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Abstract

When two manipulators cooperate to clamp a rigid object, a closed chain with both ends fixed is formed. The singularity of a single manipulator causes the closed chain to be at a singularity, which results in uncertainty in the velocity mapping between the terminal and joint space. This study proposes a closed-chain singularity processing method for dual-arm collaboration, adopting the Jacobian pseudoinverse method based on damped least squares to obtain an approximate solution. During the solution process, the adaptive damping factor is improved by cubic polynomials, and the optimal damping factor is determined by quadratic sequence programming. Furthermore, this study analyzes the kinematic constraints of the closed-chain system and constructs a closed-chain Jacobian to judge whether the system is in a singular region. Finally, the effectiveness of the proposed method is verified through simulations.

Keywords

Dual-arm coordination closed-chain system singularity damped least squares

Introduction

A dual-arm system outperforms a traditional tandem robotic manipulator system in terms of target object operations. For example, grasping of large or irregularly shaped objects can be completed through the cooperation of dual arms, whereas a single arm may not be able to perform a similar job. When the dual arms cooperate to carry a rigid object, the object and the two manipulators form a chain structure with both ends fixed, which is called a closed-chain system.

However, both a single-arm robot and a closed-chain system have a singularity, which manifests as uncertainty in the expected terminal velocity at the singularity when it is mapped to the joint space.¹ This uncertainty is due to the non-full rank of the Jacobian of the manipulator under the above joint configuration, resulting in the inability to solve the Jacobian inverse matrix. Redundant manipulators can rely on redundancy features to directly avoid singularities.² Conversely, nonredundant manipulators primarily deal with singularities by obtaining an approximate solution, and the damped least squares (DLS) method is the most typical approximation method applied to this problem.^3,4 In such methods, DLS is used to construct the pseudoinverse of the Jacobian, and then, a special velocity inverse mapping formula is used to solve the problem that the dimension of the driving joint is greater than the degree of freedom (DOF).⁵

The selection of the adaptive damping factor in the constructed pseudoinverse has the greatest impact on the accuracy of the solution results of this method. According to the scope of the damping factor, the adaptive damping factor can be roughly divided into two categories. The first type of singular area does not define the effect of the damping factor. For example, the damping factor of the Gaussian distribution with a singularity as the expected value is used so that the damping term is small when it is far away from the singularity and has no significant impact on the entire approximation process.^6,7 The other most commonly used method is to determine the scope of the damping factor by defining a singular region so that the exact solution can be calculated using the normal method without having to solve the approximate value and the error in the non-singular region can be reduced.^8,9

When changing the adaptive damping factor, it is necessary to find an optimal value, which is known as the optimal damping factor and affects the performance and efficiency of the algorithm. The purpose of using an approximate solution is to make the terminal velocity norm as close to the expected value as possible while also making the velocity norm of the joint space a reasonable value. Thus, the problem becomes a problem of optimizing one nonlinear function under the constraints of another nonlinear function. Existing work has obtained the best damping factor through optimization functions, genetic algorithms and neural networks, all of which can obtain good results.^7,10
–12

The singularity of a closed-chain system is similar to that of a single manipulator. Park and Kim¹³ described the basic singularity of a closed-chain system, where the singularity of the end effector is manifested by the inability of the end of the manipulator to generate velocity in a certain direction. This type of singularity depends on where the end coordinate system is established in the closed-chain system. Figure 1 shows two different positions of the end coordinate system of a planar closed-chain, where the X-axis points parallel to the end link, and the Y-axis follows the right-hand rule and is orthogonal to the X-axis in the plane.

Figure 1.

Typical treatment of closed-chain singularity: (a) closed-chain end-effector singularity and (b) relocating the end coordinate system to the adjacent link.

When the end coordinate system of the closed chain is established at the position shown in Figure 1(a), the end coordinate system cannot generate velocity along the link direction under this configuration, and the system is at the singular point of the end effector. While the end-effector coordinate system is relocated at the position shown in Figure 1(b), the open-chain Jacobian of the manipulator becomes full rank, and the system can generate velocity in any direction along the plane; thus, the configuration is no longer the singular point of the closed chain to realize the traversal of the closed-chain singularity.

The above methods perform well in the singularity processing of a closed-chain system under the conditions of a simple environment and no obstacles. However, in recent years, more related studies have been conducted in complex environments, which require objects being grasped to have stricter obstacle constraints in the task space. Existing research can effectively solve such problems at the planning level. For example, by keeping a certain manipulator in a configuration and fixing the grasped object, the other manipulator generates a new joint configuration to realize the regrasp action, ensuring that the trajectory of the grasped object during the entire movement process coincides with the planned original expected trajectories.^14

–17 Furthermore, some studies have focused on the actual running time of the robot. They replan the path under environmental constraints based on known closed-chain singularities to avoid singularities.^18

–21 A summary of the above work shows that there are two optimization objectives for such problems: first, the manipulator needs to satisfy the complex obstacle constraints in the task space but does not necessarily require the system to strictly follow the expected trajectory and, second, to minimize the redundant motion of the system to reduce the total running time.

