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
Given Frame trajectory
[
T
]
(
t
)
=
(
[
Q
]
(
t
)
,
R
⃗
p
(
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\left[ T \right] \left( t \right) =\left( \left[ Q \right] \left( t \right) ,\vec{R}_p\left( t \right) \right)
[T](t)=([Q](t),Rp(t)) wrt
{
O
}
\left\{ O \right\}
{O}![[足式机器人]Part4 南科大高等机器人控制课 Ch05 Instantaneous Velocity of Moving Frames,足式机器人,机器人,学习笔记](https://imgs.yssmx.com/Uploads/2023/12/753805-1.png)
Question : What is the velocity (twist/spatial velocity) of frame at time
t
t
t
[
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]
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t
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∈
R
4
×
4
\left[ T \right] \left( t \right) \in \mathbb{R} ^{4\times 4}
[T](t)∈R4×4 :
4
×
4
4\times 4
4×4 matrix
S
E
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3
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SE\left( 3 \right)
SE(3)
log
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t
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)
→
S
θ
\log \left( \left[ T \right] \left( t \right) \right) \rightarrow \mathcal{S} \theta
log([T](t))→Sθ : is
S
\mathcal{S}
S (unit) the velocity of
[
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]
(
t
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\left[ T \right] \left( t \right)
[T](t)? —— no
R
⃗
p
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t
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\vec{R}_p\left( t \right)
Rp(t) : position vector ;
R
⃗
˙
p
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t
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\dot{\vec{R}}_p\left( t \right)
R˙p(t) velocity vector;
S
θ
↔
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\mathcal{S} \theta \leftrightarrow \left[ T \right] \left( t \right)
Sθ↔[T](t) : position coordination (what velocity?——
[
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˙
]
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\left[ \dot{T} \right] \left( t \right)
[T˙](t)?)
[
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˙
]
(
t
)
∈
R
4
×
4
∉
S
E
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3
)
\left[ \dot{T} \right] \left( t \right) \in \mathbb{R} ^{4\times 4}\notin SE\left( 3 \right)
[T˙](t)∈R4×4∈/SE(3)
1.Instantanenous Velocity of Rotating Frames
{ A } \left\{ A \right\} {A} frame is rotating with orientation [ Q A ] ( t ) \left[ Q_A \right] \left( t \right) [QA](t) and velocity ω ⃗ A ( t ) \vec{\omega}_A\left( t \right) ωA(t) at time t t t (Note: everything is wrt { O } \left\{ O \right\} {O} frame)
Let ω ^ θ = log ( [ Q A ] ( t ) ) \hat{\omega}\theta =\log \left( \left[ Q_A \right] \left( t \right) \right) ω^θ=log([QA](t)) can be obtained from the reference frame (say { O } \left\{ O \right\} {O} frame) by rotating about ω ^ \hat{\omega} ω^ by θ \theta θ degree
- ω ^ θ \hat{\omega}\theta ω^θ only describes the current orientation of { A } \left\{ A \right\} {A} relative to { O } \left\{ O \right\} {O} , it does not contain info about how the frame is rotating at time t t t
What is the relation between ω ⃗ A ( t ) \vec{\omega}_A\left( t \right) ωA(t) and [ Q A ] ( t ) \left[ Q_A \right] \left( t \right) [QA](t)
d d t [ Q A ] ( t ) = ω ⃗ ~ A ( t ) [ Q A ] ( t ) ⇒ ω ⃗ ~ A ( t ) = [ Q ˙ A ] ( t ) [ Q A ] − 1 ( t ) 1 \frac{\mathrm{d}}{\mathrm{d}t}\left[ Q_A \right] \left( t \right) =\tilde{\vec{\omega}}_A\left( t \right) \left[ Q_A \right] \left( t \right) \Rightarrow \tilde{\vec{\omega}}_A\left( t \right) =\left[ \dot{Q}_A \right] \left( t \right) \left[ Q_A \right] ^{-1}\left( t \right) 1 dtd[QA](t)=ω~A(t)[QA](t)⇒ω~A(t)=[Q˙A](t)[QA]−1(t)1
What is ω ⃗ A A \vec{\omega}_{\mathrm{A}}^{A} ωAA?
