Recurrent Neural Networks-Based Collision-Free Motion Planning for Dual Manipulators Under Multiple Constraints

Dual robotic manipulators are robotic systems that are developed to imitate human arms, which shows great potential in performing complex tasks. Collision-free motion planning in real time is still a challenging problem for controlling a dual robotic manipulator because of the overlap workspace. In this paper, a novel planning strategy under physical constraints of dual manipulators using dynamic neural networks is proposed, which can satisfy the collision avoidance and trajectory tracking. Particularly, the problem of collision avoidance is first formulated into a set of inequality formulas, whereas the robotic trajectory is then transformed into an equality constraint by introducing negative feedback in outer loop. The planning problem subsequently becomes a Quadratic Programming (QP) problem by considering the redundancy, the boundaries of joint angles and velocities of the system. The QP is solved using a convergent provable recurrent neural network that without calculating the pseudo-inversion of the Jacobian. Consequently, numerical experiments on 8-DoF modular robot and 14-DoF Baxter robot are conducted to show the superiority of the proposed strategy.


I. INTRODUCTION
Robot manipulators are programmable automatic devices with multiple DoFs and are designed to imitate human arms. These robots are built to effectively complete tasks and reduce the workload of human beings [1]. However, it is difficult for a single robot to fulfill complex and flexible tasks due to limited flexibility and payload [2]. Thus, dual robotic manipulator systems have been developed to imitate human arms [3]. Despite the great potential of dual robotic manipulators, the realization of motion planning with acceptable performance in real time remains challenging for several The associate editor coordinating the review of this manuscript and approving it for publication was Choon Ki Ahn .
reasons. Firstly, for better flexibility, the mechanical structure of each robot is usually designed to be redundant, which should be solved in the planning process [4]. Secondly, in the coordination process, the physical interference between manipulators should be considered. Because the workspace of two manipulators are overlapped, there is a potential risk of collision. Therefore, collisions must be avoided at all times [5]. Thirdly, the states of manipulators, such as joint angles and velocities, must satisfy their constraints [6], [7]. Although conventional off-line programming methods can achieve collision-free trajectories, they are usually inefficient and inflexible [8]. Therefore, the optimal approach to design an online method that fulfills convergent motion plan while simultaneously avoiding collisions and physical constraints is still unresolved. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Many schemes have been reported for avoiding collisions in real time. The artificial potential field method, which has the advantage of provable stability, is widely used in industrial applications. The artificial potential field methods use an attractive pole for the position to be reached and repulsive surfaces for the obstacles [9]. Thus, the robot can be programmed to avoid collision. In [10], Csiszar et al. improved the method by introducing multiple geometrical forms to characterize obstacles, and this method was verified to be effective in a more generalized environment. In the field of the local minimum problem in the planning process, a dual minima scheme [11]- [14] has been developed by using two attractive poles to avoid the local minima.
