## Abstract

Triangular frequency-modulated continuous-wave (FMCW) laser radars (ladars) are extremely sensitive to vibration errors. An FMCW ladar 3D imaging system may suffer from severe vibrations and can use only one-period echoes for the ranging of each observation spot; consequently, it can provide only few measurement results. These vibrations may cause large errors because conventional vibration compensation methods are ineffective when applied to fast disturbances with limited measurement results. To solve this problem, we analyze the influence of vibrations on FMCW ladar ranging and propose a vibration compensation method based on an instantaneous ranging model for one-period triangular FMCW ladar signals. We first use a synchrosqueezing wavelet transform to extract time-frequency curves of the up- and down-dechirp signals and then build an instantaneous ranging model that can characterize local vibration errors. Based on the instantaneous ranges, we remove the disturbance vibration errors by taking the mean values of the instantaneous ranges and obtain the target range by using the triangular relations of the up and down observations. Experiments based on synthetic and real data verify the effectiveness of the proposed method and its superiority over the three-point method and Doppler shift method in compensating for vibrations with different frequencies and noise levels.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The triangular frequency-modulation continuous-wave (FMCW) laser radar (ladar) is a new coherent detection system that has broad applicability in long-range detection [1–4]. However, because the laser wavelength is on the order of microns, the ladar system is very sensitive to vibrations generated by engines and other mechanical equipment. The Doppler frequency shift introduced by micron-scale vibration displacements can yield an increased offset error in the measurement range, which eventually reduces the accuracy of target recognition and three-dimensional (3D) imaging. Therefore, vibration compensation methods are highly important for FMCW ladar 3D imaging systems.

Vibration compensation methods are mainly used in frequency-scanning interferometry (FSI), which has a similar principle to FMCW ladars in the application of ranging. Broadly, there are two main approaches to compensate for vibration errors. The first approach eliminates vibration errors by adding additional hardware such as another laser or speed detectors. Seiichi et al. proposed an FSI system immune to vibrations by employing a pair of laser diodes, the frequencies of which are swept in opposite directions with equal magnitudes [5]. Krause et al. added a single-frequency laser to an FMCW ladar system and eliminated vibration errors by comprehensively calculating the echoes generated by the two lasers [6]. Lu et al. established a novel absolute range measurement system that incorporated a basic range measurement system and a laser Doppler velocimeter to compensate for the Doppler effect introduced by vibration errors [7]. The above methods can compensate for vibration errors by using the symmetric relations of up and down frequency sweeps, but they increase the complexity of the ladar system, especially in an airborne platform.

The second approach first collects consecutive echoes through up and down frequency sweeps and then suppresses the vibration errors by establishing mathematical relations among these echoes. Swinkels et al. indicated that the range of a target can be measured with only one laser and proposed a three-point algorithm to eliminate vibration errors by combining subsequent measurements [8]. The three-point algorithm does not require multiple-period measurements of up and down frequency sweeps, but it performs poorly in an industrial environment with fast disturbances such as turbulence or vibration. The vibration error can be removed in one-period of a triangular FMCW by measuring the Doppler shift of the dechirp frequency for the up-chirping and down-chirping state when the vibration velocity can be assumed to be constant [9]. However, the fast vibration is time-varying, which degrades the performance of the Doppler shift method. Tao et al. proposed a range measurement method using the Kalman filter technique, which only needs a single tunable laser driven by consecutive up and down optical frequency scanning [10]. This method can predict the actual range of a target from a sufficient number of measurement ranges containing vibration errors, and multiple measurement results can be obtained in FSI by counting the interference fringes and measuring the frequency scanning range. Jia et al. applied a time-varying Kalman filter to the basic range measurement system and improved the performance of range measurement, but this system also needs a sufficient number of measurement results [11]. The above methods are good examples for vibration error compensation using measurement data without adding another laser.

However, the vibrations in signals of FMCW ladar imaging have specific features different from those in FSI. First, for vibration compensation, we usually prefer not to add additional hardware to the FMCW ladar system to avoid increasing the system complexity and hardware cost. Second, in FMCW ladar systems, especially those installed on airborne platforms, the engine and other mechanical equipment generate much more severe vibrations with frequencies of up to several hundred hertz. Finally, 3D imaging adopts a laser scanner for dynamic range measurement, in which only one-period echoes can be used for the ranging of each observation spot; thus, it provides relatively few measurement results. Some previous methods may not work well with fast disturbances such as turbulence or vibration when limited measurement results are available.

