Directed Triangular Formations

Directed Triangular Formations

A. S. Morse

Yale University

Gif – sur - Yvette May 24, 2012

Supelec

EECI Graduate School in Control

FORMATION CONTROL

FORMATION CONTROL

with Leader – Follower Constraints

FORMATION CONTROL

2

1

31 follows 2 and 3


FORMATION CONTROL

Closed cycles can present problemsbecause of miss-registration of agents’positions

d13

d12

Agent i maintains distance dij

This plus rigidity maintains the formation


FORMATION CONTROL

1 follows 2 and 3

Closed cycles can present problemsbecause of miss-registration of agents’positions

Agent i maintains distance dij

This plus rigidity maintains the formation

d13

d12

d3

d2

d1

1

2

3

d1 > d2 + d3

d2 > d1 + d3

d3 > d1 + d2

yij = position of agent j in local coordinate system of agent i

xi = position of agent i in global coordinates

_yi i = ui i 2 f1;2;3g

ei = jjyi i ¡ yi [i ]jj2 ¡ d2i i 2 f1;2;3g

_yi i = ¡ (yi i ¡ yi [i ])ei i 2 f1;2;3gyi j =R ixj +¿i i 2 f1;2;3g

ei = jjxi ¡ x[i ]jj2 ¡ d2i i 2 f1;2;3g_xi = ¡ (xi ¡ x[i ])ei i 2 f1;2;3g

[ i ] = i + 1 mod 3Can this formation be maintained?

d3

d2

d1

1

2

3


[ i ] = i + 1 mod 3

Equilibrium set =

d3

d2

d1

1

2

3


[ i ] = i + 1 mod 3

x =24x1x2x3

35

E[fx : (xi ¡ x[i ]) = 0 i 2 f1;2;3gg

3[

i=1f x : (xi ¡ x[i ]) = 0 e[i ] =0 e[[i ]] = 0g

All agents at same point Two agents at same point;third correctly positioned

Target set: E = fx : ei =0 i 2 f1;2;3gg

Target

All trajectories starting outside of equilibrium set converge to target set E

2 disconnected subsets

d3

d2

d1

1

2

3

Equilibrium set ¾Can’t have just one ei = 0Can’t have all ei = 0 with two agents at same position

So E is disjoint from

d3

d2

d1

1

2

3


E[fx : (xi ¡ x[i ]) = 0 i 2 f1;2;3gg

3[

i=1f x : (xi ¡ x[i ]) = 0 e[i ] =0 e[[i ]] = 0g

Equilibrium set =


ei = jjzi jj2 ¡ d2i i 2 f1;2;3g_xi = ¡ ziei i 2 f1;2;3g

E[fx : zi =0 i 2 f1;2;3gg

3[

i=1f x : zi =0 e[i ] =0 e[[i ]] =0g

x1 , x2 , x3 are co- linear iff rank [z1 z2 z3]2£3 < 2 verify

Co-linear set: d1 > d2 + d3

d2 > d1 + d3

d3 > d1 + d2

If all three agents are properly positioned, they cannot be in a line

E and N are disjoint

d3

d2

d1

1

2

3

Equilibrium set =



E[fx : zi =0 i 2 f1;2;3gg

3[

i=1f x : zi =0 e[i ] =0 e[[i ]] =0g

Co-linear set:

E and N are disjoint

= {x: det [z1 z2] = 0}

verify

N is an invariant set

d3

d2

d1

1

2

3


All trajectories of the z system exist and are bounded on [0, 1)

All trajectories of the x system exist on [0, 1)

self-contained

but not necessarily bounded!

d3

d2

d1

1

2

3

ei = jjzi jj2 ¡ d2i i 2 f1;2;3g

transpose of rigiditymatrix of {G , x}

self-contained

d3

d2

d1

1

2

3


transpose of rigiditymatrix of {G , x}

closedcompact

as fast as

Pick ½* small so that for T = closure of S (½*)

d3

d2

d1

1

2

3


as fast as

V decreasing if z is not in MIf z is bounded away from M, even in the limit, then becauseV is decreasing, z must enter S (½*) in some finite time.

_ If z starts outside of N it cannot enter N and thus M in finite time

_

d3

d2

d1

1

2

3


as fast as

If z is bounded away from M, even in the limit, then becauseV is decreasing, z must enter S (½*) in some finite time.

_ If z starts outside of N it cannot enter N and thus M in finite time

_

Claim: for any z 2 M, e1 + e2 + e3 < 0

Suppose:

Beyond some time T, z must be close enough to M that for t ¸ T, e1 + e2 +e3 < 0

Thus

Therefore if

Implication of claim:

d3

d2

d1

1

2

3


verify

Suppose z 2 M1

1. ||z2|| = ||z3|| = 0

2. ||z2|| = 0 and ||z3|| 0

||z1|| = 0 ei = -di2 , i = 1,2,3

||z1|| 0 e1 = e3 = 0 and e2 = -d22

3. ||z2|| 0 and ||z3|| = 0 Same type reasoning as case 2.

4. ||z2|| 0 and ||z3|| 0


Four cases to consider:

d3

d2

d1

1

2

3


Suppose z 2 M1

4. ||z2|| 0 and ||z3|| 0


d3

d2

d1

1

2

3


Suppose z 2 M1

4. ||z2|| 0 and ||z3|| 0


If any ei = 0 then all ei = 0 because of def. M

||z1|| > ||z2|| so all ||zi|| 0

So all ei 0

|e2| > |e1| |e3| > |e1|

If e1 < 0 then e2 > 0 and e3 > 0d1 > ||z1|| d2 < ||z2|| d3 < ||z3||

e1 > 0 e2 < 0 e3 < 0

e1 + e2 + e3 < 0

What about the xi ?


We already know that the xi and zi exist globally and that the zi are bounded .

If the ei tend to 0 exponentially fast, the xi then to finite constants

If the ei do not tend to 0, the xi drift off to 1.

Summary


[ i ] = i + 1 mod 3

Target set: E = fei =0 i 2 f1;2;3gg8<:

24x1x2x3

35 : rank [(x1 ¡ x[1]) (x2 ¡ x[2]) (x3 ¡ x[3]) ]< 2

9=;Co-linear set:

All x trajectories start outside of the co-linear set converge to the target set exponentially fast.

All trajectories start inside of the co-linear set which are not in the equilibrium set drift off to infinity as t ! 1.

d3

d2

d1

1

2

3

So there is a pretty good understanding of directed triangular formations

Results like the preceding also exist for more complicated graphs which are cycle free.

However there is no general theory for directed graphs which have cycles.

For example:

Remark

We will {probably not} talk about such formations next

directed

For undirected formations however, more is known.

We will talk about this later.

Documents

Directed Triangular Formations