Problem Of Motion Planning Research Articles

State classification for humanoid robots

In this paper, we decouple the motion-planning problem of humanoid robots into two sub-problems, namely topological state planning and detailed motion planning. The state classification plays a key...

Exact Minkowksi Sums of Polyhedra and Exact and Efficient Decomposition of Polyhedra into Convex Pieces

We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small number of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r 2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron.

Traveling Salesperson Problems for the Dubins Vehicle

In this paper, we study minimum-time motion planning and routing problems for the Dubins vehicle, i.e., a nonholonomic vehicle that is constrained to move along planar paths of bounded curvature, without reversing direction. Motivated by autonomous aerial vehicle applications, we consider the traveling salesperson problem for the Dubins vehicle (DTSP): given <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> points on a plane, what is the shortest Dubins tour through these points, and what is its length? First, we show that the worst-case length of such a tour grows linearly with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and we propose a novel algorithm with performance within a constant factor of the optimum for the worst-case point sets. In doing this, we also obtain an upper bound on the optimal length in the classical point-to-point problem. Second, we study a stochastic version of the DTSP where the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> targets are randomly and independently sampled from a uniform distribution. We show that the expected length of such a tour is of order at least <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> and we propose a novel algorithm yielding a solution with length of order <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> with probability one. Third and finally, we study a dynamic version of the DTSP: given a stochastic process that generates target points, is there a policy that guarantees that the number of unvisited points does not diverge over time? If such stable policies exist, what is the minimum expected time that a newly generated target waits before being visited by the vehicle? We propose a novel stabilizing algorithm such that the expected wait time is provably within a constant factor from the optimum.

Classifying the multi robot path finding problem into a quadratic competitive complexity class

We explore two motion planning problems where a group of mobile robots has to reach a target located in an a priori unknown environment while on-line planning the next step. In the first problem the target position is unknown and should be found by the robots, while in the second problem the target position is known and only a path to it should be found. We focus on optimizing the cost of the task in terms of motion time, which, under the assumption of uniform velocity of all the robots, correlates to the path length passed by the robot which reaches the target. The performance of an on-line algorithm is usually expressed in terms of Competitiveness, the constant ratio between the on-line and the optimal off-line solutions. Specifically, the ratio between the lengths of the actual path made by the robot which reached the target to the shortest path to the target. We use generalized competitiveness, i.e., the ratio is not necessarily constant, but could be any function. Classification of a motion planning task in the sense of performance is done by finding an upper and a lower bounds on the competitiveness of all algorithms solving that task. If the two bounds belong to the same functional class this is the Competitive Complexity Class of the task. We find the two bounds for the aforementioned common on-line motion planning problems, and classify them into competitive classes. It is shown that in general any on-line motion planning algorithm that tries to solve these problems must have at least a quadratic competitive performance. This is a lower bound of the problems. This paper describes two new on-line navigation algorithm which solve the problems under discussion. The first is called MRSAM, short for Multi-Robot Search Area Multiplication, and the second is called MRBUG, short for Multi-Robot BUG which extends Lumelsky famous BUG algorithm. Both algorithms have quadratic upper bounds, which prove that the problems they solve have quadratic upper bounds. Thus it is shown that navigation in an unknown environment by a group of robots belongs to a quadratic competitive class. MRSAM and MRBUG have a quadratic competitive performance and thus have optimal competitiveness. The algorithms' performance is simulated in office-like environments.

A linear-time algorithm for computing collision-free path on reconfigurable mesh

The reconfigurable mesh (RMESH) is an array of mesh-connected processors equipped with a reconfigurable bus system, which can dynamically connect the processors in various patterns. A 2D reconfigurable mesh can be used to solve motion planning problems in robotics research, in which the 2D image of robot and obstacles are digitized and represented one pixel per processor. In this paper, we present an algorithm to compute a collision-free path between two points in an environment containing obstacles. The time complexity of the algorithm is O ( k ) for each pair of source/destination points, with O ( log 2 N ) preprocessing time, where k is the number of obstacles in the working environment, and N is the size of the reconfigurable mesh.

