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Trajectory planning for second-order underactuated mechanical systems in presence of obstacles
... Both of these methods are widely used in practice today, and have been adapted to handle many extensions of the motion planning problem. RRTs have been used in several applications, and many variants have been developed [23, 30, 33, 42, 45, 49, 56, 71, 75, 97, 99, 98, 103, 127, 132, 133, 190, 191, 198, 208, 212]. The original PRM, along with its numerous extensions and variants [1, 6, 16, 17, 27, 34, 38, 49, 93, 120, 125, 126, 154, 158, 181, 199, 207, 208, 215], have been applied to problems in robotics, computer animation, and computational biology [105, 157, 186]. ...
... Kinodynamic planning was introduced by [61] to refer to the problem in which both velocity and acceleration bounds need to be satisfied. Other works on this topic include [36, 43, 45, 60, 59, 68, 84, 89, 124]. Trajectory planning approaches decompose the problem into two parts: first planning a path for the system without considering the constraints; then, transforming the path to satisfy the differential constraints [87, 168, 174, 175, 176, 177, 178]. ...
In its original formulation, the motion planning problem considers the search of a robot path from an initial to a goal configuration. The study of motion planning has advanced significantly in recent years, in large part due to the development of highly successful sampling and searching techniques. Recent advances have influenced sampling-based motion planning algorithms to be used in disparate areas such as humanoid robotics, automotive manufacturing, architecture, computational geography, computer graphics, and computational biology. Many of these methods work well on a large set of problems, however, they have weaknesses and limitations. This thesis expands the basic motion planning techniques to include critical concerns that are not covered by the motion planning algorithms that are in widespread use now. The technical approach is organized around three main thrusts: 1) the development of efficient nearest neighbor searching techniques for spaces arising in motion planning; 2) the development of uniform sampling techniques on these spaces to allow resolution completeness in sampling-based planning algorithms; and 3) the development of guided sampling techniques for efficient exploration on such spaces. We show that most of the modern motion planners incorporate one or more of these components; therefore, addressing these core issues in motion planning does not only lead to a more fundamental understanding of the problem, but also to more efficient practical algorithms. Our results include algorithms addressing the issues, theoretical analysis of their performance and experimental evaluation on motion planning problems.
... Some path planning problem which have a holonomic constraints is solved using RRT [10]. Moreover with a successful application of RRT, problems with complicated geometry was addressed using RRT [11] [12]. Extension of RRT with kinodynamic problems are also existed in [13] [14][15] [16]. ...
In this paper, we present a path planning algorithm of a 7DOF manipulator with RRT(Rapidly-exploring Random Trees). The method presents a occlusion-free and collision-free path planning algorithm. First, we present a basics of RRT algorithm which is a basic searching method for a path planning of this paper. Second, the implementation of a occlusion-free path constraints is presented. Third, the collision-avoiding path planning constraints is presented. Fourth, the integration of RRT with these two constraints is shown. The proposed method is verified by a numerical simulation with Matlab and v-rep robotics simulation tool.
... Randomlyexploring Random Trees (RRTs) [18], [19], mainly based on single-query methods, have gained popularity from their good performances. This has lead to a number of extensions specifically targeted to the solution of complicated geometrical problems [20], such as the deterministic resolution-complete alternatives that have been proposed to replace the random sampling methods in [21]. ...
A novel method which combines an optimised global
path planner with real-time sensor-based collision avoidance
capabilities in order to avoid moving obstacles (e.g. people) in
a complex environment is presented. The strategy is based on
a time efficient one step path planning algorithm for navigating
a large robotic platform in indoor environments. The planner,
which has been proved to compare favourably to currently
available path planning algorithms such as Randomly-exploring
Random Trees (RRTs) and Probabilistic Road Maps (PRMs)
in known static conditions, is enhanced here with a modified
Variable Speed Force Field (V SF 2 ) mechanism to accommodate
for dynamic changes of the environment. The basic concept
of the modified DV SF 2 is to generate a continually changing
parameterised familiy of virtual force fields for the robot based
on characteristics such as location, travelling speed, heading and
dimension of all the objects present in the vicinity, static and
dynamic. The interactions among the repulsive forces associated
with the various obstacles provide a natural way for local collision
avoidance and situational awareness. This is harnessed here by
locally modifying the planned behaviour of the moving platform
in real time, whilst preserving as much as possible the optimised
nature of the global path. Furthermore, traversability of the
path is continually monitored by the global planner to trigger
a complete re-planning from the robot’s current location in the
case of major changes to the environment, most notably when the
path is completely blocked by an obstacle. Overall, a complete
solution to the navigational problem in partially known cluttered
environments is provided.
