1. SES 2007 A Multiresolution Approach for Statistical Mobility Prediction of Unmanned Ground Vehicles 44 th Annual Technical Meeting of the Society of Engineering Science Puneet Singla Dept. of Mech. & Aerospace Engg. University at Buffalo Buffalo, NY 14260 Sesha Sai Vaddi Optimal Synthesis Inc. Palo Alto, CA 94303-4622

2. SES 2007 Objective Unmanned Ground Systems often operate with some degree of uncertainty. Poorly known parameters –variation in suspension stiffness and damping characteristics Uncertain inputs –Rough terrain, soil properties in vehicle-terrain interaction. For realistic predictions of the system behavior and performance dynamic models must account for these uncertainties. Given the uncertain nature of the terrain and the parameters of the vehicle, predict the ability of the vehicle to negotiate a terrain while satisfying certain performance metrics. –Main Challenge: propagation of high dimensional uncertainty through a nonlinear dynamic system.

3. SES 2007 Uncertainty Propagation: Continuous System

4. SES 2007 Uncertainty Propagation: Continuous System Approximate methods for uncertainty propagation: –Monte Carlo: Computationally heavy esp. in high dimensions –Gaussian Closure, Higher order closures –Statistical linearization, Stochastic averaging Not preferred for highly nonlinear systems and long time durations of propagation All the above methods provide an approximate description of the uncertainty propagation problem White-noise excitation The Fokker-Planck equation (FPE) provides the exact description of the uncertainty propagation problem under white-noise excitation

5. SES 2007 Uncertainty Propagation: Continuous System System dynamics: The following linear PDE, called the Fokker-Planck equation describes the time evolution of for the system given by (1) : (1) (2) (Fokker-Planck operator) (Drift Vector) (Diffusion Matrix)

6. SES 2007 Probability Density Function Approximation Let us assume that underlying pdf can be approximated by a finite sum of Gaussian pdfs. Question is how to find unknown parameters of this Gaussian Sum Mixture?

7. SES 2007 Uncertainty Propagation: Continuous System EKF Now, update the weights of Gaussian Sum Mixture such that FPK equation error is minimized.

8. SES 2007 Solving Fokker-Planck Equation Fokker Planck Equation Error: Minimize:Subject to Necessary Conditions:

9. SES 2007 Solving Fokker-Planck Equation Let us assume: We have designed a mean to update the weights of Gaussian Mixture Model to capture non-Gaussian behavior.

10. SES 2007 Uncertainty Propagation: Black-Box Model For most of practical applications, it is difficult to describe the system by a set of ODE.

11. SES 2007 Stochastic GLO-MAP Basic Idea: express the output as a function of input random variables. Specially designed weight functions gives us the freedom to choose independent local approximations. Local models Y i can be chosen judiciously to reduce computational burden. –Gaussian  Hermite Polynomials. –Uniform  Legendre Polynomials.

12. SES 2007 Stochastic GLO-MAP There is a choice of weighting function that will guarantee piecewise global continuity while leaving freedom to fit local data by any desired local functions. Arbitrary Local Approximations

13. SES 2007 Stochastic GLO-MAP

14. SES 2007 Half-Car Suspension Model m1m1 m2m2 y1y1 y2y2 y3y3 y4y4 y5y5 y6y6 y k2k2 k3k3 k5k5 k6k6 c2c2 c3c3 c5c5 c6c6 L1L1 L2L2 Uneven Terrain X Y y 1 (x), y 4 (x)

15. SES 2007 Validating Key Ideas

16. SES 2007 Validating Key Ideas

17. SES 2007 Monte Carlo Simulations Input Parameters –Terrain Constants, Mass(M), Inertia(I) –Stiffness(k) and Damping(c) Constants Performance Metrics –Maximum bounce of the wheels –Maximum attitude angle –RMS value of the wheel vertical velocities

18. SES 2007 Monte Carlo Simulations True State Histogram with 10000 Monte Carlo Simulations

19. SES 2007 Monte Carlo Simulations Estimated State Histogram (from model using 3000 Monte Carlo)

20. SES 2007 Monte Carlo Simulations Estimated State Histogram (from model using 5000 Monte Carlo)

21. SES 2007 Conclusions A robust uncertainty propagation method has been developed for UGV mobility prediction. –Can qualitatively capture the dynamics for multiple attractor states. Allows an accurate treatment of nonlinear dynamics and of non-Gaussian probability densities. –Does not rely on the assumption that uncertainties are small. –more efficient than sequential Monte-Carlo methods. Finally, the simulation results presented in this paper merely illustrate usefulness of the uncertainty propagation algorithm. –further testing would be required to reach any conclusions about the efficacy of the mobility prediction algorithm.