James D. Turner

Generalizations and Applications of the Lagrange Implicit Function Theorem

The implicit function theorem due to Lagrange is generalized to enable high order implicit rate calculations of general implicit functions about pre-computed solutions of interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about an existing root (for algebraic systems) or trajectory (in case of dynamical systems). The generalization to dynamical systems, as a special case, enables the calculation of high order time varying sensitivities of the solutions of boundary value problems with respect to the parameters of the system model and/or functions describing the boundary condition. The generalizations thus realized are applied to various problems arising in trajectory optimization. It was found that useful information relating the neighboring extremal paths can be deduced from these implicit rates characterizing the behavior in the neighborhood of the existing solutions. The accuracy of solutions obtained is subsequently enhanced using an averaging scheme based on the Global Local Orthogonal Polynomial (GLO-MAP) weight functions developed by the first author to blend many local approximations in a continuous fashion. Example problems illustrate the wide applicability of the presented generalizations of Lagrange’s classical results to static and dynamic optimization problems.

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.

Near-minimum-time control of asymmetric rigid spacecraft using two controls

Abstract In general, spacecraft are designed to be maneuvered to achieve pointing objectives. To reorient the spacecraft with zero angular velocity at the end of the maneuver, a three-axis control design is usually used. When an actuator fails among three actuators, one must achieve these objectives using two control inputs, so that new control laws need to be considered. A simple and novel control law, which is based on the sequential Euler angle rotation strategy, is addressed. This paper explores a near minimum time control problem with constrained control input magnitudes. By introducing the three sequential sub-maneuvers with Euler-angle transformations, the governing nonlinear equations become rigorously linear, which permits a closed-form solution to be obtained for the switch times and final time, where the three sub-maneuvers are coupled through the unknown switch times. A numerical example demonstrates that the three-dimensional maneuver for an asymmetric spacecraft with two constrained control inputs can be successfully performed using the proposed closed-form solution.

The dynamic human immune response to cancer: It might just be rocket science

The immune-mediated eradication of cancer is made difficult both by immunosuppressive changes that occur within the tumor micro- environment as well as by systemic effects that lead to chronic inflammation and immunologic exhaustion. At present, the temporal dynamics and interconnectivity of these local and systemic impediments to antitumor immunity are poorly understood. Over the past several years, our laboratory has identified methods to concep - tualize and study the complex immunobiologic interactions driving the balance between toler- ance and immune activation over time in can- cer patients. We have recognized that dynamic immune responses in cancer may mirror natural self-organizing processes, such as species disper- sal into new ecological environments (1) and suburban sprawl (2) and, as such, are amenable to hypothesis testing and data-driven predictive modeling. In the following sections, we describe our approach to understanding alterations in immunity that occur as a result of the malig- nancy, as well as challenges posed by animal models to study this phenomenon. We show that tumor-induced immune modulation facilitates metastases and how comparative models for tol- erance may help us better understand this pro- cess. Finally, we discuss a method for discovery of immune biorhythms and predictive modeling of immune responses developed with the help of aerospace engineers. With our multidisciplinary approach, we are developing methods of dynamic immune network recognition that embrace the complexities of tumor/immune interactions with the goal of real-time, individualized, curative immunotherapy for cancer patients. Many immunotherapeutic strategies have been investigated in the treatment of advanced malignancies, including our laboratorys primary

Modeling, Control and Simulation of a Novel Mobile Robotic System

We are developing an autonomous mo- bile robotic system to emulate six degree of freedom relative spacecraft motion during proximity opera- tions. A mobile omni-directional base robot provides x, y, and yaw planar motion with moderate accuracy through six independently driven motors. With a six degree of freedom micro-positioning Stewart platform on top of the moving base, six degree of freedom spacecraft motion can be emulated with high accu- racy. This paper presents our approach to dynamic modeling, control, and simulation for the overall sys- tem. Compared with other simulations that intro- duced significant simplifications, we believe that our rigorous modeling approach is crucial for the high fi- delity hardware in-the-loop emulation.

