2010 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Montréal, Canada, July 6-9, 2010
Adding Segmented Feet to Passive Dynamic Walkers Yan Huang, Baojun Chen, Qining Wang and Long Wang
Abstract— This paper presents a passive dynamic walking model with segmented feet. The model extends the Simplest Walking Model with the addition of flat feet and torsional springs based compliance on ankle joints and toe joints, to achieve stable walking on a slope driven by gravity. The pushoff phase includes foot rotations around toe joint and around toe tip, which shows a great resemblance to human normal walking. The segmented foot model is compared with single rigid foot model in simulations to study the advantages of adding toe joints to passive dynamic walkers. We investigate the effects of segmented foot structure on dynamic bipedal walking. Experimental results show that the model achieves efficient and stable walking on even or uneven slope.
I. I NTRODUCTION Recently, increasing studies on segmented foot models have been proposed to evaluate the effects of multisegmented foot structures on human normal walking [1]. Different from the traditional methods that represent the foot as a single rigid bar, multi-segmented foot models have been studied for investigations of clinical and research applications [3], adolescent gaits [2] and pediatric gaits [4]. The investigation of segmented foot has been extended to engineering areas. Several studies added segmented foot to humanoid robots to improve walking performance. Simulations and experiments on prototypes showed that adding toe joints could increase walking speed of biped robots [7], [8]. These works were carried on the humanoid robots based on the trajectory-control approach [5]. By controlling joint angles precisely, the robots can achieve static equilibrium postures at any time during motion. However, this kind of bipedal walking has low resemblance to human normal gaits and high energy consumption [10]. In contrast to the activecontrol bipedal walking mentioned above, passive dynamic walking [9] has been developed as a possible explanation for the efficiency of the human gait, which showed that a mechanism with two legs can be constructed so as to descend a gentle slope with no actuation and no active control. Several studies reported that these kinds of walking machines work with reasonable stability over a range of slopes [10]. Most studies of passive dynamic walking are based on the Simplest Walking Model [11], which consists of two rigid massless connected by a frictionless hinge at the hip, with a large point mass at the hip and a small mass at each point foot. Recently, most studies of passive dynamic walking focus on bipedal models with point feet or round feet which have clear disadvantages of being unable to achieve the start and stop of walking. This work was supported by the 985 Project of Peking University. The authors are with the College of Engineering, Peking University, Beijing 100871, China
[email protected]
978-1-4244-8030-2/10/$26.00 ©2010 IEEE
Several studies have been done on a flat foot shape in passive dynamic models [15], [14], [18]. Some of the research studied the effects of flat-foot structure on passive dynamic walking. However, the flat foot was modeled as a single rigid bar. Only a few studies have investigated passive dynamic bipedal walking model with segmented feet. Recently, [20] proposed a passive dynamic walking model with toed feet. The authors contributed to the investigation of the passive bipedal walking behavior under toe joint rotation. The toerotation phase is initiated by ankle-strike. Simulation results showed that the advantage of the proposed walker come from its relation to arc-feet walker. However, the effects of heelstrike and toe-trike during normal walking are ignored, which may influence the characteristics of bipedal walking [14]. In addition, the phase of rotation of the stance foot about the toe tip is ignored in this model, which makes the bipedal walking gait is far from natural human-like gait. In this paper, we extend the passive dynamic walking model with the addition of segmented flat feet and passive ankle joints and toe joints. Phase switching is determined by the direction of ground reaction force. The push-off phase includes rotation around toe joint and rotation around toe tip, which show a great resemblance to natural human gait. The effects of foot structure on motion characteristics including energetic efficiency and walking stability is investigated through simulation experiments. The proposed segmented foot model is compared with the rigid foot model to reveal the advantages of adding toe joint to flat foot. This paper is organized as follows. Section II describes the model in detail. In Section III, we show the experimental results. We conclude in Section IV. II. M ODEL A. Biped Model To obtain further understanding of real human walking, we propose a passive dynamic bipedal walking model that is more close to human beings. We add compliant ankle joints and flat segmented feet with compliant toe joints to the model. As shown in Fig. 1, the two-dimensional model consists of two rigid legs interconnected individually through a hinge. Each leg contains segmented foot. The mass of the walker is divided into several point masses: hip mass, leg masses, masses of foot without toe, toe masses. Each point mass is placed at the center of corresponding stick. Torsional springs are mounted on both ankle joints and toe joints to represents joint stiffness. To simplify the motion, we have several assumptions, including that legs suffering no flexible deformation, hip joint with no damping or friction, the friction between walker and ground is enough, thus the
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flat feet do not deform or slip, and strike are modeled as an instantaneous, fully inelastic impact where no slip and no bounce occurs. The passive walker travels on a flat slope with a small downhill angle. The process of push-off is dissipated
Fig. 2. Model of segmented foot. Torsional springs are added on ankle and toe joints. ms is toe mass and mf is the mass of whole foot.
