Real-time capable model inversion of underactuated multibody systems using servo-constraints

欠驱动多体系统是控制输入比自由度少的机械系统。逆模型提供了给定的输出轨迹、相关的控制输入和系统所有坐标的轨迹。采用逆模型作为前馈控制和反馈控制相结合的方法对多体系统进行轨迹跟踪。该项目的目标是开发具有实时能力的欠驱动多体系统逆模型。在此基础上，采用了源于多体动力学的伺服约束方法。这是一种适用于多种系统的系统数值方法。多体系统的数学模型由具有控制输入的运动方程和定义的输出方程组成。对于逆模型，定义了输出的附加伺服约束，使得这些约束遵循预定义的轨迹。本项目中考虑的系统要么是差分平坦系统，要么是具有稳定内部动力学的系统。该项目中最大的挑战是逆模型的设计和实时求解，这些逆模型具有较高的微分指数和内动态。在本项目结束时，应对这种系统进行处理，特别是使用几个输入和输出。在这个项目的范围内，还应该调查坐标的选择，即最小坐标或多余坐标的选择是否对反模型的解和效率有影响。目前，这是一个悬而未决的问题，因为由于伺服约束，总是产生一个微分代数方程。用起重机的简单例子已经证明，在这里，多余的坐标不一定比使用广义坐标的效率低。对于更复杂的三维系统来说，这将特别有趣。在给定的时间步长约束下，需要检验能够进行实时模型反演的极限。因此，对于具有内部动力学的系统，必须考虑它们的特性可能是比正向动力学快的根本。最后，利用基础实验对反演模型的精度进行了检验。因此，将建立一个模块化试验台，允许对4个结构不同的系统进行测试。

Underactuated multibody systems are mechanical systems which haves less control inputs than degrees of freedom. An inverse model provides for a given output trajectory the associated control inputs and the trajectories of all coordinates of the system. Inverse models are used as feedforward control in combination with feedback control for trajectory tracking of multibody systems. The aim of the proposed project is the development of real-time capable inverse models of underactuated multibody systems. Thereby the servo-constraint approach is used, which originates in multibody dynamics. This is a systematic numerical approach which is applicable to a wide range of systems. The mathematical model of multibody systems consists of the equation of motion with control inputs and the defined output equations. For the inverse model additional servo-constraints for the output are defined, such that these follow predefined trajectories. The systems considered in this project are either differentially flat systems or systems with stable internal dynamics. The biggest challenge in this project is the design and real-time solution of inverse models, which have a higher differential index and an internal dynamics. At the end of this project such systems should be treated, especially with several inputs and outputs. In the scope of this project it should also be investigated if the choice of coordinates, i.e minimal coordinates or redundant coordinates, has an influence on the solution and efficiency of the inverse model. Currently this is an open question since due to the servo-constraints always a differential algebraic equation arises. Using the simple example of a crane it has been already demonstrated that here redundant coordinates are not necessarily less efficient than using generalized coordinates. This will be especially interesting for more complex three-dimensional systems. Under given time step constraints the limits of real-time capable model inversion should be tested. Hereby for systems with internal dynamics it has to be considered that their characteristic might be fundamental faster than the forward dynamics. Finally, using basic experiments the accuracy of inverse models should be tested. Therefore a modular testbed will be set-up, which allows the testing of 4 structurally different systems.