Therefore, this study aimed to compensate for severe vibrations without additional hardware for one-period FMCW ladar signals. For this purpose, we investigate the impact of vibrations on FMCW ladar ranging and propose a vibration compensation method based on an instantaneous ranging model. We use the localized characteristics of dechirp signals to extend the measurement ranges to instantaneous ranges that can characterize the local vibration errors of the ladar platform and provide sufficient measurement results to predict the actual target range. Based on the instantaneous ranges, we compensate for the vibration errors by using the symmetric relations of triangular FMCW signals. The proposed vibration compensation method is verified through numerical tests on synthetic data of point targets and 3D targets as well as real data. Additionally, the proposed method is compared with the three-point method [8] and Doppler shift method [9], which are also effective for one-period triangular FMCW signals. The proposed method has two advantages. First, it can compensate for the vibration errors without changing the system design, which prevents any increase in the system complexity and hardware cost. Second, it can characterize the local vibrations of the ladar platform. Therefore, it can compensate for not only mild but also severe vibration errors while maintaining effectiveness for one-period triangular FMCW signals.

## 2. Impact of vibrations on ladar ranging

Figure 1 schematically shows the design of FMCW coherent ladar system. The laser source is generated in two steps. We first generate a single-frequency laser signal and a microwave frequency modulation signal. Subsequently, we applied a Mach-Zehnder modulator to the laser signal by using the driven signal and obtained the FMCW laser source [12]. The tunable laser source is divided into two beams by coupler 1. One beam is transmitted by an optical antenna and reflected to the antenna upon encountering a target. The other beam passes through a delay fiber and then interferes with the received signal through coupler 2, following which the dechirp signal is coherently acquired by the measurement detector (D_{M}). The dechirp signals are sampled by the digital acquisition card (DAQ), following which the data are imported into a computer for subsequent signal processing [13–15].

The ideal transmitted signal of the FMCW ladar is expressed as follows:

where ${s_t}(t)$ is the transmitted signal, ${f_c}$ is the initial frequency of the frequency-swept laser, $w(t)$ is the signal envelope, and*K*is the frequency modulation rate.

The reflected echo at range ${R_\textrm{0}}$ is expressed as

*c*is the velocity of light.

Then, the dechirp signal can be obtained by coherently mixing the transmitted signal with the reflected echo:

By applying the Fourier transform to Eq. (4), we can obtain the target range by using the following equation:

where ${R_0}$ is the range of target and ${f_0} = K\tau$ is the ideal dechirp frequency.The above derivation assumes that the relative range between the platform and target is fixed. However, airborne platforms usually generate vibrations, which change the relative range between the platform and target. The displacement between the platform and the target is denoted as $\Delta \varepsilon$. We use the vibration velocity *v* to indicate the speed of displacement change, and a Doppler shift is introduced into the dechirp signal by the vibration velocity *v*. Assume that the vibration velocity *v* is constant. Then the dechirp signal with vibration can be expressed as [16,17]

*v*is the vibration velocity. The term $2v/\lambda$ represents the Doppler frequency shift, where $\lambda$ is the laser wavelength. Hereafter, $2v/\lambda$ will be denoted as ${f_d}$. The second-order term of Eq. (6) broadens the spectrum in the range profiles, and this term can be ignored because the FMCW ladar has a short period. Then, the instantaneous dechirp frequency with vibration $\tilde{f}$ is expressed as

Further, the measured range with vibration can be expressed as

*v*in a time interval $\Delta t$; and $\Delta {f_c} = K\Delta t$ is the difference in frequency modulation of the laser scan in the time interval $\Delta t$. Equation (8) shows that the displacement $\Delta \varepsilon$ is amplified by a factor ${f_c}/\Delta {f_c}$. The amplified errors seriously affect the ranging accuracy because the factor is usually on the order of several thousands in the laser band. Therefore, it is essential to suppress the vibration errors to improve the ranging accuracy of the coherent ladar.

## 3. Vibration compensation based on instantaneous ranging model for one-period triangular FMCW ladar signals

This section first introduces the instantaneous ranging model and vibration compensation for one-period triangular FMCW ladar signals and then summarizes the workflow of the vibration compensation method based on the instantaneous ranging model.

