State-time Space Research Articles

Online Bidirectional Trajectory Planning for Mobile Robots in State-Time Space

In this paper, we propose a computationally efficient online local motion planning algorithm for mobile robots in unknown cluttered dynamic environments. The algorithm plans a trajectory incrementally up to the finite horizon in state-time space. Incremental planning method is capable of fast computation but has poor obstacle avoidance performance. To compensate for the drawbacks of incremental planning, a partial trajectory modification scheme is used that sets an interim goal and then plans a trajectory to pass through the interim goal. The interim goal is a temporary desired goal to prevent the robot from falling into the inevitable collision state. By using incremental planning and partial trajectory modification, it is possible to plan collision-free trajectory with small computation even in cluttered dynamic environment. To generate smooth trajectories around given waypoints and goal, we systematize bidirectional trajectory planning with three kinds of trajectories: 1) a forward trajectory from the current robot state; 2) a backward trajectory from the state of current target waypoint; 3) and a connecting trajectory between the forward and backward trajectories. A smooth trajectory is generated around the way point and goal by setting the state of the current target waypoint (or goal), while taking into account the positional relationship among the planned forward trajectory, the target waypoint and the next waypoint. Performances of the proposed algorithm are validated through extensive simulations and experiment with two types of mobile robots: 1) a holonomic mobile robot and 2) a differential drive mobile robot.

Roadmap search based motion planning for MIRADAS probe arms

MIRADAS is a near-infrared multiobject echelle spectrograph operating at spectral resolution R = 20,000 over the 1 to 2.5 μm bandpass for Gran Telescopio Canarias. It possesses a multiplexing system with 12 cryogenic robotic probe arms, each capable of independently selecting a user-defined target in the instrument field of view. The arms are distributed around a circular bench, becoming a very packed workspace when all of them are in simultaneous operation. Therefore, their motions have to be carefully coordinated. We propose here a motion planning method for the MIRADAS probe arms. Our offline algorithm relies on roadmaps comprising alternative paths, which are discretized in a state-time space. The determination of collision-free trajectories in such space is achieved by means of a graph-search technique. The approach considers the constraints imposed by the particular architecture of the probe arms as well as the limitations of the commercial off-the-shelf motor controllers used in the mechanical design. We test our solution with real science targets and a typical MIRADAS scenario presenting some instances of the two identified collision conflicts that can arise between any pair of probe arms. Experiments show the method is versatile enough to compute trajectories fulfilling the requirements.

State−Time−Space Superstructure-Based MINLP Formulation for Batch Water-Allocation Network Design

The mathematical technique presented in this work deals with one step design of single- and multicontaminant batch water-allocation network (WAN), where batch production, water-reuse subsystems, and wastewater-treatment subsystems are all taken into account. In the first place, a flexible schedule model combining the merits of discrete and continuous time formulations is introduced to integrate batch production and WAN. Then, two novel state−time−space (STS) superstructures incorporating all basic elements (i.e., states, tasks, equipment, and time) are adopted to capture all production schemes and batch WAN configurations. Specifically, by adding novel components in the original superstructure, a series of optimal network structures with multistage splitting and mixing options which have never been contained within previous superstructure can be easily generated. Finally, a reliable optimization strategy, where deterministic and stochastic searching techniques are combined, is suggested to deal with the resulting mixed integer nonlinear programming (MINLP) model. Two illustrative examples are presented to demonstrate the effectiveness of the proposed approach.

Design of Generalized Hysteresis Controllers for DC–DC Switching Power Converters

This paper proposes a novel uniform controller design approach for switching power converters. The procedure allows the deriving of both clock-driven and event-driven switching laws for controlling the stationary converter operation. A generic control scheme is introduced, which generalizes the concept of hysteresis control and executes a mode change, whenever the augmented state trajectory intersects with a switching plane in the state-time space. In contrast to previous approaches, the switching planes are systematically computed by means of results on periodic control systems. Consequently, desired loop properties such as orbital stability of a limit cycle and a fast transient response are guaranteed at least in a local neighborhood around a nominal set point. All results apply to a large class of switching converters and impose no restrictions on the number of energy storage devices or switches.

Roadmap-based motion planning in dynamic environments

In this paper, a new method is presented for motion planning in dynamic environments, that is, finding a trajectory for a robot in a scene consisting of both static and dynamic, moving obstacles. We propose a practical algorithm based on a roadmap that is created for the static part of the scene. On this roadmap, an approximately time-optimal trajectory from a start to a goal configuration is computed, such that the robot does not collide with any moving obstacle. The trajectory is found by performing a two-level search for a shortest path. On the local level, trajectories on single edges of the roadmap are found using a depth-first search on an implicit grid in state-time space. On the global level, these local trajectories are coordinated using an A/sup */-search to find a global trajectory to the goal configuration. The approach is applicable to any robot type in configuration spaces with any dimension, and the motions of the dynamic obstacles are unconstrained, as long as they are known beforehand. The approach has been implemented for both free-flying and articulated robots in three-dimensional workspaces, and it has been applied to multirobot motion planning, as well. Experiments show that the method achieves interactive performance in complex environments.

Trajectory planning in a dynamic workspace: a 'state-time space' approach

Trajectory planning in a dynamic workspace: a 'state-time space' approach

Trajectory planning in a dynamic workspace: a 'state-time space' approach

This paper addresses trajectory planning in a dynamic workspace, i.e. motion planning for a robot subject to dynamic constraints and moving in a workspace with moving obstacles. First the novel concept of state-time space is introduced, i.e. the state space of the robot augmented of the time dimension. Like configuration space which is a tool to formulate path-planning problems, state-time space is a tool to formulate trajectory planning in dynamic workspace problems. It permits us to study the different aspects of dynamic trajectory planning, i.e. moving obstacles and dynamic constraints, in a unified way. Then this new concept is applied to the case of a car-like robot subject to dynamic constraints and moving along a given path on a dynamic planar workspace. A near-time-optimal approach that searches the solution trajectory over a restricted set of canonical trajectories is presented. These canonical trajectories are defined as having discrete and piecewise constant acceleration. Under these assumptions, it is possible to transform the problem of finding the time-optimal canonical trajectory to finding the shortest path in a directed graph embedded in the state-time space.

N-point, win-by-k games

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

N-point, win-by-k games

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

Local regularization of the Magnetic-Binary problem

The author's aim is to achieve global regularization in the Magnetic-Binary problem by suitably transforming the state-time space of the system. The functions which perform the change of the physical time and the geometrical figures of the system, are connected by a special relation leaving the form of the equations of motion invariant. Additionally, a proposition for generalization of the process is discussed in an aspect as well, of how much such a regularization is profitable.

Some aspects of control tolerances and first-order sensitibity in optimal control systems

This paper considers the effects of small control function variations of bounded magnitude. Two types of variations are considered: one, called continual, is infinitesimally small over the control interval; the other, called intermittent, is a pulse-type variation. If the control task is to take the system from an initial state to some target set, it is shown that, no matter how tight the control tolerances, there is always in general some control within tolerances which will cause the target to be missed. An exception to this rule is the case where the target is of dimensionality one less than that of the state-time space. Target sets not falling in this latter class are used in design as idealizations of actual target sets (e.g. a point is used instead of a small sphere). If a nominal control is found which takes the state to the ideal target, tolerances on this control are found such that the actual target will always be reached. If the nominal control is optimal with respect to the ideal set, giving some nominal cost, it is shown how one can find the possible cost variations resulting from the use of controls wthin the tolerance limits.