## Provably good approximation algorithms for optimal kinodynamic planning for cartesian robots and open chain manipulators (1995)

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Venue: | Algorithmica |

Citations: | 74 - 9 self |

### BibTeX

@ARTICLE{Donald95provablygood,

author = {Bruce R. Donald and Patrick Xavier},

title = {Provably good approximation algorithms for optimal kinodynamic planning for cartesian robots and open chain manipulators},

journal = {Algorithmica},

year = {1995},

volume = {14},

pages = {958--963}

}

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### Abstract

shortest path, kinodynamics, polyhedral obstacles Abstract: We consider the following problem: given a robot system, nd a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety-margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in R 3). This algorithm di ers from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error term �(2) to polynomially bound the computational complexity of our algorithm � and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error term, the algorithm returns a solution that is-close to optimal and requires only a polynomial (in ( 1)) amount of time. We extend the results of [1] in two ways. First, we modifyittohalve the exponent inthe polynomial bounds from 6d to 3d, so that that the new algorithm is O c d N 1 3d, where N is the geometric complexity of the obstacles and c is a robot-dependent constant. Second, the new algorithm nds a trajectory that matches the optimal in time with an factor sacri ced in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments. The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.

### Citations

1337 |
Combinatorial optimization: Algorithms and complexity
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- 1982
(Show Context)
Citation Context ...states. The algebraic complexity of the kinodynamic bounds is the number of bits necessary to encode the kinodynamic bounds (a max ; v max ; c 1 ; c 0 ). In the language of combinatorial optimization =-=[8]-=-, we show that our algorithm is an ffl-approximation scheme that is fully polynomial in the combinatorial and algebraic complexity of the geometry, and pseudo-polynomial in the kinodynamic bounds. We ... |

706 |
Algorithms in Combinatorial Geometry
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- 1987
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Citation Context ...Gamma 2)-face bounds exactly two (d \Gamma 1)-faces) of the Minkowski sum and constructing their incidence relation. We conjecture that applying and extending work from computational geometry such as =-=[25, 26]-=- would be fruitful. In turn, this extended d-dimensional algorithm would apply, via robot-dependent constant linear transforms, to other robots with constant, decoupled dynamics equations and decouple... |

509 |
The complexity of robot motion planning
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- 1987
(Show Context)
Citation Context ...to a safety-checking method similar to that described in [1, 3] and reviewed above. It would be convenient if we could apply an algebraic collision detection predicate such as that described by Canny =-=[12, 24]-=-. (We shall soon describe why we cannot.) In fact, our ffi v -safety predicate uses the structure of his predicate and the same logical evaluation method for each pair of polyhedra that could possibly... |

372 | Spatial planning: A configuration space approach
- Lozano-Perez
- 1983
(Show Context)
Citation Context ...l robot of geometric complexity m and a set of polyhedral obstacles with geometric complexitysn, the number of configuration space constraints N = O(m(m + n)), since an arm must avoid self collisions =-=[11]-=-. Finally, we assume that all linear (i.e., non-revolute) degrees of freedom are bounded from above by a length l. As before, we define a trajectory to be ffi v -safe if and only if for all times t in... |

192 |
New lower bound techniques for robot motion planning problems
- Canny, Reif
- 1987
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Citation Context ...cted dynamics and dynamics bounds, we encourage the reader to see our companion paper [7]. Appendices A Kinodynamic Planning Lower Bounds This appendix sketches how to extend Canny's and Reif's proof =-=[4]-=- of the NP-hardness of the 3D shortest-path problem to show that Optimal Cartesian Kinodynamic Planning in 3D is also NP-hard. This claim was made in [1], but without proof. While the general descript... |

135 |
Time-optimal control of robotic manipulators along specified paths
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Citation Context ...l control in the control theory and robotics literature. For example, see [12--16]. Much of this work provides partial analytic characterizations of time-optimal solutions. Among significant results, =-=[12, 13]-=- show how to time-rescale the velocity profile of given a particular trajectory to obtain a trajectory that is time-optimal with respect to dynamics constraints. This flavor of theoretical work has le... |

