1 Introduction

Recent applications of robotics in the medical field (computer-assisted surgical operations, remote control) require precision in the execution of increasingly advanced maneuvers, particularly in heavy procedures (neurosurgery, cardiac surgery, etc.). Performance specifications in robotic applications are becoming more and more severe, and increasingly accurate robots are required. However, one must first define precision for robotic systems. Several aspects are to be considered: material aspect (assembly, choice of materials),—control aspect (control strategy, quality of the corrector),—metrological aspect (sensor resolution, accuracy of measurement, tolerance of components robot) [1,2,3,4].

To improve accuracy (i.e., reduce laying errors), it is necessary to evaluate the "true" values of the geometric parameters of the robot and to insert them later into the equations of inverse geometric model (MGI). This operation is called the geometric calibration (EG) of the robots. Since it is not always possible to perform direct measurements to obtain the actual values of the parameters, the calibration approaches proposed in the literature are based mainly on optimization models. The values of the parameters identified by these calibration methods do not necessarily correspond to the true values of the robot parameters. Instead, they represent the values that satisfy the objective functions of the optimization models used in the calibration procedure.

These functions generally consist of minimizing the residual errors of poses or joint movements.

Numerous research studies have focused on the correction of the error on the accuracy and the repeatability. We also mention the work on the online correction of the positioning defects. The kinematic chain faults of an industrial robot are nevertheless, largely responsible for the degradation of positioning accuracy in static and dynamics its working volume. It depends on the play and mechanical inaccuracies [9].

The design techniques of experiments (DOE) allow designers to simultaneously determine the individual and interaction effects of many factors that could affect the output results in any design.

The DOE also provides a comprehensive overview of the interaction between design elements; therefore, it helps to transform any standard model into a robust framework. Simply, DOE helps to spot sensitive parts and sensitive factors in drawings that cause problems. Designers are then able to solve these problems and produce robust, higher-performance models before the production and manufacture of robots.

In terms of fracture, Ep error is a parameter characterizing the significant increase in position and rotation error of the robot, which is what we tried to develop in this study [11, 12], we use the method DOE to see the interaction between factors (uncertainties on DH parameters) and their influence on the response (EP position accuracy error).

A detailed organization chart that illustrates Operational process in Fig. 1.

Fig. 1
figure 1

Operational process

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2 Literature survey

Industry today is heavily dependent on robots for a wide range of applications such as in the medical field, assembly and welding of materials. Robotic machining applications. However, improving the accuracy and repeatability of materials to meet desired process parameters is still the primary goal. For this, we have chosen to implement two methods studied separately. First of all, a calibration of the geometric parameters of its piloting model makes it possible to improve the pose precision of the robot. Then a mirror correction method [5] corrects its trajectory from the error determined between the measured and programmed trajectories (Fig. 2).

Fig. 2
figure 2

The parameters D–H

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According to the sources of imprecision to be compensated, Elatta et al. (2004) present two main families of robot calibration, namely geometric calibration (GC), intended to improve the kinematic model of the robot, and non-geometric calibration, which aims to process errors of non-geometric origin. As for articular errors, they are usually included in the GC. Three levels of calibration are presented and discussed in [6]. Level 1 is joint-level calibration, Level 2 is full robot kinematic model calibration, and Level 3 is non-kinematic (non-geometric) calibration. In this work, the limits of the improvement of the precision of the robot are attributed to the limits of repeatability and precision of the robot and to the precision of the measurement system.

Sid-Ahmed et al. [7], in this work the main objective is to optimize the position and orientation error of the platform of a plane parallel manipulator robot (3RPR) using the direct geometric model. The main disadvantage of parallel manipulators is the existence of singularities within its workspace, methods are used to determine the optimal solution. The neuro-fuzzy adaptive solution is proposed in this study to find the optimal solution.

