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## Sampling-Based Motion Planning with Reachable Volumes: Theoretical Foundations

### Citations

1277 | Probabilistic roadmaps for path planning in high dimensional configuration spaces
- Kavraki, ˇSvestka, et al.
- 1996
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Citation Context ... parallel robotics [20], grasping and manipulation [22], computational biology and molecular simulations [2], and animation [15]. Randomized motion planning methods such as the graph-based PRM method =-=[16]-=- and the tree-based RRT method [17] have had a good deal of success solving traditional motion planning problems. Unfortunately, these methods are poorly suited for constrained problems where the prob... |

1160 |
Convex bodies: the Brunn-Minkowski theory
- Schneider
- 1993
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Citation Context ...olume of a chain can be computed by calculating the Minkowski sum of the reachable volumes of the links in the chain. These may be easily computed (see [19]). Note that Minkowski sums are commutative =-=[23]-=-, which implies that the order in which the links occur in a linkages has no impact on the reachable volume of the chain. CRV i1 i {l ,...,l } (a) CRV + RV(ls)i+1 l i+1 1{l ,...,l }i (b) i+1 l 1{l ,..... |

845 | OBB-Tree: A hierarchical structure for rapid interference detection
- Gottschalk, Lin, et al.
- 1996
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Citation Context ... to the root. RV-space is analogous to C-space in that it does not include any obstacles and that it can be used to generate configurations that are tested for validity using a validity checker (e.g. =-=[5]-=-). The reachable volume of a chain is the set of points in RVspace that can be reached by the other end of the chain while satisfying any constraints associated with the problem (Figure 3(a)). First n... |

626 | Randomized kinodynamic planning”.
- LaValle, Kuffner
- 2001
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Citation Context ...nd manipulation [22], computational biology and molecular simulations [2], and animation [15]. Randomized motion planning methods such as the graph-based PRM method [16] and the tree-based RRT method =-=[17]-=- have had a good deal of success solving traditional motion planning problems. Unfortunately, these methods are poorly suited for constrained problems where the probability of randomly generating a sa... |

111 | A kinematics-based probabilistic roadmap method for closed chain systems.”
- Han, Amato
- 2000
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Citation Context ...re the probability of randomly generating a sample satisfying the constraints approaches zero [18]. Previous methods have developed specialized samplers that generate samples that satisfy constraints =-=[6, 11, 25]-=-. Such samplers can be used in combination with existing PRM-based methods to solve problems with constraints, however these methods are either unable to handle high degree of freedom (dof) systems or... |

99 |
Introduction to Robotics: Mechanics and Control, 2nd ed.
- Craig
- 1989
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Citation Context ...s possible to produce accurate reachability grids in real time; errors in estimates are almost always biased towards optimistic ones. B. Reachable workspace and reachable distance Reachable workspace =-=[3]-=- is the volume of workspace that can be reached by the center point of the end effector of a fixed base manipulator. It differs from reachable volumes in that it is only defined for serial linkages an... |

83 | A probabilistic roadmap approach for systems with closed kinematic chains.”
- LaValle, Yakey, et al.
- 1999
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Citation Context ...onal motion planning problems. Unfortunately, these methods are poorly suited for constrained problems where the probability of randomly generating a sample satisfying the constraints approaches zero =-=[18]-=-. Previous methods have developed specialized samplers that generate samples that satisfy constraints [6, 11, 25]. Such samplers can be used in combination with existing PRM-based methods to solve pro... |

78 |
Randomized path planning for linkages with closed kinematic chains.
- Yakey, LaValle, et al.
- 2001
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Citation Context ...ptations of the PRM and RRT methods PRMs and RRTs have been adapted for use in spatially constrained systems. Gradient decent methods push randomly generated configurations onto a constrained surface =-=[18, 30]-=-. They are capable of solving problems with single-loop, articulated joint, closed chains. PRM-MC combines PRM and Monte Carlo methods to generate samples that satisfy closure constraints for single l... |

75 |
A combined optimization method for solving the inverse kinematics problems of mechanical manipulators,”
- Wang, Chen
- 1991
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Citation Context ... and a family of samplers that uses them to generate constraint satisfying configurations of linkages, closed chains and tree-like robots. Unlike many previous methods (e.g., cyclic coordinate decent =-=[28]-=-, reachable distances [25], and inverse kinematics [6]) which focus on end effector constraints, our method allows for constraints on internal joints as well as end effectors. Moreover, our method all... |

