Error calibration of pitching mechanism motion based on improved ant colony algorithm

Abstract

A method for calibrating the motion error of the pitching mechanism based on an improved ant colony algorithm is proposed in this article. First, the error model for the pitching mechanism is built based on the three main error sources that affect the accuracy of the pitching mechanism, and the influence of these three error sources on the pitch motion accuracy is analyzed using the control variable method. Second, a mathematical model for the calibration of the pitch mechanism motion error is established, which transforms the error calibration problem into an optimization problem with multiple objective functions, and the error is then analyzed using an improved ant colony algorithm model. Compared with traditional numerical methods and ant colony optimization, it is proven that the algorithm has strong global optimization ability and can effectively avoid the impact of the initial value error source on the calculation results, and the accuracy can reach a level of 10⁻⁵ mm. Finally, the simulation results show that the proposed method is efficient and practical.

Keywords

Error calibration pitching mechanism simulation ant colony algorithm

Introduction

When designing the pitching mechanism for the separation of a wind tunnel body,¹ a linear variable arc mechanism is proposed to replace the rotating pair in order to avoid movement inaccuracies of the mechanism and capture the test trajectory so that the pitching motion of the separation body can be realized. The mechanism can be positioned within the wind tunnel flow field and can effectively avoid the cumulative error caused by rotation of the rotating pair. The length error of the mechanism arm is reduced, and the kinematic accuracy of the mechanism is extended. The mechanism has good kinematic characteristics and a fast response and it can significantly improve the ability of the wind tunnel to capture the trajectory.

Due to limitations in the structural space, errors caused by manufacturing and installation cannot be accurately identified. Depending on the measurement conditions that exist, the accuracy of the calibration algorithm will depend on the error model that is used, as well as the measurement and calculation accuracies.² Most of the current calibration models are based on the parallel mechanism with multiple degrees of freedom.^3–5

Ruaf and Ryu⁶ constructed an error model for the mechanical structure based on the mechanical constraints of the movement of the mechanical device. Liu et al.⁷ proposed a type of method that uses the perturbation method to establish an error model for the parallel mechanism and simulated and analyzed the calibration method for the inverse solution of the mechanism. Chen et al.⁸ synthesized a new type of error mathematical model for the parallel mechanism based on the inverse kinematics of the mechanisms.

Pei et al.⁹ used tricept to detect redundant sensor data, which is an effective method to improve the accuracy of the self-calibration. Chanala et al.¹⁰ used the external geometric calibration method to decrease the influence of the error on the motion precision processing and manufacturing. Sun¹¹ studied the virtual axis mechanism and obtained an equation based on the position vector of the driving rod length, the static platform, and the center of the six hinges of a moving platform with an unknown quantity. This equation was used to establish a simple model, but the selection of the initial value is too large which leads to low calculation precision. Kong et al.¹² built a visual space target positioning algorithm in an active binocular visual monitoring platform to improve the motion precision. Dany et al.¹³ studied an interval arithmetic method for the mathematical parallel mechanism and proved its validity. Xiao^14,15 proposes a hybrid optimization algorithm HACO (a Hybrid Ant continuous ant colony Colony, Optimization) combined with differential evolution and population growth of learning to generate dynamic Gauss probability density function based on slow fall into local optimum, improves the optimization performance of the algorithm. These improved algorithms can improve the searching ability of ACOR algorithm to a certain extent, but there is still room for improvement. Generally, regardless of the method used to calibrate the mechanism, the position and attitude error at the end of the mechanism must be obtained, and then a corresponding calibration model must be used to analyze the parameters of the mechanism.

For this pitching mechanism, the accuracy of the three error sources which affect the pitching movement process was studied and a relatively complete model calibration error is proposed using a closed vector method. The calibration model for the nonlinear transcendental equations of the multi-objective problem is transformed using an appropriate method into a weighted scalar problem. An improved ant colony algorithm is proposed to solve the multi-objective optimization problem.

Mathematical model of error of pitching mechanism

The main components of a pitching mechanism include a drive screw servo motor, a linear guide rail slider, a lead screw nut, an arc guide rail slider, a drive connecting rod, and a set of linear slide blocks. In order to verify the analytic results and find a solution, a simulation mechanism for this pitching mechanism with relevant coordinate systems is constructed by CAD variation geometry (see Figure 1).

