Nonlinear Uncertainty Propagation Research Articles

Stochastic optimal control for autonomous driving applications via polynomial chaos expansions

AbstractModel‐based methods in autonomous driving and advanced driving assistance gain importance in research and development due to their potential to contribute to higher road safety. Parameters of vehicle models, however, are hard to identify precisely or they can change quickly depending on the driving conditions. In this paper, we address the problem of safe trajectory planning under parametric model uncertainties motivated by automotive applications. We use the generalized polynomial chaos expansions for efficient nonlinear uncertainty propagation and distributionally robust inequalities for chance constraints approximation. Inspired by the tube‐based model predictive control, an ancillary feedback controller is used to control the deviations of stochastic modes from the nominal solution, and therefore, decrease the variance. Our approach allows reducing conservatism related to nonlinear uncertainty propagation while guaranteeing constraints satisfaction with a high probability. The performance is demonstrated on the example of a trajectory optimization problem for a simplified vehicle model with uncertain parameters.

Double-time-scale non-probabilistic reliability-based controller optimization for manipulator considering motion error and wear growth

Motion error and wear growth are two crucial factors that determine the performance of the manipulator system. The two factors are distinct from the sensitivity to time and respectively indicate its control performance and lifespan level. In this paper, a double-time-scale non-probabilistic reliability (DTSNPR)-based optimization method, which considers time-sensitive factor motion error as time-dependent reliability (TDR) and time-insensitive factor clearance wear growth as time-independent reliability (TIR), is proposed to comprehensively evaluate and optimize the controller of the manipulator system in consideration of these multi-scale factors. Meanwhile, for a highly nonlinear response problem of a manipulator system, the adaptive subinterval collocation method (ASICM) which transfers the highly nonlinear uncertainty propagation problem into a sequence of small subintervals, is adopted to obtain the response range more precisely. The proposed DTSNPR-based optimization method is applied to three numerical manipulator systems and ensures each of them under a predefined level of reliability upon both motion error and wear growth. The results also indicate that the proposed ASICM owns an advantage over computational cost and precision, with only 0.4% error and 1% computational cost compared with the Monte-Carlo methods.

Directional State Transition Tensors for Capturing Dominant Nonlinear Effects in Orbital Dynamics

Higher-order methods such as state transition tensors (STTs) have shown promise for a variety of applications in astrodynamics, but can be limited in their use due to their potentially large computational and storage requirements. In this paper we present a method for approximating STTs by aligning the STTs with particularly unstable directions. This strategy allows us to isolate the most important terms in the higher-order STTs and ignore any less important terms. This method, which we call directional state transition tensors (DSTTs), is applied to several examples in two- and three-body dynamic systems, and it is shown to perform well for both nonlinear state and state uncertainty propagation. The method produces similar results to the full STTs, but requires significantly fewer elements to be stored, which can result in up to improvements in speed for nonlinear uncertainty propagation computations. DSTTs are promising for expanding the use of STTs to situations with highly nonlinear dynamics but limited computational resources, such as on-board a spacecraft.

An adaptive Gaussian mixture method for nonlinear uncertainty propagation in neural networks

Using neural networks to address data-driven problems often entails dealing with uncertainties. However, the propagation of uncertainty through a network’s nonlinear layers is usually a bottleneck, since the existing techniques designed to transmit Gaussian distributions via moment estimation are not capable of predicting non-Gaussian distributions. In this study, a Gaussian-mixture-based uncertainty propagation scheme is proposed for neural networks. Given that any input uncertainty can be characterized as a Gaussian mixture with a finite number of components, the developed scheme actively examines each mixture component and adaptively split those whose fidelity in representing uncertainty is deteriorated by the network’s nonlinear activation layers. A Kullback–Leibler criterion that directly measures the nonlinearity-induced non-Gaussianity in post-activation distributions is derived to trigger splitting and a set of high-precision Gaussian splitting libraries is established. Four uncertainty propagation examples on dynamic systems and data-driven applications are demonstrated, in all of which the developed scheme exhibited exemplary fidelity and efficiency in predicting the evolution of non-Gaussian distributions through both recurrent and multi-layer neural networks.

Reduced Nonlinear Model for Orbit Uncertainty Propagation and Estimation

This paper presents a novel method for nonlinear uncertainty propagation and estimation in orbital dynamics. The proposed technique relies on a Taylor series expansion of the integral flow to model the dynamics around the reference solution and introduces an approximation of the high-order variational equations that reduces the complexity of evaluating the series. In particular, the high-order state-transition tensors (STTs) are approximated by capturing the dominant secular terms. Simple expressions to compute them are provided. The approximation stems from confining the Lyapunov instability of the motion to the time domain. The result is a time-explicit approximation of the STTs that can be used to predict the evolution of the uncertainty distribution accounting for nonlinear effects with minimal overhead. Finally, a high-order version of the extended Kalman filter is developed by implementing the approximation of the nonlinear terms of the Taylor series into an estimation scheme. The performance of the algorithm is evaluated with several practical examples.

