Control of soft robots remains nowadays a big challenge, as it does in the larger category of continuum robots. In this paper a direct and inverse kinetics models are described for a non-constant curvature structure. A major effort has been put recently in modelling and controlling constant curvature structures, such as cylindrical shaped manipulators. Manipulators with non-constant curvature, on the other hand, have been treated with a piecewise constant curvature approximation. In this work a non-constant curvature manipulator with a conical shape is built, taking inspiration from the anatomy of the octopus arm. The choice of a conical shape manipulator made of soft material is justified by its enhanced capability in grasping objects of different sizes. A different approach from the piecewise constant curvature approximation is employed for direct and inverse kinematics model. A continuum geometrically exact approach for direct kinetics model and a Jacobian method for inverse case are proposed. They are validated experimentally with a prototype soft robot arm moving in water. Results show a desired tip position in the task-space can be achieved automatically with a satisfactory degree of accuracy.