## 1 Introduction  
The six-hardpoint positioning mechanism is applied to the active support system of ground-based large aperture telescopes, which is used for primary mirror positioning and pose adjustment. It has been applied to some large aperture telescopes, such as the Large Binocular Telescope [1, 2], Vera C. Rubin Observatory (formerly known as the Large Synoptic Survey Telescope) [3, 4], and the Giant Magellan Telescope [5, 6]. Ensuring the accuracy of the pose of the primary mirror is very important for telescopes, which directly affects the imaging quality of telescopes. Once the pose error of the primary mirror exceeds the error range, it will not only lead to various aberrations, but also cause misalignment of the optical and mechanical axes, leading to the instability of the imaging position, which has a disastrous impact on the performance of the telescope [7, 8]. Therefore, the six-hardpoint positioning mechanism must have very high kinematic accuracy, so as to ensure that the primary mirror can be accurately positioned in a predetermined pose during telescope observation. However, in practice, the accuracy of six-hardpoint positioning mechanisms can significantly deteriorate due to the existence of kinematic parameter errors. Due to the complex structure of the six-hardpoint positioning mechanism, it is inevitable to introduce kinematic parameter errors during the manufacturing and assembly process. Kinematic parameter errors lead to deviations between the actual and theoretical kinematic models of the mechanism, which then leads to the pose error of the primary mirror, and reduces the imaging quality of the telescope. Therefore, measures must be taken to correct the kinematic parameter errors or compensate the kinematic parameter errors in the kinematic model. There are up to 42 kinematic parameter errors in the six-hardpoint positioning mechanism, which are difficult to measure directly, and therefore difficult to achieve direct correction. This paper attempts to solve this problem via the proposed calibration method for the six-hardpoint positioning mechanism. Generally, kinematic calibration includes four steps: (i) error modeling, (ii) pose measurement, (iii) error identification, and (iv) error compensation [9; 10]. Kinematic calibration is widely used for various robots, which can effectively and economically improve the kinematic accuracy of robots without changing hardware. Firstly, the relationship between the kinematic parameter errors of the robot and the pose errors of the robot end effector is established through an error model, so the kinematic parameter errors can be solved and identified by measuring the actual poses of the end effector. Then, the theoretical kinematic parameters are corrected using kinematic parameter errors. Finally, a new kinematic model is established based on the corrected kinematic parameters. The new kinematic model is closer to the actual kinematic model and achieves error compensation. Kinematic calibration solves the problem of kinematic parameter errors and improves the accuracy of the robot. Calibration is applied to various robots, including tri-pyramid parallel robot [11], industrial robot [12], airborne heavy-duty mechanical arm [13], Stewart platform [14; 15], hybrid manipulators [16], 7-DoF hybrid robots, etc. [17]. To perform kinematic calibration of six-hardpoint positioning mechanisms, the measurement of the poses of the primary mirror is an important step. The accuracy of calibration can only be guaranteed by obtaining accurate poses of the primary mirror. However, the measurement values of the pose always contain some measurement noise. The causes of the noise include not only errors in the measurement equipment, but also errors in the process of pose resolving. The inevitable measurement noise will reduce the accuracy of calibration. In response to the problem of measurement noise, five observability indices have been proposed in succession, namely \(O_{1}\), \(O_{2}\), \(O_{3}\), \(O_{4}\), and \(O_{5}\)[18; 19; 20; 21; 22; 23].