Dynamic simulation of flexible robots with prismatic joints.

Abstract

A dynamic model, taking into account geometric nonlinearity, for flexible manipulators with prismatic joints is presented. Experiments on a spherical coordinate robot arm, with a flexible last link, were performed to validate the model. The global motion of the flexible manipulator is treated as a nominal rigid body motion combined with an elastic motion. The elastic motion is assumed to be a small perturbation of the nominal motion. Floating frames following the nominal motion are introduced to describe the kinematics of the flexible links. The geometry of the robot is first described w.r.t. the floating frames and then referred to the inertial frame by coordinate transformations through rotation matrices. A Lagrangian approach is used in deriving the equations of motion. The work done by the rigid body axial force through the axial shortening of the link due to transverse deformations is included in the Lagrangian function. The compatibility conditions associated with revolute joints and prismatic joints are expressed as kinematic constraint equations and incorporated into the equations of motion by Lagrange multipliers. The small displacement due to the flexibility of the links is then discretized by a displacement based finite element method. Equations of motion are derived for the cases of prescribed rigid body motion as well as prescribed joint torques/forces (for two dimensions) through application of Lagrange's equations. The equations of motion and the constraint equations result in a set of differential algebraic equations. A numerical procedure combining a constraint stabilization method and a Newmark direct integration scheme is then applied to obtain the system response. The constraint stabilization method feeds back the violations of the position and velocity constraints to "damp out" the violations at the current step. The Newmark method is used to ensure the stability of the integration. Examples are presented to justify the solution methods used in this study. Experiments on a spherical coordinate robot are performed to validate the proposed dynamic model. Once the model was validated, simulations were performed to point out the coupling effects between the rigid body motions and the flexible motions, the effects of the flexible motion on a rigid body controller, and the axial stiffening effect.