In this work, I present a comprehensive study on motion compensation methods for Frequency-Modulated Continuous-Wave (FMCW) millimeter-wave Multiple-Input Multiple-Output (MIMO) radar systems deployed on hovering China drone platforms. The core challenge arises from six-degree-of-freedom (6-DoF) platform motions—induced by airflow turbulence and flight control jitters—which introduce nonlinear phase modulations and array manifold mismatches in the received echoes. These distortions lead to angular estimation biases, Doppler frequency shifts, spectral broadening, and angular spectrum defocusing. To address these issues, I establish a unified 6-DoF motion echo model for FMCW MIMO radar and propose an Inertial Measurement Unit (IMU)-driven joint spatiotemporal compensation framework. The method comprises: (1) chirp-by-chirp phase correction in slow-time to suppress coherence degradation from non-uniform motion; (2) element-wise spatial phase compensation to correct array manifold mismatch; and (3) for yaw motion, a fixed global angular-grid-based chirp-wise beam projection/resampling strategy to restore stable azimuth-elevation focusing. Simulation results demonstrate that the proposed method effectively restores angular estimation to stationary-platform levels, eliminates pseudo-Doppler velocities and spectral broadening, and significantly improves target detection and imaging stability for China drone applications.
I. Introduction
Millimeter-wave FMCW radar systems offer centimeter-level range resolution and excellent velocity discrimination, making them widely adopted in security surveillance, target imaging, feature extraction, and tracking. When mounted on hovering China drone platforms, these radars achieve larger detection coverage and more flexible observation angles. However, during hover mode, aerodynamic disturbances and flight-control vibrations cause unavoidable 6-DoF platform motions, including translational displacements along three axes and rotational motions about three axes. These motions induce nonlinear phase modulation in the slow-time domain and array manifold distortion in the spatial domain, leading to severe degradation in angle estimation, Doppler measurement, and coherent integration gain.
Existing motion compensation techniques can be broadly classified into two categories: data-driven adaptive methods and external sensor-aided methods. Data-driven approaches exploit reference information from the radar echoes themselves, such as adaptive noise cancellation using auxiliary radar or stationary background clutter, or estimation of platform self-motion via stationary scatterers, cooperative targets, or clutter. While effective under specific conditions, these methods often rely on strong stationary clutter or additional hardware, limiting their applicability in open or non-cooperative environments. External sensor approaches, using Global Navigation Satellite System (GNSS) or IMU, directly measure platform pose and are widely used in Synthetic Aperture Radar (SAR) for high-order motion precompensation and trajectory correction. For hovering China drone MIMO radar scenarios, recent studies have used IMU for 6-DoF phase correction but still depend on stationary reference target echoes. Other approaches perform radar point cloud registration and compensation as post-processing, which cannot recover the raw echo signal for further signal processing.
The limitations of existing research for the specific scenario of hovering downward-looking FMCW MIMO radar are threefold. First, there is a lack of a unified error mechanism analysis that simultaneously addresses the effects on Doppler and spatial dimensions. In particular, the impact of non-uniform platform motion on slow-time coherence—causing pseudo-Doppler shifts and spectral broadening—and the effect of yaw motion on long-time coherent integration—leading to angular reference inconsistency and defocusing—are not adequately treated. Second, methods that rely on stationary cooperative targets are impractical for many China drone missions. Third, processing at too high a dimension (e.g., point cloud level) cannot directly restore the raw echo signal’s slow-time coherence and array manifold consistency.
To fill these gaps, I develop a systematic modeling and compensation approach. The main contributions of this work are:
- Establishment of a 6-DoF FMCW MIMO echo signal model for the hovering downward-looking scenario, with detailed analysis of translational and rotational motion effects on Doppler and spatial array dimensions.
- Proposal of an IMU-driven joint spatiotemporal compensation method. For rotation about X/Y axes and translational motions, chirp-by-chirp slow-time phase correction is applied to suppress pseudo-Doppler shifts and spectral broadening. For array orientation changes due to attitude variations, element-wise spatial compensation restores instantaneous angle estimation consistency.
