1. Introduction

2. Modeling and identification of friction

2.2. Multi-parameter identification of friction model

The parameters of the LuGre model can be accurately identified by addressing the limitations of conventional iterative algorithms, as shown in Eq. (4). However, these algorithms struggle to yield satisfactory results for the multi-parameters synchronous identification. To overcome these challenges, a new fly optimization algorithm (FOA) is proposed. The FOA enables group optimization, making it suitable for identifying the parameters of the LuGre model. By employing dynamic step and adaptive cross mutation, the FOA effectively addresses the issue of the traditional Drosophila algorithm falling into local extremum, thereby improving the accuracy of parameter identification. The FOA mimics the predatory process of drosophila, utilizing its olfactory superiority to quickly distinguish complex odors in the air. FOA stands out from other optimization algorithms due to its simplicity in structure, ease of programming, faster convergence, and fewer parameters. The foraging process of FOA is depicted in Figure 1.

Figure 1. Iterative foraging process of fruit flies.

Figure 1. Iterative foraging process of fruit flies.

The adaptive crossover factor is introduced to regulate the crossover of the population to search for the global optimal value of the population. This effectively improves the adaptability of the Drosophila individual to environmental changes. The main steps involve randomly pairing the Drosophila population and selecting any pair as the parent. After crossover and mutation, the same number of offspring is generated as the parent. The cross-mutation process enhances the population diversity and the ability to jump out of local extremum.

The fitness of the N-th population of drosophila can be expressed as $f_i\left(i=1,2\dots N\right)$. The crossover probability q and is shown as follows:

$$q=\{\begin{array}{l}{q}_1-\frac{q_2-q_1}{1+e^{-2A_{con}\frac{Smel{l}_{mid}-Smel{l}_{avg}}{Smel{l}_{\max }-Smel{l}_{avg}}}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Smel{l}_{mid}\ge Smel{l}_{avg}\hfill \\ q_2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}Smel{l}_{mid}<Smel{l}_{avg}\hfill \end{array}$$

(5)

$$\{\begin{array}{l}{X}_i^{*}\left(k\right)=q{X}_i\left(k\right)+\left(1-q\right)X_i\left(k+1\right)\hfill \\ X_i^{*}\left(k+1\right)=q{X}_i\left(k+1\right)+\left(1-q\right)X_i\left(k\right)\hfill \end{array}$$

(6)

$$X_i\left(k\right)=X_0\left(1+\frac{\tan \left(rand\left(0,1\right)-0.5\right)\pi }{2}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1,2,\dots N$$

(7)

$q_1\mathrm{、}{q}_2$ are range of crossover probability, Smell_mid represents the larger fitness value of two cross particles, Smell_max is the maximum fitness in population and Smell_avg is the average value in population.

To maintain a certain crossover probability and obtain more excellent genetic genes, the function of constant A_con is used to slow down the variation trend of crossover probability q when the fitness value is high. This ensures the number of excellent offspring and improves the heritability of dominant genes (5) ~ (7). Therefore, the drosophila population, which is close to the highest quality part of the whole world, can benefit from maintaining a certain crossover probability and acquiring more excellent genetic genes. As shown in Figure 2, the parameter identification process of LuGre modeling using the improved drosophila algorithm is as follows:

(1) The calculation of the determination value of concentration S requires the initial pop ulation parameters, such as population size, the number of iterations (tmax), initial positions (X_pos and Y_ pos), fitness threshold (δ), search radius (Rvalue), mutation probability (q), and search direction.

To enhance both global search efficiency and local search accuracy, the proposed approach incorporates the strategy of dynamically reducing step length. Initially, a larger step length is employed to facilitate a wide range search of the entire population, enabling quick identification of the target area. As the target area is approached, the search step length and speed are gradually decreased to encourage the drosophila particles to explore the vicinity of the optimal solution in depth. Simultaneously, the issue of the population being prone to getting trapped in local minima is addressed through the utilization of an exponential distribution function. The improved formula is as follows:

$$\{\begin{array}{l}{X}_i=X_{\_pos}+L_s\cdot R_{Value}\hfill \\ Y_i=Y_{\_pos}+L_s\cdot R_{Value}\hfill \end{array}$$

(8)

$$L_s=\frac{L_{\max }}{Smellbes{t}_{i-1}}\cdot R_i\cdot a^c$$

(9)

$$c=\frac{t_i+t_{\max }}{t_{\max }}-\cos \frac{t_i}{t_{\max }}$$

(10)

$$R_i\left(x\right)=1-e^{-\alpha x}$$

(11)

Where Smellbest_i-1 is the last generation optimal flavor concentration value, L_max is the maximum step length, α=0.01 is the exponential factor, t_i is the current iterations and R_i is the exponential distribution function that the power exponent is 0.5.

