By Christian Ott
This monograph is dedicated to the classical subject of impedance keep an eye on, which has lately noticeable renewed curiosity following advances within the mechanical layout of light-weight robot platforms with enhanced actuation and sensing services. After a basic creation into the themes of impedance keep watch over, the ebook makes a speciality of key matters, particularly the therapy of joint flexibility and kinematic redundancy. a number of keep watch over legislation are built according to mature ways equivalent to the singular perturbation concept, cascaded keep an eye on idea, and passivity. The controllers are in comparison in accordance with their conceptual capability in addition to useful implementation concerns. The assessment used to be played via numerous experiments with the DLR palms and the humanoid manipulator 'Justin'.
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Additional info for Cartesian Impedance Control of Redundant and Flexible-Joint Robots
Example text
Clearly, by this choice the system can be written in the state variable z = x in the following form Q(x0 )˜ ¨ + 2 diag(ξi z −T ˙ + diag(λΛ λΛ F ext . 21) It should be mentioned again that the above approximations are only used for the design of the damping matrix and do not affect the stability properties of the system, because it is ensured that the damping matrix is always positive definite. 9 10 11 It can even be chosen time-varying. The author would like to thank Udo Frese for pointing out this feature of positive definite matrices.
From a mathematical point of view the kinematic singularities are the singularities of the body Jacobian. 16) in general cannot be achieved exactly but will be distorted. 18) might vanish. One method for the control of manipulators at kinematic singularities was proposed in [CK95]. In this work a factorization of the determinant of the body Jacobian is required, in which each term corresponds to one of the different singular configurations of the robot. When approaching such a singular configuration the controller splits up into two parts.
22, the term Λ ˙ − μ − μT = d J −T M J −1 + J −T M d J −1 Λ dt dt −T −1 ˙ −1 −T ˙ T −T + J M J JJ + J J J M J −1 =( T d −T d J + J −T J˙ J −T )M J −1 + J −T M ( J −1 + J −1 J˙ J −1 ) . dt dt From the equality d d (I)J −1 = (J −1 J )J −1 dt dt d = J −1 + J −1 J˙ J −1 dt 0= ˙ − μ − μT = 0 which completes the proof. 1) written in another set of coordinates. 11). 2. 11) is obtained, if it is computed via the that the same form of μ(x, x) Christoffel symbols of Λ(x). 10) clearly is the classical coordinate transformation for a covariant tensor of rank 2 [Fra97, Boo03].