Chapter 1 Introduction These exercises are designed to give you practice with the concepts, the calcula- tions, and the software associated with the book. To get the most out of these practice exercises, you are strongly encouraged not to look at the solutions untilyou have given your best effort to solve them. You are more likely to retain what you have learned when you work through the problem yourself instead ofjust reading the solution. Practice exercises for Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org 2 3 Chapter 2 Practice Exercises on Configuration Space 2.1 Practice Exercises Practice exercise 2.1 The experimental surgical manipulator shown in Fig-ure 2.1, developed at the National University of Singapore, is a parallel mech- anism with three identical legs, each with a prismatic joint and two universal joints (the joints are marked for one of the legs). Use Gru¨bler’s formula to calculate the number of degrees of freedom of this mechanism. Practice exercise 2.2 (a) Three rigid bodies move in space independently. How many degrees of freedom does this system of three bodies have? U Figure 2.1: A miniature parallel surgical manipulator with three PUU legs. 4 2.1. Practice Exercises Practice exercises for Modern Robotics, Lynch and Park, Cambridge U. Press, 2017. http://modernrobotics.org Figure 2.2: A scissor jack (also known as a scissor lift). Image courtesy of Wikipedia. (b) Now you constrain them so that each body must make contact with at least one of the other two bodies. (The bodies are allowed to slide androll relative to each other, but they must remain in contact.) How many degrees of freedom does this system of three bodies have? Practice exercise 2.3 Figure 2.2 shows a scissor jack. As you turn the screw, the jack goes up and down. The mechanical advantage provided by the mechanism allows a single person to jack up a car to change a tire. Think about what rigid bodies and joints must be present in the scissor jack. You may not be able to see all of them in the image. Use Gru¨bler’s formula to calculate the number of degrees of freedom. Does your answer agree with what you know about how a scissor jack works? If not, can you explain why? Practice exercise 2.4 Figure 2.3 shows a table lamp that moves only in the plane of the page. Use Gru¨bler’s formula to calculate the number of degrees of freedom. Practice exercise 2.5 A unicycle is controlled moving on a rigid balance beam as shown in Figure 2.4. Suppose the wheel is always touching the beam with no sliding, answer the following questions in terms of R, S, T , and I (a onedimensional closed interval).