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## Sampling-Based Motion Planning with Differential Constraints (2005)

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Citations: | 29 - 4 self |

### Citations

2470 |
Robot Motion Planning.
- Latombe
- 1991
(Show Context)
Citation Context ... been extensively used to provide a practical solution by sacrificing completeness, in which a set of sampling points are used to represent the configuration space and construct solutions. Refer 9to =-=[18; 19]-=- for a complete review of methods for the Piano Movers’ Problem. Classical planning methods have been successfully applied in many areas. In Fig. 1.8, the methods have been used to automatically desig... |

1249 | Probabilistic roadmaps for path planning in highdimensional configuration spaces. - Kavraki, Svestka, et al. - 1996 |

1101 | Planning Algorithms.
- LaValle
- 2006
(Show Context)
Citation Context ...er differential constraints in the planning process, such as nonholonomic planning [13] and kinodynamic planing [7], which are together called motion planning with differential constraints (MPD) [4], =-=[14]-=-. While a large amount of practical MPD algorithms have obtained considerable success, there is very little known about existence of exact algorithms that decide whether a solution exists in finite ti... |

758 |
A Decision Method for Elementary Algebra and Geometry, 2nd Edition,
- Tarski
- 1951
(Show Context)
Citation Context ...LE MPD PROBLEMS WITH QUANTIFIER ELIMINATION Computation in Step 9 in Fig. 2 can be achieved through evaluation of the following quantified formula: ∃ũ, ũ ∈U k free ∩U k goal. (17) By Tarski’s theorem =-=[21]-=-, if both Ufree and Ugoal are semialgebraic sets, then quantifier elimination will be able to compute (17) (i.e., Step 9) in finite time. Therefore, we have the following result. Theorem 2: If both Uf... |

611 | Randomized kinodynamic planning,”
- LaValle, Kuffner
- 2001
(Show Context)
Citation Context ...ontrol ũnew to extend the state of ncur to a new state while considering differential constraints. Local planners could sample a piecewise constant acceleration [22; 43], a piecewise-constant control =-=[13; 47]-=-, and non-constant controls [37; 38; 45; 48–50] which could drive systems between two states either in the state space [45], or in a subspace [37; 38; 48; 49]. The search graph is updated with a given... |

586 |
The complezify of robot motion planning.
- Canny
- 1987
(Show Context)
Citation Context ...ral Piano Movers’ Problem in which geometric primitives in the configuration space are polynomials, two complete algorithms were provided based on cylindrical algebraic decomposition [10] and roadmap =-=[11]-=-, whose upper bounds on running time are respectively doubly- and singly- exponential in the dimension of the configuration space. Since complete algorithms for path planning problems take a large amo... |

481 | Spatial planning: A configuration space approach.
- Lozano-Perez
- 1983
(Show Context)
Citation Context ...nitial configuration to the goal configuration 8Figure 1.7: An example of the Piano Movers’ Problem without colliding the rectangular boxes. By the introduction of the concept of configuration space =-=[8]-=-, the robotic system is reduced into a point, the work environment is mapped into a set that the point cannot enter, and the path planning problem is transformed into a pure geometry problem. The solu... |

475 | An Introduction to the Conjugate Gradient Method Without the Agonizing Pain [Online]. Available: http://www.cs.cmu.edu/~jrs/jrspapers.html
- Shewchuk
- 1994
(Show Context)
Citation Context ...he starting point for the steepest descent method might have a good chance to converge faster using other optimization techniques. Therefore, the convergence rate 1 α the starting point is calculated =-=[82]-=- as follows: of the steepest descent method for the nonlinear program at α 2 =1− ( ∑ ( ∑ i ς2 i λ 2 i ) 2 i ς2 i λ3i )(∑i ς2 i λi) , (5.25) in which {λi} are eigenvalues of Jacobian matrix J of the fi... |

448 |
On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents.
- Dubins
- 1957
(Show Context)
Citation Context ...wn about existence of exact algorithms that decide whether a solution exists in finite time. The decidability of MPD problems is unknown, except for 1D [16] and 2D problems [2], and for a Dubin’s car =-=[8]-=- with the point body moving in a polygonal environment [9]. These exact algorithms will improve understanding of the complexity of MPD problems, which provides a sound reason to pursue practical incom... |

429 | Principles of Robot Motion - Theory, Algorithms, and Implementations.
- Choset, Lynch, et al.
- 2005
(Show Context)
Citation Context ...l “or” and “and” operators, {Am,i(q)} are polynomial functions 1 of q, and no and ne are constant positive integers. For the details about computation of Cobs and semi-algebraic sets, please refer to =-=[6]-=-, [14]. 2. Finite-dimensional control space Generally, the control space can be formulated as a semigroup 2 generated from a finite-dimensional control space generator set, denoted as Ū [4], which inc... |

