In modern drone technology, hovering unmanned aerial vehicles (UAVs) equipped with frequency-modulated continuous-wave (FMCW) millimeter-wave multiple-input multiple-output (MIMO) radar are widely used for ground surveillance, target detection, and imaging. However, during hovering, the platform inevitably experiences six-degree-of-freedom (6-DoF) motion induced by airflow disturbances and flight-control jitters. Such motion introduces nonlinear phase modulation and array manifold mismatch in the received radar echoes, leading to angular estimation bias, Doppler frequency shift, spectral broadening, and angular spectrum defocusing. To address these challenges, we establish a comprehensive echo signal model incorporating 6-DoF platform motion and propose an inertial measurement unit (IMU) data-driven joint spatial-temporal motion compensation method. This method compensates for translational and rotational effects in both the slow-time Doppler dimension and the spatial array dimension, ensuring coherent processing and stable angle estimation. Our simulation results demonstrate that the proposed approach effectively restores velocity and angle estimates to their stationary-platform equivalents, significantly mitigating motion-induced artifacts. This work advances drone technology by enabling robust radar sensing under realistic hovering conditions.
Introduction
High-resolution FMCW millimeter-wave radar offers centimeter-level range resolution and excellent velocity discrimination, making it a key sensor in security, target imaging, feature extraction, and tracking. When mounted on a UAV, the radar can achieve a larger detection area and more flexible observation angles. Nevertheless, in hovering scenarios, platform perturbations from wind and control system jitter cause non-ideal motion in six degrees of freedom. These motions corrupt the received signal by introducing phase errors that degrade the coherence of slow-time samples and distort the array manifold. Consequently, angular estimates are biased, Doppler spectra are broadened, and long-term coherent integration yields defocused angle images.
Existing motion compensation methods can be broadly classified into two categories: data-driven adaptive compensation and external-sensor-assisted compensation. Data-driven methods rely on stationary clutter, cooperative targets, or auxiliary radars to estimate platform motion and cancel its effects. However, they often require specific environmental conditions or additional hardware. External-sensor methods use IMU or GNSS measurements to pre-compensate platform motion, as commonly done in synthetic aperture radar (SAR). Although effective, these techniques either depend on stationary reference targets or operate only in the data post-processing stage, failing to recover the raw signal coherence needed for further processing.
In the context of drone technology, a unified framework that jointly compensates for Doppler and spatial distortions without relying on cooperative targets remains highly desirable. In this work, we systematically analyze the impact of 6-DoF motion on FMCW MIMO radar signals. We then propose an IMU-driven compensation chain that corrects slow-time phase errors element-by-element, compensates array manifold mismatches, and introduces a chirp-wise beam projection and resampling strategy to handle yaw-induced angle reference inconsistency. Our approach directly acts on the raw intermediate-frequency (IF) signal, preserving the signal structure for downstream processing. Extensive simulations validate that the proposed method restores angle estimation accuracy, eliminates pseudo-Doppler velocities, and sharpens the angle spectrum focus, thereby enhancing the reliability of drone-borne radar systems.
Radar Signal Model
We consider an FMCW radar that transmits a linear frequency-modulated signal. Within one processing frame, $N$ equally spaced chirps are transmitted. The transmitted signal for a single chirp can be expressed as:
$$
S_{tx}(t) = A_t \exp\left[ j2\pi \left( f_c t + \frac{1}{2} \mu t^2 \right) + \varphi_0 \right], \quad 0 \leq t < T_c
$$
where $f_c$ is the center frequency, $\mu = B/T_c$ is the chirp rate with bandwidth $B$ and chirp duration $T_c$, and $A_t$ is the amplitude. For a target at initial range $R_0$ moving with radial velocity $v$, the received signal is a delayed version of the transmitted signal. After down-conversion and neglecting the residual video phase and micro-motion terms, the IF signal for the $n$-th chirp is:
$$
S_{if}(\hat{t}, n) = A_{if} \exp\left[ j2\pi \left( \frac{2\mu R_0}{c} \hat{t} + \frac{4\pi}{\lambda} (R_0 + v n T_c) \right) \right]
$$
where $\hat{t}$ is the fast time within the chirp, $c$ is the speed of light, $\lambda = c/f_c$ is the wavelength, and $A_{if}$ is the amplitude factor. The first phase term contains the range information, extracted via a fast Fourier transform (FFT) along $\hat{t}$ to obtain the beat frequency $f_b = 2\mu R_0 / c$. The second term represents the Doppler phase accumulation across chirps due to target motion.
