An energy-saving optimization method for robot laser welding systems based on multi-objective metaheuristic algorithms

2.1. Description of the problem

A typical robot laser welding system primarily consists of an industrial robot, a laser, and a chiller, as shown in Fig. 1. Fig. 1 additionally illustrates a case of workpiece welding utilizing the robotic laser welding system. $\left\{\mathrm{C}{\mathrm{S}}_0\right\}$ denotes the coordinate system of the robot base, $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{T}\mathrm{o}\mathrm{o}\mathrm{l}}\right\}$ represents the tool coordinate system, and $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$ signifies the workpiece coordinate system. The robot's motion from one stationary point to the next is defined as a sub-motion, which can be programmed using a standard robot motion instruction, during the workpiece welding process. Based on the operational state of the laser, the sub-motion is categorized into two types: the welding sub-motion and the idle sub-motion.

In reference [20], we have studied the characteristics and modeling of energy consumption in robot laser processing systems in detail and analyzed energy-saving approaches for the processing of complex parts. For a given workpiece, the energy consumption and execution time of welding sub-motions are primarily determined by the planned welding process parameters, which remain invariant throughout the welding procedure. However, it is important to note that the minimum energy consumption and execution time for each idle sub-motion, obtained through the time-scaling method under given constraints, are closely related to the layout pose of workpieces in $\left\{\mathrm{C}{\mathrm{S}}_0\right\}$, the posture of the laser tool along the tool center point’s motion path, and the inverse kinematics solutions of 6-axis robots at each path point of the end tool. A typical case is illustrated in Fig. 19 of reference [20]. In this case, the energy consumption of idle sub-motions under four different scenarios exhibits a difference of 42.13 % between the minimum and maximum values. Therefore, optimizing these three factors can significantly reduce the energy consumption of idle sub-motions during the welding of a workpiece.

Fig. 1The basic set-up of a typical robot laser welding system and a case of workpiece welding

Furthermore, to illustrate how these three factors influence the minimum energy consumption of idle sub-motions obtained through the time-scaling method under constraints, the example of the linear path task in Fig. 18 of reference [20] is utilized. The energy consumption is calculated using the model established in reference [20], and the calculation result of the total energy consumption of the robot laser welding system (represented by the red solid line) is plotted in the figure. The new figure is designated as Fig. 2. In Fig. 2, EC represents the energy consumption, IR represents the industrail robot, and RLWS represents the robot laser welding system. $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$ represents the minimum execution time of the linear path task while satisfying kinematic constraints, $T_{\mathrm{E}-\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{R}\mathrm{L}\mathrm{W}S}$ represents the energy-optimal time of the RLWS, and $T_{\mathrm{E}-\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{o}\mathrm{t}}$ represents the energy-optimal time of the robot.

Fig. 2The changes of the energy consumption of each subsystem and the total energy consumption with execution time

As illustrated in Fig. 2, the huge power consumption of the laser and the chiller results in the $T_{\mathrm{E}-\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{R}\mathrm{L}\mathrm{W}\mathrm{S}}$ being smaller than $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$; obviously, a shorter execution time for tasks can lead to a reduction in overall system energy consumption. If the layout pose of workpieces, the posture of laser tools, and the inverse kinematics solution of robots are unreasonable, the corresponding $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$ will be large, thus resulting in a high energy consumption. Therefore, optimizing these three factors to effectively reduce the energy consumption of idle sub-motions is a critical energy-saving strategy for robot laser processing systems. Next, a comprehensive mathematical model will be developed.

2.2. Problem modeling

Assume that a workpiece with $N_{\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{d}}$ welding seams needs to be processed, and there are $N_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$ non-welding strokes during the processing. According to [12], when the reference trajectory is pre-planned, the energy consumption of each sub-motion can be expressed as a polynomial function with respect to execution time, known as the task energy characteristic model, based on the time-scaling method. The minimum energy consumption (${E\text{'}}_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$) of idle sub-motions, which is obtained by the time-scaling method under constraints, can be expressed as follows [20]:

where $N_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$ represents the number of idle sub-motions, the polynomial coefficients ${\mu }_{0,\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}^{\text{'}}$, ${\mu }_{1,\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}^{\text{'}}$, ${\mu }_{2,\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}^{\text{'}}$, ${\mu }_{3,\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}^{\text{'}}$, ${\mu }_{4,\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}^{\text{'}}$ are the task energy consumption coefficients of the $j$-th idle sub-motion. Further details of the polynomial coefficients are provided by Eqs. (13-34) in reference [20]. $P_{\mathrm{L}\mathrm{D}\_\mathrm{p}\mathrm{r}\mathrm{e}}$ is the input power of lasers in the lasing-preparation state, ${\stackrel{-}{P}}_{\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}}$ is the average power of the chiller. ${T\text{'}}_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}$ is the execution time corresponding to minimum energy consumption of the $j$-th idle sub-motion under constraints, and $T_{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}$ is the minimum execution time of the $j$-th idle sub-motion with satisfying constraints. $T_{\mathrm{E}-\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}$ is the energy-optimal time of the $j$-th idle sub-motion. $Pos{e}_{\mathrm{w}}$ represents the workpiece layout pose. $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}^{j,i}$ ($i$ = 1, …, $N_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$) and $IK{S}^{j,i}$ represent the tool posture and the inverse kinematics solution of robots at the sampling point $i$ along the $j$-th idle sub-motion path curve, respectively. $N_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$ is the sampling point number.

