2.1 Obstacle avoidance and trajectory planning

Traditional methods for obstacle avoidance often rely on reactive approaches, such as potential fields [9] and vector field histograms [10]. While these methods are computationally efficient, they suffer from limitations such as local minima and suboptimal trajectories in complex environments. To overcome these challenges, model predictive control (MPC) has emerged as a promising framework for trajectory planning and collision avoidance. For example, in [11], MPC is employed for path tracking as a constrained control problem, maximizing the vehicle’s trajectory while conforming to various constraints on control actions and output. Similarly, NMPC has been effectively employed in robotic systems, taking advantage of its ability to handle non-linear dynamics and incorporate constraints directly into the optimization problem [12]. CasADi and IPOPT [5], popular numerical tools to solve optimization problems, have further facilitated the implementation of NMPC in real-time applications. The work in [13] shows how NMPC can achieve real-time performance in dynamic environments by leveraging these tools, particularly in systems with limited computational resources. Dynamic obstacle avoidance remains critical for autonomous systems in unpredictable environments [14]. Liu et al. [15] proposed a reinforcement learning (RL)-based strategy for real-time dynamic obstacle avoidance in cable-driven parallel robots, effectively integrating trajectory tracking controllers with RL for real-world applications. Similarly, Xu et al. [16] designed a gradient-based B-spline trajectory optimization algorithm for vision-aided UAV navigation, allowing real-time obstacle prediction and avoidance in dynamic environments [16]. Another innovative approach by Koutras et al. [17] integrated compliance and accuracy to enforce dynamic obstacle avoidance in collaborative robots, maintaining task execution safety.

2.2 Fluid dynamics integration in robotics

The integration of fluid dynamics into robotic systems has been explored in domains such as underwater robotics and aerial vehicles. LBM has been widely adopted for fluid simulations due to its ability to efficiently model complex flow patterns and interactions with solid bodies [18]. In [19], LBM was used to simulate underwater robotic motion, showcasing the method’s accuracy in capturing fluid-structure interactions. Another study [20] combined LBM with RL to optimize the motion of underwater robots, highlighting the potential of hybrid approaches. Zhao et al. [21] coupled LBM with a pore network model to simulate fluid flow behavior, emphasizing computational efficiency and accuracy. Sedaghat et al. [22] advanced this by integrating the immersed boundary technique utilizing the lattice Boltzmann method to model viscoelastic fluid flows around complex structures, highlighting its applicability in industrial and biological domains. Additionally, Wang et al. [23] demonstrated LBM’s effectiveness in real-time interactive 3D fluid simulations, enabling adjustments during runtime without compromising physical relevance. NMPC is widely recognised for trajectory optimization in dynamic and constrained environments. Hedjar [24] transformed NMPC optimization into a constrained quadratic problem, demonstrating its efficacy in tracking constrained spaces. Meanwhile, Zhu et al. [25] combined NMPC with a recurrent fuzzy neural network (FNN) for robust navigation in dynamic environments, enhancing optimization and collision avoidance [26]. In autonomous mobile robots, Villemazet et al. [27] validated NMPC’s potential for precise navigation through orchards using multi-camera GPS-free systems. Our method offers a robust instrument for comprehending the impact of fluid dynamics on robotic movement in intricate settings.

2.3 Hybrid computational approaches

Recent works have attempted to combine various methodologies to achieve robust performance in complex environments. For example, the hybridization of MPC and artificial intelligence techniques, such as RL, has shown promise in dynamic obstacle-rich environments [28]. Another example shows the design of obstacle avoidance controller with NMPC. This approach allows for real-time trajectory tracking while considering dynamic obstacles in the vehicle’s path [29]. Similarly, NMPC is implemented on a rapid prototyping system for vehicle dynamics as it can handle the complexities of nonlinear systems effectively [30]. However, these methods rarely incorporate physical environmental factors such as fluid dynamics. In [31], a hybrid method combining LBM for fluid simulation and traditional control techniques was proposed, but the focus was limited to static obstacles. This paper addresses this gap by presenting our two-stage analytical framework, which integrates NMPC and LBM.

