## 1 Introduction  
Linear delta robots (LDRs) for industrial applications are superior to series manipulators, because they have a large carrying capacity and are highly precise and fast [1, 2], so they are widely used for assembly detection, pick-and-place, positioning, and packaging [3]. They are constructed using multiple independent closed-loop mechanisms, each of which has an arm that connects a fixed base to a moving platform. LDRs require three linear actuators to drive the arms and to translate one-dimensional linear motion into three-degrees-of-freedom (3-DOF) movements. Linear actuators can be installed on or near to a fixed base, so that most LDRs use lightweight material for the arms and have a high rigidity-to-weight ratio. LDRs also feature excellent positioning accuracy, because the position error from one single closed-loop kinematic chain is spread among the other chains. However, LDRs have some intrinsic drawbacks, such as a small workspace and complicated kinematics, but there has been increased demand for them in recent years. Pneumatic actuators (PAs) are low-cost and widely used devices for manufacturing. They use compressed air as the operating fluid to push a piston and produce linear movement for payloads [4, 5, 6]. PAs are highly suited to industrial environments, because they are clean, safe, and easier to maintain and have a high power-to-weight ratio. However, PAs are less accurate in terms of position control and less versatile and flexible for engineering applications than electrical motors with the same power, due to the compressibility of air, air leakage, a low natural frequency and high nonlinearity. Despite these inherent drawbacks, PA-driven facilities are still commonly used in manufacturing. [PERSON] et al. [7] presented three different LDRs with actuated arms at specific angles to each other: a vertical type, a tri-pyramid type and a horizontal type. For a specific sensitivity, proper design for LDRs can fulfil a larger workspace, but cause more difficult control problems in real time, because of the multi-input and multi-output system dynamics involving various nonlinearities and coupling effects. [PERSON] et al. [8] presented a hybrid serial-parallel pneumatic-driven robot with a high ratio of strength-to-moving-weight for mounting window glass and fixing pixels. [PERSON] et al. [9] developed a pneumatic-driven LDR with three pneumatic actuators on three vertical axes at 120 degrees to each other, to achieve complex 3-DOF motion control. Furthermore, [PERSON] et al. [10] developed a three-axial tripod-type pneumatic parallel LDR. A geometric approach was used to derive the kinematic model and drive the LDR in a 3-DOF space with an acceptable output tracking error. In [11], [PERSON] et al. designed a 3-DOF tripod-type pneumatic LDR for municipal waste recycling that demonstrated excellent accuracy and repeatability in pick-and-place tasks. [PERSON] et al. [12] presented a 6-DOF pneumatic Gough-Stewart parallel robot. In order to determine an accuracy model for the 6-DOF pneumatic robot, a genetic algorithm (GA) was used to identify unknown parameters within the dynamic model of the cylinder and the proportional valve. [PERSON] et al. [13] presented a multi-objective optimization methodology for the LDRs that optimizes dexterity, stiffness and space utilization, and showed that the methodology gives optimal solutions that are better than the results using current optimization methodology. To address the problems of high nonlinearity, low accuracy and low robustness in LDRs, numerous control strategies have been proposed. [PERSON] [14] determined the kinematic reliability and sensitivity of LDRs. [PERSON] et al. [15] designed an LDR with three arms and showed that problems with kinematic accuracy are mostly due to U joint errors, clearances and driving errors.