# State and Force Estimation on a Rotating Helicopter Blade through a Kalman-Based Approach

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## Abstract

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## 1. Introduction

## 2. Methodologies: Theoretical Background

#### 2.1. Multibody Structural Model

#### 2.1.1. Finite Segment Beam Formulation

#### 2.2. Aerodynamic Model

#### 2.2.1. Aerodynamic Loads

- $\rho $ is the air density.
- ${\dot{\mathbf{q}}}_{\mathbf{r}}$ is the velocity vector evaluated at the stage $\mathbf{r}$. Considering the blade modeled as a rigid body, ${\dot{\mathbf{q}}}_{\mathbf{r}}$ is expressed as:$$\begin{array}{c}\hfill {\dot{\mathbf{q}}}_{\mathbf{r}}=\mathsf{\Omega}{\widehat{i}}_{3}\times \mathbf{r}+\dot{\beta}\left|\mathbf{r}\right|{\widehat{i}}_{3}+({u}_{1}{\widehat{i}}_{1}+{u}_{2}{\widehat{i}}_{2}+{u}_{3}{\widehat{i}}_{3})\end{array}$$$$\begin{array}{c}\hfill {\dot{\mathbf{q}}}_{\mathbf{r}}=\mathsf{\Omega}\left|\mathbf{r}\right|{\widehat{i}}_{2}+\dot{\beta}\left|\mathbf{r}\right|{\widehat{i}}_{3}+({u}_{1}{\widehat{i}}_{1}+{u}_{2}{\widehat{i}}_{2}+{u}_{3}{\widehat{i}}_{3}).\end{array}$$
- ${C}_{L}$ and ${C}_{D}$ are the aerodynamic coefficients; in general, they are tabular values for a given airfoil. Important for the evaluation of the total thrust produced by the rotating blades is the expression of the lift coefficient ${C}_{L}$. This contribution focuses on small angles of attack $\alpha $ and low values of the Mach number, such that a linear relation ${C}_{L}=a\alpha $ is guaranteed, with a representing the lift slope of the section, and $\alpha =\theta -\Phi $ (Figure 2). $\theta $ and $\mathsf{\Phi}$ are respectively the pitch and inflow angles of the section. More details about the derivation of $\mathsf{\Phi}$ are given in Section 2.2.2.
- c indicates the chord of the blade at stage r.

#### 2.2.2. Uniform Inflow Model

#### 2.3. Kalman Filter for Implicit Scheme

^{th}-order dynamics, i.e., zero time-derivative of $\mathbf{p}$, which leads to $\dot{\mathbf{p}}=\mathbf{0}$. This model has been widely adopted and validated in the literature [25,27,40,41], and a clearer explanation will be provided in the following. In this contribution, the estimation process is performed in discrete-time domain, and thus the discretization of the continuous-time GGL formulation expressed in Equation (2) is needed. The multibody solver used in the present work executes this step by applying the implicit backward differentiation formulas (BDF) integration method [42]:

^{th}-order input dynamic is also included in the third row of Equation (18), where $\dot{\mathbf{p}}\in {\mathbf{R}}^{p}$ and p is the number of inputs. The final goal of this work is to estimate or predict jointly the state vectors $\mathbf{x}$ and $\dot{\mathbf{x}}$, and the external loads identified in $\mathbf{p}$, which will not be considered anymore as an input to the system, but another set of quantities to be estimated. This is the field of application of the augmented Kalman filter (AKF), which defines an augmented state vector ${\mathbf{x}}_{\mathbf{a}}$ and its time-derivative:

- Correction—updating of the solution of the previous step with the available observations $\mathbf{y}$:$$\begin{array}{c}\hfill {\overline{{\mathbf{x}}_{\mathbf{a}}}}_{k+1}^{+}={\overline{{\mathbf{x}}_{\mathbf{a}}}}_{k+1}^{-}+{\mathbf{K}}_{k+1}({\mathbf{y}}_{k+1}-\mathbf{k}\left({\overline{{\mathbf{x}}_{\mathbf{a}}}}_{k+1}^{-}\right))\end{array}$$

## 3. Workflow for Estimation of States and Loads on a Rotor Blade

- Writing the vector $\mathbf{p}$ in Equation (19) as a function of $\mathbf{q}$ and $\dot{\mathbf{q}}$, i.e., the orientation and velocity of a given point;
- Modeling the aerodynamic forces as external loads without explicit dependence on the states.

- Definition of the set of loads. This decision should be made in conjunction with the previous step. For multiple distributed loads, the definition of a subset of loads $\mathbf{p}$ could be useful to reduce the number of observations needed. For the case in Figure 5, the full input vector is $\mathbf{p}=[{L}_{1},{D}_{1},{M}_{1},{L}_{2},{D}_{2},{M}_{2}]$; a good choice could be the selection of loads close to the tip ${\mathbf{p}}_{s}=[{L}_{2},{D}_{2},{M}_{2}]$, because their contributions to the dynamics of the blade are higher.

- Extrapolation of $\dot{\mathbf{q}}$ along the blade span. The components of the velocity vector along the blade can be approximated as a linear function of the position r:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \dot{q}\left(r\right){i}_{2}=\mathsf{\Omega}r\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \dot{q}\left(r\right){i}_{3}=\dot{\beta}r\hfill \end{array}$$
- Evaluation of the lift distribution along the blade span. Recalling Equation (5), and given a certain flight condition with known pitch angle $\theta $, the only unknown in the elemental lift will be the inflow angle $\varphi $:$$\begin{array}{c}\hfill dL=dL\left({\lambda}_{i}\right)\end{array}$$The iterative process in Section 2.2 can thus be applied to find the exact value of ${\lambda}_{i}$ at each time-step.

