ABSTRACT
Thus far in our discussion of robot control problems in the preceding 12 chapters, we have tacitly assumed that the requirements of the end effector of robot manipulators are specified exclusively in terms of tracking the motion along a reference trajectory. Yet, in most significant numbers of applications, in certain constrained degrees of freedom, there is generally a requirement that there is a certain level of tracing of a reference force or the regulation of a contact force. Contact forces arise due to the fact that the tip elements of an end effector are compliant and generate forces due to deformation which must be regulated. Consider, for example, a typical task executed by a human being in the morning such as brushing one’s teeth. This task involves grasping a tooth brush and executing a horizontal movement of the tooth brush, while the bristles make contact with the teeth and maintain a certain contact force. Furthermore, the head of the tooth brush must execute local vibratory motions so that the bristles move while in contact with the teeth and lower gums in a rhythmic fashion, a motion that can possibly be generated by local ultrasonic motors. So, the problem reduces to one of meeting a set of constraints. It can be seen that one set of constraints are natural as they arise from the geometrical constraints imposed by the environment in which the task is carried out. Considering the motion constraints, the stem of the tooth brush must be moved in the horizontal direction in an oscillatory manner. All other motions of the stem are prohibited, as long as the bristles are in contact with surface that needs cleansing. So barring the horizontal axis along which the task may require either translational or small rotational oscillatory motions, no motions are permitted in the other three directions, either rotational or translational. These constitute four natural constraints. There are indeed certain motion requirements imposed in the other two directions by the task at hand and these constitute artificial constraints. On the other hand, the motion requirements imposed by the artificial constraints can only be realized if certain forces and torques are imposed in the associated directions. These forces and torques are governed by natural physical laws, and consequently, the associated constraints on the forces and torques are also classified as natural constraints. Moreover, to regulate and track the motions imposed by the natural constraints, the head of the tooth brush must be made to meet certain force and torque constraints which are therefore artificially imposed. Thus, they constitute a set of artificial forces or torque constraints. In the example mentioned earlier, it is clear that the axes involved in establishing both the natural constraints and the artificial constraints are orthogonal to each other in both the motion and force domains. Moreover, the axes involved in establishing the natural motion constraints and the artificial force constraints are the same, while the axes defining the natural force constraints and the artificial motion constraints are the same. This is probably no accident and is a direct consequence of our intuitive choice of the reference frame to describe these constraints. The reference frame bears some relationship to the base
frame associated with the simple ‘manipulator’ we are considering and to the associated end-effector frame. This frame is therefore referred to as the constraints frame in artificial intelligence terminology or the C-frame and must be defined systematically so that the natural and artificial constraints can be defined in two sets of orthogonal direction. The orthogonality of the forces and motions in these directions follows from the principle of virtual work as the virtual displacements, which by definition must satisfy all of the physical constraints, correspond to the displacements in the motion space. From the principle of virtual work, the virtual forces, which correspond to the forces in the motion space axes, must add up to zero. On the other hand, due to the imposed motion constraints, the displacements in the associated axes are zero and the corresponding forces are determined by the natural physical laws. The implications are that along the motion space axes, one must prescribe the necessary reference motions while along the axes defining the force domain, one must track the desired forces.
