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This paper presents a novel algorithm about the industrial robot contouring control based on the NURBS (non-uniform rational B-spline) curve. First, aiming at the error between the industrial robot’s actual trajectory and the desired trajectory, the contour error is proposed as the trajectory evaluation index, and the estimation algorithm of contour error based on the tangent approximation is proposed. Based on the tangent approximation algorithm, the estimation algorithm of contour error in the local task coordinate frame is proposed to realize the transformation from the Cartesian coordinate frame to the local task coordinate frame. Second, according to the configuration of the industrial robot, a modified cross-coupling control scheme based on the local task coordinate frame is designed. Finally, the Bernoulli’s lemniscate curves are constructed by NURBS curve and five-order polynomial curve, respectively, and they are symmetrical. The contrast experiment is designed using the two types of constructed Bernoulli’s lemniscate curves as the incentive trajectory. Through the analysis and comparison between the obtained uniaxial tracking error and the contour error curve of the two incentive trajectories, it is concluded that the incentive trajectory constructed by the NURBS curve has better contour control performance than that constructed by the five-order polynomial curve. The results drawn from this paper lay a certain foundation for the future high-precision contouring control of industrial robots.

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

Bi-level programming is widely used in processing various questions, but it cannot deal with the complex and fuzzy information contained in problems. In order to solve such problems better with intricate and vague information that can be efficiently handled by bifuzzy theory, a bifuzzy–bilevel programming model that sets the parameters to bifuzzy variables is proposed in this paper, which can process complex realistic data more accurately and improve the feasibility and validity of bi-level programming models. To ensure the solvability of the model, the equivalent form of the bifuzzy–bilevel programming model is obtained by utilizing the expected value operator. According to the linear and nonlinear characteristics of the model, the Karush–Kuhn–Tucker condition and particle swarm optimization algorithm are employed to handle the problem, respectively. Finally, by taking the distribution center location problem of the supplier as an example, the bifuzzy–bilevel programming model is applied in practice to balance highly intricate customer demands and corporate cost minimization, obtaining the feasible solution of functions at the upper and lower levels, and the bifuzzy information in the problem can also be processed well, which proves the effectiveness of the proposed methodology.