Robust Trajectory Tracking for Omnidirectional Mobile Robots with Input Time Delay: An ADRC Approach

Abstract

In this article, the problem of control of the kinematic model of an omnidirectional robot with time delay in the control input is tackled through an Active Disturbance Rejection Control (ADRC) with a disturbance predictor-based scheme, which consists in predicting the generalized forward disturbance input in order to cancel it and then using a feedforward linearization approach to control the system in trajectory tracking tasks. The novelties of the scheme are to demonstrate that using the proposed extended state disturbance estimation leads to a forward estimation following the Taylor series approximation, and, to avoid using additional pose predictions, a feedforward input as an exact linearization approach is used, in which the remaining dynamics can be lumped into the generalized disturbance input. Thus, the use of extended states in prediction improves the robustness of the predictor while increasing the prediction horizon for larger time delays. The stability of the proposal is demonstrated using the second method of Lyapunov, which shows the closed-loop estimation/tracking ultimate bound behavior. Additionally, numerical simulations and experimental tests validate the robustness of the approach in trajectory-tracking tasks.

1. Introduction

Any system, process, or plant can experience delays in information transmission. In most systems, delays can be neglected; however, if the delay is significant, it can cause the system to become unstable, and therefore, the specified task cannot be correctly performed. For instance, if a mobile robot is navigating a warehouse and detects a potential collision due to a communication delay, it is highly likely to collide. In this context, in 1957, Smith developed a control technique that allows systems with constant delays in control inputs to be controlled. The significant disadvantage of this approach is that the open-loop system must be stable, and its performance depends on the accuracy of the no-delay model.

On the other hand, omnidirectional mobile robots have been widely used in industry and academia due to their diverse range of applications and their ability to move in any direction without needing to orient themselves.

In [1], the authors have demonstrated that passivity-based controller techniques are robust against communication delays. In [2], the authors proposed a decentralized output-feedback controller for fully-actuated Euler–Lagrange networks subject to time–varying delays. In the same context, a torque-controlled approach for nonholonomic mobile robots under time-varying communication delays has been developed in [3]. A nonlinear prediction-observer scheme based on a sub-prediction strategy is designed in [4] to address the trajectory-tracking problem for a differential-drive mobile robot subject to a constant time delay. Similarly, in [5], the authors estimate the future value of a particular class of linear systems with time delay at the input and output by employing a full-information predictor-observer.

Works that have tackled communication delays in the omnidirectional mobile robots can be found in [6]. The authors proposed a control framework for multiple vehicles to cooperatively transport an object in a cluster environment, accounting for time-varying communication delays and networks with directed spanning trees. A model predictive control scheme has been designed in [7] to address steering delays in a four-wheel independent-steering robot during high-speed trajectory tracking. In [8], the authors have simulated the remote control of an omnidirectional robot over a wireless network, accounting for time-varying delays. Additionally, in [9], authors have proposed a modification of the Smith Predictor to remove the disturbances originated by a terrain of an unknown slope.

A critical problem with the omnidirectional kinematic model subject to input time delays is that the control gain depends on the orientation variable, requiring a prediction-based approach to the state. However, this prediction can be complex in a disturbed nonlinear model such as the disturbed omnidirectional robot. Besides, the stability analysis of the disturbed control gain may imply several hypotheses that can involve some operation bounds or the possible use of H infinity approaches that, in contrast with Active Disturbance Rejection Control (ADRC) schemes, involve an extensive knowledge of the system and the external disturbances. To reduce the complexity of the analysis while ensuring a simple control scheme, the differential flatness property of the system can be used along with a feedforward linearization approach, which consists of using the feedforward gain term (depending on the desired predicted pose term, which is available if the reference trajectory is predefined) instead of the actual predicted state. For example, in [10], the authors proved that some nonlinear systems with the differential flat property are linearizable by a nominal feedforward strategy when the initial condition is known. As the authors present, this scheme is exact when the distance between the same pose and the desired pose is not too large. Still, possible errors can be compensated for via a robust scheme that can be formulated by robustly compensating the predicted generalized disturbance input (including the feedforward linearization error) using an Extended State Observer (ESO), as in the scheme proposed in [11]. Further information concerning convergence, robustness, and its successful application in mobile robotics can be found in [12,13]. In that sense, in [12], a methodology for analyzing robustness to parametric uncertainty in the design of control for nonlinear flat systems, as a specific combination of a nominal feedforward input and a simple feedback stabilizing control, is studied.

In this work, robust control of the kinematic model of a time-delayed omnidirectional robot is achieved using an ESO-based disturbance prediction scheme based on the Taylor series approximation. This prediction requires knowledge of the constant time delay and a set of finite-time derivatives of the disturbance. Since the control gain involves the vehicle’s rotation matrix, the orientation angle is compensated with its desired term to obtain a feedforward linearization, with the resulting linearization error lumped into the generalized disturbance input. The stability of the equilibrium (tracking and estimation errors) can be ensured in an ultimate bounded form using a quadratic Lyapunov functional. Some experimental tests validate the proposal’s correct behavior using a set of performance metrics.

The article is divided into the following sections: Section 2 includes the mathematical model of the system and the problem formulation. Section 3 presents the robust controller approach and the stability test. Section 4 presents a numerical simulation comparison and experimental results from an actual omnidirectional robot that illustrate the effectiveness of the proposal. Finally, Section 5 states some concluding remarks and offers future insights.

