Cosine Matrix Research Articles

Direction Cosine Matrix Estimation from Vector Observations using a Matrix Kalman Filter

This work presents several algorithms that use vector observations in order to estimate the direction cosine matrix (DCM) as well as three constant biases and three time-varying drifts in body-mounted gyro output errors. All the algorithms use the matrix Kalman filter (MKF) paradigm, which preserves the natural formulation of the DCM state-space model equations. Focusing on the DCM estimation problem, the assumption of white noise in the gyro and in the vector observations errors yields reduced and efficient filter covariance computations. The orthogonality constraint on the DCM is handled via the technique of pseudomeasurement, which is naturally embedded in the MKF. Two additional known "brute-force" procedures are implemented for the sake of comparison. Extensive Monte-Carlo simulations illustrate the performances of the different estimators. When estimating only the DCM, it is shown that all the proposed orthogonalization procedures accelerate the estimation convergence. Nevertheless, the pseudomeasurement technique shows a smoother and shorter transient than the brute-force procedures, which on the other hand yield more accurate steady-states. The reduced covariance computations yield a more accurate steady-state than the full covariance computations but show a slower transient. When estimating the DCM as well as the gyro biases and drifts, enforcing orthogonalization seems to penalize the DCM estimation as long as the biases are not correctly identified. For the sake of computation savings during long duration missions, a mixed estimator, switching between long periods of DCM-only estimation and short periods of DCM-biases estimation, appears to be a promising strategy.

Field Programmable Gate Array-Based Attitude Stabilization

Abstract : A system for determining vehicle attitude using a field programmable gate array (FPGA) and low cost gyroscopes is presented. The method is intended to support the stabilization of a short duration, unmanned aerial vehicle. Using a microelectronmechanical system (MEMS) inertial sensor for the calibration and serial interface, the algorithm sidesteps concerns related to electronmagnetic interference and the impact of embedded, proprietary filters. An Allan variance analysis is used to characterize the sensor errors and predict system performance. A floating point representation using a direction cosine matrix is hosted on the FPGA alongside the platform stabilization feedback loops. Although prone to drifting without additional aiding, the derived attitude has been demonstrated to be effective in stabilizing a remotely piloted quadrotor.

Interior Parameters, Exterior Parameters, and a Cayley-Like Transform

A RGUABLY, the most fundamental representation for describing the relative orientation of a rigid body is the orientation matrix, also called the direction cosine matrix. Few believe, however, that this matrix provides the most convenient parameterization of orientation. For that, many instead turn to one of a collection of three-parameter sets, even though it is known that every three-parameter orientation set will encounter some kind of singularity, a fact shown by Stuelpnagel [1]. Others turn to the Euler parameters, which are a nonsingular four-parameter set. The direct relationship between each of the various parameter sets and the orientationmatrix is considered to be of basic importance. So, too, is the time derivative of such relationships, although the angular velocity vector or matrix is usually substituted for the time derivative of the orientationmatrix. The purpose of this note is to investigate the direct relationship between a particular three-parameter orientation set, viz., the modified Rodrigues parameters (MRP), and the orientation matrix. The modified Rodrigues parameters were originally developed by Wiener [2], rediscovered by Marandi and Modi [3], and investigated quite fully by Tsiotras and Longuski [4] and Schaub and Junkins [5]. Nevertheless, there is more to say.

Transfer alignment considering measurement time delay and ship body flexure

This paper deals with the transfer alignment problem of strap-down inertial navigation systems (SDINS), using electro-magnetic (EM) log velocity information and gyrocompass attitude information of the ship. Major error sources for velocity and attitude matching are lever-arm effect, measurement time-delay, and ship-body flexure (flexibility). To reduce these alignment errors, an error compensation method based on delay state augmentation and DCM (direction cosine matrix) partial matching is devised. A linearized error model for a velocity and attitude matching transfer alignment system is devised by first linearizing the nonlinear measurement equation with respect to its time delay, and augmenting the delay state into conventional linear state equations. DCM partial matching is then properly combined with velocity matching to reduce the effects of a ship’s Y-axis flexure. The simulation results show that this method decreases azimuth alignment errors considerably.

A DCM Based Orientation Estimation Algorithm with an Inertial Measurement Unit and a Magnetic Compass

In this paper, Direction Cosine Matrix (DCM) method for attitude and orientation estimation is discussed. DCM method was chosen due to some advantages over the popular methods such as namely Euler Angle, Quaternion in light of reliability, accuracy and computational efforts. Proposed model for each method is developed for methodology comparison. It is shown that normal Kalman Filter in DCM method is better than extended Kalman Filter in Euler and Quaternion based method because it helps avoid the first order approximation error. Methodology errors are verified using Aerospace Blockset of Matlab Simulink.

