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 structures with more suitable actuation and sensing services. After a basic advent into the themes of impedance regulate, the publication specializes in key matters, particularly the remedy of joint flexibility and kinematic redundancy. numerous regulate legislation are built in response to mature techniques reminiscent of the singular perturbation idea, cascaded regulate concept, and passivity. The controllers are in comparison in keeping with their conceptual power in addition to useful implementation concerns. The evaluate used to be played via numerous experiments with the DLR palms and the humanoid manipulator 'Justin'.
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Extra info for Cartesian Impedance Control of Redundant and Flexible-Joint Robots
The desired stiﬀness usually is given by the application. 16) then brings up the problem of how to choose the damping matrix. This problem is treated in the next section. 8) the desired inertia matrix Λd is constant and can be designed in principle such that its eigenvectors coincide with the eigenvectors of K d . The design of the damping matrix can then be reduced to the design of the damping coeﬃcients for n decoupled linear and time-invariant second order systems, each describing the dynamics of the system along one of the eigenvectors.
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. 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.
Com 46 Nullspace Stiﬀness [Bai85, Sic90], or for the generalized velocities (via nullspace velocity coordinates v n ∈ Rr ), as in the Joint Space Decomposition Method [Par99, OCY98]. The problem of ﬁnding appropriate coordinates and the decoupling of the task dynamics from the nullspace dynamics will be discussed for both methods. The additional nullspace coordinates will be constructed using a nullspace base matrix Z(q) ∈ Rr×n which is composed of r row vectors that are linearly independent and span the (right) nullspace of the Jacobian J(q).