Keywords

1 Introduction

Robots are replacing humans in various fields in order to achieve accuracy, efficiency, and speed as part of modernization. The terrestrial robots are classified as wheeled, stationary, and legged robots. Wheeled mobile robots (WMR) are less complex, cheaper, and have less balance issues as compared with legged robots. Motion control algorithms are required to track the trajectory planned by the guidance systems. Highly robust control algorithms ensure a safe navigation in restricted environments.

In literature, a number of control algorithms have been proposed. In [1], an output feedback controller is proposed using adaptive sliding mode controller and is robust against un-modeled dynamics. But the tracking speed is affected by the measurement noise. In [2], the desired trajectory is exponentially converged with minimum tracking error but model uncertainties, effect of saturation, and actuator failures during motion are not considered. A controller in which the gain of the proportional controller is adaptively tuned is developed in [3]. The tracking accuracy is poor in the presence of noise and also stability is not ensured. An adaptive PID controller [4] is used to increase the flexibility of the control system. Only the tracking accuracy is focused rather than considering the stability of the vehicle body control system. Fuzzy logic is used to design a controller for the safe Robotino navigation [5]. A fuzzy-based PID controller [6] is proposed to track the path of a WMR. Tracking accuracy is improved while compared to the PID controller but the time taken to track the desired trajectory is more. Fractional order calculus and sliding mode controller are combined in a F-AIHMC [7]. High accuracy and high precision motion of the robot are ensured here. A combination of PI controller and neural network controller is used in [8] to control the omni-directional WMRs and the tracking error is reduced as compared to conventional PID. A backstepping and fuzzy sliding mode controller (BFSMC) is proposed in [9]. The desired trajectory is tracked even in the presence of external disturbances and model uncertainties.

In this work, an NN-based TID controller is developed for trajectory tracking of WMRs and the performance is evaluated. PID controllers are widely used in industrial process control owing to its simplicity and easy implementation. TID controllers are popular due to better capability for disturbance rejection, simpler tuning method, and lesser control effort as compared to PID controllers. This paper compares the performance of the proposed controller with these two popular controllers. WMR kinematics is explained in Sect. 2. The proposed controller is described in Sect. 3 followed by the results in Sect. 4 and finally ends up with conclusion in Sect. 5.

2 Kinematic Model of a Robot

A differential drive mobile robot (DDMR) consisting of two driving wheels and one castor wheel is considered. Let \(v_l\) and \(v_r\) be the left and right wheel velocities, respectively. Then the robot velocity is the average of the two wheel velocities and the angular velocity of the DDMR is

$$\begin{aligned} \omega = \frac{v_r-v_l}{L} \end{aligned}$$
(1)

where L is the distance between two wheels. From Fig. 1 and from the definition of angular velocity, the kinematic equations of a DDMR can be obtained as

Fig. 1
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Differential drive kinematics

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$$\begin{aligned} \begin{bmatrix} \dot{x}\\ \dot{y}\\ \dot{\theta } \end{bmatrix}=\begin{bmatrix} v\cos \theta \\ v\sin \theta \\ \omega \end{bmatrix} \end{aligned}$$
(2)

Consider a DDMR with angular velocity \(\omega \) and the linear velocity v, respectively. The point \((x_r(t), y_r(t))\) on the reference trajectory is the desired point. The robot which is currently at (x(t), y(t)) should track this reference point. The error in position and orientation is defined as

$$\begin{aligned} \begin{bmatrix} x_e(t)\\ y_e(t)\\ \theta _e(t) \end{bmatrix}=\begin{bmatrix} x_r(t)-x(t)\\ y_r(t)-y(t)\\ \theta _r(t)-\theta (t) \end{bmatrix} \end{aligned}$$
(3)

3 Proposed NN-Based TID Controller

Fig. 2
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Proposed controller for trajectory tracking

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The proposed methodology uses a neural network (NN)-based TID controller for the effective trajectory tracking of a WMR. The optimal parameters for the TID are generated using a neural network. The proposed structure of the neural network-based non-linear TID is shown in Fig. 2. The objective of trajectory tracking problem is to make the errors in distance and the deviation angle zero. The error in distance and deviation in azimuth angle are obtained from (4) and (5), respectively

$$\begin{aligned} d_e=\sqrt{(x_{r}-x)^2+(y_{r}-y)^2} \end{aligned}$$
(4)

The errors in position and orientation are fed back to two TID controllers (one for x and other for y) and to a proportional controller, respectively.

$$\begin{aligned} \phi _e = \tan ^{-1}\frac{y-y_{r}}{x-x_{r}} \end{aligned}$$
(5)

The errors in x- and y-positions are provided as the input signals to the proposed NN controllers. The deviation in azimuth is adjusted to zero by a proportional controller. The tracking errors are described by \(e=\begin{bmatrix} x_e&y_e&\theta _e \end{bmatrix}\). A TID controller whose parameters are optimized using the neural networks is used to track the desired trajectory with minimum control input and tracking error. The TID is similar to that of a PID [10], but the proportional gain is substituted with a tilted mode consisting of a transfer function \(1/s^{1/n}\).

