Investigation on a New Discrete-Time Synchronous Motion Planning Scheme for Dual-Arm Robot Systems

Recently, the synchronous motion planning (SMP) has been viewed as a crucial issue in dual-arm robot systems. In this study, a new SMP scheme based on the Jacobian pseudoinverse formulation is presented and investigated for dual-arm robot systems. Such a scheme is established by discretizing the existing SMP scheme via a special difference formula. The presented discrete-time SMP (DTSMP) scheme is then theoretically analyzed to possess the biquadrate pattern in the end-effector planning error of each arm, thereby guaranteeing the synchronous planning precision of dual-arm robot systems. Simulation results under the system composed of two four-link planar robot arms are provided to further substantiate the effective and superior performance of the presented DTSMP scheme.


I. INTRODUCTION
In recent years, the robot systems composed of two arms, i.e., dual-arm robot systems, have played a remarkable role in numerous fields [1]- [5]. In comparison with the single-arm robot system, the dual-arm robot system has more dexterity and flexibility, and thus can conduct more tasks requiring sophisticated motion and/or manipulation in complex environments. There are many benefits for dual-arm robot systems in industrial applications, such as synchronous motion for handling heavy loads, cooperative manipulation for completing multiple tasks, and cycle time improvement when keeping safe manipulation speed. Different types of dual-arm robot systems have been reported and investigated [6]- [10]. For example, a dual-arm robot system that consists of two 6-DOF (degrees-of-freedom) arms was developed in [7] for precision assembly. Another dual-arm robot system that is composed of two 7-DOF arms was designed in [8] for cell manufacturing process.
With regard to dual-arm robot systems, the synchronous motion planning (SMP), which can also be referred to as The associate editor coordinating the review of this manuscript and approving it for publication was Jianyong Yao . synchronous/cooperative kinematic control, is an interesting and active topic [1], [5], [9]- [13]. This topic is related to many practical application in industry, such as coordinate welding, cooperative assembling, and dextrous grasping. In general, the SMP of dual-arm robot systems can be described as that, given the desired path for the end-effector of each arm (to cooperatively perform, e.g., welding, assembling, and grasping), the related joint trajectory of each arm needs to be calculated accurately and determined simultaneously. Because of its significance, the SMP of dual-arm robot systems has recently been studied by many scientific researchers and engineering practitioners [1], [5], [9]- [25].
There are two general approaches for the SMP of dual-arm robot systems. The first one is based on the (relative) Jacobian pseudoinverse formulation [10], [15]- [21], where the SMP schemes are depicted as the summation of a minimumnorm particular solution and a homogeneous solution. The typical characteristic of this approach has made the related study popular in the past decades. For example, Lewis [16] introduced a relative Jacobian scheme for dual-arm robot systems, and a more compact expression of such scheme was presented by Jamisola Jr. and Roberts [17]. Hu et al [18] investigated another relative Jacobian scheme for dual-arm robot systems under task conflicts and joint constraints. Freddi et al [10], [19], [20] developed many effective schemes with the Jacobian pseudoinverse formulation for dual-arm robot systems. The second approach for the SMP of dualarm robot systems is based on the quadratic programming formulation [22]- [25]. In the study of this approach, the common way is to design an optimization performance index, formulate the end-effector planning requirement and physical limits as constraints, and thus establish the corresponding SMP schemes for dual-arm robot systems. These schemes can further be reformulated and unified into a quadratic program (QP) problem that can be solved by neural networks. By effectively calculating the QP problem, the SMP of dual-arm robot systems can eventually be achieved [22]- [25]. Notably, although remarkable success has been realized in the SMP of dual-arm robot systems, the existing literature mainly focuses on the SMP scheme design but the further improvement of the synchronous planning precision is generally neglected.
With regard to the SMP of dual-arm robot systems, an important issue that requires consideration is to guarantee and further improve the synchronous planning precision [7], [14]. For example, during the assembly process, we need to ensure the precision of cooperative manipulation by the dualarm robot system [7]. By using the SMP scheme that has a potential of realizing high precision, the dual-arm robot system can cooperatively and accurately perform the particular task (e.g., assembling, welding, and grasping). Generally speaking, if the end-effector planning error of each arm can be kept within the region of a (sufficient) small value, then the (high) synchronous planing precision of the whole robot system can be guaranteed. Focus on improving the end-effector planning precision of one robot arm, Li et al [26] developed a novel recurrent neural network, and Guo et al [27] used a typical difference formula. Differing from such two methods, the methods based on auto disturbance rejection controller (ADRC) to improve the robot planning precision were investigated by Liu et al [28] and Humaidi et al [29]. In addition to the ADRC, the planning precision can also be effectively improved by designing and using other controllers, such as the extended-state-observer-based adaptive controller and the barrier function-based asymptotic tracking controller [30]- [34].
In this study, following the existing successful work [10], [14], [15], [27], [35], we show a further investigation on the SMP of dual-arm robot systems. That is, a new discretetime SMP (DTSMP) scheme based on the pseudoinverse formulation is presented and investigated for dual-arm robot systems, where the end-effector planning precision of each arm can be guaranteed. The main contributions of this study are summarized and listed as follows.
i) A new DTSMP scheme for dual-arm robot systems is established by using a special difference formula [36]- [38] to discretize a previous SMP scheme. To the best of our knowledge, the presented DTSMP scheme has not been reported in existing literature.
ii) Theoretical analysis is provided to indicate the characteristic of the presented DTSMP scheme. Computer simulations are performed to verify the effectiveness of the presented DTSMP scheme. iii) Results reveal that the presented DTSMP scheme possesses the biquadrate pattern in the end-effector planning error of each arm. Specifically, the planning error changes in the mode of O(σ 4 ), where σ denotes the sampling gap. iv) This study is the first attempt to provide an effective SMP scheme with biquadrate pattern in synchronous planning precision for dual-arm robot systems. This study is crucial, because it shows a new insight into the design of different SMP schemes for dual-arm robot systems with guaranteed precision. The rest of this study is organized as follows: Section II presents some preliminaries about the SMP of dual-arm robot systems. Section III describes the new DTSMP scheme in this study for dual-arm robot systems, and provides the corresponding theoretical analysis. Section IV shows the simulation results under the robot system composed of two four-link planar arms, which are obtained with the presented DTSMP scheme. Section V concludes this study and presents the final remarks.

