Motion planning under uncertainty for an autonomous system can be formulated as a Markov Decision Process. In this paper, we propose a solution to this decision theoretic planning problem using a continuous approximation of the underlying discrete value function and leveraging finite element methods. This approach allows us to obtain an accurate and continuous form of value function even with a small number of states from a very low resolution of state space. We achieve this by taking advantage of the second order Taylor expansion to approximate the value function, where the value function is modeled as a boundary-conditioned partial differential equation which can be naturally solved using a finite element method. We have validated our approach via extensive simulations, and the evaluations reveal that our solution provides continuous value functions, leading to better path results in terms of path smoothness, travel distance and time costs, even with a smaller state space.
To extend the discrete-time actions to be continuous, the Semi-Markov Decision Processes (SMDP) @cite_10 @cite_27 have been designed based on the temporal abstraction concept so that actions can be performed in a more flexible manner. However, without relying on function approximation, the basic form of state space is still assumed to be discrete and tabular. Though the framework of continuous stochastic control can provide continuous value function and continuous actions, it requires Brownian motion driven dynamics @cite_16 . Nevertheless, the Hamilton-Jacobi-Bellman equation for infinite horizon stochastic control shares a similar diffusion-type of equations with our approaches.
@cite_27 @cite_16 @cite_10
2171212676 2169997910 2109910161 2399169217
This paper focuses on a continuous-time, continuous-space formulation of the stochastic optimal control problem with nonlinear dynamics and observation noise. We lay the mathematical foundations to construct, via incremental sampling, an approximating sequence of discrete-time finite-state partially observable Markov decision processes (POMDPs), such that the behavior of successive approximations converges to the behavior of the original continuous system in an appropriate sense. We also show that the optimal cost function and control policies for these POMDP approximations converge almost surely to their counterparts for the underlying continuous system in the limit. We demonstrate this approach on two popular continuous-time problems, viz., the Linear-Quadratic-Gaussian (LQG) control problem and the light-dark domain problem. Although many real-world stochastic planning problems are more naturally formulated by hybrid models with both discrete and continuous variables, current state-of-the-art methods cannot adequately address these problems. We present the first framework that can exploit problem structure for modeling and solving hybrid problems efficiently. We formulate these problems as hybrid Markov decision processes (MDPs with continuous and discrete state and action variables), which we assume can be represented in a factored way using a hybrid dynamic Bayesian network (hybrid DBN). This formulation also allows us to apply our methods to collaborative multiagent settings. We present a new linear program approximation method that exploits the structure of the hybrid MDP and lets us compute approximate value functions more efficiently. In particular, we describe a new factored discretization of continuous variables that avoids the exponential blow-up of traditional approaches. We provide theoretical bounds on the quality of such an approximation and on its scale-up potential. We support our theoretical arguments with experiments on a set of control problems with up to 28-dimensional continuous state space and 22-dimensional action space. Learning, planning, and representing knowledge at multiple levels of temporal ab- straction are key, longstanding challenges for AI. In this paper we consider how these challenges can be addressed within the mathematical framework of reinforce- ment learning and Markov decision processes (MDPs). We extend the usual notion of action in this framework to include options—closed-loop policies for taking ac- tion over a period of time. Examples of options include picking up an object, going to lunch, and traveling to a distant city, as well as primitive actions such as mus- cle twitches and joint torques. Overall, we show that options enable temporally abstract knowledge and action to be included in the reinforcement learning frame- work in a natural and general way. In particular, we show that options may be used interchangeably with primitive actions in planning methods such as dynamic pro- gramming and in learning methods such as Q-learning. Formally, a set of options defined over an MDP constitutes a semi-Markov decision process (SMDP), and the theory of SMDPs provides the foundation for the theory of options. However, the most interesting issues concern the interplay between the underlying MDP and the SMDP and are thus beyond SMDP theory. We present results for three such cases: 1) we show that the results of planning with options can be used during execution to interrupt options and thereby perform even better than planned, 2) we introduce new intra-option methods that are able to learn about an option from fragments of its execution, and 3) we propose a notion of subgoal that can be used to improve the options themselves. All of these results have precursors in the existing literature; the contribution of this paper is to establish them in a simpler and more general setting with fewer changes to the existing reinforcement learning framework. In particular, we show that these results can be obtained without committing to (or ruling out) any particular approach to state abstraction, hierarchy, function approximation, or the macro-utility problem. We introduce a highly efficient method for solving continuous partially-observable Markov decision processes (POMDPs) in which beliefs can be modeled using Gaussian distributions over the state space. Our method enables fast solutions to sequential decision making under uncertainty for a variety of problems involving noisy or incomplete observations and stochastic actions. We present an efficient approach to compute locally-valid approximations to the value function over continuous spaces in time polynomial (O[n4]) in the dimension n of the state space. To directly tackle the intractability of solving general POMDPs, we leverage the assumption that beliefs are Gaussian distributions over the state space, approximate the belief update using an extended Kalman filter (EKF), and represent the value function by a function that is quadratic in the mean and linear in the variance of the belief. Our approach iterates towards a linear control policy over the state space that is locally-optimal with respect to a user defined cost function, and is approximately valid in the vicinity of a nominal trajectory through belief space. We demonstrate the scalability and potential of our approach on problems inspired by robot navigation under uncertainty for state spaces of up to 128 dimensions.