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Continuum or hyper-redundant robots are robots which exhibit behavior similar to biological trunks, tentacles and snakes. Unlike traditional robots, continuum robot manipulators do not have rigid joints, hence the manipulators are compliant, extremely dexterous, and capable of dynamic, adaptive manipulation in unstructured environments; however, the development of high-performance control algorithms for these manipulators is a challenging problem. In this paper, we present an approach to whole arm grasping control for continuum robots. The grasping controller is developed in two stages; high level path planning for the grasping objective, and a low level joint controller using a neural network feedforward component to compensate for dynamic uncertainties. These techniques are used to enable whole arm grasping without using contact force measurements and without using a dynamic model of the continuum robot. Experimental results using the OCTARM, a soft continuum robotic manipulator are included to illustrate the efficacy of the proposed low level joint controller.