Chunche et al. [8], to achieve the stringent accuracy of some robotic applications such as robotic measurement systems, it is essential to compensate for non-geometric errors and thermal errors in addition to geometric errors. In this work article studies the effect of geometric errors, links and temperature variations on the positioning accuracy of the robot. A general methodology is developed to identify these errors simultaneously. A laser tracker is applied to calibrate these errors by an inverse calibration method. The geometric errors of the robot are calibrated at room temperature while the thermal errors of the robot parameters are calibrated at different temperatures as the robot warms up and cools down. Empirical thermal error models are established. These models can be integrated into the controller and used to compensate for thermal errors due to heat sources. The results show that the accuracy is improved after calibration [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].

This work focus on the concepts of accuracy and repeatability. A literature review on the topic is presented; the concepts of accuracy and repeatability are developed with the both contexts: homogeneous and differential transformation.

The main goal of this paper is evaluation and discussion of the DH parameters influence on the accuracy and repeatability of robots, using the techniques of experiments (DOE) methodology. To validate this study, a program is developed using the symbolic calculation tools, allowing the calculation of errors in the position of the robot starting from given uncertainties. And numerical example using published values for the PUMA 560 robot focused on the aforementioned concepts is presented and discussed.

3 Accuracy analysis

The manufacturers of industrial robots generally provide the values of the resolutions and of the repeatability measured according to the experimental protocol described in the international standard ISO 9283 [1]. It can be considered as an empirical rule that robots of large dimensions exhibit high errors of positioning repeatability.

3.1 Kinematic analysis

The kinematic structure of the manipulator robots is mathematically described by a representation Compact of the position and orientation of a given joint with respect to the articulation that precedes it. The most used is the modified Denavit–Hartenberg notation which allows to model any robotic structure [10,11,12,13]

The parameters D-H are grouped in a 4 × 4 matrix called a homogeneous transformation matrix \({}_{i}^{i - 1} T\), which describes the position and the orientation of the reference point linked to the joint (i) relative to the reference point linked to the articulation (i-1). It is given by Eq. (1).

$${}_{i}^{i - 1} T = \left[ {\begin{array}{*{20}c} {\cos (\theta_{i} )} & { - \sin (\theta_{i} )} & 0 & {a_{i - 1} } \\ {\sin (\theta_{i} )\cos (\alpha_{i - 1} )} & {\cos (\theta_{i} )\cos (\alpha_{i - 1} )} & { - \sin (\alpha_{i - 1} )} & { - d_{i} \sin (\alpha_{i - 1} )} \\ {\sin (\theta_{i} )\sin (\alpha_{i - 1} )} & {\cos (\theta_{i} )\sin (\alpha_{i - 1} )} & {\cos (\alpha_{i - 1} )} & {d_{i} \cos (\alpha_{i - 1} )} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(1)

In the case of the Puma 560 robot, the global homogeneous transformation matrix (from the referential linked to the tool relative to the referential linked to the robot base) is written (2–4)

$${}_{n}^{0} T = \prod\limits_{i = 1}^{n} {{}_{i}^{i - 1} T = {}_{1}^{0} T{}_{2}^{1} T{}_{3}^{2} T.....{}_{n}^{n - 1} T}$$
(2)
$${}_{0}^{n} T = \left[ {\begin{array}{*{20}c} {R_{3x3} } & {P_{3x1} } \\ {0_{1x3} } & 1 \\ \end{array} } \right]$$
(3)
$$R_{3x3} = f(\theta_{i} ,\alpha_{i - 1} ),\,\,\,\,i = 1,6\; {\text{is}}\;{\text{the}}\;{\text{orientation}}\;{\text{matrix}}$$
$$P_{3x1} = f(\theta_{i} ,a_{i} ,d_{i} ),\,\,\,\,i = 1,6\;{\text{is}}\;{\text{the}}\;{\text{position}}\;{\text{vector}}\;{\text{of}}\;{\text{the}}\;{\text{marker}}\;{\text{linked}}\;{\text{to}}\;{\text{the}}\;{\text{terminal}}\;{\text{organ}}\left( {{\text{TCP}}} \right)$$
$${}_{n}^{0} T = = \left[ {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {r_{13} } & {p_{x} } \\ {r_{21} } & {r_{22} } & {r_{23} } & {p_{y} } \\ {r_{31} } & {r_{32} } & {r_{33} } & {p_{z} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]$$
(4)

