Content
    Analogies
    Rotation Matrix and Homogeneous Translation Matrix
    Inverse
    Change of Coordinate Frame
    Rotating or Displacing a Frame
    Fixed Frame
    Body Frame
    Velocities
    Axis
    Angular Velocity and Twist
    Matrix Representation of Angular Velocity and Twist
    Idenities for Angular Velocities
    Adjoint Representation of a Homogeneous Translation Matrix
    Identities for Adjoint Representation
    Change of Coordinate Frame
    Exponential Coordinates
    Matrix Representation
    Exponential and Logarithm
    Exponential
    Logarithm
    Change Coordinate frame for Moment and Wrench
    Literature

Comparison of Rotations and Rigid-Body Motions

Analogies

RotationsRigid-Body Motions
Rotation Matrix RRHomogeneous Tranformation Matrix TT
Rotation Axis ω^\hat\omegaScrew Axis SS
Angular Velocity ω=ω^θ˙\omega = \hat\omega \dot\thetaTwist V=Sθ˙V = S\dot\theta
Exponential Coordinates ω^θ∈R3\hat\omega \theta \in \mathbb{R}^3Exponential Coordinates Sθ∈R6S\theta \in \mathbb{R}^6

Rotation Matrix and Homogeneous Translation Matrix

RotationsRigid-Body Motions
R∈SO(3)R \in SO(3)T∈SE(3)T \in SE(3)
SO(3)SO(3): 3×33 \times 3 matricesSE(3)SE(3): 4×44 \times 4 matrices
RTR=1R^TR = 1 and det⁡R=1\det R =1T=[Rp01]T = \begin{bmatrix}R & p \\ 0 & 1 \end{bmatrix} where R∈SO(3)R \in SO(3), p∈R3p \in \mathbb{R}^3

Inverse

RotationsRigid-Body Motions
R−1=RTR^{-1} = R^TT−1[RT−RTp01]T^{-1}\begin{bmatrix}R^T & -R^Tp \\ 0 & 1 \end{bmatrix}

Change of Coordinate Frame

RotationsRigid-Body Motions
matrixRab̸Rb̸c=RacR_{a\not{b}}R_{\not{b}c} = R_{ac}Tab̸Tb̸c=TacT_{a\not{b}}T_{\not{b}c} = T_{ac}
vectorRab̸pb̸=paR_{a\not{b}}p_{\not{b}} = p_{a}Tab̸pb̸=paT_{a\not{b}}p_{\not{b}} = p_{a}

Rotating or Displacing a Frame

Rotating frame {b}\{b\}Displacing frame {b}\{b\}
R=Rot(ω^,θ)R = Rot(\hat\omega, \theta)T=[Rot(ω^,θ)p01]T=\begin{bmatrix} Rot(\hat\omega, \theta) & p \\ 0 & 1 \end{bmatrix}
Fixed Frame
  • ω^s=ω^\hat\omega_s = \hat\omega: Rotation axis defined in {s}\{s\} coordinates
  • pp: Translation vector defined in {s}\{s\} coordinates
Rotating frame {b}\{b\}Displacing frame {b}\{b\}
Rsb′=RRsbR_{sb'} = RR_{sb}Tsb′=TTsbT_{sb'} = TT_{sb}
rotate θ\theta about ω^s=ω^\hat\omega_s = \hat\omegarotate θ\theta about ω^s=ω^\hat\omega_s = \hat\omega (moves origin of {b}\{b\})
then translate pp in {s}\{s\}
Body Frame
  • ω^s=ω^\hat\omega_s = \hat\omega: Rotation axis defined in {s}\{s\} coordinates
  • pp: Translation vector defined in {s}\{s\} coordinates
Rotating frame {b}\{b\}Displacing frame {b}\{b\}
Rsb′′=RsbRR_{sb''} = R_{sb}RTsb′′=TsbTT_{sb''} = T_{sb}T
rotate θ\theta about ω^b=ω^\hat\omega_b = \hat\omegatranslate pp in {b}\{b\}
then rotate θ\theta about ω^\hat\omega in new body frame

Velocities

Axis

Unit Rotation Axis“Unit” Screw Axis
ω^∈R3\hat{\omega} \in \mathbb{R}^3S=(Sω,Sv)=[ωv]∈R6S = (S_\omega, S_v) = \begin{bmatrix} \omega \\ v\end{bmatrix} \in \mathbb{R}^6
∥ω^∥=1\| \hat{\omega} \| = 1either ∥Sω∥=1\|S_\omega\| = 1
or Sω=0,∥Sv∥=1S_\omega = 0, \|S_v\| = 1

Screw axis {q,s^,h}\{q, \hat s, h\} with finite hh:

S=[ωv]=[s^−s^×q+hs^]S = \begin{bmatrix} \omega \\ v \end{bmatrix} = \begin{bmatrix} \hat s \\ - \hat s \times q + h \hat s \end{bmatrix}

