## 1 Introduction
Additionally, the modeling and coordinated control of multiple mobile manipulators with closed-chain constraints enables more efficient and safe trajectory planning, ensuring that the manipulators can operate synchronously without mutual interference [22]. Reinforcement learning, which has been explored in various studies, allows robots to learn from their environment and continuously improve their ability to avoid obstacles and plan optimal trajectories, adapting to changing and complex environments [23]. Collision detection and identification based on vibration analysis are other significant advancements that demonstrate how vibration analysis can be used to detect and prevent collisions in real time, thereby improving the safety and efficiency of robots in collaborative environments [24]. Finally, a comprehensive overview of decentralized approaches for collision avoidance in multi-robot systems is provided, highlighting the best practices and current challenges in the implementation of decentralized techniques and offering a solid foundation for future research and applications in trajectory planning and obstacle avoidance [25]. In the cooperative robotics literature, numerous studies have been conducted on control algorithms in robotic manipulators, some of which are detailed below. Predictive control has been widely used to coordinate cooperative robots, allowing anticipatory trajectory planning and movement optimization in a shared space. One study highlighted the use of predictive control to manage the interactions between multiple manipulators in a shared environment, improve efficiency, and avoid collisions [26]. Additionally, predictive control has been applied to cooperative robotic systems in manufacturing, using one-touch memory to optimize injection molding processes [27]. Another investigation focused on predictive control in cooperative systems applied to bipedal robots and UAV-UGV coupling, prioritizing tasks to enhance cooperation between different types of robots [28]. Decentralized control allows multiple manipulators to cooperate without the need for a central controller, thereby improving system robustness and scalability. One study described a decentralized approach to force and motion control in cooperative manipulators,demonstrating that robots can maintain effective cooperation through local communication and distributed control strategies [29]. Sliding mode control (SMC) is a robust technique used to handle nonlinear dynamics and uncertainties in cooperative systems. Research has demonstrated the effectiveness of sliding mode control in cooperative welding tasks, providing precise and robust control of multiple manipulators [30]. In addition, finite-time cooperative control using the integral sliding mode was implemented to improve the stability and performance of the cooperative manipulators [31]. Neural networks offer advanced adaptive capabilities, which can be leveraged to control cooperative robots. One study proposed synchronized position control based on neural networks for multiple manipulators, achieving high cooperation precision [32]. Deep learning has been integrated into the control of cooperative robots to enhance their real-time adaptation and learning capability. A study presented a collaborative control method for dual-arm robots based on deep learning, highlighting the improvements in robot cooperation and performance [33]. Adaptive control is used to dynamically adjust controller parameters in response to changes in the environment or system characteristics. An adaptive variable impedance control approach with a Fuzzy-PI composite controller has proven to be effective in managing the interaction between cooperative robots and their environment [34]. In addition, adaptive obstacle avoidance for collaborative robots has been studied to enhance their ability to adapt to dynamic environments [16]. Another example is the application of half-derivative damping in Cartesian space position control for SCARA-type manipulators [35].