2. System Model and Problem Formulation

Figure 1 shows a top view of the omnidirectional robot configuration. The fixed frame $X-Y$ provides the position of an arbitrary point. It is addressed as the global position coordinate frame. Denote $x,y\in \mathbb{R}$ as the center of mass position coordinates in the axes X and Y respectively, $\psi \in \mathbb{R}$ is the robot orientation angle (which denotes Z axis rotation of the mobile frame concerning the global coordinate frame). The following equations represent the kinematic model of the omnidirectional robot with input delay:

$$\left[\begin{array}{c}\dot{x}\left(t\right)\\ \dot{y}\left(t\right)\\ \dot{\psi }\left(t\right)\end{array}\right]=\mathbf{R}\left(\psi \right)\left[\begin{array}{c}{V}_x(t-\tau )\\ V_y(t-\tau )\\ W(t-\tau )\end{array}\right]+\left[\begin{array}{c}{F}_x\left(t\right)\\ F_y\left(t\right)\\ F_{\psi }\left(t\right)\end{array}\right],$$

(1)

where the pose vector of the robot is defined as $\mathbf{\xi }={\left[\begin{array}{ccc}x& y& \psi \end{array}\right]}^{⊺}\in {\mathbb{R}}^3$, which is formed by the cartesian coordinates and the robot planar orientation coordinate. The system is supposed to be exposed to external disturbances of different nature that are lumped into a generalized disturbance vector represented by $\mathbf{F}\left(t\right)={\left[\begin{array}{ccc}{F}_x\left(t\right)& F_y\left(t\right)& F_{\psi }\left(t\right)\end{array}\right]}^{⊺}\in {\mathbb{R}}^3$. Last vector, though unknown, is assumed to be bounded for all time $t\ge 0$.

Since the orientation of the robot is regarding the Z axis, the rotation matrix of the omnidirectional robot is of the form:

$$\mathbf{R}\left(\psi \right)=\left[\begin{array}{ccc}cos\psi & -sin\psi & 0\\ sin\psi & cos\psi & 0\\ 0& 0& 1\end{array}\right].$$

The time delayed control input assuming a constant time delay $\tau \in \mathbb{R}$, $\mathbf{U}(t-\tau )={\left[\begin{array}{ccc}{u}_1(t-\tau )& u_2(t-\tau )& u_3(t-\tau )\end{array}\right]}^{⊺}={\left[\begin{array}{ccc}{V}_x(t-\tau )& V_y(t-\tau )& W(t-\tau )\end{array}\right]}^{⊺}$, includes the cartesian linear velocity inputs, represented by the variables $V_x,V_y\in {\mathbb{R}}^2$, and the angular velocity (normal to the Z axis), say $W\in \mathbb{R}$. Besides, the angular velocity of the robot wheels vector is given by $\Omega ={\left[\begin{array}{ccc}{\omega }_1& {\omega }_2& {\omega }_3\end{array}\right]}^{⊺}\in {\mathbb{R}}^3$, whose relation with the control input is computed as follows [14]:

$$\left[\begin{array}{c}{V}_x\\ V_y\\ W\end{array}\right]={\displaystyle \frac{r}{2}}\left[\begin{array}{ccc}-{\displaystyle \frac{1}{sin\delta +1}}& -{\displaystyle \frac{1}{sin\delta +1}}& {\displaystyle \frac{2}{sin\delta +1}}\\ {\displaystyle \frac{1}{cos\delta }}& {\displaystyle \frac{1}{cos\delta }}& 0\\ {\displaystyle \frac{1}{Lsin\delta +L}}& {\displaystyle \frac{1}{Lsin\delta +L}}& {\displaystyle \frac{2sin\delta }{Lsin\delta +L}}\end{array}\right]\left[\begin{array}{c}{\omega }_1\\ {\omega }_2\\ {\omega }_3\end{array}\right],$$

where the parameter $L\in \mathbb{R}$ denotes the distance between the robot center of mass and each wheel center. The wheel radius is given by $r\in \mathbb{R}$, $\delta \in \mathbb{R}$ is the angle between each wheel and the normal to the adjacent wheel, here $\delta =\pi /6$.

2.1. Flatness of the System and Feedforward Linearization

The system (1) without external disturbances is differentially flat with flat outputs denoted by the pose vector, since all the system variables can be directly expressed in terms of the flat outputs and their first time derivative. This property allows the use of the feedforward linearization procedure, which is given by the following lemmas [15,16]:

Lemma 1.

The application of the nominal input ${\mathbf{U}}^{*}$ to the system (1) with a consistent set of initial conditions, that is, $\mathbf{\xi }\left(t_0\right)={\mathbf{\xi }}^{*}\left(t_0\right)$, leads to a trajectory $\tilde{\mathbf{\xi }}$ of (1) which exists on a time interval $I=[t_0,t_f)$, $t_f\in \mathbb{R}>0$; this trajectory corresponds to that of a linear system, in Brunovský’s canonical form, for which the input is composed of appropriate time derivatives of the nominal flat output.

Lemma 2.

The application of the nominal input

$${\mathbf{U}}^{*}(t-\tau )={\left[\begin{array}{ccc}{V}_x^{*}(t-\tau )& V_y^{*}(t-\tau )& W^{*}(t-\tau )\end{array}\right]}^{⊺}$$

to the system (1) with a non-consistent initial condition, in other words, $\mathbf{\xi }\left(t_0\right)\ne {\mathbf{\xi }}^{*}\left(t_0\right)$, which is sufficiently close to ${\mathbf{\xi }}^{*}\left(t_0\right)$, leads to a trajectory $\tilde{\mathbf{\xi }}$ of (1), which exists on an interval of finite amplitude $I_1=[t_0,t_1)$ of I. This trajectory remains in the neighborhood of ${\mathbf{\xi }}^{*}\left(t\right)$ on $I_1$, i.e., the trajectory remains in the neighborhood of the nominal trajectory ${\mathbf{\xi }}^{*}\left(t\right)$ on the interval $I_1$.