Based on the methodology for rigid body pose description, $Pos{e}_{\mathrm{w}}$ can be represented by six variables: $x_{\mathrm{w}}$, $y_{\mathrm{w}}$, $z_{\mathrm{w}}$, ${\theta }_{\mathrm{C}}$, ${\theta }_{\mathrm{B}}$, and ${\theta }_{\mathrm{A}}$. ${\theta }_{\mathrm{C}}$, ${\theta }_{\mathrm{B}}$, and ${\theta }_{\mathrm{A}}$ are the rotational angles of $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$ about the $X$, $Y$, and $Z$ axes of $\left\{\mathrm{C}{\mathrm{S}}_0\right\}$, respectively, which describe the posture of the workpiece in $\left\{\mathrm{C}{\mathrm{S}}_0\right\}$. $(x_{\mathrm{w}},y_{\mathrm{w}},z_{\mathrm{w}}{)}^T$ is the position vector of the origin of $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$ in $\left\{\mathrm{C}{\mathrm{S}}_0\right\}$, describing the position of the workpiece in $\left\{\mathrm{C}{\mathrm{S}}_0\right\}$. Similarly, the laser tool pose in $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$ can be represented by six variables: $x$, $y$, $z$, $C$, $B$, and $A$.$C$, $B$, and $A$ represent the rotational angles of $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{T}\mathrm{o}\mathrm{o}\mathrm{l}}\right\}$ about the $X$, $Y$, and $Z$ axes of $\left\{C{S}_W\right\}$, respectively, which describe the laser tool posture in $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$. $(x,y,z{)}^T$ is the position vector of the origin of $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{T}\mathrm{o}\mathrm{o}\mathrm{l}}\right\}$ (namely the tool centre point) in $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$, which represents the position of the laser tool in $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$.

For a specified workpiece, the target positions that the tool centre point (TCP) of the laser tool needs to achieve in $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$ are specific. However, at each target position, the posture of the laser tool can be adjusted. Moreover, for a given pose of the laser tool, the inverse kinematics of robots have eight possible solutions (${\mathit{q}}^{\#1}$, ${\mathit{q}}^{\#2}$, ${\mathit{q}}^{\#3}$, ${\mathit{q}}^{\#4}$, ${\mathit{q}}^{\#5}$, ${\mathit{q}}^{\#6}$, ${\mathit{q}}^{\#7}$, ${\mathit{q}}^{\#8}$), which are automatically numbered during the solving process. However, optimizing the posture of laser tools and the inverse kinematic solution at each target point of continuous path motion will introduce countless decision variables. Therefore, only the postures of laser tools and the inverse kinematic solutions at the starting and ending points of continuous path motion are considered, thus ensuring a reasonable number of decision variables.

Therefore, for the problem of optimizing the minimum energy consumption of robot laser welding systems obtained by the time-scaling method, the decision variable set is {$x_{\mathrm{w}}$, $y_{\mathrm{w}}$, $z_{\mathrm{w}}$, ${\theta }_{\mathrm{A}}$, ${\theta }_{\mathrm{B}}$, ${\theta }_{\mathrm{C}}$, $A_{j,\mathrm{s}}$, $B_{j,\mathrm{s}}$, $C_{j,\mathrm{s}}$, $A_{j,\mathrm{e}}$, $B_{j,\mathrm{e}}$, $C_{j,\mathrm{e}}$, $c{f}_{j,\mathrm{s}}$, $c{f}_{j,\mathrm{e}}|j$ = 1, …, $N_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$}. $(A_{j,\mathrm{s}},B_{j,\mathrm{s}},C_{j,\mathrm{s}})$ and $(A_{j,\mathrm{e}},B_{j,\mathrm{e}},C_{j,\mathrm{e}})$ are the attitude angles of the laser tool at the starting and ending points of the $j$-th idle sub-motion, respectively. $c{f}_{j,\mathrm{s}}$ and $c{f}_{j,\mathrm{e}}$ are the decision variables of inverse kinematic solutions at the starting and ending points, respectively. The general mathematical model can be expressed as follows:

