In recent years, the global drone industry has rapidly developed, with consumer-grade quadrotor drones becoming increasingly prevalent. However, these quadrotor drones pose potential threats to national and public security, especially near critical infrastructure such as airports and nuclear power plants. Effective detection and identification of quadrotor drones are essential, and radar is a key technological means for this purpose. Although specialized drones surveillance radars have emerged, their performance in complex environments still needs improvement. Consumer-grade quadrotor drones, characterized by their low altitude, small size, and slow speed, are challenging to detect and easily confused with birds or wind-blown trees. The micro-Doppler features, which reflect the internal local motion of targets, have become a crucial technique for radar-based detection and discrimination of quadrotor drones, as the rotation of their rotors differs from bird wing flapping or tree motion. Research on radar detection and classification of quadrotor drones requires extensive test data, particularly for classifier training in classification studies. However, measured data are limited in covering various quadrotor drone models and motion states, and parameters are difficult to control. Therefore, radar echo simulation for quadrotor drones is a vital tool for advancing research in drone radar detection and identification.
Existing radar echo simulation methods for rotor drones often model the rotating blades and drone body separately, then superimpose them, without considering the correlation between the drone body velocity and rotor rotation speeds. This approach is inconsistent with real-world scenarios and fails to fully capture the motion characteristics of quadrotor drones. In this work, I propose a novel radar echo simulation method for quadrotor drones that integrates the drone body motion with rotor rotation speeds by leveraging flight control characteristics. This method can provide accurate, parameter-controllable, and abundant test data with rich micro-Doppler features, supplementing measured data and addressing issues of low coverage in drone types and motion states for training samples in drone identification.
My approach involves three main steps. First, based on the parameters required for radar echo simulation, I develop a simplified motion model that associates the drone body speed with rotor rotation speeds using the flight control principles of quadrotor drones. Second, I derive and present a radar echo signal model for quadrotor drones under linear frequency modulated (LFM) radar, considering rotor tilt. Finally, I combine the simplified motion model with the echo signal model using coordinate system transformations. The simulation results demonstrate that the time-frequency analysis of the simulated echoes reflects micro-Doppler features linked to both the drone body velocity and rotor rotation speeds, highlighting the necessity of jointly considering these factors in radar echo simulation for varying drone motion states.
The layout of quadrotor drones typically includes “plus” and “X” configurations, with the “X” configuration being more common in consumer-grade quadrotor drones. In this configuration, diagonally opposite rotors rotate in the same direction, while adjacent rotors rotate in opposite directions to cancel out reactive torque. In hover mode, all four rotors rotate at the same speed. To control the quadrotor drone, the rotation speeds of the four rotors are adjusted to change its motion state. For instance, increasing or decreasing all rotor speeds simultaneously controls ascent or descent. By making the speeds of rotors 1 and 2 greater than those of rotors 3 and 4, the quadrotor drone pitches forward, generating forward thrust; a larger pitch angle results in greater forward force. Similarly, backward and lateral flights are controlled. Yaw rotation is achieved by making the speeds of diagonally opposite rotors 1 and 3 greater or less than those of rotors 2 and 4.
Modern quadrotor drones incorporate systematic flight control algorithms to maintain stable motion states. The remote controller typically has two joysticks controlling four channels: ascent/descent, yaw rotation, pitch motion (forward/backward), and roll motion (left/right). The common flight control process involves: (1) using the remote controller to set desired states (total thrust, pitch angle, roll angle, yaw angular velocity); (2) computing desired total thrust and moments for attitude control via flight control algorithms; (3) distributing these to individual rotor speeds through control allocation; (4) converting rotor speeds to motor throttle commands; (5) controlling actual rotor speeds with throttle commands; (6) generating actual total thrust and moments from rotor speeds; (7) controlling acceleration, velocity, angular acceleration, and angular velocity with thrust and moments; and (8) determining position and attitude from these kinematic quantities.
This control process shows that rotor speeds are the direct output of flight control, closely linked to the quadrotor drone’s motion state parameters like velocity and acceleration. For radar echo simulation, I simplify this process with the following assumptions: (1) The attitude change of the quadrotor drone is instantaneous, as the controller rapidly achieves desired pitch, roll, and yaw rates, with stable attitude thereafter. (2) Disturbances from environmental factors like wind and fine-tuning by the flight control system are neglected. (3) Actual values for pitch angle, roll angle, yaw rate, total thrust, moments, and rotor speeds are assumed equal to desired values, allowing direct use of remote controller commands for dynamics modeling. (4) At steady-state attitude, the net moment on the quadrotor drone is zero, with rotor moments proportional to remote controller commands (desired pitch angle, roll angle, and yaw rate). Thus, I derive a simplified motion model that associates rotor speeds with the quadrotor drone body motion state, as shown in the conceptual diagram.
