〈一般研究課題〉

協同現象を活用した二脚受動歩行機械の引き込み領域の拡大

助 成 研 究 者

名古屋大学

石黒章夫

Enlarging Attractor Basin of a Passive Dynamic Walker
by Exploiting Cooperative Phenomena
Akio Ishiguro(Dept. of Computational Science and Engineering, Nagoya University)
Recently, passive dynamic walkers have been attracting lots of attention in the field of legged robots,in the hope that they allow us to realize surprisingly natural and high energy efficient walking forhumanoid bipeds. Despite these interesting properties, current passive dynamic walkers have been
faced with serious issues that have to be resolved, particularly in their potential for coping withenvironmental perturbation: they are significantly sensitive to the change of gradient of slope andinitial condition. In light of these facts, this study intensively discusses the enlargement of the attractor
basin of a biped passive dynamic walker, exploiting "cooperative phenomena" by carefully increasing
its degrees of freedom.

1. Introduction
Since the pioneering work of McGeer in the late 1980s, the concept of passive dynamic walking(hereafter PDW) has been attracting lots of attention in the field of legged robots 1) A significant featureof PDW robots is that they have no control system; they merely exploit their intrinsic mechanicaldynamics carefully designed, and show surprisingly natural and human-like walking on a gentle slope
without any actuation except for gravity. Due to this, PDW robots are expected to largely contribute to
the realization of energy- efficient humanoid bipeds, which has been totally difficult with theembodiment of current biped robots.
As observed in many nonlinear dynamical systems, it is widely recognized that there exists
entrainment phenomena in PDW robots 2) : they continue stable locomotion as long as their state iswithin the basin of attraction. Recently, however, Schwab et al. have pointed out that the basin ofattraction of PDW robots are significantly small. Furthermore, they have clarified that this basin ofattraction becomes smaller as the gradient of slope increases; consequently, this will disappear when
the gradient exceeds a critical value 3).

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On the other hand, it is well known that mutually-interacting many-body systems often show
cooperative phenomena, i.e., temporary and/or spatial ordered patterns emerge as a result of theinteraction among the elements. A typical example of the cooperative phenomena can be observed in
the paramagnetic-ferromagnetic phase transition of an Ising ferromagnet . Despite that a small-scaled
system --- a system with small degrees of freedom --- does not show a "clear" phase transition due tothe thermal fluctuations , the transition from one phase to another becomes "sharper" and "clearer" as
the system size increases. This is because when several elements --- or degrees of freedom --- deviate
from the ordered state, the interaction with the rest of elements will immediately bring them back to
the original ordered state. In sum, cooperative phenomena work to stabilize the ordered state over thewhole system.

Robustness(Size of Attractor Basin)

Emergence of cooperative phenomena
Emergence of cooperative phenomen
aTransition point
Transition
point

Degrees of Freedom (DOF)
Figure 1: A schematic diagram representing the concept of this study.

Looking back at the PDW robots developed so far from the above viewpoint, one may say that they
have been built with relatively small degrees of freedom. This will automatically lead to the following
conclusion: the cause for the brittleness against perturbations observed in the current PDW robots
might mainly stem from their extremely simple structure --- their degrees of freedom might not be
sufficient to elicit cooperative phenomena.
Based on the above considerations, this study is intended to deal with the enlargement of the basinof attraction of a PDW robot by exploiting cooperative phenomena. To this end, the following methods
are employed: (1) increasing the degrees of freedom of the embodiment; and (2) implementing passiveelements --- springs and dampers --- into some of the joints, the latter of which is also expected toefficiently elicit cooperative phenomena. This idea is schematically illustrated in Figure 1.
Simulations have been carried out in a physical simulator developed for this purpose in order toverify the feasibility of our idea. The preliminary results obtained support this idea, and have clarified
some interesting phenomena for further investigation.

