###### Abstract  
An optimal control design for the uncertain Delta robot is proposed in the paper. The uncertain factors of the Delta robot include the unknown dynamic parameters, the residual vibration disturbances and the nonlinear joints friction, which are (possibly fast) time-varying and bounded. A fuzzy set theoretic approach is creatively used to describe the system uncertainty. With the fuzzily depicted uncertainty, an adaptive robust control, based on the fuzzy dynamic model, is established. It designs an adaptation mechanism, consisting of the leakage term and the dead-zone, to estimate the uncertainty information. An optimal design is constructed for the Delta robot and solved by minimizing a fuzzy set-based performance index. Unlike the traditional fuzzy control methods (_if-then_ rules-based), the proposed control scheme is deterministic and fuzzily optimized. It is proven that the global solution in the closed form for this optimal design always exists and is unique. This research provides the Delta parallel robot a novel optimal control to guarantee the system performance regardless of the uncertainty. The effectiveness of the proposed control is illustrated by a series of simulation experiments. The results reveal that the further applications in other robots are feasible.  
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linear disturbance observation, an active disturbance rejection controller was constructed to ease the presence of the possible noise effects in Delta robot [16]; Ref. [17] discussed the tracking problem for Delta robot with uncertain parameters by using an adaptive control algorithm; Ref. [18] analyzed the mechanical vibrations issue in the Par2-Delta parallel robot and proposed an adaptive dual mode controller to solve it. Furthermore, some intelligent control methods have also been adopted for controlling the Delta robot by function approximation or fitness selection, for example the neural network control in [19] and the fuzzy control method in [20,21].  
Most of the present control approaches devote to tackle the uncertainty of Delta robot from the perspectives of investigating the stochastic properties of the uncertain terms (such as, [14,15]) or on-line estimating the uncertain parameters (see, i.e., [16, 17, 18]). However, in this paper, a fuzzy set theoretic approach is proposed to explore the uncertainty in Delta robot, and can be considered as the third perspective. The uncertainty information of Delta robot is depicted by the fuzzy sets theory, which is unknown but is within a fuzzy threshold. Based on the fuzzy description of uncertainty, a deterministic control design is constructed. It is a new way to combine the fuzzy set theory and control theory of Delta robot, and does not belong to the traditionally _if-then_ rules-based fuzzy control design framework in [22, 23, 24, 25, 26, 27, 28].  
The main contributions of this work are threefold. First, the uncertain factors of Delta robot are novelly described by a fuzzy set theoretic approach. Based on the fuzzy uncertainty, we reformulate the dynamic modeling of Delta robot, which is nonlinear and not Takagi-Sugeno type modeling (using fractional linearization). Second, we propose a fuzzy performance index for Delta robot to optimize the adaptive robust control gain. The performance index considers the impacts of the transient performance, the steady state performance and the control cost. The Delta robot controller with optimized gain is deterministic and is not the _if-then_ rules-based or Mamdani-type fuzzy control. Third, through the defuzzification of the performance index, the solution of the optimal control design is proven to exist and be unique, which is resolved in the closed form. Under the optimal control design, both deterministic (uniform boundedness and uniform ultimate boundedness) and fuzzy (minimizing the fuzzy performance index) performance of Delta robot could be guaranteed.