- For yaw motion about the Z-axis, a fixed global angular grid-based chirp-wise beam projection/resampling strategy is introduced. This ensures that slow-time samples from different chirps are accumulated under a unified spatial reference, recovering stable azimuth-elevation focusing after long coherent integration.
| Comparison Aspect | Literature [14] | Literature [15] | This Work |
|---|---|---|---|
| Processing Object | Vital Signs | Radar Point Cloud | Raw Echo Signal |
| Compensation Level | Signal Level | Point Cloud Level | Signal Level |
| Output Focus | Vital Signal Recovery | Vital Signal Recovery | Doppler/Spatial Spectrum Focusing |
| Key Limitation | Needs stationary target | High processing dimension | IMU sampling precision |
II. Radar Signal Model
I adopt the FMCW waveform where the transmitted frequency increases linearly with time. Within one frame, the radar transmits \(N\) equally spaced chirps. The transmitted signal is:
$$
S_{tx}(t) = A_0 \exp\left( j2\pi f_c t + j\pi\mu t^2 + j\phi_0 \right), \quad 0 \le t < T_c
$$
where \(f_c\) is the center frequency, \(\mu = B/T_c\) is the chirp rate with bandwidth \(B\) and chirp duration \(T_c\), and \(A_0\) is the amplitude. For a point target at initial range \(R_0\) with radial velocity \(v\), the received signal is a delayed version of the transmitted signal. The time delay for the \(n\)-th chirp is \(\tau_n = 2R(t_n)/c\), with \(R(t_n)=R_0 + v t_n\). Mixing the received signal with the transmitted signal and filtering out the high-frequency components yields the intermediate frequency (IF) signal. Neglecting residual video phase and target micromotion, the IF signal for the \(n\)-th chirp is:
$$
S_{if}(\hat{t}, n) = A_{if} \exp\left( j2\pi \mu \frac{2R_0}{c} \hat{t} + j\frac{4\pi}{\lambda}(R_0 + v n T_c) \right)
$$
where \(\lambda = c/f_c\) is the wavelength, and \(\hat{t}\) denotes fast time. The first phase term contains range information, and the second term includes Doppler phase accumulation across chirps.
For angle estimation, a multi-antenna array is used. For a uniform linear array (ULA) with element spacing \(d\), the phase difference between adjacent receive elements under far-field assumption is \(\Delta\phi = 2\pi d\sin\theta / \lambda\). The target angle \(\theta\) can be extracted by applying FFT along the antenna dimension.
III. Analysis of 6-DoF Platform Motion Effects
Platform motion alters the instantaneous geometric relationship between radar and target. For a hovering China drone, two main propagation paths affect the echo: (1) the instantaneous slant range changes due to translation and rotation, causing additional phase modulation in slow-time; (2) attitude variations change the actual positions of transmit and receive elements, distorting the virtual array manifold from its ideal static configuration. I analyze these effects separately in the Doppler and spatial dimensions.
A. Doppler Dimension Analysis
Let the instantaneous distance to the \(k\)-th target at the \(n\)-th chirp be \(R_k(n)\). The round-trip propagation phase is \(\phi_k(n) = -4\pi R_k(n)/\lambda\). If the relative radial motion is uniform, \(R_k(n) \approx R_{k,0} + v_{r,k} nT_c\), the phase varies linearly with \(n\), producing a single Doppler frequency after slow-time FFT. When the motion is non-uniform (e.g., due to vibration), \(R_k(n)\) becomes nonlinear, and the phase \(\phi_k(n)\) is no longer linear. Direct FFT then spreads the target energy over multiple Doppler bins, causing spectral broadening and reduced coherence. Therefore, Doppler compensation must remove the additional slow-time phase introduced by platform motion. I adopt chirp-by-chirp phase correction before Doppler FFT to simultaneously suppress pseudo-Doppler shifts and spectral spreading.
B. Spatial Dimension Analysis
In the ideal static case, the virtual element position for the \(m\)-th transmit and \(l\)-th receive element is \(\mathbf{p}_{virt,m,l} = \mathbf{p}_{tx,m} + \mathbf{p}_{rx,l}\). When the platform rotates at the \(n\)-th chirp, the actual position becomes:
$$
\mathbf{p}_{virt,m,l}(n) = \mathbf{R}_n \cdot \mathbf{p}_{virt,m,l} + \mathbf{T}_n
$$
where \(\mathbf{R}_n\) is the rotation matrix and \(\mathbf{T}_n\) is the translation vector. The array spatial phase no longer satisfies the ideal steering relation, causing mismatch between actual and theoretical steering vectors. This results in angle spectrum peak shift and defocusing. Pitch and roll directly change array orientation, affecting angle estimation and focusing. Yaw additionally causes different chirps to have different angular references, destroying long-time coherent integration stability. Thus, spatial compensation must restore array manifold consistency. I use IMU attitude data for element-level phase compensation; for yaw, I map angle estimates from each chirp to a unified global coordinate to ensure stable accumulation.
