Nonlinear Model Predictive Control with Enhanced Actuator Model for Multi-Rotor Aerial Vehicles with Generic Designs

Abstract

In this paper, we propose, discuss, and validate an online Nonlinear Model Predictive Control (NMPC) method for multi-rotor aerial systems with arbitrarily positioned and oriented rotors which simultaneously addresses the local reference trajectory planning and tracking problems. This work brings into question some common modeling and control design choices that are typically adopted to guarantee robustness and reliability but which may severely limit the attainable performance. Unlike most of state of the art works, the proposed method takes advantages of a unified nonlinear model which aims to describe the whole robot dynamics by explicitly including a realistic physical description of the actuator dynamics and limitations. As a matter of fact, our solution does not resort to common simplifications such as: (1) linear model approximation, (2) cascaded control paradigm used to decouple the translational and the rotational dynamics of the rigid body, (3) use of low-level reactive trackers for the stabilization of the internal loop, and (4) unconstrained optimization resolution or use of fictitious constraints. More in detail, we consider as control inputs the derivatives of the propeller forces and propose a novel method to suitably identify the actuator limitations by leveraging experimental data. Differently from previous approaches, the constraints of the optimization problem are defined only by the real physics of the actuators, avoiding conservative – and often not physical – input/state saturations which are present, e.g., in cascaded approaches. The control algorithm is implemented using a state-of-the-art Real Time Iteration (RTI) scheme with partial sensitivity update method. The performances of the control system are finally validated by means of real-time simulations and in real experiments, with a large spectrum of heterogeneous multi-rotor systems: an under-actuated quadrotor, a fully-actuated hexarotor, a multi-rotor with orientable propellers, and a multi-rotor with an unexpected rotor failure. To the best of our knowledge, this is the first time that a predictive controller framework with all the valuable aforementioned features is presented and extensively validated in real-time experiments and simulations.

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Acknowledgements

We thank Anthony Mallet (LAAS-CNRS) for his contribution to the development of the software architecture exploited for the experiments and Yutao Chen (University of Padova – Eindhoven University) for the development of the MATMPC framework, integrated in our setup.

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Correspondence to Davide Bicego.

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This research was partially supported by the cooperation program “INTERREG Deutschland-Nederland” as part of the SPECTORS project number 143081 and by the European Union’s Horizon 2020 research and innovation program grant agreement ID: 871479 AERIAL-CORE.

Appendix : Allocation matrix identification

Appendix : Allocation matrix identification

The nominal values of the entries of the allocation matrix G can be calculated from the system’s geometrical properties, consistently with Eq. 7. However, the real physical parameters of the robot could be quite different from the ideal ones, due to mechanical inaccuracies unavoidably associated with the manufacturing and the assembly of the robot parts. This may dramatically affect the control system performances. For this reason, in this work the entries of the allocation matrix are identified from experimental data. In the following we briefly outline the used identification method, which is extensively used in the literature and very well-known from the community, so it is not considered as a contribution.

First of all, we used the nominal allocation matrix to design a simple but robust controller, applied on the platform. Accordingly, the so-obtained control system is used to track suitable persistently exciting 6D trajectories. To this purpose chirp signals are used, i.e., sinusoidal trajectories with increasing frequencies. While doing this, we collected the measured data p and R thanks to the MoCap system, used as ground truth. In particular, we made the assumption to be able to measure the CoM location. Then, thanks to a properly tuned post-processing of the data which mainly consisted in a constant frame-rate signal re-sampling, an anti-causal low-pass filtering and the computation of numerical derivatives, we were able to retrieve a precise-enough estimation of \(\ddot {{p}}, \boldsymbol {\omega }, \dot {\boldsymbol {\omega }}\) defined in Eq. 2. On the other hand, γ was reconstructed by collecting the measured spinning rates of the motors wi and using the thrust model Eq. 5. Finally, m was directly measured and J estimated by a precise CAD model of the robot. At this point, we re-wrote Eq. 2 as