109 |
Collision detection for moving polyhedra
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- 1986
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Citation Context ..., exact collision detection for quadratic paths requires the solution of mixed trignometric equations that cannot be transformed into algebraic equations using the usual substitution methods, such as =-=[12]-=-. Furthermore, for these manipulators, trajectory segments corresponding to constant extremal controls are solutions to systems of ordinary trignometric differential equations, and the trajectories fo... |

105 |
Robot Analysis and Control
- Asada, Slotine
- 1986
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Citation Context ...n our previous work. The robot motion is governed by a dynamics law, which relates applied generalized forces f to states, accelerations, and forces G(p) induced by gravity. For open kinematic chains =-=[9, 10]-=-: f(t) = M(p(t))a(t) + [sp T (t)C(p(t))sp(t)] +G(p(t)): (1) M(p(t)); the robot inertia tensor, is orthogonal, symmetric, and positive-definite. C(p(t)) is a tensor of rank three, and [sp T (t)C(p(t))s... |

98 |
Planning Smooth Paths for Mobile Robots
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- 1989
(Show Context)
Citation Context ...extend to the 2- and 3D cases as well. Kinodynamic planning in 2D is related to the problem of planning with non-holonomic constraints, as studied by Fortune and Wilfong [20, 21] and Jacobs and Canny =-=[22]-=-. In this problem, a robot with wheels and a bounded minimum turning radius must be moved. To make the analogy clear, in our case, the minimum turning radius is 1 amax kspk 2 . [1] (see also its revis... |

92 | Kinodynamic motion planning
- Donald, Xavier, et al.
- 1993
(Show Context)
Citation Context ...eral). Finally, we note that the True Extremal Algorithm is more theoretical than the Near-Extremal Algorithm, and that for a given robot it yields a larger c D or c E . 2.3 Previous and Related Work =-=[1, 2]-=- address the problem of kinodynamic planning for a point robot with L1-norm velocity bound v and acceleration bound a in an environment with polyhedral obstacles. [1] provide the first formulation of ... |

82 | A search algorithm for motion planning with six degrees of freedom - Donald - 1987 |

70 |
Global time-optimal motions of robotic manipulators in the presence of obstacles
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- 1988
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Citation Context ...n a trajectory that is time-optimal with respect to dynamics constraints. This flavor of theoretical work has led to algorithms that attempt to find nearly time-optimal trajectories, notably [17] and =-=[18]-=-. None of these results provided analytically guaranteed closeness to global optimality, and assuring one could control the accuracy of these algorithms by increasing the number of gridpoints required... |

66 |
Computing convolutions by reciprocal search
- Guibas, Seidel
- 1986
(Show Context)
Citation Context ...Gamma 2)-face bounds exactly two (d \Gamma 1)-faces) of the Minkowski sum and constructing their incidence relation. We conjecture that applying and extending work from computational geometry such as =-=[25, 26]-=- would be fruitful. In turn, this extended d-dimensional algorithm would apply, via robot-dependent constant linear transforms, to other robots with constant, decoupled dynamics equations and decouple... |

58 |
An algorithm for shortest-path motion in three dimensions
- Papadimitriou
- 1985
(Show Context)
Citation Context ...timal safe solution except at its endpoints. In this respect, these trajectory planning algorithms are similar to Papadimitriou's fully polynomial approximation scheme for 3D Euclidian shortest paths =-=[9]-=-. Again, the closeness of the approximation is strictly in terms of the optimization measure, so the optimal solution might not appear spatially similar to the truly optimal. In fact, the results of [... |

50 |
On the complexity of kinodynamic planning
- Canny, Reif, et al.
- 1988
(Show Context)
Citation Context ...stem, we must find a minimaltime trajectory that goes from a start state to a goal state while avoiding obstacles by a speeddependent safety-margin and respecting dynamics bounds. With Canny and Reif =-=[1]-=-, we approached this problem from an ffl-approximation standpoint and introduced a provably-good approximation algorithm for optimal kinodynamic planning for a robot obeying particle dynamics. If a so... |

