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Abstract
Highly-time-resolved and precise tracking of position, velocity, and acceleration is urgently required when highly dynamic legged robots are walking, trotting, and jumping. Frequency-modulated continuous-wave (FMCW) laser ranging is able to provide precise measurement in short distance. However, FMCW light detection and ranging (LiDAR) suffers from a low acquisition rate and poor linearity of laser frequency modulation in wide bandwidth. A sub-millisecond-scale acquisition rate and nonlinearity correction in the wide frequency modulation bandwidth have not been reported in previous studies. This study presents the synchronous nonlinearity correction for a highly-time-resolved FMCW LiDAR. The acquisition rate of 20 kHz is obtained by synchronizing the measurement signal and the modulation signal of laser injection current with a symmetrical triangular waveform. The linearization of laser frequency modulation is conducted by resampling of 1000 intervals interpolated in every up-sweep and down-sweep of 25 µs, while measurement signal is stretched or compressed in every period of 50 µs. The acquisition rate is demonstrated to be equal to the repetition frequency of laser injection current for the first time to the best of authors’ knowledge. This LiDAR is successfully used to track the foot trajectory of a jumping single-leg robot. The high velocity up to 7.15 m/s and high acceleration of 365 m/s2 are measured during the up-jumping phase, while heavy shock takes place with high acceleration of 302 m/s2 as the foot end strikes the ground. The measured foot acceleration of over 300 m/s2, which is more than 30 times gravity acceleration, is reported on a jumping single-leg robot for the first time.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, plenty of attention for light detection and ranging (LiDAR) has been paid to the development of autonomous vehicles [1], drones [2] and robots [3,4], which requires the ability to quickly recognize and classify objects under fast-changing and highly dynamic conditions. Robotic intelligence in challenging natural environments relies on its own blind kinematic estimation, reinforcement learning, and adaptive actuation [5]. For example, quadrupedal robots can run at high speed and jump in a dynamic pose with millisecond-scale actuation [6], and they require precise perception to operate in sub-millisecond scale, just as instinct perception of their bionic counterparts, like dogs, cats, etc [7–9]. However, to the best of authors’ knowledge, it is nearly impossible for state-of-the-art perceptive sensors, such as LiDAR [4,10], camera [11], fiducial marker [12], etc., to operate with sub-millisecond response for highly dynamic targets.
Coherent ranging, also known as frequency-modulated continuous-wave (FMCW) laser-based LiDAR [13] is extensively used for precise perception on autonomous systems. Compared with the conventional time-of-flight (ToF) LiDAR, FMCW LiDAR will not require high-speed electronics to measure ultra-short flight time, which is quite suitable for precise measurement of the distance of <1 m for robots’ perception. However, FMCW LiDAR requires highly coherent [14] and precisely chirped [15] laser sources with a wide modulation bandwidth, and usually has a low acquisition rate. The coherence length of laser sources is correlated with the maximum detectable range, which is determined by the linewidth of the laser. The maximum measurement resolution is inversely proportional to the frequency chirp bandwidth. This coherent LiDAR requires ideally linear chirp modulation, but in practice it is difficult for lasers, especially for compact and low-cost semiconductor laser diodes (LD), to deliver linear modulation in sub-THz wide chirp bandwidth. Nonlinearities in frequency modulation widen the target spectrum, thereby decreasing the measurement accuracy and sometimes providing detection errors. Distributed feedback (DFB) lasers can provide a large chirp bandwidth, but the laser linewidth is usually limited by the short cavity length [16]. With self-injection locking, the linewidth can be effectively narrowed, and yet the chirp bandwidth is also decreased [17,18]. Recently, several integrated external cavity lasers have been demonstrated, which possess a wide chirp bandwidth and a narrow linewidth, quite suitable for the FMCW LiDAR applications [19–21]. With synchronous tuning by pre-distortion algorithms, the chirp nonlinearity is reduced to 1.02 × 10−7 in a 7.68-GHz chirp bandwidth at the 