II. SYNCHRONOUS MOTION PLANNING
In this section, the description for the SMP of dual-arm robot systems is presented, and the SMP scheme with the pseudoinverse-based formulation is provided to lay a basis for further discussion.

A. PRELIMINARY
The SMP is a crucial issue in the study of dual-arm robot systems. Mathematically, the SMP of dual-arm robot systems can be realized by simultaneously and effectively solving the following nonlinear kinematic equation: where θ {L,R} ∈ R n denote the joint-angle vectors of the left and right arms, φ {L,R} ∈ R m denote the given end-effector desired paths, and f {L,R} (·) : R n → R m denote the nonlinear mappings. In other words, given φ L and φ R , we need to solve for θ L and θ R via (1) simultaneously and accurately, such that the end-effectors of dual-arm robot systems can synchronously track the desired paths and cooperatively perform the specific tasks (e.g., welding, assembling, and grasping). Notably, in this study, we focus on the situation of m < n for each arm in (1), which means that each arm considered in the robot system is a redundant one. In this situation, the possible joint-angle solution of each arm is infinite for a given endeffector desired path.
Because (1) is a nonlinear and underdetermined equation, it is generally difficult to obtain the solution of θ L and θ R by directly solving (1). Differentiating (1) with respect to time t results in the linear relationship as follows: where J {L,R} ∈ R m×n andθ {L,R} ∈ R n denote the Jacobian matrices and the joint-velocity vectors of the left and right arms, andφ {L,R} ∈ R m denote the time derivatives of φ {L,R} . Notably, (2) is also underdetermined because of n > m, which means that the possibleθ L andθ R solutions are infinite for the given φ L and φ R .

B. PSEUDOINVERSE-BASED SMP SCHEME
The approach based on the Jacobian pseudoinverse formulation has been developed and used to determine theθ L anḋ θ R solutions for realizing the SMP of dual-arm robot systems [10], [15]- [21]. Specifically, with regard to (2), by introducing the feedback, the Jacobian pseudoinverse approach for dual-arm robot systems is formulated as follows: where J † {L,R} ∈ R n×m denote the pseudoinverse matrices of J {L,R} , k > 0 ∈ R denotes the feedback gain, I ∈ R n×n denotes the identity matrix, and c {L,R} ∈ R n denote the vectors that are selected on the basis of some criteria [15]. Different selections of c L and c R in (3) lead to different schemes with specific characteristics (e.g., repetitive motion, singularity handling, or joint limit avoidance) for the SMP of dual-arm robot systems [10], [15].
In particular, by selecting c L = c R = 0, (3) is reduced to the following pseudoinverse-based SMP scheme for determining theθ L andθ R solutions of (2) simultaneously [35]: For presentation convenience, the pseudoinverse-based SMP scheme (4) is transformed into the unified form as follows: where the augmented vectors and matrix are given bẏ With regard to the pseudoinverse-based SMP scheme (5), an alternative for this scheme is to use the MATLAB routine ''ODE'' [39], thereby obtaining the corresponding solutions of θ L and θ R in (1). Furthermore, since (5) is depicted in a continuous-time form, the Euler difference formula [40], which is a widely-used formula, can be utilized to discretize (5). The resultant scheme with discrete-time form is developed to determine the θ L and θ R solutions at each time instant for realizing the SMP of dual-arm robot systems. However, this scheme may be ineffective in guaranteeing the (high) planning precision for dual-arm robot systems, which will be presented in Section IV.