3.1 Overall simulation framework

Our simulation system utilizes a two-stage, decoupled methodology to evaluate the efficacy of NMPC trajectories within a high-fidelity fluid environment. This approach directly addresses the distinction between a kinematically planned trajectory and its physical performance under fluid-dynamic influences. The separation of the kinematic NMPC planning from LBM fluid analysis is a deliberate and necessary technical choice. This work is not intended as a real-time experiment on fluid-mechanics informed controller design implementation, but rather it serves as a foundational simulation based validation study. By first isolating the kinematically optimal NMPC path and then subjecting it to LBM simulation, we can quantify the maximum impact of fluid dynamics specifically instantaneous drag forces and total energy dissipation that a standard kinematic planner inherently ignores. Consequently, the resulting data establishes a critical performance benchmark required for the subsequent development of fully coupled, fluid-flow aware NMPC algorithms.

• Stage 1: NMPC Trajectory Generation: The NMPC controller, executed in a Simulink environment, calculates a comprehensive and collision-free trajectory from a specified initial condition to the target. This phase functions without real-time fluid feedback, producing a pre-established dataset of the robot’s and obstacles’ positions and velocities during the whole simulation period. This dataset functions as the “sensor inputs” for the following fluid simulation.
• Stage 2: Offline LBM Analysis: The pre-computed position and velocity data from the NMPC output serve as dynamic boundary conditions for an offline lattice Boltzmann method (LBM) simulation. The LBM model does not affect the trajectory; rather, it is designed to conduct a comprehensive post-hoc examination of the fluid’s reaction to the intended path. This facilitates the calculation of critical fluid-dynamic parameters, including drag forces and vorticity, which influence the robot’s mobility in a realistic environment.

This decoupling framework offers a reliable tool for assessing control techniques by evaluating their performance in realistic physical settings. A practical application necessitates a physical sensor suite, e.g., visual-inertial odometry or sonar, that provides the real-time state information that is predetermined in our simulation.

3.2 Numerical validation and grid convergence

To ensure the numerical stability and physical accuracy of the LBM solver, a systematic grid resolution study was conducted. We evaluated lattice sizes ranging from to . The quantitative results of this study, highlighting the convergence of physical metrics and computational requirements, are summarized in Table 2.

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Table 2. Grid independence test: evaluation of numerical accuracy and computational performance.

https://doi.org/10.1371/journal.pone.0333056.t002

Based on the data presented in Table 2, the following observations justify our framework configuration:

• Accuracy: As evidenced by the force error reduction from 43.2% to 0%, increased grid densities were essential to minimize numerical diffusion and accurately resolve high-frequency vortex shedding patterns in the robot’s wake.
• Computational Cost: While the grid is the most demanding in terms of runtime (300.00 s), it provides the essential benchmark for high-fidelity Fluid-Structure Interaction (FSI) analysis, ensuring the reference solution is free from artifacts.
• Selection: The resolution was selected because it achieves the optimal 3–5 lattice units required for boundary layer resolution. This is critical for the convergence of hydrodynamic force computations used by the NMPC during trajectory optimization.

The results reveal a definitive trade-off between computational overhead and physical fidelity. To verify the numerical reliability, the relative error was monitored across the four resolutions. As shown in Fig 1, the error curve plateaus at the benchmark, suggesting that further mesh refinement would yield diminishing returns. This confirms that the selected resolution has achieved grid independence, providing a stable and trustworthy foundation for all subsequent NMPC evaluations.

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Fig 1. Grid convergence analysis for the LBM solver.

Relative errors are calculated against the benchmark resolution.

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3.3 Lattice Boltzmann Method (LBM)

• Derivation from the Boltzmann Equation: The LBM is derived from the Boltzmann equation [32], which models the statistical behavior of a gas at the mesoscopic level. In this context is the particle distribution function that describe the distribution of particles at position x, velocity v, and time t [33]. The term v denotes the particle velocity, f represents external forces acting on the particles, and is the collision operator that models particle interactions. The Boltzmann equation is expressed as:

(1)

The collision operator is simplified using the Bhatnagar-Gross-Krook (BGK) model, which assumes relaxation of the distribution function f towards an equilibrium distribution f^eq. The BGK approximation is given as:

(2)

where, denotes the relaxation time associated with the fluid’s viscosity and f^eq is the equilibrium distribution function derived from the Maxwell-Boltzmann distribution.

