13 research outputs found
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
Refined Analysis of Asymptotically-Optimal Kinodynamic Planning in the State-Cost Space
We present a novel analysis of AO-RRT: a tree-based planner for motion
planning with kinodynamic constraints, originally described by Hauser and Zhou
(AO-X, 2016). AO-RRT explores the state-cost space and has been shown to
efficiently obtain high-quality solutions in practice without relying on the
availability of a computationally-intensive two-point boundary-value solver.
Our main contribution is an optimality proof for the single-tree version of the
algorithm---a variant that was not analyzed before. Our proof only requires a
mild and easily-verifiable set of assumptions on the problem and system:
Lipschitz-continuity of the cost function and the dynamics. In particular, we
prove that for any system satisfying these assumptions, any trajectory having a
piecewise-constant control function and positive clearance from the obstacles
can be approximated arbitrarily well by a trajectory found by AO-RRT. We also
discuss practical aspects of AO-RRT and present experimental comparisons of
variants of the algorithm
Scalable and Probabilistically Complete Planning for Robotic Spatial Extrusion
There is increasing demand for automated systems that can fabricate 3D
structures. Robotic spatial extrusion has become an attractive alternative to
traditional layer-based 3D printing due to a manipulator's flexibility to print
large, directionally-dependent structures. However, existing extrusion planning
algorithms require a substantial amount of human input, do not scale to large
instances, and lack theoretical guarantees. In this work, we present a rigorous
formalization of robotic spatial extrusion planning and provide several
efficient and probabilistically complete planning algorithms. The key planning
challenge is, throughout the printing process, satisfying both stiffness
constraints that limit the deformation of the structure and geometric
constraints that ensure the robot does not collide with the structure. We show
that, although these constraints often conflict with each other, a greedy
backward state-space search guided by a stiffness-aware heuristic is able to
successfully balance both constraints. We empirically compare our methods on a
benchmark of over 40 simulated extrusion problems. Finally, we apply our
approach to 3 real-world extrusion problems
Near-Optimal Multi-Robot Motion Planning with Finite Sampling
An underlying structure in several sampling-based methods for continuous
multi-robot motion planning (MRMP) is the tensor roadmap (TR), which emerges
from combining multiple PRM graphs constructed for the individual robots via a
tensor product. We study the conditions under which the TR encodes a
near-optimal solution for MRMP---satisfying these conditions implies near
optimality for a variety of popular planners, including dRRT*, and the discrete
methods M* and CBS when applied to the continuous domain. We develop the first
finite-sample analysis of this kind, which specifies the number of samples,
their deterministic distribution, and magnitude of the connection radii that
should be used by each individual PRM graph, to guarantee near-optimality using
the TR. This significantly improves upon a previous asymptotic analysis,
wherein the number of samples tends to infinity, and supports guaranteed
high-quality solutions in practice, within bounded running time. To achieve our
new result, we first develop a sampling scheme, which we call the staggered
grid, for finite-sample motion planning for individual robots, which requires
significantly less samples than previous work. We then extend it to the much
more involved MRMP setting which requires to account for interactions among
multiple robots. Finally, we report on a few experiments that serve as a
verification of our theoretical findings and raise interesting questions for
further investigation.Comment: Submitted to the International Conference on Robotics and Automation
(ICRA), 202
The Critical Radius in Sampling-based Motion Planning
We develop a new analysis of sampling-based motion planning in Euclidean
space with uniform random sampling, which significantly improves upon the
celebrated result of Karaman and Frazzoli (2011) and subsequent work.
Particularly, we prove the existence of a critical connection radius
proportional to for samples and dimensions:
Below this value the planner is guaranteed to fail (similarly shown by the
aforementioned work, ibid.). More importantly, for larger radius values the
planner is asymptotically (near-)optimal. Furthermore, our analysis yields an
explicit lower bound of on the probability of success. A
practical implication of our work is that asymptotic (near-)optimality is
achieved when each sample is connected to only neighbors. This is
in stark contrast to previous work which requires
connections, that are induced by a radius of order . Our analysis is not restricted to PRM and applies to a
variety of PRM-based planners, including RRG, FMT* and BTT. Continuum
percolation plays an important role in our proofs. Lastly, we develop similar
theory for all the aforementioned planners when constructed with deterministic
samples, which are then sparsified in a randomized fashion. We believe that
this new model, and its analysis, is interesting in its own right
A randomized kinodynamic planner for closed-chain robotic systems
Kinodynamic RRT planners are effective tools for finding feasible trajectories in many classes of robotic systems. However, they are hard to apply to systems with closed-kinematic chains, like parallel robots, cooperating arms manipulating an object, or legged robots keeping their feet in contact with the environ- ment. The state space of such systems is an implicitly-defined manifold, which complicates the design of the sampling and steering procedures, and leads to trajectories that drift away from the manifold when standard integration methods are used. To address these issues, this report presents a kinodynamic RRT planner that constructs an atlas of the state space incrementally, and uses this atlas to both generate ran- dom states, and to dynamically steer the system towards such states. The steering method is based on computing linear quadratic regulators from the atlas charts, which greatly increases the planner efficiency in comparison to the standard method that simulates random actions. The atlas also allows the integration of the equations of motion as a differential equation on the state space manifold, which eliminates any drift from such manifold and thus results in accurate trajectories. To the best of our knowledge, this is the first kinodynamic planner that explicitly takes closed kinematic chains into account. We illustrate the performance of the approach in significantly complex tasks, including planar and spatial robots that have to lift or throw a load at a given velocity using torque-limited actuators.Peer ReviewedPreprin