where Eqs. (4-5) address the basic constraints of workspace and laser tool postures. Since the robotic workspace is a non-ideal cube, the flexibility outside the flexible space is limited, and the intricate spatial interferences exist, any alteration in $x_{\mathrm{w}}$, $y_{\mathrm{w}}$, $z_{\mathrm{w}}$, ${\theta }_{\mathrm{A}}$, ${\theta }_{\mathrm{B}}$, ${\theta }_{\mathrm{C}}$, $A_{j,\mathrm{s}}$, $B_{j,\mathrm{s}}$, $C_{j,\mathrm{s}}$, $A_{j,\mathrm{e}}$, $B_{j,\mathrm{e}}$, or $C_{j,\mathrm{e}}$ will inevitably lead to variations in the feasible domains of other variables. Therefore, the setting values for parameters $x_{\mathrm{w},\mathrm{m}\mathrm{i}\mathrm{n}}$, $y_{\mathrm{w},\mathrm{m}\mathrm{i}\mathrm{n}}$, $z_{\mathrm{w},\mathrm{m}\mathrm{i}\mathrm{n}}$, ${\theta }_{\mathrm{A},\mathrm{m}\mathrm{i}\mathrm{n}}$, ${\theta }_{\mathrm{B},\mathrm{m}\mathrm{i}\mathrm{n}}$, ${\theta }_{\mathrm{C},\mathrm{m}\mathrm{i}\mathrm{n}}$, $A_{j,\mathrm{s},\mathrm{m}\mathrm{i}\mathrm{n}}$, $B_{j,\mathrm{s},\mathrm{m}\mathrm{i}\mathrm{n}}$, $C_{j,\mathrm{s},\mathrm{m}\mathrm{i}\mathrm{n}}$, $A_{j,\mathrm{e},\mathrm{m}\mathrm{i}\mathrm{n}}$, $B_{j,\mathrm{e},\mathrm{m}\mathrm{i}\mathrm{n}}$, $C_{j,\mathrm{e},\mathrm{m}\mathrm{i}\mathrm{n}}$, $x_{\mathrm{w},\mathrm{m}\mathrm{a}\mathrm{x}}$, $y_{\mathrm{w},\mathrm{m}\mathrm{a}\mathrm{x}}$, $z_{\mathrm{w},\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\theta }_{\mathrm{A},\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\theta }_{\mathrm{B},\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\theta }_{\mathrm{C},\mathrm{m}\mathrm{a}\mathrm{x}}$, $A_{j,\mathrm{s},\mathrm{m}\mathrm{a}\mathrm{x}}$, $B_{j,\mathrm{s},\mathrm{m}\mathrm{a}\mathrm{x}}$, $C_{j,\mathrm{s},\mathrm{m}\mathrm{a}\mathrm{x}}$, $A_{j,\mathrm{e},\mathrm{m}\mathrm{a}\mathrm{x}}$, $B_{j,\mathrm{e},\mathrm{m}\mathrm{a}\mathrm{x}}$, and $C_{j,\mathrm{e},\mathrm{m}\mathrm{a}\mathrm{x}}$ are redundant. The purpose is to encompass all feasible domains of the corresponding variables. Due to the dependence of the space required for processing a workpiece on the geometric shape and size of the tool, it cannot be ensured that the space required will entirely fall within the robotic workspace, even if the workpiece is located within the robotic workspace. Therefore, satisfying Eqs. (4-5) does not guarantee the attainability of all processing poses. Eq. (7) represents the joint position constraints. If there exists at least one inverse solution that satisfies all the joint position constraints for a processing pose, then the processing pose is deemed reachable; otherwise, it is deemed unreachable. Eq. (8) represents the constraints of joint velocities, accelerations, and jerks. Eq. (9) describes the limitations of the tool velocity, acceleration, and jerk. Eq. (10) signifies the constraint of processing time.

3.1. Objective function deconstruction based on system energy consumption characteristics

The calculation of Eq. (1) is highly time-consuming due to the requirement for intricate 6-axis robot dynamics calculations during the optimization process. Therefore, this paper deconstructs the energy consumption objective into multiple feature objectives, which are computationally efficient, based on the characteristics and key influencing factors of energy consumption of robot laser welding systems. Then, the single-objective optimization of energy consumption is transformed into a multi-objective optimization of these feature objectives. Finally, we will only compute the energy consumption of the Pareto optimal solutions resulting from multi-objective optimization, to identify the solution with the lowest energy consumption.

We conducted a detailed study on the energy consumption characteristics of a typical robot laser welding system, which forms the basis for the deconstruction of energy consumption objectives in this paper. The key analyses are presented as follows:

1) During the idle sub-motion, because the total power of the laser and the chiller is still very large, the energy-optimal point of the energy characteristic curve for the whole system is still significantly shifted to the left compared to the robot, as depicted in Fig. 2. The results indicate that, for the idle sub-motion, the energy consumption of the whole system is positively correlated with execution time to a large extent. Therefore, the idle motion time is a very important feature objective.