• Lattice Boltzmann Equation (LBE): The Boltzmann equation is discretized in time, space, and velocity space, resulting in LBE. In LBE, represents the particle distribution function along the discrete velocity direction [34] i, e_i is the lattice velocity vector, and is the time step. The LBE is expressed as:

(3)

• Equilibrium Distribution Function: The equilibrium distribution function is defined to satisfy the conservation laws and is given as [35]:

(4)

In this equation w_i are weighting factors that depend on the lattice structure and represent the fluid density, u while the macroscopic velocity vector, and c_s is the speed of sound in the lattice. For the D2Q9 lattice, a specific, popular way to set up a 2D grid for fluid simulations using the LBM. D2 determines two dimensions, and Q9 means nine velocities/directions. The weights are w₀ = 4/9 for the rest particle, for the cardinal directions, and for the diagonal directions [36].

• Lattice Boltzmann simulation steps:

Initialization: The fluid density , macroscopic velocity u, and distribution function f_i are initialized.

Collision Step: The collision term is computed using:

(5)

Streaming Step: The updated distributions are propagated to neighboring lattice nodes:

(6)

Boundary Conditions: No-slip boundary conditions at walls (u = 0).

Macroscopic Variable Updates: The fluid density and velocity are calculated as:

(7)

3.3.1 LBM implementation and reproducibility parameters.

The essential lattice and physical parameters are defined in Table 3 as follows:

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Table 3. Lattice Boltzmann Method (LBM) implementation parameters.

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3.3.3 Instantaneous drag computation.

The transient hydrodynamic drag force F_D exerted by the fluid on the robot is computed using the Momentum Exchange Method (MEM). This method measures the change in momentum across the links connecting the solid domain to the surrounding fluid domain. For a link between a fluid node x and a solid node (x + e_i) in direction i, the momentum change is calculated from the pre- and post-collision distribution functions, f_i and fʹ_i as:

(8)

where, e_i is the lattice velocity vector in direction i, and the sum is performed over all nodes x adjacent to the solid boundary.

3.3.4 Energy consumption calculation.

The instantaneous energy consumption rate (Power, P) required by the robot to overcome the fluid resistance is computed as the mechanical power dissipated by the robot against the hydrodynamic drag force. This is given by the dot product of the instantaneous drag force F_D(t) and the robot’s instantaneous velocity u_R(t):

(9)

The total energy consumption over a trajectory T is then calculated by integrating the power over time.

4.1 Nonlinear Model Predictive Control (NMPC)

• Robot Kinematics: The motion of the robot is governed by a kinematic model, which relates the position (x,y), orientation , linear velocity v, and angular velocity as follows [37]:(11)(12)
• Optimization Problem: NMPC framework predicts the future behavior of the system over a finite time horizon and computes the optimal control inputs at each step, as shown in Fig 2 [39]. The control horizon spans from t = 0 to , while the predictive horizon extends from t = 1 to t = N [40]. The optimization problem minimizes the following cost function [6]:(13)

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Fig 2. The fundamental components of model predictive control (MPC): the receding horizon principle.

At each time step, the controller calculates an optimal sequence of control actions across a finite prediction horizon. Only the initial control action is executed, and the procedure is replicated at the subsequent time step [38].

https://doi.org/10.1371/journal.pone.0333056.g002

In this expression, denotes the operational cost, which quantifies the deviation of the robot state x_k and the control input u_k from the reference state and control input . The term is the terminal cost, defining the deviation of the final predictive state x_N from the reference state , ensuring the stability of the controller. This ensures that new controls are capable of being computed in each decision instance, have the ability to handle complex dynamics and constraints, and be an effective local optimizer, balancing immediate actions and future behavior [41].

NMPC optimization is subjected to several constraints. The initial state constraint ensures that the robot begins at a known position and is defined as:

(14)

The system dynamics constraint guarantees that the evolution of the state variables x_k adheres to the robot’s kinematic model:

(15)

To ensure safety, collision avoidance is maintained by requiring the distance between the robot and obstacles to remain greater than the sum of their radii [15]:

(16)

Additionally, the robot’s state variables are constrained to stay within physical or environmental limits:

(17)

Similarly, the control inputs are bound to ensure feasible velocities and angular velocities:

(18)

4.2 Cost function for the control system

The cost function explicitly balances the state and control inputs across the prediction horizon [18]:

(19)

In this equation, X_k represents the robot’s state variables at time step k, while X_ref is the desired reference state. The term U_k refers to the control inputs and the matrices Q, R, and Q_f weights of the costs of the state deviation, control effort, and terminal state deviation, respectively.

4.3 NMPC tuning and parameter rationale

The performance and computational cost of the NMPC planner are determined by the tuning parameters: the prediction horizon N, the state weighting matrix Q, and the control input weighting matrix R.