#### 3.1 Instantaneous ranging model

We extended the traditional measurement ranges to instantaneous ranges that can characterize the local vibration errors of the ladar platform and provide sufficient measurement results to predict the actual target range. As the synchrosqueezing wavelet transform (SST) is a time-frequency analysis method that can depict the local characteristics of dechirp signals, we use the SST method, rather than the short-time Fourier transform, to indicate the local vibration errors. The one-period triangular FMCW signal is composed of up- and down-dechirp signals. In this subsection, we regard the up-dechirp signal as an example to derive the instantaneous ranging model. First, we apply the continuous wavelet transform (CWT) method to the dechirp signal ${\tilde{s}_{if}}(t)$ and obtain the wavelet coefficients of the dechirp signal:

*b*denote the scale factor and translation factor, respectively; $\psi (t)$ is the mother wavelet function; and * represents the conjugate. The effective signal after CWT contains a wide energy band because the wavelet coefficients calculated using Eq. (9) are dispersed on the scale. To clearly detect the effective curve, we must squeeze the dispersed energy band to a more distinct one with high precision. Thus, the criterion of energy rearrangement should be established. The derivative of the wavelet coefficients is considered the criterion of energy rearrangement, and the phase transform of the dechirp signal at every moment can be calculated as [18,19]

Now, we can rearrange the wavelet coefficients from the time-scale domain to the time-frequency domain based on the obtained phase transform. Given a frequency division ${\omega _l}$ and a frequency interval $[{{\omega_l} - 0.5\Delta \omega ,{\omega_l} + 0.5\Delta \omega } ]$, the frequencies in this interval are on the verge of ${\omega _l}$ than any other frequency divisions. Therefore, we can rearrange the wavelet coefficients distributed in this interval to the exact frequency ${\omega _l}$, and the SST results can be obtained by converting ${W_f}({a,b} )$ to ${W_f}[{{\omega_f}({a,b} ),b} ]$:

Based on the extracted time-frequency curve, we obtain the instantaneous range $\tilde{R}(t)$:

#### 3.2 Vibration compensation for triangular FMCW

Figure 2 schematically shows the triangular FMCW coherent detection under a vibration environment, where the green line is the instantaneous frequency of the transmitted signal, the red solid line is the instantaneous frequency of the received signal containing vibration error, and the red dotted line is the instantaneous frequency of the received signal containing only the constant error [9]. The yellow line represents the dechirp frequencies ${F_{up}}({t_1})$ and ${F_{down}}({t_2})$ containing the constant error and disturbance error, and the blue line represents the dechirp frequencies ${f_{up}}$ and ${f_{down}}$ containing only constant error. ${t_1}$ and ${t_2}$ represent the observation time of up and down observations, respectively. Hereafter, we use the positive-frequency forms of dechirp frequencies ${F_{down}}({t_2})$ and ${f_{down}}$ that are obtained by symmetrically reversing the negative frequencies around zero frequency according to [9].

The relations of the instantaneous frequencies can be expressed as follows:

where ${f_0}$ is the ideal dechirp frequency which is positive value; ${f_d}$ is the Doppler frequency introduced by the constant velocity; and $\Delta {f_1}({t_1})$ and $\Delta {f_2}({t_2})$ are the frequency errors of up and down observations introduced by the disturbance velocity, respectively. The corresponding instantaneous ranges of up and down observations can be expressed as follows:We assume that the constant vibration velocities of up and down observation are equal owing to the short frequency sweep period, which implies that the constant error $\Delta {\varepsilon _{v0}}$ in the up observation equals that in the down observation. If the disturbance errors $\Delta {\varepsilon _{ve1}}({t_1})$ and $\Delta {\varepsilon _{ve\textrm{2}}}({t_2})$ in the up and down observation are equal as well, we can compensate for the vibration errors and estimate the target range by using Eqs. (15) and (16). However, the disturbance errors in the up and down observation are different because a half-period time delay exists between the two observations, and an estimation error occurs if we estimate the range of the target by directly adding half of ${\tilde{R}_{up}}({t_1})$ and half of ${\tilde{R}_{down}}({t_2})$ from Eqs. (15) and (16), respectively.

