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Continuum robots have strong potential for application in Space environments. However, their modeling is challenging in comparison with traditional rigid-link robots. The Kinematic-Model-Free (KMF) robot control method has been shown to be extremely effective in permitting a rigid-link robot to learn approximations of local kinematics and dynamics (“kinodynamics”) at various points in the robot’s task space. These approximations enable the robot to follow various trajectories and even adapt to changes in the robot’s kinematic structure. In this paper, we present the adaptation of the KMF method to a three-section, nine degrees-of-freedom continuum manipulator for both planar and spatial task spaces. Using only an external 3D camera, we show that the KMF method allows the continuum robot to converge to various desired set points in the robot’s task space, avoiding the complexities inherent in solving this problem using traditional inverse kinematics. The success of the method shows that a continuum robot can “learn” enough information from an external camera to reach and track desired points and trajectories, without needing knowledge of exact shape or position of the robot. We similarly apply the method in a simulated example of a continuum robot performing an inspection task on board the ISS.