text: This paper presents an equivalence between feasible kinodynamic planning and optimal kinodynamic planning, in that any optimal planning problem can be transformed into a series of feasible planning problems in a state-cost space whose solutions approach the optimum. This transformation gives rise to a meta-algorithm that produces an asymptotically optimal planner, given any feasible kinodynamic planner as a subroutine. The meta-algorithm is proven to be asymptotically optimal, and a formula is derived relating expected running time and solution suboptimality. It is directly applicable to a wide range of optimal planning problems because it does not resort to the use of steering functions or numerical boundary-value problem solvers. On a set of benchmark problems, it is demonstrated to perform, using the EST and RRT algorithms as subroutines, at a superior or comparable level to related planners.
  @cite_7: 
  @cite_8: The RRT* algorithm has recently been proposed as an optimal extension to the standard RRT algorithm [1]. However, like RRT, RRT* is difficult to apply in problems with complicated or underactuated dynamics because it requires the design of a two domain-specific extension heuristics: a distance metric and node extension method. We propose automatically deriving these two heuristics for RRT* by locally linearizing the domain dynamics and applying linear quadratic regulation (LQR). The resulting algorithm, LQR-RRT*, finds optimal plans in domains with complex or underactuated dynamics without requiring domain-specific design choices. We demonstrate its application in domains that are successively torque-limited, underactuated, and in belief space.
  @cite_21: Incremental sampling-based motion planning algorithms such as the Rapidly-exploring Random Trees (RRTs) have been successful in efficiently solving computationally challenging motion planning problems involving complex dynamical systems. A recently proposed algorithm, called the RRT*, also provides asymptotic optimality guarantees, i.e., almost-sure convergence to optimal trajectories (which the RRT algorithm lacked) while maintaining the computational efficiency of the RRT algorithm. In this paper, time-optimal maneuvers for a high-speed off-road vehicle taking tight turns on a loose surface are studied using the RRT* algorithm. Our simulation results show that the aggressive skidding maneuver, usually called the trail-braking maneuver, naturally emerges from the RRT* algorithm as the minimum-time trajectory. Along the way, we extend the RRT* algorithm to handle complex dynamical systems, such as those that are described by nonlinear differential equations and involve high-dimensional state spaces, which may be of independent interest. We also exploit the RRT* as an anytime computation framework for nonlinear optimization problems.
  @cite_19: Some of the latest developments in motion planning methods have addressed the merging of optimal control with sampling-based approaches, to handle the problem of optimal kinodynamic motion planning for complex robot systems in cluttered environments. These include embedding the Linear Quadratic Regulator method in an RRT* context, or solving the kinematic problem with an RRT algorithm first and then feeding the solution to an NLP solver. An alternative approach is presented here, in which NLP is embedded in an RRT* context from the start. The resulting methodological features are illustrated with numerical examples. These include problems in which differential constraints play a fundamental role.
  @cite_15: Sampling-based algorithms such as the Rapidly-exploring Random Tree (RRT) have been recently proposed as an effective approach to computationally hard motion planning problem. However, while the RRT algorithm is known to be able to find a feasible solution quickly, there are no guarantees on the quality of such solution, e.g., with respect to a given cost functional. To address this limitation, the authors recently proposed a new algorithm, called RRT*, which ensures asymptotic optimality, i.e., almost sure convergence of the solution returned by the algorithm to an optimal solution, while maintaining the same properties of the standard RRT algorithm, both in terms of computation of feasible solutions, and of computational complexity. In this paper, the RRT* algorithm is extended to deal with differential constraints. A sufficient condition for asymptotic optimality is provided. It is shown that the RRT* algorithm equipped with any local steering procedure that satisfies this condition converges to an optimal solution almost surely. In particular, simple local steering procedures are provided for a Dubins' vehicle as well as a double integrator. Simulation examples are also provided for these systems comparing the RRT and the RRT* algorithms.
  @cite_11: Sampling-based motion planning algorithms, such as the Probabilistic RoadMap (PRM) and the Rapidly-exploring Random Tree (RRT), have received a large and growing amount of attention during the past decade. Most recently, sampling-based algorithms, such as the PRM* and RRT*, that guarantee asymptotic optimality, i.e., almost-sure convergence towards optimal solutions, have been proposed. Despite the experimental success of asymptotically-optimal sampling-based algorithms, their extensions to handle complex non-holonomic dynamical systems remains largely an open problem. In this paper, with the help of results from differential geometry, we extend the RRT* algorithm to handle a large class of non-holonomic dynamical systems. We demonstrate the performance of the algorithm in computational experiments involving the Dubins' car dynamics.
summary: Several authors have extended RRT* to dynamically-constrained systems. It is relatively easy to apply RRT* to dynamically-constrained systems if a steering function is available @cite_15 . Proving convergence is harder, requiring analysis of small-time controllability conditions @cite_11 . Other authors have extended RRT* to systems whose dynamics and costs are (or can be approximated) by linear and quadratic functions, respectively, by definition of a suitable steering function based on the LQR principle @cite_8 @cite_7 . When more complex differential constraints are involved, it may not be possible to devise a suitable steering functions. One method generated complex maneuvers using RRT* and performed each rewiring step by numerically solving a two-point boundary value problems @cite_21 . This adds greatly to computational expense. A similar method performed rewiring using a spline-based trajectory representation that is optimized via a nonlinear program solver @cite_19 .