1 kHz repetition rate [22]. In order to get further wider chirp bandwidth, multiple distributed feedback lasers (DFBs) were combined to realize a 300 GHz swept frequency bandwidth [23]. An all-digital phase locked loop was proposed to achieved a sweep bandwidth of more than 200 GHz with mode-hop-free [24]. Moreover, the auxiliary interference beat signal was proposed to correct the nonlinearity in the beat frequency of each laser source by resampling the measurement signal [25,26]. Afterwards, signal stitching and resampling for multiple laser sources are conducted, where synchronous phase matching between measurement signals is critical [27]. Therefore, we can find that it is quite complicated and time-consuming for the effective nonlinearity correction in wide chirp bandwidth of sub-THz. Moreover, in order to get a large beam steering angle for optical phased array (OPA) based LiDAR, the wavelength range for laser sources are required to go beyond 200 nm, but it will be very difficult to simultaneously obtain linearly-chirped frequency signals for all these wavelengths [28–32]. Nonlinearity correction for FMCW ranging has been mostly demonstrated with pre-distorted laser chirps or phase locked loop [22–27]. So far, there has been no enough attention paid to post-processing algorithms for nonlinearity correction during data acquisition. FMCW laser ranging has been used to measure the distance of a dynamic vibration, but the measured speed and acceleration for vibrations are very low with low time resolution of 25 ms [33,34]. Low-noise integrated laser has been used to demonstrate a FMCW LiDAR at high acquisition rate of 800 kHz, but the distance resolution is limited to 12.5 cm at 10-m ranging because the frequency chirp bandwidth is only 1 GHz [35]. Up to now, wide frequency chirp bandwidth and acceptable linearity for FMCW LiDAR at a high data acquisition rate of sub-millisecond scale have not been achieved.
In this paper, a simplified scheme of synchronous nonlinearity correction for FMCW measurement signals is demonstrated. A triangular waveform of injection current is used to drive a DFB laser source, involving nonlinearity in the frequency chirp bandwidth of 160 GHz. Meanwhile, this triangular waveform of laser injection current is feedforwarded to a high-speed data acquisition module (DAQ). We use this high-speed DAQ to accommodate measurement signals, extract low-frequency signals by low-pass filter (LPF), and then synchronize this low-frequency signals and the signals of laser injection current. Afterwards, nonlinearities of frequency chirp are effectively corrected by resampling of 1000 intervals interpolated in each up-sweep/down-sweep frequency modulation. Therefore, the highly-time-resolved beat signals are achieved with the high acquisition rate of 20 kHz, which is equal to repetition rate of frequency-modulated signals. Lastly, this FMCW LiDAR is mounted on a single-leg robot to measure the locomotion during the highly dynamic jumping. The highly-time-resolved ranging results clearly show the sub-millisecond-scale time evolution of the height, velocity and acceleration of foot, for the first time.
2. Principle
The frequency of the FMCW laser ranging system is generally modulated in the form of symmetrical triangular waves or saw-tooth waves and the basic interferometric optical path is the Mach-Zender interferometer (MZI). If the laser frequency is not ideally linear modulated, the laser modulation form can be expressed as Fig. 1(a).
Fig. 1. (a) Principle of beat frequency formation, (b) schematic scheme of synchronous nonlinearity correction for FMCW LiDAR.
The principle of nonlinearity correction for laser frequency sweeping interference to measure the distance is presented below.
The linear frequency modulation function of the tunable laser can be expressed as follows:
where f0 is the initial laser frequency, k is the laser frequency modulation slope, and T is the laser frequency modulation period. The difference between the measurement optical path and the reference optical path causes the two signals to arrive at the photodetector at different times. The delay between the two optical paths is the time the laser light propagates in air, and R is the distance along the laser path in air. The air refractive index is n, and the speed of light is c, whereConsider a nonlinear term in the frequency modulation laser output G(t): the instantaneous frequency f(t) of the laser at time t is then
The frequency functions of the reference and measurement circuits are shown in Fig. 1(a).