III. NEW DTSMP SCHEME
In this section, a special difference formula in [36]- [38] is presented and used to discretize the pseudoinverse-based scheme (5). The new DTSMP scheme is thus established for dual-arm robot systems with guaranteed precision. Theoretical analysis of such a new scheme is provided as well.

A. SCHEME FORMULATION
In [36]- [38], the difference formula is established for firstorder derivative approximation, which is written as follows: It follows from the difference formula (6) that the approximation of the combined joint velocityθ k =θ(t k = kσ ) in (5) at time instant t k is given bẏ with ϑ k = ϑ(t k = kσ ) and the initial time instant t 0 = 0 s. Using (7) to discretize the pseudoinverse-based scheme (5) results in the following formulation: whereφ k =φ(t k = kσ ), ϕ k = ϕ(t k = kσ ), and h = 2σ k > 0 ∈ R denotes the step size. The elimination of the error term O(σ 4 ) in (8) further yields which is called the new DTSMP scheme for dual-arm robot systems in this study. By following the above derivation, the presented DTSMP scheme (9) has a truncation error of O(σ 4 ). With regard to (9), we need five states (i.e., ϑ 0 , ϑ 1 , ϑ 2 , ϑ 3 , and ϑ 4 ) to start the recursion. In this situation, given the initial joint states θ L0 and θ R0 of both arms, we have ϑ 0 = [θ L0 ; θ R0 ], and further have the other states by the following VOLUME 8, 2020 calculation: In summary, with five pre-calculated states, the presented DTSMP scheme (9) would generate the corresponding sequence {ϑ k } (with k = 0, 1, · · · , T /σ , and T denoting the task duration), which constitutes the θ L and θ R solutions of (1), to realize the SMP of dual-arm robot systems.

B. THEORETICAL ANALYSIS
In this subsection, the presented DTSMP scheme (9) is theoretically analyzed to possess the biquadrate pattern in the endeffector planning error of each arm, thereby guaranteeing the synchronous planning precision of dual-arm robot systems. Lemma 1: The presented DTSMP scheme (9) has the property of zero stability and consistency, thus having the property of convergence.
Proof: The zero-stability property of (9) is analyzed by the following characteristic polynomial [41]: As presented in [36]- [38], the roots of ψ(ρ) = 0 are on or in the unit circle, which indicates that (9) is zero stable. As analyzed in Section III-A, (9) has a truncation error of O(σ 4 ), thereby indicating its consistency property. On the basis of [41], the properties of being simultaneously zero stable and consistent will eventually lead to the convergence property of the presented DTSMP scheme (9). The proof is thus completed.
In mathematics, Lemma 1 shows that the solution of ϑ k generated by (9) would converge to a theoretical solution denoted by ϑ * k = ϑ * (t k = kδ) (which satisfies F(ϑ * k ) − ϕ k = 0). Furthermore, the following relationship between such two solutions is obtained with a sufficiently large k value: In this sense, Lemma 1 confirms the effective performance of the presented DTSMP scheme (9) in dual-arm robot systems.

Lemma 2: Considering dual-arm robot systems with no singularity and collision happened, the end-effector planning error of each arm via the presented DTSMP scheme (9) changes in the mode of O(σ 4 ).
Proof: Let us recall (10). When the value of k is sufficiently large, the following result is obtained: 4 ). (11) Let us recall the definitions of F(ϑ) and ϕ in Section II-B. Then, the reformulation of (11) is given by where ε Lk and ε Rk denote the end-effector planning errors of the left and right arms computed by (9) at time instant t k . It follows from (12) that the end-effector planning error of each arm via (9) changes in the mode of O(σ 4 ). This statement also indicates that the presented DTSMP scheme (9) possess the biquadrate pattern in the end-effector planning error of each arm. The proof is thus completed. Remarks: The above theoretical analysis confirms that the end-effector planning error of each arm (i.e., ε Lk and ε Rk ) computed by the presented DTSMP scheme (9) occurs without divergence. Furthermore, because of the O(σ 4 ) mode in (12), the magnitude of ε Lk and ε Rk can be maintained within the region of a (very) small value by using an appropriate value of σ in (9). This point will be presented and discussed in Section IV. In summary, Lemmas 1 and 2 indicate that the presented DTSMP scheme (9) can theoretically guarantee the precision of end-effector planning for each arm, and thus achieve the (high) synchronous planning precision of dualarm robot systems.