This study mainly aims at the application scenarios with complex obstacle constraints in the task space. Because there are many environmental constraints during the movement of object, it is difficult for the closed-chain system to avoid singularity by changing its trajectory when passing the constraints. For example, the object needs to pass through a narrow space close to its own size, or task constraints prevent the use of replanning to avoid singularity. Considering the above two optimization objectives, the main contributions of this study are as follows:

Analyzing and constructing the Jacobian of the closed-chain system, and providing the discrimination method of the end-effector singularity according to the Jacobian rank.

Improving the selection method of the adaptive damping factor and the optimization method of the best damping factor based on the DLS Jacobian pseudoinverse.

Designing a closed-chain singularity avoidance algorithm based on the DLS Jacobian pseudoinverse.

Problem statement

This study addresses the singularity problem of a dual-arm closed-chain system, where the task space of the closed-chain system is no longer described according to the end of the manipulator but needs to be described according to the grasped object in a Cartesian coordinate system because of the existence of definite kinematic constraints at the end of the manipulators and the grasped object. Let the left manipulator end position q_L be expressed in the Cartesian coordinate system as follows

q_{L} = [x_{L} y_{L} z_{L} R_{L} P_{L} Y_{L}]

where x_L , y_L , and z_L describe the positions of the end of the left manipulator under Cartesian space, and R_L , P_L , and Y_L describe the poses of the end of the left manipulator under Cartesian space. The feasible set of these poses constitutes the workspace Q_L of the end of the left manipulator

Q_{L} = {q_{L}}

Similarly, the workspace Q_R at the end of the right manipulator can be obtained. Assuming that the unreachable space at the end of the two manipulators due to obstacles or the volume of the clamped object is Q_E , the workspace Q of the clamped object in the closed-chain system can be described as follows

Q = Q_{L} \cap Q_{R} - Q_{E}

Given the desired path R of the clamped object in the task space Q, which contains information about the position and attitude of the clamped object, with a total of n path points, the desired path R is expressed as follows

R = {S_{1} S_{2} \dots S_{n - 1} S_{n}}

where S_i denotes the ith path point. Assume that the kth path point S_k in the given desired path R is a closed-chain singularity when at least one of the manipulators has a Jacobi matrix singularity. This leads to uncertainty in mapping the end velocity of the manipulator to the joint velocity. A definite joint drive velocity cannot be calculated. Therefore, the purpose of closed-chain singularity avoidance is to bypass the singularity S_k in the task space and replace S_k it with a new path point $S_{k}^{'}$ , thereby avoiding uncertainty in the joint velocity caused by the Jacobi singularity. In addition, to ensure the smoothness of the updated path and speed continuity, we replace the first m path points and last m path points of the singular point S_k with the new points. Then, the replacement path $R^{'}$ in the task space after the singularity avoidance process is as follows

R^{'} = {S_{1} \dots S_{k - m}^{′} \dots S_{k - 1}^{′} S_{k}^{′} S_{k + 1}^{′} \dots S_{k + m}^{′} \dots S_{n}}

where $S_{i}^{'}$ denotes the ith path point after replacement. The purpose of this study is to design a method for solving $R^{'}$ that should have the following characteristics:

No need for full replanning of desired paths and no need for dual-arm systems to perform regripping actions, improving the efficiency in the planning and control stages.

Avoidance is performed near singularities, and solutions in the joint space are acquired away from the singularities to avoid unnecessary errors between the actual path and the desired path in the task space.

When performing singularity avoidance, the error between the derived replacement path and the desired path should be as small as possible to ensure overlap with the original desired path when the system is in motion.

Figure 2.

A simplified model of closed-chain system and main coordinate systems.

Proposed approach

Kinematic constraints on closed-chain systems

The closed-chain kinematic constraint is also known as the tight coordination constraint for dual-arm collaboration. It should be strictly observed by the closed-chain system during motion. In this study, the closed-chain system formed by two isomorphic 6-DOF manipulators clamping a rigid object is taken as the basic model. A simplified model of the closed-chain system is depicted in Figure 2.