The two signals mix during every sweep cycle and interfere at the detector to produce an electrical signal with a beat frequency fb.
The phase of mixed signal can be written as
The pre-corrected signal involving nonlinear frequency chirp is presented in Eq. (5) [36–38]:
Then, we have
The beat frequency extracted from Eq. (6):
where fb will be a time-dependent variable that involves nonlinear errors, as shown in Fig. 1(a). Because of the nonlinear fb as function of time, the waveform of pre-corrected signals in Eq. (6) is deformed and becomes non-uniform in time domain as indicated by “Pre-corrected signals” in Fig. 1(b).If the injection current for laser source is strictly set as a form of symmetrical triangular waves, I(t), the nonlinearity of laser frequency modulation can be formulated as:
where I(t) has the same modulation period T as f(t), and G(I) is presented as the nonlinear correlation of laser frequency with injection current. The pre-corrected signal in Eq. (6) has low-frequency component, whose repetition frequency is equal to the frequency of I(t). Then, the signal in Eq. (6) can be synchronized with synchronous signal I(t) as shown in Fig. 1(b), and the up-sweep period and down-sweep period can be identified in every period T of pre-corrected signal E(t).Within every up-sweep and down-sweep of laser modulation, the nonlinearities can be corrected by linearizing G(t)=kl × t with over 1000 intervals interpolated, where kl is a time-independent constant. In this work, the correlation of laser frequency versus different injection currents, G(I), is measured by an offline method. The laser frequency has nonlinear relationship with injection current and the injection current strictly follows linear relationship with time. Then, the nonlinear relationship of laser frequency with time, G(t), is obtained by numerically fitting the laser frequency versus injection current G(I). Lastly, the linearization of G(t) will be conducted by numerical resampling of laser frequency versus time with G(t)=kl × t. Afterwards, the frequency modulation waveform is achieved as a triangular waveform, as indicated by a dashed red curve in Fig. 1(b). Meanwhile, E(t) in Eq. (6) is resampled according to the linearizing relationship of G(t), and some signals are stretched in time domain while other signals are compressed with linearized G(t). Then the measurement signals are recovered to be uniform in time domain, as indicated by “Corrected signals” in Fig. 1(b). Accordingly, the corrected signals El(t) are presented as below:
where, fbl = (k + kl)τ=Bτ/T becomes a time-independent constant, and B is the laser frequency modulation bandwidth with synchronously nonlinearities corrected in every period T. It can be further expressed as:In principle, the corrected signals El(t) should be uniform in time domain, except for the transition time from up-sweep to down-sweep, as schematically presented in Fig. 1(b). However, in the real implementation of a FMCW LiDAR system, the corrected signals El(t) include the beat frequency, frequency components of internal reflections, and phase noises, which leads to a non-uniform waveform of El(t) in time domain.
3. Experimental setup and results
3.1 Experimental setup
The optical path of the highly time-resolved FMCW LiDAR is shown in Fig. 2. This setup is based on a frequency-modulated continuous-wave laser ranging system with synchronously nonlinearity corrected on data acquisition module. Optical frequency modulation is achieved by modulating the injection current of a compact DFB laser diode (Sacher Lasertechnik DFB-1556-100-FC/APC). Current modulation is performed by a function generator with a form of symmetrical triangular waves of laser injection current and a repetition frequency of 20 kHz. The function generator also provides a synchronous voltage signal with the same waveform of current modulation toward DAQ in Fig. 2, in order to trigger the synchronization between the measurement signal and injection current signal. The measurement signal is emitted to the target through a collimating lens (CL, Thorlabs PAF2-A7C) and is reflected back along the original path. The reflected signal is collected by CL, and coupled with the reference signal in a 2 × 2 coupler (50:50). Then, the optical mixing signal is directed to a balanced photodetector (BPD, Thorlabs PDB450C-AC). A data acquisition card (DAQ, ART Technology PCIe8914) with two channels is used to simultaneously record the measurement signal and synchronous signal at a rate of 100 MS/s with an accuracy of 14 bits. Finally, the synchronous nonlinearity correction is conducted on DAQ programmed by a computer.
