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This study addresses the problem of nonlinear error predictive compensation to achieve high positioning accuracy for advanced industrial applications. An improved calibration method based on the generalisation performance evaluation is proposed to enhance the stability and accuracy of robot calibration. With the development of technology, a deep neural network (DNN) optimised by a genetic algorithm (GA) is applied to predict the nonlinear error of the calibrated robot. To address the change of external payload, an extra compliance error model is established with a linear piecewise method. A global compensation method combining the GA-DNN nonlinear regression prediction model and the compliance error model is then proposed to achieve the robot’s high-precision positioning performance under any external payload. Experimental results obtained on a Staubli RX160L robot with a FARO laser tracker are introduced to demonstrate the effectiveness and benefits of our proposed methodology. The enhanced positioning accuracy can reach 0.22 mm with 98% probability (i.e., the maximum positioning error in all test data).

With the emergence and subsequent development of digital processing and intelligent manufacturing, the breadth and depth of industrial robotic applications (i.e., application fields and task requirements) are increasing [

Robot calibration can be divided into robot kinematic calibration and nonkinematic calibration [

Guo et al. [

As we have shown, model-based kinematic calibration or nonkinematic calibration is essential for uncalibrated robots (i.e., the uncalibrated robots fresh out of a factory or the robots with numerous errors after being used for a certain period of time). Besides, the nonlinear errors including the intrinsic errors (i.e., elastic deformation of the link and joint caused by the link self-gravity of the robot, gear backlash, hysteresis, and friction) and the extrinsic alterations (i.e., the variable external payloads) of the robot cannot be eliminated by model-based robot calibration. In the current study, the predictive compensation method for the nonlinear errors of robots, especially the residual errors of calibrated robots, is yet to be fully studied in the literature. Therefore, the goal of this work is the enhancement of calibration and error compensation techniques for achieving the robot’s high-precision positioning performance under any external payload. Three key problems that should be considered are as follows:

There is no unified uniform standard for the determination of the parameters in the calibration model, so it is of great significance to propose a parameter optimisation method for the calibration quality

Given the aforementioned nonlinear errors, a global compensation method should be presented to predict effectively and efficiently these positioning errors

Due to the large range of external payload variation of the heavy robot, the nonlinear deformation error caused by the change of the external payload should be solved

In this work, an improved calibration method, which takes into account the generalisation performance and robustness of geometric parameter correction, is introduced to enhance the essential positioning accuracy of robots. DNN has received substantial attention in both the signal processing field and the machine learning field with its strong regression capabilities [

To address the aforementioned problems and the corresponding solutions, the rest of this paper is organized as follows: Section

In our preliminary work [

A 6-DOF serial robot. (a) The Staubli RX160L robot. (b) The kinematic model of the Staubli RX160L robot.

Because the actual robot base frame (_{0}) cannot be touched, the small transformation matrix ^{B}_{0} is introduced to present the error vector _{B}) and the actual robot base frame (_{0}):

Then, the error equation is expressed as follows:

Hence, the robot kinematic position error model is established as follows:_{k} represents the position errors; _{k} is the geometric identification matrix; _{k} denotes the geometric error parameters; and _{θ}, _{d}, _{a}, _{α}, _{β}, and _{σ} are the coefficient matrices corresponding to the error parameters.

By using the kinematic error model, all defined geometric parameters can be identified theoretically. However, the model is not the best choice to correct all geometric parameters in the process of robot calibration. Generalisation performance, also known as scalability, refers to the performance of the classifiers trained on a certain training set to adapt to the data outside the training set. In this work, a _{2}–_{5}, _{3}, _{4}, and _{1}–_{3}), and the maximum number is 24 (each link frame contains 4 parameters). Therefore, according to the limited range, different calibration models can be formed by the arbitrary combination of different geometric error parameters; from this, the number of calibration models can be calculated:

Then, the training data used for calibration model identification are divided into _{K} obtained by averaging

It can be seen that the optimised calibration model is obtained by using these various calibration models with different geometric error parameters and the generalisation performance evaluation method. Moreover, the improved calibration method based on the parameter optimisation method can also be combined with engineering practice, thus increasing or reducing the effective constraints on the range of parameter correction so as to obtain the required calibration model. Hence, the improved calibration method proposed in this section can enhance the stability and accuracy of robot model-based calibration. Note that, however, some errors are unable to be established by the robot kinematic position error model, which also affect the quality of the absolute positioning accuracy of robots. To tackle this problem, the nonlinear errors of the calibrated robot will be discussed later.