2. Related WorkIn this section, some notable conventional research on PDW robots are introduced. Collins et al.

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developed a 3D PDW biped robot by exploiting the movement of its arms as yaw compensation 4).
Although they successfully demonstrated that this robot walked in the real world, it is stillquestionable how much this robot is robust against environmental perturbation as well as the changeof initial condition.
Wisse et al. proposed a design for a 3D PDW biped with a pelvic body, which was employed tocompensate the undesired roll and yaw motion 5). They effectively showed that this compensator
allowed the robot to move in a similar manner of a 2D PDW biped. They, however, found that thisrobot is less stable compared to the 2D PDW biped they have developed.
Fujimoto et al. implemented a torso and two arms to a PDW robot, and showed that thesynchronization between the leg and arm movement contributes to enlarge the range of walkable
gradient 6). However, their robot is not a purely mechanical system; they artificially controlled theposture of torso and the phase relationship between the legs and arms.
As far as the authors have been informed, there are no studies in existence so far which explicitly
deal with the enlargement of attractor basin of purely mechanical PDW robots from the viewpoint of"cooperative phenomena".

3. Proposed Method
3.1 The model
Since there still remains much to be understood about which additional degrees of freedom andwhich additional passive elements implemented will play an essential role to elicit cooperative
phenomena, this paper particularly focuses on the following two different embodiments for PDW asa first step of investigation:
Embodiment 1: A kneed biped with an upper body (i.e. torso, shoulder, head), and a waist joint
rotatable around the yaw and roll axes (see Figure 2). This embodiment is employed for theintensive investigation of the role of upper body.

HeadShoulder

Waist(Yaw)

Torso
Waist(Roll)
Hip
Thigh

Hip

Shank
Foot

Knee
[Front View] [Side View] [DOF]
Figure 2: A schematic of embodiment 1 (3D model).

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Embodiment 2: A kneed biped with ankle joints which are allowed to rotate depending on thefoot contact condition (see Figure 3). This embodiment is dedicated to intensively investigate
the role of ankle joints for the emergence of cooperative phenomena.

HipBody
ThighKnee
ShankAnkle

FootFigure 3: A schematic of embodiment 2 (2D model).

Note that passive elements (i.e. springs and dampers) were implemented to the hip and waist
joints around the yaw and roll axes in embodiment 1, and the hip, knee and ankle joints inembodiment 2.
3.2 Evolving PDW Robots
As there are no current existing theory about how such PDW can be created, a synthetic approachis the method of choice in optimizing the body parameters of these embodiments. Therefore, agenetic algorithm (hereafter, GA) was employed for this purpose. In addition, in order to efficiently
create such PDW bipeds, an incremental evolutionary scheme is adopted. In the following, it isshown how each stage of the evolution was conducted:
The First Stage of Evolution
The aim of this stage is to create the basic structure of PDW bipeds, which can perform a stable
walk on a gentle slope. More specifically, the body parameters --- mass/length of each body part andspring/damper constants of each joint --- and the initial torque applied impulsively to the hip forlaunching the biped were encoded as a binary-bit string (i.e. chromosome) for the evolutionary
process. Each individual was tested, and allowed to move for the prespecified evaluation period. The
following fitness function was employed for the evaluation:

where D and S are the resultant walking distance traveled, and the number of steps performed
during the evaluation period. This fitness function encourages the generation of stable andsuccessful walking by alternatively stepping forward. One hundred individuals were randomly
generated as the initial population; they were evolved through 50 generations by applying genetic
operators (i.e. mutation and crossover). Note that in this stage the gradient of slope is prespecified
and remains unchanged during the entire evolutionary process.

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The Second or Higher Stage of Evolution
The aim of this stage is to create PDW bipeds that can negotiate environmental perturbation. In
other words, this stage is dedicated to enlarge the basin of attraction of the PDW bipeds obtained in
the previous stage. To do so, the population obtained at the last generation in the previous stage wasused as the initial population of this stage; these individuals were successively evolved through 50
generations under the existence of environmental perturbation. More specifically, each individual
was tested on a slope, the gradient of which was randomly changed in every Δt [sec] around thegradient in the first stage within a prespecified range. Note that all individuals in a generation wereevaluated under the same pattern of gradient changes, while a different pattern was employed in thenext generation. In the following simulations, the gradient of slope θi in generation i wasdynamically fluctuated, which is given by

where Δt and T denote the time interval of gradient changes and the evaluation period,respectively. ai,j is a constant determined randomly within a range of [amin, amax].

