Discrete Deformation Models for Real-Time Computation of Compliant Mechanisms in Two- and Three-Dimensional Space

Motivated by the need to develop a real-time computation method for simultaneous real-time visualization and force/torque feedback for manipulating of compliant mechanisms, this paper presents a general formulation of a reduced-order discrete state space model and its solution as a function of path lengths for a three-dimensional (3-D) curvature-based beam model (CBM). Unlike a compliant beam model where the boundary value problem is solved using a shooting method, the state-space representation decouples the 13th order CBM into two sets of reduced-order ordinary differential equations; the first solves for the orientation and moment whereas the second describes the deformed beam shape. Thus, it enables parallel computation of the deformed shape from the solutions to the orientation and moments. As illustrative examples, the state-space formulation and real-time computation method have been applied to analyze two flexure-based mobile-sensing node (FMN) designs. The new design, which overcomes several kinematic limitations and practical implementation problems commonly encountered in FMN navigation in tight 3-D space, permits bending and twisting of the compliant beam in 3-D space. The discrete linear CBM for the two FMN designs has been validated experimentally as well as verified by comparing computed results against published data and simulations using multishooting method and finite-element analysis.