# Tolerance Synthesis of Delta-like Parallel Robots Using a Nonlinear Optimisation Method  
[PERSON]  
LGMM Laboratory, University of 20 August 1955-Sikida, PB N' 26 Route Elhadaik, Sikkida 21000, Algeria; [EMAIL_ADDRESS] (A.B.); [EMAIL_ADDRESS] (A.K.); [EMAIL_ADDRESS] (A.L.)  
[PERSON]  
LGMM Laboratory, University of 20 August 1955-Sikida, PB N' 26 Route Elhadaik, Sikkida 21000, Algeria; [EMAIL_ADDRESS] (A.B.); [EMAIL_ADDRESS] (A.K.); [EMAIL_ADDRESS] (A.L.)  
[PERSON]  
Because of the random nature of the joint clearances, [PERSON] et al. [11] developed a stochastic technique to determine the reliability of the open-loop mechanism. With the premise that all kinematic parameters are normally distributed random variables, they proposed a stochastic model of links with dimensional tolerances and revolute joints with clearances. Based on an advanced first-order second-moment method, the kinematic reliability for positioning and orientation repeatability was then computed analytically. [PERSON] and [PERSON] [12] employed an experimental design strategy to reduce performance variability by employing a probabilistic approach for the selection tolerance specification of robot kinematic and dynamic parameters. The performance metric, i.e., signal-to-noise ratio, was used to validate Monte Carlo simulations. [PERSON] and [PERSON] [13] presented manipulator reliability to represent its kinematic and dynamic performance using a probabilistic technique, where the manipulator reliability was defined as the probability of an end-effector pose to fall within a certain range of the desired pose.  
Many researchers consider tolerance allocation to be an optimisation problem in which part-tolerance values are used as control variables and machining costs are used as the objective function to be minimised. However, robust tolerance design of mechanisms has been discussed in depth by creating multiobjective optimisation models [14; 15; 16]. [PERSON] [17] described an application in which system tolerances were chosen using an optimisation approach. [PERSON] et al. [18] proposed a local uniqueness hypothesis that would allow them to safely compute the pose error upper bounds, using a parametric version of [PERSON]'s theorem and certified nonlinear global optimisation. To determine optimal parameter tolerance, [PERSON] [19] used a genetic algorithm, differential evolution, and optimisation techniques. A local uniqueness hypothesis for reliably obtaining pose error upper limits using nonlinear optimisation was proposed in [20]. [PERSON] et al. [21] introduced a technique using the generalised reduced gradient algorithm optimisation to allocate tolerances into robot parts. By using the Denavit-Hartenberg (DH) rule for modelling the kinematic problem in that study, a mathematical model for tolerance allocation was developed and used in the nonlinear multivariable optimisation problem. The generalised reduced gradient method was then used to solve this optimisation problem.  
This paper proposes a new approach to determine tolerances of linear and angular dimensions. The allocation of tolerances is represented as an optimisation problem, and a new mathematical model is established and solved using the \"interior-point\" algorithm. The modelling method and the allocation of tolerances for a Delta parallel robot case study are established using the proposed technique, and the results show that the method can be used in designing the optimal tolerance values of the parts of the mechanism and can be generalised for other types of robots.  
The rest of this paper is organised as follows: In Section 2, the modelling of the Delta parallel robot is detailed. In Section 3, a model of geometric error is introduced. A new tolerance synthesis method is proposed in Section 4. Finally, the article ends with some concluding remarks in Section 5.