tailieunhanh - Crc Press - Mechanical Engineering Handbook - Robotics 1 Part 3

Tham khảo tài liệu 'crc press - mechanical engineering handbook - robotics 1 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14-20 Section 14 The Jacobian can be used to perform inverse kinematics at the velocity level as follows. If we define J 11 to be the inverse of the Jacobian assuming J is square and nonsingular then q J -1 q T and the above expression can be solved iteratively for q and hence q by numerical integration given a desired end effector trajectory Y and the current state q of the manipulator. This method for determining joint trajectories given desired end effector trajectories is known as Resolved Rate Control and has become increasingly popular. The technique is particularly useful when the positional inverse kinematics is difficult or intractable for a given manipulator. Notice however that the above expression requires that J is both nonsingular and square. Violation of the nonsingularity assumption means that the robot is in a singular configuration and if J has more columns than rows then the robot is kinematically redundant. These two issues will be discussed in the following subsections. Example By direct differentiation of the forward kinematics derived earlier for our example x y -l1 sin 01 - d3 sin 01 02 -d3 sin 01 02 l1cos 01 d3 cos 01 02 d3cos 01 02 1 1 cos 01 sin 01 0 02 01 02 02 d3 Notice that each column of the Jacobian represents the instantaneous effect of the corresponding joint on the end effector motions. Thus considering the third column of the Jacobian we confirm that the third joint with variable d3 cannot cause any change in the orientation f of the end effector. Singularities A significant issue in kinematic analysis surrounds so-called singular configurations. These are defined to be configurations qs at which J qs has less than full rank Spong and Vidyasagar 1989 . Physically these configurations correspond to situations where the robot joints have been aligned in such a way that there is at least one direction of motion the singular direction s for the end effector that physically cannot be achieved by the mechanism. .

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