Date of Award


Document Type

Thesis - Open Access

Degree Name

Master of Science in Aerospace Engineering


Aerospace Engineering

Committee Chair

Dr. Troy Henderson

First Committee Member

Dr. Morad Nazari

Second Committee Member

Dr. William MacKunis


Due to the popularity of libration points, many satellites are being maintained on their desired trajectory. Indian space research organization has planned to launch the Aditya-L1 spacecraft to study about the Sun by 2021. James Webb Space Telescope has also been designed to observe deep space at L2 in the Sun-Earth system by 2021. The combined gravity of the Earth and the Sun keep satellite’s orbit locked at libration points. Though satellites enjoys an uninterrupted view of Sun and Earth all the time, they are affected by the solar radiation pressure (SRP) continuously. Due to the instability of collinear libration points, the certain amount of thrust is required to maintain the desired trajectory. This thesis introduces the Udwadia-Kalaba (UK) formulation of constrained dynamics as applied to the restricted three-body problem of the Sun-Earth-Spacecraft. A dynamic model of the restricted three-body system is presented to analyze the unconstrained motion of spacecraft. The results show the instability due to perturbation from the SRP. Then, the Udwadia-Kalaba formulation is applied to derive the equation of motion of spacecraft with additional constraints such that spacecraft maintains the desired trajectory at libration points. The results of exact amount of control acceleration with the directions are provided for spacecraft for the following three cases: i) maintaining position at the L1 considering the Earth’s circular orbit (i.e. CR3BP) ii) maintaining position at the L1 and L4 or L5 considering the Earth’s elliptical orbit (i.e. ER3BP) and iii) maintaining the three-dimension halo orbit around the L1 and L2 in CR3BP. The UK formulation is modified using Baumgarte’s stabilization method to allow to compute the results for the incorrect initial conditions (i.e. initial state deviations). In this case, the results are analyzed for the underdamped, overdamped and critically-damped systems. In addition, the delta-v is compared for the transient response with time-varying linear quadratic regulator (LQR). For fully constrained system, the control accelerations required to maintain the desired trajectory obtained by the UK technique are shown identical to those obtain via feedforward part of the time-varying LQR, as expected.