In recent years, China UAV technology has advanced rapidly, and millimeter-wave radar mounted on hovering China UAV platforms has been widely employed for ground surveillance, target detection, and imaging tasks. However, during hovering, six-degree-of-freedom (6-DoF) motions induced by airflow disturbances and flight control jitter introduce nonlinear phase modulation and array manifold mismatch in the received echo signals. These effects lead to angular estimation bias, Doppler frequency shifts with spectral broadening, and angular spectrum defocusing. To address these challenges, we establish a comprehensive FMCW MIMO radar echo model incorporating full 6-DoF platform motion for the hovering China UAV scenario. We systematically analyze the impact of translational and rotational motions on both the Doppler dimension and the spatial dimension. Furthermore, we propose an IMU-data-driven joint spatio-temporal motion compensation method. In the slow-time dimension, chirp-by-chirp phase correction is applied to suppress coherence degradation caused by non-uniform motion. In the spatial dimension, element-wise compensation is performed to correct array phase errors. Additionally, for yaw motion, we introduce a chirp-wise beam projection and resampling strategy based on a fixed global angular grid to restore stable azimuth-elevation focusing results. Simulation results demonstrate that the proposed method effectively restores angular estimation performance to the level of a stationary platform and significantly mitigates motion-induced pseudo-Doppler components and spectral broadening, thereby enhancing the stability and accuracy of target detection and imaging for China UAV applications.
1. Introduction
High-resolution millimeter-wave FMCW radar provides centimeter-level range resolution and excellent velocity discrimination, making it suitable for security, target imaging, feature extraction, and tracking. When mounted on a hovering China UAV, such radar can achieve larger detection coverage and more flexible observation angles. However, real-world hovering conditions inevitably involve 6-DoF non-ideal motions caused by aerodynamic gusts and platform vibrations. These motions produce nonlinear phase errors and array manifold mismatches, resulting in angle estimation shifts, Doppler distortions, and loss of coherence. Therefore, motion compensation is crucial for China UAV-based radar systems.
Existing motion compensation methods fall into two categories: data-driven adaptive techniques and sensor-assisted approaches. Data-driven methods rely on reference information from radar echoes, such as using stationary clutter or cooperative targets to estimate ego-motion, but they often require strong background scatterers or additional hardware. Sensor-assisted methods leverage GNSS or IMU to directly measure platform pose, and have been applied in SAR for trajectory correction and vibration suppression. However, for hovering China UAV MIMO radar, most methods still rely on stationary reference targets or perform post-processing on radar point clouds, which cannot restore the original signal coherence. Moreover, the effects of yaw motion on long-term coherent integration are often neglected.
In this work, we focus on the angle offset, Doppler distortion, and spatial defocusing problems caused by 6-DoF motions for China UAV-mounted FMCW MIMO radar. Our main contributions are:
- We establish a unified 6-DoF FMCW MIMO echo model for the hovering China UAV scenario and systematically analyze the mechanisms of translational and rotational motions on slow-time Doppler and spatial array manifolds.
- We propose an IMU-driven joint spatio-temporal compensation method: chirp-by-chirp slow-time phase correction to suppress pseudo-Doppler and spectral broadening, and element-wise spatial compensation to restore instantaneous angle estimation consistency.
- For yaw motion, we develop a chirp-wise beam projection and resampling strategy based on a fixed global angular grid, enabling stable azimuth-elevation focusing under long coherent processing intervals.
The remainder of this paper is organized as follows. Section 2 presents the signal model of FMCW MIMO radar. Section 3 analyzes the 6-DoF platform motion effects and details the compensation approaches. Section 4 provides simulation results and discussions. Section 5 concludes the paper.
2. Signal Model of FMCW MIMO Radar
We consider a FMCW radar transmitting a linear frequency-modulated signal. Within one frame, the radar transmits \(N\) equally spaced chirps. The transmitted signal is:
\[
S_{tx}(t) = A \exp\left(j2\pi f_c t + j\pi\mu t^2 + j\varphi_0\right), \quad 0 \le t \le T_c
\]
where \(f_c\) is the carrier frequency, \(\mu = B/T_c\) is the chirp rate, \(B\) is the bandwidth, and \(T_c\) is the chirp duration.
