Swarms, Curvature, and Convergence

1
Swarms, Curvature, and Convergence
Eric W. Justh, P.S. Krishnaprasad
Institute for Systems Research ECE
Department University of Maryland College Park,
MD 20742
CNCS MURI Review Meeting, Boston University,
October 20-21, 2003
2
Acknowledgements
Collaborators Leveraging
Jeff Heyer, Larry Schuette, David Tremper
Naval Research Laboratory 4555 Overlook Ave.,
SW Washington, DC 20375
Fumin Zhang
Institute for Systems Research University of
Maryland College Park, MD 20742

• NRL Motion Planning and Control of Small Agile
  Formations
• AFOSR Dynamics and Control of Agile Formations

3
Outline

• Motivation UAV formation control
• Planar model based on unit-speed motion with
  steering control
• - Equilibrium formations
• - Two-vehicle laws and Lyapunov functions
• - Connection to gyroscopic systems
• Implementation considerations
• Future research directions

4
UAV Modeling

• Features of UAV model
• - High speed ? sluggish maneuvering.
• - Turning ? significant energy penalty.
• - Autopilot takes into account detailed vehicle
  kinematics.
• Vehicles modeled as point particles moving at
  unit speed and subject to steering control.
• A formation control law is a feedback law which
  specifies these steering controls.
• Modeling may be appropriate in other settings
  with high speeds and penalties associated with
  turning (e.g., loss of dynamic stability).

Dragon Eye
(Photo credit Jonathan Finer, The Washington
Post)
Dragon Runner (Photo from U.S. Marine Corps
website)
5
Planar Model (Frenet-Serret Equations)
Unit speed assumption
x2
x1
xn
y1
y2
yn
r2

rn
r1
u1, u2,..., un are curvature (i.e., steering)
control inputs.
Specifying u1, u2,..., un as feedback functions
of (r1, x1, y1), (r2, x2, y2),..., (rn, xn, yn)
defines a control law.
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Characterization of Equilibrium Shapes
Proposition (Justh, Krishnaprasad) For
equilibrium shapes (i.e., relative equilibria of
the dynamics on configuration space), u1 u2
... un, and there are only two
possibilities (a) u1 u2 ... un 0 all
vehicles head in the same direction (with
arbitrary relative positions), or (b) u1 u2
... un ? 0 all vehicles move on the same
circular orbit (with arbitrary chordal distances
between them).
g5
(b)
g3
(a)
g4
g2
g1
g1
g5
g2
g3
g4
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Equilibrium Formations of Two Vehicles
Rectilinear formation (motion perpendicular to
the baseline)
Collinear formation
Circling formation (vehicle separation equals the
diameter of the orbit)
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Planar Formation Laws for Two Vehicles
r
9
Shape Variables for Two Vehicles
Dot products can be expressed as sines and
cosines in the new variables
?2
? r
?1
System after change of variables
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Lyapunov Functions for Two Vehicles

• Rectilinear formation (perpendicular to
  baseline) or collinear formation

• Circling formation or collinear formation

• Impose further conditions on the ?jk to
  stabilize specific formations while destabilizing
  others.

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Biological Analogy
Choice of coefficients
Steering control
Align each vehicle perpendicular to the baseline
between the vehicles.
Steer toward or away from the other vehicle to
maintain appropriate separation.
Align with the other vehicles heading.
Biological analogy (swarming, schooling)
- Decreasing responsiveness at large separation
distances. - Switch from attraction to
repulsion based on separation distance
or density. - Mechanism for alignment of
headings.
D. Grünbaum, Schooling as a strategy for taxis
in a noisy environment, in Animal Groups in
Three Dimensions, J.K. Parrish and W.M. Hamner,
eds., Cambridge University Press, 1997.
12
Gyroscopically Interacting Particles

• Note Vpair and Vcir are not to be thought of as
  a synthetic potential (commonly used in robotics
  for directing motion toward a target or away from
  obstacles).
• Vpair and Vcir are Lyapunov functions for the
  shape dynamics.
• The kinetic energy of each particle is conserved
  (because they interact via gyroscopic forces),
  and initial conditions are such that they all
  move at unit speed.
• There is an analogy with the Lorentz force law
  for charged particles in a magnetic field.
• In mechanics, gyroscopic forces are associated
  with vector potentials.
• References
• - L.-S. Wang and P.S. Krishnaprasad, J.
  Nonlin. Sci., 1992.
• - J.E. Marsden and T.S. Ratiu, Intro.to
  Mechanics and Symmetry, 2nd ed,
• Springer, 1999.