Fig. 2. Schematic diagram of the experimental setup. CIR: circulator; CL: collimating lens; BPD: balanced photodetector; TIA: trans-impedance amplifier; DAQ: data acquisition card.
The real-time program by LabVIEW software is run by a parallel computing platform (CUDA) on graphics processing units (GPU, RTX3060). It includes five steps of real-time signal processing: low-frequency components extraction from measurement signals by low-pass filtering, synchronization of low-pass filtered signals and the modulation signals of laser injection current, linearization and resampling of laser frequency modulation according to the relationship of laser frequency with injection current, real-time Discrete Fourier transform (DFT) of resampled signals by FIFO method within every up-sweep and down-sweep of 25 µs, and smoothing/filtering of all obtained DFT spectra. The details for real-time program can be found in Supplement 1.
3.2 Results and discussion
The nonlinear properties of DBF laser source are characterized in Fig. 3. The superimposed lasing spectra are measured by an optical spectrum analyzer (OSA, AQ6370D, YOKOGAWA). The wavelength resolution is 0.02 nm and the dynamic range is 90 dB, which ensures the measurement accuracy of the side mode suppression ratio (SMSR). When the injection currents change from 20 mA to 500 mA and the TEC temperature was fixed at 25°C in Fig. 3(a), the laser wavelength is tuned from 1548.12 nm to 1549.76 nm, covering a 205 GHz frequency continuous sweep range. The nonlinearity of laser frequency versus injection current of laser diode is obtained by numerical fitting of the experimental data, as shown in Fig. 3(b). The nonlinear modulation of laser frequency is accordingly shown in Fig. 3(c), and a down-sweep laser frequency modulation bandwidth of 160 GHz is obtained by modulating injection current from 160 mA to 480 mA. A symmetrical triangular waveform of injection current modulation is schematically shown in the inset of Fig. 3(c), with the range of modulating injection current of 160-480 mA and a repetition frequency of 20 kHz. With a high sampling rate of 100 MS/s at DAQ, the low-frequency component is simultaneously extracted when recording the pre-corrected measurement signal (red line) in Fig. 3(d). And the low-pass filtered signal (green line) is synchronized with the synchronous voltage signal (black line) fed by the injection current modulation of laser source.
Fig. 3. Nonlinearity characteristic for the DFB laser source. (a) Measured wavelength tuning when changing the injection currents from 20 mA to 500 mA. (b) The nonlinearity of laser frequency versus injection current. (c) The nonlinear laser frequency modulation with down-sweep 160 GHz. (d) Synchronous FMCW measurement signal with injection current signal. Black line represents the synchronous voltage signal from laser source, red line represents the pre-corrected measurement signal and green line is low-pass filtered measurement signal.
