# Estimating the Characteristic Curve of a Directional Control Valve in a Combined Multibody and Hydraulic System Using an Augmented Discrete Extended Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Parameter Estimation Methodology

#### 2.1. Multibody Dynamic Formulations

#### 2.1.1. Double-Step Semi-Recursive Formulation

#### 2.1.2. Hydraulic Lumped Fluid Theory

#### 2.1.3. Monolithic Approach: Coupling MBS and Hydraulic Dynamic Systems

#### 2.2. Estimation Algorithm: ADEKF with a Curve-Fitting Method

#### Covariance Matrices of Process and Measurement Noises

## 3. Case Example: Hydraulically Actuated System

#### 3.1. Dynamic Model of the System

#### 3.1.1. Real and Estimation Models

#### 3.1.2. Sensor Measurements

#### 3.2. Parameter Estimation Algorithm

## 4. Results and Discussion

#### 4.1. Estimating the Characteristic Curve of the Valve

#### 4.2. Convergence of the Vector Data Control Points

#### 4.3. Accuracy Requirements of State Estimations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Estimation of the Curve Using Second-Order (Linear) B-Spline Interpolation

**Figure A1.**The estimation of the characteristic curve of the directional control valve by using the ADEKF algorithm with second-order (linear) B-spline interpolation. (

**a**) Three-point B-spline estimation. (

**b**) Four-point B-spline estimation. (

**c**) Five-point B-spline estimation. (

**d**) Six-point B-spline estimation.

## Appendix B. Estimation of the Pressure Flow Coefficient and the Flow Gain

**Figure A2.**Requirements for the accuracy in the system states for parameter estimation. (

**a**) Error in ${k}_{p}$ with $95\%$ CI. (

**b**) Error in ${k}_{0}$ with $95\%$ CI.

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**Figure 2.**Hydraulically actuated four-bar mechanism. The mechanism is actuated by a differential pressure pump. ${\mathbf{C}}_{{\mathbf{v}}_{a}},{\mathbf{C}}_{{\mathbf{v}}_{b}},{\mathbf{C}}_{{\mathbf{v}}_{c}},$ and ${\mathbf{C}}_{{\mathbf{v}}_{d}}$ represent the semi-empiric flow rate coefficients at the a, b, c, and d ports of the 4/3 directional control valve. Grey rectangles indicate the pressure sensors on the control volumes ${V}_{p}$ and ${V}_{1}$.

**Figure 3.**The estimation of the characteristic curves of the directional control valve by using the ADEKF with third-order B-spline interpolation. (

**a**) Three-point B-spline estimation. (

**b**) Four-point B-spline estimation. (

**c**) Five-point B-spline estimation. (

**d**) Six-point B-spline estimation.

**Figure 4.**Data control points between ${\mathrm{c}}_{\mathrm{min}}$ and ${\mathrm{c}}_{\mathrm{max}}$ on the characteristic curve of the directional control valve.

**Figure 5.**Convergence of the control points in the vector ${\mathbf{C}}_{{\mathbf{v}}_{a}}$ in the case of Spline 2. (

**a**) Convergence of ${\mathrm{c}}_{1}$ in the three-point estimation process. (

**b**) Convergence of ${\mathrm{c}}_{1}$ and ${\mathrm{c}}_{2}$ in the four-point estimation process. (

**c**) Convergence of ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$, and ${\mathrm{c}}_{3}$ in the five-point estimation process. (

**d**) Convergence of ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$, ${\mathrm{c}}_{3}$, and ${\mathrm{c}}_{4}$ in the six-point estimation process.

**Figure 6.**Requirements for the accuracy in the system states for parameter estimation. (

**a**) Error in s with $95\%$ CI. (

**b**) Error in $\dot{s}$ with $95\%$ CI. (

**c**) Error in ${p}_{p}$ with $95\%$ CI. (

**d**) Error in ${p}_{1}$ with $95\%$ CI. (

**e**) Error in ${p}_{2}$ with $95\%$ CI. (

**f**) Error in parameter ${\mathbf{C}}_{\mathbf{v}}$ with $95\%$ CI.

Parameter | Symbol | Value |
---|---|---|

Pump flow rate | ${Q}_{p}$ | 0.001 ${\mathrm{m}}^{3}/\mathrm{s}$ |

Tank pressure | ${p}_{T}$ | 0.1 MPa |

Volume of the hose p | ${V}_{p}$ | 3.42 $\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ |

Volume of the hose 1 | ${V}_{{h}_{1}}$ | 3.42 $\times {10}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ |

Volume of the hose 2 | ${V}_{{h}_{2}}$ | 3.42 $\times {10}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ |

Oil density | $\rho $ | 869 $\mathrm{k}\mathrm{g}$/$\mathrm{m}$${}^{3}$ |

Hydraulic parameter | ${k}_{p}$ | 1600 MPa |

Hydraulic parameter | ${k}_{0}$ | 0.5 |

Area of the piston | ${A}_{1}$ | 2 $\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ |

Area of the piston-rod | ${A}_{2}$ | 1.8 $\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ |

Length of the cylinder/piston | l | $\sqrt{3}$$\mathrm{m}$ |

Area of pressure relief valve | ${A}_{r}$ | 2.24 $\times {10}^{-12}{\mathrm{m}}^{2}$ |

