Mathematical Problems in Engineering

Volume 2018, Article ID 3085930, 16 pages

https://doi.org/10.1155/2018/3085930

## Generating the Isocurve Representation for Configuration Space of Mechanisms

School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Zhengdong Huang; nc.ude.tsuh@gnauhdz

Received 17 November 2017; Accepted 4 April 2018; Published 14 May 2018

Academic Editor: Andras Szekrenyes

Copyright © 2018 Weirui Kang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes an approach for generating a set of isocurves as a representation for configuration space (CS) of a mechanism. An isocurve here is a curve in CS with some parameters fixed. Compared with conventional methods like box approximations, sampling points, and boundary tessellations, the isocurve-based representation has some advantages in space parameterization and data management. This approach directly formulates the joint loop equations in the form of kinematic matrices, which does not need any extra conversions and solves the equations with an isocurve-tracing method applying ODE solvers. Since the isocurves are connected in certain orders with the guide isocurves in a lower-dimensional space, tracing all the isocurves only needs one initial solution point for an isolated solution component. In addition, the proposed approach includes an interference-handling step, which trims off the collision portions of the isocurves by checking their feasibility according to the previously defined half-space constraints, and a measure for identifying the forward direction at singular points where the first-order derivatives vanish. The approach is implemented through programming and the results for a few examples show its effectiveness.

#### 1. Introduction

The configuration of a part in mechanism is its position and orientation in a world coordinate system and the configuration variables for a single part are simply its three-position coordinates and three-orientation angles, which usually serve as its configuration representation [1]. Configuration space (CS) of a mechanism is a representation for all the feasible positions and orientations of all parts in the mechanism. Taking into account the motion constraints on closed chains and the collision-free requirements between different parts, the CS is often one or more regions of certain manifolds in a high-dimensional space [2] and it reflects overall kinematic characteristics of the mechanism. Usually, the CS coordinates are the kinematic parameters, its manifold dimensionality shows the degree of parameter coupling imposed by loop constraints, and its boundaries indicate the range of feasibility from interference-free requirements. Therefore, configuration space is frequently used in mechanical design for kinematical identification, function verification, and design optimization [1, 3]. Operation planning and control is another important application of configuration space after the design is finished [2, 4]. In this case, a subset of mechanism’s CS can be utilized to describe the feasible motion ranges or the reachable space of a specific point of the mechanism after considering its interference with environmental obstacles and other actuation requirements, which is known as workspace in robotics.

Generally, there exist three kinds of representation schemes for mechanism’s CS; they are box approximations, sampling points, and boundary tessellations. Box approximation methods, which describe the range of CS with a number of boxes with various sizes, are mostly adopted in the design research community because the workspace volume as an important performance metric can be calculated conveniently from this representation [5, 6]. Sampling points methods are often used in path planning area, in which the points randomly generated within CS together with the edges connecting them serve as a representation for the feasible configurations and their transition relations [7, 8]. The graph form of the sampling points representation is well suitable for the purpose of optimal path search and there is no special difficulty in handling higher-dimensional problems. Boundary tessellation methods that only focus on the positions of boundary points of CS are also preferred by some researchers [9, 10] because of their simplified target in generating the representation and probably more precise results. Although the above CS representations can support the functions in design and planning in some ways, they cannot meet the requirements in more sophisticated design analyses and process planning that involves frequent queries of the CS points, especially for the CSs trimmed from lower-dimensional manifolds in higher-dimensional spaces because the identification of each CS point requires solving nonlinear equations with a large number of unknowns. For example, when one is planning the motion of a complex parallel mechanism, he may need to frequently retrieve the neighboring CS points in order to identify an optimal motion step. In this case, the well-prepared CS representation can greatly reduce the computation cost spent on the CS point retrievals.

This paper proposes to represent a CS with a set of trimmed isocurves in certain orders. An isocurve here is a curve in the CS manifolds with some parameters fixed. Compared with the conventional methods mentioned above, the isocurve-based representation has some advantages in space parameterization and data management, and thus it is convenient for querying individual points in CS. However, identifying these points in large number needs efficient and precise algorithms for solving constraint equations.

