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.
Due to the energy-efficient of passive dynamic walking, the study of the passive biped walker is a popular area of scientific research [
The Simple Cell Mapping (SCM) method is usually used to estimate the DOAs of different passive biped walkers [
For making some concepts clearly, the basic procedures of the SCM method are introduced as follows [
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 [
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 [
The rest of the paper is organized as follows. The basic ideas and detailed procedures of the proposed method are introduced in Section
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
The domains in a two-dimensional system.
The DOA of the passive biped walker is usually small, which means a lot of regular cells are finally mapped into the sink zone. Since it only takes one step for the failure cell to map into the sink zone, we can find and delete the DOF which contains all failure cells from the whole DOI before cell classification. This will reduce the number of cells whose classified states need to be stored in cell classification. In order to find these failure cells, the image cell of each regular cell should be firstly obtained. Considering that the Poincare mapping result of a cell is totally independent of other cells, the image cells of all regular cells can be obtained together by any practicable parallel algorithm to improve the computing efficiency. Because these failure cells are no longer involved in cell classification, they have no effect on estimating the DOA. Therefore, the DOF can be found and deleted from the DOI at a lower discretization level for decreasing the total number of Poincare mappings. To compensate for the accuracy loss caused by a low discretization level, the DOA will be obtained from the remaining domains at a higher discretization level. From a low discretization level to a high discretization level, the whole method is divided into two procedures. The former is called cell deletion procedure, and the latter is called cell refinement procedure. How to determine two different discretization levels is a main challenge. The stability of domain is introduced to solve it. If the volume of a domain changes very little with discretization level increasing, the domain can be considered stable. The volume of a domain is equal to the number of contained cells multiplied by the volume of one cell. When a domain just enters the stable state, the discretization level is the optimal value. To verify the stability of a domain, the DOI should be discretized twice at two different discretization levels and the image cells of all regular cells also should be obtained twice. If this approach is applied, the total number of Poincare mappings will increase rather than decrease. We noted that the regular cell has a boundary like the DOI, so the cell also can be discretized into many smaller cells as the discretization of DOI. If these discrete smaller cells exactly have the same central points as the regular cells which are generated by discretizing the DOI at a higher discretization level, the effect of cell discretization is equal to discretizing whole DOI from a low discretization level to higher. Then, the cell discretization technique can be used to verify the stability of domain. Through discretizing all cells in a domain, the discrete smaller cells will form a new domain after deleting the cells that no longer belong to the domain. By comparing two volumes of the domain before and after cell discretization, a change rate of domain’s volume is obtained. The iteration algorithm is used to judge the criterion of domain stability for avoiding repeated calculation. In our improved method, it needs to verify the stability of DOF in cell deletion procedure and the stability of DOA in cell refinement procedure respectively. Through all iterations in two procedures, the discretization level of DOI will gradually increase with the number of iterations. When the iteration stops, the obtained DOA is stable and the discretization level of DOI is confirmed.
Based on above ideas, the implementation steps of improved method are presented in the following.
The scope of the DOI
The DOI is discretized with the initial discretization level
After a cell classification procedure, all failure cells in the approximate DOC are further deleted and the initial DOC
Because there are many iterations to verify the stability of domain in two procedures, this improved method could be called Multiple Iterations Cell Mapping (MICM) method. Figure
Flowchart of the MICM method.
The cell discretization is the key technique of the MICM method, so its detailed algorithm is introduced in this section. For avoiding confusion, the cells at different discretization levels must have a unified representation method before and after cell discretization, so the cell index and cell vector are introduced firstly.
For a
Obviously, the central point of the interval is
From (
Then, an arbitrary point
Before cell classification, all regular cells should be numbered sequentially to a positive integer sequence. The number of a cell is called cell index. There is a one-to-one consistent match between each cell index and each cell vector. Specially, if anyone component of a cell vector is out of the boundary of DOI, this cell is the sink cell and its cell index is appointed to 0.
Suppose that the first cell vector in DOI is
Now, there are three attributes to describe a regular cell, e.g., the cell index
Because the regular cell can be represented by the cell vector, the cell discretization is simplified to a process of obtaining cell vectors of all discrete cells at a high discretization level from the cell vector of a cell at a low discretization level.
Suppose that the edge
In following sections, the Simplest Passive Biped Walker (SPBW) is taken as an example to demonstrate the MICM method. The structure and basic motion of SPBW are shown in Figure
The simplest passive biped walker is walking on a slope. (a) The swing phase; (b) the collision event.
The slope angle
Obviously, the initial state of next step
The simulation parameters of the MICM method are listed in Table
Simulation parameters of the MICM method.
Parameter | Symbol | Value |
---|---|---|
Slope angle | | 0.004 |
Scope of the DOI | | |
| ||
Initial discretization level of the DOI | | 6×6 |
Deletion accuracy | | 0.2 |
Maximum deletion iteration number | | 5 |
Refinement accuracy | | 0.02 |
Maximum refinement iteration number | | 5 |
Discretization level of each cell discretization | | 3×3 |
Figure
The processes of estimating the DOA of SPBW by the MICM method. (a) Initial discretization of DOI; (b) initial DOF; (c) DOF after the first iteration of the cell delete procedure; (d) stable DOF; (e) approximate DOC and stable DOF; (f) initial DOA and initial DOC; (g) DOA and DOC after the first iteration of the cell refinement procedure; (h) stable DOA and its corresponding DOC.
