This section provides a comprehensive description of the preprocessing and analysis of the sensor data utilized in our simulation study. In consideration of the EKF converging process and RNN learning and training time samples, we utilized 70 sampling epochs as the training dataset, using data recursively, and the remaining epochs for testing.

The sensor data is simulated as flag data, which detects obstacles from a distance of 0.4 m. If obstacles are located in front of the sensor, the flag is set to 1, otherwise, it is set to 0.

To generate the input data for the learning algorithm, we used a camera mounted on the ceiling of the room to detect the position and color of obstacles in the environment. The position of the robot and obstacles were labeled as global positions, and the obstacles were represented as binary values, with 1 indicating the presence of an obstacle and 0 indicating no obstacle at that position. We also added normal random noise to the positions of the robot and obstacles to simulate the measuring noise in the sensor data.

To analyze the sensor data, we first pre-processed the data to remove noise and outliers and normalized the data to ensure that the learning algorithm could effectively learn from the data. We also used data augmentation techniques to increase the size of the dataset and improve the generalization ability of the learning algorithm.

Next, we used the Extended Kalman Filter (EKF) algorithm to estimate the position and velocity of obstacles in the environment and the robot. The EKF algorithm considered the position and velocity of the obstacles and estimated their next position by predicting one step ahead. Obstacles were designed with random velocity and shape to simulate a realistic environment. If any new obstacles were detected, a new EKF was started to estimate its position. If the predicted location and sensor data were not compatible, the sensor data was given priority, and the robot changed its direction by 45° in each step while the sensor flag was set to zero to show the lack of obstacles nearby. Finally, we used the pre-processed sensor data as input to the learning algorithm and trained the algorithm to predict the position and velocity of obstacles in the environment.

To simulate realistic dynamic behavior of a mobile robot, a real three-wheeled OMR Dynamic model parameters and internal tracking control law, given by (Peñaloza-Mejía et al., 2015) are utilized. The dynamic model of OMR is described as follows: M q ∙ ∙ q + C q ∙ , q ∙ q = B q τ (1)

The inertial matrix, the Coriolis and centrifugal force vector, and the input matrix, defined as follows: M q = m + 3 J 2 r 2 0 0 0 m + 3 J 2 r 2 0 0 0 I z + 3 J L 2 r 2 C q ∙ , q = 0 3 J 2 r 2 ∙ φ w 0 − 3 J 2 r 2 ∙ φ w 0 0 0 0 0 (2) B q = − 1 2 r sin φ w − 3 cos φ w 1 r sin φ w − 1 2 r sin φ w + 3 cos φ w 1 2 r sin φ w + 3 cos φ w − 1 r cos φ w 1 2 r sin φ w − 3 cos φ w L r L r L r

The parameters of the model are stated in Table 1, and the tracking control law is described in (Peñaloza-Mejía et al., 2015).

This paper employs m+1 Kalman filters concurrently to forecast the future states of m obstacles, as well as the robot’s position and velocity, based on the mean square error of the estimation. The Kalman Filter Parameter estimation laws are derived from (Blanchard, 2007), and the regressor for the obstacles is defined as follows: X o i k = x o i k − 1   v o i k − 1 (3)

The estimated future states of the robot are utilized to predict the future screen of the environment. Moreover, to predict future states of the robot, a Kalman filter is implemented to estimate the parameters of an Auto-Regressive Exogenous Model, which is considered as the dynamic constraint of the robot in the proposed MPC navigator.

This section addresses a common challenge in obstacle avoidance, which involves dynamic and kinematic constraints within a log-convex MPC framework. In order to tackle this issue, we have identified several key rules that are typically applied in navigation problems. These rules have been translated into mathematical expressions that can be incorporated into the MPC cost function and constraints.

Rule one emphasizes the importance of taking the shortest route possible, while rule two highlights the value of maintaining a safe distance from obstacles. Rule three prioritizes the use of sensor information over image processing for obstacle detection, and rule four stresses the need for accurate tracing at all times. Rule five reminds us to adhere to the maximum allowable speed, and rule six emphasizes the importance of ensuring system stability.

