Angle Estimation Research Articles

Flying Projectile Attitude Determination from Ground-Based Monocular Imagery with a Priori Knowledge

This paper discusses using ground-based imagery to determine the attitude of a flying projectile assuming prior knowledge of its external geometry. It presents a segmentation-based approach to follow the object and evaluates it quantitatively with simulated data and qualitatively with both simulated and real data. Two experimental cases are considered: One assumes reliable target distance measurement from an auxiliary range sensor, while the other assumes no range information. The results show that in the case of an unknown projectile–camera distance, with projectile dimensions of 1.378 m and 0.08 m in length and diameter, the estimated distance, in-plane location, and pitch angle accuracies are about 50 m, 0.15 m, and 6 degrees, respectively. Yaw angle estimation is ambiguous. In the second case, assuming that the projectile–camera distance is known resolves the ambiguity of yaw estimation, resulting in accuracies of about 0.15 m, 3 degrees, and 20 degrees for in-plane location, pitch, and yaw angles, respectively. These accuracies were normalized to a 1-km projectile–camera distance.

A method for estimating ship berthing angle based on 2D bird's eye view point cloud

The accurate estimation of berthing angle is essential for the parallel berthing of inshore ships. Herein, a new method for calculating the ship berthing angle using a shore-based LiDAR system is proposed. First, the LiDAR-scanned point cloud data is preprocessed, including coordinate conversion, noise filtering, and data downsampling. Then, the processed point cloud data is projected onto a two-dimensional (2D) plane, and the height value is used as the pixel value to generate bird's eye view (BEV) images. Finally, an improved Hough transform is proposed to detect the straight line representing the ship's nearshore profile in the BEV images and replace the ship midline with the detected line for calculating the berthing angle. The performance of the proposed method is verified by applying it on simulated and real ship data. Simulation experiment results show that the maximum absolute error of our method is only 0.151°. Real ship experiments were conducted at Dalian Port (China) to verify the reliability and stability of the method compared with other berthing angle estimation methods and the compass equipment. Overall, the proposed method can effectively obtain the berthing angle of ships, assist in parallel berthing, and improve berthing efficiency.

Efficient symbol level precoding design for full-duplex integrated sensing and communication systems

Full-duplex (FD) integrated sensing and communication (ISAC) system attracts much attention because adopting the FD mode can enhance the transmission efficiency and reduce the transmission time of the ISAC system. However, the inevitable self-interference (SI) in FD mode would result in severe performance degradation. This paper considers the FD ISAC system and investigates a symbol-level precoding (SLP) design, which minimizes the Cramér-Rao Bound of target angle estimation while considering the constant-envelope power restraint, SI suppression and the quality of service requirement. Due to the non-convex fractional problem and the unit-modulus constraint, the optimization problem is reformulated by implementing the Schur complement, where the optimal waveform is obtained via the semidefinite programming algorithm. The SLP design that optimizes each symbol vector for all users may suffer high computational complexity of matrix multiplication with high numbers of users. A grouped SLP strategy in which the communication users are divided into several groups on the basis of channel correlation is proposed to reduce the complexity. The proposed grouped SLP design only suppresses group-to-group interference and converts the intra-group interference into a constructive interference, which can reduce the dimension of each precoded signal significantly, resulting in a low computational complexity. Simulation results illustrate that the proposed SLP design can effectively mitigate SI, which can improve the accuracy of target detection. Moreover, the proposed grouped-SLP design can further reduce computational complexity without dramatic performance degradation.

The dispersion of the Ep, i–Liso correlation of long gamma-ray bursts is partially due to assembling different sources

