Research on Motion Trajectory Planning and Impedance Control for Dual-Arm Collaborative Robot Grinding Tasks

Abstract

In robot grinding tasks, dual manipulators possess improved flexibility, which can cooperate to complete different tasks with higher efficiency and satisfactory effect. In collaborative robot grinding tasks, the critical issues lie in the motion trajectory planning of the two manipulators and trajectory tracking with satisfactory accuracy under the condition that the two manipulator ends apply force on each other. In order to accomplish the goals in a more concise and feasible way, a complete scheme for dual-arm robot grinding tasks is essential. To address this issue, taking the motion trajectory planning and impedance control into consideration, a novel scheme for dual manipulators to complete collaborative grinding tasks is presented in this paper. To this end, a dual-arm grinding system is first constructed, and the kinematic constraints in the cooperative motion are analyzed, based on which the motion trajectories of the dual manipulators are planned according to the grinding task objectives. Then, an impedance controller is designed to achieve accurate tracking of the motion trajectory in the grinding process. Finally, dual-arm collaborative simulations and grinding experiments are carried out, and the results show that the proposed method can achieve good motion results and better flexibility compared to the single-arm motion, which demonstrates the effectiveness of the proposed method.

1. Introduction

With the continuous development of the manufacturing industry, the demand for grinding tasks for specific workpieces is increasing. Not only has the quantity of grinding developed rapidly, but the requirements for the quality of grinding have also increased significantly. Nowadays, although a considerable part of such grinding tasks is completed by humans, more and more grinding tasks are performed by robots. In the field of grinding and polishing processing, compared with manual operation, the grinding quality of robots is more stable and can work continuously for a long time, which can obtain better economic and social benefits. Thus, research on robot grinding and polishing has drawn increasing attention and been a popular research field during the past decades.

In recent years, the rapid development of intelligent technologies has led to remarkable advancements in the field of intelligent manufacturing [1,2]. A number of new technologies have been proposed for robot grinding and polishing, particularly with regards to trajectory planning and force controlling. A novel selected force controlling method (SFC) with consideration of regional division (RD) based on the machining allowance is proposed by Wang et al. [3] for improving the robotic grinding accuracy of complex curved blades, on the basis of the self-developed adaptive impedance controller. Xie et al. [4] presented an adaptive intelligent human-robot collaborative approach to facilitate trajectory planning for robotic belt grinding of complex parts, which bridges the experience of skilled operators through an immersive virtual reality. Zhao et al. [5] proposed a mobile robotic grinding system for large-scale workpieces and a vision-based grinding strategy for the mobile robot. Li et al. [6] proposed a novel path accuracy enhancement strategy and different evaluation methods for a six-degree-of-freedom industrial robot used for grinding an aero-engine blade.

In order to improve robot grinding performance, there have been cases where two robots have been used to work together for grinding tasks in recent years, usually with one manipulator clamping the workpiece to be polished, and the other installing a grinding tool for grinding, which has the advantages of more flexibility, high efficiency, and good performance. However, this scheme brings challenges for both planning the movement path of the manipulators and the contact problem handling in grinding, which have drawn increasing attention and been a popular research field over the years. Xian Z and Lertkultanon et al. [7] proposed an “IK switch” multi-arm trajectory planning method, which can solve complex closed-chain operation tasks. Smith et al. [8] used the potential function method to make the system avoid collisions in the whole motion by taking advantage of the redundancy of the degrees of freedom of the dual robotic arms.

Due to the complexity of the dual-arm systems and particularity of scenarios, most of the existing research is too complicated to be practically applied to robotic grinding tasks. Therefore, in order to accomplish the goals in a more concise and feasible way, it is necessary to simplify the task planning and control process of collaborative robotic grinding in practice. In this paper, we develop a dual-arm collaborative system for robotic grinding tasks, where the motion trajectory of the dual arms is planned based on the kinematic constraint and an impedance control scheme is designed. To be specific, the dual-arm system is first modeled according to the kinematic constraints. In order to reduce the complexity of the movement of the two robotic arms, we decompose the trajectory of the workpiece surface to different directions orthogonally, so that the two robots can track the decomposed trajectory separately. Meanwhile, an impedance controller is designed to keep two manipulators tracking the desired trajectory in the interaction. Finally, the grinding experiments of the dual-arm collaborative system are carried out, which proves the feasibility and effectiveness of the method.