Generalized Frequency Domain Modeling and Analysis For A Flexible Rotating Spacecraft

A flexible rotating spacecraft is modeled as a three body hybrid system consisting of a rigid hub a flexible appendage following the Euler-Bernoulli beam assumptions and a tip mass and inertia. Hamilton’s extended principle is used to derive the equations of motion and the boundary conditions of the system. This work compares the frequency domain accuracy provided by series approximation methods versus analytical models. Applying the Laplace transform to the integro-partial derivation equations of motion model, leads to a generalized state space model for the frequency domain representation of the system. Both approximate and exact transfer function models are developed and compared. Eigen decomposition is used to solve the flexible appendage sub-problem and then to find the solution for the full system of equations. The analytic frequency domain model is manipulated by introducing a spatial domain state space, where a standard convolution integral representation is used to invoke the boundary conditions that act at the tip mass for the free end of the beam. Closed-form solutions are obtained for the convolution integral forcing terms. The closed form solution is used to generate transfer functions for both the rigid and the flexible modes of the system in terms of the input torque. A numerical example is presented to compare the frequency response of the closed form solution transfer function to the numerical assumed modes solution. The difference resulting from in the natural frequencies resulting from the series truncation is highlighted and discussed. The closed form solution proves to be more accurate with no truncation errors and is suitable for control design iterations.

High-Order Uncertainty Propagation Enabled by Computational Differentiation

Modeling and simulation for complex applications in science and engineering develop behavior predictions based on mechanical loads. Imprecise knowledge of the model parameters or external force laws alters the system response from the assumed nominal model data. As a result, one seeks algorithms for generating insights into the range of variability that can be the expected due to model uncertainty. Two issues complicate approaches for handling model uncertainty. First, most systems are fundamentally nonlinear, which means that closed-form solutions are not available for predicting the response or designing control and/or estimation strategies. Second, series approximations are usually required, which demands that partial derivative models are available. Both of these issues have been significant barriers to previous researchers, who have been forced to invoke computationally intensive Monte-Carlo methods to gain insight into a system’s nonlinear behavior through a massive sampling process. These barriers are overcome by introducing three strategies: (1) Computational differentiation that automatically builds exact partial derivative models; (2) Map initial uncertainty models into instantaneous uncertainty models by building a series-based state transition tensor model; and (3) Compute an approximate probability distribution function by solving the Liouville equation using the state transition tensor model. The resulting nonlinear probability distribution function (PDF) represents a Liouville approximation for the stochastic Fokker-Planck equation. Several applications are presented that demonstrate the effectiveness of the proposed mathematical developments. The general modeling methodology is expected to be broadly useful for science and engineering applications in general, as well as grand challenge problems that exist at the frontiers of computational science and mathematics.

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.

Variable Step-Size Control for Analytic Power Series Solutions for Orbit Propagation

Recent trajectory modeling work using the continuation method has demonstrated that high accuracy trajectory models can be obtained for two-body and J2 perturbations. A high accuracy power series solution is achieved by introducing a nonlinear variable transformation for handling kinematic variables appearing in trajectory motion applications. Arbitrary order time derivatives are developed for the nonlinear kinematic variables, by recasting the variables in a form that leads to bilinear products that are easily handled by Leibnitz product rule, which leads to a highly efficient and accurate recursive numerical solution algorithm. The series solution works by computing multiple analytic continuation solutions for propagating the state trajectories. As with any numerical integration process, one must select a time step for advancing the system response. This is particularly important for highly eccentric orbits. To this end, this work presents a scheme for determining a variable step size in time for the continuation method. It is further shown that the optimally selected variable step size leads to a near linear step-size in true anomaly. Many numerical examples are presented for demonstrating the effectiveness of the analytic continuation method and the optimal step-size selection algorithm.