# Development of Power-Assist Device for a Manual Wheelchair Using Cycloidal Reducer
## 2 Materials and Methods  
### Generating Cycloidal Tooth Profile  
In this section, we present a practical method for the design of the cycloidal reducer for the PAD that can be installed on a manual wheelchair. The input shaft drives an eccentric bearing that in turn drives the cycloidal disk in an eccentric. The perimeter of this disk is geared with several pin rollers and has a series of output shaft pins in the holes on the disk. These output shaft pins directly drive the output shaft when the cycloidal disk rotates. The direction of rotation of the disk and output is opposite to that of the input shaft. In general, the cycloidal reducer, cycloidal disks and pin rollers are in contact at the same number of points as the number of pin rollers. However, in the proposed reducer, the cycloidal disk is engaged with an epitrochoid curve called the housing gear corresponding to the housing. In addition, the cycloidal disk and housing gear have a single contact point, similar to a general involute gear (Figure 1). An epicycloid is a path traced by a point on the rolling circle that rotates outside the base circle. Similarly, an epitrochoid is a geometric curve traced by a fixed point inside a rolling circle that rotates around the perimeter of the base circle. The parametric equations for an epitrochoid can be given:  
\[x(t)=(r_{b}+r_{r})\cos t-k\cos\left(\frac{r_{b}+r_{r}}{r_{r}}t\right) \tag{1}\]  
\[y(t)=(r_{b}+r_{r})\sin t-k\sin\left(\frac{r_{b}+r_{r}}{r_{r}}t\right)\]  
where \(r_{b}\) is the radius of the interior circle called the base circle, \(r_{r}\) is the radius of the exterior circle called the rolling circle, \(k\) is the distance from the center of the rolling circle, and \(t\) is geometrically the polar angle of the center the rolling circle. When the number of teeth on the cycloidal disk is \(Z\), the ratio of the radius of the interior and exterior circles is \(r_{b}/r_{r}=Z\). The first parametric derivatives of Equation (1) with respect to \(t\) are given by:  
\[x^{\prime}(t)=\frac{dx}{dt}=-(r_{b}+r_{r})\sin t+k\left(\frac{r_{b}+r_{r}}{r_{ r}}\right)\sin\left(\frac{r_{b}+r_{r}}{r_{r}}t\right) \tag{2}\]  
\[y^{\prime}(t)=\frac{dy}{dt}=(r_{b}+r_{r})\cos t-k\left(\frac{r_{b}+r_{r}}{r_{ r}}\right)\cos\left(\frac{r_{b}+r_{r}}{r_{r}}t\right)\]  
The equidistant curve, frequently called a parallel curve, is a curve that is displaced from Equation (1) by a constant offset. That curve is calculated as follows [18]:  
\[x_{D}(t)=x(t)-\frac{hy^{\prime}(t)}{\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}} \tag{3}\]  
\[y_{D}(t)=y(t)+\frac{hx^{\prime}(t)}{\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}}\]  
where \((x(t),\ y(t))\) is a point on the epitrochoid curve and \((x_{D}(t),\ y_{D}(t))\) is a point on the parallel curve (equidistant curve) by an offset distance \(h\). \(r_{p}\) is the radius of the pitch circle of the parallel curve and is calculated using \((r_{a}+r_{d})/2\). Where \(r_{a}\) and \(r_{d}\) are the radius of the addendum and dedendum circle of the equidistant curve, respectively (Figure 2). Figure 1: Comparison of cycloidal reducer (**left**: Traditional method, **right**: proposed method). Equation (3) can be replaced with Equations (1) and (2) as follows:  
\[x_{D}(t)=(r_{b}+r_{r})\cos t-k\cos\left(\frac{r_{b}+r_{r}}{r_{r}}t\right)-\frac{ h\left(r_{r}\cos t-k\cos\left(\frac{r_{b}+r_{r}}{r_{r}}t\right)\right)}{\sqrt{k^{2}-2 kr_{r}\cos\left(\frac{r_{b}}{r}t\right)+r_{r}^{2}}} \tag{4}\]  
\[y_{D}(t)=(r_{b}+r_{r})\sin t-k\sin\left(\frac{r_{b}+r_{r}}{r_{r}}t\right)+ \frac{h\left(-r_{r}\sin t+k\sin\left(\frac{r_{b}+r_{r}}{r_{r}}t\right)\right)} {\sqrt{k^{2}-2 kr_{r}\cos\left(\frac{r_{b}}{r_{r}}t\right)+r_{r}^{2}}}\]  
The tooth profile can have various shapes of gear curves having the same pitch circle according to the offset distance. To realize this design approach, the concept of tooth thickness ratio (TTR) was introduced. The TTR is defined as the ratio the tooth occupies on the pitch circle, and is calculated by dividing the tooth thickness by the sum of the tooth thickness and the space width.