Context. Long gamma-ray burst (GRB) prompt emission shows a correlation between the intrinsic peak energy, Ep, i, of the time-average νFν spectrum and the isotropic-equivalent peak gamma-ray luminosity, Lp, iso, as well as the total released energy, Eiso. The same correlation is found within individual bursts, when time-resolved Ep, i and Liso are considered. These correlations are characterised by an intrinsic dispersion, whose origin is still unknown. Discovering the origin of the correlation and of its dispersion would shed light on the still poorly understood prompt emission and would propel GRBs to powerful standard candles. Aims. We studied the dispersion of both isotropic-equivalent and collimation-corrected time-resolved correlations. We also investigated whether the intrinsic dispersion computed within individual GRBs is different from that obtained including different bursts into a unique sample. We then searched for correlations between key features, such as the Lorentz factor and jet opening angle, and intrinsic dispersion, when the latter is treated as one of the characterising properties. Methods. We performed a time-resolved spectral analysis of 20 long type-II or collapsar-candidate GRBs detected by the Fermi Gamma-ray Burst Monitor with a known redshift and estimates of the jet opening angle and/or the Lorentz factor. Time intervals were determined using Bayesian blocks. Then we carried out a statistical analysis starting from distributions of simulated values of the intrinsic dispersion of each burst in the sample. Results. The collimation-corrected correlation appears to be no less dispersed than the isotropic-equivalent one. Also, individual GRBs are significantly less dispersed than the whole sample. We excluded (at a 4.2σ confidence level) the difference in samples’ sizes as the possible reason, thus confirming that individual GRBs are intrinsically less dispersed than the whole sample. No correlation was found between intrinsic dispersion and other key properties for the few GRBs with available information. Conclusions. The contribution to the dispersion by the jet opening angle is not relevant. Moreover, our results prove that the intrinsic dispersion that affects the Ep, i − Liso correlation is partially, though not entirely, due to assembling different GRBs. We therefore conclude that the presence of different GRBs significantly contributes to the observed dispersion of both time-average Ep, i − Lp, iso and Ep, i − Eiso correlations.

Autonomous flight performance optimization of fixed-wing unmanned aerial vehicle with morphing wingtip

PurposeThe purpose of this paper is to improve autonomous flight performance of a fixed-wing unmanned aerial vehicle (UAV) via simultaneous morphing wingtip and control system design conducted with optimization, computational fluid dynamics (CFD) and machine learning approaches.Design/methodology/approachThe main wing of the UAV is redesigned with morphing wingtips capable of dihedral angle alteration by means of folding. Aircraft dynamic model is derived as equations depending only on wingtip dihedral angle via Nonlinear Least Squares regression machine learning algorithm. Data for the regression analyses are obtained by numerical (i.e. CFD) and analytical approaches. Simultaneous perturbation stochastic approximation (SPSA) is incorporated into the design process to determine the optimal wingtip dihedral angle and proportional-integral-derivative (PID) coefficients of the control system that maximizes autonomous flight performance. The performance is defined in terms of trajectory tracking quality parameters of rise time, settling time and overshoot. Obtained optimal design parameters are applied in flight simulations to test both longitudinal and lateral reference trajectory tracking.FindingsLongitudinal and lateral autonomous flight performances of the UAV are improved by redesigning the main wing with morphing wingtips and simultaneous estimation of PID coefficients and wingtip dihedral angle with SPSA optimization.Originality/valueThis paper originally discusses the simultaneous design of innovative morphing wingtip and UAV flight control system for autonomous flight performance improvement. The proposed simultaneous design idea is conducted with the SPSA optimization and a machine learning algorithm as a novel approach.

Estimation of Tire Side-Slip Angles Based on the Frequency Domain Lateral Acceleration Characteristics Inside Tires

The identification and control of tire side slip angle is the key to vehicle stability control. Intelligent tire technology based on the sensing of side-slip acceleration inside the tire provides a novel method for estimating the tire side-slip angle. This study proposed a method to estimate the tire side-slip angle by using the frequency domain lateral acceleration of the tire. First, an intelligent tire testing system was constructed by independently developing a special rim assembly and data collector. A three-axis accelerometer was placed on the right side of the tire, and the acceleration value was acquired by using a wired method with a sampling frequency of 50 kHz. Second, based on the constructed test system, a tire side deflection test was carried out on the Flat Trac bench. Through data analysis, it was found that the lateral acceleration was in the frequency domain of 400 Hz. As the side-slip angle increased from −4° to 4°, the vibration amplitude gradually decreased. Moreover, the vibration amplitude within 0.5~2 kHz was highly correlated with the side-slip angle. Subsequently, the vibration amplitude of the lateral acceleration within 2 kHz was extracted at an interval of 20 Hz as the feature point, and a frequency domain data set FDAy3 was established together with the vertical load and tire pressure. Finally, the support vector machine (SVM) algorithm was employed to make predictions on the data set. The grid search method was utilized to find the optimal parameter values of the model penalty factor c and radial basis kernel function coefficient g, which were 1.4142 and 0.0884, respectively. The results suggested that the root mean square error of the model prediction was 0.0806°, and the maximum estimated angle deviation of the prediction was 0.4587°. Meanwhile, the optimal prediction accuracy and real-time performance were achieved when the number of feature points and the feature frequency band were 25 and within 500 Hz, respectively. The findings of this study confirm that it is feasible to estimate the tire side-slip angle based on the frequency domain lateral acceleration of the tire, which provides a new method for tire side-slip angle estimation.