The operational coordinates calculated as follows:

$$P = \left[ \begin{gathered} p_{x} = \cos (\theta_{1} )\left[ {a_{2} \cos (\theta_{2} ) + a_{3} \cos (\theta_{2} + \theta_{3} ) - d_{4} \sin (\theta_{2} + \theta_{3} )} \right] - d_{3} \sin (\theta_{1} ) \hfill \\ p_{y} = \sin (\theta_{1} )\left[ {a_{2} \cos (\theta_{2} ) + a_{3} \cos (\theta_{2} + \theta_{3} ) - d_{4} \sin (\theta_{2} + \theta_{3} )} \right] + d_{3} \cos (\theta_{1} ) \hfill \\ p_{z} = - a_{3} \sin (\theta_{2} + \theta_{3} ) - a_{2} \sin (\theta_{2} ) - d_{4} \cos (\theta_{2} + \theta_{3} ) \hfill \\ \end{gathered} \right]$$
(5)

3.2 Analysis of errors

Only the resulting error on the position of the tool-related mark will be considered, and subsequent analysis of the error occurring on the rotation matrix will be carried out. The precision and repeatability of a manipulator can be deduced from (1) by substituting the values of the parameters D-H given on Table 1 and the values of the angles of the corresponding joints. However, the actual positions of the robot are not only a function of the D-H parameters, but also of their possible uncertainties. Which come either from the construction, mounting tolerances or limitations of devices such as position sensors (resolution of optical encoders).By limiting ourselves to the first-order errors which will be denoted by, the theory of differential transformations applied to matrix functions, allows us to write, if we put:

$$T_{i} = T_{i} (\alpha ,a,\theta ,d)$$
(6)
$$T_{i} = T_{i} \pm \Delta T_{i}$$
(7)
$$\Delta T_{i} = \frac{{\partial T_{i} }}{\partial \alpha }\Delta \alpha_{i} + \frac{{\partial T_{i} }}{\partial a}\Delta a_{i} + \frac{{\partial T_{i} }}{\partial \theta }\Delta \theta_{i} + \frac{{\partial T_{i} }}{\partial d}\Delta d_{i}$$
(8)
$$T = T_{i} \pm (\frac{{\partial T_{i} }}{\partial \alpha }\Delta \alpha_{i} + \frac{{\partial T_{i} }}{\partial a}\Delta a_{i} + \frac{{\partial T_{i} }}{\partial \theta }\Delta \theta_{i} + \frac{{\partial T_{i} }}{\partial d}\Delta d_{i} )$$
(9)
Table 1 DH of the robot Puma 560
Full size table

It is then possible to deduce the generalized matrix defining the orientation and the current positions of the mark of the tool with respect to the base mark either in analytical form or in digital form provided that the error associated with each parameter is known. Generally these values are not provided by the robot constructor, however, a reasonable estimate of the maximum tolerance corresponding to the structural parameters (\(\alpha ,a,d\)) and the angular error related to the tolerance of the angular position sensor can be given.

3.3 Accuracy calculation

The boundaries of the current position vector \(\left[ {P_{cx} ,P_{cy} ,P_{cz} } \right]\) of the tool-related coordinate system can be obtained if it is assumed that there is a deviation associated with each parameter D-H. The current position sought will then be the vector associated with the generalized transformation matrix \(T_{c}\).The resulting error \(e_{i}\) corresponds to the difference between the current position and the theoretical position for the three spatial coordinates (10)

$$e_{i} = P_{ci} - P_{ti} \,,\,\,\,\,\,\,i = \left\{ {x,y,z} \right\}$$
(10)

The global error will be defined as the Euclidean norm of the maximum errors committed on the three spatial coordinates:

$$e_{g} = \sqrt {\max (e_{x} )^{2} + \max (e_{y} )^{2} + \max (e_{z} )^{2} }$$
(11)