There exists plenty of research work on solving constraints in mechanisms and a good brief review could be found in [11]. Generally, the approaches developed in the past can be classified into two categories: algebraic methods and numerical methods. The algebraic methods include the elimination method [12, 13] and the Gröebner basis method [14]. The methods have advantages in finding all the solutions with high precision, but they usually suffer from the difficulties emerging in implementation that involves the equation form conversion and symbolic computation. Most of the numerical methods reported in literature are the continuation method [15–17] and the interval-based methods [11, 18–20]; the latter is more frequently used in practical cases, which generate box approximations for a CS. To avoid missing a solution, the interval-based methods check all the boxes in a CS whether they include a solution point and then minimize the boxes to achieve an appropriate precision. Although the computation cost may not be low, the calculation process is relatively simple compared to approaches involving the inverse computations. Usually, the forward computations from CS to constraint function values are carried out with the interval arithmetics [11, 18]. But partially using the inverse computations, such as the extreme point identification with optimization approaches [19] and the region exclusion by checking proper conditions for branch-and-bound approaches [20], is helpful for improving the efficiency. Newton-Raphson method, as a popular numerical method for solving general nonlinear equations, was also used for solving constraints in mechanisms [21–23]. While it usually produces the solutions with higher precision compared to the interval-based methods, it can only identify the solutions point by point. In this paper, an ODE-based approach is proposed to obtain the isocurves in the solution space of mechanism constraints.

This paper is organized into seven sections. In Section 2, the concept and representation of isocurves for a CS are presented. After this, the formulas for the ODE-based approach that generates isocurves are derived in Section 3. Section 4 focuses on the method for trimming the isocurves with interference-free constraints, while Section 5 gives a second-order derivative approach for identifying the forward direction at singular points. Section 6 presents the computational results of three examples obtained with the proposed methods. Finally, the paper is ended with some conclusions and outlooks in Section 7.

#### 2. Isocurve Representation for CS of Mechanism

For a given mechanism, let be the vector of motion parameters consisting of all its joint variables and assume that is constrained by equality constraints, which are mainly composed of the loop equations on all the closed kinematic chains. Then its configuration space without considering the collision-free requirement between parts in the mechanism isHere,is the real number set and is the* p*-dimensional space.

Partitionis, whereis the vector for* active parameters* whileis the vector composed of* passive parameters*. Here, active parameters are the motion parameters whose values can be changed independently while passive parameters are those whose values are determined by solving the equality constraints for given active parameters. Then, the domain of active parameters can be defined as follows:In addition, useto denote the set of-points in that are associated with a -pointin . For a given,consists of all the-points, made of , which satisfy the equality constraints. Generally,includes* m* points for a mechanism with* m* solution components.

For a fixed point , define the set extended fromby making its first* k* coordinates varying and the CS’s subset . An* isocurve* of is a curve which has all the active parameters fixed except one of them. Let be the variable active parameter and then the - at can be expressed withHere, defines the starting point of an isocurve whilegives its variable. It is worth noting that the* isocurve variable*may not vary monotonically on the curve like the parameter of an ordinary parametric curve. Furthermore, we define to be an isocurve* cluster*, whereis called the* guide* of the cluster and it specifies all the starting points of the isocurves in. Actually,forms* an isocurve representation of configuration space * because

Therefore, , that is, , is represented with the-clusterat guide . Recursively,is represented with the-clusterat guide, and, finally,is represented with a single-at guide . Summarily, they areAccording to the above formulation, starting with a given feasible point or a point in, a hierarchical isocurve representation can be created in the order implied in (5). The representation mainly has three characteristics (see Figure 1): (1) it is parametric because the points incan be arranged in the hierarchical order; (2) the elemental entities of the representation are curves in; (3) the curves are grouped and connected in a specific way, which facilitates the generation and query of them.