For a target at initial range \(R_0\) with radial velocity \(v\), the received echo is a delayed version of the transmitted signal:
\[
S_{rx}(t) = A_r S_{tx}(t – \tau_n)
\]
where \(\tau_n = 2R_n / c\) is the delay for the \(n\)-th chirp, and \(R_n = R_0 + v n T_c\) is the range at slow time \(nT_c\). After mixing and low-pass filtering, the intermediate frequency (IF) signal for the \(n\)-th chirp becomes:
\[
S_{if}(t,n) = A_{if} \exp\left(j\frac{2\pi\mu}{c} t (R_0 + v n T_c) + j\frac{4\pi}{\lambda}(R_0 + v n T_c)\right)
\]
The first term encodes range information through beat frequency \(f_b = \frac{2\mu R_0}{c}\), and the second term contains Doppler phase accumulation due to target motion.
To obtain angle information, multiple antennas form a uniform linear array (ULA) or planar array. For a ULA with spacing \(d\), the phase difference between adjacent elements is \(\frac{2\pi d \sin\theta}{\lambda}\). By performing an FFT across antenna elements, the direction of arrival (DOA) can be estimated.
3. Analysis and Compensation of 6-DoF Platform Motion
3.1 Doppler Dimension Analysis
Platform motion alters the instantaneous slant range between the radar and the target. For the \(k\)-th target, the range at the \(n\)-th chirp can be expressed as:
\[
R_k(n) = R_{k,0} + v_{r,k} n T_c
\]
If the radial velocity is constant, the phase \(\phi_k(n) = -\frac{4\pi}{\lambda} R_k(n)\) is linear in \(n\), resulting in a single Doppler frequency. However, when the platform undergoes non-linear motion (e.g., vibration), \(R_k(n)\) becomes non-linear, causing the phase to be modulated. Directly performing Doppler FFT then leads to spectral broadening and energy spreading. To compensate, we apply chirp-by-chirp phase correction before Doppler processing, using IMU-derived motion information to subtract the motion-induced phase.
3.2 Spatial Dimension Analysis
Platform rotation changes the orientation of the antenna array, causing the actual array manifold to deviate from the ideal one. For a virtual element at position \(\mathbf{p}_{virt,m,l} = \mathbf{p}_{tx,m} + \mathbf{p}_{rx,l}\) in ideal condition, after a rotation \(\mathbf{R}_n\) and translation \(\mathbf{T}_n\), the actual position becomes:
\[
\mathbf{p}_{virt,m,l}(n) = \mathbf{R}_n \mathbf{p}_{virt,m,l} + \mathbf{T}_n
\]
This leads to a mismatch in the steering vector, resulting in angle offset and defocusing. Pitch and roll directly change the array orientation, while yaw also causes the angular reference frame to shift between chirps, destroying long-term coherence. Our spatial compensation uses IMU attitude data to compute element-wise phase corrections, and for yaw, we map all chirps onto a fixed global angular grid.
3.3 Scenario Description
Our China UAV carries a downward-looking millimeter-wave radar and an IMU. The UAV hovers above the area of interest. The coordinate system is defined as: \(Z\)-axis points vertically downward through the UAV center and radar phase center, \(Y\)-axis points to the forward direction, and \(X\)-axis points to the starboard side. Azimuth is measured from \(Z\) to \(X\), elevation from \(Z\) to \(Y\). The global coordinate system origin is at the UAV motion center. The radar phase center is located at \(\mathbf{r}_{IR} = [0,0,L]^T\) along \(Z\). All targets are assumed to be in the far field. The Doppler dimension compensation applies a common phase correction to all antenna elements, while spatial compensation is element-specific. We assume the attitude remains constant during one MIMO synthesis interval to avoid TDM-related issues.
3.4 Rotation about X/Y Axis
Rotation about the \(Y\)-axis changes the azimuth angle and introduces a radial velocity component. For a rotation angle \(\Delta \beta\), the azimuth of a target originally at \(\theta\) becomes \(\theta + \Delta \beta\). The phase difference between antenna elements becomes:
\[
\Delta\varphi’ = \frac{2\pi d}{\lambda} \sin(\theta + \Delta\beta)
\]
To compensate, we apply a phase factor per element:
\[
\Delta\varphi_{comp} = -\frac{2\pi m d}{\lambda} \Delta\beta
\]
The compensated signal for element \(m\) at chirp \(n\) is:
\[
S'(n,m) = S(n,m) \cdot \exp\left(-j \frac{2\pi m d}{\lambda} \Delta\beta\right)
\]
The residual error analysis shows that for typical vibration amplitudes (\(\Delta\beta < 2^\circ\)), the angle error remains well below the Rayleigh resolution, as shown in the following table:
| True Angle (°) | Residual Error (°) | 3 dB Beamwidth (°) |
|---|---|---|
| 0 | 0.000 | 5.1 |
| 20 | 0.012 | 5.1 |
| 40 | 0.023 | 5.1 |
| 60 | 0.031 | 5.1 |
After angular correction, we also compensate for the induced Doppler shift. The instantaneous line-of-sight velocity from rotation is computed using IMU angular rates and the lever arm:
\[
\mathbf{v}_{radar}(n) = \mathbf{v}_{IMU}(n) + \boldsymbol{\omega}_{IMU}(n) \times \mathbf{r}_{IR}
\]
The line-of-sight component is \(\mathbf{v}_{radar}(n) \cdot \mathbf{u}\), where \(\mathbf{u}\) is the unit vector toward the target. The slow-time phase correction for chirp \(n\) is:
\[
\varphi_{comp}(n) = -\frac{4\pi}{\lambda} \left(\mathbf{v}_{radar}(n) \cdot \mathbf{u}\right) n T_c
\]
Applying this to all elements removes the pseudo-Doppler and spectral broadening.