Figure 1.

Model of pitching movement mechanism.

During movement of the pitching mechanism, rotation of the servo motor drive screw moves the nut in a linear motion. The ball screw nut is connected to the drive connecting rod which moves the linear slider along a straight line on the guide rail, which is connected to the driving rod which is installed on arc sliding block machetes. This causes a roll mechanism along the arc guide which rolls the tail end of the strut toward the center of the circle turn, eventually making stores (machine tools) around a Z-axis rotation, where the system origin is the center of the coordinate system and angle $α$ is the pitching angle.

The greatest influencers of the movement accuracy are the main geometrical parameters, which are the arc radius of the guide rail R, the length of the connecting rod L, and the location of the linear guide installation $y_{Oa}$ , and the three major errors affecting the pitching mechanism motion are set as $Δ R$ , $Δ L$ , and $Δ y_{Oa}$ , respectively.

A kinematic sketch of the pitching mechanism is shown in Figure 2. A and B are the positions of the location of the linear slide block and the arc sliding block when the pitching angle $α = 0 .$ . $AB = L$ is the length of the connecting rod and the position of $A'$ and $B'$ changes depending on the size of the pitching angle $α$ . The arc guide center of rotation is set at the origin O and an absolute coordinate system is established as Oxy. The local coordinate system $O ξ η$ is established at the origin of A, and the Oxy coordinates of A in the absolute coordinate system are $(x_{Oa}, y_{Oa} + y_{Oa})$ . Let S be the displacement of the linear slide block, $R + Δ R$ is the arc of the guide rail radius, and $L + Δ L$ is the length of the connecting rod. The angle $ϕ$ is the angle between OB and the y-axis, the angle $α$ is the pitching angle between OB and $OB'$ , $(S, 0)$ is the coordinates of $A'$ in the local coordinate system $O ξ η$ , and the Oxy coordinates of $A'$ in the absolute coordinate system can be expressed as follows

\vec{OA'} = T \vec{A {A'}_{O'}} + \vec{OA}

(1)

where T is a rotational transformation matrix from ${O ξ η}$ to ${Oxy}$ and leads to

T = [\begin{matrix} \cos ϕ & - \sin ϕ \\ \sin ϕ & \cos ϕ \end{matrix}]

Figure 2.

Diagrams of the pitching mechanism.

As shown in Figure 2, the angle $ϕ$ between the absolute coordinate system Oxy and the local coordinate system $O ξ η$ is 0, T is

T = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

the coordinates of $A'$ in the absolute coordinate system Oxy can be taken as $(x_{Oa} + S, y_{Oa} + Δ y_{Oa})$ , $((R + Δ R) \sin ϕ, (R + Δ R) \cos ϕ)$ are the coordinates of B in the absolute coordinate system Oxy, and the coordinates of the $B'$ in the absolute coordinate system Oxy are $((R + Δ R) \sin (ϕ + α), (R + Δ R) \cos ϕ)$ . The vector equations are compiled using the closed vector method¹⁶ as

\begin{matrix} \vec{OA} + \vec{AB} = \vec{OB} \\ \vec{OA'} + \vec{A' B'} = \vec{OB'} \end{matrix}}

(2)

The relationship between the pitching angle and the linear sliding block displacement is solved

\begin{matrix} {(L + Δ L)}^{2} = {[(R + Δ R) \sin ϕ - x_{Oa}]}^{2} + {[(R + Δ R) \cos ϕ - (y_{Oa} + Δ y_{Oa})]}^{2} \\ {(L + Δ L)}^{2} = {[(R + Δ R) \sin (ϕ + α) - (x_{Oa} + S)]}^{2} + {[(R + Δ R) \cos (ϕ + α) - (y_{Oa} + Δ y_{Oa})]}^{2} \end{matrix}}

