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Design of a robust self-excited biped walking mechanismV. Sangwan, A. Taneja, S. Mukherjee*Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, IndiaAccepted 25 May 2004Available online 2 October 2004AbstractA self-excited biped walking mechanism consisting of two legs that are connected in a series at the hipjoint through a servomotor is studied to determine range of stability. A torque proportional to anglebetween the shank and the vertical is seen to sustain a gait. Each leg has a thigh and a shank connectedat a passive knee joint that has a knee stopper restricting the forward motion like the human knee. Whilea torque proportional to the angle between the shank and the vertical stabilises, the optimum proportion-ality constant is to be determined. A mathematical model for the dynamics of the system including theimpact equations is used to analyze the stability of the system through examination of phase plane plots.For a specified proportionality constant, the range of physical parameters like leg-length and mass of leg forwhich the system is stable is determined. Using the stability data, a robust design has been made.? 2004 Elsevier Ltd. All rights reserved.Keywords: Biped mechanism; Self-excited walking; Feedback; Stability1. IntroductionThe laws of Newtonian mechanics bind the human body, like anything else that is large enoughand slow enough. If we can gain better insight into how humans walk, perhaps we could improveprosthetics for the gait-impaired, help correct neuro-muscular deficiencies, or build better two-leg-0094-114X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2004.05.023*Corresponding author.E-mail address: sudiptomech.iitd.ernet.in (S. Mukherjee)./locate/mechmtMechanism and Machine Theory 39 (2004) 13851397MechanismandMachine Theoryged walking robots 1. McGeer?s results 2 with passive dynamic walking machines suggest thatthe mechanical parameters of the human body (e.g., lengths, mass distributions) have a greatereffect on the existence and quality of gait than is generally recognized. That is, one needs to studymechanics, not just activation and control, to fully understand walking 3.Biped walking machines fabricated consume a large amount of power through use of quasi sta-tic equilibrium postures. It is very important to improve the efficiency of biped walking. The self-excited walking four-link biped mechanism considered in this paper based on Ono 4 possesses anactuated hip joint and passive knee joints with stoppers as shown schematically in Fig. 1. As thisNomenclatureaidistance of center of gravity of ith link from its toe, metersIimass moment of inertia of ith link about its C.G., kgm2kproportionality constant for negative position feedback, Nm/radlilength of ith link, mmimass of ith link, kgT2torque at link 2, Nmc3damping coefficient between link 2 and link 3, Nm/radhiangle of ith link with downward vertical in anti-clockwise direction, rad_hangular velocity of ith link, rad/s_h?angular velocity of ith link just before impact, rad/s_hangular velocity of ith link just after impact, rad/shangular acceleration of ith link, rad/s2smoment impulse at the knee, kgrad/sFig. 1. Schematic of the biped mechanism.1386V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 13851397model uses only one actuator at the hip joint and the two legs have a passive synchronized swingmotion, the efficiency is much higher.There are two approaches for designing a walking machine: one is where a controller tries toforce a motion on the system 510 and the other uses the models? inherent dynamics for a givenset of parameters 1113. McGeer 2 demonstrated, by both computer simulation and physical-model construction, stable, human like walking on a range of shallow slopes with no actuationand no control (energy lost in friction and collisions is recovered from gravity). He just usedthe inherent dynamics of the system under the action of gravity. This paper attempts at usingminimum actuation to replace the role of gravity in McGeer?s case.A mathematical model for the dynamics of the system including the impact equations is used toanalyze the stability of the system through examination of phase plane plots. For a specified pro-portionality constant, the range of physical parameters like leg-length and