50 |
Planning constrained motion
- Fortune, Wilfong
- 1988
(Show Context)
Citation Context ...c planning. These methods may extend to the 2- and 3D cases as well. Kinodynamic planning in 2D is related to the problem of planning with non-holonomic constraints, as studied by Fortune and Wilfong =-=[20, 21]-=- and Jacobs and Canny [22]. In this problem, a robot with wheels and a bounded minimum turning radius must be moved. To make the analogy clear, in our case, the minimum turning radius is 1 amax kspk 2... |

48 | An exact algorithm for kinodynamic planning in the plane
- Canny, Rege, et al.
- 1991
(Show Context)
Citation Context ...ffl j 3d ' , where N is the geometric complexity of the environment and c depends on the dynamics and safety margin parameters; this halves the previous exponent of the ( 1 ffl ) term. Furthermore, 2 =-=[2]-=- have recently provided an exact algorithm for the 2D L1 case. The algorithm runs in exponential time and polynomial space. 3 [5] is the journal revision of [1]. we show that if there exists an optima... |

44 |
Dynamic scaling of manipulator trajectories
- Hollerbach
- 1984
(Show Context)
Citation Context ...n our previous work. The robot motion is governed by a dynamics law, which relates applied generalized forces f to states, accelerations, and forces G(p) induced by gravity. For open kinematic chains =-=[9, 10]-=-: f(t) = M(p(t))a(t) + [sp T (t)C(p(t))sp(t)] +G(p(t)): (1) M(p(t)); the robot inertia tensor, is orthogonal, symmetric, and positive-definite. C(p(t)) is a tensor of rank three, and [sp T (t)C(p(t))s... |

43 | Planning of minimum-time trajectories for robot-arms
- Sahar, Hollerbach
- 1985
(Show Context)
Citation Context ... to obtain a trajectory that is time-optimal with respect to dynamics constraints. This flavor of theoretical work has led to algorithms that attempt to find nearly time-optimal trajectories, notably =-=[17]-=- and [18]. None of these results provided analytically guaranteed closeness to global optimality, and assuring one could control the accuracy of these algorithms by increasing the number of gridpoints... |

33 | Simplified Voronoi diagrams
- Canny, Donald
- 1988
(Show Context)
Citation Context ...tion method for each pair of polyhedra that could possibly collide. The non-overlap condition for two convex polyhedra is given by the non-overlap predicate i j k l (C ijkl (x) ? 0) (57) described in =-=[14]-=-, with the exact form of constraint functions C ijkl : C ! R given in [12]. When the real-space obstacles (polyhedra) are grown affinely with speed, the overlap predicate has the same structure as (57... |

28 |
Algorithmic motion planning
- Yap
- 1987
(Show Context)
Citation Context ...wo dimensions. In Section 5 we briefly report on this and describe extensions to the main result. 2.4 Previous and Related Work For a review of issues in robotics and algorithmic motion planning, see =-=[10, 11]-=-. There exists a large body of work on optimal control in the control theory and robotics literature. For example, see [12--16]. Much of this work provides partial analytic characterizations of time-o... |

27 |
Motion Planning for an Autonomous Vehicle
- Wilfong
- 1988
(Show Context)
Citation Context ...c planning. These methods may extend to the 2- and 3D cases as well. Kinodynamic planning in 2D is related to the problem of planning with non-holonomic constraints, as studied by Fortune and Wilfong =-=[20, 21]-=- and Jacobs and Canny [22]. In this problem, a robot with wheels and a bounded minimum turning radius must be moved. To make the analogy clear, in our case, the minimum turning radius is 1 amax kspk 2... |