a/c [19]. It is easy to convert the result of a/b in segmented foot model to a/c if toe length is known. Fig. 1. Model of the passive dynamic walker with flat segmented feet and compliant ankle joints.
into foot rotation around toe joint and around toe tip, which is the main difference between the passive walking models with rigid flat feet and with segmented flat feet. The toe and foot are restricted into a straight line during the swing phase. When the flat foot strikes the ground, there are two impulses, ”heel-strike” and ”foot-strike”, representative of the initial impact of the heel and the following impact as the whole foot contacts the ground. After foot-strike, the stance leg and the swing leg will be swapped and another walking cycle will begin. The passive walking is restricted to stop in two cases, including falling down, running. We deem that the walker falls down if the angle of either leg exceeds the normal range. And the model is considered to running when the stance leg lifts up while the swing foot has no contact with ground. And foot-scuffing at mid-stance is neglected since the model has no knee joints. We suppose that the x-axis is along the slope while the y-axis is orthogonal to the slope upwards. The configuration of the walker is defined by the coordinates of the point mass on hip joint and six angles (swing angles between vertical coordinates and each leg, foot angles between horizontal coordinates and each foot, toe angles between horizontal coordinates and each toe), which can be arranged in a generalized vector q = (xh , yh , α1 , α2 , α1f , α2f , α1t , α2t )T (see Fig. 1). The positive direction of all the angles are counter-clockwise. The segmented foot structure used in this paper is shown in Fig. 2. The foot mass is distributed at two point masses: one at the center of toe, and the other at the center of the rest part of the foot (ms and mf − ms in Fig. 2). We define foot ratio as the ratio of distance between heel and ankle joint to distance between ankle joint and toe tip, namely a/b in Fig. 2, but not a/c, to make it convenient to compare the proposed model with rigid foot model. Note that there is another definition of foot ratio with implications for the evolution of the foot in biomechanical studies, which is calculated by
B. Dynamics of Walking In the following paragraphs, we will focus on the Equation of Motion (EoM) of the bipedal walking dynamics of the proposed model. The model can be defined by the rectangular coordinates x, which can be described by the x-coordinate and y-coordinate of the mass points and the corresponding angles (suppose leg 1 is the stance leg): x = [xh , yh , xc1 , yc1 , θ1 , xc2 , yc2 , θ2 , xc1f , yc1f , θ1f , xc2f , yc2f , θ2f , xc1s , yc1s , θ1s , xc2s , yc2s , θ2s ]T
(1)
The walker can also be described by the generalized coordinates q as mentioned before: q = [xh , yh , α1 , α2 , α1f , α2f , α1t , α2t ]T
(2)
We defined matrix T as follows: dq T = (3) dx Thus T transfers the independent generalized coordinates q˙ into the velocities of the rectangular coordinates x. ˙ The mass matrix in rectangular coordinate x is defined as: M = diag(mh , mh , ml , ml , Il , ml , ml , Il , mf − ms , mf − ms , If , mf − ms , mf − ms , If , ms , ms , Is , ms , ms , Is )
(4)
Denote F as the active external force vector in rectangular coordinates. The constraint function is marked as ξ(q), which is used to maintain foot contact with ground and detect impacts. Note that ξ(q) in different walking phases may be different since the contact conditions change. We can obtain the Equation of Motion (EoM) by Lagrange’s equation of the first kind: Mq q¨ = Fq + ΦT Fc
(5)
where Fc is the contact force acted on the walker by the ground to meet the constraint of the stance foot.
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ξ(q) = 0
(6)
where Φ = coordinates:
∂ξ ∂q .
