# Model Predictive Control of Running Biped Robot
###### Abstract  
With the feet of a biped robot attached insecuvely to a terrain, its stability is strongly affected by the characteristics of the terrain on which it runs. Therefore, for stable bipedal running, online motion control based on the states of the robot and the environment is needed. This paper proposes a method for online motion control of a running biped robot on an uneven terrain based on a dual linear inverted pendulum model (D-LIPM) and hierarchical control which consists of linear model predictive control (MPC) and quadratic-problem (QP) based momentum control. The D-LIPM, which splits the nonlinear dynamics model of the running biped robot into two linear models under some assumptions, is proposed to generate the running motion through linear MPC. The D-LIPM is applied to the proposed hierarchical control for stable bipedal running. In the first stage of hierarchy, linear MPC is employed to generate the trajectory of the center of mass (COM) based  
only considered instantaneous situations [25], and thus the information about the future states of the robot cannot be considered, resulting that recursive feasibility cannot be guaranteed. To address the issue of QP-based methods, a trajectory optimization-based method was proposed in [26], which has the drawback of a long computational time. An online re-planning method can be used as a solution to the problem, and model predictive control (MPC), which is one of the re-planning methods, has gained broad interest [27]. MPC is one of the control methods used to generate online motion. Its most important feature is its ability to handle constraints on the states and inputs of a dynamic system to select an optimized behaviour using model-based prediction [28]. As a result, MPC would have a robust and flexible behavior against unmodelled dynamics based on the predictive horizon with its states [29,30]. For the stable locomotion of a biped robot, a stable trajectory generation under the constraints on its workspace with robustness is critical. Therefore, several studies used MPC for bipedal locomotion. [PERSON] et al. generated a trajectory for walking of a humanoid robot based on the MPC with zero moment point (ZMP) preview control and showed in simulations that stable walking was achieved despite some disturbances [31]. [PERSON] proposed an intrinsically stable MPC framework for humanoid gait generation that incorporated a stability constraint based on the velocity of ZMP in the formulation and realized omnidirectional motion in real time [32]. Many MPC-based studies have focused on bipedal walking, but there is limited work on bipedal running. Because the dynamics of a running biped robot are nonlinear, it is difficult to generate running motion with linear MPC. A method to control a nonlinear motion such as hopping with a nonlinear MPC was proposed in [33]. However, the nonlinear MPC suffers from drawbacks of a non-convex optimization problem such as long computational time and local minimum [34]. As one of the methods for generating a nonlinear motion including a time-varying vertical trajectory, MPC with a divergent component of motion (DCM) has been proposed under the assumption of a predefined vertical trajectory, thus making it difficult to generate a real-time running trajectory [35,36]. To address the problem of MPC with a nonlinear dynamics model and to generate a stable online running motion of a biped robot, this paper proposes a method to control a running biped robot based on a dual linear inverted pendulum model (D-LIPM) and hierarchical control with linear MPC and momentum control. D-LIPM, which splits the nonlinear dynamics model of the running biped robot into two linear models under some assumptions, is proposed to generate the running motion through linear MPC. The D-LIPM is applied to the proposed hierarchical control for stable bipedal running.