Dynamic Error Model Research Articles

Dynamic Modeling for Machine Tool Thermal Error Compensation

This paper proposes a new thermal error modeling methodology called the Dynamic Thermal Error Modeling which improves the accuracy and robustness of the machine tool thermal error model. The characteristics of the thermoelastic system are investigated from the dynamic system viewpoint. The pseudo-hysteresis effect is revealed to be the major factor causing poor robustness of the conventional static thermal error model. System identification theory is applied to build the dynamic thermal error model for machine tool thermal error on-line prediction. The modeling procedure for the linear Output Error (OE) model is illustrated using simulation work for both one-dimensional spindle and two-dimensional machine structure thermal deformations. Model performance evaluation through spindle experiments shows that the thermal error dynamic model has advantages over the conventional static model in terms of model accuracy and robustness.

Spacecraft Attitude Rate Estimation from Geomagnetic Field Measurements

Methods are developed for estimating the rotation rate of a spacecraft using only measured magnetic e eld data. The goal is to provide rate information for use in applications such as detumbling, nutation damping, and momentum management without using gyroscopes. Two algorithms are developed, a deterministic algorithm and an extended Kalman e lter. Both algorithms employ the magnetic e eld direction kinematics equation and Euler’ s equation for attitude motion of a rigid body with momentum wheels. Neither algorithm requires a model of the Earth’ s magnetic e eld. The deterministic algorithm solves a nonlinear least-squares problem for the unknown angular momentum component along the magnetic e eld direction. The extended Kalman e lter estimates the attitude rate vector, corrections to e ve of the six inertia matrix elements, and two error states of the measured magnetic e eld direction. It uses an initial rate estimate from the deterministic algorithm to avoid divergence. The algorithms have been tested using data from a spinning sounding rocket. They achieve initial accuracies in the range 2‐7 deg/s when the rocket spins at about 80 deg/s, and their accuracies improve to 1 ‐2 deg/s after the spin rate decays to 20 deg/s. These results indicate a lower bound on the ratio of the error to the nominal spin rate, which suggests that dynamic modeling error is the dominant source of uncertainty. I. Introduction M ANY spacecraft need estimates of their rotation rates. The spin rate may be used to apply a control that stops tumbling, to manage the total system angular momentum, to aid a star tracker, or as part of an attitude determination system. The most common method of determining attitude rate is by direct rate-gyro measurement. This paper develops two attitude rate estimation methods that rely on Earth magnetic e eld measurements rather than on rate-gyro

An experimental study of active control of wind-induced vibration of super-tall buildings

A control strategy named Sinusoidal Reference Strategy is developed for adaptive feedforward control of wind-induced vibration of super-tall buildings. In this approach a high-frequency sinusoidal signal is adopted as the reference signal for the strategy. Some shortcomings of the conventional adaptive feedforward control method can be overcome. A MDOF general building aeroelastic model with a square cross-section was tested in a wind tunnel. The testing results show that the present strategy can reduce vibration effectively, and can adapt to wind-induced vibration control of tall buildings with dynamic uncertainties and modeling errors.

State Estimation in Nonlinear Processes. Application to pH Process Control

In the field of process engineering, there are many factors that demand the knowledge of one part of the state vector, if not the full one. Among others, the implementation of nonlinear control methods as well as monitoring of some relevant process variables can be mentioned. The purpose of this paper is to introduce a nonlinear state observer of full order (i.e., the order of the observer matches the order of the nonlinear system observed) in order to estimate all of the state variables of the process. Though the estimated states can be used for many purposes, in this work we aim at using the estimations for output reference tracking. In particular, we deal with the pH neutralization process, which is widely recognized as a difficult problem for the purpose of control. To achieve the pH control goal, a simple algorithm is proposed which lies on a stable reference error dynamics model. Finally, computer simulations are developed for showing the performance of both the nonlinear observer and the control strategy based on pH measurement. For this purpose, a neutralization reaction between a strong acid and a strong base in the presence of a buffer agent is dealt with.

The L5/S1 joint moment sensitivity to measurement errors in dynamic 3D multisegment lifting models

The purpose of this study was to compare the sensitivity of two dynamic tridimensional (3D) multisegment lifting models (upward and downward) to different errors. Two types of lifting tasks (sagittal and asymmetric) were considered. Five cameras, two force platforms and a dynamometric box provided inputs to the models. Known errors were induced in (1) joint centres, (2) the orientation of the anatomical coordinate system (ACS) of the pelvis, (3) external forces and (4) their point of application, (5) linear accelerations of segments' centre of mass and (6) body segment parameters (BSP) data to study their effect on the L5/S1 moments of both models. The results revealed that errors in joint centres and in the orientation of ACS axes were the principal sources of variability for both models. The upward model also require precise measurement of the point of application of the external forces while a downward model needs an accurate estimation of the BSP of trunk segments and an adequate estimation of upper segment accelerations. Since combined errors in the three moment components at L5/S1 generated an average of 5–9% incertitude in lumbar compression force estimates, it was concluded that the reduction of moment errors is important in spine modelling. PsycINFO classification: 2220; 2240; 4010

Probability of Collision Error Analysis

The decision for the International Space Station (ISS) to maneuver to avoid a potential collision with another space object will be based on the probability of collision, P C. The calculation of P C requires the covariance of both objects at conjunction. It is well known that the covariance computed by US Space Command is optimistic (too small), especially at altitudes where atmospheric drag is the dominant perturbation, because its computation assumes there are no dynamic model errors. In this paper the effect of errors in the covariance on P C and the sensitivity of P C to the encounter geometry are investigated.