The last Lemma provides an alternative procedure for computing the predicted state-dependent control gain of the system using the nominal state, which is available at all times, while compensating for feedforward linearization errors via the ESO.

2.2. Problem Formulation

Consider the disturbed omnidirectional robot model with a simplified controlled representation, as shown in (1). It is desired to move the generalized coordinates position according to a reference trajectory ${\mathbf{\xi }}^{*}\left(t\right)$ via a suitable robust control $\mathbf{U}(t-\tau )$ despite the non-modeled dynamics, external disturbances, and the constant time delay in the control input, respectively.

3. A Disturbance Predictor-Based ADRC Approach

From (1), a forward system can be defined a forward system denoted with the sub-index f as follows:

$$\begin{array}{cc}\hfill {\dot{\mathbf{\xi }}}_f\left(t\right)& =\mathbf{R}\left({\psi }_f\right)\mathbf{U}\left(t\right)+\mathbf{F}(t+\tau ),\hfill \end{array}$$

(2)

where $\mathbf{F}(t+\tau )$ is denoted as the predicted generalized disturbance input to be estimated through the estimated states of the original system and the estimated extended states associated with the actual generalized disturbance input. Equation (2) can be rewritten as

$$\begin{array}{cc}\hfill {\dot{\mathbf{\xi }}}_f\left(t\right)& =\left(\mathbf{R}\left({\psi }_f\right)\pm \mathbf{R}\left(\psi \right)\right)\mathbf{U}\left(t\right)+\mathbf{F}(t+\tau ),\hfill \\ \hfill & =\mathbf{R}\left(\psi \right)\mathbf{U}\left(t\right)+(\mathbf{R}\left({\psi }_f\right)-\mathbf{R}\left(\psi \right))\mathbf{U}\left(t\right)+\mathbf{F}(t+\tau ),\hfill \\ \hfill & =\left(\mathbf{R}\right(\psi )+\Delta R)\mathbf{U}\left(t\right)+\mathbf{F}(t+\tau ),\hfill \end{array}$$

(3)

with $\Delta R=\mathbf{R}\left({\psi }_f\right)-\mathbf{R}\left(\psi \right)$. In this case, the Taylor series expansion of the disturbance input [11] is used to propose an ESO for forward disturbance estimation. The main idea is to develop an extended-state Luenberger-like observer that constructs a forward disturbance estimate via generalized disturbance estimation and a set of finite-order disturbance time-derivative estimates. The following procedure involves a stability test of the complete observer-based predictor control scheme.

Extended State Approximation

The generalized disturbance input $\mathbf{F}\left(t\right)$ evaluated on the system trajectories can be approximated by the following polynomial form:

$$\begin{array}{cccc}\hfill \mathbf{F}\left(t\right)& =a^{⊺}\kappa \left(t\right)+\tilde{F},\hfill & \hfill \hspace{1em}\hspace{1em}\kappa \left(t\right)& ={\left[\begin{array}{ccccc}1& t& t^2& \dots & t^p\end{array}\right]}^{⊺},\hfill \end{array}$$

with $a\in {\mathbb{R}}^{(p+1)\times 3}$, $\tilde{F}$ denotes the approximation error. Last polynomial approximation leads to the following ultra-local model of the disturbance estimation [17,18]:

$$\begin{array}{cc}\hfill a^{⊺}\kappa \left(t\right)& ={\rho }_0\left(t\right),\hfill \\ \hfill {\dot{\rho }}_i\left(t\right)& ={\rho }_{i+1}\left(t\right),i=0,1,\dots ,p-1,\hfill \\ \hfill {\dot{\rho }}_p& =0,\hfill \\ \hfill {\rho }_j\left(0\right)& =a_j,a_j\in {\mathbb{R}}^{3\times 1}j=0,1,\dots ,p.\hfill \end{array}$$

The extended state disturbance approximation dynamics can be written in the following form:

$$\begin{array}{cc}\hfill \rho \left(t\right)& :={\left[\begin{array}{cccc}{\rho }_0& {\rho }_1& \cdots & {\rho }_p\end{array}\right]}^{⊺},\hfill \\ \hfill \dot{\rho }\left(t\right)& =\Phi \rho \left(t\right),\Phi \in {\mathbb{R}}^{3(p+1)\times 3(p+1)},\hfill \\ \hfill \Phi & =\left[\begin{array}{ccccc}{0}_{3\times 3}& I_{3\times 3}& 0_{3\times 3}& \cdots & 0_{3\times 3}\\ 0_{3\times 3}& 0_{3\times 3}& I_{3\times 3}& \cdots & 0_{3\times 3}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0_{3\times 3}& 0_{3\times 3}& 0_{3\times 3}& \cdots & I_{3\times 3}\\ 0_{3\times 3}& 0_{3\times 3}& 0_{3\times 3}& \cdots & 0_{3\times 3}\end{array}\right],\hfill \\ \hfill {\rho }_0\left(t\right)& =\left[\begin{array}{cc}{I}_{3\times 3}& 0_{3\times 3p}\end{array}\right]\rho \left(t\right):={\Phi }_1\rho \left(t\right),\hfill \end{array}$$

where $I_{3\times 3}$ is the $3\times 3$ identity matrix and $0_{3\times 3}$ is the $3\times 3$ zero matrix. The approximation error $\tilde{F}$ is assumed to satisfy the following condition:

$$\begin{array}{c}\hfill \parallel \tilde{F}\left(t\right)\parallel \le {\xi }_{max}\in {\mathbb{R}}^{+}<\infty ,\forall t\ge 0.\end{array}$$

