## 2 LI Device Design for Fast Motion
### Transmission Design for Omnidirectional Motion  
Figures 1 and 2c show the conceptual design of the proposed 2D translational motion with geared-pulley transmission. This simple holonomic design allows generation of infinite motion in both axes. In addition, this actuation method only drives the segment belts that are on the active surface where the user can locate and walk. This can reduce the required motor power as less than half of the segment belts are actuated at any given time. However, when the proposed mechanism creates 2-dimensional ground by actuating both X and Y axes, the coupled surfaces of the geared-pulleys and segment belts get frictional forces because the segment belt should also move in a direction perpendicular to the direction of the transmitted force of the input geared pulley. Thus, the geared pulley needs to be modified to include a passive rotation mechanism for reducing this friction. In this paper, we implement the Y-axis motion using a geared omni-pulley set (GOPS) equipped with toothed rollers instead of the conventional geared pulley, as shown in Figure 3a. The design concept of GOPS is organized by an omni-wheel with gear. The high transmission efficiency of the gear is used for motion in the \(\overset{\rightarrow}{\mathbf{A}_{Y}}\) direction, while the friction on the tooth surface caused by motion in the \(\overset{\rightarrow}{\mathbf{A}_{X}}\) direction is reduced by the passive rotation of the toothed rollers, similar to an omni-wheel. The toothed roller for reducing the frictional force is designed by body of rotation of an involute toothed part of a normal geared pulley along the rotational vector \(\overset{\rightarrow}{N}\) shown in Figure 3b. This rotational vector also represents the axis of the passive rotation of the toothed rollers when segments move along the X-axis. In the presented system, the design of the GOPS uses the specifications of the commercial product called the T10 type geared pulley that works with the T10 Urethane belt. Figure 3: (**a**) GOPS design including toothed rollers, (**b**) The front projection of GOPS design with a normal geared-pulley parameter, and GOPS configuration with its geometric analysis. The parameters of the normal geared pulley, which are independent of the number of teeth, pitch (\(p=10\) mm), tooth height (\(t_{h}=3.2\) mm), tooth width (\(w_{h}=2.76\) mm) and input angle of the tooth (\(\lambda=25\) deg). Moreover, GOPS design also considers the parameters depended on the number of teeth (\(Z=36\)) such as radius of the pitch circle (\(r_{p}=57.295\) mm), total radius (\(r=56.375\) mm) of the GOPS and the radius of the base circle (\(r_{b}=53.175\) mm) because the frontal projection of the GOPS is identical to a normal geared-pulley profile. Among the GOPS configuration parameters shown in Figure 3b, the number, \(n\), of toothed rollers is the main determinant. Figure 4 shows the example of how to properly setup the GOPS configuration in the case that the number of toothed rollers is 6. The relationship between the total number of teeth (\(Z\)) and the number of toothed rollers (\(n\)) defines to the module of the GOPS (\(m_{GOP}\)) as follows:  
\[m_{GOP}=Z/n(n=2a\geq 6,\ a\geq 3),\ (a,Z,n,\ m_{GOP}\in\mathbb{N}), \tag{1}\]  
where \(a\)\(\in\)\(\mathbb{N}\) is a positive natural number greater than or equal to 3. Thus, \(m_{GOP}\) represents the number of teeth in one toothed roller. It can also define an appropriate number for \(n\) because the number of teeth must be a natural number. All toothed rollers have an index number (\(i_{th}\)) according to the range \(1\leq i_{\in\mathbb{N}}\leq n\), as shown in Figure 4. Odd and even numbers in the toothed roller index (\(i\)) make up each separate geared omni-pulley (GOP) (see also Figure 3), and two GOPs are combined to form one GOPS. In the design of GOPS, the total number of teeth (\(Z\)) of the GOPS is 36, and its 6 toothed rollers are configured with a uniform angular spacing of \(60^{\circ}\).