IV. SIMULATION COMPARISON AND VALIDATION
In this section, comparative simulation results under a dualarm robot system are provided to substantiate the effectiveness of the presented DTSMP scheme (9). For better understanding, the robot system that is composed of two four-link planar arms is shown in Fig. 1, where both arms are located at the same plane.

A. SYNCHRONOUS MOTION
In this subsection, the presented DTSMP scheme (9) is investigated with two illustrative examples. Example A.1: In this example, we simulate the presented DTSMP scheme (9) under the dual-arm robot system, where the end-effectors of both arms are expect to simultaneously track the same tricuspid path. In the simulations, the joint initial states of the left and right arms are set respectively as θ L0 = [9π/10; −π/10; −π/10; −π/10] rad and θ R0 = [π/10; π/10; π/10; π/10] rad, and the motion duration is set as T = 10 s. For comparison, the following SMP scheme, which can be derived from the discretization of (5) using the Euler difference formula [40], is also simulated: Fig. 2 shows the simulation results of the dual-arm robot system using the SMP scheme (13) with σ = 0.01 and h = 0.4, where time t ∈ {0, σ, 2σ, · · · , 10} s, and the endeffector planning errors of the left and right arms at each time instant are obtained with ε Lk = f L (θ Lk ) − φ Lk ∈ R 2 and ε Rk = f R (θ Rk ) − φ Rk ∈ R 2 . As shown in Fig. 2, the endeffector of each arm successfully tracks the desired path with the maximal planning error being less than 3 × 10 −4 m. This result indicates that (13) effectively enables the SMP of the dual-arm robot system. By contrast, Fig. 3 presents the simulation results of the dual-arm robot system using the presented DTSMP scheme (9) with σ = 0.01 and h = 0.4. As presented in Fig. 3, the end-effector of each arm also successfully tracks the desired path with the maximal planning error being less than 4 × 10 −7 m. Figs. 3(c) and (d) further present that the endeffector planning errors of both arms do not exhibit divergence. These results reflect the effective performance of (9) in realizing the SMP of the dual-arm robot system. Moreover, comparing Figs. 2 and 3 reveals that the maximal end-effector planning error of each arm via (9) is approximately 1, 000 times smaller than that via (13), i.e., 10 −7 versus 10 −4 . Therefore, the presented DTSMP scheme (9) is advantageous over the SMP scheme (13) for the dual-arm robot system.
By decreasing the value of σ from 0.01 to 0.001, both the SMP schemes (9) and (13) are simulated, and the corresponding results are shown in Figs. 4 and 5. Evidently, these simulation results verify again the effectiveness of (9) and (13) in terms of a small end-effector planning error. In addition, a comparison of Figs. 2 and 4 reveals that the maximal planning error of each arm via (13) is reduced by 100 times with decrease in σ by 10 times (i.e., from 10 −4 m to 10 −6 m). Another comparison of Figs. 3 and 5 reveals that the maximal planning error of each arm via (9) is reduced by 10, 000 times with decrease in σ by 10 times (i.e., from 10 −7 m to 10 −11 m). These comparative results show that the  presented DTSMP scheme (9) can generate a (much) smaller end-effector planning error than the SMP scheme (13) for the dual-arm robot system. A high planning precision of the dualarm robot system using (9) is thus guaranteed and achieved.
In summary, the above simulation results (i.e., Figs. 2-5) verify the effective and superior performance of the presented DTSMP scheme (9) on the dual-arm robot system.
Example A.2: In this example, we simulate the presented DTSMP scheme (9) under the dual-arm robot system, where one end-effector is expect to track the tricuspid path and the other is expect to track the astroid path. In the simulations, the joint initial states of the left and right arms are set respectively as θ L0 = [11π/12; −π/12; −π/12; −π/12] rad and θ R0 = [π/12; π/12; π/12; π/12] rad, and the motion duration is set as T = 10 s. Fig. 6 shows the simulation results of the dual-arm robot system using the presented DTSMP scheme (9) with σ = 0.01 and h = 0.4. As shown in Fig. 6, the end-effector of each arm effectively tracks the corresponding desired path. The maximal planning errors of the left and right arms are less 201550 VOLUME 8, 2020   than 2.5 × 10 −7 m and 8 × 10 −7 m, respectively. In addition, the end-effector planning errors of both arms do not encounter the divergence issue. These results confirm the effectiveness of the presented DTSMP scheme (9).
For further investigation, the presented DTSMP scheme (9) with different values of σ and h is simulated, and the related data are presented in Table 1. These data show that the SMP of the dual-arm robot system using (9) is successfully realized in the light of the small planning error of each arm. Moreover, the observed results from Table 1 are summarized as follows.
i) The sampling gap σ mainly affects the maximal endeffector planning error of each arm computed by the presented DTSMP scheme (9). With a fixed h, the maximal planning error is effectively reduced by decreasing σ .
ii) The end-effector planning precision of each arm via the presented DTSMP scheme (9) changes in the mode of O(σ 4 ). Specifically, the decrease in σ by 10 times leads to the planning precision improvement by 10, 000 times. This observation coincides with the result stated in Lemma 2. iii) A non-zero value of the step size h is to guarantee the end-effector planning error of each arm computed by the presented DTSMP scheme (9) with no divergence.
Increasing h within an appropriate range can further improve the end-effector planning precision of each arm. On the basis of these results, a small value of σ and a relatively large value of h in (9) can guarantee and achieve a high planning precision of dual-arm robot system, thereby  indicating the effective and superior performance of the presented DTSMP scheme (9).