ω ⃗ A A = [ Q O A ] ω ⃗ A O \vec{\omega}_{\mathrm{A}}^{A}=\left[ Q_{\mathrm{O}}^{A} \right] \vec{\omega}_{\mathrm{A}}^{O} ωAA=[QOA]ωAO
velocity of A A A relative to { O } \left\{ O \right\} {O} , expressed in { A } \left\{ A \right\} {A}
ω ⃗ ~ A A = [ Q O A ] ω ⃗ A O ~ = [ Q O A ] ω ⃗ ~ A O [ Q O A ] T = [ Q O A ] [ Q ˙ A O ] [ Q A O ] − 1 [ Q O A ] T = [ Q O A ] [ Q ˙ A O ] = [ Q A O ] − 1 [ Q ˙ A O ] \tilde{\vec{\omega}}_{\mathrm{A}}^{A}=\widetilde{\left[ Q_{\mathrm{O}}^{A} \right] \vec{\omega}_{\mathrm{A}}^{O}}=\left[ Q_{\mathrm{O}}^{A} \right] \tilde{\vec{\omega}}_{\mathrm{A}}^{O}\left[ Q_{\mathrm{O}}^{A} \right] ^{\mathrm{T}}=\left[ Q_{\mathrm{O}}^{A} \right] \left[ \dot{Q}_{\mathrm{A}}^{O} \right] \left[ Q_{\mathrm{A}}^{O} \right] ^{-1}\left[ Q_{\mathrm{O}}^{A} \right] ^{\mathrm{T}}=\left[ Q_{\mathrm{O}}^{A} \right] \left[ \dot{Q}_{\mathrm{A}}^{O} \right] =\left[ Q_{\mathrm{A}}^{O} \right] ^{-1}\left[ \dot{Q}_{\mathrm{A}}^{O} \right] ω~AA=[QOA]ωAO =[QOA]ω~AO[QOA]T=[QOA][Q˙AO][QAO]−1[QOA]T=[QOA][Q˙AO]=[QAO]−1[Q˙AO]
2.Instantanenous Velocity of Moving Frames
{ A } \left\{ A \right\} {A} moving frame with configuration [ T A ] ( t ) \left[ T_A \right] \left( t \right) [TA](t) at t ime t t t undergoes a rigid body motion with velocity V A ( t ) = ( ω ⃗ , v ⃗ ) \mathcal{V} _A\left( t \right) =\left( \vec{\omega},\vec{v} \right) VA(t)=(ω,v) (Note: everything is wrt { O } \left\{ O \right\} {O} frame)
The exponential coordinate S ^ ( t ) θ ( t ) = log ( [ T A ] ( t ) ) \hat{\mathcal{S}}\left( t \right) \theta \left( t \right) =\log \left( \left[ T_A \right] \left( t \right) \right) S^(t)θ(t)=log([TA](t)) only indicates the current configuration of { A } \left\{ A \right\} {A} , and does not tell us about how the frame is moving at time t t t
What is the relation between
V
A
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t
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\mathcal{V} _A\left( t \right)
VA(t) and
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\left[ T_A \right] \left( t \right)
[TA](t) ?
