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Research Papers

# Periodic-Node Graph-Based Framework for Stochastic Control of Small Aerial Vehicles

[+] Author and Article Information

Laboratory for Information
and Decision Systems,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: aliagha@mit.edu

Saurav Agarwal

Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: sauravag@tamu.edu

Suman Chakravorty

Associate Professor
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: schakrav@tamu.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 2, 2014; final manuscript received June 18, 2014; published online October 21, 2014. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 137(3), 031002 (Oct 21, 2014) (14 pages) Paper No: DS-14-1004; doi: 10.1115/1.4027888 History: Received January 02, 2014; Revised June 18, 2014

## Abstract

This paper presents a strategy for stochastic control of small aerial vehicles under uncertainty using graph-based methods. In planning with graph-based methods, such as the probabilistic roadmap method (PRM) in state space or the information roadmaps (IRM) in information-state (belief) space, the local planners (along the edges) are responsible to drive the state/belief to the final node of the edge. However, for aerial vehicles with minimum velocity constraints, driving the system belief to a sampled belief is a challenge. In this paper, we propose a novel method based on periodic controllers, in which instead of stabilizing the belief to a predefined probability distribution, the belief is stabilized to an orbit (periodic path) of probability distributions. Choosing nodes along these orbits, the node reachability in belief space is achieved and we can form a graph in belief space that can handle higher order dynamics or nonstoppable systems (whose velocity cannot be zero), such as fixed-wing aircraft. The proposed method takes obstacles into account and provides a query-independent graph, since its edge costs are independent of each other. Thus, it satisfies the principle of optimality. Therefore, dynamic programming (DP) can be utilized to compute the best feedback on the graph. We demonstrate the method's performance on a unicycle robot and a six degrees of freedom (DoF) small aerial vehicle.

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## Figures

Fig. 1

A simple PNPRM with three orbits, twelve nodes, and two edges

Fig. 2

bαl≡(vαl,Pkαl) is the center of belief nodes corresponding to the nodes shown in Fig. 1, where Pkαl's are shown by their 3σ-ellipse. As an example of a FIRM node, the magnified version of B2j, which is a small neighborhood centered at b2j, is shown in the dotted box, where the shaded region (blue) around the outer ellipse boundary depicts the covariance neighborhood and the shaded circular region (green) around the center point denotes the mean neighborhood.

Fig. 3

(a) Five orbits (T = 100) and corresponding periodic estimation covariances as the SPPS solution of DPRE in Eq. (14). (b) Sample covariance convergence on an orbit (T = 20) under PLQG. Red ellipses are the solution of DPRE and green ellipses are the evolution of estimation covariance. The initial covariance is three times bigger than the SPPS solution of DPRE, i.e., P0 = 3P0.

Fig. 4

A sample PNPRM with circular orbits. Number of each orbit is written at its center. Nine landmarks (black stars) and obstacles (gray polygons) are also shown. The directions of motion on orbits and edges are shown by little triangles with a cross in their heading direction. (a) Orbits 2 and 7 (distinguished in black) are the start and goal orbits, respectively. Shortest path (in green, flowing through Orbits 2,3,4,5,6,7) and the most-likely path (in red, flowing through orbit 2,1,9,10,8,20,19,14,15,16,7) under the FIRM policy are also shown. (b) Assuming on each orbit, there exists a single node, the feedback πg is visualized for all FIRM nodes.

Fig. 5

The aircraft attached body-fixed frame and ground frame

Fig. 6

The PNPRM in 3D showing the orbits and edges

Fig. 7

Feedback πg is shown with orbit 2 as the goal orbit. (a) Starting from orbit 1, the shortest path (in green, flowing through orbits 1,4,5,3,2) and the most-likely path (in red, flowing through orbits 1,7,8,10,16,15,2) are shown from the top view (b) The shortest path (green) and the most-likely path (red) are shown in the 3D environment.

Fig. 8

Feedback πg is shown with orbit 3 as the goal orbit. Starting from orbit 2, the most-likely path (in red, flowing through orbits 2, 3) is shown (a) from the top view and (b) in the 3D environment.

Fig. 9

Feedback πg is shown with orbit 7 as the goal orbit. Starting from orbit 3, the most-likely path (in red, flowing through orbits 12,9,8) is shown (a) from the top view and (b) in the 3D environment.

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