To incorporate these rules into our MPC framework, we have developed a set of linear constraints and kernel functions that allow us to evaluate their relative importance on a scale of zero to one. Rules one to five are incorporated into a constrained MPC structure, which is defined as follows: min ε , ϑ J ∞ = ∑ n = k ∞ ∫ n Δ t n + 1 Δ t J t d t ⏟ J n = ∑ n = k k + N J n + ∑ n = k + N + 1 ∞ J n = ∑ n = k k + N J n + I N + 1 Δ t (4-1) S . t . Y ^ k + 1 = γ ^ T Y k U k , P k − O i k > 2 2 r O i k = I 3 × 3 0 3 × 3 X o i k + 1 ⋯ X o i k + N 6 × N Y p = Y ^ k + 1 ⋯ Y ^ k + N 6 × N P k = I 3 × 3 0 3 × 3 Y p φ ^ w k ≥ F l a g × π 4 0 ⃗ 3 × 3 1 ⃗ 3 × 3 Y k 6 × 1 ≤ V max . cos φ w k V max . sin φ w k ω max (4-2) where J t is defined as follows: J t = 1 − exp − 0.5 Y k Δ t − Y f T Λ − 1 Y k Δ t − Y f + exp − 0.5 Y k Δ t − X o i T Λ − 1 Y k Δ t − X o i + 1 − exp − 0.5 U k Δ t T Λ − 1 U k Δ t (5) where Y t = Y k Δ t = Y k is discrete-time sampling. In which Λ − 1 is defined based on the constraint of the robot dynamic and environment limitations. Λ − 1 = 1 d x 0 0 0 1 d y 0 0 0 1 d θ 0 3 × 3 0 3 × 3 1 V max 0 0 0 1 V max 0 0 0 1 ω max (6)

Where d indicates maximum possible movement in the Environment toward the (x-y) axis. In addition, the maximum possible rotation angle is shown by d θ . V _max denotes the maximum possible velocity for the robot, according to the environment and robot dynamic. ω max is the maximum possible angular velocity. Y k = q q ˙ T indicates position and velocity vector of the robot dynamic at time step k. U _k is the control law, i.e., the generated path. γ ^ T is the estimated model parameter by KF. φ ^ w k is the estimated robot angle and Y ^ k + 1 is the one-step-ahead prediction of Y k . Flag is 1 if the built-in sensors detect obstacles around the robot, otherwise, it is zero, to deal with uncertainties and possible faults of environment predictions. This item is considered to avoid the robot collision with obstacles, in the KF failure situations.

The cost function used in this study is a key component of the model predictive controller (MPC) algorithm that controls the navigation of the robot. It is designed to balance the tradeoff between achieving the desired goal (i.e., reaching the target point) and avoiding obstacles while minimizing the energy consumption of the robot. The position cost function is given by the term 1 − exp − 0.5 Y k Δ t − Y f T Λ − 1 Y k Δ t − Y f . This term penalizes the distance between the robot’s current position Y k Δ t and the target point Y f . The collision cost function is given by exp − 0.5 Y k Δ t − X o i T Λ − 1 Y k Δ t − X o i , and the energy cost function is given by 1 − exp − 0.5 U k Δ t T Λ − 1 U k Δ t , here is the command requests to the robot to change its position and orientations. Constraints of the navigation problem, given in Eq. 4-2 are break down to system dynamic constraint, local and global collision avoidance, using sensor data, and kinematic constraint.

This paper utilizes ARMAX model to estimate velocity and position of the robot as follows: Y k + 1 = a Y k + b U k + e k , e k = N 0,1 E Y k + 1 = a b γ T Y k U k φ = γ φ (7)

Where Y_k denotes the position and velocity of the robot at kth sample time, and e k is a disturbance with normal distribution. φ is the input regressor, and the next states of the robot are calculated as below: C 1 : Y ^ k + 1 = γ : , 1 T Y k + γ : , 2 T U k (8)

Considering constant control input in each sample time, we define N-horizon prediction of states vector as follows: Y p = Y k + 1 ⋯ Y k + N 6 × N P k = I 3 × 3 0 3 × 3 Y p (9)

Where P _k is the prediction of N next positions of the robot.

