Chandeok Park

Solving Optimal Continuous Thrust Rendezvous Problems with Generating Functions

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Starting from the 1st order necessary conditions for optimality, we derive a Hamiltonian system for the state and adjoints with split boundary conditions. Then, recognizing the two point boundary value problem as a canonical transformation, we employ generating functions to find the optimal feedback control as well as the optimal trajectory. Though we formulate the optimal control problem in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a non-linear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two point boundary value problem using a forward shooting method.

Determination of optimal feedback terminal controllers for general boundary conditions using generating functions

Given a nonlinear system and a performance index to be minimized, we present a general approach to expressing the finite time optimal feedback control law applicable to different types of boundary conditions. Starting from the necessary conditions for optimality represented by a Hamiltonian system, we solve the Hamilton-Jacobi equation for a generating function for a specific canonical transformation. This enables us to obtain the optimal feedback control for fundamentally different sets of boundary conditions only using a series of algebraic manipulations and partial differentiations. Furthermore, the proposed approach reveals an insight that the optimal cost functions for a given dynamical system can be decomposed into a single generating function that is only a function of the dynamics plus a term representing the boundary conditions. This result is formalized as a theorem. The whole procedure provides an advantage over methods rooted in dynamic programming, which require one to solve the Hamilton-Jacobi-Bellman equation repetitively for each type of boundary condition. The cost of this favorable versatility is doubling the dimension of the partial differential equation to be solved.

STATUS AND PROGRESS OF ARGO-M SYSTEM DEVELOPMENT

KASI (Korea Astronomy and Space Science Institute) has developed an SLR (Satellite Laser Ranging) system since 2008. The name of the development program is ARGO (Accurate Ranging system for Geodetic Observation). ARGO has a wide range of applications in the satellite precise orbit determination and space geodesy research using SLR with mm-level accuracy. ARGO-M (Mobile, bistatic 10 cm transmitting/40 cm receiving telescopes) and ARGO-F (Fixed stationary, about 1 m transmitting/receiving integrated telescope) SLR systems development will be completed by 2014. In 2011, ARGO-M system integration was completed. At present ARGO-M is in the course of system calibration, functionality, and performance tests. It consists of six subsystems, OPS (Optics System), TMS (Tracking Mount System), OES (Opto-Electronic System), CDS (Container-Dome System), LAS (Laser System) and AOS (ARGO Operation System). In this paper, ARGO-M system structure and integration status are introduced and described.

Extended applications of generating functions to optimal feedback control problems

As a natural extension of our recent work on finding optimal feedback control laws based on generating functions of a Hamiltonian system, we consider an optimal control problem with control constraints and a singular optimal control problem. For the problem with control constraints, we consider the time optimal control of the double integrator, and show that our approach can recover the necessary and sufficient conditions of optimal feedback control laws directly. For the singular optimal control problem, we study the linear quadratic problem and show that our method reproduces the conventional solution satisfying the necessary conditions for optimality. The current study is used to more fully understand our approach with the goal of defining a method that is applicable to more general systems.

Development of Integrated Orbit and Attitude Software-in-the-loop Simulator for Satellite Formation Flying

An integrated orbit and attitude control algorithm for satellite formation flying was developed, and an integrated orbit and attitude software-in-the-loop (SIL) simulator was also developed to test and verify the integrated control algorithm. The integrated algorithm includes state-dependent Riccati equation (SDRE) control algorithm and PD feedback control algorithm as orbit and attitude controller respectively and configures the two algorithms with an integrating effect. The integrated SIL simulator largely comprises an orbit SIL simulator for orbit determination and control, and attitude SIL simulator for attitude determination and control. The two SIL simulators were designed considering the performance and characteristics of related hardware-in-the-loop (HIL) simulators and were combined into the integrated SIL simulator. To verify the developed integrated SIL simulator with the integrated control algorithm, an orbit simulation and integrated orbit and attitude simulation were performed for a formation reconfiguration scenario using the orbit SIL simulator and the integrated SIL simulator, respectively. Then, the two simulation results were compared and analyzed with each other. As a result, the user satellite in both simulations achieved successful formation reconfiguration, and the results of the integrated simulation were closer to those of actual satellite than the orbit simulation. The integrated orbit and attitude control algorithm verified in this study enables us to perform more realistic orbit control for satellite formation flying. In addition, the integrated orbit and attitude SIL simulator is able to provide the environment of easy test and verification not only for the existing diverse orbit or attitude control algorithms but also for integrated orbit and attitude control algorithms.

