Ball-End Cutting Tool Posture Optimization for Robot Surface Milling Considering Different Joint Load

Abstract

Robots with openness and flexibility have attracted a large number of researchers to conduct in-depth studies in the field of surface machining. However, there is a redundant degree of freedom (DOF) in 6-DOF robot machining: when a ball end milling cutter is used to process curved parts, the tool point needs to strictly follow the planned milling trajectory, but the tool axis vector only needs to be within a certain range. During the machining process, the rotation of the tool around its axis is not constrained. Therefore, it is necessary to optimize the redundant DOF. Aiming at the redundant DOF of the tool axis vector in ball end milling for surface parts, a Redundancy Optimization strategy for Minimum Joint trajectory (ROMJ) is proposed. It takes the shortest trajectory of robot joints as the optimization objective, and the numerical optimization method is adopted to carry out the optimal design of tool axis vector trajectory in the milling process. Before optimization, to decrease the data volume, the number of track points is sampled and adjusted based on curve characterization errors. In the optimization process, considering the obvious difference in the load quality characteristics of the robot joints, a Redundancy Optimization strategy for Minimum Joint trajectory considering the different Load of joints (ROMJ-L) is proposed. The load difference coefficients of each joint are introduced into the optimization objective of the trajectory of robot joints. By using this method, the optimal design of each joint trajectory of the robot is realized. In order to verify the methods proposed in this paper, a comparison experiment is carried out. The results show that under the same tool point trajectory, the proposed methods can significantly reduce the robot joint trajectory, and the joint trajectory is influenced by the load difference of each joint. Finally, an Eflin-10 robot is used to process the butterfly trajectory tool path by the trajectory planned by the ROMJ-L method, and the results show that the method is practical.

1. Introduction

Because of the advantages of 6-DOF industrial robots, such as high DOFs and open structure, the use of 6-degrees of freedom (DOF) robots for complex workpiece surface milling has become one of the popular topics in industrial robots in recent years [1,2,3,4,5].

In surface milling, only five DOFs need to be restricted, of which three are used to locate the tool’s tip and the other two are used to determine the tool axis vector. Therefore, there will be a redundant DOF, i.e., the rotation around the tool axis, when using a 6-DOF robot for surface machining. This redundant DOF results in one tool pose corresponding to infinite robot postures. When planning the motion of the 6-DOF robot to perform the ball-end milling task, the redundant DOF can lead to trouble in solving the inverse kinematics, but it also provides an opportunity to optimize the robot posture for a given tool posture. Therefore, in the trajectory plan of robot milling, the main concern is the elimination of the redundant DOF, i.e., optimizing the robot posture according to a specific index [6,7,8,9,10,11].

In the process of robot milling, it is essential for the tool point at the end of the robot to accurately reach the trajectory points specified in the task space. However, due to the existence of the redundant DOF, the motion of the robot joints between the trajectory points is free. Therefore, how to make a reasonable plan for the robot posture of the milling task has become a problem. In order to solve this robot machining problem, scholars have proposed different optimization objectives. Zargarbashi et al. utilized the Jacobi condition number as the optimization objective of flexible optimization, leveraging redundant degrees of freedom to obtain smoother joint-rate time history [12]. In order to improve the milling accuracy of the robot and achieve the best machining performance, Lin et al. proposed the evaluation index of the end deformation and the main stiffness index of the robot [13]. Wang et al. selected the ultimate cutting depth of regenerative chatter in robot milling as the optimization objective to prevent regenerative chatter during robot milling and enhance the quality and efficiency of machining [14]. Lee et al. proposed the minimum energy deformation index based on the static deformation energy generated by machining forces and optimized the robot attitude and workpiece position [15]. Xie et al. used the relationship between average absolute machining error and joint parameter error to reduce machining error and improve stiffness [16]. Chen et al. considered the deformation caused by the weight of the spindle and the deformation caused by the cutting force and proposed a comprehensive deformation index to evaluate the stiffness performance of robot milling [17]. Ni et al. combined joint constraint, stiffness performance index, and dexterity performance index as optimization objectives to conduct comprehensive optimization of redundant robot posture [18]. Guo et al. considered the relationship between the translational displacement of the machine-effector of the robot and the applied force and established an optimization model using the volume of the flexibility ellipsoid as the stiffness performance index [19]. However, the redundant robot posture optimization related to the joint trajectory and joint loads has not been published. This paper argues that the robot joint trajectory should be minimized when eliminating the redundant DOF in the milling process. Meanwhile, the joint load coefficients are introduced into each joint to reduce the trajectory of the joints which have large loads.