A Model-Based Space Teleoperation System Using Mixed Force and Motion Commands.

It is well known that bilateral control becomes unstable in teleoperation with time delay. To tackle this problem, a model-based teleoperation system was introduced. Forces, reflected from the model on the ground-site, do not depend upon the existing time delay in the system. Unfortunately, such model-based systems strongly depend on the accuracy of the model. In our previous work, we proposed a teleoperation system which is robust with respect to geometric modeling errors. However, our system could not deal with dynamic modeling errors. In this paper, we propose a new approach to the model-based teleoperation, based on mixed force and motion commands. With this approach, both geometric and dynamic modeling errors are tolerable. The experimental results confirm the effectiveness of the approach.

Factor Bias under Stochastic Technical Change

AbstractTime‐series procedures are employed to determine the influence of technological change in inducing factor bias in U.S. agricultural production between 1948 and 1983. A dynamic measurement error model is used to link research expenditures to the unobserved technological change variable. Biasedness in labor and material factor shares is established.

Recursive single-layer nets for output error dynamic models

An algorithm for training recursive single-layer nets that has been shown to exhibit rapid convergence is presented. Convergence is not guaranteed, but a sufficient condition is given to justify the method. The method is demonstrated on a difficult modeling problem from bioengineering.

Attitude control with realization of linear error dynamics

An attitude control law is derived to realize linear unforced error dynamics with the attitude error defined in terms of rotation group algebra (rather than vector algebra). Euler parameters are used in the rotational dynamics model because they are globally nonsingular, but only the minimal three Euler parameters are used in the error dynamics model because they have no nonlinear mathematical constraints to prevent the realization of linear error dynamics. The control law is singular only when the attitude error angle is exactly pi rad about any eigenaxis, and a simple intuitive modification at the singularity allows the control law to be used globally. The forced error dynamics are nonlinear but stable. Numerical simulation tests show that the control law performs robustly for both initial attitude acquisition and attitude control.

Optimalpi‐coefficients of second‐order systems

Abstract Pi‐coefficients [4] provide a quantitative measure of system stability. With the goal of finding the largest stability range of the feedback gain in a system, the max‐p problem [9] can be formulated so that its solution gives an optimal set of pi‐coefficients. This paper simplifies the max‐p problem by analyzing the special structure of a negative semi‐definite matrix, called the M‐matrix. To find the optimal pi‐coefficients of the general second‐order systems, seven solutions corresponding to different conditions are derived. In the examples we apply the results to a model reference adaptive system, calculate the pi‐coefficients of its error dynamics model, and use the optimal pi‐coefficient curves to determine a good sampling period.

Optimal projection control of an experimental truss structure

Optimal projection reduced-order control theory is applied experimentally to a controlled structure testbed. The test structure has 25 disturbed modes, and the controller uses four strain sensors to command four noncollocated stress actuators. The best optimal projection design, an 18th-order controller derived from a 58th-order structural model, is experimentally found to reduce the broad-band vibration of five independent pointing error measures by as much as 66% without saturating the actuators and without destabilizing high-frequency modes. A similar reduced-order linear quadratic Gaussian controller is found to always destabilize high-frequency modes. The homotopy algorithm used to solve the optimal projection synthesis equations is described and its convergence is discussed. Analytical and experimental closed-loop performance for a series of optimal projection controllers are compared to equivalent linear quadratic Gaussian controllers to illustrate the effect of structural dynamic modeling errors on the results.

Stability and robustness of a high gain nonlinear robot manipulator controller

A nonlinear controller is proposed for the trajectory control of an n degree of freedom rigid link robotic manipulator, using task space feedback of the end-effector position and orientation. With a Lyapunov approach, a proof of the local asymptotic stability of the proposed nonlinear control law is given. The closed-loop system is shown analytically to be robust to two separate classes of parameter uncertainty. This parameter uncertainty includes dynamic (i) parameter perturations caused by manpulation of unknown payloads and manipulator operation in the presence of unknown viscous joint friction and (ii) controller parameter error arising from uncertainty in the kinematic and dynamic parameters of the manipulator. Through the choice of a particular high gain nonlinear controller term, it is shown that sufficient conditions are satisfied for local asymptotic stability in the presence of large dynamic parameter perturbations caused principally by manipulation of unknown payloads. Hence, for sufficiently high gain the closed-loop system is robust to large parameter perturbations. In addittion to an analytic proof of the asymptotic trajectory capability for constant reference input signals, close trajectory tracking is also demonstrated analytically for a particular class of input trajectories, again a result of the high gain nonlinear controller term. Finally, a numerical simulation of a two degree of freedom robot demonstrates both the asymptotic trajectory tracking capability for constant reference inputs and the close trajectory tracking capability for time varying trajectories with high gain feedback in the presence of kinematic and dynamic controller modelling errors and large dynamic parameter uncertainty.