2) In addition, on the right side of the energy-optimal point, the smaller the time is, the smaller the rate of energy consumption reduction is. Therefore, it is always hoped that the speed of each idle sub-motion is as high as possible, while also being as close as possible, to save energy through fully utilizing the energy consumption optimization features of the system.

3) Moreover, for a specified task, robotic energy consumption is not only related to execution time, but also strongly related to the angle of joint rotation. The total joint rotation angle directly reflects the energy consumption coefficients of the task. And the larger the energy consumption coefficients are, the higher the overall energy consumption of the robot is. Therefore, the total joint rotation angle of idle motion is also an important feature objective. Considering that the energy consumption of the unit angle of each joint is different, a weighted sum is generally used.

Therefore, this paper replaces the energy consumption objective with the total time ($f_{\mathrm{t}}$) of idle sub-motions, the weighted sum ($f_{\mathrm{\theta }}$) of joint rotation angles in all idle sub-motions, and the maximum difference ($f_{\dot{\mathrm{q}}}$) among the speeds of idle sub-motions, as the criteria to determine the individual performance in the optimization process. The equivalent joint velocity (${\stackrel{-}{\dot{q}}}_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e},j}$, $j$= 1,…, $N_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$), obtained by dividing the weighted sum of joint rotation angles by the motion time, is used for each idle sub-motion. Finally, only the energy consumption of individuals in the Pareto set, which are found through multi-objective optimization, is calculated, thus greatly reducing the calculation amount required for the energy consumption optimization process. The new objective function formulas are as follows:

where ${\theta }_i$ is the rotation angle of joint $i$ in all idle sub-motions, ${\omega }_i$ is the corresponding weight value of joint $i$. The weight value of each joint is obtained by the Analytic Hierarchy Process method [25].

3.2. NSGA-II algorithm for optimizing the energy consumption of robot laser welding systems

The NSGA-II algorithm proposed by Deb et al. [26] has both good distribution and fast convergence rate, while it is widely used by scholars at home and abroad for multi-objective optimization problems. The NSGA-II algorithm with an elite strategy is very suitable for solving the problem in this paper.

Different robot poses have different eight inverse kinematic solutions and the amount and sequence number of feasible inverse kinematic solutions are also different. Therefore, the solution of the entire problem is divided into two parts: the basic solution, composed of decision variables {$x_{\mathrm{w}}$, $y_{\mathrm{w}}$, $z_{\mathrm{w}}$, ${\theta }_{\mathrm{A}}$, ${\theta }_{\mathrm{B}}$, ${\theta }_{\mathrm{C}}$, $A_{j,\mathrm{s}}$, $B_{j,\mathrm{s}}$, $C_{j,\mathrm{s}}$, $A_{j,\mathrm{e}}$, $B_{j,\mathrm{e}}$, $C_{j,\mathrm{e}}$, $c{f}_{j,\mathrm{s}}$, $c{f}_{j,\mathrm{e}}|j=$1,…, $N_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$} of the workpiece layout pose and laser tool posture, and the subordinate solution composed of decision variables {$c{f}_{j,\mathrm{s}}$, $c{f}_{j,\mathrm{e}}|j$= 1,…, $N_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$} of eight inverse kinematic solutions at the starting and ending points of all idle sub-motions.

The optimization of the basic solution is used as the main loop. Because the basic solution is only obtained under the basic workspace constraint (4) and the laser tool posture constraint (5), it cannot be guaranteed that all processing poses, which are determined by the workpiece layout pose satisfying constraints (4) and the laser tool posture satisfying constraints (5), are reachable. For each basic solution, the reachability of each processing pose should be firstly judged to determine whether it is feasible. The basic solution with unreachable processing poses is called an unreachable solution, while the set of unreachable solutions is directly used as the worst non-dominated level.

For a basic solution which satisfies the accessibility constraint, it is necessary to determine the corresponding subordinate solution. The method is as follows: firstly, based on the three feature objectives in Eq. (11), the NSGA-II algorithm is used to obtain a set of Pareto-optimal solutions about the selection of eight inverse kinematic solutions at the starting and ending points of each sub-motion; then, the Analytic Hierarchy Process method is used to select a satisfactory solution from the Pareto set obtained as the subordinate solution corresponding to the basic solution. This process is called configuration decision-making. In the configuration decision-making, the following method is used to dimensionless three objectives. Let $Uni{t}_{i,j}$ represent the dimensionless value of the objective $j$ ($j$ can be taken as $f_{\mathrm{t}}$, $f_{\mathrm{\theta }}$, and $f_{\dot{\mathrm{q}}}$) corresponding to the Pareto-optimal solution $i$, and then the following function is defined:

where $f_{i,j}$ is the function value of the objective $j$ of the Pareto-optimal solution $i$, $N_{\mathrm{P}\mathrm{a}}$ is the number of Pareto-optimal solutions.