390 | Rapidly-Exploring Random Trees: A New Tool for Path Planning,
- LaValle
- 1998
(Show Context)
Citation Context ... that the PRM-based method could be used to solve MPD problems even an analytical steering method is not available. 145Chapter 6 Reducing Metric Sensitivity for RRT-Based Planners RRT-based planners =-=[61; 83]-=- as sampling-based algorithms have been proposed recently for MPD problems. An RRT achieves rapid exploration by iteratively sampling a random state in the state space and extending the nearest state ... |

363 |
Nonlinear Systems: Analysis, Stability, and Control,
- Sastry
- 1999
(Show Context)
Citation Context ...ch the value of the control ũ at any time t1 in [0,tf] isin the input space U. Foranyũ1 and ũ2 in U, their distance is defined as ρ(ũ1, ũ2) = sup ‖ũ1(s) − ũ2(s)‖. (2.88) s∈[0,min{¯t(ũ1),¯t(ũ2)}] 3 In =-=[52]-=-, the term control is also used to represent an input function. 41U u ~ u t 0 1 f Figure 2.6: An example of the control ũ as a function from a time interval to the input space To define the sample sp... |

353 | Nonholonomic motion planning: Steering using sinusoids,”
- Murray, Sastry
- 1993
(Show Context)
Citation Context ...clude the Dubins’ car [26], Reeds-Sheep Car [27], differential drive [28], differentially flat systems [29], kinematically controllable system [30], nilpotent systems [31], and chained15form systems =-=[32]-=-. Specifically, optimal solutions even exist for steering problems for the Dubins’ car [26], Reeds-Sheep Car [27], and differential drive [28]. However, for general steering problems, numerical method... |

332 | Rapidly-exploring random trees: Progress and prospects.
- Lavalle, Jr
- 2000
(Show Context)
Citation Context ... a single-tree approach, such as the planner in [35], only the initial state node ninit associated with xinit exists in N. In a bi-directional approach, such as the bidirectional RRT-based planner in =-=[55]-=-, a goal state node ngoal associated with a goal state xgoal is also included in N. For planners in [22], a state in the ɛr neighborhood of xinit is associated with ninit. One could even place thousan... |

286 |
Complexity of the mover’s problem and generalizations
- Reif
- 1979
(Show Context)
Citation Context ...nfiguration space from the initial configuration to the goal configuration that does not intersect the set corresponding to the work environment. The Piano Movers’ Problem was shown to be PSPACE-hard =-=[9]-=-, in which geometries of the robot and the work environment are characterized by a finite collection of plane primitives. For general Piano Movers’ Problem in which geometric primitives in the configu... |

267 | Optimal paths for a car that goes both forwards and backwards,”
- Reeds, Shepp
- 1990
(Show Context)
Citation Context ...nholonomic planning problems, have been actively studied in the last two decades. Analytical solutions have been designed for many robotic systems, which include the Dubins’ car [26], Reeds-Sheep Car =-=[27]-=-, differential drive [28], differentially flat systems [29], kinematically controllable system [30], nilpotent systems [31], and chained15form systems [32]. Specifically, optimal solutions even exist... |

256 | Randomized Kinodynamic Motion Planning with Moving Obstacles. - Hsu, Kindel, et al. - 2002 |

240 |
Foundations of differential geometry. Vol
- Kobayashi, Nomizu
- 1996
(Show Context)
Citation Context ...[51] for details. Only a general description of symmetry of robotic systems will be given, and then the scope of the method is limited to a class of systems with symmetries on principle fiber bundles =-=[80]-=-, and finally use the car system in Eq. (2.159) to show as an example of a system in this class. 125τũ(x0, ¯t(ũ)) Ψg Ψg(τũ(x0, ¯t(ũ))) τũ(Ψg(x0), ¯t(ũ)) Ψg x0 Ψg(x0) Figure 5.2: Group actions commute... |

225 | On the ‘piano movers’ problem: II. General techniques for computing topological properties of real algebraic manifolds
- Schwartz, Sharir
- 1983
(Show Context)
Citation Context ...mitives. For general Piano Movers’ Problem in which geometric primitives in the configuration space are polynomials, two complete algorithms were provided based on cylindrical algebraic decomposition =-=[10]-=- and roadmap [11], whose upper bounds on running time are respectively doubly- and singly- exponential in the dimension of the configuration space. Since complete algorithms for path planning problems... |

219 | Real-time motion planning for agile autonomous vehicles
- Frazzoli, Dahleh, et al.
- 2002
(Show Context)
Citation Context ...celeration [22; 43], a piecewise-constant control [13; 47], and non-constant controls [37; 38; 45; 48–50] which could drive systems between two states either in the state space [45], or in a subspace =-=[37; 38; 48; 49]-=-. The search graph is updated with a given updating policy. If 16u~ ( e2) xinit ~ 1 u ( e ) x ( ncur u~ ( enew ) ) x( nx ) x( nnew ) < ε g xgoal ninit e1 ncur n enew x e2 ngoal Figure 1.15: An exampl... |

178 |
Time-optimal control of robotic manipulators along specied paths,”
- Bobrow, Dubowsky, et al.
- 1985
(Show Context)
Citation Context ...lties in escaping local minima when the given metric provides a poor approximation of the true cost-to-go. Instead of designing a good metric, collision information is collected online and a value in =-=[0, 1]-=- is assigned to each node in the search graph. The higher value a node is associated, the more is the number of trajectories, which are extended from the state of the node and have been detected in co... |