... Randomlyexploring Random Trees (RRTs) [12], [18] are mainly based on single-query methods. They have gained popularity from their good performances, which has lead to a number of extensions specifically targeting the solution to complicated geometrical problems [17], such as the deterministic resolution-complete alternatives that have been proposed to replace the random sampling methods in [19]. In many cases, an optimal and not just a feasible path is required. ...
A novel method which combines an optimised global
path planner with real-time sensor-based collision avoidance
capabilities in order to avoid moving obstacles (e.g. people) in
a complex environment is presented. The strategy is based on
a time efficient one step path planning algorithm for navigating
a large robotic platform in indoor environments. The planner,
which has been proved to compare favourably to currently
available path planning algorithms such as Randomly-exploring
Random Trees (RRTs) and Probabilistic Road Maps (PRMs)
in known static conditions, is enhanced here with a modified
Variable Speed Force Field (V SF^2) mechanism to accommodate
for dynamic changes of the environment. The basic concept
of the modified DV SF^2 is to generate a continually changing
parameterised familiy of virtual force fields for the robot based
on characteristics such as location, travelling speed, heading and
dimension of all the objects present in the vicinity, static and
dynamic. The interactions among the repulsive forces associated
with the various obstacles provide a natural way for local collision
avoidance and situational awareness. This is harnessed here by
locally modifying the planned behaviour of the moving platform
in real time, whilst preserving as much as possible the optimised
nature of the global path. Furthermore, traversability of the
path is continually monitored by the global planner to trigger
a complete re-planning from the robot’s current location in the
case of major changes to the environment, most notably when the
path is completely blocked by an obstacle. Overall, a complete
solution to the navigational problem in partially known cluttered
environments is provided.
... Randomly-exploring Random Trees (RRTs) [16,17] are mainly based on single-query methods. They have gained popularity from their good performances, which has lead to a number of extensions specifically targeting the solution to complicated geometrical problems [18], such as the deterministic resolution-complete alternatives that have been proposed to replace the random sampling methods in [19]. In many cases, an optimal and not just a feasible path is required. ...
In this paper we present a time efficient one step path planning
algorithm for navigating a large robotic platform in indoor
environments. The proposed strategy, based on the generation
of a novel search space [1], relies on non-uniform
density sampling of the free areas to direct the computational
resources to troubled and difficult regions, such as narrow
passages, leaving the larger open spaces sparsely populated.
A smoothing penalty is also associated to the nodes to encourage
the generation of gentle paths along the middle of
the empty spaces. Collision detection is carried out off-line
during the creation of the configuration space to speed up
the actual search for the path, which is done on-line. Results
compared to currently available path planning algorithms
such as Randomly-exploring Random Trees (RRTs)
and Probabilistic Road Maps (PRMs) proved that the proposed
approach considerably reduces the searching time and
produces smoother paths with less jagged path segments than
those from randomized planners.
... RRTs (LaValle and Kuffner, 2000); Cheng and LaValle, 2002) mainly based on single-query methods, have gained popularity from their good performances. This has lead to a number of extensions specifically targeted to the solution of complicated geometrical problems (Choudhury and Lynch, 2002), such as the deterministic resolution-complete alternatives that have been proposed to replace the random sampling methods in Kim and Ostrowski (2003). ...
A novel method which combines an optimised global path planner with a real-time sensor-based collision avoidance mechanism to accommodate for dynamic changes in the environment (e.g., people) is presented. The basic concept is to generate a continually changing parameterised family of virtual force fields for the robot based on characteristics such as location, travelling speed and dimension of the objects in the vicinity, static and dynamic. The interactions among the repulsive forces associated with the various obstacles provide a natural way for local collision avoidance in a partially known cluttered environment. This is harnessed by locally modifying the planned behaviour of the moving platform in real-time, whilst preserving the optimised nature of the global path. Furthermore, path traversability is continually monitored by the global planner to trigger a complete path re-planning from the current location in case of major changes, most notably when the path is completely blocked by obstacles.
