We determine the energy spectrum of the baryons by treating each of them as a three-body system with the interacting forces coming from a set of two-body potentials that depend on both the distance between the quarks and the spin and orbital angular momentum coupling terms. We first review constraint dynamics for a relativistic two-body system in order to assemble the necessary two body framework for the three-body problem. We review the different types of covariant two-body interactions involved in constraint dynamics, including vector and scalar, and solve the problem of energy eigenstates using constraint dynamics. We use the Two Body Dirac equations of constraint dynamics derived by Crater and Van Alstine, matched with the quasipotential formalism of Todorov as the underlying two-body formalism. We then use the three-body constraint formalism of Sazdjian to integrate the three two-body equations into a single relativistically covariant three body equation for the bound state energies. The results are analyzed and compared to experiment using a best fit method and several different algorithms, including a gradient approach, and Monte Carlo method.