(4)

Thus, the system with the extended state approximation can be defined as follows:

$$\begin{array}{cc}\hfill \dot{\mathbf{\xi }}\left(t\right)& =(\mathbf{R}\left(\psi \left(t\right)\right)+\Delta R)\mathbf{U}(t-\tau )+{\Phi }_1\rho \left(t\right)+\tilde{F}\left(t\right),\hfill \\ \hfill \dot{\rho }\left(t\right)& =\Phi \rho \left(t\right).\hfill \end{array}$$

(5)

Let ${\mathbf{\xi }}^{*}={\left[\begin{array}{ccc}{x}^{*}& y^{*}& {\psi }^{*}\end{array}\right]}^{⊺}\in {\mathbb{R}}^3$ be the flat outputs reference trajectory vector, which is assumed to be continuous and smooth with, at least, one well defined nested time derivative (which allows to define ${\dot{\mathbf{\xi }}}^{*}\left(t\right)$). The feedforward input ${\mathbf{U}}^{*}(t-\tau )$ is defined such that the following relation is obtained (under suitable initial conditions and free of disturbances):

$$\begin{array}{cc}\hfill {\dot{\mathbf{\xi }}}^{*}\left(t\right)& =\mathbf{R}\left({\psi }^{*}\left(t\right)\right){\mathbf{U}}^{*}(t-\tau ).\hfill \end{array}$$

Since the reference trajectory ${\mathbf{\xi }}^{*}\left(t\right)$ is assumed to be known, defining the following dynamics of the forward system is not an impediment:

$$\begin{array}{cc}\hfill {\dot{\mathbf{\xi }}}^{*}(t+\tau )& =\mathbf{R}\left(\psi (t+\tau )\right){\mathbf{U}}^{*}\left(t\right).\hfill \end{array}$$

The feedforward input for the forward system can be given by

$$\begin{array}{c}\hfill {\mathbf{U}}^{*}\left(t\right)={\mathbf{R}}^{⊺}\left({\psi }^{*}(t+\tau )\right){\dot{\mathbf{\xi }}}^{*}(t+\tau ).\end{array}$$

The disturbance prediction is based on the Taylor series-based approximation of the term $\mathbf{F}(t+\tau )$ through the ESO given as follows [18]:

$$\begin{array}{cc}\hfill \mathbf{F}(t+\tau )& =\sum _{i=0}^N{\rho }_i\left(t\right){\displaystyle \frac{{\tau }^i}{i!}}+{\tilde{F}}_{pred},\hfill \\ \hfill \widehat{\mathbf{F}}(t+\tau )& =\sum _{i=0}^N{\hat{\rho }}_i\left(t\right){\displaystyle \frac{{\tau }^i}{i!}}={\Phi }_2\hat{\rho }\left(t\right),\hfill \\ \hfill {\Phi }_2& =\left[\begin{array}{ccccc}{I}_{3\times 3}& \tau \left(I_{3\times 3}\right)& {\displaystyle \frac{{\tau }^2}{2!}}\left(I_{3\times 3}\right)& \cdots & {\displaystyle \frac{{\tau }^p}{p!}}\left(I_{3\times 3}\right)\end{array}\right],\hfill \end{array}$$

where ${\tilde{F}}_{pred}$ is a bounded signal denoting the prediction error, that is $\parallel {\tilde{F}}_{pred}(·)\parallel <F_{max1}$, with $F_{max1}\in \mathbb{R}$ and $F_{max1}<\infty$, $\forall t\ge 0$, and ${\hat{\rho }}_i\left(t\right)$ is the estimation of ${\rho }_i\left(t\right)$. Then, the predictor-based ADRC control for the forward system is given as follows

$$\begin{array}{cc}\hfill \mathbf{U}\left(t\right)& ={\mathbf{U}}^{*}\left(t\right)-{\mathbf{R}}^{⊺}\left(\psi (t+\tau )\right)\left[K^{⊺}\epsilon (t+\tau )+\widehat{\mathbf{F}}(t+\tau )\right],\hfill \\ \hfill \epsilon \left(t\right)& =\widehat{\mathbf{\xi }}\left(t\right)-{\mathbf{\xi }}^{*}\left(t\right),\hfill \\ \hfill \epsilon (t+\tau )& =e^{I_{3\times 3}\tau }\epsilon \left(t\right),\hfill \end{array}$$

with $K\in {\mathbb{R}}^{3\times 3}$, and $\widehat{\mathbf{\xi }}\left(t\right)$ is the estimation of $\mathbf{\xi }\left(t\right)$. For the delayed input, one has

$$\begin{array}{cc}\hfill \mathbf{U}(t-\tau )& ={\mathbf{U}}^{*}(t-\tau )-{\mathbf{R}}^{⊺}\left(\psi \left(t\right)\right)\left[K^{⊺}\epsilon \left(t\right)+\widehat{\mathbf{F}}\left(t\right)\right],\hfill \\ \hfill \epsilon \left(t\right)& =e^{I_{3\times 3}\tau }\epsilon (t-\tau ),\hfill \end{array}$$

(6)

where the predicted states are computed according to the procedure described in [19]. Notice that the extended state prediction $\hat{\rho }\left(t\right)$ has the structure of the truncated Taylor series prediction approximation in which all the extended states are used to perform the truncated approximation.