To formalize, I define a coordinate system attached to the quadrotor drone body, with the origin at its center. The attitude control model is simplified as:
$$
\tau_x = C_x \phi \\
\tau_y = C_y \theta \\
\tau_z = C_z \omega_\psi
$$
where $\phi$, $\theta$, and $\omega_\psi$ are the desired roll angle, pitch angle, and yaw rate, respectively; $\tau_x$, $\tau_y$, and $\tau_z$ are the rotor moments about the roll, pitch, and yaw axes; and $C_x$, $C_y$, $C_z$ are proportionality constants.
The total thrust $f$ and moments $\tau_x$, $\tau_y$, $\tau_z$ for a quadrotor drone are related to the rotor rotation speeds $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ by:
$$
\begin{bmatrix} f \\ \tau_x \\ \tau_y \\ \tau_z \end{bmatrix} =
\begin{bmatrix}
c_T & c_T & c_T & c_T \\
\frac{\sqrt{2}}{2} d c_T & -\frac{\sqrt{2}}{2} d c_T & -\frac{\sqrt{2}}{2} d c_T & \frac{\sqrt{2}}{2} d c_T \\
\frac{\sqrt{2}}{2} d c_T & \frac{\sqrt{2}}{2} d c_T & -\frac{\sqrt{2}}{2} d c_T & -\frac{\sqrt{2}}{2} d c_T \\
c_M & -c_M & c_M & -c_M
\end{bmatrix}
\begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix} = M \begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix}
$$
where $c_T$ and $c_M$ are constants related to rotor thrust and moment, respectively; $d$ is the distance from the rotor center to the drone body center; and $M$ is the control allocation matrix. For the “X” configuration quadrotor drone, the effective moment arm is $(\sqrt{2}/2)d$. The control allocation model is derived as:
$$
\begin{bmatrix} \omega_1^2 \\ \omega_2^2 \\ \omega_3^2 \\ \omega_4^2 \end{bmatrix} = M^{-1} \begin{bmatrix} f \\ \tau_x \\ \tau_y \\ \tau_z \end{bmatrix} = \frac{1}{4} \begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1 \\
1 & 1 & -1 & -1
\end{bmatrix} \begin{bmatrix} f / c_T \\ (\sqrt{2} \tau_x) / (d c_T) \\ (\sqrt{2} \tau_y) / (d c_T) \\ \tau_z / c_M \end{bmatrix}
$$
Assuming instantaneous attitude changes and stable attitude, the dynamics model focuses on translational motion, neglecting rotational effects from body rotation. The dynamics in the ground coordinate system are:
$$
a_x = (f_x – f_{rx}) / m \\
a_y = (f_y – f_{ry}) / m \\
a_z = (f_z – f_{rz}) / m – g
$$
where $a_x$, $a_y$, $a_z$ are accelerations along the ground coordinate axes; $f_x$, $f_y$, $f_z$ are components of rotor thrust in these axes; $f_{rx}$, $f_{ry}$, $f_{rz}$ are air drag components; $m$ is the quadrotor drone mass; and $g$ is gravitational acceleration. Using the transformation matrix from body to ground coordinates, the thrust components are:
$$
\begin{bmatrix} f_x \\ f_y \\ f_z \end{bmatrix} = f \begin{bmatrix}
\cos \phi \sin \theta \cos \psi + \sin \phi \sin \psi \\
\cos \phi \sin \theta \sin \psi – \sin \phi \cos \psi \\
\cos \phi \cos \theta
\end{bmatrix}
$$
From acceleration, the velocity at any time $t$ can be computed, and position can be obtained via integration. For radar echo simulation, I compute velocity first and later determine position after coordinate transformation to the radar system.
For general quadrotor drone motion, I sample time and compute rotor moments and accelerations at each sample, then determine rotor speeds from total thrust and moments, and update body velocity from acceleration. For common straight-line flights such as forward/backward, left/right level flight, or oblique ascent/descent, the model can be further simplified.
