## 1 Introduction  
The Gravity Recovery and Climate Experiment (GRACE) [1] is a low-low satellite-to-satellite tracking (LL-SST) mission which has provided information on long-term temporal variations in Earth's gravity field for \(15\,\mathrm{years}\). The twin satellites were launched on \(17\,\mathrm{March}\) 2002, using a K-band ranging (KBR) system (or microwave ranging instrument, MWI) to measure the intersatellite distance variations caused by the Earth's gravity field during the mission. Since non-gravitational forces also affect the intersatellite range, both GRACE satellites host an accelerometer (ACC) to measure the non-gravitational accelerations acting on the satellites. GRACE ended its mission in 2017 and its successor mission GRACE Follow-On (GRACE-FO/GFO) [2], which hosts two intersatellite ranging instruments, the KBR (or MWI) and a high-precision laser ranging interferometer (LRI), was successfully launched on \(22\,\mathrm{May}\) 2018. The KBR and LRI of GRACE-FO work in parallel and both of their range measurements can be used for gravity field recovery. Gravity field recovery requires the intersatellite range (or range rate) between the center of mass (COM or CM, the latter of which is used in this paper) of each satellite. However, the ranging instrument can only provide distance measurement between reference points of the two satellites. The reference point is the antenna phase center (APC) for KBR, and for LRI is a point called the vertex point (VP), which is defined by a triple mirror assembly (TMA), where the VP is the intersection point of three mirror planes [3], as shown in Figure 1. In each GRACE-FO satellite, the KBR antenna phase center is placed about \(1.5\,\mathrm{m}\) away from the CM, while the TMA VP is close to the CM within \(0.5\,\mathrm{mm}\)[4]. As long as an offset exists between the reference point and the CM, an angle-dependent geometric pathlength error arises--when the satellite attitudes change, the distance between the reference points of the twin satellites varies accordingly. In other words, satellite attitude variations cause an error in the range measurement for both KBR and LRI. For the LRI case, this effect is commonly referred to as tilt-to-length (TTL) coupling. Tilt-to-length coupling is commonly seen in laser interferometers with movable components. For space-based laser interferometers, such as those in the LISA mission and the LISA Pathfinder (LPF) mission, the tilt-to-length coupling is one of the main error sources. A series of studies on TTL coupling have been carried out for these missions [5; 6; 7; 8; 9]. For gravity missions like GRACE-FO, the tilt-to-length coupling in LRI is mainly due to the CM-VP offset as mentioned above. Other TTL error sources, like the beam splitters and the wavefront curvature, were discussed in [10]. The angle-dependent error generated by the CM-APC offset in KBR is also called the TTL error in this paper since its principle is the same as the error caused by the CM-VP offset in LRI. For the KBR in GRACE and GRACE-FO, in-orbit maneuvers, called KBR calibration, are performed to determine the APC positions. Each KBR calibration maneuver consists of four different submanuevers: positive pitch maneuver, negative pitch maneuver, negative pitch maneuver, positive yaw maneuver, and negative yaw maneuver [11]. For each submanuever, the TTL error signals are enlarged by wiggling the satellite with a certain angle offset. For KBR observations generated by the maneuvers, precision orbit determination (POD) results are applied to eliminate the orbital effects and polynomials are fitted to reduce the low-frequency POD error. After estimating the APC position of both satellites, a term called the antenna offset correction (AOC), which is provided in the Level-1B data product [12], is computed with the satellite attitudes determined by the star cameras.