Given the complexity of the sources of the robot’s residual errors (i.e., elastic deformation of links and joints caused by the link self-gravity of the robot and the change of external payload, gear backlash, hysteresis, and friction) after model-based calibration, these residual errors generally have strong nonlinearity. At present, most of the learning methods, such as regression, are shallow structure algorithms [

DNN can achieve complex function approximation by learning a deep nonlinear network structure (5, 6, or even 10 hidden layers are usually present) and show a strong capacity to learn the effective features of data from a small number of samples. Two types of joints of industrial robots exist, namely, rotational joints and prismatic joints, which correspond to the joint variables _{i} and _{i}, respectively. In this work, all of the robot’s joint variables are regarded as feature objects, i.e., the input data set of the DNN. For a serial robot with six rotational joints (i.e., joints _{1}–_{6}), the input layer of the DNN has six neuron nodes.

To realise the prediction and compensation of the robot positioning error, the output of the DNN is the predicted position error with a three-dimensional (3D) vector _{r} (_{r} = [_{1} [

Optimised measurement configurations with 46 groups of different orientations.

The MSE is selected as the loss function to evaluate the degree of inconsistency between the predicted value and the real value of the DNN regression prediction model, that is, the criterion to measure the predictive capacity of the model. As the DNN contains more hidden layers and adopts the backpropagation algorithm, the traditional activation functions (i.e., sigmoid function and tan-sigmoid function) can lead to gradient vanishing. To tackle this problem, the parametric rectified linear unit (PReLU) function [

Activating functions: (a) PReLU function; (b) tan-sigmoid function.

Finally, the GA is applied to optimise the number of hidden layers, the number of nodes per layer, the learning rate, the training data, the training method (including the specific distribution of the activation functions), and the validation data in the training process for building the optimal DNN architecture. The loss function (MSE) of the DNN model is treated as the fitness function for the GA, and the size of the initial population of the GA is 100. Furthermore, the crossover operation-mutation operation has 90% probability, and the mutation rate is 0.05. Hence, the GA-DNN nonlinear regression prediction method is developed as an effective solution for the high-performance compensation of the robot’s nonlinear errors, as shown in Figure

Flowchart of the GA-DNN nonlinear regression prediction method.

It is important to note that the training data set for the GA-DNN model is always measured with a constant end-effector of the calibrated robot, and the nonlinear regression prediction model does not consider the effect of the change of external payload on the positioning error. To overcome this difficulty, the robot’s positioning errors under variable external payloads will be analysed in detail in the next section.

According to the relation between force and torque, the formula of torque for each joint can be derived based on the robot’s kinematic model as follows:_{θ} represents the kinematic Jacobian matrix; equation (^{0}_{m} is the force vector of the external payload; ^{0}_{6} is the rotation matrix between the actual robot base frame (_{0}) and the flange frame (_{6}); ^{6}_{m} is the vector between the centroid of the external payload and the origin of the flange frame (_{6}); and _{3} is a 3 × 3 identity matrix.

Ψ(

To analyse the compliance error caused by the external payload, the robot joint is usually assumed to be a linear torsional spring [

The third-order polynomial satisfies two conditions in equation (_{i} is present. Then, combined with the torque obtained from equation (

Schematic of the linear piecewise method based on torque division.

Based on equation (

The torque ranges of joints 2–6 of the Staubli RX160L robot under (a) 0.4 kg and (b) 14 kg of the external payload.

As can be observed from Figure

Based on the linear piecewise method with the two subsections, the position errors of the origin of the tool frame (_{T}) for 6-DOF series robots can be given as follows:^{6}_{T} is the vector between the origin of the robot tool frame (_{T}) and the origin of the flange frame (_{6}).

