Quaternion-Based Attitude Estimation of an Aircraft Model Using Computer Vision.

Investigating aircraft flight dynamics often requires dynamic wind tunnel testing. This paper proposes a non-contact, off-board instrumentation method using vision-based techniques. The method utilises a sequential process of Harris corner detection, Kanade-Lucas-Tomasi tracking, and quaternions to identify the Euler angles from a pair of cameras, one with a side view and the other with a top view. The method validation involves simulating a 3D CAD model for rotational motion with a single degree-of-freedom. The numerical analysis quantifies the results, while the proposed approach is analysed analytically. This approach results in a 45.41% enhancement in accuracy over an earlier direction cosine matrix method. Specifically, the quaternion-based method achieves root mean square errors of 0.0101 rad/s, 0.0361 rad/s, and 0.0036 rad/s for the dynamic measurements of roll rate, pitch rate, and yaw rate, respectively. Notably, the method exhibits a 98.08% accuracy for the pitch rate. These results highlight the performance of quaternion-based attitude estimation in dynamic wind tunnel testing. Furthermore, an extended Kalman filter is applied to integrate the generated on-board instrumentation data (inertial measurement unit, potentiometer gimbal) and the results of the proposed vision-based method. The extended Kalman filter state estimation achieves root mean square errors of 0.0090 rad/s, 0.0262 rad/s, and 0.0034 rad/s for the dynamic measurements of roll rate, pitch rate, and yaw rate, respectively. This method exhibits an improved accuracy of 98.61% for the estimation of pitch rate, indicating its higher efficiency over the standalone implementation of the direction cosine method for dynamic wind tunnel testing.

Adaptive Control for Gravitational Wave Detection Formation Considering Time-Varying Communication Delays.

A distributed six-degree-of-freedom (6-DOF) cooperative control for multiple spacecraft formation is investigated considering parametric uncertainties, external disturbances, and time-varying communication delays. Unit dual quaternions are used to describe the kinematics and dynamics models of the 6-DOF relative motion of the spacecraft. A distributed coordinated controller based on dual quaternions with time-varying communication delays is proposed. The unknown mass and inertia, as well as unknown disturbances, are then taken into account. An adaptive coordinated control law is developed by combining the coordinated control algorithm with an adaptive algorithm to compensate for parametric uncertainties and external disturbances. The Lyapunov method is used to prove that the tracking errors converge globally asymptotically. Numerical simulations show that the proposed method can realize cooperative control of attitude and orbit for the multi-spacecraft formation.

Low-Cost Orientation Determination System for CubeSat Based Solely on Solar and Magnetic Sensors.

CubeSats require accurate determination of their orientation relative to the Sun, Earth, and other celestial bodies to operate successfully and collect scientific data. This paper presents an orientation system based on solar and magnetic sensors that offers a cost-effective and reliable solution for CubeSat navigation. Solar sensors analyze the illumination on each face to measure the satellite's orientation relative to the Sun, while magnetic sensors determine the Earth's magnetic field vector in the satellite's reference frame. By combining the measured data with the known ephemeris of the satellite, the satellite-Sun vector and the magnetic field orientation can be reconstructed. The orientation is expressed using quaternions, representing the rotation from the internal reference system of the satellite to the selected reference system. The proposed system demonstrates the ability to accurately determine the orientation of a CubeSat using only two sensors, making it suitable for installations where more complex and expensive instruments are impractical. Additionally, the paper presents a mathematical model of a low-cost CubeSat orientation system and a hardware implementation of the sensor. The technology, using solar and magnetic sensors, provides a reliable and affordable solution for CubeSat navigation, supporting the increasing sophistication of miniature payloads and enabling accurate satellite positioning in space missions.

A Kalman Filter for Nonlinear Attitude Estimation Using Time Variable Matrices and Quaternions.

The nonlinear problem of sensing the attitude of a solid body is solved by a novel implementation of the Kalman Filter. This implementation combines the use of quaternions to represent attitudes, time-varying matrices to model the dynamic behavior of the process and a particular state vector. This vector was explicitly created from measurable physical quantities, which can be estimated from the filter input and output. The specifically designed arrangement of these three elements and the way they are combined allow the proposed attitude estimator to be formulated following a classical Kalman Filter approach. The result is a novel estimator that preserves the simplicity of the original Kalman formulation and avoids the explicit calculation of Jacobian matrices in each iteration or the evaluation of augmented state vectors.

ALICE: Conceptual Development of a Lower Limb Exoskeleton Robot Driven by an On-Board Musculoskeletal Simulator.