(3)

\begin{matrix} x_{Oa} = (R + Δ R) \sin ϕ - \sqrt{{(L + Δ L)}^{2} - {[(R + Δ R) \cos ϕ - y_{Oa}]}^{2}} \\ α = \arcsin \frac{{(R + Δ R)}^{2} + {(y_{Oa} + Δ y_{Oa})}^{2} + {(x_{Oa} + S)}^{2} - {(L + Δ L)}^{2}}{2 (R + Δ R) \sqrt{{(x_{Oa} + S)}^{2} + {(y_{Oa} + Δ y_{Oa})}^{2}}} - \arcsin \frac{(y_{Oa} + Δ y_{Oa})}{\sqrt{{(x_{Oa} + S)}^{2} + {(y_{Oa} + Δ y_{Oa})}^{2}}} - ϕ \end{matrix}}

(4)

Expression (4) provides a model for the motion error of the pitching mechanism which is related to the three error sources of R, L, and $y_{Oa}$ . The control variable method can then be used to analyze the influence of each error source on the accuracy of the pitching angle $α$ .

The pitching angle error can be obtained as shown in Figure 3 based on the actual pitch motion. In this example, the arc guiding radius error is $Δ R = 0.5 mm$ , the length error of the connecting rod is $Δ L = 0.5 mm$ , and the installation position error of the linear guide is $Δ y_{Oa} = 0.5 mm$ .

Figure 3.

Error of the movement of the pitching angle: (a) $Δ R = 0.5 mm$ , (b) $Δ L = 0.5 mm$ , and (c) $Δ y_{Oa} = 0.5 mm$ .

The results from the simulation and analysis show that the three error sources all have a strong influence on the movement regulation of the pitching mechanism. Therefore, an effective method needs to be adopted to reduce their influence on the mechanism in order to improve the working accuracy requirements for practical engineering applications.

Error calibration and solution

The error calibration model for the pitching mechanism is a function of the measurement parameters and the error model based on the visible error sources, since each error source is identified using corresponding mathematical methods. The identified result is used to modify the control program of the kinematics model parameter, in order to reduce the influence of the original error on the accuracy of the final pitching mechanism.

There are three error sources existing for the pitch motion error model. Since the measurement of a set of data requires at least one kinematic equation, it is necessary to have at least three measurements to determine the value of each error source. However, there are usually many locations selected to measure the effect of noise on the calibration results in practical engineering applications. The mathematical model for the error calibration of the pitching mechanism is established as follows

\begin{matrix} {(L + Δ L)}^{2} = & {[(R + Δ R) \sin (ϕ + α_{i}) - (x_{Oa} + S_{i})]}^{2} \\ + {[(R + Δ R) \cos (ϕ + α_{i}) - (y_{Oa} + Δ y_{Oa})]}^{2} \\ (i = 1, 2, \dots, n) \end{matrix}

(5)

Equation (5) is a nonlinear transcendental equation which includes the measurement information, driving information, geometric parameters, and the error sources in the position and attitude of the pitching. The error parameter identification for the pitching mechanism can be attributed to a set of nonlinear transcendental equations, which can be transformed into an optimization problem of the nonlinear model parameters. An improved ant colony algorithm is used to solve the nonlinear transcendental equation and equation (5) is simplified as

\begin{matrix} {[(R + Δ R) \sin (ϕ + α_{i}) - (x_{Oa} + S_{i})]}^{2} \\ + {[(R + Δ R) \cos (ϕ + α_{i}) - (y_{Oa} + Δ y_{Oa})]}^{2} \\ - {(L + Δ L)}^{2} = 0 \\ (i = 1, 2, \dots, n) \end{matrix}

(6)

The nonlinear transcendental equations are transformed to a minimum value to solve the multi-objective optimization problem

\begin{matrix} F_{i} (dR, dL, d y_{Oa}) = {[(R + Δ R) \sin (ϕ + α_{i}) - (x_{Oa} + S_{i})]}^{2} \\ + {[(R + Δ R) \cos (ϕ + α_{i}) - (y_{Oa} + Δ y_{Oa})]}^{2} \\ - {(L + Δ L)}^{2} (i = 1, 2, \dots, n) \end{matrix}

(7)

The power and law method can be used here to transform the multi-objective optimization problem into a scalar problem of weighted targets

min f (x) = \sum_{i = 1}^{m} ω_{i} \cdot F_{i} {(x)}^{2} (i = \begin{matrix} 1 & 2 & \begin{matrix} \dots & n \end{matrix} \end{matrix})

(8)

where $ω_{i}$ is a weighted coefficient.

