Three DOF Rotation Compensation in Floating HFSWR for Ocean Current Inversion

High-frequency surface wave radar (HFSWR) has been widely used for ocean surface current inversion. Floating this radar can realize the ocean surface inversion among large areas far away from the coastline. However, six degree-of-freedom (DOF) movements will set the barrier for the ocean current inversion on its radial velocity estimation and direction of arrival estimation. Among these movements, three DOF rotations can cause both Doppler modulation and azimuth modulation while three DOF translations only cause Doppler modulation which is dependent on the arrival direction of echoes. Hence, the analysis and compensation for three DOF rotations are first to be solved. In this paper, we analyse the Doppler modulation and azimuth modulation of three DOF rotations and further develop the global adaptive beamforming method to compensate for these rotations. Adaptive beamforming is used to compensate for rotations to make the real-time beams fixed to the ground when the platform is rotating. Simulations indicate that the global adaptive beamforming can perfectly compensate for the reasonable rotations, among which the yaw rotation is the dominant factor for both Doppler spectrum modulation and azimuth modulation. The rotation compensation using global adaptive beamforming can greatly improve the precision of ocean surface inversion.


Introduction
Shore-based high-frequency surface wave radar (HFSWR) equipment has been widely used in ocean surface current inversion of coastal exclusive economic zones [1].In order to further expand the detection range of HFSW, further development of current inversion based on mobile platforms has farreaching significance [2,3].However, six degree-of-freedom (DOF) will cause complex modulations which will set the barrier for later ocean surface current inversion [4].All these modulations are dependent on the arrival direction, hence in this paper, we focus on the three DOF rotations that may influence the direction-of-arrival estimation.
For one floating HFSWR, the steering vector of the receive array changes with the movement of the platform, which will lead to spatial spectrum modulation, and then affect the estimation of ocean current orientation.We are only interested in the direction-of-arrival (DOA) estimation on the ocean surface, hence the spatial modulation can also be called azimuth modulation.Previous investigation [5] shows that for a platform moving forward with six degrees of freedom fluctuation, yawing has the most significant impact on DOA estimation.In the coherent integration time (CIT), the platform translational distance is generally much lower than the range resolution of HFSWR, and the phenomenon of distance migration will not be generated.The translational distance is much smaller than the distance of one farfield signal, and the azimuth modulation effect of the translations is very small.Platform rotation plays a dominant role in azimuth spectrum modulation.Previous investigation summarizes the characteristics of three degree-of-freedom rotations in the application scenario of unstable floating platforms [6]: 1) Pitch and roll are interchangeable, and their motion is manifested as small amplitude oscillation motion near the horizontal plane.2) The yaw is a slow, irregular motion, and its range of change can cover the whole possible yaw angle of 360 degrees.The azimuth spectrum modulation of yaw is the most obvious, and the azimuth spectrum modulation of pitch and roll is obviously smaller than that of yaw.
There are three methods for compensation of spatial spectrum modulation: the first method is to select the reference yaw.In the process of platform rotation, select one appropriate yaw angle as the reference yaw, which can effectively reduce the DOA estimation error.In literature [7], the mean yaw angle is taken as the reference yaw, which effectively reduces the DOA estimation error caused by yaw rotation.However, theoretically, this method cannot guarantee unbiased DOA estimation.The second method is to adopt the time-domain steer vector compensation, which compensates the steering vector of echoes in each direction in the slow time domain and then performs coherent processing in the slow time domain and spatial domain.The maximum coherent output azimuth is the target azimuth.In theory, this method can achieve unbiased DOA estimation of the target echo signal, but for the application scenario of ocean current inversion, the mutual interference of ocean echo from different directions will seriously restrict the effectiveness of this method.The third method is to adopt the adaptive beamforming method.The Sea State Laboratory of Wuhan University [8] is committed to converting antenna channel data into beam channel data through adaptive beamforming, ensuring that the beam shape of the beam channel in the horizontal plane remains fixed while the platform is rotating, to achieve compensation for rotating motion in the beam domain.This method shows good compensation performance on uniform circular arrays with isotropic elements and can achieve unbiased DOA estimation of targets in any direction for any yaw rotation.However, the compensation performance considering all three DOF rotations is not sure.In this paper, we further develop the global adaptive beamforming for all three DOF rotations and analyse the Doppler modulation and azimuth modulation of these rotations.The results of ocean surface currents before and after the compensation for these rotations are also discussed.

