# Transmission Angle Analysis of a Type of Parallel Manipulators
## 1 Introduction  
With the booming development of robotic technology, parallel manipulators are widely employed in the industry field, such as high-speed pick-and-place applications, stacking and handing operations, robotic end effectors, etc. [1, 2], due to the advantages of high stiffness, high speed, strong flexibility and light weight compared with serial manipulators [3, 4]. Especially, the parallel manipulators that are capable of spatial translation have drawn much concern in industrial applications, and the orientation of the traveling plate in space is not required [5, 6]. Up to now, several manipulators have been developed for such motions. One of the most famous is the Delta robot proposed in [7] for maintaining high-speed pick-and-place applications, which could realize three-DOF translations in space. High performance has been achieved by means of keeping the mass of the moving components to a minimum, as the three identical parallel chains \(\,\underline{\mathrm{R}}-\left(\mathrm{SS}\right)^{2}\) (where \(\,\underline{\mathrm{R}}\) represents an actuated revolute joint, while \(\left(\mathrm{SS}\right)^{2}\) designates a four-bar mechanism consisting of four spherical joints S) are used and all the actuators in the robot system are fixedly mounted on its base. Another typical parallel manipulator is the Diamond robot [8], which mainly consists of a planar parallel robot with two-DOF translations on a plane. The three-DOF translational hybrid robot can be developed with the help of a linear actuator, which could have a mechanism such as a servomotor lead screw assembly, a tooth-belt transmission manipulator or some other equivalent mechanism. Apart from these two main structures, it also introduces other parallel manipulators that can produce translation motion. For example, [PERSON] and [PERSON] proposed the Y-STAR and H-ROBOT parallel robot [9]. [PERSON] [10] presented the 3-UPU manipulator (where P stands for prismatic joint), which has three five-DOF limbs and is topologically similar to the Delta robot embodiment patented by [PERSON] et al. [11] exhibiting a 3-PUU architecture. [PERSON] and [PERSON] [12] put forward theorthograde robot with 3-DOF translations for machining applications, etc. In addition, in order to achieve better acceleration performance for pick-and-place manipulators such as Delta robots, actuation redundancy consisting of additional actuated branches was taken into account as an approach for reaching the goal in [13]. Enlightened by the practicable requirement, many studies investigated different issues containing the type enumerations [14], kinematics and dynamics models [15; 16] and controller design of the developed parallel manipulators with three-DOF translational performance [17]. Despite many research works concerning the kinematic design of this type of parallel manipulators exist, previous works related to the kinematic performance still mainly focused on a conditioning index [18] obtained from the Jacobian matrix, such as the condition number, determinant and singular values, etc. Though these indices are widely used in the design of robotics, some problems still exist when they are taken as performance indices. (a) For the studied out-parallel manipulator, it is inappropriate to utilize the condition number as the distance measure for the singularity. The study [19] indicated that the singular configuration of a two-DOF out-parallel robot could still be obtained, even though the condition number of the Jacobian matrix equaled one. As a consequence, a comprehensive index was adopted combining both the area of the workspace and the condition number of the Jacobian matrix to carry out the optimal design, employing the minimum transmission angle as the extra constraint. (b) It seems that the conditioning indices own no exact physical meaning. As a result, it is difficult to define the reasonable limited value of the conditioning indices as the design criterion.