1. Optimal Trajectory for Network Establishment of Remote UAVs –1–1 Prachya Panyakeow, Ran Dai, and Mehran Mesbahi American Control Conference June 2013

2. Motivation Cooperative control of multi-vehicle systems –2–2

3. Motivation Cooperative control of multi-vehicle systems –3–3

4. Motivation Cooperative control of multi-vehicle systems –4–4

5. Motivation Reconnaissance, surveillance, monitoring, imaging, data processing –5–5

6. Motivation Why disperse? Coverage issue Limit field of view Why form a connected network? Energy efficiency of a formation Information sharing –6–6

7. Outline Background and Problem Formulation Optimal path planning for target tree-graph connectivity  Control law (PMP with end-point manifold)  Control sequence (Nonlinear Opt Control necessary conditions)  Eliminate candidates (Geometry Estimation Method)  Computational issues Nonlinear Programming Method Pros/Cons for each approach Future work –7–7

8. Background and Problem Formulation –8–8 Objective: Find paths that bring scattered UAVs into proximity to form a connected network at terminal time with minimum total control effort. Nonlinear Dynamics of Each UAVs: Assumptions: UAVs are first far from each other The connected network at final time is denoted as graph The mobile agents represent the vertex set in the final connected network The communication or relative sensing channel represents edge set The initial states are given as and terminal time is given as

9. Elements in Adjacency matrix A are determined by edges of Euclidean distance based connection: Laplacian Matrix, Network connectivity constraint: Review and Background –9–9 Related works:  Spanos and Murray, 2004, Robust connectivity of networked vehicles.  Zavlanos and Pappas, 2007, Maintaining connectivity of mobile networks  Kim and Mesbahi, 2006, Maximizing the second smallest eigenvalue of a state-dependent graph Laplacian.  Dai, Maximoff, and Mesbani, 2012, Formation of connected network for fractionated spacecraft Euclidean distance Based connection d 1 0 D 1010

10. UAV Network Establishment –10 Problem Formulation:

11. UAV Network Establishment –11 Problem Formulation: Logical Constraint

12. UAV Network Establishment –12 Problem Formulation: Logical Constraint Nonlinear Constraint

13. Optimal Target Tree-Graph Connectivity –13 Problem Formulation:

14. Optimal Target Tree-Graph Connectivity –14 Direct Method: Solve the Nonlinear Optimal Control Problem Hamiltonian: End-point Manifold:

15. Optimal Target Tree-Graph Connectivity –15 Direct Method: Solve the Nonlinear Optimal Control Problem PMP with End-Point Manifold Control Law

16. Dubin’s Problem –16

17. Optimal Target Tree-Graph Connectivity –17 Direct Method: Solve the Nonlinear Optimal Control Problem Control Sequence

18. Optimal Target Tree-Graph Connectivity –18 Direct Method: Solve the Nonlinear Optimal Control Problem Proposition 1: (Path-Synthesis) found from intermediate/final conditions Control law/sequence Necessary conditions 5n-2 unknowns Final Constraints Intermediate Constraints 2n-2 n-1 n 1 n 5n-2 nonlinear equations Substitute

19. Optimal Target Tree-Graph Connectivity –19 Direct Method: Solve the Nonlinear Optimal Control Problem Optimal Trajectories Candidates: Candidates RLL-RRLCandidates RRL-RLL(Global Sol.)

20. Optimal Target Tree-Graph Connectivity –20 Direct Method: Solve the Nonlinear Optimal Control Problem Geometry Estimation Method for eliminating the candidates

21. Optimal Target Tree-Graph Connectivity –21 Direct Method: Solve the Nonlinear Optimal Control Problem Geometry Estimation Method for eliminating the candidates Direction to turn Choose initial guess for Solve for Using proposition 1 Check result Terminate No Yes

22. Optimal Target Tree-Graph Connectivity –22 Nonlinear Optimal Control with Geometry Estimation Method Pros: Optimal Solution Provides the solution (switching time) for a given graph Cons: Computational issues (NP Hard) Number of final network configurations is exponential Cayley’s Theorem: Number of distinct labeled trees on n agents is Global search of all tree graphs with five agents: 1~2 minutes Global search of all tree graphs with eight agents: 1~2 days! An efficient algorithm is required to approach the problem

23. Nonlinear Programming Method –23 Parameterized Optimization Problem: Transform the original problem to Using the same 3 segment bang-bang control scheme,as unknown Same control law/sequence Relaxation of Logical On/Off Constraint Relaxation of Connectivity Nonlinear Constraint

24. Nonlinear Programming Method –24 Model of communication/relative sensing link: The entry of weight adjacency matrix is assigned as An exponential function for power of communication link Relaxation of the logical constraints The communication efficacy drops off continuously as the distance between the agents increases d 1 0 D

25. NLP Method (Parameter Optimization Problem) –25 Relaxation of connectivity constraints via Matrix similarity transformation: Original mixed integer problem with nonlinear constraints Semi-definite constraintNonlinear constraint Parameter Optimization Problem with semi-definite constraints = small positive number to guarantee The weight network is connected

26. –26 Optimal Target-Tree Method NLP Method VS Scalability of the two methods

27. –27 Optimal Target-Tree Method NLP Method VS Global Optimal Solution Exact results for a given graph Have to search for all possible graphs Not scalable for large-scale systems Sub-Optimal Solution (NP Hard) Base from the same bang-bang control law/sequence Faster Convergence without going through all possible graph Scalable for large-scale systems Scalability of the two methods

28. Future Work –28 Vary UAV speed between stall/max Optimal dispersion Considering scenarios that combine  Optimal path planning for network connectivity  Maintain the formation  Collision Avoidance  Optimal path planning for network dispersion