The rest of this paper is organized as follows: In Section 2, we review the recent research on robotic grinding. In Section 3, the details and construction of the two-robot arm sanding system are described. In Section 4, the trajectory planning method for the collaboration of the two robotic arms is described, as well as the use of the impedance control method for the control of the two robotic arms. Finally, simulation comparison experiments as well as two-arm collaborative sanding experiments are conducted to prove the rationality of the proposed task planning method.

2. Literature Review

With the multi-arm working together to complete different tasks, planning the movement path of the multi manipulators is a more demanding task. Sun et al. proposed a classical model-based [9] and model-free [10] Cartesian spatial position synchronization controller to realize the synchronization of multiple robots through cross-coupling techniques. Rodri-guez-Angeles et al. defined the position and velocity synchronization errors of the robot joint space and proposed a proportional derivative (PD) synchronization controller [11]. Wong et al. [12] designed a motion planning method based on the Soft Actor-Critic (SAC) for a dual-arm robot with two 7-Degree-of-Freedom (7-DOF) arms so that the robot can effectively avoid self-collision and at the same time avoid the joint limits and singularities of the arm. Kim et al. [13] applied a transformer, a variant of self-attention architecture to deep imitation learning to solve dual-arm manipulation tasks in the real world. Li et al. [14] proposed a deep reinforcement learning-based motion planning method to solve the complex constrained motion planning problem of a free-floating dual-arm space manipulator.

In the collaborative task of multiple robotic arms, regardless of the existence of mutual constraints at the end of each robotic arm, it is necessary to solve the relative relationship between the base coordinate system of each robotic arm in order to enable the representation of each part of the system under the world coordinate system, and to enable the components of the system to establish a connection, which lays the foundation of the modeling and motion planning of the system. Gan et al. [15] proposed a four-point handshake method, in which the calibration needle is installed at the end of two robotic arms, respectively, and then the end attitude of the dual robotic arms is adjusted so that the tip of the calibration needle is not coplanar in space at four points of contact to realize the calibration. Jiang et al. [16] proposed a novel dual-robot accurate calibration method that uses convex optimization and Lie derivative to solve the dual-robot calibration problem simultaneously.

In the collaborative task of manipulators, there are some tasks that require force interaction between the arms, such as grinding, assembling, and handling, etc. In these tasks, if the force between the manipulators is too large, it will damage the workpiece and the manipulators; if it is too small, it will not be able to complete the task. Therefore, force control must be introduced in collaborative grinding tasks. The typical synchronous strategy includes online force/position-regulated control [17], which coordinates the forces of a group of robots during a cooperative manipulation task by regulating the robots’ position [18]. The force cross-coupling synchronous controller is first studied in [19]. Impedance control is one of the most widely used methods. The impedance control strategy was proposed by Hogan [20] and others in 1985, and its basic idea is to equate the robot control system to a second-order “inertia-damping-spring” model, and establish the relationship between the robot’s end displacement and the contact force in order to achieve different control effects. Yao B et al. [21] proposed an adaptive conductance control without sensors to improve the pliability of human-robot interaction by updating the damping coefficients in real time to enable the lifting force to vary with the environment. Yu et al. [22] proposed an adaptive impedance controller for human–robot co-transportation in the task space, which offers a safe interaction between the human and the robot and achieves a smooth control behavior along the different phases of the co-transportation task. Zhang et al. [23] proposed an inverse reinforcement learning (IRL)-based approach to recover both the variable impedance policy and the reward function from expert demonstrations. Cao et al. [24] proposed a novel passive model-predictive impedance control method, including two control loops, which contains a variable impedance controller to achieve the desired compliant interaction behavior and a model-predictive control (MPC) to ensure that the robot states satisfy the passivity constraint. Caccavale et al. proposed a unified impedance approach for dual-robot collaboration [25] to endow the robot with pliable behavior.

