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NASA NTRS1988

Recursive algorithms for vector extrapolation methods

Three classes of recursion relations are devised for implementing some extrapolation methods for vector sequences. One class of recursion relations can be used to implement methods like the modified minimal polynomial extrapolation and the topological epsilon algorithm; another allows implementation of methods like minimal polynomial and reduced rank extrapolation; while the remaining class can be employed in the implementation of the vector E-algorithm. Operation counts and storage requirements for these methods are also discussed, and some related techniques for special applications are also presented. Included are methods for the rapid evaluations of the vector E-algorithm.

Ford, William F.↗

NASA NTRS1987

Fast and stable recursive algorithms for continuous-time and discrete-time model conversions

Based on the Newton-Raphson method, this paper presents recursive algorithms that are rapidly convergent and more stable for modeling the equivalent continuous-time (discrete-time) model from the available discrete-time (continuous-time) model for a fixed sampling period. The newly developed recursive algorithms relax the constraints imposed upon the existing model conversion algorithms, and, thus, enhance the applications of microprocessors and associated microelectronics to digital control systems. A practical example is presented to demonstrate the effectiveness of the proposed procedures.

Shieh, L. S.↗

NASA NTRS1989

Recursive forward dynamics for multiple robot arms moving a common task object

Recursive forward dynamics algorithms are developed and presented for an arbitrary number of robot arms moving a commonly held object. The multiarm forward dynamics problem is to find the angular accelerations at the joints and the contact forces that the arms impart to the task object. The problem also involves finding the acceleration of this object. The multiarm forward dynamics solutions provide a thorough physical and mathematical understanding of the way several arms behave in response to a set of applied joint moments. Such an understanding simplifies and guides the subsequent control design and experimentation process. The forward dynamics algorithms also provide the necessary analytical foundation for conducting analysis and simulation studies. The multiarm algorithms are based on the filtering and smoothing approach recently advanced for single-arm dynamics, and they can be built up modularly from the single-arm algorithms. The algorithms compute recursively the joint angle accelerations, the contact forces, and the task-object accelerations. Algorithms are also developed to evaluate in closed form the linear transformations from the active joint moments to the joint angle accelerations, to the task object accelerations, and to the task-object contact forces. A possible computing architecture is presented as a precursor to a more complete investigation of the computational performance of the dynamics algorithms.

Rodriguez, Guillermo↗

NASA NTRS1989

A decoupled recursive approach for constrained flexible multibody system dynamics

A variational-vector calculus approach is employed to derive a recursive formulation for dynamic analysis of flexible multibody systems. Kinematic relationships for adjacent flexible bodies are derived in a companion paper, using a state vector notation that represents translational and rotational components simultaneously. Cartesian generalized coordinates are assigned for all body and joint reference frames, to explicitly formulate deformation kinematics under small deformation kinematics and an efficient flexible dynamics recursive algorithm is developed. Dynamic analysis of a closed loop robot is performed to illustrate efficiency of the algorithm.

Lai, Hao-Jan↗

NASA NTRS1989

Recursive multibody dynamics and discrete-time optimal control

A recursive algorithm is developed for the solution of the simulation dynamics problem for a chain of rigid bodies. Arbitrary joint constraints are permitted, that is, joints may allow translational and/or rotational degrees of freedom. The recursive procedure is shown to be identical to that encountered in a discrete-time optimal control problem. For each relevant quantity in the multibody dynamics problem, there exists an analog in the context of optimal control. The performance index that is minimized in the control problem is identified as Gibbs' function for the chain of bodies.

Deleuterio, G. M. T.↗

NASA NTRS1990

Attitude determination and calibration using a recursive maximum likelihood-based adaptive Kalman filter

An adaptive Kalman filter design that utilizes recursive maximum likelihood parameter identification is discussed. At the center of this design is the Kalman filter itself, which has the responsibility for attitude determination. At the same time, the identification algorithm is continually identifying the system parameters. The approach is applicable to nonlinear, as well as linear systems. This adaptive Kalman filter design has much potential for real time implementation, especially considering the fast clock speeds, cache memory and internal RAM available today. The recursive maximum likelihood algorithm is discussed in detail, with special attention directed towards its unique matrix formulation. The procedure for using the algorithm is described along with comments on how this algorithm interacts with the Kalman filter.

Kelly, D. A.↗

NASA NTRS1991

Implementation of 3-D isoparametric finite elements on supercomputer for the formulation of recursive dynamical equations of multi-body systems

A computational algorithmic procedure is developed and implemented for the dynamic analysis of a multibody system with rigid/flexible interconnected bodies. The algorithm takes into consideration the large rotation/translation and small elastic deformations associated with the rigid-body degrees of freedom and the flexibility of the bodies in the system respectively. Versatile three-dimensional isoparametric brick elements are employed for the modeling of the geometric configurations of the bodies. The formulation of the recursive dynamical equations of motion is based on the recursive Kane's equations, strain energy concepts, and the techniques of component mode synthesis. In order to minimize CPU-intensive matrix multiplication operations and speed up the execution process, the concepts of indexed arrays is utilized in the formulation of the equations of motion. A spin-up maneuver of a space robot with three flexible links carrying a solar panel is used as an illustrative example.

