This article presents the dynamic interpolation problem for locomotion systems evolving on a trivial principal bundle $Q$. Given an ordered set of points in $Q$, we wish to generate a trajectory which passes through these points by synthesizing suitable controls. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on $Q$. The squared $L^2-$norm of the covariant acceleration is considered as the cost function, and its first order variations are taken for generating the trajectories. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. The result is applied to generate a trajectory for the generalized Purcell's swimmer - a low Reynolds number microswimming mechanism.

中文翻译：

---

平凡主丛上运动系统的变分动态插值

本文介绍了在平凡主丛 $Q$ 上演化的运动系统的动态插值问题。给定 $Q$ 中的一组有序点，我们希望通过合成合适的控件来生成穿过这些点的轨迹。平凡丛的全局乘积结构用于在 $Q$ 上获得诱导黎曼乘积度量。协变加速度的平方$L^2-$norm 被认为是成本函数，并且采用其一阶变化来生成轨迹。非完整约束通过主连接的局部形式强制执行，并采用群对称性进行约简。平凡丛的黎曼联系的显式形式被用来达到成本函数的极值。