Shape Sensing for Continuum Robotics Using Optoelectronic Sensors with Convex Reflectors

Three-dimensional shape sensing in soft and continuum robotics is a crucial aspect for stable actuation and control in fields such as minimally invasive surgery, engine repairs and search and rescue operations, as the estimation of complex curvatures while using continuum robotic tools is required to manipulate through fragile paths. This challenge has been addressed using a range of different sensing techniques, for example, Fibre Bragg grating (FBG) technology, inertial measurement unit (IMU) sensor networks, or stretch sensors. Previously, an optics-based method using optoelectronic sensors was explored, offering a simple and cost-effective solution for shape sensing in a flexible tendon-actuated manipulator in two orientations. This was based on proximity-modulated angle estimation and has been the basis for the shape sensing method addressed in this paper. The improved and miniaturised technique demonstrated in this paper is based on the use of a convex shaped reflector with optoelectronic sensors integrated into a tendon-actuated robotic manipulator. Upgraded sensing capability is achieved using optimisation of the convex reflector shape in terms of sensor range and resolution, and improved calibration is achieved through the integration of spherical bearings for friction-free motion. Shape estimation is achieved in two orientations upon calibration of sensors, with a maximum Root-Mean-Square Error (RMS) of 3.37°.

Enhanced angle estimation in MIMO radar: Combine RD-MUSIC and SDP optimization

Multiple Input Multiple Output (MIMO) radars are increasingly employed. However, with a growing array of antenna components, the dimension of the sampling covariance matrix and the computational complexity of traditional Direction of Arrival (DOA) and Direction of Departure (DOD) estimation algorithms increase exponentially. In order to accurately and effectively identify and locate signal sources and to optimize the computational complexity, this paper presents an innovative approach for detecting target numbers based on a Semi-Definite Programming (SDP) problem, which generates efficient solutions even for higher dimensions. In addition, the SDP’s robustness is developed through shrinkage adaptation using Large Covariance Matrices (LCMs), which are crucial for angle estimation. Following the effective detection of a target, the tapering estimation is applied to the LCMs under the computationally efficient Reduced-Dimension Multiple Signal Classification (RD-MUSIC) algorithm, which is more accurate and less computationally intensive than 2D search algorithms. Simulation outcomes demonstrate that the efficacy of our proposed model achieves high success rates in target detection and enhanced precision in DOA and DOD estimations.

AUV instantaneous velocity estimation method based on azimuth and Doppler shift

Autonomous underwater vehicle (AUV), as a new underwater operation platform, has been widely used in the field of ocean engineering. AUV underwater condition monitoring is an important guarantee for the safety and effective execution of AUV operations. To meet the requirement of real-time and high-precision underwater instantaneous velocity monitoring of AUV, this paper proposed an AUV instantaneous velocity estimation method. The method takes the target azimuth and Doppler shift as the observation information. It comprehensively uses the geometric relationship among the target azimuth, heading angle, radial velocity angle, and the physical relationship between Doppler and radial velocity to construct an AUV instantaneous velocity estimation model based on distributed dual acoustic observation nodes. We derive the analytical solution of AUV speed and heading angle and analyze the influence of each observation error on the estimation precision of the algorithm. The research and practical data processing results show that the proposed method can effectively estimate the instantaneous velocity of AUV. Compared with the traditional method based on position difference, it has the advantages of high estimation precision, is not limited by frame interval, and is adaptive to many kinds of motion modes. When AUV moves in a straight line at a uniform speed, the precision of speed estimation and heading angle estimation of the algorithm proposed in the paper is 0.506 m/s and 6.146°, which is equivalent to that of the traditional method based on position difference with 50 frame intervals.