In Figure 2, the coordinate system ${W}$ is the world coordinate system and the reference coordinate system of the closed-chain system. The coordinate systems ${L}$ and ${R}$ are the base coordinate systems of the left and right manipulators, respectively. The coordinate systems ${L E}$ and ${R E}$ are the end coordinate systems of the left and right manipulators, respectively. The blue line segment indicates the rigid object being grasped, and the coordinate system ${A}$ is the center-of-mass coordinate system of the rigid object A. The transformation relationship between the world coordinate system ${W}$ and the coordinate system ${A}$ is obtained from the kinematic equation of the left manipulator as follows

T_{A}^{W} = T_{A}^{L}

Similarly, another variation relation is obtained from the rotational variation between the left and right manipulator base coordinate systems and the kinematics of the right manipulator

T_{A}^{W} = T_{R}^{L} T_{A}^{R} = T_{R}^{W} T_{A}^{R}

Both equations (6) and (7) describe the transformation between the world coordinate system ${W}$ and the coordinate system ${A}$ . The kinematic constraint of the closed-chain system expressed in equation (8) can be obtained by associating the two equations. The conformity of the dual-arm collaboration to this constraint is the most essential condition for the formation of the closed-chain system

T_{R}^{L} T_{A}^{R} - T_{A}^{L} \equiv 0

Construction of closed-chain Jacobian

According to the conclusion proposed by Park and Kim,¹³ when the closed-chain system is at the end-effector singularity, the manipulator of the system located at the singularity is in the state of Jacobian under rank, whereas the other manipulator has a Jacobian full rank. Based on the above conditions, this study constructs a closed-chain system Jacobian that can determine whether the system is in a singular region.

Long et al.²² analyzed and constructed a Jacobian for a dual-arm robot with the end velocities of the two manipulators in the closed-chain system being equal. According to the principle of vector superposition, it divides the terminal velocity into two identical new vectors with the same direction as the original vector, but half the size of the original vector. Two new vectors are used as end velocities for each two manipulators

w v_{A} = \frac{1}{2} w v_{L E} + \frac{1}{2} w v_{R E}

where $v W_{A} \in R^{6 \times 1}$ denotes the velocity of the clamped object in the task space, and $v W_{L E} \in R^{6 \times 1}$ and $v W_{R E} \in R^{6 \times 1}$ denote the velocities of the left and right manipulator ends in the world coordinate system, respectively. The mapping between the velocity of the clamped object and the joint space velocity of the closed-chain system can be obtained according to the velocity mapping relationship of the manipulator, and then the Jacobian of the closed-chain system can be obtained. In practice, there must be a volume of the rigid object being grasped so that the end velocity of the two manipulators cannot be equal to $v W_{A}$ . Because the objectA is a rigid object with uniform mass, the actual mapping relationship between the velocity of A and the velocity in the joint space can be obtained from the velocity distribution of the rigid object as follows

w A v = [\frac{1}{2} \begin{matrix} X_{L} J_{L} & \frac{1}{2} X_{R} J_{R} \end{matrix}] q.

where $X_{L} \in R^{6 \times 6}$ and $X_{R} \in R^{6 \times 6}$ denote the mapping matrices of the left and right manipulator end velocities with the center-of-mass velocity of A, respectively; $J_{L} \in R^{6 \times 6}$ and $J_{R} \in R^{6 \times 6}$ denote the left and right single-arm Jacobian, respectively; $\dot{q} \in R^{6 \times 1}$ denotes the joint velocity of the closed-chain system. Thus, the closed-chain Jacobian can be constructed according to equation (10)

J = [\frac{1}{2} \begin{matrix} X_{L} J_{L} & \frac{1}{2} X_{R} J_{R} \end{matrix}]

where $J \in R^{6 \times 12}$ denotes the closed-chain system Jacobian.

As mentioned before, when the closed-chain system is at the singular point of the end effector, one manipulator is at the singular point but the other manipulator is not. At this time, the closed-chain Jacobian J is full rank, but J_L and J_R are not full rank. Therefore, we can determine whether the system is singular based on the Jacobian rank. When the system Jacobian satisfies

{\begin{matrix} Rank (J) = 6 \\ Rank (J_{L}) < 6 \\ Rank (J_{R}) = 6 \end{matrix} or {\begin{matrix} Ran k (J) = 6 \\ Rank (J_{L}) = 6 \\ Rank (J_{R}) < 6 \end{matrix}

the closed-chain system is singular.

