We prove a homotopy invariance result for a certain covering space of the space of ordered configurations of two points in M × X where M is a closed smooth manifold and X is any fixed aspherical space which is not a point.

Let $W$ be a compact simply connected triangulated manifold with boundary and let $K\,\subset \,W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of $W\text{ }\!\!\backslash\!\!\text{ K}$ out of a model of the map of pairs $\left( K,\,K\cap \partial W \right)\,\to \,\left( W,\,\partial W \right)$ under some high codimension hypothesis.

We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

We present an algorithm for offsetting the workspace obstacles of a circular robot. Our method has two major steps: It finds the raw offset curve for both lines and circular arcs, and then removes the global invalid loops to find the final offset. To generate the raw offset curve and remove global invalid loops, O(n) and O((n+k)log m) computational times are needed respectively, where n is the number of vertices in the original polygon, k is the number of self-intersections and m is the number of segments in the raw offset curve, where m ≤ n. Any local invalid loops are removed before generating the raw offset curve by invoking a pair-wise intersection detection test (PIDT). In the PIDT, two intersecting entities are checked immediately after they are computed, and if the test is positive, portions of the intersecting segments are removed. Our method works for conventional polygons as well as the polygons that contain circular arcs. Our algorithm is simple and very fast, as each sub-process of the algorithm can be completed in linear time except the last one, which is nearly linear. Therefore, the overall complexity of the algorithm is nearly linear. By applying our simple and efficient approach, offsetting obstacles of any shape make it possible to construct a configuration space that ensures optimized motion planning.

A topological approach to the theory of equilibrium phase transitions in statistical physics is based on the topological hypothesis, which claims that phase transitions are due to changes of the topology of suitable submanifolds in the configuration space. In this paper we examine in detail the antiferromagnetic mean-field XY model and study the topology of the subenergy manifolds. The latter can be interpreted mechanically as the configuration space of a linkage with one telescopic leg. We use methods of Morse theory to describe explicitly the Betti numbers of this configuration space. We apply these results to the antiferromagnetic mean-field XY model and compute the exponential growth rate of the total Betti number. Previous authors studied the Euler characteristic rather than the total Betti number. We show that in the presence of an external magnetic field the model undergoes a single ‘total Betti number phase transition’.

Collision-free path planning for static robots is a demanding manifold of contemporary robotics research, vastly due to the growing industrial applications. In this paper, a novel ‘visibility map’-based heuristic algorithm is used to generate near-optimal safe path for a three-dimensional congested robot workspace. The final path is obtainable in terms of joint configurations, by considering the Configuration Space of the task space. The developed algorithm has been verified initially by considering representative 2D workspaces, cluttered with different obstacles with regular geometries and then after with the spatial endeavour. A case study reveals the effectiveness of the developed modules of the configuration space mapping, pertaining to a five degrees-of-freedom low payload articulated robot.

Many motion planning methods use Configuration Space to represent a robot manipulator's range of motion and the obstacles which exist in its environment. The Cartesian to Configuration Space mapping is computationally intensive and this paper describes how the execution time can be decreased by using parallel processing. The natural tree structure of the algorithm is exploited to partition the computation into parallel tasks. An implementation programmed in the occam2 parallel computer language running on a network of INMOS transputers is described. The benefits of dynamically scheduling the tasks onto the processors are explained and verified by means of measured execution times on various processor network topologies. It is concluded that excellent speed-up and efficiency can be achieved provided that proper account is taken of the variable task lengths in the computation.

This paper presents a new neural networks-based method to solve the motion planning problem, i.e. to construct a collision-free path for a moving object among fixed obstacles. Our ‘navigator’ basically consists of two neural networks: The first one is a modified feed-forward neural network, which is used to determine the configuration space; the moving object is modelled as a configuration point in the configuration space. The second neural network is a modified bidirectional associative memory, which is used to find a path for the configuration point through the configuration space while avoiding the configuration obstacles. The basic processing unit of the neural networks may be constructed using logic gates, including AND gates, OR gates, NOT gate and flip flops. Examples of efficient solutions to difficult motion planning problems using our proposed techniques are presented.

The minimum-time motion of robot manipulators is solved by defining a suitable time history for the arm end-effector to traverse. As the planning is performed in the configuration space, the uniqueness of the proposed algorithm emerges from the combination of both cubic and quadratic polynomial splines. Furthermore, the highly efficient time optimisation procedure could be applied to local segments of each joint trajectory, leading to a significant reduction of the travelling time. In addition, the ability to perform a search in the work space is granted, exploiting all possible options for an optimum motion. The method proposed considers all realistic physical limitations inherent in the manipulator design, in addition to any geometric constraints imposed on the path. Simulation programs have been written, and results are reported for the Unimation PUMA 560 robot manipulator.

Collision-Avoidance is a key issue in planning trajectories for dual robots whose workspaces overlap. In this paper, we develop a new trajectory planning method by proposing a traffic control schemes. The traffic controller determines the next positions for each robot based on the motion priority and path direction subject to the collision-avoidance conditions and the robots' physical limits. The problem of determining the next positions is formulated and optimized. Algebraic expressions for collision avoidance between every-pair of links – one from the first robot and the other from the second robot – are derived in configuration space. These algebraic expressions are then used to solve the problem of determining “optimal” (in the sense of path direction and motion priority) robots' trajectories. A solution procedure is developed using a nonlinear programming (NLP) solver. The main advantage of our approach is that the two robots' trajectories can be determined simultaneously without requiring any a priori path information. Several numerical examples are presented to demonstrate the validity and effectiveness of the proposed approach.

We describe an approach that uses causal and geometric reasoning to construct explanations for the purposes of the geometric features on the parts of a mechanical device. To identify the purpose of a feature, the device is simulated with and without the feature. The simulations are then translated into a “causal-process” representation, which allows qualitatively important differences to be identified. These differences reveal the behaviors caused and prevented by the feature and thus provide useful cues about the feature's purpose. A clear understanding of the feature's purpose, however, requires a detailed analysis of the causal connections between the caused and prevented behaviors. This presents a significant challenge because one has to understand how a behavior that normally takes place affects (or is affected by) another behavior that is normally absent. This article describes techniques for identifying such elusive relationships. These techniques employ a set of rules that can determine if one behavior enables or disables another that is spatially and temporally far away. They do so by geometrically examining the traces of the causal processes in the device's configuration space. Using the results of this analysis, our program can automatically generate text output describing how the feature performs its function.

We present a new algorithm for fine motion planning in geometrically complex situations. Geometrically complex situations have complex robot and environment geometry, crowded environments, narrow passages and tight fits. They require complex robot motions with coupled degrees of freedom. The algorithm constructs a path by incrementally building a graph of linearized convex configuration space cells and solving a series of linear optimization problems with varying objective functions. Its advantages are that it better exploits the local geometry of narrow passages in configuration space, and that its complexity does not significantly increase as the clearance of narrow passages decreases. We demonstrate the algorithm on examples which other planners could not solve.