In this paper, we present a Lyapunov-based framework for the analysis of unconstrained and constrained coupled rotational and translational control problems. Autonomous control algorithms for constrained coupled rotational and translational motions are challenging to design as they involve dependent variables that evolve in distinct configuration spaces. In the meantime, the unit dual quaternion parameterization can capture this dependency. Using this parameterization, we develop an array of globally stable control laws for unconstrained and constrained rigid body dynamics via a convex energy like Lyapunov function on dual quaternions. Furthermore, we characterize the convex representable subset of unit dual quaternions that corresponds to translational and rotational states that satisfy a predefined field of view constraints with respect to the body frame. We then proceed to construct a convex logarithmic barrier function on these constrained sets, which is subsequently used for the synthesis of the translational and rotational control laws for a wide range of applications. Powered-descent guidance problem is discussed as a potential application for the proposed framework.