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NASA NTRS1994

Comment on 'Guidance for an Aerocapture Maneuver

It is stated that the aerodynamic forces on the vehicle being aerocaptured are controlled by "altering the angle of attack" and thereby controlling the lift coefficient. Furthermore, the resulting variation of drag coefficient with angle of attack was ignored. The purpose of this Comment is to point out that an aerodynamic control method that is much more effective than the pitch modulation has been studied and utilized during entries for many years. During aerocapture, it is desirable to have a large range of lift coefficients available, while keeping the vehicle's ballistic coefficients constant. This is accomplished by modulating the vehicle's bank angle, i.e., by rolling the vehicle about its velocity vector. By this method, the angle of attack can be held constant (at the trim angle, if desired), and the C(sub D) and the ballistic coefficient remain constant. Furthermore, the vertical component of the normal force vector (essentially the lift) can be varied over its entire range, from maximum positive to maximum negative values. Reaction controls, rather than aerodynamic ones, are usually utilized to change the bank angle of the vehicle, thus requiring the use of fuel. However, the fuel expenditure that is required to change the bank angle is far less than the amount that would have to be used to continuously hold the vehicle at pitch angles that differ significantly from its trim angle of attack. Also, it has been shown that bank angle modulation to vary the lift can enlarge the entry corridor by increasing the entry angle for the undershoot boundary, where both the heating rate and deceleration reach a maximum. Finally, the crew's deceleration tolerance can be increased somewhat when the bank angle is varied, as opposed to the pitch angle. For bank modulation, the deceleration force vector can be kept at a constant angle with respect to the occupants whose tolerance to g loads is highest when the force is applied in a direction normal to the upper torso. The advantages of bank angle variation to modulate the lift vector were recognized long ago, and this method of control was used successfully on the Apollo command module during lunar return' and, more recently, for the Space Shuttle Orbiter.

Tauber, Michael E.↗

NASA NTRS2001

Comparison of Fault Detection Algorithms for Real-time Diagnosis in Large-Scale System

In this paper, we present a review of different real-time capable algorithms to detect and isolate component failures in large-scale systems in the presence of inaccurate test results. A sequence of imperfect test results (as a row vector of I's and O's) are available to the algorithms. In this case, the problem is to recover the uncorrupted test result vector and match it to one of the rows in the test dictionary, which in turn will isolate the faults. In order to recover the uncorrupted test result vector, one needs the accuracy of each test. That is, its detection and false alarm probabilities are required. In this problem, their true values are not known and, therefore, have to be estimated online. Other major aspects in this problem are the large-scale nature and the real-time capability requirement. Test dictionaries of sizes up to 1000 x 1000 are to be handled. That is, results from 1000 tests measuring the state of 1000 components are available. However, at any time, only 10-20% of the test results are available. Then, the objective becomes the real-time fault diagnosis using incomplete and inaccurate test results with online estimation of test accuracies. It should also be noted that the test accuracies can vary with time --- one needs a mechanism to update them after processing each test result vector. Using Qualtech's TEAMS-RT (system simulation and real-time diagnosis tool), we test the performances of 1) TEAMSAT's built-in diagnosis algorithm, 2) Hamming distance based diagnosis, 3) Maximum Likelihood based diagnosis, and 4) HidderMarkov Model based diagnosis.

Kirubarajan, Thiagalingam↗

NASA NTRS1994

Some Problems and Solutions in Transferring Ecosystem Simulation Codes to Supercomputers

Many computer codes for the simulation of ecological systems have been developed in the last twenty-five years. This development took place initially on main-frame computers, then mini-computers, and more recently, on micro-computers and workstations. Recent recognition of ecosystem science as a High Performance Computing and Communications Program Grand Challenge area emphasizes supercomputers (both parallel and distributed systems) as the next set of tools for ecological simulation. Transferring ecosystem simulation codes to such systems is not a matter of simply compiling and executing existing code on the supercomputer since there are significant differences in the system architectures of sequential, scalar computers and parallel and/or vector supercomputers. To more appropriately match the application to the architecture (necessary to achieve reasonable performance), the parallelism (if it exists) of the original application must be exploited. We discuss our work in transferring a general grassland simulation model (developed on a VAX in the FORTRAN computer programming language) to a Cray Y-MP. We show the Cray shared-memory vector-architecture, and discuss our rationale for selecting the Cray. We describe porting the model to the Cray and executing and verifying a baseline version, and we discuss the changes we made to exploit the parallelism in the application and to improve code execution. As a result, the Cray executed the model 30 times faster than the VAX 11/785 and 10 times faster than a Sun 4 workstation. We achieved an additional speed-up of approximately 30 percent over the original Cray run by using the compiler's vectorizing capabilities and the machine's ability to put subroutines and functions "in-line" in the code. With the modifications, the code still runs at only about 5% of the Cray's peak speed because it makes ineffective use of the vector processing capabilities of the Cray. We conclude with a discussion and future plans.