159 |
Nonholonomic multibody mobile robots: Controllability and motion planning in the presence of obstacles
- Barraquand, Latombe
- 1993
(Show Context)
Citation Context ...ach iteration, a node ncur in the search graph is firstly chosen by a node selection algorithm. Node selection could use systematic search, such as breadth first search [22; 43], Dijkstra’s algorithm =-=[35; 37]-=-, or non-systematic search, such as randomized search [13; 37; 38; 44–46]. Then a local planner uses a sampled control ũnew to extend the state of ncur to a new state while considering differential co... |

133 | Minimum-Time Control of Robotic Manipulators with Geometric Path Constraints,'' - Shin, McKay - 1985 |

132 |
Geometric control of mechanical systems
- Bullo, Lewis
- 2005
(Show Context)
Citation Context ...th the Newton-Euler mechanics, Lagrangian mechanics, or Hamiltonian mechanics. In this thesis, the dynamics for general systems will be provided without the details of the derivation. Please refer to =-=[51]-=- for details. Assume that the state bundle of a general system has nc-dimensional configuration space C and nv-dimensional velocity space, and a configuration c is represented as a configuration vecto... |

126 | Robust hybrid control for autonomous vehicles motion planning
- Frazzoli, Dahleh, et al.
- 1999
(Show Context)
Citation Context ...he smallest distance between the reachable states and the goal state is larger than the given gap tolerance. In cases in which the set of reachable states is everywhere dense [74], or even continuous =-=[75]-=-, it is possible in principle to add a sequence of sampled controls to move the final state arbitrarily close to the goal state, but this is done at the expense of the efficiency of the trajectory and... |

124 |
A mobile automaton: An application of artificial intelligence techniques
- Nilsson
- 1969
(Show Context)
Citation Context ... model are considered to time-parameterize the path from the first step into a trajectory. Path planning problems appeared around 1970 during the study for experimental mobile robots and manipulators =-=[7]-=-. An example of the path planning problem is shown in Fig. 1.7. The piano is considered as a mobile robot that can move freely in any direction on the ground, the work environment includes the room an... |

123 | Stable pushing: Mechanics, controllability, and planning, - Lynch, Mason - 1996 |

109 | A new class of decidable hybrid systems
- Lafferriere, Pappas, et al.
- 1999
(Show Context)
Citation Context ...ormulation are semialgebraic, then the problem is decidable using the proposed decision algorithm. While this sufficient condition verifies decidability of MPD problems with a class of linear systems =-=[12]-=-, we also obtain a new decidability result for a non trivial MPD problem using similar symbolic computation techniques in [12], [19], in which the Dubin’s car with a polygonal rigid body moves between... |

108 |
Planning smooth paths for mobile robots
- Jacobs, Canny
- 1989
(Show Context)
Citation Context ...em can be incorporated into this formulation. The proposed formulation can be considered as a generalization of the unconventional “configuration space” specially designed for the Dubin’s car in [9], =-=[10]-=- to characterize the trajectories connecting two vertices, instead of the conventional configurations. In this paper, we design an incremental decision algorithm to check the existence of a solution f... |

105 | Choosing Good Distance Metrics and Local Planners for Probabilistic Roadmap Methods”, - Amato, Bayazit, et al. - 2000 |

89 | Billboard clouds for extreme model simplification
- Décoret, Durand, et al.
(Show Context)
Citation Context ...he function. In computer graphics, when there are too many geometric primitives for a model, volumetric visualization [64] could be used to represent the model by a set of points, or billboard clouds =-=[65]-=- could be used to model the geometry by a set of planes. In motion planning, the configuration space of the Piano Movers’ Problem is only a continuous search space and the exact representations, such ... |

83 | The ariadne’s clew algorithm. - Mazer, Ahuactzin, et al. - 1998 |

80 | Nonholonomic motion planning for mobile robots
- Laumond
- 1998
(Show Context)
Citation Context ...ates for the local planner might greatly change the trajectory connecting them. Examples of such local planners include many optimal control based local planners for nonholonomic systems [26; 28]. In =-=[50; 67]-=-, topology properties of steering methods were introduced to reduce completeness of nonholonomic planning to that of path planning. A complete planner using the sinusoidal steering method as the local... |

78 | Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems
- Bullo, Lynch
- 2001
(Show Context)
Citation Context ...ons have been designed for many robotic systems, which include the Dubins’ car [26], Reeds-Sheep Car [27], differential drive [28], differentially flat systems [29], kinematically controllable system =-=[30]-=-, nilpotent systems [31], and chained15form systems [32]. Specifically, optimal solutions even exist for steering problems for the Dubins’ car [26], Reeds-Sheep Car [27], and differential drive [28].... |

77 | Randomized path planning for linkages with closed kinematic chains. - Yakey, LaValle, et al. - 2001 |

