Problem Of Motion Planning Research Articles

Improved Results for Minimum Constraint Removal

Given a set of obstacles and two designated points in the plane, the Minimum Constraint Removal problem asks for a minimum number of obstacles that can be removed so that a collision-free path exists between the two designated points. It is a well-studied problem in both robotic motion planning and wireless computing that has been shown to be NP-hard in various settings. In this work, we extend the study of Minimum Constraint Removal. We start by presenting refined NP-hardness reductions for the two cases: (1) when all the obstacles are axes-parallel rectangles, and (2) when all the obstacles are line segments such that no three intersect at the same point. These results improve on existing results in the literature. As a byproduct of our NP-hardness reductions, we prove that, unless the Exponential-Time Hypothesis (ETH) fails, Minimum Constraint Removal cannot be solved in subexponential time 2o(n), where n is the number of obstacles in the instance. This shows that significant improvement on the brute-force 2O(n)-time algorithm is unlikely. We then present a subexponential-time algorithm for instances of Minimum Constraint Removal in which the number of obstacles that overlap at any point is constant; the algorithm runs in time 2O(√N), where N is the number of the vertices in the auxiliary graph associated with the instance of the problem. We show that significant improvement on this algorithm is unlikely by showing that, unless ETH fails, Minimum Constraint Removal with bounded overlap number cannot be solved in time 2o(√N). We describe several exact algorithms and approximation algorithms that leverage heuristics and discuss their performance in an extensive empirical simulation.

Connectivity-based optimization of vehicle route and speed for improved fuel economy

Traditionally, vehicle route planning problem focuses on route optimization based on traffic data and surrounding environment. This paper proposes a novel extended vehicle route planning problem, called vehicle macroscopic motion planning (VMMP) problem, to optimize vehicle route and speed simultaneously using both traffic data and vehicle characteristics to improve fuel economy for a given expected trip time. The required traffic data and neighbouring vehicle dynamic parameters can be collected through the vehicle connectivity (e.g. vehicle-to-vehicle, vehicle-to-infrastructure, vehicle-to-cloud, etc.) developed rapidly in recent years. A genetic algorithm based co-optimization method, along with an adaptive real-time optimization strategy, is proposed to solve the proposed VMMP problem. It is able to provide the fuel economic route and reference speed for drivers or automated vehicles to improve the vehicle fuel economy. A co-simulation model, combining a traffic model based on SUMO (Simulation of Urban MObility) with a Simulink powertrain model, is developed to validate the proposed VMMP method. Four simulation studies, based on a real traffic network, are conducted for validating the proposed VMMP: (1) ideal traffic environment without traffic light and jam for studying the fuel economy improvement, (2) traffic environment with traffic light for validating the proposed traffic light penalty model, (3) traffic environment with traffic light and jam for validating the proposed adaptive real-time optimization strategy, and (4) investigating the effect of different powertrain platforms to fuel economy using two different vehicle platforms. Simulation results show that the proposed VMMP method is able to improve vehicle fuel economy significantly. For instance, comparing with the fastest route, the fuel economy using the proposed VMMP method is improved by up to 15%.

Online Trajectory Planning and Force Control for Automation of Surgical Tasks

Automation of surgical tasks is expected to improve the quality of surgery. In this paper, we address two issues that must be resolved for automation of robotic surgery: online trajectory planning and force control under dynamic conditions. By leveraging demonstrations under various conditions, we model the conditional distribution of the trajectories given the task condition. This scheme enables generalization of the trajectories of spatial motion and contact force to new conditions in real time. In addition, we propose a force tracking controller that robustly and stably tracks the planned profile of the contact force by learning the spatial motion and contact force simultaneously. The proposed scheme was tested with bimanual tasks emulating surgical tasks that require online trajectory planning and force tracking control, such as tying knots and cutting soft tissues. Experimental results show that the proposed scheme enables planning of the task trajectory under dynamic conditions in real time. In addition, the performance of the force control schemes was verified in the experiments. Note to Practitioners —This paper addresses the problem of motion planning and control for automation of surgical tasks. In surgical tasks, it is necessary to manipulate objects under conditions where positions or shapes of objects often change during the task. Thus, trajectories for surgical tasks need to be planned and updated according to the change in the conditions in real time. In this paper, we propose a framework for learning both spatial motion and force profile from human experts. The proposed system can plan and update task trajectories in real time and robustly control the contact force under dynamic conditions. On the other hand, generalization of trajectories is limited to the conditions, which are close to the conditions where the demonstrations were performed. In the future work, we will investigate reinforcement learning approaches in order to enable autonomous improvement of the performance.

