Figure 1.15 The ST Robotics R19 cylindrical robot (left) and the symbolic representation showing a portion of its workspace (right). Cylindrical robots are often used in materials transfer tasks. (Photo courtesy of ST Robotics.)
1.3.5 Cartesian Manipulator (PPP)
A manipulator whose first three joints are prismatic is known as a Cartesian manipulator. The joint variables of the Cartesian manipulator are the Cartesian coordinates of the end effector with respect to the base. As might be expected, the kinematic description of this manipulator is the simplest of all manipulators. Cartesian manipulators are useful for table-top assembly applications and, as gantry robots, for transfer of material or cargo. The symbolic representation of a Cartesian robot is shown in Figure 1.16.
Figure 1.16 The Yamaha YK-XC Cartesian robot (left) and the symbolic representation showing a portion of its workspace (right). Cartesian robots are often used in pick-and-place operations. (Photo courtesy of Yamaha Robotics.)
1.3.6 Parallel Manipulator
A parallel manipulator is one in which some subset of the links form a closed chain. More specifically, a parallel manipulator has two or more kinematic chains connecting the base to the end effector. Figure 1.17 shows the ABB IRB360 robot, which is a parallel manipulator. The closed-chain kinematics of parallel robots can result in greater structural rigidity, and hence greater accuracy, than open chain robots. The kinematic description of parallel robots is fundamentally different from that of serial link robots and therefore requires different methods of analysis.
Figure 1.17 The ABB IRB360 parallel robot. Parallel robots generally have higher structural rigidity than serial link robots. (Photo courtesy of ABB.)
1.4 Outline of the Text
The present textbook is divided into four parts. The first three parts are devoted to the study of manipulator arms. The final part treats the control of underactuated and mobile robots.
1.4.1 Manipulator Arms
We can use the simple example below to illustrate some of the major issues involved in the study of manipulator arms and to preview the topics covered. A typical application involving an industrial manipulator is shown in Figure 1.18. The manipulator is shown with a grinding tool that it must use to remove a certain amount of metal from a surface. Suppose we wish to move the manipulator from its home position to position A, from which point the robot is to follow the contour of the surface S to the point B, at constant velocity, while maintaining a prescribed force F normal to the surface. In so doing the robot will cut or grind the surface according to a predetermined specification. To accomplish this and even more general tasks, we must solve a number of problems. Below we give examples of these problems, all of which will be treated in more detail in the text.
Figure 1.18 Two-link planar robot example. Each chapter of the text discusses a fundamental concept applicable to the task shown.
Chapter 2: Rigid Motions
The first problem encountered is to describe both the position of the tool and the locations A and B (and most likely the entire surface S) with respect to a common coordinate system. In Chapter 2 we describe representations of coordinate systems and transformations among various coordinate systems. We describe several ways to represent rotations and rotational transformations and we introduce so-called homogeneous transformations, which combine position and orientation into a single matrix representation.
Chapter 3: Forward Kinematics
Typically, the manipulator will be able to sense its own position in some manner using internal sensors (position encoders located at joints 1 and 2) that can measure directly the joint angles θ1 and θ2. We also need therefore to express the positions A and B in terms of these joint angles. This leads to the forward kinematics problem studied in Chapter 3, which is to determine the position and orientation of the end effector or tool in terms of the joint variables.
It is customary to establish a fixed coordinate system, called the world or base frame to which all objects including the manipulator are referenced. In this case we establish the base coordinate frame o0x0y0 at the base of the robot, as shown in Figure 1.19. The coordinates (x, y) of the tool are expressed in this coordinate frame as
Figure 1.19 Coordinate frames attached to the links of a two-link planar robot. Each coordinate frame moves as the corresponding link moves. The mathematical description of the robot motion is thus reduced to a mathematical description of moving coordinate frames.
(1.1)
(1.2)
in which a1 and a2 are the lengths of the two links, respectively. Also the orientation of the tool frame relative to the base frame is given by the direction cosines of the x2 and y2 axes relative to the x0 and y0 axes, that is,
(1.3)
which we may combine into a rotation