$$ \begin{array}{@{}rcl@{}} \underbrace{ \begin{bmatrix} m\textbf{R}^{\top} (\ddot{{p}} + g{e}_{3}) \\ \textbf{J} \dot{\boldsymbol{\omega}} + \boldsymbol{\omega}\!\times\!\textbf{J}\boldsymbol{\omega} \end{bmatrix} }_{:={y}} = \underbrace{ \begin{bmatrix} f_{1} \textbf{I}_{3}\ \dots\ f_{n}\textbf{I}_{3} & \boldsymbol{0}_{3 \times 3n} \\ 0_{3 \times 3n} & f_{1}\textbf{I}_{3}\ \dots\ f_{n}\textbf{I}_{3} \end{bmatrix} }_{:=\mathbf{A}} \boldsymbol{\beta} \end{array} $$
(41)

with \(\mathbf {A} \in \mathbb {R}^{6 \times 6n}\). In such form, the equation allows to express the vector of measurable quantities \({y} \in \mathbb {R}^{6 \times 1}\) as a linear function of a vector of parameters \(\boldsymbol {\beta } \in \mathbb {R}^{6n \times 1}\), obtained re-arranging the entries of G

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\beta} := \begin{bmatrix} {\mathbf{G}_{1}(:,1)}^{\top} \dots {\mathbf{G}_{1}(:,n)}^{\top} {\mathbf{G}_{2}(:,1)}^{\top} {\dots} {\mathbf{G}_{2}(:,n)}^{\top} \end{bmatrix} ^{\top} \end{array} $$
(42)

Collecting a large number of measurements p >> 6n and stacking them in vectorial form, we obtained

$$ \begin{array}{@{}rcl@{}} (\boldsymbol{\xi} = \boldsymbol{\Lambda} \boldsymbol{\beta} ) := \left( \begin{bmatrix} {y}_{1} \\ {\vdots} \\ {y}_{p} \end{bmatrix} = \begin{bmatrix} A_{1} \\ {\vdots} \\ A_{p} \end{bmatrix} \boldsymbol{\beta} \right) \end{array} $$
(43)

At this point, applying the standard least-squares identification method, the vector of parameters which minimizes the 2-norm of the error ||Λβξ||2 is obtained as

$$ \begin{array}{@{}rcl@{}} \hat{\boldsymbol{\beta}} = \boldsymbol{\Lambda}^{\dagger} \boldsymbol{\xi} \end{array} $$
(44)

Finally, re-arranging the element of the vector \(\hat {\boldsymbol {\beta }}\) using the convention of Eq. 42, we obtained the identified allocation matrix \(\hat {{G}}\) that we used in the presented experiments.

Comparing the entries of the nominal and the identified allocation matrices in the hexarotor (Tilt-Hex) case, notice that the difference between some elements is pretty consistent. This confirms that the physical parameters of the real robot can be very dissimilar from the nominal ones.

$$ \begin{array}{@{}rcl@{}} e_{G,<percent>} = 100\ \!\left[\frac{g_{i,j} - \hat{g}_{i,j}}{g_{i,j}}\right] = \!\left[\begin{array}{llllll} 4 & 21 & 25 & 4 & 80 & 104 \\ 4 & 6 & 11 & 10 & 5 & 2 \\ 72 & 31 & 30 & 58 & 25 & 28 \\ 26 & 24 & 31 & 27 & 29 & 28 \\ 9 & 15 & 16 & 12 & 14 & 13 \end{array}\right] \end{array} $$
(45)

To conclude, we would like to point out that using the identified matrix in the controller instead of the nominal one allowed to consistently reduce both the position and the orientation errors in all the experiments that we performed. This happens already in hovering condition, as it is shown in the box-plots of Fig. 24.

Fig. 24
figure24

Box-plots for the position error (above) and the orientation error (below) of the Tilt-Hex when hovering using the nominal and the identified allocation matrices. The results for the latter case have been highlighted with yellow bands. We can appreciate how the error mean and variance is reduced

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Bicego, D., Mazzetto, J., Carli, R. et al. Nonlinear Model Predictive Control with Enhanced Actuator Model for Multi-Rotor Aerial Vehicles with Generic Designs. J Intell Robot Syst 100, 1213–1247 (2020). https://doi.org/10.1007/s10846-020-01250-9

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Keywords

  • Model predictive control
  • Multi-rotor aerial vehicles
  • Multi-directional thrust
  • Actuator constraints