... Randomlyexploring Random Trees (RRTs) [12], [18] are mainly based on single-query methods. They have gained popularity from their good performances, which has lead to a number of extensions specifically targeting the solution to complicated geometrical problems [17], such as the deterministic resolution-complete alternatives that have been proposed to replace the random sampling methods in [19]. In many cases, an optimal and not just a feasible path is required. ...
This paper presents a one step smooth and efficient path planning algorithm for navigating a large robotic platform in known cluttered environments. The proposed strategy, based on the generation of a novel search space, relies on non-uniform density sampling of the free areas to direct the computational resources to troubled and difficult regions, such as narrow passages, leaving the larger open spaces sparsely populated. A smoothing penalty is also associated to the nodes to encourage the generation of gentle paths along the middle of the empty spaces. Collision detection is carried out off-line during the creation of the configuration space to speed up the actual search for the path, which is done on-line. Results prove that the proposed approach considerably reduces the search space in a meaningful and practical manner, improving the computational cost of generating a path optimised for fine and smooth motion
... The performance success of the RRTs on many motion planning problems has led to broad extensions and applications of this approach. For example, problems with complicated geometries are handled with the RRT-based planners in [7], [10]. Extensions of RRTs for manipulation problems and motions of closed articulated chains are developed in [8], [17], [18], [33]. ...
Sampling-based planners have solved difficult problems in many applications of motion planning in recent years. In particular, techniques based on the Rapidly-exploring Random Trees (RRTs) have generated highly successful single-query planners. Even though RRTs work well on many problems, they have weaknesses which cause them to explore slowly when the sampling domain is not well adapted to the problem. In this paper we characterize these issues and propose a general framework for minimizing their effect. We develop and implement a simple new planner which shows significant improvement over existing RRT-based planners. In the worst cases, the performance appears to be only slightly worse in comparison to the original RRT, and for many problems it performs orders of magnitude better.
... In particular, techniques based on the Rapidly-exploring Random Trees (RRTs) have been highly successful in many applications of motion planning [19], [20], [20]. They have been used to handle complicated geometries [5], [7] , manipulation problems and motions of closed articulated chains [6], [14], [15], kinodynamic and nonholonomic planning [8], [17], [18], [20]. Even though RRTs work well on many problems, there are cases when they perform poorly. ...
Sampling based planners have become increasingly efficient in solving the problems of classical motion planning and its applications. In particular, techniques based on the rapidly-exploring random trees (RRTs) have generated highly successful single-query planners. Recently, a variant of this planner called dynamic-domain RRT was introduced by Yershova et al. (2005). It relies on a new sampling scheme that improves the performance of the RRT approach on many motion planning problems. One of the drawbacks of this method is that it introduces a new parameter that requires careful tuning. In this paper we analyze the influence of this parameter and propose a new variant of the dynamic-domain RRT, which iteratively adapts the sampling domain for the Voronoi region of each node during the search process. This allows automatic tuning of the parameter and significantly increases the robustness of the algorithm. The resulting variant of the algorithm has been tested on several path planning problems.
... Finally, we describe Rapidly-exploring Random Trees (RRTs) [42, 44], which were developed for problems with differential constraints, such as kinodynamic planning and nonholonomic planning. Its introduction has stimulated a flurry of recent applications and extensions (e.g., [13, 16, 19, 20, 21, 27, 36, 37, 39, 47, 67, 70]). In its basic form, the RRT attempts to grow a tree from the initial configuration to the goal configuration as follows: take a random sample, and find its nearest neighbor in the search tree. ...
In this paper, we discuss the fleld of sampling-based motion planning. In contrast to methods that con- struct boundary representations of conflguration space obstacles, sampling-based methods use only information from a collision detector as they search the conflguration space. The simplicity of this approach, along with in- creases in computation power and the development of ef- flcient collision detection algorithms, has resulted in the introduction of a number of powerful motion planning algorithms, capable of solving challenging problems with many degrees of freedom. First, we trace how sampling- based motion planning has developed. We then discuss a variety of important issues for sampling-based motion planning, including uniform and regular sampling, topo- logical issues, and search philosophies. Finally, we ad- dress important issues regarding the role of randomiza- tion in sampling-based motion planning.