Since the delayed input depends on a forward state $\psi \left(t\right)$ which is not available for measurement, the rotation matrix will be substituted by its corresponding feedforward term, by applying a feedforward linearization procedure. That is, the control input is reformulated as follows to deal with the multiplicative non accessible term:

$$\begin{array}{cc}\hfill \mathbf{U}(t-\tau )& ={\mathbf{U}}^{*}(t-\tau )-{\mathbf{R}}^{⊺}\left({\psi }^{*}\left(t\right)\right)\left[K^{⊺}\epsilon \left(t\right)+\widehat{\mathbf{F}}\left(t\right)\right],\hfill \\ \hfill \epsilon \left(t\right)& =\widehat{\mathbf{\xi }}\left(t\right)-{\mathbf{\xi }}^{*}\left(t\right),\hfill \\ \hfill \epsilon (t+\tau )& =e^{I_{3\times 3}\tau }\epsilon \left(t\right).\hfill \end{array}$$

The states and the predicted disturbance input are estimated through the following ESO:

$$\begin{array}{cc}\hfill \dot{\widehat{\mathbf{\xi }}}\left(t\right)& =\mathbf{R}\left(\widehat{\psi }\left(t\right)\right)\mathbf{U}(t-\tau )+{\Phi }_1\hat{\rho }\left(t\right)+L_1(\mathbf{\xi }\left(t\right)-\widehat{\mathbf{\xi }}\left(t\right)),\hfill \\ \hfill \dot{\hat{\rho }}\left(t\right)& =\Phi \hat{\rho }\left(t\right)+L_2(\mathbf{\xi }\left(t\right)-\widehat{\mathbf{\xi }}\left(t\right)),\hfill \end{array}$$

(7)

with $L_1\in {\mathbb{R}}^{3\times 3}$ a diagonal matrix, and $L_2\in {\mathbb{R}}^{3(p+1)\times 3}$ i.e.,

$$L_2=\left[\begin{array}{c}{L}_{2_1}\\ L_{2_2}\\ ⋮\\ L_{2_p}\end{array}\right]$$

such that each $L_{2_i}$ is a $3\times 3$ diagonal matrix. The conditions for the estimation and trajectory tracking convergence are provided in the following proposition:

Proposition 1.

Consider the omnidirectional robot with time delay input (1), with a corresponding disturbed extended state representation (5), whose approximation error satisfies condition (4). For a smooth trajectory reference vector ${\mathbf{\xi }}^{*}\left(t\right)$, an output observer based control (6) and (7) with feedforward exact linearization satisfying that the matrix $A_0$, given by

$$\begin{array}{cc}\hfill A_0& =\left[\begin{array}{ccc}-L_1& {\Phi }_1& 0_{3\times 3}\\ -L_2& \Phi & 0_{3(p+1)\times 3}\\ L_1& 0_{3\times 3(p+1)}& -K^{⊺}\end{array}\right]\hfill \end{array}$$

is Hurwitz, then the state and extended state estimation errors ($\Delta \left(t\right)$, $\delta \left(t\right)$), and the corresponding tracking error for the estimated state ($\epsilon \left(t\right)$) are uniformly ultimately bounded.

Proof. 

The state estimation error is defined as $\Delta \left(t\right):=\mathbf{\xi }-\widehat{\mathbf{\xi }}$, the extended state estimation error is denoted as $\delta \left(t\right):=\rho \left(t\right)-\hat{\rho }\left(t\right)$. Then, the tracking error is obtained as $\mathbf{\xi }\left(t\right)-{\mathbf{\xi }}^{*}\left(t\right)=\Delta \left(t\right)+\epsilon \left(t\right)$. The dynamics of the auxiliary error variables are:

$$\begin{array}{cc}\hfill \dot{\Delta }\left(t\right)& =-L_1\Delta \left(t\right)+{\Phi }_1\delta \left(t\right)+{\tilde{F}}_2\left(t\right),\hfill \\ \hfill \dot{\delta }\left(t\right)& =\Phi \delta \left(t\right)-L_2\Delta \left(t\right),\hfill \\ \hfill \dot{\epsilon }\left(t\right)& =-K^{⊺}\epsilon \left(t\right)+L_1\Delta \left(t\right)+F_R{\mathbf{U}}^{*}\left(t\right)+\tilde{F}-K^{⊺}{e}^{I_{3\times 3}\tau },\hfill \end{array}$$

where $F_R{\mathbf{U}}^{*}\left(t\right)$ is the exact feedforward linearization error, which is assumed to be bounded, and $\tilde{F}$ is the disturbance prediction approximation error, which is also bounded, while the finite order time derivatives of the disturbance estimation remain bounded, and $-K^{⊺}{e}^{I_{3\times 3}\tau }$ is a disturbance term derived from the predictor design.