Fig. 3 shows the integrated optimization process based on the NSGA-II algorithm. $Gen$ is the current number of iterations. $N_{\mathrm{i}\mathrm{t}\mathrm{e}}$ is the maximum number of iterations. ${\mathbf{P}}_0$ and ${\mathbf{Q}}_0$ are the initial parent and offspring population of the basic solution, respectively. ${\mathbf{P}}_{\mathrm{t}}$ and ${\mathbf{Q}}_{\mathrm{t}}$ are the contemporary parent and offspring population of the basic solution, respectively. ${\mathbf{R}}_{\mathrm{t}}$ is the combined population of ${\mathbf{P}}_{\mathrm{t}}$ and ${\mathbf{Q}}_{\mathrm{t}}$. ${\mathbf{P}}_{\mathrm{t}+1}$ and ${\mathbf{Q}}_{\mathrm{t}+1}$ are the new parent and offspring population of the basic solution, respectively. ${\mathcal{F}}_{\mathrm{k}}$ is the non-dominated level obtained by the $k$-th fast non-dominated sorting. The dominant relationship adopted is that if individual U and individual V meet any of the following conditions, then individual U dominates individual V:

1) Individual U satisfies all constraints while individual V does not.

2) Both individual U and individual V meet the accessibility constraint, but each objective function value of individual U is not bigger than the corresponding value of individual V and at least one objective function value is less than the corresponding value of individual V.

3) Individual U satisfies the accessibility constraint, while individual V does not satisfy the accessibility constraint.

4) Both individual U and individual V do not satisfy the accessibility constraint, but the unreachable points of individual U are less than those of individual V.

Fig. 3The integrated optimization process based on the NSGA-II algorithm

4.1. Case introduction

The workpiece is a bicycle frame. The processing platform is the robot laser welding system used in [20], and the welding process parameters is: laser power of 2.75 kW, welding speed of 3 m/min (50 mm/s), and defocusing distance of 0 mm. Fig. 4 shows the three-dimensional model of the bicycle frame and the sub-motions corresponding to a welding process selected. The welding process selected has 12 idle sub-motions and 11 welding sub-motions. The idle sub-motions and welding sub-motions are alternating and the sequence is shown in Fig. 4. The path lengths of welding sub-motion □1 to □11 are 19.70 mm, 58.74 mm, 22.20 mm, 62.02 mm, 52.98 mm, 61.11 mm, 33.60 mm, 31.98 mm, 23.69 mm, 37.60 mm, and 34.80 mm, respectively. The idle sub-motions, which are represented by the number with a circle, adopt the point-to-point motion.

By analyzing the characteristics of the selected welding process, the following knowledges can be easily obtained. Considering the fixed configuration of robots at the Home position, only the laser tool orientation and inverse kinematic solution at the end-point need to be taken into account for the initial idle sub-motion, while only the laser tool orientation and inverse kinematic solution at the start-point need to be taken into account for the final idle sub-motion. The laser tool orientation and the selection of eight inverse kinematic solutions at the start-point of the subsequent sub-motion are consistent with those at the end-point of the preceding sub-motion. It is necessary for all target points in continuous path motion to select the inverse kinematic solution with the same index, to ensure continuous motion along the specified path. Consequently, only one variable, which indicates the selection of inverse kinematic solutions at the start-point is required for continuous path motion. The orientation of the laser tool axis keeps the value predetermined along the whole trace of the tool centre point for a welding sub-motion, and therefore the attitude angle $B$ and $C$ of the laser tool in $\left\{\mathrm{C}{\mathrm{S}}_{\mathrm{W}}\right\}$ are constant.

Therefore, the basic solution is {$x_{\mathrm{w}}$, $y_{\mathrm{w}}$, $z_{\mathrm{w}}$, ${\theta }_{\mathrm{A}}$, ${\theta }_{\mathrm{B}}$, ${\theta }_{\mathrm{C}}$, $A_{1,\mathrm{e}}$, $A_{2,\mathrm{s}}$, $A_{2,\mathrm{e}}$, $A_{3,\mathrm{s}}$, $A_{3,\mathrm{e}}$, $A_{4,\mathrm{s}}$, $A_{4,\mathrm{e}}$, $A_{5,\mathrm{s}}$, $A_{5,\mathrm{e}}$, $A_{6,\mathrm{s}}$, $A_{6,\mathrm{e}}$, $A_{7,\mathrm{s}}$, $A_{7,\mathrm{e}}$, $A_{8,\mathrm{s}}$, $A_{8,\mathrm{e}}$, $A_{9,\mathrm{s}}$, $A_{9,\mathrm{e}}$, $A_{10,\mathrm{s}}$, $A_{10,\mathrm{e}}$, $A_{11,\mathrm{s}}$, $A_{11,\mathrm{e}}$, $A_{12,\mathrm{s}}$} and the subordinate solution is {$c{f}_{1,\mathrm{e}}$, $c{f}_{2,\mathrm{e}}$, $c{f}_{3,\mathrm{e}}$, $c{f}_{4,\mathrm{e}}$, $c{f}_{5,\mathrm{e}}$, $c{f}_{6,\mathrm{e}}$, $c{f}_{7,\mathrm{e}}$, $c{f}_{8,\mathrm{e}}$, $c{f}_{9,\mathrm{e}}$, $c{f}_{10,\mathrm{e}}$, $c{f}_{11,\mathrm{e}}$} for the problem corresponding to the welding process selected of the bicycle frame. The values of attitude angles $B$ and $C$ are determined by the value of the attitude angle $A$ and the orientation of the laser tool axis planned. The value range of each decision variable in the basic solution is shown in Table 1.

