Combined dynamics and kinematics networked fuzzy task priority motion planning for underwater vehicle-manipulator systems

The underwater vehicle-manipulator systems (UVMS) face significant challenges in trajectory tracking and motion planning because of external disturbance (current and payload) and kinematic redundancy. Former algorithms can finish the tracking of end-effector (EE) and free of singularity redundancy solution alone. However, only a few analytical studies have been conducted on coordinated motion planning of UVMS considering the dynamics controller. This article introduces a combined dynamics and kinematics networked fuzzy task priority motion planning method to solve the above problems. It avoids the assumption of perfect dynamic control. Firstly, to eliminate the kinematics error, a dynamic transformation method from joint space to task space is proposed. Without chattering, an outer loop sliding mode controller is designed for tracking EE’s trajectory. Further, to ensure the underwater vehicle’ posture stability and joint constraint, a task priority frame with kinematics error is used to planning the coordinated motion of UVMS, in which the posture and joint limits map into the null space of prioritized tasks, and weight gains are adopted to guarantee orthogonality of secondary tasks. On top of that, the gain weighted are updated by the networked fuzzy logic. The proposed algorithm achieves better coordinated motion planning and tracking performance. Effectiveness is validated by numerical simulation.


Introduction
The vehicle-manipulator system has a more complex external environment and force characteristics in the space and ocean. It realizes more effective motion control and grasping accurately, which highlighted its research significance. [1][2][3] The underwater vehicle-manipulator systems (UVMS), remote operating vehicle (ROV), or autonomous underwater vehicle (AUV) with manipulators is a vital tool for underwater tasks such as marine science, marine engineering, and military applications. 4,5 However, given the kinematics redundancy, processing the constraints of each joint (inequality condition) and the motion stability of AUV is still not available in the unified form. 6,7 When facing parameter uncertainty and external disturbance, the controller still has some problems, mainly reflected in poor convergence, insufficient stability and chattering. [8][9][10] In summary, the practical solutions to the abovementioned issues are rarely examined in the research.
Although the redundancy of the UVMS increases manipulability, it causes the kinematics solution to be nonunique and leads to the necessity of redundant decomposition in trajectory planning. This redundancy seriously affects online trajectory planning and motion coordination. 6 Offline trajectory planning tasks cannot be effective in a harsh environment. There are two solutions to the UVMS redundancy: one approach is the augmented Jacobin, which may bring the algorithm singularity, and the other is based on pseudoinverse approaches to assume the end-effector (EE) path, which brings space inconsistencies.
Once the trajectory of the EE is given, the motion of AUV and each joint needs to be considered. Antonelli 2 proposed a fuzzy redundancy coordinated motion control method and adopted the inequality condition to activate secondary tasks. Based on the pseudoinverse approach, Santos 11 used a fuzzy expert system to avoid singularity posture on the UVMS. Both overcome the kinematics singularity effectively. However, they still fail to solve optimal redundancy.
To the best of the authors' knowledge, researchers have proposed a variety of trajectory planning methods for UVMS at different criteria, such as obstacle avoidance, 12 free of singularity, 13 shortest time, 2,14 optimal torque, and lowest energy consumption. 15 Kim et al. 13 solved kinematics singularity by dynamic distributing tasks based on task priority (TP) method. Under this idea, it is easy to obtain the TP refactoring criteria and evaluate its performance. For minimizing systems and restoring torque, Jonghui et al. 15 designed an optimal controller with a nonlinear disturbance observer to resist parameter perturbation, external disturbance, and actuator nonlinearity. Sadly, the coordinated motion is not concerned. In the literature, 16 the resistance optimization function was added in inverse kinematic (IK) solution, and a coordinated motion planning algorithm was used to minimize energy consumption. After the closed-loop IK combined with redundancy has given a stability analysis, Antonelli 17 proposed priority IK control algorithm mapping tasks as the unified formula of the firstorder form.