Tuning parameters are the key numerical values set by the control system designer that directly influence the planner’s behavior. Performance refers to how well the planner controls the system (e.g., speed, accuracy, stability), while computational cost refers to the time and processing power required to calculate the optimal control sequence in real-time, which must be faster than the system’s sampling time ().

The selected values for N, Q, and R were chosen based on a sensitivity analysis that balanced path efficiency, convergence speed, and control smoothness. The sampling time T_s was fixed at 0.1 s. The prediction horizon N was set to 20 steps, with the control horizon (N_c) set equal to N to ensure maximum trajectory flexibility. This horizon length was empirically determined as the minimum duration required for the robot, operating within its velocity limits, to effectively detect and execute an avoidance maneuver around the given dynamic obstacles. The objective function utilizes the weighting matrices Q and R to define the penalty for state error and control effort, respectively.

• State Weighting Q: The matrix was set to diag(1, 1, 0.001). The equally high weight on the positional errors () ensures that the controller prioritizes driving the robot directly to the goal, minimizing path length. Conversely, the significantly lower weight on orientation error () reflects the focus on achieving the target position over strict angular alignment, which is typical for a kinematically constrained robot.
• Control Weighting R: The matrix was set to diag(1, 1). These equal weights penalize changes in both linear velocity (r_v = 1) and angular velocity (). This promotes smooth, physically realistic control inputs throughout the trajectory, ensuring low control and implicitly minimizing unnecessary energy expenditure.
• Terminal Cost Q_f: A high terminal cost, , was applied to the final state to guarantee stability and convergence toward the target.

The NMPC formulation relies on explicit constraints to ensure safety and physical realism. States were bounded by the simulation domain ( dimensions in units), and control inputs were bounded: linear velocity m/s and angular velocity rad/s.

The underlying IPOPT solver was configured with a tolerance (tol) of and a maximum iteration limit of 100 steps per control cycle. To handle potential solver failures or non-convergence within the iteration limit, the system was configured to ‘warm-start’ the next cycle using the previous optimal solution. If a feasible solution was not found, the controller was programmed to implement the first control command from the previously calculated optimal sequence to maintain safety while re-initializing the solver.

5.1 The effect of Reynolds number

The Reynolds number(Re) is a dimensionless quantity that characterizes the ratio of inertial forces to viscous forces. The specific orders of magnitude used in this study, Re = 100 (laminar) and Re = 2000 (turbulent), reflect the typical operating regimes for small, subaqueous autonomous vehicles in near-surface or enclosed fluid environments. The transition from 10² to 10³ allows for a robust assessment of the NMPC strategy across fundamentally different flow behaviors.

5.1.1 Laminar Flow (Re = 100):.

Laminar flow occurs when the Reynolds number (Re) is low, typically below 2000, resulting in smooth, orderly fluid layers [19].

• Reynolds number calculation is carried out as:

(20)

where:

1. -. : Fluid density (set to 1000 kg/m³ for water).
2. -. U: Velocity of the robot relative to the obstacle (m/s).
3. -. L_c: Length, defined as the obstacle diameter ().
4. -. : Dynamic viscosity of the fluid () [42].

For example for Re = 100, the flow exhibits minimal turbulence, which is modeled using LBM framework. The velocity field is initialized to ensure uniform flow conditions, and the simulation verifies the formation of layered flow patterns around static and moving obstacles. The simulation was designed to achieve a flow regime characteristic of slow navigation. The key dimensional values established were:

1. -. Velocity (U): 0.005 m/s
2. -. Obstacle Diameter (L_c): 0.02 m
3. -. Dynamic Viscosity ():

Substituting these values yields the operating Reynolds number:

The simulation operates consistently at Re = 100.

• Drag force on the robot is given by:

(21)

The drag force acting on the robot is influenced by several factors [43]. The drag coefficient C_d, which is a dimensionless number, depends on the shape and surface characteristics of the robot. The cross-sectional area A refers to the area that faces the fluid flow, which plays a significant role in determining the amount of force experienced. Lastly, the relative velocity u_rel represents the speed of the fluid in relation to the robot, affecting how much drag force is exerted on it.

5.1.2 Turbulent Flow (Re = 2000):.

Turbulent flow arises at high Reynolds numbers, leading to chaotic and irregular fluid motion. This phenomenon is clearly observed at Re = 2000 within the simulation.