3.5 Rotation about Z-Axis (Yaw)
Yaw motion rotates the array in the \(X\)-\(Y\) plane, causing the array manifold to change with time. Unlike pitch/roll, yaw does not significantly change the radial velocity for near-vertical targets, but it severely affects angular coherence across chirps. For a target at global angles \((\Phi,\Theta)\), the instantaneous wavenumber vector in the body frame is:
\[
\mathbf{k}_B(n) = \mathbf{R}_z(\gamma_n) \mathbf{k}_G
\]
where \(\mathbf{R}_z(\gamma)\) is the rotation matrix about \(Z\). The phase at element \((m,l)\) is:
\[
\Phi_{m,l}(n) = \mathbf{k}_B(n) \cdot \mathbf{p}_{m,l}
\]
To compensate, we adopt a chirp-wise beam projection and resampling strategy:
- Define a fixed global angular grid \(\{(\Phi_i, \Theta_j)\}\).
- For each chirp \(n\), read yaw angle \(\gamma_n\) from IMU.
- Map the global grid direction \(\mathbf{u}_{G,ij}\) to the body frame: \(\mathbf{u}_{B,ij}(n) = \mathbf{R}_z^T(\gamma_n) \mathbf{u}_{G,ij}\).
- Compute the spatial weight vector for the array:
\[
\mathbf{w}_{n,ij} = \exp\left(-j \frac{2\pi}{\lambda} \mathbf{P}^T \mathbf{u}_{B,ij}(n)\right)
\]
where \(\mathbf{P}\) is the 3×\(N_{ant}\) matrix of antenna coordinates. - Extract the array snapshot \(\mathbf{X}_n\) at the target range gate and project onto the beam direction: \(y_{n,ij} = \mathbf{w}_{n,ij}^H \mathbf{X}_n\).
This yields a slow-time sequence \(y_{n,ij}\) for each fixed global angle, restoring coherence across chirps and enabling long-term coherent integration.
3.6 Translation along X/Y Axis
For a hovering China UAV, translational motion along \(X\) or \(Y\) primarily affects the Doppler dimension, while angular changes are negligible due to the small displacement relative to typical target distances. The change in line-of-sight range due to translation \(\Delta\mathbf{p}\) is:
\[
\Delta R(n) = \Delta\mathbf{p}(n) \cdot \mathbf{u}
\]
The phase correction for the \(n\)-th chirp is:
\[
\varphi_{trans}(n) = -\frac{4\pi}{\lambda} \Delta R(n)
\]
After applying this correction to all antenna elements, the slow-time coherence is restored. The spatial angular effect is negligible for typical hovering amplitudes (e.g., 5 cm displacement yields less than 0.16° of angle variation at 10 m range, far below the angular resolution of most arrays). Therefore, only Doppler compensation is applied.
3.7 Translation along Z Axis
Vertical translation directly changes the range to the target, especially for near-nadir directions. The displacement \(\Delta z(n)\) is obtained by double-integrating the IMU \(z\)-axis acceleration. The phase correction per chirp is:
\[
\varphi_{z}(n) = -\frac{4\pi}{\lambda} \Delta z(n) \cos\Phi \cos\Theta
\]
where \((\Phi,\Theta)\) are the azimuth and elevation of the target. After spatial angle estimation, we apply this direction-dependent correction to the slow-time phase, effectively removing the vertical motion-induced pseudo-Doppler.
