# Reconfigurable/Foldable Overconstrained Mechanism and Its Application
## 2 Method for the Kinematic Analysis  
### Method for 6R Overconstrained Mechanism  
The classical position of the [PERSON] symmetric mechanism presents one fixed linkage, while the other five describe a spatial motion with a full mobility range. The mechanism can be analyzed with the help of Denavit-Hartenberg parameters [19]. Figure 1 shows three revolute joints and three links connecting the joints. For this linkage, the Denavit and Hartenberg notations are as follows: one kinematical variable \(\theta_{i}\)--the rotation angle of the joint, and three geometrical parameters: ai the bar lengths, \(\alpha_{i}\) the twist angle between two successive joints (i) and (i + 1), and di the offset distance between two elements (i - 1) and (i).  
To obtain a closed loop, six joints will be linked, but special conditions must be set to the linkage in order to have a mobile mechanism. To obtain a closed loop mechanism, the condition is that the result of the six transfer matrices product must be equal to the unity matrix:  
\[{}^{6}T_{1}\cdot{}^{1}T_{2}\cdot{}^{2}T_{3}\cdot{}^{3}T_{4}\cdot{}^{4}T_{5} \cdot{}^{5}T_{6}=I\,, \tag{1}\]  
where the homogeneous form for the transfer matrix is:  
\[{}^{i-1}T_{i_{i}}=\left[\begin{array}{cccc}\cos\theta_{i}&-\cos\alpha_{i} \cdot\sin\theta_{i}&\sin\alpha_{i}\cdot\sin\theta_{i}&a_{i}\cdot\cos\theta_{i} \\ \sin\theta_{i}&\cos\alpha_{i}\cdot\cos\theta_{i}&-\sin\alpha_{i}\cdot\cos \theta_{i}&a_{i}\cdot\sin\theta_{i}\\ 0&\sin\alpha_{i}&\cos\alpha_{i}&d_{i}\\ 0&0&0&1\end{array}\right]\,, \tag{2}\]  
In the particular case of a threefold-symmetric mechanism (Figure 2), six joints are linked with six links, with special geometrical conditions.  
The geometrical conditions for the case of a threefold-symmetric mechanism are:  
\[\left\{\begin{array}{l}d_{1}=d_{2}=d_{3}=d_{4}=d_{5}=d_{6}=0\\ \alpha_{1}=\alpha_{3}=\alpha_{5}=\alpha\\ \alpha_{2}=\alpha_{4}=\alpha_{6}=2\pi-\alpha\\ a_{1}=a_{2}=a_{3}=a_{4}=a_{5}=a_{6}=a\end{array}\right.\,, \tag{3}\]  
Figure 1: Denavit–Hartenberg parameters for three revolute joint linkage.  
The input and the output angles are:  
\[\left\{\begin{array}{l}\theta_{1}=\theta_{3}=\theta_{5}=\theta\\ \theta_{2}=\theta_{4}=\theta_{6}=\phi\end{array}\right., \tag{4}\]  
The input-output equation between the input angle \(\theta\) and the output angle \(\varphi\) is [20]:  
\[\cos\theta\cdot\cos\phi\cdot\left(1+\cos^{2}\alpha\right)+(\cos\theta+\cos \phi)\cdot\sin^{2}\alpha-2\cdot\sin\theta\cdot\sin\phi\cdot\cos\alpha+\cos^{2 }\alpha=0, \tag{5}\]  
Figure 3 shows the output angle \(\varphi\) according to the input angle \(\theta\), when the twist angle \(\alpha\) varies from \(\pi/2\) to \(2\pi/3\). The link length is \(a=170\) mm.  
Figure 2: Threefold symmetric mechanism.  
### Method for 6R Overconstrained Mechanism  
An estimated automotive industry application for these 6R overconstrained mechanisms would be obtained by setting the mechanism in a particular position.  
Three non-adjacent joints are constrained to remain in a fixed horizontal plane (joints on the vertexes of an equilateral triangle), while the other three joints will be disposed in a parallel plane (joints on the vertexes of another equilateral triangle). Such a device will describe a parallel motion of three non-adjacent joints according to the other three non-adjacent joints as in Figure 4.  
The distance \(H\) between the two planes can be evaluated according to the input-output angles [20]:  
\[H=\frac{\left|a\cdot\sin\frac{\theta}{2}\cdot\sin\frac{\phi}{2}\cdot\sin \alpha\right|}{\sqrt{\sin^{2}\frac{\phi}{2}\cdot\sin^{2}\alpha+\left(\sin\frac {\theta}{2}\cdot\cos\frac{\phi}{2}+\cos\frac{\theta}{2}\cdot\sin\frac{\phi}{2} \cdot\cos\alpha\right)^{2}}}\,, \tag{6}\]  
Starting from this constrained configuration, new possible applications can be obtained, imposing new movements for the actuated joints and combining many overconstrained mechanisms [21].