Optimal Perception Planning with Informed Heuristics Constructed from Visibility Maps

In this paper we consider the problem of motion planning for perception of a target position. A robot has to move to a position from where it can sense the target, while minimizing both motion and perception costs. The problem of finding paths for robots executing perception tasks can be solved optimally using informed search. In perception path planning, the solution when considering a straight line without obstacles is used as heuristic. In this work, we propose a heuristic that can improve the search efficiency. In order to reduce the node expansion using a more informed search, we use the robot Approximate Visibility Map (A-VM), which is used as a representation of the observability capability of a robot in a given environment. We show how the critical points used in A-VM provide information on the geometry of the environment, which can be used to improve the heuristic, increasing the search efficiency. The critical points allow a better estimation of the minimum motion and perception cost for targets in non-traversable regions that can only be sensed from further away. Finally, we show the contributed heuristic with improvements dominates the base PA* heuristic built on the euclidean distance, and then present the results of the performance increase in terms of node expansion and computation time.

High-dimensional stochastic optimal control using continuous tensor decompositions

Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately, most existing algorithms that guarantee convergence to optimal solutions suffer from the curse of dimensionality: the run time of the algorithm grows exponentially with the dimension of the state space of the system. We propose novel dynamic programming algorithms that alleviate the curse of dimensionality in problems that exhibit certain low-rank structure. The proposed algorithms are based on continuous tensor decompositions recently developed by the authors. Essentially, the algorithms represent high-dimensional functions (e.g. the value function) in a compressed format, and directly perform dynamic programming computations (e.g. value iteration, policy iteration) in this format. Under certain technical assumptions, the new algorithms guarantee convergence towards optimal solutions with arbitrary precision. Furthermore, the run times of the new algorithms scale polynomially with the state dimension and polynomially with the ranks of the value function. This approach realizes substantial computational savings in “compressible” problem instances, where value functions admit low-rank approximations. We demonstrate the new algorithms in a wide range of problems, including a simulated six-dimensional agile quadcopter maneuvering example and a seven-dimensional aircraft perching example. In some of these examples, we estimate computational savings of up to 10 orders of magnitude over standard value iteration algorithms. We further demonstrate the algorithms running in real time on board a quadcopter during a flight experiment under motion capture.

General Lagrange-Type Jacobian Inverse for Nonholonomic Robotic Systems

This paper discusses the nonholonomic robotic systems whose motion constraints assume the Pfaffian form, and the equations of motion are represented by driftless control systems with outputs. By reference to the end point map of such a control system, we define the system's Jacobian and study Jacobian motion-planning algorithms. A new Lagrange-type Jacobian inverse, referred to as the General Lagrangian Jacobian Inverse (GLJI), is designed as the solution of an optimal control problem with a Lagrange-type objective function. Singularities of GLJI are examined. A special choice of the objective function illustrates features of GLJI. A new motion-planning algorithm based on GLJI is proposed. Theoretical arguments are illustrated with a motion-planning problem of a space robot.

FFRob: Leveraging symbolic planning for efficient task and motion planning

Mobile manipulation problems involving many objects are challenging to solve due to the high dimensionality and multi-modality of their hybrid configuration spaces. Planners that perform a purely geometric search are prohibitively slow for solving these problems because they are unable to factor the configuration space. Symbolic task planners can efficiently construct plans involving many variables but cannot represent the geometric and kinematic constraints required in manipulation. We present the FFRob algorithm for solving task and motion planning problems. First, we introduce extended action specification (EAS) as a general purpose planning representation that supports arbitrary predicates as conditions. We adapt existing heuristic search ideas for solving strips planning problems, particularly delete-relaxations, to solve EAS problem instances. We then apply the EAS representation and planners to manipulation problems resulting in FFRob. FFRob iteratively discretizes task and motion planning problems using batch sampling of manipulation primitives and a multi-query roadmap structure that can be conditionalized to evaluate reachability under different placements of movable objects. This structure enables the EAS planner to efficiently compute heuristics that incorporate geometric and kinematic planning constraints to give a tight estimate of the distance to the goal. Additionally, we show FFRob is probabilistically complete and has a finite expected runtime. Finally, we empirically demonstrate FFRob’s effectiveness on complex and diverse task and motion planning tasks including rearrangement planning and navigation among movable objects.