The mathematical optimization model for the error parameter identification of the pitching mechanism can be derived as

\begin{array}{l} f (d R, d L, d y_{O a}) = \sum_{i = 1}^{3} ω_{i} \cdot F_{i} {(d R, d L, d y_{O a})}^{2} \\ (i = \begin{matrix} 1 & 2 & \dots & n \end{matrix}) \end{array}

(9)

The weight coefficients of each sub-objective function are equal and so can be easily calculated. Set $ω_{i} = 1$ .

Basic ant colony algorithm

Based on a study of the optimization process used by ant colonies to find food, the famous Italian scholars Colornia et al.¹⁷ proposed a type of intelligent bionic algorithm in 1992 which was the basic ant colony algorithm mathematical model for solving discrete problems, such as the TSP traveling salesman problem, assignment problem, and job scheduling problem. The method is characterized by the use of positive feedback and distributed cooperation to find the optimal path. The basic ant colony algorithm is composed of a pheromone updating rule and a state transition rule. For instance, the movement probability for a single ant k moving from position i to position j at time t can be expressed as

p_{i j}^{k} = {\begin{matrix} τ_{i j}^{α} (t) η_{i j}^{β} (t) / \underset{R \in λ_{k}}{Σ} τ_{i R}^{α} (t) η_{i R}^{β} (t) & (j \in λ_{k}) \\ 0 & (o t h e r) \end{matrix}

(10)

where $τ_{ij}$ is the pheromone strength between i and j, $η_{ij}$ is the visibility factor which remains unchanged during the moving process, $λ_{k}$ is the next movement position of ant k, and $α$ and $β$ are the importance of the information and the heuristic information about the ants’ walking paths when the ants are walking.

Each time the ant completes a cycle of movement, the intensity of the pheromone at a given time is changed and the path information of each ant is updated as

τ_{ij} (t + 1) = κ τ_{ij} (t) + Δ τ_{ij} (t, t + 1)

(11)

Δ τ_{ij} (t, t + 1) = \sum_{k = 1}^{m} Δ τ_{ij}^{k} (t, t + 1)

(12)

where $Δ τ_{ij} (t, t + 1)$ is the amount of information at the $(i, j)$ positions of the loop, $Δ τ_{ij} (t, t + 1)$ is the amount of pheromone of ant k at the $(i, j)$ position at time $(t, t + 1)$ whose value depends on the superiority of the ants, and $κ$ is the residual pheromone concentration, which is set at $κ < 1$ .

Improved ant colony algorithm

The basic ant colony algorithm has some problems due to a local optimum and a slower speed of convergence. Since the optimization problem for the objective function is a continuous domain optimization problem, a new continuous domain for the ant colony algorithm is proposed to solve this problem,¹⁸ which can be used for continuous space discretization before using the ant colony algorithm to solve the problem. Dero and Siam¹⁹ proposed a continuous domain which can be used for solving the ant colony algorithm which was the first method to use the pheromones of transfers and direct communication in two ways to guide the ant colony optimization^20–22 and also synthesized several ant colony algorithms for the continuous domain. Based on previous studies, this article proposes a relatively simple improved ant colony algorithm. The basic scheme for the improved ant colony algorithm can be expressed in the following.

First, using the machining and installation precision of the pitching mechanism, the constraint space for the three error sources can be set as $d R_{min} < dR < d R_{max}$ , $d L_{min} < dL < d L_{max}$ , $d {y_{Oa}}_{min} < d y_{Oa} < d {y_{Oa}}_{max}$ . Second, the initial positions of the ants in the feasible region are obtained using a random distribution method and the pheromone size of the ants in their initial locations is calculated and the best value from the record is selected in the process. A global search and a local search for the ants are performed based on the transition probabilities. Each movement is completed when a pheromone value gets an update, which is repeated until the optimal solution is obtained. The feasible domain of each variable is $d R_{min} < dR < d R_{max}$ , $d L_{min} < dL < d L_{max}$ , $d {y_{Oa}}_{min} < d y_{Oa} < d {y_{Oa}}_{max}$ and the initial positions of the ant colony are determined by random distribution.

