Providing smooth reference trajectories can effectively increase performance and accuracy of tracking control applications while overshoot and unwanted vibrations are reduced. Trajectory planning computations can often be simplified significantly by transforming the system dynamics into decoupled integrator chains using methods such as feedback linearization, differential flatness or the Brunovsky normal form. We present an efficient method to plan trajectories for integrator chains subject to derivative bound constraints. Therefore, an algebraic precomputation algorithm formulates the necessary conditions for time optimality in form of a set of polynomial systems, followed by a symbolic polynomial elimination using Gröbner bases. A fast online algorithm then plans the trajectories by calculating the roots of the decomposed polynomial systems. These roots describe the switching time instants of the input signal and the full trajectory simply follows by multiple integration. This method presents a systematic way to compute fast, for low system orders even time optimal trajectories exactly via algebraic calculations without numerical approximation iterations. It is applied to various trajectory types with different continuity order, asymmetric derivative bounds and non-rest initial and final states.

中文翻译：

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使用多项式消元法对积分链动力学进行高效在线轨迹规划


提供平滑的参考轨迹可以有效提高跟踪控制应用的性能和精度，同时减少超调和不必要的振动。通过使用反馈线性化、微分平坦度或布鲁诺夫斯基范式等方法将系统动力学转换为解耦积分器链，通常可以显着简化轨迹规划计算。我们提出了一种有效的方法来规划受导数边界约束的积分器链的轨迹。因此，代数预计算算法以一组多项式系统的形式制定时间最优性的必要条件，然后使用 Gröbner 基进行符号多项式消去。然后，快速在线算法通过计算分解多项式系统的根来规划轨迹。这些根描述了输入信号的切换时刻，并且完整的轨迹简单地通过多重积分来遵循。该方法提出了一种快速计算的系统方法，对于低系统阶数甚至时间最优轨迹，精确地通过代数计算而无需数值近似迭代。它适用于具有不同连续阶、不对称导数界和非静止初始状态和最终状态的各种轨迹类型。