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Lie Group Setting
Dynamics
Group variables
Frenet-Serret Equations
g1, g2, ..., gn ? G SE(2), the group of rigid
motions in the plane.
Configuration space
Assume the controls u1, u2, ..., un are functions
of shape variables only.
Shape variables capture relative vehicle
positions and orientations.
Shape variables
Shape space
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Formation Control for n vehicles
Generalization of the two-vehicle formation
control law to n vehicles
At present, it is conjectured (based on
simulation results) that such control laws
stabilize certain formations. However,
analytical work is ongoing.
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Rectilinear Control Law Simulations
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Rectilinear Control Law Simulations
Simulations with 10 vehicles (for different
random initial conditions).
Leader-following behavior the red vehicle
follows a prescribed path (dashed line).
Normalized Separation Parameter vs. Time
3
ro
On-the-fly modification of the separation
parameter.
1
time
17
Circling Control Law Simulations
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Circling Control Law
Beacon-circling behavior the vehicles respond
to a beacon, as well as to each other.
Simulations with 10 vehicles (for different
random initial conditions).
Normalized Separation Parameter vs. Time
On-the-fly modification of the separation
parameter.
3
ro
1
time
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Convergence Result for n gt 2

• We consider rectilinear relative equilibria, and
  the Lyapunov function

• Convergence Result (Justh, Krishnaprasad) There
  exists a sublevel set ? of V and a control law
  (depending only on shape variables) such that
  on ?.
• With this Lyapunov function, we cannot prove
  global convergence for n gt 2.
• Although we obtain an explicit formula for the
  controls uj, j1,...,n, there is no guarantee
  that this particular choice of controls will
  result in convergence to a particular desired
  equilibrium shape in ?.

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Performance Criteria

• Faithful following of waypoint-specified
  trajectories

• Sufficient separation between vehicles (to avoid
  collisions)

Intervehicle distances
Waypoints
time
Steering controls

• Minimize steering for UAVs, turning requires
  considerably more energy than straight, level
  flight. Maneuverability is also limited.

Steering Energy
umax
time
0
time
-umax
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Choice of Parameters

• Basic parameters
• - rsep separation while circling.
• - n number of vehicles.
• Derived parameters
• - Rectilinear law
• - Circling law
• For the circling law, we precisely control the
  equilibrium formation.
• For the rectilinear law, we only approximately
  achieve the desired equilibrium vehicle
  separations.
• The steady-state vehicle separation for the
  rectilinear law is chosen to be half that of the
  circling law, although other choices are possible.

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Motion Description Language Approach
Each vehicle simulates the evolution of the
entire formation in real time i.e., the vehicles
all run the same motion plan. - Disturbances
(e.g., wind for UAVs) lead to estimation
errors. - GPS and communication used to
reinitialize the estimators. The motion plan
can be changed on the fly. - Interrupts, due to
the environment or human intervention, can
change the motion plan (e.g., dynamical system
parameters). - The communication protocol must
ensure that all vehicles update their motion
plans simultaneously. This approach is
consistent with motion description language
formalism V. Manikonda, P.S. Krishnaprasad, and
J. Hendler, Languages, Behaviors, Hybrid
Architectures, and Motion Control, in
Mathematical Control Theory, J. Baillieul and
J.C. Willems, eds., Springer, pp. 199-226, 1999.
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Time Discretization
Control laws specify u1(t), u2(t), ..., un(t)
at each time instant t. Instead, compute
u1(tm), u2(tm), ..., un(tm), where tmmT for m1,
2, ..., and let Maximum value of T is
determined by the control law. T ½ seems to
be a reasonable choice (for ?, ?, ? ? 1).
Piecewise constant controls allow the vehicle
positions to be computed using simple formulas
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Limited Steering Authority
umax maximum (absolute) value the steering
control is permitted to take. umax is
determined either by the minimum radius of
curvature or by the steering rate.
steering rate
minimum radius of curvature
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Finite Steering Rate Effects