The neural networks trained using backpropagation algorithm are used to find the optimal \(K_i,K_d\) and \(K_t \) of a TID controller. The inputs of the neural networks are the reference trajectory, control input, and current position of the robot. There are 15 neurons in the hidden layer and the output layer consists of three neurons which corresponds to \(K_t, K_i \) and \(K_d\). Inputs to the TID controller are the errors in x- and y-positions given by \(x_e(t) \) and \(y_e(t)\), respectively. The control inputs are the linear velocity v and angular velocity \(\omega \). The accuracy of the proposed NN-based TID controller is evaluated using different performance analysis indices such as integral squared error and maximum of absolute tracking error. The integral squared error is calculated by

$$\begin{aligned} ISE=\int \limits _0^t e(t)^2 dt \end{aligned}$$
(6)

The integral of absolute error denotes the accumulated error and is computed as

$$\begin{aligned} IAE=\int \limits _0^t |e(t)| dt \end{aligned}$$
(7)

4 Results and Discussions

The effectiveness and the feasibility of the NN-based TID controller for trajectory tracking of a WMR are evaluated by MATLAB/SIMULINK simulations. This section describes the key findings. The model of the differential WMR described in Sect. 2 and the NN-based controller presented in Sect. 3 are developed using MATLAB 2020a. Neural networks are trained using backpropagation algorithm. Simulations are carried out by considering various trajectories such as circular trajectory, lemniscate trajectory, B spline trajectory, and straight line trajectory as the reference trajectories.

4.1 Case 1: Circular Trajectory

A circle of radius 1m is considered as the reference trajectory. The trajectories are tracked using PID-, TID-, and NN-based TID controllers are shown in Fig. 3a. It is evident from this figure that the NN-based TID controller tracks the desired trajectory exactly whereas the tracked trajectories by other controllers slightly deviate from the desired trajectory. Figure 3b shows the control history of each controllers. From Table 1, it is very clear that the maximum control, total control effort, ISE, and IAE are very small as compared to PID and TID controllers. Thus, for a circular trajectory, the proposed NN-based TID requires lesser control and achieves negligible small displacement error and so is superior to PID and TID controllers.

Fig. 3
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Performance evaluation of controllers (circular trajectory)

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Table 1 Comparison of controllers (circular trajectory)
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Fig. 4
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Performance evaluation of controllers (lemniscate trajectory)

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4.2 Case 2: Lemniscate Trajectory

The trajectories tracked by PID-, TID-, and NN-based TID controllers are shown in Fig. 4a. The displacement errors made by PID and TID controllers are high as compared with the proposed NN-based TID controller. As shown in Table 2, controls required for PID and TID are 3.1 and 2.8, respectively, as compared to a smaller value of 1.12 for the proposed method. In Fig. 4b, the control histories for each of these controllers are shown. It is obvious from Table 2 that the total control effort, ISE, and IAE also are very small for the proposed method.

Table 2 Comparison of controllers (lemniscate trajectory)
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4.3 Case 3: B Spline Trajectory

A DDMR cannot take sharp turn. The performance of DDMR with a sharp turn can be analyzed with a B spline trajectory. The B spline trajectory is tracked by using the three control algorithms. As can be observed from Fig. 5a, the displacement error with the NN-based TID controller is less as compared to other controllers. The initial part of the trajectory is a straight line and all the three control algorithms are able to track the desired trajectory. However, due to the sharp turn the PID and TID controllers deviate from the desired trajectory as it reaches the goal position. The NN-based TID controller reports a highest tracking accuracy. From Table 3, it can be concluded that the proposed method possesses superior performance in terms of control input and displacement error.

Fig. 5
figure 5

Performance evaluation of controllers (B spline trajectory)

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Table 3 Comparison of controllers (B spline trajectory)
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4.4 Case 4: Straight Line Trajectory

This section considers a straight line trajectory. The trajectories plotted by the NN-based TID controller, the PID controller, and the TID controller are shown in Fig. 6a. The tracking error is least for the NN-based TID controller. The error in tracking a straight line is small compared to the error occurred while tracking circular trajectory, B spline trajectory, and lemniscate trajectory. The control history shown in Fig. 6b and the performance comparison as per Table 4 clearly illustrate that the proposed NN-based TID comes with a better performance. The tracking error is reduced by nearly 50\(\%\) by proposed controller when compared to the PID and TID controllers. The maximum control input is only 1.12 in the proposed controller which is very small as compared to TID (2.8) and PID (3.1). Analyzing the performance on simulation, NN-based TID controller is more advantageous in terms of accuracy, control input, and total control effort.

Fig. 6
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Performance evaluation of controllers (straight line trajectory)

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Table 4 Comparison of controllers (straight line trajectory)
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5 Conclusion

In this paper, a neural network-based TID controller for the trajectory tracking of a differential drive mobile robot is presented. A neural network is used to tune the TID parameters. Simulations are done for circular, lemniscate, straight line, and B spline trajectories. The performance of the proposed controller is evaluated with the conventional TID and PID controllers. The simulation results illustrated that the proposed method exhibited superior performances with respect to tracking error, total control effort, and maximum control input.