The development of controllers for underactuated systems with nonholonomic constraints has been a topic of significant interest for many researchers in recent years. These systems are hard to control because their linearization transform them into uncontrollable systems. The proposed approaches involve the use of a permanent excitation in the reference trajectory; coordinate transformation; discontinuities; or complex calculations. This paper proposes the design of the controller of the second-order chained form system for trajectory tracking by using a simpler approach based on linear algebra. Up to the present time, no controllers based on this approach have been designed for that system. The control problem is solved by setting two of the three systems variables as a reference, while the remaining variable is calculated imposing the condition that the equations system has an exact solution to ensure that tracking errors go to zero. The stability of the proposed controller is theoretically demonstrated, and simulations results show a suitable control system performance. Also, no coordinate transformation is necessary.

中文翻译：

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基于线性代数的二阶链表系统轨迹跟踪与定位控制

近年来，对于具有非完整约束的欠驱动系统的控制器的开发已经成为许多研究者的重大兴趣课题。这些系统很难控制，因为它们的线性化将它们转换为不可控制的系统。所提出的方法涉及在参考轨迹中使用永久激励。坐标变换 不连续；或复杂的计算。本文提出了一种基于线性代数的简单方法，用于轨迹跟踪的二阶链表系统的控制器设计。到目前为止，还没有为该系统设计基于此方法的控制器。通过将三个系统变量中的两个设置为参考来解决控制问题，而剩下的变量是在方程系统具有精确解以确保跟踪误差为零的条件下计算的。理论上证明了所提出控制器的稳定性，仿真结果表明该控制器具有合适的控制性能。而且，不需要坐标变换。