d
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⇒
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\frac{\mathrm{d}}{\mathrm{d}t}\left[ T_A \right] \left( t \right) =\left[ \mathcal{V} _A \right] \left( t \right) \left[ T_A \right] \left( t \right) \Rightarrow \left[ \mathcal{V} _A \right] \left( t \right) =\left[ \dot{T}_A \right] \left( t \right) \left[ T_A \right] ^{-1}\left( t \right)
dtd[TA](t)=[VA](t)[TA](t)⇒[VA](t)=[T˙A](t)[TA]−1(t)
3.Review/Summary of Rigid Body Velocity & Operation
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spatial velocity / twist V = [ ω ⃗ v ⃗ O ] \mathcal{V} =\left[ \begin{array}{c} \vec{\omega}\\ \vec{v}_{\mathrm{O}}\\ \end{array} \right] V=[ωvO] , v ⃗ O \vec{v}_{\mathrm{O}} vO reference point O O O may/may not move with the body
ω ⃗ \vec{\omega} ω : angular velocity
v ⃗ O \vec{v}_{\mathrm{O}} vO : velocity of the body-fixed partical currently coincides with O O O
Given V = [ ω ⃗ v ⃗ O ] \mathcal{V} =\left[ \begin{array}{c} \vec{\omega}\\ \vec{v}_{\mathrm{O}}\\ \end{array} \right] V=[ωvO] , any body fixed point P P P , its velocity is v ⃗ P = v ⃗ O + ω ⃗ × R ⃗ O P \vec{v}_{\mathrm{P}}=\vec{v}_{\mathrm{O}}+\vec{\omega}\times \vec{R}_{\mathrm{OP}} vP=vO+ω×ROP -
Twist in frames : Given frame { B } \left\{ B \right\} {B} , { O } \left\{ O \right\} {O} with relation
V O = [ ω ⃗ O v ⃗ O O ] , V B = [ ω ⃗ B v ⃗ O B ] \mathcal{V} ^O=\left[ \begin{array}{c} \vec{\omega}^O\\ \vec{v}_{\mathrm{O}}^{O}\\ \end{array} \right] ,\mathcal{V} ^B=\left[ \begin{array}{c} \vec{\omega}^B\\ \vec{v}_{\mathrm{O}}^{B}\\ \end{array} \right] VO=[ωOvOO],VB=[ωBvOB] —— V \mathcal{V} V is a twist of some rigid body
V O = [ X B O ] V B \mathcal{V} ^O=\left[ X_{\mathrm{B}}^{O} \right] \mathcal{V} ^B VO=[XBO]VB —— change of coordinate for twist [ X B O ] ∈ R 6 × 6 \left[ X_{\mathrm{B}}^{O} \right] \in \mathbb{R} ^{6\times 6} [XBO]∈R6×6
[ X B O ] ∈ R 6 × 6 [ X B O ] = [ [ Q B O ] 0 R ⃗ ~ B O [ Q B O ] [ Q B O ] ] \left[ X_{\mathrm{B}}^{O} \right] \in \mathbb{R} ^{6\times 6}\left[ X_{\mathrm{B}}^{O} \right] =\left[ \begin{matrix} \left[ Q_{\mathrm{B}}^{O} \right]& 0\\ \tilde{\vec{R}}_{\mathrm{B}}^{O}\left[ Q_{\mathrm{B}}^{O} \right]& \left[ Q_{\mathrm{B}}^{O} \right]\\ \end{matrix} \right] [XBO]∈R6×6[XBO]=[[QBO]R~BO[QBO]0[QBO]]
For given [ T ] = ( [ Q ] , R ⃗ ) ⇒ [ X ] = [ A d T ] = [ [ Q ] 0 R ⃗ ~ [ Q ] [ Q ] ] \left[ T \right] =\left( \left[ Q \right] ,\vec{R} \right) \Rightarrow \left[ X \right] =\left[ Ad_T \right] =\left[ \begin{matrix} \left[ Q \right]& 0\\ \tilde{\vec{R}}\left[ Q \right]& \left[ Q \right]\\ \end{matrix} \right] [T]=([Q],R)⇒[X]=[AdT]=[[Q]R~[Q]0[Q]] -
