Applied Mechanics and Materials Submitted: 204-09-9 ISSN: 662-7482, Vols. 687-69, pp 279-284 Accepted: 204-09-27 doi:0.4028/www.scientific.net/amm.687-69.279 Online: 204--27 204 Trans Tech Publications, Switzerland Effects of a Passive Dynamic Walker s Mechanical Parameters on Foot- Ground Clearance Xinyu Liu, a, Xizhe Zang, b, Jie Zhao, c State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, China a wo-shiliuxinyu@63.com, b zangxizhe@hit.edu.cn, c jzhao@hit.edu.cn Keywords: Foot-ground clearance, passive dynamic walking, mechanical parameters, biped robot Abstract. Unlike human beings, a robot will fall without a sufficient walking foot-ground clearance, which is essential to a success walking. However, to obtain less calculating time and simpler analysis, foot scuffing is ignored in most studies during numerical simulations of passive dynamic walkers which can walk down a gentle slope and are actuated only by their own gravity. So this paper initials a study on the effects of a passive dynamic walker s mechanical parameters on foot-ground clearance and the results can be used to make a further parameters optimization based on walking stability analysis. A passive dynamic walking model with a hip joint, knee joints, ankle joints and an upper body and a prototype were built and numerical simulations were implemented to analyze the effects of mechanical parameters on foot-ground clearance. Finally, the results were validated in prototype experiments. Introduction Humanoid robots are an appealing research field for researchers as they can move and act as human beings to some extent. The traditional biped walking robots, such as Asimo[] and HRP-2[2], have very versatile gaits as they can jump with one leg, walk among obstacles or even run. However, they consume high energy and look quite unnatural when walking[3] due to their ZMP control method[9]. Inspired by passive toys, McGeer first proposed the concept of Passive Dynamic Walking [4]. Walking robots built based on this concept can walk down a gentle slope stably with human-like gaits only under the effect of gravity. Some successful prototypes[5,6] built recent years have proved that fully passive dynamic walkers can walk on level ground by adding controls to some joints. The main reason for their low energy-consumption walking is that they walk to reach dynamic equilibrium but not to maintain static equilibrium all the time. To maintain dynamic equilibrium and walk without falling, four conditions must be satisfied: () The swing leg swings to the front of the stance leg rapidly[7], (i.e. prevent falling forward). (2) The robot s center of mass(com) swings across the highest position of the COM, (i.e. prevent falling backward). (3) The knees joints can be locked timely before foot-ground impact, (i.e. prevent the swing leg bending when it make a collision with the ground). (4) Sufficient foot-ground clearance, (i.e. prevent foot scuffing). So, sufficient foot-ground clearance is essential to a successful walking. However, to obtain less calculating time and simpler analysis, foot scuffing is ignored in most studies during numerical simulations of passive dynamic walkers. So this paper aims to study the effects of a passive dynamic walker s mechanical parameters on foot-ground clearance. Though by using fine control scheme the passive could obtain sufficient foot-ground clearance, it s necessary to obtain sufficient foot-ground clearance only by optimized mechanical parameters to decrease the energy consumption in walking. A passive dynamic walking model with an upper body was built and numerical simulations were implemented. The factors that can affect the foot-ground clearance are given in this paper. Finally, the results were validated in prototype experiments. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 30.203.36.75, Pennsylvania State University, University Park, USA-2/05/6,23:22:)
