Several future spacecraft missions are envisioned that have small ride-along satellites to inspect and service a large parent satellite. The so called escort or inspector satellites orbit around the parent satellite and with the help of onboard imaging capabilities and other sensors assess spacecraft damage in the event of an on-orbit mishap. The design of relative periodic orbits for such missions is not difficult but at the same time is not straightforward. The traditional method to design periodic relative orbits around other spacecraft is by using the Clohessy-Wiltshire (CW) equations. However, the usage of the CW equations is limited to the parent satellite's orbital eccentricity. Also the input parameters to achieve such closed relative orbits are not linked directly to the orbital elements of the parent and escort satellites in the CW method. This paper deals with the design of closed periodic orbits with the help of the analytical method developed by Balaji and Tatnall et al. (2003). The usefulness of this method is that it directly links the design parameters like the size and shape of the relative orbit with the orbital properties of the spacecraft. Moreover there is no limitation on the eccentricity of the parent satellite. A few test cases are considered and simulations are performed to demonstrate the validity of the method.