Skiles, J. W.↗

NASA NTRS2002

Tracking Avian Reservoirs of Arboviruses using Remote Sensing and Radiotelemetry

Encephalitis is caused by a virus that is transmitted by mosquitoes between mammalian hosts. The virus is closely related to the West Nile virus (WNV), which started in New York in 1999, and has since spread to 25 states. Like encephalitis, WNV is vectored by mosquitoes, and the primary hosts are birds; humans are accidental, or'dead-end' hosts. Very little is understood about the behavior of these bird populations, and how they intersect - both in time and in space - with mosquito populations. Exploring these relationships is the first step in developing models for encephalitis and WNV transmission risk. This project combines remotely sensed data with radiotelemetry to create a spatiotemporal map of encephalitis viral activity in bird and mosquito populations in the Sacramento Valley of California. Specifically, remote sensing (RS) and geographic information system (GIS) technologies were used to characterize habitats utilized by both avian viral reservoirs and the mosquitoes that vector encephalitis. Radiotelemetry and serosurveys (blood) were then used to spatially and temporally track the patterns of infection. The project uses Landsat ETM+ multitemporal satellite data to characterize habitats utilized by both birds and the mosquito vectors. Mist nets were used to sample members of individual flocks of blackbirds and cowbirds over a period of several months; these birds were then bled to assess their viral status, banded, and fitted with transmitters. Radiotelemetry was used to spatially and temporally track the distribution of banded birds and their associated flocks. The movement of these indicator flocks were compared with the location of remotely sensed (adult and larval) mosquito habitats to determine the intersection of bird's and vectors; this is key in understanding where and when transmission occurs from bird to bird, as well as from bird to mammal, via mosquito. The relationships found during the project are being used to generate a model of encephalitis transmission risk in California.

Beck, L.↗

NASA NTRS2003

Masked Proportional Routing

Distributed approach for determining a path connecting adjacent network nodes, for probabilistically or deterministically transporting an entity, with entity characteristic mu from a source node to a destination node. Each node i is directly connected to an arbitrary number J(mu) of nodes, labeled or numbered j=jl, j2, .... jJ(mu). In a deterministic version, a J(mu)-component baseline proportion vector p(i;mu) is associated with node i. A J(mu)-component applied proportion vector p*(i;mu) is determined from p(i;mu) to preclude an entity visiting a node more than once. Third and fourth J(mu)-component vectors, with components iteratively determined by Target(i;n(mu);mu),=alpha(mu).Target(i;n(mu)-1;mu)j+beta(mu).p* (i;mu)j and Actual(i;n(mu);+a(mu)j. Actual(i;n(mu)-l;mu)j+beta(mu).Sent(i;j'(mu);n(mu)-1;mu)j, are computed, where n(mu) is an entity sequence index and alpha(mu) and beta(mu) are selected numbers. In one embodiment, at each node i, the node j=j'(mu) with the largest vector component difference, Target(i;n(mu);mu)j'- Actual (i;n(mu);mu)j'. is chosen for the next link for entity transport, except in special gap circumstances, where the same link is optionally used for transporting consecutively arriving entities. The network nodes may be computer-controlled routers that switch collections of packets, frames, cells or other information units. Alternatively, the nodes may be waypoints for movement of physical items in a network or for transformation of a physical item. The nodes may be states of an entity undergoing state transitions, where allowed transitions are specified by the network and/or the destination node.

