Symmetry

Robotic platoons must maintain geometric formation integrity while navigating complex environments, yet existing performance metrics primarily evaluate tracking accuracy or inter-vehicle spacing without directly quantifying the evolution of formation symmetry. This paper presents a real-time symmetry monitoring framework based on the proposed Turn-Induced Lateral Dispersion (TILD) metric and the Symmetry Stability Index (SSI), a normalized indicator that continuously evaluates geometric symmetry degradation during platoon operation. The methodology estimates the lateral geometric error of each follower, accumulates it through the TILD metric, and transforms the resulting dispersion into the bounded SSI. Using predefined thresholds, the framework classifies platoon operation into NORMAL, DEGRADED, and CRITICAL states, enabling continuous online supervision without modifying the underlying controller. The framework was implemented in a ROS 2 and Gazebo environment and validated with a robotic platoon navigating representative BARN benchmark worlds of increasing geometric complexity. The results reveal a consistent inverse relationship between accumulated lateral dispersion and symmetry stability, allowing progressive formation degradation to be quantitatively characterized across operating conditions. The framework thus offers a lightweight, interpretable, and continuously operating symmetry assessment that complements conventional platoon metrics, establishing a foundation for future symmetry-aware supervisory, diagnostic, and cooperative recovery control strategies.

Does square-lattice symmetry determine the search parameters selected by a fractional random walk with resetting? We compare a walk that updates one coordinate per movement with a walk generated by a fractional power of the full square-torus Laplacian. Both have the same lattice symmetry and nearest-neighbor limit. For the coordinate-updated walk, we prove that coordinate exchange maps the complete first-passage distribution to that of the parameter-swapped target problem. This identity pairs asymmetric local minima and separates the Hessian at a symmetric stationary point into even and odd sectors. For the full-torus walk, we derive the probability of simultaneous two-coordinate motion and relate it exactly to the ratio of the two lazy-chain spectral gaps. Matched searches on a 31-by-31 torus resolve one equal-index candidate for the coordinate-updated walk and two for the full-torus walk at axial target distances 3–5. One-sided derivatives support the reported boundary candidates as constrained solutions, not estimates of an unconstrained optimal index. A fixed-relative-geometry comparison at side lengths 31, 62, and 93 shows that these candidates remain sensitive to lattice resolution and the index cutoff. Thus exchange symmetry constrains relations among search problems, while the generator, encounter rule, and reset timing determine the candidates recovered in a specified finite system.

The study of hybrid multiterminal high-voltage direct current (HVDC) systems revealed different outcomes regarding their performance under fault conditions. Previous research showed that under the fault conditions, the high-voltage direct current (HVDC) voltage experienced a drop to zero, accompanied by a reverse overshoot. In this paper, a model that integrates hybrid multiterminal line commutated converters (LCCs) and a voltage source converter (VSC) with an HVDC network is presented. The model has been mathematically formulated and implemented using Matlab/Simulink (R2018b) software to examine its fault behaviors and locations, particularly on the DC line to ground fault, line to line fault, and single line to ground fault across various fault resistance levels. The system consists of a wind energy conversion system, a photovoltaic array, protection scheme, LCC rectifier station, VSC inverter station, inverter control, AC filters, a distributed parameter transmission line for the HVDC transmission line, a three-phase inductor-capacitor (LC) filter, and a three-phase transformer. The findings indicated that during the scenario of an 8 ohms of resistance under a single line to ground fault, phase A of the inverter AC grid voltage decreased from its operational level. Meanwhile, the voltage across the DC line increased, accompanied by a rise in DC line current. Additionally, calculations of the fault location were determined to be 1.1561 km from the point of reference. This study contributes to the understanding of fault dynamics in HVDC systems, providing essential insights that enhance operational reliability and efficiency in power transmission networks.