DLS Jacobian pseudoinverse and its optimization

This section introduces the DLS method for handling manipulator singularities and improves the method of varying the adaptive damping factor and the optimization finding method in it. First, equation (10) is described in general terms using the end velocity variable x and the joint angle variable q of the manipulator. Let $J (q)$ be the mapping relationship between the first-order differentials of the two variables at configuration q, which is the Jacobian of the manipulator. Then, for any differential change, the relationship between $Δ x$ and $Δ q$ is defined as follows

Δ x = J (q) Δ q

Typically, the Jacobian pseudoinverse is used to find the differential components of the joint angles when the number of driven joints is greater than the DOF of the system²³

Δ q = J^{+} (q) Δ x + (I_{n} - J^{+} (q) J (q)) ϕ

where $J^{+} (q) \in R^{n \times n}$ denotes the pseudoinverse of $J (q) \in R^{n \times n}$ , n represents the number of columns of $J (q)$ , I_n denotes an n-dimensional unit matrix, and $ϕ$ denotes an arbitrary n-dimensional column vector. When the system is at a singularity, $J (q)$ is non-row full rank and its pseudoinverse $J^{+} (q)$ has no solution in this configuration. Therefore, it is impossible to determine the joint velocity in this configuration.²⁴ The approximate solution is achieved by changing the construction of the pseudoinverse through DLS by adding damping terms near the singularities

J_{D}^{+} = J^{T} {(J J^{T} + λ I)}^{- 1}

where $λ$ denotes the damping coefficient. Substituting equation (15) into equation (14), the approximate mapping relationship between the end velocity and the joint velocity is obtained

Δ q = J_{D}^{+} (q) Δ x + (I_{n} - J_{D}^{+} (q) J (q)) ϕ

In equation (15), the most important thing is to find the optimal damping factor, which directly affects the performance of the entire algorithm and its tracking error in practical applications.²⁵ When the damping factor $λ$ is 0, the algorithm can guarantee the minimum task space error. When the damping factor $λ$ is sufficiently large, the joint velocity near the singularity can be effectively reduced by equation (15) to generate a feasible solution for velocity control. Therefore, when the algorithm always has a damping term throughout its operation, it is desirable to make the term as small as possible when it is significantly different from the singularity and reach an expected value that enables the algorithm to perform best when it is in the singular region by an adaptive damping factor.¹⁰

This work requires defining the singular region of the manipulator to determine the master and slave manipulators; thus, the adaptive damping factor is used,⁸ and the variation in the damping factor is divided into three segments

λ = {\begin{matrix} 0, σ \geq σ_{d} \\ \sqrt{σ (σ_{d} - σ)}, \frac{σ_{d}}{2} \leq σ < σ_{d} \\ \frac{σ_{d}}{2}, σ < \frac{σ_{d}}{2} \end{matrix}

where $σ_{d}$ denotes the maximum singular region, which is used to define the size of the singular region, and $σ$ denotes an implicit variable related to the joint configuration. In equation (17), $σ$ moves from the non-singular region into the singular region, and the damping factor size is changed from 0 through a quadratic polynomial to reach the optimal damping factor with the value $σ_{d} / 2$ . The change in the quadratic polynomial changes the damping factor $λ$ abruptly during the second stage of the change. This abrupt change causes an abrupt change in the joint speed solved by DLS, resulting in a larger torque in the actual operation of the manipulator. To overcome this drawback, the change phase of the quadratic polynomial in equation (17) is replaced by a cubic polynomial in this study to make the change process as smooth as possible

λ = {\begin{matrix} 0, σ \geq σ_{d} \\ \frac{σ_{d}}{2} - \frac{6}{σ_{d}} {(σ - \frac{σ_{d}}{2})}^{2} + \frac{8}{σ_{d}^{2}} {(σ - \frac{σ_{d}}{2})}^{3}, \frac{σ_{d}}{2} \leq σ < σ_{d} \\ \frac{σ_{d}}{2}, σ < \frac{σ_{d}}{2} \end{matrix}

In the original method, $σ_{d}$ is obtained from the ratio of the terminal velocity parametrization to the joint velocity parametrization, and $σ_{d} / 2$ is used as the optimal damping factor. However, this method does not guarantee that the obtained $σ_{d}$ value is optimal. The purpose of using a variable damping factor when dealing with manipulator singularities by DLS is to find a damping factor that minimizes end velocity error parameterization during motion. In other words, damping factors under different configurations are determined by velocity-dependent nonlinear constraints. To obtain the optimal damping coefficient and the maximum singular region under the constraint, the end velocity error parametrization is used as the optimized function, and the joint velocity parametrization is made to be less than a threshold value so that it is used as a nonlinear constraint to solve the problem by sequential quadratic programming (SQP)

\begin{matrix} Min f (λ) = \begin{matrix} ∥ \dot{x} - {\dot{x}}_{d} ∥ \end{matrix} \\ Subject to g (λ) = \begin{matrix} ∥ \dot{q} ∥ \end{matrix} \leq Δ \\ Output σ_{d} \end{matrix}

where f represents the constrained function, g represents the nonlinear constraint and $Δ$ denotes the joint velocity change threshold. The optimal damping factor and the maximum singular region are obtained by solving the quadratic programming problem in equation (19).

