114 research outputs found
Completeness of Randomized Kinodynamic Planners with State-based Steering
Probabilistic completeness is an important property in motion planning.
Although it has been established with clear assumptions for geometric planners,
the panorama of completeness results for kinodynamic planners is still
incomplete, as most existing proofs rely on strong assumptions that are
difficult, if not impossible, to verify on practical systems. In this paper, we
focus on an important class of kinodynamic planners, namely those that
interpolate trajectories in the state space. We provide a proof of
probabilistic completeness for these planners under assumptions that can be
readily verified from the system's equations of motion and the user-defined
interpolation function. Our proof relies crucially on a property of
interpolated trajectories, termed second-order continuity (SOC), which we show
is tightly related to the ability of a planner to benefit from denser sampling.
We analyze the impact of this property in simulations on a low-torque pendulum.
Our results show that a simple RRT using a second-order continuous
interpolation swiftly finds solution, while it is impossible for the same
planner using standard Bezier curves (which are not SOC) to find any solution.Comment: 21 pages, 5 figure
Probabilistic completeness of RRT for geometric and kinodynamic planning with forward propagation
The Rapidly-exploring Random Tree (RRT) algorithm has been one of the most
prevalent and popular motion-planning techniques for two decades now.
Surprisingly, in spite of its centrality, there has been an active debate under
which conditions RRT is probabilistically complete. We provide two new proofs
of probabilistic completeness (PC) of RRT with a reduced set of assumptions.
The first one for the purely geometric setting, where we only require that the
solution path has a certain clearance from the obstacles. For the kinodynamic
case with forward propagation of random controls and duration, we only consider
in addition mild Lipschitz-continuity conditions. These proofs fill a gap in
the study of RRT itself. They also lay sound foundations for a variety of more
recent and alternative sampling-based methods, whose PC property relies on that
of RRT
Joint Exploration of Local Metrics and Geometry in Sampling-based Planning
This thesis addresses how the local geometry of the workspace around a system state can be combined with local metrics describing system dynamics to improve the connectivity of the graph produced by a sampling-based planner over a fixed number of configurations. This development is achieved through generalization of the concept of the local free space to inner products other than the Euclidean inner product. This new structure allows for naturally combining the local free space construction with a locally applicable metric. The combination of the local free space with two specific metrics is explored in this work. The first metric is the quadratic cost-to-go function defined by a linear quadratic regulator, which captures the local behavior of the dynamical system. The second metric is the Mahalanobis distance for a belief state in a belief space planner. Belief space planners reason over distributions of states, called belief states, to include modeled uncertainty in the planning process. The Mahalanobis distances metric for a given belief state can be exploited to include notions of risk in local free space construction. Numerical simulations are provided to demonstrate the improved connectivity of the graph generated by a sampling-based planner using these concepts
Robust-RRT: Probabilistically-Complete Motion Planning for Uncertain Nonlinear Systems
Robust motion planning entails computing a global motion plan that is safe
under all possible uncertainty realizations, be it in the system dynamics, the
robot's initial position, or with respect to external disturbances. Current
approaches for robust motion planning either lack theoretical guarantees, or
make restrictive assumptions on the system dynamics and uncertainty
distributions. In this paper, we address these limitations by proposing the
robust rapidly-exploring random-tree (Robust-RRT) algorithm, which integrates
forward reachability analysis directly into sampling-based control trajectory
synthesis. We prove that Robust-RRT is probabilistically complete (PC) for
nonlinear Lipschitz continuous dynamical systems with bounded uncertainty. In
other words, Robust-RRT eventually finds a robust motion plan that is feasible
under all possible uncertainty realizations assuming such a plan exists. Our
analysis applies even to unstable systems that admit only short-horizon
feasible plans; this is because we explicitly consider the time evolution of
reachable sets along control trajectories. Thanks to the explicit consideration
of time dependency in our analysis, PC applies to unstabilizable systems. To
the best of our knowledge, this is the most general PC proof for robust
sampling-based motion planning, in terms of the types of uncertainties and
dynamical systems it can handle. Considering that an exact computation of
reachable sets can be computationally expensive for some dynamical systems, we
incorporate sampling-based reachability analysis into Robust-RRT and
demonstrate our robust planner on nonlinear, underactuated, and hybrid systems.Comment: 16 pages of main text + 5 pages of appendix, 5 figures, submitted to
the 2022 International Symposium on Robotics Researc
Collaborative Motion Planning
Planning motion is an essential component for any autonomous robotic system.
An intelligent agent must be able to efficiently plan collision-free paths in order to move through its world. Despite its importance, this problem is PSPACE-Hard which means that even planning motions for simple robots is computationally difficult.
State-of-the-art approaches trade completeness (always able to provide a solution if one exists or report none exists) for probabilistic completeness (probabilistically guaranteed to find a solution if one exists but cannot report if none exists) and improved efficiency. These methods use sampling-based techniques to design a sequence of motions for the robot. However, as these methods are random in nature, the probability of their success is directly related to the expansiveness, or openness, of the underlying planning space. In other words, narrow passages, complex systems, and various constraints make planning with these methods difficult. On the other hand, humans can often determine approximate solutions for these difficult solutions quickly.
In this research, we explore user-guided planning in which a human operator works together with a sampling-based motion planner. By having a human-in-the-loop, a human can steer a sampling-based planner towards a solution. This strategy can provide benefits to many applications such as computer-aided design and virtual prototyping, to name a few.
We begin by classifying and creating simple models of common user-guided and heuristic-guided motion planning methods. Our models encompass three forms of user input: configuration-based, path-based, and region-based input. We compare and contrast these approaches and motivate our choice of a region-based collaborative framework. Through this analysis, we gain insight into user-guided planning and further motivate methods that harness low interface complexity and work entirely in workspace, which is most natural to a human operator. Further, we extend the theory of expansiveness to analyze the various types of user inputs.
Our novel region-based collaboration framework takes advantage of human intuition by allowing a user to define regions in the workspace to bias and/or constrain the search space of a sampling-based motion planner. This approach allows a user to bias a high dimensional search with low dimensional input, supports intermittent user hints, and empowers a user to customize motion solutions. Finally, we extend region steering to both non-holonomic robotic systems and a human-inspired approach to motion planning.
Our results show that this region-based framework can aid many variants of sampling-based planning, reduce computation time, support solution customization, and can be used to develop advanced heuristic methods for solving motion planning problems. We provide experiments exemplifying our approach in planning motions for complex robotic applications such as mobile manipulators, car-like, and free-flying robots
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