Dynamic hovering for uncrewed underwater vehicles via an error-separation-based cooperative strategy

The Earth’s surface is overwhelmingly aquatic with 71% submerged in oceans, rivers, and lakes, most of which remain unexplored. These underwater bodies host resource-rich ecosystems that significantly impact human life, both directly and indirectly^1,2. As the depletion of land-based nonrenewable resources accelerates, the imperative to comprehensively explore and exploit these ocean resources is intensifying for human advancement. To unlock these ocean resources sustainably, researchers and scholars have developed advanced technologies such as underwater vehicles named “Jiaolong” equipped with intelligent systems^3,4. However, the deep-sea environment is characterized by unstructured terrain, extreme hydrostatic pressures, and unpredictable disturbances, which inevitably compromise operational efficiency and pose significant safety risks for manned submersible vehicles. Consequently, uncrewed underwater vehicles (UUVs) have garnered growing attention owing to their enhanced safety and cost-efficiency in the exploration and sustainable utilization of underwater resources^5,6.

Up to now, UUVs have been widely applied in many marine tasks such as ocean exploration, seafloor mapping, deep water mining, etc^7,8. In the above-mentioned applications, control, planning, and localization are all important^9,10. Among them, control is the core because no operators are inside the UUV to make adjustments^11,12. In the above-mentioned applications, designing a reliable controller is essential for successful marine tasks due to the absence of operators inside the UUV to adjust the vehicles to arrive at the desired position. Some efforts have been made to integrate the advanced control strategies into UUVs¹³. For instance, a 6-degree-of-freedom (DOF) UUV has been successfully implemented with a classical proportional-integral-derivative (PID) control algorithm to accomplish precise trajectory tracking tasks¹⁴. Following a similar framework, researchers also proposed an enhanced PID algorithm and demonstrated its efficacy in depth-tracking missions for the hovering UUV¹⁵. Moreover, a novel PID approach coupled with computational fluid dynamics is developed for UUVs to enable precise trajectory tracking while ensuring constant velocity¹⁶. Despite the straightforward implementation of PID algorithms in UUVs, their broader application is hindered by the intricate challenges of parameter optimization and their inherent limitations in complex tasks.

To address these limitations, the linear quadratic regulator (LQR) is introduced for UUVs to solve the control inputs by optimizing a cost function, subject to the performance objectives of complex missions. The LQR method is proposed for UUVs, yielding a 32.04% improvement in tracking performance compared to the traditional PID algorithm¹⁷. Given the excessive optimization time per control step in complex nonlinear systems, a novel path-following LQR approach is proposed, utilizing a simplified deep deterministic policy gradient algorithm to enhance computational efficiency and control precision¹⁸. Furthermore, a prescribed-performance fuzzy LQ fault-tolerant control framework is presented for UUVs, tailored for three-dimensional trajectory tracking in complex marine environments¹⁹. The developed scheme effectively mitigates unknown actuator faults, thereby ensuring precise control performance under adverse operational conditions.

The primary focus of the discussed control strategies has been to maintain the vehicles as closely as possible to their desired trajectories under ideal conditions. However, UUVs often navigate in highly uncertain and unknown circumstances, which may reduce the efficacy of these approaches and even result in complete operational failure. In particular, thrusters’ performance variations, payload changes, and tether drag altering will lead to high degree uncertainties of UUVs’ parameters. Then, the effectiveness of control strategies, depending on accurate models, is drastically limited. On the other hand, UUVs inevitably suffer from the currents and waves caused by tidal energy and topographical undulations in the deep sea. These disturbances can also result in unregulated and unpredictable movements of UUVs^20,21. Therefore, designing a high-performance controller to compensate for the influence of uncertainties and disturbances is crucial for UUVs motion.

To handle the above challenges, the sliding mode control (SMC) based control framework emerges as a highly suitable approach for high-precision missions of UUVs²². In SMC, the control law is designed and imposed through a sliding manifold²³. The high-frequency switching input mechanism inherent in SMC enables rapid compensation for perturbations, ensuring system stability and consistent performance despite parametric variations and external disturbances. Given this advantage, SMC is widely applied in UUVs control^24,25. Liu and Zhang proposed a terminal SMC by integrating a piecewise error transformation into the sliding manifold, effectively compensating for hydrodynamic disturbances in marine systems²⁶. For the precise position regulation of remotely operated UUVs, an SMC scheme integrating time-delay estimation is presented to address hydrodynamic disturbances effectively²⁷. Aiming to compensate for the bounded disturbances while minimizing control chattering, a hybrid control strategy combining integral sliding mode and super-twisting algorithm for precise 3D trajectory tracking of 4-DOF UUVs is developed²⁸. Despite the demonstrated efficacy of SMC algorithms in UUVs, it requires high control efforts. This expensive control cost is primarily driven by the necessity for continuous high-gain switching to maintain system trajectories on the sliding manifold. Moreover, the tracking accuracy often experiences significant degradation under extreme hydrodynamic loading conditions due to the fixed feedback gain parameters inherent in SMC, which lack adaptability to dynamic environmental variations.

In fact, UUVs commonly need to perform ocean missions for several hours in different regions. Throughout this period, UUVs are subjected to a variety of operational environments, including disturbance from ocean currents, modifications in seabed topography, the tranquility of calm sea surfaces, etc. Each environment imposes different primary control objectives. For instance, strong robustness is required when the ocean current occurs, and high control accuracy and low control cost are preferable in the calm deep sea. However, it is difficult for a single control strategy to satisfy all the control objectives mentioned above during the whole mission period. Therefore, how integrating multiple control methodologies into a unified hybrid framework to preserve the strengths and complement its drawbacks could be a very promising solution for UUVs missions. This is the primary motivation of this study.