David H Wolpert↗

NASA NTRS2003

James Webb Space Telescope Station-keeping

The James Webb Space Telescope (JWST) is planned to be launched in 2011 to the Sun- Earth L2 libration point. The resultant delta-Vs (dV) from momentum unloads will perturb the orbit and necessitate frequent station-keeping maneuvers. The station-keeping dV budget is highly sensitive to the direction of the resultant dV vector. A simple spacecraft reorientation prior to each momentum unload will allow some control over the direction of the resultant dV vector. For each inertial momentum vector direction, an optimum spacecraft attitude is determined which gives a resultant dV vector that requires the least amount of station-keeping dV. Using this procedure, the station-keeping dV budget for JWST can be reduced by 60%.

Beckman, Mark↗

NASA NTRS2004

Prediction of Active-Region CME Productivity from Magnetograms

We report results of an expanded evaluation of whole-active-region magnetic measures as predictors of active-region coronal mass ejection (CME) productivity. Previously, in a sample of 17 vector magnetograms of 12 bipolar active regions observed by the Marshall Space Flight Center (MSFC) vector magnetograph, from each magnetogram we extracted a measure of the size of the active region (the active region s total magnetic flux a) and four measures of the nonpotentiality of the active region: the strong-shear length L(sub SS), the strong-gradient length L(sub SG), the net vertical electric current I(sub N), and the net-current magnetic twist parameter alpha (sub IN). This sample size allowed us to show that each of the four nonpotentiality measures was statistically significantly correlated with active-region CME productivity in time windows of a few days centered on the day of the magnetogram. We have now added a fifth measure of active-region nonpotentiality (the best-constant-alpha magnetic twist parameter (alpha sub BC)), and have expanded the sample to 36 MSFC vector magnetograms of 31 bipolar active regions. This larger sample allows us to demonstrate statistically significant correlations of each of the five nonpotentiality measures with future CME productivity, in time windows of a few days starting from the day of the magnetogram. The two magnetic twist parameters (alpha (sub 1N) and alpha (sub BC)) are normalized measures of an active region s nonpotentially in that they do not depend directly on the size of the active region, while the other three nonpotentiality measures (L(sub SS), L(sub SG), and I(sub N)) are non-normalized measures in that they do depend directly on active-region size. We find (1) Each of the five nonpotentiality measures is statistically significantly correlated (correlation confidence level greater than 95%) with future CME productivity and has a CME prediction success rate of approximately 80%. (2) None of the nonpotentiality measures is a significantly better CME predictor than the others. (3) The active-region phi shows some correlation with CME productivity, but well below a statistically significant level (correlation confidence level less than approximately 80%; CME prediction success rate less than approximately 65%). (4) In addition to depending on magnetic twist, CME productivity appears to have some direct dependence on active-region size (rather than only an indirect dependence through a correlation of magnetic twist with active-region size), but it will take a still larger sample of active regions (50 or more) to certify this. (5) Of the five nonpotentiality measures, L(sub SG) appears to be the best for operational CME forecasting because it is as good or better a CME predictor than the others and it alone does not require a vector magnetogram; L(sub SG) can be measured from a line-of-sight magnetogram such as from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO).

Falconer, D. A.↗

NASA NTRS2004

A New Approach to Attitude Stability and Control for Low Airspeed Vehicles

This paper describes an approach for controlling the attitude of statically unstable thrust-levitated vehicles in hover or slow translation. The large thrust vector that characterizes such vehicles can be modulated to provide control forces and moments to the airframe, but such modulation is accompanied by significant unsteady flow effects. These effects are difficult to model, and can compromise the practical value of thrust vectoring in closed-loop attitude stability, even if the thrust vectoring machinery has sufficient bandwidth for stabilization. The stabilization approach described in this paper is based on using internal angular momentum transfer devices for stability, augmented by thrust vectoring for trim and other "outer loop" control functions. The three main components of this approach are: (1) a z-body axis angular momentum bias enhances static attitude stability, reducing the amount of control activity needed for stabilization, (2) optionally, gimbaled reaction wheels provide high-bandwidth control torques for additional stabilization, or agility, and (3) the resulting strongly coupled system dynamics are controlled by a multivariable controller. A flight test vehicle is described, and nonlinear simulation results are provided that demonstrate the efficiency of the approach.