• Adjustments for Turbulent Flow:

Step 1: Increase U or reduce the Reynolds number in Eq (20) to simulate high-velocity or low-viscosity conditions.

Step 2: LBM collision term adapts to capture vortex formation:

(22)

Step 3: Here, the relaxation time controls the viscosity and is given by:

(23)

Here is the kinematic viscosity and c_s the lattice speed of sound.

• Vorticity Calculation: Vorticity measures local rotation in the fluid.

(24)

where, represent the velocity components in the X and Y directions. The simulation visualizes turbulent eddies and their impact on robot navigation.

5.2 Boundary condition variations

Boundary conditions significantly affect the flow and interaction between fluid and obstacles. Two primary configurations are studied:

• Flow from the Left Wall: A unidirectional flow is simulated by imposing a constant velocity at the left boundary, given that U_inlet is the prescribed velocity at the left wall, and all other boundaries enforce no-slip conditions (u = 0). i.e.,

(25)

LBM handles this boundary condition through streaming and reflection steps, ensuring fluid continuity and stability.

• Radial Flow from Obstacles: This scenario models fluid emanating radially from a circular obstacle, where U_r the radial velocity is also , i.e., a unit vector in the radial direction.

(26)

Dynamic obstacles are initialized with prescribed motion:

(27)

(28)

The obstacle’s position and velocity are updated in each step, affecting the fluid flow and robot’s trajectory.

5.3 Whole system simulation framework

Robot Kinematics and NMPC: The NMPC controller calculates the optimal trajectory by solving the optimization problem [44]:

(29)

Subject to Dynamics: To accurately compute the state X_k+1 at the next time step, the fourth-order Runge-Kutta method (RK4) is employed. This method improves accuracy as compared to simpler numerical approaches like Euler integration. The intermediate steps of the RK4 integration are calculated as [45,46]:

(30)

(31)

(32)

(33)

The state at the next time step is updated as:

(34)

Here, represents the robot’s dynamic model, X_k is the current state, U_k is the control input, and is the integration time step. By averaging the weighted contributions from the four intermediate steps, RK4 provides a highly accurate prediction of the next state [47].

Obstacle avoidance can be achieved by [9]:

(35)

Goal check and stop condition: Equation (35) details the logic to compute the robot’s distance to the goal and terminate the simulation once that distance is below a specified threshold. We compute the Euclidean distance as follows:

(36)

5.4 Fluid-structure interaction

The robot’s position and velocity influence fluid dynamics through the boundary conditions:

(37)

5.5 Visualization of the spatial fields

Velocity fields , vorticity, and robot trajectories are visualized. The robot’s trajectory is updated as [48]:

(38)

(39)

The robot’s ability to achieve smooth trajectory planning, maintain safe distances from obstacles, and reduce the error toward the goal are analyzed. The use of descriptive statistics and graphical insights provides a comprehensive understanding of the robot’s behavior under dynamic environmental conditions under the influence of dynamic fluid. The results are structured to highlight the robot’s path, error reduction trends, velocity control, and obstacle avoidance capabilities.

7.1 Descriptive statistics

The robot’s performance during the simulation demonstrates effective navigation, obstacle avoidance, and goal convergence, supported by the descriptive statistics of the key parameters are shown in Tables 6 and 7. In Table 6, the robot’s X_Position and Y_Position values indicate that the robot predominantly operated near its initial region, navigating slightly to the left and below the origin on average. The theta (orientation) parameter, with an average value of 0.600 radians, suggests the robot primarily maintained a forward-facing direction, with smooth orientation adjustments over time. However, a range of to 1.360 radians and a standard deviation 0.447 indicate that the robot had to reorient itself periodically to navigate effectively around the obstacles. The Euclidean error shows the robot’s progress towards its goal. The mean error of 1.679 m, along with a gradual decrease reflected in the range 2.129 to 2.828 m, which confirms steady movement towards the goal. A standard deviation of 0.619 m suggests manageable deviations from the optimal path, with minor fluctuations likely due to obstacle avoidance. The linear velocity and angular velocity metrics reflect the robot’s controlled motion during the simulation.

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Table 6. Descriptive statistics of robot position, orientation, and Euclidean error during simulation.

https://doi.org/10.1371/journal.pone.0333056.t006

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Table 7. Descriptive statistics of robot obstacle avoidance performance in simulation and distance metrics.

https://doi.org/10.1371/journal.pone.0333056.t007

The linear velocity was highly stable, with a mean of 0.042 m/s and a negligible standard deviation of 0.001 m/s. This consistency underscores the robot’s ability to maintain smooth forward motion. The angular velocity, with a mean of 0.023 rad/s and a broader standard deviation of 0.356 rad/s, reveals occasional sharp turns during obstacle avoidance.