Attitude Determination for MAVs Using a Kalman Filter

This paper presents a Kalman filter to effectively and economically determine the Euler angles for micro aerial vehicles (MAVs), whose size and payload are severely limited. The filter uses data from a series of micro-electro mechanical system sensors to determine the selected 3 variables of the direction cosine matrix and the bias of the rate gyro sensors as state elements in a dynamic model, with the gravitational acceleration to build a measurement model. For high speed maneuvers, rigid motion equations are used to correct the measurements of the gravitational acceleration. The filter is designed to automatically tune its gain based on the dynamic system state. Simulations indicate that the Euler angles can be determined with standard deviations less than 3°. The algorithm was successfully implemented in a miniature attitude measurement system suitable for MAVs. Aerobatic flights show that the attitude determination algorithm works effectively. The attitude determination algorithm is effective and economical, and can also be applied to bionic robofishs and land vehicles, whose size and payload are also greatly limited.

1222 配向分布を有する強化基材を含む複合材料の巨視的熱伝導率のマイクロメカニックス解析(J10-2 マイクロ・ナノ材料システムの力学と強度・機能評価(2) 熱伝導・粘弾性特性評価,ジョイントセッション,21世紀地球環境革命の機械工学:人・マイクロナノ・エネルギー・環境)

Micromechanical analysis is performed on the macroscopic thermal conductivity of a composite material with various misoriented elliptic reinforcements by using the equivalent inclusion method. The disturbance of heat flux occured in the material due to the reinforcement is estimated in its local-coodinate along the semi-axies of reinfocemant. By transforming the disturbance of heart flux into the global coodinate along the applied heat flux and by adopting the Mori-Tanaka theorem to the model, the volumetric average of the disturbance of heat flux can be estimated successfully. As a result, the macroscopic thermal conductivity of the composite can be expressed as a fuction of the direction cosine matrix of the orientaion of reinforcement.

Roundoff error analysis of fast DCT algorithms in fixed point arithmetic

Discrete cosine transforms (DCT) are essential tools in numerical analysis and digital signal processing. Processors in digital signal processing often use fixed point arithmetic. In this paper, we consider the numerical stability of fast DCT algorithms in fixed point arithmetic. The fast DCT algorithms are based on known factorizations of the corresponding cosine matrices into products of sparse, orthogonal matrices of simple structure. These algorithms are completely recursive, are easy to implement and use only permutations, scaling, butterfly operations, and plane rotations/rotation-reflections. In comparison with other fast DCT algorithms, these algorithms have low arithmetic costs. Using von Neumann–Goldstine’s model of fixed point arithmetic, we present a detailed roundoff error analysis for fast DCT algorithms in fixed point arithmetic. Numerical tests demonstrate the performance of our results.

Symmetric stereographic orientation parameters applied to constrained spacecraft attitude control

The full kinematic properties of a minimal set of rigid body attitude coordinates called Symmetric Stereographic Orientation Parameters (SSOPs) are developed. These coordinates result from a stereographic projection of the Euler parameter constraint hypersphere onto a three-dimensional hyper-plane. As discussed in previous work [5], this family contains the well-known classical and modified Rodrigues parameters. Considering general SSOP projection points, transformations to the Euler parameters and the direction cosine matrix are discussed. The set of three SSOP coordinates have the unique feature that the associated singularity can be placed at a desired principal rotation angle by adjusting the projection point. In contrast to the Rodrigues parameters, the SSOP coordinates do not represent a unique orientation. The impact of this non-uniqueness on the constrained spacecraft attitude control problem is discussed. An attitude feedback control law in terms of SSOPs will inherently avoid reaching this singular attitude description, and thus constrain the attitude error response to be within a well-defined cone. Lyapunov’s direct method is used to illustrate how a SSOP-based control law can be derived to drive the spacecraft attitude away from the singularity and towards a desired orientation. This control law generalizes the previously developed classical and modified Rodrigues parameter-based attitude control laws for general stereographic projection points.