Singularity avoidance algorithm for closed-chain system

This section presents the algorithm for closed-chain singularity processing, particularly closed-chain singularities, using equation (20). The algorithm first determines whether the system is singular and then determines which manipulator is in the singular region. Based on the singularity of the closed-chain system, the manipulator located in the singular region is considered the master manipulator, and the other manipulator is considered the slave manipulator. Then, the approximate solution of the master manipulator is solved by DLS. Finally, the slave manipulator velocity is

[\begin{matrix} \dot{q_{L}} \\ \dot{q_{L}} \end{matrix}] = D_{1} (λ_{1} [\begin{matrix} J_{D L}^{+} \\ J_{R}^{+} \end{matrix}] v_{A}^{W} + λ_{2} [\begin{matrix} J_{L}^{+} \\ J_{D R}^{+} \end{matrix}] v_{A}^{W} + [\begin{matrix} (I_{n} - J_{D L}^{+} J_{L}) ϕ λ_{1} \\ (I_{n} - J_{D R}^{+} J_{R}) ϕ λ_{2} \end{matrix}]) + D_{2} [\begin{matrix} J_{L}^{- 1} \\ J_{R}^{- 1} \end{matrix}] v_{A}^{W}

where D ₁ and D ₂ denote the singular region judgment coefficients, and when the system is in the singular region, $D_{1} = 1$ and $D_{2} = 0$ . $λ_{1}$ and $λ_{2}$ denote the judgment coefficients of the master–slave manipulator, and when the left manipulator is the master manipulator, $λ_{1} = 1$ and $λ_{2} = 0$ , and vice versa, $λ_{1} = 0$ and $λ_{2} = 1$ . $\dot{q_{L}}$ and $\dot{q_{R}}$ denote the joint velocities of the left and right manipulators, respectively, $J_{D L}^{+}$ and $J_{D R}^{+}$ denote the Jacobian pseudoinverse of the left and right manipulators based on DLS, respectively, and $J_{L}^{+}$ and $J_{R}^{+}$ denote the Jacobian pseudoinverse of the left and right manipulators obtained as slave manipulators based on the approximate solution of the master manipulator, respectively. The flowchart of the algorithm is displayed in Figure 3.

Figure 3.

Closed-chain singularity avoidance algorithm flows.

The optimal damping factor was obtained by SQP in the previous section, and the method optimizes a function of the end velocity error in two parametric terms; thus, the process is one of error reduction through parameter optimization. To further reduce the error between the desired and solved paths and improve the accuracy of the solved path, this study adopts an accuracy improvement method that uses the velocity vector error as the correction quantity of the approximate solution. As depicted in Figure 4, the expected velocity v_d of the grasped object at the singularity and the recomputed velocity v_e are simplified and represented as two vectors, and the error $v_{error}$ of the two vectors can be obtained.

Figure 4.

Velocity vector relationship at singularity.

Define the joint velocity vector j_e , which describes the joint velocity vector of the entire closed-chain system capable of yielding an end velocity v_e

j_{e} = [{\dot{θ}}_{1} {\dot{θ}}_{2} \dots {\dot{θ}}_{11} {\dot{θ}}_{12}]

where ${\dot{θ}}_{n}$ represents the joint velocity of the nth driving joint of the closed-chain system. The joint velocity vector yielding $v_{error}$ is defined as $j_{error}$ , and this joint velocity vector can also be found by DLS. From Figure 4, when v_e and $v_{error}$ are superimposed, the expected velocity v_d can be obtained. In other words, when the joint velocity is the superimposition of j_e and $j_{error}$ , a velocity vector j that is closer to the expected value with respect to v_e can be obtained as follows

j = j_{e} + j_{e r r o r} = [({\dot{θ}}_{1} + {\dot{θ}}_{e 1}) ({\dot{θ}}_{2} + {\dot{θ}}_{e 2}) \dots ({\dot{θ}}_{11} + {\dot{θ}}_{e 11}) ({\dot{θ}}_{12} + {\dot{θ}}_{e 12})]

The above method calculates the error velocity vector $v_{error}$ and its joint velocity $j_{error}$ based on v_e and v_d , and the solution accuracy is further improved by feedback information. Therefore, throughout the algorithm, the method also ensures, to a certain extent, that the accuracy of the solution results will not fluctuate significantly with different sizes of damping factors or different configurations.

Experiments

In this study, Webots R2020a is used as the simulation platform. We call two ABB Irb4600-40 6-DOF manipulators from its robot model library so that the ends of both manipulators operate the same rigid object to form a closed chain. In the simulation environment, the rigid object is set as a 5 kg cuboid, and the robot and physical parameters are completely set according to the real world. Among them, the acceleration of gravity is set to $g = 9.8 m / s^{2}$ and simulation step is 32. The D–H parameter and joint angle limits of the manipulator can be acquired from ABB’s official website.²⁶

The four aspects of (1) adaptive damping factor improvement, (2) optimal damping coefficient and maximum singularity region finding, (3) singularity processing for the closed-chain system, and (4) cost time comparison are experimentally verified in this simulation environment. The numerical solution is obtained using MATLAB R2018a, and the calculated parameters such as joint angle and drive joint speed are written into the controller of the simulation platform. The simulation platform used in this study is depicted in Figure 5. The world coordinate system in the set task space is marked in the figure, with the red axis as the X-axis, the green axis as the Y-axis, and the blue axis as the Z-axis.