Shareef, N. H.↗

NASA NTRS1992

Recursive Inversion Of Externally Defined Linear Systems

Technical memorandum discusses mathematical technique described in "Recursive Inversion by Finite-Impulse-Response Filters" (ARC-12247). Technique is recursive algorithm yielding finite-impulse-response approximation of unknown single-input/single-output, causal, time-invariant, linear, real system, response of which is sequence of impulses. Useful in such diverse applications as medical diagnoses, identification of military targets, geophysical exploration, and nondestructive testing.

Bach, Ralph E., Jr.↗

NASA NTRS1990

Recursive dynamics for geared robot manipulators

The authors consider the dynamical modeling of robot manipulators whose joint actuators consist of motors driving the joints through gears. The dynamical models for such manipulators are significantly more complex than those for direct drive manipulators. The authors develop recursive O(n) inverse and forward dynamics algorithms as well as recursive O(n2) algorithms for the computation of the mass matrix for geared manipulators. It is shown that, despite the added complexity of the dynamical models for geared manipulators, the algorithms closely resemble the corresponding algorithms for direct drive manipulators, and that the additional algorithmic or computational complexity is relatively insignificant. As a consequence, with little additional cost, existing direct drive algorithms can be easily extended to handle the effects of gearing at the joints.

Jain, A.↗

NASA NTRS1992

Recursive formulation of operational space control

A recently developed spatial operator algebra approach to modeling and analysis of multibody robotic systems is used to develop O(n) recursive algorithms that compute the operational space mass matrix and the operational space coriolis/centrifugal and gravity terms of an n-link serial manipulator. These algorithms enable an O(n) recursive implementation of operational space control.

Kreutz-Delgado, K.↗

NASA NTRS1990

On recursive least-squares filtering algorithms and implementations

In many real-time signal processing applications, fast and numerically stable algorithms for solving least-squares problems are necessary and important. In particular, under non-stationary conditions, these algorithms must be able to adapt themselves to reflect the changes in the system and take appropriate adjustments to achieve optimum performances. Among existing algorithms, the QR-decomposition (QRD)-based recursive least-squares (RLS) methods have been shown to be useful and effective for adaptive signal processing. In order to increase the speed of processing and achieve high throughput rate, many algorithms are being vectorized and/or pipelined to facilitate high degrees of parallelism. A time-recursive formulation of RLS filtering employing block QRD will be considered first. Several methods, including a new non-continuous windowing scheme based on selectively rejecting contaminated data, were investigated for adaptive processing. Based on systolic triarrays, many other forms of systolic arrays are shown to be capable of implementing different algorithms. Various updating and downdating systolic algorithms and architectures for RLS filtering are examined and compared in details, which include Householder reflector, Gram-Schmidt procedure, and Givens rotation. A unified approach encompassing existing square-root-free algorithms is also proposed. For the sinusoidal spectrum estimation problem, a judicious method of separating the noise from the signal is of great interest. Various truncated QR methods are proposed for this purpose and compared to the truncated SVD method. Computer simulations provided for detailed comparisons show the effectiveness of these methods. This thesis deals with fundamental issues of numerical stability, computational efficiency, adaptivity, and VLSI implementation for the RLS filtering problems. In all, various new and modified algorithms and architectures are proposed and analyzed; the significance of any of the new method depends crucially on specific application.

Hsieh, Shih-Fu↗

NASA NTRS1993

Recursive dynamics algorithm for multibody systems with prescribed motion

This paper uses spatial operator techniques to develop a new algorithm for the dynamics of multibody systems with hinges undergoing prescribed motion. This algorithm is spatially recursive, and its computational complexity grows only linearly with the number of degrees of freedom in the system. Its structure is a hybrid of known recursive forward and inverse dynamics algorithms for regular multibody systems. Changes to the prescribed/nonprescribed nature of hinges can be implemented during run time since they are handled with very low overhead in the algorithm.

Jain, Abhinandan↗

NASA NTRS1993

Development of a recursion RNG-based turbulence model

Reynolds stress closure models based on the recursion renormalization group theory are developed for the prediction of turbulent separated flows. The proposed model uses a finite wavenumber truncation scheme to account for the spectral distribution of energy. In particular, the model incorporates effects of both local and nonlocal interactions. The nonlocal interactions are shown to yield a contribution identical to that from the epsilon-renormalization group (RNG), while the local interactions introduce higher order dispersive effects. A formal analysis of the model is presented and its ability to accurately predict separated flows is analyzed from a combined theoretical and computational stand point. Turbulent flow past a backward facing step is chosen as a test case and the results obtained based on detailed computations demonstrate that the proposed recursion -RNG model with finite cut-off wavenumber can yield very good predictions for the backstep problem.

