Teleoperation systems require a human-centered approach in which the kinematic mapping is intuitive and straightforward for the operators. However, a mismatch in the degrees-of-freedom (DoFs) between master and slave could result in an asymmetrical teleoperation system. That is an obstacle for intuitive kinematic mapping. In particular, it is even more challenging when the missing DoF is a pure rotational DoF, since the rotation group $SO(3)$ is a nonlinear Riemannian manifold. This letter is concerned with an asymmetric teleoperation system, where the master subsystem can provide 6-DoF pose sensing while the slave subsystem only has 5 DoFs. The rotation along the missing DoF, which is configuration-dependent, is mapped to a geodesic curve in $SO(3)$. We define and prove the closed-form solution of the perpendicular curve to the geodesic curve. Based on the perpendicular curve, we develop a novel Incomplete Orientation Mapping (IOM) approach to avoid the motion in the missing DoF. By comparing with two baseline methods, the experimental results demonstrate that the proposed method can discard the rotational motion along the missing DoF for all configurations, while preserving the remaining rotations.

中文翻译：

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单自由度主从不对称遥操作的不完全方向映射

远程操作系统需要一种以人为中心的方法，其中运动学映射对操作员来说是直观和直接的。但是，主从之间的自由度 (DoF) 不匹配可能会导致远程操作系统不对称。这是直观运动学映射的障碍。尤其是当缺失的 DoF 是纯旋转 DoF 时更具挑战性，因为旋转组$SO(3)$是非线性黎曼流形。这封信与非对称遥操作系统有关，其中主子系统可以提供 6-DoF 姿态感知，而从子系统只有 5 个自由度。沿缺失 DoF 的旋转（依赖于配置）被映射到$SO(3)$. 我们定义并证明了垂直曲线到测地曲线的闭式解。基于垂直曲线，我们开发了一种新颖的不完全方向映射 (IOM) 方法，以避免丢失 DoF 中的运动。通过与两种基线方法的比较，实验结果表明，所提出的方法可以丢弃所有配置沿缺失的自由度的旋转运动，同时保留剩余的旋转。