## 2 Non-Inertial Reference in Turning Walk  
In the translational movement, the robot can be seen as a mass point because when it is treated as a rigid, the motion of the COM can represent the motion of the whole body. However, in the turning walk, there is not only translational movement but also rotation around the axis perpendicular to the horizontal plane. So, the motion of the COM cannot represent the motion of the whole robot which will make the robot fall down. In this paper, we introduce the non-inertial reference in which the motion of the turning walk is the straight-line walk. The motion of the non-inertial reference in the turning walking is shown in Figure 1, where the green curve represents the robot trajectory and the red curve represents the non-inertial reference. The non-inertial trajectory is defined as involute. The reference frame not only moves along the curve but also rotates which makes the Y-axis always point to the robot. In this paper, we call this the \"involute reference frame\" (IRF). In the IRF, the motion of the robot can be described similarly to straight-line walk. For the IRF, its motion has tangential acceleration, normal acceleration, and centripetal acceleration that rotates synchronously with circular motion. The motion of the object moving in the IRF is affected by inertial forces. The motion of the IRF and the inertial forces acting on the robot are shown in Figure 2. \(a_{t}\) is the tangential acceleration; \(a_{n}\) is the normal acceleration; and \(\omega\) is the angular velocity. \(f_{a_{n}}\) is generated due to normal acceleration, \(f_{a_{t}}\) is generated due to tangential acceleration and \(f_{rin}\) is generated due to centripetal acceleration. \(f_{CoF}\) is the coriolis force. \(f_{\omega}\) is the force generated due to the changing rate of angular velocity. All the inertial force expressions can be obtained as shown in Formula (1). \(l\) is the distance that the robot walks in the forward direction and is also the length between the origin of the IRF and the tangent point of the circle. \[\left\{ \begin{array}{l}f_{\rm a_{n}}=m\omega^{2}r\\ f_{\rm a_{t}}=m(\dot{\omega}l+\omega\dot{r})\\ f_{\rm rim}=m\omega^{2}r\\ f_{\rm CoF}=2m\omega v\\ f_{\omega}=m\dot{\omega}r.\end{array} \right. \tag{1}\]  
Figure 1: Non-inertial Reference in turning walking. Figure 2: Force analysis in a non-inertial frame. where \(m\) is the mass of the object moving in the non-inertial frame, and \(r\) is the turning radius. The directions of the inertial centrifugal force (\(f_{\rm{fin}}\)) and normal force (\(f_{\rm{a_{n}}}\)) are opposite, so the effect of the two forces cancels out. It is only necessary to calculate the \(f_{\omega}\), \(f_{\rm{fin}}\) and \(f_{\rm{CoF}}\) in the IRF. Through the relationship between the physical quantities shown in Formula (2), we can simplify this force based on the involute properties in Formula (3). \[l=\alpha r,v=\omega r,\dot{\alpha}=\omega \tag{2}\]  
\[\left\{ \begin{array}{l}f_{\rm{CoF}}=2m\omega^{2}r\\ f_{\omega}=m\omega\\ f_{\rm{a_{t}}}=m(\dot{v}\alpha+\frac{v^{2}}{r}).\end{array} \right. \tag{3}\]  
where \(v\) is the velocity in the forward direction related to the IRF and it is also the tangential velocity of circular motion related to the global reference. \(\alpha\) is the angle that the robot turns. \(f_{a_{t}}\) is a changing force which is determined by the acceleration and velocity of the object. In Formula (4), through the the vector superposition of these inertial forces, \(f_{\rm{sum}}\) is the resultant of inertial force applied to an object moving in the IRF, and its direction is the same as that of \(f_{a_{t}}\)  
\[ \begin{array}{c}\mathbf{f}_{\rm{sum}}=\mathbf{f}_{\rm{CoF}}+\mathbf{f}_{\rm {a_{t}}}+\mathbf{f}_{\rm{a_{n}}}+\mathbf{f}_{\rm{rin}}+\mathbf{f}_{\omega}\\ f_{\rm{sum}}=m(\frac{v^{2}}{r}).\end{array}  \tag{4}\]  
The discussion above centers around a scenario in which the turning radius is constant. During turning walk, however, the radius must change so that the robot can reach its target destination.