The steps of the algorithm are proposed as follows:

1. Determine the initial position of the ant colony by random distribution within the feasible region of the variable.

2. Determine the size of the pheromones based on the objective function

τ = - f (α, β, z)

(13)

The minimum value of the objective function needs to be solved since the monotonicity of the minimum value is the opposite of the objective function, which is the maximum pheromone value. When $f (dR, dL, d y_{Oa})$ is close to 0, the pheromone value is negative and also close to 0, resulting in a concentration of pheromones that is difficult to assess. Therefore, the average value of $τ_{mean}$ is used to measure the value of the information

τ_{mean} = mean (- f (α, β, z))

(14)

where mean is the average value of the function.

3. Determine the size of the pheromone value when the ant colony is in its initial positions, record the optimal value $τ_{best}$ , and calculate the average value $τ_{mean}$ .

4. Perform global searches and local searches to try to locate the ants. The search is mainly performed using the transition probability of the ant colony as shown in equation (15)

P_{i, j} = \frac{| τ_{i, j} | - | τ_{best} |}{| τ_{i, j} |}

(15)

where i is the ith movement of the ant colony, j is the jth ant, $τ_{best}$ is the ant pheromone optimal value after the last movement, and $τ_{i, j}$ is the pheromone value of ant j after movement of the ant i times.

5. Determine the pheromone value of each ant after each search.

If $τ_{i, j}^{new} > τ_{i, j}^{old}$ , update the pheromone values of the ants; if $τ_{i, j}^{new} < τ_{i, j}^{old}$ , the pheromone values of the ants remain the same.

6. Update the pheromone value of each ant each time a movement is completed

τ_{i + 1, j} = κ \cdot τ_{i, j}^{old} + τ_{i, j}^{new}

(16)

7. Repeat the iterative calculation by returning to step 4, unless the number of iterations is greater than the maximum number of cycles and then end the cycle and output the final results.

Example simulation

In order to verify that the improved ant colony algorithm can meet the accuracy requirements for the calibration of the pitching mechanism, a simulation experiment for the pitching mechanism was performed in this article. According to wind tunnel test requirements, a 3D model of the actual pitching mechanism is shown in Figure 4.

Figure 4.

View of the pitching mechanism.

The accuracy of the mechanical machining and assembly are used to give the range of accuracy requirements for the three error sources as $- 1 mm < dR < 1 mm$ , $- 1 mm < dL < 1 mm$ , and $- 1 mm < d y_{Oa} < 1 mm$ . The radius error of the arc guide is set as $Δ R = 0.8 mm$ , the length of the connecting rod is set as $Δ L = - 0.4 mm$ , and the installation error of the linear guide rail is set as $Δ y_{Oa} = - 0.7 mm$ to verify the effectiveness of the improved ant colony algorithm for the calibration of the pitching mechanism. The angles are set at [−25°, −20°, −15°, −10°, −5°, 0, 5°, 10°, 15°, 20°, 25°] successively and the corresponding linear slider displacement S is obtained for each value. The error calibration of the pitching mechanism can be transformed into a minimum value in the cube (see Figure 5). This can be observed more easily by opening the cube map and removing the color, as shown in Figure 6.

Figure 5.

Range of the objective function.

Figure 6.

Initial position distributions of ants.

This displays the random distributions of the ant colonies. Using the improved ant colony algorithm, all the ants gather at the minimum value of the objective function, which is not affected by a local minimum, as shown in Figures 7 and 8.

Figure 7.

Ants’ position distribution after moving 200 times based.

Figure 8.

Ants’ position distribution after moving 400 times.

As shown in Figure 8 for the ants movement after 400 times, all of the ants are concentrated in a point location; the convergence effect is very good. The problem is to be solved by the basic ant colony algorithm in the same conditions, as shown in Figure 9, the ant movement after 400 times is still relatively scattered, they did not achieve the convergence effect, it means that the ants continue to move in order to meet the requirements of precision. When every ant movements 800 times, eventually all the ants gathered at one point, as shown in Figure 10.

Figure 9.

Ants’ position distribution after moving 400 times based ACO.

Figure 10.

Ants’ position distribution after moving 800 times based ACO.