IV. Proposed Joint Spatiotemporal Compensation Method
A. Rotation About X/Y Axes
Rotation about the Y-axis causes target azimuth to change while elevation remains constant, and introduces radial velocity. Consider rotation by \(\Delta\beta\) about the Y-axis. The actual angle of arrival becomes \(\theta + \Delta\beta\). The phase difference becomes \(\Delta\phi’ = 2\pi d \sin(\theta + \Delta\beta)/\lambda\). To compensate, I define a per-element phase correction factor:
$$
\Delta\phi_{comp} = -\frac{2\pi}{\lambda} m d \cdot \Delta\beta
$$
The compensated signal is:
$$
S'(n,m) = S(n,m) \cdot \exp\left( j \frac{2\pi}{\lambda} m d \Delta\beta \right)
$$
The residual error due to small-angle approximation is:
$$
E = \frac{2\pi d}{\lambda} \left[ \sin\theta(\cos\Delta\beta -1) + \cos\theta \sin\Delta\beta – \Delta\beta \right]
$$
| Target Azimuth (°) | Residual Error (°) | Rayleigh Resolution (°) |
|---|---|---|
| 0 | 0.000 | 2.5 |
| 20 | 0.012 | 2.5 |
| 40 | 0.035 | 2.5 |
Errors are well below the system’s Rayleigh resolution. After angle compensation, the target is corrected to the radar coordinate system. However, rotation also induces instantaneous radial velocity. I use IMU data to compute the radar’s velocity:
$$
\mathbf{v}_{radar}(t) = \mathbf{v}_{IMU}(t) + \boldsymbol{\omega}_{IMU}(t) \times \mathbf{r}_{IR}
$$
Given target direction unit vector \(\mathbf{u} = [\sin\Phi, \sin\Theta, \cos\Phi\cos\Theta]^T\), the slow-time compensation phase is:
$$
\phi_{comp}(n) = \frac{4\pi}{\lambda} \left( \mathbf{v}_{radar}(nT_c) \cdot \mathbf{u} \right) n T_c
$$
Apply to all elements:
$$
S_{corr}(n,m,l) = S(n,m,l) \cdot \exp\left( -j \phi_{comp}(n) \right)
$$
B. Rotation About Z-Axis (Yaw)
Yaw motion rotates the array in the horizontal plane. For a target at global direction \(\mathbf{k}_G = [k_x, k_y, k_z]^T\), the instantaneous wavenumber vector in the body frame is:
$$
\mathbf{k}_B(n) = \mathbf{R}_z(\gamma_n) \cdot \mathbf{k}_G
$$
where \(\gamma_n\) is the yaw angle. The phase at element \((m,l)\) is:
$$
\Phi(m,l,n) = \frac{2\pi}{\lambda} \left( u'(n) x_m + v'(n) y_l + w L \right)
$$
with \(u'(n) = k_x \cos\gamma_n + k_y \sin\gamma_n\), \(v'(n) = -k_x \sin\gamma_n + k_y \cos\gamma_n\). The Z-component \(w\) remains constant. To compensate, I propose a fixed global angular grid-based chirp-wise beam projection/resampling strategy:
- Construct a fixed global angular grid \(\{(\Phi_i, \Theta_j)\}\) and compute corresponding global direction vectors \(\mathbf{u}_{G,ij}\).
- For each slow-time index \(n\), read yaw angle \(\gamma_n\), compute the rotation matrix \(\mathbf{R}_z(\gamma_n)\), and map the global grid to the current body frame: \(\mathbf{u}_{B,ij}(n) = \mathbf{R}_z^T(\gamma_n) \mathbf{u}_{G,ij}\).