Figure 5.

Webots simulation platform

Experiment on the effect of adaptive damping factor improvement

An improvement in the Jacobian pseudoinverse based on DLS is presented in section “Proposed approach,” and the results before and after the improvement are compared in this experiment. Let the maximum singular region be $σ_{d}$ and the optimal damping factor be $σ_{d} / 2$ . According to the expressions of the two adaptive damping factors, two curves of damping factor variation with the position in the singular region can be obtained, as depicted in Figure 6.

Figure 6.

Adaptive damping factor change curve before and after improvement.

Figure 6 shows that the improved damping coefficient has a significantly lower abrupt change in the change phase. In this experiment, assuming $σ_{d} = 0.1$ , the optimal damping factor is $0.05$ , tested at a single manipulator with three rear joint angles of $[0 0 0]$ . At this point, the manipulator is at an end-effector singularity with the loss of freedom at the wrist. Let the end of the manipulator pass the singularity with a linear speed of $0.01 m/s$ in $0.5 s$ , and the velocity change curve in the joint space is obtained, as depicted in Figure 7.

Figure 7.

Variation of joint velocity at singularity.

In Figure 7, the vertical axis is the joint velocity in $rad/s$ , and the horizontal axis is the path point serial number. This shows the changes in the angular velocities of the second, third, and fifth joints occurring at this singularity.

The solution effect before and after the adaptive damping factor improvement is compared in the singular region, and the comparison curve of the velocity change is obtained, as depicted in Figure 8.

Figure 8.

Comparison of joint angular velocity variation curves at singular regions: (a) comparison of angular velocity variation in the second joint; (b) comparison of angular velocity variation in the third joint; and (c) comparison of angular velocity variation in the fifth joint.

The key parameters of the three joint angle variation curves before and after the improvement in Figure 8 are listed in the following three tables, containing the maximum velocity variations at the adjacent path points and the joint angle error caused by the velocity variation.

Table 1.

Comparison of parameters before and after the improvement in the second joint.

	Speed change maximum ( $rad/s$ )	Joint angle error ( $rad$ )
Before improvement	0.0018	0.0015
After improvement	0.001	0.0012

Table 2.

Comparison of parameters before and after the improvement in the third joint.

	Speed change maximum ( $rad/s$ )	Joint angle error ( $rad$ )
Before improvement	0.038	0.021
After improvement	0.025	0.0177

Table 3.

Comparison of parameters before and after the improvement in the fifth joint.

	Speed change maximum ( $rad/s$ )	Joint angle error ( $rad$ )
Before improvement	0.04	0.0228
After improvement	0.028	0.0169

From Tables 1, 2, and 3, after changing the variation curve of the damping factor from a quadratic polynomial to a cubic polynomial, the variation in the joint angular velocity at the singularity is smoother than that before the improvement. In addition, the improved method produces less error in the singular region, as indicated by the joint angle error caused by the velocity change in the table.

Optimal damping coefficient and maximum singularity region search experiments

Based on the existing work, the optimal damping factor is set to $σ_{d} / 2$ . This experiment uses different quadratic planners in MATLAB to solve this problem, including Active Set, Interior-Point, and SQP, and compares their results. In this experiment, the joint configuration of the manipulator singularity in section “Experiment on the effect of adaptive damping factor improvement” is used, and the optimal value of the best damping coefficient $σ_{d} / 2$ is output by quadratic programming of end velocity error parametrization according to this nonlinear constraint. The iterative process of the three methods is depicted in Figure 9.

Figure 9.

Optimization effects of three quadratic programming methods on damping factor: (a) Active Set optimization effect; (b) Interior-Point optimization effect; and (c) SQP optimization effect.

In Figure 9, the horizontal axis represents the number of iterations and the vertical axis represents the value of the function being optimized. To more intuitively compare the optimization effects of the three types of secondary planning, the parameters in the above figure are presented in the form of a table. The parameter types include the resulting optimal damping coefficient, end velocity error parametrization, and the number of iterations, as listed in Table 4.

Table 4.

Comparison of parameters of three secondary planning methods.

Name	Optimal damping factor	Speed error range	Number of iterations
Active Set	0.0234	0.5	3
Interior-Point	0.0173	0.0447	10
SQP	0.0173	0.0447	7

According to the data in Table 4, although the Active Set has fewer iterations, the results obtained from its iterations are not optimal solutions. Both Interior-Point and SQP can obtain the optimal solution, but the number of iterations of SQP is relatively small. This means that its solution is more efficient than that of Interior-Point.

