reply to comments on the discrete periodic radon transform this paper presents the reply for the comment made on the discrete periodic radon transform by a m grigoryan this comment presents a series of paper about tensor and paired transform as applied to the fast realization of two dimensional discrete fourier transform the reply indicates that the claim made by a m grigoryan was incorrect the contributions of t hsung and d p k lun are far more than that described by a m grigoryan the dprt is not only a forward transform for computing 2 d dft it is a complete discrete transformation method that includes efficient inverse transform

minimum energy filtering for attitude estimation in this work we study minimum energy filtering for attitude kinematics with vectorial measurements using mortensen s approach the exact form of a minimum energy attitude observer is derived and is shown to depend on the hessian of the value function of an associated optimal control problem a suitably chosen matrix representation of the hessian operator leads to a riccati equation that approximates a minimum energy attitude filter an extended version of the proposed approximate filter is included for a situation where there is slowly time varying bias in the gyro measurements a unit quaternion version of the proposed filter is derived and shown to outperform the multiplicative extended kalman filter mekf for situations with large initialization errors or large measurement errors

nonlinear estimator design on the special orthogonal group using vector measurements directly the convergence properties of popular nonlinear attitude estimators can be traced to the choice of an attitude error function this paper considers a nonlinear deterministic direction cosine matrix estimator whose form is derived from an alternate attitude error function while the resulting estimator shares several properties with those previously presented in the literature the careful selection of an attitude error function results in an estimator with superior convergence properties the attitude estimate is propagated using a rate gyroscope measurement and corrected using two or more vector measurements simulation and experimental results are presented that highlight the desirable properties of the proposed estimator

attitude estimation with intermittent measurements we propose a framework for attitude estimation on the special orthogonal group so 3 using intermittent body frame vector measurements we consider the case where the vector measurements are synchronously intermittent all measurements are received at the same time and the case where the vector measurements are asynchronously intermittent not all measurements are received at the same time the proposed observers have a measurement triggered structure where the attitude is predicted using the continuously measured angular velocity when the vector measurements are not available and adequately corrected upon the arrival of the vector measurements a hybrid framework is proposed to capture the behaviour of the closed loop system by extending the state with timers that are reset at each jump of the observer state almost global asymptotic stability is shown using rigorous lyapunov techniques for hybrid systems

hybrid global exponential stabilization on so 3 we propose a central synergistic hybrid approach for global exponential stabilization on the special orthogonal group so 3 so 3 we introduce a new switching concept relying on a central family of possibly non differentiable potential functions that enjoy the following properties 1 being quadratic as well as their gradients with respect to the euclidean attitude distance and 2 being synergistic with respect to the gradient s singular and or critical points the proposed approach is used to solve the attitude tracking problem leading to global exponential stability results

involutive automorphisms and iwasawa decomposition of some hyperbolic kac moody algebras the involutive automorphisms of hyperbolic kac moody algebras are computed from satake diagrams of these algebras which are then used to furnish a general procedure of their iwasawa decomposition in particular 1 1 and 2 1 are considered as representative examples to illustrate the efficacy of the underlying treatment

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

satake superdiagrams real forms and iwasawa decomposition of classical lie superalgebras satake superdiagrams corresponding to the lie superalgebras c n together with the exceptional ones are constructed from their dynkin diagrams with a view to determine the real forms of these superalgebras the involutive automorphisms are computed from a modified formula to account for all pertinent cases in a consistent way this mechanism is also used to furnish a general treatment of the iwasawa and langlands decompositions of these superalgebras in particular we compute these decompositions for for illustrational purposes and obtain the corresponding induced representations using the method of schmidt construction

attitude estimation using biased gyro and vector measurements with time varying reference vectors we present two results on attitude estimation using vector and rate gyro measurements the first result concerns an observer previously presented by hamel mahony and pflimlin with proven stability results when i the reference vectors are stationary or ii the gyro measurements are unbiased we prove semiglobal stability without either of these assumptions when a parameter projection is added and convergence from all initial attitudes when using a resetting strategy the second result is an algorithm for estimation of bias in the body fixed vector measurements which is analyzed in combination with the attitude and gyro bias observer

principle of superposition by direction images this paper discusses the decomposition of the image by direction images which is based on the concept of the tensor representation and its advanced form the paired representation the 2 d image is considered of the size n n where n is prime a power of two and a power of odd primes the tensor and paired representations in the frequency and time domain define the image as a set of 1 d signals which we call splitting signals each of such splitting signals is calculated as the sum of the image along the parallel lines and it defines the direction image as a component of the original image the unique decomposition of the image by direction images is described and formulas for the inverse tensor and paired transforms are given these formulas can be used for image reconstruction from projections when splitting signals or their direction images are calculated directly from the projection data the number of required projections is uniquely defined by the tensor representation of the image