To estimate the target range ${R_0}$, we need to performs two steps based on the symmetric relations of the triangular FMCW. First, we must track the up and down ranges ${R_{up}}$ and ${R_{down}}$, respectively, with the constant error $\Delta {\varepsilon _{v0}}$ by removing the disturbance errors $\Delta {\varepsilon _{ve1}}({t_1})$ and $\Delta {\varepsilon _{ve\textrm{2}}}({t_2})$. Second, the constant error $\Delta {\varepsilon _{v0}}$ is removed according to the triangular relations of the up and down observation, as described below.

In the first step, we take the mean values of the up and down instantaneous ranges ${\tilde{R}_{up}}({t_1})$ and ${\tilde{R}_{down}}({t_2})$, respectively, to remove the disturbance errors and obtain ${R_{up}}$ and ${R_{down}}$:

As the vibration amplitude is on the order of micron, which is much smaller than the ladar detection limit. Therefore, the up and down ranges ${R_{up}}$ and ${R_{down}}$, respectively, within a single period can be expressed as the sum of an approximate constant value, which includes the ideal range and the vibration error caused by the constant velocity.

After removing the disturbance errors, we can obtain the range of the target ${R_0}$ by combining Eq. (17) with Eq. (18):

Thus, the vibration errors can be eliminated based on the instantaneous ranging model for triangular FMCW signals, and the actual range of the target can be estimated accurately.

#### 3.3 Workflow of the proposed method

The workflow of the proposed vibration compensation method based on the instantaneous ranging model consists of three main steps. First, we obtain the instantaneous ranges of the up- and down-dechirp signals by using the instantaneous ranging model. Second, we compensate for the disturbance vibration errors to obtain the up and down ranges containing only the constant vibration error by using the instantaneous ranging model. Finally, we use the up and down ranges of the up and down observation in a triangular FMCW period to estimate the actual range of the target. The detailed steps of the method are summarized in Table 1.

In Table 1, the subscript *j* is used to distinguish the signals, frequencies, and ranges of up and down observations, in which *j*=1 corresponds to the up observation and *j*=2 corresponds to the down observation.

## 4. Experimental analysis

We conducted several experiments to compare the proposed method with the three-point method in a mild vibration environment (subsection 4.1), severe vibration environment (subsection 4.2), and 3D imaging (subsection 4.4). Additionally, we conducted two experiments (subsection 4.3) to compare the proposed method with the three-point method [8] and Doppler shift method [9] for vibration compensation with different frequencies and different SNRs. Finally, we verified the effectiveness of the proposed method in a real environment (subsection 4.5) by using a set of real data from a simulated airborne ladar scene. In the simulations (subsections 4.1–4.4), the bandwidth and period of frequency modulation were 1 GHz and 4 ms, respectively, the frequency modulation rate is $5 \times {10^{11}}Hz/s$, and the laser wavelength was 1.55 µm.

#### 4.1 Mild vibration

In this experiment, we added a sinusoidal vibration with an amplitude of 20 µm and frequency of 30 Hz as well as Gaussian white noise with a signal-to-noise ratio (SNR) of 3 dB with respect to the ideal dechirp signal. Further, the ideal range of the target is 500 m. The ideal dechirp signal and the up-dechirp and down-dechirp signals with vibrations are shown in Figs. 3(a)–3(c), respectively. Fourier transform was applied to the dechirp signals, and the range of the target was calculated using the frequency spectra. The range profiles are shown in Fig. 3(d). The range extracted from the up-dechirp signal is much larger than the ideal range, while that extracted from down-dechirp signal is much smaller. This phenomenon is consistent with the influence of the Doppler effect on ranging, as described by Eq. (8). Next, the proposed method is used to estimate the range of the target.

First, we use CWT to transform the dechirp signals shown in Figs. 3(b)–3(c) into the time-frequency domain, and the results are shown in Figs. 4(a)–4(b), respectively. Figure 4(c) shows the CWT result for the ideal dechirp signal. The energy bands in Fig. 4 are wide and obscure, which makes it difficult to trace precise instantaneous frequency curves of the effective dechirp signals (the solid lines indicated by the arrows). Second, the energy rearrangement operation is applied to the CWT results to obtain synchrosqueezing results, which are shown in Figs. 5(a) and 5(b). The SST results shown in Fig. 5 have a higher time-frequency resolution and more distinct energy bands than the CWT results shown in Fig. 4. Moreover, noise affects the effective energy in the CWT result, whereas the SST method squeezes the random noise to a more concentrated state, allowing for clear tracing of the instantaneous frequency curves of the dechirp signals with low noise interruption.