Angular Velocity and Twist

Rotations (angular velocity)Rigid-Body Motions (twist)
ω=ω^θ˙\omega = \hat{\omega} \dot{\theta}V=Sθ˙V = S \dot{\theta}
Matrix Representation of Angular Velocity and Twist
Rotations (angular velocity)Rigid-Body Motions (twist)
for ω∈R3\omega \in \mathbb{R}^3for V=[ωv]∈R6V = \begin{bmatrix} \omega \\ v\end{bmatrix} \in \mathbb{R}^6
[ω]=[0−ω3ω2ω30−ω1−ω2ω10]∈so(3)[\omega] = \begin{bmatrix} 0 & -\omega_3 & \omega_2 \\ \omega_3 & 0 & -\omega_1 \\ -\omega_2 & \omega_1 & 0\end{bmatrix} \in so(3)[V]=[[ω]v00]∈se(3)[V] = \begin{bmatrix} [\omega] & v \\ 0 & 0 \end{bmatrix} \in se(3)
R˙R−1=[ωs]\dot{R}R^{-1} = [\omega_s]T˙T−1=[Vs]\dot{T}T^{-1} = [V_s]
R−1R˙=[ωb]R^{-1}\dot{R} =[\omega_b]T−1T˙=[Vb]T^{-1}\dot{T} = [V_b]
Idenities for Angular Velocities

With ω,x∈R3\omega, x \in \mathbb{R}^3 and R∈SO(3)R \in SO(3):

  • [ω]=−[ω]T[\omega] = -[\omega]^T
  • [ω]x=−[x]ω[\omega]x = -[x]\omega
  • [ω][x]=([x][ω])T[\omega][x] = ([x][\omega])^T
  • R[ω]RT=[Rω]R[\omega]R^T = [R\omega]

Adjoint Representation of a Homogeneous Translation Matrix

With a Homogeneous Translation Matrix T=[Rp01]T = \begin{bmatrix}R & p \\ 0 & 1 \end{bmatrix} where R∈SO(3)R \in SO(3), p∈R3p \in \mathbb{R}^3 the adjoint representation is:

[AdT]=[R0[p]RR]∈R6×6[Ad_T] = \begin{bmatrix} R & 0 \\ [p]R & R \end{bmatrix} \in \mathbb{R}^{6\times 6}
Identities for Adjoint Representation
  • [AdT]−1=[AdT−1][Ad_T]^{-1} = [Ad_{T^{-1}}]
  • [AdT1][AdT2]=[AdT1T2][Ad_{T_1}][Ad_{T_2}] = [Ad_{T_1T_2}]

Change of Coordinate Frame

RotationsRigid-Body Motions
ω^a=Rab̸ω^b̸\hat{\omega}_a = R_{a\not{b}}\hat{\omega}_{\not{b}}Sa=[AdTab̸]Sb̸S_a = [Ad_{T_{a\not{b}}}]S_{\not{b}}
ωa=Rab̸ωb̸\omega_a = R_{a\not{b}} \omega_{\not{b}}Va=[AdTab̸]Vb̸V_a = [Ad_{T_{a\not{b}}}]V_{\not{b}}

Exponential Coordinates

RotationsRigid-Body Motions
exponential coordinates forR∈SO(3)R \in SO(3)T∈SE(3)T \in SE(3)
areω^θ∈R3\hat\omega\theta \in \mathbb{R}^3Sθ∈R6S\theta \in \mathbb{R}^6

Matrix Representation

RotationsRigid-Body Motions
[ω^]θ∈so(3)[\hat{\omega}]\theta \in so(3)[S]θ∈se(3)[S]\theta \in se(3)

Exponential and Logarithm

RotationsRigid-Body Motions
exponentialso(3)→SO(3)so(3) \rightarrow SO(3)se(3)→SE(3)se(3) \rightarrow SE(3)
logarithmSO(3)→so(3)SO(3) \rightarrow so(3)SE(3)→se(3)SE(3) \rightarrow se(3)
Exponential
RotationsRigid-Body Motions
[ω^]θ∈so(3)→R∈SO(3)[\hat\omega]\theta \in so(3) \rightarrow R \in SO(3)[S]θ∈se(3)→T∈SE(3)[S]\theta \in se(3) \rightarrow T \in SE(3)
R=Rot(ω^,θ)=e[ω^]θ=I+sin⁡θ[ω^]+(1−cos⁡θ)[ω^]2R = Rot(\hat\omega, \theta) = e^{[\hat{\omega}]\theta} = I + \sin\theta[\hat\omega]+(1 - \cos\theta)[\hat\omega]^2T=e[S]θ=[e[ω]θ(Iθ+(1−cos⁡θ)[ω]+(θ−sin⁡θ)[ω]2)v01]T = e^{[S]\theta} = \begin{bmatrix} e^{[\omega]\theta} & (I\theta + (1 - \cos \theta) [\omega] + (\theta - \sin \theta) [\omega]^2) v \\ 0 & 1 \end{bmatrix}
Logarithm
RotationsRigid-Body Motions
R∈SO(3)→[ω^]θ∈so(3)R \in SO(3) \rightarrow [\hat\omega]\theta \in so(3)T∈SE(3)→[S]θ∈se(3)T \in SE(3) \rightarrow [S]\theta \in se(3)
see here for algorithmsee here for algorithm

Change Coordinate frame for Moment and Wrench

  • Moment: ma=Rabmbm_a = R_{ab}m_b
  • Wrench: Fa=(ma,fa)=[AdTba]TFbF_a = (m_a, f_a) = [Ad_{T_{ba}}]^TF_b

Literature

Notes taken from:

Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017

  • Category

  • Mechanics

  • Tags

  • Robotics

  • Created

  • 5. January 2021

  • Modified

  • 3. June 2023