an algorithm for redundant binary bit pipelined rational arithmetic the authors introduce a redundant binary representation of the rationals and an associated algorithm for computing the sum difference product quotient and other useful functions of two rational operands using this representation the algorithm extends r w gosper s 1972 partial quotient arithmetic algorithm and allows the design of an online arithmetic unit with computations granularized at the signed bit level each input or output port can be independently set to receive produce operands result in either binary radix or the binary rational representation the authors investigate by simulation the interconnection of several such units for the parallel computation of more complicated expressions in a tree pipelined manner with particular regard to measuring individual and compounded online delays

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

graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

feedback particle filter on riemannian manifolds and matrix lie groups this paper is concerned with the problem of continuous time nonlinear filtering of stochastic processes evolving on connected riemannian manifolds without boundary the main contribution of this paper is to derive the feedback particle filter fpf algorithm for this problem in its general form the fpf is shown to provide an intrinsic description of the filter that automatically satisfies the geometric constraints of the manifold the particle dynamics are encapsulated in a stratonovich stochastic differential equation that retains the feedback structure of the original euclidean fpf the implementation of the filter requires a solution of a poisson equation on the manifold and a numerical algorithm is described for this purpose for the special case when the manifold is a matrix lie group explicit formulae for the filter are derived using the matrix coordinates filters for two example problems are presented the attitude estimation problem on so 3 and the robot localization problem in se 3 comparisons are also provided between the fpf and popular algorithms for attitude estimation namely the multiplicative extended kalman filter ekf the invariant ekf the unscented quaternion estimator the invariant ensemble kalman filter and the bootstrap particle filter numerical simulations are presented to illustrate these comparisons

error free solution to a toeplitz system of equations a numerical method which allows error free computation of the solution to a toeplitz system of equations is developed the method requires that the system of equations have rational entries to avoid error that is inherent in floating point arithmetic multiple modulus residue arithmetic is applied to a modified version of the levinson algorithm and to an algorithm presented by s y kung and y h hu 1983 the error free method has a highly parallel structure and can be implemented with existing software and hardware the exact method presented here serves as an outline for the general approach to the development of error free solution methods

riemannian consensus for manifolds with bounded curvature consensus algorithms are popular distributed algorithms for computing aggregate quantities such as averages in ad hoc wireless networks however existing algorithms mostly address the case where the measurements lie in euclidean space in this work we propose riemannian consensus a natural extension of existing averaging consensus algorithms to the case of riemannian manifolds unlike previous generalizations our algorithm is intrinsic and in principle can be applied to any complete riemannian manifold we give sufficient convergence conditions on riemannian manifolds with bounded curvature and we analyze the differences with respect to the euclidean case we test the proposed algorithms on synthetic data sampled from the space of rotations the sphere and the grassmann manifold

invariant extended kalman filter on matrix lie groups abstract we derive symmetry preserving invariant extended kalman filters iekf on matrix lie groups these kalman filters have an advantage over conventional extended kalman filters as the error dynamics for such filters are independent of the group configuration which in turn provides a uniform estimate of the region of convergence the proposed iekf differs from existing techniques in literature on the account that it is derived using minimal tools from differential geometry that simplifies its representation and derivation to a large extent the filter error dynamics is defined on the lie algebra directly instead of identifying the lie algebra with an euclidean space or defining the error dynamics in local coordinates using exponential map and the associated differential riccati equations are described on the corresponding space of linear operators using tensor algebra the proposed filter is implemented for the attitude dynamics of the rigid body which is a benchmark problem in control and its performance is compared against a conventional extended kalman filter ekf numerical experiments support that the iekf is computationally less intensive and gives better performance than the ekf

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

area efficient multipliers based on multiple radix representations in this paper we shall introduce several new algorithms for integer multiplication that are based on specific multiple radix representation of one of the multiplicands we provide extensive theoretical analysis and experimental results for multipliers based on the new representations on 0 18 m cmos technology they provide a clear picture about the advantages of the new method in 64 bit hardware implementations compared to array based classical multiplier and radix 8 based multiplier the proposed multipliers have better area and power consumption compared to reference multipliers

symmetry preserving observers this paper presents the theory of invariant observers i e symmetry preserving observers we consider an observer to consist of a copy of the system and a correction term and we propose a constructive method based on the cartan moving frame method to find all the symmetry preserving correction terms the construction relies on an invariant frame a classical notion and on an invariant output error a less standard notion precisely defined here using the theory we build three non linear observers for three examples of engineering interest a non holonomic car a chemical reactor and an inertial navigation system for each example the design is based on physical symmetries and the convergence analysis relies on the use of invariant state errors a symmetry preserving way to define the estimation error