In terms of the redundant DOFs, many redundancy resolutions have been proposed to address the redundant resolution problem [15], [16], [34]. They are usually based on calculating the Jacobian-matrix pseudo-inverse (JMPI). Specifically, the velocity command can be formulated asθ = J +ẋ r + (I − J + J )γ , and it composes a minimum norm solution J +ẋ r and a homogeneous solution (I − J + J )γ in the null space [18]. Given that there may exist multiple solutions in the joint space that produce the same end-effector response, γ can be defined according to particular redundancy resolution schemes, such as minimizing the velocity norm or the acceleration norm. Based on the above methods, a self-motion component in the null space can be designed to avoid collisions in [17]. However, in previous studies, boundaries of robot states have been rarely considered.
In redundancy resolution problems, the selection of a homogeneous component follows a particular rule to obtain optimal solutions of predefined objective functions, and the physical constraints can usually be defined by inequality formulas. Thus, it is highly suitable to define the redundancy resolution problem as an optimization problem with constraints [19]. According to the nonlinearity of robotic systems, it is difficult to obtain the solutions of these optimization problems in real time [28]. In [20], a recurrent neural network (RNN) was introduced to calculate the control command online. In [21], a modified RNN was proposed to eliminate the drifting of end-effectors at the speed domain.
Based on the previous research, Cai expanded the RNN to different domains [22], i.e., the velocity domain and acceleration domain. Taking non-convex objective functions into account, Jin studied a manipulability optimization strategy [23] in which the objective function is rebuilt according to its gradient, and a convexification operator was also proposed. Recently, this method has been successfully used in robotic force control [24], [25], adaptive control with model uncertainties [26], [35], noisy systems [29], and visual servo systems [36]. Guo and Zhang have done a lot of work on single manipulators to address the obstacle avoidance problem. For example, in [30], obstacles were described as a group of critical points, and the distance between these points and the robot was calculated on the basis of possible obstacle positions. In [31], Guo improved the method by introducing a novel distance-based smoothing function, and it was verified that the modified method could effectively restrain dithering. In [32], Xu extended the smoothing function to Class-K functions. After discussing the influence of different Class-K functions, the method of function selection was summarized. However, the research mainly focused on motion planning methods for single manipulators, and the obstacles are usually regarded as slow time-varying.
Motivated by the above observations, in this paper, we propose a RNN based motion planning method for dual robotic arms. By understanding the basic principle of obstacle avoidance method, a collision avoidance scheme is developed in form of inequality constraints, where one arm is described as a group of dynamic obstacles to the other arm. By selecting the minimum-velocity-norm scheme for redundancy resolution, a quadratic programming (QP)-type problem formulation in which the physical constraints are also considered. Then, a dynamic neural network is proposed to solve the QP problem online. Numerical results show that the proposed method is efficient. The main contributions of this paper are as: • The proposed method can simultaneously avoid the collision of dual robots, and ensure convergent planning to predefined trajectories.
• The proposed RNN based method could handle physical constraints online, such as the boundedness of joint angles and velocities, which is of great significance in realtime motion planning.
• Different with traditional JMPI based methods, the scheme in this paper does not require the calculation of pseudo-inversion, which is capable of enhancing computing efficiency.