Lim, K. B.↗

NASA NTRS1997

Relative Navigation In Elliptical Orbits Using An Iterative Nonlinear Filter

The two step filter is applied to process intersatellite radar measurements to determine the motion of one satellite relative to another in close elliptical orbits. This filter breaks a nonlinear estimation problem into two state vectors. The 'first step' state is chosen so as to have a linear measurement equation. This is nonlinearly related to the 'second step' state which describes the dynamics. Two different forms are used. In one, the first step state is the second step state vector augmented by the measurement equation. In the other, the first step and second step state vectors are of equal dimension. The two step filter is compared against an iterated extended Kalman filter and a Kalman filter using a change of variables. Analytical differences between the two step estimator and these conventional filters are highlighted. Special concerns for initializing the first step state covariance matrix and handling the possibility of numerically rank deficient covariance matrices are addressed. Numerical simulations are performed which show that the Two Step estimator produces a lower estimation bias under two circumstances; large apriori initial error; and small dimension observation vectors which require a longer arc of measurements to generate observability of the state.

Garrison, James L.↗

NASA NTRS2005

Spatial orientation of optokinetic nystagmus and ocular pursuit during orbital space flight

On Earth, eye velocity of horizontal optokinetic nystagmus (OKN) orients to gravito-inertial acceleration (GIA), the sum of linear accelerations acting on the head and body. We determined whether adaptation to micro-gravity altered this orientation and whether ocular pursuit exhibited similar properties. Eye movements of four astronauts were recorded with three-dimensional video-oculography. Optokinetic stimuli were stripes moving horizontally, vertically, and obliquely at 30 degrees/s. Ocular pursuit was produced by a spot moving horizontally or vertically at 20 degrees/s. Subjects were either stationary or were centrifuged during OKN with 1 or 0.5 g of interaural or dorsoventral centripetal linear acceleration. Average eye position during OKN (the beating field) moved into the quick-phase direction by 10 degrees during lateral and upward field movement in all conditions. The beating field did not shift up during downward OKN on Earth, but there was a strong upward movement of the beating field (9 degrees) during downward OKN in the absence of gravity; this likely represents an adaptation to the lack of a vertical 1-g bias in-flight. The horizontal OKN velocity axis tilted 9 degrees in the roll plane toward the GIA during interaural centrifugation, both on Earth and in space. During oblique OKN, the velocity vector tilted towards the GIA in the roll plane when there was a disparity between the direction of stripe motion and the GIA, but not when the two were aligned. In contrast, dorsoventral acceleration tilted the horizontal OKN velocity vector 6 degrees in pitch away from the GIA. Roll tilts of the horizontal OKN velocity vector toward the GIA during interaural centrifugation are consistent with the orientation properties of velocity storage, but pitch tilts away from the GIA when centrifuged while supine are not. We speculate that visual suppression during OKN may have caused the velocity vector to tilt away from the GIA during dorsoventral centrifugation. Vertical OKN and ocular pursuit did not exhibit orientation toward the GIA in any condition. Static full-body roll tilts and centrifugation generating an equivalent interaural acceleration produced the same tilts in the horizontal OKN velocity before and after flight. Thus, the magnitude of tilt in OKN velocity was dependent on the magnitude of interaural linear acceleration, rather than the tilt of the GIA with regard to the head. These results favor a 'filter' model of spatial orientation in which orienting eye movements are proportional to the magnitude of low frequency interaural linear acceleration, rather than models that postulate an internal representation of gravity as the basis for spatial orientation.

Flight Experiment↗

NASA NTRS2005

Attitude Estimation for Large Field-of-View Sensors

The QUEST measurement noise model for unit vector observations has been widely used in spacecraft attitude estimation for more than twenty years. It was derived under the approximation that the noise lies in the tangent plane of the respective unit vector and is axially symmetrically distributed about the vector. For large field-of-view sensors, however, this approximation may be poor, especially when the measurement falls near the edge of the field of view. In this paper a new measurement noise model is derived based on a realistic noise distribution in the focal-plane of a large field-of-view sensor, which shows significant differences from the QUEST model for unit vector observations far away from the sensor boresight. An extended Kalman filter for attitude estimation is then designed with the new measurement noise model. Simulation results show that with the new measurement model the extended Kalman filter achieves better estimation performance using large field-of-view sensor observations.