Fig. 4The bicycle frame model and welding paths

In general, the workers mainly focus on the accessibility of the processing posture, the singularity of robotic configurations, the continuity of continuous path motion, and collision avoidance. The relevant decision variables of the original scheme are determined according to the general principle. Usually, the workpiece is installed in the middle of the robot workspace to achieve maximum processing flexibility. According to the principle of no bias, the attitude angles of the laser tool at the starting and ending points of each sub-motion are all taken as zero based on the corresponding value ranges in Table 1. Under the layout pose of the bicycle frame and laser tool postures aforementioned, except for the start-point of the first idle sub-motion and the end-point of the final idle sub-motion, the #7 inverse kinematic solution is selected at other starting and ending points of sub-motions, to achieve the minimum change in joint position relative to Home position. Table 2 shows the decision variable values corresponding to the original scheme.

4.2. Results and analysis

The independent optimization and integrated optimization of three factors: the workpiece layout pose, the laser tool posture, and the inverse kinematics solution are performed. In all optimizations, the population size is $N_{\mathrm{P}}$= 100, the crossover probability is 0.7, the mutation probability is 0.2. All optimization calculations are performed in the MATLAB software on a computer with an Intel i3-8100 CPU 3.60 GHz and 8-GB DDR2. In the following text, to simplify the expression, $Pos{e}_{\mathrm{W}}$ is used to represent the workpiece layout pose, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$ is used to represent the laser tool posture, and $IKS$ is used to represent the inverse kinematics solution.

4.2.1. Independent optimization of $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$

To demonstrate the impact of three factors: $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$ on the energy consumption and excution time of idle sub-motions, the independent optimization of each factor is carried out with the optimization order of $Pos{e}_{\mathrm{W}}$→$Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$→$IKS$. The maximum iteration number is 200 for the optimization of each factor. Two types of optimization situations are considered, as shown in Table 3.

Table 3Two optimization situations

Optimization order	Optimization situation I			Optimization situation II
	$Pos{e}_{\mathrm{W}}$	$Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$	$IKS$	$Pos{e}_{\mathrm{W}}$	$Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$	$IKS$
First	Optimizing	Original scheme	Original scheme	Optimizing	Original scheme	Original scheme
Second	Original scheme	Optimizing	Original scheme	Optimized scheme	Optimizing	Original scheme
Third	Original scheme	Original scheme	Optimizing	Optimized scheme	Optimized scheme	Optimizing

Furthermore, we conducted a single-objective optimization with using energy consumption as the cost function under identical conditions. During this procedure, we adopted a genetic algorithm which incorporates an elitism preservation strategy. Table 4 provides a comparative analysis of the results and computational efficiency between the energy consumption-based single-objective optimization and the multi-objective optimization proposed. $E_{\mathrm{m}\mathrm{i}\mathrm{n}}^{*}$ is the energy consumption of the best individual obtained by the optimization, and $T^{*}$ is the execution time corresponding to $E_{\mathrm{m}\mathrm{i}\mathrm{n}}^{*}$.

According to the data presented in Table 4, compared to the single-objective optimization focused on energy consumption, the multi-objective optimization method proposed achieves a significant reduction in optimization computation time, exceeding 95 %. Furthermore, it yields remarkably similar optimization results, exhibiting deviations of less than 1.5 %. Additionally, it is essential to exhibit the correlation characteristics between the three feature targets and energy consumption. When each of the three feature objectives $f_{\mathrm{t}}$, $f_{\mathrm{\theta }}$, and $f_{\dot{\mathrm{q}}}$ and the energy consumption objective is minimum or maximum, the results of the corresponding Pareto-optimal solution, which is obtained by the optimization of $Pos{e}_{\mathrm{W}}$, are shown in Table 5.

According to the data in Table 5, we observe that the minima of any one of three feature objectives do not align with the minimum energy consumption scenario. This suggests a coupled relationship among these feature objectives and energy consumption. Specifically, the fluctuation in energy consumption is not solely determined by the variation in a single feature objective; it is also impacted by the degree of change in the other two feature objectives. In summary, among the three feature objectives examined, $f_{\mathrm{t}}$ exhibits the most substantial impact on energy consumption, followed by $f_{\mathrm{\theta }}$, while $f_{\dot{\mathrm{q}}}$ has a comparably smaller influence. The results suggest that the selection of these three feature objectives is appropriate, and they can collectively function as cost functions for assessing energy consumption. The solving approach proposed is reliable for the energy consumption optimization problem.