(2) The paragraph calculates the average taste concentration (Smell_avg), and the variance in taste concentration (${\sigma }^2$). A smaller variance indicates a higher concentration and lower diversity within the population.

Figure 2. Iterative foraging process of fruit flies.

Figure 2. Iterative foraging process of fruit flies.

When the current fitness value Smellbest does not reach the target requirement and σ^2≤δ, it can be inferred that the current population is trapped in a local extremum. In this case, if the random number (0~1) is less than the mutation probability q, the optimal positions undergo Cauchy mutation for optimization. Subsequently, the new optimal positions of the drosophila individuals are obtained through secondary optimization of the mutated particles.

(3) Re-estimate the distance between the new position, the origin and the determination value of concentration.

$$Dis{t_j}^{\prime }=\sqrt{X_{posbj}^2+Y_{posbj}^2}$$

(13)

$${S_j}^{\prime }=\frac{1}{Dis{t^{\prime }}_j}$$

(14)

Substituting ${\mathrm{S}}_j^{\mathrm{\text{'}}}$ into taste concentration function and calculating the new taste concentration.

$$Smel{l}_j=function\left({S_j}^{\prime }\right)$$

(15)

If ${Smell}_j^{\mathrm{\text{'}}}<Smellbest$, then ${Smellbest=Smell}_j^{\mathrm{\text{'}}}$，$X_{pos}=X_{posbj}$，$Y_{pos}=Y_{posbj}$. The flowchart shown in Figure 2 represents the iterative process described above, which is repeated until either the maximum number of iterations is reached or the desired level of accuracy is achieved. This process incorporates the adaptive crossover of drosophila, which helps to make the distribution of individuals more realistic during the optimization process. Additionally, the inclusion of a threshold and the implementation of Cauchy mutation serve to enhance the overall diversity of the population, addressing the issue of particles becoming prematurely trapped in local optima.

The advantages of the improved drosophila algorithm are demonstrated through the iterative evolution curve shown in Figure 3 and the parameter identification results displayed in Table 1. Figure 3 reveals that the improved drosophila algorithm exhibits faster convergence speed and higher efficiency. To further highlight these advantages, three traditional algorithms are compared. As demonstrated in Tab.1, the identification algorithm proposed in this paper achieves higher accuracy, with the maximum error being only 0.96%. This method is more suitable for the multi-parameter synchronous identification of the LuGre model compared to traditional iterative algorithms. Finally, the identified LuGre parameters are utilized to calculate friction and fit the friction feed rate curve, as depicted in Figure 4.

Figure 3. Iterative foraging process of fruit flies.

Figure 3. Iterative foraging process of fruit flies.

Figure 4. Relationship diagram of friction and velocity.

Figure 4. Relationship diagram of friction and velocity.

Table 1. Identification results of friction parameters.

Table 1. Identification results of friction parameters.

	Design values	Least square method	Genetic algorithm	Ant colony algorithm	Proposed algorithm
${\mathit{F}}_{\mathit{s}}$ (N)	30.00	30.51	30.63	30.21	30.12
${\mathit{v}}_{\mathit{s}}$ (rad/s)	0.500	0.479	0.485	0.490	0.498
${\mathit{\sigma }}_0$ (Nm/s)	560.0	549.4	554.1	556.7	550.6
${\mathit{\sigma }}_1$ (Nm/s)	300.0	291.0	294.2	303	299.8

3. Friction compensation and control strategy

3.1. Friction feed-forward control design

The problem of error compensated signal has been a major obstacle in the traditional method of controlling the motion accuracy of linear motors through feedback. To overcome this difficulty and improve precision machining, a feed-forward control compensator strategy has been designed. This strategy estimates and adjusts the tracking error caused by friction before it is input into the motor, thus avoiding cycle delay and improving the motion tracking accuracy of a single axis. To illustrate this strategy, a schematic diagram of the X-axis is shown in Figure 5.

Figure 5. Friction compensation model.

Figure 5. Friction compensation model.

