DEFORMATION GRADIENT ELEMENT FOR FLEXIBLE BEAMS WITH LARGE DEFORMATION IN A FLOATING FRAME REFERENCE

The dynamic modeling of robotic arm vibration control often adopts the floating frame reference method to ensure appropriate control variables for joint motion and vibration suppression. However, the floating frame reference method is limited by small deformations and lacks precision in solving large deformation dynamic problems. The commonly used modeling theories for large deformation problems, such as the absolute nodal coordinate formulation (ANCF) which can accurately solve large rotation and large deformation problems in robotic arms, fail to provide suitable control variables for vibration control. To address this issue, this paper proposes a novel discretization scheme for large deformation fields - the deformation gradient element. This element employs nodal deformation and its gradient as nodal coordinates without any small deformation assumptions, and its nodal coordinates can be accurately transformed into absolute nodal coordinates and their gradients. It can be viewed as an equivalent element of ANCF, capable of precisely describing the large deformation motion of flexible bodies. The following conclusions are drawn: (1) When solving large deformation static problems in beam structures, the results obtained using the deformation gradient element are identical to those using the reduced beam element of ANCF, with comparable computational efficiency. (2) The solutions of the proposed model and ANCF for large deformation dynamic problems are almost identical, validating the correctness of the proposed model. When solved using the generalized-α method, the slight differences in dynamic solutions between the two methods stem from the different forms of their dynamic equations. (3) For motion control of robotic arms undergoing large deformations, the first-order approximation rigid-flexible coupling model based on small deformation assumptions is prone to yielding erroneous or even divergent predictions. (4) The proposed model naturally distinguishes between large rotations of joints and large deformations of beams. By combining PD control strategies to apply joint driving torques and multiple transverse control forces to the system, the derivative of the system's Lyapunov function with respect to time is non-positive, ensuring that system energy gradually decays over time and the system remains stable. Moreover, the more transverse control forces applied, the better the suppression of residual vibrations in the robotic arm.