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The purpose of this paper is to propose a smooth double-spline interpolation method for six-degree-of-freedom rotational robot manipulators, achieving the global C2 continuity of the robot trajectory.

This paper presents a smooth double-spline interpolation method, achieving the global C2 continuity of the robot trajectory. The tool center positions and quaternion orientations are first fitted by a cubic B-spline curve and a quartic-polynomial-based quaternion spline curve, respectively. Then, a parameter synchronization model is proposed to realize the synchronous and smooth movement of the robot along the double spline curves. Finally, an extra u-s function is used to record the relationship between the B-spline parameter and its arc length parameter, which may reduce the feed rate fluctuation in interpolation. The seven segments jerk-limited feed rate profile is used to generate motion commands for algorithm validation.

The simulation and experimental results demonstrate that the proposed method is effective and can generate the global C2-continuity robot trajectory.

The main contributions of this paper are as follows: guarantee the C2 continuity of the position path and quaternion orientation path simultaneously; provide a parameter synchronization model to realize the synchronous and smooth movement of the robot along the double spline curves; and add an extra u-s function to realize arc length parameterization of the B-spline path, which may reduce the feed rate fluctuation in interpolation.

This paper introduces a new mathematical model for analyzing the economic benefits of incorporating the fourth party logistics (4PL), which is a contractor (i.e. agent) for the supply chain coordination and construction based on the division of community and the outsourcing development. Based on the physical theory and the wave-particle duality, a supply chain is the special organization whose characteristic has wave-particle duality. The mathematical model enriches the connotation of 4PL and it broadens the thought for 4PL development. Secondly, the proposed mathematical model predicated on transaction costs, is supported by Transaction Cost Theory (TCT) and acts as the theoretical analysis tool of 4PL for coordinating 3-party generic supply chain. Through the model, some trendy conclusions can be drawn to provide theoretical support for 4PL’s practices. Finally, a case illustrates our conclusions.

In the development processes of a product’s market life cycle, there are three phases of an enterprise’s innovation: new product development, production processes, and product management. In this article, the analyses of benefit and costs, value, and profit to companies are discussed in different stages. New logistics features that appear in an enterprise’s supply chain based on innovative modeling are discussed. Then a logistics model and its technical system based on the classified logistics center are established, which are appropriate for innovative modeling within an agile supply chain. Using the basic theory and techniques of ‘extenics’, the formal conception of innovative modeling-based manufacture in logistics is presented, and the matter-element models are established. Finally, a case study demonstrates the results.