The DAQ has a high sampling rate of 100 MS/s with a high accuracy of 14 bits, and it can realize the real-time resampling of measurement signals in every period. Figure 4(a) shows the measurement signal in up-sweep frequency chirp before synchronous correction. Here, the laser frequency nonlinearly changes as time goes in an up-sweep, as presented in the right axis of Fig. 4(a). The DAQ simultaneously records the measurement signal and synchronous signal, and accurately synchronizes the low-frequency measurement signal and synchronous signal as shown in Fig. 3(d). With identifying up-sweep and down-sweep in each period in time, the measurement signal can be resampled to correct the nonlinearity of frequency modulation in every up-sweep and down-sweep. Figure 4(b) demonstrates a typical resampling process of the measured FMCW signal on DAQ, according to the nonlinearity correction of an up-sweep frequency modulation. The linearization of frequency modulation versus time is conducted with 1000 highly time-resolved intervals in the up-sweep of 25 µs. The measurement signal is also interpolated with 1000 intervals in time domain and resampled according to linearization of laser frequency. Compared with signal stitching and resampling for multiple laser sources in [23,27], this LiDAR system is simplified with similar frequency modulation bandwidth of sub THz and phase mismatching for signal stitching is eliminated. Therefore, this synchronous acquisition with nonlinear chirps corrected in every 25 µs can deliver the precise measurement results at a high data rate of 20 kHz, which is equal to the repetition frequency of laser frequency chirps. The amplitude waveform in Fig. 4(b) at 0-15 µs is relatively uniform compared to that at 15-25 µs, close to the adjacent down-sweep, because the waveform is dominated by low-frequency amplitude modulation at the transition time from up-sweep to down-sweep. Also, the corrected signal should be non-uniform in time domain in the real implementation of a LiDAR, since it consists of many frequency components in frequency domain including the beat frequency, the signal internally reflected by CL, and phase noise.
Fig. 4. (a) The Pre-corrected measurement signal (blue curve on the left axis) with a nonlinear frequency modulation in up-sweep period (black curve on the right axis). (b) The corrected signal is resampled (red curve on the left axis), according to a linear up-sweep frequency modulation (black curve on the right axis).
Figure 5(a) shows the synchronous correction for measured signals of 1-m distance in time domain. As previously discussed, the measurement signal is corrected based on the synchronous linearization of laser frequency, with signal stretched or compressed in time domain (blue curve). Discrete Fourier transform (DFT) of time-domain signal is used to extract the beat frequency signal. The pre-corrected waveform in time domain is deformed and non-uniform at 0-15 µs in Fig. 5(a), because nonlinear laser frequency chirps are involved. The beat signal cannot be exacted on the pre-corrected frequency spectrum in Fig. 5(b). With nonlinearity corrected, the waveform is stretched and becomes uniform at 0-15 µs. Clear and robust beat frequency is obtained on the corrected spectrum in Fig. 5(b).
Fig. 5. Measured signals for 1-m distance in time domain (a) and frequency domain (b). (b) is obtained by DFT of (a).
This highly-time-resolved FMCW LiDAR is validated by a series of measurement experiments. The moving target is mounted on a guide rail with accurate pre-defined position. Typical frequency spectra after DFT of nonlinearity-corrected signals are superimposed in Fig. 6(a), corresponding to the distance measurement for 0.05-1 m. The distance versus beat frequency is shown in Fig. 6(b). Each measurement point involves 40,000 pieces of beat signal curve in two seconds. And a visualized probability density function of 1-m distance measurement is shown in the inset of Fig. 6(b). The beat signal is clearly presented even though the background noise involving thermal noise and phase noise is much brighter. The statistical distribution of 1-m distance measurement presents a Gaussian function (dashed line) as in Fig. 6(c), with an average distance of 0.997 m and a standard deviation of 2.5 cm. Error bars are marked for each point of Fig. 6(b) to provide the deviations for all measurements. The theoretical resolution for FMCW LiDAR ranging is related to the laser frequency modulation bandwidth [16,27], which is about millimeter scale for 160 GHz modulation bandwidth. Here, the ranging accuracy is deteriorated by phase noises and thermal noise during this fast measurement as show in the inset of Fig. 6(b). The accuracy of this fast FMCW laser ranging is good enough for the measurement of highly dynamic locomotion of robots.
Fig. 6. (a) Measured frequency spectra of 0.05-1 m distance with nonlinearity corrected. (b) Highly-time-resolved distance measurement by FMCW LiDAR. (c) Statistical distribution of 1-m distance measurement with total 40,000 counts.