global optimal attitude estimation using uncertainty ellipsoids a deterministic attitude estimation problem for a rigid body in a potential field with bounded attitude and angular velocity measurement errors is considered an attitude estimation algorithm that globally minimizes the attitude estimation error is obtained assuming that the initial attitude the initial angular velocity and measurement noise lie within given ellipsoidal bounds an uncertainty ellipsoid that bounds the attitude and the angular velocity of the rigid body is obtained the center of the uncertainty ellipsoid provides point estimates and the size of the uncertainty ellipsoid measures the accuracy of the estimates the point estimates and the uncertainty ellipsoids are propagated using a lie group variational integrator and its linearization respectively the attitude and angular velocity estimates are optimal in the sense that the sizes of the uncertainty ellipsoids are minimized

optimum nonlinear filtering this paper is composed of two parts the first part surveys the literature regarding optimum nonlinear filtering from the continuous time stochastic analysis point of view and the other part explores the impact of recent applications of neural networks in a discrete time context to nonlinear filtering in particular the results obtained by using a regularized form of radial basis function rbf networks are presented in fair detail

reduction techniques for strongly graded rings and finite representation type abstract we present reduction techniques for studying the category of lattices over strongly graded orders in particular we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring

automorphisms of tiled orders abstract let be a tiled r order we give a description of autr as the semidirect product of inn and a certain subgroup of aut q where q is the link graph of additionally we give criteria for determining when an element of aut q belongs to this subgroup in terms of the exponent matrix for

a p p bit fraction model of binary floating point division and extremal rounding cases we introduce the ordered series fp p of irreducible p p bit fractions as a model of p bit precision binary floating point division we employ and extend results from the number theoretic literature on farey fractions and continued fractions to provide a foundation for generation and analysis of the series fp p an algorithm for ordered on the fly enumeration of a consecutive subsequence of fp p over a selected interval is introduced which requires only a couple of integer additions and or subtractions per p p bit fraction enumerated we characterize two extremal rounding boundary sets rnp respectively rdp of irreducible p p bit fractions over the standard binade 1 2 whose 2p o 1 member fractions have rational values that are each comparably close to a boundary for rounding to a normalized p bit floating point number by round to nearest respectively by a directed rounding a transformation is shown allowing either set rnp rdp to be simply computed from the other we determine properties of these extremal rounding boundary sets rnp rdp and describe their use in the testing of floating point division implementations

optimal estimation for the satellite attitude using star tracker measurements an optimal estimation scheme is presented which determines the satellite attitude using the gyro readings and the star tracker measurements of a commonly used satellite attitude measuring unit the scheme is mainly based on the exponential fourier densities that have the desirable closure property under conditioning by updating a finite and fixed number of parameters the conditional probability density which is an exponential fourier density is recursively determined simulation results indicate that the scheme is more accurate and robust than extended kalman filtering it is believed that this approach is applicable to many other attitude measuring units as no linearization and approximation are necessary in the approach it is ideal for systems involving high levels of randomness and or low levels of observability and systems for which accuracy is of overriding importance

crossed product algebras defined by separable extensions abstract we generalize the classical construction of crossed product algebras defined by finite galois field extensions to finite separable field extensions by studying properties of rings graded by groupoids we are able to calculate the jacobson radical of these algebras we use this to determine when the analogous construction of crossed product orders yield azumaya maximal or hereditary orders in a local situation thereby we generalize results by haile larson and sweedler

neat idempotents and tiled orders having large global dimension abstract we study neat primitive idempotents in a semiperfect noetherian ring and as an application we improve an example of a tiled order having large global dimension given by jansen and odenthal moreover another two tiled orders having large global dimension are added and two questions on tiled orders of finite global dimension are posed

finite precision rational arithmetic an arithmetic unit the foundations of an arithmetic unit performing the add subtract multiply and divide operations on rational operands are developed the unit uses the classical euclidean algorithm as one unified algorithm for all the arithmetic operations including rounding binary implementations are discussed based on techniques known from srt division and utilizing ripple free borrow save and carry save addition average time behavior is investigated

