steering laws and continuum models for planar formations
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Steering Laws and Continuum Models for Planar Formations. Eric W. Justh, P.S. Krishnaprasad. Institute for Systems Research & ECE Department University of Maryland College Park, MD 20742. 42nd IEEE Conference on Decision and Control, Dec 9-12, 2003. Acknowledgements. - PowerPoint PPT PresentationTRANSCRIPT
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Steering Laws and Continuum Models for Planar Formations
Eric W. Justh, P.S. Krishnaprasad
Institute for Systems Research& ECE Department
University of MarylandCollege Park, MD 20742
42nd IEEE Conference on Decision and Control, Dec 9-12, 2003
Boston University--Harvard University--University of Illinois--University of Maryland
Acknowledgements
Collaborators: Jeffrey Heyer, Larry Schuette, David Tremper atNaval Research Laboratory, Washington D.C., and Fumin Zhang
at the University of Maryland, College Park.
• NRL: “Motion Planning and Control of Small Agile Formations”
• AFOSR: “Dynamics and Control of Agile Formations”
• ARO: “Communicating Networked Control Systems”
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References1. 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).
2. E.W. Justh and P.S. Krishnaprasad, “Steering laws and continuum models for planar formations,” Proc. 42nd IEEE Conf. Decision and Control, in press, 2003.
3. E.W. Justh and P.S. Krishnaprasad, “Equilibria and steering laws for planar formations,” Systems and Control Letters, in press, 2003.
4. E.W. Justh and P.S. Krishnaprasad, “Steering laws and convergence for planar formations,” Proc. Block Island Workshop on Cooperative Control, to appear, 2003.
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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
• Further developments
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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 Runner
(Photo from U.S. Marine Corps website)
Dragon Eye(Photo credit: Jonathan Finer, The Washington Post)
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Planar Model (Frenet-Serret Equations)
x1
y1
r1
111
111
11
uuxy
yxxr
u1, u2,..., un are curvature (i.e., steering) control inputs.
x2
y2
r2222
222
22
uu
xyyxxr
xn
yn
rn
nnnnnn
nn
uu
xyyxxr
•••
Unit speed assumption
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 ShapesProposition (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).
g1g5
g4
g2
g3
g1
g3g5
g2
g4
(a)(b)
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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
111
111
11
uuxy
yxxr
222
222
22
uu
xyyxxr
122111 ||
|)(|),,,,( yrrryxyxr fFu
12 rrr
211222 ||
|)(|),,,,( yrrryxyxr fFu
x1y1
r1
x2
y2r2
kj
j kjk
F
yrrx
rrr
yxyxr
|||||)(|
),,,,(2
1
2
1
1122
2
||1|)(|
rr orf
|r|
f(|r|)
0 ro
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Shape Variables for Two Vehicles
= |r| 1
21111 cos
||sin
|| y
rrx
rr
2222 cos||
sin||
yrrx
rr
12 sinsin
)cos)(cos/1(cos)()cos,sin,cos,sin,(
12122111
f
F
)cos)(cos/1(cos)()cos,sin,cos,sin,(
12211222
fF
System after change of variables:
Dot products can be expressed as sines and cosines in the new variables:
)sin()sin(
2121
1212
yxyx
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• Rectilinear formation (perpendicular to baseline) or collinear formation:
Lyapunov Functions for Two Vehicles
)())cos(1ln( 12 hVpair
)())cos(1ln( 12 hVcirc
0,)()()()( 22112112 0,0)()()()( 22211211
or
)(h
)(f
• Circling formation or collinear formation:
0,)()()()( 22112112 0,0)()()()( 22211211
• Impose further conditions on the jk to stabilize specific formations while destabilizing others.
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Biological Analogy
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 vehicle’s heading.
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.
• 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.
Steering control: 212222 |||)(|
||||yxy
rrry
rrx
rr
fu
22211211 ,,,0Choice of coefficients:
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Gyroscopically Interacting Particles• Note: Vpair and Vcir are not to be thought of as synthetic potentials (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, Introduction to Mechanics and Symmetry, Springer, 1999, (2nd edition)
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Lie Group Setting
....,,2,1 nj
u
u
jjj
jjj
jj
xy
yx
xr
....,,2,1
100
nj
g
jjj
j
ryx
....,,2,1),2(000
001
010
000
000
100
, njseg
ugg
jjj
jjj
DynamicsGroup variablesFrenet-Serret Equations
g1, g2, ..., gn G = SE(2), the group of rigid motions in the plane.
copies n
GGGS
Configuration space
.,...,2,~ 11 njggg jj
Shape variables Shape space
copies 1
n
GGGR
Assume the controls u1, u2, ..., un are functions of shape variables only.Shape variables
capture relative vehicle positions and orientations.
