## Group VDOL

Group Description |
Optimal control problems, Vehicle Dynamics & Optimization Lab, UF Anil Rao and Begum Senses, University of Florida http://vdol.mae.ufl.edu Each optimal control problem is described below. Each of these problems gives rise to a sequence of matrices of different sizes when they are being solved inside GPOPS, an optimal control solver created by Anil Rao, Begum Senses, and others at in VDOL lab at the University of Florida. The matrices are all symmetric indefinite. Rao, Senses, and Davis have created a graph coarsening strategy that matches pairs of nodes. The mapping is given for each matrix, where map(i)=k means that node i in the original graph is mapped to node k in the smaller graph. map(i)=map(j)=k means that both nodes i and j are mapped to the same node k, and thus nodes i and j have been merged. Each matrix consists of a set of nodes (rows/columns) and the names of these rows/cols are given for each matrix. Anil Rao, Begum Sense, and Tim Davis, 2015. ---------------------------------------------------------------------- VDOL/dynamicSoaring ---------------------------------------------------------------------- Dynamic soaring optimal control problem is taken from Ref.~\cite{zhao2004optimal} where the dynamics of a glider is derived using a point mass model under the assumption of a flat Earth and stationary winds. The goal of the dynamic soaring problem is to determine the state and the control that minimize the average wind gradient slope that can sustain a powerless dynamic soaring flight. The state of the system is defined by the air speed, heading angle, air-realtive flight path angle, altitude, and the position of the glider and the control of the system is the lift coefficient. The specified accuracy tolerance of $10^{-7}$ were satisfied after eight mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 647 to 3543. @article{zhao2004optimal, title={Optimal Patterns of Glider Dynamic Soaring}, author={Zhao, Yiyuan J}, journal={Optimal Control applications and methods}, volume={25}, number={2}, pages={67--89}, year={2004}, publisher={Wiley Online Library} } ---------------------------------------------------------------------- VDOL/freeFlyingRobot ---------------------------------------------------------------------- Free flying robot optimal control problem is taken from Ref.~\cite{sakawa1999trajectory}. Free flying robot technology is expected to play an important role in unmanned space missions. Although NASA currently has free flying robots, called spheres, inside the International Space Station (ISS), these free flying robots have neither the technology nor the hardware to complete inside and outside inspection and maintanance. NASA's new plan is to send new free flying robots to ISS that are capable of completing housekeeping of ISS during off hours and working in extreme environments for the external maintanance of ISS. As a result, the crew in ISS can have more time for science experiments. The current free flying robots in ISS works are equipped with a propulsion system. The goal of the free flying robot optimal control problem is to determine the state and the control that minimize the magnitude of thrust during a mission. The state of the system is defined by the inertial coordinates of the center of gravity, the corresponding velocity, thrust direction, and the anglular velocity and the control is the thrust from two engines. The specified accuracy tolerance of $10^{-6}$ were satisfied after eight mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 798 to 6078. ---------------------------------------------------------------------- VDOL/goddardRocketProblem ---------------------------------------------------------------------- Goddard rocket maximum ascent optimal control problem is taken from Ref.~\cite{goddard1920method}. The goal of the Goddard rocket maximum ascent problem is to determine the state and the control that maximize the final altitude of an ascending rocket. The state of the system is defined by the altitude, velocity, and the mass of the rocket and the control of the system is the thrust. The Goddard rocket problem contains a singular arc where the continuous-time optimality conditions are indeterminate, thereby the nonlinear programming problem solver will have difficulty determining the optimal control during the singular arc. In order to prevent this difficulty and obtain more accurate solutions the Goddard rocket problem is posed as a three-phase optimal control problem. Phase one and phase three contains the same dynamics and the path constraints as the original problem, while phase two contains an additional path constraint and an event constraint. The specified accuracy tolerance of $10^{-8}$ were satisfied after two mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 831 to 867. @article{goddard1920method, title={A Method of Reaching Extreme Altitudes.}, author={Goddard, Robert H}, journal={Nature}, volume={105}, pages={809--811}, year={1920} } ---------------------------------------------------------------------- VDOL/hangGlider ---------------------------------------------------------------------- Range maximization of a hang glider optimal control problem is taken from Ref.~\cite{bulirsch1993combining}. The goal of the optimal control problem is to determine the state and the control that maximize the range of the hang glider in the presence of a thermal updraft. The state of the system is defined by horizontal distance, altitude, horizontal velocity, and the vertical velocity and the control is the lift coefficient. The specified accuracy tolerance of $10^{-8}$ were satisfied after five mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 360 to 16011. This problem is sensitive to accuracy of the mesh and it requires excessive number of collocation points to be able to satisfy the accuracy tolerance. Thus, the size of the KKT matrices changes drastically. @book{bulirsch1993combining, title={Combining Direct and Indirect Methods in Optimal Control: Range Maximization of a Hang Glider}, author={Bulirsch, Roland and Nerz, Edda and Pesch, Hans Josef and von Stryk, Oskar}, year={1993}, publisher={Springer} } ---------------------------------------------------------------------- VDOL/kineticBatchReactor ---------------------------------------------------------------------- ---------------------------------------------------------------------- VDOL/lowThrust ---------------------------------------------------------------------- Low-thrust orbit transfer optimal control problem is taken from Ref.