Time-Optimal Control of Robotic Manipulators Along Specified Paths

Bobrow, J.E.; Dubowsky, Steven; Gibson, J. S.

doi:10.1177/027836498500400301

“…Moreover, is polynomial in x , v , 1 vmax , a n d 1 amax . S p eci cally, can be chosen as the largest such that a max divides v max and min x 5v max v 2a max : (13) As in 1], it will be su cient to consider the one-dimensional case, since we are using the L 1 -norm for dynamics bounds. Assume that trajectory ; r obeys the velocity and acceleration bounds.…”

Section: The Strong Tracking Lemma 421 Preliminary Discussion Of Thmentioning

confidence: 99%

Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open-chain manipulators

Donald

¹

,

Xavier

²

1995

View full text Add to dashboard Cite

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 w ays. First, we m o d i f y i t t o h a l v e the exponent i n t h e 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.

show abstract

“…Again, we s h o w h o w to modify the \tail" of ; r to get a trajectory that satis es the hypotheses of Lemma 4.5. To get the desired bounds, we m ust choose so that using (18) yields a maximal as given by (13). Let us therefore de ne for 0…”

Section: Removing Restrictionsmentioning

confidence: 99%

Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds

Donald

¹

,

Xavier

²

1995

View full text Add to dashboard Cite

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 w ays. First, we m o d i f y i t t o h a l v e the exponent i n t h e 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.

show abstract

“…2) Kinodynamic planning: Here, there are both velocity and acceleration bounds, and the system is fully actuated [2]. 3) Trajectory planning: This problem has been pursued for several decades [3,4,5] and typically involves computing an open loop control for a manipulator while satisfying the kinematics and dynamics expressed as a control system. See Chapter 14 of [6] for a detailed presentation of this unified class of problems.…”

Section: Introductionmentioning

confidence: 99%

Sufficient Conditions for the Existence of Resolution Complete Planning Algorithms

Yershov

¹

,

LaValle

²

2010

Springer Tracts in Advanced Robotics

View full text Add to dashboard Cite

This paper addresses theoretical foundations of motion planning with differential constraints in the presence of obstacles. We establish general conditions for the existence of resolution complete planning algorithms by introducing a functional analysis framework and reducing algorithm existence to a simple topological property. First, we establish metric spaces over the control function space and the trajectory space. Second, using these metrics and assuming that the control system is Lipschitz continuous, we show that the mapping between open-loop controls and corresponding trajectories is continuous. Next, we prove that the set of all paths connecting the initial state to the goal set is open. Therefore, the set of open-loop controls, corresponding to solution trajectories, must be open. This leads to a simple algorithm that searches for a solution by sampling a control space directly, without building a reachability graph. A dense sample set is given by a discrete-time model. Convergence of the algorithm is proven in the metric of a trajectory space. The results provide some insights into the design of more effective planning algorithms and motion primitives.

show abstract

Time-Optimal Control of Robotic Manipulators Along Specified Paths

Cited by 1,173 publications

References 4 publications

Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open-chain manipulators

Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open-chain manipulators

Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds

Sufficient Conditions for the Existence of Resolution Complete Planning Algorithms

Contact Info

Product

Resources

About