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Unformatted text preview: Introduction to Robotics, H. Harry Asada 1 Chapter 6 Statics Robots physically interact with the environment through mechanical contacts. Mating work pieces in a robotic assembly line, manipulating an object with a multi-fingered hand, and negotiating a rough terrain through leg locomotion are just a few examples of mechanical interactions. All of these tasks entail control of the contacts and interference between the robot and the environment. Force and moment acting between the robot end-effecter and the environment must be accommodated for in order to control the interactions. In this chapter we will analyze the force and moment that act on the robot when it is at rest. A robot generates a force and a moment at its end-effecter by controlling individual actuators. To generate a desired force and moment, the torques of the multiple actuators must be coordinated. As seen in the previous chapter, the sensitivities of the individual actuators upon the end-effecter motion, i.e. the Jacobian matrix, are essential in relating the actuator (joint) torques to the force and moment at the end-effecter. We will obtain a fundamental theorem for force and moment acting on a multi degree-of-freedom robot, which we will find is analogous to the differential kinematics discussed previously. 6.1 Free Body Diagram We begin by considering the free body diagram of an individual link involved in an open kinematic chain. Figure 6.1.1 shows the forces and moments acting on link i, which is connected to link i-1 and link i+1 by joints i and i+1 , respectively. Let O i be a point fixed to link i located on the joint axis i+1 and O i-1 be a point fixed to link i-1 on the joint axis i . Through the connections with the adjacent links, link i receives forces and moments from both sides of the link. Let f i-1,i be a three-dimensional vector representing the linear force acting from link i-1 to link i . Likewise let f i,i+1 be the force from link i to link i+1 . The force applied to link i from link i+1 is then given by f i,i+1 . The gravity force acting at the mass centroid C i is denoted m i g , where m i is the mass of link i and g is the 3x1 vector representing the acceleration of gravity. The balance of linear forces is then given by n i m i i i i i , , 1 , 1 , , 1 " = = + + g f f (6.1.1) Note that all the vectors are defined with respect to the base coordinate system O-xyz. Next we derive the balance of moments. The moment applied to link i by link i-1 is denoted N i-1,i , and therefore the moment applied to link i by link i+1 is N i,i+1 . Furthermore, the linear forces f i-1,i and f i,i+1 also cause moments about the centroid C i . The balance of moments with respect to the centroid C i is thus given by n i i i Ci i i i Ci i i i i i i i , , 1 , ) ( ) ( ) ( 1 , , , 1 , , 1 1 , , 1 " = = + + + + f r f r r N N (6.1.2) where r i-1,i is the 3x1 position vector from point O i-1 to point O i with reference to the base coordinate frame, and...
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- Fall '05