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Formation Control for n vehiclesGeneralization 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.
jkj
kj
kjkjkkjjkjj fF
nu y
rrrr
rryxyxrr||
|)(|),,,,(1
jjj
jjj
jj
u
u
xy
yx
xr
nj ,,2,1
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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. Time3
1time
roOn-the-fly modification of the separation parameter.
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Circling Control Law Simulations
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Circling Control Law
On-the-fly modification of the separation parameter.
Normalized Separation Parameter vs. Time3
1time
ro
Simulations with 10 vehicles (for different random initial conditions).
“Beacon-circling” behavior: the vehicles respond to a beacon, as well as to each other.
jkj
kj
kjkjj
kj
kjj f
nu y
rrrr
rrxrrrr
|||)(|
||1
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Convergence Result for n > 2
• 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 > 2.
• Although we obtain an explicit formula for the controls uj, j=1,...,n, there is no guarantee that this particular choice of controls will result in convergence to a particular desired equilibrium shape in .
0V
n
j jkkjkj hV
1|)(|)cos(1ln rr
• We consider rectilinear relative equilibria, and the Lyapunov function
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Performance Criteria
Waypoints
time
Steering controlsumax
-umax
0
Steering “Energy”
time
time
Intervehicle distances
• Faithful following of waypoint-specified trajectories
• Sufficient separation between vehicles (to avoid collisions)
• Minimize steering: for UAVs, turning requires considerably more energy than straight, level flight. Maneuverability is also limited.
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• 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.
Choice of Parameters
1,22 nro
sepr
nrsepno orr 2
2 cos12,
1
.5
02 50 100
n
n
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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 tm=mT for m=1, 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:
).,[),()( 1 mmmjj ttttutu
)(
)(cos1
)(sin)()(
)(1)( 1 mj
mj
mjmjmj
mjmj t
Ttu
Ttutt
tut ryxr
TtuTtu
TtuTtutttt
mjmj
mjmjmjmjmjmj
)(cos)(sin
)(sin)(cos)()()()( 11 yxyx
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Limited Steering Authority
minimum radius of curvature
steering rate
• 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.
....,,2,1
if , if , if ,
nj
uuuuuuu
uuuu
maxidealjmax
maxidealjmax
idealj
maxidealjmax
j
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• 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 Simpson’s Rule in the numerical integration.
Finite Steering Rate Effects
umax
-umax
uj
t
transition governed by steering rate limitation
T 2T 3T 4T 5T
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Sensor-Based Implementationtransmit antenna
receive antennas
• One pair of antennas gives a sinusoidal function of angle of arrival.
0 -
• Range is inversely related to received power./4s1(t) s2(t)
• Antenna separation and transmission frequency are related to UAV dimensions.
• Two pairs of antennas, used for both transmitting and receiving, can provide all the terms in the control law.
• GPS is not required.
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3-Dimensional Frenet-Serret Equations
xzy
r
r - position vectorx - tangent y - normal z - binormal wv
wuvu
yxzzxy
zyxxr
u, v, w are control inputs (two of which uniquely specify the trajectory)
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
unit speed assumption
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Continuum Model
• Continuity equation (Liouville equation):
.sin
cos
ru
t
• Vector field (in polar coordinates):
.sincos
/
/
udtd
dtd
r
• Conservation of matter:
.,1,, tddtG
rr
• Energy functional:
.~~)~,~,(),,(|)~(|)~cos(1ln21)( ddddtthtV
G Gc rrrrrr
• 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.
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References1. 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).
2. E.W. Justh and P.S. Krishnaprasad, “Steering laws and continuum models for planar formations,” Proc. 42nd IEEE Conf. Decision and Control, in press, 2003.
3. E.W. Justh and P.S. Krishnaprasad, “Equilibria and steering laws for planar formations,” Systems and Control Letters, in press, 2003.
4. E.W. Justh and P.S. Krishnaprasad, “Steering laws and convergence for planar formations,” Proc. Block Island Workshop on Cooperative Control, to appear, 2003.