Using relative analytic equations and MATLAB, the three error values of the pitching mechanism are obtained: the radius error of the arc guide $Δ R = 0.8017 mm$ , the length error of the connecting rod $Δ L = - 0.3986 mm$ , and the position error of the linear guide rail $Δ y_{Oa} = - 0.6997 mm$ . The convergence rate of the algorithm can be seen from the change in the pheromone values and the average pheromone values (see Figure 11).

Figure 11.

The maximum value of pheromone and the average value of pheromone.

Compared with the initial values, the precision of the errors produced after calibration is 0.0017, 0.0014, and 0.0003 mm. Thus, this method to calibrate the pitching mechanism will not be affected by the initial values, and the calculation accuracy can meet the requirements of the wind tunnel test.

Comparing Tables 1 –3, calculation results easily affected by the initial impact of the traditional Newton–Raphson method. The initial value of the different calculation results will make a substantial deviation. The improved ant colony algorithm is proposed in this article can effectively avoid the initial value impact on the calculation results and obtain a more accurate computational solution.

Table 1.

Results of improved ant colony algorithm.

$Δ R$ (mm)			$Δ L$ (mm)			$Δ y_{a}$ (mm)
Target value	Calculated value	Absoluted error	Target value	Calculated value	Absoluted error	Target value	Calculated value	Absoluted error
−1	−1.00001	1 × 10⁻⁵	−1	−1.00002	2 × 10⁻⁵	−1	−0.99985	1.5 × 10⁻⁵
−0.5	−0.50003	6 × 10⁻⁵	−0.5	−0.49985	3 × 10⁻⁴	−0.5	−0.50002	4 × 10⁻⁵
0	0.00003	3 × 10⁻⁵	0	0.00004	4 × 10⁻⁵	0	0.00000	0
0.5	0.50000	0	0.5	0.49986	2.8 × 10⁻⁴	0.5	0.50001	2 × 10⁻⁵
1	0.99998	2 × 10⁻⁵	1	1.00004	4 × 10⁻⁵	1	0.99999	1 × 10⁻⁵

Table 2.

Results of Newton–Raphson method.

Initial values (mm)			Calculated value (mm)
$Δ R$	$Δ L$	$Δ y_{a}$	Calculated value	Absoluted error	Calculated value	Absoluted error	Calculated value (mm)	Absoluted error
−1	−1	−1	−1.1134	0.1134	−1.3132	0.3132	−1.538	0.538
−0.5	−0.5	−0.5	−0.4427	0.1146	−0.6290	0.258	−0.725	0.45
0	0	0	0.1432	0.1432	0.1562	0.1562	0.2032	0.2032
0.5	0.5	0.5	0.4425	0.115	0.6342	0.1342	0.4286	0.1428
1	1	1	0.8892	0.1202	1.252	0.252	0.352	0.352

Table 3.

Results of basic ACO method.

Initial values (mm)			Calculated value (mm)
$Δ R$	$Δ L$	$Δ y_{a}$	Calculated value	Absoluted error	Calculated value	Absoluted error	Calculated value (mm)	Absoluted error
−1	−1	−1	−1.0933	0.0933	−1.0835	0.0835	−1.0738	0.0738
−0.5	−0.5	−0.5	−0.4635	0.0730	−0.5795	0.1590	−0.5825	0.1650
0	0	0	0.0834	0.0834	0.0962	0.0962	0.0825	0.0825
0.5	0.5	0.5	0.4650	0.0700	0.5758	0.1516	0.4895	0.0210
1	1	1	0.9265	0.0735	1.0572	0.0572	0.0265	0.0735

Finally, the error source is identified by introducing the error source into the kinematic equation, and the pitching angle error calibrated is shown in Figure 12. Compared with Figure 13, the error of the error of the pitching angle is obviously reduced, and the aim of reducing the influence of the original error on the motion accuracy of the mechanism is achieved.

Figure 12.

Error of the pitching angle after calibration.

Figure 13.

Error of the pitching angle before calibration.

