Chapter 5, Problem 62P

(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…

One image show problem c.1 and c.2 that I need help with. The second image shows the lyapunov function and its derivative but it is NOT the same function that is given in problem. I have attached that image as an example.

This is a tilt and rotation question. Here are notes attached for reference.