## 3 LI Control for Omnidirectional Running
### Performance of Curvature Radius  
An ODT allows for a curved path of locomotion by combined motion of the X and Y axes in the treadmill frame. It is necessary to determine the lowest radius of curvature of a user, which can be used as the performance index for changes in direction during locomotion [9]. This range can be obtained based on the linear and angular velocities of a user according to the acceleration limits of the proposed ODT. When a user changes direction at an arbitrary angular speed in the user coordinate frame, the velocity command, \(\mathbf{v}_{c}\), in the treadmill frame can be approximated as follows:  
\[\mathbf{v}_{c}(t)\cong\mathbf{R}_{z,\theta}\mathbf{v}_{w}^{user},\left(\theta =\dot{\theta}t+\theta_{initial}\right), \tag{10}\]  
where \(\dot{\theta}\) is the angular velocity of the user, \(\theta_{initial}\) is the initial user orientation angle, and \(\mathbf{v}_{w}^{user}\) is the user's intended velocity that is assumed to be a positive real value along the \(X^{user}\)-axis (see Figure 12). Therefore, the maximum velocity and acceleration that the ODT can generate relates to the intended velocity and the angular speed of a user as follows (here, \(\theta_{initial}\) = 0):  
\[\begin{array}{l}\mathbf{v}_{c}(t)=\|\mathbf{v}_{w,given}^{user}\|\cos\!\left( -\dot{\theta}t\right)\overset{\rightarrow}{i}-\|\mathbf{v}_{w,given}^{user} \|\sin\!\left(-\dot{\theta}t\right)\overset{\rightarrow}{j}\\ \|\frac{\partial\mathbf{v}_{c}}{dt}\|=\|\mathbf{v}_{w,given}^{user}\|\times \left|\dot{\theta}\right|\leq 3m/s^{2}\text{ s.t. }\|\mathbf{v}_{w,given}^{user}\|\leq 3m/s\end{array} \tag{11}\]  
where \(i\) and \(j\) are the unit vectors along the \(X^{user}\) and \(Y^{user}\) axes. If the intentional user velocity is less than 3 m/s and the proposed ODT has an acceleration limitation of 3 m/s\({}^{2}\), the allowable range of the angular velocity \(\dot{\theta}\) of the user, shown in Figure 13a, can be calculated from Equation (11). Given that the intended velocity (\(\mathbf{v}_{w,given}^{user}\)) is not directly measurable and is instead an estimated value, the user's travel trajectory (\(\overset{\rightarrow}{L^{\text{VR}}}\)), defined in VR coordinates (\(X^{\text{VR}}\), \(Y^{\text{VR}}\)), can be calculated by integrating the \(\mathbf{v}_{c}\) into Equation (12), as follows:  
\[\overset{\rightarrow}{L}^{VR}(t)\simeq\int_{0}^{t}\mathbf{v}_{c}(\tau)\;d \tau=\int_{0}^{t}\mathbf{R}_{Z}(\theta)\mathbf{v}_{w}^{user}(\tau)\;d\tau, \tag{12}\]Since \(\overset{\rightarrow}{L}^{\text{VR}}\) has arbitrary curved paths in the VR, the curvature radius of \(\overset{\rightarrow}{L}^{\text{VR}}\) can be calculated as follows:  
\[r_{c}=\|\mathbf{v}_{c}(t)\|^{3}/\ \|\mathbf{v}_{c}(t)\times\dot{\mathbf{v}}_{c}(t) \|=\|\mathbf{v}_{c}(t)\|\ /\ \Big{|}\dot{\theta}\Big{|}, \tag{13}\]  
When a user changes direction without moving, their allowable angular velocity can be infinite. Faster user speeds tend to decrease the allowable range of angular velocity. From Equations (12) and (13), it is possible to determine the minimum radius of curvature of the proposed ODT corresponding to any acceleration (0.001-3 m/s\({}^{2}\)), as shown in Figure 13b. It should be noted that due to the limited acceleration performance defined in Equation (11), the minimum radius of curvature is dependent on both the user's speed and the allowable angular velocity.