Chapter 1 Rigid Body Kinematics
Chapter 1 Rigid Body Kinematics 1.1 Introduction This chapter builds up the basic language and tools to describe the motion of a rigid body – this is called rigid body kinematics. This material will be the foundation for describing general mech- anisms consisting of interconnected rigid bodies in various topologies that are the focus of this book. Only the geometric property of the body and its evolution in time will be studied; the re- sponse of the body to externally applied forces is covered in Chapter ?? under the topic of rigid body dynamics. Rigid body kinematics is not only useful in the analysis of robotic mechanisms, but also has many other applications, such as in the analysis of planetary motion, ground, air, and water vehicles, limbs and bodies of humans and animals, and computer graphics and virtual reality. Indeed, in 1765, motivated by the precession of the equinoxes, Leonhard Euler decomposed a rigid body motion to a translation and rotation, parameterized rotation by three principal-axis rotation (Euler angles), and proved the famous Euler’s rotation theorem [?]. A body is rigid if the distance between any two points fixed with respect to the body remains constant. If the body is free to move and rotate in space, it has 6 degree-of-freedom (DOF), 3 trans- lational and 3 rotational, if we consider both translation and rotation. If the body is constrained in a plane, then it is 3-DOF, 2 translational and 1 rotational. We will adopt a coordinate-free ap- proach as in [?, ?] and use geometric, rather than purely algebraic, arguments as much as possible.
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