By Christian Ott
This monograph is dedicated to the classical subject of impedance regulate, which has lately visible renewed curiosity following advances within the mechanical layout of light-weight robot platforms with more desirable actuation and sensing services. After a common advent into the subjects of impedance keep an eye on, the booklet specializes in key matters, specifically the therapy of joint flexibility and kinematic redundancy. numerous keep watch over legislation are built in keeping with mature ways similar to the singular perturbation concept, cascaded regulate idea, and passivity. The controllers are in comparison according to their conceptual strength in addition to useful implementation concerns. The overview was once played via a number of experiments with the DLR hands and the humanoid manipulator 'Justin'.
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Extra info for Cartesian Impedance Control of Redundant and Flexible-Joint Robots
These issues, though practically relevant, are only rarely addressed in the robotics literature. Finally, an alternative orientation stiﬀness implementation which avoids the analytical singularities inherent to any minimal representation of SO(3) was brieﬂy discussed for further reference. 4 Nullspace Stiﬀness In the previous chapter the non-redundant case was treated in which the number m of task coordinates is equal to the number n of conﬁguration coordinates (m = n). The DLR lightweight robots, instead, have n = 7 joints, while the end-eﬀector motion is described only by m = 6 degrees-of-freedom.
Since these (n, i)th co-factors are computed from the submatrix J (q) alone and do not depend on z(q), one can choose z(q) = cn (q)T . 11) where J i (q) ∈ Rm×m is the matrix J (q) with the ith column omitted. 2 Projection Based Approaches The superposition principle for impedances provides a suitable framework for the combination of diﬀerent impedance behaviors. e. 4). 2 gives the following joint torque as an output, cf. 18), τ c = g(q) + J (q)T F imp , ˜ − Dd x ˙ x˙ d − K d x ˜˙ . 1. In this section it will be shown, how such a nullspace impedance with output τ n can be designed in an intuitive way.
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) . x Notice that, although these quantities become time-varying now, these substitutions do not alter the stability statements presented in this book. Explicit formulas of J x (q, t) and v t (q, t) for the diﬀerent sets of local coordinates can be given in terms of the body Jacobian J b (q).
Cartesian Impedance Control of Redundant and Flexible-Joint Robots by Christian Ott