Impulsive Mechanical System Research Articles

State observers for the time discretization of a class of impulsive mechanical systems

AbstractIn this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman–Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.

On the Dynamic Stability of Impulsive Mechanical Systems with Delay

We consider impulsive mechanical systems with delay and establish sufficient conditions for asymptotic stability of the equilibrium positions of a system with periodic parameters in the linear approximation. It is also assumed that the value of delay is equal to the period of the system. The conditions of parametric stability of the lower equilibrium position of a mathematical pendulum are deduced with regard for the impulsive perturbations and delay.

Influence of the System of Forces on the Stability of Impulsive Mechanical Gyroscopic Systems

The linear stability of an impulsive mechanical system under dissipative, potential, circulation forces, and gyroscopic forces of high intensity is analyzed. Coefficient conditions for the linear stability of the equilibrium state of the system when the eigendecomposition of the matrix of gyroscopic forces is known explicitly are established. The conditions for the parametric resonance of a plane gyroscopic pendulum to occur are established

Analysis of limit-cycle walking for a compass-like biped robot

We consider an impulsive mechanical system modeling dynamics of a planar two-link walker commonly known as a compass-gait biped. It is assumed that there is actuation in the hip but that the desired periodic trajectory describes an unstable passive walking gait. We recall and apply a recently developed technique for design of orbitally stabilizing feedback controllers. After that, we illustrate how to assess various properties of the closed-loop system. In particular, sensitivity to perturbations of the slope of the walking surface is analyzed and an estimate for possible deviations from the nominal trajectory is computed analytically.

Transverse Linearization for Impulsive Mechanical Systems With One Passive Link

A general method for planning and orbitally stabilizing periodic motions for impulsive mechanical systems with underactuation one is proposed. For each such trajectory, we suggest a constructive procedure for defining a sufficient number of nontrivial quantities that vanish on the orbit. After that, we prove that these quantities constitute a possible set of transverse coordinates. Finally, we present analytical steps for computing linearization of dynamics of these coordinates along the motion. As a result, for each such planned periodic trajectory, a hybrid transverse linearization for dynamics of the system is computed in closed form. The derived impulsive linear system can be used for stability analysis and for design of exponentially orbitally stabilizing feedback controllers. A geometrical interpretation of the method is given in terms of a novel concept of a moving Poincare section. The technique is illustrated on a devil stick example.

A Passive 2-DOF Walker: Hunting for Gaits Using Virtual Holonomic Constraints

A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.

Stability in critical cases of a holonomic mechanical system with two degrees of freedom subject to impacts

Stability conditions for a holonomic impulsive mechanical system with two degrees of freedom in the critical case of one pair of complex-conjugate multipliers are established. An analog of the first Lyapunov exponent is calculated

Constraints in impulsive mechanics and gauss's minimum principle

We construct a geometric setup based on the theory of left and right jet bundles of a fiber bundle over ℝ appropriate to model the presence of constraints for a time-dependent impulsive mechanical system. A detailed classification of constraints is presented and all types of constraints are proved to be naturally framed in the setup. The concepts of reactive impulse and constitutive characterization of constraints can be defined very naturally according to the nature of the constraint. A minimum principle strongly related to Gauss's principle of minimal constraint is introduced to propose an ideality criterion suitable for all the classified constraints.