## 3 Kinematic Analysis
\begin{array}{l}A_{1}=\cos\beta\ \ \ B_{1}=0\ \ \ C_{1}=-\sin\beta\ \ \ D_{1}=-r\cos\varphi_{1}\\ A_{l}^{\prime}=0\ \ \ \ \ B_{1}^{\prime}=1\ \ \ C_{1}^{\prime}=0\ \ \ \ \ \ D_{1}^{\prime}=-r\sin\varphi_{1}\\ A_{1}^{\prime\prime}=\cos\theta_{y}tg\varphi_{0}\ \ \ \ B_{1}^{\prime}=\sin \theta_{x}\sin\theta_{y}tg\varphi_{0}-\cos\theta_{y}\ \ \ \ \ C_{1}^{\prime\prime}=-\cos \theta_{x}\sin\theta_{y}tg\varphi_{0}-\sin\theta_{x}\\ D_{1}^{\prime}=-\rho\cos\theta\sin 3\varphi_{1}tg\varphi_{0}+\rho\cos\theta \cos 3\varphi_{0}\\ A_{1}^{\prime\prime\prime}=\sin\theta_{y}\ \ \ \ \ B_{1}^{\prime\prime\prime}=\sin \theta_{x}\cos\theta_{y}\ \ \ \ \ C_{1}^{\prime\prime\prime}=\cos\theta_{x}\cos\theta_{y}\ \ \ \ \ D_{1}^{\prime\prime\prime}=\rho\sin\theta\end{array} \]  
According to Equation (14), a solution exists, so the condition must be satisfied. \[\left| \begin{array}{cccc}A_{1}&B_{1}&C_{1}&D_{1}\\ A_{1}^{\prime}&B_{1}^{\prime}&C_{1}^{\prime}&D_{1}^{\prime}\\ A_{1}^{\prime\prime}&B_{1}^{\prime\prime}&C_{1}^{\prime\prime}&D_{1}^{\prime \prime}\\ A_{1}^{\prime\prime}&B_{1}^{\prime\prime}&C_{1}^{\prime\prime}&D_{1}^{\prime \prime}\\ \end{array} \right|=0 \tag{15}\]  
Substitute each coefficient into Equation (15), expand it, and obtain a complex relationship involving arbitrary angular variables \(\varphi_{1}\) and \(\varphi_{0}\). It can be written in the following form:  
\[A(\varphi_{0})\sin\varphi_{1}+B(\varphi_{0})\cos\varphi_{1}+C(\varphi_{0})=0 \tag{16}\]  
Suppose that:  
\[ \begin{array}{rl}A(\varphi_{0})=&r\big{[}\cos\beta\big{(}\sin\theta_{x}\sin \theta_{y}tg\varphi_{0}-\cos\theta_{y}\big{)}\cos\theta_{x}\cos\theta_{y}+\sin \beta\cos\theta_{y}tg\varphi_{0}\sin\theta_{x}\cos\theta_{y}\\ &+\sin\theta_{y}\big{(}\sin\theta_{x}\sin\theta_{y}tg\varphi_{0}-\cos\theta_{y }\big{)}\sin\beta\\ &-\sin\theta_{x}\cos\theta_{y}\big{(}\cos\theta_{x}\sin\theta_{y}tg\varphi_{0} +\sin\theta_{x}\big{)}\cos\beta\big{]}\\ B(\varphi_{0})=&r\big{[}\cos\theta_{y}tg\varphi_{0}\cos\theta_{x}\cos\theta_{y }+\sin\theta_{y}\big{(}\cos\theta_{x}\sin\theta_{y}tg\varphi_{0}+\sin\theta_{ x}\big{)}\big{]}\\ C(\varphi_{0})=&\cos\beta\big{[}\varphi\sin\theta\big{(}\cos\theta_{x}\sin \theta_{y}tg\varphi_{0}+\sin\theta_{x}\big{)}-\cos\theta_{x}\cos\theta_{y}( \rho\cos\theta\sin 3\varphi_{0}tg\varphi_{0}\\ &+\rho\cos\theta\cos 3\varphi_{0}\big{)}\big{]}+\sin\beta\big{[}\rho\sin\theta\cos \theta_{y}tg\varphi_{0}\\ &+\sin\theta_{y}\big{(}\cos\theta\sin 3\varphi_{0}tg\varphi_{0}+\rho\cos\theta \cos 3\ \varphi_{0}\big{)}\big{]}\end{array} \]  
The citation for the half-angle formula can be obtained as follows:  
\[\varphi_{1}=2 arctg\frac{A\Big{(}\varphi_{0}\pm\sqrt{A^{2}(\varphi_{0})+B^{2} (\varphi_{0})-C^{2}(\varphi_{0})}\Big{)}}{B(\varphi_{0})-C(\varphi_{0})} \tag{17}\]  
From this, the functional relationship between the input angle \(\varphi_{1}\) and the output angle \(\varphi_{0}\) can be obtained. When \(r/l\) is small, we can assume that \(\rho/l=tg\theta\approx\sin\theta\approx\theta\) or, in other words, point \(O^{\prime\prime}\) lies within the \(oxyz\) coordinate plane. At the same time, terms containing \(r/l\) or \(\rho/l\) squared, as well as higher-order terms, can be neglected. Thus, we have the following simplified calculation relationship:  
\[\varphi_{0}-\varphi_{1}\approx\frac{r(1-\cos\beta)}{2l(1+\cos\beta)}tg\beta \cos 3\varphi_{1} \tag{18}\]  
\[\frac{d\varphi_{0}}{d\varphi_{1}}=\omega_{0}/\omega_{1}=1-\frac{3r(1-\cos\beta) }{2l(1+\cos\beta)}tg\beta\cos 3\varphi_{1} \tag{19}\]  
The theoretical angular deviation curve calculated according to Equation (18) is shown in Figure 5. From Equation (18), it can be seen that the angular deviation varies with the deflection angle \(\beta\). When the input shaft rotates one turn, the angular deviation undergoesthree cycles of variation. At the deflection angle \(10^{\circ}\), only a fraction of the maximum angular deviation occurs at point \(1^{\prime}\); even when the deflection angle is \(30^{\circ}\), the maximum angular deviation does not exceed \(10^{\prime}\).