22 |
Time-optimal trajectories for a robotic manipulator: A provably good approximation algorithm
- Heinzinger, Jacobs, et al.
- 1990
(Show Context)
Citation Context ...ic ideas behind two general algorithms for finding near-optimal kinodynamic trajectories for Cartesian robots with L 2 -norm dynamics bounds and for open-chain 3 We will refer to this body of work as =-=[6, 8]-=-. 4 The [6, 8] result preceded the [16, 17] result. manipulators. The first algorithm searches a reachability graph corresponding to piecewiseconstant, extremal forces and torques, and we will refer t... |

21 |
Motion-planning with inertial constraints
- O'Dunlaing
- 1987
(Show Context)
Citation Context ...at in 3D, optimal kinodynamic planning is NP-hard� a proof sketch isgiven in the Appendix of this paper. [9] gives a fully-polynomial approximation algorithm for the shortest path problem. O'Dunlaing =-=[19]-=- provides an exact algorithm for one-dimensional kinodynamic planning. These methods may extend to the 2- and 3D cases as well. Kinodynamic planning in 2D is related to the problem of planning with no... |

21 |
Spatial Planning: A Con guration Space Approach
- Lozano-Perez
- 1983
(Show Context)
Citation Context ...mentation� additional, though limited, parallelism can be extracted in safety-checking. 5.3 Extensions Our results can be directly extended in several ways. Via a transformation to con guration space =-=[24]-=-, our results can be applied to a rigid, non-rotating robot whose geometry is given by a union R of convex polyhedra. This con guration space transformation has been discussed extensively in the liter... |

17 | Time-Optimal Control of Manipulators - Sontag, Sussmann - 1962 |

15 |
O’Dunlaing: Motion planning with inertial constraints, Algorithmica 2(4
- unknown authors
- 1987
(Show Context)
Citation Context ...n 3D, optimal kinodynamic planning is NP-hard; a proof sketch is given in the Appendix of this paper. [9] gives a fully-polynomial approximation algorithm for the shortest path problem. ' O'D'unlaing =-=[19]-=- provides an exact algorithm for one-dimensional kinodynamic planning. These methods may extend to the 2- and 3D cases as well. Kinodynamic planning in 2D is related to the problem of planning with no... |

10 |
Near-optimal kinodynamic planning for robots with coupled dynamics bounds
- Donald, Xavier
- 1989
(Show Context)
Citation Context ...r finding near-optimal kinodynamic trajectories for Cartesian robots with L 2 -norm dynamics bounds and for open-chain 3 We will refer to this body of work as [6, 8]. 4 The [6, 8] result preceded the =-=[16, 17] result. m-=-anipulators. The first algorithm searches a reachability graph corresponding to piecewiseconstant, extremal forces and torques, and we will refer to it as "the True-Extremal Algorithm ". The... |

9 | Approximate Kinodynamic Planning Using L2-norm Dynamics Bounds
- Reif, Tate
- 1990
(Show Context)
Citation Context ... to being locally constant. Our approach (see the early description in [15{17]) toward statedependent dynamics is similar, but we obtain better complexity results. 4 In concurrent work, Reif and Tate =-=[18]-=- used a parameter-dependent acceleration-space discretization implicitly to obtain a polynomial-time approximation algorithm for robots with decoupled dynamics, L2 dynamics bounds, and polyhedral C-sp... |

8 |
Provably-good approximation algorithms for optimal kinodynamic robot motion plans
- Xavier
- 1992
(Show Context)
Citation Context ...advantage over the adversary's allowable accelerations. This would imply a provably good polynomial-time approximation algorithm for kinodynamic planning using (approximately) bang-bang controls. See =-=[4]-=-. 2. Since the tracking lemmas do not require the force bounds be state-invariant, it should be possible to extend the results to relax this requirement. 3. Because of the use ofsl acceleration advant... |

8 |
editors. Robot Motion: Planning and Control
- Brady
- 1982
(Show Context)
Citation Context ...wo dimensions. In Section 5 we briefly report on this and describe extensions to the main result. 2.4 Previous and Related Work For a review of issues in robotics and algorithmic motion planning, see =-=[10, 11]-=-. There exists a large body of work on optimal control in the control theory and robotics literature. For example, see [12--16]. Much of this work provides partial analytic characterizations of time-o... |