4. Results
For efficient investigation, a simulator has been developed. The following simulations have been
conducted with the use of a physics-based, three-dimensional simulation environment named Open
Dynamics Engine 7). This environment simulates both the internal and external forces acting on theagent and objects in its environment, as well as various other physical properties such as contact
between the agent and the ground within an acceptable time limit.
4.1 Preliminary Experiments
Experiments under Embodiment 1
Figure 4 explains a view of the simulator developed for embodiment 1. The following
simulations were conducted under the condition of the gradient of slope 3 [deg]; and theevaluation period T = 30[sec].
Shown in Figure 5 represents a typical evolutionary run leading to successful PDW. As thefigure indicates, the fitness value increases as the evolutionary process proceeds. Figure 6
denotes a representative example of the evolved PDW biped. This snapshot was obtained byrecording every 0.2 [sec], and only the right leg is shown for clarity. From the figure, it isrecognized that the robot smoothly enters stable and periodical locomotion after a short transitionperiod.

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(a) Front view(b) Side view

Figure 4: A view of the simulator developed for embodiment 1.

1400
1200Fitness

1000
800
600
400
200
005101520253035404550

GenerationFigure 5: Transition of the fitness in the first stage of embodiment 1.

Figure 6: A representative locomotion of the best evolved PDW biped
obtained in the first stage of evolution (see from left to right).

Experimentsunder Embodiment 2
Figure 7 illustrates a view of the simulator for embodiment 2. Since this embodiment isdedicated for the intensive investigation of the role of ankle joints in the enlargement of attractor
basin, the collision between the left and right legs was not considered for simplicity; they moved
independently on the same sagittal plane. The following simulations were conducted under thesame condition as that of embodiment 1 except that the gradient of slope was set to 4 [deg]. A
typical example of the fitness transition in the first stage under this embodiment is depicted inFigure 8. Figure 9 shows a representative locomotion pattern of the best evolved PDW biped
obtained in the first stage. This snapshot was obtained in the same way as shown in Figure 6.

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Figure 7: A view of the simulator developed for embodiment 2.

1200
1000

Fitness

800600400
200005101520253035404550

Generation
Figure 8: Transition of the fitness in the first stage of embodiment 2.

Figure 9: A representative locomotion of the best evolved PDW biped
obtained in the first stage of evolution (see from left to right).

4.2 Verification of Robustness
Experiments under Embodiment 1
In order to verify the robustness of the best evolved PDW biped obtained in the first stage, its
ability for coping with the gradient changes was investigated. More specifically, the number ofsteps successfully traveled was recorded under a different gradient of slope never explored in its
evolutionary process. Notice that this PDW biped was evolved on the slope, the gradient of which
was set to 3 [deg]. The result is shown in Figure 10. As the figure indicates, the number of steps
observed fluctuates drastically between 3 and 6 [deg]. In sum, the PDW biped obtained in the firststage does not show a consistent performance in terms of robustness.

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Based on the above consideration, the incremental evolutionary scheme described in section
3.2 was conducted in order to enlarge the basin of attraction of this PDW biped. The last
population obtained in the first stage was further subjected to the following two-staged
evolutionary processes --- the second and third stage of evolution. During these stages, eachindividual of the population was tested under the existence of environmental perturbations: amin = 0[deg] and amax = 6 [deg] for the second stage; followed by amin = 0 [deg] and amax = 8 [deg] for thethird stage (see section 3.2). Δt was set to 0.2[sec] for both stages.
The verification of robustness of the best evolved PDW biped obtained in the third stage wasperformed in a similar manner of Figure 10. (see Figure 11). As the figure explains, the range inwhich the biped can cope with the change of gradient is significantly expanded. Furthermore, incontrast to Figure 10, the fluctuation of its performance is also improved.
For a detailed investigation of the role of degrees of freedom, this embodiment was also evolvedin the same manner (i.e. from the first to the third stage of evolution) under the condition where
its waist joints was forcibly fixed (see Figure 12). As the figure indicates, the PDW biped without
these degrees of freedom shows less performance. This strongly suggests that the degrees of5050

454540403535

Number of steps

Number of steps

freedom in upper body significantly contributes to enlarge the basin of attraction of PDW bipeds.

