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网站宣传创意视频,南极电商是做什么的,秦皇岛市建设局,雄安移动网络本文仅供学习使用 本文参考#xff1a; B站#xff1a;CLEAR_LAB 笔者带更新-运动学 课程主讲教师#xff1a; Prof. Wei Zhang 南科大高等机器人控制课 Ch03 Operator View of Rigid-Body Transformation 1. Rotation Operation via Differential Equation1.1 Skew Symmetr… 本文仅供学习使用 本文参考 B站CLEAR_LAB 笔者带更新-运动学 课程主讲教师 Prof. Wei Zhang 南科大高等机器人控制课 Ch03 Operator View of Rigid-Body Transformation 1. Rotation Operation via Differential Equation1.1 Skew Symmetric Matrices1.2 Rotation Operation via Differential Equation 1.3 Raotation Matrix as a Rotation Operator2. Rotation Operation in Different Frames2.1 Rotation Martix Properties2.2 Rotation Operation in Different Frames 3. Rigid-Body Operation via Diffeential Equation3.1 SE(3)3.2 se(3) 4. Homogeneous Transformation Matrix as Rigid-Body Operator5. Rigid-Body Operation of Screw Axis 1. Rotation Operation via Differential Equation 1.1 Skew Symmetric Matrices Recall that cross product is a special linear transformation. For any ω ⃗ ∈ R 3 \vec{\omega}\in \mathbb{R} ^3 ω ∈R3, there is a matrix ω ⃗ ~ ∈ R 3 × 3 \tilde{\vec{\omega}}\in \mathbb{R} ^{3\times 3} ω ~∈R3×3 such that ω ⃗ × R ⃗ ω ⃗ ~ R ⃗ \vec{\omega}\times \vec{R}\tilde{\vec{\omega}}\vec{R} ω ×R ω ~R ω ⃗ [ ω 1 ω 2 ω 3 ] ⟷ ω ⃗ ~ [ 0 − ω 3 ω 2 ω 3 0 − ω 1 − ω 2 ω 1 0 ] \vec{\omega}\left[ \begin{array}{c} \omega _1\ \omega _2\ \omega _3\ \end{array} \right] \longleftrightarrow \tilde{\vec{\omega}}\left[ \begin{matrix} 0 -\omega _3 \omega _2\ \omega _3 0 -\omega _1\ -\omega _2 \omega 1 0\ \end{matrix} \right] ω ​ω1​ω2​ω3​​ ​⟷ω ~ ​0ω3​−ω2​​−ω3​0ω1​​ω2​−ω1​0​ ​ Note that a ⃗ ~ − a ⃗ ~ T \tilde{\vec{a}}-\tilde{\vec{a}}^{\mathrm{T}} a ~−a ~T (called skew stmmetric) ω ⃗ ~ \tilde{\vec{\omega}} ω ~ is called a skew-symmetric matrix representation of the vector ω ⃗ \vec{\omega} ω The set of skew-symmetric matrices in : s o ( n ) ≜ { S ∈ R n × n : S T − S } so\left( n \right) \triangleq \left{ S\in \mathbb{R} ^{n\times n}:S^{\mathrm{T}}-S \right} so(n)≜{S∈Rn×n:ST−S} We are interested in case n 2 , 3 n2,3 n2,3 Rotation matrix ∈ S O ( 3 ) { R T R E , det ⁡ ( R ) 1 } \in SO\left( 3 \right) \left{ R^{\mathrm{T}}RE,\det \left( R \right) 1 \right} ∈SO(3){RTRE,det®1} 1.2 Rotation Operation via Differential Equation Consider a point initially located at R ⃗ p 0 \vec{R}{p_0} R p0​​ at time t 0 t0 t0Rotate the point with unit angular velocity ω ^ \hat{\omega} ω^. Assuming the rotation axis passing through the origin, the motion is describe by R ⃗ ˙ p ( t ) ω ^ × R ⃗ p ω ^ ~ R ⃗ p ( t ) , p ( 0 ) p 0 \dot{\vec{R}}_p\left( t \right) \hat{\omega}\times \vec{R}_p\tilde{\hat{\omega}}\vec{R}_p\left( t \right) ,p\left( 0 \right) p_0 R ˙p​(t)ω^×R p​ω^~R p​(t),p(0)p0​ linear velocity at time t t t, recall x ˙ A x , x ( 0 ) x 0 ⇒ x ( t ) e A t x 0 \dot{x}Ax,x\left( 0 \right) x_0\Rightarrow x\left( t \right) e^{At}x_0 x˙Ax,x(0)x0​⇒x(t)eAtx0​This is a linear ODE with solution : R ⃗ ˙ p ( t ) ω ^ ~ R ⃗ p ( t ) ⇒ R ⃗ ˙ p ( t ) e ω ^ ~ t R ⃗ p 0 \dot{\vec{R}}_p\left( t \right) \tilde{\hat{\omega}}\vec{R}_p\left( t \right) \Rightarrow \dot{\vec{R}}p\left( t \right) e^{\tilde{\hat{\omega}}t}\vec{R}{p_0} R ˙p​(t)ω^~R p​(t)⇒R ˙p​(t)eω^~tR p0​​After t θ t\theta tθ, the point has been rotated by θ \theta θ degree. Note R ⃗ ˙ p ( θ ) e ω ^ ~ θ R ⃗ p 0 \dot{\vec{R}}p\left( \theta \right) e^{\tilde{\hat{\omega}}\theta}\vec{R}{p0} R ˙p​(θ)eω^~θR p0​​, e ω ^ ~ θ e^{\tilde{\hat{\omega}}\theta} eω^~θ is rotation operator R o t ( ω ^ , θ ) ≜ e ω ^ ~ θ \mathrm{Rot}\left( \hat{\omega},\theta \right) \triangleq e^{\tilde{\hat{\omega}}\theta} Rot(ω^,θ)≜eω^~θ can be viewed as a rotation operator that rotates a point about ω ^ \hat{\omega} ω^ through θ \theta θ degree The discussion holds for any reference frame 1.3 Raotation Matrix as a Rotation Operator Theorem: Every rotation matrix R R R can be written as R R o t ( ω ^ , θ ) ≜ e ω ^ ~ θ ∈ S O ( 3 ) { R T R E , det ⁡ ( R ) 1 } R\mathrm{Rot}\left( \hat{\omega},\theta \right) \triangleq e^{\tilde{\hat{\omega}}\theta}\in SO\left( 3 \right) \left{ R^{\mathrm{T}}RE,\det \left( R \right) 1 \right} RRot(ω^,θ)≜eω^~θ∈SO(3){RTRE,det®1}, i.e. , it represents a rotation operation about ω ^ \hat{\omega} ω^ by θ \theta θWe have seen how to use R R R to represent frame orientation and change of coordinate between different frames. They are quite different from the operator interpretation of R R RTo apply the rotation operation, all the vectors / matrices have to be expressed in the same refrence frame For example, assume R [ 1 0 0 0 0 − 1 0 1 0 ] R o t ( x ^ , π 2 ) R\left[ \begin{matrix} 1 0 0\ 0 0 -1\ 0 1 0\ \end{matrix} \right] \mathrm{Rot}\left( \hat{x},\frac{\pi}{2} \right) R ​100​001​0−10​ ​Rot(x^,2π​) Consider a relation q R p qRp qRp: Change reference frame interpretation : two frames { A } , { B } \left{ A \right} ,\left{ B \right} {A},{B} , one physical Point a a a R R R : orientation of { B } \left{ B \right} {B} relative to { A } \left{ A \right} {A} : i.e. R [ Q B A ] R\left[ Q{\mathrm{B}}^{A} \right] R[QBA​] then : p a B , q a A , q R p ⇒ a A [ Q B A ] a B pa^B,qa^A,qRp\Rightarrow a^A\left[ Q{\mathrm{B}}^{A} \right] a^B paB,qaA,qRp⇒aA[QBA​]aBRotation operator interpretation : one frame and two points a → a ′ , p a A , q a ′ A ⇒ a ′ A R a A a\rightarrow a^{\prime},pa^A,q{a^{\prime}}^A\Rightarrow {a^{\prime}}^ARa^A a→a′,paA,qa′A⇒a′ARaA Consider the frame operation: Change of reference frame : Have one “frame object”, two reference frames Frame object { A } \left{ A \right} {A}, orientation in { O } \left{ O \right} {O} , is R A O , R A B R{\mathrm{A}}^{O},R{\mathrm{A}}^{B} RAO​,RAB​, ⇒ R A O R B O R A B R R A B \Rightarrow R{\mathrm{A}}^{O}R{\mathrm{B}}^{O}R{\mathrm{A}}^{B}RR_{\mathrm{A}}^{B} ⇒RAO​RBO​RAB​RRAB​Rotation a frame : R ′ A R R A {R^{\prime}}_ARR_A R′A​RRA​ two frame objects, one refrence frame more precisely : R ′ A R R A ⇒ R A ′ O R R A O {R^{\prime}}_ARRA\Rightarrow R{\mathrm{A}^{\prime}}^{O}RR_{\mathrm{A}}^{O} R′A​RRA​⇒RA′O​RRAO​ 2. Rotation Operation in Different Frames 2.1 Rotation Martix Properties [ Q ] [ Q ] T E \left[ Q \right] \left[ Q \right] ^{\mathrm{T}}E [Q][Q]TE [ Q 1 ] [ Q 2 ] ∈ S O ( 3 ) , i f [ Q 1 ] , [ Q 2 ] ∈ S O ( 3 ) \left[ Q_1 \right] \left[ Q_2 \right] \in SO\left( 3 \right) ,if\,\,\left[ Q_1 \right] ,\left[ Q_2 \right] \in SO\left( 3 \right) [Q1​][Q2​]∈SO(3),if[Q1​],[Q2​]∈SO(3) product of two rotation matrices is also a rotation matrix p , q ∈ R 3 , ∥ [ Q ] R ⃗ p − [ Q ] R ⃗ q ∥ 2 ∥ Q ∥ 2 ( R ⃗ p − R ⃗ q ) T [ Q ] T Q ∥ R ⃗ p − R ⃗ q ∥ 2 p,q\in \mathbb{R} ^3,\left| \left[ Q \right] \vec{R}p-\left[ Q \right] \vec{R}{\mathrm{q}} \right| ^2\left| \left[ Q \right] \left( \vec{R}p-\vec{R}{\mathrm{q}} \right) \right| ^2\left( \vec{R}p-\vec{R}{\mathrm{q}} \right) ^{\mathrm{T}}\left[ Q \right] ^{\mathrm{T}}\left[ Q \right] \left( \vec{R}p-\vec{R}{\mathrm{q}} \right) \left| \vec{R}p-\vec{R}{\mathrm{q}} \right| ^2 p,q∈R3, ​[Q]R p​−[Q]R q​ ​2 ​Q ​2(R p​−R q​)T[Q]TQ ​R p​−R q​ ​2 ∥ R ⃗ p − R ⃗ q ∥ ∥ [ Q ] R ⃗ p − [ Q ] R ⃗ q ∥ \left| \vec{R}p-\vec{R}{\mathrm{q}} \right| \left| \left[ Q \right] \vec{R}p-\left[ Q \right] \vec{R}{\mathrm{q}} \right| ​R p​−R q​ ​ ​[Q]R p​−[Q]R q​ ​ : rotation operator preserves distance Q [ Q ] v ⃗ × [ Q ] w ⃗ \left[ Q \right] \left( \vec{v}\times \vec{w} \right) \left[ Q \right] \vec{v}\times \left[ Q \right] \vec{w} Q[Q]v ×[Q]w [ Q ] w ⃗ ~ [ Q ] T [ Q ] w ⃗ ~ \left[ Q \right] \tilde{\vec{w}}\left[ Q \right] ^{\mathrm{T}}\widetilde{\left[ Q \right] \vec{w}} [Q]w ~[Q]T[Q]w ​ —— important 2.2 Rotation Operation in Different Frames Consider two frames { A } \left{ A \right} {A} and { B } \left{ B \right} {B} the actual numerical values of the operator R