Creating Better Collision-Free Trajectory for Robot Motion Planning by Linearly Constrained Quadratic Programming

Many algorithms in probabilistic sampling-based motion planning have been proposed to create a path for a robot in an environment with obstacles. Due to the randomness of sampling, they can efficiently compute the collision-free paths made of segments lying in the configuration space with probabilistic completeness. However, this property also makes the trajectories have some unnecessary redundant or jerky motions, which need to be optimized. For most robotics applications, the trajectories should be short, smooth and keep away from obstacles. This paper proposes a new trajectory optimization technique which transforms a polygon collision-free path into a smooth path, and can deal with trajectories which contain various task constraints. The technique removes redundant motions by quadratic programming in the parameter space of trajectory, and converts collision avoidance conditions to linear constraints to ensure absolute safety of trajectories. Furthermore, the technique uses a projection operator to realize the optimization of trajectories which are subject to some hard kinematic constraints, like keeping a glass of water upright or coordinating operation with dual robots. The experimental results proved the feasibility and effectiveness of the proposed method, when it is compared with other trajectory optimization methods.

Finding a High-Quality Initial Solution for the RRTs Algorithms in 2D Environments

In this paper, we propose a bioinspired path planning algorithm for finding a high-quality initial solution based on the pipeline of the Rapidly exploring Random Tree (RRT) method by modifying the sampling process. The modification mainly includes controlling the sampling space and using the probabilistic sampling with the two-dimensional Gaussian mixture model. Inspired by the tropism of plants, we use a Gaussian mixture model to imitate the tree’s growth in nature. In a 2D environment, we can get an approximate moving point’s probabilistic distribution, and the initial path can be found much quickly guided by the probabilistic heuristic. At the same time, only a small number of nodes are generated, which can reduce the memory usage. As a meta-algorithm, it can be applicable to other RRT methods and the performance of underlying algorithm is improved dramatically. We also prove that the probabilistic completeness and the asymptotic optimality depend on the original algorithm (other RRTs). We demonstrate the application of our algorithm in different simulated 2D environments. On these scenarios, our algorithm outperforms the RRT and the RRT* methods on finding the initial solution. When embedded into post-processing algorithms like the Informed RRT*, it also promotes the convergence speed and saves the memory usage.

New perspective on sampling-based motion planning via random geometric graphs

Roadmaps constructed by many sampling-based motion planners coincide, in the absence of obstacles, with standard models of random geometric graphs (RGGs). Those models have been studied for several decades and by now a rich body of literature exists analyzing various properties and types of RGGs. In their seminal work on optimal motion planning, Karaman and Frazzoli conjectured that a sampling-based planner has a certain property if the underlying RGG has this property as well. In this paper, we settle this conjecture and leverage it for the development of a general framework for the analysis of sampling-based planners. Our framework, which we call localization–tessellation, allows for easy transfer of arguments on RGGs from the free unit hypercube to spaces punctured by obstacles, which are geometrically and topologically much more complex. We demonstrate its power by providing alternative and (arguably) simple proofs for probabilistic completeness and asymptotic (near-)optimality of probabilistic roadmaps (PRMs) in Euclidean spaces. Furthermore, we introduce three variants of PRMs, analyze them using our framework, and discuss the implications of the analysis.

Sparsification of motion-planning roadmaps by edge contraction

Roadmaps constructed by the recently introduced PRM* algorithm for asymptotically-optimal motion planning encode high-quality paths yet can be extremely dense. We consider the problem of [Formula: see text] the roadmap, i.e. reducing its size, while minimizing the degradation of the quality of paths that can be extracted from the resulting roadmap. We present a simple, effective sparsifying algorithm, called [Formula: see text][Formula: see text] (RSEC). The primitive operation used by RSEC is [Formula: see text]—the contraction of a roadmap edge ( v′, v″) to a new vertex [Formula: see text] and the connection of the new vertex [Formula: see text] to the neighboring vertices of the contracted edge’s vertices (i.e. to all neighbors of [Formula: see text] and [Formula: see text]). For certain scenarios, we compress more than 97% of the edges and vertices of a given roadmap at the cost of degradation of average shortest path length by at most 4%.

Sparse roadmap spanners for asymptotically near-optimal motion planning

Asymptotically optimal planners, such as PRM*, guarantee that solutions approach optimal as the number of iterations increases. Roadmaps with this property, however, may grow too large for storing on resource-constrained robots and for achieving efficient online query resolution. By relaxing optimality, asymptotically near-optimal planners produce sparser graphs by not including all edges. The idea stems from graph spanners, which produce sparse subgraphs that guarantee near-optimality. Existing asymptotically optimal and near-optimal planners, however, include all sampled configurations as roadmap nodes, meaning only infinite-size graphs have the desired properties. To address this limitation, this work describes SPARS, an algorithm that returns a sparse roadmap spanner. The method provides the following properties: (a) probabilistic completeness, (b) asymptotic near-optimality and (c) the probability of adding nodes to the spanner converges to zero as iterations increase. The last point suggests that finite-size data structures with asymptotic near-optimality in continuous spaces may indeed exist. The approach builds simultaneously a dense graph similar to PRM* and its roadmap spanner, meaning that upon construction an infinite-size graph is still needed asymptotically. An extension of SPARS is also presented, termed SPARS2, which removes the dependency on building a dense graph for constructing the sparse roadmap spanner and for which it is shown that the same desirable properties hold. Simulations for rigid-body motion planning show that algorithms for constructing sparse roadmap spanners indeed provide small data structures and result in faster query resolution. The rate of node addition is shown to decrease over time and practically the quality of solutions is considerably better than the theoretical bounds. Upon construction, the memory requirements of SPARS2 are significantly smaller but there is a small sacrifice in the size of the final spanner relative to SPARS.