The stiffness model plays a crucial role in improving the performance of robots. During the operation of an underground mining cable-driven parallel robot (UMCDPR), insufficient stiffness can lead to motion instability, posing safety hazards. Additionally, the complexity of the underground mining environment, which is often accompanied by external disturbances, leads to offline stiffness indices failing when used underground as an optimal criterion. To address these problems, this article proposes a robust optimal stiffness direction (ROSD) index grounded in Rayleigh's theorem, which is characterized by three primary features: (1) strong robustness, (2) suitable for multi-trajectory optimization engineering problems, and (3) global visualization. Firstly, considering the influence of pulleys on the end-effector, the stiffness model of UMCDPR is modified. Secondly, a trajectory optimization method utilizing ROSD is introduced, incorporating the Kepler Conjecture and stiffness model correction. Finally, the characteristics of ROSD are validated through numerical simulations. Based on two numerical simulations, the ROSD index can serve as an optimal criterion for guiding stiffness optimization of UMCDPR. Furthermore, an optimal stiffness trajectory is obtained to meet the task objectives of UMCDPR.

Metro disruptions due to unexpected events reduce transit system reliability, resulting in significant productivity loss and long passenger delays. Bus bridging strategy is often used to connect stations affected by metro disruptions such that passengers could continue their journey. The literature usually designed bridging routes and then allocated buses to designed routes with specific frequencies. The restriction that each bus can only operate on a route greatly limits the service flexibility and decreases operation efficiency. We propose a flexible bus bridging strategy to deal with the disruptions of metro networks. The proposed strategy optimizes a tailored bridging path for each bus. The path dictates the stations that a bus should visit in sequence once it is dispatched from the depot. A two-stage model that balances the needs of transit agency and passengers is developed to optimize the tailored bridging paths based on affected metro stations, reserved buses, bus capacity, passenger demands and bus travel times. The Stage I model produces schematic bridging paths by minimizing the maximum bus bridging time. The Stage II model further details the paths by minimizing average passenger delay. The superiority of the proposed strategy to a traditional strategy is demonstrated in a case study in Rotterdam, The Netherlands.