arXiv subjects

Riccardo Bonalli

Publications and source records attributed to Riccardo Bonalli.

35 records · Page 2Linked to original sources

arXiv2023

First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems

In this paper, we derive first-order Pontryagin optimality conditions for risk-averse stochastic optimal control problems subject to final time inequality constraints, and whose costs are general, possibly non-smooth finite coherent risk measures. Unlike preexisting contributions covering this situation, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. In addition, it presents the advantages of neither involving second-order adjoint equations, nor leading to the so-called weak version of the PMP, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian.

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arXiv2023

Statistical Linearization for Robust Motion Planning

The goal of robust motion planning consists of designing open-loop controls which optimally steer a system to a specific target region while mitigating uncertainties and disturbances which affect the dynamics. Recently, stochastic optimal control has enabled particularly accurate formulations of the problem. Nevertheless, despite interesting progresses, these problem formulations still require expensive numerical computations. In this paper, we start bridging this gap by leveraging statistical linearization. Specifically, through statistical linearization we reformulate the robust motion planning problem as a simpler deterministic optimal control problem subject to additional constraints. We rigorously justify our method by providing estimates of the approximation error, as well as some controllability results for the new constrained deterministic formulation. Finally, we apply our method to the powered descent of a space vehicle, showcasing the consistency and efficiency of our approach through numerical experiments.

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arXiv2022

Sequential Convex Programming For Non-Linear Stochastic Optimal Control

This work introduces a sequential convex programming framework for non-linear, finite-dimensional stochastic optimal control, where uncertainties are modeled by a multidimensional Wiener process. We prove that any accumulation point of the sequence of iterates generated by sequential convex programming is a candidate locally-optimal solution for the original problem in the sense of the stochastic Pontryagin Maximum Principle. Moreover, we provide sufficient conditions for the existence of at least one such accumulation point. We then leverage these properties to design a practical numerical method for solving non-linear stochastic optimal control problems based on a deterministic transcription of stochastic sequential convex programming.

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arXiv2022

Using Spectral Submanifolds for Nonlinear Periodic Control

Very high dimensional nonlinear systems arise in many engineering problems due to semi-discretization of the governing partial differential equations, e.g. through finite element methods. The complexity of these systems present computational challenges for direct application to automatic control. While model reduction has seen ubiquitous applications in control, the use of nonlinear model reduction methods in this setting remains difficult. The problem lies in preserving the structure of the nonlinear dynamics in the reduced order model for high-fidelity control. In this work, we leverage recent advances in Spectral Submanifold (SSM) theory to enable model reduction under well-defined assumptions for the purpose of efficiently synthesizing feedback controllers.

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arXiv2022

Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control

Sequential Convex Programming (SCP) has recently gained significant popularity as an effective method for solving optimal control problems and has been successfully applied in several different domains. However, the theoretical analysis of SCP has received comparatively limited attention, and it is often restricted to discrete-time formulations. In this paper, we present a unifying theoretical analysis of a fairly general class of SCP procedures for continuous-time optimal control problems. In addition to the derivation of convergence guarantees in a continuous-time setting, our analysis reveals two new numerical and practical insights. First, we show how one can more easily account for manifold-type constraints, which are a defining feature of optimal control of mechanical systems. Second, we show how our theoretical analysis can be leveraged to accelerate SCP-based optimal control methods by infusing techniques from indirect optimal control.

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arXiv2022

Risk-sensitive safety analysis using Conditional Value-at-Risk

This paper develops a safety analysis method for stochastic systems that is sensitive to the possibility and severity of rare harmful outcomes. We define risk-sensitive safe sets as sub-level sets of the solution to a non-standard optimal control problem, where a random maximum cost is assessed via Conditional Value-at-Risk (CVaR). The objective function represents the maximum extent of constraint violation of the state trajectory, averaged over a given percentage of worst cases. This problem is well-motivated but difficult to solve tractably because the temporal decomposition for CVaR is history-dependent. Our primary theoretical contribution is to derive computationally tractable under-approximations to risk-sensitive safe sets. Our method provides a novel, theoretically guaranteed, parameter-dependent upper bound to the CVaR of a maximum cost without the need to augment the state space. For a fixed parameter value, the solution to only one Markov decision process problem is required to obtain the under-approximations for any family of risk-sensitivity levels. In addition, we propose a second definition for risk-sensitive safe sets and provide a tractable method for their estimation without using a parameter-dependent upper bound. The second definition is expressed in terms of a new coherent risk functional, which is inspired by CVaR. We demonstrate our primary theoretical contribution via numerical examples.