77 | A provably good approximation algorithm for optimal-time trajectory planning
- Donald, Xavier
- 1989
(Show Context)
Citation Context ...ontrol ũnew to extend the state of ncur to a new state while considering differential constraints. Local planners could sample a piecewise constant acceleration [22; 43], a piecewise-constant control =-=[13; 47]-=-, and non-constant controls [37; 38; 45; 48–50] which could drive systems between two states either in the state space [45], or in a subspace [37; 38; 48; 49]. The search graph is updated with a given... |

75 | In the piano mover’s problem: III. Coordinating the motion of several independent bodies: The special case of circular bodies moving amidst polygonal barriers
- Schwartz, Sharir
- 1983
(Show Context)
Citation Context ...er decompose complexity of motion planning problems and solve them in two steps [1–5]. In the first stage, the motion planning problem is reduced into a path planning problem or Piano Movers’ Problem =-=[6]-=- by considering only geometries of the 4In some variation of motion planning problems, uncertainties are considered. However, in this thesis, only the basic motion planning problem is considered, in w... |

75 | Control of mechanical systems with second-order nonholonomic constraints: underactuated manipulators,”
- Oriolo, Nakamura
- 1991
(Show Context)
Citation Context ...2 are examples of first-order nonholonomic and inequality constraints, respectively. Second-order constraints include non-integrable equality constraints, called second-order nonholonomic constraints =-=[20]-=-, and inequality constraints over second-order time derivatives of configuration variables. The second-order nonholonomic constraints normally come from underactuated systems for which the number of i... |

75 |
Coordinated motion planning for multiple car-like robots using probabilistic roadmaps
- Svestka, Overmars
- 1995
(Show Context)
Citation Context ... planners in [22], a state in the ɛr neighborhood of xinit is associated with ninit. One could even place thousands of initial nodes in N, as in the case of initializing a probabilistic roadmap (PRM) =-=[14; 56]-=- with uniform random samples from Xfree, and the methods using such initialization are called PRM-based methods. Select node The node selection selects a node from the search graph by giving nodes dif... |

71 |
Feasible trajectories for mobile robots with kinematic and environmental constraints
- Laumond
- 1986
(Show Context)
Citation Context ...dered in the planning process, their solutions are more reliable and efficient. Specifically, if only first-order nonholonomic constraints are considered, it is generally called nonholonomic planning =-=[21]-=- in robotics literature. If second-order constraints are considered, it is called kinodynamic planning in [22]. Although in [22] only fully actuated holonomic systems are considered, kinodynamic plann... |

69 | Kinodynamic motion planning amidst moving obstacles - Kindel, Hsu, et al. - 2000 |

65 | Visibility based probabilistic roadmaps for motion planning - Simeon, Laumond, et al. |

63 |
Dynamic Scaling of Manipulator Trajectories,"
- Hollerbach
- 1984
(Show Context)
Citation Context ... steering angle, denoted as u. The input space is [−0.6, 0.6]. The control space generator set includes all continuous functions from a time interval to the input space with control duration range be =-=[0, 2]-=-. This generator set is continuous 55and uncountable. The motion equation is ⎧ ˙vy = −vxω +(fyf + fyr)/M ⎪⎨ ⎪⎩ ˙ω = (fyfa − fyrb)/I ˙x = vx cos(θ) − vy sin(θ) ˙y = vx sin(θ)+vy cos(θ) ˙θ = ω, (2.159)... |

61 | Reactive path deformation for nonholonomic mobile robots.
- Lamiraux, Bonnafous, et al.
- 2004
(Show Context)
Citation Context ... and inputs could be naturally enforced in the trajectory after symmetry-based perturbations. Besides our work, a similar method is presented in [78] by tailoring the reactive path deformation method =-=[79]-=-. However, their methods did not use special geometric structures of the robotic systems such that their cost function evaluation could be very expensive and it is difficult to enforce the local const... |

56 |
Planning constrained motion
- Fortune, Wilfong
- 1991
(Show Context)
Citation Context ... a solution exists in finite time. The decidability of MPD problems is unknown, except for 1D [16] and 2D problems [2], and for a Dubin’s car [8] with the point body moving in a polygonal environment =-=[9]-=-. These exact algorithms will improve understanding of the complexity of MPD problems, which provides a sound reason to pursue practical incomplete algorithms and helps design of novel verification al... |

55 | An exact algorithm for kinodynamic planning in the plane. Discrete Computational Geometry, 6:461–484
- Canny, Rege, et al.
- 1991
(Show Context)
Citation Context ...s, there is very little known about existence of exact algorithms that decide whether a solution exists in finite time. The decidability of MPD problems is unknown, except for 1D [16] and 2D problems =-=[2]-=-, and for a Dubin’s car [8] with the point body moving in a polygonal environment [9]. These exact algorithms will improve understanding of the complexity of MPD problems, which provides a sound reaso... |