For forward level flight of the quadrotor drone, I set roll angle $\phi = 0$ and yaw angle $\psi = 0$. Air drag acts only in the forward direction (X-axis). The dynamics simplify to:
$$
a_x = (f \sin \theta – f_r) / m \\
a_y = 0 \\
a_z = (f \cos \theta) / m – g = 0
$$
where $f_r$ is the total air drag. The vertical component of thrust balances gravity. Air drag is modeled as $f_r = K \nu^2$, where $\nu$ is the quadrotor drone forward speed and $K$ is an air drag coefficient. Thus, the forward acceleration is:
$$
a_x = \frac{d\nu}{dt} = g \tan \theta – \frac{K \nu^2}{m}
$$
Solving this differential equation, with initial speed $\nu_0$ at $t=0$, the speed at time $t$ is:
$$
\nu = \begin{cases}
\sqrt{\frac{mg \tan \theta}{K}} \tanh \left( \sqrt{\frac{g K \tan \theta}{m}} t + \text{arctanh} \left( \frac{\nu_0 \sqrt{K}}{\sqrt{mg \tan \theta}} \right) \right), & \text{if thrust > drag at } t=0 \\
\sqrt{\frac{mg \tan \theta}{K}} \left[ \tanh \left( \sqrt{\frac{g K \tan \theta}{m}} t + \text{arctanh} \left( \frac{\sqrt{mg \tan \theta}}{\nu_0 \sqrt{K}} \right) \right) \right]^{-1}, & \text{if thrust < drag at } t=0
\end{cases}
$$
The maximum speed at pitch angle $\theta$ is:
$$
\nu_{\text{max}}(\theta) = \sqrt{\frac{mg \tan \theta}{K}}
$$
The air drag coefficient $K$ can be calculated from aerodynamic formulas or using known maximum speed-pitch angle relationships from quadrotor drone manuals.
For oblique ascent or descent of the quadrotor drone from hover, vertical force balance does not hold. Considering motion starting from hover ($\nu_0 = 0$), the acceleration direction has an elevation angle $\xi$ relative to horizontal:
$$
\xi = \arctan \left( \frac{f \cos \theta – mg}{f \sin \theta} \right)
$$
The acceleration magnitude in this direction is:
$$
a_\xi = \frac{F – K \nu^2}{m}
$$
where $F = \sqrt{(f \cos \theta – mg)^2 + (f \sin \theta)^2}$ is the magnitude of net force excluding air drag. Solving the differential equation gives the speed at time $t$:
$$
\nu_\xi = \sqrt{\frac{F}{K}} \tanh \left( \sqrt{\frac{K F}{m^2}} t \right)
$$
with maximum speed:
$$
\nu_{\xi \text{max}}(\theta, f) = \sqrt{F / K}
$$
Next, I derive the radar echo signal model for the quadrotor drone. For an LFM pulse radar, the transmitted signal for the $m$-th pulse is:
$$
p(\hat{t}) = \text{rect} \left( \frac{\hat{t}}{T} \right) \exp \left\{ j 2 \pi \left( f_c \hat{t} + \frac{1}{2} k \hat{t}^2 \right) \right\}
$$
where $T$ is pulse width, $f_c$ is carrier frequency, $\hat{t}$ is fast time, $k$ is chirp rate, and $\text{rect}(u) = 1$ for $|u| \leq 1/2$, else $0$. Under the stop-and-go assumption, after demodulation and matched filtering, the echo from a scatterer $P$ is:
$$
s_P(\hat{t}, t_m) = \sigma_P T \frac{\sin[\pi k T (\hat{t} – 2 R_P(t_m)/c)]}{\pi k T (\hat{t} – 2 R_P(t_m)/c)} \exp \left\{ -j \frac{4 \pi R_P(t_m)}{\lambda} \right\}
$$
where $t_m = m T_{\text{PRI}}$ is slow time, $T_{\text{PRI}}$ is pulse repetition interval, $\sigma_P$ is scattering coefficient, $R_P(t_m)$ is distance from scatterer to radar at $t_m$, $c$ is light speed, and $\lambda$ is wavelength.