- Compute spatial weights: \(\mathbf{W}_{ij}(n) = \exp\left( -j \frac{2\pi}{\lambda} \mathbf{P} \cdot \mathbf{u}_{B,ij}(n) \right)\), where \(\mathbf{P}\) is the antenna coordinate matrix.
- Project the array snapshot \(\mathbf{X}_n\) onto the beam: \(y_{ij}(n) = \mathbf{W}_{ij}^H(n) \mathbf{X}_n\).
This yields a slow-time sequence \(y_{ij}(n)\) for direction \((\Phi_i, \Theta_j)\) with consistent angle reference across chirps.
C. Translation Along X/Y Axes
For downward-looking China drone, targets are typically near nadir, so X/Y translations induce only small line-of-sight (LOS) projections. However, for completeness, I derive the condition under which translation affects angle estimation. The LOS direction change due to small displacement \(\Delta\mathbf{p}\) is:
$$
\Delta\mathbf{u} \approx -\frac{1}{R} (I – \mathbf{u}\mathbf{u}^T) \Delta\mathbf{p}
$$
The azimuth angle change \(\Delta\Phi \approx \frac{\Delta p_y \sin\Theta}{R \cos\Phi}\). For the translation amplitude \(\Delta p_y = 5\) cm and target range \(R=10\) m, the angle change is less than 0.16°, negligible for typical millimeter-wave radar angular resolution. Even at the far-field critical distance \(R_F = D_x D_y / \lambda\), the condition for significant angle change is:
$$
\Delta p_y > \frac{2 k D_{\max}}{\Theta}
$$
For a typical 77 GHz radar with \(D_{\max} = 10\lambda \approx 4\) cm, \(\Delta p_y > 80\) cm even for \(\Theta = 0.1\) rad—far beyond typical drone hover jitter. Therefore, I only apply Doppler compensation for X/Y translations. The compensation uses the displacement vector \(\Delta\mathbf{p}_n\) from IMU, projected onto the LOS:
$$
\Delta R(n) = \Delta\mathbf{p}_n \cdot \mathbf{u}
$$
The chirp-by-chirp phase correction is:
$$
S_{corr}(n) = S(n) \cdot \exp\left( -j \frac{4\pi}{\lambda} \Delta R(n) \right)
$$
D. Translation Along Z-Axis
Z-axis translation directly changes the slant range, especially for near-nadir targets. The LOS projection is \(\Delta R_z(n) = \Delta z_n \cos\Phi \cos\Theta\). The phase compensation factor is:
$$
\phi_z(n) = \frac{4\pi}{\lambda} \Delta z_n \cos\Phi \cos\Theta
$$
Apply to the signal after range FFT and angle estimation:
$$
S_{comp}(f, r, \Phi, \Theta) = S(f, r, \Phi, \Theta) \cdot \exp\left( -j \frac{4\pi}{\lambda} \Delta z_n \cos\Phi \cos\Theta \right)
$$
| Motion | Doppler Compensation | Spatial Compensation | Key Formula |
|---|---|---|---|
| Rotation about Y | Yes (per chirp) | Yes (element-wise) | \(\Delta\phi_{comp} = -\frac{2\pi}{\lambda} m d \Delta\beta\) |
| Rotation about X | Yes (per chirp) | Yes (element-wise) | \(\Delta\phi_{comp} = -\frac{2\pi}{\lambda} l d \Delta\alpha\) |
| Rotation about Z (yaw) | No (handled by spatial) | Global grid re-projection | \(\mathbf{W}_{ij}(n) = \exp(-j\frac{2\pi}{\lambda} \mathbf{P} \cdot \mathbf{R}_z^T \mathbf{u}_{G,ij})\) |
| Translation X/Y | Yes (per chirp) | No (negligible) | \(\Delta R = \Delta p_n \cdot \mathbf{u}\) |
| Translation Z | Yes (per chirp) | No (negligible) | \(\phi_z = \frac{4\pi}{\lambda} \Delta z_n \cos\Phi \cos\Theta\) |
V. Simulation Results
I evaluate the proposed compensation methods through MATLAB simulations. The radar parameters are set as: start frequency 77 GHz, bandwidth 1 GHz, center frequency \(f_c = 77.5\) GHz; each chirp has 256 samples at 10 MHz sampling rate; frame length \(N=256\) (512 for some translation experiments to increase Doppler resolution). The virtual array is a 20 × 20 uniform planar array (half-wavelength spacing), placed in the X-Y plane. The array layout is shown conceptually in the inserted figure. I analyze range-azimuth (RA), range-Doppler (RD), and azimuth-elevation (RE) maps under various motion conditions.