At present, there are many optimization algorithms for the robot redundant DOF, including genetic algorithm [20,21,22,23,24], particle swarm optimization [25] and so on. However, these algorithms still have some limitations and defects in different application scenarios and problems. For example, the genetic algorithm requires more computing resources and time, while the particle swarm optimization algorithm is easy to fall into the local optimal solution when the search space is large. Therefore, in this paper, the Sequential Quadratic Programming (SQP) algorithm is selected to optimize the robot’s redundant DOF [26]. SQP is an iterative algorithm based on optimization theory, which can find the global optimal solution in complex nonlinear optimization problems. Compared with other optimization algorithms, the SQP algorithm has efficient and reliable optimization performance, and it is suitable for the optimization model in this paper.

Therefore, the Redundancy Optimization for Minimum Joint trajectory method (ROMJ) and Redundancy Optimization for Minimum Joint trajectory considering the joint Load method (ROMJ-L), which are the global optimal robot joint trajectory planning methods, are proposed in this paper. The ROMJ method combines with the feature that the tool’s tip should strictly follow the planned trajectory, but the tool axis vector only needs to be limited to a certain range when processing surface by a ball end milling cutter. The robot joint trajectory is taken as the optimization objective. The optimization model is established by combining the tool pose and joint limit constraints with the optimization objective. ROMJ-L method is based on the ROMJ method, which considers different joint loads. Load coefficients are introduced to redesign the optimization objective of the ROMJ method. Before optimization, parametric curves are discretized according to the error between chord length and arc length to reduce the amount of data and shorten the processing time. Finally, SQP in the MATLAB toolbox is used to search for the optimal trajectory, and the given tool axis vector is optimized to meet the requirements of processing technology.

Compared with the traditional methods to optimize the redundant DOF of 6-DOF robots, the innovation of these methods is that the traditional method to optimize the redundant DOF of 6-axis robots mainly focuses on the posture of the end of the robot so as to achieve better motion trajectory and precision control. The methods proposed in this paper take the trajectory of robot joints as the optimization objective and realizes smoother and more efficient robot motion control by optimizing the motion of each joint. In addition, the load of each joint is also taken as a reference factor for optimization. The trajectory of each joint is weighted so as to achieve a more stable and reliable robot motion control.

The structure of this paper is as follows: Section 2 introduces a method for selecting trajectory points based on the error between linear and curvilinear distances is developed. In Section 3, the kinematics model of the robot is established. Section 4 provides the definition and expression of optimization variables, optimization objectives, and constraints of ROMJ and ROMJ-L models. Section 5 introduces how to choose initial value optimization, build the optimization model and explain the principle of SQP. In Section 6, the butterfly curve is used to verify that the methods(ROMJ and ROMJ-L) are feasible. Finally, the conclusion is drawn in Section 7.

2. Generation of Discrete Trajectory Points

In surface machining, the tool-tip trajectory is typically a parametric curve. While optimizing the tool axis vector along this curve, it is necessary to discretize it. However, the resulting volume of discrete trajectory points can significantly affect the efficiency of the optimization methods. Therefore, it is important to design a data source for tool axis vector optimization that reduces the amount of data while maintaining accuracy.

The essence of discretization is to replace curves with straight lines. There are many discretization criteria to discretize the curve, such as the distance between the chord and the curve. However, considering the calculation burden, a method based on the error between the chord length and arc length of the curve is proposed. (The curve used in this paper is a B-spline interpolation curve, and the node vector $u_i$ is a vector composed of multiple nodes which represent how to segment the B-spline curve.)