As an intelligent algorithm, the fuzzy logic can compensate the model and find the control parameters. Proportional integral derivative (PID) tuning parameters are found by fuzzy logic to have a higher motion precision. 11,18 Fuzzy logic 6,19,20 and neural networks 9 are usually used to solve the task weights in real time. Fuzzy logic, as a higher-level supervisor, can possess applicability instead of constant.
Solving the lumped uncertainty for the controller of UVMS has attracted many scholars to study. [21][22][23][24][25][26] In a complex environment, the variety of disturbances include (a) ocean current, (b) payload changes, (c) lift or resistance, (d) noise (noise caused by the sensor and actuator error), (e) time delay or joint lag, and (f) friction and other disturbances. Simultaneously, uncertainty generally includes model parameter uncertain item (mainly generated by model inaccuracy) and parameter uncertain item (often not directly reflected as model parameters).
It generally divides into two categories to solve this problem. The first solution is to consider the disturbance and uncertainty as to the lumped uncertainty; the second solution is to deal with them separately by subdividing each characteristics. In the first solution, Ahmadi and Fateh 27 approximated the lumped uncertainty through the Taylor series, and Han et al. 10 designed an inertial delay controller to estimate it and improved the control effect using a fuzzy compensator. Although this method resulted in a relatively simplified controller, a bottleneck was encountered to control accuracy. The second solution, more targeted, is often based on the characteristics of disturbance and uncertainty. Considering (b) and (d), Mohan and Kim 8 adopted the extended Kalman filter (EKF) algorithm to construct disturbance compensation; Salloom et al. 9 analyzed (c) as an unknown external disturbance and considered the nonparametric uncertainties as (f); the literature 5,9-13 designed EKF, adaptive algorithm, and observer compensation, respectively, for (d). Before using the time delay estimator, Yang et al. 22 summarized (e) as (d). The abovementioned classification of disturbance terms is the basis to provide a compelling for the optimization of the controller. Similarly, the parameter uncertainty term in the uncertainty is passed through Legendre polynomial, 22 EKF, 8,10 respectively, achieved online estimation. Sadly, the kinematics redundancy of UVMS is less reflected in the controller designed with lumped uncertainty.
Under the highly coupling of UVMS dynamics, Kim et al. 8,28 proposed a sliding mode control (SMC) with UVMS motion redundancy considered to ensure task completion successfully when the target object's shape, inertia, and others are unknown. Combined kinematics and dynamics control method for UVMS was proposed for underwater swimming manipulator by Borlaug et al. 29 It allows us to design the kinematic and dynamic subsystems together without the assumption of perfect dynamic control. In this article, the SMC in position needs to be further strengthened. A condition is set that model can be decoupled in dynamics and kinematics separately. 7,13 The singularly robust multitask priority framework (SRMTP) eliminates this condition. 17 It brings us an idea. However, in the process of combined kinematics and model uncertainty error, the quality of control still needs to be improved. Inspired by the above studies, a novel motion planning and coordinated controller for UVMS is proposed in this article. This contribution to this article is that the proposed method within the multitasks can achieve precise and robust performance under the disturbances. First, an outer-loop sliding mode controller is proposed, where the dynamics transform from joint space to task space for avoiding joint error accumulation, and the outer loop eliminates the kinematics error in the task space. Second, the multiple TP motion planning and coordinated frame of UVMS is constructed, where weighted gains are adopted in the null space of prioritized tasks to guarantee orthogonality of secondary tasks, and the task error is feedback to the controller. Third, to determine secondary tasks' contribution, a networked fuzzy logic is employed, instead of traditional fuzzy redundant rules. Last, the effectiveness and feasibility of the proposed method are verified by numerical simulations.
In this article, the model of UVMS kinematics and dynamics is derived in the second section, and the third section is concerned with a tracking controller combined with kinematics error. In the fourth section, the proposed coordinated motion planning method is constructed. The fifth section verifies the effectiveness of the proposed method by simulation experiments. Conclusions are presented in the sixth section.