For robot collaboration tasks, their control requires more aspects to be considered than for single robots, such as the assignment of tasks to each robot and the planning of motion trajectories. In task scenarios that require force interaction, the way the robots collaborate with each other requires even more careful consideration. Zhang et al. [26] proposed a multi-objective synchronization control scheme using a nonlinear model-predictive policy, which is verified by dual-robot mirror grinding. Kornmaneesang et al. [27] proposed an innovative manufacturing system based on a dual-arm robot, and the method of equivalent errors is employed to design a robust contouring controller. Lv et al. [28] studied the precise single-arm and dual-arm robot manipulation control of deformable linear objects (DLOs).

However, due to the complexity of the dual-arm systems and the particularity of the scenarios, only limited studies have been conducted on the motion trajectory planning and precise tracking of the dual-arm robot system, while a relatively complete scheme for dual-arm robot grinding tasks is lacking in the existing research. To address the difficult problems, including both the motion trajectory planning and impedance control of the dual-arm robot collaborative system, a novel scheme for dual manipulators to complete collaborative grinding tasks is presented in this paper. In order to be able to reduce the complexity of collaborative task planning for the motion of the two robotic arms, after planning the trajectory points on the surface of the workpiece, the trajectory is decomposed, and the trajectory curves are decomposed to the orthogonal directions, so that the two robots track the decomposed trajectory, respectively. Moreover, in order for the two robotic arms to be able to keep tracking the given motion trajectory in the interaction, impedance controllers are designed for the control of the two robotic arms.

3. Modeling of the Dual-Arm Collaborative Grinding System

To illustrate our planning strategy, a system consisting of two seven-degree-of-freedom force-controlled robotic arms will be studied first. In the process of two robotic arms working together to perform the grinding task, one robotic arm is set to install a fixture and clamp the workpiece, and the other robotic arm installs a grinding tool. For a dual-arm system, it contains a world coordinate system $\left\{W\right\}$. The workpiece clamping arm is set to manipulator 1, its base coordinate system is expressed as $\left\{B_1\right\}$, and the end coordinate system is expressed as $\left\{E_1\right\}$. The grinding manipulator is set to manipulator 2, while its base coordinate system $\left\{B_2\right\}$ and end coordinate system $\left\{E_2\right\}$ are set. As well as the installed clamping tool coordinate system, represented as $\left\{G\right\}$, the workpiece coordinate system $\left\{O\right\}$, and grinding tool coordinate system $\left\{T\right\}$ are set. The relationship between the coordinate systems is shown in Figure 1.

3.1. Calibration of the Base Coordinate System of the Dual-Arm System

Here, the regular calibration method is used; after connecting the ends of two robotic arms by a connecting tool to form a closed kinematic chain, and it can then be obtained according to the closed kinematic chain:

$${}_{B2}{}^{B1}T={}_{E1}{}^{B1}T{}_{E2}{}^{E1}T{}_{B2}{}^{E2}T$$

(1)

where ${}_{E2}{}^{E1}T$ is the transformation matrix between the ends of the two robotic arms, it is considered constant after the ends are connected. ${}_{E1}{}^{B1}T$ and ${}_{B2}{}^{E2}T$ can be calculated by the robots. ${}_{B2}{}^{B1}T$ is the desired matrix. Then, in n measurements, since ${}_{E2}{}^{E1}T$ is a constant, n−1 sets of equations can be listed accordingly:

$${}_{E1}{}^{B1}T_{n-1}^{-1}{}_{B2}{}^{B1}T{}_{E2}{}^{B2}T_{n-1}={}_{E1}{}^{B1}T_n^{-1}{}_{B2}{}^{B1}T{}_{E2}{}^{B2}T_n$$

(2)

The above equation can be organized into the form of AX = XB:

$${}_{E1}{}^{B1}T_n{}_{E1}{}^{B1}T_{n-1}^{-1}{}_{B2}{}^{B1}T={}_{B2}{}^{B1}T{}_{E2}{}^{B2}T_n{}_{E2}{}^{B2}T_{n-1}^{-1}$$

(3)

where $A={}_{E1}{}^{B1}T_n{}_{E1}{}^{B1}T_{n-1}^{-1}$, $B={}_{E2}{}^{B2}T_n{}_{E2}{}^{B2}T_{n-1}^{-1}$, and $X={}_{B2}{}^{B1}T$. And solving the calibration equations yields the relationship for the base coordinate system [29].

