A novel approach for coverings transformation to define a robot workspace

The article discusses the application of optimization algorithms to solve the problem of determining the workspace of robots. The development of a numerical method for approximating the set of solutions of a system of nonlinear inequalities that describe the restrictions on the geometric parameters of a robot based on the concept of non-uniform coverings is considered. An approach is proposed that allows you to reduce the number of boxes and numbers describing each of the boxes. Reducing the number of boxes is achieved by combining the boundary boxes of the covering set and the area between them along one of the axes. The dimensions along the other axes are equal for all boxes, which allows them to be described by one number. The transition to the integer space is described. Thus, converting non-uniform covering sets to a partially ordered set of integers reduces computational complexity. An algorithm for converting boxes of a covering set is presented. The approach has been tested for a 3-RPS robot.


Introduction
Workspace is one of the most important characteristic of robots. The issues of structural synthesis, methods for studying the workspace, movement trajectory optimization of parallel robots are discussed in detail in [1][2][3][4]. The task of the workspace determination can be solved using various methods, in particular, deterministic ones. The actual problem for applying these methods is often significant computational complexity. One of the most effective deterministic methods is the nonuniform coverings method. The application of this method to determine the workspace is considered in works [5][6][7]. In [8], a comparison of two approaches is considered, one of which is based on the use of a system of inequalities to describe the structural constraints of the robot, and the other on the use of a system of equations.
One of the tools for implementing the method is iterative bisection. With each division, the box decreases by 2 times, respectively, the ratio of the sizes of the initial box and one of the boxes forming an approximation of the workspace can be estimated by the degree of division d, and the ratio itself is . With an increase in the degree of division d, the number of boxes, the combination of which describes the workspace, increases. Due to the increase in computational complexity for processing an approximated workspace of higher accuracy, the problem arises of reducing it by converting the resulting set of boxes describing the workspace. As part of this work, an approach is proposed for converting a covering set obtained using the method of non-uniform coverings into a partially ordered set of integers. It includes two components: reducing the number of boxes and the transition from the space of real to the space of integers. Application of the proposed approach and assessment of its effectiveness is considered on the problem of determining the workspaces of the 3-RPS robot.

Materials and methods
Consider the set of boxes which described the workspace. Using the system of inequalities , the workspace is described by the union of two sets , where is an internal approximation set that is included in the set of solutions to the system of inequalities, is the boundary set. For each from , the condition .
(1) For each from , the system of inequalities is fulfilled: (2) We introduce the following notation: is the boundary of the set , is the outer region for which the condition .
(3) The elements of the set are necessarily located between the elements of the sets and . Therefore, it is possible to describe the workspace as a set of boxes using the following approach. We introduce the following notation for the boundaries of the box: , .
(4) We denote two subsets in the set as and . The subset includes only those boxes at the boundary of which there is a point for which the system is satisfied: The condition for the subset is similar, but for it . It should be noted that the number m of boxes in the subsets and is equal to each box from corresponds to the box from with equal values , and , while for all points for which the condition is satisfied, is true. is the set of boxes and .
The n-dimensional box is described by 2n real numbers. The proposed approach to the conversion of boxes allows us to describe boxes with a smaller quantity of numbers, while integers. First, we consider this concept in the general case of converting the set of real numbers Y into the set of integers Z with the approximation accuracy δ (Fig. 1). For each of the set points, its coordinates in the space of integers are calculated: Moreover, the Hausdorff distance between the sets depends on δ: (7) The integers of one of the coordinates are likewise combined into intervals, for each of which the values of the remaining coordinates are equal. Let us consider the application of this concept to the transformation of the set of boxes to . For and the following holds: (8) The accuracy of determining the workspace allows us to estimate the Hausdorff distance between the sets and (Fig. 2a). The maximum distance is defined as (9) In order to reduce the Hausdorff distance, we modify the formula (7): (10) where -bias coefficients, which are determined by the formula: In this case, (Fig. 2b).
a b Figure 2. The Hausdorff distance: a) between the sets Y and , b) between the sets Y and taking into account the displacement coefficients.
We introduce the following variables: , It is worth noting that an additional coefficient of 0.5 is added to exclude rounding of the upper boundary value to the next integer. We synthesize a box conversion algorithm for n = 2, taking into account formulas (10) - (12) As the dimension increases, we add external loops and similar to actions with the coordinate.

Results and discussion
Consider the application of the method of non-un iform coverings to determine the workspace of one of the robot modules -a planar 3-RPS mechanism (Fig. 5), which consists of three kinematic chains containing variable-length rods pivotally attached to a fixed base at the vertices of an equilateral triangle (Fig. 3) The other ends of the rods are pivotally mounted at the vertices of an equilateral triangle on a movable platform. The input coordinates are the rod lengths ( 1 2 3 ,, l l l ), the output coordinates are the position of the geometric center of the moving platform in Cartesian coordinates (x, y) associated with the center of the base of the mechanism, and its rotation angle (φ) relative to the axis perpendicular to the plane of the base. R and r are the radii of circles describing triangles and, respectively. Define the workspace of the 3-RPS mechanism. To do this, we introduce restrictions on the geometric parameters of the mechanism (13) where are determined by the design parameters of the mechanism, is the current length of the i-th rod, If the points and are located at the vertices of equilateral triangles, then the change in the length of the rods is determined by the formulas (14) (15) (16) Algorithms for approximating the set of solutions of nonlinear inequalities were considered earlier in the authors' work [7]. To speed up the calculations, multithreaded calculations using the OpenMP library are used. This is considered in more detail in [8].
We use the proposed approach to transforming the covering set. In fig. 4a shows a visualization of the workspace symmetrical about the Y axis. The left half is described by the sets and , the right half is described by the set . The simulation results for mm, mm, , are presented in Fig. 4b. The calculation time for approximation accuracy mm, the grid dimension for calculating 8x8 functions using parallelization of computations into 8 flows on a personal computer was 6 seconds. a b Figure 4. The 3-RPS mechanism workspace: a) before and after the transformation of the covering set, b) fixed angle φ = 0 °: the blue area is the internal approximation, the yellow is the boundary.