I model a rotating blade as a rigid line with length $L$. Summing echoes from all scatterers on the blade, the baseband signal for a single blade after matched filtering is approximated as:
$$
s_{\text{blade}}(\hat{t}, t_m) = \sigma T \frac{\sin[\pi k T (\hat{t} – 2 R_0 / c)]}{\pi k T (\hat{t} – 2 R_0 / c)} L \cdot \text{Sa} \left[ \frac{2 \pi}{\lambda} L \cos \beta \cos(\theta_0 + \omega t_m – \alpha) \right] \cdot \exp \left\{ -j \frac{4 \pi}{\lambda} \left( R_0 + \frac{L}{2} \cos \beta \cos(\theta_0 + \omega t_m – \alpha) \right) \right\} \exp \left\{ -j \frac{4 \pi}{\lambda} \nu t_m \right\}
$$
where $\text{Sa}(x) = \sin x / x$, $\sigma$ is blade scattering coefficient, $R_0$ is distance from blade rotation center to radar, $\omega$ is blade rotation speed, $\beta$ and $\alpha$ are elevation and azimuth angles of the rotation center relative to radar, $\theta_0$ is initial blade phase at $t=0$, and $\nu$ is radial velocity of the quadrotor drone body relative to radar (positive away from radar). If the rotor plane is tilted relative to horizontal, $\beta$ and $\alpha$ are adjusted accordingly.
For a rotor with $N$ blades, initial phases are $\theta_n = \theta_0 + 2\pi n / N$ for $n=0,1,\dots,N-1$. For a quadrotor drone, each rotor typically has two blades ($N=2$), with initial phases $\theta_0$ and $\theta_0 + \pi$. Summing echoes from both blades, the rotor echo is:
$$
s_{\text{rotor}}(\hat{t}, t_m) = 2 \sigma T \frac{\sin[\pi k T (\hat{t} – 2 R_0 / c)]}{\pi k T (\hat{t} – 2 R_0 / c)} L \cdot \text{Sa} \left[ \frac{2 \pi}{\lambda} L \cos \beta \cos(\theta_0 + \omega t_m – \alpha) \right] \cdot \exp \left\{ -j \frac{4 \pi}{\lambda} R_0 \right\} \exp \left\{ -j \frac{4 \pi}{\lambda} \nu t_m \right\} \cosh \left[ j \frac{2 \pi L}{\lambda} \cos \beta \cos(\theta_0 + \omega t_m – \alpha) \right]
$$
where $\cosh(\cdot)$ is hyperbolic cosine.
Let the rotation speeds, initial phases, and distances to radar for the four rotors of the quadrotor drone be $\omega_i$, $\theta_i$, $R_i$ for $i=1,2,3,4$. Let $R_d$ be the distance from the projection point $O$ of the drone center onto the rotor plane to the radar. Since deviations between rotor centers and point $O$ are small, I ignore them in simulation. Modeling the drone body as a point scatterer with fixed scattering coefficient $\sigma_d$, the total echo from the quadrotor drone is the sum of body and rotor echoes:
$$
s_{\text{drone}}(\hat{t}, t_m) = T \frac{\sin[\pi k T (\hat{t} – 2 R_d / c)]}{\pi k T (\hat{t} – 2 R_d / c)} \exp \left\{ -j \frac{4 \pi}{\lambda} R_d \right\} \exp \left\{ -j \frac{4 \pi}{\lambda} \nu t_m \right\} \cdot \left\{ \sigma_d + \sum_{i=1}^{4} 2 \sigma L \text{Sa} \left[ \frac{2 \pi}{\lambda} L \cos \beta \cos(\theta_i + \omega_i t_m – \alpha) \right] \cosh \left[ j \frac{2 \pi L}{\lambda} \cos \beta \cos(\theta_i + \omega_i t_m – \alpha) \right] \right\}
$$
where $\beta$ and $\alpha$ are elevation and azimuth angles of point $O$ relative to radar. If rotor planes are tilted, adjustments are made as described.
To combine the motion model with the echo model, I transform velocities from ground to radar coordinates. Let the radar coordinate system have a horizontal偏转角 $\gamma$ relative to the ground system. The velocity transformation is:
$$
\begin{bmatrix} \nu_{rx} \\ \nu_{ry} \\ \nu_{rz} \end{bmatrix} = \begin{bmatrix} \cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \nu_x \\ \nu_y \\ \nu_z \end{bmatrix}
$$
The radial velocity is:
$$
\nu = \nu_{rx} \cos \beta \cos \alpha + \nu_{ry} \cos \beta \sin \alpha + \nu_{rz} \sin \beta
$$
For experimental validation, I simulate a quadrotor drone based on parameters similar to the DJI Phantom 4 Pro, a common consumer-grade quadrotor drone. The parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Number of blades per rotor | 2 |
| Blade length $L$ | 120 mm |
| Drone wheelbase $2d$ | 350 mm |
| Drone mass $m$ | 1.388 kg |
| Thrust coefficient $c_T$ | 9.4461 × 10^{-4} N/rps² |
| Pitch moment coefficient $C_y$ | 2 N·m/rad |
| Air drag coefficient $K$ | 0.0334 N/(m/s)² |
| Blade scattering coefficient $\sigma$ | 1 |
| Body scattering coefficient $\sigma_d$ | 0.76 |
The radar parameters are based on a typical drone surveillance radar using LFM pulses, as shown in Table 2.