3.2. Dual-Arm Sanding System Constraints

After determining the relationship between the two manipulators, the constraint relationships between the other components of the system need to be further determined. The position of the workpiece coordinate system $\left\{O\right\}$ on the workpiece surface is determined first, then, according to the shape of the workpiece surface, a series of processing trajectory points ${}^{O}T_i(i=1,2...m)$ are planned out below $\left\{O\right\}$. The processing point here comprises both the position and the attitude of the coordinate system on each point, which is represented by a $4\times 4$ homogeneous transformation matrix. At the same time, for the convenience of calculation, the coordinate system of the clamping tool $\left\{G\right\}$ coincides with the end flange coordinate system of manipulator 1 $\left\{E_1\right\}$, i.e., ${}_G{}^{B1}T={}_{E1}{}^{B1}T$. Then, the processing point information is represented in the world coordinate system, as follows:

$${}^{W}T_i={}_{E1}{}^{B1}T_i{}_O{}^{E1}T{}^{O}T_i$$

(4)

where the installation posture ${}_O{}^{E1}T$ from the end of the mechanical arm to the workpiece coordinate system can be measured.

When performing a grinding task, the grinding tool coordinate system needs to be in contact with the trajectory point at different times. After the grinding tool is installed at the end of manipulator 2, a series of tool motion points corresponding to the machining trajectory points are expressed in the world coordinate system as:

$${}^{W}T_{Ti}={}_{B2}{}^{B1}T{}_{E2}{}^{B2}T_i{}_T{}^{E2}T$$

(5)

We set the end point of the grinding tool and the corresponding machining track point to coincide at each moment, that is ${}^{W}p_i={}^{W}p_{Ti}$, and further obtain:

$${}_{E1}{}^{B1}T_i{}_O{}^{E1}T{}^{O}T_i={}_{B2}{}^{B1}T{}_{E2}{}^{B2}T_i{}_T{}^{E2}T$$

(6)

Therefore, after setting the end trajectory of manipulator arm 1, the trajectory of the movement of manipulator 2 can be calculated according to this formula:

$${}_{E2}{}^{B2}T_i={}_{B2}{}^{B1}T^{-1}{}_{E1}{}^{B1}T_i{}_O{}^{E1}T{}^{O}T_i{}_T{}^{E2}T^{-1}$$

(7)

4. Trajectory Planning and Tracking Methods

In Section 3, the two-arm constraint in the grinding scenario is described, and the respective motion trajectories of the two arms need to be designed under this constraint. In order to reduce the difficulty of planning the respective motion trajectories of the two arms, here we developed a design to decompose the machining trajectories to the two mechanical arms in different directions according to the shape of the workpiece surface.

When planning a machining trajectory on the surface of a workpiece, the surface of the workpiece is divided into planar slices along the X-direction of the world coordinate system, and curves in YZ-coordinates are created along the surface of the workpiece in each plane. The curve trajectory is decomposed along the Y-direction and Z-direction, viz:

$$\overrightarrow{P_{YZ}}=\overrightarrow{P_Y}+\overrightarrow{P_Z}$$

(8)

where $\overrightarrow{P_{YZ}}$ is the trajectory curve, $\overrightarrow{P_Y}$ is the trajectory in the Y-direction, and $\overrightarrow{P_Z}$ is the trajectory in the Z-direction. Therefore, the trajectories in these two directions are input to the two robotic arms, respectively, and the relative movement of the sanding tool along the surface of the workpiece is realized by the movement of the two robotic arms to achieve sanding.

After designing the respective trajectories of the manipulators, in order to be able to track the set trajectory of the manipulators in contact with each other during the grinding task, the two robots were controlled using impedance control. The implementation principle of the impedance control algorithm is as follows:

$$M_d({\ddot{X}}_d-\ddot{X})+D_d({\dot{X}}_d-\dot{X})+K_d(X_d-X)=F_{ext}$$

(9)

where $M_d$, $D_d$, and $K_d$ are the inertia coefficient matrix, damping coefficient matrix, and stiffness coefficient matrix of the desired impedance model, respectively. $X_d$, ${\dot{X}}_d$, $X_d$ are the reference position, the reference velocity, and the reference acceleration at the end of the manipulator in the operating space, respectively. $F_{ext}$ is the force in contact with the environment.