o t ( ω ^ , θ ) \mathrm{Rot}\left( \hat{\omega},\theta \right) Rot(ω^,θ) depend on both the reference frame to repersent ω ^ \hat{\omega} ω^ and the reference frame to represent the operator itself Consider a rotation axis ω ^ \hat{\omega} ω^ (coordinate free vector), with { A } \left{ A \right} {A} - frame coordinate ω ^ A \hat{\omega}^A ω^A and { B } \left{ B \right} {B} - frame. We know ω ⃗ A [ Q B A ] ω ⃗ B \vec{\omega}^A\left[ Q{\mathrm{B}}^{A} \right] \vec{\omega}^B ω A[QBA​]ω B Let B R o t ( B ω ^ , θ ) ^B\mathrm{Rot}\left( ^B\hat{\omega},\theta \right) BRot(Bω^,θ) and A R o t ( A ω ^ , θ ) ^A\mathrm{Rot}\left( ^A\hat{\omega},\theta \right) ARot(Aω^,θ) be the two rotation matrices, representing the same rotation operatioon R o t ( ω ^ , θ ) \mathrm{Rot}\left( \hat{\omega},\theta \right) Rot(ω^,θ) in frames { A } \left{ A \right} {A} and { B } \left{ B \right} {B} We have the relation : A R o t ( A ω ^ , θ ) [ Q B A ] B R o t ( B ω ^ , θ ) [ Q A B ] ^A\mathrm{Rot}\left( ^A\hat{\omega},\theta \right) \left[ Q{\mathrm{B}}^{A} \right] ^B\mathrm{Rot}\left( ^B\hat{\omega},\theta \right) \left[ Q{\mathrm{A}}^{B} \right] ARot(Aω^,θ)[QBA​]BRot(Bω^,θ)[QAB​] Approach 1 : two points p → p ′ ⇒ R ⃗ p ′ A A R o t ( A ω ^ , θ ) R ⃗ p A p\rightarrow p^{\prime}\Rightarrow \vec{R}{\mathrm{p}^{\prime}}^{A}^A\mathrm{Rot}\left( ^A\hat{\omega},\theta \right) \vec{R}{\mathrm{p}}^{A} p→p′⇒R p′A​ARot(Aω^,θ)R pA​ { B } \left{ B \right} {B} - frame : R ⃗ p ′ B B R o t ( B ω ^ , θ ) R ⃗ p B \vec{R}{\mathrm{p}^{\prime}}^{B}^B\mathrm{Rot}\left( ^B\hat{\omega},\theta \right) \vec{R}{\mathrm{p}}^{B} R p′B​BRot(Bω^,θ)R pB​ ⇒ [ Q B A ] R ⃗ p ′ B [ Q B A ] B R o t ( B ω ^ , θ ) R ⃗ p B ⇒ R ⃗ p ′ A [ Q B A ] B R o t ( B ω ^ , θ ) [ Q B A ] R ⃗ p A ⇒ A R o t ( A ω ^ , θ ) [ Q B A ] B R o t ( B ω ^ , θ ) [ Q B A ] \Rightarrow \left[ Q{\mathrm{B}}^{A} \right] \vec{R}{\mathrm{p}^{\prime}}^{B}\left[ Q{\mathrm{B}}^{A} \right] ^B\mathrm{Rot}\left( ^B\hat{\omega},\theta \right) \vec{R}{\mathrm{p}}^{B}\Rightarrow \vec{R}{\mathrm{p}^{\prime}}^{A}\left[ Q{\mathrm{B}}^{A} \right] ^B\mathrm{Rot}\left( ^B\hat{\omega},\theta \right) \left[ Q{\mathrm{B}}^{A} \right] \vec{R}{\mathrm{p}}^{A}\Rightarrow ^A\mathrm{Rot}\left( ^A\hat{\omega},\theta \right) \left[ Q{\mathrm{B}}^{A} \right] ^B\mathrm{Rot}\left( ^B\hat{\omega},\theta \right) \left[ Q{\mathrm{B}}^{A} \right] ⇒[QBA​]R p′B​[QBA​]BRot(Bω^,θ)R pB​⇒R p′A​[QBA​]BRot(Bω^,θ)[QBA​]R pA​⇒ARot(Aω^,θ)[QBA​]BRot(Bω^,θ)[QBA​] Approach 2 : for a ⃗ ∈ R 3 , a ⃗ ~ ∈ s o ( 3 ) , [ Q ] ∈ S O ( 3 ) \vec{a}\in \mathbb{R} ^3,\tilde{\vec{a}}\in so\left( 3 \right) ,\left[ Q \right] \in SO\left( 3 \right) a ∈R3,a ~∈so(3),[Q]∈SO(3) ⇒ [ Q ] a ⃗ ∈ R 3 , [ Q ] a ⃗ ~ [ Q ] a ⃗ ~ [ Q ] T \Rightarrow \left[ Q \right] \vec{a}\in \mathbb{R} ^3,\widetilde{\left[ Q \right] \vec{a}}\left[ Q \right] \tilde{\vec{a}}\left[ Q \right] ^{\mathrm{T}} ⇒[Q]a ∈R3,[Q]a ​[Q]a ~[Q]T R o t ( ω ⃗ A , θ ) e ω ⃗ ~ A θ e [ Q B A ] ω ⃗ B ~ θ e [ Q B A ] ω ⃗ ~ B [ Q B A ] T θ [ Q B A ] e ω ⃗ ~ B θ [ Q B A ] T ← e P A P − 1 P e A P − 1 Rot\left( \vec{\omega}^A,\theta \right) e^{\tilde{\vec{\omega}}^A\theta}e^{\widetilde{\left[ Q{\mathrm{B}}^{A} \right] \vec{\omega}^B}\theta}e^{\left[ Q{\mathrm{B}}^{A} \right] \tilde{\vec{\omega}}^B\left[ Q{\mathrm{B}}^{A} \right] ^{\mathrm{T}}\theta}\left[ Q{\mathrm{B}}^{A} \right] e^{\tilde{\vec{\omega}}^B\theta}\left[ Q{\mathrm{B}}^{A} \right] ^{\mathrm{T}}\gets e^{PAP^{-1}}Pe^AP^{-1} Rot(ω A,θ)eω ~Aθe[QBA​]ω B ​θe[QBA​]ω ~B[QBA​]Tθ[QBA​]eω ~Bθ[QBA​]T←ePAP−1PeAP−1

1. Rigid-Body Operation via Diffeential Equation Recall: Every [ Q ] ∈ S O ( 3 ) \left[ Q \right] \in SO\left( 3 \right) [Q]∈SO(3) can be viewed as the state transition matrix associated with the rotation ODE(1). It maps the initial position to the current position(after the rotation motion) p ⃗ ( θ ) R o t ( ω ⃗ , θ ) p ⃗ 0 \vec{p}\left( \theta \right) Rot\left( \vec{\omega},\theta \right) \vec{p}_0 p ​(θ)Rot(ω ,θ)p ​0​ viewed as solution to p ⃗ ˙ ( t ) ω ⃗ ~ p ⃗ ( t ) , p ⃗ ( 0 ) p ⃗ 0 \dot{\vec{p}}\left( t \right) \tilde{\vec{\omega}}\vec{p}\left( t \right) ,\vec{p}\left( 0 \right) \vec{p}0 p ​˙​(t)ω ~p ​(t),p ​(0)p ​0​ at t θ t\theta tθThe above relation requires that the rotation axis passes through the origin. We can obtain similar ODE characterization for [ T ] ∈ S E ( 3 ) \left[ T \right] \in SE\left( 3 \right) [T]∈SE(3) , which will lead to exponential coordinate of SE(3) Recall : Theorem (Chasles): Every rigid body motion can be realized by a screw motion Consider a point p p p undergoes a screw motion with screw axis S \mathcal{S} S and unit speed ( θ ˙ 1 \dot{\theta}1 θ˙1). Let the cooresponding twist be V θ ˙ S ( ω ⃗ , v ⃗ ) \mathcal{V} \dot{\theta}\mathcal{S} \left( \vec{\omega},\vec{v} \right) Vθ˙S(ω ,v ) . The motion can be described be the following ODE. p ⃗ ˙ ( t ) ω ⃗ × p ⃗ ( t ) v ⃗ ⇒ [ p ⃗ ˙ ( t ) 0 ] [ ω ⃗ ~ v ⃗ 0 0 ] [ p ⃗ ( t ) 1 ] \dot{\vec{p}}\left( t \right) \vec{\omega}\times \vec{p}\left( t \right) \vec{v}\Rightarrow \left[ \begin{array}{c} \dot{\vec{p}}\left( t \right)\ 0\ \end{array} \right] \left[ \begin{matrix} \tilde{\vec{\omega}} \vec{v}\ 