In this paper, we propose a novel speeding-up method for biped walking using a swinging-arms motion based on the principle of an up-and-down wobbling mass. We have shown that biped robots with a wobbling mass can achieve fast walking using an active up-and-down motion of the wobbling mass. We have also shown that the active up-and-down motion increases walking speed of biped robots. We apply this principle to a biped robot with two linked arms like humans for achieving high-speed limit cycle walking. We show that the proposed method achieves high-speed limit cycle walking of biped robots with arms.2015 IEEE International Conference on Robotics and Automation (ICRA), 26-30 May 2015, Seattle, WA, US

Asano  Fumihiko

Hanazawa Yuta

Kyutacar : Kyushu Institute of Technology Academic Repository

High-Speed Biped Walking Using Swinging-ArmsBased on Principle of Up-and-Down WobblingMass著者 Hanazawa Yuta, Asano  Fumihikojournal orpublication title2015 IEEE International Conference on Roboticsand Automation (ICRA)page range 5191-5196year 2015-07-02その他のタイトル High-speed biped walking using swinging-armsbased on principle of up-and-down wobblingmassURL http://hdl.handle.net/10228/00008238doi: https://doi.org/10.1109/ICRA.2015.7139922High-Speed Biped Walking Using Swinging-Arms Based on Principle ofUp-and-Down Wobbling MassYuta Hanazawa1 and Fumihiko Asano2Abstract— In this paper, we present a speeding-up methodfor biped walking using swinging-arm motion based on theprinciple of an up-and-down wobbling mass. We have shownthat biped robots with a wobbling mass can achieve fast walkingusing an active up-and-down motion of the wobbling mass.We have also shown the principle that the active up-and-downmotion increases walking speed of biped robots. We apply thisprinciple to a biped robot with two linked arm like humansfor achieving high-speed limit cycle walking. We show that theproposed method achieves high-speed limit cycle walking forbiped robots with arms.I. INTRODUCTIONBiped robot have high moving performance in variousenvironments and we expect that they work in various field[1], [2], [3]. These biped robot fast and stable dynamicwalking. However, energy-efficiency of the many biped robotis bad in dynamic walking and this problem degrades theusability of the biped robot. We have studied active dynamicwalking based on the principle of passive dynamic walking[4]. This type walking is known as limit cycle walking andmany limit cycle walkers have been proposed [5], [6], [7],[8], [9].Limit cycle walkers achieve energy-efficient walking buttheir walking speed are typically slow. Speeding-up methodsfor limit cycle walkers have thus developed. Asano et al.have shown fast limit cycle walking using property of arc-feet [10]. They have also proposed high-speed limit cyclewalking using principle of parametric excitation mechanismusing telescoping legs [11]. Hobbelen et al. have developedspeeding-up method using torso posture [12]. Hanazawa etal. have shown speeding-up method using ankle springs andinerters [9]. We have also proposed a speeding-up methodfor limit cycle waking using an active up-and-down controlof a wobbling mass [13]. We have also shown a principlefor the speeding-up biped walking by the wobbling massmethod mathematically. This principle is a regular up-and-down wobbling mass according to the stance leg angle andthe regular motion of the wobbling mass generates propulsiveeffects.We consider that biped robots with arms achieving limitcycle walking work at various field in the future. We alsoexpect that the biped robots with arms serve for many tasks1Y. Hanazawa is with the Dept. of Applied Science for IntegratedSystem Engineering, Graduate School of Engineering, Kyushu Insti-tute of Technology, 1-1 Sensui, Tobata, Kitakyushu, Fukuoka 804-8550,JAPAN hanazawa-y@ise.kyutech.ac.jp 