In Table 7, the robot’s ability to avoid obstacles is confirmed by the distance-to-obstacles metrics. For example, the average distance to the obstacle 1 was 1.334 meters (std = 0.171 m), while for obstacle 2, it was 0.609 meters (std = 0.354 m). Although the robot occasionally came close to the obstacles, as indicated by minimum distances as low as 0.005 meters, it maintained a safe buffer zone, with the average minimum distance to obstacles being 0.377 meters. The robot and obstacles were defined with a physical radius of (for a combined minimum separation distance of 0.10 m). All distances reported in Table 7 are in meters. The NMPC constraint was enforced . The non-zero, small minimum distance (0.005 m) is attributed to lattice interpolation artifacts inherent to the Immersed Boundary Method (IBM) used for the LBM coupling. When the solid boundary is mapped onto the discrete lattice nodes, a slight, instantaneous penetration can occur before the no-slip condition is enforced at the next time step. This is a common numerical effect, where the true continuous position of the robot momentarily falls slightly inside the collision boundary defined by the NMPC. This small residual distance is a numerical artifact equivalent to a error relative to the defined 0.10 m safety margin. Crucially, the presence of this small gap confirms that no sustained or problematic collision occurred, and the NMPC system successfully prevented the robot from achieving a zero or negative distance, i.e., complete overlap. Positive skewness values for all distance metrics show that closer proximity was less frequent, while flat kurtosis values suggest the distances were evenly distributed without extreme outliers.

The linear and angular velocity plots in Fig 3 below show the robot’s speed and rotational adjustments during navigation. The linear velocity remains constant for most of the journey, suggesting steady forward motion. The angular velocity shows the changes when the robot adjusts its orientation, particularly during obstacle avoidance towards the end. During turbulent region traversal, the required corrective angular velocity reached the maximum allowable limit of the robot model, suggesting that in a real-world scenario, the robot would have lost control of its orientation and path-following accuracy.

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Fig 3. Comparison of the robot’s linear and angular velocities over time, illustrating the stability of linear motion versus the extreme fluctuations in angular velocity.

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The Euclidean error shows the straight-line distance between the robot and the goal position over time. As expected in Fig 4, the error decreases steadily as the robot moves closer to the target. This trend indicates that the control algorithm effectively guides the robot toward the target. Minor errors in obstacle avoidance maneuvers.

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Fig 4. Euclidean goal-directed error reduction illustrates the decreasing linear distance between the robot and the target location over time.

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The plot of the robot’s trajectory in the x-y plane demonstrates the path followed by the robot during the simulation, as shown in Fig 5. The clear path shown in Fig 6 shows that the robot starts at the initial position (marked in green) and progresses to the goal position (marked in red). The trajectory reflects the robot’s ability to navigate while avoiding the obstacles. The path appears smooth, indicating efficient trajectory planning with minimal deviations.

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Fig 5. Fluid-robot interaction in a dynamic environment: The color map represents fluid velocity magnitude (blue = high, white = low).

https://doi.org/10.1371/journal.pone.0333056.g005

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Fig 6. The robot’s trajectory from its initial point (green) to the target goal (red), demonstrating its ability to navigate while successfully avoiding the obstacles.

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The minimum distance to obstacles in Fig 7 tracks the robot’s proximity to the nearest obstacle during the simulation. The robot consistently maintains a safe distance from obstacles, with no values approaching zero, indicating successful collision avoidance. This behavior validates the obstacle avoidance strategy employed in the control algorithm.

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Fig 7. Minimum distance to obstacle, which represents the obstacle avoidance safety margin.

It illustrates that the robot is maintaining distance from surrounding obstacles during the simulation.

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Table 8 shows the robot’s performance in simulation, demonstrating high different path efficiency before 0.887. The actual path length was 2.828 m, which was longer than the optimal straight-line distance of 2.508 m, indicating that the robot performed well.

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Table 8. Robot performance metrics of the robot in simulation.

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The shortest path distance, d_optimal = 2.508 m, is defined as the straight-line Euclidean distance between the robot’s starting position (P_start) and the target position (P_target). This distance serves as a non-collision baseline ideal for evaluating the kinematic cost of the NMPC trajectory. It is important to note that this d_optimal value does not account for the obstacle constraints and it is the absolute theoretical minimum distance in free space.