Let us define the following vector:

$$\begin{array}{cc}\hfill z\left(t\right)& ={\left[\begin{array}{ccc}{\Delta }^{⊺}\left(t\right)& {\delta }^{⊺}\left(t\right)& {\epsilon }^{⊺}\left(t\right)\end{array}\right]}^{⊺}\in {\mathbb{R}}^{9+3p}.\hfill \end{array}$$

The error vector $z\left(t\right)$ satisfies the following dynamics:

$$\begin{array}{cc}\hfill \dot{z}\left(t\right)& =A_{0}z\left(t\right)+{\xi }_0\left(t\right),\hfill \end{array}$$

with:

$$\begin{array}{cc}\hfill {\xi }_0& =\left[\begin{array}{c}{\tilde{F}}_2\\ 0_{3(p+1)\times 1}\\ -K^{⊺}{e}^{I_{3\times 3}\tau }+{\tilde{F}}_R+\tilde{F}\end{array}\right].\hfill \end{array}$$

Since the terms $-K^{⊺}{e}^{I_{3\times 3}\tau }$, and the feedforward linearization error are bounded, let us define the following condition for a suitable bound ${\xi }_{max2}\in {\mathbb{R}}^{+}$ as follows:

$$\begin{array}{c}\hfill \parallel -K^{⊺}{e}^{I_{3\times 3}\tau }\parallel +\parallel {\tilde{F}}_R\parallel \le {\xi }_{max2}.\end{array}$$

(8)

Now, let us consider the following Lyapunov candidate function given by:

$$\begin{array}{cc}\hfill V\left(z\right)& =z^{⊺}Pz,\hfill \end{array}$$

with $P=P^{⊺}$ as a positive definite matrix. The time derivative of $V\left(z\right)$ is given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{dV\left(z\right)}{dt}}& =2z^{⊺}P\dot{z}=2z^{⊺}P{A}_{0}z+2z^{⊺}P{\xi }_0,\hfill \\ \hfill & =z^{⊺}\left(A_0^{⊺}P+P{A}_0\right)z+2z^{⊺}P{\xi }_0.\hfill \end{array}$$

(9)

If matrix $A_0$ is Hurwitz, then for any P positive definite matrix, there exists a matrix $Q=Q^{⊺}>0$ such that $A_0^{⊺}P+PA=-Q$. Then, (9) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{dV\left(z\right)}{dt}}& =-z^{⊺}Qz+2z^{⊺}P{\xi }_0.\hfill \end{array}$$

From (4), (8), $\parallel {\xi }_0\parallel \le \sqrt{{\xi }_{max1}^2+{\xi }_{max2}^2}$. Let ${\xi }_{max}=\sqrt{{\xi }_{max1}^2+{\xi }_{max2}^2}$. Using last assumptions, the time derivative of $V\left(z\right)$ satisfies the following inequality.

$$\begin{array}{cc}\hfill {\displaystyle \frac{dV\left(z\right)}{dt}}& \le -z^{⊺}Qz+2\parallel z\parallel \parallel P\parallel \parallel {\xi }_0\parallel ,\hfill \\ \hfill & \le -z^{⊺}Qz+2{\xi }_{max}\parallel z\parallel \parallel P\parallel .\hfill \end{array}$$

Using the Rayleigh inequality:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dV\left(z\right)}{dt}}& \le -{\lambda }_{min}{\left(Q\right)\parallel z\parallel }^2+2\mu {\lambda }_{max}\left(P\right)\parallel z\parallel ,\hfill \\ \hfill & ={\lambda }_{min}\left(Q\right)\parallel z\parallel \left[-\parallel z\parallel +2{\xi }_{max}{\displaystyle \frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(Q\right)}}\right],\hfill \end{array}$$

(10)

where ${\lambda }_{min}$ and ${\lambda }_{max}$ corresponds to the minimun and maximum eigenvalue, respectively; and $\mu >0$. From (10), the time derivative of the Lyapunov candidate function is negative outside the set $\parallel z\parallel \le 2{\xi }_{max}{\displaystyle \frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(Q\right)}}$. Thus, the vector z is uniformly ultimately bounded with ultimate bound $2{\xi }_{max}{\lambda }_{max}\left(P\right)$, implying that the estimation and tracking errors are ultimately bounded. □

Note that the estimation error $\Delta$ satisfies the following equation

$${\Delta }^{(p+1)}+L_{2_p}{\Delta }^{\left(p\right)}+\dots +L_{2_1}\dot{\Delta }+L_1\Delta ={\displaystyle \frac{d^p\rho }{d{t}^p}}\approx 0,$$

which is matched with a Hurwitz polynomial [20]

$$\begin{array}{c}{I}_3{s}^{p+1}+L_{2_p}{s}^p+\dots +L_{2_1}s+L_1=\left(I_3{s}^2+2I_3\zeta {\omega }_{n}s+I_3{w}_n^2\right){(s+p_d)}^{p-1},\end{array}$$

(11)

where $I_3$ is a $3\times 3$ identity matrix; $\zeta$ is the desired damping factor; ${\omega }_n$ is the desired undamped natural frequency and $p_d$ is a desired non-dominant pole.

Figure 2 illustrates a schematic diagram of the developed approach.

4. Results

In the first part of this section, a comparison is made, through numerical simulations, between the proposed strategy and the one given in [4]. Subsequently, in the second part, the results obtained on an experimental platform are presented.

