Dual quaternion synthesis of constrained robotic systems

Constrained robotics systems are serial or parallel robots with less than six degrees of freedom. Dimensional synthesis is defined as the process of dimensioning a robot, that is, designing the link dimensions for a given task or set of tasks. In finite-position synthesis, we define the task as a series of positions that the robot must reach. Dimensional synthesis of planar mechanisms was first solved using graphic methods, and later those methods were transformed into algebraic equations that described the constraints on the movement of the mechanism. This approach was successfully applied to spherical mechanisms and simple cases of spatial mechanisms. The methodology was not extended to general constrained robots due to the difficulty in stating the geometric constraints for robots with more than three links. A systematic approach for the synthesis of spatial robots was developed based on using the kinematics equations of the robot. The kinematics equations are spatial transformations from a fixed frame to the end-effector of the robot, parameterized by both the dimensions of the links and the joint variables. In this dissertation, a method for the kinematic synthesis of constrained robots is presented. It is based on the use of dual quaternions to construct the kinematics equations of the robot from a reference position and to equate them to a set of task positions. A calculation was devised to compute the maximum number of task positions for each robot topology, and a classification of constrained robots was obtained according to this. The design equations produced using this methodology have been solved numerically for both the link dimensions and the joint variables, and also a scheme has been introduced to eliminate the joint variables in order to obtain algebraic equations. These have been further simplified to closed algebraic expressions in several cases. The dual quaternion synthesis methodology provides with a tool for the systematic design of constrained robots. Some of these results have been implemented in computer-aided design systems.