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

在工程领域中使用的几乎所有的动力学系统是多体动力学（MBD）系统，如机器人，飞机，假肢等解决这些系统可能是具有挑战性的，因为系统通常是多维的，可以是非线性和复杂的变量之间的高度耦合。因此，在实践中，人们可能不得不依赖于数值或近似方法来求解系统，因为MBD系统的解析解如果不是不可能的话也难以实现。此外，在用现有方法求解此类系统时，线性化和简化通常是不可避免的，这对准确性和可靠性产生负面影响，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MBD系统。 申请人及其研究团队在开发求解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，该方法计算时间短，计算精度高，结果可靠。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。所提出的研究计划的目的是开发一种分段线性化方法，适用于准确可靠地求解和分析非线性MBD系统，这些系统难以使用现有方法以所需的精度和可靠性求解。为了实现这一目标，申请人先前开发的P-T方法和其他方法将被修改和扩展。本研究将结合联合收割机、修正P-T法、泰勒展开、拉普拉斯变换及留数定理，以理论及数值方式求解多分支系统。线性MBD系统的解析解有望实现，连续的半解析解，沿着数值解，有望得到非线性MBD系统。一个面向工程应用的算法求解非线性MBD系统也有望开发使用上述分段线性化方法。 开发的方法，沿着的计算算法和它们的工业应用的方向，将显着推进当前的方法来解决，分析和优化线性和非线性MBD系统。因此，拟议的研究计划的结果将切实改善MBD系统的设计以及操作和控制质量。这对于工业中需求不断增长的复杂，精密，高速和AI控制的设备和机器尤为重要。

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.