Table 4A comparison of results and computational efficiency between the single-objective optimization and the multi-objective optimization proposed

		Optimization situation I			Optimization situation II
		$Pos{e}_{\mathrm{W}}$	$Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$	$IKS$	$Pos{e}_{\mathrm{W}}$	$Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$	$IKS$
Single-objective optimization	Computation time (min)	635.57	717.38	951.20	635.57	779.16	844.62
	$E_{\mathrm{m}\mathrm{i}\mathrm{n}}^{*}$ (J)	29075.84	29445.11	37948.05	29075.84	27630.73	26497.63
	$T^{*}$ (s)	4.31	4.38	5.46	4.31	4.16	4.03
Multi-objective optimization proposed	Computation time (min)	26.85	22.14	7.94	26.85	22.75	3.89
	$E_{\mathrm{m}\mathrm{i}\mathrm{n}}^{*}$ (J)	29239.43	29753.34	38497.56	29239.43	27486.89	26601.83
	$T^{*}$ (s)	4.33	4.43	5.54	4.33	4.13	4.05
Deviation in computational time		–95.78 %	–96.91 %	–99.16 %	–95.78 %	–97.08 %	–99.54 %
Deviation in energy consumption		0.56 %	1.04 %	1.45 %	0.56 %	–0.52 %	0.39 %
Note: The deviation = (Multi-objective optimization value - Single-objective optimization value)/Single-objective optimization value *100 %

Table 5Corresponding results for minimum and maximum of each objective, respectively

Category	$f_{\mathrm{t}}$ (s)	$f_{\mathrm{\theta }}$ (°)	$f_{\dot{\mathrm{q}}}$ (°/s)	Energy consumption (J)
Minimum $f_{\mathrm{t}}$	4.12	159.30	53.08	32728.18
Maximum $f_{\mathrm{t}}$	5.36	141.80	27.27	35436.97
Minimum $f_{\mathrm{\theta }}$	4.62	119.71	39.30	30110.89
Maximum $f_{\mathrm{\theta }}$	4.12	159.53	52.57	32707.60
Minimum $f_{\dot{\mathrm{q}}}$	5.36	141.80	27.27	35436.97
Maximum $f_{\dot{\mathrm{q}}}$	4.12	158.91	53.21	32662.03
Minimum energy consumption	4.33	127.58	41.30	29239.43
Maximum energy consumption	5.36	141.80	27.27	35436.97
Original scheme	6.00	188.38	41.82	41242.79

To illustrate the influence of three factors on energy consumption, Figs. 5-6 show the Pareto-optimal solutions obtained by the optimization of three factors and the corresponding energy consumption simulation values for the optimization situation I and II, respectively. $N_{\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{o}}$ is the number of Pareto-optimal solutions. $E_{\mathrm{i}\mathrm{d}\mathrm{l}\mathrm{e}}$ represents the total energy consumption of idle sub-motions.

According to the results in Figs. 5-6, each of the three factors $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$ has a significant impact on the energy consumption and execution time of idle sub-motions. The effect of optimizing multiple factors is better than that of optimizing a single factor. Compared to only optimizing one of $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$, the optimizition of all three factors leads to reductions in energy consumption by 9.02 %, 10.59 %, and 30.90 %, respectively. Notably, the optimization effects of $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$ and $IKS$ considerably differ for different $Pos{e}_{\mathrm{W}}$ or $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$. The significant variance suggests that $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$ are interconnected and mutually dependent. Therefore, it is imperative to consider the combined influence of $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$.

Fig. 5The Pareto-optimal solutions obtained and the corresponding energy consumption simulation values for the optimization situation I

a) The optimization of workpiece layout poses

b) The optimization of laser tool postures

c) The optimization of inverse kinematics solutions

Fig. 6The Pareto-optimal solutions obtained and the corresponding energy consumption simulation values for the optimization situation II

a) The optimization of workpiece layout poses

b) The optimization of laser tool postures

c) The optimization of inverse kinematics solutions

4.2.2. Integrated optimization of $Pos{e}_{\mathrm{W}}$, $Postur{\mathrm{e}}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$

For the independent optimization of three factors, the coupling impact of three factors on the optimization results is not considered. Therefore, we also do the integrated optimization of $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$ from a random initial population. In order to ensure that the iteration number of the basic solutions for two ways are basically the same, the maximum iteration number for the integrated optimization is 400. Fig. 7 shows the Pareto-optimal solutions obtained after 10, 200, and 400 iterations and the corresponding energy consumption simulation values.