Zhou, YE↗

NASA NTRS1993

Non-recursive augmented Lagrangian algorithms for the forward and inverse dynamics of constrained flexible multibodies

A technique is presented for solving the inverse dynamics of flexible planar multibody systems. This technique yields the non-causal joint efforts (inverse dynamics) as well as the internal states (inverse kinematics) that produce a prescribed nominal trajectory of the end effector. A non-recursive global Lagrangian approach is used in formulating the equations for motion as well as in solving the inverse dynamics equations. Contrary to the recursive method previously presented, the proposed method solves the inverse problem in a systematic and direct manner for both open-chain as well as closed-chain configurations. Numerical simulation shows that the proposed procedure provides an excellent tracking of the desired end effector trajectory.

Bayo, Eduardo↗

NASA NTRS1994

Adaptive model reduction for continuous systems via recursive rational interpolation

A method for adaptive identification of reduced-order models for continuous stable SISO and MIMO plants is presented. The method recursively finds a model whose transfer function (matrix) matches that of the plant on a set of frequencies chosen by the designer. The algorithm utilizes the Moving Discrete Fourier Transform (MDFT) to continuously monitor the frequency-domain profile of the system input and output signals. The MDFT is an efficient method of monitoring discrete points in the frequency domain of an evolving function of time. The model parameters are estimated from MDFT data using standard recursive parameter estimation techniques. The algorithm has been shown in simulations to be quite robust to additive noise in the inputs and outputs. A significant advantage of the method is that it enables a type of on-line model validation. This is accomplished by simultaneously identifying a number of models and comparing each with the plant in the frequency domain. Simulations of the method applied to an 8th-order SISO plant and a 10-state 2-input 2-output plant are presented. An example of on-line model validation applied to the SISO plant is also presented.

Lilly, John H.↗

NASA NTRS1995

Development of Fast Algorithms Using Recursion, Nesting and Iterations for Computational Electromagnetics

In the first phase of our work, we have concentrated on laying the foundation to develop fast algorithms, including the use of recursive structure like the recursive aggregate interaction matrix algorithm (RAIMA), the nested equivalence principle algorithm (NEPAL), the ray-propagation fast multipole algorithm (RPFMA), and the multi-level fast multipole algorithm (MLFMA). We have also investigated the use of curvilinear patches to build a basic method of moments code where these acceleration techniques can be used later. In the second phase, which is mainly reported on here, we have concentrated on implementing three-dimensional NEPAL on a massively parallel machine, the Connection Machine CM-5, and have been able to obtain some 3D scattering results. In order to understand the parallelization of codes on the Connection Machine, we have also studied the parallelization of 3D finite-difference time-domain (FDTD) code with PML material absorbing boundary condition (ABC). We found that simple algorithms like the FDTD with material ABC can be parallelized very well allowing us to solve within a minute a problem of over a million nodes. In addition, we have studied the use of the fast multipole method and the ray-propagation fast multipole algorithm to expedite matrix-vector multiplication in a conjugate-gradient solution to integral equations of scattering. We find that these methods are faster than LU decomposition for one incident angle, but are slower than LU decomposition when many incident angles are needed as in the monostatic RCS calculations.

Chew, W. C.↗

NASA NTRS1996

REQUEST: A Recursive QUEST Algorithm for Sequential Attitude Determination

In order to find the attitude of a spacecraft with respect to a reference coordinate system, vector measurements are taken. The vectors are pairs of measurements of the same generalized vector, taken in the spacecraft body coordinates, as well as in the reference coordinate system. We are interested in finding the best estimate of the transformation between these coordinate system.s The algorithm called QUEST yields that estimate where attitude is expressed by a quarternion. Quest is an efficient algorithm which provides a least squares fit of the quaternion of rotation to the vector measurements. Quest however, is a single time point (single frame) batch algorithm, thus measurements that were taken at previous time points are discarded. The algorithm presented in this work provides a recursive routine which considers all past measurements. The algorithm is based on on the fact that the, so called, K matrix, one of whose eigenvectors is the sought quaternion, is linerly related to the measured pairs, and on the ability to propagate K. The extraction of the appropriate eigenvector is done according to the classical QUEST algorithm. This stage, however, can be eliminated, and the computation simplified, if a standard eigenvalue-eigenvector solver algorithm is used. The development of the recursive algorithm is presented and illustrated via a numerical example.

Bar-Itzhack, Itzhack Y.↗

NASA NTRS1998

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes: A Recursive Maximum Likelihood Decoding - Part 3

The Viterbi algorithm is indeed a very simple and efficient method of implementing the maximum likelihood decoding. However, if we take advantage of the structural properties in a trellis section, other efficient trellis-based decoding algorithms can be devised. Recently, an efficient trellis-based recursive maximum likelihood decoding (RMLD) algorithm for linear block codes has been proposed. This algorithm is more efficient than the conventional Viterbi algorithm in both computation and hardware requirements. Most importantly, the implementation of this algorithm does not require the construction of the entire code trellis, only some special one-section trellises of relatively small state and branch complexities are needed for constructing path (or branch) metric tables recursively. At the end, there is only one table which contains only the most likely code-word and its metric for a given received sequence r = (r(sub 1), r(sub 2),...,r(sub n)). This algorithm basically uses the divide and conquer strategy. Furthermore, it allows parallel/pipeline processing of received sequences to speed up decoding.

Lin, Shu↗