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arXiv2022

A Simple and Efficient Sampling-based Algorithm for General Reachability Analysis

In this work, we analyze an efficient sampling-based algorithm for general-purpose reachability analysis, which remains a notoriously challenging problem with applications ranging from neural network verification to safety analysis of dynamical systems. By sampling inputs, evaluating their images in the true reachable set, and taking their $ε$-padded convex hull as a set estimator, this algorithm applies to general problem settings and is simple to implement. Our main contribution is the derivation of asymptotic and finite-sample accuracy guarantees using random set theory. This analysis informs algorithmic design to obtain an $ε$-close reachable set approximation with high probability, provides insights into which reachability problems are most challenging, and motivates safety-critical applications of the technique. On a neural network verification task, we show that this approach is more accurate and significantly faster than prior work. Informed by our analysis, we also design a robust model predictive controller that we demonstrate in hardware experiments.

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arXiv2021

Convex Optimization for Trajectory Generation

Reliable and efficient trajectory generation methods are a fundamental need for autonomous dynamical systems of tomorrow. The goal of this article is to provide a comprehensive tutorial of three major convex optimization-based trajectory generation methods: lossless convexification (LCvx), and two sequential convex programming algorithms known as SCvx and GuSTO. In this article, trajectory generation is the computation of a dynamically feasible state and control signal that satisfies a set of constraints while optimizing key mission objectives. The trajectory generation problem is almost always nonconvex, which typically means that it is not readily amenable to efficient and reliable solution onboard an autonomous vehicle. The three algorithms that we discuss use problem reformulation and a systematic algorithmic strategy to nonetheless solve nonconvex trajectory generation tasks through the use of a convex optimizer. The theoretical guarantees and computational speed offered by convex optimization have made the algorithms popular in both research and industry circles. To date, the list of applications includes rocket landing, spacecraft hypersonic reentry, spacecraft rendezvous and docking, aerial motion planning for fixed-wing and quadrotor vehicles, robot motion planning, and more. Among these applications are high-profile rocket flights conducted by organizations like NASA, Masten Space Systems, SpaceX, and Blue Origin. This article aims to give the reader the tools and understanding necessary to work with each algorithm, and to know what each method can and cannot do. A publicly available source code repository supports the provided numerical examples. By the end of the article, the reader should be ready to use the methods, to extend them, and to contribute to their many exciting modern applications.

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arXiv2021

Composable Geometric Motion Policies using Multi-Task Pullback Bundle Dynamical Systems

Despite decades of work in fast reactive planning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This work presents a principled method for designing and fusing desired robot task behaviors into a stable robot motion policy, leveraging the geometric structure of non-Euclidean manifolds, which are prevalent in robot configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS) framework drives desired task behaviors and prioritizes tasks using separate position-dependent and position/velocity-dependent Riemannian metrics, respectively, thus simplifying individual task design and modular composition of tasks. For enforcing constraints, we provide a class of metric-based tasks, eliminating local minima by imposing non-conflicting potential functions only for goal region attraction. We also provide a geometric optimization problem for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces to a simple least-squares problem, and we show that our approach is geometrically well-defined. We demonstrate the PBDS framework on the sphere $\mathbb S^2$ and at 300-500 Hz on a manipulator arm, and we provide task design guidance and an open-source Julia library implementation. Overall, this work presents a fast, easy-to-use framework for generating motion policies without unwanted potential function local minima on general manifolds.

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arXiv2020

Refined Analysis of Asymptotically-Optimal Kinodynamic Planning in the State-Cost Space

We present a novel analysis of AO-RRT: a tree-based planner for motion planning with kinodynamic constraints, originally described by Hauser and Zhou (AO-X, 2016). AO-RRT explores the state-cost space and has been shown to efficiently obtain high-quality solutions in practice without relying on the availability of a computationally-intensive two-point boundary-value solver. Our main contribution is an optimality proof for the single-tree version of the algorithm---a variant that was not analyzed before. Our proof only requires a mild and easily-verifiable set of assumptions on the problem and system: Lipschitz-continuity of the cost function and the dynamics. In particular, we prove that for any system satisfying these assumptions, any trajectory having a piecewise-constant control function and positive clearance from the obstacles can be approximated arbitrarily well by a trajectory found by AO-RRT. We also discuss practical aspects of AO-RRT and present experimental comparisons of variants of the algorithm.

