Swarms in Three Dimensions
E.W. Justh1, P.S. Krishnaprasad1,2
for Systems Research, 2ECE Department
1Institute
Rectilinear Control Law
Naval Research Lab (NRL) collaborators:
Jeff Heyer, Larry Schuette, David Tremper
öæ r jk
ö
1 é æ r jk
×x j ÷ç
×y j ÷+ f (| r jk
ê- h çç
å
֍
÷
n k ¹ j êë è | r jk |
øè | r jk |
ø
uj =
ISR collaborator: Fumin Zhang
Align vehicle j perpendicular
to the baseline between
vehicles j and k.
Abstract
• Modeling and analysis of novel control laws for vehicles moving at
constant speed.
• Practical motivation: coordinating the flight of meter-scale UAVs
(unmanned aerial vehicles). Possible implications for UUV, UGV,
or USV swarms, or biological swarming/schooling systems.
• Objective: UAV formation demo in collaboration with NRL.
Model: Unit-Speed Motion r1 = x1
z2
y2
x2
z1
r2
xn yn
x 1 = y1u1 + z1v1
y 1 = - x1u1
x1
z 1 = - x1v1
r2 = x 2
r1
$
zn
y1
unit-speed
assumption
vj =
r jk = rk - r j ,
ù
æ r jk
ö
×y + mx k ×y j ú
|) ç
ç | r jk | j ÷
÷
úû
è
ø
Steer toward or away from
vehicle k to maintain
appropriate separation.
öæ r jk
ö
1 é æ r jk
×z + f (| r jk
å ê- h ç ×x j ÷ç
÷ç | r jk | j ÷
÷
n k ¹ j ëê çè | r jk |
øè
ø
é
f (| r jk |) = a ê1 ê
ë
æ ro
çç
è | r jk
2
ö
÷
|÷
ø
Gyroscopic Interaction Laws
Align vehicle j
with vehicle k.
ù
æ r jk
ö
×z + mx k ×z j ú
|) ç
ç | r jk | j ÷
÷
ú
è
ø
û
(u1,v1), (u2,v2),..., (un,vn)
are controls.
Global convergence result for n = 2 (Justh, Krishnaprasad 2004)
using the Lyapunov function: V = - ln (1 + x 2 ×x1 )+ h(| r2 - r1 |)
penalizes heading-direction differences
z2
y2
x2
z1
r2
y1
x1
r1
h( r)
Goals: boundary
following and
non-collision.
f ( r)
r
F. Zhang, E.W. Justh, and P.S. Krishnaprasad, “Boundary
following using gyroscopic control,” submitted to IEEE Conf.
Decision and Control, 2004.
#
rn = x n
x n = y nun + z n vn
References
y n = - x nun
z n = - x n vn
• The control laws are assumed to be invariant under rigid motions
in three-dimensional space.
• Therefore, we can define appropriate shape variables (which
capture relative distances and angles between vehicles).
• Shape equilibria correspond to steady-state formations.
g4
g1
g2
g4
g4
g2
g5
g1
g3
g5
g1
g2
g3
g5
g3
rectilinear formation
circling formation
helical formation
Idea: control inputs for the
moving vehicle are determined
by the trajectory of the closest
point on the obstacle surface.
penalizes inter-vehicle distances which are too large or small
x 2 = y 2u2 + z 2 v2
y 2 = - x 2u2
Equilibrium Shapes
Future Research: Boundary-Following
ù
ú, m > h > 0, a > 0.
2
ú
û
z 2 = - x 2 v2
rn
• In our formation control model, particles (i.e., vehicles) interact
through gyroscopic forces.
• Gyroscopic forces do no mechanical work: the kinetic energy (and
hence the speed) of each particle remains constant.
• The physically relevant quantities are rj (the position) and xj (the
unit tangent vector to the trajectory), j=1,…,n, which imposes
constraints on the form of (uj,vj), j=1,…,n.
E.W. Justh and P.S. Krishnaprasad, “Formation control in three
dimensions,” submitted to IEEE Conf. Decision and Control, 2004.
Circling Control Law
uj =
öæ r jk
ö
1 ìï æ r jk
×y + f (| r jk
å í - h ç ×x j ÷ç
÷ç | r jk | j ÷
÷
n k ¹ j îï çè | r jk |
øè
ø
ær
ö
|) ç jk ×y j ÷
ç | r jk |
÷
è
ø
é
ær
öæ r
öùü
ï
+ m ê- x k ×y j + 2 ç jk ×x k ÷ç jk ×y j ÷úý
ç
֍
÷
|
|
|
|
r
r
úþï
è jk
øè jk
øû
ëê
Global convergence result for n = 2
(Justh, Krishnaprasad 2004).
E.W. Justh and P.S. Krishnaprasad, “Equilibria and steering laws
for planar formations,” Systems and Control Letters, in press, 2004
(see also ISR TR 2002-38, 2002).
E.W. Justh and P.S. Krishnaprasad, “Steering laws and continuum
models for planar formations,” Proc. IEEE Conf. Decision and
Control, pp. 3609-3614, 2003.
Acknowledgements
- Naval Research Laboratory Grants N00173-02-1G002,
N00173-03-1G001, and N00173-03-1G019.
- Army Research Office ODDR&E MURI01 Program Grant No.
DAAD19-01-1-0465.
- Air Force Office of Scientific Research AFOSR Grant No.
F49620-01-0415.