# Dual-Rate Extended Kalman Filter Based Path-Following Motion Control for an Unmanned Ground Vehicle: Realistic Simulation

^{*}

## Abstract

**:**

^{®}/Simulink) model has been developed for a realistic simulation, considering the contact forces between the wheels and the ground, not included in the kinematic and dynamic UGV representation. Non-linear behavior of the motors and limited resolution of the encoders have also been included in the model for a more accurate simulation of the real vehicle. The simulation model has been experimentally validated from the real process. Simulation results reveal the benefits of the control solution.

## 1. Introduction

- Consideration of DREKF, which enables one to:
- -
- Design a fast-rate dynamic controller capable of reaching the desired specifications for the UGV and precisely following the predefined path.
- -
- Generate fast-rate state estimates from slow-rate measurements to be supplied to the dynamic controller.
- -
- Face non-linear UGV dynamics and possible Gaussian-like modeling and measurement uncertainties.

- Development of a powerful simulation tool, which takes into account complex modeling aspects to realistically represent the UGV behavior.

## 2. Problem Scenario

- Robot simulator, which contains some complex UGV modeling aspects as a result of using the features of the specialized Simscape Multibody simulation tool.
- Control structure, which includes a path tracking controller (in this case, the Pure Pursuit algorithm), an inverse kinematics computation block, a dynamic proportional integral (PI) controller, and a state estimator (in this case, the proposed DREKF).

- At the current instant $kT$, the Pure Pursuit path tracking algorithm [23,24,25] generates velocity references ${({v}^{ref},{w}^{ref})}_{k}^{T}$ from a set of waypoints and the current pose estimation ${(\widehat{X},\widehat{Y},\widehat{\psi})}_{k}^{T}$. The set of waypoints is composed of reference positions $({X}^{ref},{Y}^{ref})$, a velocity constant reference ${V}^{ref}$, and a look ahead distance ${L}^{ref}$.
- The UGV inverse kinematics block transforms the velocity references ${({v}^{ref},{w}^{ref})}_{k}^{T}$ into dynamic references ${({w}_{r}^{ref},{w}_{l}^{ref})}_{k}^{T}$.
- From this dynamic reference ${({w}_{r}^{ref},{w}_{l}^{ref})}_{k}^{T}$ and the estimated angular velocities ${({\hat{w}}_{r},{\widehat{w}}_{l})}_{k}^{T}$, the dynamic PI controller computes the control signal to be applied to the UGV ${({u}_{r},{u}_{l})}_{k}^{T}$, which respectively are the control actions at period T for the right and left motors. These control actions will be applied under Zero Order Hold (ZOH) conditions.
- The UGV is equipped with a virtual beacon. Four fixed beacons are additionally placed on the walls of the simulation environment, emulating a beacon based indoor positioning system. The measurements ${({d}_{1},{d}_{2},{d}_{3},{d}_{4})}_{k}^{NT}$ are the distances between the virtual mobile beacon and the four fixed beacons, which are located in a known place with respect to the world system of the simulator. The simulation tool is able to work from these distances or, alternatively, from the direct pose information $(X,Y,$${\psi )}_{k}^{NT}$. Distances and pose can be equivalently deduced by applying the Pithagorean theorem (see, e.g., [26] and more details in Section 3).
- Any UGV output measurement ${({w}_{r},{w}_{l})}_{k}^{T}$, ${({d}_{1},{d}_{2},{d}_{3},{d}_{4},\psi )}_{k}^{NT}$, or $(X,Y,$${\psi )}_{k}^{NT}$ may be disturbed by Gaussian noises, which will be created from a set of independent seeds in order to generate a reproducible pseudo-random noise. As a consequence, the experiments developed with the simulator will be reproducible under the same conditions.
- The system state estimate ${({\hat{w}}_{r},{\hat{w}}_{l},\widehat{X},\widehat{Y},\widehat{\psi})}_{k}^{T}$ is computed via the DREKF. The prediction step is generated at period T from the control actions ${({u}_{r},{u}_{l})}_{k}^{T}$. The correction step is also obtained at period T, but from data sensed at the two different periods, that is, ${({w}_{r},{w}_{l})}_{k}^{T}$, and ${(X,Y,\psi )}_{k}^{NT}$, or ${({d}_{1},{d}_{2},{d}_{3},{d}_{4},\psi )}_{k}^{NT}$. More details can be found in Section 3.