53 |
Convergence of discretization procedures in dynamic programming,” Automatic Control,
- Bertsekas
- 1975
(Show Context)
Citation Context ...ons are applied on two existing algorithms in Section 4.5. 4.1 Related Results Our results are related to many techniques and concepts from a large amount of existing research [22; 35; 50; 66–68]. In =-=[66]-=-, convergence of value functions computed by dynamic programming for discrete-time systems with infinite control set and continuous state space was analyzed. It is shown that the value function could ... |

52 | P.: On the complexity of kinodynamic planning - CANNY, REIF, et al. - 1988 |

52 |
Random number generation and quasi-Monte-Carlo methods
- NIEDERREITER
- 1992
(Show Context)
Citation Context ...d their different runs, i.e, these characterizations will be invariant when the sampling is used in different algorithms and their different runs. The characterization needs the concept of dispersion =-=[63]-=-. For a set B with a norm ‖·‖,asetA consists of sampled points from set B. The dispersion of set A with respect to set B is defined as sup inf ‖a − b‖. (3.22) b∈B a∈A Intuitively, it is the furthest a... |

49 | Time optimal trajectories for bounded velocity differential drive vehicles,
- Balkcom, Mason
- 2002
(Show Context)
Citation Context ...ems, have been actively studied in the last two decades. Analytical solutions have been designed for many robotic systems, which include the Dubins’ car [26], Reeds-Sheep Car [27], differential drive =-=[28]-=-, differentially flat systems [29], kinematically controllable system [30], nilpotent systems [31], and chained15form systems [32]. Specifically, optimal solutions even exist for steering problems fo... |

45 | Resolution complete rapidlyexploring random trees.
- Cheng, LaValle
- 2002
(Show Context)
Citation Context ... 67Check termination condition The termination condition tells whether the algorithm should stop. Some algorithms, such as [13], will stop after a given number of iterations. Some algorithm, such as =-=[35; 36]-=-, will stop when all nodes in the search graph have been explored. 3.2 Descriptions of Sampling-Based Algorithms in the Template In this section, three types of planners, which are the single-directio... |

44 |
On the reachability of quantized control systems,”
- Bicchi, Marigo, et al.
- 2002
(Show Context)
Citation Context ...Usually, the number of states in R∞(xinit) is infinite because Ū is infinite. Intuitively, when Ū is finite as in some problems [53], it seems that R∞(xinit) should be finite. However, it is shown in =-=[53; 54]-=- that R∞(xinit) could be infinite for a class of discrete-time chained-form systems, even with a finite control space generator set. In the following part, Ū is always assumed to be uncountable for co... |

41 | Reducing Metric Sensitivity in Randomized Trajectory Design”,
- Cheng, LaValle
- 2001
(Show Context)
Citation Context ...which the dashed lines discretize the state space. Otherwise, some planners, e.g., [35], discard xnew and no edge is added as shown in the bottom left picture in Fig. 3.4, while other planners, e.g., =-=[62]-=-, discard xnew, but a new edge is inserted as shown in the bottom middle picture from ncur to node nx whose state is in the discretization set . Implicit state space discretization is achieved by requ... |

36 | Steering nonholonomic systems using sinusoids. Twenty-ninth CDC
- Murray, slstry
- 1990
(Show Context)
Citation Context ...celeration [22; 43], a piecewise-constant control [13; 47], and non-constant controls [37; 38; 45; 48–50] which could drive systems between two states either in the state space [45], or in a subspace =-=[37; 38; 48; 49]-=-. The search graph is updated with a given updating policy. If 16u~ ( e2) xinit ~ 1 u ( e ) x ( ncur u~ ( enew ) ) x( nx ) x( nnew ) < ε g xgoal ninit e1 ncur n enew x e2 ngoal Figure 1.15: An exampl... |

28 |
Topological property for collision-free nonholonomic motion planning: The case of sinusoidal inputs for chained form systems
- Sekhavat, Laumond
- 1998
(Show Context)
Citation Context ...ates for the local planner might greatly change the trajectory connecting them. Examples of such local planners include many optimal control based local planners for nonholonomic systems [26; 28]. In =-=[50; 67]-=-, topology properties of steering methods were introduced to reduce completeness of nonholonomic planning to that of path planning. A complete planner using the sinusoidal steering method as the local... |

26 |
Motion Planning with Inertial Constraints”,
- O’Dunlaing
- 1987
(Show Context)
Citation Context ...yhedral obstacles has been proved to be NP-hard [23]. Exact solutions to kinodynamic motion planning problems only exist from point masses with ℓ∞ bounds on velocity and acceleration in one-dimension =-=[24]-=- and two-dimensions [25]. Steering problems, which are obtained by ignoring the work environments in nonholonomic planning problems, have been actively studied in the last two decades. Analytical solu... |

26 | Non-uniform discretization approximations for kinodynamic motion planning and its applications
- Reif, Wang
- 1996
(Show Context)
Citation Context ...ms. For an n-dimensional fully-actuated system, there are n independent inputs such that the theorem in [22] could be proved by n simultaneous independent tracking games in one-dimensional spaces. In =-=[42]-=-, the planner for robots with L2 dynamics bounds was improved to give the first known polynomial approximation algorithm for the curvature-constrained shortest-path problem in three or more dimensions... |

