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At least 487 records · Page 27

The dynamics and control of large flexible space structures - 12, supplement 11

The rapid 2-D slewing and vibrational control of the unsymmetrical flexible SCOLE (Spacecraft Control Laboratory Experiment) with multi-bounded controls is considered. Pontryagin's Maximum Principle is applied to the nonlinear equations of the system to derive the necessary conditions for the optimal control. The resulting two point boundary value problem is then solved by using the quasilinearization technique, and the near minimum time is obtained by sequentially shortening the slewing time until the controls are near the bang-bang type. The tradeoff between the minimum time and the minimum flexible amplitude requirements is discussed. The numerical results show that the responses of the nonlinear system are significantly different from those of the linearized system for rapid slewing. The SCOLE station-keeping closed loop dynamics are re-examined by employing a slightly different method for developing the equations of motion in which higher order terms in the expressions for the mast modal shape functions are now included. A preliminary study on the effect of actuator mass on the closed loop dynamics of large space systems is conducted. A numerical example based on a coupled two-mass two-spring system illustrates the effect of changes caused in the mass and stiffness matrices on the closed loop system eigenvalues. In certain cases the need for redesigning control laws previously synthesized, but not accounting for actuator masses, is indicated.

Bainum, Peter M.↗

Factorization and the synthesis of optimal feedback gains for distributed parameter systems

An approach based on Volterra factorization leads to a new methodology for the analysis and synthesis of the optimal feedback gain in the finite-time linear quadratic control problem for distributed parameter systems. The approach circumvents the need for solving and analyzing Riccati equations and provides a more transparent connection between the system dynamics and the optimal gain. The general results are further extended and specialized for the case where the underlying state is characterized by autonomous differential-delay dynamics. Numerical examples are given to illustrate the second-order convergence rate that is derived for an approximation scheme for the optimal feedback gain in the differential-delay problem.

Milman, Mark H.↗

Sensitivity of control-augmented structure obtained by a system decomposition method

The verification of a method for computing sensitivity derivatives of a coupled system is presented. The method deals with a system whose analysis can be partitioned into subsets that correspond to disciplines and/or physical subsystems that exchange input-output data with each other. The method uses the partial sensitivity derivatives of the output with respect to input obtained for each subset separately to assemble a set of linear, simultaneous, algebraic equations that are solved for the derivatives of the coupled system response. This sensitivity analysis is verified using an example of a cantilever beam augmented with an active control system to limit the beam's dynamic displacements under an excitation force. The verification shows good agreement of the method with reference data obtained by a finite difference technique involving entire system analysis. The usefulness of a system sensitivity method in optimization applications by employing a piecewise-linear approach to the same numerical example is demonstrated. The method's principal merits are its intrinsically superior accuracy in comparison with the finite difference technique, and its compatibility with the traditional division of work in complex engineering tasks among specialty groups.

Sobieszczanski-Sobieski, Jaroslaw↗

Fundamentals of Mold Free Casting: Experimental and Computational Studies

Researchers are developing the technology of 'Ballistic Particle Manufacturing' (BPM) in which individual drops are precisely layered onto a substrate, and the drops are deposited so as to prevent splatting. These individual drops will ultimately be combined to form a net-shape, three-dimensional object. Our understanding of controlled drop deposition as applied to BPM is far from complete. Process parameters include the size and temperature of the liquid metal drop, its impact velocity and trajectory, and the condition and temperature of the substrate. Quantitative knowledge of the fluid mechanics and heat transfer of drop deposition and solidification are necessary to fully optimize the manufacturing process and to control the material microstructure of the final part. The object of this study is to examine the dynamics of liquid metal drops as they impinge upon a solid surface and solidify under conditions consistent with BPM (i.e. conditions which produce non-splatting drops). A program of both numerical simulations and experiments will be conducted. Questions this study will address include the following: How do the deformation and solidification of the drop depend on the properties of the fluid drop and the solid substrate? How does the presence of previously deposited drops affect the impingement and solidification process? How does the impingement of the new drop affect already deposited material? How does the cooling rate and solidification of the drops influence the material microstructure?

Tryggvason, Gretar↗

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.