68 | Dynamicdomain RRTs: Efficient exploration by controlling the sampling domain,” in
- Yershova, Jaillet, et al.
- 2005
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Citation Context ...rical, Prismatic Yes Inverse Kinematics [6] End Effector Yes Yes Yes Planar Yes Constrained Dynamics [4] End Effector Yes Yes No Planar Yes I-CD None Yes No Yes Planar, Spherical, Prismatic Yes DDRRT =-=[33, 32]-=-, Atlas RRT [14], Only End Effect- Not shown Yes Not shown Only Planar No Tangent Bundle RRT [24] tor Shown Shown CCD [28] End Effector No Yes No Planar, Spherical Yes CHOMP [35] Hard and Soft Yes Not... |

53 | Sampling-based motion planning under kinematic loop-closure constraints,”
- Cortes, Simeon
- 2005
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Citation Context ...ontact with each other (e.g., closed chains). Motion planning with constraints is applicable to parallel robotics [20], grasping and manipulation [22], computational biology and molecular simulations =-=[2]-=-, and animation [15]. Randomized motion planning methods such as the graph-based PRM method [16] and the tree-based RRT method [17] have had a good deal of success solving traditional motion planning ... |

50 | Planning collision-free reaching motions for interactive object manipulation and grasping
- Kallmann, Aubel, et al.
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Citation Context ...her (e.g., closed chains). Motion planning with constraints is applicable to parallel robotics [20], grasping and manipulation [22], computational biology and molecular simulations [2], and animation =-=[15]-=-. Randomized motion planning methods such as the graph-based PRM method [16] and the tree-based RRT method [17] have had a good deal of success solving traditional motion planning problems. Unfortunat... |

35 | The geometry of configuration spaces for closed chains in two and three dimensions,” Homology Homotopy and Applications,
- Milgram, Trinkle
- 2002
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Citation Context ...h single-loop, articulated joint, closed chains. PRM-MC combines PRM and Monte Carlo methods to generate samples that satisfy closure constraints for single loop closed chains up to 100 links [8]. In =-=[26, 21]-=-, Trinkle and Milgram develop a method that uses C-space analysis for path planning while ignoring self collisions. They show results for a set of planar parallel star-shaped manipulators. Alternative... |

34 | Complete path planning for closed kinematic chains with spherical joints,”
- Trinkle, Milgram
- 2002
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Citation Context ...h single-loop, articulated joint, closed chains. PRM-MC combines PRM and Monte Carlo methods to generate samples that satisfy closure constraints for single loop closed chains up to 100 links [8]. In =-=[26, 21]-=-, Trinkle and Milgram develop a method that uses C-space analysis for path planning while ignoring self collisions. They show results for a set of planar parallel star-shaped manipulators. Alternative... |

30 |
Path planning with general end-effector constraints
- Yao, Gupta
- 2007
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Citation Context ... manipulators. Alternative Task-space and Configuration-space Exploration (ATACE) for path planning with constrained manipulators uses a randomized gradient decent method for constrained manipulators =-=[31]-=-. They present results for a 9 dof manipulator robot with a set of end effector constraints. In [34] Zhang and Hauser present a Monte Carlo method for generating closed chain samples. Atlas-RRT [14] a... |

25 |
M.: Constraint-based motion planning using voronoi diagrams
- GARBER, LIN
- 2002
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Citation Context ...t Multi-Query Types Robots Chains Robots Types Problems Basic PRM [16] None Yes No Yes Planar, Spherical, Prismatic Yes Inverse Kinematics [6] End Effector Yes Yes Yes Planar Yes Constrained Dynamics =-=[4]-=- End Effector Yes Yes No Planar Yes I-CD None Yes No Yes Planar, Spherical, Prismatic Yes DDRRT [33, 32], Atlas RRT [14], Only End Effect- Not shown Yes Not shown Only Planar No Tangent Bundle RRT [24... |

23 | Still a long way to go on the road for parallel mechanisms”, - Merlet - 2002 |