mass of leg for whichthe system is stable has been determined. Phase plane plots for the stable trajectories did not havecrossovers, while the unstable trajectories show crossovers. This indicates the presence of constantenergy surfaces for stable controllers. There is hence scope for tuning controllers for optimumenergy consumption. Using the stability data, a robust design has been made.2. Biped mechanismThe biped mechanism to be treated in this study is shown in Fig. 1. The biped mechanism doesnot have a prominent upper body and consists of only two legs that are connected in a series at thehip joint through a motor. Each leg has a thigh and a shank connected at a passive knee joint thathas a knee stopper. By the knee stopper, an angle of the knee rotation is restricted like the humanknee. The legs have no feet, and the tip of the shank has a small roundness. To avoid sidewaystopple of the mechanism while walking the legs are in pairs which provides a larger span of thelegs on the ground. The objective of this study is to make the biped mechanism perform its inhe-rent natural walking locomotion on level ground by active energy through the hip motor.The necessary conditions 4 for the biped mechanism to be able to walk on level ground are asfollows:1. The inverted pendulum motion of the support leg must synchronize with the swing legmotion.2. The swing leg should bend so that the tip does not touch the ground.3. The motor should supply the dissipated energy of the mechanism through collisions at theknee and the ground, as well as friction at joints.4. The internal force of the knee stopper should not bend the knee of the support leg.5. The synchronized motion between the inverted pendulum motion of the support leg and thetwo DOF pendulum motion of the swing leg, as well as the balance of the input and outputenergy, should have stable characteristics against deviations from the synchronized motion.The swing leg motion that can satisfy the necessary conditions (2) and (3) is generated by apply-ing the self-excitation control by applying negative feedback 4 from the shank joint angle h3tothe input torque T at the thigh joint. This makes the system?s stiffness matrix asymmetrical,V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 138513971387changing the swing motion so that the shank motion delays at about 90? from the thigh motion.Through this feedback, it is also expected that the kinetic energy of the swing leg increases andthat the reaction torque (?T) will make the support leg rotate in the forward direction in a regionwhere h3 0.3. Analytical modelA mathematical model of biped locomotion is presented here. From an aspect of the differenceof the basic equation, one-step process can be divided into two phases 4 as shown in Fig. 2.1. From the start of the swing leg motion to the collision at the knee.2. From the knee collision to the touch down of the straight swing leg.We assume that the change of the support leg to the swing leg occurs instantly and that the endof the second phase is the beginning of the first phase. The self-excitation feedback is applied onlyduring the first phase. Also the support leg is kept straight because of internal reaction torque atthe knee.So with the assumption of a straight support leg the biped can be modeled as a three DOF linksystem as shown in Fig. 3. Viscous damping is applied to the knee joint of the swing leg. The equa-tions of motion of the system and the equations governing the impacts are derived below.3.1. Derivation of equations of motion for phase 1During phase 1, the biped walker has three degrees of freedom.Force Balance equations on link 3:Fig. 2. Phases of biped locomotion 4.1388V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 13851397m3l1a3cosh3? h1h1 m3l2a3cosh3? h2h2 I3 m3a23h3 m3l1a3sinh3? h1_h1?_h1 m3l2a3sinh3? h2_h2? c3?_h2 c3_h3 m3a3gsinh3 01Moment balance about center of mass of link 2:m3l2 m2a2l1cosh2? h1h1 I2 m2a22 m3l22h2 m3a3l2cosh3? h2h3 m3l2 m2a2l1sinh2? h1_h21 c3_h2 ?m3a3l2sinh3? h2_h3? c3?_h3 m3l2 m2a2gsinh2 T22Taking moments about the ground pivot:I1 m1a21 m2l21 m3l21h1 m3l2 m2a2l1cosh2? h1h2 m3a3l1cosh3? h1h3? m3l2 m2a2l1sinh2? h1_h22? m3a3l1sinh3? h1_h23 m1a1 m2l1 m3l1g? sinh1 ?T23Collecting Eqs. (1)(3) together in matrix form, we get:M11M12M13M21M22M23M31M32M3326643775h1h2h3266437750C12_h2C13_h3?C12_h1c3C23_h3? c3?C13_h1?C23_h2? c3c326643775h1h2h326643775K1K2K326643775?T2T20266437754where T2= kh3; M11 I1 m1a21 m2l21 m3l21; M12= (m2a2+ m3l2)l1cos(h2? h1); M13= m3a3l1cos(h3? h1);M22 I2 m2a22 m3l22;M23= m3a3l2cos(h3? h2);M33 I3 m3a23;C12= ?