25 |
Kinodynamic planning
- Donald, Xavier, et al.
- 1993
(Show Context)
Citation Context ...der nonholonomic constraints are considered, it is generally called nonholonomic planning [21] in robotics literature. If second-order constraints are considered, it is called kinodynamic planning in =-=[22]-=-. Although in [22] only fully actuated holonomic systems are considered, kinodynamic planning can also take into account underactuated systems. With these new techniques, problems in Fig. 1.13, in whi... |

24 |
Onder, "Time-Optimal Motions of Robots in Assembly Tasks
- Geering, Guzzella, et al.
- 1986
(Show Context)
Citation Context ...use the performance of these numerical techniques depends very much on initial guesses of the solutions and has slow convergence on nonlinear problems, this method is only practical for limited cases =-=[33; 34]-=-. Considering the work environments, most of current algorithms use samplingbased techniques [13; 22; 35–42], in which a search graph is incrementally built to construct solutions with sampled control... |

23 | Time-optimal trajectories for a robot manipulator: a provably good approximation algorithm - Heinzinger, Jacobs, et al. - 1990 |

23 | motion planning and trailer systems - Flatness - 1993 |

20 | Multiple query probabilistic roadmap planning using single query primitives - Bekris, Chen, et al. - 2003 |

19 | Optimum Path Planning for Mechanical Manipulators - Luh, Lin - 1981 |

17 |
Bicchi: Motion planning through symbols and lattices
- Pancanti, Pallottino, et al.
(Show Context)
Citation Context ...tudied. Conditions for the reachable set to be either dense or a lattice structure were studied for linear systems and nonholonomic chained-form systems with different discrete control set. Recently, =-=[68]-=- shows that with enough controls to achieve a reachable set with the lattice structure, any trajectories could be tracked with a given tolerance. However, the results are limited to driftless nilpoten... |

15 |
An exact algorithm for kinodynamic planning
- Canny, Rege, et al.
- 1990
(Show Context)
Citation Context ...en proved to be NP-hard [23]. Exact solutions to kinodynamic motion planning problems only exist from point masses with ℓ∞ bounds on velocity and acceleration in one-dimension [24] and two-dimensions =-=[25]-=-. Steering problems, which are obtained by ignoring the work environments in nonholonomic planning problems, have been actively studied in the last two decades. Analytical solutions have been designed... |

15 |
Efficient Algorithm for Time Optimal Control o a Two-Link Manipulator",
- Meier, Bryson
- 1990
(Show Context)
Citation Context ...use the performance of these numerical techniques depends very much on initial guesses of the solutions and has slow convergence on nonlinear problems, this method is only practical for limited cases =-=[33; 34]-=-. Considering the work environments, most of current algorithms use samplingbased techniques [13; 22; 35–42], in which a search graph is incrementally built to construct solutions with sampled control... |

15 |
Randomized searches and nonlinear programming in trajectory planning
- Karatas, Bullo
(Show Context)
Citation Context ...a piecewise constant acceleration [22; 43], a piecewise-constant control [13; 47], and non-constant controls [37; 38; 45; 48–50] which could drive systems between two states either in the state space =-=[45]-=-, or in a subspace [37; 38; 48; 49]. The search graph is updated with a given updating policy. If 16u~ ( e2) xinit ~ 1 u ( e ) x ( ncur u~ ( enew ) ) x( nx ) x( nnew ) < ε g xgoal ninit e1 ncur n ene... |

14 | Kinodynamic motion planning: connecting exploration trees using trajectory optimization methods
- Lamiraux, Ferré, et al.
- 2004
(Show Context)
Citation Context ... of our method is that many local constraints on states and inputs could be naturally enforced in the trajectory after symmetry-based perturbations. Besides our work, a similar method is presented in =-=[78]-=- by tailoring the reactive path deformation method [79]. However, their methods did not use special geometric structures of the robotic systems such that their cost function evaluation could be very e... |

14 |
Motion planning for nonlinear underactuated vehicles using H-infinity techniques
- Toussaint, Başar, et al.
- 2000
(Show Context)
Citation Context ...RT toward the random state with a set of sampled controls. Various RRT-based planners have been designed recently for autonomous vehicles motion planning [84] and for nonlinear underactuated vehicles =-=[85]-=-. In spite of the successes of RRTs, one of the key shortcomings is the sensitivity of their performance with respect to a chosen metric. The metric helps to select a node in the search graph in each ... |

14 | Sampling-based falsification and verification of controllers for continuous dynamic systems
- Cheng, Kumar
- 2006
(Show Context)
Citation Context ...xact algorithms will improve understanding of the complexity of MPD problems, which provides a sound reason to pursue practical incomplete algorithms and helps design of novel verification algorithms =-=[5]-=- using motion planning techniques. The concept of the configuration space established a fundamental abstract formulation for general path planning problems [15]. Workspace obstacles are mapped into co... |

12 | Trajectory planning for second-order underactuated mechanical systems in presence of obstacles - Choudhury, Lynch - 2002 |