Mq is the mass matrix in the generalized Mq = T T M T
(7)
Fq is the active external force in the generalized coordinates: Fq = T T F − T T M
∂T q˙q˙ ∂q
(8)
We choose parameters of the passive walker in normal range which lead to natural human-like gait. The motion will stop if the walker moves to a phase not included in this gait (For example, the heel of stance leg rises up before the swing foot contacts ground). One step can be described by several phases (see Fig. 3(a)-(h), each subfigure represents a walking phase during one step):
Equation (6) can be transformed to the followed equation: Φ¨ q=−
∂(Φq) ˙ q˙ ∂q
(9)
Then the EoM in matrix format can be obtained from Equation (5) and Equation (9): Fq q¨ Mq −ΦT (10) = q) ˙ Fc Φ 0 − ∂(Φ ∂q q˙
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The equation of strike moment can be obtained by integration of Equation (5): Mq q˙+ = Mq q˙− + ΦT Ic +
(11)
−
where q˙ and q˙ are the velocities of generalized coordinates after and before the strike, respectively. Here, Ic is the impulse acted on the walker which is defined as follows: t+ Fc dt (12) Ic = lim t− →t+
t−
where Ic is the impulse acted on the walker by ground. Since the strike is modeled as a fully inelastic impact, the walker satisfies the constraint function ξ(q). Thus the motion is constrained by the followed equation after the strike: ∂ξ + q˙ = 0 ∂q
(13)
Then the equation of strike in matrix format can be derived from Equation (11) and Equation (13): + Mq q˙− q˙ Mq −ΦT (14) = Ic 0 Φ 0 C. Walking Phases Different parameters of the walking model (the length and mass of each part and the stiffness of ankle and toe joints) may result in different gaits. For example, if the step length and the ankle stiffness is large enough, the heel of stance leg will rise up before the swing foot contacts ground, which is called premature heel rise in previous research[15], otherwise, the whole stance foot will keep contact with ground during single-support phase. Dynamic switching of walking phases has been investigated in previous work[21]. We focus on the effect of segmented feet on walking characteristics with natural human-like gait in this paper. In this study, the contact of stance foot is modeled by two ground reaction forces act on the two endpoints of the foot, respectively. If one of the forces decreases below zero in the direction orthogonal to the slope, the corresponding endpoint of the stance foot will lose contact with ground and the stance foot will rotate around the other endpoint.
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Fig. 3. The phases of one step of the passive walking model with segmented feet. Each subfigure represents a walking phase during on step
1) Phase a: The phase of foot rotation around toe joint. The rear foot rotates around toe joint while the toe keeps contact with ground. The toe will lose contact with ground when the ground force acted on the toe joint in the direction orthogonal to slope decreases to zero. Then the walker will move to phase b. 2) Phase b: The phase of foot rotation around toe tip. The toe rotates around the contact point. There is no constraint at toe joint in this phase. The model will move to phase c when the toe rotates to the direction same to that of the foot, which leads to toe impact. If the toe tip of the stance foot loses contact with ground before toe impact, which means that the walker performs a non-natural gait, the motion will stop and the parameters will be retuned. Similar cases in other phases are treated in the same way. 3) Phase c: Toe-strike phase. There is an impact at the toe joint when the toe rotates to the direction same with the foot. After the strike, the toe joint is locked and the toe and the foot are restricted into a straight line. The toe joint will not be released until the leg performs push-off effect again two steps later. 4) Phase d: The phase of foot rotation around toe tip. The whole foot rotates around the toe tip as a rigid stick. Push-off consists of phase a, b, c and d. The walker will move to phase e when the foot loses contact with ground,
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which is determined by the ground reaction force acted on the toe tip of the trailing leg. 5) Phase e: Single-support phase. The swing leg has no contact with ground and swings freely. Foot-scuffing at midstance will not be taken into consider since the model has no knee joints. When the heel of the swing leg strikes ground the walker will move to phase f. 6) Phase f: Heel-strike phase. There is an impact between the swing leg and ground when the heel of swing leg contacts ground. After the impact, the toe joint of the trailing leg is released. The constraint of trailing foot after impact is determined by the impulsive forces act on heel, toe joint and toe tip. If non-natural phase (for example, the whole foot loses contact with ground immediately after impact) appears, the walking will stop and parameters will be retuned, otherwise, after the strike the walker moves to phase g. 7) Phase g: Double-support Phase. The toe joint of the trailing leg is released after heel-strike. The toe of rear leg keeps contact with ground while the other part of the foot rotates around toe joint. And the foot of leading leg rotates around the heel. When the toe of the leading leg strikes ground the walker will move to phase f. 8) Phase h: Foot-strike phase. There is an impact between the whole foot of leading leg and ground. After foot-strike, the stance leg and the swing leg are swapped and the model moves back to phase a, which means another walking cycle begins.