... Whereas path planning is more common, planning under differential constraints has been addressed since Laumond introduced nonholonomic planning to the community [3]. Some approaches to planning under differential constraints are based on decomposing the problem into two parts: first, planning a path for the system without considering the constaints; then, transforming the path to satisfy the constraints [4] [8]. These planners have the same completeness properties as the underlying path planner. ...
In this paper, we present an incremental, multiresolution motion planning algorithm designed for systems with differential constraints. Planning for these systems is more difficult than ordinary path planning due to the presence of momentum (drift) or nonholonomic velocity constraints. Given a motion planning problem for such a system and that a solution to the problem exists, then a finite reachability graph containing a solution trajectory is guaranteed to exist, under very reasonable conditions. In general, this graph can be generated using sufficiently dense input space sampling, sufficiently small time step, and sufficiently large tree depth. We show how to find and search such a tree in an incremental, multiresolution way. We prove the completeness of our algorithm, discuss related practical concerns, and show experimental results for several systems
... The final state of the trajectory is x new . The sampled controls could be constant [2], [6], [11], or kinematically decoupling vector fields [8]. The durations of these sampled controls could be a fixed value [6], [11] or a varying value [9]. ...
Sampling-based nonholonomic and kinodynamic planning iteratively constructs solutions with sampled controls. A constructed trajectory is returned as an acceptable solution if its ldquogaps,rdquo including discontinuities within the trajectory and mismatches between the terminal and goal states, are within a given gap tolerance. For a given coarseness in the sampling of the control space, finding a trajectory with a small gap tolerance might be either impossible or extremely expensive. In this paper, we propose an efficient trajectory perturbation method, which complements existing steering and perturbation methods, enabling these sampling-based algorithms to quickly obtain solutions by reducing large gaps in constructed trajectories. Our method uses system symmetry, e.g., invariance of dynamics with respect to certain state transformations, to achieve efficient gap reduction by evaluating trajectory final state with a constant-time operation, and, naturally, generating the admissible perturbed trajectories. Simulation results demonstrate dramatic performance improvement for unidirectional, bidirectional, and PRM-based sampling-based algorithms with the proposed enhancement with respect to their basic counterparts on different systems: one with the second-order dynamics, one with nonholonomic constraints, and one with two different modes.
In applications of the navigation and control of unmanned ground vehicles in a cross-country environment and tele-driving a rover on the unknown lunar surface for scientific exploration, human-computer interactive path planning and planned path tracking is a significant way of teleoperation. In this paper, we use the method based on Rapidly-Exploring Random Trees (RRTs) to solve the robot path planning problems in a complex environment with crowded obstacles. Aiming at the problem of efficiency reduction that original RRTs planners have difficulties in automatically finding a resolution path in crowded regions of the robot's configuration space, A novel artificial-guided RRTs (AG-RRTs) planner based on multiple RRTs will be introduced. The AG-RRTs planner improves the efficiency of connection and mergence of multiple RRTs by artificial-guided waypoints which mark the regions of narrow passages. In our experiments, the AG-RRTs planner yields much better performance in complex environments than original RRTs planners, which validates the effectiveness and practicality of this new method.
Motion planning with both obstacles and differential constraints, which includes both nonholonomic and kinodynamic motion planning, has attracted substantial interest in recent years. The most dramatic trend in the last couple of decades has been the rise of sampling-based techniques. Since time, control spaces, and state spaces of most robotic systems are continuous, most approaches will first sample time and possibly the control space of a given problem to obtain a discretized problem, then a search graph is incrementally constructed by integrating the system for selected controls applied from reached states. It is essential to guarantee resolution completeness for a given discretized problem, that is, to carefully design these algorithms and choose sampling parameters so that in finite time they will either find a solution or correctly report that one does not exist at that resolution. For the design of the algorithms, we introduce an algorithmic framework describing characteristics of resolution completeness. For choosing parameters, we present an analysis framework based on the relationship between the reachability graph, which encodes the set of all reachable states for the system, and a search graph, which are states reached by a planning algorithm. Due to complications such as control space sampling, state space sampling, and numerical computation errors, the search graph might not necessarily be even a subset of the reachability graph. Both frameworks are combined to prove resolution completeness conditions for general systems and sampling-based planning algorithms. Advantages of our conditions include: 1) precise bounds on the algorithm parameters are given to decide whether the desired solution exists in finite time, 2) bounds are based primarily on Lipschitz conditions without assumptions on the controllability of the system, 3) assistance can be provided with practical issues, such as choosing of numerical integration methods and arithmetic precision. To demonstrate the general approach, we apply the theorems of the paper to improve the analysis of both RRT-based search and the nonholonomic planner of Barraquand and Latombe.
