Solving and Analyzing Nonlinear Multibody Dynamic Engineering Systems with a Piecewise Linearization Approach

工程领域中使用的几乎所有动力学系统都是多体动力学(MBD)系统，如机器人、飞机、假肢等。这类系统的求解可能是具有挑战性的，因为系统通常是多维的，并且可能是非线性的、变量之间高度耦合的复杂系统。因此，在实践中，人们可能不得不依靠数值或近似方法来求解系统，因为MBD系统的解析解即使不是不可能，也很难获得。此外，在用现有方法求解这类系统时，线性化和简化往往是不可避免的，这对求解的精度和可靠性有不利影响，甚至可能导致错误的解。因此，需要一种具有更高精度和可靠性的创新方法来解决和分析总体上的MBD系统。申请人和他的研究团队在开发解决非线性复杂动力系统的新的解析和数值方法方面取得了重大的研究进展。与其他数值方法(如最流行的龙格-库塔法)相比，用较少的计算时间获得了高精度和可靠的结果。申请人开发的方法和技术为推进拟议的研究计划提供了坚实的基础和工具。提出的研究方案的目的是发展一种分段线性化方法，适用于准确和可靠地求解和分析非线性MBD系统，这些系统很难用现有的方法以期望的精度和可靠性求解。为了达到这一目的，将对申请人以前开发的P-T方法和其他方法进行修改和推广。该研究将结合修正的P-T方法、泰勒展开、拉普拉斯变换和留数定理对MBD系统进行理论和数值求解。线性MBD系统的解析解有望得到，非线性MBD系统的连续半解析解和数值解有望得到。利用上述分段线性化方法，还有望开发出一种面向工程应用的求解非线性MBD系统的算法。即将开发的方法以及计算算法及其面向工业应用的方向，将极大地推动当前求解、分析和优化线性和非线性MBD系统的方法。因此，拟议的研究计划的结果将明显改善MBD系统的设计以及运行和控制质量。这对于复杂、精密、高速和人工智能控制的设备和机器来说尤其重要，这些设备和机器在工业中的需求越来越大。

Almost all dynamic systems used in engineering fields are multibody dynamic (MBD) systems, such as those in robotics, aircrafts, artificial limbs, etc. Solving for such systems can be challenging as the systems are typically multidimensional and can be nonlinear and complex with a high degree of coupling between variables. In practice, therefore, one may have to rely on numerical or approximate approaches for solving the systems as analytical solutions of the MBD systems is difficult to achieve, if not impossible. Moreover, in solving such systems with existing methods, linearization and simplifications are usually unavoidable, which negatively affect the accuracy and reliability and may even lead to erroneous solutions. An innovative methodology with higher degree of accuracy and reliability is therefore desired for solving and analyzing MBD systems in general. Significant research progresses have been achieved by the applicant and his research team in developing new analytical and numerical methods for solving nonlinear complex dynamic systems. Highly accurate and reliable results have been obtained with less computational time, when compared against other numerical methods such as the most popular Runge-Kutta method. The methods and techniques developed by the applicant provide a solid foundation and tools for advancing the proposed research program. The aim of the proposed research program is to develop a piecewise linearization approach suitable for accurately and reliably solving and analyzing nonlinear MBD systems which are difficult to solve with desired accuracy and reliability using existing methods. To achieve the aim, the P-T method previously developed by the applicant and other methods will be modified and extended. The proposed research will combine the modified P-T method, Taylor expansion, Laplacian transformation, and residue theorem to theoretically and numerically solve for MBD systems. Analytical solutions for linear MBD systems are expected to be achieved and continuous semi-analytical solutions, along with numerical solutions, are expected to be derived for nonlinear MBD systems. An engineering application oriented algorithm for solving nonlinear MBD systems is also expected to be developed using the aforementioned piecewise linearization approach. The approach to be developed, along with the computational algorithm and their orientation towards industrial applications, will significantly advance current approaches for solving, analyzing and optimizing linear and nonlinear MBD systems. The results of the proposed research program will therefore tangibly improve the design as well as operation and control quality of MBD systems. This is particularly significant for complex, precise, highspeed and AI controlled equipment and machines that are in increasing demand in industry.