This paper introduces a novel directional control method of self-propelled oil droplets. Oil droplets locomote spontaneously with surfactant action. This self-propulsion is caused by Marangoni convection within the oil droplet due to differences in the surfactant concentration at the droplet surface. Recent studies have reported that self-propelled oil droplets change their locomotion style depending on their shapes. We confirm that spherical oil droplets move randomly, including straight motion, bending motion, and rotation. In particular, we discover that boomerang-shaped oil droplets exhibit only straight motion. In this study, we introduce an exoskeleton for the directional and velocity control of oil droplets. A droplet shaped as a boomerang by an exoskeleton locomotes in the direction from a concave region to a convex region. Through experimental studies, we found that the stability of the velocity and locomotion direction depended on the boomerang shape. Self-propelled oil droplets with exoskeletons were then applied to a transporting robot driven only by the energy obtained from chemical reactions. We demonstrate the robot pushes and transports an object floating on water.