(b): Next, we consider minimum-time transfers. For this problem, use the maximum control magnitude constraint as |u2 ≤umax, where Umax = 0.1 in the non-dimensional unit. (b.1): Derive and report the necessary conditions of optimality including the transversality conditions for this optimal transfer problem. (b.2): Implement the indirect method with single shooting and find a minimum-time transfer by using Ao = [5, 2, 2, 7] and T = 6.5 (non-dimensional) as the initial guess. Report the numerical value of the obtained Ao and T, and plot the obtained optimal transfer, control time history, and costate time history. (b.3): Plot the time history of (H(t) - Ho), and discuss the result. (b.4): Perform the same analysis with random initial guesses (N = 20), and discuss the convergence properties in comparison to the minimum-energy problem. For this problem, use the distributions of U(-10, 10) for the elements of Ao and U(5, 10) for T.
(b): Next, we consider minimum-time transfers. For this problem, use the maximum control magnitude constraint as |u2 ≤umax, where Umax = 0.1 in the non-dimensional unit. (b.1): Derive and report the necessary conditions of optimality including the transversality conditions for this optimal transfer problem. (b.2): Implement the indirect method with single shooting and find a minimum-time transfer by using Ao = [5, 2, 2, 7] and T = 6.5 (non-dimensional) as the initial guess. Report the numerical value of the obtained Ao and T, and plot the obtained optimal transfer, control time history, and costate time history. (b.3): Plot the time history of (H(t) - Ho), and discuss the result. (b.4): Perform the same analysis with random initial guesses (N = 20), and discuss the convergence properties in comparison to the minimum-energy problem. For this problem, use the distributions of U(-10, 10) for the elements of Ao and U(5, 10) for T.
Applications and Investigations in Earth Science (9th Edition)
9th Edition
ISBN:9780134746241
Author:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Publisher:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Chapter1: The Study Of Minerals
Section: Chapter Questions
Problem 1LR
Related questions
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![Problem 2 Optimal Low-thrust Orbit Transfer
In this problem, we consider designing optimal low-thrust orbit transfers by applying optimal control the-
ory. For simplicity, we consider planer (i.c., two-dimensional) interplanetary transfers from Earth to Mars
with no perturbation (other than the control thrust), where the heliocentric orbits of Earth and Mars are
modeled by circular orbits. Use the following values for their semi-major axes and true anomalies at t = 0:
{aEarth, VEarth (t = 0)} {1.0 AU, 0.0 rad} and {aMars, Mars (t = 0)} = {1.524 AU, T rad}.
=
We represent the state vector æ by the position and velocity vectors in the heliocentric inertial frame,
i.c., x = [r¹, v™]™ € R4, and the control vector u € R² by acceleration generated by a low-thrust engine.
The orbital dynamics of the spacecraft in Cartesian coordinates are given by:
02x2
= [], B = [02²]
12
f(x, u) =
) = fo(x) + Bu, fo(x) =
(6)
where is the Sun's gravitational parameter.
Use non-dimensional values for all the physical quantities with the characteristic length 1* = 1 AU
and the characteristic time t* = √*/μ so that the non-dimensional value of µ becomes unity. You
may use the non-dimensional unit for the answers in the problem, including plots. To solve two-point
boundary value problems, you may use an existing nonlinear root-finding solver, such as fsolve in Matlab,
scipy.optimize.newton in python, etc. Use 1 × 10-8 for the optimality tolerance and function tolerance
in your nonlinear root-finding solver.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fb67b7bab-e045-4d39-a296-ee701ad6108e%2Fojgo0d3_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2 Optimal Low-thrust Orbit Transfer
In this problem, we consider designing optimal low-thrust orbit transfers by applying optimal control the-
ory. For simplicity, we consider planer (i.c., two-dimensional) interplanetary transfers from Earth to Mars
with no perturbation (other than the control thrust), where the heliocentric orbits of Earth and Mars are
modeled by circular orbits. Use the following values for their semi-major axes and true anomalies at t = 0:
{aEarth, VEarth (t = 0)} {1.0 AU, 0.0 rad} and {aMars, Mars (t = 0)} = {1.524 AU, T rad}.
=
We represent the state vector æ by the position and velocity vectors in the heliocentric inertial frame,
i.c., x = [r¹, v™]™ € R4, and the control vector u € R² by acceleration generated by a low-thrust engine.
The orbital dynamics of the spacecraft in Cartesian coordinates are given by:
02x2
= [], B = [02²]
12
f(x, u) =
) = fo(x) + Bu, fo(x) =
(6)
where is the Sun's gravitational parameter.
Use non-dimensional values for all the physical quantities with the characteristic length 1* = 1 AU
and the characteristic time t* = √*/μ so that the non-dimensional value of µ becomes unity. You
may use the non-dimensional unit for the answers in the problem, including plots. To solve two-point
boundary value problems, you may use an existing nonlinear root-finding solver, such as fsolve in Matlab,
scipy.optimize.newton in python, etc. Use 1 × 10-8 for the optimality tolerance and function tolerance
in your nonlinear root-finding solver.
![(b): Next, we consider minimum-time transfers. For this problem, use the maximum control magnitude
constraint as ||||2 ≤umax, where umax = 0.1 in the non-dimensional unit.
(b.1): Derive and report the necessary conditions of optimality including the transversality conditions
for this optimal transfer problem.
(b.2): Implement the indirect method with single shooting and find a minimum-time transfer by using
Ao = [5, 2, 2, 7]T and T = 6.5 (non-dimensional) as the initial guess. Report the numerical
value of the obtained Ao and T, and plot the obtained optimal transfer, control time history, and
costate time history.
(b.3): Plot the time history of (H(t) – Ho), and discuss the result.
-
(b.4): Perform the same analysis with random initial guesses (N = 20), and discuss the convergence
properties in comparison to the minimum-energy problem. For this problem, use the distributions
of U(-10, 10) for the elements of Ao and U(5, 10) for T.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fb67b7bab-e045-4d39-a296-ee701ad6108e%2F00ph67_processed.png&w=3840&q=75)
Transcribed Image Text:(b): Next, we consider minimum-time transfers. For this problem, use the maximum control magnitude
constraint as ||||2 ≤umax, where umax = 0.1 in the non-dimensional unit.
(b.1): Derive and report the necessary conditions of optimality including the transversality conditions
for this optimal transfer problem.
(b.2): Implement the indirect method with single shooting and find a minimum-time transfer by using
Ao = [5, 2, 2, 7]T and T = 6.5 (non-dimensional) as the initial guess. Report the numerical
value of the obtained Ao and T, and plot the obtained optimal transfer, control time history, and
costate time history.
(b.3): Plot the time history of (H(t) – Ho), and discuss the result.
-
(b.4): Perform the same analysis with random initial guesses (N = 20), and discuss the convergence
properties in comparison to the minimum-energy problem. For this problem, use the distributions
of U(-10, 10) for the elements of Ao and U(5, 10) for T.
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