The concrete algorithm, as shown in Figure 1, is provided as follows:

Step 1: Set the initial node vector of the parametric curve $u_1=0$. Set node vector step length $\Delta u=0.0001$ and the variable $b=\Delta u$. Go to Step 2;

Step 2: Using Equation (1) to obtain the node vector of the n-th trajectory point, if $u_i\ge 1$, then $u_i=1$, get the coordinates of the last trajectory point $P_i$, and end the generation of discrete trajectory points. If $u_i<1$, go to Step 3;

$$\begin{array}{c}{u}_i=u_{i-1}+\Delta u\end{array}$$

(1)

Step 3: According to the node vector $u_i$, $P_i$ is calculated. Judge whether the straight-line distance $L_{i-1}$ and the distance $S_{n-1}$ along the curve between point $P_i$ and $P_{i-1}$ satisfy the condition. If yes, the coordinates of the trajectory point $P_i$ are saved and return to step 2. If not, go to step 4;

Step 4: Use the dichotomy method to reassign $\Delta u$: Determine whether $L_{i-1}$ and $S_{i-1}$ satisfy condition 2. If yes, set $b=\frac{1}{2}b$, $\Delta u=\Delta u-b.$ If not, set $\Delta u=\Delta u+b$. Return to Step 2;

3. Robot Kinematic Equation Construction

This paper takes the Eflin-10 robot of Han’s Company as an example to propose the methods. In addition, these methods can also be applied to other 6-DOF robots. The joint variables are set as ${\theta }_j$ $\left(j=1~6\right)$, and the joint coordinate systems are established, as shown in Figure 2.

The transformation matrixes of each joint coordinate system are obtained by substituting the known parameters of the robot in

Table 1. Distance between the origin of the joint $i$ coordinate system and the joint $i-1$ coordinate system (direction is in accordance with the joint $i-1$ coordinate system.

Joint $\mathit{i}$	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
x/(mm)	0	0	0	0	0	0
y/(mm)	0	−100.6	29.6	71	−61	61
z/(mm)	79.4	180.6	480	145	375	109

.

$$\begin{gathered} {}_1{}^{0}T=\left[\begin{array}{cccc}\cos {\theta }_1& -\sin {\theta }_1& 0& 0\\ \sin {\theta }_1& \cos {\theta }_1& 0& 0\\ 0& 0& 1& 79.4\\ 0& 0& 0& 1\end{array}\right] \\ {}_2{}^{1}T=\left[\begin{array}{cccc}\cos {\theta }_2& 0& -\sin {\theta }_2& 0\\ 0& 1& 0& -100.6\\ \sin {\theta }_2& 0& \cos {\theta }_2& 180.6\\ 0& 0& 0& 1\end{array}\right] \\ {}_3{}^{2}T=\left[\begin{array}{cccc}\cos {\theta }_3& 0& \sin {\theta }_3& 0\\ 0& 1& 0& 29.6\\ -\sin {\theta }_3& 0& \cos {\theta }_3& 480\\ 0& 0& 0& 1\end{array}\right] \\ {}_4{}^{3}T=\left[\begin{array}{cccc}\cos {\theta }_4& -\sin {\theta }_4& 0& 0\\ \sin {\theta }_4& \cos {\theta }_4& 0& 71\\ 0& 0& 1& 145\\ 0& 0& 0& 1\end{array}\right] \\ {}_5{}^{4}T=\left[\begin{array}{cccc}\cos {\theta }_5& 0& \sin {\theta }_5& 0\\ 0& 1& 0& -61\\ -\sin {\theta }_5& 0& \cos {\theta }_5& 375\\ 0& 0& 0& 1\end{array}\right] \\ {}_6{}^{5}T=\left[\begin{array}{cccc}\cos {\theta }_6& -\sin {\theta }_6& 0& 0\\ \sin {\theta }_6& \cos {\theta }_6& 0& 61\\ 0& 0& 1& 109\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$

where ${}_1{}^{0}T$ represents the transformation matrix of robot joint $1$ coordinate system with respect to the robot coordinate system $\left[RCS\right]$, length in mm.

When the poses of each joint are known, the pose of the joint at the end of the robot with respect to the robot coordinate system $\left[RCS\right]$ can be obtained by right multiplication sequentially.

$$\begin{array}{c}{}_6{}^{0}T{}_{\theta }={}_1{}^{0}T·{}_2{}^{1}T·{}_3{}^{2}T·{}_4{}^{3}T·{}_5{}^{4}T·{}_6{}^{5}T\end{array}$$

(2)

4. Establishing Optimization Objectives and Constraints

The joint tool path for surface machining with a ball end cutter is optimized by using a numerical optimization method. In the optimization process, the total joint distance is minimized by optimizing the tool vector. To achieve this, the optimization variables, optimization objectives and constraints should be established. The definitions and formulas are provided in this section.