Suppose that D _oi = [x _oi , v _oid ] ^T denotes to the i_th obstacle’s position and velocity vector in the environment at the global coordination. Obstacles’ states predications are calculated below: X o i k + 1 = I 3 × 3 Δ t × I 3 × 3 0 3 × 3 I 3 × 3 X o i k O i k = I 3 × 3 0 3 × 3 X o i k + 1 ⋯ X o i k + N 6 × N (10)

To show the constraint of the robot and obstacles collision avoidance, we define the below criteria: C 2 : P k − O i k > Ω , Ω > 0 (11)

Where Ω is the conservative distance parameter, it should be determined based on the maximum estimated velocity of the robot, and obstacles to handling prediction’s uncertainties. Where Ω is the conservative distance parameter, it should be determined based on the maximum estimated velocity of the robot, and obstacles to handling prediction’s uncertainties, here is considered 2 2 r .

Due to possible prediction faults and uncertainties, the importance of checking robot obstacle detection sensors increases, so we consider a Flag to detect an obstacle. Flag = 1 indicates an obstacle is nearby the robot, and Flag = 0 reveals the lack of obstacle around the robot. C 3 : Y 3 ≥ F l a g × π 4 (12)

This inequality can also be considered as a fitness neighborhood searching function, instead of intending it as another constraint of the MPC problem. This substitution helps to preserve convexity of the problem, when convex optimization solver is utilized.

Kinematic constraints should be regarded in the navigation strategy. This study considers a mere velocity constraint. C 5 : 0 1 Y ≤ V max . cos θ V max . sin θ ω max (13)

This section provides an error feedback control law, U t = F Y t − Y f which is able to ensure the stability of MPC problem, (4). To this aim, consider the candidate Lyapunov function as follows: J ∞ = ∫ t ∞ 1 − exp − 0.5 Y t − Y f T Λ − 1 Y t − Y f + exp − 0.5 Y t − X o i T Λ − 1 Y t − X o i + 1 − exp − 0.5 Y t − Y f T F T Λ − 1 F Y t − Y f d t (14)

F is discrete-time error-feedback gain In order to ensure stability in a navigation system, it is necessary to determine the criterion for stability guarantee. This can be achieved by calculating the derivative of the defined Lyapunov function, which should be negative. By simplifying the Lyapunov inequality constraint of stability, it is possible to determine whether the navigation system is stable.

Stability in navigation is crucial for ensuring that a robot can reach its intended destination without any collisions. To simplify the stability criterion, a linear approximation of the cost function is used, taking into account the limited arguments of the kernel functions, which are restricted to values of 0 and 1. By following these steps, it is possible to ensure that the navigation system is stable and reliable, allowing for safe and efficient navigation. ∂ J ∞ t ∂ t = H + H T H = 0.5 Y ˙ T − Λ − 1 X o i − Y f + F T Λ − 1 F Y − Y f ∂ J ∞ t ∂ t ≤ ∫ t ∞ H t + H T t ∂ t < 1 − − Y T Λ − 1 X o i − Y f + Y T F T Λ − 1 F Y − Y f ⏟ I t < 0 (15) I t = 1 − − Y t Y t T Λ − 1 0 0 F T Λ − 1 F X o i − Y f Y − Y f < 0 , Λ − 1 > 0 , F T Λ − 1 F > 0

This section is assigned to map the newly defined optimization problem into the searching-based recurrent Neural Network structure. To this aim, consider a general Lagrangian constrained optimization problem as follows (Ramezani et al., 2023): L x 1 , x 2 , ⋯ , x n , λ 1 , ⋯ , λ m = f x 1 , x 2 , ⋯ , x n + ∑ i = 1 m λ i r i ∂ L ∂ x i = 0 , i = 1 , ⋯ , n ∂ L ∂ λ j = 0 , j = 1 , ⋯ , m (16)

A new and innovative RNN structure has been proposed to find a near-optimal solution in possible movement directions. This structure is made up of newly designed Gaussian spiking neurons that construct the cost function criteria concept in their membrane function. The synaptic neurons are defined in such a way that the network can search for the best position.

