Euler Angles Research Articles

Attitude control of a 3-DoF quadrotor platform using a linear quadratic integral differential game approach

In this study, a linear quadratic integral differential game approach is applied to regulate and track the Euler angles for a quadrotor experimental platform using two players. One produces commands for each channel of the quadrotor and another generates the worst disturbance based on the mini-maximization of a quadratic criterion with integral action. For this purpose, first, the attitude dynamics of the platform are modeled and its parameters are identified based on the Nonlinear Least Squares Trust-Region Reflective method. The performance of the proposed controller is evaluated for regulation and tracking problems. The ability of the controller is also examined in the disturbance rejection. Moreover, the influence of uncertainty modeling is studied on the obtained results. Then, the performance of the proposed controller is compared with the classic Proportional Integral Derivative, Linear Quadratic Regulator, and Linear Quadratic Integral Regulator. The results demonstrate the effectiveness of the Game Theory on the Linear Quadratic Regulator approach when the input disturbance occurs.

Low-Thrust Trajectory Optimization with Averaged Dynamics Using Analytical Switching Detection

This paper presents an efficient indirect optimization method for the time- and fuel-optimal low-thrust many-revolution transfers with averaged dynamics. Compared with the previous indirect method, the proposed method involves a new set of variables and several near-optimal control laws to formulate the two-point boundary value problem based on three practical assumptions. Then, the costate transformations between two reference frames are first derived with respect to the the corresponding Eulerian angles, such that the switching function can be expressed with a six-degree polynomial function and the number of parameters used to determine the energy-optimal solutions is reduced to 6. Finally, a neural network is trained to generated the initial guesses, and the solution algorithms for solving the time- and fuel-optimal problem are formulated. Numerical simulations demonstrate that the convergence ratio of initial guesses is higher than 98%, the proposed assumptions are reasonable for guaranteeing the near optimality, and the presented method can obtain convergent solutions within tenths of a second for the Earth- and moon-centered external/interior transfers considering the shadow J2 and third-body perturbations.

Modelling the brittle rock failure by the quaternion-based bonded-particle model in DEM

This paper presents an investigation of brittle rock failure by the quaternion-based bonded-particle model in discrete element method (DEM). Unlike traditional approaches that utilize Euler angles or rotation matrices, this model employs unit quaternions to represent the spatial rotations of particles. This method simplifies the representation of 3D rotations, providing a more intuitive framework for modelling complex interactions in granular materials. The numerical model was validated by the uniaxial compression tests on rock, with good agreement with well-documented experimental data in terms of the rock uniaxial compression strength (UCS) and failure mode. During loading, the rock sample demonstrated a linear-elastic response at an axial strain of smaller than 0.45%. However, as internal bond breakage accumulated, this linear relationship weakened, and the stress-strain curve began to deviate from its initial linear trajectory. The bond breakage and the overall deformation of the rock were primarily controlled by the shear bonding force. The UCS was achieved at an axial strain of 0.625%, at which point the internal shear bonding force chains were predominantly aligned vertically. The brittle failure occurred when the internal damage of solids nucleated to form an interconnected failure plane, accompanied by a sharp rise in the internal damage ratio. The area of failure plane increased with the loading strain rate, gradually transforming the failure pattern from the local damage to a complete fragmentation.

Real-time analysis of star sensor attitude accuracy

In the conventional methods of star sensor accuracy evaluation, neither the accuracy calculation formula nor the Monte Carlo method can comprehensively reflect the attitude measurement accuracy of star sensors in real-time. In this paper, a real-time analysis and calculation model for the attitude measurement accuracy of star sensors is proposed. Firstly, the basic attitude measurement model of the star sensor is established through the pinhole imaging model. Moreover, a moving frame of star sensor attitude measurement is established to avoid the defects of attitude representation by the Euler angle in the star sensor accuracy analysis. Then, combined with the error transfer of implicit overdetermined equations proposed in this paper, the error transfers of starspot extraction random error, star catalog random error, and intrinsic parameter system error are provided. Finally, simulation and field experiments are conducted to verify the accuracy of the proposed error analysis theory. The experimental results show that the attitude measurement accuracy of star sensors can be accurately estimated by using single-frame data of guide stars.

