[PPT] - AUTONOMOUS PERPENDICULAR AND PARALLEL PARKING USING MULTI-SENSOR PowerPoint Presentation

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AUTONOMOUS PERPENDICULAR AND PARALLEL PARKING USING MULTI-SENSOR BASED CONTROL

David Pérez-Morales, Olivier Kermorgant, Salvador Domínguez-Quijada and Philippe Martinet Updated: 2017/09/18

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Overview

1. Modeling and Notation

1.1 Car-like robot model and notation 1.2 Multi-sensor modeling

2. Perception
3. Interaction Model

3.1 Task 3.2 Constraints

4. Control
5. Results

5.1 Unconstrained cases - MATLAB 5.2 Constrained cases

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MODELING AND NOTATION

SLIDE 4

Car-Like Robot Rear-Wheel Driving

x0 y0 ϕ y

m

x

m

lwb ltr lfo lro lls lrs θ M y x

Figure: Kinematic model diagram for a car-like rear-wheel driving robot

          

˙ x ˙ y ˙ θ ˙ φ

          



         

cos θ sin θ tan φ/lwb

          

v +

          

1

          

˙ φ (1) Where v and ˙ φ are the driving and steering velocities.

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Experimental Setup

Velocity, direction of travel, steering and turning signals can be controlled by computer.

Figure: Robotized Renault ZOE 4

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Multi-sensor modeling

F0 Fm F1

F2

FO

S1 S2

control frame Object of interest

bject

frame

1Wm 2Wm

5 10 15 20 25 30 35 40 45

6
4
2

2 4 5 10 15 20 25 30 35 40 45

6
4
2

2 4

sensor signal sensor signal

Figure: Multi-sensor model

In a static environment, the sensor feature deriva- tive can be expressed as 1: ˙ si Livi Li

(di×6) iWm (6×6)

vm

(6×1)

(2) Ls

(d×6)

LWm          L1 . . . . . . ... . . . . . . Lk         

(d×6k)

        

1Wm

. . .

kWm

        

(6k×6)

(3) ˙ s Lsvm (4) Under a planar world assumption: ˙ si Livi Li

(di×3) iWm (3×3)

vm

(3×1)

(5) where vm [vxm , vym , ˙ θ]T

1Kermorgant and Chaumette, “Dealing with constraints in sensor-based robot control”

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Multi-sensor modeling

x0 y0 ϕ ym xm lwb ltr lfo lro lls lrs θ M y x

Figure: Kinematic model diagram for a car-like rear-wheel driving robot

Assuming vym 0 (no slipping nor skidding) vm [vxm , ˙ θ]T (6) dim(Ls) (d × 2) where vxm v.

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Weighted error

The weighted multi-sensor error signal is defined as: eH He (7) where e s − s∗ is the difference between the current sensor signal s and its desired value s∗ and H is a diagonal positive semi-definite weighting matrix that depends on the current value of s. Its associated interaction matrix is LH HLs.

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PERCEPTION

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Perception

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Extraction of empty parking place

h t p e D t

p

S

3

d Spot Length d1 d2 d

1

/ 2 d

2

/ 2 c11 c12 c13 c14 c21 c22 c23 c24

ls

10

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Parking spots

Figure: ⊥ parking spot model Figure: Parking spot model for reverse parking maneuvers Figure: Parking spot model for forward parking maneuvers Table: Pair of points through which each line passes Line Perpendicular Parallel (reverse) Parallel (forward)

iL1

(ip5, ip6) (ip5, ip6) (ip5, ip6)

iL2

(ip1, ip4) (ip3, ip4) (ip1, ip2)

iL3

(ip3, ip4) (ip1, ip4) (ip1, ip4)

iL4

(ip1, ip2) (ip1, ip2) (ip3, ip4) 11

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INTERACTION MODEL

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Interaction Model

Figure: Sensors’ configuration and sensor features

The sensor signals siLj and reduced inter- action matrix LiLj are defined respectively as: siLj

iuj(1), iuj(2), ihj(3) T

(8) LiLj

        

iuj(2)

−iuj(1) −iuj(2)

iuj(1)

        

(9)

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Task sensor features

Figure: Sensors’ configuration and sensor features

Task sensor features

st [siL1 , siL2]T (10) st is obtained from S1 for forward maneu- vers and from S2 for reverse ones. The corresponding interaction matrix is defined as: Lt LL + L∗

L

2 (11) where LL [LiL1 , LiL2]T and L∗

L is equal

to the value of LL at the desired pose.

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Weighting of the task sensor features

Figure: Sensors’ configuration and sensor features

The associated weighting matrix Ht is defined as: Ht diag(ht

1, ht 2, htconst 3

, ht

4, ht 5, htconst 6

)

(12) where the values htconst

3 and htconst

3 6 are con- stant while the values of ht

i

∀ i 1, 2, 4, 5 are computed using the following smooth weighting function:

s-

i

s*

i

s

i

ss-

i

h-

i

ht

i

h+

i

ss

+

i

s

i

s*

i

s+

i

h-

i

ht

i

h+

i

et(i) et(i)

Figure: Weighting function ht

i

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Constraints (reverse perpendicular case)

Figure: Lateral constraint d

Constrained sensor features

sc [s3, s4, s5]T (13)

The corresponding interaction matrix:

Lc [L3, L4, L5]T (14)

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CONTROL

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Control

Control law

vm argmin||LHtvm + λeHt||2 s.t. Avm ≤ b (15)

with:

A [Lc, −Lc]T (16) b [α(s+

c − sc), −α(s− c − sc)]T

(17)

where α is a gain constant, λ is the control gain and [s−

c , s+ c ] is the desired interval in

which we want to keep sc.

