Real-Time Kinematically Synchronous Planning for Cooperative Manipulation of Multi-Arms Robot Using the Self-Organizing Competitive Neural Network

This paper presents a real-time kinematically synchronous planning method for the collaborative manipulation of a multi-arms robot with physical coupling based on the self-organizing competitive neural network. This method defines the sub-bases for the configuration of multi-arms to obtain the Jacobian matrix of common degrees of freedom so that the sub-base motion converges along the direction for the total pose error of the end-effectors (EEs). Such a consideration ensures the uniformity of the EE motion before the error converges completely and contributes to the collaborative manipulation of multi-arms. An unsupervised competitive neural network model is raised to adaptively increase the convergence ratio of multi-arms via the online learning of the rules of the inner star. Then, combining with the defined sub-bases, the synchronous planning method is established to achieve the synchronous movement of multi-arms robot rapidly for collaborative manipulation. Theory analysis proves the stability of the multi-arms system via the Lyapunov theory. Various simulations and experiments demonstrate that the proposed kinematically synchronous planning method is feasible and applicable to different symmetric and asymmetric cooperative manipulation tasks for a multi-arms system.


Introduction
The development of artificial intelligence technology has facilitated the research on the autonomous manipulation of robot manipulators. Meanwhile, the increase in requirement of using robots to replace human hands' manipulation has made the cooperative motion of the robot important in the autonomous operations (e.g., manipulating the rudder, using pliers or a wrench, carrying large objects, or other similar manual tasks in daily life) [1]. Various multi-arm robots, such as Baxter [2], YUMI [3], Justin [4], and Robonaut [5], have been proposed to satisfy the requirement because of their outstanding capability of cooperative manipulation in replacing humans.
The force-based strategies include two types: One relies on dynamics control, but the highly complicated nonlinear dynamic model makes it difficult to apply in robot system. The other one depends on position compensation control that adjusts the position of the end-effector (EE) to maintain a certain interaction force and that has been studied coupling base, the dynamic coupling base is set to be stationary to ensure the cooperation between arms [24], and this is caused by the uncoordinated movement between the EEs of redundant arms. Few studies have been devoted to the kinematics-based planning of the redundant multi-arms with a dynamic coupling base to ensure the cooperation between the arms. Therefore, on this basis of the previous studies [24,30,31], this paper proposed a novel kinematics-based synchronous planning for collaborative manipulation of the redundant multi-arms with dynamic coupling, which involves the inverse kinematics based on the subbase method and the self-organizing competitive neural network. The following aspects differ from those in the existing literature.
A class of cooperative manipulation tasks of multi-arm robot described by generalized coordinate transformation matrix are summarized, such as carrying, manipulating the rudder, using a wrench, manipulating pliers, multi-station manipulation, and other similar cooperative manipulation. The configuration branch division of the multi-arm robot based on the sub-base method is proposed to identify each branch of the robot, and the inverse kinematics is calculated based on the damped least square method. Then, the multiarms robot system can synchronously converge along the reducing direction of the total error. The self-organizing competitive neural network is proposed to promote motion synchronization between multiple arms, and it regards the cooperative movement between arms as the competitive relationship of neurons instead of relying on a defined arm motion as a reference in existing research. The inner star learning rule is used to change the neuron weight, and all neuron weight values are updated to adjust the motion of multi-arms in every instance of competitive learning. Thus, the multi-arm robot motion planning method is formed and realizes the synchronization of the arms' motion state. The stability of the motion-planning algorithm is analyzed by using the Lyapunov theory and the inner star learning rule principle. The feasibility of the proposed method, the synchronization of motion state, and its applicability are demonstrated by dual-arm and three-arm robots with a dynamic base in different symmetric and asymmetric cooperative manipulations.
The remainder of this paper is organized as follows. Section 2 presents a type of cooperative manipulation, the sub-base description, and the Jacobian matrix definition for the multi-arms with physical coupling. Section 3 discusses the real-time kinematically synchronous planning method of collaborative manipulation based on the self-organizing competitive neural network and the stability. Sections 4 and 5 provide the simulation and experimental results in different cases, respectively. Section 6 presents the conclusion.

