Mathematical Formulations for Robot Modelling: Serial Versus Parallel Structures

Abstract

AbstractThis chapter reviews the mathematical formulations of serial and parallel robots. The main objective is to show the complexity of parallel robots models compared to those obtained for serial manipulators. The diversity of designs found in parallel robots makes a standard representation almost impossible. For serial robots, the structure, as the name indicates, is an open kinematic chain and hence all the links have two joints except the first one, the base, and the last one, the end-effector (EE). This type of structure allows a simple and standard geometric description, where the most famous one was introduced by Denavit-Hartenberg in the 50s. This standard geometric description allows a systematic mathematical formulation of the kinematic model, which could be easily automated. However, parallel robots have different structures, specific for each robot, where closed kinematic chains are used to connect the base to the EE. The use of closed kinematic chains made the use of a standard geometric description almost impossible. Each case has its particularity and requires a specific description. Therefore, the resulting kinematic model is also specific to the robot. Moreover, two challenges were added in the case of parallel robots. In their serial counterpart, all the joints are active and limited to Revolute or Prismatic joints. However, parallel robots usually have several passive joints, which adds extra variables in the equations of the resulting geometric model. Due to this complexity, researchers proposed several parallel robots with less than 6 degrees of freedom (DoF) and, in particular, translational ones and purely rotational ones (spherical). In this chapter, examples selected from both types were modelled and the forward and inverse problems were solved analytically, when possible.