A complete analytical solution for the dimensional synthesis of 3-DOF delta parallel robot for a prescribed workspace

Highlights

• Analytic Geometrical workspace calculation of Delta Parallel Robot for the first time.

• Obtaining the largest possible box which can be inserted in the robot workspace.

• Analytical dimensional synthesis of Delta parallel robot for a prescribed workspace.

• Optimal design by Lagrange multiplier based on the smallest robot criterion.

• Industrial Delta Parallel Robot investigating and rating from the workspace viewpoint.

Abstract

This paper presents an analytical approach for the dimensional synthesis of the 3-DOF Delta parallel robot for a prescribed workspace. The dimensional synthesis of parallel robots is a challenging problem for which obtaining an analytical solution is a cumbersome task and no appropriate analytical closed-form solution has been developed for the DPR so far. First, the workspace is analytically calculated based on inverse kinematic and it showed that the workspace of DPR comes from some geometric conditions. Afterward, the largest possible box which can be inserted in the workspace is obtained. Optimal analytical dimensional design for a prescribed workspace is calculated by considering three types of construction constraints. Optimization objectives based on subjective and the smallest robot criterion have been defined which is optimized by resorting to the so-called Lagrange multiplier method. The results revealed that in the best case of the existing industrial robots, workspace expression is only close to %87.29 of the optimal workspace of this paper and in optimal design it could have up to %51.70 smaller robot with the same given workspace.