Experiments on singularity processing for closed-chain systems

This experiment implements the proposed singularity avoidance algorithm of the closed-chain system in the simulation platform and demonstrates the simulation effects of the closed-chain system for singularity avoidance. A sine function path in the world coordinate system in the task space is selected so that its midpoint is at the singularity of the left manipulator, as depicted in Figure 10.

Figure 10.

Expected path in task space.

All axes in Figure 10 are in meters. In the task space, the starting point is $[- 0.5 1.25 1.5]$ , the ending point is $[0.5 1.35 1.5]$ , and the singularity position is $[0.5 1.3 1.5]$ . The joint angles of the closed-chain system at each path point are obtained by the joint sensors during the entire operation process, introduced into the positive kinematic equations of the closed-chain system to obtain three views of the actual running path of the grasped object in the task space, as depicted in Figure 11.

All axes in Figure 11 are in meters. Because the algorithm makes the error at the singularities small, the path in Figure 11 is almost identical to the expected path. Taking the singularity as the center to observe its neighborhood, the six-dimensional position change of the path is enlarged and compared with the desired path, and the result is obtained, as presented in Figure 12.

Figure 11.

Three views of the actual path of the object: (a) X-Y plane view; (b) X-Z plane view; and (c) Y-Z plane view.

Figure 12.

Comparison of the expected and actual poses in singularity neighborhood: (a) Y-axis position; (b) X-axis rotation; and (c) Z-axis rotation.

Although more pronounced, the errors in the 3-DOFs shown in Figure 12 are still of a minute order of magnitude. The average error of the Y-axis position in Figure 12(a) is $0.3 mm$ , and the average error of the X- and Z-axis turning angles is $5 \times 10^{- 5} rad$ approximately. The remaining 3-DOFs have almost no error. Figure 13 depicts the process of completing the path in this experiment by the singularity processing algorithm for the closed-chain system in the simulation environment.

Figure 13.

Closed-chain system operation process in simulation environment

Cost time comparison

To further verify the advantages of our method, we carry out the comparative experiments for the following three methods:

The DLS-based singularity avoidance algorithm proposed in this article.

The method of avoiding singularities through path replanning:

The robot strictly follows the expected path while it moves in a non-singular region. When running to a singular area, we first delete some path points belonging to the singular area in the expected path, and regard the singular area as an obstacle in the task space, then replan the path in this area and its neighborhood through the RRT algorithm, and detect whether the planned path intersects with the singular area until a feasible path is generated.

The method of traversing singular points by regrasping:

When the manipulator reaches the singular point, the end effector cannot generate velocity due to the loss of a DOF in one direction. Based on the method in the study of Xian et al.,¹⁸ the configuration of a certain manipulator in the closed chain is changed by regrasping the object, thus leading to the variation of the closed-chain Jacobian to realize the traversal of the singularity.

The time spent on closed-chain singularity processing mainly includes the motion planning time at the singular point and the execution time of the manipulator. The time spent by the above three methods to deal with the closed-chain singularity is shown in Table 5.

Table 5.

Cost time comparison.

Method	Planning time(s)	Execution time(s)	Total time (s)
A	1.35	6.73	8.08
B	1.87	8.04	9.91
C	1.31	10.63	11.94

It can be seen from Table 5 that the planning time of method B is relatively long, and the execution time of the robot increases due to the increase in the length of the path generated by replanning compared to the expected path. Method C costs slightly less time in the planning phase, but the execution time of the robot increases due to the configuration variation of a certain manipulator during the regrasping process. The total time cost by method A is the shortest.

Conclusion

This article investigates and solves the problem of end-effector singularities during the motion of a grasped object in a closed-chain system that should follow an expected path. First, for the singularity judgment problem of the closed-chain system, a method of discriminating the singularity point of the end effector according to the rank of this Jacobian is presented. Then, the adaptive damping factor is improved by a cubic polynomial, and the parameters are optimized by SQP. Singularity avoidance is finally achieved by designing a closed-chain singularity avoidance algorithm based on the DLS Jacobian pseudoinverse. Future work includes:

In this work, the optimal damping factor and the maximum singular region are determined by quadratic programming, where the threshold value of joint velocity variation parametrization as a constraint is selected empirically. We would like to find the optimal value of this threshold by optimization which allows a more accurate iteration of the optimal damping factor and the minimum singular region.

We verified the feasibility and effectiveness of the proposed method in the experimental validation phase through a simulation platform. In the future, the effectiveness of the method needs to be verified on a physical object.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the Shandong Provincial Natural Science Foundation (ZR2022MF296), the National Natural Science Foundation of China (62073191), and a Project of Shandong Province Higher Educational Youth and Innovation Talent Introduction and Education Program.