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arXiv2019

Trajectory Optimization on Manifolds: A Theoretically-Guaranteed Embedded Sequential Convex Programming Approach

Sequential Convex Programming (SCP) has recently gained popularity as a tool for trajectory optimization due to its sound theoretical properties and practical performance. Yet, most SCP-based methods for trajectory optimization are restricted to Euclidean settings, which precludes their application to problem instances where one must reason about manifold-type constraints (that is, constraints, such as loop closure, which restrict the motion of a system to a subset of the ambient space). The aim of this paper is to fill this gap by extending SCP-based trajectory optimization methods to a manifold setting. The key insight is to leverage geometric embeddings to lift a manifold-constrained trajectory optimization problem into an equivalent problem defined over a space enjoying a Euclidean structure. This insight allows one to extend existing SCP methods to a manifold setting in a fairly natural way. In particular, we present a SCP algorithm for manifold problems with refined theoretical guarantees that resemble those derived for the Euclidean setting, and demonstrate its practical performance via numerical experiments.

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arXiv2019

Optimal Control of Endo-Atmospheric Launch Vehicle Systems: Geometric and Computational Issues

In this paper we develop a geometric analysis and a numerical algorithm, based on indirect methods, to solve optimal guidance of endo-atmospheric launch vehicle systems under mixed control-state constraints. Two main difficulties are addressed. First, we tackle the presence of Euler singularities by introducing a representation of the configuration manifold in appropriate local charts. In these local coordinates, not only the problem is free from Euler singularities but also it can be recast as an optimal control problem with only pure control constraints. The second issue concerns the initialization of the shooting method. We introduce a strategy which combines indirect methods with homotopies, thus providing high accuracy. We illustrate the efficiency of our approach by numerical simulations on missile interception problems under challenging scenarios.

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arXiv2019

GuSTO: Guaranteed Sequential Trajectory Optimization via Sequential Convex Programming

Sequential Convex Programming (SCP) has recently seen a surge of interest as a tool for trajectory optimization. However, most available methods lack rigorous performance guarantees and they are often tailored to specific optimal control setups. In this paper, we present GuSTO (Guaranteed Sequential Trajectory Optimization), an algorithmic framework to solve trajectory optimization problems for control-affine systems with drift. GuSTO generalizes earlier SCP-based methods for trajectory optimization (by addressing, for example, goal-set constraints and problems with either fixed or free final time) and enjoys theoretical convergence guarantees in terms of convergence to, at least, a stationary point. The theoretical analysis is further leveraged to devise an accelerated implementation of GuSTO, which originally infuses ideas from indirect optimal control into an SCP context. Numerical experiments on a variety of trajectory optimization setups show that GuSTO generally outperforms current state-of-the-art approaches in terms of success rates, solution quality, and computation times.

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arXiv2018

Continuity of Pontryagin extremals with respect to delays in nonlinear optimal control

Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. We also discuss the sharpness of our assumptions.

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arXiv2017

Solving nonlinear optimal control problems with state and control delays by shooting methods combined with numerical continuation on the delays

In this paper we introduce a new procedure to solve nonlinear optimal control problems with delays which exploits indirect methods combined with numerical homotopy procedures. It is known that solving this kind of problems via indirect methods (which arise from the Pontrya-gin Maximum Principle) is complex and computationally demanding because their implementation is faced to two main difficulties: the extremal equations involve forward and backward terms, and besides, the related shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, delays are introduced by a numerical continuation. This creates a sequence of optimal delayed solutions that converges to the desired solution. We establish a convergence theorem ensuring the continuous dependence w.r.t. the delay of the optimal state, of the optimal control (in a weak sense) and of the corresponding adjoint vector. The convergence of the adjoint vector represents the most challenging step to prove and it is crucial for the well-posedness of the proposed homotopy procedure. Two numerical examples are proposed and analyzed to show the efficiency of this approach.

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arXiv2017

Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation

- In this paper we introduce a new method to solve fixed-delay optimal control problems which exploits numerical homotopy procedures. It is known that solving this kind of problems via indirect methods is complex and computationally demanding because their implementation is faced with two difficulties: the extremal equations are of mixed type, and besides, the shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, the delay is introduced by numerical homotopy methods. Convergence results, which ensure the effectiveness of the whole procedure, are provided. The numerical efficiency is illustrated on an example.

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arXiv2017

Analytical Initialization of a Continuation-Based Indirect Method for Optimal Control of Endo-Atmospheric Launch Vehicle Systems

In this paper, we propose a strategy to solve endo-atmospheric launch vehicle optimal control problems using indirect methods. More specifically, we combine shooting methods with an adequate continuation algorithm, taking advantage of the knowledge of an analytical solution of a simpler problem. This procedure is resumed in two main steps. We first simplify the physical dynamics to obtain an analytical guidance law which is used as initial guess for a shooting method. Then, a continuation procedure makes the problem converge to the complete dynamics leading to the optimal solution of the original system. Numerical results are presented.

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