An accuracy estimate is obtained by calculating the global position error assuming that all D-H parameters are affected by errors. On the other hand, the error due to the repeatability considers only the error coming from the generalized angular coordinates (case of the Puma 560).This is because once the robot is assembled, the only changes that can occur are the position instructions (angular rotations of the joints). All other parameters are static and are not likely to vary during an operating sequence, as long as the environmental influence quantities do not vary (temperature, pressure, humidity, load bending). Indeed once the robot is assembled, the static errors will always be present and will not influence the repeatability. Precision errors will depend on the D-H parameters and their corresponding variations. If we assume that the joints are all of the rotoid type (case of the Puma 560), we can write then:

* For precision, the error of position is written:

$$\varepsilon_{P} = f(\alpha_{i} ,a_{i} ,\theta_{i} ,d_{i} ,\Delta \alpha_{i} ,\Delta a_{i} ,\Delta \theta_{i} ,\Delta d_{i} )$$
(12)

Precision is a function of the static (constant) parameters \(\alpha_{i} ,a_{i} ,d_{i}\).

Static uncertainties \(\Delta \alpha_{i} ,\Delta a_{i} ,\Delta d_{i}\) are due to construction and assembly tolerances, while dynamic uncertainties \(\Delta \theta_{i}\) arise from the resolution of the position sensors. Since the robot control scheme requires the calculation of the inverse kinematic model, only the nominal values of the static D-H parameters can be used, since the uncertainties cannot be taken into account since they are not known. This results in a systematic uncertainty of positioning which requires prior calibration.

On the other hand, repeatability is easier to predict and control, since it expresses the robot's ability to return to the same imposed position. The static quantities do not affect the repeatability, since the robot is assembled, the nominal values of the parameters D-H provided by the manufacturer can be adopted. When a position is assigned to the manipulator, all that the control system needs is the current value of the dynamic variables, in this case the angle set points. This is the only useful information to return to the imposed position. It follows that repeatability depends only on generalized dynamic coordinates \(\Delta \theta_{i}\).

4 Numerical example

We reproduce here below (Table 1) the values of the parameters DH of the robot Puma 560. The choice was on this Robot because of the numerous works that were devoted to him, which make it a prototype of study very documented.

After proposing a study of the modeling and the characteristics of the robot "Puma560", the next step consists in implementing the solution proposed in order to be able to simulate the behavior of the robot for the pursuit of the positions desired by the proposed control laws. In order to optimize and facilitate this essential step for a better precision and repeatability of use, we have developed a simulator which will be detailed in the following section.

5 Results obtained

We reproduce on (Table 1) the values of the modified DH parameters of the Puma 560 robot. The choice was made on this Robot because of the numerous works that have been dedicated to it, which make it a prototype of study very documented.

In Tables 2 and 3, two series of tests were carried out, one for absolute uncertainties on the static parameters corresponding to lengths (Δa, Δd), the second for absolute uncertainties on the angles.

Table 2 Result of the first test
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Table 3 Result of the second test
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(Δ \(\alpha\), Δ \(\theta\)). The results of the Puma 560 robot validated in simulation are given in Figs. 4 and 5 and Tables 2 and 3.

6 Comparison of proposed approaches

From the results obtained (see Figs. 3, 4 and 5), it was found that the latter make it possible to control the position of the robot and a convergence toward the desired poses. The simulations based on the robot model are used to estimate the position error. From Table 1, a quantitative comparison is made by calculating the precision error along the position trajectories. According to this Table 2 and 3, the reduction of tolerance values implies better accuracy and repeatability at the desired position. If we are interested in the last case where the tolerances are considerable, a comparison can be made between the two tables starting from Table 2. The accuracy error shows good tracking of the reference positions. The same is true of the repeatability error (see Fig. 4a) The tolerance on the parameters Δa and Δd must be reduced in order to obtain a good accuracy and better follow-up of the input set points. This can be seen in Figs. 4a and 5a, whereby errors in accuracy and repeatability are reduced because of the low tolerances adopted for the Puma 560 robot.

