# Tolerance Synthesis of Delta-like Parallel Robots Using a Nonlinear Optimisation Method
## 5 Conclusions  
In this study, a position error modelling procedure and tolerance synthesis of the Delta parallel robot are proposed. In the position error modelling part, Khalil-Kleinfiger parameterisation was used, and the geometric errors were introduced into the homogeneous transformation matrices. The direct geometric model equations are used to calculate geometric errors. Concerning the tolerance synthesis part, a three-step sequential method was proposed to calculate the geometric tolerances of the Delta parallel robot. The mathematical model used does not require the calculation of the inverse of the Jacobian matrix. Therefore, the convergence is faster in this case.  
The calculation and simulation results show the efficiency and accuracy of the proposed method for determining linear and angular geometric tolerances for the Delta parallel robot. The obtained dimensional tolerance values are optimal. Therefore, these values maintain accuracy at a level that is lower than or equal to the imposed value (0.2 mm).  
A verification calculation of the position error for different values of geometric tolerances was carried out. The verification results show that the value of the position error is always less than the value of the accuracy, which means that the proposed method is very effective for the design of mechanism geometric tolerances.  
In addition, the proposed design method allows us to obtain high-accuracy robots. This is obtained by calculating geometric tolerances according to a given accuracy (required value) of the robot's end-effector. In this way, the repeatability of the robot can be improved.  
When high accuracy is required, it is necessary to achieve a robust design. In this case, other constraints on the dimensional parameters must be added to the optimisation problem (Equation (13)). To determine these constraints, a sensitivity analysis must be performed. The robust design allows us to obtain a mechanism that meets the requirements mentioned in the specifications at a minimum cost.  
The proposed method presents an interesting advantage in the number of calculations of geometric tolerances. As mentioned in Section 4.2, among the fifteen robot configurations, only four are needed to calculate all the geometric tolerances, which means that the calculation algorithm quickly converges to the desired solution.  
The proposed method allows for the calculation of all types of linear and angular errors in a robot and can be generalised to more complex types of serial and parallel robots. In future works, we may consider the study of orientation tolerances (parallelism, perpendicularity, and inclination) to quantify their impact on the robot's accuracy and validate the calculation on a real robot. We can improve the proposed tolerance method and achieve a robust design based on the accuracy criterion. We are also interested in exploring other methods of calculating geometric tolerances, especially statistical methods.  
Research concept and design: A.B., A.K., R.K. and A.L.; collection and/or assembly of data: A.B., A.K., R.K. and A.L.; data analysis and interpretation: A.B., A.K., R.K. and A.L.; writing the article: A.B., A.K., R.K. and A.L.; critical revision of the article: A.B., A.K., R.K. and A.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement:** No indication of ethics is given.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data that support the findings of this study are available upon request from the authors.  
**Acknowledgments:** This research was supported by the Directorate General for Scientific Research and Technological Development (DGRSDT) in Algeria (Project PRFU No. A01L09 UN210120220001).  
**Conflicts of Interest:** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.