The paper explores the challenging task of performing a non-prehensile manipulation of several balls synchronously rolling on the curved hands of Butterfly robots. Each Butterfly robot represents a standard benchmark hardware setup, comprising a DC motor rotating a butterfly-shaped frame in a vertical plane, with a ball moving freely upon it, equipped with integrated computer vision, communication, programmable control, and computation interfaces. The combined dynamics of the considered system, consisting of N ≥ 2 such robots, is inherently underactuated, characterized by N active and N passive degrees of freedom, as well as N independent unilateral constraints that model the interactions between the frames and the balls, assuming no slipping. We focus on designing a model-based centralized feedback controller to achieve synchronized rotations of the balls. We assume the accuracy of our mathematical model and the feasibility of implementing a discretized version of the proposed continuous-time controller with a sufficiently small sampling time, that, in particular, is necessary for numerical differentiation. Relying on orbital stability of nominal periodic solution of the closed-loop system, we will experimentally check robustness to various inevitable challenges such as noises, disturbances, uncertainties, and communication delays. Hence, our concentration lies in designing an orbitally stabilizing controller for the underactuated models. The primary contribution is proposing one set of transverse coordinates, enabling transverse-linearization-based controller design, accompanied by pertinent closed-loop system analysis tools, thereby enhancing the efficacy of solving the manipulation task. Analytical and model-based arguments are validated through successful simulations and experiments conducted on two Butterfly robots, thereby emphasizing the validity and practicality of the proposed approach.

We approach a problem of motion planning and stabilization for a benchmark example, known as the “Butterfly” robot. It was proposed as a benchmark challenge for developing systematic techniques for nonprehensile rolling manipulation. A dynamical model of the underactuated system with a non-unilateral contact is derived. The recently proposed methodologies, known as virtual-holonomic-constraints-based motion planning and transverse-linearization-based orbital stabilization, are appropriately extended to suit the task. Finally, the feasibility is demonstrated through a hardware implementation and an experimental validation of the concept.

We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.

The stabilization of the motion of a nonholonomic vehicle is considered. The control system developed has a two-level architecture. The lower control level operates within the kinematic model of the vehicle to stabilize its motion to a desired trajectory. The upper control level uses the dynamic model of the vehicle and stabilizes the feedback obtained on the lower control level. The operation of the control system is studied when unknown bounded disturbances affect the motion. The adaptive motion control is proposed to deal with uncertain dynamic parameters of the vehicle.