## 3 Analysis of Special Positions of the Mechanism  
Analysis of special positions of the mechanism. When solving the system (7) in small movements, it is necessary to satisfy the condition (8), i.e., it is necessary to exclude special positions of the mechanism. A technique that allows for establishing special positions of the mechanism based on the analysis of the functional from equation is proposed (8):  
\[F(\varphi_{2},\varphi_{3},\psi)=I_{BC}\text{sin}\varphi_{3}\text{sin}(\psi+ \alpha_{6}-\varphi_{2})-I_{6}\text{sin}\psi\text{sin}(\varphi_{3}-\varphi_{2} )=\mathbf{0} \tag{9}\]  
Let us consider a trivial solution to Equation (9) with geometric interpretation. Below are the options for these solutions:  
(A) let \(\text{sin}\varphi_{3}=\mathbf{0}\), then the Equation (9) is performed in two versions: (1) \(\text{sin}\psi=\mathbf{0}\) or (and) (2) \(\text{sin}(\varphi_{3}-\varphi_{2})=\mathbf{0}\). In the first version, we have at \(\varphi_{3}=\mathbf{0}\)**or \(\pi\)** result \(\psi=\mathbf{0}\)**or \(\pi\)**; in the second version at \(\varphi_{3}=\mathbf{0}\) or \(\pi\)** turns out \(\varphi_{3}=\varphi_{2}\). The first version is interpreted as follows: link \(Q\)**P** and links \(C\mathbf{D}\), \(F\mathbf{E}\) occupy mutually parallel positions. In the second case links \(Q\)**P** and \(AB\) are parallel.  
(B) let \(\text{sin}(\psi+\alpha_{6}-\varphi_{2})=\mathbf{0}\), hence there is a \(\varphi_{2}=\psi+\alpha_{6}\) which happens when links \(AB\) and \(C\mathbf{B}\) are parallel. For a trivial solution to Equation (9), one of two variants must also be fulfilled here: (3) \(\text{sin}\psi=\mathbf{0}\) or (and) (4) \(\text{sin}(\varphi_{3}-\varphi_{2})=\mathbf{0}\). In the third version the links \(C\mathbf{D}\), \(F\mathbf{E}\) occupy a perpendicular position relative to the slide stroke. In the fourth version the links \(Q\)**P** and \(AB\) are parallel.  
In order to visualize the functional (9) in a three-dimensional system, a fixed value of one the parameters should be set. Such parameter is \(\psi\), which has the following domain of variability \(\frac{\pi}{2}-\boldsymbol{tan}^{-1}\boldsymbol{t}\leq\psi\leq\frac{\pi}{2}+ \boldsymbol{tan}^{-1}\boldsymbol{t}\), where \(\boldsymbol{\tau}\)--crank length, \(\boldsymbol{l}\)--rod length BC.  
Further, for clarity the functional \(F(\varphi_{2},\varphi_{3},\psi)\) is presented in Figure 2 at following values \(\psi=1.8165\); \(1.7182\); \(1.6199\); \(1.5216\); \(1.3251\) [rad] in the system of coordinates \(\{F(\varphi_{2},\varphi_{3}),\varphi_{2},\varphi_{3}\}\). This case corresponds to a second version.  
Figure 2: Functional graphs \(F(\varphi_{2},\varphi_{3},\psi)\) in the three-dimensional system \(\{F(\varphi_{2},\varphi_{3}),\varphi_{2},\varphi_{3}\}\).  
Figure 2a shows the functional \(F(\varphi_{2},\varphi_{3},\psi=const)=F_{1}+F_{2}\). That is, with a fixed \(\psi=1.8165\) as the intersection of two functions \(F_{1}=I_{BC}\text{sin}\varphi_{3}\text{sin}(\psi+\alpha_{6}-\varphi_{2})\) and \(F_{2}=-I_{6}\text{sin}\psi\text{sin}(\varphi_{3}-\varphi_{2})\). Further built were \(F_{1}\) and \(F_{2}\) at other values \(\psi=1.7182;1.6199;1.5216;1.3251\) [rad] which are shown in Figure 2b in the form of intersection curves of these two surfaces. In Figure 2c the intersection curves are projected on a plane \(\{\varphi_{2},F\}\). Thus, in Figure 2, the special position of the mechanism corresponds to \(\varphi_{3}=0\) and \(\varphi_{2}=0\) or \(\varphi_{3}=\pi\) and \(\varphi_{2}=\pi\).