Global Solution for the Optimal Feedback Control of the Underactuated Heisenberg System

We present a global solution for an optimal feedback controller of the underactuated Heisenberg system or nonholonomic integrator. Employing a recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory, we circumvent a singularity caused by under actuation to develop a nonlinear optimal feedback control in an implicitly analytical form. The systematic procedure to deal with under actuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems.

Globally Optimal Feedback Control Law of the Underactuated Heisenberg System by Generating Functions

We present a global solution for an optimal feedback control problem of the underactuated Heisenberg system or nonholonomic integrator. Set in the general framework of the Hamilton-Jacobi theory, this work demonstrates the potential applicability of our methodology to general underactuated optimal control problems. We incorporate the Heisenberg system into a typical optimal control formulation called the hard constraint problem, and transform into a two point boundary value problem for a Hamiltonian system. It is viewed as a canonical transformation in itself, to which we apply our recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory. It is first recognized that our previously developed procedure for solving fully-actuated optimal control problems is not directly applicable due to a singularity caused by underactuation. However, within the same framework of generating functions we are provided with a way to circumvent this singularity by algebraic manipulations linked with the underactuated coordinate. This results in a scalar transcendental equation whose solution ultimately leads to a nonlinear optimal feedback control law in an analytical form. We illustrate our solution by numerical examples

Solutions of the optimal feedback control problem using Hamiltonian dynamics and generating functions

We show that the optimal cost function that satisfies the Hamilton-Jacobi-Bellman (HJB) equation is a generating function for a class of canonical transformations for the Hamiltonian dynamical system defined by the necessary conditions for optimality. This result allows us to circumvent the final time singularity in the HJB equation for a finite time problem, and allows us to analytically construct a nonlinear optimal feedback control and cost function that satisfies the HJB equation for a large class of dynamical systems. It also establishes that the optimal cost function can be computed from a large class of solutions to the Hamilton-Jacobi (HJ) equation, many of which do not have singular boundary conditions at the terminal state.

Non-recursive estimation using a batch filter based on particle filtering

In this paper, the non-recursive estimation algorithm using a batch filter based on particle filtering is developed and utilized for one-dimensional nonlinear example. For comparison study, algorithms of a batch filter based on unscented transformation and generic particle filtering are briefly reviewed and new algorithm of a batch filter based on particle filtering is presented. For verification of presented batch filters performance, the numerical simulations and the accuracy assessment are achieved and results are compared with those of a batch filter based on unscented transformation for various nonlinear and non-Gaussian environments. The root mean square value of differences between observational values and computed values after convergence is used for precision check of filtering process, estimated initial state value and the difference between true initial state value and estimated initial state value are used for state accuracy check of nonlinear estimation. Large initial state and various type of measurement noise are used for nonlinear and non-Gaussian environments, respectively. Under large initial state error or large non-Gaussian measurement noise, the developed non-recursive estimation algorithm give more robust and accurate estimation results than those of a batch filter based on unscented transformation easily. Finally, the non-recursive batch filter based on particle filtering is effectively applicable for batch estimation problems under nonlinear and non-Gaussian environments.

Laser-based Relative Navigation Using GPS Measurements for Spacecraft Formation Flying

This study presents a precise relative navigation algorithm using both laser and Global Positioning System (GPS) measurements in real time. The measurement model of the navigation algorithm between two spacecraft is comprised of relative distances measured by laser instruments and single differences of GPS pseudo-range measurements in spherical coordinates. Based on the measurement model, the Extended Kalman Filter (EKF) is applied to smooth the pseudo-range measurements and to obtain the relative navigation solution. While the navigation algorithm using only laser measurements might become inaccurate because of the limited accuracy of spacecraft attitude estimation when the distance between spacecraft is rather large, the proposed approach is able to provide an accurate solution even in such cases by employing the smoothed GPS pseudo-range measurements. Numerical simulations demonstrate that the errors of the proposed algorithm are reduced by more than about 12% compared to those of an algorithm using only laser measurements, as the accuracy of angular measurements is greater than 0.001° at relative distances greater than 30 km.