We present Kinodynamic RRT*, an incremental sampling-based approach for asymptotically optimal motion planning for robots with linear differential constraints. Our approach extends RRT*, which was introduced for holonomic robots ( 2011), by using a fixed-final-state-free-final-time controller that exactly and optimally connects any pair of states, where the cost function is expressed as a trade-off between the duration of a trajectory and the expended control effort. Our approach generalizes earlier work on extending RRT* to kinodynamic systems, as it guarantees asymptotic optimality for any system with controllable linear dynamics, in state spaces of any dimension. Our approach can be applied to non-linear dynamics as well by using their first-order Taylor approximations. In addition, we show that for the rich subclass of systems with a nilpotent dynamics matrix, closed-form solutions for optimal trajectories can be derived, which keeps the computational overhead of our algorithm compared to traditional RRT* at a minimum. We demonstrate the potential of our approach by computing asymptotically optimal trajectories in three challenging motion planning scenarios: (i) a planar robot with a 4-D state space and double integrator dynamics, (ii) an aerial vehicle with a 10-D state space and linearized quadrotor dynamics, and (iii) a car-like robot with a 5-D state space and non-linear dynamics.
The term kinodynamic planning was first introduced in 1993 in @cite_7 , which presented a resolution-complete algorithm for optimal planning of robots with discretized double integrator dynamics in low-dimensional workspaces. Kinodynamic planning has since been an active area of research. Incremental sampling-based algorithms, in particular the rapidly-exploring random tree (RRT) approach @cite_6 , proved to be effective in state spaces of high dimensionality, and is applicable to general dynamics systems as it builds a random tree of trajectories, and complex dynamics can be forward integrated to expand the tree. Unfortunately, RRT does not produce optimal trajectories. In fact, the probability that it finds an optimal path is zero @cite_18 . Recently, RRT* was introduced to overcome this problem and guarantees asymptotic optimality @cite_5 ; it iteratively builds a tree of trajectories through the state space whose probability of containing an optimal solution approaches 1 as the number of iterations of the algorithm approaches infinity. However, RRT* requires for its step critical to achieving asymptotic optimality that any pair of states can be connected. Therefore, RRT* was introduced for holonomic systems, where any pair of states can be optimally connected by a straight-line trajectory through the state space.
@cite_5 @cite_18 @cite_6 @cite_7
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We present Kinodynamic RRT*, an incremental sampling-based approach for asymptotically optimal motion planning for robots with linear dynamics. Our approach extends RRT*, which was introduced for holonomic robots [10], by using a fixed-final-state-free-final-time controller that optimally connects any pair of states, where the cost function is expressed as a trade-off between the duration of a trajectory and the expended control effort. Our approach generalizes earlier work on RRT* for kinodynamic systems, as it guarantees asymptotic optimality for any system with controllable linear dynamics, in state spaces of any dimension. In addition, we show that for the rich subclass of systems with a nilpotent dynamics matrix, closed-form solutions for optimal trajectories can be derived, which keeps the computational overhead of our algorithm compared to traditional RRT* at a minimum. We demonstrate the potential of our approach by computing asymptotically optimal trajectories in three challenging motion planning scenarios: (i) a planar robot with a 4-D state space and double integrator dynamics, (ii) an aerial vehicle with a 10-D state space and linearized quadrotor dynamics, and (iii) a car-like robot with a 5-D state space and non-linear dynamics. Kinodynamic RRT planners are considered to be general tools for effectively finding feasible trajectories for high-dimensional dynamical systems. However, they struggle when holonomic constraints are present in the system, such as those arising in parallel manipulators, in robots that cooperate to fulfill a given task, or in situations involving contacts with the environment. In such cases, the state space becomes an implicitly-defined manifold, which makes the diffusion heuristic inefficient and leads to inaccurate dynamical simulations. To address these issues, this paper presents an extension of the kinodynamic RRT planner that constructs an atlas of the state-space manifold incrementally, and uses this atlas both to generate random states and to dynamically steer the system towards such states. To the best of our knowledge, this is the first randomized kinodynamic planner that explicitly takes holonomic constraints into account. We validate the approach in significantly-complex systems. Kinodynamic planning algorithms like Rapidly-Exploring Randomized Trees (RRTs) hold the promise of finding feasible trajectories for rich dynamical systems with complex, nonconvex constraints. In practice, these algorithms perform very well on configuration space planning, but struggle to grow efficiently in systems with dynamics or differential constraints. This is due in part to the fact that the conventional distance metric, Euclidean distance, does not take into account system dynamics and constraints when identifying which node in the existing tree is capable of producing children closest to a given point in state space. We show that an affine quadratic regulator (AQR) design can be used to approximate the exact minimum-time distance pseudometric at a reasonable computational cost. We demonstrate improved exploration of the state spaces of the double integrator and simple pendulum when using this pseudometric within the RRT framework, but this improvement drops off as systems' nonlinearity and complexity increase. Future work includes exploring methods for approximating the exact minimum-time distance pseudometric that can reason about dynamics with higher-order terms. In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm FMT*. The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a 'lazy' dynamic programming recursion on a predetermined number of probabilistically drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As such, this algorithm combines features of both single-query algorithms chiefly RRT and multiple-query algorithms chiefly PRM, and is reminiscent of the Fast Marching Method for the solution of Eikonal equations. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds-the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order On −1 d+? , where n is the number of sampled points, d is the dimension of the configuration space, and ? is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.