Recently, advances have been made to enhance robustness and adaptability under complex underwater conditions. For example, a distributed tube model predictive control (tube-MPC) is used to handle communication delays and disturbances of multiple UUVs under dynamic marine environments^29,30. To address the low-visibility conditions in controlling heterogeneous multi-UUV systems, researchers used graph-based models and relative states to achieve robustness and real-time performance in simultaneous localization and mapping^31,32. Recent papers also combine integrating event-triggered MPC, nonlinear Lyapunov control, and collision avoidance to enhance tracking accuracy, safety, and computational efficiency when facing current disturbance^16,33.

Motivated by these developments, this study proposes a cooperative control (COC) framework that combines the SMC algorithm’s strong robustness and rapid responsiveness with the high accuracy and low control cost offered by LQR. Specifically, a deviation separation approach is developed for the first time to segregate the errors during UUVs mission. Based on the real-time error separation result, the deviation from the current disturbance is rejected by the SMC, and LQR achieves the precise hovering task. Through this approach, UUVs equipped with the proposed COC framework can better address varying operational challenges and fulfill diverse mission objectives. Contributions of this work are summarized as follows:

1. (1)

   A novel COC protocol is developed for UUVs to adeptly maintain the superior robustness of SMC architecture while meticulously leveraging the efficacy of LQR to mitigate its inherent high control efforts and frequent switching. With the designed cooperative framework, the controlled UUVs can perform underwater missions efficiently and precisely with low control consumption for long operation duration.

2. (2)

   A deviation separation strategy is introduced for the first time to address the challenge of coupling underwater missions. By leveraging the influence function, we establish an analytical relationship between variations in disturbances and corresponding changes in control performance, enabling the estimation of disturbances based on control outcomes. Consequently, the deviation in hovering within the marine environment is decoupled into two sub-deviations: one corresponding to the hovering task and the other to the anti-disturbance task. This strategy facilitates the design of a tailored controller for each decoupled task, thereby enhancing overall system performance.

3. (3)

   This proposed controller can mitigate a wide range of disturbances without requiring prior knowledge, such as the size or source of the ocean disturbance. The disturbance is estimated at each point in real-time using the deviation separation strategy. Consequently, the controller can effectively compensate for the effects of the disturbance without the need for additional information. This approach ensures that the controlled UUVs remain stable, even in the event of a failure of one of the sub-controllers.

The hovering of UUV plays a crucial role in various underwater missions. As shown in Fig. 1, the primary objective is to maintain the position (x, y, z) and orientation (ϕ, θ, ψ) at a desired pose while performing tasks such as object retrieval, device connection, or underwater docking. During these processes, the UUV’s previous motion state and the ocean currents’ movement can significantly impact the pose in the next moment. As a result, two primary challenges emerge for UUV underwater operations: maintaining hovering at the desired pose and counteracting current disturbances. We address the coupled tasks of hovering and disturbance suppression. A variety of complex underwater missions, including military reconnaissance, hull inspection, scientific exploration, and fisheries, can be analyzed as different combinations of these fundamental control subtasks, with UUV hovering and anti-disturbance being a consistent requirement across them. This classification framework distinguishes itself from traditional, application-centric approaches by prioritizing the fundamental control of the UUV’s motion states.

Current disturbances can be attributed to two primary sources. The first is wind-driven currents, generated by wind forces acting on the ocean surface. These currents typically weaken with increasing depth, with the affected depths extending several hundred meters. The second source of disturbance arises from changes in seawater temperature and salinity, which alter the density distribution of the ocean. These variations cause discrepancies between the ocean’s isobaric and equipotential surfaces, thus generating ocean currents. Such disturbances primarily affect the forces and moments acting on the UUV, leading to deviations in the control inputs generated by the thrusters. In addition to the current disturbance, random noise can also arise from various independent events, such as the collapse of tiny bubbles produced by the thrusters. Although this kind of noise typically has a minimal impact on the UUV’s overall performance, continuous correction is necessary to prevent the accumulation of errors over time.

Related research indicates that the abovementioned disturbances can be mitigated using SMC, which applies continuous high-gain thrust to counteract current disturbances. On the other hand, optimal strategies such as the LQR can effectively address the hovering tasks influenced by thruster noise. However, SMC treats all deviations as significant disturbances, applying high-gain thrust continuously, resulting in inefficient energy consumption. While LQR is more thrust-efficient for hovering tasks, it fails to suppress current disturbance. Therefore, in practical scenarios, relying on a single control strategy cannot simultaneously meet the multiple critical objectives of low power consumption, robust anti-disturbance performance, and high precision for UUV hovering tasks.

To address this, we propose a COC strategy, as illustrated in Fig. 2, which combines the robust and rapid responsiveness of the SMC algorithm with the energy efficiency of LQR. By employing a deviation separation mechanism, we decompose the complex task of maintaining the UUV’s desired pose in dynamic ocean environments into two distinct sub-tasks: the traditional hovering and anti-disturbance tasks. We then design two sub-controllers (LQR and SMC) tailored to each task based on their strengths. Finally, the thrust outputs from both sub-controllers are integrated and constrained through engineering techniques to form the cooperative controller, ensuring optimal performance while maintaining robustness.

To evaluate the engineering feasibility of the proposed COC strategy, a computational analysis was performed on a standard desktop platform (Intel Core i7-13700F, 16GB RAM). All algorithms—including deviation separation, LQR, and SMC sub-controller computations—were implemented discretely with a fixed control cycle corresponding to a 1 Hz sampling frequency over 2000 steps. Under these conditions, the total execution time of the entire simulation was 20.687 s, while the cumulative execution time for all control loop computations was measured to be 0.124 s. This duration includes all control matrix operations, sliding surface evaluations, and separation matrix computations, amounting to less than 0.5994% of the total available time budget. These results confirm the strategy’s suitability for real-time implementation.