| Parameter | Value |
|---|---|
| RF frequency | 5.5 GHz |
| Bandwidth | 20 MHz |
| Sampling rate | 80 MHz |
| Pulse width $T$ | 20 μs |
| Pulse repetition interval $T_{\text{PRI}}$ | 60 μs |
| Coherent processing interval | 192 ms |
In the level flight experiment, the quadrotor drone starts from hover at position (4000, 0, 100) m in radar coordinates, then flies away from the radar along the X-axis. The pitch angles are set to 10°, 30°, and 20° during time intervals 0–5 s, 5–10 s, and 10–15 s, respectively. Using my simplified motion model, I compute the motion parameters and rotor speeds over time. The results show that the quadrotor drone’s motion state varies with pitch angle, affecting rotor speeds and body velocity.
The simulated radar echoes are analyzed using time-frequency methods, specifically Short-Time Fourier Transform (STFT) and Reassigned Spectrogram (RSP), to examine micro-Doppler features. The time-frequency plots reveal two sets of flashes (vertical lines) corresponding to the two pairs of rotors with different rotation speeds. The repetition periods of these flashes differ due to varying rotor speeds. At 5 s and 10 s, when the motion state changes, the rotor speeds adjust, causing shifts in the maximum micro-Doppler frequencies. For instance, at 5 s, rotors 1 and 2 change from 66.35 rps to 79.96 rps, with blade tip maximum Doppler frequency changing from 1799 Hz to 1888 Hz; rotors 3 and 4 change from 53.94 rps to 43.78 rps, with frequency changing from 1463 Hz to 1034 Hz. At 10 s, further adjustments occur. The horizontal lines in the plots represent the quadrotor drone body scatter, with positions varying according to body velocity.
These results demonstrate that even during simple forward level flight, the micro-Doppler features of the quadrotor drone can differ significantly due to changes in motion state controlled by pitch angle. The correlation between body velocity and rotor speeds is evident in the time-frequency patterns.
I also analyze the relationship between pitch angle and motion parameters during level flight at 2.5 s. As the pitch angle increases from 0° to 42° (the maximum for this quadrotor drone model), the rotor speeds, body velocity, and maximum micro-Doppler frequencies change non-linearly, emphasizing the interdependence in quadrotor drone dynamics.
Furthermore, I investigate the effect of radar operating frequency on the micro-Doppler features of the quadrotor drone. The maximum micro-Doppler frequency scales linearly with frequency, but the slope varies with rotor speed. For example, at different frequencies, the blade tip micro-Doppler frequencies shift proportionally, highlighting the importance of frequency selection in radar design for drone detection.
In the oblique flight experiment, the quadrotor drone starts from hover and ascends along the X-axis with an elevation angle of 35° relative to horizontal. Pitch angles are set similarly to the level flight case. The time-frequency analysis shows micro-Doppler flashes with higher maximum frequencies and shorter repetition periods compared to level flight, due to greater rotor thrust required for ascent. At 5 s, rotors 1 and 2 change from 70.12 rps to 96.01 rps, with frequencies shifting from 1901 Hz to 2263 Hz; rotors 3 and 4 change from 58.51 rps to 68.83 rps, with frequencies from 1586 Hz to 1622 Hz. At 7.5 s, slight variations occur due to changes in relative elevation angle to the radar.
The integration of flight control characteristics into the radar echo simulation for quadrotor drones effectively captures the correlation between body motion and rotor dynamics. This approach generates realistic micro-Doppler signatures that vary with flight states, providing valuable data for radar algorithm development. The use of simplified models balances accuracy and computational efficiency, making it suitable for large-scale simulation tasks.
In conclusion, my proposed method for quadrotor drone radar echo simulation, which leverages flight control principles to link body velocity and rotor speeds, offers a more accurate representation of real-world quadrotor drone behavior compared to existing methods. The simulated echoes exhibit rich micro-Doppler features that are essential for radar-based detection and classification of quadrotor drones. This work underscores the importance of considering the coupled dynamics of quadrotor drones in radar simulation, especially for scenarios involving motion state transitions. Future work could extend this approach to other drone configurations, incorporate more detailed aerodynamic effects, and validate simulations with measured data from various quadrotor drone models.