# Online Foot Location Planning for Gait Transitioning Using Model Predictive Control
## 2 Introduction
[PERSON] et al. use the MPC to complete a terrain adaptive gait transitioning between bounding and trotting [11]. MPC-based foot location planning has also been widely studied. A number of researchers achieve better robustness and interference resistance on quadruped robots based on the LIP model for foot location planning using MPC [12; 13; 14]. [PERSON] et al. design LIP-based foot location planning using MPC [15]. The method applies the LIP model to predict the COM position and velocity. Furthermore, the prediction result is used as the initial state of MPC for planning the foot location. It is successfully applied not only to quadruped robots but also to bipedal robots, and has good anti-interference capability. However, the calculating time grows rapidly with increasing prediction length. This paper aims at addressing the transitioning of walking, trotting and flight trotting with different gait periods at the same speed. Walking (four beat or two beat) has a duty cycle greater than 50%. The duty cycle is the proportion of the support period to the entire gait period. Furthermore, the four-legged support period can exist in this gait. The two diagonally opposite legs move together for trotting. The duty cycle of trotting is equal to 50%. The two diagonal opposite leg of the flight trotting gait also moves simultaneously, but differs from trotting in that the duty cycle is less than 50%. There is a period of quadrupedal vacancy. The gait cycles for three gaits are shown in Figure 1. Walking is a common gait and employed by all terrestrial vertebrates [16]. Walking is stable with little change in height and posture. And the quadruped robot has little contact force with the ground. The results in [1; 2] indicate that either two-beat walking or four-beat walking has less energy expenditure at low speeds than the other gaits. Trotting is widely adaptable to a greater range of speeds and has high energy efficiency when the speed is below 1.2 m/s [4]. Trotting with a vacant period (flight trotting) has a better performance in terms of energy utilization at high speeds [2]. Furthermore, these three gaits are representative on quadruped robots that can adapt to a variety of environments and task needs. During the gait transitioning process, the quadruped robot posture and speed will experience large changes. By choosing a suitable foot location, the effect of the foot location on posture and speed can be reduced. The quadruped robot can return to the stable position more quickly. The three gaits which are used for gait transitioning, with widely varying characteristics, have different gait periods. The HF gain parameters required for each gait are also different. There is no fixed order for transitioning. The difference in gait characteristics leads to large changes in posture after transitioning, making it difficult for the fixed parameter HF to cope with. The MPC-based foot location planning method can select a foot location that best meets the desired future state based on the actual situation during the transitioning process and the prediction results. Thus, the quadruped robot can better complete the gait transitioning and quickly correct the state deviation caused by the transitioning, with better robustness and adaptability. Therefore, it was decided to use UPMPC as the foot location planning method for quadruped robots. During the motion of the quadruped robot, the position of the foot in the world coordinate system remains unchanged. By unifying the foot positions at each time step, the final optimization variables are ensured to be independent of the predicted horizon length. The matrix dimensions to be computed are greatly reduced, which can significantly improve the computational speed of UPMPC to satisfy the demand of high-frequency foot location planning. The gait transitioning results given at the end of the article well demonstrate that UPMPC has better practical performance than HF and can be applied to the foot location planning for fixed gait.