6 |
Approximate Kinodynamic Planning Using L 2 -norm Dynamics Bounds
- Reif, Tate
- 1990
(Show Context)
Citation Context ...to being locally constant. Our approach (see the early description in [15--17]) toward statedependent dynamics is similar, but we obtain better complexity results. 4 In concurrent work, Reif and Tate =-=[18]-=- used a parameter-dependent acceleration-space discretization implicitly to obtain a polynomial-time approximation algorithm for robots with decoupled dynamics, L 2 dynamics bounds, and polyhedral C-s... |

6 | Remarks on the time-optimal control of two-link manipulators - Sontag, Sussmann - 1985 |

6 |
Simpli ed voronoi diagrams
- Canny, Donald
- 1988
(Show Context)
Citation Context ... freedom and polyhedral workspace obstacles, the only change in the algorithm would again be in the safety-checking step. For a description of the modi ed safety-checking step, which extends [27] and =-=[28]-=-, see our companion paper [7] or [3], which present our results for robots with coupled dynamics. Finally, the [1] approach and its descendents reduce the problem of nding an approximately optimal tra... |

5 | On the Optimality of Bang-Bang Trajectories in R 3 - Schaettler - 1987 |

4 |
An Introduction to Optimal Control
- Leitman
- 1966
(Show Context)
Citation Context ...it would be desirable for an algorithm to possibly find a truly optimal trajectory with respect ffi 0 v -safety, even for a "nearby" problem (as in numerical analysis). In robotics and contr=-=ol theory [19] there is -=-a family of results (e.g., [20--22]) on the feasibility of planning and approximating optimal trajectories using only piecewise-extremal controls; these results are often called "bang-bang" ... |

4 | On the Optimality of Bang-Bang Trajectories in R - Schaettler - 1987 |

2 |
Planning guaranteed near-timeoptimal planning in a cluttered workspace
- Jacobs, Heinzinger, et al.
- 1989
(Show Context)
Citation Context ...ic ideas behind two general algorithms for finding near-optimal kinodynamic trajectories for Cartesian robots with L 2 -norm dynamics bounds and for open-chain 3 We will refer to this body of work as =-=[6, 8]-=-. 4 The [6, 8] result preceded the [16, 17] result. manipulators. The first algorithm searches a reachability graph corresponding to piecewiseconstant, extremal forces and torques, and we will refer t... |

2 |
Time-safety trade-offs and a bang-bang algorithm for kinodynamic planning
- Donald, Xavier
- 1991
(Show Context)
Citation Context ...ng time. Furthermore, while a worst-case analysis is necessary when considering safety, an expected case analysis would be appropriate for measuring time-optimality versus algorithmic complexity. See =-=[26]-=- and [4]. We have presented provably good approximation algorithms for optimal kinodynamic planning with the lowest known complexity for robots obeying coupled dynamics bounds. While optimal kinodynam... |

2 |
Time-safety trade-o s and a bang-bang algorithm for kinodynamic planning
- Donald, Xavier
- 1991
(Show Context)
Citation Context ...namic trajectory, we can increase the timestep size. This single change in the algorithm dramatically reduces the size of the reachability graph and the running time. (See Figure 14.) The analysis in =-=[3, 23]-=- closely parallels the one described here. Finally, we note that because of the particular graph-search nature of the algorithm, we could greatly exploit parallelism in a practical implementation� add... |

1 |
Bounds on robot dynamics
- Heinzinger, Paden
- 1989
(Show Context)
Citation Context ...s arising from @F(f ;p;sp) @p and @F(f ;p;sp) @sp . This can be done loosely by inspection because all terms are bounded. However, simple expressions bounding the tensor norms have been calculated by =-=[23], an-=-d derivations of C r , C q0 , C q1 , and C q2 can be found in Appendix A. Recalling (47) and (50), we therefore choose �� 0 = min i ff 5Cr ; ff 5C q0 j ; j x0 = ff 10C q1 ; j v0 = ff 10C q2 : (56)... |