II. PROBLEM FORMULATION A. KINEMATIC FORMULATION AND TRAJECTORY TRACKING
In this paper, we consider motion planning problem in velocity level, that is, design joint velocity commands to ensure the convergence of planning errors. Firstly, the kinematic model of each manipulator of a dual robotic system is given as where x i ∈ R m is the positional description of the ith robot in Cartesian space, and θ i ∈ R n is the vector of joint angles. Without loss of generality, every single robot is consider as redundant. Therefore, we have n > m. The kinematic equation at the speed level can be described bẏ where J i is the Jacobian matrix of the ith robot. In the motion planning problem, a fundamental objective is to generalize joint commands such that the 1 st and 2 nd arm could track the predefined trajectories x 1d , x 2d . In velocity level, the equality constraints about the above problem can be described as The envelope shape-based collision avoidance scheme. When the robots are described by a set of envelope shapes and the distance between critical points is greater than a specified value, collision will be avoided.
where k|e i | ρ sgn(e i ) is a negative feedback of planning errors It can be easily proved that the planning error e i would converge to 0 in . That is, if the joint velocity commandsθ i satisfies (3), the fundamental motion planning problem can be realized.

B. COLLISION AVOIDANCE SCHEME
In the process of robot cooperation, there is a risk of collisions between two arms. Therefore, another key issue in the motion planning problem of dual robotic manipulators is to avoid potential collisions between robots. In [31], collision detection methods based on envelope modeling have been used to identify whether a collision will happen between a robot and the external environment. The robot manipulator can be simply described by a set of envelope shapes, such as cylindrical or spherical. Thus, a collision avoidance method based on the spherical envelope shapes to describe the robots is proposed in this paper. Figure 1 shows the basic idea of the collision-free scheme used in this paper. Let A 1 = [A 1 (1), . . . , A 1 (a)] ∈ R m×a be a set of centers of spherical envelope shapes of the 1 st robot, with the radius being selected uniformly as r 1 . The principle of choosing A 1 and r 1 is that the union of spheres could completely surround the 1 st robot s body. Similarly, A 2 = [A 2 (1), . . . , A 2 (a)] ∈ R m×a and r 2 are defined for 2 nd robot. When no collision occurs, the sufficient condition can be described as follows: for arbitrary point pairs A 1 (k)A 2 (l) from the 1 st and 2 nd robot, the distance ||A 1 (k)A 2 (l)|| 2 2 between A 1 (k) and A 2 (l) satisfies the inequality ||A 1 (k)A 2 (l)|| 2 2 ≥ D, where D = r 1 + r 2 + d is a called the safety distance, and d > 0 is the distance allowance. Remark 1: Although different envelope shapes can be selected, the description methods are similar. In this paper, spherical envelope shapes are used to show the central idea of collision avoidance between robots; the description can be readily extended to other envelope shapes.
The inequality ||A 1 (k)A 2 (l)|| 2 2 ≥ D defines a basic description of collision avoidance in the position level, which is independent of joint commands. Therefore, by considering the dynamic formula where K is a positive parameter, it can be readily appreciated that the original inequality ||A 1 (k)A 2 (l)|| 2 2 ≥ D can be guaranteed by ensuring (4).
It is notable that the kinematic description of A 1 (k) can be calculated according to forward kinematics, similar to (1) and (2). Note thatȦ 1 (k) = J 1k (θ 1 )θ 1 , where J 1k is the corresponding Jacobian matrix of the kth spherical center of the ith manipulator. Similarly,Ȧ 2 (l) = J 2l (θ 2 )θ 2 . Calculating the time derivative of ||A 1 (k)A 2 (l)|| 2 2 yields d dt Substituting (5) to (4) yields is a unit vector from It is noteworthy that the derived inequality in (6) is capable of avoiding a potential collision between A 1 (k) and A 2 (l). The collision can be avoided effectively by ensuring all inequalities hold for k = 1, . . . a, l = 1, . . . b. Thus, the inequality form in the velocity level can be formulated as where

Remark 2:
The inequality in (7) describes the cascaded form of the collision avoidance scheme by considering all VOLUME 8, 2020 pairs of spherical centers. It is noted that the minus sign in the first column in J ob plays an important role in collision avoidance, which shows that the corresponding velocities of the centers of the 1 st and 2 nd robots are opposite. The proposed collision avoidance schemes considers the relative movement of the two arms, which is an important improvement than methods in [30], [31]. It is also notable that from the description in (7), the self-collision formulation can be similarly derived for the joint-speed level. Since this paper mainly focuses on the collision between robots, it is not addressed here.