Estimating external exposure from different source geometries in soil contaminated by gamma-ray emitting radionuclides: A computational model combining Monte Carlo simulation and mathematical transformation

A computational model is developed to estimate the external exposure to an arbitrary scenario describing the spatial distribution of the source in a contaminated soil by gamma-ray emitting radionuclides released after nuclear activities. This model welds together a Monte Carlo simulation approach and a mathematical transformation based on Tait–Bryan angles formalism. For Monte Carlo simulation, a source depth dependent optimized geometry describing a semi-infinite soil-cover-air medium of propagation is mathematically derived based on the physical attenuation of the gamma ray in matter. For each primary energy of interest, the proposed mathematical transformation is applied to the output of the Monte Carlo simulation to define the overlapping region between the contaminated soil volume and the proposed optimized volume. This approach made it possible to estimate the human external exposure associated with the scenario under investigation. Several scenarios were studied. The flux energy distribution of the detected radiation and the corresponding dose quantities are calculated. A benchmarking test with a direct Monte Carlo simulation and a comparison with published results in the literature show good agreement validating therefore the presented model.

SUGAR: Spherical ultrafast graph attention framework for cortical surface registration

Cortical surface registration plays a crucial role in aligning cortical functional and anatomical features across individuals. However, conventional registration algorithms are computationally inefficient. Recently, learning-based registration algorithms have emerged as a promising solution, significantly improving processing efficiency. Nonetheless, there remains a gap in the development of a learning-based method that exceeds the state-of-the-art conventional methods simultaneously in computational efficiency, registration accuracy, and distortion control, despite the theoretically greater representational capabilities of deep learning approaches. To address the challenge, we present SUGAR, a unified unsupervised deep-learning framework for both rigid and non-rigid registration. SUGAR incorporates a U-Net-based spherical graph attention network and leverages the Euler angle representation for deformation. In addition to the similarity loss, we introduce fold and multiple distortion losses to preserve topology and minimize various types of distortions. Furthermore, we propose a data augmentation strategy specifically tailored for spherical surface registration to enhance the registration performance. Through extensive evaluation involving over 10,000 scans from 7 diverse datasets, we showed that our framework exhibits comparable or superior registration performance in accuracy, distortion, and test-retest reliability compared to conventional and learning-based methods. Additionally, SUGAR achieves remarkable sub-second processing times, offering a notable speed-up of approximately 12,000 times in registering 9,000 subjects from the UK Biobank dataset in just 32 min. This combination of high registration performance and accelerated processing time may greatly benefit large-scale neuroimaging studies.

Deep-learning-based head pose estimation from a single RGB image and its application to medical CROM measurement

For human beings, neck movement will be degraded due to aging, trauma, musculoskeletal disorders, or degenerative diseases. Cervical range of motion (CROM) measurement is one of the popular quantitative neck examinations. Despite radiography is considered as the gold standard, it suffers from invasiveness, radiation exposure, and expensiveness. Recently, vision-based methods have been applied for CROM measurement but achieve large errors and require depth camera. On the other hand, deep neural networks provide good performances on head pose estimation (HPE) from a single image, thus promising for medical CROM measurement. We propose to use CNN networks to extract pyramidal or multi-level image features, which are passed to cross-level attention modules for feature fusion and then to a modified ASPP module and a multi-bin classification/regression module for spatial-channel attention and Euler angle conversion/prediction, respectively. The proposed technique was evaluated on public datasets, such as 300W_LP, AFLW2000, and BIWI, to verify its superior performances (with mean MAE = 3.50°, 3.40°, and 2.31° for different experimental protocols) than state-of-the-art methods. Our pre-trained model was also evaluated with our own collected dataset from hospital for CROM measurement. It also achieved the lowest MAE of 4.58° among other methods and conformed with a medical standard of 5 degrees except the pitch angle (which has a MAE of 5.70°, larger than the standard and the yaw (MAE = 3.60°) and roll angles (MAE = 4.44°)). In general, HPE technique is feasible for CROM measurement and shows its advantages of speed, non-invasiveness, free of anatomical landmark and low cost of operation.

Skin marker-based versus bone morphology-based coordinate systems of the hindfoot and forefoot

Segment coordinate systems (CSs) of marker-based multi-segment foot models are used to measure foot kinematics, however their relationship to the underlying bony anatomy is barely studied. The aim of this study was to compare marker-based CSs (MCSs) with bone morphology-based CSs (BCSs) for the hindfoot and forefoot. Markers were placed on the right foot of fifteen healthy adults according to the Oxford, Rizzoli and Amsterdam Foot Model (OFM, RFM and AFM, respectively). A CT scan was made while the foot was loaded in a simulated weight-bearing device. BCSs were based on axes of inertia. The orientation difference between BCSs and MCSs was quantified in helical and 3D Euler angles. To determine whether the marker models were able to capture inter-subject variability in bone poses, linear regressions were performed. Compared to the hindfoot BCS, all MCSs were more toward plantar flexion and internal rotation, and RFM was also oriented toward more inversion. Compared to the forefoot BCS, OFM and RFM were oriented more toward dorsal and plantar flexion, respectively, and internal rotation, while AFM was not statistically different in the sagittal and transverse plane. In the frontal plane, OFM was more toward eversion and RFM and AFM more toward inversion compared to BCS. Inter-subject bone pose variability was captured with RFM and AFM in most planes of the hindfoot and forefoot, while this variability was not captured by OFM. When interpreting multi-segment foot model data it is important to realize that MCSs and BCSs do not always align.