280 Manufacturing Technology, Electronics, Computer and Information Technology Applications Walking Model To study the effects of the passive dynamic walker s mechanical parameters on foot-ground clearance, a planar passive dynamic walking model consists of an torso, flat feet, a hip joint, knee and ankle joints was built in this study which could walk on slope without any actuation or walk on level ground by adding hip actuation, as shown in Fig.,whereθ, θ 2, θ 2 s and θ3 denote the angle of the stance leg, swing upper leg, swing lower leg and torso between the vertical direction respectively.the torso is constrained in the middle of the two legs all the time by a bisecting mechanism [8]. One step of walking can be divided into four phases, as shown in Fig.2. Phase (a): 4-link swing phase. After impact of the swing leg with ground, the former stance leg changes into the new swing leg, and the knee joint is unlocked to rotate freely. Phase (b): 4-link impact phase. When the swing leg (dashed line in Fig.2) has rotated to a certain angle (before foot impact), the knee joint is locked to prevent falling caused by knee-bending. Phase (c): 3-link swing phase. After the knee is locked, the thigh and shank change into one leg and the 4-link model becomes to the 3-link model. The swing leg will still rotate forward. Phase (d): 3-link impact phase. The swing leg impacts the ground and changes into the new stance leg. By now, one cycle of walking is ended. Fig. Passive dynamic walking model Fig.2 Walking phases Equations of Motion For the reason of easier calculating and comparability between different robots, all parameters are turned into dimensionless form by dividing the parameters by leg length l or total mass of the robot M, as shown in table. Talbe Non-dimensionless and dimensionless mechanical parameters Parameter Description Dimensionless process Dimensionless parameter l t Thigh length l t/l k lt b t Thigh center of mass(com) b t/l t (l t/l) k bt b s Shank center of mass b s/l s (l s/l) k bs b b Body center of mass b b/l k bb m b Body mass m b/m k mb m t Thigh mass m t/m k mt m s Shank mass m s/m k ms j t Thigh moment of inertia J t/(m tl 2 t ) (m tl 2 t /Ml 2 ) k jt j s Shank moment of inertia J s/(m sl 2 s ) (m sl 2 s /Ml 2 ) k js t time t/(l/g) /2 τ The hip mass mh has nearly no effect on the motion of the swing leg, so is not considered here. The foot length lf will lead to foot scuffing inevitably with the increasing of its value, so the value of lf is kept at 0.25 in this paper. The model can be described by the generalized coordinates q.there are three DOF at phase (a), so the generalized coordinate q = [ θ, θ2, θ2 ] T s. The upper body is constrained at the middle of the two legs all the time as θ3 = ( θ + θ2) / 2.
Applied Mechanics and Materials Vols. 687-69 28 The dynamics of walking can be obtained by the method of Lagrange. Dynamics of phase (a) is written in the form Mf ( θ, θ, θ, & θ, & θ, & θ )&& θ = Ff ( θ, θ, θ, & θ, & θ, & θ ) + Sτ Ⅰ 2 2s 2 2s Ⅰ 2 2s 2 2s () In Eq.(), MfⅠ ( θ ) denotes the inertia matrix, FfⅠ ( θ) denotes the Coriolis force, centrifugal force and gravity matrix, and τ denotes the torques applied on each DOF. For the fully-passive robot, the coefficient matrix S=0. Eq.() is a general form of Lagrange differential equation, which can be solved directly by the ODE45 method in Matlab. The dynamics of phase (b) is obtained under the assumption that: both knee impact and foot impact are modeled as an instantaneous fully inelastic impact where no slip or bounce occurs. System variables reduced from 6 to 4 as the knee has been locked. Angular momentum is conserved during impact for the swing leg about the hip H and for the whole robot about the stance leg contact point O as uuur uuur + + + LO( θ, θ2, θ2, & s θ, & θ2, & + + θ2s ) = LO( θ, θ2, & θ, & θ2 ) uur uur L ( θ, θ, θ, & θ, & θ, & θ ) = L ( θ, θ, & θ, & θ ) + + + + + H 2 2s 2 2s H 2 2 + + The impact process should also satisfyθ = θ, θ 2 = θ 2 = θ 2 s.rearrange Eq.(2) to the general form (2) & θ + af af2 & θ bf bf2 bf3 = & θ + 2 af2 af 22 θ bf 2 2 bf22 bf & 23 & θ 2s (3) The state variable values after impact can be used as the initial conditions of phase(c). At phase (c) the DOF of the system has reduced to 2, so the generalized coordinate q = [ θ, θ ] T 2. Dynamics of phase (c) is written in the form mf mf2 && θ ff( θ, θ2, & θ, & θ2) mf mf = && θ ff ( θ, θ, & θ, & θ ) 2 22 2 2 2 2 (4) During phase (d) Angular momentum is conserved during impact for the new swing leg about the hip H and for the whole robot about the impact point O2 as uuur uuur + + + LO 2( θ, θ2, & θ, & + + θ2 ) = LO 2( θ, θ2, & θ, & θ2 ) uur uur L ( θ, θ, & θ, & θ ) = L ( θ, θ, & θ, & θ ) + + + + + H 2 2 H 2 2 + + + + + The impact process should also satisfy θ = θ2, θ2s = θ2 = θ, & θ2s = & θ2 as the swing leg becomes to the new stance leg and vise versa. Rearrange Eq.(5) to the general form (5) + af af2 & θ bf bf2 & θ + af2 af = 22 θ bf 2 2 bf & 22 & θ 2 (6) Details of all coefficient matrix are omitted due to space limitation.