A Type of Kinematically Cooperative Manipulation
In Figure 1, a type of cooperative manipulation for multi-arms is considered for this paper, such as carrying, manipulating rudder, using pliers and multi-station manipulation, etc. Figure 2 presents the common features for the kinematically cooperative manipulations that can be concluded as follows: Sensors 2023, 23, x FOR PEER REVIEW 3 of 22 common and fixed base. Even if there is an application in the robot with a dynamic coupling base, the dynamic coupling base is set to be stationary to ensure the cooperation between arms [24], and this is caused by the uncoordinated movement between the EEs of redundant arms. Few studies have been devoted to the kinematics-based planning of the redundant multi-arms with a dynamic coupling base to ensure the cooperation between the arms. Therefore, on this basis of the previous studies [24,30,31], this paper proposed a novel kinematics-based synchronous planning for collaborative manipulation of the redundant multi-arms with dynamic coupling, which involves the inverse kinematics based on the sub-base method and the self-organizing competitive neural network. The following aspects differ from those in the existing literature.
A class of cooperative manipulation tasks of multi-arm robot described by generalized coordinate transformation matrix are summarized, such as carrying, manipulating the rudder, using a wrench, manipulating pliers, multi-station manipulation, and other similar cooperative manipulation. The configuration branch division of the multi-arm robot based on the sub-base method is proposed to identify each branch of the robot, and the inverse kinematics is calculated based on the damped least square method. Then, the multi-arms robot system can synchronously converge along the reducing direction of the total error. The self-organizing competitive neural network is proposed to promote motion synchronization between multiple arms, and it regards the cooperative movement between arms as the competitive relationship of neurons instead of relying on a defined arm motion as a reference in existing research. The inner star learning rule is used to change the neuron weight, and all neuron weight values are updated to adjust the motion of multi-arms in every instance of competitive learning. Thus, the multi-arm robot motion planning method is formed and realizes the synchronization of the arms' motion state. The stability of the motion-planning algorithm is analyzed by using the Lyapunov theory and the inner star learning rule principle. The feasibility of the proposed method, the synchronization of motion state, and its applicability are demonstrated by dual-arm and three-arm robots with a dynamic base in different symmetric and asymmetric cooperative manipulations.
The remainder of this paper is organized as follows. Section 2 presents a type of cooperative manipulation, the sub-base description, and the Jacobian matrix definition for the multi-arms with physical coupling. Section 3 discusses the real-time kinematically synchronous planning method of collaborative manipulation based on the self-organizing competitive neural network and the stability. Sections 4 and 5 provide the simulation and experimental results in different cases, respectively. Section 6 presents the conclusion.