2F. Asano is withthe School of Information Science, Japan Advanced Institute of Sci-ence and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, JAPANfasano@jaist.ac.jpSwing!Up and down! Fig. 1. Schematic illustration of analogy between up-and-down motion ofwobbling mass and swinging-armsusing their arms instead of humans. We thus apply theprinciple of the up-and-down wobbling mass to biped robotwith two linked arms for speeding-up limit cycle walking.In this paper, we propose a novel speeding-up methodfor biped robots using their arms based on the principle ofthe up-and-down wobbling mass. Fig. 1 shows schematicillustration of analogy between up-and-down motion ofa wobbling mass and swinging-arms. We design controlmethod for biped robot with arms from the viewpoint ofthis analogy. We show the validity of the proposed methodthrough numerical simulations.II. MODEL OF BIPED ROBOTA. Dynamic equationWe use a model of a biped robot with two linked arms(Fig. 2). This robot also has six actuators for active controlof the arms, the torso and the swing-leg (thigh). Dynamicequation of the robot is given byM(q)q̈ +H(q, q̇) = Su+ Jc(q)Tλ, (1)where q = [θ1, θ2, θ3, θ4, θ5, θ6, θ7, x1, z1]T isthe generalized coordinate vector, M(q) ∈ R9×9 is theinertia matrix, H(q, q̇) ∈ R9 is the vector that consistsof Coriolis, centrifugal force and gravitational vector, u =[u1, u2, u3, u4, u5, u6]T is the input vector, S ∈ R9×6 isthe driving matrix and is detailed asS =0 −1 0 0 0 01 0 0 0 0 0−1 1 −1 0 −1 00 0 1 −1 0 00 0 0 1 0 00 0 0 0 1 −10 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0.2u hm1u3u4u5u6u1m 1m2m3m3m1θ 2θ3θ4θ5θ6θ7θ4m4m1a1a1b 1b1l1l4b2a2b1d4a4a4b4l4l3l3l3a3b 3a3bzxO ),( 11 zxg+2lFig. 2. Model of biped robot with two linked armsJc(q) ∈ RN×9 is the Jacobian matrix and is determinedaccording to the constraint conditions of the robot and N isthe number of constraint conditions. λ ∈ RN is the constraintforce vector given byλ=−X(q)−1(Jc(q)M(q)−1Γ(q,q̇,u)+J̇c(q, q̇)q̇), (2)X(q)=Jc(q)M(q)−1Jc(q)T, (3)Γ(q,q̇,u)=Su−H(q, q̇). (4)Although the swing leg of the biped robot scuffs since thebiped robot does not have knees, we ignore this foot-scuffingin the simulation.B. Constraint conditionsSince the contact point of the biped robot is constrainedwith ground, constraint equations are expressed asẋ1 = 0, (5)ż1 = 0. (6)From these equations, we obtain Jc(q) ∈ R2×9 andJ̇c(q, q̇) ∈ R2×9 asJc(q)q̇ =[0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1]q̇ = 02×1,(7)J̇c(q, q̇) = 02×9 (8)C. Impact equationWe assume that the collision of the swing-leg with theground is inelastic and instantaneous. We can derive thevelocity immediately after impact by solving the impactequations described in the following [14]. Since the contactpoint of the biped robot is constrained with the ground at theTABLE IMECHANICAL PARAMETERSSymbol Unit Value Symbol Unit Valuem1 kg 5.0 a1 m l1/2m2 kg 5.0 a2 m l2/2m3 kg 1.5 a3 m l3/2m4 kg 1.5 a4 m l4/2mh kg 5.0 b1 m l1/2l1 m 0.8 b2 m l2/2l2 m 0.8 b3 m l3/2l3 m 0.4 b4 m l4/2l4 m 0.4 d1 m 0.6I1 kg·m2 m1l21/12 I2 kg·m2 m2l22/12I3 kg·m2 m3l23/12 I4 kg·m2 m4l24/12collision of the swing-leg, constraint equations are expressedasl1 cos θ1θ̇1 − l2 cos θ2θ̇2 + ẋ1 = 0, (9)−l1 sin θ1θ̇1 + l2 sin θ2θ̇2 + ż1 = 0. (10)From these equations, the instantaneous constraint matrixJI(q) ∈ R2×9 is given byJI(q)=[l1 cos θ1 −l2 cos θ2 0 0 0 0 0 1 0−l1 sin θ1 l2 sin θ2 0 0 0 0 0 0 1]. (11)Impulsive force vector, λI ∈ R2, and a velocity vector, q̇+ ∈R9, immediately after the impact are given byλI = −XI(q)−1JI(q)q̇−, (12)XI(q) = JI(q)M(q)−1JI(q)T, (13)q̇+ = (I −M(q)−1JI(q)TXI(q)−1JI(q))q̇−, (14)where q̇− ∈ R9 is the velocity vector immediately beforeimpact. The state vector of the robot immediately after theimpact are then reset to[qTq̇T]=[θ2 θ1 θ3 θ6 θ7 θ4 θ5 x1 0θ̇+2 θ̇+1 θ̇+3 θ̇+6 θ̇+7 θ̇+4 θ̇+5 0 0],(15)where the velocities with super script, ”+”, indicate thoseimmediately after impact due to Eq. (14). Table I lists themechanical parameters of the biped robot. We use theseparameters in our simulations.III. PRINCIPLE OF UP-AND-DOWN WOBBLINGMASSHere, we show the principle of an up-and-down wobblingmass for high-speed limit cycle walking [13]. Fig. 3 andFig. 