4.1. Optimization Objectives Considering the Distance of the Joints

The optimization objective is affected by variations in optimization variables, which serve as inputs to the optimization problem. The shortest robot joint trajectory is taken as the optimization objective, so the optimization variables are the joint variables ${\theta }_j$ $\left(j=1~6\right)$ of all trajectory points in the robot milling process.

According to optimization variables, the joint trajectory of the robot can be obtained as follows:

$$\begin{array}{c}J{T}_1\left(ROMJ\right)={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}\sqrt[2]{{\displaystyle {\displaystyle \sum }_{j=1}^6}{\left({\theta }_{j, i+1}-{\theta }_{j, i}\right)}^2}\end{array}$$

(3)

Additionally, larger joint loads in a robot require more energy consumption and can impact the robot’s accuracy and stability due to the increased inertia. Hence, considering the varying loads of each joint, weighted processing is performed for each joint trajectory in the optimization objective $J{T}_2\left(ROMJ-L\right)$.

$$\begin{array}{c}J{T}_2\left(ROMJ-L\right)={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}\sqrt[2]{{\displaystyle {\displaystyle \sum }_{j=1}^6}{w}_j{\left({\theta }_{j, i+1}-{\theta }_{j, i}\right)}^2}\end{array}$$

(4)

$w_j$ is the load difference coefficient of the robot joint $j$, which depends on the load mass of the robot end and the mass of the robot connecting rod.

4.2. Constraint

In numerical optimization, it is necessary to limit the optimization variables, i.e., the joint variables ${\theta }_j$. In this method, joint variables are restricted from two perspectives of joint limits and tool pose.

4.2.1. Join Limit Constraint

In robot milling, the robot joints are restricted from reaching areas beyond the limit range. Therefore, the limit range of the joint is taken as a constraint for optimization. This means that the joint variables in the optimized machining process must meet the corresponding range of joint values. Furthermore, the joint limit range can also determine whether the joint trajectory after an inverse solution is reasonable during the initial value optimization process. The robot joint constraint is as follows:

$$\begin{array}{c}min {\theta }_j\le {\theta }_{j,i}\le max {\theta }_j \left(j=1~6,i=1~n\right)\end{array}$$

(5)

4.2.2. Tool Pose Constraint

The change in the tool’s posture and the position of the tool’s tip will result in a change in the robot’s posture. Therefore, in this section, the mapping relationship between the tool pose and the joint variables ${\theta }_j$ of the robot is established, and the tool position is to limit the joint variables.

The tool-tip of the ball-end milling cutter, i.e., the origin of the tool coordinate system $\left(\left[TCS\right]\right),$ must strictly follow the machining curve during the machining process to ensure the accuracy of the milling trajectory. According to Equation (6), the tool pose in workpiece coordinate system $\left(\left[WCS\right]\right)$ is obtained through homogeneous transformation.

$$\begin{array}{c}{}_{tool}{}^{w}T\left(\theta \right)={}_0{}^{w}T{}_6{}^{0}T\left(\theta \right){}_{tool}{}^{6}T=\left[\begin{array}{cc}{}_{tool}{}^{w}R\left(\theta \right)& {}_{tool}{}^{w}P\left(\theta \right)\\ 0 0 0& 1\end{array}\right]\end{array}$$

(6)

${}_0{}^{w}T$ is the homogeneous coordinate transformation of $\left[WCS\right]$ with respect to $\left[RCS\right]$, and ${}_{tool}{}^{6}T$ is the homogeneous coordinate transformation of $\left[TCS\right]$ with respect to the end of the robot.