B. COOPERATIVE MANIPULATION
In this subsection, the presented DTSMP scheme (9) is investigated under the dual-arm robot system, where the end-effectors of both arm are expect to cooperatively track the astroid path. In the simulations, the joint initial states of the left and right arms are set respectively as θ L0 = [2π/3; −π/6; −π/6; 0] rad and θ R0 = [π/3; π/6; π/6; 0] rad, and the motion duration is set as T = 5 s. Fig. 7 presents the simulation results of the dual-arm robot system using the presented DTSMP scheme (9) with σ = 0.01 and h = 0.4. As presented in Fig. 7(a)-(c), the end-effector of each arms effectively track its corresponding path, and both arm finish tracking cooperatively the desired astroid path. As presented in Fig. 7(d)-(f), the maximal end-effector planning error of each arm computed by (9) is less than 4 × 10 −7 m. This statement means that the astroid path tracking task is completed successfully via the dual-arm robot system with the cooperative manipulation. Evidently, the task execution time can be shorten greatly by the dual-arm robot system, as compared with the single-arm robot system (e.g., see the simulation result of the right arm in Fig. 6). These results show the advantage of the dual-arm robot system, and indicate the effectiveness of the presented DTSMP scheme (9).
With the decrease in σ (i.e., from 0.01 to 0.001), the presented DTSMP scheme (9) is simulated. The corresponding results are shown in Fig. 8, which verify again that (9) effectively enables the SMP of the dual-arm robot system. Moreover, a detailed inspection on Figs. 7 and 8 reveals that the end-effector planning precision of each arm via (9) changes in the mode of O(σ 4 ). Specifically, with regard to each arm, its end-effector planning error is reduced by 10, 000 times; i.e., from 10 −7 m to 10 −11 m, when decreasing σ by 10 times. Therefore, the high planning precision of the dual-arm robot system using (9) is guaranteed and achieved.
In summary, these simulation results (i.e., Figs. 7 and 8) substantiate again the effective and superior performance of the presented DTSMP scheme (9) on the dual-arm robot system.

V. CONCLUSION
In this study, the SMP of dual-arm robot systems is investigated by presenting the new DTSMP scheme (9). Such a scheme is established from the discretization of the SMP scheme (5) using the special difference formula (7). It is then theoretically analyzed that the presented DTSMP scheme (9) possesses the biquadrate pattern in the end-effector planning error of each arm. Specifically, the planning error changes in the mode of O(σ 4 ), which can guarantee the synchronous planning precision of dual-arm robot systems. On the basis of under the robot system composed of two four-link planar arms, comparative simulation results with different illustrative examples further indicate the effectiveness and superiority of the presented DTSMP scheme (9).
As for dual-arm robot systems, one future research direction can be the study of designing the SMP schemes by using the approach of neural networks [42]- [44]. Another future research direction can be the study of designing the SMP schemes by considering the handle of additive noise, environmental obstacle, and/or physical limits [45]- [47]. By following this study, the presented DTSMP scheme (9) is expected to be implemented on the real dual-arm robot system that consists of two practical EFORT robot manipulators.