4 show the schematic illustration of the moment due toan up-and-down wobbling mass. We thus see that the reactionforce due to the wobbling mass generates moment at thecontact point of the stance-leg that is given byM = r × F , (16)where r is the vector from the contact point to the hipjoint and F is the reaction force vector due to the wobblingsθ−rFFrM ×=+xzTorsoWobblingmassFig. 3. Schematic illustration of moment due to wobbling mass at thenegative stance leg anglesθTorsoWobblingmassrFFrM ×=+xzFig. 4. Schematic illustration of moment due to wobbling mass at thepositive stance leg anglemass. Fig. 3 shows the moment when the wobbling massis dropped at the negative stance leg angle. Then, Fig. 4shows the moment when the wobbling mass is raised at thepositive stance leg angle. We can see that biped walkingspeed increases by up-and-down of the wobbling mass basedon the principle.To achieve high-speed limit cycle walking, we designcontrol method for biped robots with arms based on theprinciple.IV. CONTROL METHODSA. Swing-leg and torso controlWe first show the control methods of the swing-leg andtorso posture for level ground walking. We achieve levelground walking of the biped robot by using the followingcontrol methods:u1 = −KP1(θ2 − θ1 − ϕd)−KD1(θ̇2 − θ̇1), (17)u2 = −KP2(θ3 − θ3d)−KD2θ̇3, (18)where KP1, KP2, KD1 and KD2 are the control gains, ϕdis the desire hip joint angle. The biped robot can achievethe desired hip-joint angle by Eq. (17) and the desired torsoposture by Eq. (18).B. Swing-arms controlWe design control method for the swinging-arms based onthe principle of the up-and-down wobbling mass. To generatethe propulsive moment, biped robots drop the CoM (centerof mass) of the arms when the stance leg angle is negative.Biped robots then raise the CoM of the arms to generate thepropulsive moment when the stance leg angle is positive.These motions are given byu3 = −KP3(θ4 −Kaθ1)−KD3(θ̇4 −Kaθ̇1), (19)u4 = −KP4(θ5 −Kaθ1)−KD4(θ̇5 −Kaθ̇1), (20)u5 = −KP5(θ6 +Kaθ1)−KD5(θ̇6 +Kaθ̇1), (21)u6 = −KP6(θ7 +Kaθ1)−KD6(θ̇7 +Kaθ̇1), (22)where KP3, KP4, KP5, KP6, KD3, KD4, KD5 and KD6are the control gains and Ka is the control parameterfor amplitude of swinging-arms. Table II lists the controlparameters.V. WALKING ANALYSISFig. 5 shows the kinetic, potential and mechanical energy(KE, PE and ME) in limit cycle walking at Ka = 1.0. Fig. 6show the torque in the limit cycle walking. We can see thatthe biped robot achieves periodic walking. Fig. 7 shows thestance leg angle and Height of the CoM of the arms from themass point of the torso with respect to time. We see that theCoM of the arms goes up and down based on the principleof an up-and-down wobbling mass.Fig. 8 shows the walking speed and Fig. 9 shows thespecific resistance (SR) in limit cycle walking with respectTABLE IICONTROL PARAMETERSSymbol Value Symbol ValueKP1 100 KD1 10KP2 1000 KD2 100KP3 600 KD3 60KP4 600 KD4 60KP5 600 KD5 60KP6 600 KD6 60ϕd −0.60 [rad] θ3d 0.00 [rad]0 0.5 1 1.5 202040Time [s]KE [J]0 0.5 1 1.5 2190200210Time [s]PE [J]0 0.5 1 1.5 2180200220Time [s]ME [J]Fig. 5. Kinetic, potential and total mechanical energy with respect to time0 0.5 1 1.5 2−150−100−50050100150Time [s]Torque [Nm]  u1u2u3u4u5u6Fig. 6. Torque with respect to time0 0.5 1 1.5 2−0.500.5Time [s]Angle [rad](a) Stance leg angle0 0.5 1 1.5 2−0.21−0.2−0.19−0.18Time [s]Height [m](b) Height of CoM of armsFig. 