OBJECTIVE: In this article, we present the conceptual development of a robotics platform, called ALICE (Assistive Lower Limb Controlled Exoskeleton), for kinetic and kinematic gait characterization. The ALICE platform includes a robotics wearable exoskeleton and an on-board muscle driven simulator to estimate the user's kinetic parameters. BACKGROUND: Even when the kinematics patterns of the human gait are well studied and reported in the literature, there exists a considerable intra-subject variability in the kinetics of the movements. ALICE aims to be an advanced mechanical sensor that allows us to compute real-time information of both kinetic and kinematic data, opening up a new personalized rehabilitation concept. METHODOLOGY: We developed a full muscle driven simulator in an open source environment and validated it with real gait data obtained from patients diagnosed with multiple sclerosis. After that, we designed, modeled, and controlled a 6 DoF lower limb exoskeleton with inertial measurement units and a position/velocity sensor in each actuator. SIGNIFICANCE: This novel concept aims to become a tool for improving the diagnosis of pathological gait and to design personalized robotics rehabilitation therapies. CONCLUSION: ALICE is the first robotics platform automatically adapted to the kinetic and kinematic gait parameters of each patient.

Foundations of the Quaternion Quantum Mechanics.

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary 'i' should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

Evaluation of Pattern Recognition Methods for Head Gesture-Based Interface of a Virtual Reality Helmet Equipped with a Single IMU Sensor.

The motivation of this paper is to examine the effectiveness of state-of-the-art and newly proposed motion capture pattern recognition methods in the task of head gesture classifications. The head gestures are designed for a user interface that utilizes a virtual reality helmet equipped with an.

A Tangible Solution for Hand Motion Tracking in Clinical Applications.

Objective real-time assessment of hand motion is crucial in many clinical applications including technically-assisted physical rehabilitation of the upper extremity. We propose an inertial-sensor-based hand motion tracking system and a set of dual-quaternion-based methods for estimation of finger segment orientations and fingertip positions. The proposed system addresses the specific requirements of clinical applications in two ways: (1) In contrast to glove-based approaches, the proposed solution maintains the sense of touch. (2) In contrast to previous work, the proposed methods avoid the use of complex calibration procedures, which means that they are suitable for patients with severe motor impairment of the hand. To overcome the limited significance of validation in lab environments with homogeneous magnetic fields, we validate the proposed system using functional hand motions in the presence of severe magnetic disturbances as they appear in realistic clinical settings. We show that standard sensor fusion methods that rely on magnetometer readings may perform well in perfect laboratory environments but can lead to more than 15 cm root-mean-square error for the fingertip distances in realistic environments, while our advanced method yields root-mean-square errors below 2 cm for all performed motions.

A novel kinematics analysis method using quaternion interpolation-a case study in frog jumping.

Spherical Linear Interpolation (SLERP) has long been used in computer animation to interpolate movements between two 3D orientations. We developed a forward kinematics (FK) approach using quaternions and SLERP to predict how frogs modulate jump kinematics between start posture and takeoff. Frog limb kinematics have been studied during various activities, yet the causal link between differences in joint kinematics and locomotor variation remains unknown. We varied 1) takeoff angle from 8 to 60°; 2) turn angle from 0 to 18°; and 3) initial body pitch from 0 to 70°. Simulations were similar to experimentally observed frog kinematics. Findings suggest a fundamental mechanism whereby limb elevation is modulated by thigh and shank adduction. Forward thrust is produced by thigh and proximal foot retraction with little contribution from the shank except to induce asymmetries for turning. Kinematic shifts causing turns were subtle, marked only by slight counter-rotation of the left versus right shank as well as a 10% timing offset in proximal foot adduction. Additionally, inclining initial body tilt influenced the centre of mass trajectory to determine direction of travel at takeoff. Most importantly, our theory suggests firstly that the convergence of leg segment rotation axes toward a common orientation is crucial both for limb extension and for coordinating jump direction; and, secondly, the challenge of simulating 3D kinematics is simplified using SLERP because frog limbs approximately follow linear paths in unit quaternion space. Our methodology can be applied more broadly to study living and fossil frog taxa as well as to inspire new control algorithms for robotic limbs.

[Local helix parameters fitting of proteins based on dual quaternions registration method].

A fitting method of calculating local helix parameters of proteins based on dual quaternions registration fitting (DQRFit) is proposed in this paper. First, the C and N atom coordinates of each residue in the protein structure data are extracted. Then the unregistered data and reference data are constructed using the sliding windows. The square sum of the distance of the data points before and after registration is regarded as an optimization goal. We calculate the optimal rotation matrix and the translation vector using the dual quaternion registration algorithm, and get the helix parameters of the secondary structure which contain the number of residues per turn( τ), helix radius( ρ)and helix pitch( p). Furthermore, we can achieve the fitting of three-helix parameters of τ, ρ, p simultaneously with the dual quaternion registration, and can adjust the sliding windows to adapt to different error levels. Compared with the traditional helix fitting method, DQRFit has some advantages such as low computational complexity, strong anti-interference, and high fitting accuracy. It is proven that the precision of proposed DQRFit for α helix detection is comparable to that of the dictionary of secondary structure of proteins (DSSP), and is better than that of other traditional methods. This is of great significance for the protein structure classification and functional prediction, drug design, protein structure visualization and other fields in the future.

