# Kinematic and Workspace Analysis of RRU-3 RSS: A Novel 2T2R Parallel Manipulator
## 1 Introduction  
From industrial machine tools and pick-and-place to rehabilitation programs, many applications require less than 6 degrees of freedom (DOF). For these cases, it is interesting to use lower-mobility parallel manipulators. Lower-mobility parallel manipulators are closed-looped mechanisms with less than 6 DOF. They consist of a mobile platform and a fixed platform connected by at least two kinematic chains, also known as limbs [1]. In contrast to serial manipulators, parallel manipulators (PMs) provide better position accuracy, large payload capacity, high stiffness and low inertia [2]. Moreover, in comparison to 6 DOF PMs, lower-mobility PMs bring several advantages including simpler kinematics, enlarged workspace, fewer driving elements and lower cost [3]. Due to these outstanding features, studies on lower-mobility PMs have drawn significant interest in the research community [4]. Nevertheless, type synthesis of lower-mobility PMs is challenging and the design process relies mainly on the experience and inspiration of the designer [5]. To tackle this problem, several approaches have been proposed in the past decades for the systematic type synthesis of lower-mobility PMs. Methods based on the screw theory [6] or the Lie group theory are the most well known ones [7]. Although a wide range of PMs have been synthesized with the previous methods, early research on lower-mobility PM synthesis was mainly focused on 3-DOF PMs. The 3-RPS proposed by [PERSON] [8], the 3-PRS presented by [PERSON] et al. [9] and the 3-DOF DELTA robot introduced by [PERSON] [10] are some of the most famous examples. Compared to their 3-DOF counterparts, 4-DOF PMs have received less attention [11]. However, interest in synthesizing 4-DOF PMs has recently increased due to the needfor this type of motion in micromachining [12], industrial handling [13], flight-motion simulators [14] and medical devices [15; 16; 17]. Overall, depending on the motion capabilities, all 4-DOF PMs can be classified into three groups: 3T1R, 2T2R and 1T3R manipulators, where T and R stand for translation and rotation, respectively. Out of these three groups, 2T2R PMs are the least studied [18], and thus, the number of PMs in the literature with 2T2R motion turns out to be scarce. There exists a high demand for mechanical designs with 2T2R motion because a large variety of applications used in the aerospace industry [19; 20; 21], manufacturing [22; 23] and rehabilitation [24; 25] fields require mechanisms capable of performing two translations in a vertical plane and two rotations about horizontal axes. Researchers have made efforts to design 2T2R type PMs using different synthesis approaches such as the Lie group theory [26], linear transformation theory [27], equivalent chain method [28] and screw theory [29]. Nonetheless, most of the proposed designs are over-constrained mechanisms and/or must fulfill strict geometric conditions to provide the desired motion [30], which makes them difficult to assemble and hinders the development of a prototype. Consequently, this paper presents a novel 2T2R PM that can be used in the aforementioned applications, but with the following advantages:  
* Simple architecture. Each kinematic chain consists of only two links and three joints. Additionally, from a manufacturing point of view, the geometric conditions that the joints' axes must meet to provide 2T2R motion are not difficult to achieve. * Non-over-constrained mechanism and, thus, easier to assemble. Nevertheless, if a high stiffness is required, the proposed design has the flexibility to reach a symmetric configuration with kinematic redundancies by simply changing two joints. * Fully PM with all the actuators attached to the fixed frame. This means lower inertial loads, lower energy consumption, and a larger amount of power available for increasing the speed of operations or the load-carrying capacity. * The movements in the 4-DOF are free of parasitic motions.