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# Optimal Trajectory Generation With Probabilistic System Uncertainty Using Polynomial Chaos

[+] Author and Article Information
James Fisher

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141mzungu@tamu.com

Raktim Bhattacharya

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141raktim@aero.tamu.edu

J. Dyn. Sys., Meas., Control 133(1), 014501 (Nov 23, 2010) (6 pages) doi:10.1115/1.4002705 History: Received August 31, 2008; Revised June 22, 2010; Published November 23, 2010; Online November 23, 2010

## Abstract

In this paper, we develop a framework for solving optimal trajectory generation problems with probabilistic uncertainty in system parameters. The framework is based on the generalized polynomial chaos theory. We consider both linear and nonlinear dynamics in this paper and demonstrate transformation of stochastic dynamics to equivalent deterministic dynamics in higher dimensional state space. Minimum expectation and variance cost function are shown to be equivalent to standard quadratic cost functions of the expanded state vector. Results are shown on a stochastic Van der Pol oscillator.

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## Figures

Figure 1

Comparison of trajectories obtained from gPC expansions and Monte Carlo simulations

Figure 2

Evolution of the probability density function of the state trajectories and the mean trajectory due to μ(Δ). The solid (red) line denotes the expected trajectory of (x1,x2). The circles (blue) denote time instances, on the mean trajectory, for which the snapshots of pdf are shown.

Figure 3

PDF at final time due to the terminal constraint based on covariance

Figure 4

Comparison of trajectories obtained from gPC expansions and Monte Carlo simulations

Figure 5

Comparison of mean trajectories obtained from Monte Carlo simulations and gPC theory for Van der Pol system

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