C. QP-TYPE PROBLEM FORMULATION AND RNN DESIGN
In previous subsections, we have obtained the equality and inequality constraints about the fundamental plan objective and the collision avoidance scheme. In this subsection, by considering the physical constraints and cost function to be optimized, the original motion planning problem is transferred into a QP one. Then some proper modification is done to formulate the QQP problem in the same level, and finally a RNN is established to solve the QP online.
In real applications, the physical states of manipulators are limited, i.e., the joint angles θ 1 and θ 2 are limited by the mechanical design, and the joint speedsθ 1 andθ 2 are also constrained by the limitation of actuators. Therefore, during the planning process, both joint angles θ 1 , θ 2 and joint velocitiesθ 1 andθ 2 must not exceed their limits, in the sense and θ + 2 are the lower and upper bounds of the two manipulators, andθ − 1 ,θ − 2 ,θ + 1 , anḋ θ + 2 are the limits of the joint velocities. Due to the redundancy of the manipulators, in addition to completing the original task, it can also achieve the secondary target simultaneously. Therefore, without loss of generality, the secondary task is selected to minimize the norm of joint velocities, i.e.,θ T 1θ 1 +θ T 2θ 2 . On the basis of the previous descriptions, the QP-type problem formulation is derived as d2 ] T , and define J = J 1 0 0 J 2 as a block diagonal matrix of J 1 and J 2 , (8)can be reformulated as The original QP (9) gives the basic idea to transfer the original trajectory tracking problem into an equality constraint, which is essential to rebuild the trajectory tracking into QP type problem. In this paper, (9) is capable of ensuring fast convergence of tracking error.
Given the angular constraints in (9c) and (9d), a unified description can be obtained according to the escape velocity method as belowθ ) play important roles in ensuring the inequality constraints in (9c) and (9d). Here, α is a positive constant for weighting the feedback gain to ensure (9c). Now, the final form of the QP-type formulation is formulated as According to the Karush-Kuhn-Tucker conditions, the optimal solution of the QP-type formulation in (11) satisfiesθ where L is a Lagrange function, which is defined by (13) in which λ 1 ∈ R 2m and λ 2 ∈ R ab are dual state variables corresponding to the constraints in (11b) and (11c), respectively. P is a projection operator to a closed set = [θ − r ,θ + r ] and P (x) = argmin y∈ ||y − x|| is the saturation function defined as Equation (12) is typically nonlinear, and it is timeconsuming to solve. Moreover, because of the time-varying characteristic, the solution of (12) also differs with time. Therefore, in this paper, a dynamic neural network which has the ability of parallel computing is proposed as 54228 VOLUME 8, 2020 where the operator (•) + is a defined as y The structure diagram of the developed planning method is shown in figure 2. Given the desired task x d andẋ d , the scheme can be built based on the sensor feedbacks θ,θ of the dual manipulator system. The collision avoidance scheme has been established by setting a proper safety distance considering the joint configurations of both manipulators. With the consideration of physical constraints and optimization metrics, the motion planning problem is transferred into a QP type description, and an RNN is developed to obtain the joint commandθ in real time.
Remark 3: Compared to conventional pure data driven neural networks that are used to approximate unknown nonlinear items in reference [19], [27], the RNN established in this paper uses model information to determine weights of the neural network, and thus it has a simpler structure than conventional ones.
Remark 4: Using the proposed RNN based controller, the joint velocity commandθ can be obtained by the updating law Eq. (14). It can be shown in the latter section that the output of Eq.(14) would ensure the convergent tracking of the dual manipulator systems, and avoiding the possible collisions. It is notable that based on the proposed controller Eq. (14), dynamic controllers in the inner loop can be further developed, in which the robot dynamics such as inertia, Coriolis-centripetal force, external disturbances can be considered. Relative studies can be found in references such as [39]- [41].

D. STABILITY ANALYSIS
In this subsection, the stability of the proposed RNN based planning method is discussed. Firstly, several lemmas are given.
Definition 1: A continuously differentiable function F ∈ R k is monotonic if its gradient ∇F satisfies the following skew symmetry property: Calculate matrices J ob and D dis in the collision avoidance scheme according to (7) 6: Calculate the reference velocity limitsθ − r ,θ + r 7: Update joint commandθ byθ according to (14a) 8: Update state variable λ 1 according to (14b) 9: Update state variable λ 2 according to (14c) 10: until (t > T ) network is ensured to converge to its equilibrium point if its dynamics satisfies the following equation: where κ > 0 is a positive constant that describes the convergence speed of the network, and F(x) is a monotonic function, which is introduced later. P S (x) = argmin y∈S ||y − x|| is a projection operator to the convex set S. The stability is analyzed and proved by three parts. Firstly, the neural network is able to converge to its equilibrium point. Then this equilibrium point is proved to be equivalent to the optimal solution of the QP-type description of the problem. Finally, the convergence of the planning error is analyzed.
Theorem 1: The proposed recurrent neural network (14) will converge to its optimal solution globally.
Proof: If a new augmented vector ξ composed ofθ, λ 1 , and λ 2 is defined as then equation (14) is reformulated as which can be further presented as where