7. Stance leg angle and height of CoM of arms from mass point ofthe torso with respect to timeto Ka. The SR is an index of energy-efficiency in dynamicwalking given bySR :=pMgv, (23)where p [J] is the average input power, M [kg] is the totalweight of the robot and v [m/s] is the average walking speed.Average input power, p, is given by0 0.5 1 1.5 2 2.50.40.450.50.550.60.650.70.750.80.850.9KaSpeed [m/s]Fig. 8. Walking speed with respect to Ka0 0.5 1 1.5 2 2.50.40.450.50.550.60.65KaSR [−]Fig. 9. SR with respect to Kap :=1T∫ T0(|u1(θ̇2−θ̇3)|+|u2(θ̇3−θ̇1)|+ |u3(θ̇4 − θ̇3)|+|u4(θ̇5 − θ̇4)|+|u5(θ̇6 − θ̇3)|+|u6(θ̇7 − θ̇6)|)dt,(24)where T [s] is the total walking time (n-steps).We can see that walking speed and SR monotonically in-creases with respect to increasing Ka. Since Ka is the controlparameter for amplitude of swinging arms, the amplitudeof up-and-down CoM of the arms increase by increasingKa. The waking speed thus increases and SR increases bythe increasing impact effects due to the increasing walkingspeed. We can considerably improve walking speed of thebiped robots by our proposed method. However, we cannotset overlarge Ka since biped robots have limit of motionrange of the shoulder joints. we did not also use the elbowjoint effectively for limit cycle walking yet. we expect thatmore high-speed walking by using the elbow joints. In thenext section, we show a speeding-up method using the elbowjoints effectively.VI. WALKING ANALYSIS WITH BENDING ARMSFig. 10 shows schematic illustration of biped robot withbending arms. When the biped robot bend arms vertically atthe elbow, the CoM of the arms generates the moment forCoM of the armsaFaraMFig. 10. Schematic illustration of biped robot with bending armsFig. 11. Schematic illustration of swinging arms with bending armspropulsive effect. The moment is given byMa = ra × Fa, (25)where ra is the vector from the contact point of the stanceleg to the CoM of the arms and Fa is the force vector due tothe CoM of the arms. We expect that biped robot achievesmore high-speed walking by using swinging bending armsbased on the principle of an up-and-down wobbling mass.We thus change two control method for swinging bendingarms as shown in Fig. 11.u4 = −KP4(θ5 −Kaθ1 + π/2)−KD4(θ̇5 −Kaθ̇1), (26)u6 = −KP6(θ7 +Kaθ1 + π/2)−KD6(θ̇7 +Kaθ̇1). (27)Fig. 12 shows the kinetic, potential and mechanical energy(KE, PE and ME) in limit cycle walking with bending arms atKa = 1.0. Fig. 13 show the torque in the limit cycle walking.Fig. 14 shows the stance leg angle and height of the CoMof the arms from mass point of the torso with respect totime. We see that the biped robot achieves periodic walkingand the CoM of the arms goes up and down based on theprinciple of an up-and-down wobbling mass.0 0.5 1 1.5 202040Time [s]KE [J]0 0.5 1 1.5 2190200210Time [s]PE [J]0 0.5 1 1.5 2200220240Time [s]ME [J]Fig. 12. Kinetic, potential and total mechanical energy with respect totime0 0.5 1 1.5 2−300−250−200−150−100−50050100150Time [s]Torque [Nm]  u1u2u3u4u5u6Fig. 13. Torque with respect to time0 0.5 1 1.5 2−0.500.5Time [s]Angle [rad](a) Stance leg angle0 0.5 1 1.5 2−0.11−0.1−0.09−0.08Time [s]Height [m](b) Height of CoM of armsFig. 14. Stance leg angle and height of CoM of arms from mass point ofthe torso with respect to timeFig. 15 shows the walking speed of the limit cycle walkingwith respect to Ka. Fig. 16 shows the SR of the limit cyclewalking with respect to Ka We see that walking speed andSR monotonically increase with respect to increasing Ka.Fig. 17 shows comparison of waking speed between thebiped robot with straight arms (Fig. 8) and the biped robotwith bending arms (Fig. 15). We can see that the walkingspeed of the biped robot with bending arms is faster thanthat of the biped robot with straight arms. Fig. 18 showscomparison of SR between the biped robot with straight arms(Fig. 9) and the biped robot with bending arms (Fig. 16).0 0.5 1 1.5 2 2.50.860.880.90.920.940.960.981KaSpeed [m/s]Fig. 15. Walking speed in bending arm with respect to Ka0 0.5 1 1.5 2 2.50.350.40.450.50.550.6KaSR [−]Fig. 16. SR in bending arm with respect to Ka0 0.5 1 1.5 2 2.50.40.50.60.70.80.91KaSpeed [m/s]  straightbendingFig. 