We used the ridge detection method to extract the squeezed time-frequency curves from Fig. 5, and the instantaneous ranges of the up- and down-dechirp signals were calculated using Eq. (12). The ranges are shown in Fig. 6, in which the red and green lines represent the up and down instantaneous ranges, respectively. Subsequently, we calculated the mean values of the measurement ranges and obtained up and down mean ranges containing only constant errors. Finally, the target range was estimated to be 499.97 m by applying Eq. (19). Thus, the vibration errors caused by the Doppler effect are eliminated by the proposed method, and the estimated range is close to the ideal one (500 m).

The root-mean-square error (RMSE) and the mean value are commonly used to judge the quality of ranging measurements [21]. To calculate the RMSE and mean value achieved with the proposed method, we repeated the above experiment 200 times, and the estimated ranges are shown in Fig. 7(a). The RMSE and the mean value of the proposed method are 0.0294 m and 500.01 m, respectively. As the three-point method [8] and the Doppler shift method [9] can also compensate for the vibration error by using only one-period echo, we applied these two methods to the dechirp signals shown in Figs. 3(b) and 3(c) for comparison. We also repeated the experiment 200 times using the Doppler shift method and the three-point method, and the resulting estimated ranges are shown in Fig. 7(b) and 7(c), respectively. Please find the introduction of the three-point method in the appendix. The RMSEs of the three-point method and the Doppler shift method are 0.171 m and 0.05 m, respectively. The mean values of the three-point method and the Doppler shift method are 499.83 m and 500.05 m, respectively. The RMSE and estimated error of the three-point method are much larger than those of the proposed method and Doppler shift method, indicating that the proposed method and the Doppler shift method are more effective and stable when compensating for mild vibration errors. As the three-point method uses three specific points to establish mathematical relations that are used to compensate for the vibrations, the jitter introduced by random noise negatively impacts the vibration compensation result. Thanks to the coherent accumulation in frequency domain, the Doppler shift method has an anti-noise property. Moreover, the proposed method uses all the measurement points in one period and reduces the effect of random noise to a certain extent by using SST and the mean value method. Thus, the proposed method and the Doppler shift method perform better than the three-point method when applied to mild vibration errors with noise.

#### 4.2 Severe vibration

In the second experiment, we added two sinusoidal vibrations with frequencies of 40 Hz (mild vibration) and 850 Hz (severe vibration) and amplitudes of 20 µm and 1 µm to the ideal dechirp signals to validate the proposed method in a severe vibration environment. Further, we added Gaussian white noise with an SNR of 0 dB to the dechirp signals with vibration, and the ideal range of the target was 500 m. The ideal dechirp signal and the up-dechirp and down-dechirp signals with vibration are shown in Figs. 8(a)–8(c), respectively. Moreover, the corresponding vibration errors are shown in Fig. 8(d), in which the blue and red lines represent the up and down observation, respectively. As shown in Fig. 8(d), the vibration displacements are on the micron scale.

Fourier transform was performed on the ideal dechirp signal as well as the up- and down-dechirp signals with severe vibration error, and the range profiles are shown in Fig. 9. The figure shows that the severe vibration greatly reduces the range resolution of the target, making it difficult to accurately determine the peak range of the target by using the up and down ranges. Thus, the micron-scale vibration displacement could induce large errors in the range detection.

We first applied SST to the up- and down-dechirp signals with severe vibration errors, and the results are shown in Figs. 10(a) and 10(b), respectively. The figure shows that the time-frequency information of the dechirp signals with severe vibration can be tracked owing to the high time-frequency resolution of SST. Moreover, the noise after squeezing has a granular distribution, which reduces the impact of noise on the extraction of time-frequency curves.

Next, the instantaneous ranges were calculated using Eq. (12), and the results are shown in Fig. 11(a). The mean ranges containing only constant errors were obtained by taking the mean value of the instantaneous ranges, and the range of the target was estimated to be 499.89 m. These results show that the proposed method can obtain good ranging results in a severe vibration environment and effectively reduce the influence of noise on ranging.

