Swarms in Three Dimensions
E.W. Justh1, P.S. Krishnaprasad1,2
1Institute for Systems Research, 2ECE Department
Rectilinear Control Law
Naval Research Lab (NRL) collaborators:
Jeff Heyer, Larry Schuette, David Tremper
⎞ ⎛ r jk
⎞
1 ⎡ ⎛ r jk
⋅x j ⎟⎜
⋅ y j ⎟ + f (| r jk
⎢ −η ⎜⎜
∑
⎟
⎜
⎟
n k ≠ j ⎢⎣ ⎝ | r jk |
⎠ ⎝ | r jk |
⎠
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
Steer toward or away from
vehicle k to maintain
appropriate separation.
⎞⎛ r
⎞
1 ⎡ ⎛ r jk
∑ ⎢−η ⎜ ⋅ x j ⎟⎟ ⎜⎜ | rjk | ⋅ z j ⎟⎟ + f (| r jk
n k ≠ j ⎣⎢ ⎜⎝ | r jk |
⎠ ⎝ jk
⎠
r jk = rk − r j ,
⎤
⎛ r jk
⎞
|) ⎜
⋅ y + µ xk ⋅ y j ⎥
⎜ | r jk | j ⎟⎟
⎥⎦
⎝
⎠
⎡ ⎛ r
f (| r jk |) = α ⎢1 − ⎜ o
⎢ ⎜⎝ | r jk
⎣
⎞
⎟
| ⎟⎠
2
Align vehicle j
with vehicle k.
⎤
⎛ r jk
⎞
|) ⎜
⋅ z + µ xk ⋅ z j ⎥
⎜ | r jk | j ⎟⎟
⎝
⎠
⎦⎥
⎤
⎥ , µ > η > 0, α > 0.
2
⎥
⎦
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
penalizes inter-vehicle distances which are too large or small
(u1,v1), (u2,v2),..., (un,vn)
are controls.
circling formation
x2
z1
r2
y1
x1
Goals: boundary
following and
non-collision.
References
Circling Control Law
uj =
helical formation
z2
y2
F. Zhang, E.W. Justh, and P.S. Krishnaprasad, “Boundary
following using gyroscopic control,” submitted to IEEE Conf.
Decision and Control, 2004.
y n = − x n un
z n = − x n vn
3
Idea: control inputs for the
moving vehicle are determined
by the trajectory of the closest
point on the obstacle surface.
ρ
#
rn = x n
x n = y nun + z 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
g
Future Research: Boundary-Following
f (ρ)
x 2 = y 2u2 + z 2 v2
y 2 = −x 2u2
Equilibrium Shapes
• 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.
r1
h( ρ )
z 2 = − x 2 v2
rn
rectilinear formation
vj =
Gyroscopic Interaction Laws
⎞⎛ r
⎞
1 ⎪⎧ ⎛ r jk
∑ ⎨−η ⎜ ⋅ x j ⎟⎟ ⎜⎜ | r jk | ⋅ y j ⎟⎟ + f (| rjk
n k ≠ j ⎩⎪ ⎜⎝ | r jk |
⎠ ⎝ jk
⎠
⎛ r
⎞
|) ⎜ jk ⋅ y j ⎟
⎜ | r jk |
⎟
⎝
⎠
⎡
⎛ r
⎞⎛ r
⎞ ⎤ ⎪⎫
+ µ ⎢ − x k ⋅ y j + 2 ⎜ jk ⋅ x k ⎟ ⎜ jk ⋅ y j ⎟ ⎥ ⎬
⎜|r |
⎟⎜ | r |
⎟⎥
⎝ jk
⎠ ⎝ jk
⎠ ⎦ ⎭⎪
⎣⎢
Global convergence result for n = 2
(Justh, Krishnaprasad 2004).
E.W. Justh and P.S. Krishnaprasad, “Formation control in three
dimensions,” submitted to IEEE Conf. Decision and Control, 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.