Motion Camou

7/27/2019 Motion Camou

1/15

Motion Camouflage for Unicycle Robots using

Optimal Control

Inaki Rano and Chris Burbridge

[email protected]

Computer Sciences and Systems Engineering Dept

University of Zaragoza

Spain

TAROS 2010

Rano & Burbridge UniZar & ISRC-Ulster

Unicycle Motion Camouflage
http://find/


2/15

Table of Contents

1 Motion Camouflage for

2 Unicycle Robots using

3 Optimal Control

4 All together

5 Results


http://find/http://goback/


3/15

Everybody knows what Motion is!!

We see motion on static images. Can we see motion as somethingstatic?


http://find/


4/15

Camouflage

A method of crypsis (avoidance of observation) that allows anotherwise visible organism or object to remain indiscernible fromthe surrounding environment through deception.




5/15

Motion Camouflage? Are You Kidding?

Thats impossible!


http://find/


6/15

Motion Camouflage

Stealth behaviour ofdragonflies and hover-flies.

Eyes (cameras) onlymeasure angles (rays).

The Shadowee sees theShadower always on theFocal Point.

Blooming; the visual size of

the Shadower increases.

Camouflage for attack orretreat.


http://find/


7/15

Unicycle Robots

Unicycle motion model

x = vcos

y = vsin

=

3D state space (x, y, ) and twocontrol inputs (v, ).

Some robots follow this model;dual-drive, synchro-drive, tricycle,(Ackerman).


http://find/


8/15

Why Unicycle Robots?

Existing 2D and 3D MotionCamouflage techniques treat theshadower as a point

Animals move in 2D/3D but they

have a heading (theyre not points)

Restriction to motion (2D)

dxsin dycos = 0

animals usually dont moveperpendicular to their heading.

Unless youre a crab. . .




9/15

Calculus of Variations

Find the function y(t) that minimises the functional

J[y] =

tfti

g(t, y(t), y(t))dt

y(t) can be obtained solving the Euler-Lagrange equation

g

y

d

dt

g

y= 0

with boundary conditions. But we have a system to be controlledx = F(x, u), where x(t) n and u(t) m.


http://find/


10/15

Optimal Control

Find the control functions u(t) that minimise the functional

J[x, u] = h(x(tf)) +

tfti

g(t, x(t), u(t))dt

with the dynamic constrain x = F(x, u) and the appropriateboundary conditions.

Constrained Minimisation? This is a job for the LagrangeMultipliers!!

J[x, u] = h(x(tf)) +

tf

ti

[g(t, x(t), u(t))+

(t)(F(x(t), u(t)) x(t))] dt


http://find/


11/15

Optimal Control Solution

If we define the Hamiltonian

H(x(t), u(t), (t)) = g(t, x(t), u(t)) + (t)tF(x(t), u(t))

the solution can be obtained from the equations

x(t) =H

(x(t), u(t), (t))

(t) = H

x(x(t), u(t), (t))

0 = Hu

(x(t), u(t), (t))

as the solution to a Boundary Value Problem.




12/15

All together

f x=(f , f )y

=(x , y )p pxp

=(x , y )t tx t

CCL

Y

X

p

Dynamic constrains

xpyp

pxtyt

=

vcos pvsin p

vxtvyt

Control inputs u=

v

Function to minimise

g(x) =1

2|(xp f) (xt f)|

2

+1

2uTRu


http://find/


13/15

Results

Focal Point

Shadower

Shadowee

CC

L

Trajectory

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Error

0

2

4

6

8

10

12

14

16

18

0 20 40 60 80 100

Trajectory

1.5

1.0

0.5

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Error

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160 180 2 00


http://goback/http://find/http://goback/


14/15

Discussion

The good news

Motion Camouflage can be achieved for non-holonomicmotion...

. . . for any focal point.

It is extensible to any shadowee trajectory. . .

. . . and velocities seem to stay bounded

The bad news

Need to know too much information (time, positions, velocity)No certainty on bounded velocities




15/15

Thanks for your attention

Better to remain silent and

be thought a fool than to speak outand remove all doubt

Abraham Lincoln


http://find/

Motion Camou

Documents

Transcript of Motion Camou