By Christian Ott
This monograph is dedicated to the classical subject of impedance keep watch over, which has lately visible renewed curiosity following advances within the mechanical layout of light-weight robot platforms with greater actuation and sensing functions. After a normal advent into the themes of impedance keep an eye on, the booklet makes a speciality of key concerns, specifically the therapy of joint flexibility and kinematic redundancy. a number of keep watch over legislation are built in keeping with mature methods akin to the singular perturbation conception, cascaded keep watch over concept, and passivity. The controllers are in comparison in keeping with their conceptual strength in addition to sensible implementation matters. The evaluate was once played via numerous experiments with the DLR fingers and the humanoid manipulator 'Justin'.
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Extra info for Cartesian Impedance Control of Redundant and Flexible-Joint Robots
K c = 0. 25) corresponds to a vector from the origin of the desired frame D to the origin of the tool frame T , represented in frame D. From this one can get an appropriate set of translational end-eﬀector coordinates by rotating this vector into the base frame S pdt (q, t) = Rsd (t)d pdt (q, t) = pst (q) − psd (t) . 26) Then, one can choose xt = pst (q) as translational coordinates and psd (t) as their corresponding virtual equilibrium position. By this choice the translational part of the stiﬀness matrix corresponds to a stiﬀness, which is represented in the base frame S, and consequently the eigenvectors of K t , corresponding to the principal axes of the stiﬀness matrix, are constant vectors expressed in the base frame.
16). e. x˙ d = 0) it represents a passive mapping from external ˙ This property is especially important generalized forces F ext to the velocity x. if the interaction of the robot with passive8 environments is considered. 16) also represents a passive mapping from the external force ˙ F ext to the velocity x. 5. 16) with symmetric and positive deﬁnite matrices K d and Dd is time-invariant and represents a passive mapping ˙ from the external force F ext to the velocity x. 16). 18) is more convenient.
A unit quaternion (η, ) consists of a scalar part η ∈ R and a vector part ∈ R3 , which fulﬁll the condition η 2 + T = 1. The relation between a rotation matrix and unit quaternions is given as follows. Suppose that a rotation matrix is speciﬁed by a rotation of an angle α about an axis r (with ||r||2 = 1), then the corresponding unit quaternion is given by η = cos(α/2), = r sin(α/2). With the restriction of α to the interval [−π, π] the set of unit quaternions is a one-to-one covering of SO(3).
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