## 3. Dual-Rate Extended Kalman Filter

#### 3.1. Kinematic and Dynamic UGV Modeling

#### 3.2. DREKF Algorithm

- Prediction of the next state ${\widehat{\xi}}_{k|k-1}^{T}$ and propagation of the covariance ${P}_{k|k-1}^{T}$:$$\begin{array}{cc}\hfill {\widehat{\xi}}_{k|k-1}^{T}& =f\left({\widehat{\xi}}_{k-1|k-1}^{T},{\left({n}_{1}\right)}_{k-1}^{T},{u}_{k-1}^{T}\right)\hfill \\ \hfill {P}_{k|k-1}^{T}& ={A}_{k}^{T}{P}_{k-1|k-1}^{T}{\left[{A}_{k}^{T}\right]}^{\top}+{L}_{k}^{T}{Q}_{k-1}^{T}{\left[{L}_{k}^{T}\right]}^{\top}\hfill \end{array}$$$\mathrm{for}\phantom{\rule{4pt}{0ex}}k\in {\mathbb{N}}_{\ge 1}$, where ${\widehat{\xi}}_{0}^{T}=E\left[{\xi}_{0}^{T}\right]$, $E[\xb7]$ being the expectation, and ${P}_{0}^{T}=E[\left({\xi}_{0}^{T}-E\left[{\xi}_{0}^{T}\right]\right)$${\left[\left({\xi}_{0}^{T}-E\left[{\xi}_{0}^{T}\right]\right)\right]}^{\top}]$, and where ${A}_{k}^{T}$ and ${L}_{k}^{T}$ are Jacobian matrices computed in order to respectively linearize the process model about the current state and about the process noise:$$\begin{array}{cc}\hfill {A}_{k}^{T}& ={\left.\frac{\partial f}{\partial \xi}\right|}_{{\widehat{\xi}}_{k-1|k-1}^{T},{\left({n}_{1}\right)}_{k-1}^{T},{u}_{k-1}^{T}}\hfill \\ \hfill {L}_{k}^{T}& ={\left.\frac{\partial f}{\partial {n}_{1}}\right|}_{{\widehat{\xi}}_{k-1|k-1}^{T},{\left({n}_{1}\right)}_{k-1}^{T},{u}_{k-1}^{T}}\hfill \end{array}$$
- Prediction of the future output ${\widehat{z}}_{k}^{T}$, being ${\widehat{z}}_{k}^{T}={\left[{\left({\widehat{\omega}}_{r},{\widehat{\omega}}_{l}\right)}_{k}^{T}\right]}^{\top}$ for $k\ne NT$, and ${\widehat{z}}_{k}^{T}={\left[{({\widehat{\omega}}_{r},{\widehat{\omega}}_{l},\widehat{X},\widehat{Y},\widehat{\psi})}_{k}^{T}\right]}^{\top}$ for $k=NT$:$${\widehat{z}}_{k}^{T}=h\left({\widehat{\xi}}_{k|k-1}^{T},{\left({n}_{2}\right)}_{k}^{T}\right)$$
- Computation of the Kalman filter gain ${K}_{k}^{T}$:$${K}_{k}^{T}={P}_{k|k-1}^{T}{\left[{H}_{k}^{T}\right]}^{\top}{\left({H}_{k}^{T}{P}_{k|k-1}^{T}{\left[{H}_{k}^{T}\right]}^{\top}+{M}_{k}^{T}{R}_{k}^{T}{\left[{M}_{k}^{T}\right]}^{\top}\right)}^{-1}$$$$\begin{array}{cc}\hfill {H}_{k}^{T}& ={\left.\frac{\partial h}{\partial \xi}\right|}_{{\widehat{\xi}}_{k|k-1}^{T},{\left({n}_{2}\right)}_{k}^{T}}\hfill \\ \hfill {M}_{k}^{T}& ={\left.\frac{\partial h}{\partial {n}_{2}}\right|}_{{\widehat{\xi}}_{k|k-1}^{T},{\left({n}_{2}\right)}_{k}^{T}}\hfill \end{array}$$
- Correction of the state ${\widehat{\xi}}_{k|k}^{T}$ and correction of the covariance ${P}_{k|k}^{T}$:$$\begin{array}{cc}\hfill {\widehat{\xi}}_{k|k}^{T}& ={\widehat{\xi}}_{k|k-1}^{T}+{K}_{k}^{T}({z}_{k}^{T}-{\widehat{z}}_{k}^{T})\hfill \\ \hfill {P}_{k|k}^{T}& ={K}_{k}^{T}{R}_{k}^{T}{\left[{K}_{k}^{T}\right]}^{\top}+(I-{K}_{k}^{T}{H}_{k}^{T}){P}_{k|k-1}^{T}{\left[(I-{K}_{k}^{T}{H}_{k}^{T})\right]}^{\top}\hfill \end{array}$$

