(b): Let us first consider controlling the orbit of deputy spacecraft to rendezvous with chief spacecraft. Define =[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 8x=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 semi-major axis eccentricity inclination Deputy ad = ed id=35 Chief Unit 11500 ac= 10000 km = 0.15 % = 0.3 i = 50 degree Ως = 50 degree Wd = 40 vd = 0 We = 40 degree Ve=0 degree right ascension of ascending node d = 50 argument of periapsis true anomaly at epoch (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 = ½r K,dr+dv¹ dv, 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 = -8v Pôv where P > 0, and discuss the stability property of the controlled system based on V (Lyapunov/asymptotic? local/global?). (b.4): Show the asymptotic stability of the system by applying either Theorem 1 or Theorem 2. (b.5): For diagonal K, and P, design three types of controller gains to achieve critically, under, or over damped systems, and report the relation K, and P needs to satisfy for each controller. (b.6): Perform the numerical integrations of the controlled system with each of the (i) critically- damping, (ii) over-damping, and (iii) under-damping controllers, where assume K, = 13 and scale the length and time units appropriately as discussed in class. Simulate the dynamics for at least 5 revolutions of the chief Keplerian orbits. Discuss the results with relevant plots. Derivative: 1 V 2 = 1 ½ dr K, dr + ½ 8v 8v, δν KT > 0 From nonlinear relative dynamics: V = dr Kdv + dv³dv rd rc δύ =u- μ ||rd|| 3 ||rc|| 3 ◆ Step 2: Desired Lyapunov Rate We want to enforce: = V-8vP8v, where P > 0 ◆ Step 3: Derive Control Law Substitute dynamics into V: rd rc = ν - δι κ δν + δν μ |||rd|| 3 ||rc||³ To make this match the desired rate -v Pdv, propose the control: u = fl ( rd ||rd|| 3 - rc .). Κ.δι - Ρόν