282 Manufacturing Technology, Electronics, Computer and Information Technology Applications Effects of Mechanical Parameters on Foot-ground Clearance Numerical simulations were performed using Matlab to calculate the minimum foot-ground clearance each step under different mechanical parameters. By giving proper initial conditions of the walking process, the passive dynamic robot could walk successfully showing a periodic gait. The instant immediately after impact is chosen as the beginning and ending of one step, as the number of initial independent variables could be reduced to 3 which lead to a less calculating time. In one step, the equations of motion are integrated numerically until the impact event is detected, and then impact will be calculated based on the principle of angular momentum conservation. After foot impact, one step simulation is ended. Parameters related to total robot mass M are plotted in fig.3. The value of minimum foot-ground clearance increases with the increase of kmt, kmb and decreases with the increase of kms.the main reason is that the COM of the swing leg drop to a lower position with the increase of kms, which makes the swing leg harder to rotate about the hip joint as the moment of inertia becomes larger. This will lead to a smaller bending angle of the knee joint and thus lead to a smaller foot-ground clearance. The swing leg is easier to rotate about the hip joint with the increase of kmt, kmb and thus lead to a larger foot-ground clearance. As shown in fig.3, the parameter of the shank is more sensitive to the foot-ground clearance than the thigh. It is because that the change of the parameter of the shank not only influence the motion of the shank but also the motion of the whole leg, but the change of the parameter of the thigh only influence the motion of the whole leg but not the shank. As analyzed above, the foot-ground clearance is mainly affected by two factors: the bending angle of the knee joint and the angle between the stance leg and the vertical direction. The more bending of the knee joint, the larger foot-ground clearance could obtain. When the stance leg is perpendicular to the ground, the largest foot-ground clearance could obtain. Consequently, to get larger knee joint bending angle and thus lager foot-ground clearance, the swing leg must swing faster than the shank. Parameters related to the leg length l are plotted in fig.4. The value of minimum foot-ground clearance increases with the increase of kbb, kbs and decreases with the increase of kbt, klt. kbb is the least sensitive parameter to the foot-ground clearance as it has very little effect on the swing of the swing leg. Other parameters effects on foot-ground clearance are due to the same reason mentioned above. Fig.3 Effects of kms, kmt and kmb on foot-ground Fig.4 Effects of kbb, kbt, kbs and klt on foot-ground clearance clearance Parameters related to the moment of inertia are plotted in fig.5. The value of minimum foot-ground clearance increases with the increase of kjs, increases and then decreases with the increase of kjt. Moment of inertia of the two legs are less sensitive than other parameters as they only influence motion of the swing leg but have little effect on the motion of the whole robot about the stance leg. However, other parameters (i.e., center of mass and mass of each part) can affect both the motion of the swing leg and the stance leg. Fig.6 shows periodic angle-time curves of the two legs and upper body, which means the passive robot could walk successfully and stably in numerical simulations.
