## 5 Mechanism Description and Inverse Position Analysis  
According to the type discussion in Section 2, the planar part is set as a redundantly actuated parallel mechanism, and the purpose is to increase the active stiffness and enlarge the workspace. Besides, the driving force will be reduced through the effective optimization of driving force distribution. In order to increase constraint stiffness of the parallel part, each kinematic chain is set as an over-constrained mechanism. The kinematic model of the 4-(2 PRR) mechanism is shown in Figure 6a; MP has the capability of x-direction translation, y-direction translation, and z-direction rotation in a planar movement space. The mechanism is composed of four kinematic chains, and each kinematic chain contains two driving chains. As shown in Figure 6b, each kinematic chain has two hydraulic motors used as the driving actuator, and these hydraulic motors push a slider to achieve the translation in the y-direction. The slider is connected with the leg through the hinge, and the two legs are also connected with the MP through the hinge. In Figure 6a, F1 and F2 are fixed bases. F1 is parallel to F2. A fixed Cartesian reference frame, O-xyz, is assigned at the middle of the upper part of the F1 plate. Axis x, y, and z can be attached to point O in the fixed base; \(x\)-axis is perpendicular to F1, and \(y\)-axis parallel to F1. A moving frame, P-x'y'z', is set at the centroid of the MP. In the x-y plane, \(\mathrm{A}_{0j}\) is the middle point of the connecting line between the two hydraulic motors of a kinematic chain, \(\mathrm{C}_{0j}\) is the middle point of the connecting line between the two rotating joints in a kinematic chain, and \(\mathrm{B}_{0j}\) is the middle point of the connecting line between the hinge points of the MP in a kinematic chain. For the convenient calculation of stiffness, we make the following agreements: \(\mathbf{n}_{i}\) = [1,0,0] T, \(\mathbf{s}_{1,J}\) = [0,1,0] T, \(\mathbf{s}_{2,I}\) = \(\mathbf{s}_{3,J}\) = [0,0,1] T.  
Similarly, the kinematic model of the 4-(2 RPR) mechanism is shown in Figure 7b. Only the form of driving chain is different, that is, the position of the MP and fixed base, and the reference frame is the same as those in the 4-(2 RPR) mechanism. Hereinafter, we will use typel to stand for the 4-(2 PRR) mechanism and type II to stand for the 4-(2 RPR) mechanism. In Figure 6a, the single kinematic chain closed vector of type I can be expressed as follows:  
\[\mathbf{p}+\mathbf{B}_{0,j}=\mathbf{h}_{0,j}+\mathbf{q}_{0,j}\mathbf{L}_{0,j}, \mathbf{B}_{0,j}=\mathbf{R}\mathbf{b}_{0,j},\]  
\[\mathbf{h}_{0,j}=\mathbf{a}_{0,j}+\mathbf{v}_{0,j},\]  
\[\mathbf{q}_{0,j}\mathbf{L}_{0,j}=\mathbf{p}_{0,j}+\mathbf{R}\mathbf{b}_{0,j}- \mathbf{a}_{0,j}-\mathbf{v}_{0,j},\ \ (j=1,2,3,4). \tag{27}\]  
In Figure 7a, the single kinematic chain closed vector of type II can be expressed as:  
\[\mathbf{q}_{0,j}\mathbf{L}_{0,j}=\mathbf{p}_{0,j}+\mathbf{R}\mathbf{b}_{0,j}- \mathbf{c}_{0,j},\ (j=1,2,3,4) \tag{28}\]  
Figure 6: Kinematic models of 4-(2 PRR) parallel planar mechanism. (**a**) Kinematic model of the 4-(2 PRR) redundant parallel manipulator; (**b**) topology structure of 4-(2 PRR) kinematic chain. Figure 7: Kinematic models of 4-(2 RPR) parallel planar mechanism. (**a**) Kinematic model of the 4-(2 RPR) redundant parallel manipulator; (**b**) topology structure of 4-(2 RPR) kinematic chain. In Equations (27) and (28), \(\mathrm{L}_{0,j}\) is the length of the leg, \(j\) is the number of kinematic chain, \(\mathbf{q}_{0,j}\) is the unit vector of the leg, \(\mathbf{R}\) is the orientation matrix of the mobile platform, \(\gamma\) is the rotation angle of MP around the \(z\)-axis. \[\mathrm{R}=\left[ \begin{array}{ccc}\mathrm{c}\gamma&-\mathrm{s}\gamma&0\\ \mathrm{s}\gamma&\mathrm{c}\gamma&0\\ 0&0&1\end{array} \right]\]  
The reachable workspace of the MP can be calculated according to the length of the leg, the constraint of the rotational angle, and the interference condition.