#### 3.3. Beacon Measurement Model

## 4. UGV Modeling. Simulation Tool

#### 4.1. Preliminary Considerations

- Vision-based system: Using a zenithal camera the position and orientation of the vehicle can be measured. The camera sees what the vehicle is doing from above and can provide information about the time evolution of the $(X,Y)$ coordinates and angular position $\psi $. The main drawback is that the vehicle must lie under the camera, which is a great limitation for the desired trajectory.
- GPS: This is probably the best way to measure the real trajectory to be compared with the desired one. The main drawback is that it must be used outdoors, and the resolution is not suitable to be used with small vehicles as the one is being used in this work.
- Beacons: Suitable to be used indoors it will probably be the best solution. As mentioned in previous sections, position $(X,Y)$ can be measured based on the distance ${d}_{i},(i=1\dots n)$ from the vehicle to several (n) fixed beacons. Orientation $\psi $ can be measured based on information provided by an inertial measurement unit. The drawback is the measurement noise and the lack of precision when using small vehicles.

#### 4.2. Modeling Aspects. Tool Description

^{®}Mindstorms

^{®}EV3. This model has been developed in two stages. The first one involves modeling the Lego DC motor, used in the real vehicle without considering the friction of the wheels with the ground (Section 4.2.1). Simscape Electrical library could have been used to build this model. However, as the behavior of the motor is quite linear, it can be fairly well modeled with some conventional Simulink blocks to include some minor non-linearities (saturation, dead zone, quantization). The output of this electrical part is the torque applied by the motor to the wheel. This torque has been applied to a Simscape Multibody revolute joint to generate the angular velocity of the wheel. This simulated velocity can be compared with the real one, measured by the encoder in the real vehicle. In the second stage, mechanical aspects like forces and torques generated by the wheels against the ground have been modeled using Simscape Multibody library (Section 4.2.2). By means of Simulink Sensitivity Analysis Tool, the value of every parameter included in the simulation model has been tuned (Section 4.3).

#### 4.2.1. Wheels-on-the-Air Simulation Model

#### 4.2.2. Wheels-on-the-Ground Simulation Model

#### 4.3. Parameter Adjustments

- -
- The geometric parameters of the UGV in (1), that is, the wheel radius ${r}_{r}={r}_{l}=0.028$ m and the half track width between wheels $b=0.068$ m.
- -
- The parameters used for the pure pursuit algorithm: velocity constant reference ${V}^{ref}=0.1$ m/s, and look ahead distance ${L}^{ref}=0.2$ m.
- -
- The Gaussian noises are generated by considering zero mean $\mu =0$ and variance ${\sigma}^{2}={10}^{-4}$.
- -
- The solver chosen is Ode4 Runge–Kutta, running at a fixed step of $0.1$ ms.