For each robot, the dynamics are described as:

$$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G+d=\tau -J^T(q)F$$

(10)

where $q$, $\dot{q}$, and $\ddot{q}\in {ℝ}^m$ are the position, velocity, and acceleration vectors of m joints, respectively. $M(q)\in {ℝ}^{m\times m}$ is the inertial matrix. $C(q,\dot{q})\dot{q}\in {ℝ}^m$ represents the coriolis force and centrifugal force. $G\in {ℝ}^m$ is the gravity vector. $d\in {ℝ}^m$ is the distraction, such as friction. $\tau \in {ℝ}^m$ indicates the joint driving torque. $F\in {ℝ}^n$ is an n-dimensional external force vector, and $J^T(q)\in {ℝ}^{m\times n}$ is the transposition of the Jacobian matrix of the arm. The Jacobian matrix is used to perform the mapping of the robotic arm joint space velocities into Cartesian space, which is denoted as:

$$J(q)=\left[\begin{array}{cccc}{\displaystyle \frac{\partial {\mathit{f}}_1}{\partial {\mathit{q}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_1}{\partial {\mathit{q}}_2}}& \cdots & {\displaystyle \frac{\partial {\mathit{f}}_1}{\partial {\mathit{q}}_7}}\\ {\displaystyle \frac{\partial {\mathit{f}}_2}{\partial {\mathit{q}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_2}{\partial {\mathit{q}}_2}}& \cdots & {\displaystyle \frac{\partial {\mathit{f}}_2}{\partial {\mathit{q}}_7}}\\ \cdots & & & \\ {\displaystyle \frac{\partial {\mathit{f}}_7}{\partial {\mathit{q}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_7}{\partial {\mathit{q}}_2}}& \cdots & {\displaystyle \frac{\partial {\mathit{f}}_7}{\partial {\mathit{q}}_7}}\end{array}\right]$$

(11)

where $f_i$ denotes the functional relationship between the joint position q and the spatial orientation x, and ${\displaystyle \frac{\partial {\mathit{f}}_i}{\partial {\mathit{q}}_j}}$ denotes the differentiation of the ith function with respect to the jth joint.

From the dynamic description equation, it can be seen that the driving torque input to the arm can be divided into the torque ${\tau }_{ext}=J^T(q)F$ generated by the interaction between the arm and the environment and the torque generated by the arm itself. Then, according to the impedance control model, for the whole manipulator system, Equation (9) is substituted into Equation (10), and the driving torque $\tau$ that needs to be input is obtained:

$$\begin{array}{l}\tau =M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G+d+J^T(q)(M({\ddot{X}}_d-\ddot{X})\\ +D({\dot{X}}_d-\dot{X})+K(X_d-X))\end{array}$$

(12)

When the input torque is applied, in order to realize the trajectory tracking effect, the expected force is set to 0; when the actual motion speed is designed, the speed change is generally too slow for safety, and the acceleration is very small. Thus, the acceleration term in the driving torque calculation can be ignored, then the simplified input torque ${\tau }_d$ is:

$${\tau }_d=C(q,\dot{q})\dot{q}+G+d+J^T(q)\left(D({\dot{X}}_d-\dot{X})+K(X_d-X)\right)$$

(13)

At this point, the manipulator can collect position and speed information in real time, and drive the arm to move according to the input torque calculated.

5. Experimental Results and Discussion

In order to test the effectiveness of the proposed method, the following simulation comparison experiments as well as physical grinding and polishing experiments are conducted.

5.1. Simulation

In order to validate the effectiveness of the proposed method, dual-arm co-simulation is carried out in Matlab R2023b and compared with the conventional single-arm sanding motion. The simulation scenario is as shown in Figure 2:

Since there are motion errors in the robotic arms themselves, Gaussian noise with a mean of 0 and a variance of 0.01 mm is added to each arm here. The motion between the synthesized motion in the two-armed scenario and the individual motion in the one-armed scenario was compared and the respective motion errors in both cases were obtained, as shown in Figure 3:

From the simulation results, it can be seen that using the method of decomposing the motion trajectories into two orthogonal directions and then assigning them to the two collaborating arms has similar results to directly inputting the machining trajectories to the individual arms. However, the two-arm collaboration provides greater flexibility and the method of decomposing the trajectories is simpler.

