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Non-prehensile manipulation of a devil-stick: planar symmetric juggling using impulsive forces

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Abstract

Juggling a devil-stick can be described as a problem of non-prehensile manipulation. Assuming that the devil-stick remains confined to the vertical plane, the problem of juggling the stick between two symmetric configurations is considered. Impulsive forces are applied to the stick intermittently and the impulse of the force and its point of application are modeled as control inputs to the system. The dynamics of the devil-stick due to the impulsive forces and gravity is described by half-return maps between two Poincaré sections; the symmetric configurations are fixed points of these sections. A coordinate transformation is used to convert the juggling problem to that of stabilization of one of the fixed points. Inclusion of the coordinate transformation in the dynamic model results in a nonlinear discrete-time system. A dead-beat design for one of the inputs simplifies the control problem and results in a linear time-invariant discrete-time system. Standard control techniques are used to show that symmetric juggling can be achieved from arbitrary initial conditions.

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Notes

  1. Impulsive inputs cause discontinuous jumps in the velocity coordinates but no change in the position coordinates [24]. The dynamics of underactuated systems subjected to impulsive inputs is discussed in [25,26,27,28,29].

  2. Poincaré sections have been previously used for design of gaits for bipedal robots [30, 31]. and orbital stabilization of underactuated systems [27].

  3. It is assumed that the initial conditions of the devil-stick are such that its trajectory intersects one of the two Poincaré sections before the first impulsive control input is applied.

  4. Half-return maps have been used to analyze the behavior of dynamical systems such as the van der Pol oscillator [33, 34].

  5. It should be noted that dead-beat inputs have been used in earlier works on juggling [8, 11, 17].

  6. A detailed discussion of MPC design for discrete-time systems can be found in Chapters 1-3 in [36].

  7. The augmented state-space model is controllable; this was verified using Theorem 1.2 in [36].

  8. It should be noted that the state variables are shown in the reference frame of the right-handed juggler.

  9. The quadprog MATLAB function was used.

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Acknowledgements

The authors acknowledge the support provided by the National Science Foundation, Grant CMMI-1462118.

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  1. Mechanical Engineering Department, Michigan State University, East Lansing, MI, 48824, USA

    Nilay Kant & Ranjan Mukherjee

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  1. Nilay Kant
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  2. Ranjan Mukherjee
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Correspondence to Ranjan Mukherjee.

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Kant, N., Mukherjee, R. Non-prehensile manipulation of a devil-stick: planar symmetric juggling using impulsive forces. Nonlinear Dyn 103, 2409–2420 (2021). https://doi.org/10.1007/s11071-021-06254-0

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