Control accuracy comparisons under different control policies

This section mainly considers the horizontal non-rotational ocean current during the UUVs hovering mission described in Fig. 1. Results and analysis under other interference conditions could be found in Sections S.4–S.6 of the supplementary file. Referred to reference³⁴, the total disturbance is a combination of current disturbance (a sine function with an amplitude of 1 and a frequency of 5 Hz) and thruster noise (a standard Gaussian distribution signal). The initial pose of the UUV is set as [45, 45, 110, 0, 0, 0]^T, representing its starting position and orientation in the global coordinate system, where the first three values denote the position in meters along the x, y, and z axes, and the last three values indicate roll, pitch, and yaw angles in radians. This initial pose simulates the UUV’s offset from the desired hovering location at the start of the mission. The target hovering pose is set as [50, 50, 100, 0, 0, 0]^T., representing the desired location and orientation the UUV aims to reach and maintain during the hovering task. Assuming the current disturbance affects the system at the 1000th step. To illustrate the effectiveness and superiority of our proposed COC algorithm, the control performance of SMC and LQR under parameters R = I, Q = 100I, and z = 2 is compared. Detailed mathematical derivations of SMC and LQR algorithms can be found in Sections S.2–3 of the supplementary note.

Thrusts computed by LQR, SMC, and COC are applied to UUVs. Figure 3 shows the current and desired pose deviations. The embedded inset figures on certain phases provide a detailed view of a more precise comparison of tracking performance. To better analyze the results, we roughly divide the process into three stages: initial value convergence, stabilization, and anti-disturbance. In particular, during the initial value convergence stage (within 650 steps), the gray lines reach the desired point (zero in this experiment) faster and closer than the blue lines. The above trend indicates that the COC framework has a faster convergence speed and higher steady-state accuracy than LQR. This enhanced performance arises from integrating SMC’s advantages during the initial phase, where COC leverages significant, high-frequency thrust adjustments to rapidly drive the UUV’s posture toward the desired state. By combining the robustness of SMC with the precision of LQR, COC achieves superior dynamic response and stability.

In the stabilization phase (between 650 and 1000 steps), the blue lines in all subfigures converge to within at least 5% of the maximum deviation from the desired state. In contrast, red lines in all subfigures of Fig. 3 other than Fig. 3f remain closer to the desired state but oscillate within 1% of the maximum deviation. This phenomenon indicates a static error exists between the UUVs hovering pose and the desired pose under the control of LQR. Although SMC has no static error, Fig. 3f shows some situations in which the pose states oscillate. During the initial 1000 steps in Fig. 3e, the slow convergence of the blue lines prevents it from reaching a stable state, indicating that LQR struggles to stabilize the system rapidly in the presence of significant initial deviations or sudden disturbances. This limitation stems from LQR’s reliance on an accurate model.

The COC framework overcomes the drawbacks of both LQR and SMC, as evidenced by the gray lines’ steady trends in Fig. 3. These lines exhibit smoother trajectories, with reduced oscillation amplitude and frequency compared to SMC, while maintaining more minor pose deviations than LQR. These trends demonstrate that when the deviation state of COC achieves a high accuracy, it does not overreact to minor deviations, which ensures the system will not oscillate. Further, the cooperative mechanism ensures that, once the pose deviation converges within a desired level, the thrust input transfers from a severely oscillating one to an efficient and stable one. Therefore, the COC system only needs a small thrust input to stabilize the system, thereby minimizing control consumption while ensuring accuracy.

During the anti-disturbance phase (after 1000 steps), the gray and red lines remain closer to the desired state than the blue lines, demonstrating that both SMC and COC achieve higher steady-state accuracy than LQR. However, the lines of SMC exhibit occasional high peaks and persistent oscillations. For instance, between 1280 and 1360 steps in Fig. 3a, the red line peaks above the blue line, indicating that SMC can induce significant deviations in the pose state. Similarly, in Fig. 3f, the fluctuation of the red line reaches a peak of 70°, indicating that the roll angle of the UUV hull is approaching a critical lateral capsizing threshold. Moreover, the red line in this subfigure oscillates, failing to reach a steady state before subsequent disturbances occur. These issues arise because SMC treats all disturbances as maximum disturbances, often applying high-frequency and high-thrust corrections even for minor disturbances, leading to severe overshoot.

However, the gray lines in the same period do not exceed the blue lines, indicating that COC effectively avoids the overshoot problem associated with aggressive control inputs. By utilizing the deviation separation mechanism, the COC framework dynamically modulates the weight ratio between LQR and SMC, effectively mitigating overshoot in the presence of minor disturbances. Ensures robust and precise control, overcoming the inherent limitations of both SMC and LQR in dynamic and uncertain environments.

As discussed in the previous section, the effectiveness of the COC framework relies heavily on the performance of the deviation separation mechanism. To vividly illustrate the impact of this mechanism, Fig. 4 compares the separated sub-deviations. It delineates the relative contributions of thruster noise and current disturbance to the total deviation, offering critical insights into the system’s dynamic behavior. From the beginning to the 650th step, the purple region dominates, indicating that the sub-deviation caused by the current disturbance occupies most of the total deviation. This dominance arises because the initial values deviate from the desired state, similar to a sudden current disturbance affecting the UUV. As a result, the purple region predominates, and according to the collaborative control mechanism, the proposed COC algorithm is primarily governed by SMC. Due to SMC’s superior control accuracy and dynamic response, the COC framework achieves faster convergence and more minor steady-state deviations than LQR during this phase.

Upon focusing on the stabilization phase (from step 650 to 1000), the purple region steadily diminishes, signifying that the sub-deviations stemming from the initial pose have progressively converged. At this juncture, the COC strategy transitions to an LQR-dominated regimen, which persists until the onset of the current disturbance. As previously discussed, the aggressive thrust application of SMC is primarily responsible for the oscillations in the pose state. COC addresses this issue by integrating the thrust constraints of LQR. With the pose state that has converged and oscillated within a narrow range, COC achieves the best control accuracy with limited thrust and mitigates the oscillation cases. During the initial stabilization process, COC harnesses SMC’s rapid convergence and high precision while circumventing the drawbacks of substantial overshoot and protracted oscillations, offering a refined control solution for the UUV.

