## 1 Introduction  
In the past three decades, the use of parallel manipulators [1; 2; 3] has been increased in industrial and academic fields [4; 5; 6], especially in the fields of product transportation and classification, motion simulators, multi-dimensional force sensors, micro-operation consoles, precision positioning devices, etc., for their relative features like superior rigidity, higher accuracy, lighter mechanical construction, resulting in a homogenous and symmetric force/torque distribution, easier inverse kinematic calculations [7; 8; 9]. In recent years, researchers are increasingly inclined to concentrate on three translational degree-of-freedom (DOF) parallel manipulator [10; 11; 12; 13]. As we all know, the Delta parallel robots [14; 15] have effective mechanical structures [16], which makes the solution of its forward kinematic simpler. Because of its good characteristics, the Delta robot is often applied for high-speed pick-and-place applications with light-weight objects [9; 17]. Pick-and-place is considered as a commonly used action [18] in palletizing and sorting in which objects are collected in an orderly manner and then put in their corresponding locations [19], however, due to the geometrically complicated surroundings, planning the robot's pick-and-place movement is often difficult. Moreover, the research on optimizing the trajectories of parallel robots is relatively less than that of serial robots. Some work scenarios have requirements on the pick-and-place operation (PPO) path to avoid collisionsbetween the workpiece and surroundings. Therefore, the study of trajectory planning for a parallel Delta robot with prescribed geometrical constraint is more challenging problem. The PPO path of the Delta robot often includes two vertical segments and one horizontal segment [19]. Because of the existence of the right angle between the vertical and horizontal direction, the Delta robot is prone to shock and vibration, which has a bad effect on the rapid PPO. So, substituting the right angle by smooth curves is commonly used in related references [20]. It has the following advantages: (1) the overall route of smooth curves are shorter than the right angles; (2) Since the smooth curves have limited extreme curvature, it can maintain a relatively stable performance throughout its entire range rather than having a shock in the case of the right angle [21]. The above two advantages can decrease the PPO's motion period and produce smoother motion characteristics. However, the degree of curvature of the smooth curve is important, since too much rounding could result in interference with objects in the robot workspace. Therefore, the PPO path should be transitioned through the smooth curve which deviates from the apex of the right angle by no more than a prescribed geometrical constraint. The strategy of replacing the right angle by a smooth curve needs to meet some conditions, such as real-time interpolation calculation, smooth changes of the velocity, and smooth curve should match the PPO path, etc. Pythagorean-Hodograph curve [22] is very suitable for the above situation. The total arc length of the Pythagorean-Hodograph curves can be expressed as a polynomial with parameters [23], which makes real-time interpolation possible [21]. In addition, the Pythagorean-Hodograph (PH) curve has other advantages, for example, the curve has finite extremum curvature [24; 25]. Due to the special property of the PH curve, it is widely used in path planning [26; 27], obstacle avoidance [28], continuum manipulators [29], highways and railways design [30; 31], and many other fields [32]. Researchers proposed interpolation methods of a PH curve of degree nine, however, it puts more emphasis on geometric issues, and the calculation of real-time interpolators is rather complicated [33]. This paper proposes a trajectory planning method for Delta parallel robot. (1) By utilizing quintic PH curves, the prescribed geometrical constraint is satisfied in PPO.