For angle estimation, we employ a MIMO array. Under the far-field assumption, a uniform linear array (ULA) with element spacing $d$ produces a phase difference between adjacent receivers given by $2\pi d \sin\theta / \lambda$, where $\theta$ is the angle of arrival. A spatial FFT along the array dimension then extracts the angle. For a uniform planar array (UPA) with elements on the $X$-$Y$ plane, both azimuth and elevation angles can be resolved simultaneously.
Analysis of 6-DoF Motion Effects
Doppler Dimension
The impact of platform motion on the Doppler dimension originates from the variation of the instantaneous slant range between the radar and the target over slow time. Let $R_k(n)$ be the instantaneous distance to the $k$-th target at the $n$-th chirp. The two-way propagation phase is $\phi_k(n) = -4\pi R_k(n)/\lambda$. If the relative radial motion is uniform, $R_k(n)$ is linear in $n$, and the Doppler spectrum exhibits a sharp peak. However, when $R_k(n)$ varies nonlinearly (e.g., due to platform oscillations), the phase becomes non-linear, causing spectral broadening and pseudo-Doppler shifts. Compensating for this requires removing the platform-induced slow-time phase before Doppler FFT. Rather than simply shifting the spectrum, we adopt a chirp-by-chirp phase correction to suppress both bias and broadening.
Spatial Dimension
Platform rotation alters the actual positions of the virtual array elements. The ideal virtual element location for a transmit element $m$ and receive element $l$ is $\mathbf{p}_{virt,m,l} = \mathbf{p}_{tx,m} + \mathbf{p}_{rx,l}$. When the platform undergoes rotation, the actual position becomes:
$$
\mathbf{p}_{virt,m,l}^{(n)} = \mathbf{R}_n \mathbf{p}_{virt,m,l} + \mathbf{T}_n
$$
where $\mathbf{R}_n$ is the rotation matrix and $\mathbf{T}_n$ the translation vector at chirp $n$. This mismatch between actual and theoretical steering vectors causes the angle spectrum peak to shift or defocus. Pitch and roll rotations change the array orientation, leading to angular bias; yaw rotations cause the angle reference to vary across chirps, destroying coherence in long integrations. Therefore, spatial compensation must both correct array manifold mismatches per element and align angle references across chirps.
| Degree of Freedom | Primary Effect on Doppler | Primary Effect on Angle | Compensation Approach |
|---|---|---|---|
| Roll / Pitch (X/Y axis rotation) | Pseudo-Doppler shift and broadening | Azimuth/Elevation bias | Chirp-by-chirp phase correction + element-wise spatial phase compensation |
| Yaw (Z axis rotation) | Minimal direct effect | Angle reference inconsistency across chirps, defocusing in coherent integration | Chirp-wise beam projection onto fixed global angle grid |
| Translation along X/Y axis | Pseudo-Doppler (significant only for targets with large elevation angle) | Negligible (below angular resolution for small motion amplitudes) | Chirp-by-chirp range displacement correction via velocity projection |
| Translation along Z axis | Strong pseudo-Doppler (particularly for near-nadir targets) | Negligible (below angular resolution) | Chirp-by-chirp Z-displacement correction with directional projection |
Proposed Motion Compensation Method
Our compensation framework is IMU-data-driven and operates on the raw IF signal before range compression or angle estimation. The IMU provides three-axis accelerations and angular rates. We integrate these to obtain instantaneous linear velocity, displacement, and orientation. The compensation is applied separately for translational and rotational effects.
Compensation for Rotation about X/Y Axis
When the platform rotates about the Y-axis by a small angle $\Delta\beta$, the apparent azimuth angle of a target shifts by the same amount. The phase difference between adjacent array elements becomes $2\pi d \sin(\theta + \Delta\beta)/\lambda$. To compensate, we apply an element-wise spatial phase correction:
$$
S'(n,m) = S(n,m) \cdot \exp\left(-j \frac{2\pi}{\lambda} m d \Delta\beta\right)
$$
where $m$ is the element index along the X-axis. For a target with actual azimuth $\theta$, this correction restores the phase difference to $2\pi d \sin\theta/\lambda$, provided $\Delta\beta$ is small. The residual angle error due to the approximation $\sin(\theta+\Delta\beta) \approx \sin\theta + \Delta\beta\cos\theta$ is negligible compared to the Rayleigh resolution.
