## 1 Introduction  
The basic functionality required for humanoid robots is the ability to achieve various human movements. There are several existing methods for the planning and control of walking without revolving around the axis perpendicular to the horizontal plane [1; 2; 3; 4; 5]. The more challenging problem inherent to controlling a biped robot is maintaining its stability when it is moving. A widely used method for determining the stability of the robots is whether the zero moment point (ZMP) is in the supporting area [6; 7]. In the trajectory planning method, the ZMP is seen as a linear inverted pendulum (LIP) which is a simplified model. The robot is regarded as a point, and the entire mass is concentrated at the center of the mass. Another trajectory planning method considers the humanoid robot as a seven-link model. The position of the ZMP can be calculated by the state of the COM of every link rod including the position, the velocity and the acceleration [8]. By comparing the ZMP trajectories, the trajectory with the highest stability margin is selected as the off-line trajectory. These methods have unique advantages and disadvantages. Recently, newer gait-planning methods [9; 10; 11; 12; 13] have been developed that do not rely on the ZMP and have indeed produced marked improvements in humanoid robot walking. However, it is also very important for humanoid robots to be able to turn while walking at a high speed. For the robot to achieve various types of locomotion and efficiently complete any given task, it must be capable of turning while walking at a high speed. When the robot turns at a very low speed, the trajectory of the center of mass can be generated by the traditional LIP. However, the robot is likely to tip over during fast turns, but only if the trajectory is only generated by the LIP. Unlike straight-line walk (or \"linear walk\"), the so-called \"turning walk\" requires highly complex dynamical systems. During turning walk, the COM moves as it does during linear walk but while simultaneously performing circular motions, so in the world reference, the robot can not be seen as a particle. There are two choices that can be chosen to deal with the turning walk. One is to establish the humanoid robot whole body dynamics model in the world reference. The other is to translate the turning walk into the straight-line walk by choosing an appropriate non-inertial reference, and, in the non-inertial reference, the LIP model can also be used to generate the trajectory while the dynamics of the LIP are changed. There is an extended model of the LIP that is applied to the trajectory planning of the robot. The most popular model is the spring loaded inverted pendulum (SLIP) [14; 15; 16] which is used in robot running. In robot running, the LIP model causes big collisions to the ground, so in order to reduce the force that is generated by the collisions, the SLIP is proposed and has produced a good effect. Previous researchers have indeed explored dynamic turning. The most famous example is the ASIMO robot, which can turn while walking and even while running. [PERSON] [17] achieved a quasi-passive turn walk by utilizing a mechanical oscillator. There has also been research into slip turns; [PERSON] presented a novel hierarchical controller for walking torque-controlled humanoid robots [18; 19] which is capable of executing quick slip-turns on a HRP-4C on its toes. [PERSON] at al. achieved slip-turns in the HOAP-2 robot [20; 21; 22; 23]. Their method minimizes the turning angle based on variations in friction across the floor. Despite these valuable contributions, there has been relatively little research on turning while the robot is walking. Most research has applied straight-line walking models directly to the turning walks [24]. This method does not take into account the effect of the robot's own rotation on the actual ZMP so in the lateral direction, the actual ZMP will be different form the planned ZMP. Due to the large model error, the robot will be easily unstable.