MOON PHASES AS GIS COMPUTER APPLICATION

In this paper, Moon phases will be investigated as a computer application in GIS. Moon phases depend on the position of moon and sun relative to the earth observation points. Objective of this study will be based on the following: (1) The concept of GIS and SDB, where the data measured to determine the moon phases have features as spatial data and can be treated as GIS. (2)The spatial data will be measured at any point on the surface of earth (local view), but all the data models as well as queries will be handled relative to the center of earth. The center of earth can be considered as the global point of view for all the spatial data of the moon phases. Therefore, a special transformation (direction cosine matrix) will be used to transform from local view to global view and vise versa. (3) The moon and sun position modeling which have a lot of work done during the history. The most recent model data will be used in this work. As a result of this work, we can overcome the difficulties due to the local view. Crescent as one of the most critical moon phases is seen in some parts of the earth, and not others. Therefore in this paper we will try to determine the locus of points on earth surface which has the scene of the moon phases of the global view at earth center, i.e. the locus of points at the local view which has the same scene of moon phases as global view.

Energy consistent time integration of planar multibody systems

AbstractThe planar motion of rigid bodies and multibody systems can be easily described by coordinates belonging to a linear vector space. This is due to the fact that in the planar case finite rotations commute. Accordingly, using this type of generalized coordinates can be considered as canonical description of planar multibody systems. However, the extension to the three‐dimensional case is not straightforward. In contrast to that, employing the elements of the direction cosine matrix as redundant coordinates makes possible a straightforward treatment of both planar and three‐dimensional multibody systems. This alternative approach leads in general to differential‐algebraic equations (DAEs) governing the dynamics of rigid body systems. The main purpose of the present paper is to present a comparison of the two alternative descriptions. In both cases energy‐consistent time integration schemes are applied. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A comparison of nonlinear beam finite element formulations

AbstractTwo alternative nonlinear finite element formulations emanating from the Simo‐Reissner beam theory are considered. The orientation of the beam cross section is characterised by a director frame which can be either represented by means of rotational parameters or the direction cosine matrix. In the planar case the use of a single angle can be considered as the canonical formulation. The corresponding finite element approximation relies on the interpolation of the nodal angles. However, the extension of this approach to the three‐dimensional case is nontrivial and often leads to element formulations beeing not frame‐indifferent. On the other hand, the interpolation of the nodal direction cosines in general yields frame‐indifferent element formulations [1, 2]. The present talk focuses on a comparison of the two aforementioned finite element beam formulations for planar problems. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Research on transmission principle and kinematics analysis for involute spherical gear

A new kind of transmission-type spherical gear called ‘ring involute spherical gear mechanism’ is introduced. Compared to the famous TRALLFA spherical gear, this new spherical gear has an involute tooth profile, and a ring tooth that is distributed continuously on the surface of the sphere. This allows the gear to overcome two disadvantages of the TRALLFA spherical gear: the drive principle error and the manufacturing difficulty. The new transmission and the formation principle of the tooth’s surface of the new spherical gear mechanism are first introduced, then another mechanism, called disk rack, is introduced, which is derived from the spherical gear mechanism when one of the spherical gear’s tooth number reaches infinity. To make the research more convenient, every part of the new spherical gear mechanism is named. In the following sections, some problems are discussed, such as the assembly form, the construction characteristics, the correct meshing condition, the continuum transmission condition and so on. Furthermore, the paper deduces the surface formula of the conjugate teeth profiles, which proves that the conjugate teeth profiles is also one of the ringed involute spherical gear. In order to analyze the relationship between two coordinate systems, which is attached respectively to the diving spherical gear and the driven spherical gear, the orientation cosine matrix method is utilized. By series rotational transformation, the kinematics model and inverse kinematics model are deduced. Using the method for calculating the transmission ratio of planet gear train, the relationship of the two oscillating angle between output axis and the bracket is established. Based on the research, the kinematics graphic simulation of spherical gear mechanism and disk rack are made respectively. The results prove correctness of the kinematics model.

Field-free one-dimensional alignment of ethylene molecule

We report an experimental study of nonadiabatic laser-induced molecular alignment of ethylene $({\mathrm{C}}_{2}{\mathrm{H}}_{4})$ using a linearly polarized short laser pulse of moderate intensity. The information about the confinement of the $\mathrm{C}\mathrm{C}$ bond axis along the direction of the applied electric field is obtained by measuring the depolarization of a second short pulse of weak intensity interacting with the molecules after they have been exposed to the first pulse. The experimental data are compared with the numerical simulation of the Schr\"odinger equation written for the nonresonant interaction of an asymmetric top rigid rotor with a linearly polarized electric field. The field-free alignment is described by analyzing the time evolution of the angular distribution of the molecular axes together with the expectation value of the relevant squared direction cosine matrix elements.