Signal model considering six DOF Motions
The definition of the instantaneous state of a rigid body with six DOF motions is shown in Figure 1. f ,  f and  f represent the eastward displacement, the northward displacement and the vertical displacement respectively. f ,  f , and  f represent the yaw angle, pitch angle, and roll angle, respectively, that is, the Euler angle of Z-X-Y sequential rotation.The X-Y-Z coordinate system is fixed to the floating platform and moves with the platform, which is called the moving coordinate system.The E-N-S (east-north-sky) coordinate system does not move with the platform and always remains stationary, which is called the reference coordinate system.
The rotation motion of the three DOF is described by the rotation matrix described by the Euler angles of the Z-X-Y sequence rotation, and the final rotation matrix ( f ,  f ,  f ) is obtained by the product of the rotation matrix corresponding to the three Euler angles.

𝑹(𝜑
For one transmitting antenna, the real-time coordinate  t with respect to the global coordinate origin is (3)  p ,  p0 and  t0 are the attitude measurement centre vector, the vector pointing from the attitude measurement centre to the local coordinate origin and the local antenna coordinate vector.
For one receiving antenna, the real-time coordinate  r with respect to the global coordinate origin is r0 is the local antenna coordinate for the k th receive antenna.Suppose that the transmitting signal is (), and the transmitted signal is scattered by the target and received by the k th receiving antenna, then the signal form of the echo is ( −   ()), and  () is the delay of the whole process, defined as   () =  to () +  ro () to () is the delay from the transmitting antenna to the target,  ro () is the delay from the target to the receiving antenna,  o () is the real-time coordinate of the target, and c is the propagation speed of the electromagnetic wave.

Doppler modulation
For one far-field target, the delay from the transmitting antenna to the k th receiving antenna is approximate as ̂o = (cos(Ψ o ) cos(θ o ) , cos(Ψ o ) sin(θ o ) , sin(Ψ o ) (14)  ̂o is the unit vector of the orientation of the target with respect to the receiving array, θ o and Ψ o are the azimuth and pitch angles of the target with respect to the receiving array, respectively.For ocean surface inversion, the pitch angle is always zero. po ()is the delay corresponding to the distance between the target and the initial position of the attitude measurement centre. T () is the delay corresponding to the projected distance of the translation vector in the target direction, and  Rp () is the delay corresponding to the projected distance of the vector pointing to the local coordinate origin from the attitude measurement centre in the target direction. t0 () is the delay corresponding to the projected distance of the local coordinate of the transmitting antenna in the target direction, and  r0 () is the delay corresponding to the projected distance of the local coordinate vector of the k th receiving antenna in the target direction.For the application scenario where the transmit and receive are arranged on the same floating platform, the time delay described by  po (), T () and  R () is twice, and  po () is twice the general conclusion of the self-collecting mode of operation.The whole delay will cause the Doppler modulation, and all six DOF motions will contribute to this kind of modulation.

Azimuth modulation
The normalized steer vector obtained by using one receiving channel as the reference is the decisive factor in the DOA estimation, defined as If we don't consider the influence of antenna pattern distortions, the normalized steer vector is only related to the antenna coordinates and the three DOF rotations, which will influence the precision of DOA estimation.Because Doppler modulation is dependent on the arrival direction of echoes, this paper mainly focuses on the analysis and the compensation for the three DOF rotations to realize unbiased DOA estimation.

Adaptive beamforming for three DOF rotation compensation
In the platform rotation scenario, the algorithm optimization framework of adaptive beamforming for spatial spectrum compensation needs to meet two constraints, one is that the error between the real-time beam and the reference beam should be as small as possible, and the other is that the modulus of the adaptive weight vector should be smaller than that of the reference weight vector.The former is the requirement of beam shape control, and the latter is the requirement of noise control, so the weight vector of global adaptive beamforming for compensation of the three DOF rotations should meet the following constraints: )  is the azimuth in the horizontal plane,  BG ( f ,  f ,  f ) is the weight vector of global adaptive beamforming,   (,  f ,  f ,  f ) is the realizable real-time beam in the horizontal plane, and b 0 (,  f0 ,  f0 ,  f0 ) is the reference beam in the horizontal plane. 0 is the reference weight vector for obtaining b 0 (,  f0 ,  f0 ,  f0 ), and   real-time weight vector for obtaining   (,  f ,  f ,  f ).The two constraints can also be defined as The non-negative ξ is the balance factor that balances the ability between beam shape control the noise control, so we can get the optimal weight vector as with 4. Simulations

