The next frontier in solar system exploration will be missions targeting extreme and rugged environments such as caves, canyons, cliffs and crater rims of the Moon, Mars and icy moons. These environments are time capsules into early formation of the solar system and will provide vital clues of how our early solar system gave way to the current planets and moons. These sites will also provide vital clues to the past and present habitability of these environments. Current landers and rovers are unable to access these areas of high interest due to limitations in precision landing techniques, need for large and sophisticated science instruments and a mission assurance and operations culture where risks are minimized at all costs. Our past work has shown the advantages of using multiple spherical hopping robots called SphereX for exploring these extreme environments. Our previous work was based on performing exploration with a human-designed baseline design of a SphereX robot. However, the design of SphereX is a complex task that involves a large number of design variables and multiple engineering disciplines. In this work we propose to use Automated Multidisciplinary Design and Control Optimization (AMDCO) techniques to find near optimal design solutions in terms of mass, volume, power, and control for SphereX for different mission scenarios.