John L. Junkins

A Simultaneous Structure/Controller Design Iteration Method

Several methods are presented for placement/constrained optimization of the closed loop eigenvalues and eigenvectors of linear dynamical systems. A unified approach is taken to (i) iterate the design parameters in the plant being controlled, (ii) the location of sensor and actuators, (iii) the elements of a direct output feedback gain matrix, and (iv) the weight matrices in a time-domain LQG performance index, or (v) a combination of the foregoing, to accomplish a constrained, simultaneous optimization of the systems closed loop eigenvalues, eigenvectors, and their sensitivities. A low dimensioned discrete system and an order 42 model of a flexible structure controlled via direct output feedback are used to illustrate the approach.

Perturbation methods based upon varying action integrals

Abstract Direct perturbation solution procedures to non-linear systems are developed from variational statements derived from the principle of invariance of the action integral under infinitesimal transformations. Solution procedures that are the variational equivalent of the classical perturbation methods of strained parameters, the KBM method of averaging and the method of multiple time scales are presented.

Dynamic Analysis and Control of a Stewart Platform Using A Novel Automatic Dierentiation Method

This paper presents a kinematic based Lagrangian approach to generate the equations of motion and design an adaptive control law for multi-body systems. This methodology is applied to dynamic analysis and controller design study for a Stewart platform. Novel means of utilizing automatic dierentiation are employed to generate and solve the equations of motion, using only high level geometric and kinematic descriptions of the system. Based on deriving and coding only the kinematic descriptions of the system, the nonlinear motion of the platform is solved automatically and the analyst is freed from deriving, coding, and validating the lengthy nonlinear equations of motion. Lyapunov stability theory and concepts from adaptive control are used to formulate a nonlinear feedback control law. The control law is of the model reference adaptive structure, designed to track a prescribed smooth trajectory. By designing an adaptive update rule for the system mass and inertia parameters, the tracking errors are proven to be asymptotically stable for arbitrary parameter errors. Also, a PID adaptive control law is designed to guarantee bounded stability in the presence of bounded disturbances. Numerical results are included to illustrate the performance of the algorithm in the presence of large parameter errors and external disturbance.

Jth Moment Extended Kalman Filtering for Estimation of Nonlinear Dynamic Systems

Two flavors of an analytical approach (called the Jth Moment Extended Kalman Filtering, JMEKF) to estimate the state of a nonlinear dynamical system from vector measurements, developed by the authors recently are compared. The main distinction in the two flavors lies in the method of computing the time evolution of arbitrarily high order statistical moments between the classical Kalman update stages of the filter. The first flavor involves the state transition tensor approach (originally due to Park and Scheeres[1]) and the second flavor explicitly derives the statistical moment evolution equations using perturbation theory. Updating all, not only the first two, of the propagated statistical moments constitutes the JMEKF framework for estimation of nonlinear systems. This brief review is followed by a discussion outlining the assumptions in the assumed structure and associated convergence issues. The connection between probability theory and the associated statistical propagation developed in this paper is explored by an elementary exposition to entities called cumulants and characteristic functions. The tensor transformation between moments and cumulants is presented in a vector matrix form for ease in computations. To overcome the local nature of the Taylor expansions involved in the propagation and update of the JMEKF framework, a smooth particle approach is suggested. Several building blocks required for such a scheme are developed. The generation of the local density function process from evolved moments is detailed where multiple nominal expansion nominal solutions in the phase space are considered to reconstruct the evolved density function.

Dynamics and Controls of a Generalized Frequency Domain Model Flexible Rotating Spacecraft

Modeling a flexible rotating spacecraft as a distributed parameters system of a rigid hub attached to a flexible appendage is very common. When considering large angle maneuvers the same model applies to flexible robotic manipulators by adding a tip mass at the end of the flexible appendage to account for the payload. Following Euler-Bernoulli beam theory the dynamics for both no tip mass and tip mass models are derived. A Generalized State Space (GSS) system is constructed in the frequency domain to completely solve for the input-output transfer functions of the models. The analytical solution of the GSS is obtained and compared against the classical assumed modes method. The frequency response of the system is then used in a classical control problem where a Lyapunov stable controller is derived and tested for gain selection. The assumed modes method is used to obtain the time response of the system to verify the gain selections and draw connections between the frequency and the time domains. The GSS approach provides a powerful tool to test various control schemes in the frequency domain and a validation platform for existing numerical methods utilized to solve distributed parameters models.

A New Method for Space Objects Probability of Collision

The state of a dynamical system and its uncertainty, as defined by its probability density function (PDF), are valuable for numerous fields in science and engineering. There have been numerous methods proposed to estimate and quantify this uncertainty. In astrodynamics, space situational awareness (SSA) is a major area that relies on uncertainty quantification to estimate a space object’s state and its associated uncertainty. This data is invaluable for making informed decisions regarding probability of collision, tracking, and catalog maintenance. A new method for uncertainty quantification based on orthogonal polynomials and the application of Liouville’s theorem is developed. The method identifies the region of extreme probability at the time of interest and populates that region with structured points. The associated PDF is computed based on the a-priori PDF of the initial conditions and/or the nominal values of the system parameters (e.g. drag coefficient). High dimension orthogonal polynomials are used to approximate the PDF at the target time. Having an analytical expression for the propagated PDF enables rigorous probabilistic analysis. The present method is applied to several problems to compute the probability of collision between two objects. Numerical experiments show orders of magnitude improvement in computational cost versus classical Monte Carlo Methods. The new approach is easy to implement, extensible to higher dimensions, computationally efficient and provides a rigorous approach to address probability of collision problems in SSA.

Digital Pulse Processing Methods for Range Refinement of LADAR Systems

This paper presents several data processing techniques to resolve the time of flight from digitized light pulses received by a state of the art scanning LADAR system called HD6D. The challenges in the wave form identification and peak-to-peak distance quantification are outlined. Algorithms to identify appropriate cases from digitized samples are outlined and the most challenging case of overlapping wave forms is detailed. For the challenging case of overlapping waves, two distinct algorithms are proposed to estimate the peak to peak distance. First approach known as XCURV uses a sum of two exponential functions to model the over-lapping wave forms. A nonlinear least squares algorithm is used to estimate the unknown distance between the peaks along with the amplitude ratios of the waves recorded. Second approach uses a cubic spline fit to the data to estimate the area under the curve. It is shown that the centroid computation of the spline fit is equivalent to a weighted average of the measurement points of interest. Numerical simulations and experimental results are presented for both algorithms.