Unit Quaternion Research Articles

Event-triggered attitude synchronization of multiple rigid-body systems

Abstract In this paper, an attitude synchronization problem of multiple rigid-body systems is investigated by using an event-based approach. The leader and followers are described by unit quaternions. A nonlinear distributed observer with event-triggered observations is proposed to estimate the attitude and angular velocity of the leader without continuous information exchange. The triggering mechanism is intermittent and asynchronous; and a positive lower bound of inter-event triggering times is given to show that Zeno behavior can be excluded in the intermittent communication sequence for any agent. Based on the estimated attitude and angular velocity of the leader, a distributed controller is synthesized for each follower to achieve attitude synchronization with the leader via intermittent communication. Finally, an example is provided to illustrate the effectiveness of the theoretical results.

Quaternion‐based adaptive control for trajectory tracking of quadrotor unmanned aerial vehicles

SummaryThe problem of trajectory tracking of aerial vehicles with four rotors and dynamic parameter uncertainties is addressed in this article. By exploiting the cascade structure of the translational and attitude dynamics of the quadrotor, a nonlinear hierarchical adaptive control is proposed. The attitude control law is designed based on the unit quaternion, therefore, the singularities of the Euler angles and angle‐axis representation are avoided. Instead of relying on the stability properties of time‐varying cascade systems, the stability of the whole closed‐loop system is proven by an appropriate Lyapunov function. In addition, a novel method to compute the desired orientation based on the desired Cartesian trajectories and position control inputs is proposed. A set of simulations results are presented to assess the performance of the proposed adaptive controller.

Global exponential attitude tracking for spacecraft with gyro bias estimation

Relying on contraction analysis, this paper addresses the global attitude tracking problem of a spacecraft when angular velocity measurements are corrupted by bias. A nonlinear observer with exponential convergence is designed firstly to estimate the bias in gyro sensors. Then an exponentially convergent attitude tracking controller with gyro bias correction is devised. Next, to remove the topological constraints of unit quaternions for global stability, a switching variable with hysteresis is incorporated in the control loop, enhancing the robustness in the presence of measurement noise and energy efficiency by preventing the unwinding phenomenon. Numeric simulations are shown to illustrate the performance and compare with other similar controllers in terms of tracking error, estimation error and energy efficiency, as well as the robustness to noisy measurements and time-varying bias in gyro sensors.

Modeling of Inverse Kinematic of 3-DoF Robot, Using Unit Quaternions and Artificial Neural Network

SUMMARYThis paper presents a novel method for modeling a 3-degree of freedom open kinematic chain using quaternions algebra and neural network to solve the inverse kinematic problem. The structure of the network was composed of 3 hidden layers with 25 neurons per layer and 1 output layer. The network was trained using the Bayesian regularization backpropagation. The inverse kinematic problem was modeled as a system of six nonlinear equations and six unknowns. Finally, both models were tested using a straight path to compare the results between the Newton–Raphson method and the network training.

Modelling and control of a spherical pendulum via a non–minimal state representation

ABSTRACT A spherical pendulum is a 2 degree-of-freedom mechanism consisting on a rod whose tip moves on the surface of a sphere. It is common to use two angular coordinates to describe such a system. This paper proposes the use of a non-minimal set of coordinates for modelling and controlling a fully-actuated torque-driven spherical pendulum. These coordinates is merely for the purpose of showing the application of unit quaternions as a useful tool for dealing with the orientation of rigid bodies. First, we recall the properties of unit quaternions, and explain how they can be employed for the definition of such non-minimal pendulum coordinates. Later, the control objective for orientation regulation is established and an inverse-dynamics controller, which uses joint displacement and velocity measurements but also some non-minimal states for the orientation error, is proposed. The stability analysis shows the fulfilment of the control objective and is validated through simulations.

Attitude Estimation Using Low-Cost MARG Sensors With Disturbances Reduction

Attitude information obtained from low-cost magnetic, angular rate, and gravity (MARG) sensors is prone to be deteriorated by disturbances such as external acceleration and magnetic anomaly in operational environments. To address the problem, a robust attitude estimation algorithm using low-cost MARG sensors is proposed. An error state Kalman filter (ESKF) is used for data fusing, in which the attitude errors are interpreted as a small rotation vector, and the gyro bias variation is also estimated. The nominal state and the error state are propagated with unit quaternion and rotation vector, respectively. An attitude correction is followed which combines the nominal and error parts by quaternion multiplication. The robustness of the proposed algorithm is embodied in resisting accuracy degradation of the attitude estimation caused by both external acceleration and magnetic anomalies. While external accelerations and magnetic disturbances are detected online, the accuracy reduction in attitude estimation is prevented by adaptively adjusting the related measurement noise covariance matrix. Besides, a two-step measurement update strategy is designed to guarantee that the roll and pitch update is separated from the yaw update. Various rotation and land vehicle dynamic tests have been conducted to validate the effectiveness and robustness of the proposed algorithm.