## 5. Simulation

#### 5.1. Cases Evaluated

- Direct pose and angular velocity: in this experiment, output measurements $({w}_{r},{w}_{l},$${X,Y,\psi )}_{k}^{T}$ are directly sampled from the simulator block at different periods T = 0.1 s, T = 0.2 s, and T = 0.5 s. No noise is considered.
- Odometry: in this case, only angular velocities ${({w}_{r},{w}_{l})}_{k}^{T}$ are sampled at T = 0.1 s. Pose data ${(X,Y,\psi )}_{k}^{T}$ are estimated by odometry. Ideally, no noise may be considered. However, in real environments, where for instance an encoder may be needed to take the velocities, some measurement noise may appear. In this simulation, two cases are considered: without noise and with noise (assuming additive Gaussian noises and limited encoder resolution).
- Dual-rate Extended Kalman Filter: this is the case exposed in Section 2. Gaussian noises are assumed in pose ${(X,Y,\psi )}_{k}^{NT}$ and velocities ${({w}_{r},{w}_{l})}_{k}^{T}$. Two options are simulated: $N=10$ and $N=50$.
- Dual-rate Extended Kalman Filter with beacon distances: this is the case stated in Section 3.2. Gaussian noises are assumed in pose $({d}_{1},{d}_{2},{d}_{3},$${d}_{4}{)}_{k}^{NT}$ and velocities ${({w}_{r},{w}_{l})}_{k}^{T}$. For the sake of clarity, only the option for $N=10$ will be presented.

#### 5.2. Cost Indexes for Performance Assessment

- ${J}_{1}$, which is based on the ${\ell}_{2}$-norm, and its goal is to provide a measure (in meters) about how accurately the path is followed:$${J}_{1}=\frac{1}{l}\sum _{k=1}^{l}\underset{1\le {k}^{\prime}\le l}{min}\sqrt{{\left({X}_{k}^{T}-{\left({X}_{ref}\right)}_{{k}^{\prime}}^{T}\right)}^{2}+{\left({Y}_{k}^{T}-{\left({Y}_{ref}\right)}_{{k}^{\prime}}^{T}\right)}^{2}}$$
- ${J}_{2}$, which is based on the ${\ell}_{\infty}$-norm and is defined to know the maximum difference (in meters) between the desired path and the current UGV position:$${J}_{2}=\underset{1\le k\le l}{max}\left\{\underset{1\le {k}^{\prime}\le l}{min}\sqrt{{\left({X}_{k}^{T}-{\left({X}_{ref}\right)}_{{k}^{\prime}}^{T}\right)}^{2}+{\left({Y}_{k}^{T}-{\left({Y}_{ref}\right)}_{{k}^{\prime}}^{T}\right)}^{2}}\right\}$$
- ${J}_{3}$, which measures the total amount of time (in seconds) elapsed to arrive at the final destination:$${J}_{3}=lT$$

## 6. Results

- The direct pose case at $T=0.2$ s (D.P. 0.2) worsens its behavior with respect to the nominal case, since the trajectory presents oscillations, which is confirmed by the clear increase of every cost index. On average, ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ increase their values $333\%$, $118\%$, and $45\%$, respectively.
- The direct pose case at $T=0.5$ s (D.P. 0.5) presents an unstable response, not being able to track the path in any case.
- The odometry case without noise at $T=0.1$ s (Odom) does not show so many oscillations like the D.P. 0.2 case, but the tracking seems to be not so accurate as in the nominal (D.P. 0.1) case. This analysis is corroborated by the cost indexes, which show a worsening with respect to the nominal case: ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ are respectively increased $90\%$, $59\%$, and $2\%$, which are lower increases than in the D.P 0.2 case. The worsening is due to the addition of a systematic numerical error over time that typically appears when odometry is used.
- The odometry case with noise at $T=0.1$ s (Odom N) depicts a considerable path tracking worsening for the square reference, and is incapable of following the Lissajous curve. The cost indexes confirm this statement, since they are highly increased or ∞, respectively.
- The DREKF case with $N=10$ (DREKF 10) shows an accurate path tracking, despite having scarce pose measurements (10 times less), and it assumes noise and non-linearities. The cost indexes indicate the achievement of satisfactory control properties, since ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$ are slightly worsened with respect to the nominal case ($32\%$, $23\%$, and $12\%$, respectively), and even ${J}_{1}$ and ${J}_{2}$ outperform the Odom case.
- The DREKF case with $N=50$ (DREKF 50) presents a worse response than DREKF 10, as expected (DREKF 50 is provided with 5 times less measurements). The cost indexes ${J}_{1}$ and ${J}_{2}$ with respect to the DREKF 10 case are increased $207\%$ and $179\%$, respectively; ${J}_{3}$ is very similar. Despite having five times fewer measurements, if DREKF 50 is compared with D.P. 0.2, both cases reach similar cost indexes. The main difference between them obeys the way of path tracking: whereas the D.P. 0.2 case presents oscillations, the DREKF 50 case depicts a smooth trajectory with underdamped response.
- The DREKF case with $N=10$ using beacon distances (DREKF-D) depicts a quite close response to the DREKF 10 case, which is confirmed by achieving very similar cost index values.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CAD | Computer Aided Design |

