Unit Quaternions: A Mathematical Tool for Modeling, Path Planning and Control of Robot Manipulators

Robot manipulators are thought of as a set of one or more kinematic chains, composed by rigid bodies (links) and articulated joints, required to provide a desired motion to the manipulator’s end–effector. But even though this motion is driven by control signals applied directly to the joint actuators, the desired task is usually specified in terms of the pose (i.e., the position and orientation) of the end–effector. This leads to consider two ways of describing the configuration of the manipulator, at any time (Rooney & Tanev, 2003): via a set of joint variables or pose variables. We call these configuration spaces the joint space and the pose space, respectively. But independently of the configuration space employed, the following three aspects are of interest when designing and working with robot manipulators: • Modeling: The knowledge of all the physical parameters of the robot, and the relations among them. Mathematical (kinematic and dynamic) models should be extracted from the physical laws ruling the robot’s motion. Kinematics is important, since it relates joint and pose coordinates, or their time derivatives. Dynamics, on the other hand, takes into account the masses and forces that produce a given motion. • Task planning: The process of specifying the different tasks for the robot, either in pose or joint coordinates. This may involve from the design and application of simple time trajectories along precomputed paths (this is called trajectory planning), to complex computational algorithms taking real–time decisions during the execution of a task. • Control: The elements that allow to ensure the accomplishment of the specified tasks in spite of perturbances or unmodeled dynamics. According to the type of variables used in the control loop, we can have joint space controllers or pose space controllers. Robot control systems can be implemented either at a low level (e.g. electronic controllers in the servo–motor drives) or via sophisticated high–level programs in a computer. Fig. 1 shows how these aspects are related to conform a robot motion control system. By motion control we refer to the control of a robotic mechanism which is intended to track a desired time–varying trajectory, without taking into account the constraints given by the environment, i.e., as if moving in free space. In such a case the desired task (a time function along a desired path) is generated by a trajectory planner, either in joint or pose variables. The motion controller can thus be designed either in joint or pose space, respectively.