Particle swarm optimization for inverse kinematics solution and trajectory planning of 7-DOF and 8-DOF robot manipulators based on unit quaternion representation

Finding the inverse kinematic solution of a serial manipulator has always attracted the attention of optimization enthusiasts, as the solution space is highly nonlinear and, depending on the number of degrees of freedom, has multiple solutions. In the literature, one can find several proposed solutions using heuristic techniques; however, for highly redundant manipulators, e.g., seven or more, the discussions focused on minimizing the positional error. In this paper, a metaheuristic approach is presented to solve not only the inverse kinematics of a 7 and 8 DOF manipulators but the proposed algorithm is used to find the robot's poses for trajectory planning where the robot is required to meet the desired position and orientation based on quaternion representation of each point along the path. The metaheuristic approach used in this paper is particle swarm optimization (PSO), where the unit quaternion is used in the objective function to find the orientation error. The results prove that the use of the unit quaternion representation improved the performance of the algorithm and that our approach can be used not only for individual poses but for trajectory planning.

Real and complex hedgehogs, their symplectic area, curvature and evolutes

Classical (real) hedgehogs can be regarded as the geometrical realiza-tions of formal di¤erences of convex bodies in R n+1. Like convex bodies, hedgehogs can be identi…ed with their support functions. Adopting a pro-jective viewpoint, we prove that any holomorphic function h : C n ! C can be regarded as the 'support function' of a complex hedgehog H h in C n+1. In the same vein, we introduce the notion of evolute of such a hedgehog H h in C 2 , and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a 'Brunn-Minkowski theory for complex hedgehogs' (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give …rst two results in this direction. We next return to real hedgehogs in R 2n endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on R 4 endowed with a linear Kahler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider brie ‡y the convolution of hedgehogs, and the particular case of hedgehogs in R 4n regarded as a hyperkahler vector space.

Optimization of building model based on 5G virtual reality technology in computer vision software.

The 5G virtual reality system needs to interact with the user to draw the scene in real time. The contradiction between the complexity of the scene model and the real-time interaction is the main problem in the operation of the virtual reality system. The model optimization strategy of architectural scene in virtual reality design is studied, and the method of architectural scene model optimization is summarized. This article aims to study the optimization of computer vision software modeling through 5G virtual reality technology. In this paper, the optimization of the architectural model is studied by the method of image gray scale transformation, computer vision detection technology and virtual modeling technology. The four experiments are comprehensive evaluation and quantitative evaluation, comparison of channel estimation performance of different pilot structures, comparison of calculated and true values of external azimuth elements, and the effect of window-to-wall ratio on energy consumption per unit of residential building. The results show that hollow bricks of building materials have a great impact on the environment. The values of the three pixel coordinates X, Y, and Z calculated by the unit quaternion method are 1.27, 1.3, and -6.11, respectively, while the actual coordinate positions are 1.25, 1.37, and -6.22, respectively. It can be seen that the outer orientation element value calculated by the quaternion-based spatial rear intersection method is not much different from the actual value, and the correct result can be accurately calculated.

Spherical kinematics in 3-dimensional generalized space

In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space [Formula: see text]. It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in [Formula: see text] space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in [Formula: see text] space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for [Formula: see text] in 3-dimensional Eulidean space [Formula: see text] in 3-dimensional Lorentzian space [Formula: see text] respectively.

Dynamic rotational trajectory planning of a cable-driven parallel robot for passing through singular orientations

This paper presents a dynamic trajectory planning scheme for passing through singular orientations of a three degree-of-freedom (DOF) rotational cable-driven parallel robot (CDPR) that uses a rigid link to restrict its translational DOFs. The kinematics, dynamics and actuation singularity loci of this CDPR are analyzed. Singularity loci divide the workspace into four independent parts, which significantly limit the motion performance of the mechanism. To this end, consistency conditions that can allow the robot to steadily pass through singular orientations are studied. The vector inner product form of consistency conditions is derived, which can simplify the determination of the conditions of a more complex parallel robot. A sequence of target orientations are reached by planning dynamic trajectories using the unit quaternion. A modified spherical linear interpolation with a fifth degree polynomial as the time function is used for the generation of a trajectory which does not contain a singular orientation. Moreover, a transition segment using a seventh degree polynomial is designed to merge into the fifth degree polynomial in order to satisfy the consistency conditions and pass through singular orientations. Simulations are provided to verify the proposed technique.