global attitude and gyro bias estimation based on set valued observers a b s t r a c t the problem of attitude and rate gyro bias estimation is addressed by resorting to measurements acquired from rate gyros and vector observations a set valued observer svo is proposed that has no singulari ties and that for any initial conditions provides a bounding set with guarantees of containing the actual unknown rotation matrix the sensor readings are assumed to be corrupted by bounded measurement noise and constant gyro bias conditions for the boundedness of the estimated sets are established and implementation details are discussed the feasibility of the technique is demonstrated in simulation

regularization in the selection of radial basis function centers subset selection and regularization are two well known techniques that can improve the generalization performance of nonparametric linear regression estimators such as radial basis function networks this paper examines regularized forward selection rfs a combination of forward subset selection and zero order regularization an efficient implementation of rfs into which either delete 1 or generalized cross validation can be incorporated and a reestimation formula for the regularization parameter are also discussed simulation studies are presented that demonstrate improved generalization performance due to regularization in the forward selection of radial basis function centers

finite precision rational arithmetic slash number systems fraction number systems characterized by fixed slash and floating slash formats are specified the structure of arithmetic over such systems is prescribed by the rounding obtained from best rational approximation multitiered precision hierarchies of both the fixed slash and floating slash type are described and analyzed with regards to their support of both exact rational and approximate real computation

intrinsic filtering on lie groups with applications to attitude estimation this paper proposes a probabilistic approach to the problem of intrinsic filtering of a system on a matrix lie group with invariance properties the problem of an invariant continuous time model with discrete time measurements is cast into a rigorous stochastic and geometric framework building upon the theory of continuous time invariant observers we introduce a class of simple filters and study their properties without addressing the optimal filtering problem we show that akin to the kalman filter for linear systems the error equation is a markov chain that does not depend on the state estimate thus when the filter s gains are held fixed the noisy error s distribution is proved to converge to a stationary distribution under some convergence properties of the filter with noise turned off we also introduce two novel tools of engineering interest the discrete time invariant extended kalman filter for which the trusted covariance matrix is shown to converge and the invariant ensemble kalman filter the methods are applied to attitude estimation allowing to derive novel theoretical results in this field and illustrated through simulations on synthetic data

fast rounding in multiprecision floating slash arithmetic a computational algorithm is described which quickly rounds large fractions into a fixed length multiprecision floating slash representation using single precision operations where possible an easily calculated criterion for terminating the rounding process is given

the group of automorphisms of certain subalgebras of matrix algebras the purpose of this note is to find and characterize the full group of automorphisms of certain prime algebras which are subalgebras of matrix algebras over commutative rings here we extend the methods which we used in 6 to characterize automorphisms of certain upper triangular matrix rings for a central simple k algebra a the group of k automorphisms of a is extensively treated in the literature 5 theorem 6 21 other cases where the automorphism group of a ring is known are the automorphism group of a polynomial ring in two indeterminates over a field and the free algebra in two variables over a field 4 chap 61 finally if a is a prime p z algebra with center c then every c automorphism of a is x inner 7 example 3 91

a nonlinear position and attitude observer on se 3 using landmark measurements this paper addresses the problem of position and attitude estimation based on landmark readings and velocity measurements a derivation of a nonlinear observer on se 3 is presented using a lyapunov function conveniently expressed as a function of the difference between the estimated and the measured landmark coordinates the resulting feedback laws are explicit functions of the landmark measurements and velocity readings exploiting the sensor information directly in the observer the proposed observer yields almost global asymptotic stabilization of the position and attitude errors and exponential convergence in any closed ball inside the region of attraction also it is shown that the asymptotic convergence of the estimation error trajectories is shaped by the landmark geometry and observer design parameters the problem of non ideal velocity readings is also considered and the observer is augmented to compensate for bias in the angular and linear velocity measurements the resulting position attitude and bias estimation errors are shown to converge exponentially fast to the desired equilibrium points for bounded initial estimation errors simulation results are presented to illustrate the stability and convergence properties of the observer

on filtering with observation in a manifold reduction to a classical filtering problem the current paper deals with filtering problems where the observation process conditioned on the unknown signal is an elliptic diffusion in a differentiable manifold precisely the observation model is given by a stochastic differential equation in the underlying manifold the main new idea is to use a le jan watanabe connection instead of a usual levi civita connection in performing the operation of antidevelopment both connections can be constructed from a stochastic differential equation as in the observation model the following results are obtained first it is shown that antidevelopment reduces the original filtering problem to a classical one i e an additive white noise model second a new form of the zakai and filtering equations is derived which has a structure very similar to that of classical equations currently there are few well understood general numerical solution methods for filtering problems with observation in a manifold reduction to classical filtering problems seems de