• Why steering rate matters
• The transition time should be a small fraction
  of the interval T.
• If the transition times are not trivial, they
  can be taken into account by using Simpsons Rule
  in the numerical integration.

transition governed by steering rate limitation
uj
umax
t
T
2T
3T
4T
5T
-umax
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Sensor-Based Implementation
transmit antenna
One pair of antennas gives a sinusoidal
function of angle of arrival.
?
0
?
-?
?
s1(t)
Range is inversely related to received power.
?/4
s2(t)
receive antennas
Two pairs of antennas, used for both
transmitting and receiving, can provide all the
terms in the control law.
Antenna separation and transmission frequency
are related to UAV dimensions.
GPS is not required.
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3-Dimensional Frenet-Serret Equations
r - position vector x - tangent y - normal z -
binormal
unit speed assumption
z
x
u, v, w are control inputs (two of which uniquely
specify the trajectory)
y
r
Frenet-Serret v 0 u curvature
w torsion
Note the Frenet-Serret frame applies to the
trajectory, and is not a body-fixed frame for the
UAV
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Continuum Model

• Vector field (in polar coordinates)

• This continuum formulation only involves two
  scalar fields the density ?(t,r,?) and the
  steering control u(t,r,?).
• However, the underlying space is 3-dimensional
  (for planar formations).
• Incorporating time and/or spatial derivatives in
  the equation for u yields a coupled system of
  PDEs for ? and u.

• Continuity equation (Liouville equation)

• Conservation of matter

• Energy functional

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Presentations
1 Poster at AFOSR Dynamics and Control
Workshop, Pasadena, CA, August 12-14, 2002 (Justh
and Krishnaprasad). 2 Intelligent Automation
Inc., Rockville, MD, September 23, 2002 (Justh
and Krishnaprasad). 3 Dynamics and Control of
Agile Formations, Review of Annual Progress,
AFOSR Theme Project on Cooperative Control, Univ.
of Maryland, Oct. 25, 2002 (Justh). 4 Naval
Research Lab, Washington, DC, November 25, 2002
(Justh). 5 Multi-Robot Systems Workshop, Naval
Research Lab, Washington, DC, March 17-19, 2003
(Justh). 6 Poster at Research Review Day, Univ.
of Maryland, March 21, 2003 (Justh). 7 Caltech
CDS Seminar, April 16, 2003 (Krishnaprasad). 8
ISR Student-Faculty Colloquium, Univ. of
Maryland, April 29, 2003 (Justh). 9 SIAM Conf.
on Applications of Dynamical Systems, Snowbird,
UT, May 27-31, 2003 (Krishnaprasad). 10 Block
Island Workshop on Cooperative Control, Block
Island, RI, June 10-11, 2003 (Krishnaprasad). 11
Institute for Pure and Applied Mathematics,
UCLA, Oct. 3, 2003 (Krishnaprasad). 12 Workshop
on Future Directions in Nonlinear Control of
Mechanical Systems, Univ. of Illinois,
Urbana-Champaign, Oct. 4, 2003 (Justh).
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References
E.W. Justh and P.S. Krishnaprasad, A simple
control law for UAV formation flying, Institute
for Systems Research Technical Report TR 2002-38,
2002 (see http//www.isr.umd.edu). E.W. Justh
and P.S. Krishnaprasad, Steering laws and
continuum models for planar formations, Proc.
IEEE Conf. Decision and Control, to appear,
2003. E.W. Justh and P.S. Krishnaprasad,
Equilibria and steering laws for planar
formations, Systems and Control Letters, to
appear, 2003. E.W. Justh and P.S. Krishnaprasad,
Steering laws and convergence for planar
formations, Proc. Block Island Workshop on
Cooperative Control, to appear, 2003. See also
http//www.isr.umd.edu/justh