By Christian Ott
This monograph is dedicated to the classical subject of impedance regulate, which has lately noticeable renewed curiosity following advances within the mechanical layout of light-weight robot structures with enhanced actuation and sensing features. After a basic advent into the subjects of impedance regulate, the publication makes a speciality of key matters, particularly the remedy of joint flexibility and kinematic redundancy. a number of keep watch over legislation are built in line with mature methods reminiscent of the singular perturbation concept, cascaded keep watch over conception, and passivity. The controllers are in comparison according to their conceptual strength in addition to useful implementation matters. The review was once played via numerous experiments with the DLR palms and the humanoid manipulator 'Justin'.
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Additional resources for Cartesian Impedance Control of Redundant and Flexible-Joint Robots
Q, t) are given by The ﬁrst and second derivatives of the error vector x ˜ (q, t) ˜ (q, t) ∂x ∂x q˙ + , x ˜˙ = ∂q ∂t ˜ (q, t) ˜ (q, t) d ∂x ∂x ¨ ¨+ q x ˜ = ∂q dt ∂q 15 q˙ + ˜ (q, t) d ∂x . dt ∂t For unit quaternions also the term Euler parameters is sometimes used. 3). 2. For vector x the ease of presentation the following substitutions are made ˜ (q, t) ∂x = J x (q, t) , ∂q ˜ (q, t) ∂x = −v t (q, t) . 11). 12), one can see that the same controllers as in the previous sections can be used, when the following substitutions are made J (q) → J x (q, t) , x˙ d → −vt (q, t) , ¨ d → −v˙ t (q, t) .
In this section it will be shown, how such a nullspace impedance with output τ n can be designed in an intuitive way. 15) Robot Dynamics q, q˙ τn Fig. 1. Superposition of the outputs of the Cartesian impedance controller and the nullspace impedance controller Projection Based Approaches 49 where the symmetric and positive deﬁnite matrices K n ∈ Rn×n and D n ∈ Rn×n represent a desired stiﬀness and damping behavior with respect to the conﬁguration coordinates.
Flexible-Joint Robots, STAR 49, pp. 29–44, 2008. 1 Task Formulation The goal of impedance control is to realize a particular desired dynamical relationship between the robot motion and the external torques. In case of the Cartesian impedance control problem, this relationship is speciﬁed in terms of coordinates which describe the motion of the end-eﬀector. e. 3). For the purpose of controller design, instead, a minimal representation in terms of m end-eﬀector coordinates x ∈ Rm is often preferred.