VOLUME 8, 2020
It is noteworthy that for (20), P R (•) can be regarded as a projection operator of λ 1 to a special convex R, with the lower and upper bounds being ±∞. P + (•) = (•) + is also a special projection operator to a non-negative real set. For example, the lower and upper bounds of the set are 0 and +∞, respectively.
The gradient of F is calculated as Thus ∇F(ξ ) + ∇F T (ξ ) = It can be concluded that the dynamics of the established dynamic neural network in (14) satisfies the property described in (21). This is equivalent to the sufficient condition for the convergence of neural networks based on projection operators, such as (15). According to Lemma 1, the convergence of the proposed planning method is guaranteed, i.e., let ξ * = [θ * ; λ * 1 ; λ * 2 ] be the equilibrium point, ξ → ξ * as t → ∞.
Theorem 2: The optimal solution ξ * is equivalent to the QP problem in (11) and ensures the convergence of the planning error.
From theorem 1, it can be concluded that the outputθ of the established dynamic neural network will eventually satisfy the equality constraint in (22b). From the definition of the planning error e, its gradientė can be formulated aṡ e =ẋ d −ẋ =ẋ d − Jθ. By defining a Lyapunov function V as V = e T e/2, it can be readily observed thaṫ and then the convergence of the planning error is ensured. Remark 5: It is notable that the RNN based planning method could only ensure the global convergence of planning error, rather than finite convergent. Since the proposed Eq. (14) is global convergent, therefore, the planning error considering the whole system is global convergent. However, whether the planning error is finite-convergent or global convergent is no the key issue of the proposed method. The main contributions of the paper is that the method could ensure convergence of planning error and avoid collision between robots, while it is capable of handling physical constraints in real time.
Remark 6: In this section, the stability of the proposed control scheme is proved. Compared with other neural networkbased methods in which neural networks are used to learn nonlinear items, the method proposed in this paper is much simpler. This is mainly because robotic models are introduced to the neural networks, while conventional networks only use error information.
Remark 7: So far, the proposed method is proved to be capable of ensuring the convergent planning of the endeffectors of both manipulators, and the collision between robots is avoided. By perceiving the configuration of the manipulators, collision avoidance is achieved by adjusting the robots online, which results in better performance than methods with offline programming.

III. NUMERICAL RESULTS
In this section, numerical simulations are presented to show the effectiveness of the proposed method. The coordination tasks of dual manipulator systems require robots to cooperate with each other. In other words, the manipulators receive control commands that end-effectors x 1 and x 2 track their reference trajectories x 1d and x 2d , respectively, where x 1d are 2d can be the same or different in real applications. Firstly, simulations on a 8DOF planar modular robot system are done. An interesting case is analyzed in which one end-effector is required to stay at a fixed point while the other end-effector is planned to track a certain trajectory. Then a case in which the robots are planed to track different trajectories is presented. Secondly, simulation on a 14DOF dual arm Baxter is carried out. Finally, comparisons are made with previous methods to demonstrate the superiority of the proposed motion planning method.