12 | Steering driftless nonholonomic systems by control quanta - Marigo, Bicchi - 1998 |

12 | Improving the performance of sampling-based planners by using a symmetry-exploiting gap reduction algorithm
- Cheng, Frazzoli, et al.
- 2004
(Show Context)
Citation Context ...rk environment, and in other local planners 62u new ~ x = f x new u new ~ ≤ εl x f xnew x ( ncur ) x( ncur ) Figure 3.2: The trajectories generated from two types of connecting local planners (e.g., =-=[58; 59]-=-) constraints from the work environment are considered. The trajectories generated by these local planners may either: 1) succeed in exactly reaching xnew, 2) succeed in reaching a given neighborhood ... |

12 | On computing time-optimal motions of robotic manipulators in the presence of obstacles - Shiller, Dubowsky - 1991 |

12 | Reachability analysis for a class of quantized control systems,” in
- Marigo, Piccoli, et al.
- 2000
(Show Context)
Citation Context ... path could be found if the smallest distance between the reachable states and the goal state is larger than the given gap tolerance. In cases in which the set of reachable states is everywhere dense =-=[74]-=-, or even continuous [75], it is possible in principle to add a sequence of sampled controls to move the final state arbitrarily close to the goal state, but this is done at the expense of the efficie... |

11 | Completeness in robot motion planning
- Goldberg
- 1995
(Show Context)
Citation Context ...f real numbers are approximated with floating point numbers in the computer, then state mismatches will also be induced. The floating point approximation is an important factor affecting completeness =-=[72]-=-. When floating point numbers are used, it is too complicated to explicitly manipulate floating point calculation error; therefore, it is assumed that these errors happen only when the state is stored... |

10 |
B ,(2001) Controllability for discrete systems with a finite control set
- Chitour, Piccoli
(Show Context)
Citation Context ...ibe the sampling set for the state space sampling in Section 3.3.1. k=0 Usually, the number of states in R∞(xinit) is infinite because Ū is infinite. Intuitively, when Ū is finite as in some problems =-=[53]-=-, it seems that R∞(xinit) should be finite. However, it is shown in [53; 54] that R∞(xinit) could be infinite for a class of discrete-time chained-form systems, even with a finite control space genera... |

10 | Exploiting group symmetries to improve precision in kinodynamic and nonholonomic planning
- Cheng, LaValle
- 2003
(Show Context)
Citation Context ...rollable system [30], the system needs to stop completely when switching between decoupling vector fields during the steering process. Alternatively, a symmetry-based perturbation method is developed =-=[59; 77]-=- to reduce the gaps by perturbing the solution candidate. The comparison of gap reduction 121xinit The trajectory after perturbation xgoal xf The original trajectory with the gap Gap reduction by ste... |

9 |
Motion planning for controllable systems without drift
- Laffierriere, Sussman
- 1991
(Show Context)
Citation Context ...or many robotic systems, which include the Dubins’ car [26], Reeds-Sheep Car [27], differential drive [28], differentially flat systems [29], kinematically controllable system [30], nilpotent systems =-=[31]-=-, and chained15form systems [32]. Specifically, optimal solutions even exist for steering problems for the Dubins’ car [26], Reeds-Sheep Car [27], and differential drive [28]. However, for general st... |

8 | Heuristics (Addison-Wesley - Pearl - 1984 |

7 | Minimum-Time Along the Path for a Mechanical Arm - Luh, Walker - 1977 |

7 | Vehicle rollover on smooth surface - Bernard, Shannan, et al. - 1991 |

7 | Lowdiscrepancy curves and efficient coverage of space
- Ramamoorthy, Rajagopal, et al.
- 2006
(Show Context)
Citation Context ... done on sampling techniques in the state space. It is expected that research on sampling techniques in the control space will also improve design and analysis of MPD algorithms [1], [4], [14], [17], =-=[18]-=-. For example, the structure of the reachable set of quantized control systems under a given discrete set of sample controls [1] leads to an efficient MPD algorithm in [17]. Research on the relationsh... |

7 | Semilinear motion planning in REDLOG
- Weispfenning
- 2001
(Show Context)
Citation Context ...ction (see [12] for details). Therefore, the corresponding Uobs, Ufree, and Ugoal will be semi-algebraic. As a special case of [12], it can also be shown that Uobs, Ufree, and Ugoal of the problem in =-=[22]-=- are also semi-algebraic because the trajectory is piecewise-linear. B. A decidable MPD problem for the Dubin’s car The MPD problem for the Dubin’s car with a point body moving in a polygonal environm... |

6 | A catalog of inverse-kinematics planners for underactuated systems on matrix groups
- Martínez, Cortés, et al.
- 2003
(Show Context)
Citation Context ...e durations of the coasting trajectories to achieve the desired group action is called trajectory planning via inverse kinematics in [30], which generally has numerical solutions except for few cases =-=[81]-=-. 5.2.5 Selection of a subspace for optimization Generally, if a gap exists in a space of dimension ng, perturbing ng parameters of the final state could eliminate the gap. If the number of parameters... |