0 0\ \end{matrix} \right] \left[ \begin{array}{c} \vec{p}\left( t \right)\ 1\ \end{array} \right] p ​˙​(t)ω ×p ​(t)v ⇒[p ​˙​(t)0​][ω ~0​v 0​][p ​(t)1​] Solution in homogeneous coordinate is : [ p ⃗ ( t ) 1 ] exp ⁡ ( [ ω ⃗ ~ v ⃗ 0 0 ] t ) [ p ⃗ ( 0 ) 1 ] \left[ \begin{array}{c} \vec{p}\left( t \right)\ 1\ \end{array} \right] \exp \left( \left[ \begin{matrix} \tilde{\vec{\omega}} \vec{v}\ 0 0\ \end{matrix} \right] t \right) \left[ \begin{array}{c} \vec{p}\left( 0 \right)\ 1\ \end{array} \right] [p ​(t)1​]exp([ω ~0​v 0​]t)[p ​(0)1​] 3.1 SE(3) For any twist V ( ω ⃗ , v ⃗ ) \mathcal{V} \left( \vec{\omega},\vec{v} \right) V(ω ,v ), let [ V ] \left[ \mathcal{V} \right] [V] be its matrix representation of twist V \mathcal{V} V [ V ] [ ω ⃗ ~ v ⃗ 0 0 ] ∈ R 4 × 4 \left[ \mathcal{V} \right] \left[ \begin{matrix} \tilde{\vec{\omega}} \vec{v}\ 0 0\ \end{matrix} \right] \in \mathbb{R} ^{4\times 4} [V][ω ~0​v 0​]∈R4×4 The abve definition also applies to a screw axis S ( ω ⃗ , v ⃗ ) , [ S ] [ ω ⃗ ~ v ⃗ 0 0 ] \mathcal{S} \left( \vec{\omega},\vec{v} \right) ,\left[ \mathcal{S} \right] \left[ \begin{matrix} \tilde{\vec{\omega}} \vec{v}\ 0 0\ \end{matrix} \right] S(ω ,v ),[S][ω ~0​v 0​] With this notation, the solution is [ p ⃗ ( t ) 1 ] e [ S ] t [ p ⃗ ( 0 ) 1 ] \left[ \begin{array}{c} \vec{p}\left( t \right)\ 1\ \end{array} \right] e^{\left[ \mathcal{S} \right] t}\left[ \begin{array}{c} \vec{p}\left( 0 \right)\ 1\ \end{array} \right] [p ​(t)1​]e[S]t[p ​(0)1​] Fact: e [ S ] t ∈ S E ( 3 ) e^{\left[ \mathcal{S} \right] t}\in SE\left( 3 \right) e[S]t∈SE(3) is always a valid homogeneous transformation matrix. [ T ] e [ S ] t [ [ Q ] R ⃗ 0 1 ] \left[ T \right] e^{\left[ \mathcal{S} \right] t}\left[ \begin{matrix} \left[ Q \right] \vec{R}\ 0 1\ \end{matrix} \right] [T]e[S]t[[Q]0​R 1​] Fact: Any T ∈ S E ( 3 ) T\in SE\left( 3 \right) T∈SE(3) can be written as [ T ] e [ S ] t \left[ T \right] e^{\left[ \mathcal{S} \right] t} [T]e[S]t, i.e. , it can be viewed as an operator that moves a point/frame along the screw axis S \mathcal{S} S at unit speed for time t t t
   3.2 se(3) ∀ ω ⃗ ∈ R 3 → ω ⃗ ~ ∈ s o ( 3 ) → e ω ⃗ ~ θ ∈ S O ( 3 ) ∀ S ∈ R 6 → [ S ] ∣ 4 × 4 [ ω ⃗ ~ v ⃗ 0 0 ] ∈ s e ( 3 ) → e [ S ] θ ∈ S E ( 3 ) \forall \vec{\omega}\in \mathbb{R} ^3\rightarrow \tilde{\vec{\omega}}\in so\left( 3 \right) \rightarrow e^{\tilde{\vec{\omega}}\theta}\in SO\left( 3 \right) \ \forall \mathcal{S} \in \mathbb{R} ^6\rightarrow \left. \left[ \mathcal{S} \right] \right|{4\times 4}\left[ \begin{matrix} \tilde{\vec{\omega}} \vec{v}\ 0 0\ \end{matrix} \right] \in se\left( 3 \right) \rightarrow e^{\left[ \mathcal{S} \right] \theta}\in SE\left( 3 \right) ∀ω ∈R3→ω ~∈so(3)→eω ~θ∈SO(3)∀S∈R6→[S]∣4×4​[ω ~0​v 0​]∈se(3)→e[S]θ∈SE(3) Similar to s o ( 3 ) so\left( 3 \right) so(3) , we can define s e ( 3 ) se\left( 3 \right) se(3) : s e ( 3 ) { ( ω ⃗ ~ , v ⃗ ) , ω ⃗ ~ ∈ s o ( 3 ) , v ⃗ ∈ R 3 } se\left( 3 \right) \left{ \left( \tilde{\vec{\omega}},\vec{v} \right) ,\tilde{\vec{\omega}}\in so\left( 3 \right) ,\vec{v}\in \mathbb{R} ^3 \right} se(3){(ω ~,v ),ω ~∈so(3),v ∈R3} s e ( 3 ) se\left( 3 \right) se(3) contains all matrix representation of twists or equivalently all twists.In some references, [ V ] \left[ \mathcal{V} \right] [V] is called a twistSometimes, we may abuse notation by writing V ∈ s e ( 3 ) \mathcal{V} \in se\left( 3 \right) V∈se(3)
2. Homogeneous Transformation Matrix as Rigid-Body Operator ODE for rigid motion under V ( ω ⃗ , v ⃗ ) \mathcal{V} \left( \vec{\omega},\vec{v} \right) V(ω ,v ) p ⃗ ˙ v ⃗ ω ⃗ × p ⃗ ⇒ [ p ⃗ ˙ 0 ] [ ω ⃗ ~ v ⃗ 0 0 ] [ p ⃗ 1 ] ⇒ [ p ⃗ ˙ 0 ] e [ V ] t [ p ⃗ 1 ] \dot{\vec{p}}\vec{v}\vec{\omega}\times \vec{p}\Rightarrow \left[ \begin{array}{c} \dot{\vec{p}}\ 0\ \end{array} \right] \left[ \begin{matrix} \tilde{\vec{\omega}} \vec{v}\ 0 0\ \end{matrix} \right] \left[ \begin{array}{c} \vec{p}\ 1\ \end{array} \right] \Rightarrow \left[ \begin{array}{c} \dot{\vec{p}}\ 0\ \end{array} \right] e^{\left[ \mathcal{V} \right] t}\left[ \begin{array}{c} \vec{p}\ 1\ \end{array} \right] p ​˙​v ω ×p ​⇒[p ​˙​0​][ω ~0​v 0​][p ​1​]⇒[p ​˙​0​]e[V]t[p ​1​] Consider “unit velocity” V S \mathcal{V} \mathcal{S} VS, then time t t t means degree if not unit speed : V θ ˙ S \mathcal{V} \dot{\theta}\mathcal{S} Vθ˙S [ p ⃗ ′ 1 ] [ T ] [ p ⃗ 1 ] \left[ \begin{array}{c} \vec{p}^{\prime}\ 1\ \end{array} \right] \left[ T \right] \left[ \begin{array}{c} \vec{p}\ 1\ \end{array} \right] [p ​′1​][T][p ​1​] : “rotate” p ⃗ \vec{p} p ​ about screw axis S \mathcal{S} S by θ \theta θ degree [ T ] e [ S ] θ \left[ T \right] e^{\left[ \mathcal{S} \right] \theta} [T]e[S]θ, two points : [ p ⃗ 1 ] → [ p ⃗ ′ 1 ] \left[ \begin{array}{c} \vec{p}\ 1\ \end{array} \right] \rightarrow \left[ \begin{array}{c} \vec{p}^{\prime}\ 1\ \end{array} \right] [p ​1​]→[p ​′1​] , more precisely : [ p ⃗ O ′ 1 ] [ T O ] [ p ⃗ O 1 ] \left[ \begin{array}{c} {\vec{p}^O}^{\prime}\ 1\ \end{array} \right] \left[ T^O \right] \left[ \begin{array}{c} \vec{p}^O\ 1\ \end{array} \right] [p ​O′1​][TO][p ​O1​] For [ T ] ∈ S E ( 3 ) \left[ T \right] \in SE\left( 3 \right) [T]∈SE(3) config representative [ T