Discrete Dynamics in Nature and Society

Volume 2019, Article ID 2868543, 11 pages

https://doi.org/10.1155/2019/2868543

## An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China

Correspondence should be addressed to Yu Wang; moc.621@megnik

Received 22 January 2019; Accepted 21 March 2019; Published 7 April 2019

Academic Editor: Charalampos Skokos

Copyright © 2019 Yu Wang 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

An indicator of a passive biped walker’s global stability is its domain of attraction, which is usually estimated by the simple cell mapping method. It needs to calculate a large number of cells’ Poincare mapping result in the estimating process. However, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of the passive biped walker. How to estimate the domain of attraction efficiently and reliably is a problem to be solved. Based on the simple cell mapping method, an improved method is proposed to solve it. The proposed method uses the multiple iteration algorithm to calculate a stable domain of attraction and effectively decreases the total number of Poincare mappings. Through the simulation of the simplest passive biped walker, the improved method can obtain the same domain of attraction as that calculated using the simple cell mapping method and reduce calculation time significantly. Furthermore, this improved method not only proposes a way of rapid estimating the domain of attraction, but also provides a feasible tool for selecting the domain of interest and its discretization level.

#### 1. Introduction

Due to the energy-efficient of passive dynamic walking, the study of the passive biped walker is a popular area of scientific research [1, 2]. These passive biped walkers can walk stably on a downhill slope only in the interaction between the leg and environment. The stable gait is usually a periodic or cyclic gait, which is shown as a manifold in the state space or a limit cycle in the two-dimension phase diagram [3]. However, these passive biped walkers will lose their stability when they are moving from a wrong initial state or encountering a very small disturbance [4]. An indicator of a passive biped walker’s global stability is its Domain Of Attraction (DOA), which is a set of all appropriate initial states that lead the walker to the perpetual walking. The walker with a larger DOA can walk stably in more complex walking conditions and increase tolerance to more disturbances in the uncertain environments. The DOA of the passive biped walker can be enlarged by actuating some joints with a control algorithm [5]. It is an important way to evaluate the performance of different control algorithms by comparing the sizes of their DOAs. Besides, the DOA can effectively guide to build some robust prototypes [6, 7]. The detailed structure of DOA is an important aspect of studying the nonlinear properties of the passive biped walker [8, 9].

The Simple Cell Mapping (SCM) method is usually used to estimate the DOAs of different passive biped walkers [6, 7, 10]. The SCM method is an effective way to study the global stability of nonlinear dynamical systems, and its basic idea is to obtain the global properties of system by using the ordered information of local finite cells rather than investigating the properties of all states in the infinite state space [11–13]. However, there are still some issues to be solved when estimating the DOA of passive biped walker by the SCM method.

For making some concepts clearly, the basic procedures of the SCM method are introduced as follows [11, 13]. Firstly, a feasible continuous domain with boundary is chosen from the infinite state space as the Domain Of Interest (DOI). The state space outside of the DOI is sink zone. Secondly, the DOI is discretized into a large number of () small uniform cells which are called regular cells. These regular cells are sequentially numbered with positive integers 1 to . The sink zone is regarded as the sink cell and numbered as 0. Thirdly, these numbered regular cells are classified to different groups by the classification algorithm. The cells in the same group have similar global properties. Suppose that the dynamics of an entire regular cell are represented by the dynamics of its central point. By taking the central point of a regular cell numbered as the initial condition of the Poincare mapping , the cell that contains its image point is called the image cell of regular cell . The mapping procedure is called cell-to-cell mapping or cell mapping procedure. Specially, the image cell of the sink cell is itself since the system’s evolution outside the DOI is out of our interest. At the same time, if the evolution of a cell cannot reach the Poincare section in a limited time, the image cell of this cell is appointed to the sink cell. The classification algorithm of these regular cells is to generate many mapping sequences based on the relationship of the regular cells and their image cells. A typical mapping sequence starts with an unclassified cell and continuously increases by adding the image cell of the sequence’s last cell to the sequence, like . The mapping sequence will stop when the image cell is a sink cell or a repetitive cell in the mapping sequence. According to the classified state of the end cell, the cells in the mapping sequence are classified to different groups. If the mapping sequence ends in the sink cell, all cells in the mapping sequence are classified to the sink group. If the end cell is unclassified, all cells in the mapping sequence are classified to a new group. If the mapping sequence is encountered with an already classified cell, all cells in the mapping sequence are classified to the exist group which contains the already classified cell. The mapping sequence will be generated repeatedly until all regular cells are classified. Each group contains some attractor cells and their basins of attraction. The DOA consists of all groups except the sink group.

