非完整约束运动

˙ = f (q) q
q( ) = q
f
⇒ ow: ϕt (q),
f
d f f ϕ (q) = f (ϕt (q)) dt t
Taylor series expansion: ˙( ) + ε q ¨ ( ) + O(ε ) q(ε) = ϕ ε (q ) = q + εq
q( ε) = ϕ ε ○ ϕ ε ○ ϕ ε ○ ϕ ε (q )
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
Consider the previous example, let f (q), f (q) be a basis of ∆q, then ˙ = f (q)u + f (q)u q parameterizes a feasible path of the system. For example, let ˙, then ˙, u = θ u =ϕ ˙ ⎤ ⎡ ρ cos θ ⎤ ⎡ x ⎡ ⎤ ⎢ y ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ˙ ⎥ ⎢ ρ sin θ ⎥ ⎢ ⎥u = ⎢ ⎥ ⎢ θ ⎥ u + ˙ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ϕ ⎥ ˙ ⎣ ⎦ ⎦ ⎣ ⎦ ⎣ Note: A(q)fi (q) = , i = ,
−f −f f f
◻ Lie bracket of vector elds f , f :
q( ε) f εf f εf
−εf
˙( ) + q(ε) = q + εq
ε
¨ ( ) + O(ε ) q
q
−εf
ε ∂f = q + εf (q ) + ∂q
f (q ) + O(ε )
Chapter
Nonholonomic Motion Planning
= ϕ ε (q + εf (q ) +
ε ∂f ∂q
q
f (q ) + O(ε ))
= q + εf (q ) + + ε
ε ∂f f (q) + εf (q + εf (q )) ∂q
∂f f (q ) + O(ε ) ∂q ∂f ∂f f (q ) + εf (q ) + ε f (q ) ∂q ∂q
Chapter Nonholonomic Motion Planning
Introduction Controllability and Frobenius eorem
Examples of Nonholonomic Systems Nonholonomic Motion Planning
Chapter
q( ε) = ϕ ε (q + ε(f (q ) + f (q )) ∂f ∂f f (q ) + + ε ( f (q ) + ∂q ∂q = q + ε(f (q ) + f (q )) ∂f ∂f + ε ( f (q ) + f (q )ห้องสมุดไป่ตู้+ ∂q ∂q
−f
∂f f (q )) + O(ε )) ∂q
Con guration:
˙= ˙ − cos θρ ϕ x ˙= ˙ − sin θρ ϕ y
q = (x , y , θ , ϕ) ∈ E Subspace of Permissible velocities ˙ ∈ Tq E a T ˙ = ,i = , } ∆q = {q i (q)q
Chapter
− ρ cos θ − ρ sin θ
A(q)
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
˙ x ˙ y ˙ θ ˙ ϕ
⎤ ⎥ ⎥ ⎥= ⎥ ⎥ ⎦
(∗)
Q : Is it possible to move between any two points in E while satisfying the constraints (∗)? Q : Is it possible to nd two constraint functions hi (q) = , i = , s.t ker A(q) = ker Dq h?
Nonholonomic Motion Planning
8.1 Introduction ◻ Examples of nonholonomic system:
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
θ
φ
Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
(x, y ) Figure 6.2: Disk rolling on a plane.
No slippage constraints:
φ
θ (x, y ) Figure 6.2: Disk rolling on a plane.
Chapter
Nonholonomic Motion Planning
8.1 Introduction ◻ Examples of nonholonomic system:
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
8.1 Introduction
q( ε) = ϕ ε ○ ϕ ε (q )
f f f
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
Nonholonomic Motion Planning
8.1 Introduction
q( ε) = ϕ ε (q + εf (q ) + ε (
−f
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
◇ Example: Lie bracket of two linear vector elds
Let f (q) = Aq, f (q) = Bq, A, B ∈ Rn×n , then [f , f ](q) = (AB − BA)q
f (q) f (q)
◻ Nonholonomic constraints:
Chapter
Nonholonomic Motion Planning
8.1 Introduction ◻ Flows of di erential equations:
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
∂f f (q ) ∂q
∂f ∂f f (q ) − f (q )) + O(ε )) ∂q ∂q ∂f ∂f ∂f = q + εf (q ) + ε ( f (q ) + f (q ) − f (q )) ∂q ∂q ∂q ∂f − εf (q + εf (q )) + ε f (q ) + O(ε ) ∂q ∂f ∂f =q +ε ( f − f ) = q + ε [f , f ](q ) + O(ε ) ∂q ∂q De nition: Lie bracket of two vector elds f (q), f (q) + [f , f ](q ) = ∂f ∂q
Nonholonomic Motion Planning
8.1 Introduction
˙= Pfa an constraints A(q)q
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
Chapter 8 Nonholonomic Motion Planning
Summer School-Math. Methods in Robotics@TU-BS.DE
-
July
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
8.1 Introduction
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
Chapter
Nonholonomic Motion Planning
8.1 Introduction
∂h ∂h ∂h ∂h Assume a (q) = ∂h − ρ cos θ ], ∂q = [ ∂x ∂y ∂θ ∂ϕ ] = [ ∂h ∂h ∂ h ∂ h ⇒ = − ρ cos θ , = , = = ∂ϕ ∂θ ∂θ∂ϕ ∂ϕ∂θ Since we would have ρ sin θ = which is in general impossible, this appears to be impossible.
q
f (q ) −
∂f ∂q
q
f (q )
(∗)
Chapter
Nonholonomic Motion Planning
8.1 Introduction
Chapter Nonholonomic Motion Planning Introduction Controllability and Frobenius Theorem Examples of Nonholonomic Systems Nonholonomic Motion Planning
∂f f (q )) ∂q ∂f − εf (q )(q + ε(f (q ) + f (q ))) + ε f (q ) + O(ε ) ∂q ∂f ∂f ∂f = q + εf (q ) + ε ( f (q ) + f (q ) − f (q )) + O(ε ) ∂q ∂q ∂q
Chapter
Chapter
Nonholonomic Motion Planning
8.1 Introduction
Consider a rolling disk of radius ρ, as shown in Fig. .
Chapter Nonholonomic Motion Planning Introduction
= q + εf (q ) + ε + ε
∂f f (q ) + O(ε ) = q + ε(f (q ) + f (q )) ∂q ∂f ∂f ∂f + ε ( f (q ) + f (q ) + f (q )) + O(ε ) ∂q ∂q ∂q
Chapter
Nonholonomic Motion Planning