To obtain the best piecewise critical torque values (i.e.,

Hence, the extra compliance error model (i.e., equation (_{(GA−DNN)} is the predictive position error by the GA-DNN nonlinear regression prediction model.

Figure

Experimental platform with a Staubli RX160L robot and a FARO laser tracker.

To collect the robot measurement positions for the geometric parameter identification, 90 measurement positions are selected by the intelligent selection algorithm of optimal measurement poses [

The nominal and identified MDH parameters of the Staubli RX160L robot.

_{i} (°) | _{i} mm | _{i} mm | _{i} (°) | _{i} (°) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Nom. | Iden. | Nom. | Iden. | Nom. | Iden. | Nom. | Iden. | Nom. | Iden. | |

1 | 0 | 0.0272 | 0 | −1.0050 | 150 | 150.7542 | −90 | −90 | — | — |

2 | −90 | −89.9518 | — | — | 825 | 825.4951 | 0 | 0.0135 | 0 | −0.0484 |

3 | 90 | 90.5940 | 0 | 0.5650 | 0 | 0.1124 | 90 | 90 | — | — |

4 | 0 | 0.0535 | 925 | 923.5001 | 0 | −0.0883 | −90 | −90 | — | — |

5 | 0 | 0.1623 | 0 | 0.0813 | 0 | 0 | 90 | 90 | — | — |

6 | 0 | 0 | 110 | 109.9630 | 0 | −0.0317 | 0 | 0 | — | — |

“Nom.”: nominal value; “Iden.”: identified value; “—”: does not exist.

To present the advantages of the proposed improved calibration method (_{1}–_{5}, _{1}, _{3}–_{6}, _{1}–_{6}, _{2}, and _{2}) and 23 (i.e., _{1}–_{6}, _{1}, _{3}–_{6}, _{1}–_{6}, _{1}–_{5}, and _{2}), respectively.

The identified MDH parameters by the conventional kinematic calibration method.

_{i} (°) | _{i} (mm) | _{i} (mm) | _{i} (°) | _{i} (°) | |
---|---|---|---|---|---|

1 | −0.0289 | −0.9085 | 150.7783 | −89.9964 | 0 |

2 | −89.9514 | 0 | 825.4384 | 0.0148 | −0.0828 |

3 | 90.5969 | 1.2935 | 0.0886 | 89.9786 | 0 |

4 | 0.0886 | 923.5273 | −0.1002 | −89.9877 | 0 |

5 | 0.1653 | 0.0688 | −0.0155 | 89.9507 | 0 |

6 | −0.0084 | 109.9556 | −0.0304 | 0 | 0 |

Furthermore, 40 random positions are chosen as the test data in the overall workspace of the robot, and the enhancement results of position accuracy are shown in Figure

Position errors before and after calibration using the conventional calibration method and the proposed calibration method. (a) The errors in the

Due to the influence of nonlinear errors, the calibrated robot still has large residual errors which cannot meet the requirements in high-precision field applications. The Staubli RX160L robot has a general structure of a 6-DOF serial robot, which is often used in high-precision processing, and its repeatability is about ±0.05mm. Hence, the repetitive accuracy is the limit value of the absolute positioning accuracy of the robot and also the target value of error compensation using our proposed approach.

Based on the 46 configurations of different orientations (Figure

Spatial distribution of the measurement points. (a) Distribution of the training points. (b) Distribution of the test points.

To better present the nonlinear characteristics of the error data, four sets of data are selected from 2,796 measurement positions, each with the same configuration and the same position in the

Spatial distribution of the residual errors. (a) Under the first of 46 configurations. (b) Under the seventh of 46 configurations. (c) Under the 16th of 46 configurations. (d) Under the 25th of 46 configurations.

