The Incomplete Orientation Constraint (IOC), which does not require a controlled motion constrained by all three spatial directions, exists widely in a lot of robotic tasks. However, the IOC remains a challenge for existing methods, due to the nonlinear structure of the rotation group SO(3). Moreover, the IOCs are time varying in the trajectory tracking problem, which makes it more challenging than the set-point control. To address the IOC problems, we define, identify and prove the closed-form solution of the perpendicular curve in SO(3). Based on the proposed perpendicular curve, we develop a new trajectory tracking controller considering the IOC. Compared with existing methods, the proposed method can achieve faster and more accurate tracking results. Moreover, the proposed method can be applied not only to manipulators with redundancy (including both functional and intrinsic redundancy), but also to manipulators that are non-redundant. Furthermore, it is easier to incorporate a secondary optimization objective into consideration for the intrinsic redundant case, which is difficult for existing methods. The proposed method has been implemented on both simulations and experiments. The numerous simulation and experimental results validate the effectiveness and advantages of the proposed method. Note to Practitioners—The trajectory tracking control with IOC is important and can be applied in a wide spectrum of robotic-assisted manufacturing, e.g. arc-welding, engraving, etc.. In this paper, a perpendicular curve-based trajectory tracking method is proposed to consider the IOC automatically. It is no longer necessary to carefully plan a reachable orientation trajectory. Users only need to focus the planning of the tool direction, which is determined by the tasks. In addition, the proposed method can achieve faster and more accurate tracking result. Moreover, it is universal and can be applied to both redundant and non-redundant cases.

Orientability is an important performance index to evaluate the dexterity of haptic master devices. Currently, most of the existing haptic master devices have limited workspace and limited dexterity. In this paper, we present the RemoT-ARM, a 6 Degree-of-Freedom (DOF) haptic master device that can provide larger and more dexterous workspace for operators. To evaluate its reachability of orientations, we propose a novel orientability index. Furthermore, a relative orientability index is proposed to characterize the matching degree of the workspace of a given manipulator to its target workspace. The volume, the manipulability and the condition number are also introduced as performance indices to evaluate the size and the isotropy of the workspace. According to these performance indices, all possible configurations for the RemoT-ARM have been taken into consideration, analyzed, and compared to finalize its optimal configuration.