Reliability modeling of systems under dependent competing failures using bilateral coupling variables

There are multiple dependent competing failure processes (MDCFP) under dynamic loads and constantly changing environments, which can cause humanoid robot failure, such as structural fatigue, fracture, deformation, and joint wear. This poses a serious challenge in investigating its reliability behavior, requiring the development of more effective reliability models to fully capture its failure characteristics. The proposed modeling approach first divides these failures into soft (degradation) failures and hard (shock) failures, and then fully investigates the bilateral interaction between degradation and shocks in a unique way, establishing a general reliability framework for the system under MDCFP by introducing virtual coupling variables. This model possesses a wider range of applications, and for different problems only the performance functions require modification. Subsequently, we propose a new efficient algorithm by introducing auxiliary variables and additional inequality constraints to estimate the system reliability, which reduces the problem complexity through transforming the original nested problem into two sub-problems which are solved sequentially and iteratively. Finally, this paper applies the developed model and algorithm to a numerical example of humanoid robot systems, demonstrating its effectiveness in characterizing how correlated degradation and shocks influence system reliability. © 2026 Elsevier Ltd.