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(b) Fig. 4. (a) Angular trajectory of the thigh of passive dynamic walking with rigid flat feet. (b) Angular trajectory of the thigh of passive dynamic walking with segmented flat feet. a → b: swing phase; b → c: heel-strike; d → e: foot-strike; f → g: heel-strike of the other leg; h → i: foot-strike of the other leg; i → a: push-off phase. The main difference between (a) and (b) is that passive walking with segmented feet includes the phase of foot rotation around toe joint (i → j) and toe-strike (k → l).
D. Energetic Efficiency Energetic efficiency is another important gait characteristics. The energy consumption of passive dynamic based models is usually represented in the nondimensional form of ’specific resistance’: energy consumption per kilogram mass per distance traveled per gravity [10], [12]. However, for passive walkers on a gentle slope, specific resistance is not a suitable measure of efficiency, since all walkers have the same specific resistance for a given slope [14]. Therefore, similar to [14], walking velocity is used as the measure of efficiency, such that ”most efficient” is synonymous with ”fastest”. In general, there are two gait solutions of the passive dynamic walker for a given slope: short-period solution and long-period solution. The Short-period solution results in quicker steps and is more efficient than the long-period solution. However, short-period solutions are always unstable, while most long-period solutions are stable [11], [14]. Taking the dynamic stability and leg kinematics into account, many studies have suggested that long-period solutions are more indicative of human gait than short-period solutions [13], [14]. Thus, in this study, we only discuss the passive biped with long-period gaits. III. R ESULTS All simulations and data processing were performed using Matlab 7 (The Mathworks, Inc., Natick, MA). Based on the EoMs mentioned above, we analyzed the walking motion of the biped model presented in this paper. The walking
characteristics of passive walking models with rigid feet and with segmented feet are compared in the simulation experiments to reveal the effect of adding toe joints to passive dynamic walking. Parameters values used in the analysis are obtained from Table I. All mass and length are normalized by the leg mass TABLE I PARAMETER VALUES IN SIMULATIONS . Parameter hip mass toe mass toe length ratio slope angle
Value 2 0.04 0.15 0.03rad
Parameter foot mass ankle joint stiffness toe joint stiffness
Value 0.2 0.765 0.005
and leg length respectively. The spring constants (stiffness of ankle joint and toe joint) are normalized by both the mass and the length of leg. Foot mass contains the whole mass of foot, including toe mass. Toe length ratio is the ratio of distance from toe joint to toe tip and length of the whole foot. The ankle spring reaches equilibrium position when the foot is orthogonal to leg. And the equilibrium position of toe spring is obtained when the toe and foot are in the same line. Fig. 4 shows thigh trajectories of passive walking of the two models. The main difference is push-off phase. A. Walking Velocity of Segmented Foot Model Since the specific resistance keeps a constant during the passive walking on a slope, the efficiency is evaluated
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Fig. 5. Comparison of walking velocities of rigid foot model and segmented foot model. The curved surfaces are smooth processed based on the sample data. (a) Average walking velocity versus foot length and foot ratio for rigid flat-foot passive walking model. (b) Average walking velocity versus foot length and foot ratio for segmented flat-foot passive walking model. (c) The difference of walking velocity of the two models, obtained by (b) subtracts (a). Both walking velocity and foot length are normalized by leg length. Foot ratio is defined as the ratio of distance between heel and ankle joint to distance between ankle joint and toe tip.
more efficient than the rigid foot model.
