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Distributionally robust risk map for learning-based motion planning and control: A semidefinite programming approach.
TL;DR: In this paper, the authors proposed a distributionally robust risk map (DR-risk map) for a mobile robot operating in a learning-enabled environment to reliably assess the conditional value-at-risk (CVaR) of collision with obstacles whose movements are inferred by Gaussian process regression (GPR).
Abstract: This paper proposes a novel safety specification tool, called the distributionally robust risk map (DR-risk map), for a mobile robot operating in a learning-enabled environment. Given the robot's position, the map aims to reliably assess the conditional value-at-risk (CVaR) of collision with obstacles whose movements are inferred by Gaussian process regression (GPR). Unfortunately, the inferred distribution is subject to errors, making it difficult to accurately evaluate the CVaR of collision. To overcome this challenge, this tool measures the risk under the worst-case distribution in a so-called ambiguity set that characterizes allowable distribution errors. To resolve the infinite-dimensionality issue inherent in the construction of the DR-risk map, we derive a tractable semidefinite programming formulation that provides an upper bound of the risk, exploiting techniques from modern distributionally robust optimization. As a concrete application for motion planning, a distributionally robust RRT* algorithm is considered using the risk map that addresses distribution errors caused by GPR. Furthermore, a motion control method is devised using the DR-risk map in a learning-based model predictive control (MPC) formulation. In particular, a neural network approximation of the risk map is proposed to reduce the computational cost in solving the MPC problem. The performance and utility of the proposed risk map are demonstrated through simulation studies that show its ability to ensure the safety of mobile robots despite learning errors.
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01 Jan 2022
TL;DR: In this paper, the authors introduce a neural network architecture for robustly learning the distribution of traversability costs and show that this approach reliably learns the expected tail risk given a desired probability risk threshold between 0 and 1, producing a traversability costmap which is more robust to outliers, more accurately captures tail risks, and is more computationally efficient.
Abstract: One of the main challenges in autonomous robotic exploration and navigation in unknown and unstructured environments is determining where the robot can or cannot safely move. A significant source of difficulty in this determination arises from stochasticity and uncertainty, coming from localization error, sensor sparsity and noise, difficult-to-model robot-ground interactions, and disturbances to the motion of the vehicle. Classical approaches to this problem rely on geometric analysis of the surrounding terrain, which can be prone to modeling errors and can be computationally expensive. Moreover, modeling the distribution of uncertain traversability costs is a difficult task, compounded by the various error sources mentioned above. In this work, we take a principled learning approach to this problem. We introduce a neural network architecture for robustly learning the distribution of traversability costs. Because we are motivated by preserving the life of the robot, we tackle this learning problem from the perspective of learning tail-risks, i.e. the conditional value-at-risk (CVaR). We show that this approach reliably learns the expected tail risk given a desired probability risk threshold between 0 and 1, producing a traversability costmap which is more robust to outliers, more accurately captures tail risks, and is more computationally efficient, when compared against baselines. We validate our method on data collected by a legged robot navigating challenging, unstructured environments including an abandoned subway, limestone caves, and lava tube caves.
13 citations
Posted Content•
TL;DR: In this paper, the authors considered a risk-constrained infinite-horizon optimal control problem and proposed to solve it in an iterative manner, where the safe states and samples are accumulated in previous iterations.
Abstract: This paper considers a risk-constrained infinite-horizon optimal control problem and proposes to solve it in an iterative manner. Each iteration of the algorithm generates a trajectory from the starting point to the target equilibrium state by implementing a distributionally robust risk-constrained model predictive control (MPC) scheme. At each iteration, a set of safe states (that satisfy the risk-constraint with high probability) and a certain number of independent and identically distributed samples of the uncertainty governing the risk constraint are available. These states and samples are accumulated in previous iterations. The safe states are used as terminal constraint in the MPC scheme and samples are used to construct a set of distributions, termed ambiguity set, such that it contains the underlying distribution of the uncertainty with high probability. The risk-constraint in each iteration is required to hold for all distributions in the ambiguity set. We establish that the trajectories generated by our iterative procedure are feasible, safe, and converge asymptotically to the equilibrium. Simulation example illustrates our results for the case of finding a risk-constrained path for a mobile robot in the presence of an uncertain obstacle.
3 citations
Posted Content•
TL;DR: In this paper, the authors introduce a neural network architecture for robustly learning the distribution of traversability costs and show that this approach reliably learns the expected tail risk given a desired probability risk threshold between 0 and 1.
Abstract: One of the main challenges in autonomous robotic exploration and navigation in unknown and unstructured environments is determining where the robot can or cannot safely move. A significant source of difficulty in this determination arises from stochasticity and uncertainty, coming from localization error, sensor sparsity and noise, difficult-to-model robot-ground interactions, and disturbances to the motion of the vehicle. Classical approaches to this problem rely on geometric analysis of the surrounding terrain, which can be prone to modeling errors and can be computationally expensive. Moreover, modeling the distribution of uncertain traversability costs is a difficult task, compounded by the various error sources mentioned above. In this work, we take a principled learning approach to this problem. We introduce a neural network architecture for robustly learning the distribution of traversability costs. Because we are motivated by preserving the life of the robot, we tackle this learning problem from the perspective of learning tail-risks, i.e. the Conditional Value-at-Risk (CVaR). We show that this approach reliably learns the expected tail risk given a desired probability risk threshold between 0 and 1, producing a traversability costmap which is more robust to outliers, more accurately captures tail risks, and is more computationally efficient, when compared against baselines. We validate our method on data collected a legged robot navigating challenging, unstructured environments including an abandoned subway, limestone caves, and lava tube caves.
1 citations
References
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Book•
23 Nov 2005TL;DR: The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics, and deals with the supervised learning problem for both regression and classification.
Abstract: A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.
11,357 citations
TL;DR: In this paper, the authors present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties "coherent", and demonstrate the universality of scenario-based methods for providing coherent measures.
Abstract: In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties “coherent.” We examine the measures of risk provided and the related actions required by SPAN, by the SEC=NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.
8,651 citations
TL;DR: A comprehensive description of the primal-dual interior-point algorithm with a filter line-search method for nonlinear programming is provided, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix.
Abstract: We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.
7,966 citations
TL;DR: This paper describes how to work with SeDuMi, an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints by exploiting sparsity.
Abstract: SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.
7,655 citations
TL;DR: In this paper, the authors studied the asymptotic behavior of the cost of the solution returned by stochastic sampling-based path planning algorithms as the number of samples increases.
Abstract: During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.
3,438 citations