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Swarms, Curvature, and Convergence

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. Acknowledgements. Collaborators: Leveraging:.

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Swarms, Curvature, and Convergence

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  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.

  6. 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). (b) g5 g3 (a) g4 g2 g1 g1 g5 g2 g3 g4

  7. Equilibrium Formations of Two Vehicles Rectilinear formation (motion perpendicular to the baseline) Collinear formation Circling formation (vehicle separation equals the diameter of the orbit)

  8. x2 f(|r|) x1 y1 y2  r2 ro 0 r1 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:

  10. 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.

  11. 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 vehicle’s 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.

  13. Lie Group Setting Frenet-Serret Equations Group variables Dynamics g1, g2, ..., gn G = SE(2), the group of rigid motions in the plane. Configuration space Assume the controls u1, u2, ..., unare functions of shape variables only. Shape variables capture relative vehicle positions and orientations. Shape variables Shape space

  14. 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.

  15. Rectilinear Control Law Simulations

  16. 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 On-the-fly modification of the separation parameter. ro 1 time

  17. Circling Control Law Simulations

  18. 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

  19. Convergence Result for n > 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 > 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 .

  20. Performance Criteria • Faithful following of waypoint-specified trajectories • Sufficient separation between vehicles (to avoid collisions) Intervehicle distances Waypoints time • Minimize steering: for UAVs, turning requires considerably more energy than straight, level flight. Maneuverability is also limited. Steering controls Steering “Energy” umax time 0 time -umax

  21. 1 n .5 0 2 50 100 n 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.

  22. 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.

  23. 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:

  24. 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

  25. 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 Simpson’s Rule in the numerical integration. transition governed by steering rate limitation uj umax t T 2T 3T 4T 5T -umax

  26. 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.

  27. 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

  28. 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:

  29. 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).

  30. 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

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