RAPHIA: A deep learning pipeline for the registration of MRI and whole-mount histopathology images of the prostate

Image registration can map the ground truth extent of prostate cancer from histopathology images onto MRI, facilitating the development of machine learning methods for early prostate cancer detection. Here, we present RAdiology PatHology Image Alignment (RAPHIA), an end-to-end pipeline for efficient and accurate registration of MRI and histopathology images. RAPHIA automates several time-consuming manual steps in existing approaches including prostate segmentation, estimation of the rotation angle and horizontal flipping in histopathology images, and estimation of MRI-histopathology slice correspondences. By utilizing deep learning registration networks, RAPHIA substantially reduces computational time. Furthermore, RAPHIA obviates the need for a multimodal image similarity metric by transferring histopathology image representations to MRI image representations and vice versa. With the assistance of RAPHIA, novice users achieved expert-level performance, and their mean error in estimating histopathology rotation angle was reduced by 51% (12 degrees vs 8 degrees), their mean accuracy of estimating histopathology flipping was increased by 5% (95.3% vs 100%), and their mean error in estimating MRI-histopathology slice correspondences was reduced by 45% (1.12 slices vs 0.62 slices). When compared to a recent conventional registration approach and a deep learning registration approach, RAPHIA achieved better mapping of histopathology cancer labels, with an improved mean Dice coefficient of cancer regions outlined on MRI and the deformed histopathology (0.44 vs 0.48 vs 0.50), and a reduced mean per-case processing time (51 vs 11 vs 4.5 min). The improved performance by RAPHIA allows efficient processing of large datasets for the development of machine learning models for prostate cancer detection on MRI. Our code is publicly available at: https://github.com/pimed/RAPHIA.

Guided Lamb Wave Array Time-Delay-Based MUSIC Algorithm for Impact Imaging.

Composite materials, valued in aerospace for their stiffness, strength and lightness, require impact monitoring for structural health, especially against low-velocity impacts. The MUSIC algorithm, known for efficient directional scanning and easy sensor deployment, is gaining prominence in this area. However, in practical engineering applications, the broadband characteristics of impact response signals and the time delay errors in array elements' signal reception lead to inconsistencies between the steering vector and the actual signal subspace, affecting the precision of the MUSIC impact localization method. Furthermore, the anisotropy of composite materials results in time delay differences between array elements in different directions. If the MUSIC algorithm uses a fixed velocity value, this also introduces time delay errors, further reducing the accuracy of localization. Addressing these challenges, this paper proposes an innovative MUSIC algorithm for impact imaging using a guided Lamb wave array, with an emphasis on time delay management. This approach focuses on the extraction of high-energy, single-frequency components from impact response signals, ensuring accurate time delay measurement across array elements and enhancing noise resistance. It also calculates the average velocity of single-frequency components in varying directions for an initial impact angle estimation. This estimated angle then guides the selection of a specific single-frequency velocity, culminating in precise impact position localization. The experimental evaluation, employing equidistantly spaced array elements to capture impact response signals, assessed the effectiveness of the proposed method in accurately determining array time delays. Furthermore, impact localization tests on reinforced composite structures were conducted, with the results indicating high precision in pinpointing impact locations.

Investigating the influence of neck muscle vibration on illusory self-motion in virtual reality

The illusory experience of self-motion known as vection, is a multisensory phenomenon relevant to self-motion processes. While some studies have shown that neck muscle vibrations can improve self-motion parameter estimation, the influence on vection remains unknown. Further, few studies measured cybersickness (CS), presence, and vection concurrently and have shown conflicting results. It was hypothesized that 1) neck vibrations would enhance vection and presence, and 2) CS to negatively relate to presence and vection, whereas presence and vection to positively relate to one another. Thirty-two participants were visually and audibly immersed in a virtual reality flight simulator and occasional neck muscle vibrations were presented. Vection onset and duration were reported through button presses. Turning angle estimations and ratings of vection quality, presence, and CS were obtained after completion of the flights. Results showed no influence of vibrations on turning angle estimation errors, but a medium positive effect of vibrations on vection quality was found. Presence and vection quality were positively related, and no strong association between CS and presence or vection was found. It is concluded that neck vibrations may enhance vection and presence, however, from the current study it is unclear whether this is due to proprioceptive or tactile stimulation.

Automatic estimation of hallux valgus angle using deep neural network with axis-based annotation