Problem setting
UVMS consists of the AUV and the manipulator. Figure 1 shows the UVMS in each coordinate frame, the AUV is connected to the EE through link L i and joint q i (it could be either prismatic or revolute). The following coordinate systems have been established: S I f g is the inertial frame in the earth, S b f g expresses the frame of AUV body, and S 0 f g represents the frame, where joint q 1 is connected to AUV body and S e f g is the manipulator EE frame with respect to inertial frame. The total dimension of freedom (DOF) is 6 þ n, in which q ¼ q 1 ; q 2 . . . q n ½ 2R n is the joint vector of n DOF manipulator, v 1 ¼ u v w ½ T and v 2 ¼ p q r ½ T , respectively, represent the position/orientation velocity of AUV.
The velocity kinematics equation of the UVMS is 2R 6 represents the desired EE velocity in S I f g, p d and r d are the position and orientation vectors, T 2 R 6þn is the system velocity in the joint space with respect to inertial frame, ð Þ is the Jacobian velocity matrix of EE and each joint in the AUV body frame S b f g, as the function of R I B and q, and is the skew-symmetric matrix. 2 Consider the dynamics of UVMS in joint space as follows where _ Under the corresponding accurate model, M Ã , C Ã , and D Ã are known matrices in equation (2). Due to the error from model measurement and motion in the current, the unknown/uncertain matrix terms are considered in this article as DM, DC, and DD. Gðq; R I B Þ is gravity and buoyancy term. t d represents the external disturbance (current or payload changes).
Further, equation (2) with unknown matrix terms can be expressed as Consider the unknown matrix terms abovementioned as an external disturbance. This idea can be compensated and estimated in real time using the controller designed in the next section.
So, rewrite equation (3) as follows where In the UVMS model, the AUV is underactuated in the AUV body frame, and its design parameters are given in Table 1.  The manipulator, n ¼ 3, consists of three joints q 1 , q 2 , and q 3 , as shown in Figure 1. Those revolute joints are actuated by servo motor. According to the manipulator's mechanical characteristic, the joints limit is considered in this article, as given in Table 2. The DH parameters of the manipulator are given in detail by Han et al. 10 It takes no attention to the impact of change in AUV's center of gravity.

Design of a tracking controller
It is worth noticing that the problem of uncertainty and external disturbance, as the precondition of UVMS coordinated motion planning, is based on a sliding mode controller. 22 Given the EE desired velocity _ x E , the SMC eliminates the error in velocity space. Unfortunately, the EE's desired trajectory is unachievable, reflecting in the static error of position space. So, an outer loop (position loop) is proposed to solve this problem.

Dynamics from joint space to task space
Firstly, EE error is defined in the task space Further, get the time derivation of e In equation (6), where k e is the error gain, expressed as constant. Differentiate it Combined with equations (1), (3), and (8), the dynamics in task space is as follows where t r represents external disturbance. Rewrite as equation (10) M€ where Assuming that dðx; _ xÞ meets the following conditions 24 : (2) dðx; _ xÞ is instantaneous, satisfying _ d ðx; _ xÞ ¼ 0. From the abovementioned, the tracking error of EE in the task space consists of two. The first is the accumulation, in the task space, of trajectory tracking errors from joint space, and the other comes from the kinematics of the desired pose in the joint space, as shown in equation (1). This error in the task space from the desired pose of EE is caused by IK. Rewriting equation (3) into equation (10), expression in the task space can more intuitively reflect EE's control error. Next, the controller designed is to avoid error accumulation.

Outer loop sliding mode controller
Substituting equation (11) into equation (8) The SMC is defined as follows Substituting equation (13) into equation (12) _ xÞÞ (14) where > 0, sgn Á ð Þ, as a sign function, is used to ensure the state's continuity. 23 To ensure the accuracy of the EE posture trajectory, an outer loop (position loop) is introduced based on the SMC to achieve precise control.
Assuming N.1, when _ e ! 0, there exist e > 0 for which Define the error e o in the outer loop, and its derivative form as equation (16) Further where r 2 > 0, k o is a positive constant. Then, the integrated control law combined equation (13) with equation (17). The outer-loop SMC structure is shown in Figure 2, and its stability analysis is followed in the next subsection.

Stability analysis
Further, we propose the Lyapunov-like positive definite function V as equation (18) V With the properties 23 Let us take the derivative of V as following Applying equation (16) to equation (17) yields Using equations (14) and (20), equation (19) can be rewritten as If and only if e ¼ 0, _ V ¼ 0. To satisfy equation (21), r 2 needs enough large for It is worth noticing that e is bounded at every time. Thus, the proposed controller is designed in the condition r 2 ! je j jj, and the tracking error asymptotically converges to zero as t ! 1.

Fuzzy multitask priority motion planning method
Given EE position/orientation, coordinated motion planning for joints and AUV body is required; secondly, this step is to execute the path planned through the control law reflecting in the joints and EE pose. Then, there exists an error between the desired and the actual pose, in which the designed controller eliminates. It is considered that the closed-loop form not only needs to combine dynamics and kinematics models but also designs a motion planning algorithm for multiple tasks.
When the EE position and orientation is desired, the motion of each joint and the AUV body will be planned in the general form x is calculated by IK.

