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2 results

  • 3 - Derivatives

  • Alex Gezerlis, University of Guelph, Ontario
  • Book:
    Numerical Methods in Physics with Python
    Published online:
    30 August 2023
    Print publication:
    20 July 2023, pp 89-125

  • Summary

    Chapter 3 starts out with a physics motivation, as well as a mathematical statement of the problem that will be tackled in later sections. After a brief discussion of analytical differentiation, the bulk of the chapter is devoted to increasingly better finite-difference approximations, like the forward difference and the central difference. These are explicitly derived using Taylor expansions, and also applied to second derivatives and to points on a grid. A section introduces the useful tool of Richardson extrapolation, which reappears in later chapters. The chapter also includes an original discussion of automatic differentiation, which is built up from the concept of dual numbers. The chapter is rounded out by a physics project, which studies the kinetic energy in single-particle quantum mechanics, and a problem set. The physics project involves different wave functions and provides the groundwork for the project in the integrals chapter.

  • Space Flyaround and In-orbit Inspection Coupled Control Based on Dual Numbers

  • Lijun Zhang, Hanqiu Li, Jianping Liu, Shan Qian, Yi Lu, Hengnian Li
  • Journal:
    The Journal of Navigation / Volume 71 / Issue 5 / September 2018
    Published online by Cambridge University Press:
    03 May 2018, pp. 1088-1110
    Print publication:
    September 2018

  • In this paper, both the proportional derivative feedback control and variable-structure sliding mode control approaches based on dual numbers are presented to design space flyaround and in-orbit inspection missions. Dual-number-based spacecraft kinematics and dynamics models are formulated. The integrated translational and rotational motions can be described in one compact expression, and the mutual coupling effect can be considered. A space flyaround and in-orbit inspection mission model based on dual numbers is derived. Both proportional derivative feedback control and variable-structure sliding mode control laws are designed using dual numbers. Simulation results indicate that both the proposed control system can provide high-precision control for relative position and attitude. Of the two systems, the variable-structure sliding mode control system performs the best.