dynkin diagrams for hyperbolic kac moody algebras hyperbolic algebras such as e10 are based on minkowskian root spaces that revert to finite or affine root systems upon the removal of any simple root in string theory terms finite lie affine kac moody and hyperbolic kac moody algebras are generated by tachyon vertex operators trachyon plus photon vertex operators and vertex operators for all mass levels respectively the 136 possible hyperbolic dynkin diagrams between the ranks 3 and 10 are classified and exhibited completing an earlier enumeration by kac of the 18 rank 7 10 cases the rank 2 hyperbolic algebras infinite in number have been classified by lepowsky and moody 1979

image representation via a finite radon transform a model of finite radon transforms composed of radon projections is presented the model generalizes to finite group projections in the classical radon transform theory the radon projector averages a function on a group over cosets of a subgroup reconstruction formulae that were formally similar to the convolved backprojection ones are derived and an iterative reconstruction technique is found to converge after a finite number of steps applying these results to the group z sub 2 sup p new computationally favorable image representations have been obtained a numerical study of the transform coding aspects is attached

exponential stability of an attitude tracking control system on so 3 for large angle rotational maneuvers abstract this paper studies a tracking control system for the attitude dynamics of a rigid body by selecting an attitude error function carefully we show that the proposed control system guarantees a desirable tracking performance uniformly for rotational maneuvers involving a large initial attitude error a strict lyapunov analysis is presented to show exponential stability and a sufficient condition to avoid non differentiable points of the attitude error function is also shown the proposed control system is directly developed on the special orthogonal group to avoid complexities and ambiguities associated with other attitude representations such as euler angles or quaternions these are illustrated by numerical examples

involutive automorphisms and iwasawa decomposition of affine kac moody superalgebras the involutive automorphisms of affine kac moody superalgebras are computed from satake superdiagrams corresponding to these algebras these are then used to furnish a general treatment of the iwasawa decomposition of these algebras in particular we consider a 1 0 1 and b 1 1 1 as representative examples for the purpose of illustration

image reconstruction from finite number of projections method of transferring geometry this paper introduces a novel method of image reconstruction from a finite number of projections by processing the image along parallel rays the geometry from the image plane is transferred to the cartesian lattice by means of using the original image s line integrals to calculate the line sums of the discrete image to be reconstructed such a transformation of geometry allows for the 2d discrete paired transform whose complete set of functions is defined by directions to be effectively used in the exact reconstruction of the original image the model of image reconstruction is described and both examples and experimental results of implementation of the proposed method are provided for reconstruction on the cartesian lattice of size 2r 2r where r 2

the discrete periodic radon transform in this correspondence a discrete periodic radon transform and its inversion are developed the new discrete periodic radon transform possesses many properties similar to the continuous radon transform such as the fourier slice theorem and the convolution property etc with the convolution property a 2 d circular convolution can be decomposed into 1 d circular convolutions hence improving the computational efficiency based on the proposed discrete periodic radon transform we further develop the inversion formula using the discrete fourier slice theorem it is interesting to note that the inverse transform is multiplication free this important characteristic not only enables fast inversion but also eliminates the finite word length error that may be generated in performing the multiplications

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

iwasawa decomposition of affine kac moody algebras using satake diagrams we carry out the iwasawa decomposition of the affine kac moody algebras within the general framework of involutive automorphisms which are determined with the help of the satake diagrams corresponding to these algebras the parabolic subalgebras essential for the construction of induced representations are computed from the langlands decomposition in particular we consider the untwisted versions viz a1 1 and a2 1 as representative examples for the purpose of illustration

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

comments on the discrete periodic radon transform in the above paper the new discrete periodic radon transform was proposed we would like to prove that this transform was published earlier by artyom grigoryan in 1984 and 1986 in the ussr and was known as the tensor or vector transform the applications of the tensor and more advanced paired transforms for image reconstruction of discrete images were also described by artyom and merughan grigoryan in many publications from 1986 through 1990 and later the paired transform was introduced as an orthogonal 2 d discrete transform which was completely defined by the direction binary functions the paired transform showed a way of collecting the discrete linear integrals or projection data to compose the unitary 2 d transform and reconstruct the image

2 d and 1 d multipaired transforms frequency time type wavelets a concept of multipaired unitary transforms is introduced these kinds of transforms reveal the mathematical structure of fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images considering paired transforms we analyze simultaneously the splitting of the multidimensional fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one dimensional 1 d signals that determine such splitting the main properties of the orthogonal system of paired functions are described and the matrix decompositions of the fourier and hadamard transforms via the paired transforms are given the multiplicative complexity of the two dimensional 2 d 2 sup r spl times 2 sup r point discrete fourier transform by the paired transforms is 4 sup r 2 r 7 3 8 3 12 r 3 which shows the maximum splitting of the 5 d fourier transform into the number of the short 1 d fourier transforms the 2 d paired transforms are not separable and represent themselves as frequency time type wavelets for which two parameters are united frequency and time the decomposition of the signal is performed in a way that is different from the traditional haar system of functions

