Derivation of motion equation for mobile manipulator with viscoelastic links and revolute-prismatic flexible joints via recursive Gibbs-Appell formulations

Abstract This paper investigates the dynamic equations of an N-flexible link manipulator with revolute–prismatic joints while considering the effects of manipulator locomotion by the mobile platform bound by non-holonomic kinematic constraints. Such constraints, in addition to creating dynamic interaction between manipulator and platform, cause serious motion limitations and introduction of more computational complexity. The manipulator’s flexible links are modeled by the assumed mode method, where the Timoshenko beam theory is used for the substitution of the assumed mode shapes. The internal and external damping effects are also studied for the model precision. Moreover, revolute–prismatic joints in each arm are exploited to develop the robot mobility. The new joint structure makes it possible to use mobile manipulators with long flexible links. However, in regard to the variations of links length caused by prismatic joints, time-varying dynamic equations are obtained, leading to comparatively complex and lengthy formulations. Therefore, the Gibbs–Appell formulation is utilized as an alternative to the Lagrange equations to facilitate the process of deriving the motion equations. In addition, the non-dimensional form of the Timoshenko beam theory mode shapes is recommended to circumvent the computation of time step mode shapes. It is also necessary to examine the system tip-over stability based on long and variable-length arms, lightweight base, and widespread environmental factors using the zero moment point methods. Finally, a numerical simulation for a mobile manipulator, with two flexible links and revolute–prismatic joints is carried out to demonstrate the performance of the presented model for such complex systems. Different amounts of link elasticity and the effects of internal and external damping coefficients are separately studied. The results are verified by recent fixed base flexible manipulators employing revolute–prismatic joints as well as the IUST Revolute–Prismatic joints experimental setup incorporating rigid links.