${}_{tool}{}^{w}P\left(\theta \right)={\left[{}_{tool}{}^{w}p{}_x {}_{tool}{}^{w}p{}_y {}_{ tool}{}^{w}p{}_z\right]}^T$ is the tool-tip path point in $\left[WCS\right]$ during the robot joint is in $\theta$. If the discretized curve points of the required tool path are $P_i\left(\mathit{x}, \mathit{y}, \mathit{z}\right) \left(i=1~n\right)$, the robot end joint’s position should be equal to the discretized curve points $P_i\left(\mathit{x}, \mathit{y}, \mathit{z}\right) \left(i=1~n\right).$

$${}_{tool}{}^{w}P\left(\theta \right)=P_i\left(x,y,z\right)$$

During the milling process, the tool posture needs to be confined to a certain area. In this paper, the tool posture angle $\left(\alpha , \beta \right)$ of the tool axis with the local coordinate system $\left(\left[LCS\right]\right)$ is selected to define the tool posture.

According to Figure 3, taking the trajectory point as the origin, the normal vector of the machined surface as the $Z$-axis, and the tangent vector of the machined trajectory as the $X$-axis. $\alpha$ is the angle between the tool axis and the $Z$-axis of $\left[LCS\right]$. $\beta$ is the angle between the projection of the tool axis in the $XOY$ plane of $\left[LCS\right]$ and the $X$-axis of $\left[LCS\right]$.

The pose matrix ${}_{tool}{}^{l}T{}_{\theta }$ of $\left[WCS\right]$ with respect to $\left[LCS\right]$ is obtained by homogeneous transformation.

$$\begin{array}{c}{}_{tool}{}^{l}T\left(\theta \right)={}_w{}^{l}T{}_{tool}{}^{w}T\left(\theta \right)=\left[\begin{array}{cc}{}_{tool}{}^{l}R\left(\theta \right)& {}_{tool}{}^{l}P\left(\theta \right)\\ 0 0 0& 1\end{array}\right]\end{array}$$

(7)

${}_w{}^{l}T$ is the pose matrix of $\left[LCS\right]$ with respect to $\left[TCS\right]$.

${\left[{}_{tool}{}^{l}a{}_x {}_{tool}{}^{l}a{}_y {}_{tool}{}^{l}a{}_z\right]}^T$ in ${}_{tool}{}^{l}R\left(\theta \right)$ is the component of the Z-axis of $\left[TCS\right]$ in the three directions of $\left[LCS\right].$ The tool posture angle $\left(\alpha , \beta \right)$ can be calculated according to it:

$$\begin{array}{c}\alpha \left(\theta \right)={\cos }^{-1}{}_{tool}{}^{l}a{}_z\end{array}$$

(8)

$$\begin{array}{c}\beta \left(\theta \right)={\tan }^{-1}\frac{{}_{tool}{}^{l}a{}_z}{{}_{tool}{}^{l}a{}_x}\end{array}$$

(9)

Finally, the range of tool posture angle $\left(\alpha ,\beta \right)$ is set as follows:

$$\begin{array}{c} {\alpha }_{min}\le \alpha \left(\theta \right)\le {\alpha }_{max}\\ {\beta }_{min}\le \beta \left(\theta \right)\le {\beta }_{max}\end{array}$$

(10)

5. Robot Global Posture Optimization

5.1. Optimized Initial Value $X_0$ Calculation

Selecting an appropriate initial value for robot redundant posture optimization is a crucial step, and an improper selection may result in the optimization algorithm failing to converge or converging to local optimization. To address this issue, it is important to carefully consider the initial tool posture angle $\left(\alpha ,\beta \right)$ and take into account its distribution within the feasible region interval. This approach can prevent the initial value ${ \mathit{X}}_0$ from being excessively constrained by the optimization algorithm due to being too far away from the boundary. Furthermore, careful consideration of the robot’s kinematic parameters is necessary to ensure the accuracy and effectiveness of the optimization results.

During computation, the DOFs of a 6-DOF robot are equal to the number of inverse solution equations and the optimized initial value $X_0$ of the robot can be calculated by solving this equation group. However, if the DOF of tool rotation is missing, the equation group will be unsolvable, and the initial optimization value $X_0$ cannot be obtained. Therefore, in this paper, it is necessary to pre-set the $X$-axis direction of $\left[TCS\right]$ before solving the equations.