Dual Quaternions as Constraints in 4D-DPM Models for Pose Estimation.

The goal of this research work is to improve the accuracy of human pose estimation using the Deformation Part Model (DPM) without increasing computational complexity. First, the proposed method seeks to improve pose estimation accuracy by adding the depth channel to DPM, which was formerly defined based only on red-green-blue (RGB) channels, in order to obtain a four-dimensional DPM (4D-DPM). In addition, computational complexity can be controlled by reducing the number of joints by taking it into account in a reduced 4D-DPM. Finally, complete solutions are obtained by solving the omitted joints by using inverse kinematics models. In this context, the main goal of this paper is to analyze the effect on pose estimation timing cost when using dual quaternions to solve the inverse kinematics.

Human Actions Analysis: Templates Generation, Matching and Visualization Applied to Motion Capture of Highly-Skilled Karate Athletes.

The aim of this paper is to propose and evaluate the novel method of template generation, matching, comparing and visualization applied to motion capture (kinematic) analysis. To evaluate our approach, we have used motion capture recordings (MoCap) of two highly-skilled black belt karate athletes consisting of 560 recordings of various karate techniques acquired with wearable sensors. We have evaluated the quality of generated templates; we have validated the matching algorithm that calculates similarities and differences between various MoCap data; and we have examined visualizations of important differences and similarities between MoCap data. We have concluded that our algorithms works the best when we are dealing with relatively short (2-4 s) actions that might be averaged and aligned with the dynamic time warping framework. In practice, the methodology is designed to optimize the performance of some full body techniques performed in various sport disciplines, for example combat sports and martial arts. We can also use this approach to generate templates or to compare the correct performance of techniques between various top sportsmen in order to generate a knowledge base of reference MoCap videos. The motion template generated by our method can be used for action recognition purposes. We have used the DTW classifier with angle-based features to classify various karate kicks. We have performed leave-one-out action recognition for the Shorin-ryu and Oyama karate master separately. In this case, 100 % actions were correctly classified. In another experiment, we used templates generated from Oyama master recordings to classify Shorin-ryu master recordings and vice versa. In this experiment, the overall recognition rate was 94.2 % , which is a very good result for this type of complex action.

Variability Analysis of Therapeutic Movements using Wearable Inertial Sensors.

A variability analysis of upper limb therapeutic movements using wearable inertial sensors is presented. Five healthy young adults were asked to perform a set of movements using two sensors placed on the upper arm and forearm. Reference data were obtained from three therapists. The goal of the study is to determine an intra and inter-group difference between a number of given movements performed by young people with respect to the movements of therapists. This effort is directed toward studying other groups characterized by motion impairments, and it is relevant to obtain a quantified measure of the quality of movement of a patient to follow his/her recovery. The sensor signals were processed by applying two approaches, time-domain features and similarity distance between each pair of signals. The data analysis was divided into classification and variability using features and distances calculated previously. The classification analysis was made to determine if the movements performed by the test subjects of both groups are distinguishable among them. The variability analysis was conducted to measure the similarity of the movements. According to the results, the flexion/extension movement had a high intra-group variability. In addition, meaningful information were provided in terms of change of velocity and rotational motions for each individual.

Quaternion-Based Gesture Recognition Using Wireless Wearable Motion Capture Sensors.

This work presents the development and implementation of a unified multi-sensor human motion capture and gesture recognition system that can distinguish between and classify six different gestures. Data was collected from eleven participants using a subset of five wireless motion sensors (inertial measurement units) attached to their arms and upper body from a complete motion capture system. We compare Support Vector Machines and Artificial Neural Networks on the same dataset under two different scenarios and evaluate the results. Our study indicates that near perfect classification accuracies are achievable for small gestures and that the speed of classification is sufficient to allow interactivity. However, such accuracies are more difficult to obtain when a participant does not participate in training, indicating that more work needs to be done in this area to create a system that can be used by the general population.

Keeping a Good Attitude: A Quaternion-Based Orientation Filter for IMUs and MARGs.