ObjectivesWe developed the deep neural network (DNN) model to automatically measure hallux valgus angle (HVA) and intermetatarsal angle (IMA) on foot radiographs. The objective is to assess the accuracy of the model by comparing to the manual measurement of foot and ankle surgeons.Materials and methodsA DNN was developed to predict the bone axes of the first proximal phalanx and all metatarsals from the first to the fifth in foot radiographs. The dataset used for model development consisted of 1798 radiographs collected from a population-based cohort and patients at our foot and ankle clinic. The retrospective validation cohort comprised of 92 radiographs obtained from 92 consecutive patients visiting our foot and ankle clinic. The mean absolute error (MAE) between automatic measurements by the model and the median of manual measurements by three foot and ankle surgeons was compared to 3° using one-tailed t-test and was also compared to the inter-rater difference in manual measurements among the three surgeons using two-tailed paired t-test.ResultsThe MAE for HVA was 1.3° (upper limit of 95% CI 1.6°), and this was significantly smaller than the inter-rater difference of 2.0 ± 0.2° among the surgeons, demonstrating the superior accuracy of the model. In contrast, the MAE for IMA was 0.8° (upper limit of 95% CI 1.0°) that showed no significant difference from the inter-rater difference of 1.0 ± 0.1° among the surgeons.ConclusionOur model demonstrated the ability to measure the HVA and IMA with an accuracy comparable to that of specialists.

High Precision Positioning and Rotation Angle Estimation of a Flatbed Truck Based on BDS and Vision.

Centimeter-level localization and precise rotation angle estimation for flatbed trucks pose significant challenges in unmanned forklift automated loading scenarios. To address this issue, the study proposed a method for high-precision positioning and rotation angle estimation of flatbed trucks using the BeiDou Navigation Satellite System (BDS) and vision technology. First, an unmanned forklift equipped with a Time-of-Flight (ToF) camera and a dual-antenna mobile receiver for BDS positioning collected depth images and localization data near the front and rear endpoints of the flatbed. The Deep Dual-Resolution Network-23-slim (DDRNet-23-slim) model was used to segment the flatbed from the depth image and extract the straight lines at the edges of the flatbed using the Hough transform. The algorithm then computed the set of intersection points of the lines. A neighborhood feature vector was designed to identify the endpoint of a flatbed from a set of intersection points using feature screening. Finally, the relative coordinates of the endpoints were converted to a customized forklift navigation coordinate system by BDS positioning. A rotation angle estimation was then performed using the endpoints at the front and rear. Experiments showed that the endpoint positioning error was less than 3 cm, and the rotation angle estimation error was less than 0.3°, which verified the validity and reliability of the method.

A Kalman Filtering Algorithm for Measurement Interruption Based on Polynomial Interpolation and Taylor Expansion.

Combined SINS/GPS navigation systems have been widely used. However, when the traditional combined SINS/GPS navigation system travels between tall buildings, in the shade of trees, or through tunnels, the GPS encounters frequent signal blocking, which leads to the interruption of GPS signals, and as a result, the combined SINS/GPS-based navigation method degenerates into a pure inertial guidance system, which will lead to the accumulation of navigation errors. In this paper, an adaptive Kalman filtering algorithm based on polynomial fitting and a Taylor expansion is proposed. Through the navigation information output from the inertial guidance system, the polynomial interpolation method is used to construct the velocity equation and position equation of the carrier, and then the Taylor expansion is used to construct the virtual measurement at the moment of the GPS signal interruption, which can make up for the impact of the lack of measurement information on the combined SINS/GPS navigation system when the GPS signal is interrupted. The results of computer simulation experiments and road measurement tests based on the loosely combined SINS/GPS navigation system show that when the carrier faces a GPS signal interruption situation, compared with a combined SINS/GPS navigation algorithm that does not take any rescue measures, our proposed combined SINS/GPS navigation algorithm possesses a higher accuracy in the attitude angle estimation, a higher accuracy in the velocity estimation, and a higher accuracy in the positional localization, and the system possesses higher stability.

Improving the tracking performance of space observation radar systems

This study focuses on the improvement of the tracking performance of space observation radar systems. Coherent radar systems usually provide an observation vector including four observables (range, range rate, azimuth, elevation) measured at a given epoch. Moreover, for some coherent radars, such as the TIRA system, it is possible to determine an additional parameter, the range rate rate, which is extracted from the collected complex amplitude. The tracking performance can be improved by increasing the observation parameter accuracy. The influence of each parameter on the overall tracking performance can be assessed by conducting a sensitivity analysis. The investigation shows that the best way to increase the accuracy of the state vector is by improving the angle estimates. The TIRA system is equipped with a tracking radar and an imaging radar. A powerful solution to improve the tracking performance is the realization of a multi-sensor data fusion between the two radars of the system.