As shown in Figure
Figure
The number of mappings and the volume of domain at each iteration. (a) The curves of the cell deletion procedure; (b) the curves of the cell refinement procedure.
After above processes, the DOI is finally discretized into 406×406 cells. Through 26267 Poincare mappings and 2 cell classification procedures, the DOA of SPBW contains 5127 cells. Using the same parameters and discretization level in the SCM method, the DOA consists of 5162 cells through 164836 Poincare mappings and 1 cell classification procedure. Although 35 cells missed in the DOA obtained by the MICM method, the total number of Poincare mappings sharply decreases. Because the Poincare mapping is a consuming-time process, the MICM method saves much time. For an estimation process, it is worthy to accelerate the computation with a little accuracy loss.
If there is no prior knowledge about a system, a proper DOI should be selected before formal simulation. The selection of DOI is a procedure of removing the infeasible domains from whole state space, so the cell deletion procedure, which can efficiently remove the failure domains from a given domain, will be useful to select the DOI.
The details of selecting the DOI with the cell deletion procedure are presented as follows. Firstly, a wider initial DOI is chosen to be the basic research scope. Next, a DOC can be obtained from the DOI by using a cell deletion procedure with a lower deletion accuracy. If the scope of DOC is not closed to the scope of the DOI, the scope of DOC is used to be a new DOI for next iteration. Otherwise, the scope of DOC is taken as the formal DOI.
The selection of SPBW’s DOI is taken as an example. The scope of initial angle and angular velocity of stance leg should be estimated firstly. The scope of initial angle of stance leg is easy to determine since it is only limited by the physical conditions. So the initial angle of stance leg certainly belongs to
The processes of selecting the DOI of SPBW. (a) Initial DOI and DOC (b) DOI and DOC after the first iteration; (c) DOI and DOC after the second iteration; (d) DOI and DOC after the third iteration; (e) DOI and DOC after the fourth iteration; (f) DOI and DOC after the fifth iteration; (g) the formal DOI.
Comparing with the DOA obtained by the SCM method, there are a few cells missed in the DOA obtained by the MICM method. The reason is that some domains are wrongly covered by the DOF when the discretization level is low. After many iterations in the cell deletion procedure, these domains cannot be covered by the approximate DOC at the discretization level of stable DOF. For decreasing the lost cells, it needs to find these wrong domains and add them into the approximate DOC at a higher discretization level. Since most lost cells are located near the boundaries of DOF and DOC, the domains that contain the cells on the stable DOF’s boundary should be considered firstly. These boundary cells will be discretized into smaller cells, and the discrete cells need to be verified for finding the domains where they really belong to by Poincare mapping. Because these boundary cells are discretized again for further verification, the total number of Poincare mappings will increase. This procedure mainly focuses on the boundary cells, so it can be called boundary refinement.
Figure
The processes of estimating the DOA of SPBW by the MICM method with boundary refinement. (a) The boundary cells in the stable DOF; (b) approximate DOC and stable DOF; (c) initial DOA and initial DOC; (d) the stable DOA and its corresponding DOC.
Table
Comparison between SCM method and MICM method for estimating the DOA of SPBW.
Method | Number of Poincare mappings | Number of cells in DOA | Percent of decreasing Poincare mappings | Percent of missing cells | Time[s] | Speed up |
---|---|---|---|---|---|---|
SCM method | 164836 | 5162 | - | - | 530.72 | - |
MICM method without boundary refinement | 26267 | 5127 | 84.06% | 0.68% | 89.03 | 5.96× |
MICM method with boundary refinement | 30471 | 5156 | 81.51% | 0.15% | 103.25 | 5.14× |
This paper presents a new multiple iterations cell mapping method for estimating the DOA of the passive biped walker. The MICM method consists of two procedures, i.e., the cell deletion procedure and the cell refinement procedure. The cell deletion procedure not only can delete DOF from the DOI at a low discretization level, but also can select a proper DOI from an uncertain wide state space. The cell refinement procedure can quickly find a stable DOA at a high discretization level. The major improvement of the MICM method is that the DOI is selected by the cell deletion procedure instead of many trails and the discretization level of DOI is determined by the cell refinement procedure instead of prespecified artificially. Comparing with the results of the SCM method, the new method can obtain the same DOA with much less calculation and time.
The MICM method can ensure the DOA reliable and yet avoid excessive calculation, so it is a valuable analysis tool for the passive biped walker. Although this paper only takes the simplest passive biped walker as an example to describe the usage of the MICM method, it also can be used to estimate the DOA of more complex passive biped walkers that have more global states variables. Since the MICM method is based on the SCM method, it can be used not only for the complex dynamical systems like the passive biped walker, but also for other nonlinear dynamical systems which have small DOAs.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research is supported by the National Natural Science Foundation of China