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Unformatted text preview: Introduction to Robotics, H. Harry Asada 1 Chapter 7 Dynamics In this chapter, we analyze the dynamic behavior of robot mechanisms. The dynamic behavior is described in terms of the time rate of change of the robot configuration in relation to the joint torques exerted by the actuators. This relationship can be expressed by a set of differential equations, called equations of motion, that govern the dynamic response of the robot linkage to input joint torques. In the next chapter, we will design a control system on the basis of these equations of motion. Two methods can be used in order to obtain the equations of motion: the Newton-Euler formulation, and the Lagrangian formulation. The Newton-Euler formulation is derived by the direct interpretation of Newton's Second Law of Motion, which describes dynamic systems in terms of force and momentum. The equations incorporate all the forces and moments acting on the individual robot links, including the coupling forces and moments between the links. The equations obtained from the Newton-Euler method include the constraint forces acting between adjacent links. Thus, additional arithmetic operations are required to eliminate these terms and obtain explicit relations between the joint torques and the resultant motion in terms of joint displacements. In the Lagrangian formulation, on the other hand, the system's dynamic behavior is described in terms of work and energy using generalized coordinates. This approach is the extension of the indirect method discussed in the previous chapter to dynamics. Therefore, all the workless forces and constraint forces are automatically eliminated in this method. The resultant equations are generally compact and provide a closed-form expression in terms of joint torques and joint displacements. Furthermore, the derivation is simpler and more systematic than in the Newton-Euler method. The robots equations of motion are basically a description of the relationship between the input joint torques and the output motion, i.e. the motion of the robot linkage. As in kinematics and in statics, we need to solve the inverse problem of finding the necessary input torques to obtain a desired output motion. This inverse dynamics problem is discussed in the last section of this chapter. Efficient algorithms have been developed that allow the dynamic computations to be carried out on-line in real time. 7.1 Newton-Euler Formulation of Equations of Motion 7.1.1. Basic Dynamic Equations In this section we derive the equations of motion for an individual link based on the direct method, i.e. Newton-Euler Formulation. The motion of a rigid body can be decomposed into the translational motion with respect to an arbitrary point fixed to the rigid body, and the rotational motion of the rigid body about that point. The dynamic equations of a rigid body can also be represented by two equations: one describes the translational motion of the centroid (or center of mass), while the other describes the rotational motion about the centroid. The former is Newton's mass), while the other describes the rotational motion about the centroid....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.12 taught by Professor Harryasada during the Fall '05 term at MIT.
- Fall '05