Fig. 7The Pareto-optimal solutions obtained by the integrated optimization and the corresponding energy consumption simulation values

According to the results in Fig. 7, with the search proceeding, the individuals in the population move towards the Pareto-optimal region in total, the minimum energy consumption gradually decreases, and the non-dominated points corresponding to the Pareto-optimal solutions found are well distributed in the objective vector space. The difference between two minimum energy consumption values found through 200 and 400 iterations is small (about 66.93 J), and the two minimum energy consumption values are smaller than the minimum energy consumption obtained by the independent optimization of three factors for the optimization situation II (reduced by 435.62 J and 502.55 J, respectively). It indicates that the proposed integrated optimization process based on the NSGA-II algorithm is applicable, and the energy-saving effect of integrated optimization, which considers the coupling among $Pos{e}_{\mathrm{W}}$, $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$, and $IKS$, is better than that of the independent optimization.

Table 6The best Pareto scheme searched

$x_{\mathrm{w}}$ (mm)	$y_{\mathrm{w}}$ (mm)	$z_{\mathrm{w}}$ (mm)	${\theta }_{\mathrm{A}}$ (°)	${\theta }_{\mathrm{B}}$ (°)	${\theta }_{\mathrm{C}}$ (°)	$A_{1,\mathrm{e}}$ (°)	$A_{2,\mathrm{s}}$ (°)	$A_{2,\mathrm{e}}$ (°)	$A_{3,\mathrm{s}}$ (°)
1136.28	112.68	1062.33	32.02	45.62	33.775	17.22	18	14	32
$A_{3,\mathrm{e}}\mathrm{}($°)	$A_{4,\mathrm{s}}\mathrm{}($°)	$A_{4,\mathrm{e}}\mathrm{}($°)	$A_{5,\mathrm{s}}\mathrm{}($°)	$A_{5,\mathrm{e}}\mathrm{}($°)	$A_{6,\mathrm{s}}\mathrm{}($°)	$A_{6,\mathrm{e}}\mathrm{}($°)	$A_{7,\mathrm{s}}\mathrm{}($°)	$A_{7,\mathrm{e}}\mathrm{}($°)	$A_{8,\mathrm{s}}\mathrm{}($°)
36	–72.44	4	28	10	18	32	4	8	14
$A_{8,\mathrm{e}}\mathrm{}($°)	$A_{9,\mathrm{s}}\mathrm{}($°)	$A_{9,\mathrm{e}}\mathrm{}($°)	$A_{10,\mathrm{s}}\mathrm{}($°)	$A_{10,\mathrm{e}}\mathrm{}($°)	$A_{11,\mathrm{s}}\mathrm{}($°)	$A_{11,\mathrm{e}}\mathrm{}($°)	$A_{12,\mathrm{s}}\mathrm{}($°)	$c{f}_{1,\mathrm{e}}$	$c{f}_{2,\mathrm{e}}$
8	20	14	32	32	16	22	–16	#3	#3
$c{f}_{3,\mathrm{e}}$	$c{f}_{4,\mathrm{e}}$	$c{f}_{5,\mathrm{e}}$	$c{f}_{6,\mathrm{e}}$	$c{f}_{7,\mathrm{e}}$	$c{f}_{8,\mathrm{e}}$	$c{f}_{9,\mathrm{e}}$	$c{f}_{10,\mathrm{e}}$	$c{f}_{11,\mathrm{e}}$	–
#3	#3	#3	#3	#3	#3	#3	#3	#3	–

Table 6 shows the decision variable values corresponding to the best solution scheme found. The KUKA KR60-3 robot is used to carry out the motion experiments of the original scheme and the best solution scheme, and the execution time and robotic energy consumption data were measured. The energy consumption of the laser and the chiller is calculated based on the execution time measured. The statistical average of multiple measurement results is used for relevant calculations, to reduce the impact of measurement errors on the final results. The comparisons of processing energy consumption and execution time between the two schemes are illustrated in Fig. 8. The rectangular bar represents the sample mean of each parameter, and the red line represents the confidence interval of each parameter calculated at a 95 % confidence level.

Fig. 8Processing energy consumption and execution time of two schemes

a) Comparison of energy consumption

b) Comparison of execution time

According to the results in Table 2, Table 6, and Fig. 8, there is a significant difference in $Pos{e}_{\mathrm{W}}$ and $Postur{e}_{\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{l}}$ between two schemes. Compared with the original scheme, the best solution scheme reduces the energy consumption and execution time of idle sub-motions by 34.91 % and 32.49 %, respectively, while the total energy consumption and execution time of the whole motion are reduced by 7.29 % and 13.07 %, respectively. The results show that the robot laser welding of complex components has great potential in energy conservation. In addition, due to the complexity of robot kinematics characteristics and energy consumption characteristics, there is a large optimization space in energy efficiency and efficiency for the processing scheme which is planned by operators based on the general cognition. Therefore, the optimization operation is necessary.