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Legged robots can traverse challenging terrain, use perception to plan their safe foothold positions, and navigate the environment. Such unique mobility capabilities make these platforms a perfect candidate for scenarios such as search and…
The quantum geometry arising in Loop Quantum Gravity has been known to semi-classically lead to generalizations of length-geometries. There have been several attempts to interpret these so called twisted geometries and understand their role…
The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic…
In four-dimensional symplectic maps complex instability of periodic orbits is possible, which cannot occur in the two-dimensional case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter…
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to…
Robust gait recognition requires highly discriminative representations, which are closely tied to input modalities. While binary silhouettes and skeletons have dominated recent literature, these 2D representations fall short of capturing…
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $\mathbb{R}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are…
Geometric mechanics provides valuable insights into how biological and robotic systems use changes in shape to move by mechanically interacting with their environment. In high-friction environments it provides that the entire interaction is…
A global model is presented that can be used to study attitude maneuvers of a rigid spacecraft in a circular orbit about a large central body. The model includes gravity gradient effects that arise from the non-uniform gravity field and…
Multi-legged robots offer enhanced stability in complex terrains, yet autonomously learning natural and robust motions in such environments remains challenging. Drawing inspiration from animals' progressive learning patterns, from simple to…
In this paper, we show that the constant property of the Gaussian curvature of surfaces of revolution in both $\mathbb R^4$ and $\mathbb R_1^4$ depend only on the radius of rotation. We then give necessary and sufficient conditions for the…
In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to…
The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976…
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…
We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on…
Inertia-dominated mechanical systems can achieve net displacement by 1) periodically changing their shape (known as kinematic gait) and 2) adjusting their inertia distribution to utilize the existing nonzero net momentum (known as momentum…
Free-space-oriented roadmaps typically generate a series of convex geometric primitives, which constitute the safe region for motion planning. However, a static environment is assumed for this kind of roadmap. This assumption makes it…
This paper analyses joint-space walking mechanisms and redundancies in delivering functional gait outcomes. Multiple biomechanical measures are analysed for two healthy male adults who participated in a multi-factorial study and walked…
When walking on loose terrains, possibly covered with vegetation, the ground and grass should deform, but the character's gait should also change accordingly. We propose a method for modeling such two-ways interactions in real-time. We…