Abstract
In this chapter we provide a brief review of the theoretical methodologies employed in this book for the analytical study of tethered aerial vehicles. In particular, this review covers fundamental methods to (i) model the system; (ii) analyze its dynamic properties; (iii) design nonlinear control methods to accomplish the sought autonomous behavior; and finally (iv) design state estimation methods to retrieve the state from the available sensors to close the control loop. For the modeling of the analyzed tethered aerial systems we used mainly two equivalent yet different approaches, namely the Lagrangian and the Newton-Euler formalisms (see Sect. 2.1). The combination of the two allowed us to obtain the best representation of the dynamics for our control objectives. A particular attention is given to the modeling of a rigid body. Indeed, most of the aerial vehicles are modeled as floating rigid bodies. The obtained formal description of the dynamics was firstly used to determine whether the system is differentially flat or not, and if yes, with respect to which flat outputs. The analysis of this property results very useful for both control and motion planning (see Sect. 2.2). Since there exists a strong relation between differential flatness and feedback linearization [1], we then applied the latter method, described in Sect. 2.3, to solve the tracking problem of the flat outputs previously discovered. Finally, in order to practically implement the control action based on feasible measurements, we investigated the minimal sensory configuration that makes the state observable. The applicability of a globally exponentially stable nonlinear High Gain Observer (described in Sect. 2.4) has been studied. In order to facilitate the reader understating of the theoretical results proposed in this book, in the following, we shall describe the previously mentioned methodologies.
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Notes
- 1.
The notation \(\mathbb {R}^{n \times n}_{> 0}\) denotes the set of positive-definite real matrices, i.e., \(\mathbb {R}^{n \times n}_{> 0} = \{ \mathbf {A}\in \mathbb {R}^{n \times n} \;|\; \mathbf {x}^\top \mathbf {A}\mathbf {x}> 0 \;\forall \; \mathbf {x}\in \mathbb {R}^{n} \}\).
- 2.
More in general, \(\mathbf {e}_i\in \mathbb {R}^{3}\) is the canonical vector with 1 in position ith and zero otherwise.
- 3.
In this thesis, the superscript is used to indicate the frame of references. When not present, \(\mathcal {F}_W\) has to be intended as the reference frame, if not otherwise specified.
- 4.
Notice that this representation is equivalent to the classical Roll-Pitch-Yaw representation. The latter consists in successive rotations along the fixed axes \(\mathbf {x}_B\), \(\mathbf {y}_B\) and \(\mathbf {z}_B\) (in this order) about the angles \(\phi \), \(\theta \) and \(\psi \) respectively.
- 5.
\(\mathsf {SO}(3) = \{ \mathbf {R}\in \mathbb {R}^{3 \times 3} \;| \; \mathbf {R}^\top \mathbf {R}= \mathbf {I}_{3} \}\) where \(\mathbf {I}_{n}\) is the identity matrix of dimension n. \(\mathsf {SO}(3)\) is also called special orthogonal group.
- 6.
The notation \(\mathbf {x}^{(r)}\) represents the rth time derivative of \(\mathbf {x}\), i.e., \(\mathbf {x}^{(r)} = {d^{r} \mathbf {x}}/{dt^r}\).
- 7.
If \(\mathbf {E}\) is not invertible one can always apply an invertible, state-dependent, input transformation that zeroes the maximum number of columns in \(\mathbf {E}\).
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Tognon, M., Franchi, A. (2021). Theoretical Background. In: Theory and Applications for Control of Aerial Robots in Physical Interaction Through Tethers. Springer Tracts in Advanced Robotics, vol 140. Springer, Cham. https://doi.org/10.1007/978-3-030-48659-4_2
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