Abstract
This chapter covers the aspect of stability in multi-objective whole-body impedance control. That involves both theoretical stability analyses and the experimental validation of the developed concepts.
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- 1.
Excluding mobile systems with less than three wheels such as Golem Krang [SOG10] or platforms based on the Segway technology.
- 2.
In the following experiments, only the forward/backward motion of the platform is allowed in order to simplify the experimental evaluation.
- 3.
The storage function (5.21) does not match the real physical energy due to the use of the admittance inertia instead of the real one. An overall potential energy including the other subtasks is not meaningful due to the null space projections as described in [DOAS13].
- 4.
Algorithmic singularities arise when lower-priority tasks and higher-priority tasks conflict with each other, i.e. stacking their Jacobian matrices results in a matrix that does not have full row-rank.
- 5.
Depending on the application, the damping matrix can be chosen constant or configuration-dependent as long as it is a positive definite matrix, see e.g. [ASOFH03].
- 6.
The proof can be found in [Ott08].
- 7.
A further discussion on this requirement will be given in Sect. 5.2.6. The proof of invertibility of \({\varvec{\bar{J}}}\) is provided in Appendix C.2.
- 8.
To simplify the notations, the corresponding matrices are chosen according to the solution (4.29), i.e. \(\varvec{B}_{\varvec{X}}=\varvec{M}({\varvec{q}})\) and \(\varvec{B}_{\varvec{Y}}=\varvec{I}\), without loss of generality.
- 9.
Note that the superscript “\(\text {p}\)” is also used for the main task with \(\varvec{\tau }_{1}^{\text {p}}=\varvec{\tau }_{1}\) to unify the notations.
- 10.
If the tasks do not even statically conflict with each other at all, then \({\varvec{q}}^{*}\) is actually the global minimum of all \(V_{i}\). In that case, (5.69) is fulfilled because \(\partial V_{i}(\tilde{{\varvec{x}}}_{i})/\partial {\varvec{x}}_{i} = \varvec{0}~\forall i\).
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Dietrich, A. (2016). Stability Analysis. In: Whole-Body Impedance Control of Wheeled Humanoid Robots. Springer Tracts in Advanced Robotics, vol 116. Springer, Cham. https://doi.org/10.1007/978-3-319-40557-5_5
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DOI: https://doi.org/10.1007/978-3-319-40557-5_5
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