Combined dynamics and kinematics task priority motion planning method
The general form of the TP solution of equation (22) is as follows 13 where Þ is the identity matrix, and z a represents an arbitrary vector.
The z a can be mapped to the secondary tasks, which is related to gradient of the target optimization function 6 where s is the total number of secondary tasks and H i ðqÞ is the objective function to be optimized with the scalar gain factor k i . If k i > 0, the motion will change in the direction of gradient rH i ðqÞ increasing; otherwise, it will vary in the direction of rH i ðqÞ decreasing. Combining equations (23) and (24), we obtain It can be seen from equation (25) that x includes two items, namely pseudoinverse and null-space solutions. Defining the J y _ x E by the pseudoinverse method is the prioritized task. The null space solution k i ðI N À J y J ÞDH i ðqÞ is the secondary task to complete the UVMS self-motion and optimize the movement of each joint.
The prioritized task of UVMS is h p 2 R m , where m is the dimension of this task, and the corresponding Jacobian matrix is J p ðqÞ 2 R mÂN , then the relation between _ h p and arbitrary vector of priority task z ap can be expressed as Setting the EE position/orientation as the prioritized task when facing the UVMS motion planning problem. So, in this article, it meets the following condition: Similarly, the secondary tasks h s 2 R r , where r is the dimension of the tasks, and the corresponding Jacobian matrix is J s ðqÞ 2 R rÂN , then the equation of time derivate and arbitrary vector of secondary task z as can be expressed in equation (27) _ h s ¼ J s ðqÞz as The motion planning algorithm, including secondary objectives, based on TP can be obtained by equations (25) to (27) x where _ h si ði ¼ 1; 2; :::; kÞ is the i'th secondary task and J si ðqÞði ¼ 1; 2; :::; kÞ is the Jacobian matrix of the corresponding secondary tasks.
It can be seen from equation (28) that the prioritized and secondary tasks do not conflict, the secondary task requirements can be met simultaneously in the space of the prioritized task. However, when the two tasks conflict, it is necessary to ensure that the prioritized task is completed. When the position is obtained by integrating the velocity, it may cause a numerical drift problem. So, according to the dynamics-kinematics controller and equation (28), the closed-loop form is introduced where e is the error between the EE position and orientation desired and the actual value, e s is the error between the desired and achieved values of the secondary tasks, 30 K p and K S are the corresponding gain matrix, respectively. Based on equation (29), algorithm error of secondary tasks is zero when J y and J þ s meet the orthogonality. It effectively ensures the accuracy of highest priority tasks and can control the errors of secondary priority tasks. As a result, TP avoids algorithm singularity through null space. 17 Sadly, they cannot satisfy this strict condition in real time. 30 In this article, adopting the weight form of equation (22) to achieve that each secondary task's errors are zero. And K si e si in equation (29) as a compensator under the external disturbance is uncertain.
Summly, the weighted TP with compensator, as shown in Figure 3 can be obtained, where J W y is the weighted pseudoinverse of the Jacobian matrix, 7 and a i 2 ½0; 1 is the priority scale factor of the secondary task i. As an open field, a i will be realized by networked fuzzy logic in "Networked fuzzy logic for secondary tasks gains" section.