Orientation estimation using low cost sensors is an important task for Micro Aerial Vehicles (MAVs) in order to obtain a good feedback for the attitude controller. The challenges come from the low accuracy and noisy data of the MicroElectroMechanical System (MEMS) technology, which is the basis of modern, miniaturized inertial sensors. In this article, we describe a novel approach to obtain an estimation of the orientation in quaternion form from the observations of gravity and magnetic field. Our approach provides a quaternion estimation as the algebraic solution of a system from inertial/magnetic observations. We separate the problems of finding the "tilt" quaternion and the heading quaternion in two sub-parts of our system. This procedure is the key for avoiding the impact of the magnetic disturbances on the roll and pitch components of the orientation when the sensor is surrounded by unwanted magnetic flux. We demonstrate the validity of our method first analytically and then empirically using simulated data. We propose a novel complementary filter for MAVs that fuses together gyroscope data with accelerometer and magnetic field readings. The correction part of the filter is based on the method described above and works for both IMU (Inertial Measurement Unit) and MARG (Magnetic, Angular Rate, and Gravity) sensors. We evaluate the effectiveness of the filter and show that it significantly outperforms other common methods, using publicly available datasets with ground-truth data recorded during a real flight experiment of a micro quadrotor helicopter.

Constructing Kinematic Confidence Regions With Double Quaternions.

A spatial displacement as an element of SE ( 3 ) can be approximated by a 4D rotation, which is an element of SO ( 4 ) . In this way, the problem of constructing confidence regions of uncertain spatial displacements may be studied as that of constructing confidence ellipsoids in SO ( 4 ) . In this light, a double-quaternion formulation of kinematic confidence regions is presented that approximately preserve the geometry of SE ( 3 ) . Examples are provided to demonstrate the efficacy of this approach in comparison with the dual-quaternion formulation.

Applications of the Clifford torus to material textures.

This paper introduces a new 2D representation of the orientation distribution function for an arbitrary material texture. The approach is based on the isometric square torus mapping of the Clifford torus, which allows for points on the unit quaternion hypersphere (each corresponding to a 3D orientation) to be represented in a periodic 2D square map. The combination of three such orthogonal mappings into a single RGB (red-green-blue) image provides a compact periodic representation of any set of orientations. Square torus representations of five different orientation sampling methods are compared and analyzed in terms of the Riesz s energies that quantify the uniformity of the samplings. The effect of crystallographic symmetry on the square torus map is analyzed in terms of the Rodrigues fundamental zones for the rotational symmetry groups. The paper concludes with example representations of important texture components in cubic and hexagonal materials. The new RGB representation provides a convenient and compact way of generating training data for the automated analysis of material textures by means of neural networks.

Generalization of adding angular momenta and circular potential in quaternionic quantum mechanics.

Complex numbers were created by introducing the imaginary unit i to represent the square root of -1, allowing solutions to equations that involved square roots of negative numbers. Complex numbers were further extended to quaternionic numbers by introducing additional imaginary units, namely "j" and "k". Quaternions are non-commutative and involve four components, with each component being a real number or a multiple of an imaginary unit. Usually complex numbers are used to represent wave functions in quantum mechanics. Solutions using quaternions to square well, spin and angular momentum, Dirac equations have been obtained by many researchers. In this article, we have made use of quaternions to study the generalization of adding angular momenta, the digital signal processing of a quaternionic function and circular potential of a particle in real Hilbert space and have obtained quaternionic solutions in terms of Bessel functions.

On the Construction of Confidence Regions for Uncertain Planar Displacements.

This paper studies the statistical concept of confidence region for a set of uncertain planar displacements with a certain level of confidence or probabilities. Three different representations of planar displacements are compared in this context and it is shown that the most commonly used representation based on the coordinates of the moving frame is the least effective. The other two methods, namely the exponential coordinates and planar quaternions, are equally effective in capturing the group structure of SE(2). However, the former relies on the exponential map to parameterize an element of SE(2), while the latter uses a quadratic map, which is often more advantageous computationally. This paper focus on the use of planar quaternions to develop a method for computing the confidence region for a given set of uncertain planar displacements. Principal component analysis (PCA) is another tool used in our study to capture the dominant direction of movements. To demonstrate the effectiveness of our approach, we compare it to an existing method called rotational and translational confidence limit (RTCL). Our examples show that the planar quaternion formulation leads to a swept volume that is more compact and more effective than the RTCL method, especially in cases when off-axis rotation is present.

A new multimedia cryptosystem using chaos, quaternion theory and modular arithmetic.

Based on the combination of quaternion numbers, residual matrices, and chaotic attractors, a new cryptosystem is proposed for multimedia processing files such as images and audio. The key employed in this encryption schema consists of an image with a wide and sensitive range, obtained from the Julia Quaternion set rendered using a computational tool. Due to the use of quaternion matrices mixing the information between RGB layers and audio samples was possible, whereas using XOR operation and residual matrices modulus 257, added high sensitivity to small perturbations during encryption, key preparation and decryption processes, to such an extent that a minimal change in the image or in the audio leads to a totally different encryption result. The use of dynamic programming also reduced the processing time for matrix operations on the ℤ257 ring. To corroborate security of the algorithm, different tests were performed, including the National Institute of Standards and Technology test obtaining different indicators that were compared with other scientific references of similar works, finding behavioral patterns in accordance with those referenced works.