## 3 HTP Tooth Profile Design Using IH Tooth Profile and Cycloidal Curve  
### IH Tooth Profile Theory  
The harmonic reducer achieves high reduction ratios and precision by using a structure in which the wave generator deforms the external gear (flexspline), causing it to engage with the internal gear (circular spline). When the number of teeth is large, the tooth profile \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Involute** & **Cycloid Tooth Profile** & **IH Tooth Profile** \\ \hline Tooth profile curve & Involute curve & Circular curve & Straight line and circular curve \\ \hline Contact ratio & 15 & 30 & 30 \\ \hline Efficiency (\%) & 90 & 70–80 & 80 \\ \hline Transmission error & \multirow{2}{*}{1} & \multirow{2}{*}{2–3} & \multirow{2}{*}{1} \\ (arc min) & & & \\ \hline Tooth root stress & \multicolumn{3}{c}{Reduction compared to involute tooth profile} \\ \hline Torque & \multicolumn{3}{c}{Improvement compared to involute tooth profile} \\ \hline Meshing state & Instantaneous contact & Initial contact but separation in the intermediate stage & Not in contact for just a moment butremanians in contact consistently \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of tooth profiles in a speed reducer. Figure 1: IH tooth curve of harmonic systems company (S series)—redrawn. can be approximated as straight; however, when the number of teeth is small, problems such as increased friction and backlash may occur. To address these issues, the IH tooth profile models the trajectory of a point \((p(\phi),\ q(\phi))\) on the flexspline as a function of the wave generator's rotation angle, \(\phi\), and establishes continuous meshing conditions between the internal and external gears based on this trajectory. This trajectory-based design concept is visually represented in Figure 2[25]. When the rotation angle, \(\phi\), of the wave generator in the harmonic reducer is 0, the motion trajectory of the outer tooth gear can be defined as a curve, \(l\), based on the engagement position between the inner and outer tooth gears, as \(\phi\) varies from 0 to \(\pi/2\). This curve, \(l\), is described by Equation (1) and corresponds to \(L_{1}\). In IH tooth profile design, only a portion of this curve may be used, and in such cases, the range of \(\phi\) may be narrower than \(0\leq\phi\leq\pi/2\). \[ \begin{bmatrix}x_{l}\\ y_{l}\end{bmatrix} = \begin{bmatrix}p(\phi)\\ q(\phi)\end{bmatrix} ,\ 0\leq\phi\leq\frac{\pi}{2} \tag{1}\]  
Among the endpoints of curve \(l\), point A is defined at \(\phi=0\), point B at \(\phi=\pi/2\), and point C as the midpoint between A and B. The addendum profile of the internal tooth gear can be defined using a similar curve, obtained by scaling curve \(l\) by a factor \(\lambda\ (0<\lambda<1)\) about point B. This scaled curve is represented by Equation (2) and corresponds to \(L_{2}\). \[ \begin{bmatrix}x_{\alpha}\\ y_{\alpha\alpha}\end{bmatrix} = \begin{bmatrix}\lambda p(\theta)+(1-\lambda)p (\frac{\pi}{2})\\ \lambda q(\theta)+(1-\lambda)q(\frac{\pi}{2})\end{bmatrix} ,\ 0\leq\theta\leq\frac{\pi}{2} \tag{2}\]  
By scaling curve \(l\) by \((1-\lambda)\) about point B to obtain a homothetic curve and then rotating it \(180^{\circ}\) about point C, the addendum tooth profile of the external tooth gear can be derived. This profile is expressed in Equation (3), corresponding to \(L_{3}\). \[ \begin{bmatrix}x_{f}\\ y_{f}\end{bmatrix} = \begin{bmatrix}-(1-\lambda)p(\theta)+p(0)+(1-\lambda)p (\frac{\pi}{2})\\ -(1-\lambda)q(\theta)+q(0)+(1-\lambda)q(\frac{\pi}{2})\end{bmatrix} ,\ 0\leq\theta\leq\frac{\pi}{2} \tag{3}\]  
Since the internal and external tooth gears of the harmonic reducer have a large number of teeth, they can be approximated as racks with infinite teeth. Under this rack approximation, the inclination of the tooth vanishes.