Integrating Rotations Using Nonunit Quaternions
Integrating Rotations Using Nonunit Quaternions
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DOI：
10.1109/lra.2018.2849557
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发表时间：
2018-06
影响因子：
5.2
通讯作者：
Caleb Rucker
中科院分区：
文献类型：
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作者：
Caleb Rucker
在这封信中，我们提出并证明了一种方法，使用非单位四元数的旋转积分。单位四元数通常用于表示旋转，在这种情况下，旋转操作涉及单位四元数的共轭。然而，一个冗余映射可以从所有非零四元数定义到旋转矩阵集，${\mathrm{SO}（3）}$，基于更一般的旋转操作，涉及四元数逆。从这一点上，我们表明，众所周知的公式，映射角速度的衍生物的单位四元数实际上代表的最小范数解决方案内的一组解决方案的衍生物的非单位四元数。这一事实使有效的，奇点自由，数值积分的旋转在很长的时间间隔。该方法在与任何标准例程或常微分方程（ODE）求解器包集成期间固有地保留了${SO}（3）}$的结构，而无需采用专门设计的几何积分方案、指数更新或文献中发现的许多四元数长度强制技术。我们证明了这种方法的准确性相比，其他常见的方法应用于集成一个已知的角速度函数和一个经典的拉格朗日顶部。
In this letter, we propose and demonstrate a method for integration of rotations using nonunit quaternions. Unit quaternions are commonly used to represent rotation, in which case the rotation operation involves the conjugate of the unit quaternion. However, a redundant mapping can be defined from all nonzero quaternions to the set of rotation matrices, ${\mathrm{SO}(3)}$, based on the more general rotation operation involving the quaternion inverse. From this we show that the well-known formula that maps angular velocity to the derivative of a unit quaternion actually represents the minimum-norm solution within a set of solutions for the derivative of a nonunit quaternion. This fact enables efficient, singularity free, numerical integration of rotations over long intervals. The approach inherently preserves the structure of ${\mathrm{SO}(3)}$ during the integration with any standard routine or ordinary differential equation (ODE) solver package without employing specially designed geometric integration schemes, exponential updates, or the many quaternion length enforcement techniques found in the literature. We demonstrate the accuracy of this approach compared to other common methods applied to integrate a known angular velocity function and a classic Lagrange top.