non linear symmetry preserving observers on lie groups in this technical note we give a geometrical framework for the design of observers on finite dimensional lie groups for systems which possess some specific symmetries the design and the error between true and estimated state equation are explicit and intrinsic we consider also a particular case left invariant systems on lie groups with right equivariant output the theory yields a class of observers such that the error equation is autonomous the observers converge locally around any trajectory and the global behavior is independent from the trajectory which is reminiscent of the linear stationary case

mathematical model and implementation of rational processing precision in computations is a considerable challenge to adequately addressing many current scientific or engineering problems the way in which the numbers are represented constitutes the first step to compute them and determines the validity of the results the aim of this research is to provide a formal framework and a set of computational primitives to address high precision problems of mathematical calculation in engineering and numerical simulation the main contribution of this research is a mathematical model to build an exact arithmetical unit able to represent without error rational numbers in positional notation system the functions under consideration are addition and multiplication because they form an algebraic commutative ring which contains a multiplicative inverse for every non zero element this paper reviews other specialized arithmetic units based on existing formats to show ways to make high precision computing it is proposed a formal framework of the whole arithmetic architecture in which the operators are based then the design of the addition operation is detailed and its hardware implementation is described finally extensive evaluation of this operator is performed to prove its ability for exact processing it is introduced a formal framework for processing rational numbers a representation system based on positional notation system is described a method for calculating the addition function is detailed experiments and application example have been made to validate the model

epsilon strongly graded rings separability and semisimplicity abstract we introduce the class of epsilon strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products we determine precisely when an epsilon strongly graded ring is separable over its principal component thereby we simultaneously generalize a result for strongly group graded rings by n st sescu van den bergh and van oystaeyen and a result for unital partial crossed products by bagio lazzarin and paques we also show that the class of unital partial crossed products appears in the class of epsilon strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings thereby we obtain in the special case of unital partial crossed products a short proof of a general result by dokuchaev exel and simon concerning when graded rings can be presented as partial crossed products we also provide some interesting classes of examples of separable epsilon strongly graded rings with finite as well as infinite grading groups in particular we obtain an answer to a question raised by le bruyn van den bergh and van oystaeyen in 1988

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

gradient like observers for invariant dynamics on a lie group this paper proposes a design methodology for non linear state observers for invariant kinematic systems posed on finite dimensional connected lie groups and studies the associated fundamental system structure the concept of synchrony of two dynamical systems is specialized to systems on lie groups for invariant systems this leads to a general factorization theorem of a nonlinear observer into a synchronous internal model term and an innovation term the synchronous term is fully specified by the system model we propose a design methodology for the innovation term based on gradient like terms derived from invariant or non invariant cost functions the resulting nonlinear observers have strong almost global convergence properties and examples are used to demonstrate the relevance of the proposed approach

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

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

picard groups and infinite matrix rings we describe a connection between the picard group of a ring with local units t and the picard group of the unital overring end tt using this connection we show that the three groups pic r pic fm r and pic rfm r are isomorphic for any unital ring r furthermore eaclh ele ment of pic rfm r arises from an automorphism of rfm r which yields an isomorphsm between pic rfm r and out rfm r as one applica tion we extend a classical result of rosenberg and zelinsky by showing that the group outr rfm r is abelian for any commutative unital ring r

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

the invariant extended kalman filter as a stable observer we analyze the convergence aspects of the invariant extended kalman filter iekf when the latter is used as a deterministic nonlinear observer on lie groups for continuous time systems with discrete observations one of the main features of invariant observers for left invariant systems on lie groups is that the estimation error is autonomous in this paper we first generalize this result by characterizing the much broader class of systems for which this property holds for those systems the lie logarithm of the error turns out to obey a linear differential equation then we leverage this log linear property of the error evolution to prove for those systems the local stability of the iekf around any trajectory under the standard conditions of the linear case one mobile robotics example and one inertial navigation example illustrate the interest of the approach simulations evidence the fact that the ekf is capable of diverging in some challenging situations where the iekf with identical tuning keeps converging

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

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent

semi logarithmic number systems we present a new class of number systems called semi logarithmic number systems that constitute a family of various compromises between floating point and logarithmic number systems this allows trade between the speed of the arithmetic operations and the size of the required tables we give arithmetic algorithms addition subtraction multiplication division for the semi logarithmic number systems and we compare these number systems to the classical floating point or logarithmic number systems