Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.
A finite difference scheme for solving the Timoshenko beam equations with boundary feedback