a globally convergent angular velocity observer for rigid body motion the problem of obtaining the angular velocity of a rigid body from orientation and torque measurements only without noisy numerical differentiation is considered a novel angular velocity angular momentum observer for rigid body motion is presented using euler quaternions and a mechanical energy function approach it is shown that the observer estimates converge globally and that the convergence is eventually exponential it is hoped that a mechanical energy function approach to rigid body control can be combined with the observer presented to lead to a globally stable nonlinear observer based rigid body controller in which the observer and controller errors can be separated in much the same way as one can separate controller and observer poles in the output feedback controllers of linear system theory

quantum group symmetries and non local currents in 2d qft we construct and study the implications of some new non local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions including the sine gordon theory and its generalization to affine toda theory these non local currents provide a non perturbative formulation of the theories the symmetry algebras correspond to the quantum affine kac moody algebras thes matrices are completely characterized by these symmetries formals matrices for the imaginary coupling affine toda theories are thereby derived the application of theses matrices to perturbed coset conformal field theory is studied non local charges generating the finite dimensional quantum group in the liouville theory are briefly presented the formalism based on non local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d

near optimal deterministic filtering on the rotation group this technical note considers the problem of obtaining minimum energy state estimates for a system defined on the rotation group the signals of the system are modeled as purely deterministic signals we derive a nonlinear observer filter posed directly on that respects the geometry of the group and achieves a performance that is close to optimal in the sense of minimizing an integral cost that is measuring the state energy the performance of the proposed filter is demonstrated in simulations involving large initialization process and measurement errors where the results are compared against a quaternion implementation of an extended kalman filter our results indicate that the proposed filter achieves better robustness against a range of noise levels and initialization errors

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

a coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise we propose a new algorithm for estimating constant biases in gyro measurements of angular velocity and demonstrate that the resulting estimates converge to the true bias values exponentially fast the new observer is then combined with a nonlinear attitude tracking control strategy in a certainty equivalence fashion and the combination shown via lyapunov analysis to produce globally stable closed loop dynamics with asymptotically perfect tracking of any commanded attitude sequence the analysis is then extended to consider the effects of stochastic measurement noise in the gyro in addition to the bias a simulation is given for a rigid spacecraft tracking a specified time varying attitude sequence to illustrate the theoretical claims

nonlinear complementary filters on the special orthogonal group this paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units the outputs of such systems are characterized by high noise levels and time varying additive biases we formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group so 3 driven by reconstructed attitude and angular velocity measurements lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error the approach taken leads to an observer that we term the direct complementary filter by exploiting the geometry of the special orthogonal group a related observer termed the passive complementary filter is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs both the direct and passive filters can be extended to estimate gyro bias online the passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude this leads to an observer on so 3 termed the explicit complementary filter that requires only accelerometer and gyro outputs is suitable for implementation on embedded hardware and provides good attitude estimates as well as estimating the gyro biases online the performance of the observers are demonstrated with a set of experiments performed on a robotic test bed and a radio controlled unmanned aerial vehicle

method of paired transforms for reconstruction of images from projections discrete model an effective method of discrete image reconstruction from its projections is introduced the method is based on the vector and paired representations of the two dimensional 2d image with respect to the 2d discrete fourier transform such representations yield algorithms for image reconstruction by a minimal number of attenuation measurements in certain projections the proposed algorithms are described in detail for an n spl times n image when n 2 sup r r 1 the inverse formulas for image reconstruction are given the efficiency of the algorithms is expressed in the fact that they require a minimal number of multiplications or can be implemented without such at all the problem of discrete image reconstruction is also considered in three dimensional 3d space namely on the 3d torus where the reconstruction is performed by means of the nonlinear projections that are integral over 3d spirals on the torus

hybrid attitude and gyro bias observer design on so 3 this paper presents an approach for the design of globally exponentially stable hybrid attitude and gyro bias observers on so 3 times mathbb r 3 first we propose a hybrid observer developed in a generic manner involving a generic family of potential functions with specific properties thereafter we present a design method for such potential functions via an appropriate angular warping transformation applied to a new potential function on so 3 this results in a hybrid observer that uses directly body frame measurements of known inertial vectors

a new efficient algorithm to compute the two dimensional discrete fourier transform an algorithm is presented for computation of the two dimensional discrete fourier transform dft the algorithm is based on geometric properties of the integers and exhibits symmetry and simplicity of realization only one dimensional transformation of the input data is required the transformations are independent hence parallel processing is feasible it is shown that the number of distinct n point dfts needed to calculate n n point two dimensional dfts is equal to the number of linear congruences spanning the n n grid examples for n 3 n 4 and n 10 are presented a short apl code illustrating the algorithm is given