There are three problems to be solved when using the SCM method to estimate the DOA of a passive biped walker. Firstly, there are no clear criteria to select a feasible DOI especially for the system without any prior knowledge. Because the DOI limits the scope of the feasible states and determines the evaluation result of a regular cell, it is important to select a proper DOI. Secondly, it is also uncertain to determine the discretization level of DOI. On the one hand, the DOI should be discretized at a high discretization level to satisfy the accuracy requirement. On the other hand, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of a passive biped walker. The discretization level cannot be too high for avoiding the memory space and computation time out of tolerance. So, the discretization level of DOI is a key factor that affects the accuracy of results and the efficiency of estimating method. Thirdly, the memory space is limited in the cell classification procedure. It needs a lot of memory space to store the classified state of each regular cell based on the cell classification algorithm. This will make the memory space is insufficient when the number of regular cells is large.

Some researchers have tried different ways to solve these problems. Jeon et al. selected the DOI of the simplest biped walker by calculating its theoretical falling boundary [14], but this method was difficult to apply for those complicated biped walkers which had complex falling boundaries. Zhang et al. proposed a bisection method to quickly determine the approximate edge of the simplest passive biped walker’s DOA, but this method must know the fixed point at the beginning and the obtained edge was too rough [15]. For avoiding the problem caused by a large amount of computation, the direct way is to increase the speed of calculating a single cell’s Poincare mapping. Li et al. proposed an algorithm to accelerate the speed of Poincare mapping in the heterogeneous platforms with CPU and GPU, but it is only used in some simpler walkers due to the programming method [9]. Another feasible way is to minimize the number of Poincare mappings while meeting the same accurate requirement. Liu et al. presented the Poincare-like-alter-cell-to-cell mapping method to decrease the total number of Poincare mappings [16].

This paper proposes a method to improve the SCM method for dealing with the aforementioned problems. The proposed method uses the multiple iteration algorithm to gradually increase the discretization level of DOI for decreasing the total number of Poincare mappings and uses the domain stability as the stopping criterion of iterations for ensuring the result is accuracy. The improved method mainly contains two stages, i.e., the cell deletion procedure and the cell refinement procedure. The cell deletion procedure finds and removes a stable failure domain from DOI, and the cell refinement procedure finds a stable DOA from the remaining domains of DOI. For finding a stable domain, the change rate of domain’s volume at different discretization levels, which are generated by decreasing the size of cell in each iteration, is taken as the indicator of domain stability. The DOI can be selected by repeatedly removing a stable failure domain from an uncertain wide state space until the scope of valid domain is stable. The discretization level of DOI can be determinated when the stable DOA is found. Since the cells in the failure domain have been removed before cell classification, the shortage problem of memory space is also solved. At last, the proposed method improves the overall efficiency of estimating process by reducing the total number of cell mappings without losing the accuracy of results.

In the cell mapping method, a common and popular tool named the subdivision algorithm also uses the iterative method to improve the accuracy of solutions. Dellnitz et al. introduced the subdivision algorithm for obtaining the invariant sets of nonlinear dynamical systems [17]. This algorithm can be integrated with most cell mapping methods to solve different problems, such as the multiobjective optimization problems [18] and the high-dimensional problems [13]. Although both the subdivision algorithm and the proposed method in this paper use the iterative method to decrease the number of cell mappings, they have different goals. The subdivision algorithm focuses on finding the invariant sets and the attractor cells with high accuracy and efficiency. However, the proposed method in this paper uses the iterative method to generate the domains at different discretization levels for verifying the stability of domain.

The rest of the paper is organized as follows. The basic ideas and detailed procedures of the proposed method are introduced in Section 2. In Section 3, the proposed method is applied on estimating the DOA of the simplest passive biped walker. To further improve the performance of the proposed method, some key issues are discussed in Section 4. Section 5 provides some conclusions about this paper.

#### 2. Method

##### 2.1. Introduction of Method

In the SCM method, the whole state space is divided into the DOI and the sink zone by the boundary of DOI. By discretizing the DOI, many regular cells are generated and taken as the initial conditions of Poincare mappings. According to the location of respective image cell, the regular cells can be classified into two classes. If the image cell of a regular cell is outside the DOI, the regular cell will lose opportunity to become the member cell of DOA. This regular cell is defined as failure cell. If the image cell of a regular cell is inside the DOI, the regular cell still has opportunity to become the member cell of DOA. This regular cell is defined as candidate cell. The domain covered by all failure cells is called Domain Of Failure (DOF) and the domain covered by all candidate cells is called Domain Of Candidate (DOC). Obviously, the DOI consists of the DOF and the DOC. Under different discretization levels, the DOI has its corresponding DOF and DOC.

Figure 1 illustrates the domains of state space in a two-dimensional system. The state space consists of the DOI and sink zone. The DOI contains a group of cells numbered from 1 to 30. The Poincare mapping is denoted as the arrow line, which points to the image point from the central point of a regular cell. The white cells are mapped into the sink zone, so they are the member cells of the DOF. The blue and grey cells are the member cells of the DOC, because their image cells are still inside the DOI. Especially, the grey cells are the member cells of the DOA since they have periodic relationship with each other.