21 | CHOMP: Covariant Hamiltonian optimization for motion planning.
- Zucker, Ratliff, et al.
- 2013
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Citation Context ...ismatic Yes DDRRT [33, 32], Atlas RRT [14], Only End Effect- Not shown Yes Not shown Only Planar No Tangent Bundle RRT [24] tor Shown Shown CCD [28] End Effector No Yes No Planar, Spherical Yes CHOMP =-=[35]-=- Hard and Soft Yes Not Shown Yes Planar, Spherical, Prismatic, Combinations No Reachable Distances [25] End Effector Yes Yes Not shown Planar, Prismatic Yes Reachable Volumes (this paper) End Effector... |

20 | Motion planning for mobile manipulators along given ed-effector paths
- Oriolo, Mongillo
- 2005
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Citation Context ...o require that certain joints of the robot remain in contact with each other (e.g., closed chains). Motion planning with constraints is applicable to parallel robotics [20], grasping and manipulation =-=[22]-=-, computational biology and molecular simulations [2], and animation [15]. Randomized motion planning methods such as the graph-based PRM method [16] and the tree-based RRT method [17] have had a good... |

19 |
The inverse kinematics of a serial chain with joints under distance constraints,”
- Han, Rudolph
- 2006
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Citation Context ... to large linkages [29] and multiple loops [2]. Inverse kinematic methods for 3D, 5D, and 6D end effector constraints are shown to efficiently generate samples for linkages with as many as 1000 links =-=[11, 9]-=-. It has been shown that for any planar polygonal loop there exist two special configurations such that any connectable pair of configurations can be connected by a sequence of straight line paths thr... |

11 |
Reachable distance space: Efficient sampling-based planning for spatially constrained systems”. The international journal of robotics research
- Tang, Thomas, et al.
- 2010
(Show Context)
Citation Context ...re the probability of randomly generating a sample satisfying the constraints approaches zero [18]. Previous methods have developed specialized samplers that generate samples that satisfy constraints =-=[6, 11, 25]-=-. Such samplers can be used in combination with existing PRM-based methods to solve problems with constraints, however these methods are either unable to handle high degree of freedom (dof) systems or... |

10 | A hybrid probabilistic roadmap – monte carlo motion planner for articulated robots and proteins
- Han
- 2003
(Show Context)
Citation Context ...lems with single-loop, articulated joint, closed chains. PRM-MC combines PRM and Monte Carlo methods to generate samples that satisfy closure constraints for single loop closed chains up to 100 links =-=[8]-=-. In [26, 21], Trinkle and Milgram develop a method that uses C-space analysis for path planning while ignoring self collisions. They show results for a set of planar parallel star-shaped manipulators... |

9 | Motion planning for highly constrained spaces
- Yershova, LaValle
- 2009
(Show Context)
Citation Context ...rical, Prismatic Yes Inverse Kinematics [6] End Effector Yes Yes Yes Planar Yes Constrained Dynamics [4] End Effector Yes Yes No Planar Yes I-CD None Yes No Yes Planar, Spherical, Prismatic Yes DDRRT =-=[33, 32]-=-, Atlas RRT [14], Only End Effect- Not shown Yes Not shown Only Planar No Tangent Bundle RRT [24] tor Shown Shown CCD [28] End Effector No Yes No Planar, Spherical Yes CHOMP [35] Hard and Soft Yes Not... |

8 | Path Planning with Loop Closure Constraints using an Atlas-based
- Jaillet, Porta
- 2011
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Citation Context ... Inverse Kinematics [6] End Effector Yes Yes Yes Planar Yes Constrained Dynamics [4] End Effector Yes Yes No Planar Yes I-CD None Yes No Yes Planar, Spherical, Prismatic Yes DDRRT [33, 32], Atlas RRT =-=[14]-=-, Only End Effect- Not shown Yes Not shown Only Planar No Tangent Bundle RRT [24] tor Shown Shown CCD [28] End Effector No Yes No Planar, Spherical Yes CHOMP [35] Hard and Soft Yes Not Shown Yes Plana... |

7 |
Convexly Stratified Deformation Spaces and Efficient Path Planning for Planar Closed Chains with Revolute Joints
- Han, Rudolph, et al.
- 2008
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Citation Context ...een shown that for any planar polygonal loop there exist two special configurations such that any connectable pair of configurations can be connected by a sequence of straight line paths through them =-=[7]-=-. This method has been extended to produce paths guaranteed to be self-collision free [13]. They show that any two convex configurations of a closed chain can be connected by a path comprised of two s... |