The main idea behind this proposed structure is to solve the Lagrangian problem by considering synaptic current as a term that can indicate the weighted constraints. This synaptic current is then inputted to the synaptic neurons.

Figure 3 demonstrates how the searching screen window is selected based on the location of the robot and the prediction horizon. Additionally, the figure indicates the constraints inputted to the reservoir recurrent neural network beside the selected window. The amount of four-sided shapes was based on the desired forecast range, with a single quadrangle corresponding to a forecast one step ahead, two quadrangles corresponding to a forecast two steps ahead, and so on. This facilitated the capture of pertinent data pertaining to the automaton’s immediate surroundings at various prediction intervals, which was employed as input to our RNN-based optimization methodology. The dimensions of the display were determined in accordance with the number of quadrangles required to capture the sought-after prediction horizon.

After analyzing the low-risk position corresponding to the minimum membrane potential, the activation function generates the desired path according to the calculated feedback gain corresponding to the low-risk possible location. In other words, the minimum Vn indices indicate the path error feedback indices, and consequently, the desired path is calculated.

The proposed network algorithm is defined in Eq. 17. This new RNN structure has the potential to revolutionize the way we approach movement directions and find optimal solutions. V n = ∑ i = x r − 1 x r + 1 ∑ j = y r − 1 y r + 1 s i j e − x r − i − 1 2 − y r − j − 1 2 + 2 s i j − 1 e − x f − i 2 − y f − j 2 + I n ρ n = ∑ k = 1 m W n k r k + θ n I n = tanh ρ k r 1 = Δ x r − V x max . Δ t r 2 = Δ y r − V y max . Δ t r 3 = x r − γ φ 1 r 4 = y r − γ φ 2 r 6 = tan − 1 y / x − F l a g × π 4 r 7 = 1 − − Y T Λ − 1 X o i − Y f ⏞ Z + Y T F T Λ − 1 F ⏞ Φ Y − Y f (17) where S _ij is a Boolean value that represents the state of a grid cell in the global screen. The screen is divided into a grid based on the scale of the robot’s dimensions, treating the robot as a single point in the encoded matrix of the environment. The values of x _r and y _r represent the robot’s position in the (x-y) coordinate system. The variable V _n in the equation represents the cost of each state in the neighborhood of the robot, as well as the cost of searching around it, considering one-step movements and calculating the cost of the distance from the target. S _ij is used to increase or decrease the cost if an obstacle is present or absent at that position. The first term of V _n searches for low-risk positions around the robot, while the next term evaluates the risk of one-step movements.

This searching structure is designed with the aim of reducing map size dependency and consequently its computational cost. In is the input synaptic current on the neuron n and Nv is the neuron membrane function, which indicates the proposed cost function and 1 step prediction of moving environment screen. The combination of the searching-based idea with the reservoir structure can reduce sensitivity of the proposed navigator to uncertainties and improve its robustness.

The input neurons and the size of proposed network can be considered constant and relevant to the steps of prediction horizon. Synaptic weights (W _ij ) are updated based on minimizing V _n , as an indicator of the (i, j)th Lagrangian function of the proposed algorithm. ∆ W n k = − η ∂ V n ∂ W n k = − η 1 − ρ n 2 r k ∆ θ n = − η 1 − ρ n 2 ∆ F = − μ ∂ r 7 ∂ Φ = μ Λ 1 2 Y Y T − Y Y f T 1 2 (18)

The desired error feedback gain is calculated based on the minimum-Vn neuron location. Consequently, the desired path is calculated according to the following active function: Y d = Y r o b o t + F Y f − Y r o b o t Y f − Y d > 2 V max * d t Y f e l s e (19)

The dynamics of a robot can limit its ability to follow a planned path precisely, as shown in Figure 3.