The computational efficiency of NMPC algorithm is evident, with an average computation time of seconds per step and a total computation time of 0.0004 seconds, confirming its suitability for real-time trajectory planning. These results reflect a well-balanced approach to safe, efficient, and computationally feasible navigation.

The robot trajectory series plots in Fig 8 show effective navigation from the start to the goal while avoiding the obstacles. At t = 0, the robot is stationary, and by t = 20 seconds, it begins to move along a smooth path, maintaining safe distances from obstacles.

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Fig 8. The temporal evolution of the navigation trajectory, which shows the robot’s path at various time steps.

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At t = 25, it adjusts its path to navigate through a cluster of obstacles, demonstrating precise maneuvering. By t = 60, the robot successfully achieves the objective without collisions, highlighting the robustness and efficiency of the control algorithm in dynamic environments.

7.2 CFD based spatial analysis

Table 9 presents key performance metrics for the fluid-robot interaction, highlighting how the robot navigates the fluid under different Reynolds number conditions. The Table 9 includes drag force components along the X and Y axes, representing the resistive forces acting on the robot. Energy dissipation quantifies the loss of energy due to fluid resistance, while the average velocity and the average vorticity describe the flow characteristics around the robot. Various Reynolds numbers indicate different flow regimes that affect the robot’s movement and stability.

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Table 9. Fluid dynamics parameters calculated from LBM simulations.

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This data is further analyzed in subsequent plots to evaluate robot efficiency and the impact of fluid dynamics on its movement.

The Reynolds number plot in Fig 9 reveals significant variations between successive frames, with values starting low and rising as the simulation progresses. The initial low values suggest a laminar flow regime, as shown in Fig 10, while the spike and subsequent fluctuations indicate a transition to turbulent flow. In this scenario shown in Fig 11, the boundary condition was modified to allow fluid to flow from the left wall while also generating flow from an obstacle, creating a radial flow pattern.

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Fig 9. The Reynolds number evolution during robot navigation, which shows significant variations across simulation frames, with values starting low and increasing over time.

https://doi.org/10.1371/journal.pone.0333056.g009

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Fig 10. An initial laminar flow regime (low values) resulting from the fluid interaction pattern at the bottom.

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Fig 11. Boundary-driven fluid flow patterns: Color gradients represent velocity magnitude (blue = high, white = low).

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The purpose of this adjustment was to observe how the system responds to different boundary constraints and fluid interactions. By setting the boundary condition to flow exclusively from the left wall, the fluid moves from left to the right, interacting with obstacles along its path. This creates turbulence, as shown in the simulation results. The visualization highlights areas of high velocity near the left boundary (in blue) and regions of recirculating flow around obstacles, demonstrating the impact of boundary conditions on fluid behavior. This simulation in time steps: 266 in Fig 12 shows the movement of the robot navigating through a turbulent fluid environment with obstacles. The color map represents the velocity of the fluid, with white indicating low-velocity regions and blue indicating high-velocity areas. At this stage, the robot has progressed significantly from its initial position (bottom left) towards its goal, following a trajectory (indicated by the red line). The fluid dynamics around the obstacles create complex vortices, especially near the clustered obstacles in the upper-mid region. The high-velocity zone around the robot shows that it is maneuvering through strong fluid forces, likely adjusting its path to avoid collisions with the obstacles. This evolution highlights the robot’s influence on the surrounding fluid and how it modifies the flow conditions as it moves. Such transitions may impact the robot’s performance and stability in dynamic environments.

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Fig 12. Complex fluid-robot interaction at time step 266: Color-coded velocity magnitude (white = low to blue = high) with directional vectors illustrating complex fluid dynamics, including vortices and recirculation zones around the obstacles.

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The drag forces in the X and Y directions start at zero and increase sharply during the initial frames as the robot begins to interact with the fluid. The plot in Fig 13 shows that the drag force X is significantly larger than the drag force Y, indicating that most of the resistance experienced by the robot is aligned with its primary motion direction. Following the initial steep rise, the drag forces stabilize, reflecting the robot’s steady velocity and maintaining efficient movement. The initial spike in Y is a result of transient hydrodynamic forces when the robot begins its first forward and turning, not a continuous moving in the Y direction. The fact that the line immediately flattens proves the robot does not continue moving in the Y direction.