A Type of Kinematically Cooperative Manipulation
In Figure 1, a type of cooperative manipulation for multi-arms is considered for this paper, such as carrying, manipulating rudder, using pliers and multi-station manipulation, etc. Figure 2 presents the common features for the kinematically cooperative manipulations that can be concluded as follows:  (1) The coordinate system {Ti} in or out of the specified object is referenced for the object pose, ti, of each arm EE in real time. The object pose, ti, is defined as follows: where is the desired position vector of the i-th EE in coordinate system {O0}. T is the vector from the coordinate system {O0} to the i-th coordinate system {Ti}.
is the position vector in coordinate system {O0} for ti. (2) When the arms perform collaborative operations, the arms form a closed loop, and there is a certain motion constraint relationship between the EEs. Thus, the motion states of EEs from "1, 2, …, i" to "1', 2', …, i'" are kinematically consistent and synchronous. The descriptions for movement states are mutual during the execution of these tasks, like the motion error and the motion rate for each EE. Such common motion states of cooperative manipulations are expressed as the following problem in Equation (2), and Equation (2) is used as the judgment criteria for coordinated synchronous motion and is proved in Section 3.2.
where denotes the pose error of the i-th EE, defines the i-th EE velocity, is the pose of the i-th EE, and t is the time. Equation (2a) refers to the pose errors of EEs along the direction of error convergence, which not only can make the multi-arms reach the execution position of the manipulated object at the same time but also can ensure the minimization of the movement error between the arms in the process of cooperative manipulation. Equation (2b) guarantees that the movement speeds of the EEs are synchronous and that the manipulation speed of the EE is equal/close to the set or constraint value in (1) The coordinate system {T i } in or out of the specified object is referenced for the object pose, t i , of each arm EE in real time. The object pose, t i , is defined as follows: where Q i is the desired position vector of the i-th EE in coordinate system {O 0 }. Q T i is the vector from the coordinate system {O 0 } to the i-th coordinate system {T i }. D i is the position vector in coordinate system {O 0 } for t i .
T i t i R denotes the rotation matrix from the coordinate the rotational operator about the axis directionf t i by ψ t i radians. t i ∈ R b×1 . b = 3 for planar robot. b = 6 for spatial robot. i = 1, 2, . . . , N. N is the number of arms (or EEs).
(2) When the arms perform collaborative operations, the arms form a closed loop, and there is a certain motion constraint relationship between the EEs. Thus, the motion states of EEs from "1, 2, . . . , i" to "1', 2', . . . , i'" are kinematically consistent and synchronous. The descriptions for movement states are mutual during the execution of these tasks, like the motion error and the motion rate for each EE. Such common motion states of cooperative manipulations are expressed as the following problem in Equation (2), and Equation (2) is used as the judgment criteria for coordinated synchronous motion and is proved in where e i denotes the pose error of the i-th EE, v i defines the i-th EE velocity, s i is the pose of the i-th EE, and t is the time. Equation (2a) refers to the pose errors of EEs along the direction of error convergence, which not only can make the multi-arms reach the execution position of the manipulated object at the same time but also can ensure the minimization of the movement error between the arms in the process of cooperative manipulation. Equation (2b) guarantees that the movement speeds of the EEs are synchronous and that the manipulation speed of the EE is equal/close to the set or constraint value in the manipulation. Finally, the synchronization of arms during cooperative manipulation can be achieved. The i-th EE pose, velocity, and object pose are s i , v i and t i , respectively. s i , v i , t i ∈ R b×1 . The i-th EE pose error and velocity can be calculated as follows: where T signifies the current time, (T − ∆T) is the last sampling time, ∆T denotes the sampling period, s = (s 1 , s 2 , . . . , s k ) T , and t = (t 1 , t 2 , . . . , t k ) T .

Multi-Arms Robot with Physical Coupling
This paper considers the general configuration of the multi-arm robot with physical coupling to achieve the cooperative manipulation, as shown in Figure 3. Suppose that there are r joints of the multi-arms and each value of θ j is the joint angle. The completely joint configuration of the multi-arms is defined as Θ = (θ 1 , . . . , θ r ) T , Θ ∈ R r×1 . The pose mapping of the multi-arm robot from joint space to Cartesian space can be expressed as follows: and s i = f i (Θ) for the EE. the manipulation. Finally, the synchronization of arms during cooperative manipulation can be achieved. The i-th EE pose, velocity, and object pose are , and , respectively. , , ∈R b×1 . The i-th EE pose error and velocity can be calculated as follows: where T signifies the current time, (T − ∆T) is the last sampling time, ∆T denotes the sampling period, = (s1, s2, …, sk) T , and t = (t1, t2, …, tk) T .

Multi-Arms Robot with Physical Coupling
This paper considers the general configuration of the multi-arm robot with physical coupling to achieve the cooperative manipulation, as shown in Figure 3. Suppose that there are r joints of the multi-arms and each value of θj is the joint angle. The completely joint configuration of the multi-arms is defined as Θ = (θ1, …, θr) T , ∈ ×1 . The pose mapping of the multi-arm robot from joint space to Cartesian space can be expressed as follows:   The corresponding inverse mapping is as follows: where f is a highly nonlinear operator and difficult to solve. The iterative method via Jacobian matrix is used to approach the good solution of the mapping problem. The traditional Jacobian matrix, J, is the partial derivative matrix of the whole chain system relative to the EE . The Jacobian matrix is obtained via linear approximations of inverse kinematic problems. They linearly simulate the motion of the EE with respect to the instantaneous system changes of the link translation and joint angle. The traditional Jacobian matrix, J, is a function of the joint angle, , defined as follows: The corresponding inverse mapping is as follows: where f is a highly nonlinear operator and difficult to solve. The iterative method via Jacobian matrix is used to approach the good solution of the mapping problem. The traditional Jacobian matrix, J, is the partial derivative matrix of the whole chain system relative to the EE s. The Jacobian matrix is obtained via linear approximations of inverse kinematic problems. They linearly simulate the motion of the EE with respect to the instantaneous system changes of the link translation and joint angle. The traditional Jacobian matrix, J, is a function of the joint angle, Θ, defined as follows: where I = 1, . . . , N. j = 1, . . . , r. J(Θ) ij ∈ R b×1 .

