Task space control
In the above joint space control schemes, it was assumed that the reference trajectory is available in terms of the time history of joint positions, velocities and accelerations. On the other hand, robot manipulator motions are typically specified in the
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Task space control In the above joint space control schemes, it was assumed that the reference trajectory is available in terms of the time history of joint positions, velocities and accelerations. On the other hand, robot manipulator motions are typically specified in the task space in terms of the time history of end-effector position, velocity and acceleration. This chapter is devoted to control of rigid robot manipulators in the task space. The natural strategy to achieve task space control goes through two successive stages; namely, kinematic inversion of the task space variables into the corresponding joint space variables, and then design of a joint space control. Hence this approach, termed kinematic control, is congenial to analyze the important properties of kinematic mappings: singularities and redundancy. A different strategy consists of designing a control scheme directly in the task space that utilizes the kinematic mappings to reconstruct task space variables from measured joint space variables. This approach has the advantage to operate directly on the task space variables. However, it does not allow an easy management of the effects of singularities and redundancy, and may become computationally demanding if, besides positions, also velocities and accelerations are of concern. The material of this chapter is organized as follows. The inversion of differential kinematics is discussed in terms of both the pseudoinverse and the damped least-squares inverse of the Jacobian. Inverse kinematics algorithms are proposed which are aimed at generating the reference trajectories for joint space control schemes; velocity resolution schemes are presented based on the use of either the pseudoinverse or the transpose of the Jacobian matrix, and the extension to acceleration resolution is also discussed. As opposed to the above kinematic control schemes, two kinds of
C. C. de Wit et al. (eds.), Theory of Robot Control © Springer-Verlag London Limited 1996
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CHAPTER 3. TASK SPACE CONTROL
direct task space control schemes are presented which are analogous to those analyzed in the joint space; namely, a PD control with gravity compensation scheme that achieves end-effector regulation, and an inverse dynamics control scheme that allows end-effector trajectory tracking.
3.1
Kinematic control
Control of robot manipulators is naturally achieved in the joint space, since the control inputs are the joint torques. Nevertheless, the user specifies a motion in the task space, and thus it is important to extend the control problem to the task space. This can be achieved by following two different strategies. Let us start by illustrating the more natural one, kinematic control, which consists of inverting the kinematics of the manipulator to compute the joint motion corresponding to the given end-effector motion. In view of the difficulties in finding closed-form solutions to the inverse kinematics problem, it is worth considering the problem of differential kinematics inversion which is well posed for any manipulator
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