## 4. Numerical Validation on a 2-Blade Rotor Model

- Noisy reference observation data generated in Python code by simulating the coupled structural-aerodynamic model. Estimation of a subset of lumped loads on the same structural model without aerodynamic and subsequent distributed load evaluations.
- Reference observation data generated in MBDyn on a similar, but independently-formulated structural-aerodynamic model. Estimation of a subset of lumped loads on the structural Python model and subsequent evaluation of distributed loads.

#### 4.1. Kalman-Based Estimation

- The aerodynamic loads give higher contribution in the proximity of the blade tip;
- The in-plane loads are much smaller than the out-of-plane ones, namely, $L\gg D$,

#### 4.1.1. Reference Noisy Data

^{th}-order input model in the discrete time formulation will be read as:

- The sensor layout does not include in-plane quantities, e.g., position sensors in $y\text{-}$direction.
- Only out-of-plane loads are considered in the estimation, even if the reference sensors come from a coupled multibody-aerodynamic simulation which includes lift and drag forces. The choice to not estimate the drag forces was because the resulting magnitude would be comparatively small with respect to the computed lift.

#### 4.1.2. Reference Data from Mbdyn

- Finite segment: ${n}_{bodies}=2({n}_{flex}-1)$;
- Finite volume: ${n}_{bodies}=2{n}_{flex}+1$.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Two rigid bodies connected through elastic elements. a and b are the two faces of each body where the connections are. ${k}_{12}$ is the total stiffness matrix given by the interaction of both elastic forces from faces ${B}_{1b}$ and ${B}_{2a}$.

**Figure 2.**Velocity vector and angles definition for a blade station at distance r from the hub rotating frame.

**Figure 4.**Proposed workflow for distributed aerodynamic loads, based on Kalman filtering estimation.

**Figure 5.**Example of a blade discretized with 2 elements. The lift distribution per unit length is shown above.

**Figure 6.**A 4-element blade model with all aerodynamic forces (

**left**) and a reduced set of significant forces (

**right**).

**Figure 8.**Estimation of the unmeasured velocity and position entities with reference to the sensor layout in Table 2. Reference noisy data.

**Figure 9.**Force estimation on elem3 and elem4 with reference to the sensor layout in Table 2. Reference noisy data.

**Figure 10.**Distributed lift and angle of attack along the aerodynamic span in steady-state, with 11 points of integration. Reference noisy data used in the estimation.

**Figure 11.**Absolute error of distributed lift and angle of attack—reference data in Figure 10.

**Figure 12.**Graphic comparison between finite segment and finite volume modeling of a flexible element of free-length l.

**Figure 13.**Comparison of vertical displacement at elem4 position (

**a**) without and (

**b**) with aerodynamics. The model is in rotation and under gravity.

**Figure 15.**Estimation of the unmeasured velocity and position entities with reference to the sensor layout in Table 2. Reference data from MBDyn.

**Figure 16.**Distributed lift along the aerodynamic span in steady-state (

**left**) and absolute error (

**right**), with 11 points of integration. Reference data from MBDyn.

**Figure 17.**Estimation of the velocities of elem3 and elem4 with reference to the sensor layout in Table 3. Reference data from MBDyn.

Blade Structural Properties | Blade Elastic Properties | ||||
---|---|---|---|---|---|

Total mass | 82.76 | kg | $E{A}_{elem}$ | 5.69E8 | N |

Chord | 0.537 | m | $E{I}_{yy,elem}$ | 4.00E5 | Nm${}^{2}$ |

Total Length | 6.988 | m | $E{I}_{zz,elem}$ | 4.00E5 | Nm${}^{2}$ |

Radius | 7.420 | m | $G{J}_{elem}$ | 8.40E5 | Nm${}^{2}$/rad |

Root Hinges—Distance From Hub | Root Hinges—Elastic Properties | ||||

Flap offset | 0.289 | m | Flap damping characteristic | 7.50E3 | Nms/rad |

Lag offset | 0.269 | m | Lag damping characteristic | 7.00E3 | Nms/rad |

Pitch offset | 0.432 | m |

Position Sensors | Velocity Sensors | ||
---|---|---|---|

elem3 | z | elem3 | z |

elem4 | z | elem4 | z |

Position Sensors | Velocity Sensors | ||
---|---|---|---|

elem3 | z | elem3 | z |

elem4 | z | elem4 | z |

elem4 | y |

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**MDPI and ACS Style**

Cumbo, R.; Tamarozzi, T.; Jiranek, P.; Desmet, W.; Masarati, P.
State and Force Estimation on a Rotating Helicopter Blade through a Kalman-Based Approach. *Sensors* **2020**, *20*, 4196.
https://doi.org/10.3390/s20154196

**AMA Style**

Cumbo R, Tamarozzi T, Jiranek P, Desmet W, Masarati P.
State and Force Estimation on a Rotating Helicopter Blade through a Kalman-Based Approach. *Sensors*. 2020; 20(15):4196.
https://doi.org/10.3390/s20154196

**Chicago/Turabian Style**

Cumbo, Roberta, Tommaso Tamarozzi, Pavel Jiranek, Wim Desmet, and Pierangelo Masarati.
2020. "State and Force Estimation on a Rotating Helicopter Blade through a Kalman-Based Approach" *Sensors* 20, no. 15: 4196.
https://doi.org/10.3390/s20154196