Joint limit constraint and coordinated motion of underwater vehicle-manipulator systems
It is hoped that the AUV-body move with a minor pitch and the joints cannot exceed its mechanical limit. For example, the q 2 should motion within ½À230 ; 230 . The weight matrix in equation (26) not only meet the orthogonality of J y and J þ s but also can be used to avoid the joint limit in this article. Specifically, a diagonal matrix W J 2 R N ÂN is used, when the joint i is in the middle of the allowable range, W J i;i ð Þ ¼ 1. When the joint i approaches its limit, W J i;i ð Þ ¼ 1, this DOF motion is constrained. H J ðqÞ represents the function of joint limit to be optimized. 6 Then, the partial derivative of joint i is defined as @H J ðqÞ=@q i .
There are certain problems in restricting joint constraints only by the size of the defined W J i;i ð Þ . When the median is infinitely close to the bounds or the bounds return to the median with the equal assignment, 30 the same constraint is applied to the return motion, which does not match expectations.
Introduce the norm j@H J ðqÞ=@q i j j jand its trend D j@H J ðqÞ=@q i j j j : (1) When in the situation of D j@H J ðqÞ=@q i j j j! 0, in other words, joint i is approaching its limit, (2) Only in the situation of D j@H J ðqÞ=@q i j j j< 0, we consider that the joint i is away from bounds, This article mainly studies the motion planning of UVMS combining kinematics and dynamics. There is still a key point in this topic to realize the coordinated motion of the AUV and manipulator. 31 This question is of great significance to the total system. Under the TP framework in the task space, the EE posture is the prioritized task.
In equation (26), weighted matrix W can also be used to achieve this task. In this article, the AUV-body and manipulator's coordinated motion is taken as one of the secondary tasks, so that this item meets the unified form in equation (29), and is set as the inverse form of the weighted matrix to realize the coordinated motion control of UVMS. Setting b 2 0; 1 ½ to measure the motion distribution of AUV body and manipulator, 2 b ¼ 0 corresponds to the sole AUV-body motion, while b ¼ 1 corresponds to the sole manipulator motion UVMS contains two subsystems with different characteristics. The AUV system has large inertia and slow timevarying characteristics, while the manipulator system has the characteristics of small inertia and fast time-varying. For its different characteristics, it exists certain problems in equation (31), when b ! 0, the DOF of each joint will get a smaller movement, correspondingly, the AUV body will have a larger range of movement. From an energy point of view, it still hopes that the joints can give full use to the small inertia characteristics, with small hydrodynamic disturbances, to adopt the large inertia AUV body as small as possible. A saturation factor ! v 2 0; 1 ð is used to limit the AUV body's motion and increase the DOF utilization of joints. Equation (31) can be written as The weighted matrix in equation (30) can be written as the product of the weighted matrix with joint constraints and coordinated motion The following discusses the characteristics of the W: (1) Exchangeability Both W J and W C are diagonal matrices, then (2) Free of singularity The singularity of W depends on the singularity of W C and W J . From the abovementioned, W J j j ! 1, W C exists inverse matrix, then W j j 6 ¼ 0, W À1 exist, so W is free of singularity.

Networked fuzzy logic for secondary tasks gains
"Combined dynamics and kinematics task priority motion planning method" and "Joint limit constraint and coordinated motion of underwater vehicle-manipulator systems" sections construct the UVMS multitask motion planning algorithm of equation (30). It can ensure EE's control accuracy, simultaneously complete secondary tasks (avoiding joint limit, coordinated motion of AUV body and manipulator, etc.) in its null space. Sadly, it cannot effectively determine the weight of each secondary task. Fuzzy logic 6,11,19,20 and neural networks 9 are usually used to solve this problem. Fuzzy logic, as a higher-level supervisor, considers here. Simultaneously, based on fuzzy logic, 20 the logic is networked that how secondary tasks are allocated. It is vital to avoid the joint limit when task execution, the distance to the joint limit is taken as the first secondary task Holding the position and posture stability of the AUV body during underwater operations is important for UVMS coordinated control. Considering that the AUV pitch maps a significant effect for UVMS, it is taken as second secondary task in this article Here, _ q is the time derivate of pitch q. Define that the range of the AUV pitch is ½À20 ; 20 , the range of motion for q 2 and q 3 are ½À142 ; 142 and ½À230 ; 230 , respectively.
The abovementioned secondary tasks can be activated by fuzzifier with the three inputs (q 2 ,q 3 , _ q). In this article, normalized inputs, respectively, are considered within the range of [0,1]. Two rules are considered: Joint limits (JL) ¼ fclose, not closeg and vehicle attitude (VA) ¼ fsmall, not smallg.
Then, a networked fuzzy inference engine is constructed, based on the traditional fuzzy logic, 2 a hidden layer neural network is introduced into the fuzzy inference engine layer as shown in Figure 4. Through the neural layer, weights are shared to obtain more effective outputs. The neural layer is as the following where B i corresponds to the i'th output imported into defuzzifier. A j i represents the contribution of the j'th input to the i'th output, and j i 2 0; 1 ½ is the corresponding weight. Further, the inputs normalized with the corresponding fuzzy rules, as the network's input, satisfy A j 1 ¼ A j 2 . The output B i of the network further generates a i through the de-fuzzifier.
Networked fuzzy logic has the following advantages: (a) Compared with the simplified input-output relationship in Table 3, in this networked fuzzy logic, the output characteristics under each input's combined action can be obtained, which compensate for the incompleteness of the former. (b) A more detailed fuzzy set can also solve the incompleteness mentioned in (a). It needs to subdivide the input and output rules, often reach more than 32. 2 Unlike this idea, networked fuzzy logic can guarantee the reconfigurability of fuzzy relationships by adjusting the weights, replacing artificial excessive constraints, and still achieving good flexibility and realizability.