Definition of Sub-Bases
The traditional Jacobian matrix can ensure that each EE converges along its own error reduction direction. Unlike the traditional Jacobian matrix, J, the sub-bases are defined to make the EEs converge along the reducing direction of the system's total error and guarantee that the EEs converge at the same time.
This paper defines that the nodes with multiple branches as the sub-bases for the multi-arms configuration, as shown in Figure 3. The Jacobian matrix is modified as follows: For the 1-th sub-base pose, P 1,1 , the corresponding element of the Jacobian matrix is as follows: where J 1,1 (Θ) j ∈ R b×1 . J 1,1 (Θ) ∈ R b×M 0,1 . θ j belongs to the chain P 0¯P1,1 with M 0,1 DoFs. For the n,k-th sub-base pose, P n,k , the corresponding element of the Jacobian matrix is as follows: where J n,k (Θ) j ∈ R b×1 . J n,k (Θ) ∈ R b×M n,k . θ j belongs to the chain P 1,1¯Pn,k with M n,k DoFs. By analogy, the Jacobian matrix corresponding to other sub-bases can be obtained. For J 1 , . . . , J N without common degrees of freedom, the corresponding elements can be obtained according to (6).
The velocities of 1-th sub-base and n,k-th sub-base are calculated by the following: where η 1 and η n,k are the gain coefficient. N and N k are fixed value and related to the configuration of multi-arms. The inverse kinematics is as follows: where J * denotes the pseudo-inverse of Jacobian matrix, J(Θ), based on the damped least squares method, and represents the damping factor that can handle the ill-conditioned J in the neighborhood of singular configurations for redundant manipulators and guarantee the EEs with the minimum possible deviation at all configurations. I is a unit matrix with the dimension b(k + N + 1) × b(k + N + 1). In accordance with the traditional fixed proportion-based method [31] for the real-time tracking of a given object pose, . t, the EE velocities are planned as follows: where k p is the gain coefficient.

Iteration for Multi-Arms Robot Motion
The iterative method is utilized to achieve the real-time movement of multi-arms via updating the joint angles, Θ, according to (17).
where ∆Θ deduced from (15) becomes where µ = k p ·∆T, and µ < 1. Moreover, ∆t represents the changing pose of the object at a sampling time interval, ∆T.
Then, according to the sub-base method, the movement of the multi-arm robot can be achieved. The sub-base motion facilitates the synchronous convergence, and the synchronous performance is more obvious when the common DoFs are enough. The corresponding verifications are presented in Sections 4 and 5.2.

Synchronous Planning Using Self-Organizing Competitive Neural Network
The DoFs of the sub-base are not always enough to ensure the system convergence along the reducing error direction for the pose of the EEs, thereby resulting in the asynchronous EE motion. Thus, the self-organizing competitive neural network based on the rule of inner star model is proposed to adjust the synchronism of multi-arms movement. Equation (2) indicates that the error e i and the v i will tend to a stable value. In accordance with the principle of the self-organizing competitive neural network, the kinematically synchronous planning for the multi-arm robot is designed as shown in Figure 4.
The learning rule of the inner star model defines the weight updating as follows: where η denotes the learning rate; P i is the i-th input element of the neuron and the minimum pose velocity error norm, min ; w i is the weight value; i = 1, 2, . . . , N; N is the number of arms (or EE); and Y i is the value of output neuron and is defined as where The input element, P i , and the weight value, w i , are defined as Sensors 2023, 23, 5120 The input vector P is constituted by P i and defined as

