Connected Coordinated Motion Planning with Bounded Stretch

We consider the problem of connected coordinated motion planning for a large collective of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-complete, even for deciding whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved. On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, then the total duration of our overall schedule is $\mathcal{O}(d)$, which is optimal up to constant factors.Comment: 28 pages, 18 figures, full version of an extended abstract that appeared in the proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC 2021); revised version (more details added, and typing errors corrected

Connected Coordinated Motion Planning with Bounded Stretch

We consider the problem of coordinated motion planning for a swarm of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, continuous, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-hard, even for deciding whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved. On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is ?(d), which is optimal up to constant factors

Coordinated motion planning of manipulators by evolution strategies

A method for obtaining coordinated motion plans of manipulator robots is presented. This planning can be easily implemented by programs written in any industrial robot programming language, such as VAL II. The generated programs minimize the total motion time of the robots along their paths, with some constraints directed at avoiding collision between the robots. A method based on Evolution Strategies is used for the optimization

Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Despite a broad range of other non-trivial results for multi-object motion planning, previous work has largely focused on sequential schedules, in which one robot moves at a time, with objectives such as the number of moves; attempts to minimize the overall makespan of a coordinated parallel motion schedule (with many robots moving simultaneously) have defied all attempts at establishing the complexity in the absence of obstacles, as well as the existence of efficient approximation methods. We resolve these open problems by developing a framework that provides constant-factor approximation algorithms for minimizing the execution time of a coordinated, parallel motion plan for a swarm of robots in the absence of obstacles, provided their arrangement entails some amount of separability. In fact, our algorithm achieves constant stretch factor: If all robots want to move at most d units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is O(d). Various extensions include unlabeled robots and different classes of robots. We also resolve the complexity of finding a reconfiguration plan with minimal execution time by proving that this is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Omega(N^{1/4}) may be required. On the positive side, we establish a stretch factor of O(N^{1/2}) even in this case. The intricate difficulties of computing precise optimal solutions are demonstrated by the seemingly simple case of just two disks, which is shown to be excruciatingly difficult to solve to optimality

Coordinated Motion Planning for On-Orbit Satellite Inspection using a Swarm of Small-Spacecraft

This paper addresses the problem of how to plan optimal motion for a swarm of on-orbit servicing (OOS) small-spacecraft remotely inspecting a non-cooperative client spacecraft in Earth orbit. With the goal being to maximize the information gathered from the coordinated inspection, we present an integrated motion planning methodology that is a) fuel-efficient to ensure extended operation time and b) computationally-tractable to make possible on-board re-planning for improved exploration. Our method is decoupled into first offline selection of optimal orbits, followed by online coordinated attitude planning. In the orbit selection stage, we numerically evaluate the upper and lower bounds of the information gain for a discretized set of passive relative orbits (PRO). The algorithm then sequentially assigns orbits to each spacecraft using greedy heuristics. For the attitude planning stage, we propose a dynamic programming (DP) based attitude planner capable of addressing vehicle and sensor constraints such as attitude control system specifications, sensor field of view, sensing duration, and sensing angle. Finally, we validate the performance of the proposed algorithms through simulation of a design reference mission involving 3U CubeSats inspecting a satellite in low Earth orbit

Coordinated Motion Planning for On-Orbit Satellite Inspection using a Swarm of Small-Spacecraft

This paper addresses the problem of how to plan optimal motion for a swarm of on-orbit servicing (OOS) small-spacecraft remotely inspecting a non-cooperative client spacecraft in Earth orbit. With the goal being to maximize the information gathered from the coordinated inspection, we present an integrated motion planning methodology that is a) fuel-efficient to ensure extended operation time and b) computationally-tractable to make possible on-board re-planning for improved exploration. Our method is decoupled into first offline selection of optimal orbits, followed by online coordinated attitude planning. In the orbit selection stage, we numerically evaluate the upper and lower bounds of the information gain for a discretized set of passive relative orbits (PRO). The algorithm then sequentially assigns orbits to each spacecraft using greedy heuristics. For the attitude planning stage, we propose a dynamic programming (DP) based attitude planner capable of addressing vehicle and sensor constraints such as attitude control system specifications, sensor field of view, sensing duration, and sensing angle. Finally, we validate the performance of the proposed algorithms through simulation of a design reference mission involving 3U CubeSats inspecting a satellite in low Earth orbit

Conflict Optimization for Binary CSP Applied to Minimum Partition into Plane Subgraphs and Graph Coloring

CG:SHOP is an annual geometric optimization challenge and the 2022 edition proposed the problem of coloring a certain geometric graph defined by line segments. Surprisingly, the top three teams used the same technique, called conflict optimization. This technique has been introduced in the 2021 edition of the challenge, to solve a coordinated motion planning problem. In this paper, we present the technique in the more general framework of binary constraint satisfaction problems (binary CSP). Then, the top three teams describe their different implementations of the same underlying strategy. We evaluate the performance of those implementations to vertex color not only geometric graphs, but also other types of graphs.Comment: To appear at ACM Journal of Experimental Algorithmic

Coordinated Motion Planning and Optimal Force Distribution for Robots with Multiple Cooperating Arms

This paper is concerned with the problems of inter-arm coordination which arise when multiple robot arms attach their end effectors to a single payload in order to perform a manipulation task. The approach taken is to let the arms cooperate with equal rights, and to exploit the redundancy present in the system for the minimization of a quadratic cost criterion. The main elements of the proposed control system are generators for coordinated nominal motions and force/torque interactions with the payload, and an active compliance scheme capable of resolving kinematic conflicts and inconsistencies between motion and force/torque commands. The paper gives a detailed mathematical description of both the motion and the force/torque generators. The proposed active compliance scheme is presented in summary form