Applied Mechanics and Materials Vols. 687-69 Fig.5 Effects of kjs, kjt on foot-ground clearance 283 Fig.6 Angle-time curves Prototype Experiments A 5 DOF (one at hip joint, two at knee joints and two at ankle joints) physical prototype, as shown in fig.7, was built in this study. A mechanical latch is located at each knee joint to lock and unlock the knee joint by solenoid. One DC servo motor located at the upper body is connected to two antagonistic-connected linear springs through cables to drive the hip joint. Digital servo amplifier and PCI data collection card are connected to a PC to fulfill the automatic control of walking. As we can not measure the minimum foot-ground clearance during walking, we let the passive robot walk on slope. We use the same control scheme at hip joint but change the slope angle in each walking trial. If the robot could walk on a smaller angle or even on level ground without foot scuffing, we consider that the robot could walk with larger foot-ground clearance. In physical prototype experiments, we can change the parameters of mass and position of COM by adding counterweight part to different positions of each body part and validated the results obtained from the numerical simulations. When adding a counterweight to the higher position of the thigh, the robot could walk on a slope with the minimum value of 0.02 without foot scuffing. When adding a counterweight to the lower position of the shank, the robot could walk on a slope with the minimum value of 0.06, and will fall forward if the slope angle are large than 0.08. When adding a counterweight to the upper body at a proper position, the robot could walk on level ground. The experiments show the same results obtained in numerical simulation which proves that our conclusion is correct. It is worth mentioning that, the control parameters are tuned carefully to make the robot walk successfully only on slope but not on level ground under certain mechanical parameters to validate our simulation results. The robot could also walk on level ground successfully and stably on level ground by changing the control parameters. One success walking trail from experiments video is shown in fig.8. In which the robot walk successfully on the slope angle of 0.06 by adding counterweight to the lower position of the shank. Fig.7 Mechanical prototype Fig.8 One successful walking on slope
284 Manufacturing Technology, Electronics, Computer and Information Technology Applications Conclusion To study the effects of mechanical parameters of the passive dynamic walker on foot-ground clearance, numerical simulations and physical prototype experiments were implemented in this paper. The results show that the mechanical parameters of the mass and center of mass of each part are more sensitive to foot-ground clearance than other parameters. We conclude that the foot-ground clearance is mainly determined by two factors: the bending angle of the knee joint and the angle between the stance leg and the vertical direction. The more bending of the knee joint, the larger foot-ground clearance could obtain and when the stance leg is perpendicular to the ground, the largest foot-ground clearance could obtain. To get larger knee joint bending angle, the swing leg must swing faster than the shank. So when building a passive dynamic walking robot, the mass of the thigh should be lager and the shank should be smaller, the COM of the thigh should be at a higher position and COM of the shank should be at a lower position and the mass of the upper body should be as large as possible in a reasonable range. Acknowledgement This research is funded and supported by the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.202039) and Independent Research of State Key Laboratory of Robotics and System (HIT) (Grant No. SKLRS20304B). References [] Y.Sakagami, R.Watanabe, C.Aoyama, S.Matsunaga, N.Higaki, K.Fujimura. The intelligent Asimo: system overview and integration, 2002IEEE/RSJ International Conference on Intelligent Robots and Systems, Lausanne, Switzerland, 2002:2478~2483. [2] K. Kaneko, F. Kanehiro, and S. Kajita, Humanoid robot HRP-2, in Proc. IEEE Int. Conf. Robot Automat. New Orleans, LA, 2004, pp. 083-090. [3] Collins S H, Ruina A, A bipedal walking robot with efficient and human-like gait [C]//Robotics and Automation, 2005, ICRA 2005, Proceedings of the 2005 IEEE International Conference on, IEEE, 2005: 983-988. [4] T.McGeer. Passive dynamic walking, International Journal of Robotics Research, 990,9(2):62~82. [5] Schuitema E, Hobbelen D G E, Jonker P P, et al. Using a controller based on reinforcement learning for a passive dynamic walking robot[a], Proceedings of the IEEE /RAS International Conference on Humanoid Robots [C]. New York, USA: IEEE, 2005. 232~237. [6] Narukawa, Terumasa, Masaki Takahashi, and Kazuo Yoshida. "Level-ground walk based on passive dynamic walking for a biped robot with torso."robotics and Automation, 2007 IEEE International Conference on. IEEE, 2007:3224-3229. [7] M. Wisse, A. L. Schwab, R. Q. v. d. Linde, F. C.T. v. d. Helm. How to keep from falling forward; elementary swing leg action for passive dynamic walkers. IEEE Transactions on Robotics, 2005. 2(3): 393-40. [8] M. Wisse, D. G. E. Hobbelen, A. L. Schwab. Adding the upper body to passive dynamic walking robots by means of a bisecting hip mechanism. IEEE Transactions on Robotics, 23(), 2-23. 2007. [9] M.Vukobratovic, B.Borovac. Zero-Moment Point-Thirty Five Years of Its Life. Int J Hum Robot. 2004,():57~73.
Manufacturing Technology, Electronics, Computer and Information Technology Applications 0.4028/www.scientific.net/AMM.687-69 Effects of a Passive Dynamic Walker s Mechanical Parameters on Foot-Ground Clearance 0.4028/www.scientific.net/AMM.687-69.279