Article Figures & Data
Figures
- FIGURE 1
Lidar PC showing return-light intensity (color-coded from blue to red, from low to high intensity)
- FIGURE 2
(a) 3D segmentation of a lidar PC; (b) circle fitting and point-feature measurement extraction
- FIGURE 3
Block diagram of lidar/IMU integration (prediction, estimation, and integrity risk evaluation processes)
- FIGURE 4
Initialization and EKF prediction process with an IMU
- FIGURE 5
Data association and EKF measurement update with lidar
- FIGURE 6
ADS positioning error covariance ellipses obtained by using lidar-only and IMU/lidar schemes for a two-landmark scenario
- FIGURE 7
Lateral positioning integrity risk when using the lidar-only and IMU/lidar schemes for a two-landmark scenario: analytical bound versus MC simulations over 50,000 trials
- FIGURE 8
Automated testbed setup with a sensor platform repeatedly moving on a figure-eight track
- FIGURE 9
(a) IR camera, (b) IR markers on a sensor platform, (c) lidar-VLP-16 Puck, (d) IMU-IGM-A1
- FIGURE 10
Covariance ellipses obtained by using the IMU/lidar scheme with intensity measurements
- FIGURE 11
Integrity risk bounds obtained by using the lidar-only scheme versus the lidar+ and IMU/lidar+ schemes
- FIGURE 12 Cross-track error of ADS for 100 laps
The zoomed-in panel shows the IMU-derived positioning drift corrected via lidar updates at regular 0.1-s intervals.
- FIGURE C1
Quantile-to-quantile plots of the lidar PC’s extracted feature error distribution and Gaussian overbounding model for the (a) range measurement and (b) bearing angle measurement (4,250 data points)
- FIGURE D1
Intensity measurement error distribution and Gaussian overbounding model for a retroreflective surface with a 70° incidence angle (17,500 data points)
- FIGURE D2
Mean and overbounding standard deviations (±1σ and ±3σ) of intensity measurements for black, white, and retroreflective surfaces at 0°, 30°, and 70° incidence angles (total of 166,334 data points, approximately 18,480 data points per configuration over the nine configurations)
Tables
is the 3×1 ground speed vector, i.e., the vehicle velocity vector with respect to E expressed in N, Ad/dt is a time derivative with respect to frame “A”, where A may stand for B, E, I, or N, is the 3×3 rotation matrix from B to N (Titterton et al., 2004), is the 3×1 vehicle position expressed in N, is the 3×1 IMU-measured and corrected specific force vector expressed in B, as derived in Appendix B, NωIE is the angular velocity vector of E with respect to I expressed in N, NωEN is the angular velocity vector of N with respect to E expressed in N, is the measured and corrected angular velocity vector of B with respect to I expressed in B, as derived in Appendix B, Ng is the local gravity vector at the IMU axis center expressed in N (Rogers, 2007), [a×] is the skew symmetric matrix of vector a. x is an ns × 1 state vector including ADS position, velocity, orientation, and 3D IMU gyroscope and accelerometer biases; i.e., ns = 15, v is the 2nL × 1 feature measurement error vector. is the ns × 1 initial predicted state estimate vector, is the ns × ns initial EKF state prediction covariance matrix, P(HMI0) is the initial value of the probability of HMI, Bk is the 3 × 3 rotation matrix relating navigation axes at time k – 1 to navigation axes at time k, as given in Appendix B (Titterton et al., 2004), Wk is the ns × ns process noise covariance at time step k, is the ns × 1 predicted state estimate vector at time step k, is the ns × ns prediction of the EKF covariance matrix at time step k, g is the discrete-time form of Equations (1) and (2), defined in Appendix B. sk is an nL × 1 mean return-light intensity vector for all nL visible landmarks, ŝk is the nL × 1 lidar-measured mean return-light intensity vector; we assume ŝk ~ N(sk, VS,k), is the mapped mean return-light intensity vector, is an nL × nL covariance matrix capturing the uncertainty in the mean intensity values of the mapped landmarks, are nL × nL permutation matrices similar to those in Equation (19) but for scalar permutations, for i = 0…hIA. K designates a range of time indices: K = {1…k}, J designates a range of time indices: J = {1…j}, Q() is the tail probability function of the standard normal distribution, ℓ is the specified alert limit that defines a hazardous situation, σk is the standard deviation of the estimation error for the vehicle state of interest, IALLOC,k is a predefined integrity risk allocation for feature extraction, chosen to be a fraction of the overall integrity risk requirement IREQ,k, is a chi-square distributed random variable with a number of degrees of freedom that is the sum of the number of measurements and of states at time step j, represents the minimum value of the mean landmark feature separation, including intensity separation, at time step j. System Parameters Values Standard deviation of lidar feature range data 0.12 m Standard deviation of lidar feature angular data 4° Lidar data sampling interval 0.5 s Accelerometer velocity random walk Gyroscope angle random walk Accelerometer time constant 3600 s Gyroscope time constant 3600 s Standard deviation of accelerometer GMP bias 0.05 m/s2 Standard deviation of gyroscope GMP bias 0.2° Vehicle speed 1 m/s Alert limit 0.25 m Note: GMP: Gauss–Markov process.
System Parameters Values Overbounding standard deviation of feature extraction ranging measurement 0.15 m Overbounding standard deviation of feature extraction angular measurement 3° Laser data sampling interval 0.1 s Accelerometer velocity random walk Gyroscope angle random walk Accelerometer GMP bias correlation time constant 3600 s Gyroscope GMP bias correlation time constant 3600 s Standard deviation of accelerometer GMP bias 0.67 m/s2 Standard deviation of gyroscope GMP bias 10° IMU sampling time 0.01 s Vehicle speed 0.6 m/s Alert limit ℓ 0.35 m Note: GMP: Gauss–Markov process.
R is the earth’s radius, h is the vehicle’s altitude, λ is the vehicle’s latitude, g0 is the acceleration of gravity at zero altitude. Bf is the 3×1 true specific force vector of body B with respect to I expressed in body frame B, is the measured specific force vector of body B with respect to I expressed in B, Sac, Mac are the true accelerometer calibration scale factor and misalignment matrices in B, bac is the accelerometer time-varying bias vector in B, vac is accelerometer measurement white noise error component expressed in B. τac is the GMP time constant, nac is a 3×1 vector of GMP time-uncorrelated driving noise. ts is the IMU sampling interval.
Jump to section
- Article
- Summary
- 1 INTRODUCTION
- 2 HIGH-INTEGRITY IMU AND LIDAR MEASUREMENT MODELS
- 3 INTEGRATED LIDAR/IMU ESTIMATION PROCESS
- 4 INTEGRITY RISK EVALUATION USING SIMULATED DATA
- 5 EXPERIMENTAL INTEGRITY RISK EVALUATION
- 6 CONCLUSIONS
- HOW TO CITE THIS ARTICLE
- CONFLICT OF INTEREST
- ACKNOWLEDGMENTS
- A IMU AND LIDAR MEASUREMENTS AND COEFFICIENTS
- B DISCRETE-TIME EQUATIONS FOR THE IMU
- C OVERBOUNDING OF MEASUREMENT ERROR DISTRIBUTIONS
- D LIDAR INTENSITY MEASUREMENT DISTRIBUTION
- REFERENCES
- Figures & Data
- Supplemental
- References
- Info & Metrics
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