A geometrical and physical interpretation of quaternionic generalized magnetic flux tubes

In the present paper, we give a generalization for the magnetic flux tubes. Firstly, we define the flux tube via quaternions. We obtain the magnetic flux canal surfaces and flux tubes by quaternion product of a unit quaternion and the unit normal vector of the magnetic field line. Besides, we generate these surfaces by the homothetic motion. Then, we determine the magnetic field components by using the quaternionic representation of the flux tube. Secondly, the flux tube is dictated by the magnetic vector field defined at each point on a generating curve. A local orthonormal basis is attached to every point of the generating curve then the magnetic field is given in terms of the local coordinate directions, leading to a geometric viewpoint of the flux tubes. These provide direct access to the magnetic field line and kinematics equations related to the flux tube. Moreover, the magnetic energy function and magnetic diffusion of the flux tube that has an arbitrary cross-section curve are computed. The impacts of the stretch factor of the flux tube on the magnetic trajectories are investigated. Finally, the analytical solutions of the kinematics equations and several examples of the theory are provided.

Relativistic Combination of Non-Collinear 3-Velocities Using Quaternions

Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by v=2w1+w2, so that w=v1+1−v2=v2+O(v3), and promote them to quaternions using w=wn^, where n^ is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by w1⊕2≡w1⊕w2≡(1−w1w2)−1(w1+w2), and then show that this is also equivalent to w1⊕2=(w1+w2)(1−w2w1)−1. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of w1 with w2. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle Ω and the direction of the combined velocities: eΩ=eΩΩ^=(1−w1w2)−1(1−w2w1), and w^1⊕2=eΩ/2w1+w2|w1+w2|. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the non-commutative algebra of quaternions.

Tracking Controller Design for Satellite Attitude Under Unknown Constant Disturbance Using Stable Embedding

We propose a tracking control law for the fully actuated rigid body system in the presence of any unknown constant disturbance by employing quaternions with the stable embedding technique and Lyapunov stability theory. The stable embedding technique extends the attitude dynamics from the set of unit quaternions to the set of quaternions, which is a Euclidean space, such that the set of unit quaternions is an invariant set of the extended dynamics. Such a stable extension of the system dynamics to a Euclidean space allows us to employ well studied Lyapunov techniques in Euclidean spaces such as LaSalle–Yoshizawa’s theorem. A robust tracking control law is proposed for the attitude dynamics subject to unknown constant disturbance and the convergence properties of the tracking control law is rigorously proven. It is demonstrated with the help of numerical simulations that the proposed control law has a remarkable performance even in some challenging situations.

On the global operator and Fueter mapping theorem for slice polyanalytic functions

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.

Measurement of Prostate Inter-Fraction Movement by Tracking Implanted Fiducials in Daily CBCT Projections

To develop a method for accurately determining implanted fiducial locations in 3D using template matching and sequential stereo in CBCT projection data. This technique will be used in both retrospective studies to improve motion models and to improve fiducial tracking during treatment for motion management. During planning each fiducial is contoured individually. Using non-clinical research software, a separate template is generated from the planning CT for each fiducial. In offline mode each fiducial is tracked in the CBCT projections by template matching followed by sequential stereo. In the offline mode the full scan data is available so the sequential stereo is applied in retrospective analysis mode (as opposed to real-time tracking), and by iteration outliers are removed. The match score threshold used for accepting 2D matches is initially set to a high value and reduced incrementally until a minimum of 100 trajectory points (out of > 900 projections) are produced with a range of motion below a threshold. The estimated 3D location is taken as the average of results from these trajectory points. Horn’s method for absolute orientation using unit quaternions is used to determine the 3D translation and rotation of the fiducials from the planning CT scan to the CBCT. Results from a prostate patient, with 3 implanted fiducials and 9 treatment CBCTs are shown here. The tracked fiducial locations were overlaid on the CBCT scan and evaluated by an observer. No failed detections were identified in this preliminary dataset. The recorded prostate motion was random over the course of treatment, with no clear temporal trends. The largest translations were in the Z (P-A) direction and rotations about the X (L-R) axis. Amplitude of motions were within expected range. The residual mismatch in fiducial positions after rigid 3D transformation was on the order of the voxel size, or less. CBCT projection images can be used directly for 3D fiducial localization. Further analysis is being carried out, but the technique may benefit from not being limited by the CBCT voxel size, not influenced by CBCT reconstruction artifacts and more robust to motion. The technique can be used in both retrospective applications, where projections are archived, and online for motion management.Tabled 1Abstract 2732; TableDays Post SimulationTranslation IEC (mm)Rotation IEC isocenter (deg)TxTyTzRxRyRz13-0.042.17-3.1412.02.72.316-0.67-0.20-0.435.13.80.9300.322.25-8.3124.01.81.7370.600.25-0.196.82.81.7442.710.32-0.097.42.52.6510.701.86-6.2720.04.43.851-0.57-4.064.21-15.53.4-2.061-0.622.98-6.2821.24.42.864-1.642.892.248.35.40.4Mean0.090.94-2.039.933.461.57Range-1.6, 2.7-4.1, 3.0-8.3, 4.2-15.5, 241.8, 5.4-2.0, 3.8 Open table in a new tab

