Multilevel Motion Planning: A Fiber Bundle Formulation

Motion planning problems involving high-dimensional state spaces can often be solved significantly faster by using multilevel abstractions. While there are various ways to formally capture multilevel abstractions, we formulate them in terms of fiber bundles, which allows us to concisely describe and derive novel algorithms in terms of bundle restrictions and bundle sections. Fiber bundles essentially describe lower-dimensional projections of the state space using local product spaces. Given such a structure and a corresponding admissible constraint function, we can develop highly efficient and optimal search-based motion planning methods for high-dimensional state spaces. Our contributions are the following: We first introduce the terminology of fiber bundles, in particular the notion of restrictions and sections. Second, we use the notion of restrictions and sections to develop novel multilevel motion planning algorithms, which we call QRRT* and QMP*. We show these algorithms to be probabilistically complete and almost-surely asymptotically optimal. Third, we develop a novel recursive path section method based on an L1 interpolation over path restrictions, which we use to quickly find feasible path sections. And fourth, we evaluate all novel algorithms against all available OMPL algorithms on benchmarks of eight challenging environments ranging from 21 to 100 degrees of freedom, including multiple robots and nonholonomic constraints. Our findings support the efficiency of our novel algorithms and the benefit of exploiting multilevel abstractions using the terminology of fiber bundles.Comment: Submitted to IJR

Sampling-Based Motion Planning: A Comparative Review

Sampling-based motion planning is one of the fundamental paradigms to generate robot motions, and a cornerstone of robotics research. This comparative review provides an up-to-date guideline and reference manual for the use of sampling-based motion planning algorithms. This includes a history of motion planning, an overview about the most successful planners, and a discussion on their properties. It is also shown how planners can handle special cases and how extensions of motion planning can be accommodated. To put sampling-based motion planning into a larger context, a discussion of alternative motion generation frameworks is presented which highlights their respective differences to sampling-based motion planning. Finally, a set of sampling-based motion planners are compared on 24 challenging planning problems. This evaluation gives insights into which planners perform well in which situations and where future research would be required. This comparative review thereby provides not only a useful reference manual for researchers in the field, but also a guideline for practitioners to make informed algorithmic decisions.Comment: 25 pages, 7 figures, Accepted for Volume 7 (2024) of the Annual Review of Control, Robotics, and Autonomous System

Sensory Steering for Sampling-Based Motion Planning

Sampling-based algorithms offer computationally efficient, practical solutions to the path finding problem in high-dimensional complex configuration spaces by approximately capturing the connectivity of the underlying space through a (dense) collection of sample configurations joined by simple local planners. In this paper, we address a long-standing bottleneck associated with the difficulty of finding paths through narrow passages. Whereas most prior work considers the narrow passage problem as a sampling issue (and the literature abounds with heuristic sampling strategies) very little attention has been paid to the design of new effective local planners. Here, we propose a novel sensory steering algorithm for sampling- based motion planning that can “feel” a configuration space locally and significantly improve the path planning performance near difficult regions such as narrow passages. We provide computational evidence for the effectiveness of the proposed local planner through a variety of simulations which suggest that our proposed sensory steering algorithm outperforms the standard straight-line planner by significantly increasing the connectivity of random motion planning graphs. For more information: Kod*la

Efficient Configuration Space Construction and Optimization for Motion Planning

The configuration space is a fundamental concept that is widely used in algorithmic robotics. Many applications in robotics, computer-aided design, and related areas can be reduced to computational problems in terms of configuration spaces. In this paper, we survey some of our recent work on solving two important challenges related to configuration spaces

Learning Probabilistic Generative Models For Fast Sampling-Based Planning

Due to their simplicity and efficiency in high dimensional space, sampling-based motion planners have been gaining interest for robotic manipulation in recent years. We present several new learning approaches using probabilistic generative models for fast sampling-based planning. First, we propose fast collision detection in high dimensional configuration spaces based on Gaussian Mixture Models (GMMs) for Rapidly-exploring Random Trees (RRT). In addition, we introduce a new probabilistically safe local steering primitive based on the probabilistic model. Our local steering procedure is based on a new notion of a convex probabilistically safety corridor that is constructed around a configuration using tangent hyperplanes of confidence ellipsoids of GMMs learned from prior collision history. For efficient sampling, we suggest a sampling method with a learned Q-function with linear function approximation based on feature representations such as Radial Basis Functions. This sampling method chooses the optimal node from which to extend the search tree via the softmax function of learned state values. We also discuss a novel constrained sampling-based motion planning method for grasp and transport tasks with redundant robotic manipulators, which allows the best grasp configuration and approach direction to be automatically determined. Since these approaches with the learned probabilistic models require large size data and time for training, it is essential that they are able to be adapted to environmental change in an online manner. The suggested online learning approach with the Dirichlet Process Mixture Model (DPMM) can adapt the complexity to the data and learn new Gaussian clusters with streaming data in newly explored areas without batch learning. We have applied these approaches in a number of robot arm planning scenarios and have shown their utility and effectiveness in simulation and on a physical 7-DoF robot manipulator

Enabling Motion Planning and Execution for Tasks Involving Deformation and Uncertainty

