Semi passive walking of a 7 DOF biped robot ABSTRACT 1 INTRODUCTION

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1 Semi passive walking of a 7 DOF bipe robot N.Khraief & N.K.M Siri khraief@lrv.uvsq.fr, nkms@free.fr Laboratoire e Robotique e Versailles Université e Versailles Saint Quentin en Yvelines 0, Avenue e l Europe 7840 Vélizy ABSTRACT This paper is concerne with the passive walking of an uneractuate bipe robot. First, we present the moelling of the bipe robot (that is a 7 DOF robot). Then, we focus on the stuy of the almost passive ynamic walking. In particular, we show that the application of a nonlinear feeback control to stabilize the torso an knees leas the bipe robot to perform a stable almost-passive ynamic walking when ealing with motions on a ownhill a slope. More such control also leas the system trajectories to converge towars stable limit cycles. In this context, we present some results base on both Poincaré map metho an trajectory sensitivity analysis to efficiently characterize the stability of the almost-passive limit cycles. INTRODUCTION Bipe locomotion is one of the most sophisticate forms of legge locomotion. A bipe robot is a two-legge robot, which may have knees an it most closely resembles human locomotion, which is interesting to stuy an is very complex. There has been extensive research over the past ecae to fin mechanical moels an control systems which can generate stable an efficient bipe gaits. This paper eals with the use of stabilization of perioic gaits for control of planar bipe robots. Inee, we investigate the existence an the control of passive limit cycles. The notion of obtaining passive gaits from mechanical moels was pioneere by McGeer [5]. These gaits are only powere by gravity an are thus very energy efficient. In passive human walking, a great part of the swing phase is passive, i.e. no muscular actuation is use. Activation is primarily uring the ouble contact phase an then it essentially turns off an allows the swing leg to go through like a ouble penulum, McMahon [4]. Passive walking has been stuie by several researchers, Collins, Garcia, Goswami [6], McGeer [5], M.W.Spong [8]. These have foun stable passive gaits for shallow slopes where gravity powers the walk own the slope. However, it is foun that no stable passive gaits exist for level groun. Consequently, a complementary control schemes is require. Thus, we will present some theoretical an simulation results base on the use of a recent control metho (referre to as Controlle Limit Cycle [] [] [3]), which consiers the system energy for both controller esign an system stabilization. Motivate by the work one so far [4] [5] [] [3] an potential applications of bipe locomotion, we conuct this paper on stuy of energy base control of bipe robots. A new metho is use to stabilize perioic cycles for such systems. The control objective is

2 regulation of the system energy (using a nominal energetic representation) for stabilization of the walking robot with ifferent spees. The CLC control allows to establish this task. We foun that it is possible to evelop energy mimicking control laws which can make the bipe gaits stable an robust uner various kin of environment. They can also be very energy efficient an make the bipe gait anthropomorphic. The paper will be organize as follows: First we will present the moelling of the bipe robot uner consieration (that is a knee robot with torso with 7 DOF). Then, we will focus on the stuy of the passive ynamic walking of this robot, on incline slopes. Moreover, we will show that uner a simple control applie to stabilize a posture, trajectories of the bipe robot can converge towars stable limit cycles. In this context, we will present some results base on Poincaré map metho an trajectory sensitivity analysis to efficiently characterize the stability of the almost-passive limit cycles. However, as such limit cycles may not exist for all groun configurations, a complementary control schemes is require. Thus, we will present some theoretical an simulation results base on the use of a recent control metho (referre to as Controlle Limit Cycle [] [] [3]), which consiers the system energy for both controller esign an system stabilization. A very interesting iea is to use the torso an the slope to control an stabilize the robot at a esire spee. Finally, some potential extensions for future works will be iscusse. THE BIPED ROBOT MODEL The ynamic moel of a simple planar bipe robot is consiere in this section.it s shown in figure (). The robot has five egrees of freeom. It consists of a torso, two rigi legs, with no ankles an two knees, connecte by a frictionless hinge at the hip. This linke mechanism moves on a rigi ramp of slope γ. During locomotion, when the swing leg contacts the groun (ramp surface) at heel strike, it has a plastic (no slip, no bounce) collision an its velocity jumps to zero. The motion of the moel is governe by the laws of classical rigi boy mechanics. It s assume that walking cycle takes place in the sagittal plane an the ifferent phases of walking consist of successive phases of single support. With respect to this assumption the ynamic moel of the bipe robot consists of two parts: the ifferential equations escribing the ynamic of the robot uring the swing phase, an the algebraic equations for the impact (the contact with the groun). -q L3 z3 z L q3 q3 Torso q4 q4 L4 z4 Sol y Slope x Figure A passive ynamic walking moel of a 7 of bipe robot. Swing phase moel During the swing phase the robot is escribe by ifferential equations written in the state space as follows: x& = f( x) + g( x) u ()

