A Comparison of the Effectiveness of the RRT, PRM, and Novel Hybrid RRT-PRM Path Planners

Sampling-based path planners develop paths for robots to journey to their destinations. The two main types of sampling-based techniques are the probabilistic roadmap (PRM) and the Rapidly Exploring Random Tree (RRT). PRMs are multi-query methods that construct roadmaps to find routes, while RRTs are single-query techniques that grow search trees to find paths. This investigation evaluated the effectiveness of the PRM, the RRT, and the novel Hybrid RRT-PRM methods. This novel path planner was developed to improve the performance of the RRT and PRM techniques. It is a fusion of the RRT and PRM methods, and its goal is to reduce the path length. Experiments were conducted to evaluate the effectiveness of these path planners. The performance metrics included the path length, runtime, number of nodes in the path, number of nodes in the search tree or roadmap, and the number of iterations required to obtain the path. Results showed that the Hybrid RRT-PRM method was more effective than the PRM and RRT techniques because of the shorter path length. This new technique searched for a path in the convex hull region, which is a subset of the search area near to the start and end locations. The roadmap for the Hybrid RRT-PRM could also be re-used to find pathways for other sets of initial and final positions. Keywords: Path Planning, Sampling-based algorithms, search tree, roadmap, single-query planners, multi-query planners, Rapidly Exploring Random Tree (RRT), Probabilistic Roadmap (PRM), Hybrid RRT-PRM

Basic Spatiotemporal Gait Variables of Young and Older Healthy Volunteers Walking Along a Novel Figure-of-8 Path

Background: Locomotion along curved trajectories requires fine coordination among body segments. Elderly people may adopt a cautious attitude when steering. A simple, expeditious, patient-friendly walking protocol can be a tool to help clinicians. We evaluated the feasibility of a procedure based upon a newly designed Figure-of-eight (nFo8) path and an easy measurement operation.Methods: Sixty healthy volunteers, aged from 20 to 86 years, walked three times at self-selected speed along a 20 m linear (LIN) and the 20 m nFo8 path. Number of steps, mean speed and walk ratio (step length/cadence) were collected. Data were analysed for the entire cohort and for the groups aged 20–45, 46–65, and >65 years.Results: There was no difference in mean LIN walking speed between the two younger groups but the oldest was slower. During nFo8, all groups were slower (about 16%) than during LIN. Cadence was not different across groups but lower during nFo8 in each group. Step length was about 8% shorter in the two younger groups and 14% shorter in the oldest during nFo8 compared to LIN. Walk ratio was the smallest in the oldest group for both LIN and nFo8.Conclusions: A complex nFo8 walking path, with fast and easy measurement of a simple set of variables, detects significant differences with moderate and large effects in gait variables in people >65 years. This challenging trajectory is more revealing than LIN. Further studies are needed to develop a quick clinical tool for assessment of gait conditions or outcome of rehabilitative treatments.

Determining the influencing factors of preferential flow in ground fissures for coal mine dump eco-engineering

Ground fissures (GF), appearing in front of dumps, are one of the most obvious and harmful geological hazards in coal mining areas. Studying preferential flow and its influencing factors in the ground fissures of dumps may provide basic scientific support for understanding the rapid movement of water and vegetation restoration and reconstruction in mining areas. Based on field surveys of ground fissures, three typical ground fissures were selected in the studied dump. The morphological characteristics of preferential flow for ground fissures were determined through field dye tracing, laboratory experiments, and image processing technology. The results indicated that the lengths of the three ground fissures ranged from 104.84 cm to 120.83 cm, and the widths ranged from 2.86 cm to 9.85 cm. All of the ground fissure area densities were less than 10%, and the proportion of ground fissure surface area was small in the dump. The maximum fissure depth was 47 cm, and the minimum was 16 cm. The ground fissure widths ranged from 0 cm to 14.98 cm, and the fissure width and fissure width-to-depth ratios decreased with increasing soil depth. The stained area was greater than 90% in the 0–5 cm soil layers of the three fissures, and water movement was dominated by matrix flow. The stained width decreased from 90 cm to 20 cm with increasing soil depth. The preferential flow was mainly concentrated on both sides of the fissure, which was distributed as a “T” shape. The preferential flow stained area ratios were 27.23%, 31.97%, and 30.73%, respectively, and these values decreased with increasing soil depth. The maximum stained depths of the preferential flow among the three fissures were different, and the maximum stained depth of GF II was significantly larger than that of GF I and GF III (P < 0.05). The stained path numbers of the three fissures ranged from 0 to 49. With increasing soil depth, the stained path number first increased and then decreased. The stained path widths of the three fissures ranged from 0 cm to 90 cm. With the increase in soil depth, the stained path width decreased. The stained area ratio was significantly positively correlated with ground fissure width, the ground fissure width-to-depth ratio, soil saturated hydraulic conductivity, soil organic matter, and sand content and was significantly negatively correlated with soil water content and clay content. The stained path number was significantly positively correlated with ground fissure width, the ground fissure width-to-depth ratio, soil saturated hydraulic conductivity and soil organic matter. The stained path width was significantly positively correlated with the ground fissure width-to-depth ratio, soil saturated hydraulic conductivity, soil organic matter and sand content and was significantly negatively correlated with clay content. Plant roots could significantly increase the stained area ratio, stained path number, and width and promote the formation and development of preferential flow.

Hosoya and Harary Polynomials of Hourglass and Rhombic Benzenoid Systems

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Harry Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we compute the Hosoya polynomials for hourglass and rhombic benzenoid systems and recover Wiener and hyper-Wiener indices from them.

Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.

Induced cycle path number of derived graphs

An induced cycle path partition of a simple connected graph [Formula: see text] is a partition of [Formula: see text] into subsets such that the subsets induce cycles and paths. The minimum order taken over all induced cycle path partitions is called the induced cycle path number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we investigate the induced cycle path number of some derived graphs such as line graph, block graph, subdivision graph, power graph and total graph.

Graph covers and ergodicity for zero-dimensional systems

Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.