Simultaneously, the rotation induces a radial velocity component. The instantaneous line-of-sight velocity of the radar phase center due to rotation is computed from the IMU angular velocity $\boldsymbol{\omega}_{IMU}(t)$ and the lever arm $\mathbf{r}_{IR}$ (vector from UAV center to radar phase center):
$$
\mathbf{v}_{radar}(t) = \boldsymbol{\omega}_{IMU}(t) \times \mathbf{r}_{IR}
$$
Projecting this velocity onto the target direction unit vector $\mathbf{u}$ yields the radial velocity $v_r(t)$. The slow-time phase correction for the $n$-th chirp is then:
$$
\phi_{comp}(n) = \exp\left( -j \frac{4\pi}{\lambda} v_r(nT_c) n T_c \right)
$$
This removes the pseudo-Doppler shift and suppresses spectral broadening. The same principle applies to rotation about the X-axis, with the spatial correction applied along the Y-axis elements.
Compensation for Yaw (Rotation about Z Axis)
Yaw rotation changes the global orientation of the array without altering the nadir direction. For a target at global angles $(\Phi,\Theta)$, the steering vector at chirp $n$ is modulated by the yaw angle $\gamma(n)$. We propose a chirp-wise beam projection and resampling strategy to align all slow-time samples to a fixed global angular grid.
First, define a fixed global angle grid $\{(\Phi_i,\Theta_j)\}$. For each chirp $n$, compute the rotation matrix $\mathbf{R}_z(\gamma_n)$. Map the global direction vector $\mathbf{u}_{G,ij}$ to the body frame: $\mathbf{u}_{B,ij}^{(n)} = \mathbf{R}_z^T(\gamma_n) \mathbf{u}_{G,ij}$. Then calculate the beamforming 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 $\mathbf{P} \in \mathbb{R}^{3 \times N_{ant}}$ is the matrix of antenna coordinates. The spatial snapshot at the target range gate is $\mathbf{X}_n$. The chirp-wise projection output is:
$$
y_{n,ij} = \mathbf{W}_{n,ij}^H \mathbf{X}_n
$$
This produces a slow-time sequence for each global angle bin that is free from yaw-induced reference wandering. Coherent accumulation over chirps then reinforces the true target angle and suppresses defocusing.
Compensation for Translation along X/Y Axis
For small translational motions (e.g., 5 cm amplitude) common in drone technology, the change in line-of-sight direction is well below the angular resolution of typical millimeter-wave arrays except for very close ranges or extremely large elevation angles. Therefore, we only compensate for the Doppler effect. The displacement vector $\Delta\mathbf{p}_n$ is obtained by integrating the IMU velocity. The range change projected onto the target direction is $\Delta R_n = \Delta\mathbf{p}_n \cdot \mathbf{u}$. The slow-time phase correction factor is:
$$
\phi_{trans}(n) = \exp\left( -j \frac{4\pi}{\lambda} \Delta R_n \right)
$$
This eliminates the pseudo-Doppler shift and narrows the spectral peak.
Compensation for Translation along Z Axis
Vertical translation primarily affects targets near the nadir. The displacement $\Delta z_n$ is integrated from IMU vertical acceleration. The equivalent range change is $\Delta R_n = \Delta z_n \cos\Phi \cos\Theta$. The corresponding phase correction is applied chirp-by-chirp:
$$
\phi_Z(n) = \exp\left( -j \frac{4\pi}{\lambda} \Delta z_n \cos\Phi \cos\Theta \right)
$$
After angle estimation, this correction is applied per target direction. For broadside targets ($\Phi=\Theta=0$), $\cos\Phi\cos\Theta=1$, giving maximum effect.