The complexity of snake and undirected NCL variants

Snake and Nibbler are two well-known video games in which a snake slithers through a maze and grows as it collects food. During this process, the snake must avoid any collision with its tail. Various goals can be associated with these video games, such as avoiding the tail as long as possible, or collecting a certain amount of food, or reaching some target location. Unfortunately, like many other motion-planning problems, even very restricted variants are computationally intractable. In particular, we prove the NP-hardness of collecting all food on solid grid graphs; as well as its PSPACE-completeness on general grid graphs. Moreover, given an initial and a target configuration of the snake, moving from one configuration to the other is PSPACE-complete, even on grid graphs without food, or with an initially short snake.Our results make use of the nondeterministic constraint logic framework by Hearn and Demaine, which has been used to analyze the computational complexity of many games and puzzles. We extend this framework for the analysis of puzzles whose initial state is chosen by the player.

On repelling robotic trajectories: coordination in navigation of multiple mobile robots

The paper attacks the problem of motion planning of a set of mobile robots. While artificial potential fields are the simplest methods of use, they are also locally optimal and can be easily stuck in scenarios. Probabilistic roadmap, elastic roadmaps, elastic strip and similar methods have a weak modelling of coordination between the robots. An inspiration is drawn from the artificial potential field method where the potential is computed in configuration space. In this paper the notion is extended to a ‘trajectory space’, where the complete trajectories of robots repel each other. With the added assumption of communication between the robots and higher computational costs, the resultant approach is near optimal and does not get the robot stuck or trapped. A variant of the algorithm with no direct communication is also presented. The method is experimented by using computer simulations and found to perform better over well-known approaches in the literature.

Navigation of carlike robots in an extended dynamic environment with swarm avoidance

SummaryIn this paper, we propose a new solution to the motion planning and control problem for a team of carlike mobile robots traversing in an extended dynamic environment. Motivated by the emerging necessity to avoid or defend against a swarm of autonomous robots, the wide array of obstacles in this dynamic environment for the first time includes a swarm of boids governed separately by a system of ordinary differential equations. The swarm exhibits collective emergent behaviors, whereas the carlike mobile robots safely navigate to designated targets. We present a set of nonlinear continuous controllers for obstacle, collision, and swarm avoidance. The controllers provide a collision‐free trajectory within a constrained workspace cluttered with various fixed and moving obstacles while satisfying the nonholonomic and kinodynamic constraints associated with the vehicular robotic system. An advantage of the proposed method is the ease in deriving the acceleration‐based control laws from the Lyapunov‐based control scheme. The effectiveness of the control laws is demonstrated via computer simulations. The novelty of this paper lies in the simplicity of the controllers and the ease in the treatment of an extended dynamic environment, which includes swarm avoidance.

Collision and Deadlock Avoidance in Multirobot Systems: A Distributed Approach

Collision avoidance is a critical problem in motion planning and control of multirobot systems. Moreover, it may induce deadlocks during the procedure to avoid collisions. In this paper, we study the motion control of multirobot systems where each robot has its own predetermined and closed path to execute persistent motion. We propose a real-time and distributed algorithm for both collision and deadlock avoidance by repeatedly stopping and resuming robots. The motion of each robot is first modeled as a labeled transition system, and then controlled by a distributed algorithm to avoid collisions and deadlocks. Each robot can execute the algorithm autonomously and real-timely by checking whether its succeeding state is occupied and whether the one-step move can cause deadlocks. Performance analysis of the proposed algorithm is also conducted. The conclusion is that the algorithm is not only practically operative but also maximally permissive. A set of simulations for a system with four robots are carried out in MATLAB. The results also validate the effectiveness of our algorithm.