Fig. 3
figure 3

Graphical interface (calculation of the precision error)

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Fig. 4
figure 4

Calculation of precision error of the Puma 560 robot: (a) Calculation of the precision error in XYZ, (b) Calculation of the precision error in XY, (c) Calculation of the error of precision in XZ

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Fig. 5
figure 5

Calculation of precision error of the Puma 560 robot: (a) Calculation of the precision error in XYZ, (b) Calculation of the precision error in XY, (c) Calculation of the error of precision in XZ

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7 Applying the experience design method

In this study, we apply an optimization method, the design method of experiments; we simulated the parameters influencing the precision of industrial robot manipulators. From the Denavit–Hardenberg model (standard or modified DH) and the generalized transformation matrix, the calculation of the precision is carried out in the general case then applied to the Puma 560 robot. To illustrate this work, a program developed and using the symbolic calculation tools of this language was developed for the calculation of the robot's precision for given absolute uncertainties. To first see the effect of each parameter α, a and d on the robot's precision error (Ep), and after seeing the most influential parameter or combination of parameters that affects the precision of our robot. To first see the effect of each parameter α, a and d on the robot's precision error (Ep), and after seeing the most influential parameter or combination of parameters that affects the precision of our robot.

7.1 Experimental matrix

In our case of three factors, the matrix of the experiments is represented by Table 4. The experimental points defining the experimental field of study are located on the vertices.

Table 4 The parameters selected for the design of experiments method
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The grouping of the fields of the three factors defines the “field of study “ which is the area of the experimental space chosen by the experimenter to perform the eight trials. A study, that is to say several well-defined experiments, is represented by points distributed in the study domain (Fig. 6). This way of representing an experiment by points in a logical Cartesian space is a geometric representation of the study. In this case, a plane 23 is associated with a mathematical model (Tables 5 and 6 and 7).

$$y = \, a_{0} + a_{1} .X_{1} + \, a_{2} .X_{2} + \, a_{3} .X_{3} + \, a_{12} .I_{12} + \, a_{13} .I_{13} + \, a_{23} .I_{23}$$
(13)

a0: the overall averagea1, a2, a3: Effect of α, a and d respectivelya12: Effect of the interaction between the α and aa13: Effect of the interaction between the α and da23: Effect of the interaction between the a and dI12, I13 and I23: Interaction between the different variables.

Fig. 6
figure 6

Field of study

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Table 5 Table of experiments
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Table 6 Effect of factors
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Table 7 Calculating predicted responses
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The experimental results and the simulation presented in Fig. 7 gives us the impression that our simulation result is close to the experimental results.

Fig. 7
figure 7

Representation of the effect of the three factors

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7.2 Analysis with a single effect

7.2.1 Global formula of the mathematical model

$$\begin{aligned} y & = 0, \, 0110539 + 4, \, 65661e - 010x_{1} + 0, \, 00608x2 + 0,00608x_{3} \\ & + 4,80682e - 010I_{12} + 2,103e - 010I_{13} + 0,0090441I_{23} \\ \end{aligned}$$
(14)
$$\begin{aligned} E & = 0, \, 0110539 + 4, \, 65661e - 010\alpha + 0,00608 \, a + 0,00608d \\ & + 4,80682e - 010I_{12} + 2,103e - 010I_{13} + 0,0090441I_{23} \\ \end{aligned}$$
(15)

7.3 Effects analysis

From Fig. 7, we will represent the effect of each factor [α (X1), a (X2) and d (X3)] on the response (precision error Ep).

In this section we will try to analyze the effect of the three main factors. First, the first remark is the direction of the three straight lines, this means that the transition from low to high for all factors causes precision (upward) and linear errors. It can be observed that for the value of Δd of 0.100 mm, the precision error is equal to 0.015 mm (c). The behavior is the same for the second effect (a) figure (b).The second remark concerning the slope of the straight line corresponding to the first factor, which is practically horizontal, indicates that this factor has no influence on the error (not significant).

7.4 Interaction effect for two factors

It is possible to create interaction graphs, when we select a two factor interaction, the predicted change in the response

In this case (Fig. 8a), it is a question of leaving the Δd constant and of varying the robot's Δα and the Δa to illustrate the precision error. First, we notice that increasing the robot's Δα and Δa at the same time increases the precision error. The horizontal analysis of Fig. 8b shows that for Δα of 0.075° we go from an error of 0.005 for Δa of the robot of 0.08 mm to an error of 0.0086 for a value of 0.12 mm. Going to Δα of 0.11°, the precision error goes from 0.0147 for a 0.11 mm to an error of 0.0147 for Δa of 0.11 mm. In other words, increasing one of the two factors or both at the same time causes an increase in the error.

