(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft. Define x = [r] and x = x = R to represent the deputy orbital state and its target (= chief orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the ECI frame. Denote the relative state by dx = x-x. Table 2 summarize the initial orbital elements. Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame) Orbital element Deputy Unit Chief semi-major axis ad = 11500 ac 10000 km eccentricity inclination ed = 0.15 id=35 ee = 0.3 i = 50 degree right ascension of ascending node d = 50 Ως = 50 degree argument of periapsis true anomaly at epoch Wd Vd= 0 = 40 We = 40 degree Ve=0 degree (b.1): Derive the error dynamics of our system in ECI frame under the influence of u. (b.2): Consider a candidate Lyapunov function V = ½dr¹ K₁dr+dv₁dv, where K₁ = K, and K, > 0. Discuss the positive definiteness of V, and derive the Lyapunov rate of this system. (b.3): Derive a stabilizing controller such that satisfies V = -v Pôv where P > 0, and discuss the stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?).
(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft. Define x = [r] and x = x = R to represent the deputy orbital state and its target (= chief orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the ECI frame. Denote the relative state by dx = x-x. Table 2 summarize the initial orbital elements. Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame) Orbital element Deputy Unit Chief semi-major axis ad = 11500 ac 10000 km eccentricity inclination ed = 0.15 id=35 ee = 0.3 i = 50 degree right ascension of ascending node d = 50 Ως = 50 degree argument of periapsis true anomaly at epoch Wd Vd= 0 = 40 We = 40 degree Ve=0 degree (b.1): Derive the error dynamics of our system in ECI frame under the influence of u. (b.2): Consider a candidate Lyapunov function V = ½dr¹ K₁dr+dv₁dv, where K₁ = K, and K, > 0. Discuss the positive definiteness of V, and derive the Lyapunov rate of this system. (b.3): Derive a stabilizing controller such that satisfies V = -v Pôv where P > 0, and discuss the stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?).
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
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Chapter1: Basic Modes Of Heat Transfer
Section: Chapter Questions
Problem 1.3DP
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![(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft.
Define x = [r] and x = x = R to represent the deputy orbital state and its target (= chief
orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the
ECI frame. Denote the relative state by dx = x-x. Table 2 summarize the initial orbital elements.
Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame)
Orbital element
Deputy
Unit
Chief
semi-major axis
ad =
11500
ac 10000 km
eccentricity
inclination
ed = 0.15
id=35
ee = 0.3
i = 50
degree
right ascension of ascending node d = 50
Ως = 50
degree
argument of periapsis
true anomaly at epoch
Wd
Vd= 0
=
40
We =
40
degree
Ve=0
degree
(b.1): Derive the error dynamics of our system in ECI frame under the influence of u.
(b.2): Consider a candidate Lyapunov function V = ½dr¹ K₁dr+dv₁dv, where K₁ = K, and K, > 0.
Discuss the positive definiteness of V, and derive the Lyapunov rate of this system.
(b.3): Derive a stabilizing controller such that satisfies V = -v Pôv where P > 0, and discuss the
stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F4b8ed17f-0f02-4284-943a-0000eb264bf9%2Fnt880ms_processed.png&w=3840&q=75)
Transcribed Image Text:(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft.
Define x = [r] and x = x = R to represent the deputy orbital state and its target (= chief
orbit) in Cartesian coordinates, respectively. The control input is thruster acceleration, u € R³, in the
ECI frame. Denote the relative state by dx = x-x. Table 2 summarize the initial orbital elements.
Table 2: Keplerian orbital elements at epoch (t = 0) for deputy and chief about Earth (ECI frame)
Orbital element
Deputy
Unit
Chief
semi-major axis
ad =
11500
ac 10000 km
eccentricity
inclination
ed = 0.15
id=35
ee = 0.3
i = 50
degree
right ascension of ascending node d = 50
Ως = 50
degree
argument of periapsis
true anomaly at epoch
Wd
Vd= 0
=
40
We =
40
degree
Ve=0
degree
(b.1): Derive the error dynamics of our system in ECI frame under the influence of u.
(b.2): Consider a candidate Lyapunov function V = ½dr¹ K₁dr+dv₁dv, where K₁ = K, and K, > 0.
Discuss the positive definiteness of V, and derive the Lyapunov rate of this system.
(b.3): Derive a stabilizing controller such that satisfies V = -v Pôv where P > 0, and discuss the
stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?).
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