# Tracking Control for Wheeled Mobile Robot Based on Delayed Sensor Measurements

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- the observer is developed for trajectory tracking of a WMR in the case of delayed measurements where the delay is constant and known;
- the stability of the proposed approach is treated formally;
- the proposal of a coordinate transformation for the orientation error which results in an increased feasibility region of the LMI problem and better tracking of the reference trajectory;
- validation of the approach on the real platform of a MIABOT mobile robot.

## 2. Materials and Methods

#### 2.1. Kinematic Model of a Wheeled Mobile Robot

- Calculating the control for a kinematic model (speed control) is in general simpler than for a dynamic model (toque control).
- There are no complex geometric or inertial parameters to be identified for a kinematic model.
- Finally, very often (e.g., in the case of miniature mobile robots used in our application), the inertia parameters of the robot are relatively low, while the dynamics of the actuators and the power stage are very fast.

#### 2.1.1. Kinematic Model

#### 2.1.2. Kinematic Error-Model of Trajectory Tracking

#### 2.2. Parallel Distributed Compensation Control of a WMR

#### 2.2.1. Control Problem Statement

**The control problem**: We have to design the control ${u}_{B}^{\prime}$ based on the full state ${e}^{\prime}$ for the modified system (8) taking into account the limitation on the control action $\Vert {u}_{B}^{\prime}\Vert <\sigma $ ($\sigma $ is a constant chosen according to the WMR control capabilities) and assuming $\Vert {e}^{\prime}\left(0\right)\Vert <\varsigma $ while ${k}_{t}$ is a free design parameter to be chosen during the control design.

**Implementation of the control**: After obtaining the tracking error $e$ in the original coordinates, the last component of the vector is multiplied by ${k}_{t}$ to get the tracking error ${e}^{\prime}$ used in the control law. In the control action ${u}_{B}^{\prime}$ the last component is divided by ${k}_{t}$ before applying the control to the WMR.

#### 2.2.2. TS Fuzzy Model of a WMR

#### 2.2.3. PDC Control of a WMR

#### 2.3. Nonlinear Predictor Observer

_{1}, L

_{2}, L

_{3}are constant gains, and τ is the value of the measurement delay assumed to be constant. The observer (20) is asymptotically stable if the gains L

_{i}are such that ${L}_{i}\tau <\pi /2\text{\hspace{0.17em}}\mathrm{for}\text{\hspace{0.17em}}i\in \left\{1,2,3\right\}$. The control law will now use estimated error instead of the actual one:

#### 2.4. TS Fuzzy Predictor Observer

- I is the identity matrix;
- $\varpi \left(t\right)=\left({A}_{z}-{A}_{\widehat{z}}\right)e\left(t\right)+\left({B}_{z}-{B}_{\widehat{z}}\right){u}_{B}\left(t\right)$ acts as a disturbance.

_{2}-stability of the estimated error dynamics (27) is proven. The effect of the disturbance term $\varpi $ is attenuated in a similar manner as proposed in [34] for the case of non-delayed measurements.

**Theorem**

**1.**

- ${X}_{11}={A}_{i}^{T}{P}_{2}+{P}_{2}{A}_{i}+{M}_{i}^{T}+{M}_{i}+{I}_{n}$;
- ${X}_{12}={A}_{i}^{T}{P}_{2}+{M}_{i}^{T}+{P}_{1}-{P}_{2}$;
- ${X}_{22}=-{P}_{2}-{P}_{2}^{T}+{I}_{n}+\tau R$.

_{2}-norm of the extended estimated error is upper bounded:

**Proof of Theorem**

**1.**

_{1}, P

_{2}, P

_{3}are $3\times 3$ symmetric positive definite matrices.

- ${I}_{n}$ is an $n\times n$ identity matrix;
- R is $3\times 3$ symmetric positive definite matrix.

_{3}equal to P

_{2}and to introduce ${M}_{i}={P}_{2}{\kappa}_{i}$ (i = 1, …, r). In this case, the inequality (42) can be written in the LMI form represented in (28). □

_{2}-stability of the observer is retained and the errors in the system are bounded in their L

_{2}-norm. To show the efficiency of the proposed observer some experimental results are given which is the aim of the next section. The block diagram of the whole closed loop control system using PDC control and TS fuzzy predictor observer (25) is shown in Figure 4.

## 3. Results

_{i}= x

_{f}= 0.4 m; y

_{i}= y

_{f}= 0.7 m; θ

_{i}= −90°.

#### 3.1. The Use of Original Tracking Error $e$ in the Control Law

^{2}= 5, the gains of the TS fuzzy predictor observer (25), ${\kappa}_{i}=\left(i=1,\dots ,16\right)$ (see Section 2.4) are as follows:

#### 3.2. The Use of a Modified Tracking Error ${e}^{\prime}$ in the Control Law

## 4. Discussion

_{2}-norm of the observer error is bounded. The proposed approach with the PDC control using the TS fuzzy predictor observer is compared to a PDC control with nonlinear predictor observer. Experimental results (only a few are shown in the paper) show that the TS fuzzy predictor observer can cope with delayed noisy measurements much better than a nonlinear observer presented in [24].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Differentially driven mobile robot with pose and control variables [26].

**Figure 2.**Posture error [28].

**Figure 7.**Comparison of the robot trajectories in the case of the nonlinear predictor observer (20).

**Figure 12.**Comparison of a measured, estimated and reference trajectories in the case of the nonlinear predictor observer.

**Figure 13.**Comparison of a measured, estimated and reference trajectories in the case of the TS fuzzy predictor observer.

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**MDPI and ACS Style**

Guechi, E.-H.; Belharet, K.; Blažič, S.
Tracking Control for Wheeled Mobile Robot Based on Delayed Sensor Measurements. *Sensors* **2019**, *19*, 5177.
https://doi.org/10.3390/s19235177

**AMA Style**

Guechi E-H, Belharet K, Blažič S.
Tracking Control for Wheeled Mobile Robot Based on Delayed Sensor Measurements. *Sensors*. 2019; 19(23):5177.
https://doi.org/10.3390/s19235177

**Chicago/Turabian Style**

Guechi, El-Hadi, Karim Belharet, and Sašo Blažič.
2019. "Tracking Control for Wheeled Mobile Robot Based on Delayed Sensor Measurements" *Sensors* 19, no. 23: 5177.
https://doi.org/10.3390/s19235177