5.2. Dual-Robot Collaborative Grinding Experiment

In order to validate the effectiveness of the proposed method, according to the discussion in Section 3 and Section 4, dual-arm collaborative grinding experiments are carried out in two different scenarios:

Scheme 1: the attitude of the workpiece remains unchanged in the grinding, and the workpiece installed by the manipulator 1 is set to be in a horizontal state.

Scheme 2: the attitude of the workpiece changes in the grinding, the position of the end of the workpiece clamping manipulator 1 is set to remain unchanged, and only the attitude changes.

The experimental scene for two different grinding scenarios is shown in Figure 4:

The manipulators used for the experiment were Franka Emika, whose main parameters are shown in

Table 1. Franka Emika’s main parameters.

Axes	Weight	Load	Repeat Positioning Accuracy	Maximum Reach Distance	Control Frequency
7	18 kg	3 kg	0.1 mm	855 mm	1 kHz

:

After calibration, the relationship between the coordinate systems of components in the two-arm system can be obtained as shown below (unit of length:m):

$$\begin{array}{l}{}_{B_2}{}^{B_1}T=\left[\begin{array}{cccc}-0.9985& 0.0383& 0.0396& 0.7970\\ -0.0377& -0.9992& 0.0162& 0.0059\\ 0.0401& 0.0146& 0.9991& -0.0012\\ 0& 0& 0& 1\end{array}\right],\\ {}_T{}^{E_2}T=\left[\begin{array}{cccc}1& 0& 0& -0.07071\\ 0& 1& 0& -0.07071\\ 0& 0& 1& 0.2\\ 0& 0& 0& 1\end{array}\right],\\ {}_O{}^{E_1}T=\left[\begin{array}{cccc}0.7071& 0& 0.7071& -0.0636\\ -0.7071& 0& 0.7071& 0.1485\\ 0& -1& 0& 0.1\\ 0& 0& 0& 1\end{array}\right]\end{array}$$

(14)

According to the presentation in Section 3, the impedance parameters set for the two arms are given in

Table 2. The impedance parameters selected by the two manipulators.

	Manipulator 1	Manipulator 2
Stiffness matrix  $K$	$K_1=diag(1200,1200,1200,20,20,20)$	$K_2=diag(1200,1200,1200,20,20,20)$
Damping matrix  $D$	$D_1=diag(69,69,69,9,9,9)$	$D_2=diag(85,85,85,9,9,9)$

. Table 2 shows the impedance control parameters selected for each of the two robotic arms in this experiment.

5.2.1. Longitudinal Collaborative Movement

In this scenario, the workpiece is set to a horizontal clamping state, and the grinding trajectory is designed according to the shape of the workpiece surface. Moreover, the grinding trajectory is decomposed into the X-direction and the Z-direction, which are input to the two manipulators, respectively. Here, the motion trajectories set for the two manipulators are as follows:

$$\mathrm{manipulator} 1: Z_1=0.01\times \sin (0.01\times \pi (t-2))+0.0015(t-2)$$

$$\mathrm{manipulator} 2: Y_2=0.00195(t-2), Z_2=0.0015(t-2)$$

In this experiment, a total of 203 trajectory points were planned, and the movement time of each desired position was 0.5 s, so the cooperative movement time of the two robotic arms was 101.5 s.

After the experiment, for manipulator 1 clamping the workpiece, the movement in each desired direction is shown in Figure 5:

In the process of the Z-direction, the motion error generally shows a trend of first rising and then decreasing. The error reaches the maximum in about 60 s, and the maximum error is 2.079 mm. After 60 s, the error gradually shows a decreasing trend, and the average error in the whole process is 0.99 mm.

For manipulator 2 performing grinding, the movement in the desired direction is shown in Figure 6 and Figure 7:

For the Y-direction, the average error of the whole process is 3.62 mm, the minimum error is 2.38 mm, and the maximum error in the process is 5.4 mm. The error is in a decreasing trend as a whole. For the Z-direction, the error in the whole process shows a gradual decreasing trend, the average error in the whole process is 1.88 mm; the initial error is the largest, at 3.87 mm, which then gradually decreases, reaching 0.4 mm at the lowest at the end of the movement.

For the coordinated motion of manipulator 1 and manipulator 2 in the Z-direction, the tracking effect of manipulator 1 is significantly better than that of manipulator 2.

