## 3 Kinematics and Frame Assignment  
The manipulator discussed in this paper is shown in Figure 2. It consists of a fixed base, three legs, a moving platform and several joints. The three legs and the moving platform will be referred to as bodies or links. The joints used in an order starting from the fixed base are prismatic (P) joints, revolute (R) joints which connect the sliders with the legs and other revolute (R) joints which connect the legs with the moving platform. Therefore, the manipulator is called the 3 PRR PM. The prismatic joints are actuated, have translation along \(X\)-axis and are implemented by using sliders moving along the base through guideway. As a result of this configuration, the manipulator provides three DOFs namely translation in \(X\)-axis, translation in \(Y\)-axis and rotation about the \(Z\)-axis. The placement of the three sliders in a parallel manner along the \(X\)-axis implies a non-symmetric configuration of the planar 3 PRR PM. The translational motion of the sliders along the \(X\)-axis does not only enable the moving platform to move upward and downward as well as to tilt but also creates more workspace. This is suitable for an application requiring an elongated workspace which cannot be provided by a symmetric configuration of a planar 3 PRR PM. Furthermore, two neighboring legs of the PM at hand are put in a crossing configuration to avoid singularity as well as maximize the useful workspace and the stiffness of the manipulator.  
A hybrid manipulator composed of two non-symmetric planar 3 PRR PMs, as shown in Figure 3, was developed. The two similar PMs are conjugated--one on the bottom and another one on the perpendicular side. This gives a spatial workspace with a redundancy in the translation along the sliders motion direction. This translational redundancy provides a larger workspace in the sliders motion direction. Both the lower and upper platforms use planar PMs as they are easier to design than spatial PMs and more geometrical constraints on the DOFs are expected to minimize the error due to joint errors. This hybrid manipulator can be used for five-axis machining by putting a worktable on the lower module and holding a tool spindle on the upper module. To avoid a redundancy between the rotation of the spindle and the rotation of the upper module, the tool spindle is oriented horizontally. As a result, a five-axis machine is composed. The five-axis machine has three orthogonal translational degrees of freedom as well as two rotational degrees of freedom.  
Figure 2: Planar non-symmetric 3 PRR manipulator (with \(Z\)-axis perpendicular to the paper).  
The flexible body dynamics of the manipulator at hand is formulated based on the frame assignment shown in Figure 4. As the problem is planar, the coordinate systems are given in a 2D space. The global frame is indicated by **O-XY** frame whereas four local frames corresponding to the four bodies of the manipulator are indicated **O\({}^{1}\)**-\(x_{1}{}^{i}\,x_{2}{}^{j}\) frames. Since the local frames are attached exactly at the joint positions, no intermediate joint frames are required. This accordingly simplifies the problem. The position of the joints is indicated by the letters **A**, **B**, **C**, **D**, **E** and **F**. These letters are used when defining the position constraint equations.