Parallel parking a car - - PDF document

Parallel parking a car

The states of the system

(simplified model – correct for bicycle)

✟✟✟✟✟✟✟ ✲ ✻

x y Θ Φ

• ✟✟✟✟✟✟

✟✟✟✟✟✟ ✟✟✟✟✟✟ ❆ ❆ ❆ ❆ ❆ ❆ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

• x, y coordinates of center of car

v speed Θ

• rientation of the car

Φ steering angle

                        

states a acceleration ω angular velocity of steering wheel (about a vertical axis)

                

inputs (“controls”) Equations

• f motion

                                  

˙ x = v cos Θ ˙ y = v sin Θ ˙ v = a (= u1) ˙ Θ = v sin Φ ˙ Φ = ω (= u2) This is not a standard dynamical system. Instead: This is a controlled dynamical system.

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Parallel parking a car

                                        

˙ x = v cos Θ ˙ y = v sin Θ ˙ v = a (= u1) ˙ Θ = v sin Φ ˙ Φ = ω (= u2) Written in standard as a multi-input control system

                    

˙ z = f0(z) +

2

j=1 ujfj(z)

z ∈ M 5 = R3 × S1 × S1 u ∈ [−amax, amax] × [−ωmax, ωmax] with drift vector field f0 = z3 cos z4 ∂ ∂z1 + z3 sin z4 ∂ ∂z2 + z3 sin z5 ∂ ∂z4 , and controlled fields f1 = ∂ ∂z3 , and f2 = ∂ ∂z5 . For every choice of controls u1(t), u2(t) (that are locally integrable) one obtains an ordinary dynamical system.

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Easiest example: Piecewise constant controls

in this simple cascade system Control inputs: Blue: For/backward acceleration u1 Red: Angular velocity of steering wheel u2 States: Black: For/backward speed v =

u1

Magenta: Steering angle Φ =

u2.

Brown: Orientation of the car Θ =

v sin Φ.

Green: x-component of velocity vx = v cos Θ Cyan: y-component of velocity vy = v sin Θ v(0) = Φ(0) = Θ(0) = x(0) = y(0) = 0 = v(t) = Φ(t) = Θ(t) = x(t) y(t) =

t t1

0 u1(t2) dt2

• v(t1)

· sin

Θ(t1)

• (

t1 t2

0 u1(t2) dt2

• v(t2)

· sin(

t2

0 u2(t3) dt3)

• Φ(t2)

dt2) dt1

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Noncommuting flows

Concatenate the solutions of the (standard) dynamical systems (controls normalized to ±1) ≤ t < T1 ˙ z = f0(z) +f1(x) +f2(z) T1 ≤ t < T2 ˙ z = f0(z) +f1(x) −f2(z) T2 ≤ t < T3 ˙ z = f0(z) −f2(z) T3 ≤ t < T4 ˙ z = f0(z) −f1(x) −f2(z) T4 ≤ t < T5 ˙ z = f0(z) −f1(x) +f2(z) T5 ≤ t < T6 ˙ z = f0(z) −f1(x) −f2(z) T6 ≤ t < T7 ˙ z = f0(z) −f2(z) T7 ≤ t < T8 ˙ z = f0(z) +f1(x) −f2(z) T8 ≤ t < T9 ˙ z = f0(z) +f1(x) +f2(z) The flows do not commute. Most simple picture for:

f1(x) =

     

1 −y

     

f2(x) =

     

1 x

     

1 x 1 y 1 z

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Parking model is controllable

Recall:

f0(z) =

             

z3 cos z4 z3 sin z4 z3 sin z5

             

f1(z) =

             

1

             

f2(z) =

             

1

             

[f1, f0](x) =

             

cos z4 sin z4 sin z5

             

[f2, f0](x) =

             

z3 cos z5

             

[f2, f1](x) =

                           

[f1, [f2, f0]](x) =

             

cos z5

             

[[f1, [f2, f0]], [f1, f2]](x) =

             

sin z4 cos z4

             

These iterated Lie brackets span the tangent space of R5 at the origin – hence the system is accessible from 0.

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Manipulating exponential products

Instead of working with complicated concatenations of flows like z(t) = e

t9 t8 (f0+f1+f2) dt ◦ . . . e t2 t1 (f0+f1−f2) dt ◦ e t1 0 (f0+f1+f2) dt(p)

it is desirable to rewrite the solution curve using a minimal number of vector fields fπk that span the tangent space (typically using iterated Lie brackets of the system fields f0, f1, . . . fm) Coordinates of the first kind z(t) = eb1(t,u)fπ1+b2(t,u)fπ1+b3(t,u)fπ3+...+bn(t,u)fπn(p) Coordinates of the second kind z(t) = ec1(t,u)fπ1 ◦ ec2(t,u)fπ1 ◦ ec3(t,u)fπ3 ◦ . . . ◦ ecn(t,u)fπn(p) Using the Campbell-Baker-Hausdorff formula, this is possible, but a book-keeping nightmare. Moreover, the CBH formula does not use a basis, but uses linear combinations of all possible iterated Lie brackets. Yet, by the Jacobi identity (and anticommutativity), in ever Lie algebra e.g. [X, [Y, [X, Y ]]] + [Y, [[X, Y ], X]

• ] + [[X, Y ], [X, Y ]]] = 0

and hence [X, [Y, [X, Y ]]] = [Y,

• [X, [X, Y ]]]

Plan:

• Work with bases for (free) Lie algebras.
• Find useful formulae for the coefficients bk(t, u) or ck(t, u).

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The Chen Fliess series

• K. T. Chen, 1957: Geometric invariants of curves in Rn
• M. Fliess, 1970s: adaptation to control

The formal control system ˙ S = S

  m

• i=1

uiXi

  ,

S(0) = I

• n the associative algebra ˆ

A(X1 . . . Xm) of formal power series in the noncommuting indeterminates (letters) X1, . . . Xm has the unique solution SCF(T, u) =

• I

T t1 0 · · · tp−1

uip(tp) . . . ui1(t1) dt1 . . . dtp

• ΥI(T,u)

Xi1 . . . Xip

• XI

What is the CF-series good for? For any given control system ˙ x =

m

• i=1

ui(x)fi(x), x(0) = p, with “output” y = ϕ(x) φ(x(T, u)) =

• I

T t1 0 · · · tp−1

uip(tp) . . . ui1(t1) dt1 . . . dtp

• ΥI(T,u)

(fi1 ◦ . . . ◦ fipϕ)(p) (uniform convergence for small T and IC’s on compacta [Sussmann, 1983]) The CF-series was basic tool for deriving many high-order conditions for controllability and optimality. [Hermes, Stefani, Sussmann, Kawski, ...]

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