References

Asada

Granito

. Kinematic and static characterization of wrist joints and their optimal design. In: 1985 IEEE international conference on robotics and automation, St. Louis, MO, USA, 25–28 March 1985, pp. 244–250. New York: IEEE.

Dubey

Luh

. Redundant robot control for higher flexibility. In: 1987 IEEE international conference on robotics and automation, Raleigh, NC, USA, 31 March–03 April 1987, pp. 1066–1072. New York: IEEE.

Wampler

. Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods. IEEE Trans Syst Man Cybern Syst 1986; 16(1): 93–101.

Deo

Walker

. Overview of damped least-squares methods for inverse kinematics of robot manipulators. J Intell Robot Syst 1995; 14: 43–68.

Whitney

. The mathematics of coordinated control of prosthetic arms and manipulators. J Dyn Sys Meas Control 1972; 94(4): 303–309.

Di Vito

Natale

Antonelli

. A comparison of damped least squares algorithms for inverse kinematics of robot manipulators. IFAC-PapersOnLine 2017; 50(1): 6869–6874.

Phuoc

Martinet

Lee

, et al. Damped least square based genetic algorithm with gaussian distribution of damping factor for singularity-robust inverse kinematics. J Mech Sci Technol 2008; 22: 1330–1338.

Baerlocher

. Inverse kinematics techniques for the interactive posture control of articulated figures. PhD Thesis, Ecole Polytechnique Federale de Lausanne, Switzerland, 2001.

Caccavale

Chiaverini

Siciliano

. Second-order kinematic control of robot manipulators with Jacobian damped least-squares inverse: theory and experiments. IEEE ASME Trans Mechatron 1997; 2(3): 188–194.

10.

Bai

Zhang

Huang

, et al. Dual-arm coordinated manipulation for object twisting with human intelligence. In: 2021 IEEE international conference on systems, man, and cybernetics (SMC), Melbourne, Australia, 17–20 October 2021, pp. 902–908. New York: IEEE.

11.

Wang

Liu

Chen

, et al. Deep-learning damped least squares method for inverse kinematics of redundant robots. Measurement 2021; 171: 108821.

12.

Omisore

Han

Ren

, et al. Deeply-learnt damped least-squares (DL-DLS) method for inverse kinematics of snake-like robots. Neural Netw 2018; 107: 34–47.

13.

Park

Kim

. Singularity analysis of closed kinematic chains. J Mech Des 1999; 121(1): 32–38.

14.

Völz

Graichen

. An optimization-based approach to dual-arm motion planning with closed kinematics. In: IEEE/RSJ international conference on intelligent robots and systems, Madrid, Spain, 01–05 October 2018, pp. 8346–8351. New York: IEEE.

15.

Jang

Baek

Park

, et al. Motion planning for closed-chain constraints based on probabilistic roadmap with improved connectivity. IEEE ASME Trans Mechatron 2022; 27(4): 2035–2043.

16.

Haustein

Hang

Stork

, et al. Object placement planning and optimization for robot manipulators. In: IEEE/RSJ international conference on intelligent robots and systems, Macau, China, 03–08 November 2019, pp. 7417–7424. New York: IEEE.

17.

Zhang

Jia

. Motion planning of redundant dual-arm robots with multicriterion optimization. IEEE Syst J 2023; 17(3): 4189–4199. doi: 10.1109/JSYST.2023.3292430.

18.

Xian

Lertkultanon

Pham

. Closed-chain manipulation of large objects by multi-arm robotic systems. IEEE Robot Autom Lett 2017; 2(4): 1832–1839.

19.

Bohigas

Henderson

Ros

, et al. Planning singularity-free paths on closed-chain manipulators. IEEE Trans Robot 2013; 29(4): 888–898.

20.

Bohigas

Henderson

Ros

, et al. A singularity-free path planner for closed-chain manipulators. In: IEEE international conference on robotics and automation, Saint Paul, MN, USA, 14–18 May 2012, pp. 2128–2134. New York: IEEE.

21.

Wong

Chien

Feng

, et al. Motion planning for dual-arm robot based on soft actor-critic. IEEE Access 2021; 9: 26871–26885.

22.

Long

Khalil

Caro

. Kinematic and dynamic analysis of lower-mobility cooperative arms. Robotica 2015; 33(9): 1813–1834.

23.

Nakamura

Advanced robotics: redundancy and optimization. Addison-Wesley Longman Publishing Co., Inc., 1990.

24.

Kim

Marani

Chung

, et al. Task reconstruction method for real-time singularity avoidance for robotic manipulators. Adv Robot 2006; 20(4): 453–481.

25.