Strapdown inertial navigation using dual quaternion algebra: error analysis

In a strapdown inertial navigation system (INS), the general displacement of a rigid body is traditionally separately modeled and analyzed, i.e., direction cosine matrix or quaternion for rotation analysis and vector for translation analysis. As a subsequent work of a companion paper (Wu et al., 2005), this paper adopts dual quaternion algebra, a most concise and unified mathematical tool for representing the general displacement of a rigid body, to analyze error characteristics of the strapdown INS. Two new error models in terms of quaternion algebra are developed: the additive dual quaternion error (ADQE) model and multiplicative dual quaternion error (MDQE) model. Both are expected to facilitate the future inertial navigation-based integrated navigation filter.

Worst-case error analysis of lifting-based fast DCT-algorithms

Integer DCTs have a wide range of applications in lossless coding, especially in image compression. An integer-to-integer DCT of radix-2-length n is a nonlinear, left-invertible mapping, which acts on /spl Zopf//sup n/ and approximates the classical discrete cosine transform (DCT) of length n. All known integer-to-integer DCT-algorithms of length 8 are based on factorizations of the cosine matrix C/sub 8//sup II/ into a product of sparse matrices and work with lifting steps and rounding off. For fast implementation one replaces floating point numbers by appropriate dyadic rationals. Both rounding and approximation leads to truncation errors. In this paper, we consider an integer-to-integer transform for (2/spl times/2) rotation matrices and give estimates of the truncation errors for arbitrary approximating dyadic rationals. Further, using two known integer-to-integer DCT-algorithms, we show examplarily how to estimate the worst-case truncation error of lifting based integer-to-integer algorithms in fixed-point arithmetic, whose factorizations are based on (2/spl times/2) rotation matrices.

Matrix recursive expressions of the DFT of even and odd complex sequences

AbstractThe discrete Fourier transform of even complex sequences involves, in matrix formulation, a cosine matrix and, in the same way, the discrete Fourier transform of odd complex sequences is related with a sine matrix. Using structural characteristics of the two matrices, whose order is half the length of the symmetric input data, some recursive expressions of them will be constructed.Unlike the previous classical forms of the discrete Fourier transform of an even or odd real vector, the recursive terms in the matrix expressions derived here only involve the same cosine and sine matrices, with halved order. That is, it is shown that, as for the FFT of a general complex sequence, a divide and conquer approach can also be used to compute the DFT of even or odd sequences. The recursion is closed in the family of the sine and cosine matrices used to describe the DFT of even and odd sequences and no other kind of sine or cosine based matrices are required in recursive terms.Although the obtained results give a new theoretical view on the problem of computing the DFT of even and odd complex sequences, these are not of immediate practical use since the computational cost is higher than other currently employed algorithms. Copyright © 2005 John Wiley & Sons, Ltd.

A second-order scheme for integration of one-dimensional dynamic analysis

This paper proposes a second-order scheme of precision integration for dynamic analysis with respect to long-term integration. Rather than transforming into first-order equations, a recursive scheme is presented in detail for direct solution of the homogeneous part of second-order algebraic and differential equations. The sine and cosine matrices involved in the scheme are calculated using the so-called 2 N algorithm. Numerical tests show that both the efficiency and the accuracy of homogeneous equations can be improved considerably with the second-order scheme. The corresponding particular solution is also presented, incorporated with the second-order scheme where the excitation vector is approximated by the truncated Taylor series.

Gyroless Attitude Estimation of Sun-Pointing Satellites Using Magnetometers

Deterministic attitude estimation methods based on vector observations are used for the coarse-attitude calculation of the sun-pointing satellites. The objective of the deterministic attitude estimation is to get the direction cosine matrix without considering the stochastic noises. The existing deterministic attitude estimation methods require two complete (three-axis) vector observations corresponding to two known reference vectors. This letter suggests a new deterministic attitude estimation method using only geomagnetic-field measurements on two axes for sun-pointing satellites in the sun-acquisition mode.

Multibody dynamics of closed, open, and switching loop mechanical systems

The vast mechanical systems could be classified as closed loop system, open loop system and open & closed (switching) system In the closed loop system, the kinematics and dynamics of 3-D mechanisms will be reviewed and closed form solutions using the direction cosine matrix method and reflection transformation method will be introduced In the open loop system, kinematic & dynamic analysis methods regarding the redundant system which has more degrees of freedom in joint space than those of task space are leviewed and discussed Finally, switching system which changes its phase between closed and open loop motion is investigated with the pinciple of dynamical balance Among switching systems, the human gait in biomechanics and humanoid in robotics are presented