5.2.2. Rotational Collaborative Movement

In this scenario, the workpiece is initially held horizontally, and then the workpiece clamping manipulator changes the attitude of the end to change the gripping attitude of the workpiece. At the same time, another manipulator follows the surface of the workpiece. The experimental scenario is shown in Figure 8.

The motion time of each planned trajectory point is still set to 0.5 s, the end of the manipulator 1 is set to rotate 90 degrees, and the rotation is set between each attitude to be 1 degree, so that the motion process is 45 s in total. Since there is only the end rotation, it can be seen that the workpiece rotates around the X axis in the coordinate system of manipulator 1, and the corresponding series of rotation matrices can be calculated, and the position of the trajectory point that manipulator 2 needs to reach in different periods can be described accordingly.

The movement of the two manipulators in their respective desired directions was analyzed. For manipulator 1, the movement in the desired direction is shown in Figure 9:

For manipulator 1, the error in the direction of rotation is small, the average rotation angle error is 0.0094, about 0.538 degrees, and the maximum rotation error is 0.02356, about 1.350 degrees. The error is in a small range and the distribution is relatively uniform, which indicates that the impedance control is better in the effect of tracking attitude.

For manipulator 2, the movement in the desired direction is shown in Figure 10 and Figure 11:

In the Y-direction, the overall motion error is roughly in a downward trend, with the maximum error in the Y-direction being 2.77 mm, the minimum error being 0.36 mm, and the average error being 1.37 mm. The error distribution in the Z-direction is relatively uniform; the minimum error is 0.42 mm at the beginning, increasing rapidly within 6 s, and then remaining stable; the maximum error is 2.65 mm; and the average error is 2.05 mm. It can be seen that, in the process of grinding tasks, the error in the Z-direction is relatively stable, but the error in the Y-direction becomes relatively larger with more complicated rules. In general, however, the tracking effect in the Y-direction is better than that in the Z-direction.

5.3. Discussion

From the simulation experiments, we can obtain the synthetic motion trajectories of the proposed two-arm synergistic method, which can achieve similar motion results and better flexibility with respect to the single-arm motion trajectories. Further, two scenarios of two-arm cooperative sanding experiments were conducted, and we recorded the trajectories of each of the two robotic arms in different directions for each scenario.

In the two-arm longitudinal collaboration experiments, we recorded the motion of the two robotic arms in their respective designed motion directions. From the calculated motion errors, the average error basically stays at 1–4 mm, which is still some difference from the desired trajectory, and the error of the decomposition motion will obviously affect the collaboration effect. However, under the whole length of motion, the synthesized motion basically conforms to the shape of the workpiece, and only produces an overall offset in a certain direction. Considering the final overall effect, this is a more reasonable result. For the rotational cooperative motion, it can be seen from the results that the motion accuracy in the rotational direction is slightly higher than the translational motion. Moreover, the difficulty of motion planning and trajectory tracking in this case is higher in practical situations, but the actual motion results can still be obtained with good results, which essentially proves the effectiveness of the proposed method.

6. Conclusions and Future Works

A dual-arm collaborative grinding system is developed for robot grinding tasks in this paper, where the motion trajectory of the dual arms is planned and an impedance control scheme is designed. To this end, the coordinate system transformation relationship of each component of the two-arm system is analyzed based on kinematic constraints, based on which the motion trajectories of the two manipulators are designed, combined with different application scenarios in the workpiece grinding task. In addition, an impedance controller is designed to track the respective trajectories of the two robotic arms during the grinding process. Finally, dual-arm collaborative grinding experiments for two different scenarios are performed. The experimental results show that the proposed method can achieve good performance with an acceptable stable error in both scenarios.

Although the proposed dual-arm collaborative grinding scheme proved to be effective for corresponding grinding tasks, there are still some limitations that may need to be addressed in future research work. On the one hand, the constraint relationship between the two robotic arms in cooperation is analyzed in this study, but we do not analyze the constraints of the dynamics. Therefore, future work will further analyze the dynamic constraints of the cooperative system. On the other hand, only the impedance control is considered in this study, which may not fully satisfy the demand for force or position control. Therefore, a force/position controller with higher performance will be further studied to make the two-arm collaborative grinding more advantageous and competitive in the near future.