Conditions
To verify the proposed method's performance, simulations were achieved on an AUV, as given in Tables 1 and 2, with a 3-DOF manipulator 10 shown in Figure 1. The initial pose . Networked fuzzy inference system, where the close and not close, respectively, represent the membership function close to and away from the joint bound. Small and not small, respectively, express small or large AUV-body pitch. AUV: autonomous underwater vehicle. Table 3. Fuzzy logic 2 (.).
Meanwhile, define the prioritized task is the secondary tasks are expressed as _ The proposed motion plan method is compared with the pseudoinverse method, 2 which is given in equation (22). Besides, combining equation (31), the weighted pseudoinverse method is also used as a comparison. For simple representation, the proposed outer loop controller is termed case 1 (c1). The classic sliding mode controller is termed case 2 (c2). Hence, the proposed motion planning method based on the proposed dynamic controller is termed proposed plan c1 . The proposed motion planning method based on the classic SMC is termed proposed plan c2 . The pseudoinverse method and weighted pseudoinverse method based on the proposed dynamic controller are termed Pse plan c1 and W-pse plan c1 , respectively.

Simulation results
The results are shown in Figures 5 to 11. It is assumed that the Pse plan c1 and W-pse plan c1 are free of singularity in this simulation condition. Figure 5 shows the circular trajectory desired and simulated. Given the desired EE trajectory as a circle with a radius of 1.80 m, the proposed plan c2 failed to reach the endpoint at 30 s and the error up to 0.14 m. The proposed plan c1 can track the desired trajectory well in the first 15 s, while the Pse plan c1 and W-pse plan c1 have better performance at this time. Figure 5 also shows the posture of the UVMS at 0, 15, and 30 s. Pse plan c1 and W-pse plan c1 have a larger pitch of AUV, as shown in Figure 5(a) and (b), the pitch at t ¼ 15 s is 63.9 and 56.3 , respectively. The pitch at t ¼ 30 s is 45.9 and 38.8, respectively. The angle in the roll and yaw directions under those two algorithms is very small, and the AUV still has the instability of pitch. In contrast, the proposed plan c1 and proposed plan c2 are largely in the yaw direction, and no more than 30 . As shown in Figure 5(c) and (d), the UVMS has attitude stability in the proposed plan c1 and proposed plan c2 . Figure 6 shows the error of EE in x; y; z and q 1 ; q 2 ; q 3 between the desired and actual. As shown in Figure 6(a) to (c), Pse plan c1 and W-pse plan c1 achieve the position errors in x e ; y e ; z e within 0.05 m. It embodies the excellent control effect of the outer loop SMC. The maximum error of the proposed plan c2 appears in the y direction at 27 s, reaching 0.12 m. Similarly, the tracking error of the proposed plan c1 is larger than Pse plan c1 and W-pse plan c1 . Through the What needs to be reviewed in this article is that the proposed plan c1 and proposed plan c2 , under the frame of the TP, consider the stability of the AUV attitude as second secondary task. Figure 7 shows the velocity of AUV when tracking the circle trajectory. The velocity of the proposed plan c1 and proposed plan c2 in the q; r directions is much lower than that of the others, as shown in Figure 7(e) to (f). It ensures the smaller angular displacement of AUV the maximum value does not exceed 30 , as shown in Figure 10(c). However, the redundancy is not exploited in the pse plan c1 . For example, when t ¼ 5 s, the v of AUV is planned to be 0.15 m/s and the angular velocity q is planned to be 0.16 rad/s, as shown in Figure 7(e), and the desired velocity of q 1 is 0.17 rad/s, as shown in Figure 11(a), which leads to the angular displacement of AUV in the y direction up to 28.5 .