As mobile robots operate with limited resources which they carry onboard in large obstructed environments, their success is dependent on how efficiently they move while they avoid collision with obstacles and other robots. Moving optimally is the ultimate efficiency a mobile robot can achieve. Therefore, planning optimal motions and devising optimal coordination strategies are two important and challenging fundamental problems in mobile robotics, which have received significant attention in the last couple of decades. Both of those problems can be reduced to shortest path, or equivalently {\em geodesic}, problems in appropriate geometric settings. Geodesic problems have been studied in two disciplines: 1) optimal control theory, and 2) computational geometry. Optimal control theory has focused on the differential constraints of robotic systems, while computational geometry has focused on shortest path problems in an environment with obstacles. Optimal control theory has historically disregarded obstacles in the environment, and computational geometry does not consider dynamics of the robotic system, various optimality criteria, or multi-objective optimality. While each discipline has its own powerful tools to address some geodesic problems, there is a large class of problems that cannot be solved using existing algorithms and methods. We introduce a unified approach that is inspired by main results in both disciplines. In this dissertation, we demonstrate our technique, which combines the celebrated Pontryagin Maximum Principle from optimal control theory with visibility graph methods from computational geometry, by solving three geodesic problems for mobile robots: 1) geodesics for the differential drive among obstacles, 2) geodesics for a kinematic airplane, and 3) optimal coordination of two polygonal robots moving on a predetermined network of paths. We consider the differential drive because it is ubiquitous in mobile robotics. To obtain a well-defined notion of shortest, the total amount of wheel rotation is optimized. We analytically characterize minimum wheel-rotation trajectories in the absence of obstacles, and identify 52 different minimum wheel-rotation trajectories. In the presence of obstacles, every minimum wheel-rotation trajectory is composed of two kinds of subtrajectories: on the boundary of obstacles and in the interior of collision-free space. We prove those subtrajectories that lie in the interior of collision-free space are tangent to the obstacles at both ends. The bitangency condition yields a \emph{nonholonomic bitangency graph} which is a network of collision-free trajectories in which the solution is sought. In general, our nonholonomic bitangency graph is a 2-dimensional subset of the 3-dimensional configuration space of the robot. Therefore, further optimization or a continuous search may be required to answer queries. However if obstacles are circular and far enough from one another, the graph is a 1-dimensional subset of the configuration space and any graph search algorithm, such as Dijkstra's algorithm, extracts the solution. In the second problem, we introduce a new kinematic airplane model. Our airplane is a natural extention of the Dubins car \cite{Dub57}, and extends it with an additional configuration variable for the altitude. We use the Pontryagin Maximum Principle to analytically characterize geodesics for it. To obtain a notion of shortest, time is optimized. Finally, we present an algorithm for computing optimal coordination of two polygonal robots without differential constraints. Each robot has a reference point that must lie on a network of paths called a roadmap. Each robot wants to move from its given initial location to its goal location without colliding with the other one. Rather than impose an a priori cost scalarization for choosing the best combined motion, we consider finding motions whose cost vectors are Pareto-optimal. Pareto-optimal coordination strategies are the ones for which there exists no strategy that would be better for both robots. The problem is equivalent to computing geodesics in the coordination space which is the Cartesian product of the roadmap with itself. We extend visibility graphs to solve the problem. If the roadmap is acyclic, then our algorithm has O(mn^2\log n) time complexity, in which m is the number of paths in the roadmap, and n is the number of coordination space vertices. For cyclic roadmaps, our algorithm computes solutions in time O(2^{5\alpha}m^{1+5\alpha} n^2 \log(m^{2\alpha}n)), in which \alpha = 1+\lceil (5\ell+r)/b \rceil where \ell is total length of the roadmap, r is total length of coordination space obstacle boundary, and b is the length of the shortest edge in the roadmap.
In this article, the role of motion planning in the hierarchical architecture of intelligent vehicle is discussed. The development of motion planning is reviewed. Finally a promising motion planning technology of rapidly-exploring random trees is introduced.
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