Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

Jasour, Ashkan; Wang, Allen; Williams, Brian

doi:10.48550/arxiv.2101.12490

Cited by 5 publications

(12 citation statements)

References 43 publications

(86 reference statements)

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“…In this section, we provide an approach for nonlinear moment propagation that can, in principle, work for moments up to arbitrary order [30], [42]. Given a nonlinear motion model and a random vector for control inputs, w t , this section is concerned with the problem of computing statistical moments of the uncertain position x t s.t.…”

Section: Non-gaussian Risk Assessment With Sos Programmingmentioning

confidence: 99%

“…In [42], we show that nonlinear stochastic motion dynamics can be transformed in to equivalent linear-state dynamical systems by introducing suitable new state variables. In this section, we define such equivalent augmented linear-state system for stochastic nonlinear Dubin's model (20) as follows:…”

Section: B Equivalent Augmented Linear-state Systemmentioning

confidence: 99%

“…We define the moment-state linear systems for the obtained augmented linear-state system in (22) as follows [42]:…”

Section: Moment-state Linear Systemsmentioning

confidence: 99%

“…We can use different motion dynamics with different sources of uncertainties to model the uncertain behaviour of the agent vehicles. For more information see [42] and https://github.com/jasour/Uncertainty-Propagation…”

Section: Moment-state Linear Systemsmentioning

confidence: 99%

“…In this section, we show how trigonometric moments of the form E[cos n (X)], E[sin n (X)], and E[cos m (X) sin n (X)] can be computed in terms of the characteristic function of the random variable X, denoted by Φ X [42], [43]. We begin by applying Euler's Identity to the definition of the characteristic function as follows:…”

Section: B Moments Of Trigonometric Random Variablesmentioning

confidence: 99%

See 4 more Smart Citations

Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures

Jasour¹,

Huang²,

Wang³

et al. 2021

Preprint

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This paper presents fast non-sampling based methods to assess the risk for trajectories of autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models to predict both agent positions and control inputs conditioned on the scene contexts. We show that the problem of risk assessment when Gaussian mixture models (GMMs) of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using nonlinear Chebyshev's Inequality and sums-of-squares (SOS) programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require higher order statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent control inputs as opposed to positions, we propagate the moments of uncertain control inputs through the nonlinear motion dynamics to obtain the exact moments of uncertain position over the planning horizon. To this end, we construct deterministic linear dynamical systems that govern the exact time evolution of the moments of uncertain position in the presence of uncertain control inputs. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.

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Section: Non-gaussian Risk Assessment With Sos Programmingmentioning

confidence: 99%

Section: B Equivalent Augmented Linear-state Systemmentioning

confidence: 99%

“…We define the moment-state linear systems for the obtained augmented linear-state system in (22) as follows [42]:…”

Section: Moment-state Linear Systemsmentioning

confidence: 99%

Section: Moment-state Linear Systemsmentioning

confidence: 99%

Section: B Moments Of Trigonometric Random Variablesmentioning

confidence: 99%

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Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures

Jasour¹,

Huang²,

Wang³

et al. 2021

Preprint

Self Cite

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Lie Algebraic Cost Function Design for Control on Lie Groups

Teng

Clark

Bloch

et al. 2022

2022 IEEE 61st Conference on Decision and Control (CDC)

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This paper reports a novel result: with proper robot models on matrix Lie groups, one can formulate the kinodynamic motion planning problem for rigid body systems as exact polynomial optimization problems that can be relaxed as semidefinite programming (SDP). Due to the nonlinear rigid body dynamics, the motion planning problem for rigid body systems is nonconvex. Existing global optimization-based methods do not properly deal with the configuration space of the 3D rigid body; thus, they do not scale well to long-horizon planning problems. We use Lie groups as the configuration space in our formulation and apply the variational integrator to formulate the forced rigid body systems as quadratic polynomials. Then we leverage Lasserre's hierarchy to obtain the globally optimal solution via SDP. By constructing the motion planning problem in a sparse manner, the results show that the proposed algorithm has linear complexity with respect to the planning horizon. This paper demonstrates the proposed method can provide rank-one optimal solutions at relaxation order two for most of the testing cases of 1) 3D drone landing using the full dynamics model and 2) inverse kinematics for serial manipulators.

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Non-Gaussian Risk Bounded Trajectory Optimization for Stochastic Nonlinear Systems in Uncertain Environments

Han¹,

Jasour²,

Williams³

2022

2022 International Conference on Robotics and Automation (ICRA)

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We consider the motion planning problem for stochastic nonlinear systems in uncertain environments. More precisely, in this problem the robot has stochastic nonlinear dynamics and uncertain initial locations, and the environment contains multiple dynamic uncertain obstacles. Obstacles can be of arbitrary shape, can deform, and can move. All uncertainties do not necessarily have Gaussian distribution. This general setting has been considered and solved in [1]. In addition to the assumptions above, in this paper, we consider long-term tasks, where the planning method in [1] would fail, as the uncertainty of the system states grows too large over a long time horizon. Unlike [1], we present a real-time online motion planning algorithm. We build discrete-time motion primitives and their corresponding continuous-time tubes offline, so that almost all system states of each motion primitive are guaranteed to stay inside the corresponding tube. We convert probabilistic safety constraints into a set of deterministic constraints called risk contours. During online execution, we verify the safety of the tubes against deterministic risk contours using sum-ofsquares (SOS) programming. The provided SOS-based method verifies the safety of the tube in the presence of uncertain obstacles without the need for uncertainty samples and time discretization in real-time. By bounding the probability the system states staying inside the tube and bounding the probability of the tube colliding with obstacles, our approach guarantees bounded probability of system states colliding with obstacles. We demonstrate our approach on several long-term robotics tasks.

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Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

Cited by 5 publications

References 43 publications

Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures

Fast nonlinear risk assessment for autonomous vehicles using learned conditional probabilistic models of agent futures

Lie Algebraic Cost Function Design for Control on Lie Groups

Non-Gaussian Risk Bounded Trajectory Optimization for Stochastic Nonlinear Systems in Uncertain Environments

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