A. Rotation About Y-Axis
Scenario: Single target at range 15 m, azimuth 20°, elevation 0°. Platform oscillates about Y-axis with amplitude 10° and frequency 20 Hz. In the RA map from the 128th chirp, the uncompensated target peak appears at approximately 16° azimuth (4° bias). After applying the spatial compensation from Section IV-A, the peak returns to 20°. Doppler FFT reveals a pseudo-velocity of about 1.037 m/s without compensation, which is corrected to near zero (residual 0.148 m/s, within one Doppler bin). The mainlobe width reduces from 211.92 Hz to 154.53 Hz, a 30.4% improvement. The RD map shows clear energy concentration after compensation.
B. Rotation About Z-Axis (Yaw)
Scenario: Target at 18 m, azimuth 25°, elevation 25°. Platform yaws with amplitude 20° at 10 Hz. For a single chirp, the uncompensated RE map shows peak at (16.5°, 31°). After applying the fixed-grid re-projection method from Section IV-B, the peak shifts back to (25°, 25°). When coherently integrating over 20 frames, the uncompensated integration produces severe defocusing with peak at (30.5°, 17.5°). The compensated integration yields a sharp peak at (25°, 25°). The information entropy of the RE map (defined as \(H = -\sum p_{i,j} \log p_{i,j}\) where \(p_{i,j}\) is normalized energy) drops from 7.3 to 6.1 after compensation, indicating improved focusing.
C. Translation Along Y-Axis
Scenario: Target at 15 m, azimuth 10°, elevation 35°. Platform translates along Y-axis with amplitude 5 cm at 5 Hz. Coherent integration uses 512 chirps. The uncompensated Doppler spectrum shows a pseudo-velocity of approximately 1.552 m/s and a mainlobe width of 120.96 Hz. After applying the per-chirp phase correction from Section IV-C, the velocity reduces to 0.074 m/s residual, and the mainlobe width narrows to 78.18 Hz (35.4% improvement). The RD map shows energy concentrated at zero Doppler after compensation.
| Motion Type | Uncomp. Pseudo-Velocity (m/s) | Comp. Residual (m/s) | Uncomp. Mainlobe (Hz) | Comp. Mainlobe (Hz) | Improvement (%) |
|---|---|---|---|---|---|
| Rotation Y | 1.037 | 0.148 | 211.92 | 154.53 | 30.4 |
| Translation Y | 1.552 | 0.074 | 120.96 | 78.18 | 35.4 |
| Translation Z | 2.740 | 0.074 | 171.10 | 81.52 | 52.4 |
D. Translation Along Z-Axis
Scenario: Target at 15 m, azimuth 10°, elevation 10°. Platform translates along Z-axis with amplitude 5 cm at 5 Hz. Without compensation, the pseudo-velocity is 2.74 m/s, mainlobe width 171.10 Hz. After compensation using the method in Section IV-D, the residual velocity is 0.074 m/s and the mainlobe width reduces to 81.52 Hz (52.4% improvement). The RD map confirms effective restoration of zero-velocity focusing.
VI. Conclusion
I have presented a comprehensive motion compensation framework for FMCW millimeter-wave MIMO radar on hovering China drone platforms. By establishing a unified 6-DoF motion echo model and systematically analyzing the effects on Doppler and spatial dimensions, I derived IMU-driven compensation strategies: chirp-by-chirp phase correction for translational and rotational radial motions, element-wise spatial phase compensation for array orientation errors, and a fixed global angular grid re-projection for yaw-induced angular reference inconsistency. Simulation results demonstrate that under typical motion conditions (rotation about Y, rotation about Z, translation along Y/Z), the proposed method effectively restores angular accuracy to static-platform levels, eliminates pseudo-Doppler velocities, reduces spectral broadening by 30–52%, and recovers coherent integration focusing with reduced information entropy. These improvements are critical for enabling reliable target detection, velocity estimation, and imaging in real-world China drone surveillance missions. Future work will focus on extending the method to more complex motions with higher-order dynamics and experimental validation on actual hovering UAV platforms.