A FORTRAN program for time-varying linear regression via flexible least squares

Suppose the prior theoretical beliefs concerning the generation of a time-series data set take two forms: a prior measurement specification that the data has been generated by a linear regression model; and a prior dynamic specification that the regression coefficients evolve only slowly over time, if at all. The objective is to understand the actual relationship between the observed data and the regressor variables. In particular, do the estimated regression coefficients display any systematic time-variation? Is time-constancy a reasonably satisfactory approximation? The “ flexible least squares” solution is defined to be the collection of all coefficient sequence estimates which yield vector-minimal sums of squared residual measurement and dynamic modelling errors for the given observations – i.e. which attain the “ residual efficiency frontier ”. Coefficient sequence estimates which attain the frontier have the following basic efficiency property: there exists no other coefficient sequence estimate which simultaneously lowers residual measurement error and residual dynamic error. The time-paths traced out by the frontier estimates thus indicate how the regression coefficients could have evolved over time in a manner minimally incompatible with the prior theoretical specifications. A user-friendly FORTRAN program (FLS) is now available for generating the frontier estimates. Program FLS has been extensively tested, and incorporates two basic validation tests which users can employ. Simulation experiments involving linear, quadratic, sinusoidal, and regime shift motions in the true regression coefficients are described. In each experiment the frontier estimates accurately reflect the qualitative time-variation displayed by the true regression coefficients, despite noisy observations.

Minimum model error estimation for poorly modeled dynamic systems

A novel strategy (which we call minimum model error'* estimation) for postexperiment optimal state estimation of discretely measured dynamic systems is developed and illustrated for a simple example. The method is especially appropriate for postexperiment estimation of dynamic systems whose presumed state governing equations are known to contain, or are suspected of containing, errors. The hew method accounts for errors in the system dynamic model equations in a rigorous manner. Specifically, the dynamic model error terms in the proposed method do not require the usual Kalman filter-smoother process noise assumptions of zero-mean, symmetrically distributed random disturbances, nor do they require representation by assumed parameterized time series (such as Fourier series); Instead, the dynamic model error terms require no prior assumptions other than piecewise continuity. Estimates of the state histories, as well as the dynamic model errors, are Obtained as part of the solution of a two-point boundary value problem. The state estimates are continuous and optimal in a global sense, yet the algorithm processes the measurements sequentially. The example demonstrates the method and shows it to be quite accurate for state estimation of a poorly modeled dynamic system.

Hypothesis testing in regression models with AR(1) errors and a lagged dependent variable: Bootstrapping and artificial regression compared

Serially correlated errors in dynamic models render the standard conditional estimator of the covariance matrix inconsistent. A Monte Carlo experiment confirms that the downward bias in the conventional variance estimator also exists in small samples. The results favour a consistent estimator based on an artificial regression (suggested by Davidson and Mackinnon) over bootstrapping the distribution of parameter estimates.

Practical Kalman Filter Software Performance Testing & Validation

Several tests are described which can be used for any Kalman-type filter/smoother computer program. These tests are demonstrated by a case history on a large dimensional Kalman filter/smoother program which implements a 34-state inertial navigation system dynamic error model. The execution of a large dimensional Kalman filter/smoother (KFS) on real measurement data does not represent a software test of the KFS since the right answer (the correct underlying state vector) is unknown; only ``reasonableness checks'' are actually possible. Simulated test data were used to exercise the KFS program in a Monte Carlo sense and its outputs evaluated using heuristic plot comparisons as well as rigorous statistical tests. Direct tests on the accuracy of the transition matrix, discrete process noise matrix, and covariance matrix calculations have been derived and demonstrated. Methods for testing properties of the Kalman filter innovations sequence are also covered. The approach and required auxiliary software that generates the test data can be employed to perform suboptimal modeling sensitivity studies and for evaluating analysis methods that depend on KFS estimates.

Estimating predictions, prediction errors and their standard deviations using constructed variables

Increasing use has been made of predictive tests for assessing model adequacy, but it is sometimes difficult to generate predictions and their standard errors in dynamic or simultaneous equation models. Following earlier suggestions by Salkever and Fuller, this paper shows how the requisite information may be obtained by the use of specially constructed variables in a regression framework. The main use of the method will be in those situations where prediction information is not available as a standard option in econometric packages.

Estimation of unmodeled forces on a low-thrust space vehicle

The application of a sequential estimation algorithm, which compensates for random errors in the dynamic model, to the problem of estimating the state of a continuously thrusting solar electric propulsion space vehicle is investigated. The dynamic model errors, due to random anomalies in the propulsion system, are approximated successfully by both first order and second order Gauss-Markov processes to obtain a more accurate and stable orbit determination algorithm. The importance of correct dynamic and measurement modeling in achieving accurate estimates is demonstrated.