Robotics & Automation
Lecture 05
Differential Kinematics and Rigid Body Motion SE(3)
John T. Wen
September 8, 2008
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Understanding SE(3) & Angular Velocity in Robotics: Kinematics & Rigid Body Motion - Prof., Study notes of Robotics
A lecture note from rpi ecse/csci 4480 robotics i, focusing on differential kinematics and rigid body motion in the context of se(3) and angular velocity. The lecture covers the exponential formula for rotation representation, the relationship between angular velocity and rotation, and the differentiation of vectors and transforms. It also discusses the representation jacobian and singularities in the context of unit quaternions.
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Robotics & Automation
Lecture
Differential Kinematics and Rigid Body Motion
SE
John T. Wen
September 8, 2008
RPI ECSE/CSCI 4480 Robotics I
Last Time
Matrix exponential representation of rotation:
Rot
k , (^) θ
) =
(^) e
kˆ θ .
p eters, Euler angles.Other representations: unit quaternion, vector quaternion, Euler-Rodrigues param-
is a representation of
SO
if
p
=
f (^) ( R )
is a
local
one-to-one and onto mapping
(i.e., every
R
corresponds to a unique
p
and every
p
corresponds to a unique
R
, in a
neighborhood). Locally, the mapping is therefore invertible,
R
f (^) −
1 ( p ) .
Today: differential kinematics, what if
R
is a time varying and we would like to
compute
R
(and
˙p )?
Copyrighted by John T. Wen
Page 1
RPI ECSE/CSCI 4480 Robotics I
Differentiation of
R
and Angular Velocity
Robotics literature:
R
ab
( ω b / a ) a R
ab
,
or just
R
ωˆ
R .
This is also sometimes written as
R
R
ω
) b .
Spacecraft literature:
R
ba
( ω a / b ) b R
ba
( ω b / a ) b R
ba
,
or just
R
(^) ˆω
R .
This is also sometimes written as
R
R
ω
) a , where
ω
) a
is the same
ω
as in the
The collection of skew symmetric matrices of the formrobotics literature.
ωˆ
for
ω
R
3
is called
so
(considered to be the “tangent” of
R
Copyrighted by John T. Wen
Page 3
RPI ECSE/CSCI 4480 Robotics I
Differentiation of vectors and transforms
˙v a
d ( R ab
v b )
dt
R
ab
v b (^) +
R
ab
(^) ˙v b
=
( ω b / a ) a v a
R
ab
(^) ˙v b
L ˙
a
d ( R ab
L b R ba
)
dt
( ω b / a ) a R
ab
L b R
ba
R
ab
L b R
ba
ω b / a ) a
R
ab
(^) L˙
b R ba
( ω b / a ) a L a
L
a ( ω b / a ) a +
R
ab
L˙ b R
ba
.
Copyrighted by John T. Wen
Page 4