screw axis : S = ( s ^ , R ⃗ q , h ) ↔ [ ω ⃗ v ⃗ ] = [ s ^ h s ^ − s ^ × R ⃗ q ] \mathcal{S} =\left( \hat{s},\vec{R}_{\mathrm{q}},h \right) \leftrightarrow \left[ \begin{array}{c} \vec{\omega}\\ \vec{v}\\ \end{array} \right] =\left[ \begin{array}{c} \hat{s}\\ h\hat{s}-\hat{s}\times \vec{R}_{\mathrm{q}}\\ \end{array} \right] S=(s^,Rq,h)↔[ωv]=[s^hs^−s^×Rq]
Al rigid body motion can be "thought of " screw motion rotation & linear motion along the axis
we typically write V = S θ ˙ \mathcal{V} =\mathcal{S} \dot{\theta} V=Sθ˙ ( S \mathcal{S} S - ‘unit’ normalized twist) -
rotation operation / Exp coordinate (wrt { O } \left\{ O \right\} {O})
OED for rotation : R ⃗ ˙ p = ω ⃗ × R ⃗ p = ω ⃗ ~ R ⃗ p ⇒ R ⃗ p ( t ) = e ω ⃗ ~ t R ⃗ p ( 0 ) , s o ( 3 ) = { ω ⃗ ~ : ω ⃗ ∈ R 3 } \dot{\vec{R}}_{\mathrm{p}}=\vec{\omega}\times \vec{R}_{\mathrm{p}}=\tilde{\vec{\omega}}\vec{R}_{\mathrm{p}}\Rightarrow \vec{R}_{\mathrm{p}}\left( t \right) =e^{\tilde{\vec{\omega}}t}\vec{R}_{\mathrm{p}}\left( 0 \right) ,so\left( 3 \right) =\left\{ \tilde{\vec{\omega}}:\vec{\omega}\in \mathbb{R} ^3 \right\} R˙p=ω×Rp=ω~Rp⇒Rp(t)=eω~tRp(0),so(3)={ω~:ω∈R3}
if ω ⃗ = ω ^ \vec{\omega}=\hat{\omega} ω=ω^ , unit vector , t = θ t=\theta t=θ, ω ^ θ ↔ [ Q ] = e ω ⃗ ~ θ ∈ S O ( 3 ) \hat{\omega}\theta \leftrightarrow \left[ Q \right] =e^{\tilde{\vec{\omega}}\theta}\in SO\left( 3 \right) ω^θ↔[Q]=eω~θ∈SO(3), ω ^ θ \hat{\omega}\theta ω^θ os calld exponention cooedinate of [ Q ] \left[ Q \right] [Q] denoted log ( [ Q ] ) \log \left( \left[ Q \right] \right) log([Q])
R ⃗ p ′ = e ω ⃗ ~ θ R ⃗ p \vec{R}_{\mathrm{p}^{\prime}}=e^{\tilde{\vec{\omega}}\theta}\vec{R}_{\mathrm{p}} Rp′=eω~θRp
Given a frame { A } \left\{ A \right\} {A}, [ Q A ] = [ x ^ A , y ^ A , z ^ A ] \left[ Q_A \right] =\left[ \hat{x}_A,\hat{y}_A,\hat{z}_A \right] [QA]=[x^A,y^A,z^A] then
[ Q ] [ Q A ] = e ω ⃗ ~ θ [ Q A ] \left[ Q \right] \left[ Q_A \right] =e^{\tilde{\vec{\omega}}\theta}\left[ Q_A \right] [Q][QA]=eω~θ[QA] : [ Q ] \left[ Q \right] [Q] action operation , means rotate [ Q A ] \left[ Q_A \right] [QA] about ω ^ \hat{\omega} ω^ by θ \theta θ degree
Experssion of rotation operator [ Q ] \left[ Q \right] [Q] in { O } \left\{ O \right\} {O} and { B } \left\{ B \right\} {B} : [ Q ] \left[ Q \right] [Q] in { O } \left\{ O \right\} {O} ; [ Q B O ] − 1 [ Q ] [ Q B O ] \left[ Q_{\mathrm{B}}^{O} \right] ^{-1}\left[ Q \right] \left[ Q_{\mathrm{B}}^{O} \right] [QBO]−1[Q][QBO] same rotation operator in { B } \left\{ B \right\} {B} -