Area of directional control valve | ${A}_{d}$ | 1.96 $\times {10}^{-6}{\mathrm{m}}^{2}$ |

Coulomb friction force | ${F}_{c}$ | 210 N |

Static friction force | ${F}_{s}$ | 830 N |

Stribeck velocity | ${v}_{s}$ | 1.25 $\times {10}^{-2}$ $\mathrm{m}$/$\mathrm{s}$ |

Coefficient of viscous friction | $\sigma $ | 330 $\mathrm{N}$$\mathrm{s}$/$\mathrm{m}$ |

Discharge coefficient | ${C}_{d}$ | 0.5 |

Area of throttle | ${A}_{R}$ | 2.24 $\times {10}^{-12}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ |

**Table 2.**Properties of the real model, the estimation model, and the simulation model. Errors in the simulation model and the estimation model are given in comparison to the real model. ${s}_{{1}_{0}},\phantom{\rule{3.33333pt}{0ex}}{p}_{{p}_{0}}$, and ${p}_{{1}_{0}}$ represent the initial actuator position, the initial pump pressure, and the initial pressure on the piston side as the system states. The system parameters ${\mathbf{C}}_{{\mathbf{v}}_{a}}$, ${\mathbf{C}}_{{\mathbf{v}}_{b}}$, ${\mathbf{C}}_{{\mathbf{v}}_{c}}$, ${\mathbf{C}}_{{\mathbf{v}}_{d}}$, ${k}_{0}$, and ${k}_{p}$ represent the semi-empiric flow rate coefficient at the a, b, c, and d ports of the directional control valve, the flow gain, and the pressure flow coefficients, respectively.

Errors | Symbol | Real Model | Estimation Model | Simulation Model |
---|---|---|---|---|

State | ${s}_{{1}_{0}}$ | $\sqrt{3}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $1.62\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $1.62\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |

State | ${p}_{{p}_{0}}$ | 7.6 MPa | 5.6 MPa | 5.6 MPa |

State | ${p}_{{1}_{0}}$ | 1 MPa | 2 MPa | 2 MPa |

Parameter | ${\mathbf{C}}_{{\mathbf{v}}_{a}}$ | Non-linear | Linear | Linear |

Parameter | ${\mathbf{C}}_{{\mathbf{v}}_{b}}$ | Non-linear | Linear | Linear |

Parameter | ${\mathbf{C}}_{{\mathbf{v}}_{c}}$ | Non-linear | Linear | Linear |

Parameter | ${\mathbf{C}}_{{\mathbf{v}}_{d}}$ | Non-linear | Linear | Linear |

Parameter | ${k}_{0}$ | 0.5 | 0.4 | 0.4 |

Parameter | ${k}_{p}$ | 1600 MPa | 1500 MPa | 1500 MPa |

**Table 3.**Root mean square error in the estimation of the characteristic curve. The third and fourth columns represent the root mean square errors in Spline 1 and Spline 2, respectively.

Control Points | Control Point Vector ${\mathbf{N}}_{\mathit{a}}$ | RMSE | RMSE |
---|---|---|---|

Three points | $\left[\begin{array}{ccc}0& 5& 10\\ 0& 47.5& 95\end{array}\right]$ | $0.04\%$ | $0.03\%$ |

Four points | $\left[\begin{array}{cccc}0& 3.3& 6.6& 10\\ 0& 31.6& 63.3& 95\end{array}\right]$ | $0.05\%$ | $0.01\%$ |

Five points | $\left[\begin{array}{ccccc}0& 2.5& 5& 7.5& 10\\ 0& 23.7& 47.5& 71.2& 95\end{array}\right]$ | $0.06\%$ | $0.07\%$ |

Six points | $\left[\begin{array}{cccccc}0& 2& 4& 6& 8& 10\\ 0& 19& 38& 57& 76& 95\end{array}\right]$ | $0.07\%$ | $0.08\%$ |

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**MDPI and ACS Style**

Khadim, Q.; Kiani-Oshtorjani, M.; Jaiswal, S.; Matikainen, M.K.; Mikkola, A.
Estimating the Characteristic Curve of a Directional Control Valve in a Combined Multibody and Hydraulic System Using an Augmented Discrete Extended Kalman Filter. *Sensors* **2021**, *21*, 5029.
https://doi.org/10.3390/s21155029

**AMA Style**

Khadim Q, Kiani-Oshtorjani M, Jaiswal S, Matikainen MK, Mikkola A.
Estimating the Characteristic Curve of a Directional Control Valve in a Combined Multibody and Hydraulic System Using an Augmented Discrete Extended Kalman Filter. *Sensors*. 2021; 21(15):5029.
https://doi.org/10.3390/s21155029

**Chicago/Turabian Style**

Khadim, Qasim, Mehran Kiani-Oshtorjani, Suraj Jaiswal, Marko K. Matikainen, and Aki Mikkola.
2021. "Estimating the Characteristic Curve of a Directional Control Valve in a Combined Multibody and Hydraulic System Using an Augmented Discrete Extended Kalman Filter" *Sensors* 21, no. 15: 5029.
https://doi.org/10.3390/s21155029