B A ] \left[ T{\mathrm{B}}^{A} \right] [TBA​] : config of { B } \left{ B \right} {B} relative to { A } \left{ A \right} {A} —— [ p ⃗ A 1 ] [ T B A ] [ p ⃗ B 1 ] \left[ \begin{array}{c} \vec{p}^A\ 1\ \end{array} \right] \left[ T{\mathrm{B}}^{A} \right] \left[ \begin{array}{c} \vec{p}^B\ 1\ \end{array} \right] [p ​A1​][TBA​][p ​B1​]——same physical point but two different frames [ T ] [ T A ] \left[ T \right] \left[ TA \right] [T][TA​] : “rotate” { A } \left{ A \right} {A}-frame about S \mathcal{S} S by θ \theta θ degree Rigid-Body Operator in Different Frames Expression of [ T ] \left[ T \right] [T] in another frame (other than { O } \left{ O \right} {O}): [ T O ] ↔ [ T B O ] − 1 [ T O ] [ T B O ] \left[ T^O \right] \leftrightarrow \left[ T{\mathrm{B}}^{O} \right] ^{-1}\left[ T^O \right] \left[ T_{\mathrm{B}}^{O} \right] [TO]↔[TBO​]−1[TO][TBO​] 5. Rigid-Body Operation of Screw Axis Consider an arbitrary screw axis S \mathcal{S} S , suppose the axis has gone through a rigid transformation T \left[ T \right] \left( \left[ Q \right] ,\vec{R} \right) T and the resulting new screw axis is S ′ \mathcal{S} ^{\prime} S′ , then S ′ [ A d T ] S \mathcal{S} ^{\prime}\left[ Ad_T \right] \mathcal{S} S′[AdT​]S Let’s work an arbitrary frame { A } \left{ A \right} {A} (rigidly attached to the screw axis) Let { B } \left{ B \right} {B} be the frame obtained be apply [ T ] \left[ T \right] [T] operation the coordinate of S \mathcal{S} S in { A } \left{ A \right} {A} is the same as the coordinate of S ′ \mathcal{S} ^{\prime} S′ in { B } \left{ B \right} {B} : S A S ′ B \mathcal{S} ^A{\mathcal{S} ^{\prime}}^B SAS′B We also know [ T B ] [ T ] [ T A ] , [ T ] [ T B A ] \left[ TB \right] \left[ T \right] \left[ T{\mathrm{A}} \right] ,\left[ T \right] \left[ T{\mathrm{B}}^{A} \right] [TB​][T][TA​],[T][TBA​] Multiply [ X B A ] \left[ X_{\mathrm{B}}^{A} \right] [XBA​] : ⇒ [ X B A ] S A [ X B A ] S ′ B S ′ A ⇒ S ′ A [ X B A ] S A \Rightarrow \left[ X{\mathrm{B}}^{A} \right] \mathcal{S} ^A\left[ X{\mathrm{B}}^{A} \right] {\mathcal{S} ^{\prime}}^B{\mathcal{S} ^{\prime}}^A\Rightarrow {\mathcal{S} ^{\prime}}^A\left[ X{\mathrm{B}}^{A} \right] \mathcal{S} ^A ⇒[XBA​]SA[XBA​]S′BS′A⇒S′A[XBA​]SA [ X B A ] [ [ Q B A ] 0 R ⃗ ~ B A [ Q B A ] [ Q B A ] ] [ A d T ] \left[ X{\mathrm{B}}^{A} \right] \left[ \begin{matrix} \left[ Q{\mathrm{B}}^{A} \right] 0\ \tilde{\vec{R}}{\mathrm{B}}^{A}\left[ Q{\mathrm{B}}^{A} \right] \left[ Q_{\mathrm{B}}^{A} \right]\ \end{matrix} \right] \left[ Ad_T \right] [XBA​][[QBA​]R ~BA​[QBA​]​0[QBA​]​][AdT​] 1 2 3 4 5 6 7 8 9