30
2520
153025
20
15

10

10

5

5001234567890

100

12

3

Gradient of slope [deg]4Figure 10: Robustness of the PDW biped obtained in the firststage against gradient changesembodiment 1).

4540

Number of steps

678910Figure 11: Robustness of the PDW biped obtained in the thirdstage against gradient changes (embodiment 1).

5035302520
15105005

Gradient of slope [deg]12345678910

Gradient of slope [deg]Figure 12: Robustness of the PDW biped obtained in the firststage against gradient changes (embodiment 1).

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Experiments under Embodiment 2
The incremental evolutionary scheme was conducted in the same manner as in the case ofembodiment 1 except that amin, amax was set to (1, 7) [deg] in the second stage and (0, 9) [deg] inthe third stage, respectively. Figure 13 and Figure 14 represent the robustness of the best
evolved biped obtained in the first stage and the third stage against the change of the gradient,corresponding to Figure 10 and Figure 11 of embodiment 1, respectively. In order to investigate
the role of the degrees of freedom in ankle joints, embodiment 2 with forcibly locked ankle joints
was also evolved in the same manner. Shown in Figure 15 indicates the result after the evolutionof three stages. As can be recognized from Figure 14 and Figure 15, the biped's walkable range ofthe gradient is expanded by the degrees of freedom in ankle joints. However, a significant

60605050

Number of steps

Number of steps

improvement of the robustness cannot be observed in this experiment.

4030
2040
30
20
10

1000

12345

678

90

100

123

Gradient of slope [deg]45678910

Gradient of slope [deg]Figure 13: Robustness of the PDW biped obtained in the firststage against gradient changes (embodiment 2).

Figure 14: Robustness of the PDW biped obtained in the thirdstage against gradient changes (embodiment 2).

60

Number of steps

50
40
3020
100012345678910

Gradient of slope [deg]Figure 15: Robustness of the PDW biped obtained in the third stage against
gradient changes (embodiment 2 with locked ankle joints).

5. Conclusionand Future Work
This paper has investigated the enlargement of basin of attraction of PDW bipeds. To this end, theconcept of cooperative phenomena, widely observed in mutually-interacting many- body systems, hasbeen introduced. The preliminary simulation results have clarified some interesting phenomena forfurther investigation, which can be summarized as: (1) increasing the degrees of freedom in upperbody (i.e. embodiment 1) outperforms the one in lower body (i.e. embodiment 2); (2) do the current− 109 −

degrees of freedom and the implementation of passive elements successfully lead to the emergence ofthe cooperative phenomena?; and (3) how the increase of the degrees of freedom in both upper (e.g.
pelvic body, torso, arms) and lower body (e.g. foot, ankle) affects the property of the attractor basin.

References
1) T. McGeer, "Passive Dynamic Walking", The International Journal of Robotics Research, Vol.9,
No.~2, pp.62-82, 1990.
2) K. Osuka, and K. Kirihara, "Motion Analysis and Experiment of Passive Walking Robot Quartet II",
Journal of the Robotics Society of Japan, Vol.18, No.5, pp.737-742,2000.
3) L. Schwab and M. Wisse, "Basin of Attraction of the Simplest Walking Model", ASME 2001 DesignEngineering Technical Conferences, DETC2001/VIB-21363, 2001.
4) S. H. Collins, M. Wisse, and A. Ruina, "A Three-Dimensional Passive-Dynamic Walking Robot with
Two Legs and Knees", The International Journal of Robotics Research, Vol.20, No.7, pp.607-615,2001.
5) M. Wisse, A. L. Schwab, R. Q. vd. Linde, "A 3D Passive Dynamic Biped with Yaw and Roll
Compensation", Robotica, Vol.19, pp.275-284, 2001
6) H. Fujimoto, A. Sano, and Y. Ikemata, "Motion Analysis of Passive Biped Walking System with
Upper Limb", Proc. of the 19th Annual Conference of the Robotics Society of Japan, pp.41-42, 2001.
7) http://opende.sourceforge.net/− 110 −