With pre-defined movement of target on a guide rail, this LiDAR is used to track the dynamic locomotion of a moving target. In Fig. 7, the target moves at high speed and low speed respectively. The velocity is derived by the derivative of distance as function of time, and the acceleration is the derivative of velocity. Figure 7(a) and Fig. 7(d) show the distance tracking of a fast locomotion and a slow locomotion, respectively. The fast locomotion speeds up to the velocity of about 2.7 m/s in 0.1 s and then slows down in Fig. 7(b). The slow locomotion speeds up to 1.6 m/s, keeps the fixed velocity for 0.2 s, and then slows down in Fig. 7(e). The acceleration of fast locomotion is approaching 20 m/s2 in Fig. 7(c), which is almost two times gravity acceleration. In contrast, in Fig. 7(f), the measured acceleration for slow locomotion is below the gravity acceleration. We can see that this highly-time-resolved FMCW LiDAR is able to accurately track the distance, velocity and acceleration together for dynamic locomotion.
Fig. 7. Measured data for a moving target using highly-time-resolved FMCW LiDAR. Distance (a), velocity (b), and acceleration (c) of fast locomotion; and distance (d), velocity (e), and acceleration (f) of slow locomotion, respectively.
4. Demonstration of highly-time-resolved ranging
4.1 Dynamic jumping of single-leg robot
Legged robots have powerful motion ability and are able to overcome challenging terrains. When the legged robot is walking, trotting or jumping, its feet would strike the ground and swing periodically [39]. The high velocity of the foot end in the direction perpendicular to the ground when striking the ground will cause collision and shock on the leg [40]. The internal sensors equipped on legged robots, such as encoders, accelerometers, cannot provide direct data for foot end tracking. The robot cannot respond fast enough to avoid collision and shock without the environmental perception. So, highly-time-resolved and precise tracking for the high velocity and high acceleration of the foot end is quite critical to characterize dynamic locomotion of legged robots.
As shown in Fig. 8(a), the single-legged robot platform has two joints, each joint consisting of a high-speed electric motor (Maxon EC45) equipped with a harmonic gear driven with reduction ratio of 100 and a 14-bit absolute encoder. In order to minimize the inertia of the robot’s leg, both the motors are mounted on the hip joint. Different from the hip joint in dynamic structure, the knee joint is driven by a motor mounted on the hip through a belt from which the torque is transmitted to the harmonic gear. The robot platform has a thigh length of 0.38 m and a shank length of 0.361 m, which is equipped with horizontal and vertical guides, so that the robot can move forth or back, up or down. The high reduction ratio of harmonic gear and belt make the structure absorb the torque fluctuation of the motor input more efficiently. The locomotion control system uses a computer as EtherCAT’s host with TwinCAT running at 4 kHz. The CL of FMCW LiDAR is mounted on the shank of this leg robot, with laser beam perpendicular to the ground. The repetition frequency for laser chirp is 20 kHz and the LiDAR can track the position of foot end in the direction perpendicular to the ground by every 50 µs. Figure 8(b) shows the foot trajectory in one cycle of jumping, as planned actions including leg protraction, leg retraction and foot landing.
Fig. 8. (a) The CL is mounted on shank of a single-leg robot to track the dynamic locomotion of foot end. (b) The planned foot trajectory in one cycle of jumping, including leg protraction, leg retraction and foot landing.
As shown in Fig. 9, the maximum distance measured by LiDAR is 0.42 m at around 150 ms in one cycle of jumping actions.
Fig. 9. The screenshot of the jumping experiment on the single leg at every 50 ms (see Visualization 1).