Conclusion

The three principal initial error sources which affect the motion precision of the pitching mechanism are analyzed. These initial error sources prevent the actual mechanism from behaving according to the theory of motion laws. A closed vector method was first used to establish the error mathematical model of the pitching mechanism in order to solve the effects of the three initial error source on the pitching motion. A control variable method is used to analyze the influence of the three initial error sources on the motion accuracy of the mechanism. An error parameter identification model based on an improved ant colony algorithm is then established. Finally, an improved ant colony algorithm is proposed to transform the nonlinear problem of the calibration model of the pitching mechanism into a multi-objective function to find the optimal solution. And compared with results which utilize traditional Newton–Raphson iterative method and ACO, the motion accuracy of improved ant colony algorithm was higher, and the accuracy can reach a level of 10⁻⁵ mm. The improved ant colony algorithm is used to identify the error parameters, and the recognition results are input into the kinematic formula to correct the influence of the initial error on the accuracy of the mechanism. The method can be implemented for error calibration of other complex parallel mechanisms, and thus has a universal characteristic.

Footnotes

Academic Editor: Ramoshweu Lebelo

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant no. U1530138).

References

Special test technology for wind tunnel. Beijing, China: Aviation Industry Press, 2010.

Pei

Luo

Chen

Calibration of a 6UPS parallel machine by vector method. China Mech Eng 2006; 17: 854–857.

Vischer

Clavel

Kinematic calibration of the parallel Delta robot. Robotica 1998; 16: 207–218.

Masory

Wang

Zhuang

Kinematic modeling and calibration of a Stewart platform. Adv Robotics 1996; 11: 519–539.

Tsai

Lenz

RK.

A new technique for fully autonomous and efficient 3D robotics hand/eye calibration. IEEE T Robotic Autom 1989; 5: 345–358.

Rauf

Ryu

. Fully autonomous calibration of parallel manipulators by imposing position constraint. In: Proceedings of international conference on robotics and automation, Seoul, Korea, 21–26 May 2001, pp.2389–2394. New York: IEEE.

Liu

Gong

Zhao

Error analysis and calibration of a 4-DOF parallel mechanism. Robot 2005; 27: 6–9.

Chen

Zhao

Deng

Error modeling and simulation of a novel parallel robot coordinate measuring machine. Comput Integr Manuf 2008; 14: 477–481.

Pei

Chen

Wang

Calibration and analysis of the joints position and clearance error on 6UPS parallel machine. Mach Des Res 2006; 22: 125–132.

10.

Chanala

Duca

Raya

. A new approach for the geometrical calibration of parallel kinematics machines tools based on the machining of a dedicated part. Int J Mach Tool Manu 2007; 47: 1151–1163.

11.

Sun

Research on error analysis and precision guarantee of virtual axis mechanism. Beijing, China: Tsinghua University, 2001.

12.

Kong

Wang

Zhao

Space target positioning based on binocular active vision system of parallel robot. Comput Integr Manuf 2007; 13: 2284–2288.

13.

Daney

Andreff

Chabert

Interval method for calibration of parallel robots: vision-based experiments. Mech Mach Theory 2006; 8: 929–944.

14.

Zhou

Yuan

. Artificial bee colony based ant colony optimization for continuous domains. Comput Eng Sci 2016; 38: 1156–1163.

15.

Jing

Liang-ping

A hybrid ant colony optimization for continuous domains. Expert Syst Appl 2011; 38: 11072–11077.

16.

Kumar

Waldron

KJ.

Force distribution in closed kinematic chains. IEEE Robot Autom Mag 1988; 4: 657–664.

17.

Shen

Liu

Huang

Fast ant colony algorithm for solving traveling salesman problem (Engineering and Technology Edition). J Jilin Univ 2013; 43: 147–151.

18.

Gao

Yang

Swarm intelligence algorithm and its application. Beijing, China: China Water Power Press, 2006.

19.

Dero

Siam

Continuous interacting ant colony algorithm based on dense heterarchy. Future Gener Comp Sy 2004; 20: 84l–856.

20.

TJ.

An adaptive ant colony system algorithm for continuous space optimization problems. J Zhejiang Univ: Sc A 2003; 4: 40–46.

21.

Karaboga

Kalinli

Karaboga

Designing digital IIR filters using ant colony optimization algorithm. Eng Appl Artif Intel 2004; 17: 301–309.

22.

Socha

Dorigo

Ant colony optimization for continuous domains. Eur J Oper Res 2008; 185: 1155–1173.