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B. Adaptive walking on uneven terrain
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In the following paragraphs, we evaluate adaptive walking on uneven terrain to analyze stability of the proposed model. In the simulation, we changed the level slope to a slope with uneven surface to evaluate the ability of overcoming disturbance of passive dynamic walking with rigid flat feet and with segmented flat feet. (see Fig.7) Fig. 8 shows the
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Fig. 6. Walking velocities of rigid foot model and segmented foot model with normalized foot length as 0.225. The velocities are normalized by leg length.
Fig. 7. Adaptive locomotion on uneven slope. The stick diagram is obtained every 15 frames during a continuous walking. The passive walker with segmented feet successfully performs a stable walking on the slope with 0.625 percent leg length disturbance with no active control. The foot length normalized by leg length is 0.1875, and foot ratio is 0.3.
relationship between the maximal ground disturbance the walker can overcome and the foot ratio of flat foot bipedal model. The ground disturbance decreases monotonously as −3
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by the average forward walking velocity. In the following paragraphs, the walking velocities of different foot structures of segmented foot model and rigid foot model are compared to study the effect of adding toe joint to flat-foot passive walking model on energetic efficiency. The walking velocity of rigid foot model decreases monotonously as foot length or foot ratio grows (see Fig. 5(a)). For the segmented foot model, the walker moves slower for longer foot according to the main tendency (see Fig. 5(b)). Walking velocity achieves the maximum value when foot ratio is near 0.3 for any fixed foot length. A peak appears at relative large foot length (larger than 0.2) and foot ratio near 0.3, which is similar to the foot structure of human beings (if we convert the foot ratio to the definition of a/c, the new foot ratio is about o.4, which is close to real human foot structure as reported by [19]). The comparison of the two models shows that the walker with segmented feet moves slower than rigid foot model with small foot length, however, the velocity of segmented foot model is larger when the foot is long enough, especially when foot ratio is near 0.3. Fig. 6 shows comparison of walking velocities of the two models at different foot ratios with large foot length. The passive walking with segmented feet performs much better when foot ratio is 0.3. In another word, if the segmented foot ratio is close real human foot, the segmented foot model is
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Fig. 8. Adaptability of ground disturbance of rigid foot model with different foot ratio. The normalized foot length is 0.1875. The ground disturbance is also normalized by leg length.
the foot ratio grows, which is similar to the tendency of
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curved surface of walking velocity. In the case of short hindfoot and long forefoot (foot ratio is 0.2), the walker can return to stable motion cycle after ground disturbance larger than 0.7 percent of leg length. However, the maximal disturbance the model can overcome decreases below 0.2 percent leg length when the lengths of hindfoot and forefoot are comparable (foot ratio is 0.8). The relationship between ground disturbance and foot ratio of segmented foot model also shows great resemblance to the walking velocity curved surface. As shown in Fig. 9, the maximal value is obtained when the foot ratio is 0.3. In that −3
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Fig. 9. Adaptability of ground disturbance of segmented foot model with different foot ratio. The normalized foot length is 0.1875. The ground disturbance is also normalized by leg length.
case the model can overcome the ground disturbance more than 0.6 percent leg length. The adaptability of segmented foot model decreases greatly if the foot ratio changes. The results indicate that there exists a best foot structure of segmented foot model, which achieves both excellent adaptability and walking velocity. The motion characteristics of rigid foot model changes monotonously for different foot parameters. This can be explained by the relative simple foot structure described as a single rigid stick. Segmented foot model is more complicated than rigid foot model in both foot structure and walking sequence. The experimental results show that there is a suitable foot ratio for walking velocity (energetic efficiency) and stability. In other cases the walker performs not very well. Segmented foot model moves