Knife-Edge Motion on a Surface as a Nonholonomic Control Problem

This letter studies a new formulation for the kinematics of a knife-edge moving on an arbitrary smooth surface in ${\mathbb {R}^{3}}$ . The kinematics equations for a knife-edge, viewed as a rigid body, are constrained by the requirement that the knife-edge maintain contact with the surface. They describe the constrained translation of the point of contact of the knife-edge on the surface and the constrained attitude of the knife-edge as a rigid body. These equations for the knife-edge kinematics in ${\mathbb {R}}^{3}$ are expressed in a geometric form, without the use of local coordinates; they are globally defined without singularities or ambiguities. The kinematics equations can be expressed in several simplified forms and written as a drift-free nonlinear control system. Comments are made about interesting motion planning and path planning problems. The kinematics equations are specialized for two specific surfaces defined in ${ {\mathbb {R}}^{3}}$ , namely, a flat plane and the surface of a sphere. Results for the flat plane are compared with standard results obtained using local coordinates; results for the sphere, in contrast, require full attention to the 3-D geometry.

Efficient Motion Planning for Problems Lacking Optimal Substructure

We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We suggest a natural cost function that balances path length and risk-exposure time. Specifically, we consider the discrete setting where we are given a graph, or a roadmap, and we wish to compute the minimal-cost path under this cost function. Interestingly, paths defined using our cost function do not have an optimal substructure. Namely, subpaths of an optimal path are not necessarily optimal. Thus, the Bellman condition is not satisfied and standard graph-search algorithms such as Dijkstra cannot be used. We present a path-finding algorithm, which can be seen as a natural generalization of Dijkstra’s algorithm. Our algorithm runs in O ((n B · n) log(n B · n) + n B · m) time, where n and m are the number of vertices and edges of the graph, respectively, and n B is the number of intersections between edges and the boundary of the risk zone. We present simulations on robotic platforms demonstrating both the natural paths produced by our cost function and the computational efficiency of our algorithm.

Bi-objective path planning using deterministic algorithms

Path planning has become a central problem in motion planning. The classic version of the problem aims to find an obstacle-free path with the minimum length for a given workspace containing a set of obstacles and two sources and destination points. However, some real world applications consider maximizing the path clearance (i.e., the distance between the robot and obstacles) as the secondary objective. This bi-objective path planning problem has been studied using evolutionary and other heuristic algorithms which do not guarantee achieving Pareto optimal paths. In this paper, we first study this problem using deterministic algorithms. Next, we propose an efficient algorithm for the problem in the grid workspace. We then propose an O(n3) time algorithm for the problem under the Manhattan distance in a continuous workspace containing n vertical segments as obstacles. Finally, we show the obtained solutions are proper approximation for the problem under the Euclidean distance.

Periodic motion planning and control for underactuated mechanical systems

ABSTRACTWe consider the problem of periodic motion planning and of designing stabilising feedback control laws for such motions in underactuated mechanical systems. A novel periodic motion planning method is proposed. Each state is parametrised by a truncated Fourier series. Then we use numerical optimisation to search for the parameters of the trigonometric polynomial exploiting the measure of discrepancy in satisfying the passive dynamics equations as a performance index. Thus an almost feasible periodic motion is found. Then a linear controller is designed and stability analysis is given to verify that solutions of the closed-loop system stay inside a tube around the planned approximately feasible periodic trajectory. Experimental results for a double rotary pendulum are shown, while numerical simulations are given for models of a spacecraft with liquid sloshing and of a chain of mass spring system.

