# Time Efficiency Improvement in Quadruped Walking with Supervised Training Joint Model
## 1 Introduction  
Since the 1980s [1], the locomotion of quadruped robots has been widely investigated to ascertain a stable and more effective walking motion. Studies have typically involved either the moment of inertia for the entire robot system [2; 3; 4] or the zero-moment point (ZMP) [5; 6] to optimize the trajectory for the robot's center of mass in order to allow the robot to walk in a stable fashion. To this end, the locomotion control inputs the optimized center-of-mass (COM) trajectory result into the inverse kinematics calculations to obtain the joint solution. As a promising stable walking approach for a quadruped robot, the robotic dynamic could be modeled as the optimization objective, in which the walking stability, such as the zero-moment point or the inverted pendulum, would be derived as the constraint. However, the resulting calculations are tedious, at best. For example, [PERSON] et al. argued that using an adaptive controller for quadruped locomotion (ACQL) could provide a recursive update. so that the external force of the control law could be incorporated with an inverse-dynamic-based quadratic programming (QP) method to realize an optimized trotting gait [7]. However, although the ACQL provided an optimum and stable walk for the quadruped, it could only do so at the expense of time efficiency. In another study, [PERSON] et al. proposed the implementation of quadratic programming (QP) for the optimization of the model predictive control (MPC) of the inertia of a robot that could effectively generate optimum locomotion [8]. In this case, the robot walking dynamic involved a non-linear model, thus requiring the non-linear programming (NP) optimization method. Among the NP methods for optimizing robot trajectory, the method using mixed-integer quadratic programming (MIQP) has been the most promising for an optimized solution. MIQP has been related to the problem of optimizing a quadratic function over the points in a polyhedral set containing some integers and continuous variables [9]. This optimization was also implemented by [PERSON] et al. [10] for optimizing the center-of-mass trajectory for quadrupedrobot locomotion control. This implementation demonstrated that the MIQP method could provide an optimal solution and offered a time-effective optimization process, as compared to other non-linear programming optimization methods. However, this optimization task required that every walking step be redefined, which affected the time efficiency related to the overall walking performance of the quadruped. To address this issue, this study extended the conventional locomotion manipulation method and implemented a machine-learning algorithm to simplify the control policy while generating the joint solution. By reducing the optimization time, the time efficiency of the walking simulation was noticeably improved. The simplified control policy could effectively generate results for the quadruped in a short time, which enhanced the computational time during locomotion. The advantage of this implementation was that it omitted the optimization time in the conventional method and stabilized the quadruped's walking locomotion. Machine learning involves two main learning methods: reinforcement (unsupervised) learning and supervised learning. There has been substantial research on locomotion control for legged robots using reinforcement learning, which has provided a practical and general approach in building controllers. These trained controllers have been used to perform a range of sophisticated tasks. However, using the designated behaviors to create the reward functions necessary for reinforcement training involves a tedious task-specific tuning process. In addition, a trained model based on reinforcement learning can adopt unnatural behaviors that are dangerous and infeasible in real-world applications, particularly if the reward functions have not been designed carefully.