Figure 4 exemplifies the plausible path tracking predicaments that mobile robot may face, encompassing inaccuracies and aberrations. These predicaments may culminate in the fiasco of collision evasion schemes, especially in fluctuating surroundings where barriers approach the automaton. To ensure the safe and efficacious operation of mobile automatons, it is crucial to ponder over the impact of dynamic constraints on the efficiency of path plotting. By apprehending the restrictions imposed by a robot’s dynamics, we can formulate more sturdy and dependable path plotting algorithms that take into account the possibility of inaccuracies. This approach can decrease the probability of collisions and other safety perils while augmenting the comprehensive efficiency and performance of mobile robots.

To assess the effectiveness of the proposed navigation strategy, we utilized the CVXOPT library to solve the constrained MPC problem. Our evaluation included sample environments with both static and dynamic obstacles, all within a 20 × 20 screen size.

Figure 5 provides a visual representation of the robot’s movements and the obstacles encountered every 0.3 s. As demonstrated in this figure, the robot successfully navigated to its target destination without any collisions.

To further evaluate the efficiency of our proposed strategy, we utilized Figure 6, which provides a global overview of the environment. Through these evaluations, we were able to demonstrate the effectiveness of our log-concave MPC approach in navigating mobile robots through environments with both static and dynamic obstacles.

This section evaluates the performance of RNN searching window and its efficiency in comparison to utilize CVXOPT optimization tool in another example of environment.

CVXOPT is a convex optimization library that provides solvers for a wide range of convex optimization problems, including nonlinear MPC. The specific algorithm used by CVXOPT for solving the nonlinear MPC problem depends on the formulation of the problem and the specific options selected by the user. In general, CVXOPT uses a primal-dual interior-point method for solving convex optimization problems (Dahl and Andersen, 2022). This method iteratively solves a sequence of linear systems and updates the primal and dual variables until convergence. For nonlinear MPC problems, CVXOPT can use either a direct or an indirect method to solve the problem.

Figure 7 reveals how both proposed path planning methods navigate the robot to destination, safely. This comparison has been conducted in the same dynamic environment, in which three dynamic environments start moving from the same initial points in both a and b figures and the obstacle’s velocity are also considered the same.

In Figure 7, the path passed by the obstacles is depicted by light blue circles. The current positions of obstacles are depicted in red circles. The robot passes through a green–square dash line, and the star is the target destination. Upon scrutinizing both Figures 7A, B, it becomes conspicuous that the simulation circumstances are strikingly alike. The motion of obstacles and their velocity remains consistent in both illustrations. However, a noticeable dissimilarity arises when contemplating the obstructions illustrated in Figure 7B, as they progressively shift further away from the intended target locale in comparison to Figure 7A. Therefore, the robot in Figure 7B is coerced to traverse for an extended duration in order to attain the desired position. This incongruity can be credited to the inferior pace of the robot in Figure 7B, which, in turn, is instigated by the path planner dispensing a smaller displacement by CVXOPT. In this situation, some generated paths of movements are in the range of 0.03 which cannot empower the robot efficiently, therefore, the robot stays at its position for several sequential sample times. In contrast, the initial training time of the RNN takes almost 0.05 (second) longer than the CVXOPT tool. Then, the robot is continuously navigated to the target position without any long time stopping in a particular position. Of course, we cannot regard stopping in a particular position as a disadvantage for a path planner, while in social environments, waiting for other people to pass the way and then continuing the optimal path or safe route is considered an advantage. On the other hand, this feature is not recommended in automated vehicles on highways, in air traffic, and other applications. In this case, RNN-Log-MPC can be more efficient. Figure 8 indicates the time-distance cost comparison.

Classic methods are not able to handle path planning programming in dynamic environments efficiently. Therefore, RRT and A-star as the classic methods are recommended to be utilized in static environments. On the other hand, reactive and heuristic methods are mostly utilized in dynamic environments. To evaluate the performance of the proposed Log-MPC method, the following Table 2 depicts the comparison of RNN-Log-MPC and A-star performances in the given below static environment (Figure 9).