A dϕ-Strategy: Facilitating Dual-Formation Control of a Virtually Connected Team

This paper describes the design of new centralized acceleration-based controllers for the multitask problem of motion planning and control of a coordinated lead-carrier team fixed in a dual-formation within an obstacle-ridden environment. A dϕ-strategy, where d and ϕ are Euclidean measures with respect to the lead robot, is developed to ensure virtual connectivity of the carrier robots to the lead robot. This connectivity, built into the system itself, inherently ensures globally rigid formation between each lead-carrier pair of the team. Moreover, a combination of target configuration, dϕ-strategy, orientation consensus, and avoidance of end-effector of robots results in a second, locally rigid formation (not infinitesimally rigid). Therefore, for the first time, a dual-formation control problem of a lead-carrier team of mobile manipulators is considered. This and other kinodynamic constraints have been treated simultaneously via the overarching Lyapunov-based control scheme, essentially a potential field method favored in the field of robotics. The formulation of this new scheme, demonstrated effectively via computer simulations, is timely, given that the current proposed engineering solutions, allowing autonomous vehicles on public roads, include the development of special lanes imbued with special sensors and wireless technologies.

Constrained Autonomous Precision Landing via Dual Quaternions and Model Predictive Control

The problem of powered descent guidance and control for autonomous precision landing for next-generation planetary missions is addressed. The precision landing algorithm aims to trace a fuel-optimal trajectory while keeping geometrical constraints such as the line of sight to the target site. The design of an autonomous control algorithm managing such mission scenarios is challenging due to fact that critical geometrical constraints are coupled with the translational and rotational motions of the lander spacecraft, leading to a complex motion-planning problem. This problem is approached within the model predictive control framework by representing the dynamics of the rigid body in a uniform gravity field via a piecewise affine system taking advantage of the unit dual-quaternion parameterization. Such a parameterization in turn enables a six-degree-of-freedom motion planning in a unified framework while also admitting a quadratic cost on the required control commands to minimize propellant consumption. A novel feature of the proposed approach is the development of convex representable subsets in the configuration space in terms of unit dual quaternions. The stability and reachability issues of the corresponding piecewise affine model predictive control approach are then discussed. Numerical simulations are presented to demonstrate the effectiveness of the proposed methodology for autonomous precision landing.

Probabilistic Roadmaps with Higher Expressive Power

Sampling-based motion planning had an enormous impact on robot motion planning because of its efficiency and scalability. Many sampling-based motion planners construct a probabilistic roadmap (PRM) that captures the connectivity of the robot's free configuration space. A valid node of a PRM contains a collision-free robot configuration (also known as a sample) and a valid edge of a PRM connects two valid nodes with a collision-free path. Nodes connected by an edge are usually also required to satisfy additional requirements based on the distance between them. PRM planners use PRMs. Increasing the expressive power will allow PRMs to be used for a wider set of motion planning problems. In this paper we report on increasing the expressive power of PRMs by including the following five features in PRMs-nodes with multiple samples that need not be organized as a graph, temporal intervals of validity of nodes and edges, nodes with samples of multiple robots, special edges for the state transitions performed by humans sharing a workspace with robots, and conditional validity of samples and edges. We report on motion planning problems solvable using these new features.

Optimal motion planning for parallel robots via convex optimization and receding horizon

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Energy-Optimal Motion Planning for Multiple Robotic Vehicles With Collision Avoidance

We propose a numerical algorithm for multiple-vehicle motion planning that explicitly takes into account the vehicle dynamics, temporal and spatial specifications, and energy-related requirements. As a motivating example, we consider the case where a group of vehicles is tasked to reach a number of target points at the same time ( simultaneous arrival problem ) without colliding among themselves and with obstacles, subject to the requirement that the overall energy required for vehicle motion be minimized. With the theoretical setup adopted, the vehicle dynamics are explicitly taken into account at the planning level. This paper formulates the problem of multiple-vehicle motion planning in a rigorous mathematical setting, describes the optimization algorithm used to solve it, and discusses the key implementation details. The efficacy of the method is illustrated through numerical examples for the simultaneous arrival problem. The initial guess to start the optimization procedure is obtained from simple geometrical considerations, e.g., by joining the desired initial and final positions of the vehicles via straight lines. Even though the initial trajectories thus obtained may result in intervehicle and vehicle/obstacle collisions, we show that the optimization procedure that we employ in this paper will generate collision-free trajectories that also minimize the overall energy spent by each vehicle and meet the required temporal and spatial constraints. The method developed applies to a very general class of vehicles; however, for clarity of exposition, we adopt as an illustrative example the case of wheeled robots.