optimal trajectory for network establishment of remote uavs –1–1 prachya panyakeow, ran dai,...

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Optimal Trajectory for Network Establishment of Remote UAVs

Prachya Panyakeow, Ran Dai, and Mehran Mesbahi

American Control Conference June 2013

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Motivation• Cooperative control of multi-vehicle systems

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Motivation• Cooperative control of multi-vehicle systems

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Motivation• Cooperative control of multi-vehicle systems

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MotivationReconnaissance, surveillance, monitoring, imaging, data processing

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MotivationWhy disperse?• Coverage issue• Limit field of view

Why form a connected network?• Energy efficiency of a formation• Information sharing

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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

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Background and Problem Formulation• 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

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• Elements in Adjacency matrix A are determined by edges of• Euclidean distance based connection:

• Laplacian Matrix ,• Network connectivity constraint:

Review and BackgroundRelated 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

0D

10

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UAV Network Establishment

Problem Formulation:

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UAV Network Establishment

Problem Formulation:

Logical Constraint

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UAV Network Establishment

Problem Formulation:

Logical Constraint

Nonlinear Constraint

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Optimal Target Tree-Graph Connectivity

Problem Formulation:

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Optimal Target Tree-Graph Connectivity

Direct Method: Solve the Nonlinear Optimal Control Problem

Hamiltonian:

End-point Manifold:

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Optimal Target Tree-Graph Connectivity

Direct Method: Solve the Nonlinear Optimal Control Problem

PMP with End-Point Manifold

Control Law

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Dubin’s Problem

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Optimal Target Tree-Graph Connectivity

Direct Method: Solve the Nonlinear Optimal Control Problem

Control Sequence

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Optimal Target Tree-Graph Connectivity

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

n1

n

5n-2 nonlinear equations

Substitute

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Optimal Target Tree-Graph Connectivity

Direct Method: Solve the Nonlinear Optimal Control Problem

Optimal Trajectories Candidates:

Candidates RLL-RRL Candidates RRL-RLL(Global Sol.)

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Optimal Target Tree-Graph Connectivity

Direct Method: Solve the Nonlinear Optimal Control Problem

Geometry Estimation Method for eliminating the candidates

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Optimal Target Tree-Graph Connectivity

Direct Method: Solve the Nonlinear Optimal Control Problem

Geometry Estimation Method for eliminating the candidates

Direction to turn

Choose initial guess for

Solve forUsing proposition 1

Check result

Terminate

No

Yes

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Optimal Target Tree-Graph ConnectivityNonlinear 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

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Nonlinear Programming MethodParameterized 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

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Nonlinear Programming MethodModel 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

0D

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NLP Method (Parameter Optimization Problem)Relaxation of connectivity constraints viaMatrix similarity transformation:

Original mixed integer problemwith nonlinear constraints

Semi-definite constraintNonlinear constraint

Parameter Optimization Problemwith semi-definite constraints

= small positive number to guaranteeThe weight network is connected

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Optimal Target-Tree Method NLP MethodVS

Scalability of the two methods

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Optimal Target-Tree Method NLP MethodVS

•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

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Future Work

• 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