Kennedy, Christopher A.↗

Implicit Formulation of Muscle Dynamics in OpenSim

Astronauts lose bone and muscle mass during spaceflight. Exercise countermeasure is the primary method for counteracting bone and muscle mass loss in space. New spacecraft exercise device concepts are currently being developed for the NASAs new crew exploration vehicle. The NASA Digital Astronaut Project (DAP) uses computational modeling to help determine if the new exercise devices will be effective as countermeasures. The NASA Digital Astronaut Project is developing the ability to utilize predictive simulation to provide insight into the change in kinematics and kinetics with a change in device and gravitational environment (1-g versus 0-g). For example, in space exercise the subject's body weight is applied in addition to the loads prescribed for musculoskeletal maintenance. How and where these loads are applied obviously directly impacts bone and tissue loads. Additionally, due to space vehicle structural requirements, exercise devices are often placed on vibration isolation systems. This changes the apparent impedance or stiffness of the device as seen by the user. Data collection under these conditions is often impractical and limited. Predictive modeling provides a means to have a virtual subject to test hypotheses. Predictive simulation provides a virtual subject for which we are able to perform studies such as sensitivity to device loading and vibration isolation without the need for laboratory kinematic or kinetic test data.Direct Collocation optimization provides an efficient means to perform task based optimization and predictive modeling. It is relatively straight forward to structure a physical exercise task in a Direct Collocation mathematical formulation: perform a motion such that you start at an initial pose, achieve a given amount of deflection i.e a squat, return to the initial pose, and minimize muscle activation cost. Direct Collocation is advantageous in that it does not require numerical integration to evaluate the objective function. Instead, the system dynamics are transformed to discrete time and the optimizer is constrained such that the solution is not considered to be a valid unless the dynamic equations are satisfied at all time points. The simulation and optimization are effectively done simultaneously. Due to the implicit integration, time steps can be more coarse than in a differential equation solver. In a gait scenario this means that that the model constraints and cost function are evaluated at 100 nodes in the gait cycle versus 10,000 integration steps in a variable-step forward dynamic simulation. Furthermore, no time is wasted on accurate simulations of movements that are far from the optimum. Constrained optimization algorithms require a Jacobian matrix that contains the partial derivatives of each of the dynamic constraints with respect to of each of the state and control variables at all time points. This is a large but sparse matrix. An implicit dynamics formulation requires computation of the dynamic residuals f as a function of the states x and their derivatives, and controls u:f(x, dxdt, u) 0If the dynamics of musculoskeletal system are formulated implicitly, the Jacobian elements are often available analytically, eliminating the need for numerical differentiation; this is obviously computationally advantageous. Additionally, implicit formulation of musculoskeletal dynamics do not suffer from singularities from low mass bodies, zero muscle activation, or other stiff system or

physical exercise↗

An optimal output feedback gain variation scheme for the control of plants exhibiting gross parameter changes

A concept for optimally designing output feedback controllers for plants whose dynamics exhibit gross changes over their operating regimes was developed. This was to formulate the design problem in such a way that the implemented feedback gains vary as the output of a dynamical system whose independent variable is a scalar parameterization of the plant operating point. The results of this effort include derivation of necessary conditions for optimality for the general problem formulation, and for several simplified cases. The question of existence of a solution to the design problem was also examined, and it was shown that the class of gain variation schemes developed are capable of achieving gain variation histories which are arbitrarily close to the unconstrained gain solution for each point in the plant operating range. The theory was implemented in a feedback design algorithm, which was exercised in a numerical example. The results are applicable to the design of practical high-performance feedback controllers for plants whose dynamics vary significanly during operation. Many aerospace systems fall into this category.

Moerder, Daniel D.↗

Control optimization of a lifting body entry problem by an improved and a modified method of perturbation function

A study of the solution problem of a complex entry optimization was studied. The problem was transformed into a two-point boundary value problem by using classical calculus of variation methods. Two perturbation methods were devised. These methods attempted to desensitize the contingency of the solution of this type of problem on the required initial co-state estimates. Also numerical results are presented for the optimal solution resulting from a number of different initial co-states estimates. The perturbation methods were compared. It is found that they are an improvement over existing methods.

Garcia, F., Jr.↗

Airfoil Design Using a Coupled Euler and Integral Boundary Layer Method with Adjoint Based Sensitivities

The objective of this paper is to present a control theory approach for the design of airfoils in the presence of viscous compressible flows. A coupled system of the integral boundary layer and the Euler equations is solved to provide rapid flow simulations. An adjunct approach consistent with the complete coupled state equations is employed to obtain the sensitivities needed to drive a numerical optimization algorithm. Design to target pressure distribution is demonstrated on an RAE 2822 airfoil at transonic speed.

Edwards, S.↗

Multi-Impulse to Time Optimal Finite Burn Trajectory Conversion

A novel algorithm is presented that provides an improvement over a traditional parameter optimization method when solving time-optimal, finite-burn pseudo-rendezvous spacecraft trajectory problems. A hybrid optimization procedure is described that converts a set of multiple-impulses, representing high- or low-thrust maneuvers, to an exact time-optimal finite-burn trajectory for a thrust limited, constant exhaust velocity spacecraft. The Hybrid Method applies a control law derived from the Euler-Lagrange system of equations within the classical Indirect Method to a modern Direct Method. An iterative adjoint-control transformation and an evolving constraint vector are introduced to solve the optimal control two-point boundary value problem. This method requires no prior knowledge of the solution, which adds simplicity to the trajectory design process and aids automation. Examples are shown for low-thrust apogee raise maneuvers, non-coplanar Earth orbit transfers, and a modified Deep Space 1 low-thrust trajectory. A numerically significant improvement to objective cost is shown across all application problems compared to a traditional solution method, as well as a significant improvement to convergence speed for a select class of problems.

