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.
Optimal motion planning has been a topic of renewed activity in robotics largely due to the advent of sampling-based motion planners that are proven to be asymptotically optimal @cite_1 . But the community has had a long history of interest in optimal motions. Numerical trajectory optimization techniques @cite_20 @cite_13 @cite_3 @cite_14 have been long studied, but have several drawbacks. First, they are prone to falling into local minima, and second they typically require differentiable constraint and cost representations, which are often hard to produce for complex obstacles. Grid-based planners @cite_6 @cite_5 @cite_9 are often fast in low dimensional spaces but suffer from the curse of dimensionality,'' with performance degrading rapidly in spaces of higher dimension @cite_18 . Sampling-based planners were originally developed to overcome many of these challenges, and have been shown to have excellent empirical performance in finding feasible paths in high-dimensional spaces, both without @cite_17 and with dynamic constraints @cite_22 @cite_25 . However, they tend to produce jerky paths that are far from optimal. Some hybrid approaches have combined sampling-based planning with local optimization to produce better paths @cite_4 @cite_0 .
@cite_18 @cite_14 @cite_4 @cite_22 @cite_9 @cite_1 @cite_3 @cite_6 @cite_0 @cite_5 @cite_13 @cite_25 @cite_20 @cite_17
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Finding asymptotically-optimal paths in multi-robot motion planning problems could be achieved, in principle, using sampling-based planners in the composite configuration space of all of the robots in the space. The dimensionality of this space increases with the number of robots, rendering this approach impractical. This work focuses on a scalable sampling-based planner for coupled multi-robot problems that provides asymptotic optimality. It extends the dRRT approach, which proposed building roadmaps for each robot and searching an implicit roadmap in the composite configuration space. This work presents a new method, dRRT* , and develops theory for scalable convergence to optimal paths in multi-robot problems. Simulated experiments indicate dRRT* converges to high-quality paths while scaling to higher numbers of robots where the naive approach fails. Furthermore, dRRT* is applicable to high-dimensional problems, such as planning for robot manipulators As sampling-based motion planners become faster, they can be reexecuted more frequently by a robot during task execution to react to uncertainty in robot motion, obstacle motion, sensing noise, and uncertainty in the robot's kinematic model. We investigate and analyze high-frequency replanning (HFR) where, during each period, fast sampling-based motion planners are executed in parallel as the robot simultaneously executes the first action of the best motion plan from the previous period. We consider discrete-time systems with stochastic nonlinear (but linearizable) dynamics and observation models with noise drawn from zero mean Gaussian distributions. The objective is to maximize the probability of success (i.e., avoid collision with obstacles and reach the goal) or to minimize path length subject to a lower bound on the probability of success. We show that, as parallel computation power increases, HFR offers asymptotic optimality for these objectives during each period for goal-oriented problems. We then demonstrate the effectiveness of HFR for holonomic and nonholonomic robots including car-like vehicles and steerable medical needles. Existing high-dimensional motion planning algorithms are simultaneously overpowered and underpowered. In domains sparsely populated by obstacles, the heuristics used by sampling-based planners to navigate “narrow passages” can be needlessly complex; furthermore, additional post-processing is required to remove the jerky or extraneous motions from the paths that such planners generate. In this paper, we present CHOMP, a novel method for continuous path refinement that uses covariant gradient techniques to improve the quality of sampled trajectories. Our optimization technique both optimizes higher-order dynamics and is able to converge over a wider range of input paths relative to previous path optimization strategies. In particular, we relax the collision-free feasibility prerequisite on input paths required by those strategies. As a result, CHOMP can be used as a standalone motion planner in many real-world planning queries. We demonstrate the effectiveness of our proposed method in manipulation planning for a 6-DOF robotic arm as well as in trajectory generation for a walking quadruped robot. We study the sparsity and optimality properties of crowd navigation and find that existing techniques do not satisfy both criteria simultaneously: either they achieve optimality with a prohibitive number of samples or tractability assumptions make them fragile to catastrophe. For example, if the human and robot are modeled independently, then tractability is attained but the planner is prone to overcautious or overaggressive behavior. For sampling based motion planning of joint human-robot cost functions, for @math agents and @math step lookahead, @math samples are needed for coverage of the action space. Advanced approaches statically partition the action space into free-space and then sample in those convex regions. However, if the human is into free-space, then the partition is misleading and sampling is unsafe: free space will soon be occupied. We diagnose the cause of these deficiencies---optimization happens over space---and propose a novel solution: optimize over trajectory space by using a Gaussian process (GP) basis. We exploit the "kernel trick" of GPs, where a continuum of trajectories are captured with a mean and covariance function. By using the mean and covariance as proxies for a trajectory family we reason about collective trajectory behavior without resorting to sampling. The GP basis is sparse and optimal with respect to collision avoidance and robot and crowd intention and flexibility. GP sparsity leans heavily on the insight that joint action space decomposes into free regions; however, the decomposition contains feasible solutions only if the partition is dynamically generated. We call our approach .