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Attaching Coordinate Frames

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(1)

Dongjun Lee

Rigid Body Rotation and SO(3)

Dongjun Lee (이동준)

Department of Mechanical & Aerospace Engineering Seoul National University

Dongjun Lee

Attaching Coordinate Frames

(2)

Dongjun Lee

Coordinate Frames: Examples

Dongjun Lee

Joint Space Q and Workspace W

(3)

Dongjun Lee

Configuration Space C

posture C

Dongjun Lee

Rigid Body Transformation

{B(0)}

{B(t)}

{A}

p(0)

p(t)=g(p(0))

(4)

Dongjun Lee

Free Vector

{B(0)}

{B(t)}

{A}

p

g(p)

r g(q)

s q g(r)

g(s)

Dongjun Lee

Rigid Transformation: Definition

{B(0)} {B(t)}

{A}

p

g(p) g(q) q

(5)

Dongjun Lee

Rotation Matrix R

{A}

xb

yb zb

Dongjun Lee

Three Roles of R

q(0) {A}

q(t)

(6)

Dongjun Lee

Properties of R

Dongjun Lee

Properties of R

v w x

Rv Rw x

v x w R(v x w)=Rv x Rw

R R

R

body‐frame inertial‐frame

(7)

Dongjun Lee

Properties of R

xb yb

xa ya

R

body‐frame

inertial‐frame w^

R (Rw)^

Dongjun Lee

Composition of Rotations

{A} {A}

y

z

x {C}

xa

ya za

(8)

Dongjun Lee

Composition of Rotations

xb yb

xa ya

R1

frame {B}

frame {A}

R2^b

R1 R2^a

Dongjun Lee

Special Orthogonal Group SO(3)

(9)

Dongjun Lee

Exponential of w

{A}

w q(0) {B(t)}

q(t)

Dongjun Lee

Angular Velocity and so(3)

(10)

Dongjun Lee

Rodrigues’ Formula

Dongjun Lee

Exponential Coordinates for SO(3)

(11)

Dongjun Lee

Logarithm Map for SO(3)

Dongjun Lee

Equivalent Axis Representation

{A}

w {B(t)}

(12)

Dongjun Lee

Exponential Coordinates in Different Frames

xb yb

xa ya

Rab Rab

Dongjun Lee

Geometric Meaning of exp(wat) and exp(wbt)

{A}

{B1} w t

(13)

Dongjun Lee

Rotational Velocity

{A}

g(t) w

Dongjun Lee

Parameterization of SO(3): Euler Angle

{A}

xb

yb zb

x

y z

(14)

Dongjun Lee

Parameterization of SO(3): Euler Angle

Dongjun Lee

Parameterization of SO(3): Quaternions

(15)

Dongjun Lee

Parameterization of SO(3): Quaternions

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