To demonstrate the superiority of the DNN over the shallow neural network (SNN), the performance of the proposed GA-DNN model is compared with that of a 2-layer BPNN model and another 5-layer BPNN model. Gradient descent (GD) is one of the most popular algorithms with which to perform optimisation and by far the most common way to optimise neural networks. To optimise neural networks further, certain excellent gradient descent optimisation algorithms are proposed and applied to neural network training, i.e., momentum, Nesterov accelerated gradient (NAG), Adagrad, Adadelta, RMSprop, and adaptive moment estimation (Adam). Kingma and Ba [

In our approach, the GA is applied to select the most suitable gradient descent optimisation algorithm for training the DNN model. Based on the collected training data, the optimised GA-DNN model has a network architecture with 11 hidden layers, and the corresponding optimised node numbers are listed as follows: 10, 10, 13, 13, 16, 16, 15, 13, 11, 9, and 7. The results of the iteration, which we obtained by using the joint data and error data of the 2,796 measurement positions, are shown in Figure

Prediction accuracy comparison using different NN models.

The predictive compensation and validation results of the proposed GA-DNN nonlinear regression prediction method are compared with those obtained by the popular ANN-based nongeometric correction method from [

The positioning errors of the training data before and after compensation using the four methods.

The positioning errors of the test data before and after compensation using the two better methods.

The 3,504 endpoints used in the above experiments are measured under the same end-effector (the mass is 3.6 kg), so there are no positioning errors caused by the change of external payload. To demonstrate the practical effectiveness and performance of the proposed global compensation method and to confirm the benefits of the method for the change of external payload, this section addresses the compliance error model of the Staubli RX160L robot with the change of external payload.

A measurement tool is designed for changing the external payload, which can fix different sizes and number of weight blocks and be used as the end-effector of the robot, as shown in Figure

The measurement tool with weight blocks for the Staubli RX160L robot. (a) The minimum external payload with 0.4 kg of the external payload. (b) Increasing weight blocks to change the size of the external payload.

The piecewise critical torque values and the identified joint stiffness.

_{l} (N⋅m/rad) | _{h} (N⋅m/rad) | ||
---|---|---|---|

Joint 2 | 118.236 | 2.34 × 10^{9} | 2.65 × 10^{9} |

Joint 3 | 75.625 | 9.73 × 10^{8} | 1.19 × 10^{9} |

Joint 4 | — | 8.52 × 10^{7} | 8.52 × 10^{7} |

Joint 5 | 15.382 | 8.07 × 10^{6} | 9.73 × 10^{6} |

“—”: does not exist.

To verify the accuracy of the proposed global compensation method, two groups of different orientations in 708 verification positions, which include 168 test points, are selected. Using the proposed global compensation method, which combines the compliance error model and the GA-DNN nonlinear regression prediction model, we can predict and compensate the position errors of these test points with 5.95 and 9.95 kg of the external payloads, respectively.

Figure

Frequency histograms of positioning accuracy after compensation under (a) 5.95 kg and (b) 9.95 kg of the external payload.

In this paper, a multilevel accuracy improvement method is proposed, which includes the improved kinematic calibration, the GA-DNN nonlinear regression prediction, and the extra compliance error compensation. To enhance the stability and accuracy of robot calibration, a parameter optimisation method based on the

The proposed theoretical methods have been validated via an experimental study involving a kinematic calibration and error compensation of a Staubli RX160L industrial robot. The experimental results using the GA-DNN prediction method are more accurate than the previous methods (i.e., methods from [

In the experimental study, data acquisition takes a long time in robot calibration and compensation, especially for the training of the DNN model (2,796 measurement points are used in this paper). Admittedly, the greater the amount of training data is, then the more accurate the DNN model becomes. Given that the large amount of data is needed for each robot for nonlinear compensation, the efficiency becomes relatively low. Consequently, we will focus on the application of the transfer learning method in this DNN-based prediction model, which only needs to use a small amount of data to share a similar DNN model for the same type of the robot.

No data were used to support this study.

Yilin Sun received his B.S. degree in School of Mechanical Engineering from Jiangnan University, Wuxi, China, in 2013. He is currently a Ph.D. candidate of Jiangnan University. His research is in the development of bionic grippers, e-mail:

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (no. 51575236) and Postgraduate Research and Practice Innovation Program of Jiangsu Province (no. KYCX18_1840).