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Fig 13. Directional drag forces on the robot, showing an initial increase in the X and Y components due to robot-fluid interaction at the start of the simulation.

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The energy dissipation seen in Fig 14 shows the rate at which kinetic energy is lost in the fluid due to viscous effects. The initial values are minimal, corresponding to the laminar flow regime. As the Reynolds number increases and turbulence develops, energy dissipation spikes occur dramatically, particularly at specific frames where instabilities are observed. The peak energy dissipation indicates regions of high turbulence or flow instability, which may necessitate adjustments to the robot’s design or control strategies to minimize energy losses and enhance efficiency.

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Fig 14. The fluid energy dissipation profile shows a transition from low dissipation during laminar flow to high dissipation during turbulent flow.

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The average velocity seen in Fig 15 reflects the overall speed of the fluid, which increases steadily as the simulation progresses, likely due to the movement of the robot and its impact on the fluid flow.

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Fig 15. The average fluid velocity increases over the simulation, influenced by the movement, and the average vorticity illustrates the fluid’s rotational dynamics.

The most striking observation is the time history of the average velocity, which does not follow a single path but separates into two distinct, stable ranges, a phenomenon known as bimodal oscillation. This indicates the fluid rapidly cycles between two energetic states, likely transitioning between its high-power thrust generation phase and a lower-power recovery stroke.

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The spikes in velocity suggest regions of intensified flow, possibly resulting from wake-up effects or interactions with obstacles. The average vorticity plot complements this by showing the rotational dynamics of the fluid. The higher vorticity values correspond to the formation of vortices behind the robot, indicating wake turbulence, as shown in Fig 16.

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Fig 16. Spatial visualization illustrating the formation of wake turbulence vortex behind the robot, resulting from the robot’s movement.

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The qualitative features of the vorticity contours in Fig 12, serve as a crucial proxy for the unquantified pressure distribution and drag forces acting on the robot. Specifically, the wake, the region immediately downstream of the body, is characterized by strong shear layers and the rapid diffusion of vortex structures. The size and intensity of this wake are directly proportional to the magnitude of the low-pressure region forming behind the body, which dictates the pressure drag. The NMPC actively controls the trajectory to minimize the size of this low-pressure, high-vorticity zone. The resulting small, diffuse vortices indicate that the controlled motion is effectively minimizing the time-averaged pressure difference between the front and rear faces of the robot. This successful active shaping of the wake is essential for limiting the net resistance (drag) and thereby validates the NMPC’s role in optimizing the robot’s path-following energy expenditure in highly dynamic fluidic environments.

7.3 Comparison with existing studies

This work significantly contributes to mobile robot navigation by addressing a critical gap in recent research. While existing studies have focused on disturbance rejection or effective path planning, they often overlook the impact of fluid dynamics on robot performance. To highlight the value of our analytical framework, we provide a comparison in Table 10, which delineates the techniques, key objectives, and significant outcomes of each study, enabling a clear contrast with our current research.

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Table 10. Methodological comparison of robotic navigation studies.

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Our study is distinguished by its innovative analytical framework that integrates a navigation technique with an in-depth fluid dynamics analysis, offering novel insights into the interaction between fluid environments and robotic mobility. The NMPC-planned trajectory achieves a high path efficiency of 0.887 and a short actual path length of 2.828 m, demonstrating its navigational efficiency. Its low computation time confirms NMPC’s suitability for real-time trajectory planning.

Our framework’s distinctive value lies in its capacity to achieve robust performance while also offering essential post-hoc analysis of fluid forces, a feature lacking in most similar studies. This allows us to measure the actual energetic expense and stability of the trajectory, representing a substantial improvement over purely kinematic or simplified dynamic models.

The LBM analysis of the Reynolds number progression from laminar to turbulent regimes highlights the robot’s influence on flow dynamics and the necessity for robust stability measures in high-turbulence conditions. Drag forces underscore the significant resistance faced by the robot, while their eventual stabilization demonstrates the robot’s controlled motion and its ability to maintain efficient movement. Energy dissipation trends reveal critical regions of turbulence, indicating areas where energy losses can be minimized through optimized design [21]. Additionally, the average velocity and vorticity plots illustrate intensified flow and rotational dynamics, pointing to wake turbulence and vortex formation. These findings collectively underline the significance of integrating navigation and fluid interaction analyses for designing efficient and robust robotic navigation systems [22].