Figure 9 illustrates a static environment example in which RNN-log-MPC method is compared with A^* strategy. The proposed strategy is considered with one prediction step as a global planner. The costs are compared in Table 2. Table 2 provides a comparison of both strategies in the same environment and with the same processor, significant efficiency of the proposed method, especially in the sense of time costs. In addition, the results illustrate that RRT may fail in providing a free-collision path in a dynamic environment, while the proposed method can navigate the robot efficiently.

The parameters of simulated RNN is given in the Table 3:

Another sample environment includes two sinusoidal shape movements of the obstacles and one random walker, with the target position at the point (12, 15). The robot movement path is simulated under the following conditions:

1.When the utilized path planning algorithm is RRT.

The global screen of the environment, including the movement path of the robot and obstacles are shown in Figure 10.

Figure 11 presents a more comprehensive comparison of our proposed method and RRT in a simulated environment with varying target positions. The figure showcases the success of our method and the failure of RRT. The target is represented by a violet star shape, while the robot is depicted as a green diamond. The obstacles are shown as red circles, and the passed path is indicated by the yellow line. The left side of both parts of Figure 10A demonstrates the crucial scenes, while the right side shows the entire passed trajectory.

It is evident that generating a path using RRT can be time-consuming, especially if a unique solution is obtainable in this structure. To better understand this issue, we have provided a table below that compares our proposed method with RRT in terms of distance versus time. The simulation screen size is 20*20 with a scale of 0.45 m, and the arrangement of obstacles differs in at least five similar simulation examples. The target position is located at (Limon et al., 2017; Kim et al., 2018) in Figures 10, 11.

Our proposed method offers a more efficient and effective solution for navigation in a simulated environment. With our method, the robot can navigate through the obstacles and reach the target in a shorter amount of time compared to RRT. This not only saves time but also reduces the risk of errors and increases the overall success rate of the navigation process.

Figures 10A, B demonstrates a comparison of the RRT performance (10-a) with the suggested strategy (10-b). The results show that the RRT failed in providing a safe path, and consequently collides with the obstacle in a dynamic environment.

Figure 11 presents a more comprehensive comparison of our proposed method and RRT in a simulated environment with varying target positions. The figure showcases the success of our method and the failure of RRT. The target is represented by a violet star shape, while the robot is depicted as a green diamond. The obstacles are shown as red circles, and the passed path is indicated by the yellow line. The left side of both parts of Figure 10A demonstrates the crucial scenes, while the right side shows the entire passed trajectory.

It is evident that generating a path using RRT can be time-consuming, especially if a unique solution is obtainable in this structure. To better understand this issue, we have compared our proposed method with RRT in terms of distance versus time in Table 2. The simulation screen size is 20*20 with a scale of 0.45 m, and the arrangement of obstacles differs in at least five similar simulation examples. The target position is located at (Limon et al., 2017; Kim et al., 2018) in Figures 10, 11.

Our proposed method offers a more efficient and effective solution for navigation in a simulated environment. With our method, the robot can navigate through the obstacles and reach the target in a shorter amount of time compared to RRT. This not only saves time but also reduces the risk of errors and increases the overall success rate of the navigation process.

The proposed RNN-Log-MPC algorithm presents a notable advantage by utilizing both sensor and camera data to diminish the sensitivity of the navigation system to momentary data loss or prediction error by EKF. Nonetheless, this does not eliminate the exigency to tackle potential issues such as prediction errors, sensor disconnection, and lack of accuracy in the command follower controller. Therefore, a conservative approach to adjusting and evaluating restrictions may be deemed necessary, which could constrain optimization, particularly in unfamiliar environments where speed changes occur, and there is a possibility of collision risk.

Moreover, the precision of obstacle detection, which is processed via a central server and communicated wirelessly to the robot’s internal controller, can also influence the system’s performance. Hence, it is crucial to scrutinize the algorithm in various scenarios to identify potential errors and devise solutions to rectify them. Constant monitoring of the system can also enhance the accuracy of performance and reliability of the navigation system.