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This paper presents new results for intuitive teleoperation of continuous backbone "continuum" robot manipulators. Continuum robots present unique challenges due to the nonintuitive (to humans) nature of their basic movements. We present a series of mappings, at both position and velocity levels, which provide human operators with a versatile and intuitive suite of options for joystick control of continuum robots. Results from implementations on two types of continuum manipulator hardware are summarized.