Narrow Results

Given an edge-oriented polygonal graph in ℝ³, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of path-connected open regions on the unit sphere. Within each open region, the directional writhe is constant. We obtain a closed formula which extends the formula for the writhe of a polygon in ℝ³, including the important special case of writhe of embedded open arcs.

The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications [7, 8, 11, 12, 13, 17, 18]. The first such algorithms developed were able to traverse triangulated subdivisions [10]. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope [3]. The research progress culminated to an algorithm that can traverse any planar subdivision [6, 9]. In this paper, we extend the notion of planar subdivision to quasi-planar subdivision in which we allow many edges to cross each other. We generalize the algorithm from [9] to traverse any quasi-planar subdivision that satisfies a simple geometric requirement. If we use techniques from [6] the worst case running time of our algorithm is O(|E|log|E|); matching the running time of the traversal algorithm for planar subdivisions [6].

This paper proposes a method of giving humanoid robots a natural humanlike walk, which we call the extended-knee walk. Unlike the bent-knee walk of most humanoid robots to date, this walk includes a period in which the knee is fully extended. A parallel mechanism is used in the legs and a method of calculating the walk trajectory copes with the difficulty of the singularity in achieving a humanlike walk. The advantages of this walk were verified from two aspects: good visual appearance and good energy efficiency. An experiment comparing the trajectories of the knee angle during walking showed that the walking style produced by the proposed method is more humanlike than the usual walking style of other humanoid robots. The energy efficiency was verified through power consumption and motor temperature measurements and the possibilities for practical use of this method are discussed with reference to the results of the worldwide soccer competition RoboCup 2008.

The Estrada index of a simple connected graph $G$ of order $n$ is defined as $EE(G)={\sum }_{i=1}^n{\mathrm{\text{e}}}^{{\lambda }_i}$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are the eigenvalues of the adjacency matrix of $G$. In this paper, we characterize all tetracyclic graphs of order $n$ with maximal Estrada index.