4.2 Dynamic tracking of jumping single-leg robot with high time resolution
The distance, velocity and acceleration are acquired in Fig. 10 by every 50 µs during one cycle of jumping. In the up-jumping phase, the distance increases quickly to 0.4 m at about 100 ms as marked in Fig. 9 and Fig. 10(a). In order to jump to a much higher position, the robot make a second protraction within 50 ms in the leg swing phase and reach the maximum height of 0.42 m at about 150 ms (dashed purple lines in Fig. 10). This behavior is clearly characterized by highly-time-resolved LiDAR for the first time. In the leg protraction progress in Fig. 8(b), there will be some unexpected disturbances deviated from the planned trajectory. This high-time-resolution measurement can help identify the sources of disturbances during the leg protraction. There are two peaks of velocity, 7.15 m/s at 78 ms and 2.84 m/s at 128 ms in Fig. 10(b), corresponding to the first and second leg protractions, respectively. Meanwhile, there are two valleys of acceleration, -365 m/s2 and -312 m/s2 in Fig. 10(c), corresponding to the distance peaks of 0.4 m and 0.42 m in Fig. 10(a), respectively. During the foot end landing on the ground, there is an acceleration peak of 302 m/s2 at about 200 ms. After the foot strikes the ground for the first time, it is bounced upwards, and strikes the ground again with acceleration of 78 m/s2 at about 250 ms, as shown by dashed blue lines in Fig. 10(c). This highly-time-resolved characterization of one cycle of leg jumping can also be found in details in video of Visualization 1.
Fig. 10. Measured data for one cycle of jumping experiment on the single leg using highly-time-resolved FMCW LiDAR. Distance (a), velocity (b), and acceleration (c) in one cycle are acquired, respectively.
With the high acquisition rate of 20 kHz, tracking of highly dynamic legged robot is presented with measured acceleration of over 300 m/s2, as summarized in Table 1. The high velocity of over 7 m/s during the leg protraction is recorded also. The tracking of acceleration peaks corresponds to the accurate time when the foot end strikes the ground (200 ms and 250 ms), which leads to collision and shock on the leg. The accurate measurement can provide the timely precautions for locomotion control to reduce striking collisions. This high acceleration over 30 times gravity has not been successfully characterized for legged robot so far. This highly-time-resolved FMCW LiDAR presents the clear and precise tracking of leg protraction and landing for the first time, to the best of authors’ knowledge.
Table 1. Summary of measurement in one cycle of jumping
5. Conclusion
A highly-time-resolved FMCW LiDAR is proposed to track the position, velocity and acceleration of foot for a jumping single-leg robot. A synchronously-nonlinearity-corrected acquisition is principally and experimentally validated with a high acquisition rate of 20 kHz. The linearization of laser frequency modulation is conducted in every up-sweep and down-sweep by synchronizing the measurement signal and the modulation signal of laser injection current. The laser frequency modulation bandwidth of 160 GHz is obtained by modulating injection current from 160 mA to 480 mA, with a symmetrical triangular waveform of injection current and a repetition frequency of 20 kHz. With signal stretched or compressed in every period T, the measurement signal is corrected based on the synchronous linearization of laser frequency. Then, the clear and robust beat frequency is obtained on the corrected frequency spectrum. With the LiDAR mounted on a single-leg robot, the leg swinging and landing in one cycle of jumping is recorded. The high velocity up to 7.15 m/s and high acceleration of 365 m/s2 are measured during the up-jumping phase, while heavy shock takes place with high acceleration of 302 m/s2 as the foot strikes the ground.
To the best of our knowledge, the acquisition rate is demonstrated to be equal to the repetition frequency of laser injection current for the first time. If the laser is modulated up to higher frequency, the data acquisition can be increased accordingly. The measured acceleration of over 300 m/s2, which is more than 30 times gravity acceleration, is reported on a jumping single-leg robot for the first time. This method relaxes the tight requirements for coherence and linearity of laser source and reduces the time consumed in nonlinearity correction for FMCW measurement. It paves a new way of realizing real-time tracking for autonomous systems operating in highly dynamic conditions.
Funding
National Natural Science Foundation of China (U1613223, U1813207); Shenzhen Fundamental Research and Discipline Layout project (JCYJ20180508163015880); Basic and Applied Basic Research Foundation of Guangdong Province (2021B1515120084); Tip-top Scientific and Technical Innovative Youth Talents of Guangdong Special Support Program (2019TQ05X062).
Acknowledgments
Caiming Sun thanks Peng Cheng Laboratory for providing facilities of equipment and software for this work.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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