A Lagrangian-based algorithm for a Multiple Depot, Multiple Traveling Salesmen Problem

In this manuscript, we consider the problem of motion planning of m Dubins’ vehicles through n points in a plane. The initial location and heading of the vehicles is specified and is assumed to be distinct for each vehicle. A motion plan for a vehicle is given by the sequence of points and the corresponding angles at which each point must be visited by the vehicle. We require that each vehicle return to the same initial location (depot) at the same heading after visiting the points. The objective of the motion planning problem is to choose at most q ( ≤ m ) Dubins’ vehicles and find their motion plans so that all the points are visited and the total cost of the tours of the chosen vehicles is a minimum amongst all the possible choice of vehicles and their tours. This problem is a generalization of the Multiple Depot, Multiple Traveling Salesmen Problem (MDMTSP) in two directions — the problem involves the determination of choice of vehicles and their respective headings at each of their assigned points. This problem is NP-hard. We propose a two step approach to solve this problem — (1) The combinatorial problem of choosing the vehicles and their associated tours is based on Euclidean distances between points and (2) Once the sequence of points to be visited is found, the heading at each point is determined based on a Dynamic Programming scheme. The solution to the first step is based on a generalization of Held–Karp’s method for the MDMTSP. We modify the Lagrangian heuristics, in the literature, for finding a close primal solution from the Held–Karp’s (dual) solution. Empirical results indicate that this scheme seems to provide primal solutions which are within 5 % of the optimum in the span of 25 dual iterations for instances which have about 45 points to visit and 6 vehicles.

Translating Structured English to Robot Controllers

Recently, Linear Temporal Logic (LTL) has been successfully applied to high-level task and motion planning problems for mobile robots. One of the main attributes of LTL is its close relationship with fragments of natural language. In this paper, we take the first steps toward building a natural language interface for LTL planning methods with mobile robots as the application domain. For this purpose, we built a structured English language which maps directly to a fragment of LTL.

State Classification for Humanoid Robots

In this paper, we decouple the motion-planning problem of humanoid robots into two sub-problems, namely topological state planning and detailed motion planning. The state classification plays a key role for the first sub-problem. We propose several basic states, including lying, sitting, standing and handstanding, abstracted from the daily exercises of human beings. Each basic state is classified further from the topological point of view. Furthermore, generalised function (GF) set theory is applied with the aim of analysing the kinematic characteristics of the end effectors for each state, and meaningful names are assigned for each state. Finally a topological state-planning example is given to show the effectiveness of this methodology. The results show that the large amounts of states can be described using assigned names, which leads to systematic and universal description of the states for humanoid robots.

Motion Planning Under Uncertainty: Application to an Unmanned Helicopter

A methodology is presented in this work for intelligent motion planning in an uncertain environment using a nonlocal sensor, such as a radar. This methodology is applied to an unmanned helicopter navigating a cluttered urban environment. It is shown that the problem of motion planning in an uncertain environment, under certain assumptions, can be posed as the adaptive optimal control of an uncertain Markov decision process, characterized by a known, control dependent system, and an unknown, control independent environment. The strategy for motion planning then reduces to computing the control policy based on the current estimate of the environment, also known as the certainty equivalence principle in the adaptive control literature. The methodology allows the inclusion of a nonlocal sensor into the problem formulation, which significantly accelerates the convergence of the estimation and planning algorithms. Further, the motion planning and estimation problems possess special structure which can be exploited to significantly reduce the computational burden of the associated algorithms. The methodology is applied to the problem of motion planning for an unmanned helicopter through a partially known model of the Texas A&M campus and the testing is done on a flight simulator.