Rigid body transformation and exp coordinate(wrt { O } \left\{ O \right\} {O})
ODE : R ⃗ ˙ p = v ⃗ + ω ⃗ × R ⃗ p ⇒ [ R ⃗ ˙ p 0 ] = [ ω ⃗ ~ v ⃗ 0 0 ] ∣ 4 × 4 [ R ⃗ p 1 ] ⇒ [ R ⃗ ˙ p 0 ] = e [ V ] t [ R ⃗ p 1 ] \dot{\vec{R}}_{\mathrm{p}}=\vec{v}+\vec{\omega}\times \vec{R}_{\mathrm{p}}\Rightarrow \left[ \begin{array}{c} \dot{\vec{R}}_{\mathrm{p}}\\ 0\\ \end{array} \right] =\left. \left[ \begin{matrix} \tilde{\vec{\omega}}& \vec{v}\\ 0& 0\\ \end{matrix} \right] \right|_{4\times 4}\left[ \begin{array}{c} \vec{R}_{\mathrm{p}}\\ 1\\ \end{array} \right] \Rightarrow \left[ \begin{array}{c} \dot{\vec{R}}_{\mathrm{p}}\\ 0\\ \end{array} \right] =e^{\left[ \mathcal{V} \right] t}\left[ \begin{array}{c} \vec{R}_{\mathrm{p}}\\ 1\\ \end{array} \right] R˙p=v+ω×Rp⇒[R˙p0]=[ω~0v0] 4×4[Rp1]⇒[R˙p0]=e[V]t[Rp1] e [ V ] t e^{\left[ \mathcal{V} \right] t} e[V]t - matrix representation of V \mathcal{V} V, s e ( 3 ) = { [ V ] , V ∈ R 6 } se\left( 3 \right) =\left\{ \left[ \mathcal{V} \right] ,\mathcal{V} \in \mathbb{R} ^6 \right\} se(3)={[V],V∈R6}![[足式机器人]Part4 南科大高等机器人控制课 Ch05 Instantaneous Velocity of Moving Frames,足式机器人,机器人,学习笔记](https://imgs.yssmx.com/Uploads/2023/12/753805-2.png)
S ^ θ \hat{\mathcal{S}}\theta S^θ is the exp coordinate of [ T ] \left[ T \right] [T]
[ R ⃗ p ′ 1 ] = [ T ] [ R ⃗ p 1 ] \left[ \begin{array}{c} \vec{R}_{\mathrm{p}^{\prime}}\\ 1\\ \end{array} \right] =\left[ T \right] \left[ \begin{array}{c} \vec{R}_{\mathrm{p}}\\ 1\\ \end{array} \right] [Rp′1]=[T][Rp1]
[ T ] [ T A ] \left[ T \right] \left[ T_A \right] [T][TA] : rotate frame { A } \left\{ A \right\} {A} about screw axis S ^ \hat{\mathcal{S}} S^ by θ \theta θ degree -
rigid operation of screw axis
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S ′ = [ A d T ] S \mathcal{S} ^{\prime}=\left[ Ad_T \right] \mathcal{S} S′=[AdT]S means rotate about axis S \mathcal{S} S along S ^ \hat{\mathcal{S}} S^ by θ \theta θ degree
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expression of [ T ] \left[ T \right] [T] in { B } \left\{ B \right\} {B}: [ T B ] − 1 [ T ] [ T B ] \left[ T_{\mathrm{B}} \right] ^{-1}\left[ T \right] \left[ T_{\mathrm{B}} \right] [TB]−1[T][TB]文章来源：https://www.toymoban.com/news/detail-753805.html
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velocity of moving frame [ T ] ( t ) \left[ T \right] \left( t \right) [T](t) : [ V ] = [ T ˙ ] [ T ] − 1 \left[ \mathcal{V} \right] =\left[ \dot{T} \right] \left[ T \right] ^{-1} [V]=[T˙][T]−1文章来源地址https://www.toymoban.com/news/detail-753805.html
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