A. NUMERICAL RESULTS ON MODULAR ROBOT
Firstly, two simulations are carried out on a 8DOF planar modular robot system. The physical structure and the DH parameters of single arm are given in Fig. 3. The ground coordinate system is marked in red arrows in Fig. 3(a). The 54230 VOLUME 8, 2020 key points A i , i = 1, · · · , 6 and B i , i = 1, · · · , 6, also marked in Fig. 3(a), are important points for achieving collision avoidance. The 1 st , 3 rd , and 5 th key points are located in the 2 nd , 3 rd , and 4 th joint, respectively, while the 2 nd , 4 th , and 6 th key points are located at the center of mass of the 2 nd , 3 rd , and 4 th link. The 7 th key point exactly marks the end-effector. The DH parameters are given in Fig. 3(b).
In this section, the safety distance D is defined as D = 0.15 m to eliminate collisions, and the parameter in the collision avoidance scheme in (6) is selected as K = 200. In the outer loop, the control gain of the position controller is selected as k = 8 and ρ = 1. As an important parameter that scales the convergence of the established dynamic neural network, is selected as = 0.001. For the physical constraints, the angular limits of both manipulators are defined
It is remarkable that at the beginning stage, joint velocities of the robots reach the maximum value, and then slow down quickly. Correspondingly, the planning errors converge to 0. It can be also readily observed from Fig. 4 (g) that the joints are ensured not to exceed the upper and lower bounds (black dotted line). During t = 0-0.5s and t = 3-5 s, the robots approach each other, accordingly, the state variable λ 2 become positive, which guarantees the inequality (11d), and the minimum safety distance is obviously ensured through (14c), as shown in Fig. 4 (b). The proposed planning method ensures that the safety distance between the two robots is maintained, and the collision is thus avoided. The solution errors of the established neural network are shown in Fig. 4 (c). The errors converge to zero in less than 0.5 seconds. At t = 5 s, it reaches its maximum value of 3 × 10 −3 m. The snapshots during the planning process are given in Fig. 4 (h). It can be observed that the right arm adjusts its joint configuration while maintaining the position of endeffector, and the collision is avoided in realtime.
Simulation results are shown in Fig. 5. The planning errors of both manipulators converge to 0 successfully, and stable errors are less than 2 × 10 −4 m, showing that the designed scheme can achieve stable tracking (Fig. 5(a)). The changing curve of the distance between robots is given in Fig. 5(b). Fig. 5(f) and (g) illustrate the states of the manipulators, and it VOLUME 8, 2020 can be readily observed that the boundary of the joint angles and velocities is guaranteed. And the speed curves uplift at t = 3 s, which is because of the influence of (14c). As proved in Section 2.4, the RNN-based scheme globally converges, which can be verified from the quick convergence of solution errors in Fig. 5(c). Snapshots of the planning process are given in Fig. 5(h). The end-effectors move smoothly toward the desired paths, and then they track them successfully.

3) COMPARISON WITH JMPI BASED METHOD
In this part, comparative simulations are carried out to show the superiority of the proposed controller. We compare the proposed strategy with JMPI based methods, in which the joint velocity command is obtained by calculating the pseudoinverse of Jacobian matrix in every control period, and the collision avoidance is achieved in the null space. Comparative results are shown in Fig. (6). Fig. 6(a) shows the Euclidean norm of the planning errors by methods based on RNN and JMPI, respectively, and the corresponding curves of minimum distance between the left and right arm are as given in Fig. 6(b). Both methods could guarantee the convergence of planning errors and collision avoidance. However, by comparing the curves of joint velocities (Fig. 6(c) and Fig. 5(f)) and joint angles (Fig. 6(d) and Fig. 5(g)), it can be observed that JMPI based method failed in handling physical constraints. It is remarkable that in real applications, the satisfaction of physical constraints in real time is of great significance in ensuring the safety of dual-manipulator systems.

4) WITH NOISE
In this section, we evaluate the impact of noise on neural controllers. It is notable that the angle feedback accuracy of robot joint is very high, and the noise mainly exists in the angle velocity. Therefore, we consider white Gaussian noise in the feedback of joint speed. In this simulation, four cases are considered, i.e., covariance is selected as 2, 10, 20, 50, respectively. Simulation results are shown in Fig. 7. Fig. 7 (a) shows the Euclidean norm planning errors, with the maximum value in stable state is 0.02. Fig. 7 (b) shows the minimum distance between the robots. It can be found that under the disturbance of white Gaussian noise, the proposed planning method could achieve satisfactory results.