Joshua A Fogel↗

Research on output feedback control of systems with ill-conditioned dynamics

The accuracy of a two time scale approach to the output feedback regulator design problem is examined. An approximate quadratic performance index which reflects a two time scale decomposition of the system dynamics is developed. A sequential numerical algorithm is defined which obtains output feedback gains minimizing a broad class of performance indices, including the standard LQ case. A procedure for optimally zeroing selected gain elements in an output feedback gain matrix is developed and demonstrated. A summary of conference and journal publications from this research is also provided.

Calise, A. J.↗

The use of locally optimal trajectory management for base reaction control of robots in a microgravity environment

A locally-optimal trajectory management (LOTM) approach is analyzed, and it is found that care should be taken in choosing the Ritz expansion and cost function. A modified cost function for the LOTM approach is proposed which includes the kinetic energy along with the base reactions in a weighted and scale sum. The effects of the modified functions are demonstrated with numerical examples for robots operating in two- and three-dimensional space. It is pointed out that this modified LOTM approach shows good performance, the reactions do not fluctuate greatly, joint velocities reach their objectives at the end of the manifestation, and the CPU time is slightly more than twice the manipulation time.

Lin, N. J.↗

A quasi-analytical method for non-iterative computation of nonlinear controls

An optimal control solution process was developed for a general class of nonlinear dynamical systems. The method combines control theory, perturbation methods, and Van Loan's recent matrix exponential results. A variety of applications support the practical utility of this method. Nonlinear rigid body optimal maneuvers are routinely solved. Flexible body dynamical systems of an order greater than 40 were solved. The method fails occasionally due to poor convergence of the perturbation expansion or numerical difficulties associated with computing the matrix exponential. The method is attractive because it appears to be a good candidate for semi-automation; no initial guess is required, and it usually converges at 2nd or 3rd order in minutes of machine time.

Junkins, J. L.↗

Active control law synthesis for flexible aircraft

An application of an active control law synthesis procedure is presented, to meet multiple design requirements for a flexible aircraft modeled by a large order state space system. In this synthesis procedure, a linear quadratic Gaussian type cost function is minimized by updating the free parameters of the control law, while satisfying a set of constraints on the design loads, responses and stability margins. Analytical expressions for gradients of the cost function and the constraints, with respect to the control law design variables are used to facilitate rapid numerical convergence. These gradients can also be used for sensitivity study. A stable classical control law as well as an estimator-based full or reduced order control law can be modified, in order to meet individual root-mean-square response limitations as well as minimum singular value restrictions. Both analog and digital control laws can be optimized. Low order, robust control laws were synthesized for flutter suppression of a flexible aircraft.

Mukhopadhyay, V.↗

Structural tailoring/analysis for hypersonic components - Executive system development

No direct analytical or integrated numerical tool exists today for the optimal design of a generic class of built-up actively cooled composite structure for applications in hypersonic propulsion ducts. The need exists for a numerical tool to perform the comprehensive design/analysis of a panel on the inlet wall under hypersonic flight conditions. Such a tool requires relatively complex multi-disciplinary analysis. One such numerical tool controlled by an executive system has been developed and is named as STAHYC (Structural Tailoring/Analysis for HYpersonic Components). A detailed account of the executive system development of STAHYC along with the results of one example inlet panel design problem is given in this paper.

Narayanan, G. V.↗

Approximate analysis for repeated eigenvalue problems with applications to controls-structure integrated design

A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.

Kenny, Sean P.↗

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(sub 2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(sub 2) methods.

Kennedy, Christopher A.↗

Handling qualities of large flexible control-configured aircraft

The effects on handling qualities of low frequency symmetric elastic mode interaction with the rigid body dynamics of a large flexible aircraft was analyzed by use of a mathematical pilot modeling computer simulation. An extension of the optimal control model for a human pilot was made so that the mode interaction effects on the pilot's control task could be assessed. Pilot ratings were determined for a longitudinal tracking task with parametric variations in the undamped natural frequencies of the two lowest frequency symmetric elastic modes made to induce varying amounts of mode interaction. Relating numerical performance index values associated with the frequency variations used in several dynamic cases, to a numerical Cooper-Harper pilot rating has proved successful in discriminating when the mathematical pilot can or cannot separate rigid from elastic response in the tracking task.

Swaim, R. L.↗