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Unformatted text preview: Introduction to Robotics, H. Harry Asada 1 Chapter 4 Planar Kinematics Kinematics is Geometry of Motion . It is one of the most fundamental disciplines in robotics, providing tools for describing the structure and behavior of robot mechanisms. In this chapter, we will discuss how the motion of a robot mechanism is described, how it responds to actuator movements, and how the individual actuators should be coordinated to obtain desired motion at the robot end-effecter. These are questions central to the design and control of robot mechanisms. To begin with, we will restrict ourselves to a class of robot mechanisms that work within a plane, i.e. Planar Kinematics . Planar kinematics is much more tractable mathematically, compared to general three-dimensional kinematics. Nonetheless, most of the robot mechanisms of practical importance can be treated as planar mechanisms, or can be reduced to planar problems. General three-dimensional kinematics, on the other hand, needs special mathematical tools, which will be discussed in later chapters. 4.1 Planar Kinematics of Serial Link Mechanisms Example 4.1 Consider the three degree-of-freedom planar robot arm shown in Figure 4.1.1. The arm consists of one fixed link and three movable links that move within the plane. All the links are connected by revolute joints whose joint axes are all perpendicular to the plane of the links. There is no closed-loop kinematic chain; hence, it is a serial link mechanism. x End Effecter Joint 1 Link 3 Link 2 Link 1 Joint 3 Joint 2 A O 2 A 1 A y 1 θ Link 0 3 A e φ 3 θ 2 θ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ e e y x B E Figure 4.1.1 Three dof planar robot with three revolute joints To describe this robot arm, a few geometric parameters are needed. First, the length of each link is defined to be the distance between adjacent joint axes. Let points O, A, and B be the locations of the three joint axes, respectively, and point E be a point fixed to the end-effecter. Then the link lengths are E B B A A O = = = 3 2 1 , , A A A . Let us assume that Actuator 1 driving Department of Mechanical Engineering Massachusetts Institute of Technology Introduction to Robotics, H. Harry Asada 2 link 1 is fixed to the base link (link 0), generating angle 1 θ , while Actuator 2 driving link 2 is fixed to the tip of Link 1, creating angle 2 θ between the two links, and Actuator 3 driving Link 3 is fixed to the tip of Link 2, creating angle 3 θ , as shown in the figure. Since this robot arm performs tasks by moving its end-effecter at point E, we are concerned with the location of the end-effecter. To describe its location, we use a coordinate system, O-xy, fixed to the base link with the origin at the first joint, and describe the end-effecter position with coordinates e and e . We can relate the end-effecter coordinates to the joint angles determined by the three actuators by using the link lengths and joint angles defined above: x y ) cos( ) cos( cos 3 2 1 3 2 1 2 1 1 θ θ θ θ θ θ + + +...
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- Fall '05