Robot
Inner star rule Self-organizing competitive neural network Inverse kinematics based on sub-base method The learning rule of the inner star model defines the weight updating as follows: where denotes the learning rate; is the i-th input element of the neuron and the minimum pose velocity error norm, min(‖ (T) −̇‖); is the weight value; i = 1, 2, …, N; N is the number of arms (or EE); and is the value of output neuron and is defined as The input element, , and the weight value, , are defined as The input vector is constituted by and defined as T N×1 Since the minimum of ṽ is used as an input and each EE may become the one with the minimum of ṽ , the proposed method makes the planner no longer use a manipulator as a reference as in the existing literature. In each period, the new input vectors, P ' , in neural networks are defined as where In order to make the system quickly reach the state of synchronization, the winning weight value will get more rewards in a cycle. The new weight value, ∆ŵ i , is designed as where σ i is a small positive real number; and c i denotes a positive real number to adjust the updating slope of neuron weight and contributes to the synchronization of the EE motion. The response time of converges is shorter when the parameter c i becomes larger. The i-th EE's planned related velocity is deduced according to the iterative method as follows:v Then, the planned velocity for EEs is Thus, (18) becomes where The learning rate η, η 1 and η n,k are related to the step size for each iteration. The response time of converges is shorter when the learning rate becomes larger.

Stability Analysis
Supposing that the learning rate, i.e., η, η 1 , η n,1 , . . . , η n,k , is small, based on the rule of inner star model, the weight value with the minimum EE pose error, e min , is never updated, and the minimum pose error changes in a cycle is ∆e min (T). Thus, for the EE with the minimum pose error, the Lyapunov function is defined as follows: The change of the pose error is For the EE with the minimum pose error, e min (T), the planned iterative step based on the fixed proportion method [31] is µe min (T) in each control cycle, as shown in Figure 5. However, the actual motion of EE will have a small deviation, ϕ, as follows: where 0 < µ < 1.  Due to the high-precision motion of the advanced manipulator and the small learning rate in the proposed method, the deviation, , is usually in a very small range. The norm of the deviation ‖ ‖ is less than ‖ min (T)‖ to ensure that the EE can move along the planned path, i.e., ‖ ‖ < ‖ min (T)‖. Therefore, and the minimum pose error, min , is convergent. Based on Equations (3) and (4), the derivation of ̇m in (T) is Due to the high-precision motion of the advanced manipulator and the small learning rate in the proposed method, the deviation, ϕ, is usually in a very small range. The norm of the deviation ϕ is less than µe min (T) to ensure that the EE can move along the planned path, i.e., ϕ < µe min (T) . Therefore, According to the rule of inner star model, the input and the weight ultimately become equal, as follows: Therefore, the proposed planning method based on the self-organizing competitive neural network is convergent, and Equation (2) is proved to be valid to achieve motion synchronization.

Simulation
The three-arm robot with 15-DoFs is utilized to provide the contrast simulations for inverse kinematics by using inverse kinematics based on the sub-bases and the traditional method [31], as shown in Figure 6. The common DoFs are 3 that are enough for all EEs to make the pose error converge along the decreasing direction of the total error simultaneously. The synchronization performances of EE movements are compared by tracking the static-designated location on the specified-carried object. The robot configuration parameters, initial parameters, and kinematics parameters are presented in Tables         Arm2 Arm3  Figures 7c-f and 8c-f. Because of the subbase motion, the convergence of the EE pose error is much faster than that based on the traditional method. Furthermore, the EE pose velocities form uniform motion states before complete convergence in Figure 8c,d. Thus, the proposed sub-base method is conducive to the synchronization from the initial motion state to the cooperative motion state and improves the efficiency for the carrying of the multi-arm robot.

Experimental Setup
The two-arm robot with 13-DoFs is used to compare the proposed synchronous planning for the collaborative manipulation, and 1-DoF is common in the base joint, as shown in Figure 9. The D-H parameters of the two-arm robot are presented in Table 4. In Figure  10, the principle of the experimental setup can be briefly described as follows.