To verify the superiority of the proposed method for severe vibrations, we applied the Doppler shift method and the three-point method to the dechirp signals shown in Figs. 8(b) and 8(c). The range profiles obtained after applying the Doppler shift method are shown in Fig. 11(b). Because the Doppler shift method assumes the vibration velocity is constant, it fails to compensate for the severe vibration error. The range profile after Doppler shift method still has low resolution, making it difficult to estimate the target range. We regarded the spectrum peak as the estimated range of target, and the range of target is estimated to be 498.39 m which is much smaller than the actual range (500 m). The resulting phases with vibrations for the three-point method are shown in Fig. 12(a). For comparison, the phases of the ideal dechirp signals are shown in Fig. 12(b). In Fig. 12, the blue and red lines represent the up and down observation, respectively. Compared with the ideal phases, the phases in Fig. 12(a) are distorted because of the severe vibrations and noise. The first, middle, and last point in Fig. 12(a) are used to estimate the range of the target by using the three-point method [8], and the range was obtained as 497.48 m.

To verify the robustness and superiority of the proposed method, we repeated the ranging experiment 200 times by using the proposed method, three-point method and Doppler shift method, and the estimated ranges are shown in Figs. 13(a), 13(b) and 13(c), respectively. The RMSEs of the proposed method, three-point method and Doppler shift method are 0.17 m, 2.75 m, and 1.63 m, respectively. The mean values of the proposed method, the three-point method and the Doppler shift method are 499.94 m, 497.31 m, and 499.90 m, respectively. The proposed method performed better than the three-point method for severe vibration with noise because the instantaneous ranges can characterize the local vibration errors while the SST and the mean-value method can reduce the interference of random noise. Although the Doppler shift method is robust to noise, it cannot deal with severe vibration error. Compared with the three-point method and the Doppler shift method, the proposed method is robust to not only noise but also severe vibrations.

#### 4.3 Ranging performance under different SNRs and vibration frequencies

This section presents two experiments conducted to illustrate the ranging performance of the proposed method for vibrations with different frequencies and noise levels. We first added two sinusoidal vibrations with frequencies of 40 Hz and 850 Hz and amplitudes of 20 µm and 1 µm to the ideal dechirp signals. Subsequently, we added Gaussian white noise with different SNRs to the dechirp signals with severe vibrations. We simultaneously tested the proposed method, three-point method, and Doppler shift method with the same simulation parameters. The estimated errors for these three methods with different SNRs are shown in Fig. 14(a). As shown in the figure, the overall estimated errors for the three-point method increase with the increase of the noise level, as expected from the findings in subsections 4.1 and 4.2. Thus, the three-point method cannot produce good results in a severe vibration environment with noise. In contrast, the proposed method uses all measurement points in one-period triangular FMCW signals and reduces the effect of random noise to a certain extent through two main mechanisms. First, SST can compress the noise into granular particles in the time-frequency domain, which helps separate the noise from the effective signals in different frequency bands. Second, when the effective signal and noise are in the same frequency band, the mean-value method can effectively reduce the interference of noise in the effective signal. The errors of the proposed method are all smaller than those of the three-point method for different noise levels because the SST and the mean-value method are robust to noise. In addition, the three-point method and Doppler shift method do not reduce the sensitivity to fast disturbances such as turbulence and vibration, which occur in industrial environments. Therefore, although the Doppler shift method is more robust than the three-point method to noise, it has relatively large estimation errors than the proposed method when applied to environments with severe vibrations. Furthermore, the errors of the proposed method are small and nearly constant with a decrease in the SNR from 10 dB to –5 dB, and the errors increase only when the SNR is less than –5 dB. Thus, the proposed method performs better than the three-point method and Doppler shift method in a severe vibration environment with noise.