7 | A unified geometric approach for inverse kinematics of a spatial chain with spherical joints. In: ICRA;
- Han, Rudolph
- 2007
(Show Context)
Citation Context ...re the probability of randomly generating a sample satisfying the constraints approaches zero [18]. Previous methods have developed specialized samplers that generate samples that satisfy constraints =-=[6, 11, 25]-=-. Such samplers can be used in combination with existing PRM-based methods to solve problems with constraints, however these methods are either unable to handle high degree of freedom (dof) systems or... |

4 | Voxel-based motion bounding and workspace estimation for robotic manipulators
- Anderson-Sprecher, Simmons
- 2012
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Citation Context ...ples with self-contact [12]. The reachability grid is a voxel-based representation that consists of a workspace grid in which each grid cell is denoted by the minimum time required to reach that cell =-=[1]-=-. It is possible to produce accurate reachability grids in real time; errors in estimates are almost always biased towards optimistic ones. B. Reachable workspace and reachable distance Reachable work... |

4 | Simplex-tree based kinematics of foldable objects as multi-body systems involving loops
- Han, Rudolph
- 2009
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Citation Context ...oft constraints. 3) Enforcing constraints during sampling: Some approaches explicitly enforce constraints while sampling. Closed chain problems may by transformed into a system of linear inequalities =-=[10]-=-. Constrained dynamics enforce constraints such as joint connectivity, spatial relationships, and obstacle avoidance for manipulators up to 6 dof [4]. Other planners require the end effector to traver... |

4 |
Bending and kissing : Computing self-contact configurations of planar loops with revolute joints
- Han, Rudolph, et al.
- 2009
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Citation Context ...the end effector to traverse a predefined trajectory by generating samples that satisfy the end effector constraints given by the trajectory [22]. Other methods can generate samples with self-contact =-=[12]-=-. The reachability grid is a voxel-based representation that consists of a workspace grid in which each grid cell is denoted by the minimum time required to reach that cell [1]. It is possible to prod... |

4 |
The tangent bundle RRT algorithms for constrained motion planning
- Suh, Kim, et al.
- 2011
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Citation Context ...[4] End Effector Yes Yes No Planar Yes I-CD None Yes No Yes Planar, Spherical, Prismatic Yes DDRRT [33, 32], Atlas RRT [14], Only End Effect- Not shown Yes Not shown Only Planar No Tangent Bundle RRT =-=[24]-=- tor Shown Shown CCD [28] End Effector No Yes No Planar, Spherical Yes CHOMP [35] Hard and Soft Yes Not Shown Yes Planar, Spherical, Prismatic, Combinations No Reachable Distances [25] End Effector Ye... |

3 | Unbiased, scalable sampling of closed kinematic chains
- Zhang, Hauser, et al.
- 2013
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Citation Context ...with constrained manipulators uses a randomized gradient decent method for constrained manipulators [31]. They present results for a 9 dof manipulator robot with a set of end effector constraints. In =-=[34]-=- Zhang and Hauser present a Monte Carlo method for generating closed chain samples. Atlas-RRT [14] and Tangent Bundle based RRT [24] construct RRTs along manifolds in the environment, while DDRRT [33,... |

2 |
Configurations and path planning of convex planar polygonal loops
- Han, Rudolph, et al.
- 2012
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Citation Context ...that any connectable pair of configurations can be connected by a sequence of straight line paths through them [7]. This method has been extended to produce paths guaranteed to be self-collision free =-=[13]-=-. They show that any two convex configurations of a closed chain can be connected by a path comprised of two straight line segments consisting only of convex configurations. While inverse kinematics-b... |

2 | Sampling based motion planning with reachable volumes: Application to manipulators and closed chain systems
- McMahon, Thomas, et al.
- 2014
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Citation Context ...OMPARISON OF METHOD CAPABILITIES. volumes can be computed efficiently and application to a range of systems including arbitrary linkages, closed chain systems, and mobile manipulators can be found in =-=[19]-=-. II. RELATED WORK We give an overview of previous methods that are applicable to motion planning systems with constraints. A complete overview of related work can be found in [19]. Table I summarizes... |