Experimental Setup
The two-arm robot with 13-DoFs is used to compare the proposed synchronous planning for the collaborative manipulation, and 1-DoF is common in the base joint, as shown in Figure 9. The D-H parameters of the two-arm robot are presented in Table 4. In Figure 10, the principle of the experimental setup can be briefly described as follows.          (1) The global depth camera transfers the observed frame of depth and co the computer by using a universal serial bus (USB) in real time. The ro system (ROS) node runs in the computer and extracts the position infor object from each frame image. The vision-processing procedures in the developed based on the morphology, using the Open Source Compu brary and the camera Software Development Kit. Θ d , are sent to the robot joint actuator through the RS485 bus to control the motion. The baud rate of the CAN bus, USB serial bus, SCI bus, and RS485 bus is set to 1 MHz. The control period of the whole system containing the proposed kinematically synchronous planning method is less than 10 ms.

Synchronous Planning Experiments
(1) Collaboration Carrying The collaboration carrying an object was provided to verify the feasibility of the proposed planning method, as illustrated in Figure 11. The parameters for carrying are shown in Tables 5 and 6 Table 6. The parameters of self-organizing competitive neural network. Parameters All the parameters in Tables 5 and 6, the rotation center position, t init 1 , and t init 1 , were used in Figure 4. The inverse kinematics is based on Equation (30) to obtain the joint angle, Θ. The change of the new weight value, ∆ŵ i , of the self-organizing competitive neural network is obtained according to Equation (27). The learning rate, i.e., η, η 1 , η n,1 , . . . , η n,k , is used to calculate the carrying step size of EE motion in Cartesian space. With the continuous learning and competition of the proposed planning method, the position velocities of EEs gradually form a consistent movement. Figure 12a Table 6. The parameters of self-organizing competitive neural network.

Parameters
Value 1 × 10 −5 1 × 10 −5 2.0 0.03 All the parameters in Tables 5 and 6, the rotation center position, 1 init , and 1 init , were used in Figure 4. The inverse kinematics is based on Equation (30) to obtain the joint angle, Θ. The change of the new weight value, ∆̂, of the self-organizing competitive neural network is obtained according to Equation (27). The learning rate, i.e., , 1 , ,1 , …, , , is used to calculate the carrying step size of EE motion in Cartesian space. With the continuous learning and competition of the proposed planning method, the position velocities of EEs gradually form a consistent movement. Figure 12a  (2) Manipulating the pliers Figure 13 shows that the two arms manipulated a pair of pliers. The parameters are illustrated in Tables 5 and 6 wheref 1 , ψ 1 ,f 2 , and ψ 2 can be obtained according to the following equations:  The inverse kinematics is based on Equation (30) to obtain the joint angle, Θ. The change of the new weight value, ·ŵ i , is obtained according to Equation (27) and used to adjust the EE motions according to Equation (28). The learning rate, i.e., η, η 1 , η n,1 , . . . , η n,k , is used to calculate the step size of manipulating the pilers in the Cartesian space. The proposed planning method makes the position velocities of EEs gradually form a consistent movement. Figure 14a,b show the motion paths of EEs and joint trajectories, respectively.   The inverse kinematics is based on Equation (30) to obtain the joint angle, Θ. The change of the new weight value, ∆̂, is obtained according to Equation (27) and used to adjust the EE motions according to Equation (28). The learning rate, i.e., , 1 , ,1 , …, , , is used to calculate the step size of manipulating the pilers in the Cartesian space. The proposed planning method makes the position velocities of EEs gradually form a consistent movement. Figure 14a,b show the motion paths of EEs and joint trajectories, respectively. Figure 14c wheref 1 , ψ 1 ,f 2 , and ψ 2 can be obtained according to the following equations: to Equation (2), and no arm motion is set as the reference for the other arms. Hence, the proposed planning method based on the self-organizing competitive neural network owns the feasibility, synchronism, and effectiveness in achieving the collaboration manipulations for the arms with physical coupling and has the contributions to the practical application.    Figure 16c,d, the two arms begin to manipulate the rudder at 6.75 s. In Figure 16e-h, the EE pose errors can converge to 0, and the corresponding EE motions own almost the same states ultimately by using the proposed planning method with the learning of the selforganizing competitive neural network. Considering (1)-(3) in Section 5.2 comprehensively, the pose speed and pose error converge to 0 for the successful execution of coordination manipulation and correspond to Equation (2), and no arm motion is set as the reference for the other arms. Hence, the proposed planning method based on the self-organizing competitive neural network owns the feasibility, synchronism, and effectiveness in achieving the collaboration manipulations for the arms with physical coupling and has the contributions to the practical application.