B. NUMERICAL RESULTS ON 14DOF BAXTER
In this section, numerical results on a 14-DOF robot Baxter are presented. The physical structure and the DH parameters 54232 VOLUME 8, 2020  of Baxter are given in Fig. 8. The key points A i , i = 1, · · · , 8 and B i , i = 1, · · · , 8, also marked in Fig. 8(a), are important points for achieving collision avoidance. The 1 st and 5 th key points are located in the 4 th , 6 th joint, respectively, while the 2 nd , 3 rd , 4 th , 6 th , 7 th , 8 th are evenly distributed on the 3 rd , 4 th link. The DH parameters are given in Fig. 8(b).
The safety distance D is defined as D = 0.2 m to eliminate collisions, and the parameter in the collision avoidance scheme in (6)  Numerical results are as shown in Fig. 9-11. The planning errors are shown in Fig. 9(a), the stable errors of robots A and B are about 1 × 10 −3 and 6 × 10 −4 m, respectively. At the beginning stage, joint velocities of the robots reach the maximum value, and then slow down quickly. Correspondingly, the planning errors converge to 0. During t = 1.5-7s, the robots approach each other, and the distance reaches the minimum value (0.2m). Using the proposed motion plan method, joint angular velocity curve jumps (up to about 1 rad/s), and the planning error of the robot changes slightly (about 1 × 10 −3 ), but the minimum safety distance robots is guaranteed. It is notable that the state variables λ 1 and λ 2 change according to the planning error and minimum distance, respectively. The solution errors of the established neural network are shown in Fig. 9(g). The errors converge to zero in less than 0.3 seconds. The profile of joint angles and the corresponding limits are shown in Fig. 10, in which the red filled part represents the accessible range of the joint angles. It can be readily observed that the joint angles are ensured not to exceed the limits (the 2 nd joint of left arm and 1 st joint reach the upper bound). Although existing JMPI-based online motion planning methods may obtain better performance in the suppression of regulation errors, the proposed method in this paper is still meaningful. In conventional methods, the self-motion is calculated in the null space of the Jacobian matrix: pseudo-inversion is calculated to decouple the movement of the end-effector from self-motion. In this paper, with the dynamic neural network in (14), rather than calculating pseudo-inversion, the joint command is obtained in a recurrent manner that is not decoupled. The elimination of the pseudo-inversion calculation reduces the computational cost in the planning process. Furthermore, the boundedness of robot joint states is ensured, which is difficult to do with conventional methods. The snapshots during the planning process are given in Fig. 11. It can be observed that the manipulators adjust joint configuration while maintaining the position of end-effector, and the collision is avoided in realtime.

C. COMPARISONS
In this part, comparisons with other motion planning methods are presented to show the superiority of the proposed collision-free planning strategy, as shown in Table 1. In [17], [37], JMPI-based motion planners were introduced in which collision is avoided in the null space. In [37], the JMPI method was improved by introducing a damped least-squares algorithm to solve the singularity problem. In [31], [32],   optimization-based motion planners were designed for single robots, and they are able handle physical constraints. In [38], a configuration space-based motion planning method was developed. This method uses a reachable manifold and contact manifold to realize collision avoidance. However, the research mainly focused on point-to-point collisionfree planning. In this paper, the motion planning problem for dual manipulator systems with continuous trajectories in Cartesian space and physical constraints are taken into consideration. It is noteworthy that, compared with the JMPI-based methods, the proposed scheme does not require the pseudo-inversion calculation, which could enhance realtime performance.

IV. CONCLUSION
This paper proposes a novel RNN-based motion planning strategy to avoid collisions for dual manipulator systems in real time. An outer-loop controller was built as an equality constraint, and the physical constraints are described as inequality constraints. Consequently, the collision avoidance can be defined by the inequality constraints using the envelope model-based collision detection methods, and the constraints can be reformulated in the speed level. A QP-type problem is obtained to describe the implicit solution of the motion planning problem through constraint optimization. Finally, a dynamic neural network is used to optimize the solution. This work is a significant extension of the RNN-based framework for robotic systems, and it can handle both collision avoidance and physical constraints in real time.
ZHIHAO XU received the B.E. and Ph.D. degrees from the School of Automation, Nanjing University of Science and Technology, Nanjing, China, in 2010 and 2016, respectively.
He is currently a Researcher with the Robotics Team, Guangdong Institute of Intelligent Manufacturing, Guangzhou, China. His main research interests include neural networks, force control, and intelligent information processing.