On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyse their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in Beyn, et al. [Angular values of nonautonomous and random linear dynamical systems: Part I – fundamentals, SIAM J. Appl. Dyn. Syst. 21(2) (2022), pp. 1245–1286; Angular values of nonautonomous linear dynamical systems: Part II – reduction theory and algorithm, SIAM J. Appl. Dyn. Syst. 22(1) (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

UHead

The focus of Advanced driver-assistance systems (ADAS) is extending from the vehicle and road conditions to the driver because the driver's attention is critical to driving safety. Although existing sensor and camera based methods can monitor driver attention, they rely on specialised hardware and environmental conditions. In this paper, we aim to develop an effective and easy-to-use driver attention monitoring system based on UWB radar. We exploit the strong association between head motions and driver attention and propose UHead that infers driver attention by monitoring the direction and angle of the driver's head rotation. The core idea is to extract rotational time-frequency representation from reflected signals and to estimate head rotation angles from complex head reflections. To eliminate the dynamic noise generated by other body parts, UHead leverages the large magnitude and high velocity of head rotation to extract head motion information from the dynamically coupled information. UHead uses a bilinear joint time-frequency representation to avoid the loss of time and frequency resolution caused by windowing of traditional methods. We also design a head structure-based rotation angle estimation algorithm to accurately estimate the rotation angle from the time-varying rotation information of multiple reflection points in the head. Experimental results show that we achieve 12.96° median error of 3D head rotation angle estimation in real vehicle scenes.

Vehicle Sideslip Angle estimation under critical road conditions via nonlinear Kalman filter-based state-dependent Interacting Multiple Model approach

The knowledge of Vehicle Sideslip Angle (VSA) can play an essential role in active safety vehicle control systems. However, due to the high costs of sensing instruments, this information is difficult to be directly measured onboard of series production vehicles, restricting de facto its application in practice. It follows that there is a need for online VSA estimation methods only based on available measurements from low-cost sensors. From this perspective, this work proposes a strategy based on Interacting Multiple Model (IMM) filters, which does not require tyre–road friction coefficient knowledge. By integrating the available onboard sensor data, the IMM estimates relevant information in different driving conditions leveraging a 2-Degrees Of Freedom (DOF) single-track vehicle model embedding a Dugoff tyre representation. Two alternative IMM algorithms based on the Extended (EKF) and Unscented Kalman filter (UKF) are developed. Moreover, while usually the transition probabilities among models in classical IMMs are fixed and set on prior information and/or dedicated analysis, here these conservative hypotheses are relaxed introducing a state-dependent Markov transition matrix based on a novel model switching algorithm. The effectiveness of the new proposed methods is evaluated on extensive non-trivial simulation scenarios through a Monte Carlo analysis exploiting an accurate 15-DOF vehicle model via a purposely designed high-fidelity co-simulation platform embedding the dSPACE software Automotive Simulation Model (ASM). Results provide a meaningful comparative performance analysis between the IMMEKF and IMMUKF solutions, as well as with respect to traditional IMM based on constant probabilities transition matrix, blue in both the EKF and UKF configuration. Finally, the developed IMM-based estimation strategy is tested in two realistic driving scenarios to assess the VSA estimation accuracy in case of abrupt changes in road surface conditions.

Обобщённое неравенство Крамера – Рао в случае совместного пеленгования по азимуту и углу места в условиях сложной электромагнитной обстановки

In this work, a generalized expression is obtained for the Cramer–Rao lower bound in the case of a joint estimation of azimuth and elevation angle in an arbitrary coordinate system in which the vector complex radiation pattern is determined, under conditions of a complex electromagnetic environment, including the presence of noise correlation in spatial channels and differences in its intensity. A theorem is formulated for the Fisher information of joint estimates of azimuth and elevation angle in the case of a multi-channel detector-direction finder with an antenna system described in the selected coordinate system by a vector complex radiation pattern. Dependences of the root-mean-square error of direction finding in azimuth and elevation on the signal-to-noise ratio and true directions to the radio source are obtained, calculated using known and generalized expressions for the Cramer–Rao lower bound. The condition for the applicability of the existing and proposed in the article expression for the Fisher information matrix is obtained. The concept of the “energy” center of the antenna system is given. To confirm the reliability of the obtained generalized expressions for the lower bound of the dispersion of the Cramer–Rao inequality, statistical modeling is carried out. The use of the obtained expressions in practice will allow the most optimal synthesis of antenna devices depending on the structure and characteristics of the direction finder antenna system, the number of antennas and the characteristics of the radio receiving path. Generalized expressions for the lower Cramer–Rao bound are the foundation for organizing the operation of intelligent radio monitoring systems, allowing for a well-founded choice of the optimal antenna array structure that provides the required direction-finding efficiency.