Simulation conditions
Simulation conditions are the same as in our previous work [6].The working frequency is 13.15 MHz and the sweep period is 0.5 second.There are 600 sweep periods for one coherent integration time.The receive array is one uniform circular array with eight elements, the radius of which is 6 meters.The ocean current is set as uniform with amplitude as 100 cm/s and direction from west to east.The ocean surface wind is also uniform which blows from east to west.
In this paper, we only consider the modulation of the three DOF rotations.The modulation of three DOF translations for both the receive antennas and the transmit antenna is not considered.The three DOF rotations used in the following simulations are defined as Yaw() = −45 + 90sin(  T a ),0 ≤  ≤ T a (23) T a is the duration time for one CIT, which is 300 seconds in the following simulations.The three DOF rotations are also displayed in Figure 2. Yaw changes slowly with a large yaw range.Pitch and roll have similar characteristics, and both change quickly with small oscillation amplitude.These characteristics are identical to the measured rotation data [6].

Azimuth modulation
The MUSIC spectrums for azimuth searching are shown in Figure 4.The reference yaw is set as the median value of yaw angles which is 19 o , so the true angle of the scatter patch is 19 o which is denoted by the red upward triangles.Among these rotations, yaw rotation is still the most obvious factor that causes the azimuth modulation which makes the searching peak shift from the true position and makes the searching spectrum broadened.Pitch rotation and roll rotation don't make the peak shift from the true position and just make the searching spectrum broadened slightly.The global adaptive beamforming is used to rotation compensate, and the compensation performance is excellent for all three DOF rotations.

Ocean surface inversion
In this part, we further analyze the results of radial current inversion.The radial current maps and the histograms of radial current errors are shown in Figure 5. Compared to the fixed platform, the radial current map of the rotating platform without compensation looks like just rotating counterclockwise for about twenty degrees, which indicates that the azimuth modulation of yaw rotation is the main factor that reduces the inversion reliability.Global adaptive beamforming This analysis indicates that yaw rotation is the dominant factor among the three DOF rotations that cause Doppler modulation and azimuth modulation and further influence the ocean surface current inversion.The global adaptive beamforming can be effectively used for the compensation of these rotations.Although the Doppler modulation and azimuth modulation of the pitch and roll rotation is minor under the rational simulation conditions and the compensation performance is not obvious, the compensation for the two DOF rotations will be important when the oscillation amplitude of the two DOF rotations is extremely large.

Conclusion
We investigate the modulations and the compensation method for three DOF rotations.The signal model indicates that rotations can cause both Doppler modulation and azimuth modulation.The Doppler modulation may cause the signal loss, and the azimuth modulation may make the DOA estimation shift from the true angle.Among the three DOF rotations which are set based on the measured rotation data, yaw rotation is the dominant factor for both Doppler modulation and azimuth modulation, while the modulations of pitch rotation and roll rotation are slight.The further developed global adaptive beamforming for the three DOF rotations can effectively compensate for these rotations, and the compensation for these rotations is vital for reliable ocean surface inversion.In our future work, we will further investigate the ocean surface current inversion considering the antenna distortions and the modulations of the three DOF translations.

Figure 1 .
Figure 1.Definition of the instantaneous state of a rigid body with six DOF motions.The time-domain signal model introduced by Chang et al.[9] is more favourable for the simulation of complex motion modulation, but they only analyse condition with uniform linear array.In order to extend the time domain signal model, the concept of instantaneous coordinate is introduced here, and

Figure 3 . 3 .
Doppler spectrum before and after adaptive beamforming.(a) yaw; (b) pitch; (c) roll.Considering the scatter path from the east, the Doppler spectrums before and after adaptive beamforming when separately considering the three DOF rotations are shown in Figure Among these DOF rotations, the signal loss caused by yaw rotation is most obvious which is around 1 dB.Pitch rotation and roll rotation nearly don't cause any signal loss.The global adaptive beamforming can effectively compensate for the Doppler modulation

Figure 5 .
Results of radial current inversion.(a), (b) and (c) are the inversion radial current maps for one fixed platform, the rotating platform without compensation and the rotating platform with compensation.(d), (e) and (f) are the corresponding histograms of radial current errors.