Trajectory-Tracking Control for Quadrotors Using an Adaptive Integral Terminal Sliding Mode under External Disturbances

In the face of external disturbances affecting the trajectory tracking of quadrotors, a control scheme targeted at accurate position and attitude trajectory tracking was designed. Initially, a quadrotor dynamic model, essential for control design, was derived. Adaptive integral backstepping control (AIBS) was then employed within the position loop, enabling the upper boundaries of disturbances to be estimated through adaptive estimation. Subsequently, a new adaptive backstepping fast nonsingular integral terminal sliding mode control (ABFNITSM) was proposed to enable adherence to the desired Euler angles. Rapid convergence and accurate tracking were facilitated by the incorporation of the nonsingular terminal sliding mode and an integral component. The dead zone technique was deployed to curtail estimation errors, while a saturation function was used to eradicate the phenomenon of chattering. Finally, to validate the proposed control scheme, simulation experiments were conducted in the Simulink environment, and the results were contrasted with those obtained from traditional integral terminal sliding mode control (ITSM) and integral backstepping control (IBS), providing evidence of the effectiveness of the proposed method.

Head Pose Estimation Based on Multi-Level Feature Fusion

Head Pose Estimation (HPE) has a wide range of applications in computer vision, but still faces challenges: (1) Existing studies commonly use Euler angles or quaternions as pose labels, which may lead to discontinuity problems. (2) HPE does not effectively address regression via rotated matrices. (3) There is a low recognition rate in complex scenes, high computational requirements, etc. This paper presents an improved unconstrained HPE model to address these challenges. First, a rotation matrix form is introduced to solve the problem of unclear rotation labels. Second, a continuous 6D rotation matrix representation is used for efficient and robust direct regression. The RepVGG-A2 lightweight framework is used for feature extraction, and by adding a multi-level feature fusion module and a coordinate attention mechanism with residual connection, to improve the network’s ability to perceive contextual information and pay attention to features. The model’s accuracy was further improved by replacing the network activation function and improving the loss function. Experiments on the BIWI dataset 7:3 dividing the training and test sets show that the average absolute error of HPE for the proposed network model is 2.41. Trained on the dataset 300W_LP and tested on the AFLW2000 and BIWI datasets, the average absolute errors of HPE of the proposed network model are 4.34 and 3.93. The experimental results demonstrate that the improved network has better HPE performance.

Characteristics of throwing arm movement for an elongated implement: a comparison of the javelin and baseball throwing.

In javelin, although many previous studies have examined throwing movements that can increase initial velocity, the characteristics of throwing arm movement an elongated implement have not been clarified. The purpose of the present study was to examine the characteristics of the throwing movement of an elongated implement by comparing throwing movement between a javelin and baseball. Twelve male javelin throwers were asked to perform a javelin throw (JT) and a baseball long toss (LT) twice. The three-dimensional coordinates of reflective markers attached to the athlete's body, javelin, and baseball were measured using an optical motion capture system. %Trajectory was used as an index to evaluate the degree to which the hand was moved linearly during the throw. A smaller value of this indicator meant that the hand was move closer to a straight line. The joint angles in the throwing arm were obtained by calculating the Euler angles between body segments. These data were used to compare JT and LT. %Trajectory showed that JT was significantly smaller than LT. Significant differences in the joint angles of the throwing arm were noted between JT and LT. JT showed a kinematic pattern in which the hand was moved more linearly than in baseball long toss.

Automated analysis of trapeziometacarpal joint kinematics using four-dimensional computed tomography.

