# Parallel Continuum Delta: On the Performance Analysis of Flexible Quasi-Translational Robots
## 1 Introduction  
Parallel Continuum Manipulators (PCMs) are robotic devices that, in general terms, consist of a fixed platform and a moving platform connected by several flexible elements, in a closed-chain morphology typical of parallel kinematics. Unlike conventional robots formed by rigid elements, PCMs use flexible elements such as flexible rods, cables, or springs to connect the rigid end-effector to their actuators [1]. These manipulators offer great versatility as well as flexibility and precision in object manipulation, which makes them ideal for applications in fields such as medicine, aerospace, and especially in robot-human collaborative work, as in the case of the so-called Collaborative Robots, CoBots [2, 3]. Indeed, knowing that impact mitigation is a core issue in the field of collaborative robotics, this type of continuum manipulator is gaining special attention, with the aim of developing novel flexible spatial mechanisms, such as the parallel elastic mechanism made of thin plates recently proposed in [4]. PCMs share some characteristics with classical parallel Kinematic machines and compliant mechanisms [5, 6], and attempt to combine the best features of both systems. One of the main challenges in the design and control of PCMs is to ensure stability and precision in the movements. Since flexible elements possess a certain elasticity and deformation, the mobility of those mechanisms is produced by kinematic pairs and flexibility of rods alike, and the definition of the robot's degree of freedom is not so clearly established. A common approach in the literature is to use as degrees of freedom the number of actuators needed to keep a pose in equilibrium. Indeed, many of the current PCMs under study are modifications on classical rigid kinematic machines. For example, a broad line of research on hexapod-like flexible manipulators has emerged, solving a variety of problems such as workspace evaluation, real-time position control and elastic analysis [7, 8, 9, 10, 11, 12]. In the field of rigid parallel manipulators, the family of translational parallel manipulators is noteworthy. The best known example of this class of robots is the Delta manipulator [13; 14; 15; 16]. The main applications of Delta manipulators are pick-and-place operations [17], packaging of small products (cosmetics, pharmaceutical industry, etc.) [18], high precision assembly operations [6], 3D printers [19] and haptic devices [20]. The Delta robot belongs to the family of lower-mobility parallel manipulators that can be defined as closed-loop spatial mechanisms whose end-effector has only three degrees of freedom controlled by actuation on three identical kinematic chains [21]. This type of rigid-body mechanism is subjected to geometrical constraints produced by the arrangement and morphology of the passive kinematic pairs, making it independent of the actuators' values. In some cases, those geometric constraints fix some output parameters, as is the case of the Delta manipulator, which has fixed orientation parameters, resulting in a pure 3\(D\) translational motion. The motivation of this research is the development of Continuum parallel Delta manipulators with the ability to be used in manipulation tasks in cooperative scenarios with humans, incorporating this kind of flexible mechanism in collaborative workspaces such as those proposed in the IIDEA project intended for Inclusion and Integration through Collaborative Robotics [22; 23]. For reference, we will use as design specifications the characteristics of commercial Delta robots, and specifically the Drylin robot Delta from IGUS Company [24]. Few works can be found on flexible lower-mobility parallel manipulators [25; 26; 27; 28]. It is important to note that in these flexible manipulators, it is not possible to impose strict constraints, as in the case of their rigid counterparts, precisely because of the flexibility of the rods.