5.2. Establishment of Optimization Model

Considering the above constraints, the global optimization models of ROMJ and ROMJ-L is as follows:

$$\begin{gathered} \min JT\left(\theta \right) \\ s.t.{}_{tool}{}^{w}P\left(\theta \right)=P_i\left(\mathit{x}, \mathit{y}, \mathit{z}\right) \\ {\alpha }_{min}\le \alpha \left(\theta \right)\le {\alpha }_{max} \\ {\beta }_{min}\le \beta \left(\theta \right)\le {\beta }_{max} \\ min {\theta }_j\le {\theta }_{j,i}\le max {\theta }_j \end{gathered}$$

After the optimization model is established, the SQP in MATLAB is used to find the optimal result. SQP algorithm is one of the most effective methods to solve constrained nonlinear optimization problems. It is suitable for the optimization model in this paper, so SQP is selected to optimize the robot’s redundant DOF. The specific optimization steps are as follows:

Step 1: Set Optimized initial value $X_0$ and optimized precision;

Step 2: The original problem is simplified into a quadratic programming problem by using Taylor expansion at the iteration point $X_k$, and the constraint function is simplified to a linear function;

Step 3: Set $s=x-X^k$, then solve the above quadratic programming problem and take the solution $S$ as the search direction $S^k$;

Step 4: The optimization objective is searched in one dimension in the direction $S^k$, $X^{k+1}$ is obtained;

Step 5: If $X^{k+1}$ satisfies the given precision, then $X^{k+1}$ is regarded as the optimal solution, and the calculation is terminated. Otherwise, let $k=k+1$, and go to step 2.

6. Experimental Verification

To demonstrate the effectiveness of the proposed method, a B-spline curve with a butterfly style is selected as the target curve to optimize the robot joint trajectory. The resulting optimized joint trajectory is then used to control the robot to process.

To discrete the target parameter curve, the proposed method utilizes a relationship between chord length and arc length, as shown in Figure 4. In this case, the curve is discretized into 257 points, and more points are distributed in the place where the curvature changes greatly. Compared with Figure 5, which adopts a constant node vector step size $\Delta u=0.0001$, the data volume is significantly reduced.

The experimental device is shown in Figure 6. The spindle motor and ball end milling cutter are installed at the end of the robot in sequence. In this paper, the material used for machining is titanium alloy (Ti-6Al-4V). The spindle weight is 4 kg, and the power is 1.5 KW. In the process of milling, the spindle speed is 1500 r/min. The radius of the ball end milling cutter is 1 mm.

The position relationship between the clamped workpiece and the robot is shown. Then measure the pose matrix of ${}_{tool}{}^{6}T$ and ${}_0{}^{w}T$.

$${}_0{}^{w}T=\left[\begin{array}{cccc}1& 0& 0& -612.111\\ 0& 1& 0& 283.874\\ 0& 0& 1& 214.917\\ 0& 0& 0& 1\end{array}\right] {}_{tool}{}^{6}T=\left[\begin{array}{cccc}0& 0& -1& -143.882\\ 1& 0& 0& 96.636\\ 0& 1& 0& 66.288\\ 0& 0& 0& 1\end{array}\right]$$

Because the surface geometry of the curved workpiece is complex, if the tool posture angle is not selected correctly, it is easy to cause collision and interference between the workpiece and the tool, reduce the quality of the workpiece surface processing and even scrap the workpiece. Therefore, choosing the range of tool posture angle $\left(\alpha , \beta \right)$ is very important. The constraint of the tool posture angle $\left(\alpha , \beta \right)$ in this paper is as follows:

$$\begin{array}{c}4°\le \alpha \le 16°\end{array}$$

(11)

$$0\le \beta \le 180°$$

(12)

Constrain $\beta$ in the range of $0~180°$ so that the tool axis is always to the left of the $X$-axis of $\left[LCS\right]$ to prevent local interference. The reason why $\alpha$ is limited to $4~16°$ is to avoid direct milling of the tool-tip with a milling speed of 0 and to prevent interference in surface machining due to excessive angle.

According to the parameters given by the robot, the joint limit range and the load of each joint are obtained, as shown in Equation (13) and

Table 2. Joint load and load coefficient.