As the engine and other mechanical equipment of the platform generate severe vibrations with frequencies of up to several hundred hertz, we tested the ranging performance of the three compensation methods with different vibration frequencies to examine their applicability in different vibration environments. We first added Gaussian white noise with an SNR of 0 dB to the ideal dechirp signals, following which we successively added vibration errors with frequencies ranging from 25 Hz to 800 Hz. The estimated errors obtained using the three methods are shown in Fig. 14(b). The estimated errors of the three methods are similar when the vibration frequencies are small, but the estimated errors of the proposed method become smaller than those of the three-point method and Doppler shift method as the frequency increases. The three-point method loses validity when the vibration frequencies are large for two reasons. First, it uses three points obtained from the phases to compensate for the vibration errors, which cannot fully reflect the inner fluctuation of the whole signal with fast vibrations. Second, noise leads to jitters in the phase of dechirp signals, which negatively impact the detection of the signal based on only three points. The Doppler shift method loses validity when the vibration frequencies are large because it can only estimate and compensate for relative frequency shifts, and fails to deal with severe time-varying vibration errors. In contrast, the proposed method uses SST to account for the localized characteristics of the effective signals and thereby characterize the fluctuations of severe vibrations in the dechirp signals.

#### 4.4 Applicability to 3D imaging

This subsection presents the results of applying the proposed method to 3D ladar imaging to verify its applicability. In this experiment, the ladar first collects one-period dechirp signals from one laser spot to obtain the corresponding range of target at this time. We calculate the height of the ground of this laser spot by combining the navigation information of the aircraft and the scanning angle. Through 3D scanning, we can reconstruct the 3D scene of interest.

The 3D structure in this experiment was designed to show effect of vibrations on 3D imaging. Therefore, we only provide the relative location and ignore geographic information. The ideal 3D structure in this experiment is a slowly undulating ground, which is shown in Fig. 15(a). The geometric grid of the ideal 3D structure has the following characteristics: the grid height ranges from 199 m to 201 m, and the grid plane ranges from 1 m to 10 m in both the X and Y directions. Then, we added a sinusoidal vibration with an amplitude of 30 µm and a frequency of 100 Hz to the ideal 3D dechirp signals, in addition to Gaussian white noise with an SNR of 3 dB. Here, we used the conventional ranging process to show the impact of the vibration errors on the 3D imaging results. Fourier transform was performed on the up-dechirp signals with severe vibration error, and the peak range of range profiles was considered as the range of the target. Figure 15(b) shows the 3D imaging result without vibration compensation. The result fluctuates dramatically because of the vibration errors and noise, and such fluctuation errors negatively impact target recognition and ground elevation measurement.

Next, the proposed method was applied to the 3D dechirp signals of each laser spot to compensate for the vibration errors, and the reconstructed imaging result is shown in Fig. 16(a). The estimated error between the reconstructed imaging result of the proposed method and the ideal 3D imaging result is shown in Fig. 16(b). As shown in the figure, the estimated imaging result is close to the ideal one, and the estimated error between the estimated and the ideal results is small, indicating that the proposed method can effectively compensate for the vibration errors in FMCW ladar 3D imaging. We also applied the three-point method and Doppler shift method to the 3D dechirp signals for comparison. The 3D reconstructed imaging result and the estimated error of the three-point method are shown in Fig. 17(a) and 17(b), respectively. Compared with the ideal 3D ground image shown in Fig. 15(a), the reconstructed result of the three-point method fluctuates, and the estimated error shown in Fig. 17(b) is much greater than that of the proposed method. The reconstructed result of the Doppler shift method shown in Fig. 17(c) is closer to the ideal 3D image shown in Fig. 15(a) than that of the three-point method shown in Fig. 17(a) because it is less sensitive to noise. However, minor deformation appears in the estimated result because the Doppler shift method only compensate for the vibration error caused by constant vibration velocity. Therefore, the estimated error of the Doppler shift method shown in Fig. 17(d) is larger than that of the proposed method shown in Fig. 16(b). The RMSEs of the proposed method, three-point method and Doppler shift method in this experiment are 0.1042 m, 1.1328 m, and 0.1546 m, respectively. Thus, the reconstructed result of the proposed method is smoother and closer to the ideal 3D ground image compared with the three-point method and Doppler shift method, which further demonstrates that the proposed method is more suitable to compensate for the vibrations in FMCW ladar 3D imaging.

#### 4.5 Ranging performance for real data from a simulated airborne ladar scene

We tested the proposed method on real data which is obtained from the simulated airborne ladar scene. Figure 18 shows the schematic diagram of experimental equipment. The ladar platform is fixed, and a reflective board is installed on a vibrating motor to simulate the relative vibration between the ladar platform and target. The schematic diagram of FMCW ladar platform is shown in Fig. 1. The parameters of the triangular FMCW ladar system are as follows: the bandwidth and period of frequency modulation are 2.4 GHz and 0.5 ms, respectively, the laser wavelength is 1.55 µm; and the frequency modulation rate is 9.6 × 10^{12} Hz/s. The frequency modulation rate used in the simulated experiments is smaller than that in this experiment, because we want to use the simulated experiment to simulate more complicated situations in the airborne system and to test the ability of the proposed method in dealing with severe vibration errors. The range of the target was measured as 232 m when the target was stable, which will be regarded as the true range of the target.

