Spherical pendulum

Abstract: This study examines the mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom and proposes an analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics transverse to a feasible motion. This constructive procedure is based on a particular choice of coordinates and allows explicit introduction of a moving Poincare? section associated with a nontrivial finite-time or periodic motion. In these coordinates, transverse dynamics admits analytical linearization before any control design. If the forced motion of an underactuated mechanical system is periodic, then this linearization is an indispensable and constructive tool for stabilizing the cycle and for analyzing its orbital (in)stability. The technique is illustrated with two challenging examples. The first one is stabilization of a circular motions of a spherical pendulum on a puck around its upright equilibrium. The other one is creating stable synchronous oscillations of an arbitrary number of planar pendula on carts around their unstable equilibria.

Abstract: This paper investigates tracking controls for an arbitrary number of cooperating quadrotor unmanned aerial vehicles with a suspended load. Assuming that a point mass is connected to multiple quadrotors by rigid massless links, control systems for quadrotors are constructed such that the point mass asymptotically follows a given desired trajectory and quadrotors maintain a prescribed formation, either relative to the point mass or with respect to the inertial frame. These are developed in a coordinate-free fashion to avoid singularities and complexities associated with local parameterizations. The desirable features are illustrated by several numerical examples, including a flying inverted spherical pendulum on a quadrotor.

Abstract: The equations of motion for a lightly damped spherical pendulum that is subjected to harmonic excitation in a plane are approximated in the neighborhood of resonance by discarding terms of higher than the third order in the amplitude of motion. Steady-state solutions are sought in a four-dimensional phase space. It is found that: (a) planar harmonic motion is unstable over a major portion of the resonant peak, (b) non-planar harmonic motion is stable in a spectral neighborhood above resonance that overlaps neighborhoods of both stable and unstable planar motions, and (c) no stable, harmonic motions are possible in a finite neighborhood of the natural frequency. The spectral width of these neighborhoods is proportional to the two-thirds power of the amplitude of excitation. The steady-state motion in the last neighborhood is quasisinusoidal (at the forcing frequency) with slowly varying amplitude and phase. The waveform, as determined by an analog computer, is periodic but quite complex.