(m2a2+ m3l2)l1sin(h2? h1); C13= ? m3a3l1sin(h3? h1); C23= ? m3a3l2sin(h3?h2); K1= (m1a1+m2l1+ m3l1)gsinh1; K2= (m2a2+ m3l2)gsinh2; K3= m3a3gsinh3Fig. 3. Three DOF system.V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 138513971389Phase 2 is governed by the following set of equations:I1 m1a21 m2l21m2a2l1cos h2? h1m2a2l1cos h2? h1I2 m2a22#h1h2#0?m2a2l1sin h2? h1_h2m2a2l1sin h2? h1_h10#_h1_h2#m1a1 m2l1gsinh1=h100m2a2gsinh2=h2?h1h2?00?53.2. Equations governing the impact3.2.1. Impact with the ground at the end of second phaseAfter the end of first phase a collision occurs when the shank comes in line with the thigh. Theangular velocities of the shank and thigh become same after the impact.Energy methods were used to derive the impact equations 14. Selecting the generalized coor-dinates to be X, Y, h1, h2, h3as shown in Fig. 4.The impact equations are:_h1_h2_h326643775_h?1_h?2_h?3264375 ? M?1g1g2g32643756whereg1 m2l1 m3l1 m1a1 l1_h?1 l2_h?2cos h2? h1 l3_h?3cos h3? h1hiG1G2G3XYyx123a1a2a3Fig. 4. Biped walker showing generalized coordinates X, Y, h1, h2, h3.1390V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 13851397g2 m2a2 m3l2 l1_h?1cos h2? h1 l2_h?2 l3_h?3cos h3? h2hig3 m2l1 m3l1 m1a1 l1_h?1cos h3? h1 l2_h?2cos h3? h2 l3_h?3hi3.3. Impact at the kneeDuring the second phase, the biped system can be regarded as a two DOF link system. Usinggeneralized coordinates as h1, h2and h3as shown in Fig. 5, impact equations are:_h1_h2_h326643775_h?1_h?2_h?3264375 ? M?10?ss2643757where M is given before.Similar equations have been used by Ono et al. 4 but the equations reported for the first phaseare dimensionally incorrect. The derivations presented here are complete and include several addi-tional terms missing in 4.4. Design issuesThe equations above were coded and executed with varying parameter values i.e. mass of thelinks, link lengths and the feedback gain to design a stable biped mechanism.G1G2G3123a1a2a3Fig. 5. Biped walker showing generalized coordinates h1, h2, and h3.V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 138513971391For the biped to walk steadily, the solution of the equations of motion has to be periodic or alimit cycle. If the phase plane diagram is a closed loop; it implies that after a cycle when the bipedattains the same angular configuration than its angular velocities would also be the same as in theprevious cycle. The solution was seen to be sensitive to the time step used, so the time step wasprogressively reduced till the solution stopped changing with further reduction.A sample phase diagram for one leg is shown in Fig. 6. The solid lines represent the post kneeimpact motion, classified as phase II earlier. The gaps in the phase plane diagram are due to thecollisions at the knee and at the ground. The effect of collisions at the other leg is not transmittedsignificantly. So, only two significant collisions show up in the plot. The stability has to bechecked in successive steps as the conditions at the start of the cycle change until a stable gaitis obtained. Only when the loops for consecutive steps close, the system can be called stable.As can be seen from Fig. 7, the lower the density; the larger the range for stable k values. Alu-minium was selected as the material for the link design as it is one of the lightest materials around.However, magnesium based alloys and plastics would have been more suitable if we had access tothe materials and processing methodology for them.Fig. 6. Phase plane illustration of one walking cycle.Fig. 7. Ranges of k as the density of the material vary.1392V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 13851397Fig. 8 shows that there exists an optimal shank length for a given material. In this case optimalshank length is 150mm (for aluminium). This has been chosen as the basis for design.Fig. 9 shows a model built on the basis of above study. The height of the biped (from feet tohip) is 300mm. Symmetry about the sagittal plane has been introduces to prevent a side fall whilewalking. The cross-section of one leg is 40mm 10mm with a total weight of 4kg approximately.5. Stability investigationsConvergence to the stable cycle over the consecutive steps from varying initial conditions wastraced. It was found that the initial point approaches the stable cycle tangentially on the phaseFig. 