We first extracted the up- and down-dechirp signal from the triangular FMCW ladar data, as shown in Fig. 19(a) and 19(b), respectively. We then applied fast Fourier transform to these dechirp signals to obtain the range distributions of the target, and the range profiles are shown in Fig. 19(c). As shown in Fig. 19(c), the vibration errors in the up- and down-dechirp signals introduce range displacement to the range profiles. If we regard the peak of the up range profile or that of the down range profile as the true range of the target, the ranging result will have a large error.

Thus, we applied the proposed method to the dechirp signals, and the resulting range profiles are shown in Fig. 20(a). The up and down instantaneous ranges are calculated using Eq. (12), as indicated by the red and the green lines in Fig. 20(a), respectively. The instantaneous ranges fluctuate during the frequency scan owing to the vibrations, and the instantaneous ranges are then used to obtain the mean ranges. The range of the target was calculated to be 231.92 m by using Eq. (19), which is close to the true range (232 m). For comparison, we applied the three-point method and Doppler shift method to the dechirp signals. The resulting phases of the up- and down-dechirp signals for the three-point method are shown in Fig. 20(b). In Fig. 20(b), the blue and red lines represent the up and down observation, respectively. The first, middle, and last point in Fig. 20(b) were utilized to estimate the range of the target as 231.12 m by using the three-point method. The range profiles obtained after applying the Doppler shift method are shown in Fig. 20(c), and the estimated range is 231.8 m.

To avoid random measurement errors, we collected the signals of 50 periods, and repeated the three methods to the collected dechirp signals 50 times. The RMSEs of the proposed method, three-point method and Doppler shift method are 0.10 m, 0.65 m, and 0.22 m, respectively, and the mean values of these methods are 231.94 m, 231.36 m, and 231.91 m, respectively. The RMSE and estimation error of the proposed method are smaller than those of the three-point method and Doppler shift method because the former can account for the instantaneous fluctuations of the dechirp signals and thereby accurately compensate for the vibration errors.

## 5. Conclusion

Vibration errors on the order of microns may introduce a large Doppler frequency shift in FMCW ladar ranging results because the laser wavelength is extremely short, and the large Doppler shift can seriously affect the ranging accuracy. To reduce the vibration errors in one-period FMCW ladar signals, we proposed a vibration compensation method based on an instantaneous ranging model. Numerical tests on simulated and field data showed that the proposed vibration compensation method can effectively compensate for the vibration error introduced by the Doppler frequency shift and accurately estimate the actual range of targets.

Compared with the three-point method and Doppler shift method, the proposed method produces better vibration compensation results for different frequencies ranging from 25 Hz to 800 Hz and noise with different SNRs ranging from -10 dB to 10 dB. The effectiveness of the proposed method on severe vibrations with frequencies up to several hundred hertz is particularly remarkable. Moreover, the proposed method is robust to noise with an SNR larger than –5 dB. When the SNR decreases to an extremely low level such as –10 dB, the compensation errors become large because the proposed method calculates the instantaneous ranges to characterize the inner fluctuations without coherent accumulation in the entire period. However, it still produces better results than the three-point method and Doppler shift method at a noise level of –10 dB. In addition, our experiment considered stable temperature and light conditions. In the future, we will investigate the effect of temperature and light on ranging and improve the robustness of the vibration compensation method in different scenarios.

## Appendix

Here, we briefly introduce the three-point method in [8]. According to [8], if the vibrations are smooth, the target range can be estimated by using the phases of three measurements from the dechirp signals of triangular FMCW ladar:

The algorithm assumes that the vibrations are smooth, i.e., the vibration velocity is close to a constant in a single period. Thus, the three-point method can compensate for the vibration error with constant velocity or slow vibration frequencies, but losses validity with fast vibration frequencies.

## Funding

National Natural Science Foundation of China (62073306); Youth Innovation Promotion Association.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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