slower than rigid foot model with small foot length. In the case of large foot length (above 0.2) and suitable foot ratio (near 0.3), which are both close to those of human beings, the segmented foot model performs better velocity (efficiency) and comparable stability compared with rigid foot model. IV. C ONCLUSION In this paper, we extend the passive dynamic walking model with the addition of segmented feet and ankle joints. The phase of push-off is divided into foot rotation around toe joint and around toe tip, which show a great resemblance to human normal walking. The proposed model is compared with single rigid foot model. The effect of segmented foot structure on motion characteristics is investigated. Experimental results indicates that the segmented foot model with
foot structure close to human performs better in energetic efficiency and comparable walking stability compared with rigid foot model. There are several ways to extend this work in the future. The effect of toe length and toe mass also needs to be investigated to obtain more motion characteristics. In addition, bipedal robotic prototype with segmented feet could be built based on the analysis in this paper. R EFERENCES [1] N. Okita, S. A. Meyers, J. H. Challis and N. A. sharkey, An objective evaluation of a segmented foot model, Gait and posture, vol. 30, pp. 27-34, 2009 [2] B. A. MacWilliams, M. Cowley, D. E. Nicholson, Foot kinematics and kinetics during adolescent gait, Gait and Posture, vol. 17, pp. 214-224, 2003 [3] M. C. Carson, M. E. Harrington, N. Thompson, J. J. OConnor, T. N. Theologis, Kinematic analysis of a multi-segment foot model for research and clinical applications: a repeatability analysis, Journal of Biomechanics, vol. 34, pp. 1299-1307, 2001 [4] K. A. Myers, M. Wang, R. M. Marks, G. F. Harris, Validation of a multisegment foot and ankle kinematic model for pediatric gait, IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 12, no. 1, pp. 122-130, 2004 [5] K. Hirai, M. Hirose, Y. Haikawa, T. Takenaka, The development of the Honda Humanoid robot, IEEE International Conference on Robotics and Automation Proceedings, 1998, pp. 1321-1326. [6] K. Yamamoto, T. Sugihara, Y. Nakamura, Toe joint mechanism using parallel four-bar linkage enabling humanlike multiple support at toe pad and toe tip, Proc. of the IEEE-RAS 7th International Conference on Humanoid Robots, 2007. [7] K. Nishiwakiz, S. Kagamiy, Y. Kuniyoshiz, M. Inabaz, H. Inouez, Toe joints that enhance bipedal and fullbody motion of humanoid robots, Proc. of the 2002 IEEE International Conference on Robotics & Automation, Washington, DC, America, 2002. [8] R. Sellaouti, O. Stasse, S. Kajita, K. Yokoi, A. Kheddar, Faster and smoother walking of humanoid HRP-2 with passive toe joints, Proc. of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2006. [9] T. McGeer, Passive dynamic walking, International Journal of Robotics Research, vol. 9, pp. 68-82, 1990. [10] S. Collins, A. Ruina, R. Tedrake, and M. Wisse, Efficient bipedal robots based on passive-dynamic walkers, Science, vol. 307, pp. 10821085, 2005. [11] M. Garcia, A. Chatterjee, A. Ruina, and M. Coleman, The simplest walking model: stability, complexity, and scaling, ASME Journal Biomechanical Engineering, vol. 120, pp. 281-288, 1998. [12] M. Wisse, A. L. Schwab, F. C. T. Van der Helm, Passive dynamic walking model with upper body, Robotica, vol.22, pp.681-688, 2004. [13] C. E. Bauby, A. D. Kuo, Active control of lateral balance in human walking, Journal of Biomechanics, vol. 33, pp. 1433-1440, 2000. [14] M. Kwan, M. Hubbard, Optimak foot shape for a passive dynamic biped, Journal of Theoretical Biology, vol. 248, pp. 331-339, 2007. [15] D. G. E. Hobbelen, M.Wisse, Ankle Actuation for Limit Cycle Walkers, International journal of robotics research, vol. 27, pp. 709735, 2008. [16] S. Mochon, T. A. McMahon, Ballistic walking, Journal of Biomechanics, vol. 13, no. 1, pp. 49-57, 1980. [17] A. Ruina, J. E. A. Bertram, M. Srinivasan, A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition, Journal of Theoretical Biology, vol. 237, no. 2, pp. 170-192, 2005. [18] Q. Wang, Y. Huang, L. Wang, Passive dynamic walking with flat feet and ankle compliance, Robotica, vol. 28, pp. 413-425, 2010. [19] W. J. Wang, R. H. Crompton, Analysis of the human and ape foot during, Journal of Biomechanics, vol. 37, pp. 1831-1836, 2004. [20] R. P. Kumar, J. Yoon, Christiand, G. Kim, The simplest passive dynamic walking model with toed feet: a parametric study, Robotica, vol. 27, pp. 701-703, 2009. [21] Y. Huang, Q. Wang, L. Wang, Modeling passivity-based seven-link bipeds with dynamic switching of walking phases, Proc. of the 12th International Conference on Climbing and Walking Robots, 2009
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