Fig. 8
figure 8

a Response surface with X1, X2 variables, X3. b Interaction between X1 and X2 fixed X3

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It is therefore a matter of varying the value Δα of the robot in the range from −0.005° to 0.002° as well as Δd 0.07 mm to 0.12 mm, the value Δa being taken equal to a constant value (arbitrarily chosen) Fig. 9. In Fig. 9a, usually called "Iso curve" and which is the projection of the response surface on the plane, one can easily draw values predicted by the modeling other than those measured. The vertical analysis of Fig. 9b shows that for Δd equal to 0.07 mm, the accuracy error goes from 0.00512 for an Δα of −0.004° to 0.00201 for an Δα equal to 0.002°. Also, the accuracy error changes from a value of 0.0524 for Δα equal to −0.005 to an error of 0.00291 for Δd equal to 0.12 mm.

Fig. 9
figure 9

a Response surface with X1, X3 variables, X2 constant. b Interaction between X1 and X3 fixed X2

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In Fig. 10 above, the value of Δa and the value of Δd are varied and the value of Δα of the robot is fixed. We then see that the precision error reaches a maximum with a value equal to approximately 0.0187 when Δa and Δd are equal to 0.11 mm, this angle reaches 0.00902 for Δa = 0.08 mm and Δd = 0.11 mm. We also notice that this angle reaches an average value (between 0.00677 and 0.00584) if the Δd is fixed at its low level.

Fig. 10
figure 10

Response surface with X3, X2 variables, X1 constant. b Interaction between X2 and X3 fixed X1

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7.5 Variance estimation

The statistical calculations used to determine whether the effects are significant, to calculate the confidence intervals and to validate the linearity of the model involve the residuals ei, i.e. the difference between the experimental value and the value predicted by the model (Tables 7 and 8).

Table 8 Calculation of model residuals
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7.6 Residue calculation

The absolute value of the difference between the experimental response and the predicted (estimated) response gives the value of the residual with relation (16):

$$e_{i} = \left| {Y_{i\exp } } \right. - \left. {Y_{pre} } \right|$$
(16)

These deviations show the difference between the experimental values and those described by the mathematical model. We find that they are insignificant and therefore we conclude that the model adequately describes the experimental. These deviations are visualized by (Fig. 11) in which we easily see a perfect correspondence between the linear model (right) and the tests.

Fig. 11
figure 11

Distribution of calculated responses with respect to the line

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7.7 Realization of the test of effects significance

The test used is Student's "t" test. An effect will be said to be significant (i.e. the variable or the interaction associated with it has an influence on the response), if it is, for a given risk, significantly different from 0.

We then use a Student's table with ν = n—p degrees of freedom (n is the number of experiments carried out and p the number of effects including the constant). We choose a risk of the first kind α (most often 5% or 1%) and the value tcrit(α, ν) is read from this Student table, using the part of the table relating to a bilateral test [20]. The test rule is then:

  • If |the effect of a parameter |> tcrit(α, ν) * if: The effect is significant.

  • If |the effect of a parameter |< tcrit(α, ν) *si: The effect is not significant.

The Student table gives for a risk α = 5% and v = n—p = 8—7 = 1,tcrit(0,05; 1) = 12,71.

An effect will therefore be significant at the 5% risk if it is "ti" and greater than 12.71. We get the following Table 9.

