Gravity gradiometers can measure the spatial rate of change in gravitational acceleration. The gravitational field vector is the gradient of the gravitational potential U. In the Cartesian coordinate system, the gravitational gradient tensor T can be defined as follows: T = ∇ ∇ U = ∂ 2 U ∂ x 2 ∂ 2 U ∂ x ∂ y ∂ 2 U ∂ x ∂ z ∂ 2 U ∂ y ∂ x ∂ 2 U ∂ y 2 ∂ 2 U ∂ y ∂ z ∂ 2 U ∂ z ∂ x ∂ 2 U ∂ z ∂ y ∂ 2 U ∂ z 2 = T x x T x y T x z T y x T y y T y z T z x T z y T z z (1) Only five of the nine components of the gravity gradient are independent, as the external gravity field of the earth satisfies the Laplace equation and the gradient tensor is symmetric, whereby T _xy = T _yx , T _xz = T _zx , T _yz = T _zy , and T _xx + T _yy + T _zz = 0.

The main gravity gradiometers that are currently available for mobile platform measurements can be divided into full-tensor gravity gradiometers and partial-tensor gravity gradiometers based on their measurement elements. The core measurement component of a gravity gradiometer is the gravity gradient instrument (GGI). The Air-FTG, which has three sets of GGIs mounted on an inertially stabilized platform, is a typical instrument for full-tensor measurements. The partial-tensor gravity gradiometer, on the other hand, measures either a partial component or a combination of the partial components of the gravity gradient, with the Falcon system being a typical partial-tensor instrument. The principle of our gravity gradiometer is similar to that of the Falcon system, which measures the horizontal components of the gravity gradient tensor (i.e., T _yy - T _xx and T _xy ). The x-, y-, and z-axes of the gravity gradient measurement coordinate system are defined to correspond to the local geographic coordinate system’s E (eastward), N (northward), and U (skyward) coordinates, respectively, whereby T _yy , T _xx , an T _xy (T _xy = T _yx ) correspond to T _NN , T _EE , and T _NE , respectively.

The principle of a GGI measurement is illustrated in Figure 1. Four accelerometers, a1–a4, are evenly distributed on a round plate of radius R that is centered at point O and rotates at a specified angular rate. The input axis of the accelerometer is along the tangential direction of the rotation plane and perpendicular to the rotation axis.

The output of the accelerometers is related to the gravitational gradient as follows [22]: a 1 + a 2 − a 3 + a 4 = 2 R T y y − T x x sin ⁡ 2 ω t + 4 R T x y ⁡ cos ⁡ 2 ω t (2) where ω is the rotation frequency and t is the measurement time. The sum and difference of the accelerometer output signals expressed in Eq. 2 are demodulated using twice the rotation frequency (i.e., 2 ω ) signal to extract the gravity gradient signals (T _yy – T _xx and T _xy ).

The new shipborne GGMS includes a GGI, inertial stabilization platform, buffer damping unit, temperature control unit, and power supply unit.

The shipboard GGI is the core component of the gravity gradiometer and consists of three major parts: the rotating body, rotating shaft system, and rotating support frame. The high-resolution accelerometer, accelerometer servo circuit board, gravity gradient measurement circuit board, and accelerometer temperature control board are all installed on the rotating body. The shape of the gravity gradient sensor is shown in Figure 2. The rotating body is the mounting base for the accelerometer. The four accelerometers are evenly and symmetrically arranged along the circumference of the rotating body, and centered on the rotation axis.

The inertial stabilization platform (Figure 3) is one of the key components of the system, and its main functions are to house the GGI, isolate the carrier angular motion and track the local geographic coordinate system, and provide a dynamic environment for the gravity gradient sensor to meet the gravity gradiometer requirements. The principal scheme of the stabilized platform adopts the mechanical arrangement scheme of the classical three-loop semi-analytic inertial navigation system. Here the output information of the platform accelerometer, in combination with the initial velocity and position binding information of the carrier, are integrated to solve for the carrier velocity and position, and further solve for the corrected angular velocity, which is required for the stabilized platform to track the geographic coordinate system. This corrected angular velocity is then used to correct the output data of the corresponding fiber optic gyroscope on the platform, and is also applied to the platform through the stabilization loop to make the platform coordinate system track the geographic coordinate system, without accounting for the stabilization loop error, such that the local horizontal north-pointing inertial platform is then realized.