Conclusions
This paper presents a real-time kinematically synchronous planning method for collaborative manipulation through the self-organizing competitive neural network. This method considers a type of collaborative manipulation known as the synchronization of EE motion. The sub-bases are defined for the configuration of multi-arms to obtain the Jacobian matrix of common DoFs and ensure the pose errors converging along the reducing direction of the EE total pose errors. The simulations of multi-arms with common DoFs display the consistency before the pose errors converge completely and make contributions to the collaborative manipulation of multi-arms. On this basis, an unsupervised competitive neural network is raised to regard the EE synchronous motion as the competition of neurons and adaptively increase the convergence ratio of multi-arms through the mutual learning and competition of neurons by using the inner star rules. The stability of multi-arms system is analyzed through the Lyapunov theory. Various simulations and experiments confirm that the proposed synchronous planning method is feasible, synchronous, and has the application potentiality in different cooperative manipulation tasks.
Supplementary Materials: The following supporting information can be downloaded at: www.mdpi.com/xxx/s1, Video S1: title  Similar to the previous experiments, the inverse kinematics is based on Equation (30) to obtain the joint angle, Θ. The change of the new weight value, ∆ŵ i , is obtained according to Equation (27) and used to adjust the EE motions according to Equation (28). The learning rate, i.e., η, η 1 , η n,1 , . . . , η n,k , is used to calculate the step size of manipulating the rudder in Cartesian space. Since the shape of the rudder is symmetrical at the center, the tracked locations are also symmetrical, and the EE attitude changes are the same. In Figure 16c,d, the two arms begin to manipulate the rudder at 6.75 s. In Figure 16e-h, the EE pose errors can converge to 0, and the corresponding EE motions own almost the same states ultimately by using the proposed planning method with the learning of the self-organizing competitive neural network.
Considering (1)-(3) in Section 5.2 comprehensively, the pose speed and pose error converge to 0 for the successful execution of coordination manipulation and correspond to Equation (2), and no arm motion is set as the reference for the other arms. Hence, the proposed planning method based on the self-organizing competitive neural network owns the feasibility, synchronism, and effectiveness in achieving the collaboration manipulations for the arms with physical coupling and has the contributions to the practical application.

Conclusions
This paper presents a real-time kinematically synchronous planning method for collaborative manipulation through the self-organizing competitive neural network. This method considers a type of collaborative manipulation known as the synchronization of EE motion. The sub-bases are defined for the configuration of multi-arms to obtain the Jacobian matrix of common DoFs and ensure the pose errors converging along the reducing direction of the EE total pose errors. The simulations of multi-arms with common DoFs display the consistency before the pose errors converge completely and make contributions to the collaborative manipulation of multi-arms. On this basis, an unsupervised competitive neural network is raised to regard the EE synchronous motion as the competition of neurons and adaptively increase the convergence ratio of multi-arms through the mutual learning and competition of neurons by using the inner star rules. The stability of multi-arms system is analyzed through the Lyapunov theory. Various simulations and experiments confirm that the proposed synchronous planning method is feasible, synchronous, and has the application potentiality in different cooperative manipulation tasks.

Conflicts of Interest:
No conflicts of interest exit in regard to this manuscript, and this manuscript was approved by all authors for publication. I would like to declare, on behalf of my co-authors, that the work described was original research that was not published previously and not under consideration for publication elsewhere, either in whole or in part. All the authors listed approved the manuscript that is enclosed.