Based on pse plan c1 , a coordinated motion distribution W À1 C is added to ensure the motion stability of the AUV in W-pse plan c1 . Sadly, the AUV angular velocity under this approach still reaches 0.16 rad/s, which still affect the stability of AUV. It is only as a local optimization of pseudoinverse method. As shown in Figures 10(a) to (c) and 11(a), under the proposed plan c1 and proposed plan c2 , a large displacement is used in q 1 and x,y direction of AUV. It reduces the angular velocity of the AUV to ensure its stability. Taking the pitch as a secondary task, the posture  stability of the AUV in this direction is ensured. Figure 8 shows the posture of EE. The pose of EE at the initial time is [139 , 0 , 90 ]. Pse plan c1 and W-pse plan c1 have a smaller angular displacement in the q e ; e direction. On the contrary, the proposed plan c1 and proposed plan c2 have a larger angular displacement in those directions, reaching 31.5 , as shown in Figure 8(c). This also further illustrates the effectiveness of the proposed motion planning algorithm in ensuring the stability of AUV. Considering the weight gains of secondary tasks, the idea of networked fuzzy logic is adopted in the proposed plan c1 . It further optimizes the coordinated motion of UVMS. The results show that the proposed plan c1 uses real-time inputs _ h s to adjust secondary task weights a i i ¼ 1; 2 ð Þ , continuously, achieves a smaller control error, and in the case of joint disturbances, it also ensures better stability in direction of pitch and yaw. Figure 11 demonstrates the desired joint velocity when tracking the circular trajectory. The proposed plan c1 and proposed plan c2 are similar that _ q 1 is relatively large, which can reach À4.2 rad/s. What needs to be reviewed is that the angle range of the revolute joint q 1 is [À360 to 360 ]. Therefore, the larger _ q 1 does not need to consider the limitation of q 1 . However, the redundancy of joints is not fully considered under the pseudoinverse and the weighed approach, _ q 1 is relatively small, as shown in Figure 11(a).
Besides, the conditions for q 2 and q 3 are all satisfied. Among them, q 2 reaches the maximum of 68.8 at t ¼ 30 s under the Pse plan c1 , as shown in Figure 5(b). q 3 reaches 62.0 and 57.3 under Pse plan c1 and W-pse plan c1 , respectively. As one of the secondary tasks, avoiding joint limit is easily achieved in the proposed plan c1 . Furthermore, errors caused by external disturbance (current) must be considered. Given the desired joint velocity _ q 1 ; _ q 2 ; _ q 3 , the outer loop SMC in the proposed plan c1 has reduced those errors of 1.8%, 35.0%, and À2.3%, respectively, compared with the traditional SMC in the proposed plan c2 . Furthermore, the errors at x e , y e , and z e are further reduced by À22.48%, À1.48%, and 39.64%, respectively. Therefore, the proposed outer loop SMC has a positive effect on reducing errors introduced by external disturbance. tracking the circular trajectory. Among them, the proposed plan c2 has the largest energy consumption while the proposed plan c1 is the smallest. At 3 s < t < 15 s, under the Pse plan c1 and W-pse plan c1 , the EE's tracking accuracy is enhanced, as shown in Figure 5, the jt j jj during this time is relatively high. It always keeps a high tracking accuracy at t >15 s, and jt j jj is reduced. At 15 s < t < 25 s, under the proposed plan c1 , the error of EE position is continuously reduced from 0.12 m. Its energy consumption jt j jj is relatively large. It is worth noticing that the proposed plan c2 failed to reach the endpoint. In summary, the proposed plan c1 , the EE error within 0.1 m, has engineering applicability.

Conclusions
In this article, a combined dynamics and kinematics networked fuzzy TP motion planning method is proposed. The  fully considered in which a large displacement is used in q 1 and x,y direction of AUV. However, the pse plan c1 and Wpse plan c1 are free of coordinated motion planning. Its pitch reaches 63.9 and 56.3 , respectively, which still affects the stability of AUV. Facing the external disturbance, the proposed plan c1 has reduced desired joint velocity's errors of 1.8%, 35.0%, and À2.3%, respectively, compared with the traditional SMC in proposed plan c2 . Furthermore, the errors at z e are reduced by 39.64%, respectively. The proposed plan c2 failed to reach the endpoint, while the proposed plan c1 has the smallest energy consumption. Even though the coordinated motion plan framework's effectiveness was validated through simulations, experiments should be carried out to further enhance the simulation results, which will be done in the future.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Research and Development of Key Technologies and Equipment for Underwater Life Detection and Search and Rescue of China under grant number 2020YFC1512200.