Cheng, Yang↗

NASA NTRS2006

Accommodating Sensor Uncertainty in the Cones Method: Polycones and Fuzzycones

The "cones method" is an analytical algorithm to combine a pair of angle observations into a common vector. Two new algorithms have been developed to optimize the "cones method" solutions when more than two observation angles are available and when estimates of measurement uncertainties can be made. The polycones algorithm consists of determining a simple weighted average of the solution vectors over all possible pairs of measurements with the weights determined from the measurement uncertainty. The Fuzzycones method finds the vector of maximum probability. Both of these methods have been implemented and tested and both reduce errors in computed vector positions.

Hashmall, Joseph A.↗

NASA NTRS1999

Signal processing applications of massively parallel charge domain computing devices

The present invention is embodied in a charge coupled device (CCD)/charge injection device (CID) architecture capable of performing a Fourier transform by simultaneous matrix vector multiplication (MVM) operations in respective plural CCD/CID arrays in parallel in O(1) steps. For example, in one embodiment, a first CCD/CID array stores charge packets representing a first matrix operator based upon permutations of a Hartley transform and computes the Fourier transform of an incoming vector. A second CCD/CID array stores charge packets representing a second matrix operator based upon different permutations of a Hartley transform and computes the Fourier transform of an incoming vector. The incoming vector is applied to the inputs of the two CCD/CID arrays simultaneously, and the real and imaginary parts of the Fourier transform are produced simultaneously in the time required to perform a single MVM operation in a CCD/CID array.

Fijany, Amir↗

NASA NTRS2007

Modeling Malaria Transmission in Thailand and Indonesia

Malaria Modeling and Surveillance is a project in the NASA Applied Sciences Public Health Applications Program. The main objectives of this project are: 1) identification of the potential breeding sites for major vector species: 2) implementation of a malaria transmission model to identify they key factors that sustain or intensify malaria transmission; and 3) implementation of a risk algorithm to predict the occurrence of malaria and its transmission intensity. Remote sensing and GIs are the essential elements of this project. The NASA Earth science data sets used in this project include AVHRR Pathfinder, TRMM, MODIS, NSIPP and SIESIP. Textural-contextual classifications are used to identify small larval habitats. Neural network methods are used to model malaria cases as a function of precipitation, temperatures, humidity and vegetation. Hindcastings based on these environmental parameters have shown good agreement to epidemiological records. Examples for spatio-temporal modeling of malaria transmissions in Southeast Asia are given. Discrete event simulations were used for modeling the detailed interactions among the vector life cycle, sporogonic cycle and human infection cycle, under the explicit influences of selected extrinsic and intrinsic factors. The output of the model includes the individual infection status and the quantities normally observed in field studies, such as mosquito biting rates, sporozoite infection rates, gametocyte prevalence and incidence. Results are in good agreement with mosquito vector and human malaria data acquired by Coleman et al. over 4.5 years in Kong Mong Tha, a remote village in western Thailand. Application of our models is not restricted to Southeast Asia. The model and techniques are equally applicable to other regions of the world, when appropriate epidemiological and vector ecological parameters are used as input.

Kiang, Richard↗

NASA NTRS2008

A Data Type for Efficient Representation of Other Data Types

A self-organizing, monomorphic data type denoted a sequence has been conceived to address certain concerns that arise in programming parallel computers. A sequence in the present sense can be regarded abstractly as a vector, set, bag, queue, or other construct. Heretofore, in programming a parallel computer, it has been necessary for the programmer to state explicitly, at the outset, what parts of the program and the underlying data structures must be represented in parallel form. Not only is this requirement not optimal from the perspective of implementation; it entails an additional requirement that the programmer have intimate understanding of the underlying parallel structure. The present sequence data type overcomes both the implementation and parallel structure obstacles. In so doing, the sequence data type provides unified means by which the programmer can represent a data structure for natural and automatic decomposition to a parallel computing architecture. Sequences exhibit the behavioral and structural characteristics of vectors, but the underlying representations are automatically synthesized from combinations of programmers advice and execution use metrics. Sequences can vary bidirectionally between sparseness and density, making them excellent choices for many kinds of algorithms. The novelty and benefit of this behavior lies in the fact that it can relieve programmers of the details of implementations. The creation of a sequence enables decoupling of a conceptual representation from an implementation. The underlying representation of a sequence is a hybrid of representations composed of vectors, linked lists, connected blocks, and hash tables. The internal structure of a sequence can automatically change from time to time on the basis of how it is being used. Those portions of a sequence where elements have not been added or removed can be as efficient as vectors. As elements are inserted and removed in a given portion, then different methods are utilized to provide both an access and memory strategy that is optimized for that portion and the use to which it is put.

