1. 1 Navigation Functions for Patterned Formations Daniel E. Koditschek Electrical & Systems Engineering Department School of Engineering and Applied Science, University of Pennsylvania www.swarms.org

2. SWARMS 2 Original Limitations Fully Actuated Completely Sensed Presumption of known topological model Navigation Functions Exploit Invariance under Diffeomorphism for “Simple” Topology Theorem: for every smooth compact oriented manifold with boundary there exists an NF at each point Theorem: if h: M ¼ Q is a diffeomorphism and  2 NF ( M ) then  ± h 2 NF ( Q ) We can fix these ! Perhaps realistic ? Definition: NF ( Q )  : Q ! [0,1]  -1 [0] = destination  -1 [1] = boundary no other minima (nondegenerate) [Kod & Rimon, AAM ’90] [Rimon & Kod, TAMS’91]

3. SWARMS 3 Visual “Bead Patterns” The Visible Set: Visual Landmarks: Standard Sensor Model pinhole camera:  : A 2 ! RP 1 : (  1,  2 )   2 /  1 narrow field of view:  ( A 2 ) µ [-  E,  E ] µ R landmark: P = [ p 1, p 2, p 3 ] 2 ( A 2 ) 3 camera frame transformation: H(x c,y c,  c ) 2 SE(2) camera map: c : SE(2) ! [-  E,  E ] 3 : H  [  (Hp 1 ),  (Hp 2 ),  (Hp 3 )] [ Cowan, et al., IEEE TRA’02]

4. SWARMS 4 Encoding Bead Patterns: NF (I) is convex [ Kod, Robotica ‘94] Moreover each of the q := M(M-1)/2 connected components of B := { b 2 R M | b i  b j 8 i  j } is also convex Proposition: Lemma 3 b 1 -axis b 2 -axis d1d1 d2d2 …hints toward a “syntax” for NF?

5. SWARMS 5 Gradient Vector Field Pullback The camera map is a diffeomorphism onto its image, c : V ¼ I Hence, if  2 NF( I ) then  ± c 2 NF( V ) yields a visual servo for fully actuated kinematic rigid bodies Safe initial conditions: q 0 2 c -1 ( I ) =: V ) Assure safe, convergent results: q(t) 2 V & q(t) ! c -1 (d) [ Cowan, et al., IEEE TRA’02] for fully actuated dynamical rigid bodies (q,v) 2 TSE(2); q 0 2 c -1 ( I ) & v 0 T Mv 0 < 1 ) (q,v) (t) 2 T V SE(2) & (q,v) (t) ! c -1 (d) £ {0} [ Kod, JDynMechSys’91].. but what about underactuated rigid bodies? and

6. SWARMS 6 Navigation for Nonholonomic Systems? Heisenberg System (illustrative example) Unicycle System Scalar Assembly Problem [ Kod, Robotica, 1994. 12(2):137-155]  x y Brockett’s [Springer-Verlag,’81] canonical example: completely controllable not smoothly stabilizable

7. SWARMS 7 Toward a Unified NF “Servo” Theory Ingredients Underactuated System m = # actuators < dof = n nonholonomic constraints Goal: appropriate sensor predicate Obstacle avoidance to avoid physical obstacles to maintain gravitational balance to respect sensory limitations Construction Projector onto column space: Analysis (idealized case) C enter Manifold of f 1, W c Stable Manifold of f 1, W s Flow of f 2 destabilizes W c stabilizes W s x y  Orthogonal Field: Negative Gradient Field: [Kod&Lopes, IROS04] Realistic case: automated “parallel parking” [Bloch, Kod&Lopes, in progress]

8. SWARMS 8 Encoding Disk Patterns: NF( R 2 -  ) Recent sufficient conditions for non-colliding disks [Karagoz, Bozma & Kod, UM Tech Report ’03]

9. SWARMS 9 RHex: a “Swarm” of Legs [Saranli et al, Int. J. Rob. Res, 2001. 20(7): 616-631] Bioinspiration (Full ‘98) Initial Prototype (UM ’99) Refined Mechanism (McGill ’00) Design Concept (Buehler ‘98) Commercial Prototype (Boston Dynamics Inc ’03) Well-tuned Controls (UM ’02) Joint work: Buehler & Full

10. SWARMS 10 Tracking Circular Bead Patterns TerrainWithout Coord. With Coord. Linoleum10/10 Bricks (easy)28/3015/17 Bricks (medium)11/3019/30 Bricks (hard)6/3016/30 Bricks (extreme)1/104/10 Successful Traversals at ~2 m/s [Weingarten et al., RAM’04] FF Failures Alternating with Coordinated Controller Successes: Extreme Brick Bed Empirical Value: Contrast Coordinated vs. FF Control Ease of Design: Alternating Tripod Clock Example [Klavins & Kod (2002) Int. J. Rob. Res. 21(3):257-275] Clock1Clock3Clock5 Clock2Clock4Clock6 The system corresponding to this connection graph meets the specification: it has a single, global attracting behavior. The same analysis on this system gives multiple stable orbits. The system does not perform the task specified. Clock1Clock3Clock5 Clock2Clock4Clock6 At present, operating point must be tuned for each new environment Environment 1Environment n … … [cf. Jadbabaie, et al. ]

11. SWARMS 11 Emerging Limitations of NF Tracking Trackers Arise from sections Bundle  : NF( R n -  ) ! R n (projection onto goal pattern) Section  : R n ! NF( R n -  ) such that  ±  = id R n Controllers for tracking a moving pattern, r: R ! R n -  “Moving NF”  (r,b) := (  ± r)(b) “Safe” Tracking Controller: Topological Obstructions Hirsch & Hirsch [ Mich. Math. J. 1998 ] Definitions:  NF( D 2 – {o 1, o 2, o 3 }) - the set of navigation functions on the three-point punctured 2-disk)  The Bundle  : NF( D 2 – {o 1, o 2, o 3 }) ! ( D 2 ) 3 - projection onto the obstacles Result:  : NF( D 2 – {o 1, o 2, o 3 }) ! ( D 2 ) 3 has no continuous section Farber Definition [ Disc. Comp. Geom. 2003]: Topological Complexity, TC( X ), of a topological space, X  Definition: Pathspace, P( X ), the set of continous paths between pairs of points in X  The minimal cardinality, k, of an open cover {U 1, …, U k } of X £ X such that  : P( X ) ! X £ X has a continuous section on each Ui Working Conjecture:  : NF( X ) ! X (projection onto the goal point) admits a continuous section if and only if TC( X )=1