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Bounding the control signals

Figure: Distance to stop line

The control signals v and φ and their increments are bounded as shown below: |v| < vmax (18) |φ| < φmax (19) (vn−1 − ∆dec) ≤ vn ≤ (vn−1 + ∆acc) (20) (φn−1 − ∆φ) ≤ φn ≤ (φn−1 + ∆φ) (21)

Distance to stop line

d th

stop

dstop(T=0)

vmax

v0

max

2

Figure: Deceleration profile 19

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RESULTS

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Unconstrained cases - MATLAB

2

2 4 6 8 10

2

2 4 6 8 10 Parking maneuver evolution

(a) Performed maneuver

time 5 10 15 20 25 30 35

2
1

Linear velocity evolution (km/h) time 5 10 15 20 25 30 35

40
20

Steering angle evolution (degs)

(b) Control signals

time 5 10 15 20 25 30 35

5

5 10 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(c) Task error signal

time 5 10 15 20 25 30 35 1 2 3 4 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(d) Task features’ weights Figure: Perpendicular reverse parking maneuver 21

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Unconstrained cases - MATLAB

8
6
4
2

2 4

2

2 4 6 8 10 Parking maneuver evolution

(a) Performed maneuver

time 5 10 15 20 25 30 35 1 2 Linear velocity evolution (km/h) time 5 10 15 20 25 30 35

40
20

Steering angle evolution (degs)

(b) Control signals

time 5 10 15 20 25 30 35

8
6
4
2

2 4 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(c) Task error signal

time 5 10 15 20 25 30 35 0.5 1 1.5 2 2.5 3 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(d) Task features’ weights Figure: Perpendicular forward parking maneuver 22

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Unconstrained cases - MATLAB

2

2 4 6 8 10 12

4
2

2 4 6 8 Parking maneuver evolution

(a) Performed maneuver

time 5 10 15 20 25 30 35

2
1

Linear velocity evolution (km/h) time 5 10 15 20 25 30 35

40

40 Steering angle evolution (degs)

(b) Control signals

time 5 10 15 20 25 30 35

8
6
4
2

2 4 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(c) Task error signal

time 5 10 15 20 25 30 35 1 2 3 4 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(d) Task features’ weights Figure: Parallel reverse parking maneuver 23

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Unconstrained cases - MATLAB

8
6
4
2

2 4

4
2

2 4 6 8 Parking maneuver evolution

(a) Performed maneuver

time 5 10 15 20 25 30 35 1 2 Linear velocity evolution (km/h) time 5 10 15 20 25 30 35

40

40 Steering angle evolution (degs)

(b) Control signals

time 5 10 15 20 25 30 35

2

2 4 6 8 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(c) Task error signal

time 5 10 15 20 25 30 35 1 2 3 4 5 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(d) Task features’ weights Figure: Parallel forward parking maneuver 24

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Unconstrained cases - MATLAB

2

2 4 6 8 10 12

2

2 4 6 8 10 Parking maneuver evolution

(a) Performed maneuver

time 5 10 15 20 25 30 35

2
1

Linear velocity evolution (km/h) time 5 10 15 20 25 30 35

50

50 Steering angle evolution (degs)

(b) Control signals

time 5 10 15 20 25 30 35

10
5

5 10 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(c) Task error signal

time 5 10 15 20 25 30 35 1 2 3 4 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(d) Task features’ weights Figure: Perpendicular reverse parking maneuver from far 25

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Constrained cases - Fast prototyping environment

Figure: Constrained perpendicular reverse parking maneuver 26

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Constrained cases - Fast prototyping environment

time 5 10 15 20 25

2
1

Linear velocity evolution (km/h)

control response

time 5 10 15 20 25

30
20
10

Steering angle evolution (degs)

control response

(a) Control signals

time 5 10 15 20 25

5

5 10 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(b) Task error signal

time 5 10 15 20 25 1 2 3 4 5 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(c) Task features’ weights

time 5 10 15 20 25

10
5

5 10 Constraints sensor signals sc(1) sc(2) sc(3) sc(4) sc(5)

(d) Constraints sensor signals Figure: Constrained perpendicular reverse parking maneuver 27

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Constrained cases - Real experimentation

time 5 10 15

4
2

Linear velocity evolution (km/h)

control response

time 5 10 15

20

20 Steering angle evolution (degs) control response

(a) Control signals

time 5 10 15

4
2

2 4 6 Task errors et(1) et(2) et(3) et(4) et(5) et(6)

(b) Task error signal

time 5 10 15 1 2 3 4 5 Task weights

ht

1

ht

2

htconst

3

ht

4

ht

5

htconst

6

(c) Task features’ weights

time 5 10 15

10
5

5 10 Constraints sensor signals sc(1) sc(2) sc(3) sc(4) sc(5)