Reconfiguring Smart Structures Using Approximate Heteroclinic Connections in a Spring-Mass Model

Abstract

Several new methods are proposed to reconfigure smart structures with embedded computing, sensors and actuators. These methods are based on heteroclinic connections between equal-energy unstable equilibria in an idealised spring-mass smart structure model. Transitions between equal-energy unstable (but actively controlled) equilibria are considered since in an ideal model zero net energy input is required, compared to transitions between stable equilibria across a potential barrier. Dynamical system theory is used firstly to identify sets of equal-energy unstable configurations in the model, and then to connect them through heteroclinic connection in the phase space numerically. However, it is difficult to obtain such heteroclinic connections numerically in complex dynamical systems, so an optimal control method is investigated to seek transitions between unstable equilibria, which approximate the ideal heteroclinic connection. The optimal control method is verified to be effective through comparison with the results of the exact heteroclinic connection. In addition, we explore the use of polynomials of varying order to approximate the heteroclinic connection, and then develop an inverse method to control the dynamics of the system to track the polynomial reference trajectory. It is found that high order polynomials can provide a good approximation to true heteroclinic connections and provide an efficient means of generating such trajectories. The polynomial method is envisaged as being computationally efficient to form the basis for real-time reconfiguration of real, complex smart structures with embedded computing, sensors and actuators.

Reconfiguration of a smart surface using heteroclinic connections

“…For a true heteroclinic connection, motion away from an unstable equilibrium point and towards a connected unstable equilibrium point is asymptotically slow. In practice, the actual phase trajectory must shadow the real heteroclinic connection and a controller used to initiate and terminate the heteroclinic connection [29,30]. The corresponding shape of the surface during the transition from E 1 (1, 1, 1, 1) to E 2 (−1, −1, −1, −1,) is shown in figure 9.…”

“…It is envisaged that being computationally efficient, the control strategy could form the basis of real-time reconfiguration of smart structures [29]. Then, a more complex and realistic spring-mass model has been developed to better represent a more realistic smart structure system [30,31]. Again, a set of equilibria can be found which in principle can be connected with heteroclinic paths in the phase space of the problem.…”

Reconfiguration of a four-bar mechanism using phase space connections

“…In principle, zero net energy is required to achieve transition between these unstable equilibria, unlike transitions between stable equilibria which require the addition of and then dissipation of energy [16]. Moreover, a computational optimal control method can be used to determine the required control time histories under a set of desired boundary conditions with a suitable performance index function [19]. In addition, a reconfiguration method based on a reference trajectory and an inverse control method has been applied to a simple double mass-spring model of a smart structure [20].…”