Then, focus on the post-1000 steps in Fig. 4. The total deviation increases primarily due to the growth of the pink segment, which represents the sub-deviation caused by thruster noise. This trend persists until approximately 1200 steps, when the purple area, the sub-deviation from the current disturbance, begins to emerge. The delayed onset of the purple area can be attributed to the fact that, in the early stages of current influence, the total pose deviation has not yet accumulated sufficiently to trigger adjustments in the COC’s collaborative strategy. During this phase, the COC remains predominantly governed by LQR, which is less effective in mitigating the immediate effects of current disturbances. As a result, the deviation accumulates until around 1200 steps, after which the purple area expands, and its contribution to the total deviation increases.

This observation reveals the dynamic interplay between the sub-deviations, where the dominance of one is swiftly mitigated, only for the other to become more pronounced. This cyclical process gradually diminishes the total deviation, enabling the UUV’s pose state to approximate the desired state continuously. The underlying mechanism driving this behavior can be explained as follows: when the COC transitions from an LQR-dominated to an SMC-dominated strategy, the impact of current disturbance and its associated sub-deviation begins to subside. As the sub-deviation from the current disturbance decreases, the relative contribution of the thruster-noise sub-deviation to the total deviation increases, prompting a shift back to an LQR-dominated strategy within the COC framework. This cyclical regulation results in the observed proportional fluctuations. Throughout this fluctuation period, the UUV’s pose accuracy is iteratively refined, and the system efficiently manages thruster consumption, thereby avoiding unnecessary power expenditure. This adaptive response underscores the effectiveness of the proposed COC in maintaining optimal control performance despite environmental disturbances.

Thrust and control performance comparisons under different control policies

In UUV hovering missions, thrust consumption is a critical factor, as is the system’s anti-disturbance performance and hovering error. In the above experiment, eight thrusters generate the needed force and moment, including two main thrusters, four vertical thrusters, and two lateral thrusters. This subsection presents a comparative analysis of the thrust curves under LQR, SMC, and COC control strategies. As illustrated in Fig. 5, during the initial phase of the convergence process (within 300 steps), the red lines, representing the SMC strategy, exhibit significant overshoot and oscillation. This behavior indicates that SMC utilizes an aggressive thrust output to mitigate deviations caused by initial conditions swiftly. However, this approach often leads to overshooting and thruster output saturation.

In contrast, COC uses a collaborative strategy that inherently constrains thrust output, mitigating excessively aggressive thrust responses at the expense of a slightly extended adjustment period. For instance, as illustrated in all six subfigures of Fig. 5, COC avoids the extreme thrust amplitudes ranging from  −300 N to 200 N observed in SMC. As a price, it achieves stabilization within 200 steps. During the latter phase of the convergence process, the red and gray lines exhibit high-frequency oscillations within the range of  −50 N–50 N. In contrast, the blue line, representing LQR, maintains smaller thrust values. This case indicates that SMC and COC continue applying thrust to suppress disturbance. In summary, COC effectively mitigates the issues of overshoot and thruster saturation associated with aggressive thrust outputs while maintaining precise pose control during the initial value convergence process.

After 650 steps, the lines of COC in Fig. 5 hold at a lower value and exhibit no oscillations compared to those of SMC. It indicates that, at this stage, COC achieves high control accuracy without relying on high-frequency thrust inputs. This phenomenon can be attributed to the cooperative strategy transitioning to an LQR-dominated approach once the deviations converge within a sufficiently small range. Given COC’s excellent steady-state accuracy, only minimal control inputs are required to correct residual deviations. As a result, the thrust outputs under COC are significantly reduced compared to SMC and are even lower than those of LQR, demonstrating the efficiency of the proposed control framework.

After 1000 steps, the red lines exhibit slightly more pronounced oscillations, while the gray lines resume oscillating after another 200 steps. Although COC and SMC achieve comparable control accuracy under current perturbations, the former requires a longer response time. This delay exists because the deviations are too small to prompt a shift in COC’s cooperative strategy. As deviations accumulate, the COC strategy progressively shifts towards SMC, increasing thrust outputs to counteract the effects of current disturbance quickly. Leveraging the coordination mechanism, COC dynamically adjusts the control weights between LQR and SMC in response to the disturbance, thereby maintaining high control accuracy with less consumption than only SMC.

Keep focusing on the distinct thrust variation trends observed between steps 1400 and 1500 in Fig. 5g to elucidate the interaction between the coordination and deviation separation mechanisms. The enlarged view within Fig. 5g reveals that during steps 1440–1450, the gray line remains close to zero. However, for the rest of the period between steps 1400 and 1500, the gray line oscillates within a narrow range of  −3 N to 3 N. The former case indicates that COC stabilizes the UUV with minimal thrust, while the latter suggests that COC still requires significant thrust input to maintain system stability. This cyclical behavior occurs because when the sub-deviations caused by current disturbances are significantly suppressed through an SMC-dominated strategy, the COC coordination mechanism automatically transitions to an LQR-dominated mode. Then, there is a reduction in thrust. As a result, the influence of the thruster-noise sub-deviation diminishes until the coordination mechanism reverts to an SMC-dominated approach. This self-regulating collaborative mechanism mitigates the drawbacks of SMC’s tendency to switch among significant, opposing thrusts and LQR’s overly conservative thrust application. Consequently, COC consistently selects a more minor, smoother thrust profile while maintaining control accuracy, thereby reducing thruster wear and enhancing precision to achieve superior overall control performance.

Finally, we present a comparative analysis of the three control strategies’ overall performance in Fig. 6. For each segment, deviations and thrusts are first expressed in quadratic form and then summed to derive the control cost, which is represented as a bar chart. Given the substantial control inputs associated with SMC, the logarithm ({log }_{10}) of all control costs is applied to better visualize differences across orders of magnitude. As depicted in Fig. 6, the control cost of COC in the initial two segments exceeds that of LQR but remains lower than that of SMC. This is attributed to COC’s strategy of stabilizing the UUV through high-frequency thrust adjustments with lower amplitude than SMC. In subsequent segments, where deviations have significantly converged after 1000 steps, the gray bars consistently remain the lowest, indicating that COC achieves minimal overall control costs. These findings further validate that COC attains superior control accuracy with reduced control effort through its collaborative control strategy.