Quaternion-valued recurrent projection neural networks on unit quaternions

Hypercomplex-valued neural networks, including quaternion-valued neural networks, can treat multi-dimensional data as a single entity. In this paper, we present the quaternion-valued recurrent projection neural networks (QRPNNs). Briefly, the QRPNNs are obtained by combining the non-local projection learning with the quaternion-valued recurrent correlation neural network (QRCNNs). We show that the QRPNNs overcome the cross-talk problem of the QRCNNs. Thus, they are appropriate to implement associative memories. Furthermore, computational experiments reveal that the QRPNNs exhibit greater storage capacity and noise tolerance than their corresponding QRCNNs.

Trajectory tracking with quaternion-based attitude representation for autonomous underwater vehicle based on terminal sliding mode control

This paper addresses trajectory tracking control problem for autonomous underwater vehicles (AUVs) using quaternion-based attitude representation. A quaternion based adaptive non-singular terminal sliding mode control scheme is proposed for an AUV subject to ocean current and modelling uncertainty. At first, kinematic equations of the translational and rotational motions are described by unit quaternions. And then, a class of modified non-singular terminal sliding mode surface is developed by smoothly switching into a non-singular sliding mode surface formed by sinusoidal functions when the tracking error reaches a small range of the origin. The finite-time convergences of both the position tracking error and quaternion error are proved based on Lyapunov theory. Finally, the proposed control scheme is applied on a typical fully-actuated AUV to perform simulations. And the simulation results illustrate the effectiveness of the proposed control scheme in comparison with another terminal sliding mode control algorithm.

Quaternion statistics applied to the classification of motion capture data

Unit quaternions give quite new possibilities in an analysis of motion capture data. They provide a compact, holistic axis-angle representation of 3D rotations. An application of descriptive statistics – measures of location and dispersion – is common in numerous problems related to an assessment of joint movements. For those reasons, the paper proposes new approaches to the extraction of motion descriptors on the basis of descriptive statistics of 3D rotations represented by unit quaternions as well as the appropriate classification schemes operating on motion descriptors. Mean and median values and standard deviations are calculated for time series with raw rotational data as well as angular velocities and accelerations. The problem of human gait identification is addressed in the numerical validation of the introduced methods and highly precise marker-based motion capture data are utilized. The results obtained – the accuracy of gait recognition – are compared to the ones achieved by descriptive statistics calculated for time series of Euler angles. The general conclusion is that unit quaternions are effective in the calculation of descriptive statistics. They preserve robust discriminative features of joint movements and they can be applied in numerous challenges of expert and intelligent systems.

Efficient Relative Pose Estimation for Cameras and Generalized Cameras in Case of Known Relative Rotation Angle

We propose two minimal solvers to the problem of relative pose estimation for a camera with known relative rotation angle. In practice, such angle can be derived from the readings of a 3D gyroscope. Different from other relative pose formulations fusing a camera and a gyroscope, the use of relative rotation angle does not require extrinsic calibration between the two sensors. The first proposed solver is formulated for a calibrated regular camera and requires four-point correspondences from two views. The second solver extends the problem to a generalized camera and requires five-point correspondences. We represent the rotation part of the motion in terms of unit quaternions in order to construct polynomial equations encoding the epipolar constraints. The Grobner basis technique is then used to efficiently derive the polynomial solutions. Our first solver for regular cameras significantly improves the existing state-of-the-art solution. The second solver for generalized cameras is novel. The presented minimal solvers can be used in a hypothesize-and-test architecture such as RANSAC for reliable pose estimation. Experiments on synthetic and real datasets confirm that our algorithms are numerically stable, fast, and robust.