# Online Foot Location Planning for Gait Transitioning Using Model Predictive Control
## 2 Introduction  
Legged robots have greater environmental adaptability than wheeled and tracked robots. In order to meet the needs of different tasks in various environments, users are no longer satisfied with quadruped robots moving in a fixed gait and expect legged robots to be able to switch stably between multiple gaits during moving. Quadruped robots usually perform gait transitioning to reduce energy consumption. Horses [1] have varying energy expenditure at different gaits and speeds. In order to maximize energy efficiency, the horse will choose the gait with the least energy consumption at each speed. Quadruped robots designed with reference to quadrupeds naturally have variable energy consumption at different gaits. The energy consumption of a quadruped robot at various gaits and speeds is explored in detail in [2]. Each gait has a significant energy gap at the same speed. It is also found that the long-period, large duty cycle gait and the short-period, low duty cycle gait consume less energy at low and high speeds, respectively. Therefore, it is of utmost importance to switch from a high-energy-consuming gait to a gait with relatively low energy consumption. However, in actual use, the appropriate gait is also selected according to the task needs. When traversing uneven terrain, it is expected that the impact on the ground is small and the height of the quadruped robot is basically constant while maintaining a certain speed. Thus, the gait chosen at this point may not be the less energy intensive one. The choice of gait is the result of many factors. Therefore, the quadruped robot needs to successfully switch between different gaits at a certain speed, not just from a high-energy-consuming gait to a low-energy-consuming gait. The successful locomotion of the robot cannot be separated from the choice of the foot location. A lot of researches have been done on foot location planning. HF [3] is a commonly used method. By adjusting the gain parameters, better results can be obtained. The method has been applied in several MIT's robots [4; 5]. Combining HF and MPC [4], MIT's Mini Cheetah implements multiple gaits. However, the fixed-parameter HF is difficult to cope with transitioning between various gaits with different periods. The properties of gaits are distinct, in the case of different periods, leading to the inability of the fixed-parameter HF to adapt to the large changes of posture after the gait transitioning. The capture point technique which is frequently used in bipedal robots is also a common method for foot location planning. [PERSON] et al. simplify the bipedal robot into a LIP model with a flywheel, and determine the range of foot locations and the foot location point of the bipedal robot  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Meaning** \\ \hline \(x\) & The position of the COM on the x-axis \\  & in the world coordinate system \\ \(y\) & The position of the COM on the y-axis \\  & in the world coordinate system \\ \(z\) & The position of the COM on the z-axis \\  & in the world coordinate system \\ \(\mathbf{p}\) & The position of the COM \\  & in the world coordinate system \\ \(p_{x}\) & The position of supporting foot in the LIP model \\  & on the x-axis in the world coordinate system \\ \(p_{y}\) & The position of supporting foot in the LIP model \\  & on the y-axis in the world coordinate  
by establishing the capture domain and the capture point where the robot can stop in one step [6]. [PERSON] et al. find the capture domain that can two-step stop based on the capture points and N-Step Capturability theories, and successfully verify the effect of the algorithm on simulation [7]. [PERSON] et al. [8] and [PERSON] et al. [9] both utilize capture point theory for foot location planning and achieve good results on bipedal robots. However, the capture point method requires setting the number of steps to stop, which limits the performance of the robot. MPC is capable of generating behavior with a wide range of adaptations based on real-world situations [10].