~\cite{betts2010practical}. The goal of the low-thrust orbit transfer problem is to determine the state and the control that minimize the fuel consumption during the orbit transfer of a spacecraft that starts from a low-earth orbit and terminates at the geostationary orbit via low-thrust propulsion systems. The highly nonlinear dynamics of the low-thrust orbit transfer problem is given in modified equinoctial elements (state of the system) and the thrust direction (control of the system). Furthermore, the low-thrust optimal control problem is a badly scaled problem because of the small thrust-to-initial-mass ratio, that is typically on the order of $O(10^{-4})$, and the long orbit transfer duration. Badly scaling of the problem leads to a lot of delayed pivots. The specified accuracy tolerance of $10^{-8}$ were satisfied after thirteen mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 584 to 18476. @book{betts2010practical, title={Practical Methods for Optimal Control and Estimation Using Nonlinear Programming}, author={Betts, John T}, volume={19}, year={2010}, publisher={SIAM Press}, address = {Philadelphia, Pennsylvania}, } ---------------------------------------------------------------------- VDOL/orbitRaising ---------------------------------------------------------------------- Orbit raising problem that is taken from Ref.~\cite{bryson1975applied}. The goal of the optimal control problem is to determine the state and the control that maximize the radius of an orbit transfer in a given time. The state of the system is defined by radial distance of the spacecraft from the attracting center (e.g Earth, Mars, etc.) and velocity of the spacecraft and the control is the thrust direction. The specified accuracy tolerance of $10^{-8}$ were satisfied after four mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 442 to 915. @book{bryson1975applied, title={Applied Optimal Control: Optimization, Estimation, and Control}, author={Bryson, Arthur Earl}, year={1975}, publisher={CRC Press} } ---------------------------------------------------------------------- VDOL/reorientation ---------------------------------------------------------------------- Minimum-time reorientation of an asymmetric rigid body optimal control problem is taken from Ref.~\cite{betts2010practical}. The goal of the problem is to determine the state and the control that minimize the time that is required to reorient a rigid body. The state of the system is defined by quaternians that gives the orientation of the spacecraft and the angular velocity of the spacecraft and the control of the system is torque. The vehicle data that is used to model the dynamics are taken from NASA X-ray Timing Explorer spacecraft. The specified accuracy tolerance of $10^{-8}$ were satisfied after eight mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 677 to 3108. @book{betts2010practical, title={Practical Methods for Optimal Control and Estimation Using Nonlinear Programming}, author={Betts, John T}, volume={19}, year={2010}, publisher={SIAM Press}, address = {Philadelphia, Pennsylvania}, } ---------------------------------------------------------------------- VDOL/spaceShuttleEntry ---------------------------------------------------------------------- Space shuttle launch vehicle reentry optimal control problem is taken from Ref.~\cite{betts2010practical}. The goal of the optimal control problem is to determine the state and the control that maximize the cross range (maximize the final latitude) during the atmospheric entry of a reusable launch vehicle. State of the system is defined by the position, velocity, and the orientation of the space shuttle and the control of the system is the angle of attack and the bank angle of the space shuttle. The specified accuracy tolerance of $10^{-8}$ were satisfied after two mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 560 to 2450. ---------------------------------------------------------------------- VDOL/spaceStation ---------------------------------------------------------------------- Space station attitude optimal control problem is taken from Ref.~\cite{betts2010practical}. The goal of the space station attitude control problem is to determine the state and the control that minimize the magnitude of the final momentum while the space statition reaches an orientation at the final time that can be maintained without utilizing additional control torque. The state of the system is defined by the angular velocity of the spacecraft with respect to an inertial reference frame, Euler-Rodriguez parameters used to defined the vehicle attitude, and the angular momentum of the control moment gyroscope and the control of the system is the torque. The specified accuracy tolerance of $10^{-7}$ were satisfied after thirteen mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 99 to 1640. ---------------------------------------------------------------------- VDOL/tumorAntiogenesis ---------------------------------------------------------------------- Tumor anti-angiogenesis optimal control problem is taken from Ref.~\cite{ledzewicz2008analysis}. A tumor first uses the blood vessels of its host but as the tumor grows oxygen that is carried by the blood vessels of its host cannot defuse very far into the tumor. Therefore, the tumor grows its own blood vessels by producing vascular endothelial growth factor (VEGF). This process is called angiogenesis. But blood vessels have a defense mechanism, called endostatin, that tries to impede the development of new blood cells by targeting VEGF. In addition, new pharmacological therapies that is developed for tumor-type cancers also targets VEGF. The goal of the tumor anti-angiogenesis problem is to determine the state and control that minimizing the size of the tumor at the final time. The state of the system is defined by the tumor volume, carrying capacity of a vessel, and the total anti-angiogenic treatment administered and the control of the system is the angiogenic dose rate. The specified accuracy tolerance of $10^{-7}$ were satisfied after eight mesh iterations. As the mesh refinement proceeds, the size of the KKT matrices increases from 205 to 490. @article{ledzewicz2008analysis, title={Analysis of Optimal Controls for a Mathematical Model of Tumour Anti-Angiogenesis}, author={Ledzewicz, Urszula and Sch{\"a}ttler, Heinz}, journal={Optimal Control Applications and Methods}, volume=29, number=1, pages={41--57}, year=2008, publisher={Wiley Online Library} } |
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