The aim of this study was to develop an automated approach model to define in vivo kinematics of the trapeziometacarpal (TMC) joint using four-dimensional computed tomography. A total of 15 healthy volunteers were included and their TMC joint kinematics were studied during a retropulsion-opposition-retropulsion movement. We used cardan angles estimated from transformation matrices using a ZYX-decomposition and analysed the motion of the thumb metacarpal relative to the trapezium, the thumb metacarpal relative to the index metacarpal, and the trapezium relative to the index metacarpal. The study also included an analysis of the joint hysteresis effect and a joint proximity model that estimated the joint contact area during a retropulsion-opposition-retropulsion movement. The automated approach significantly decreased the time needed to analyse each case and makes this model applicable for further research on TMC kinematics.

Quantum Euler angles and agency-dependent space-time

Abstract Quantum gravity is expected to introduce quantum aspects into the description of reference frames. Here we begin exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SUq(2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to an aspect of agency dependence: space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness.

Error analysis of inertial navigation based on quaternion approach

Abstract This research aims to establish the properties of the directional cosines and quaternion parameters to examine the proper scale, inclination, and numerical drift errors in a four-dimensional space. The research uses transformation matrices in the four-dimensional space whose elements are the directional cosines or quaternions. The kinematic equations are derived in terms of both directional cosines and quaternions. Applying the Kalman filter allows us to minimize the measurement error and refine the numerical calculation. The observation of the Euler angle covariance matrix takes on a crucial feature in the design of inertial navigation systems vital for analyzing efficient attitude.

Flatness‐based sliding mode control for stabilizing a spherical pendulum on a quadrotor

AbstractThis paper presents the problem of transporting a suspended load by a quadrotor. A full model considering the quadrotor and the dynamics of the suspended load, in a three‐dimensional space, is proposed considering as control inputs the torques and the thrust force due to the motors. The solution proposed consists of simple control strategies based on the tangent linearization of the model, which is controllable and therefore flat. A sliding mode control technique is developed for the thrust force and torques associated with the Euler angles in order to track a desired trajectory of the vehicle in a three‐dimensional space with minimum oscillation of the suspended load. The control strategy results to be relatively simple to implement and achieves local stability of the tracking errors, as well as robustness to internal nonlinearities, which are neglected in the linearization process and other external disturbances. The attractiveness of the sliding surfaces and the stability of the tracking errors are formally studied using Lyapunov stability theory. Simulation results are given to show the performance of the proposed control strategy.

Agile Attitude Maneuver Control of Micro-Satellites for Multi-Target Observation Based on Piecewise Power Reaching Law and Variable-Structure Sliding Mode Control

This paper addresses the issue of agile attitude maneuver control for low-Earth-orbit satellites during short arc segments for multi-target observations. Specifically, a configuration design for Control Moment Gyroscopes (CMGs) and a hybrid control law are provided. The control law is adept at avoiding singularities and escaping singular planes. Subsequently, an optimal time-based attitude maneuver path-planning method is presented, rooted in the relationship between Euler angles/axis and quaternions. Furthermore, a novel satellite attitude maneuver controller is developed based on a piecewise power-reaching law for variable structure sliding mode control. The paper theoretically demonstrates that the proposed piecewise power reaching law possesses two favorable properties regarding convergence time. On the other hand, the designed reaching law maintains continuity at all stages, theoretically eliminating buffeting. The simulation results demonstrate that the proposed controller achieves an Euler angle control precision of ±0.03° and angular velocity accuracy of ±0.15°/s, fulfilling the demands of multi-objective observational tasks. Compared to conventional power reaching law controllers, the convergence time is reduced by 3 s, and Euler angle accuracy is improved by 70%. This underscores the effectiveness of the proposed algorithm.

The anatomy and feeding mechanism of the Japanese giant salamander (Andrias japonicus).

