Whisker Sensing by Force and Moment Measurements at the Whisker Base

We address the theoretical question which forces and moments measured at the base of a whisker (tactile sensor) allow for the prediction of the location in space of the point at which a whisker makes contact with an object. We deal with the general case of three-dimensional deformations as well as with the special case of planar configurations. All deformations are treated as quasi-static, and contact is assumed to be frictionless. We show that the minimum number of independent forces or moments required is three but that conserved quantities of the governing elastic equilibrium equations prevent certain triples from giving a unique solution in the case of contact at any point along the whisker except the tip. The existence of these conserved quantities depends on the material and geometrical properties of the whisker. For whiskers that are tapered and intrinsically curved, there is no obstruction to the prediction of the contact point. We show that the choice of coordinate system (Cartesian or cylindrical) affects the number of suitable triples. Tip and multiple point contact are also briefly discussed. Our results explain recent numerical observations in the literature and offer guidance for the design of robotic tactile sensory devices.