In summary, the above subsections analyze the control performance of LQR, SMC and COC controllers under the influence of current disturbance. With the comparisons between three control strategies, it is shown that the COC can obtain an optimal performance while balancing the need for robustness and consumption. Due to space limitations, the further exploration of the control performance in other scenarios is presented in the supplementary file from Sections S.4–S.6, including stable ocean current disturbance, increasing current disturbance, and load changing disturbance.

In this study, we proposed a novel COC framework that synergistically integrates SMC’s robustness and rapid responsiveness with the precision and low control cost of LQR. By developing a real-time error separation strategy, we successfully decoupled the challenges of disturbance rejection and precise hovering, enabling UUVs to operate efficiently in complex underwater environments. The proposed framework demonstrated remarkable adaptability to current disturbance without requiring prior knowledge of the disturbance characteristics. Simulation results revealed that the developed COC framework effectively mitigates the impact of current disturbance while maintaining precise hovering performance. Specifically, while the COC cost function remains optimal, the state response of position and direction under COC control remains fast and accurate. This highlights the ability of the proposed framework to achieve high-precision control with low energy consumption, making it particularly suitable for long-duration underwater missions. Furthermore, the COC framework demonstrated exceptional robustness in scenarios where other controllers failed, ensuring system stability and preventing catastrophic outcomes.

These findings underscore the potential of the COC framework to revolutionize underwater robotics by addressing the dual challenges of robustness and precision in dynamic marine environments. Future research will extend this framework to multi-agent systems and explore its applicability in more complex and unstructured underwater scenarios. This work represents a significant advancement in UUV control, offering a versatile and reliable solution for the growing demand for underwater exploration and operations.

Uncrewed underwater vehicles model

Consider a UUV system in the form of (For the details of the model settings, please see Section S.1 of the supplementary file.) :

where η = [x, y, z, ϕ, θ, ψ]^T represents the pose vector of the UUV in the inertial reference frame, and v = [u, w, v, p, q, r]^T denotes the velocity vector, which includes both linear and angular velocities in the body-fixed reference frame. The control input, influenced by the current disturbance δ_c, is expressed as ({{{boldsymbol{tau }}}}_{delta }={[{F}_{x},{F}_{y},{F}_{z},{M}_{phi },{M}_{theta },{M}_{psi }]}^{{{rm{T}}}}), representing the resultant force F and moments M along the x, y, and z axes, as well as the orientation angles ϕ, θ, and ψ. The matrix M(v) denotes the inertia matrix (which includes the added mass), C(v) refers to the matrix accounting for the centripetal force and the Coriolis forces, D(v) represents the hydrodynamic damping matrix, and g(η) is the vector of hydrostatic forces. In addition, ΔM(v), ΔC(v), ΔD(v) and Δg(η) represent the model uncertainties on the above matrixes generated from the loading changing or inaccurate modeling situations. The influence of these model uncertainties can be viewed as an external disturbance term as δ_m, from where the above model (1) can be simplified to:

where δ is the integrated disturbance with δ = δ_c + δ_m. The model presented in equation (2) is referred to as the continuous model in this paper, which provides an intuitive representation of the relationship between thrust, pose, and velocity. Table 1 introduces the common mathematical expressions used in this article.

Given the widespread adoption of digital controllers, designing controllers directly in the discrete-time domain effectively mitigates the delay issues introduced by sampling devices, such as zero-order hold elements, thereby ensuring the stability of the control system. The discrete-time form of the model (2) is derived to accommodate the dynamics of digital systems more effectively:

where dt represents the sampling time, determined by the hardware requirements, and k denotes the discrete time index. v_k, v_k+1, and η_k, η_k+1 represent the velocity vector v and the orientation vector η at time steps k and k + 1, respectively. τ_δ,k and δ_k denote the control input and the current disturbance at time step k. The matrices M(v_k), C(v_k), D(v_k), g(η_k), and J(v_k) retain the same physical interpretation as in the continuous model (2). Still, they represent the discrete-time model’s corresponding functions of v_k and η_k.

In practical applications, the hydrostatic force g(η_k) can be considered negligible and effectively treated as zero. To simplify the controller design, a Taylor expansion at the time step k = 0 is applied to the model above, thereby establishing a linear relationship between the vehicle’s pose and the forces and moments generated by the thrusters:

where ({{{bf{x}}}}_{{{boldsymbol{delta }}},k}={[{{{boldsymbol{eta }}}}_{k}-{{{boldsymbol{eta }}}}_{r,k},{{{bf{v}}}}_{k}-{{{bf{v}}}}_{r,k}]}^{{{rm{T}}}}) represents the deviation between the desired pose η_r,k and velocity v_r,k, and the current pose and velocity under the influence of current disturbance only, which constitutes the system state. The following system matrix A_k and input matrix B_k are defined as:

η₀, v₀, and τ₀ represent the values of η, v, and τ_δ at the time step k = 0, respectively. I is the identity matrix with the same dimensions as J(η_k). H(v₀, τ₀) is an intermediate variable introduced to simplify the notation.

A COC is designed to ensure stability and high precision during UUV hovering under the influence of currents for the UUV system described above. As previously noted, the controller manages hovering and disturbance rejection functions. To illustrate the coupling between these two objectives, we present the mismatched UUV model, derived from the discrete model (3), as follows:

where x_k and τ_k represent the system state and thrust input of the mismatched model, respectively. Their physical meanings are identical to those of x_δ,k and τ_δ,k. The only distinction is that, in contrast to model (4), random noise ({{{bf{w}}}}_{k}in {{mathscr{N}}}(0,{{{bf{W}}}}_{k})) from the thrusters is introduced. The remaining terms are consistent with those in the model (4). The model presented in equation (6) represents the actual model in this paper, and the subsequent derivations are based on this model.