The fully aquatic Japanese giant salamander (Andrias japonicus) is a member of the Cryptobranchidae, and is currently distributed in western Japan, with other members of this group restricted to China and North America. Their feeding behaviour is characterized by a form of suction feeding that includes asymmetric movements of the jaw and hyobranchial apparatus. Previous studies on the North American species, Cryptobranchus alleganiensis, have suggested that this specialized jaw movement is produced by a flexible quadrate-articular joint combined with a loosely connected lower jaw symphysis including two small fibrocartilaginous pads. However, little is known about this feeding behaviour in the Asian species, nor have the three-dimensional asymmetric jaw movements been fully investigated in any member of Cryptobranchidae. In this study, we explore the asymmetric jaw movements in A. japonicus using three methods: (1) dissection of musculoskeletal structures; (2) filming of feeding behaviour to understand in which situations asymmetric feeding is used; (3) analysis of 3D movement of jaws and skull. In the third component, fresh (from frozen) specimens of A. japonicus were manipulated to replicate asymmetric and symmetric jaw movements, with the specimens CT scanned after each step to obtain the 3D morphology of the jaws at different positions. These positions were combined and their Euler angles from resting (closed) jaw position were calculated for asymmetric or symmetric jaw positions. Our filming revealed that asymmetric jaw movements are linked to the position of the prey in relation to the snout, with the jaw closest to the prey opening asymmetrically. Moreover, this action allows the salamander to simultaneously grasp prey in one side of the mouth while ejecting water on the other side, if the first suction attempt fails. The asymmetric jaw movements are performed mainly by rotation of the mandible about its long axis, with very limited lateral jaw movements. During asymmetric and symmetric jaw movements, the posterior ends of the maxilla and quadrate move slightly. The asymmetric jaw movements are permitted by a mobile quadrate-articular joint formed by wide, round cartilages, and by two small fibrocartilage pads within the jaw symphysis that act as cushions during jaw rotation. Some of these soft tissue structures leave traces on the jaws and skull, allowing feeding mode to be reconstructed in fossil taxa. Understanding cryptobranchid asymmetric jaw movement thus requires a comprehensive assessment of not only the symphysial morphology but also that of other cranial and hyobranchial elements.

On the representation and methodology for wide and short range head pose estimation

Head pose estimation (HPE) is a problem of interest in computer vision to improve the performance of face processing tasks in semi-frontal or profile settings. Recent applications require the analysis of faces in the full 360° rotation range. Traditional approaches to solve the semi-frontal and profile cases are not directly amenable for the full rotation case. In this paper we analyze the methodology for short- and wide-range HPE and discuss which representations and metrics are adequate for each case. We show that the popular Euler angles representation is a good choice for short-range HPE, but not at extreme rotations. However, the Euler angles’ gimbal lock problem prevents them from being used as a valid metric in any setting. We also revisit the current cross-data set evaluation methodology and note that the lack of alignment between the reference systems of the training and test data sets negatively biases the results of all articles in the literature. We introduce a procedure to quantify this misalignment and a new methodology for cross-data set HPE that establishes new, more accurate, SOTA for the 300W-LP/Biwi benchmark. We also propose a generalization of the geodesic angular distance metric that enables the construction of a loss that controls the contribution of each training sample to the optimization of the model. Finally, we introduce a wide range HPE benchmark based on the CMU Panoptic data set. code:https://github.com/pcr-upm/opal23_headpose

Serially connected strip-type acoustic wave resonator using X-40°Y LiNbO3 (XSAR)—feasibility study using macro model

This work presents a new serially connected strip-type acoustic resonator using X40°Y-LiNbO3 (LN) (XSAR). The proposed design is composed of a serial of strip-type bulk acoustic wave resonators connected together. The volume between two strips is penetrated through to eliminate the spurious response and enhance quality (Q) factor. The Euler angle (90°, 90°, 40°) of LN is optimized to obtain the maximal piezoelectric constant e 34, which determines the coupling factor of shear mode of the resonator. Simulation results show that the new design is superior in terms of impedance ratio (Z-ratio) and bandwidth (BW) and exhibits the largest BW of 32% ever reported. The slow shear mode spurious response is successfully removed by the proposed penetrated structure combined with narrow strip. Furthermore, the device was prototyped in macro scale using 350 μm thick LN, by which the ultra-wide BW and potentially good performance in HF were confirmed.

Configuration Investigation, Structure Design and Deployment Dynamics of Rigid-Reflector Spaceborne Antenna with Deviation-Angle Panel.

Rigid-reflector spaceborne antennas (RRSAs) are well-suited for high-frequency application scenarios due to their high surface accuracy. However, the low stowing efficiency of RRSAs limits the aperture diameters and further deteriorates the electromagnetic (EM) performances in terms of gain, resolution and sensitivity. After conducting systematic feature analysis with respect to several typical RRSAs, we propose a novel type of RRSA to solve the aforementioned problems. Inspired by the pose adjustment process for a higher stowing efficiency of traditional RRSAs, we also propose a new segmentation scheme of a reflective surface consisting of a deviation-angle panel that facilitates a higher stowing efficiency. Based on this scheme, its corresponding folded configuration is implemented by combining Euler's rotation theorem and the idea of parameter identification. In addition, we also compare the stowing efficiency of different schemes to verify the high stowing efficiency of the configuration. Finally, we perform mechanism/structure design and deployment dynamics to demonstrate that the antenna can be successfully deployed and exhibits excellent deployment quality. The results suggest that the proposed antenna possesses higher stowing efficiency than that of the same kind, with a stable deployment and interference-free process.