# Energy Consumption Analysis of a Rolling Mechanism Based on a Five-Bow-Shaped-Bar Linkage
## 1 Introduction  
Mobile mechanisms have a variety of locomotion modes, commonly including walking, rolling, and crawling [1; 2; 3; 4]. In the rolling mode, a class of mobile mechanisms, called structural morphing mechanisms, rolls forward by changing their shapes [5; 6; 7]. The closed five-bow-shaped-bar linkage is a structural morphing mechanism. The rolling locomotion of the mechanism is generated by making a certain offset of its CoM in the horizontal direction with respect to the contact point between the bow-shaped module and the ground. This mechanism was first investigated by our project team, and its favorable characteristics were demonstrated in speed, compliance, and continuity [8; 9]. Then, the mechanism's climbing capacity and the minimum coefficient of friction were analyzed [10]. Previous studies have mainly focused on the kinematic performance of the mechanism, neglecting the study of energy consumption. However, energy consumption is an essential research direction that can significantly reduce the operating costs of mechanisms. For the mechanism, the location of its CoM is closely related to energy consumption. Therefore, it is necessary to determine the relationship between the energy consumption of the mechanism and CoM trajectories, which can be used as a reference for locomotion planning. Locomotion planning is an essential branch of mechanism research that studies how a mechanism moves from the initial position to the target position with less energy consumption [11]. Many studies have been carried out to improve and optimize the trajectory of mechanisms. A trajectory plan was generated by [12] based on a smooth point-to-point method to obtain continuous displacement, velocity, acceleration, and jerk of each jointand end-effector. An algorithm was established by [13], as well, based on cubic smoothing B-spline curves to represent the final trajectories that are collision-free. [PERSON] et al. [14] proposed an improved lazy Probabilistic Roadmap algorithm based on the online collision-free path planning method. [PERSON] et al. [15] raised a new method for the path planning of the robotic arm mounted on a free-floating unmanned spacecraft. [PERSON] et al. [16] constructed an adaptive model predictive control trajectory tracking system that considers disturbances in the path curvature. Some studies used machine learning models [17; 18] to optimize the trajectories of mechanisms. [PERSON] et al. [19] presented a biological-inspired reinforcement learning control method to help the robot learn adaptive swimming actions. [PERSON] et al. [20] proposed a machine learning-based energy efficient trajectory planning method for industrial robots. [PERSON] et al. [21] obtained the optimal gait of a quadruped robot by using deep reinforcement learning. Furthermore, some studies have focused on reducing energy consumption by optimizing trajectories. The objective function was assumed to minimize the total consumed energy to find an optimum trajectory planning of the positioning manipulator [22]. A control architecture was developed to coordinate energy management with trajectory tracking control for hybrid UAVs [23]. [PERSON] et al. [24] raised an efficient co-optimization strategy of unmanned hybrid electric tracked vehicles to achieve accurate path tracking and optimal energy management simultaneously. [PERSON] and [PERSON] [25] presented a new trajectory optimization scheme adopting dynamic programming, which combined the joint energy consumption with a particle swarm optimization algorithm. Some researchers have paid particular attention to the problem of mechanism workspace and its trajectory planning. The end-effector motion trajectory was planned based on the workspace of the weeding mechanism [26]. [PERSON] et al. [27] studied the reachable workspace of the robot body with shank-ground interferences, which helped to enhance the efficiency of foothold selection.