For actuator design and motion simulations of slender flexible robots, planar C1-beam elements are developed for Reissner’s large deformation, shear-deformable, curbed-beam model. Internal actuation is mechanically modeled by a rate-form of beam constitutive relation, where actuation curvature is prescribed at each time. Geometrically, a curbed beam is modeled as a frame bundle, whereby at each point on beam’s curve of centroids a moving orthonormal frame is attached to a cross section. After a finite element discretization, a curve of centroids is modeled as a C1-curve, employing cubic shape functions for both planar coordinates with an arc-parameter. The cubic shape functions have already been utilized in linear Euler-Bernoulli beams for the interpolation of transverse displacement. To define the rotation angle of each cross section or the attitude of the moving frame, quadratic shape functions are used introducing a middle node, resulting in three angular nodal displacements. As a result, each beam element has total eleven nodal coordinates.
The implementation of a nonlinear finite element code is facilitated by the principle of virtual work, which yields Reissner’s large deformation curbed beam model as the Euler-Lagrange equations. For time integration, the Newmark method is utilized.
Finally, as applications of the code, a few inchworm motions induced by different actuation curvature fields are presented.
System validation via constraint modeling