The second-order linear model adopted in this study is widely used in the control design of underwater vehicles, mainly when the vehicle operates within a limited envelope under hovering or low-speed conditions. This modeling assumption is well-suited for streamlined, low-damping UUVs with symmetric hull structures and 6 degrees of freedom (6-DOF) control realized through vector thrusters or distributed actuators. As supported by prior studies^35,36, such reduced-order models preserve the dominant dynamics relevant to pose regulation and disturbance rejection while greatly simplifying controller design and real-time implementation. However, we acknowledge that more sophisticated nonlinear models may be required for UUVs performing high-speed cruising, exhibiting complex hydrodynamic interactions (e.g., crossflow effects), or undergoing large-angle maneuvers. In such cases, the proposed COC framework can be extended by incorporating nonlinear control strategies (e.g., gain scheduling, feedback linearization), as long as the core deviation separation mechanism is maintained.

In this paper, we assume the hydrostatic force g(η_k) is negligible and can be treated as zero. This assumption is valid for typical UUVs with slightly positive buoyancy operating at shallow to moderate depths, where the effect of hydrostatic restoring forces is minimal and can be safely ignored for pose control purposes. However, this assumption may no longer hold for deep-sea operations or missions involving heavy payloads due to significant variations in buoyancy and restoring torques. The term g(η_k) should be retained in the dynamic model. Nevertheless, doing so does not affect the structural validity of the proposed COC framework. The additional terms can be incorporated into the LQR and SMC components through model-based feed-forward compensation, thus compensating without affecting the cooperative framework.

Cooperative control framework for UUV

The previous section introduced the mismatched model, and the thrust required to stabilize the UUV system was calculated. Relying solely on SMC to address the coupled tasks would lead to inefficient utilization of limited power resources. In contrast, LQR alone is insufficient for handling current disturbances³⁷. Therefore, a combined control strategy is required to exploit the strengths of both approaches.

However, to enable each sub-controller to perform its respective task, it is necessary to first decouple the mismatched model into two distinct models, each representing the hovering task and the disturbance rejection task, respectively. The fundamental framework of COC relies on feedback control, with the only available feedback being the deviation between the current pose and the desired one. Therefore, a deviation separation mechanism is introduced to partition the overall deviation into sub-deviations, each corresponding to the specific characteristics of the respective tasks. Mathematically, this analysis can be expressed as follows:

where x_ε,k is the vector representing the separated sub-deviation from thruster noise, while x_δ,k is the sub-deviation from current disturbance as introduced earlier. These two sub-deviations are related by the equation x_k = x_δ,k + x_ε,k. Specifically, x_δ,k corresponds to the sub-deviation associated with the anti-disturbance task, whereas x_ε,k represents the sub-deviation related to the hovering task.

Recent researches show that the linear quadratic problem can be reformulated as an optimization problem involving probability distributions, which transforms the task of calculating optimal thrust into maximizing the posterior probability distribution of the deviation state x_k^38,39. Building on these findings, the effect of minor disturbances on the posterior distribution can be estimated by analyzing the influence function. Similarly, the anti-disturbance task can be viewed as a minor disturbance influencing the hovering task. By examining the influence function of the LQR, we can quantify the impact of the anti-disturbance task on the hovering task under the thrust calculated by the LQR. In this context, it corresponds to the influence of the current disturbance on the hovering task. Subsequently, the estimated sub-deviation caused by the current disturbance can be computed by inverting the influence function. The difference between the current deviation and the calculated sub-deviation represents the estimated sub-deviation of the hovering task, which forms the foundation for the separation mechanism.

Mathematically, the optimal thrust in LQR is computed by solving an optimization problem subject to the constraints of the nominal model, as described in Eq. (4). When current disturbances affect the UUVs, the dynamic model transitions from the nominal model (4) to the mismatched model (6). As a result, the optimization problem is modified as follows:

where Q and R are two weighting matrices defined artificially, J_k and J_ε,k represent the cost functions of the COC and LQR, respectively. ({{{boldsymbol{tau }}}}_{k}^{star }) is the optimal control thrust. ({J}_{k}^{star }) is the optimal cost function of the COC, minimized by ({{{boldsymbol{tau }}}}_{k}={{{boldsymbol{tau }}}}_{k}^{star }), which can also be expressed as ({J}_{k}^{star }={min }_{{{{boldsymbol{tau }}}}_{k}}{J}_{k}). P_k+1 is a symmetric matrix obtained through the Riccati equation (The detailed derivation process is shown in Section S.3 of the supplementary file). The bias between ({{{boldsymbol{tau }}}}_{k}^{star }) and τ_ε,k is defined as ({tilde{{{boldsymbol{tau }}}}}_{k}). According to the maximum principle, the first derivative of Eq. (8) satisfies:

For simplicity, let (f({{{boldsymbol{tau }}}}_{k}^{star })) denote the left-hand side of Eq. (9). By performing a second-order Taylor expansion of Eq. (9), we obtain:

Substituting Eq. (9) into the above equations and noting that τ_ε,k minimizes J_ε,k, it follows that the partial derivative (frac{partial {J}_{{{boldsymbol{varepsilon }}},k}}{partial {{{boldsymbol{tau }}}}_{k}}{| }_{{{{boldsymbol{tau }}}}_{k} = {{{boldsymbol{tau }}}}_{{{boldsymbol{varepsilon }}},k}}) equals to zero. By substituting ({tilde{{{boldsymbol{tau }}}}}_{k}) for the corresponding terms in Eq. (10) and neglecting higher-order terms, Eq. (10) simplifies to:

By substituting the cost function expression (8) into the above formula (11), the final expression for ({tilde{{{boldsymbol{tau }}}}}_{k}) is obtained as:

Since τ_ε,k is independent of x_δ,k, the first-order partial derivative of ({tilde{{{boldsymbol{tau }}}}}_{k}) with respect to x_δ,k can be expressed as:

As mentioned earlier, x_δ,k, which represents the sub-deviation of the anti-disturbance task, influences the cost function. This influence can be quantified by performing a partial derivative. Thus, to calculate the change in the cost function resulting from the sub-deviation x_δ,k, we take the derivative of J_k along the x_δ,k direction. The partial derivative mentioned above is denoted as L_k, and its expression is:

Thus, we establish the relationship between the partial derivative and the sub-deviation we aim to estimate. By performing an inverse integration on the partial derivative, the expression for the sub-deviation can be written as ({{{bf{x}}}}_{{{boldsymbol{delta }}},k}={{{bf{L}}}}_{k}^{-1}({J}_{k}-{{boldsymbol{beta }}})), where ({{boldsymbol{beta }}}=min {J}_{k}| forall ({{{bf{x}}}}_{k},{{{boldsymbol{tau }}}}_{k})=0). The term β is close to zero due to the optimality of LQR, and since ({{{bf{L}}}}_{k}^{-1}) is bounded, the expression for x_δ,k is approximated as ({{{bf{L}}}}_{k}^{-1}{J}_{k}). In the ocean environment, where current changes are slow, the UUV control problem can be modeled as a system with a fast control frequency and slow time-invariant disturbances. Therefore, we assume that x_δ,m ≈ x_δ,m+1 ≈ ⋯ ≈ x_δ,k, where m = k − M₂ + 1 is a different time step and M₂ represents the dimension of the sub-deviation vector x_δ,k. Consequently, the full expression for x_δ,k is given by ({{{bf{x}}}}_{{{boldsymbol{delta }}},k}={{{bf{J}}}}_{m,k}{{{bf{L}}}}_{m,k}^{-1}), where J_m,k = [J_m, J_m+1, …, J_k] and L_m,k = [L_m, L_m+1, …, L_k] are two augmented matrices of J_k and L_k, respectively.

In the previous subsections, the sub-deviation of the anti-disturbance task was separated from the total deviation. To express the COC more concisely in subsequent sections, we define the separation matrix Φ_k such that ({{{bf{x}}}}_{{{boldsymbol{delta }}},k}={{{mathbf{Phi }}}}_{k}{{{bf{x}}}}_{k}={{{bf{J}}}}_{m,k}{{{bf{L}}}}_{m,k}^{-1}). The analytical expression for Φ_k is derived as follows:

where ./ denotes element-wise division of the corresponding elements, and diag(…) represents a matrix with the vectors in (…) as its diagonal elements. If the system is stable at the desired pose or has little current disturbance, no sub-deviation is generated from the anti-disturbance task. Consequently, the value of L_k becomes very small, and its inverse grows exceedingly large. In order to avoid numerical instability when the matrix L_k is tiny, a threshold control strategy is introduced when constructing the separation matrix Φ_k. Specifically, a safety bound is set for the augmented vector L_m,k as ∣∣L_m,k∣∣ ≤ ε, where ∣∣L_m,k∣∣ is the determinant of L_m,k and ε is an artificial defined scalar. When the strategy is satisfied, the deviation caused by the disturbance is almost negligible, and the LQR controller is sufficient to stabilize the system without the need for the SMC compensation term, so the separation matrix Φ_k is set to zero.

Since the sub-deviation has been obtained, the decoupled sub-deviations are introduced into the LQR and SMC controllers separately. For the SMC sub-controller, the thrust is calculated as follows:

where η_δ,k = x_δ,k(1: 6) and v_δ,k = x_δ,k(7: 12) are the pose vector and velocity vector of the sub-deviation, respectively. s_δ,k represents the sliding mode surface, and c is the sliding mode gain, both of which are similar to the SMC. ρ is the switching gain, and sgn is the sign function, with their respective expressions are available in Section S.2 of the supplementary file. From a mechanical perspective, the thrust τ_δ,k can be viewed as a combination of two components. The second part,  −M(v_k)c⁻¹ρsgn(s_δ,k), is a discontinuous switching term that drives the UUV’s dynamics toward the sliding surface s_δ,k. This sliding surface is inherently stabilized by the feedforward term C(v_k)v_δ,k + D(v_k)v_δ,k + g(η_k) − M(v_k)c⁻¹J(η_k)v_δ,k, which ensures the system converges to the desired pose without requiring additional thrust.

For the LQR sub-controller, its thrust is calculated by:

where K_ε is the control gain of LQR. It follows the same feedback control strategy as the traditional LQR from Section S.3 of the supplementary file, with the only difference being that the deviation here corresponds to the sub-deviation of the hovering task. By combining the two control inputs, the thrust of the COC is:

From the thrust of COC in Eq. (18), it can be observed that the thrust is a combination of the thrusts from the two sub-controllers, including the LQR and SMC. By integrating both a feedforward term and a compensated switching term, the proposed COC strategy introduces SMC’s robust disturbance suppression capabilities into the COC framework from two key perspectives. First, the feedforward term ensures that the system reaches the SMC’s sliding surface within a finite time. Second, the switching term guarantees the inherent stability of the sliding surface, thereby enhancing the system’s resilience to uncertainties and disturbances. Coupled with the deviation separation mechanism, this collaborative framework allows the proposed COC approach to effectively balance robustness, precision, and energy efficiency in dynamic and uncertain environments. The overall framework is listed as Algorithm 1.