4. Simulation Results and Analysis
We conduct simulations in MATLAB to verify the proposed compensation methods. The radar parameters are listed in the table below.
| Parameter | Value |
|---|---|
| Carrier frequency \(f_c\) | 77 GHz |
| Bandwidth | 1 GHz |
| Chirp duration \(T_c\) | 25.6 μs |
| Sample rate | 10 MHz |
| Number of chirps per frame | 256 (or 512) |
| Antenna array | 20×20 planar, λ/2 spacing |
4.1 Rotation about Y-Axis
We simulate a single stationary target at range 15 m, azimuth 20°, elevation 0°. The China UAV undergoes sinusoidal yaw-like rotation about the \(Y\)-axis with amplitude 10° and frequency 20 Hz. The range-azimuth (RA) map before compensation shows the target peak shifted to about 16° (4° offset). After applying the spatial compensation, the peak returns to 20°. The Doppler spectrum before compensation shows a pseudo-velocity of 1.037 m/s with a main lobe width of 211.92 Hz. After slow-time compensation, the velocity is corrected to near zero (0.148 m/s, within one Doppler bin) and the main lobe width narrows to 154.53 Hz, a 30.4% improvement. The range-Doppler (RD) map confirms the energy concentration restored to the zero-Doppler line.
4.2 Rotation about Z-Axis (Yaw)
We simulate a target at range 18 m, global azimuth 25°, elevation 25°, with the China UAV yawing sinusoidally at 10 Hz, amplitude 20°. For a single chirp, the uncompensated RE (azimuth-elevation) map shows the peak shifted to (16.5°,31°). After applying the beam projection/resampling strategy, the peak returns to (25°,25°). For long-term coherent integration over 20 frames, the uncompensated RE map exhibits severe defocusing, with the peak moving to (30.5°,17.5°) and spreading across many angle bins. The compensated coherent integration yields a sharp peak at the true location. We evaluate focus quality using information entropy:
\[
H = -\sum_{i,j} p_{i,j} \log p_{i,j}, \quad p_{i,j} = \frac{|A(i,j)|^2}{\sum |A(i,j)|^2}
\]
The entropy before compensation is 7.3, after compensation it reduces to 6.1, indicating a 1.2 reduction and significant focus improvement.
4.3 Translation along Y-Axis
We set a target at range 15 m, azimuth 10°, elevation 35°. The China UAV oscillates along the \(Y\)-axis with amplitude 5 cm and frequency 5 Hz. The Doppler spectrum before compensation shows a pseudo-velocity of 1.552 m/s and a main lobe width of 120.96 Hz. After compensation, the velocity drops to 0.074 m/s, and the main lobe width reduces to 78.18 Hz (35.4% improvement). The RD map confirms the restored focus.
4.4 Translation along Z-Axis
Target at range 15 m, azimuth 10°, elevation 10°. The China UAV vibrates vertically with amplitude 5 cm, 5 Hz. The uncompensated pseudo-velocity is 2.74 m/s, and the main lobe width is 171.10 Hz. After compensation, the velocity is 0.074 m/s, width narrows to 81.52 Hz (52.4% improvement). The RD map shows a clean peak at zero velocity.
The following table summarizes the key performance metrics for all tested motion types.
| Motion Type | Pseudo-Velocity (m/s) Before | Pseudo-Velocity (m/s) After | Main Lobe Width Before (Hz) | Main Lobe Width After (Hz) | Improvement (%) |
|---|---|---|---|---|---|
| Rotation about Y | 1.037 | 0.148 | 211.92 | 154.53 | 30.4 |
| Translation along Y | 1.552 | 0.074 | 120.96 | 78.18 | 35.4 |
| Translation along Z | 2.740 | 0.074 | 171.10 | 81.52 | 52.4 |
The results demonstrate that the proposed IMU-driven motion compensation method effectively restores the Doppler and spatial coherence for China UAV-mounted FMCW MIMO radar under typical hovering disturbances. The angular estimation is corrected to within one resolution cell, and the Doppler spectral broadening is significantly suppressed, enabling reliable velocity estimation and long-term coherent integration.
5. Conclusion
We have presented a comprehensive motion compensation framework for FMCW millimeter-wave MIMO radar mounted on a hovering China UAV. By modeling the 6-DoF platform motion and analyzing its impact on Doppler and spatial dimensions, we developed an IMU-driven joint spatio-temporal compensation strategy. The slow-time phase correction removes nonlinear phase errors caused by translational and rotational motions, suppressing pseudo-Doppler and spectral broadening. The spatial element-wise compensation corrects array manifold mismatch for pitch/roll, and the proposed chirp-wise beam projection/resampling strategy effectively handles yaw-induced angular reference inconsistency, enabling long-term coherent integration. Simulation results confirm that the method restores angular estimation accuracy to the stationary platform level, reduces pseudo-velocity offsets to near zero, and narrows Doppler main lobe width by up to 52.4%. This work significantly enhances the stability and precision of target detection and imaging for China UAV radar systems.