The GGI, inertial stabilization platform, buffer shock absorption unit, temperature control unit and power distribution unit, gravity gradient data acquisition and processing system, and motion compensation module with mechanical and electrical interfaces are all connected in the new shipboard GGMS to ensure effective communication and connectivity among the system components, and realize the systematic gravity gradient measurement operation function (Figure 4).

A laboratory-based static accuracy test was the first performance test conducted after assembling the shipboard GGMS. The static accuracy test employs the mass gravitational excitation method. A mass gravitational uniform gradient field excitation device is used to conduct the test. The mass walls on both sides of the gravity gradiometer are high-density bodies that can generate a uniform gravitational gradient field in the space where the gradiometer is located.

A schematic diagram of the laboratory-based static test is shown in Figure 5. The static working GGMS is in the center, with the two high density mass walls that generate the uniform gravitational gradient excitation positioned on the left and right sides of the GGMS. The two mass walls move along the dotted line to either the center or both sides simultaneously during the test, such that different gravitational gradients can be excited at different positions relative to the spatial position of the GGMS. The static measurement accuracy of the GGMS can be determined by comparing the difference between the theoretical gravitational gradient excitation at two positions and the measured gradient difference of the GGMS.

The gravitational gradient excitation on the gradiometer, which is produced by two rectangular mass walls at a distance S2 from the distal end and a distance S1 from the proximal end of the gradiometer, can be calculated theoretically [23] by defining T _uv = (T _yy –T _xx )/2, such that the variation of the T _uv signal is 224 E and the variation of the T _xy signal is 0 E. The mass walls were moved repeatedly (seven times) between the distal and proximal ends during the test experiment. The output signal of the GGMS was recorded continuously for 3 and 20 min at the proximal and distal ends, respectively. The average value within 100 s of system stabilization was taken as the signal output value, and the repeatability of the output signal and accuracy of the difference between the output signals at two positions were counted for each position multiple times to determine the static measurement accuracy of the GGMS. Figure 6 shows the static accuracy test curve of the GGMS, and Tables 1, 2 list the static accuracy test data statistics. The T _uv and T _xy measurement accuracies are 7.22 E and 3.58 E, respectively. The results indicate that the static measurement accuracy of the system is better than 7.22 E.

The laboratory-based dynamic accuracy test of the GGMS was similar to the static test method. However, the GGMS was not stationary during the dynamic accuracy test, but was instead placed on a motion simulation platform to simulate the swaying motion of a ship. The test site is shown in Figure 7.

The moving process and resting position of the mass body were the same as those in the laboratory-based static test, and the stationary measurement times at the distal and proximal ends were both 6 min. A motion simulation table simulated the swaying motion for both large and medium-sized ships under four sea-state levels during the test. The swaying spectrum is shown in Table 3. Figure 8 shows the dynamic accuracy test curve of the GGMS, and Tables 4, 5 list the data statistics of the dynamic accuracy test. The measurement accuracy of the GGMS is better than 9.09 E for external gradient excitation changes.

The obtained test data included the original output data of the gravity gradient sensor, which were acquired at a continuous sampling rate of 800-Hz; the output motion information data of the inertial measurement unit on the stable platform, which were acquired at a continuous sampling rate of 800-Hz; the attitude data of the ship, which were acquired at a continuous sampling rate of 200-Hz using external fiber-optic rosettes; and GPS position data, which were acquired at a continuous sampling rate of 100-Hz. The walk-away measurement mode was adopted during the shipborne test, with the part of the navigation path that possessed a uniform speed and straight heading selected as an effective measurement line.

The output motion information data of the inertial measurement component on the stabilized platform were used during the data processing to compensate the original output data of the GGI for vertical motion, demodulation of the gravity gradient signal, horizontal angular velocity compensation, and self-gradient compensation. The gravity gradient measurement data results were then obtained after lowpass filtering the compensated gravity gradient output data with a cutoff frequency of 1/300 Hz (equivalent spectral resolution of 750 m [24]). The data processing flow chart is shown in Figure 10.