Simulation Results
We conducted numerical simulations to validate each compensation module. The radar parameters are listed in Table 2. The virtual array is a 20×20 UPA with half-wavelength spacing, located on the $X$-$Y$ plane. Targets are placed at various ranges and angles. For each degree of freedom, we compare compensated and uncompensated results in terms of Range-Angle (RA) maps, Range-Doppler (RD) maps, and Azimuth-Elevation (RE) maps.
| Parameter | Value |
|---|---|
| Center frequency | 77 GHz |
| Bandwidth | 1 GHz |
| Chirp duration $T_c$ | 60 μs |
| Sampling rate | 10 MHz |
| Number of chirps per frame | 256 (or 512 for high Doppler resolution) |
| Array size (virtual) | 20 × 20, half-wavelength spacing |
Rotation about Y-Axis
A single static target at range 15 m, azimuth 20°, elevation 0°. The platform oscillates about the Y-axis with amplitude 10° at 20 Hz. The uncompensated RA map (from the 128th chirp) shows the angle peak shifted to approximately 16° (4° error). After spatial compensation, the peak returns to 20°. The Doppler spectrum reveals a pseudo-velocity of 1.037 m/s before compensation; after slow-time phase correction, the peak moves to 0.074 m/s (within one Doppler bin). The mainlobe width narrows from 211.92 Hz to 154.53 Hz, a 30.4% improvement. The RD map confirms clearer focusing.
Rotation about Z-Axis (Yaw)
A static target at range 18 m, azimuth 25°, elevation 25°. The platform yaws with amplitude 20° at 10 Hz. For a single chirp, the uncompensated RE map shows the peak at about (16.5°,31°). After applying the beam projection/resampling method, the peak returns to (25°,25°). Coherent accumulation over 20 frames without compensation results in severe defocusing (information entropy 7.3). With compensation, the entropy drops to 6.1 (a reduction of 1.2), indicating much sharper focus. The angle spectrum peak is restored to the correct location.
Translation along Y-Axis
Target at range 15 m, azimuth 10°, elevation 35°. Platform translates along Y-axis with 5 cm amplitude at 5 Hz. Frame length extended to 512 chirps. Uncompensated Doppler spectrum shows a pseudo-velocity of 1.552 m/s and mainlobe width 120.96 Hz. After compensation, the peak shifts to 0.074 m/s and mainlobe width reduces to 78.18 Hz (35.4% improvement). The RD map demonstrates clear peak concentration at zero velocity.
Translation along Z-Axis
Target at range 15 m, azimuth 10°, elevation 10°. Platform translates along Z-axis with 5 cm amplitude at 5 Hz. The uncompensated Doppler spectrum exhibits a strong pseudo-velocity of 2.74 m/s and width of 171.10 Hz. Compensation brings the peak to 0.074 m/s and reduces width to 81.52 Hz (52.4% improvement). The RD map verifies energy concentration.
| Motion Type | Pseudo-Velocity Before (m/s) | Pseudo-Velocity After (m/s) | Mainlobe Width Before (Hz) | Mainlobe Width After (Hz) | Width Improvement (%) |
|---|---|---|---|---|---|
| Rotation about Y-axis | 1.037 | 0.074 | 211.92 | 154.53 | 30.4% |
| Translation along Y-axis | 1.552 | 0.074 | 120.96 | 78.18 | 35.4% |
| Translation along Z-axis | 2.740 | 0.074 | 171.10 | 81.52 | 52.4% |
Conclusion
We have presented a comprehensive IMU-driven joint spatial-temporal motion compensation method for FMCW millimeter-wave MIMO radar mounted on a hovering UAV. By modeling the 6-DoF platform motion and analyzing its distinct effects on Doppler and spatial dimensions, we proposed tailored compensation strategies: chirp-by-chirp phase correction for translational and rotational Doppler errors, element-wise spatial phase compensation for angular bias, and a novel chirp-wise beam projection/resampling technique for yaw-induced angle reference inconsistency. Simulations demonstrate that our method effectively restores angle estimates to their stationary-platform values, eliminates pseudo-Doppler velocities (reducing them to within a single Doppler bin), and significantly narrows spectral broadening (improvements of 30–52%). The yaw compensation reduces information entropy by 1.2, enabling stable coherent integration over multiple frames. These results confirm that the proposed approach enhances the reliability and accuracy of drone-borne radar sensing in realistic hovering conditions, contributing to the advancement of drone technology for surveillance, imaging, and target detection applications. Future work will focus on integrating the compensation chain with real-time implementations and extending it to multipath and dense clutter environments.