigid-Body Mechanics
? Rigid-Body Pose/Screw Parameters
Ways to describe general rigid-body motion,
- Plücker array + pitch + amplitude (angle)
- Orientation + displacement of 1 point
Total 6 independent parameters needed,
Define
Parameters for
Orientation
Parameters for orientation
Displacement of point A
Linear & quadratic
invariants
Natural Invariant
Pose array
Rigid-Body Mechanics
? Application Example - Pose Estimation
Rigid-Body Mechanics
? Rigid-Body Displacement Decomposition
Pure translation + pure rotation
Translation
Rotation
The three vectors are orthonormal and form a
right hand system
Rigid-Body Mechanics
? Rigid-Body Displacement Decomposition
Element (i,j) - projection on to
X
X'
Y
Z
Y'
Z'
The three columns of Q,directional cosine of
Rigid-Body Mechanics
? Rigid-Body Rotation About a Fixed Point
Take derivative
Angular velocity matrix p(t)
p0
Q(t)
O
Rigid-Body Mechanics
? General Instantaneous Motion
O
Q(t)
a0
p0
p
a
A0
P0
A
P
Rigid-Body Mechanics
? Example
Two frames F1 & F2.as shown
Unit circle defined in Y-Z plane
[p]2 =
Find
1) Q matrix and ? of F2
2) [p]1 when ? = ?/4
3) Linear velocity of P
when ? = ?/4
? ?T1,1,1
Rigid-Body Mechanics
? General rigid body motion
Properties
O
Similar to the concept of screw axis
P A
Rigid-Body Mechanics
? Instantaneous Screw
ISA axis
Pitch
General rigid-body motion defined by