1. 1 Combinatorial Specification of Collective Behavior An ARO MURI Project (with support from DARPA SToMP & RiSE) www.swarms.org D. E. Koditschek Electrical and Systems Engineering Department School of Engineering and Applied ScienceUniversity of Pennsylvania

2. SWARMS 2 Objective What? Build hybrid controllers with provable properties for specifying & controlling heterogeneous teams with various combinations of specialized agents Subject to combinatorial goals and obstacles Why? Immediate: gait controllers for multi-legged robots Short term: tight coordination of small-team maneuvers Longer term: represent & control complex swarm behavior

3. SWARMS 3 Technical Approach Scientific methodology Dynamical systems theory (coupled oscillators) Algebraic & combinatorial topology (rudimentary) Bioinspired engineering practice (empirical) Is it unique? First legged wall climbing robots (RiSE) Generalizing navigation functions (SToMP) Coordination for cyclic behaviors (e.g. climbing gaits) Obstacles (repellor as well as attractor structure) Combinatorial tasks

4. SWARMS 4 Major Breakthroughs Outline key ideas/innovation coming out of SWARMS funding over the last three years Gait controllers (+ RiSE with Rizzi & Haynes) Problem: unstructured outdoor vertical environment; under-powered robot Solution: specific hexapedal, pentapedal, tetrapedal gaits + hybrid transitions origin of combinatorial specification statement (“look ma, no hands”) Navigation Field Design (+ SToMP with Weinberger & Ferry) Problem: ( M, G, O ) – smooth complex, goal & obstacle subcomplexes Emerging Solution :  normal flow toward G (away from O ) through tubular neighborhood H ¾ G ( N ¾ O )  combinatorial vector field on 1-skeleton of dual complex M – ( H [ N )  present status: work on smoothing combinatorial into C r vector fields Combinatorial Coordination (+ SToMP with Cohen & Ghrist) Problem: gaits ½ diagonal toral complex: T n (n-1)! = Conf n ( S 1 ) t  n ( S 1 ) Emerging Solution: (modified) Young’s tableaux can index T n (n-1)!  calculus for computing cells, their boundaries, stars, and links  indexing complete system of hybrid gait controllers  present status: handle obstacles within combinatorial framework

5. SWARMS 5 DoD Benefits Why are the breakthroughs relevant to DoD? Climbing robots for previously denied mobility in vertical environments (stealthy) surveillance quick, effective escape from comms shadows Generalizations to coordination of complex (multi-specialty) teams with intricate (combinatorial) goals “always keep at least three ground contacts while climbing” “cage the evader while keeping one agent guarding each door” “engage the opposing players while maintaining at least two `sweep` defense-men in the goal zone”

6. SWARMS 6 SWARMS Collaborations Highlight why swarms funding enabled joint projects that could not have been accomplished with traditional single-investigator grants 1 kod-month per year allowed deep cross-pollination across several more narrowly specialized efforts Why did the breakthroughs require SWARMS funding, as opposed to NSF funding? NSF FIBR focused on generating refutable biological hypotheses DARPA RiSE focused on specific climbing milestones DARPA SToMP is small !

7. SWARMS 7 Success Stories Examples of achievements beyond just technical break through Numerous best paper, etc. for RiSE (www.riserobot.org) Numerous press reports on RiSE (www.riserobot.org) RiSE post-docs have recently won academic positions at Johns Hopkins GA Tech Clemson FSU …

8. SWARMS 8 Technology Transition Technology used by private industry RiSE closely connected to Boston Dynamics, Inc. Start-ups http://www.sandboxinnovations.com List any/all contacts with DoD labs Kod serves on National Academy Review Panel for ARL VTD

9. SWARMS 9 Plans for Years 4-5 Outline new unanticipated directions Proposal: ARO to help broker MOU with DARPA RiSE Technology Transfer to ARL (and RDECOM) Proposal: Increase Swarms support (add PhD slot) for climbing application of combinatorial specification ideas for herd applications of combinatorial specification ideas Planned research outlined in proposal for years 4-5 New algorithms for mobile robot team coordination with optical flow sensing only New algorithms for consensus and anti-consensus on Lie Groups