17. Comparison of walking speed between biped robot with straightarms and biped robot with bending armsFrom this figure, we see that the energy-efficiency of thebiped robot with bending arms is better than that of the bipedrobot with straight arms. We thus see that swinging bendingarms based on the principle of up-and-down wobbling massis important for achieving high-speed limit cycle walking.VII. CONCLUSION AND FUTURE WORKIn this paper, we presented a novel speeding-up methodfor biped walking using swinging-arms motion based on theprinciple of an up-and-down wobbling mass. we proposed acontrol method for swinging-arms motion based on the prin-ciple of up-and-down of a wobbling mass. By the proposed0 0.5 1 1.5 2 2.50.350.40.450.50.550.60.650.7KaSR [−]  straightbendingFig. 18. Comparison of SR between biped robot with straight arms andbiped robot with bending armsmethod, the biped robot achieves high-speed limit cyclewalking. Moreover, we have shown that the walking speedand energy-efficiency improve by using the bending arms.We plan to validate the our proposed method by experiment.Moreover, we want to extend our proposed method to 3Dbiped walking.REFERENCES[1] HONDA, “Humanoid Robot ASIMO,” [Online]. Available:http://www.honda.co.jp/ASIMO/.[2] PETMAN, “Boston dynamics: Humanoid robot petman,” [Online].Available : http://www.bostondynamics.com.[3] AIST, “Humanoid robot hrp4-c,” [Online]. Available :http://www.aist.go.jp.[4] T. McGeer, “Passive dynamic walking,” The International Journal ofRobotics Research, vol. 9, no. 2, pp. 62–82, 1990.[5] F. Asano, M. Yamakita, and K. Furuta, “Virtual passive dynamic walk-ing and energy-based control laws,” in Proc. IEEE/RSJ InternationalConference on Intelligent Robots and Systems (IROS), 2002, pp. 1149–1154.[6] M. Wisse and J. Van Frankenhuyzen, “Design and construction ofmike; a 2d autonomous biped based on passive dynamic walking,” inProc. International Symposium of Adaptive Motion and Animals andMachines (AMAM03), 2003.[7] M. Wisse, “Three additions to passive dynamic walking; actuation,an upper body, and 3D stability,” in Proc. IEEE/RAS InternationalConference on Humanoid Robots (ICHR), vol. 1, 2004, pp. 113–132.[8] D. G. E. Hobbelen and M. Wisse, “Limit cycle walking”, HumanoidRobots, Human-like Machines, chapter 14. InTech, 2007.[9] Y. Hanazawa, H. Suda, Y. Iemura, and M. Yamakita, “Active walkingrobot mimicking flat-footed passive dynamic walking,” in Proc. IEEEInternational Conference on Robotics and Biomimetics (ROBIO),2012, pp. 1281–1286.[10] F. Asano and Z. W. Luo, “Efficient dynamic bipedal walking usingeffects of semicircular feet,” Robotica, vol. 29, no. 03, pp. 351–365,2011.[11] ——, “Energy-efficient and high-speed dynamic biped locomotionbased on principle of parametric excitation,” IEEE Transactions onRobotics, vol. 24, no. 6, pp. 1289–1301, 2008.[12] D. Hobbelen and M. Wisse, “Controlling the walking speed in limitcycle walking,” The International Journal of Robotics Research,vol. 27, no. 9, pp. 989–1005, 2008.[13] Y. Hanazawa, T. Hayashi, M. Yamakita, and F. Asano, “High-speedlimit cycle walking for biped robots using active up-and-down motioncontrol of wobbling mass,” in Proc. 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High-speed biped walking using swinging-arms based on principle of up-and-down wobbling mass

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High-speed biped walking using swinging-arms based on principle of up-and-down wobbling mass

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