4.1. Numerical Simulations

The simulations were performed in MATLAB^®-Simulink^® R2023b software. For comparison purposes, the control strategy designed in [4] was adapted, where a non-linear prediction-observer scheme estimates the future values of the state at a constant time delay in the input signal. The observer initial conditions are: $\widehat{\mathbf{\xi }}\left(0\right)={\left[\begin{array}{ccc}0.2& 0.1& 0\end{array}\right]}^{\top }$, $\hat{\rho }\left(0\right)=0_{9\times 1}$, with $0_{9\times 1}$ a $9\times 1$ zero vector; and the initial conditions for the omnidirectional robot are: $\mathbf{\xi }\left(0\right)={\left[\begin{array}{ccc}0.2& 0.1& 0\end{array}\right]}^{\top }$. Considering an observer with $p=3$ states, then, Equation (11) yields

$$I_3{s}^4+L_{2_3}{s}^3+L_{2_2}{s}^2+L_{2_1}s+L_1=\left(I_3{s}^2+2I_3\zeta {\omega }_{n}s+I_3{w}_n^2\right){(s+p_d)}^2.$$

Thus, to set up the observer gains, we choose $\zeta =1.5$, $w_n=5$ rad/s, and $p_d=130$, which yields $L_1=\mathrm{diag}\left\{422500\right\}$ and

$$L_2\left[\begin{array}{ccc}26\times {10}^4& 0& 0\\ 0& 26\times {10}^4& 0\\ 0& 0& 26\times {10}^4\\ 20825& 0& 0\\ 0& 20825& 0\\ 0& 0& 20825\\ 275& 0& 0\\ 0& 275& 0\\ 0& 0& 275\end{array}\right].$$

The desired trajectory is a Lissajous curve given by the following parametric functions:

$${\mathbf{\xi }}^{*}={\left[\begin{array}{ccc}2cos\left({\displaystyle \frac{\pi }{20}}t\right)& sin\left({\displaystyle \frac{\pi }{10}}t\right)& {tan}^{-1}\left({\displaystyle \frac{{\dot{y}}^{*}}{{\dot{x}}^{*}}}\right)\end{array}\right]}^{\top }.$$

Figure 3 illustrates the motion of the omnidirectional robot as it follows the desired trajectory with a delay of $0.5$ s. Note that, using the scheme in [4], the vehicle follows the desired trajectory more effectively compared to the proposed approach. This is because no external disturbances were implemented. In this respect, the scheme we propose addresses both external disturbances and delays.

After increasing the delay to 1 s, Figure 4 shows how the vehicle, using scheme [4], is no longer able to follow the desired trajectory. Furthermore, it is evident that, at least in simulation, the proposed scheme can handle longer delays than in [4].

4.2. Real-Time Experiments

To determine the effectiveness of the proposed approach, real-time experiments with an omnidirectional mobile robot were carried out through MATLAB^®-Simulink^® R2022b software. The omnidirectional robot has reflective markers, which are used by the VICON^® Tracker software v3.10, and a set of 10 infrared cameras, Bonita, with a precision of $0.2$ mm to determine the robot’s position and orientation in a 5 m × 4 m area. The omnidirectional robot was constructed by employing three 12 V POLOLU 37D gear motors. The STM32F4 Discovery board is then used for data acquisition. Communication between the computer and the mobile robot is performed in real time via the publicly available “waijung1504” MATLAB^®-Simulink^® R2022b library over Bluetooth, using an ESP32 microcontroller with a sampling time of $0.005$ s.

Considering an observer with $p=5$ states, then, Equation (11) yields

$$I_3{s}^6+L_{2_5}{s}^5+L_{2_4}{s}^4+L_{2_3}{s}^3+L_{2_2}{s}^2+L_{2_1}s+L_1=\left(I_3{s}^2+2I_3\zeta {\omega }_{n}s+I_3{w}_n^2\right){(s+p_d)}^4.$$

Thus, to set up the observer gains, we choose $\zeta =1$, $w_n=2$ rad/s, and $p_d=20$, which yields $L_1=\mathrm{diag}\left\{640\times {10}^3\right\}$ and

$$L_2\left[\begin{array}{ccc}768\times {10}^3& 0& 0\\ 0& 768\times {10}^3& 0\\ 0& 0& 768\times {10}^3\\ 297600& 0& 0\\ 0& 297600& 0\\ 0& 0& 297600\\ 41920& 0& 0\\ 0& 41920& 0\\ 0& 0& 41920\\ 2724& 0& 0\\ 0& 2724& 0\\ 0& 0& 2724\\ 84& 0& 0\\ 0& 84& 0\\ 0& 0& 84\end{array}\right].$$

The experiments consider different time delays $(\tau =0.1,0.2,0.3)$ and different orders for the observer $(p=3,4,5)$. We assume that a higher-order observer yields a better estimate of the perturbation. Furthermore, different delays were applied to determine the highest delay the system can handle experimentally.

The desired trajectory is a Lissajous curve given by the following parametric functions:

$${\mathbf{\xi }}^{*}={\left[\begin{array}{ccc}0.5cos\left({\displaystyle \frac{2}{25}}\pi t\right)& 0.5sin\left({\displaystyle \frac{4}{25}}\pi t\right)& 0\end{array}\right]}^{\top }.$$

Based on the above, Figure 5 illustrates a comparison between the real trajectory and desired trajectory performed by the mobile robot with a delay of $0.1$ s and an observer with $p=5$ states. In this case, the delay is 20 times the sample time of $0.005$ s. The observer initial conditions are: $\widehat{\mathbf{\xi }}\left(0\right)={\left[\begin{array}{ccc}0.5& 0& 0\end{array}\right]}^{\top }$, $\hat{\rho }\left(0\right)=0_{15\times 1}$, with $0_{15\times 1}$ a $15\times 1$ zero vector; and the initial conditions for the omnidirectional robot are: $\mathbf{\xi }\left(0\right)={\left[\begin{array}{ccc}0.55& 0.061& {\displaystyle \frac{\pi }{2}}\end{array}\right]}^{\top }$.

Figure 6 and Figure 7 show the required velocities of the omnidirectional mobile robot to perform its motion and follow the desired trajectory. One can note that the control inputs are feasible and remain bounded by the system, that is, $V_x,V_y\in [-0.4,0.4]$ m/s and $W\in [-3.5,3.5]$ rad/s.

