Abstract
Robots with many DOF and several simultaneous objectives necessarily require a redundancy resolution. In most state-of-the-art approaches, one solves optimization problems for a hierarchical arrangement of the involved tasks. The highest-priority task is executed employing all capabilities of the robotic system. The second-priority task is then performed in the null space of this highest-priority task. In other words, the task on the second level is executed as well as possible without disturbing the first level. The task on level three is then executed without disturbing the two higher-priority tasks, and so forth.
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Notes
- 1.
Dealing with singular matrices or changing rank requires additional treatment, both in kinematic and torque control [DW95, Chi97, DASH12, DWASH12a]. This aspect will be addressed in Sect. 4.4.
- 2.
Since \(\{\}^{\#}\) is not unique, the particular choice for the inverse has an influence on the projected torques. That aspect will be addressed in Sect. 4.2.
- 3.
The term \(\varvec{A} \varvec{W}^{-1} \varvec{A}^{T}\) has to be of rank m, and \(\varvec{W}\) must be invertible.
- 4.
Notice that \(\ddot{\varvec{x}}_{i} = \dot{\varvec{J}}_{i}({\varvec{q}},\dot{\varvec{q}}) \dot{\varvec{q}}+ \varvec{J}_{i}({\varvec{q}}) \ddot{\varvec{q}}\) due to the dependencies in the Jacobian matrices (4.2).
- 5.
In this context, completely means that the Jacobian matrix w. r. t. a low-priority task is a linear combination of all higher-level Jacobian matrices. Then this lower-priority task is dropped completely, i.e. there is not even a null space left in which it could be partially executed.
- 6.
A unilateral constraint describes a task that is not permanently active, but it can be activated and deactivated at run time. An example is given in Fig. 4.8.
- 7.
Actually, only the first m column vectors in \(\varvec{V}({\varvec{q}})\), which span the subspace of \(\varvec{J}({\varvec{q}})\), are relevant here. Thus, a reduced SVD suffices to compute the required elements of \(\varvec{V}({\varvec{q}})\).
- 8.
As stated in [BB04], a trimmed-down or reduced SVD is sufficient to compute the projector.
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Dietrich, A. (2016). Redundancy Resolution by Null Space Projections. 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_4
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DOI: https://doi.org/10.1007/978-3-319-40557-5_4
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