A number of outstanding problems in robotic motion and manipulation involve tasks where degrees of freedom (DoF), be they part of the robot, an object being manipulated, or the surrounding environment, cannot be accurately controlled by the actuators of the robot alone. Rather, they are also controlled by physical properties or interactions - contact, robot dynamics, actuator behavior - that are influenced by the actuators of the robot. In particular, we focus on two important areas of poorly controlled robotic manipulation: motion planning for deformable objects and in deformable environments; and manipulation with uncertainty. Many everyday tasks we wish robots to perform, such as cooking and cleaning, require the robot to manipulate deformable objects. The limitations of real robotic actuators and sensors result in uncertainty that we must address to reliably perform fine manipulation. Notably, both areas share a common principle: contact, which is usually prohibited in motion planners, is not only sometimes unavoidable, but often necessary to accurately complete the task at hand. We make four contributions that enable robot manipulation in these poorly controlled tasks: First, an efficient discretized representation of elastic deformable objects and cost function that assess a ``cost of deformation\u27 for a specific configuration of a deformable object that enables deformable object manipulation tasks to be performed without physical simulation. Second, a method using active learning and inverse-optimal control to build these discretized representations from expert demonstrations. Third, a motion planner and policy-based execution approach to manipulation with uncertainty which incorporates contact with the environment and compliance of the robot to generate motion policies which are then adapted during execution to reflect actual robot behavior. Fourth, work towards the development of an efficient path quality metric for paths executed with actuation uncertainty that can be used inside a motion planner or trajectory optimizer

A Partially Randomized Approach to Trajectory Planning and Optimization for Mobile Robots with Flat Dynamics

Motion planning problems are characterized by huge search spaces and complex obstacle structures with no concise mathematical expression. The fixed-wing airplane application considered in this thesis adds differential constraints and point-wise bounds, i. e. an infinite number of equality and inequality constraints. An optimal trajectory planning approach is presented, based on the randomized Rapidly-exploring Random Trees framework (RRT*). The local planner relies on differential flatness of the equations of motion to obtain tree branch candidates that automatically satisfy the differential constraints. Flat output trajectories, in this case equivalent to the airplane's flight path, are designed using Bézier curves. Segment feasibility in terms of point-wise inequality constraints is tested by an indicator integral, which is evaluated alongside the segment cost functional. Although the RRT* guarantees optimality in the limit of infinite planning time, it is argued by intuition and experimentation that convergence is not approached at a practically useful rate. Therefore, the randomized planner is augmented by a deterministic variational optimization technique. To this end, the optimal planning task is formulated as a semi-infinite optimization problem, using the intermediate result of the RRT(*) as an initial guess. The proposed optimization algorithm follows the feasible flavor of the primal-dual interior point paradigm. Discretization of functional (infinite) constraints is deferred to the linear subproblems, where it is realized implicitly by numeric quadrature. An inherent numerical ill-conditioning of the method is circumvented by a reduction-like approach, which tracks active constraint locations by introducing new problem variables. Obstacle avoidance is achieved by extending the line search procedure and dynamically adding obstacle-awareness constraints to the problem formulation. Experimental evaluation confirms that the hybrid approach is practically feasible and does indeed outperform RRT*'s built-in optimization mechanism, but the computational burden is still significant.Bewegungsplanungsaufgaben sind typischerweise gekennzeichnet durch umfangreiche Suchräume, deren vollständige Exploration nicht praktikabel ist, sowie durch unstrukturierte Hindernisse, für die nur selten eine geschlossene mathematische Beschreibung existiert. Bei der in dieser Arbeit betrachteten Anwendung auf Flächenflugzeuge kommen differentielle Randbedingungen und beschränkte Systemgrößen erschwerend hinzu. Der vorgestellte Ansatz zur optimalen Trajektorienplanung basiert auf dem Rapidly-exploring Random Trees-Algorithmus (RRT*), welcher die Suchraumkomplexität durch Randomisierung beherrschbar macht. Der spezifische Beitrag ist eine Realisierung des lokalen Planers zur Generierung der Äste des Suchbaums. Dieser erfordert ein flaches Bewegungsmodell, sodass differentielle Randbedingungen automatisch erfüllt sind. Die Trajektorien des flachen Ausgangs, welche im betrachteten Beispiel der Flugbahn entsprechen, werden mittels Bézier-Kurven entworfen. Die Einhaltung der Ungleichungsnebenbedingungen wird durch ein Indikator-Integral überprüft, welches sich mit wenig Zusatzaufwand parallel zum Kostenfunktional berechnen lässt. Zwar konvergiert der RRT*-Algorithmus (im probabilistischen Sinne) zu einer optimalen Lösung, jedoch ist die Konvergenzrate aus praktischer Sicht unbrauchbar langsam. Es ist daher naheliegend, den Planer durch ein gradientenbasiertes lokales Optimierungsverfahren mit besseren Konvergenzeigenschaften zu unterstützen. Hierzu wird die aktuelle Zwischenlösung des Planers als Initialschätzung für ein kompatibles semi-infinites Optimierungsproblem verwendet. Der vorgeschlagene Optimierungsalgorithmus erweitert das verbreitete innere-Punkte-Konzept (primal dual interior point method) auf semi-infinite Probleme. Eine explizite Diskretisierung der funktionalen Ungleichungsnebenbedingungen ist nicht erforderlich, denn diese erfolgt implizit durch eine numerische Integralauswertung im Rahmen der linearen Teilprobleme. Da die Methode an Stellen aktiver Nebenbedingungen nicht wohldefiniert ist, kommt zusätzlich eine Variante des Reduktions-Ansatzes zum Einsatz, bei welcher der Vektor der Optimierungsvariablen um die (endliche) Menge der aktiven Indizes erweitert wird. Weiterhin wurde eine Kollisionsvermeidung integriert, die in den Teilschritt der Liniensuche eingreift und die Problemformulierung dynamisch um Randbedingungen zur lokalen Berücksichtigung von Hindernissen erweitert. Experimentelle Untersuchungen bestätigen, dass die Ergebnisse des hybriden Ansatzes aus RRT(*) und numerischem Optimierungsverfahren der klassischen RRT*-basierten Trajektorienoptimierung überlegen sind. Der erforderliche Rechenaufwand ist zwar beträchtlich, aber unter realistischen Bedingungen praktisch beherrschbar