The simulation structure diagram of the LuGre model, shown in Figure 6, utilizes the velocity feedback as input to calculate the parameters. To estimate the friction force, the average elastic deformation is obtained through simulation since the bristle z, being an intermediate state variable, cannot be accurately determined. The position tracking accuracy control performance of the linear motor is improved by superimposing the tracking error compensation, adjusted by feedforward control, with the theoretical data. Furthermore, Figure 7 illustrates the simulation block diagram of the friction feedforward controller.

Shown in Figure 8, the comparison of simulated results for friction compensation reveals that the feedforward compensation strategy effectively mitigates friction disturbance during linear motion. The maximum friction is reduced by over 50%, and the fluctuation of friction compensation remains stable. Consequently, it is evident that this control strategy enhances the tracking performance of linear motors and improves the contour accuracy of precision machining.

4. Contour error modeling and controlling

4.3. Design of cross-coupled control strategy

In this section, a three-axis cross-coupled controller is designed to adjust and control the estimated contour error. The central controller, represented by ${\mathit{K}}_{\mathit{c}\mathit{c}}$ and consisting of a Fuzzy PID controller, plays a crucial role in the entire CCC system as it determines the accuracy of compensation. K_ci and G_i（i=x, y, z）are the position controller and controlled object respectively. The estimated contour error, as shown in Figure 12, is input into the central controller, which generates an error control signal by coupling it with the parameters of each axis. The compensation of the contour error is then decoupled to each axis in proportion and superimposed with the theoretical point ${\mathit{R}}_{\mathit{d}}=\left[R_{dx},R_{dy},R_{dz}\right]$. This process enables the prediction and compensation of the contour error. The actual position ${\mathit{R}}_{\mathit{a}}=\left[R_{ax},R_{ay},R_{az}\right]$, meanwhile, serves as a reference for comparison. To facilitate the controller design and ensure consistency, the motion cycle parameters of each axis are set to the same value, $T_x=T_y=T_z$.

Firstly, it is assumed that the contour error is $∆\mathit{\epsilon }$ when cross-coupled controller is not added, that is to say ${\mathit{K}}_{\mathit{c}\mathit{c}}=0$, as known in Figure 12, where K_ci and G_i（i=x, y, z）are position controller and the controlled object. When the CCC is adopted, the contour error can be expressed as

$$\Delta \epsilon \hspace{0.17em}=\frac{1}{1+K_{cx}{G}_x}{C}_x{R}_{dx}+\frac{1}{1+K_{cy}{G}_y}{C}_y{R}_{dy}+\frac{1}{1+K_{cz}{G}_z}{C}_z{R}_{dz}$$

(31)

$$\epsilon =\frac{\Delta \epsilon }{1+\left(\frac{1}{1+K_{cx}{G}_x}{C}_x^2{G}_x+\frac{1}{1+K_{cy}{G}_x}{C}_y^2{G}_y+\frac{1}{1+K_{cz}{G}_z}{C}_z^2{G}_z\right)K_{cc}}$$

(32)

Figure 12. Central controller of cross-coupled controller.

Figure 12. Central controller of cross-coupled controller.

Figure 13. B-spline complex spline path.

Figure 13. B-spline complex spline path.

In actual machining, the main factor that affects contour accuracy is the mismatch of parameters in multi-axis. This mismatch leads to an uneven distribution of contour error throughout the whole trajectory. To address this issue, a pre-compensation strategy is proposed. The strategy takes advantage of the curvature variation characteristics of the spline path to reduce the contour error. Additionally, it considers the inertia matching and coordination of the parameters in each axis to improve the machining contour accuracy. A complex B-spline path of simulation, with a total length of 742 mm and a feed rate of 1400mm/min, is shown in Figure 13. As shown in Figure 14, the maximum tracking error of the x-axis and y-axis is reduced by 47.01% and 47.02% respectively, while the contour error is reduced by 46.5%. Consequently, the proposed strategy significantly improves the contour error.

Figure 14. Error contrastive results of pre-compensation of CCC.

Figure 14. Error contrastive results of pre-compensation of CCC.

Figure 15. Machining experiment of spherical shell.

Figure 15. Machining experiment of spherical shell.

5. Machining experiment verification

Figure 16. Compensation result of tracking error.

Figure 16. Compensation result of tracking error.

Figure 17. Compensation result of contour error

Figure 17. Compensation result of contour error

6. Conclusions

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