Ingenieria

From “Introduction to Robotics” by S.K. Saha Tata McGraw-Hill Publishing Company Limited, New Delhi, 2008

Chapter 5 Transformations
For the purpose of controlling a robot, it is necessary to know the relationships between the joints motion (input) and the end-effector motions (output), because the joint motions control the end-effector movements. Thus, the study of kinematics is important,where transformations between the coordinate frames attached to different robot links of the robot need to be performed

5.1 Robot Architecture
A robot is made up of several links connected serially by joints. The robot’s degree of freedom (DOF) depends on the number of links and joints, their types, and the kinematic chain of the robot. 5.1.1 Links and joints The individual bodies that make up arobot are called 'links.' Here, unless otherwise stated, all links are assumed to be rigid, i.e., the distance between any two points within the body does not change while it is moving. A rigid body in the three dimensional Cartesian space has six DOF. This implies that the position of the body can be described by three translational, and the orientation by three rotational coordinates. Forconvenience, certain non-rigid bodies, such as chains, cables, or belts, which when serve the same function as the rigid bodies, may be considered as links. From the kinamatic point of view, two or more members when connected together without any relative motion between them is considered as a single link. For example, an assembly of two gears connected by a common shaft is treated as one link. Linksof a robot are coupled by kinematic pairs or joints. A 'joint' couples two links and provides physical constraints on the relative motion between the links. It is not a physical entity but just a concept that allows one to specify how one link moves with respect to another one. For example, a hinge joint of a door allows it to move relative to the fixed wall about an axis. No other motion ispossible. Type of relative motion permitted by a joint is governed by the form of the contact surface between the members, which can be a surface, a line, or a point. Accordingly, they are termed as either 'lower' or 'higher' pair joint. If two mating links are in surface contact, the joint is referred to as a lower pair joint. On the contrary, if the links are in line or point contact, the joint iscalled higher pair joint. As per the definition, the hinge joint of the door is a lower pair joint, whereas a ball rolling on a plane makes a higher pair joint. Frequently used lower and higher pairs are listed in Table 5.1. (i) Revolute Joint, R A revolute joint, also known as a 'turning pair' or a 'hinge' or a 'pin joint,' permits two paired links to rotate with respect to each other about the axisof the joint, say, Z, as shown in Fig. 5.1. Hence, a revolute joint imposes five constraints, i.e., it prohibits one of 1

From “Introduction to Robotics” by S.K. Saha Tata McGraw-Hill Publishing Company Limited, New Delhi, 2008 the links to translate with respect to the other one along the three perpendicular axes, X, Y, Z, along with the rotation about two axes, X and Y. This joint hasone-degree- offreedom (DOF). X Y R

Z (a) Geometrical form (ii) Prismatic Joint, P A 'prismatic joint' or a 'sliding pair' allows two paired links to slide with respect to each other along its axis, as shown in Fig. 5.2. It also imposes five constraints and, hence, has one-DOF. (b) Representations Figure 5.1 A revolute joint

P

(a) Geometrical form (iii) Helical Joint, H

(b)RepresentationsFigure 5.2 A prismatic joint

As shown in Fig. 5.3, a helical joint allows two paired links to rotate about and translate at the same time along the axis of the joint. The translation is, however, not independent. It is related to the rotation by the pitch of the screw. Thus, the helical joint also has five constraints, and accordingly one-DOF.

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From “Introduction to Robotics” by S.K....