Optimality and Bifurcations in a Model for Collective Motion
Investigate collective robotic motion, allelomimesis behavior, and optimal paths for robots based on coupling constants. Explore relationships between robot paths, curvature, and turning rates using a model proposed by Justh and Krishnaprasad.
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Optimality and Bifurcations in a Model for Collective Motion BY APRIL ROSZKOWSKI MENTOR: PROF. ANDY BORUM, MISHA PADIDAR
Who cares about collective robotic motion? Agents copying each other s behavior arises in nature (allelomimesis a cool word!) Most robotic motion planning relies upon agents parsing others position or orientation How can allelomimesis be used in robotic motion planning? 3
Justh and Krishnaprasad 2015 Model Coupling constant ? 0 11 2?12+1 ?1= cos ?3 ?2= sin ?3 ?3= ?1 2?22+1 2 ? 2? ?1 ?2 ?4= cos ?6 minimize 0 ? ? = ?? subject to ?5= sin ?6 ? 1 = ?? ?6= ?2 ?? is the turning rate of robot i Only studied for coupling of 0 or ! 4
Make the two robots paths as similar in curvature as the coupling dictates Give us an optimal path for robot 1 11 2?12+1 2?22+1 2 ? 2? ?1 ?2 0 Give us an optimal path for robot 2 5
1 1 2?12+1 1 2? ?1 ?2 2?22 + ? 2 0 If ? is small, each robot will act more independently. 6
1 1 2? ?1 ?2 1 2?12+1 2 + ? 2?22 0 If ? is large, the robots will attempt to follow similar paths. If ? = , the robots will follow the same path. 7
Canonical solution forms 8
Bell 2 inflection points Optimal S shape 3 inflection points Suboptimal Each robot starts at the origin (green circle) and ends at time ? = 1 at (0.5,0) (red circle) More complex solutions can be described in terms of simple ones (hence, canonical) Loop 0 inflection points Optimal 9
. . . 10
Bifurcations in the solution space 11
Pitchfork Bifurcation (subcritical) A suboptimal solution splits off into two suboptimal solutions one optimal solution Fold (Saddle-node) Bifurcation Two solutions one optimal, one suboptimal collide and obliterate each other 12
What Does this Plot Mean? Insight into shape and optimality of tree for coupling between 0 and Connected trajectories represent solutions which can be moved between using continuous deformation 19
Recapitulation Multiple optimal solutions exist to our problem, with interesting results in bifurcation land There are probably more interesting, optimal solutions we haven t found that only exist for larger values of ? Provides insight to cases where there are more than two agents 21
References Justh EW, Krishnaprasad PS. 2015 Optimality, reduction and collective motion. Proc. R. Soc. A 471:20140606. http://dx.doi.org/10.1098/rspa.2014.0606 22