Introduction

Among parallel kinematic Machines (PKMs), the simplest and sophisticated parallel robot which is capable of moving in 3D space with superior speed and robustness is the so-called Delta Parallel Robot (DPR). DPR was invented in the early 1980s by Clavel [1]. The purpose of DPR was to manipulate light and small objects at a very high speed, which has a widespread of interest in sorting and collating, i.e. pick-and-place applications, or even in the machine-tool industry due to increased stiffness [2]. DPRs are also desirable for application in millimeter scale [3]. Besides several advantages of parallel mechanisms, including, among others, the DPRs, having a limited workspace which is a major drawback. Based on the motion pattern performed by the robot, different types of workspace can be defined. The workspace analysis of robotic mechanical systems falls into three types: (1) Obtaining the workspace boundary, (2) Obtaining an optimal shape inscribed within the workspace of a robot and (3) Obtaining the dimension of a robot for a prescribed workspace, which is referred to as dimensional synthesis. In the literature, different approaches have been proposed for obtaining the workspace of robotic mechanical systems, as reported by Merlet [4] namely, numerical methods, discretization methods, and geometrical methods. Numerical methods can be readily applied to determine the boundary of the workspace but they are relatively time-consuming and inaccurate and provide little insight into the workspace. There are different approaches for the numerical methods, for instance Farzaneh et al [5] have found a singularity-free workspace by a technic based on sign change of Jacobian matrix and by offsetting they have found the largest circle center which can fit in the workspace. In [6], the workspace was partitioned equivolumetrically to obtain the orientation workspace. Karimi and Mousavi [7], [8] obtained the singularity-free workspace based on the convex optimization and improved the computational time of the iterative approach. In [9] an interval analysis is proposed in order to obtain an obstacle-free workspace of parallel mechanisms containing a prismatic joint per limb. In the latter study, the workspace determination is done by finding intersection points of rays emanating from a radiating point for planner manipulators by Snyman et al. [10]. There are also other numerical methods based on Monte Carlo technique like [11], [12], or straightforward search methods such as [13], [14]. Discretization methods [15], [16], [17] allow more convenient in dealing with constraints but step vise accuracy, huge number of nodes and not easily extendable to be used in comparing manipulators or design analysis are some drawback of this method. Geometrical methods such as [18], [19], [20], [21] are faster and more accurate and they bring some insight into the problems [22], such as optimal design and collision among the links, which is a definite asset in the design stage of the under study robot. However, such formulation cannot easily be achieved for any architecture of the robot. It should be mentioned that the geometric, algebraical and closed-form solutions are more flexible and easily extendable to be merged with optimizing problems [23], [24], [25]. Therefore, finding such closed-form solutions would be very valuable for optimization purposes. The design step would require understanding the relationship governing the workspace. Optimal design is often constrained, nonlinear, multimodal and without closed-form solution which often leads to numerical optimization methods like Sequential Quadratic Programming (SQP) [26], the Controlled Random Search (CRS) [27], the Genetic Algorithm (GA) [28], [29], [30], the Differential Evolution (DE) [31], [32], [33], and the Particle Swarm Optimization (PSO) [34], [35], [36]. Convergence performance of typical numerical optimization algorithms are compared in [37]. SQP is a deterministic and local but is highly depending on an initial guess, while DE, PSO, GA, and CRS are all probabilistic and global [37]. Liu, et al. [38] have used the combination of GA and SQP methods for dimensional synthesis of a delta mechanism. They used the result of GA as the starting point of SQP. In some of the numerical methods reported in the literature [12], [38], [39] the objective consists in maximizing the workspace base on prescribed subjective constraints. However, the problem can be regarded as designing a manipulator which satisfies a prescribed workspace and not to maximize the workspace. From this viewpoint, some researchers tried to define a cost function as an objective, such as Gosselin and Boudreau [40] which used GA algorithms to obtain an actual workspace as close as possible to the prescribed one as a cost function. Based on the GA approach, Laribi et al. [28] determined the dimension of the DPR to get the smallest workspace containing a prespecified region. Bulatovic et al [32] performed the optimization by resorting to DE approach with an objective function which represents the sum of squares of deviations of the current path out of the controlled space around the given path. Shiakolas et al. [33] have minimized a cost function by a combination of DE and the geometric centroid of precision positions technique. In some literature, multi-objective optimization is used for instance, Karimi et al [41], have considered geometric, workspace, and singularity constraints for the dimensional synthesis of a 3-RPR planar parallel manipulator by using McCormick relaxations and a branch-and-prune algorithm. Rao et al [42] used Pareto-optimal solution in dual objective optimization of hexaslides considering workspace and dexterity tradeoffs by SQP. Huang et al. [25] based on a thorough understanding effect of the design parameters on the kinematic performances, optimized the design of a hexapod-based machine tools for both the workspace and dexterity. In numerical methods, due to highly non-linear objective function the process is based on iterative approaches, time-consuming, needing to set controlling parameters and there is boundary error. However, the analytical solutions are preferred because they provide a reliable answer without getting stuck in local minima, although getting such closed-form solutions is a challenging task. Liu et al. [43] proposed a geometrical design method for a DPR with prismatic actuators by introducing maximum inscribed workspace, although workspace analysis in prismatic DPR, relative to DPR with revolute actuators, is less complicated. Chablat and Wenger [44] have considered bounded velocity and force transmission factors in addition to prescribed workspace for optimizing a simpler prismatic mechanism with the purpose of machining application. Le at al. [24] presented a geometrical analysis and based on their geometric view to the workspace shape, they recommend a design method, but it can be shown that the provided answer could not be the best possible answer. Mahmoodi et al. [23] have tried to find a closed-form solution based on maximum surrounded workspace for the PDR and introduced a geometric suggestion for the dimensional synthesis. However one can readily show that the proposed method cannot results on a suitable accuracy. In [45], geometrical determination of workspace calculation and workspace optimization for a newly invented 5-DOF hybrid robot have been done. In [19], the workspace of the under study robot is modeled based on CAD Boolean intersection, and using Simulated Annealing Algorithm for optimizing the workspace of DPR, which the processing time is about 20 min. Methods for general structure manipulators are needed for robots in which no specific solution exists [46] but ad hoc methods are desirable because they are faster or simpler.

As the main contribution of this paper, an analytical approach for the dimensional synthesis of the DPR is proposed. The introduced geometrical solution would be very fast and easily programmable in a any computer algebra system, such as Matlab and Maple. The objective criteria are defined in a manner to satisfy the real request of industrial design. In the proposed closed-form solution, the prescribed workspace and construction constraint are given as subjective and minimizing the size of the manipulator are the objective.

This paper is organized as follows. In Section 2, the kinematic equations of DPR which are essential for the workspace analysis are introduced. In Section 3, the workspace of each limb in different views is geometrically discussed. In Section 4, common workspace of limbs is calculated and behavior of it has been studied. In Section 5, the workspace is calculated analytically and proved that the workspace of the DPR is composed of some particular types of geometric shapes in different conditions. In Section 6, the largest box which could be inscribed in the workspace is calculated. In Section 7, optimization design parameters for prescribed workspace are analyzed, here objective criterion is having the smallest robot which can reach a given workspace. In Section 8, some construction constraints are added to optimization problem and in Section 9, by selecting some industrial examples, it has been shown that the introduced approach is quite an agile and optimal and also singularity in the prescribed workspace was discussed. Finally, the paper concludes with some hints as ongoing works [44].