Optimal Trajectory Planning for Flexible Robots with Large Deformation

Robot arms with lighter weight can reduce unnecessary energy consumption which is desirable in robotic industry. However, lightweight arms undergo undesirable elastic deformation. In this paper, the planar motion of a lightweight flexible arm is investigated. In order to obtain a precise mathematical model, the axial displacement and nonlinear curvature of flexible arm arising from large bending deformation is taken into consideration. An in-extensional condition, the axial displacement is related to transverse displacement of the flexible beam, is applied. This leads to a robotic model with three rigid modes and one elastic mode. The elastic mode depends on time and position. An Assume Mode Method is used to remove the spatial dependence. The governing equations is derived using Lagrange Method. The effects of nonlinear terms due to the large deformation, gravity, and tip-mass are considered. Control inputs include forces and moment exerted at the joint between slider and arm (see Fig. 1). The conventional computed torque control laws cannot stabilize the system, since there are not as many control inputs as states of the system. A Particle Swarm Optimization (PSO) technique is then used to obtain a suitable trajectory with the aim of minimizing excitations of the elastic mode. Two methods are considered for generating a trajectory function, either to use a three-layer Artificial Neural Network (ANN) or to use spline interpolation. A sliding mode control strategy is proposed in which the sliding surfaces include elastic mode in order to guarantee robustness. The simulations show that the three-layer ANN technique provides arbitrary small settling time, and also the optimization algorithm converges faster and generates smooth trajectories unlike spline function technique.