Stochastic Event Capture Using Mobile Sensors Subject to a Quality Metric

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Mobile sensors cover more area over a fixed period of time than do the same number of stationary sensors. However, the quality of coverage (QoC) achieved by mobile sensors depends on the velocity, mobility pattern, number of mobile sensors deployed, and the dynamics of the phenomenon being sensed. The gains attained by mobile sensors over static sensors and the optimal motion strategies for mobile sensors are not well understood. In this paper, we consider the following event capture problem: the events of interest arrive at certain points in the sensor field and disappear according to known arrival and departure time distributions. An event is said to be captured if it is sensed by one of the mobile sensors before it fades away. We analyze how the QoC scales with velocity, path, and number of mobile sensors. We characterize cases where the deployment of mobile sensors has no advantage over static sensors, and find the optimal velocity pattern that a mobile sensor should adopt. We also present algorithms for two motion planning problems: 1) for a single sensor, what is the sensor trajectory and the <emphasis emphasistype="italic">minimum speed</emphasis> required to satisfy a bound on the event loss probability and 2) for sensors with fixed speed, what is the <emphasis emphasistype="italic">minimum number of sensors</emphasis> required to satisfy a bound on the event loss probability. When the robots are restricted to move along a line or a closed curve, our algorithms return the optimal velocity for the minimum velocity problem. For the minimum sensor problem, the number of sensors used is within a factor of 2 of the optimal solution. For the case where the events occur at arbitrary points on a plane, we present heuristic algorithms for the aforementioned motion planning problems and bound their performance with respect to the optimal. </para>

Sampling-based robot motion planning: Towards realistic applications

This paper presents some of the recent improvements in sampling-based robot motion planning. Emphasis is placed on work that brings motion-planning algorithms closer to applicability in real environments. Methods that approach increasingly difficult motion-planning problems including kinodynamic motion planning and dynamic environments are discussed. The ultimate goal for such methods is to generate plans that can be executed with few modifications in a real robotics mobile platform.

Hybrid mixed-logical linear programming algorithm for collision-free optimal path planning

Methods to solve collision-free fuel-optimal path- and motion-planning problems for the reconfiguration of spacecraft formations are presented. First, the motion-planning problem is formulated as a parameter optimisation problem in which the spacecraft are represented by unrotated cubes and the trajectory followed by each spacecraft is discretised in time using cubic splines. In other words, the trajectory is parameterised by the spacecraft positions and velocities at a set of waypoints. Big-M relaxation is then used to formulate the parameter optimisation as a mixed-integer linear program (MILP) whose solution can be obtained using standard MILP solvers. Several concepts are subsequently introduced to obtain the solutions more efficiently and reliably. In particular, the concept of a sequential linear program is introduced in which a sequence of closely-related linear programs are solved until a better local minimum cannot be found without ‘bifurcation’, that is, without qualitatively changing the nature of the solution. An example of such a change is making one spacecraft move in front of an obstacle instead of behind it. To handle these bifurcation points efficiently, the concept of a feasibility MILP (FMILP) is introduced in which the following question is answered: is there a feasible solution whose objective function value is better than that of the current bifurcation point. This is achieved by simply appending a single linear constraint to the set of constraints. A bisection search can be applied to the cost using FMILPs to further speed the proposed method. The method was applied to two standard tests of motion planning with collision avoidance such as swapping (using minimum fuel and along the major diagonals) the positions of eight spacecraft located at the corners of a cube. Preliminary simulations show significant improvement in efficiency by at least one order of magnitude.

CAN WE MAKE A ROBOT BALLERINA PERFORM A PIROUETTE? ORBITAL STABILIZATION OF PERIODIC MOTIONS OF UNDERACTUATED MECHANICAL SYSTEMS

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we define and discuss problems of motion planning and orbit planning, analysis methods such as the classical Poincaré first-return map and the transverse linearization, and exponentially orbitally stabilizing control designs. We begin with general nonlinear systems, and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically. The paper concludes with a discussion of numerical issues related to control design via periodic Riccati equations.

CAN WE MAKE A ROBOT BALLERINA PERFORM A PIROUETTE? ORBITAL STABILIZATION OF PERIODIC MOTIONS OF UNDERACTUATED MECHANICAL SYSTEMS1

This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we define and discuss problems of motion planning and orbit planning, analysis methods such as the classical Poincaré first-return map and the transverse linearization, and exponentially orbitally stabilizing control designs. We begin with general nonlinear systems, and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically. The paper concludes with a discussion of numerical issues related to control design via periodic Riccati equations.

