Propeller Motion of a Devil-Stick using Normal Forcing

The problem of realizing rotary propeller motion of a devil-stick in the vertical plane using forces purely normal to the stick is considered. This problem represents a nonprehensile manipulation task of an underactuated system. In contrast with previous approaches, the devil-stick is manipulated by controlling the normal force and its point of application. Virtual holonomic constraints are used to design the trajectory of the center-of-mass of the devil-stick in terms of its orientation angle, and conditions for stable propeller motion are derived. Intermittent large-amplitude forces are used to asymptotically stabilize a desired propeller motion. Simulations demonstrate the efficacy of the approach in realizing stable propeller motion without loss of contact between the actuator and devil-stick.

💡 Research Summary

The paper addresses the classic “devil‑stick” (or “propeller stick”) problem: how to make a slender rod rotate continuously in the vertical plane using only forces that are normal (perpendicular) to the rod. This is a non‑prehensile manipulation task in which the actuator never grasps the object, but instead pushes on it at a single contact point. Prior work has relied on controlling both normal and tangential components of the contact force, and has often assumed that the contact point instantaneously resets after each full rotation, leading to a hybrid (continuous‑discrete) model.

The authors propose a fundamentally different approach. The control inputs are reduced to (i) the magnitude of the normal force F and (ii) the distance r from the rod’s centre of mass G to the point of application of that force. No tangential force is applied, and the point of application can slide continuously along the rod. The rod is modelled as a planar rigid body with three generalized coordinates ((h_x, h_y, \theta)) – the Cartesian position of G and the orientation angle (\theta) measured counter‑clockwise from the horizontal. Using Lagrange’s equations the dynamics become

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