Algorithm 1

The framework of the cooperative controller

Require: v_k, η_k, M(v_k), C(v_k), D(v_k), g(η_k), Q, R, c, ρ, dt

Ensure: τ_k

1: ({{bf{H}}}({{{bf{v}}}}_{{{bf{0}}}},{{{boldsymbol{tau }}}}_{{{bf{0}}}})={{bf{M}}}{({{{bf{v}}}}_{0})}^{-1}{{bf{C}}}({{{bf{v}}}}_{0}){{{bf{v}}}}_{0}+{{bf{M}}}{({{{bf{v}}}}_{0})}^{-1}{{bf{D}}}({{{bf{v}}}}_{0}){{{bf{v}}}}_{0}+{{bf{M}}}{({{{bf{v}}}}_{0})}^{-1}{{bf{g}}}({{{boldsymbol{eta }}}}_{0})-{{bf{M}}}{({{{bf{v}}}}_{0})}^{-1}{{{boldsymbol{tau }}}}_{0}),              ⊳ initialization process

2: ({{{bf{A}}}}_{k}=left[begin{array}{c}(frac{partial {{bf{J}}}({{{boldsymbol{eta }}}}_{{{bf{0}}}}){{{bf{v}}}}_{0}}{partial {{{boldsymbol{eta }}}}_{{{bf{0}}}}}+{{bf{I}}})dt{{bf{J}}}({{{boldsymbol{eta }}}}_{0})dt\ -frac{partial {{bf{M}}}{({{{bf{v}}}}_{{{bf{0}}}})}^{-1}{{bf{g}}}({{{boldsymbol{eta }}}}_{0})}{partial {{{boldsymbol{eta }}}}_{{{bf{0}}}}}dt[{{bf{I}}}-frac{partial {{bf{H}}}({{{bf{v}}}}_{{{bf{0}}}},{{{boldsymbol{tau }}}}_{{{bf{0}}}})}{partial {{{bf{v}}}}_{0}}]dtend{array}right],{{{bf{B}}}}_{k}={left[0frac{partial {{bf{M}}}{({{{bf{v}}}}_{{{bf{0}}}})}^{-1}{{{boldsymbol{tau }}}}_{0}}{partial {{{boldsymbol{tau }}}}_{{{bf{0}}}}}dtright]}^{{{rm{T}}}}),

3: ({{{bf{P}}}}_{k}={{{bf{A}}}}_{k}^{T}{{{bf{P}}}}_{k+1}{{{bf{A}}}}_{k}-{{{bf{A}}}}_{k}^{T}{{{bf{P}}}}_{k+1}{{{bf{B}}}}_{k}{({{bf{R}}}+{{{bf{B}}}}_{k}^{T}{{{bf{P}}}}_{k+1}{{{bf{B}}}}_{k})}^{-1}{{{bf{B}}}}_{k}^{T}{{{bf{P}}}}_{k+1}{{{bf{A}}}}_{k}+{{bf{Q}}}quad triangleright) This equation is convergent. Considering the infinite time domain, then P_k = P_k+1 = P.

4: ({{{bf{K}}}}_{varepsilon }=-{({{bf{R}}}+{{{bf{B}}}}_{k}^{T}{{bf{P}}}{{{bf{B}}}}_{k})}^{-1}{{{bf{B}}}}_{k}^{T}{{bf{P}}}{{{bf{A}}}}_{k})

5: loop                                                 ⊳ main loop

6:  ({{{bf{L}}}}_{k}=-2{({{bf{R}}}{{{boldsymbol{tau }}}}_{k}+{{{bf{B}}}}_{k}^{{{rm{T}}}}{{{bf{P}}}}_{k+1}{{{bf{x}}}}_{k+1})}^{{{rm{T}}}}{({{bf{R}}}+{{{bf{B}}}}_{k}^{{{rm{T}}}}{{{bf{P}}}}_{k+1}{{{bf{B}}}}_{k})}^{-1}{{{bf{B}}}}_{k}^{{{rm{T}}}}{{{bf{P}}}}_{k+1}), ({J}_{k}=({{{{bf{x}}}}_{k}}^{{{rm{T}}}}{{bf{Q}}}{{{bf{x}}}}_{k}+{{{{boldsymbol{tau }}}}_{k}}^{{{rm{T}}}}{{bf{R}}}{{{boldsymbol{tau }}}}_{k})+{{{{bf{x}}}}_{k+1}}^{{{rm{T}}}}{{{bf{P}}}}_{k+1}{{{bf{x}}}}_{k+1})

7:  J_m,k = [J_k−5, J_k−4, …, J_k], L_m,k = [L_k−5, L_k−4, …, L_k]

8:  ({{{mathbf{Phi }}}}_{k}=diag({{{bf{J}}}}_{m,k}{{{{bf{L}}}}_{m,k}}^{-1})./{{{bf{x}}}}_{k}), x_δ,k = Φ_kx_k, x_ε,k = (I − Φ_k)x_k

9:  τ_ε,k = −K_εx_ε,k                                      ⊳ LQR subcontroller

10: η_δ,k = x_δ,k(1: 6), v_δ,k = x_δ,k(7: 12), s_δ,k = cv_δ,k + η_δ,k

11: τ_δ,k = C(v_k)v_δ,k + D(v_k)v_δ,k + g(η_k) − M(v_k)c⁻¹J(η_k)v_δ,k − M(v_k)c⁻¹ρsgn(s_δ,k)            ⊳ SMC subcontroller

12: τ_k = τ_δ,k + τ_ε,k                                    ⊳ Cooperative controller

13: return τ_k

14: end loop

This algorithm also raises a natural question: since the final expression of the proposed controller is a weighted form, why not directly calculate the final thrust in a weighted manner? The reason is that simply combining these two control inputs does not guarantee that their advantages are maximized or the disadvantages mitigated. Using LQR alone causes improper handling of the current disturbance, leading to divergence of the control system. On the other hand, using SMC only results in overreaction to minor deviations caused by thruster noise, leading to excessive control power consumption and potentially causing oscillations in the control system.

Thus, as depicted in Fig. 7, instead of directly combining the two controllers as done previously, we propose decomposing the deviation into two parts, each tailored to LQR and SMC. The proposed deviation separation algorithm divides the mixed deviation into two distinct parts. LQR is then used to address the sub-deviation caused by thruster noise, while SMC handles the sub-deviation arising from the current disturbance. Compared to the mixed controller, the COC allows the two controllers to complement each other’s strengths, leading to improved performance.