Impact of Nuclear Data Uncertainty in the Modeling of Neutron-Induced Recoil Atom Energy Distributions in Silicon

Recoil ion distributions in silicon and the resulting distribution of the linear energy transfer (LET) are important metrics in microdosimetric studies and in the investigation of neutron-induced single-event effects. A rigorous methodology is presented for quantifying the uncertainty in these metrics due to the underlying uncertainty contributors, including that due to the nuclear data, recoil ion electronic stopping power, and incident neutron spectrum. The methodology uses a Monte Carlo-based approach so that the nonlinear uncertainty propagation is rigorously treated as the response function folded with the full incident neutron spectrum. The uncertainty is captured in the form of both recoil energy and LET-dependent covariance matrices. The uncertainty contributions from the nuclear data are shown to have strong energy-dependent correlations which are comparable in magnitude to that from the uncertainty found in the spectrum characterization for high fidelity reference neutron fields. The uncertainty from the stopping power has the largest magnitude of the largest contributors, but it shows a very strong energy-dependent correlation that translates into a systematic uncertainty that may cancel out in many applications.

Nonlinear orbital uncertainty propagation with differential algebra and Gaussian mixture model

Nonlinear uncertainty propagation is of critical importance in many application fields of astrodynamics. In this article, a framework combining the differential algebra technique and the Gaussian mixture model method is presented to accurately propagate the state uncertainty of a nonlinear system. A high-order Taylor expansion of the final state with respect to the initial deviations is firstly computed with the differential algebra technique. Then the initial uncertainty is split to a Gaussian mixture model. With the high-order state transition polynomial, each Gaussian mixture element is propagated to the final time, forming the final Gaussian mixture model. Through this framework, the final Gaussian mixture model can include the effects of high-order terms during propagation and capture the non-Gaussianity of the uncertainty, which enables a precise propagation of probability density. Moreover, the manual derivation and integration of the high-order variational equations is avoided, which makes the method versatile. The method can handle both the application of nonlinear analytical maps on any domain of interest and the propagation of initial uncertainties through the numerical integration of ordinary differential equation. The performance of the resulting tool is assessed on some typical orbital dynamic models, including the analytical Keplerian motion, the numerical J2 perturbed motion, and a nonlinear relative motion.

Probabilistic Algorithm for Ballistic Parachute Transition Altitude Optimization

A probabilistic algorithm for the optimization of drogue-to-main parachute transition altitude is proposed for high-altitude, low-opening ballistic airdrop. In light of the significant effects of wind and model uncertainty on impact-point dispersion, the algorithm makes use of explicit nonlinear uncertainty propagation techniques to provide a probabilistically optimal transition altitude. The algorithm begins by the specification of a desired impact-point distribution, which is propagated backward in time through the nonlinear main parachute dynamics using the stochastic Liouville equation. This spatially varying probability density is uploaded to the package before release from the aircraft. During drogue descent, the package descent trajectory through these probability densities is predicted, and an optimal drogue-to-main parachute transition altitude is identified. A key aspect of the proposed algorithm is that the desired impact-point distribution is an input to the algorithm, enabling novel capabilities to shape the resulting dispersion pattern and avoid obstacles in the target area. Furthermore, the algorithm explicitly accounts for primary sources of uncertainty affecting ballistic airdrop operations. Results demonstrate that the proposed optimization algorithm has the ability to reduce dispersion, improve accuracy, and shape impact distributions around obstacles and among multiple drop zones.

Nonlinear Uncertainty Propagation in Astrodynamics Using Differential Algebra and Graphics Processing Units

In this paper, two numerical methods for nonlinear uncertainty propagation in astrodynamics are presented and thoroughly compared. Both methods are based on the Monte Carlo idea of evaluating multiple samples of an initial statistical distribution around the nominal state. However, whereas the graphics processing unit implementation aims at increasing the performances of the classical Monte Carlo approach exploiting the massively parallel architecture of modern general-purpose computing on graphics processing units, the method based on differential algebra is aimed at the improvement and generalization of standard linear methods for uncertainty propagation. The two proposed numerical methods are applied to test cases considering both simple two-body dynamics and a full -body dynamics with accurate ephemeris. The results of the propagation are thoroughly compared with particular emphasis on both accuracy and computational performances.

Stochastic Model Predictive Control for Guided Projectiles Under Impact Area Constraints

The dynamics of guided projectile systems are inherently stochastic in nature. While deterministic control algorithms such as impact point prediction (IPP) may prove effective in many scenarios, the probability of impacting obstacles and constrained areas within an impact zone cannot be accounted for without accurate uncertainty modeling. A stochastic model predictive guidance algorithm is developed, which incorporates nonlinear uncertainty propagation to predict the impact probability density in real-time. Once the impact distribution is characterized, the guidance system aim point is computed as the solution to an optimization problem. The result is a guidance law that can achieve minimum miss distance while avoiding impact area constraints. Furthermore, the acceptable risk of obstacle impact can be quantified and tuned online. Example trajectories and Monte Carlo simulations demonstrate the effectiveness of the proposed stochastic control formulation in comparison to deterministic guidance schemes.

Nonlinear Mapping of Uncertainties in Celestial Mechanics

The problem of nonlinear uncertainty propagation represents a crucial issue in celestial mechanics. In this paper, a method for nonlinear propagation of uncertainty based on differential algebra is presented. Working in the differential algebra framework enables a general approach to nonlinear uncertainty propagation that can provide highly accurate estimate with low computational cost. The nonlinear mapping of the statistics is shown here, adopting the two-body problem as the working framework, including coordinate system transformations. The general feature of the proposed method is also demonstrated by presenting long-term integrations in complex dynamic systems, such as the -body problem or the simplified general perturbation model.