A novel formulation for the description of spatial rigid body motion using six non-redundant, homogeneous local velocity coordinates is presented. In contrast to common practice, the formulation proposed here does not distinguish between a translational and rotational motion in the sense that only translational velocity coordinates are used to describe the spatial motion of a rigid body. We obtain these new velocity coordinates by using the body-fixed translational velocity vectors of six properly selected points on the rigid body. These vectors are projected into six local directions and thus give six scalar velocities. Importantly, the equations of motion are derived without the aid of the rotation matrix or the angular velocity vector. The position coordinates and orientation of the body are obtained using the exponential map on the special Euclidean group \(\mathit{SE}(3)\). Furthermore, we introduce the appropriate inverse tangent operator on \(\mathit{SE}(3)\) in order to be able to solve the incremental motion vector differential equation. In addition, we present a modified version of a recently introduced a fourth-order Runge–Kutta Lie-group time integration scheme such that it can be used directly in our formulation. To demonstrate the applicability of our approach, we simulate the unstable rotation of a rigid body.

中文翻译：

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使用非冗余统一局部速度坐标的刚体运动方程

提出了一种新颖的公式，用于描述使用六个非冗余的均匀局部速度坐标的空间刚体运动。与通常的实践相反，此处提出的公式在仅使用平移速度坐标来描述刚体的空间运动的意义上没有区分平移运动和旋转运动。我们通过使用刚体上六个正确选择的点的固定于身体的平移速度矢量来获得这些新的速度坐标。这些向量被投影到六个局部方向，从而给出六个标量速度。重要的是，在不借助旋转矩阵或角速度矢量的情况下得出运动方程。\（\ mathit {SE}（3）\）。此外，我们在\（\ mathit {SE}（3）\）上引入了适当的反正切算符，以便能够求解增量运动矢量微分方程。此外，我们介绍了最近推出的四阶Runge–Kutta Lie群时间积分方案的修改版本，以便可以直接在我们的公式中使用它。为了证明我们方法的适用性，我们模拟了刚体的不稳定旋转。