Singular Trajectories of Control-Affine Systems

When applying methods of optimal control to motion planning or stabilization problems, we see that some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, minimizing singular trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system—with respect to the Whitney topology—all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend results of [Y. Chitour, F. Jean, and E. Trelat, Comptes Rendus Math., 337 (2003), pp. 49-52 (in French); Y. Chitour, F. Jean, and E. Trelat, J. Differential Geom., 73 (2006), pp. 45-73]. As a consequence, for generic control-affine systems (with or without drift) defined by more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. Besides, we prove that, given a control-affine system satisfying the Lie algebra rank condition (LARC), singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, from both theoretical and implementational points of view.

A Neural Network Adaptive Controller for End-effector Tracking of Redundant Robot Manipulators

In this paper we propose a neural network adaptive controller to achieve end-effector tracking of redundant robot manipulators. The controller is designed in Cartesian space to overcome the problem of motion planning which is closely related to the inverse kinematics problem. The unknown model of the system is approximated by a decomposed structure neural network. Each neural network approximates a separate element of the dynamical model. These approximations are used to derive an adaptive stable control law. The parameter adaptation algorithm is derived from the stability study of the closed loop system using Lyapunov approach with intrinsic properties of robot manipulators. Two control strategies are considered. First, the aim of the controller is to achieve good tracking of the end-effector regardless the robot configurations. Second, the controller is improved using augmented space strategy to ensure minimum displacements of the joint positions of the robot. Simulation examples are also presented to verify the effectiveness of the proposed approach.

On the Motion Planning Problem, Complexity, Entropy, and Nonholonomic Interpolation

We consider the sub-Riemannian motion planning problem defined by a sub-Riemannian metric (the robot and the cost to minimize) and a non-admissible curve to be e-approximated in the sub-Riemannian sense by a trajectory of the robot. Several notions characterize the E-optimality of the approximation: the "metric complexity" MC and the "entropy" E (Kolmogorov-Jean). In this paper, we extend our previous results. 1. For generic one-step bracket-generating problems, when the corank is at most 3, the entropy is ssrelated to the complexity by E = 2πMC. 2. We compute the entropy in the special 2-step bracket-generating case, modelling the car plus a single trailer. The e-minimizing trajectories (solutions of the "e-nonholonomic interpolation problem"), in certain normal coordinates, are given by Euler's periodic inflexional elastica. 3. Finally, we show that the formula for entropy which is valid up to corank 3 changes in a wild case of corank 6: it has to be multiplied by a factor which is at most 3/2.

A continuation method for motion-planning problems

A continuation method for motion-planning problems

A continuation method for motion-planning problems

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned singularities are the abnormal extremals associated to the dynamics of the control system and the Wazewski equation is an o.d.e. on the control space called the Path Lifting Equation (PLE). We first show elementary facts relative to the maximal solution of the PLE such as local existence and uniqueness. Then we prove two general results, a finite-dimensional reduction for the PLE on compact time intervals and a regularity preserving theorem. In a second part, if the Strong Bracket Generating Condition holds, we show, for several control spaces, the global existence of the solution of the PLE, extending a previous result of H.J. Sussmann.

Maneuver-based motion planning for nonlinear systems with symmetries

In this paper, we introduce an approach for the efficient solution of motion-planning problems for time-invariant dynamical control systems with symmetries, such as mobile robots and autonomous vehicles, under a variety of differential and algebraic constraints on the state and on the control inputs. Motion plans are described as the concatenation of a number of well-defined motion primitives, selected from a finite library. Rules for the concatenation of primitives are given in the form of a regular language, defined through a finite-state machine called a Maneuver Automaton. We analyze the reachability properties of the language, and present algorithms for the solution of a class of motion-planning problems. In particular, it is shown that the solution of steering problems for nonlinear dynamical systems with symmetries and invariant constraints can be reduced to the solution of a sequence of kinematic inversion problems. A detailed example of the application of the proposed approach to motion planning for a small aerobatic helicopter is presented.