DC | Direct Current |

DREKF | Dual-Rate Extended Kalman Filter |

PI | Proportional Integral |

UGV | Unmanned Ground Vehicle |

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**Figure 3.**DC motor response: real vs. simulated. (

**a**) Input–output characteristic response. (

**b**) Open–loop step response.

**Figure 9.**Sensitivity Analysis Tool. (

**a**) Uniform random distribution generation. (

**b**) Results of the experiment.

**Figure 13.**Results for a Lissajous reference. (

**a**) D.P and Odom options. (

**b**) Odom N and DREKF options.

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

Torque constant (${K}_{\tau}$) | 0.00374 Nm |

Damping coefficient (B) | 5.3473 $\times {10}^{-4}$ $\frac{\mathrm{Nms}}{deg}$ |

Moment of inertia (J) | 1.38083 $\times {10}^{-5}$ kgm${}^{2}$ |

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

Static friction coeff. rubber-ground | 0.9285 |

Dynamic friction coeff. rubber-ground | 0.5059 |

Static friction coeff. steel-ground | 0.6619 |

Dynamic friction coeff. steel-ground | 0.4349 |

Reference | Square | Lissajous | ||||
---|---|---|---|---|---|---|

Experiment | ${\mathit{J}}_{1}$ | ${\mathit{J}}_{2}$ | ${\mathit{J}}_{3}$ | ${\mathit{J}}_{1}$ | ${\mathit{J}}_{2}$ | ${\mathit{J}}_{3}$ |

Direct pose $T=0.1s$ (D.P. 0.1) | 0.01251 | 0.04947 | 40.2 | 0.01259 | 0.04513 | 92.9 |

Direct pose $T=0.2s$ (D.P. 0.2) | 0.03417 | 0.09590 | 57.0 | 0.03688 | 0.10901 | 138.2 |

Direct pose $T=0.5s$ (D.P. 0.5) | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ |

Odometry w/o noise (Odom) | 0.02695 | 0.07053 | 41.1 | 0.02077 | 0.07999 | 94.3 |

Odometry w noise (Odom N) | 0.10844 | 0.36137 | 43.7 | ∞ | ∞ | ∞ |

DREKF $N=10$ (DREKF 10) | 0.01759 | 0.06215 | 45.1 | 0.01551 | 0.05451 | 105.0 |

DREKF $N=50$ (DREKF 50) | 0.03252 | 0.09978 | 46.2 | 0.03495 | 0.11363 | 106.6 |

DREKF $N=10$ beacon (DREKF-D) | 0.01739 | 0.07056 | 44.8 | 0.01577 | 0.06306 | 104.9 |

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Carbonell, R.; Cuenca, Á.; Casanova, V.; Pizá, R.; Salt Llobregat, J.J.
Dual-Rate Extended Kalman Filter Based Path-Following Motion Control for an Unmanned Ground Vehicle: Realistic Simulation. *Sensors* **2021**, *21*, 7557.
https://doi.org/10.3390/s21227557

**AMA Style**

Carbonell R, Cuenca Á, Casanova V, Pizá R, Salt Llobregat JJ.
Dual-Rate Extended Kalman Filter Based Path-Following Motion Control for an Unmanned Ground Vehicle: Realistic Simulation. *Sensors*. 2021; 21(22):7557.
https://doi.org/10.3390/s21227557

**Chicago/Turabian Style**

Carbonell, Rafael, Ángel Cuenca, Vicente Casanova, Ricardo Pizá, and Julián J. Salt Llobregat.
2021. "Dual-Rate Extended Kalman Filter Based Path-Following Motion Control for an Unmanned Ground Vehicle: Realistic Simulation" *Sensors* 21, no. 22: 7557.
https://doi.org/10.3390/s21227557