Table 9 Significance of effects results
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For this we use Student’s test given by the relationship (17):

$${\mathbf{t}}_{i} = {\raise0.5ex\hbox{$\scriptstyle {\left| {a_{i} } \right|}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {S_{i} }$}}$$
(17)

8 Calculation of the variance

To further estimate the model, we first calculate the variance s, the square of which is expressed by:

$$S^{2} = \frac{1}{n - p}\sum {e_{i}^{2} }$$
(18)
$$\mathop s\nolimits^{2} = \frac{1}{8 - 7}\left( {\text{3,6e - 17}} \right)$$

Under these conditions, we can show that all effects have the same variance given by Eq. (19):

$$S_{i}^{2} = \frac{{S^{2} }}{n}$$
(19)

where n is the number of experiments performed and p the number of model coefficients. Under these conditions, it can be shown that all the effects have the same variance given by:

$${\varvec{S}}_{i}^{2} = \frac{3.6e - 17}{8}$$

Consider our experimental design in which 23 neglecting the order of interaction 3, we collect residues and variances correspond to each effect in Table 5:

The common variance of residuals is given as follows:

$$S_{{}}^{2} = \frac{1}{n - p}\sum {e_{i}^{2} = \frac{1}{8 - 7}} \left( {0.01805} \right) = 0.01805$$

The Student test table gives, for a risk of 5%with v = n – p (v dof) = 8–7 = 1 (dof)

$$t\;(\alpha ;\nu ) = t\;\left( {0.05;\;1} \right) = 12.71$$

One effect will be significant at the 5% risk if the “\(t_{i}\)” is greater than 12.71. Table 6 is obtained (Table 10).

Table 10 The confidence interval of model
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We note that the effect of the interaction between the thickness and the yield strength (t12) is not significant. Therefore, only the variables X1, X2, X3 and I13 interactions, I23 are retained. And the model will become in the following form:

$$y \, = \, 0.0110539 + \, 0.00608X_{2} + 0.00608X_{3} + 0.0090441X_{2} X_{3}$$
(20)
$$Ep \, = \, 0.0110539 + \, 0.00608a_{{}} + 0.00608d_{{}} + 0.0090441ad$$
(21)

8.1 The confidence interval of model

We will find a confidence interval at the risk of 5%, for the significant effects: a1, a2, a3, a12 and a13, If we choose a risk α we can determine the value of t (α; v) using the Student table and the confidence interval of an effect is given by:

$${\text{Risk}}\;\alpha_{i} :\left[ {(a_{i} - \, t \, \left( {\alpha , \, v} \right)S_{i} ;\left( {a_{i} + \, t \, \left( {\alpha , \, v} \right)S_{i} } \right)} \right]$$
(22)

After replacing the values of t(α; v) and S i we obtain:

$$\alpha_{{\text{i}}} = \, \left[ {{\text{a}}_{{\text{i}}} - { 12}.{7}0{6 }* \, 0, \, 0{475};{\text{ a}}_{{\text{i}}} + { 12}.{7}0{6 }* \, 0.0{475}} \right]$$
$$\alpha i \, : \, [(ai \, - \, t \, \left( {\alpha , \, v} \right)Si \, ; \, \left( {ai \, + \, t \, \left( {\alpha , \, v} \right)Si \, } \right]$$

Table 7 shows the confidence interval of significant effects coefficients:

9 Conclusion

The work presented was devoted to the study of improving the precision and repeatability of a robotic system. For example, improving the manufacturing quality of robot components helps reduce tolerances and thus limit geometric errors. Robot calibration is the solution that improves the static precision of robot positioning without any modification of the structure. Our work is devoted to geometric calibration. The work consists in analyzing the static errors of precision and repeatability of a robotic system given by these DH parameters (standard or modified), knowing the precision on each link (Δq, Δα, Δd), and in calculating the overall systematic precision. From the choice of an initial pose (zero position), the global error is analytically evaluated.

We were then interested in the simulation of the precision and repeatability of the first link or link of the puma 560 robot. This part allowed us to draw the following conclusions:

  • The evolution of the various parameters governing the precision error converges in the same direction;

  • The higher the tolerance on the geometric parameters (Δa, Δd), the greater the precision error.

  • To improve precision, we must reduce the geometric tolerance on the first links (on machining).

Subsequently, we studied by the method of design of experiments the effect of the static errors of each parameter of the first link (predominant effect) of the puma 560 robot on the precision of the robots. The following results:

  • The precision error increases with increasing tolerances of the geometric parameters (Δa, Δd);

  • The most significant influence of these parameters on the overall error.

  • The influence of the angle (Δα) has an insignificant effect on the precision of the robot To calibrate a robot, it is necessary to intervene on the first link.