From Figure 8 and Figure 9, one can realize that the position and orientation errors are bounded and are oscillating around zero.

To visualize the behavior of the system when the delay increases, Figure 10 and Figure 11 present the trajectory in the plane of the omnidirectional robot with $\tau =0.2$ s and $\tau =0.3$ s, respectively. Comparing Figure 5 and Figure 10, one can see that the robot’s performance is quite similar, and the system can handle a delay of $0.2$ s, which is 40 times the sample time. On the other hand, when $\tau =0.3$ s, Figure 11 shows that the desired trajectory has not been followed correctly, and the position errors have increased. Therefore, for this system in particular, the maximum delay it can handle without affecting performance is $0.2$ s.

To quantitatively compare the results obtained with different delays and observer order, the mean square error MSE and the integral of the absolute error IAE were calculated as follows

$$\begin{array}{ccc}\hfill {\nu }^{RMSE}& =& \sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{e}_j^2\left(t\right)},\hfill \\ \hfill {\nu }^{IAE}& =& {\int }_0^{\infty }\left|e_j\left(t\right)\right|dt,\hfill \end{array}$$

with $j=x,y,\psi$.

Table 1. Comparison of error performance index using an observer with 3 states.

Performance Index	Delay
	0.1	0.2	0.3
$e_x^{RMSE}$	0.0083	$0.0042$	$0.0234$
$e_y^{RMSE}$	0.0056	$0.0072$	$0.251$
$e_{\psi }^{RMSE}$	0.0538	$0.014$	$0.1419$
$e_x^{IAE}$	0.0992	$0.07673$	$0.4955$
$e_y^{IAE}$	0.1009	$0.1496$	$0.5286$
$e_{\psi }^{IAE}$	0.7078	$0.2928$	$2.9226$

,

Table 2. Comparison of error performance index using an observer with 4 states.

Performance Index	Delay
	0.1	0.2	0.3
$e_x^{RMSE}$	0.0055	0.0042	0.0519
$e_y^{RMSE}$	0.0068	0.0074	$0.0385$
$e_{\psi }^{RMSE}$	0.1273	0.0142	$0.2077$
$e_x^{IAE}$	0.0942	0.0739	$0.9921$
$e_y^{IAE}$	0.1324	0.1513	$0.7941$
$e_{\psi }^{IAE}$	1.7401	0.3043	$3.7811$

and

Table 3. Comparison of error performance index using an observer with 5 states.

Performance Index	Delay
	0.1	0.2	0.3
$e_x^{RMSE}$	0.0134	0.0045	$0.0292$
$e_y^{RMSE}$	0.0065	0.0076	$0.0325$
$e_{\psi }^{RMSE}$	0.0967	0.0197	$0.2369$
$e_x^{IAE}$	0.132	0.0806	$0.5858$
$e_y^{IAE}$	0.1218	0.1555	$0.6533$
$e_{\psi }^{IAE}$	1.297	0.4293	$4.6995$

shows the results obtained.

Comparing the RMSE value in Table 1, Table 2 and Table 3, one can find out that the best behavior is obtained when the delay is of $0.2$ s and using an observer with $p=3$ states, since the RMSE is less than when using an observer of higher order. For the IAE performance index, the results vary. For the position error $e_x$, the smallest value of IAE is obtained when using an observer with $p=3$ states and a delay of $0.2$ s. On the other hand, for the position error $e_y$, the lower value of IAE is obtained when using an observer with $p=3$ states and a delay of $0.1$ s. Finally, for the orientation error $e_{\psi }$, the lower value of IAE is obtained by employing an observer with $p=3$ states and a delay of $0.1$ s. This could be due to the robot’s different initial conditions. What is clear is that having more states does not necessarily imply better trajectory-tracking performance.

Remark 1.

Although the delay is treated as constant in the controller design, in practice, the actual delay may differ from the nominal value due to estimation errors, computation and communication jitter, among others. Such a delay mismatch introduces prediction errors in the compensated control input, which can be interpreted as an additional disturbance acting on the system. For small delay mismatches, the closed-loop system exhibits robustness. The experimental results, obtained under real-time conditions where minor delay variations are unavoidable, indicate that the proposed control scheme tolerates moderate delay uncertainty. Nevertheless, for large delay uncertainties or time-varying delays, the proposed controller may experience significant performance degradation or even instability. Addressing this issue would require adaptive delay estimation or other control techniques, which are left for future work.

5. Conclusions

This paper focused on designing a control strategy for path tracking in an omnidirectional robot subject to control input delays, using the ADRC approach and a predictor. Using the second Lyapunov method, it was proven that the closed-loop system is ultimately bounded. To support the theoretical framework, real-time experiments were conducted with different observer orders to estimate disturbances and various delays. Specifically, the most considerable delay this system can handle without affecting its performance was $0.2$ s. Considering a larger delay would significantly affect the system’s behavior and, consequently, trajectory tracking.

Since the delay compensation is based on a predictor structure, minor mismatches between the assumed and actual delay result in bounded prediction errors. Consequently, the closed-loop system exhibits robustness to moderate delay uncertainties, provided that the mismatch remains within a tolerable range. Furthermore, it was noted that the controller’s performance is sensitive to the observer’s order and gains, and that performing both tasks (predicting system behavior and canceling disturbances) is difficult.

As future work, studies with variable and bounded time delays will be included, incorporating possible gain adjustments to verify the robustness of the control strategy with respect to these changes.