9. Model based on the above study.Fig. 8. Ranges of k as the shank length varies for aluminium as the material (density = 2770kg/m3).V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 138513971393plane over consecutive steps as shown in Fig. 10. The solid lines are the parts of the stable cycle.The broken line from right to left shows the direction of convergence. For different initialconditions but same value of k, the same stable cycle is obtained. This indicates that the stablecycle is unique for a particular k, though we do not have a proof for it.It has been mentioned earlier that a stable cycle was obtained for only a range of k. The phaseplane plots for different k values have no crossover between cycles (except during collisions) forvarying k (Fig. 11). Consecutive steps of the biped for k equal to 2, for which no stable cyclewas found, were plotted. The consecutive cycles now show crossovers unlike the stable phaseplane plots where there are crossovers only near the collisions (Fig. 12). This suggests the existenceof constant energy surfaces 15 for each stable controller design.For large values of k, the angle between the two legs keeps increasing, until the mechanismeventually flops down. In Fig. 13, we have superposed the phase points for the start and endof the collisions for varying stable k. The continuous curve in the middle corresponds to the stablek = 8 case. We see that the knee impact is initiated at progressively higher velocities, suggestingFig. 10. Convergence to stable cycle.Fig. 11. Stable phase plane plots for varying k (Nm/rad).1394V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 13851397larger energy losses. For further increase in k step length becomes very large, and the mechanismcollapses.To investigate the patterns of power consumption by the biped the changes in KE and PE aswell as the power supplied by the motor vs. time were plotted (Figs. 14 and 15). It was found thatFig. 12. Consecutive cycles for an unstable k (=2Nm/rad) showing crossover.Fig. 13. Phase plane diagram showing the start and end of the collisions for varying stable k (=3, 8, 13, 18, 23Nm/rad).V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 138513971395there are zones where the motor removes mechanical energy from the system instead of supplyingit, as indicated by the negative power input by the motor (Fig. 15). About 80% of the energy inputgoes into overcoming the loss due to impact and friction at the knee joints. It might be possible toreduce the energy input to the system by using a varying controller gain.6. ConclusionsA self-excited biped walking mechanism consisting of two legs that are connected in a series atthe hip joint through a servomotor is studied to determine range of stability. A mathematicalmodel for the dynamics of the system including the impact equations is used to analyze the sta-bility of the system through examination of phase plane plots. For a specified proportionality con-stant, the range of physical parameters like leg-length and mass of leg for which the system isstable is determined. Phase plane plots for the stable trajectories do not have crossovers, whilethe unstable trajectories show crossovers. This indicates the presence of constant energy surfacesFig. 14. Changes in KE and PE of the biped mechanism over a cycle fork = 6Nm/rad.Fig. 15. Power input by the motor.1396V. Sangwan et al. / Mechanism and Machine Theory 39 (2004) 13851397for stable controllers. There is hence scope for tuning controllers for optimum energy consump-tion. Using the stability data, a robust design has been made.References1 B.E. McCownMcClintick, G.D. Moskowitz, The behaviour of a biped walking gait on irregular terrain, TheInternational Journal of Robotics Research 17 (1) (1998) 4355.2 T. McGeer, Passive dynamic walking, The International Journal of Robotics Research 9 (2) (1990) 6281.3 M. Gracia, A. Chatterjee, A. Ruina, M. Coleman, The simplest walking model: stability, complexity, and scaling,Journal of Biomechanic
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