James, Mark↗

NASA NTRS2009

Hybrid NN/SVM Computational System for Optimizing Designs

A computational method and system based on a hybrid of an artificial neural network (NN) and a support vector machine (SVM) (see figure) has been conceived as a means of maximizing or minimizing an objective function, optionally subject to one or more constraints. Such maximization or minimization could be performed, for example, to optimize solve a data-regression or data-classification problem or to optimize a design associated with a response function. A response function can be considered as a subset of a response surface, which is a surface in a vector space of design and performance parameters. A typical example of a design problem that the method and system can be used to solve is that of an airfoil, for which a response function could be the spatial distribution of pressure over the airfoil. In this example, the response surface would describe the pressure distribution as a function of the operating conditions and the geometric parameters of the airfoil. The use of NNs to analyze physical objects in order to optimize their responses under specified physical conditions is well known. NN analysis is suitable for multidimensional interpolation of data that lack structure and enables the representation and optimization of a succession of numerical solutions of increasing complexity or increasing fidelity to the real world. NN analysis is especially useful in helping to satisfy multiple design objectives. Feedforward NNs can be used to make estimates based on nonlinear mathematical models. One difficulty associated with use of a feedforward NN arises from the need for nonlinear optimization to determine connection weights among input, intermediate, and output variables. It can be very expensive to train an NN in cases in which it is necessary to model large amounts of information. Less widely known (in comparison with NNs) are support vector machines (SVMs), which were originally applied in statistical learning theory. In terms that are necessarily oversimplified to fit the scope of this article, an SVM can be characterized as an algorithm that (1) effects a nonlinear mapping of input vectors into a higher-dimensional feature space and (2) involves a dual formulation of governing equations and constraints. One advantageous feature of the SVM approach is that an objective function (which one seeks to minimize to obtain coefficients that define an SVM mathematical model) is convex, so that unlike in the cases of many NN models, any local minimum of an SVM model is also a global minimum.

Rai, Man Mohan↗

NASA NTRS2010

Performance Analysis of Apollo Navigational Starter Routine

The focus of this project is to recreate and analyze the effectiveness of the original Apollo Starter Routine (ASR) which was used to generate the state vector of the Apollo spacecraft based on a series of radiometric observations. The original Apollo navigation software is unavailable in a modern programming language and the original coding has not been preserved. This necessitates its recreation using the original software documentation. Space Shuttle navigation software does not typically use the ASR or an algorithm like it since the Shuttle s state vector is easily deduced from GPS information or other sources. However, this tactic will be ineffective when trying to determine the state vector of a craft approaching, departing or in orbit around the Moon since the GPS network faces the surface of the Earth, not outer space. The recreation of the ASR from the original documentation is therefore vital as a simulation baseline for the navigation software under development for the Constellation program. The algorithms that make up the ASR will be extracted from the original documentation and adapted for and then implemented in a modern programming language; the majority of it will be coded in Matlab. The ASR s effectiveness will then be tested using simulated tracking data. The ability of the ASR to handle realistically noisy data and the accuracy with which it generates state vectors were analyzed. The ASR proved to be robust enough to process data with range and angle noise as large as 10,000 meters and 10(exp -6) radians together and 300,000 meters and 5x10(exp -4) radians separately at Lunar distances. The ASR was able to handle marginally more noise at distances closer to the Earth where the angle noise was less significant. The ASR is capable of effectively processing 40-80 data points gathered at a rate of one per 20 seconds at close Earth orbit and up to 28-40 data points gathered at a rate of one per minute at distant Earth orbit and Lunar orbit.

Ivanov, Stoyan I.↗

NASA NTRS2010

Forces Associated with Nonlinear Nonholonomic Constraint Equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.

Roithmayr, Carlos M.↗