SES 2007

A Multiresolution Approach for Statistical Mobility Prediction of

Unmanned Ground Vehicles

44th Annual Technical Meeting of the Society of Engineering Science

Puneet SinglaDept. of Mech. & Aerospace Engg.

University at BuffaloBuffalo, NY 14260

Sesha Sai VaddiOptimal Synthesis Inc.

Palo Alto, CA 94303-4622

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.

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Uncertainty Propagation: Continuous System

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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 theuncertainty propagation problem under white-noise excitation

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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)

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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?

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Uncertainty Propagation: Continuous System

EKF

Now, update the weights of Gaussian Sum Mixture such that FPK equation error is minimized.

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Solving Fokker-Planck Equation

Fokker Planck Equation Error:

Minimize: Subject to

Necessary Conditions:

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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.

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Uncertainty Propagation: Black-Box Model

For most of practical applications, it is difficult to describe the system by a set of ODE.

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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 Yi can be chosen judiciously to reduce computational burden.– Gaussian Hermite Polynomials.– Uniform Legendre Polynomials.

1

( , )N

i i ii

Y wY

1

( )M

i k kk

Y

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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.

0

(0) 1,

| 0, 0,1, ,k

xk

w

d wk m

dx

1

(1) 0,

| 0, 0,1, ,k

xk

w

d wk m

dx

( ) ( 1) 1,

, -1 x 1

I Iw x w x

x

1( )Iw x

Arbitrary Local Approximations

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Stochastic GLO-MAP

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Half-Car Suspension Model

m1

m2

y1

y2

y3

y4

y5

y6

y

k2

k3

k5

k6

c2

c3

c5

c6

L1

L2

Uneven Terrain

X

Y

y1(x), y4(x)

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Validating Key Ideas

2 41 2 3( ) 0x x x k k x k x

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Validating Key Ideas

2 41 2 3( ) 0x x x k k x k x

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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

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Monte Carlo Simulations

True State Histogram with 10000 Monte Carlo Simulations

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Monte Carlo Simulations

Estimated State Histogram (from model using 3000 Monte Carlo)

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Monte Carlo Simulations

Estimated State Histogram (from model using 5000 Monte Carlo)

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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.