6 |
SE(2)- snakes and motion control: A study of the roller racer
- Oscillations
- 1998
(Show Context)
Citation Context ...losed chains. Furthermore, the properties of motion equations are desirable, which will tell the mobile ability and limit of the robotic system. For example, the nonslipping model of the roller racer =-=[88]-=- is proved that it cannot be stopped only by the steering angle control if it starts moving, which tells that it is impossible to design a trajectory for the model to move from one configuration at re... |

5 | Flatness and defect of nonlinear systems - Fliess, Levine, et al. - 1993 |

5 |
Reducing RRT metric sensitivity for motion planning with differential constraints
- Cheng
- 2002
(Show Context)
Citation Context ...tion for topology property on the local planner is presented and used to provide precise resolution completeness conditions for general sampling-based MPD planners using connecting local planners. In =-=[36; 71]-=-, resolution completeness conditions for RRT-based planners were developed based on Lipschitz conditions of the motion equation and analysis of the reachability graph and the search graph. It provided... |

3 |
Collision free trajectory planning of a 3-DoF robot with a passive joint
- Lynch, Shiroma, et al.
- 2000
(Show Context)
Citation Context ...celeration [22; 43], a piecewise-constant control [13; 47], and non-constant controls [37; 38; 45; 48–50] which could drive systems between two states either in the state space [45], or in a subspace =-=[37; 38; 48; 49]-=-. The search graph is updated with a given updating policy. If 16u~ ( e2) xinit ~ 1 u ( e ) x ( ncur u~ ( enew ) ) x( nx ) x( nnew ) < ε g xgoal ninit e1 ncur n enew x e2 ngoal Figure 1.15: An exampl... |

3 |
Volumetric visualization of 3D data
- Russell, Miles
- 1993
(Show Context)
Citation Context ...vailable, a set of sample points could be used to approximate the closed set and the function. In computer graphics, when there are too many geometric primitives for a model, volumetric visualization =-=[64]-=- could be used to represent the model by a set of points, or billboard clouds [65] could be used to model the geometry by a set of planes. In motion planning, the configuration space of the Piano Move... |

3 | Parameter synthesis in robot motion planning using symbolic reachability computations
- Lafferriere, Pappas, et al.
- 2000
(Show Context)
Citation Context ...lem are semi-algebraic sets, then the MPD problem is decidable. The following two types of MPD problems satisfy these conditions. The first one is the MPD problems for a class of linear systems [12], =-=[11]-=-. The second one is an MPD problem for the Dubin’s car with a polygonal rigid body moving in a polygonal environment. The key to show the decidability of these problems is to find nontrivial transform... |

1 |
Planning Algorithms ([Online
- LaValle
- 2004
(Show Context)
Citation Context ... been extensively used to provide a practical solution by sacrificing completeness, in which a set of sampling points are used to represent the configuration space and construct solutions. Refer 9to =-=[18; 19]-=- for a complete review of methods for the Piano Movers’ Problem. Classical planning methods have been successfully applied in many areas. In Fig. 1.8, the methods have been used to automatically desig... |

1 |
curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents
- “On
- 1957
(Show Context)
Citation Context ...ork environments in nonholonomic planning problems, have been actively studied in the last two decades. Analytical solutions have been designed for many robotic systems, which include the Dubins’ car =-=[26]-=-, Reeds-Sheep Car [27], differential drive [28], differentially flat systems [29], kinematically controllable system [30], nilpotent systems [31], and chained15form systems [32]. Specifically, optima... |

1 |
Inverse kinematics-based motion planning for underactuated systems
- Choudhury, Stephens, et al.
- 2004
(Show Context)
Citation Context ...d to reduce completeness of nonholonomic planning to that of path planning. A complete planner using the sinusoidal steering method as the local planner was also designed for chained form systems. In =-=[70]-=-, an inverse kinematics-based planner for underactuated systems is proved to be complete using topology properties. In this thesis, a Lipschitz condition as a sufficient condition for topology propert... |

1 | Nonlinear systems (Nacmillan Pulishing - Khalil - 2002 |

1 |
Bidirectional heuristic search reconsidered.” JAIR
- Kaindl, Kainz
- 1997
(Show Context)
Citation Context ...ssible that the original planner might continue to explore the state space even there exists a solution. The exploration frontiers of two search trees passing through each other was also mentioned in =-=[86]-=-. To overcome the above problem, whether each new node in one tree is within a specified distance to any node in the other tree is test. Although costly, it generally leads to reliable performance bec... |

1 |
Multiple query probabilistic roadmap planning using single query primitives
- Heuristics
- 1984
(Show Context)
Citation Context ... algorithm is used, as in [35], then its node selection selects a node that has untried controls and the shortest distance to xinit. Other possibilities are depth-first, breadth-first [22; 43], or A∗ =-=[57]-=-. In the case of an RRT-based planner [13], a random state xrand is generated in X, and then the node whose state is the nearest (with respect to a distance function on X) to state xrand over all node... |