On-Limb Orbiting Robot: Proprioceptive Diameter Estimation and Orthogonal Grip-Orbit Control.

On-body robots that travel around a human limb must keep a firm enough grip to avoid slipping or detaching, while never pressing hard enough to hurt-a balance that is hardest to strike precisely when the robot is orbiting the limb and gravity continually redistributes the contact loads. This paper presents an open, non-anthropomorphic robot that wraps around a compliant cylindrical surface with a three-contact grasp: a central traction module with two in-line driven wheels, and two lateral spring-loaded arms with distal wheels. Its central contribution is an actuation-space decomposition in which the two lateral wheel torques, expressed in a common-mode/differential basis, simultaneously drive the orbital motion and regulate the central normal force. We show that this basis diagonalises both the rolling kinematics and the static force balance, so the differential (grip-regulating) channel is provably orthogonal to the common-mode (propulsion) channel: a single pair of actuators perform both tasks without mutual interference and without a dedicated force mechanism. A model-based feedforward law derived from the static contact model, corrected by a PI term fed back from the compliant arms-which double as the force sensor-keeps the central force within a safe band; in a full-revolution simulation the differential command reverses sign to counteract the gravitational load swing while leaving the orbit undisturbed. The same compliant arms yield a closed-form estimate of the cylinder radius and contact geometry, accurate to below one millimetre across a 45-87 mm diameter range, from proprioception alone. Preliminary prototype tests reproduce the predicted behaviour, supporting the approach for future wearable and assistive applications.