Joint	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Joint load/kg	15.24	10.17	8.5	5.97	5.2	4.12
$w_j$	1.86	1.26	1.02	0.72	0.66	0.64

. In addition, after calculation, the load coefficients $w_j$ of each joint are obtained, and the joints with a larger load are given a greater weight so that the optimization objective $J{T}_1$ and $J{T}_2$ is obtained in Figure 7.

$$\begin{array}{c}\left\{\begin{array}{c}-3.14\le {\theta }_1\le 3.14\\ -2.35\le {\theta }_2\le 2.35\\ -2.61\le {\theta }_3\le 2.61\\ -3.14\le {\theta }_4\le 3.14\\ -2.56\le {\theta }_5\le 2.56\\ -3.14\le {\theta }_6\le 3.14\end{array}\right.\end{array}$$

(13)

Given the tool posture angle $\alpha =5°$ and $\beta =45°$ in the angle range of Equation (11), the initial value of the fixed posture is solved by inverse kinematics, as shown in Figure 8.

The SQP function in MATLAB is selected for optimization. The optimized joint variables during processing are shown in Figure 8, and the processing posture angle $\left(\alpha , \beta \right)$ after optimization is shown in Figure 9. Obviously, the joint trajectory obtained by the proposed methods is obviously decreased with the tool posture angle satisfying the constraints.

Table 3. Comparison of the optimization effects of two optimization objectives.

	$\mathit{J}{\mathit{T}}_1$	$\mathit{J}{\mathit{T}}_2$
Initial	6.376	5.708
Optimized	2.629	2.308
Optimization effect	58.8%	59.6%

compares the joint trajectory values before and after optimization. Both of the selected two optimization objectives can significantly reduce the joint trajectory after optimization, and it can be seen from Figure 8 and Figure 9 that the joint variable curves of the robot are smoother after optimization. The optimized tool posture angle is restricted within the set area. The effectiveness and rationality of the ROMJ and ROMJ-L methods are proven.

Table 4. The sum of the kinematic distance of each joint for two optimization objectives.

Joint Distance (Rad)	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Initial	0.8455	1.6155	2.1608	2.7265	5.2734	8.4284
ROMJ	0.4582	0.5662	1.0169	1.0509	1.1472	1.4683
ROMJ-L	0.4384	0.5149	0.4271	1.1039	1.1634	1.4858
The effects of adding $w_j$	4.3%	9.1%	58.9%	−5.1%	−1.5%	−1.2%

respectively gives the sum of the optimized distances of each joint by using ROMJ and ROMJ-L. It can be seen from the table that the distances of the joints decreased significantly after optimization with ROMJ. Using ROMJ-L, the kinematic distances of the joints are not only significantly reduced after optimization, but also, under the influence of load coefficients, the kinematic distances of the joints with large load decrease, while the kinematic distances of the joints with small load increase, which proves the effectiveness of optimizing joint trajectory with load coefficient.

In order to verify the accuracy and effectiveness of the proposed method, the joint trajectory obtained by ROMJ-L is used as the input for processing by using the Elfin-10 robot, and the Supplementary Video S1 of the processing process is provided as proof materials. Figure 10 shows the robot processing process and the butterfly curve processed by ROMJ-L method.

7. Conclusions

Robot redundancy has a direct influence on milling. In existing technologies, robot redundancy is generally optimized to improve machining quality and efficiency. In this paper, the ROMJ method and ROMJ-L method are proposed to optimize the robot redundant DOF. The ROMJ method takes the robot’s global joint trajectory as the optimization objective, and the joint limit and the tool pose as constraints for optimization. ROMJ-L method is based on the ROMJ method, considering the joint loads of different joints and introducing load coefficients to redesign the global joint trajectory and reduce the rotation of joints with heavy loads. In addition, before optimization, the relationship between chord length and arc length is used to discretize the trajectory points, which greatly reduces the amount of data.

To prove the effectiveness of the method, the butterfly trajectory is optimized, and the joint trajectory is reduced by 58.8% and 59.6% by the ROMJ method and ROMJ-L method, respectively. In addition, the rotations of joints 1, 2 and 3 with high loads decreased by 4.3% and 9.1% and 58.9%, while the rotations of joints 4, 5 and 6 with low loads increased after the optimization using the ROMJ-L method. Experiments show that the proposed methods can significantly reduce the trajectory of the robot joints, and the introduction of load coefficients can reduce the movement distance of the joints under heavy load, to improve the robot’s performance. This method can be applied to milling robots and pave the way for their future market.

The disadvantage of these methods is that the effect on joint velocities and accelerations is not considered. Therefore, the focus of future research will be on the selection and optimization of joint velocity and acceleration.