3 Where x = ( qq, &. ) () is erive from the ynamic equation between successive impacts given by: M ( qq ) & + Hqq (, &) = Bu () Where q= ( q3, q3, q4, q4, q ), u = ( u, u, u3, u4) : u an u are the torques applie between the torso an the stance leg, an the torso an the swing leg, respectively, u 3 an u 4 are torques H( q, q &) = 5 is the coriolis applie at both knees ( ) [ 5 5] M q = is the inertia matrix an [ ] an gravity term (i.e.: Hq ( ) = Cqq (, &) + Gq ( ) ) while B is a constant matrix. The matrices M, CGB,, are evelope in [7].. Impact moel The impact between the swing leg an the groun (ramp surface) is moelle as a contact between two rigi boies. The moel use here is from [8], which is etaile by Grizzle & al. in [6]. The collision occurs when the following geometric conition is met: x = ztgγ (3) Where: x = L3(sin q3 sin q3) + L4(sin q4 sin q4) (4) z = L3 (cos q3 cos q3) + L4 (cos q4 cos q4) Yet, from bipe s behaviour, there is a suen exchange in the role of the swing an stance sie members. The overall effect of the impact an switching can be written as: h: S χ (5) + x = hx ( ) &, with h is specifie in [].The superscripts (-) an (+) enote quantities immeiately before an after impact, respectively. Where: S = {( q, q) χ/ x z tgγ = 0}.3 Overall moel The overall 7-of bipe robot has as hybri moel: x& = f ( x) + g( x) u x ( t ) S () x + = h( x ) x ( t ) S x& = f(, x Λ i) u+ g(, x Λi), x() t C( ei) (7) This can be written + x = hx (, ei), xt ( ) Ce ( i) (8) Where the iscrete state set is Λ=ss,ss { } = { Λ, Λ }. C (e i ) is state change conition event after impact for the event e i. The states are ss (leg n is the support leg) an ss (leg n is the support leg). The two events correspon to single supports on one leg an then the other. The commutation events or changes are in the set E= { e, e} = {( ss, ss)(, ss, ss) }. We can consier now this moel an transform the system by a first partial feeback, in orer to allow existence of Limit Cycles corresponing to walking ownhill a slope. 3 SEMI PASSIVE DYNAMIC WALKING 3. Outline of proceure Our objective is to examine the possibility that a knee bipe robot with torso can exhibit a passive ynamic walking in a stable gait cycle, ownhill a slope. In our previous work [4]

4 [5] [] [3] on a kneeless bipe robot with torso, we have shown that such systems can steer a passive ynamic walking on incline slopes, one time they have been transforme by a partial feeback control. Thus, the main iea of this work is to transform the 7-DOF bipe robot in a system for which there exists limit cycles. The objective is then, to obtain nearly passive Limit Cycles, which will be exploite further to get a ynamic walking behaviour. This is possible when a torque is applie to stabilize the torso an knees at fixe positions. A partial feeback will be esigne for, in the next section. 3. Non collocate input/output linearization In this section we use some results from [9] [0] [] [4], the objective is to get a control scheme able to stabilize the torso at a esire position. To show this we may write the ynamic equations system () as: Mθ& + Mθ& + h + φ = τ Mθ& + Mθ& + h + φ = τ (9) (0) Where θ = q an θ = ( q3, q3, q4, q4) T an: M( q) M( q) M( q) = M ( q) M( q) () is the symmetric, positive efinite inertia matrix, the vector functions h ( ) 4 φ ( 4 ( τ, τ ) = Bu represents the input generalize force prouce by the actuators, such that: τ = u, et τ = ( ). From (9)-(0) we obtain: M θ & + h + φ τ = Mv θ& = v () (3) In orer to steer some trajectories for the knees, we have to partition the vector v in two parts from (-3) we obtain the following expression: Mθ& + h + φ = Mθ& = Mθ& Mθ& (4) θ& = ( θ&, θ& ) = ( v, v ); M = [ M; M] (5) 5 As rank( M ( q )) = for all q R, the system (9)-(0) is sai to be strongly inertially couple. Uner this assumption we may compute a pseuo-inverse matrix T T M = M( MM) an efine vin (3): v = M( Mv + h + φ τ + Mv) (6) Or v R, et v R are aitional controls chosen as follows: v = q& + k ( q& q& ) + k ( q q ) (7) p q& 4 + k q& 4 q& 4 + kp q4 q4 q& 4 + k3 q& 4 q& 4 + kp3 q4 q4 v ( ) ( ) = ( ) ( )

5 Note also that we can choose a esire trajectory to avoi hitting the groun, for example such as: π q 4 = Fa sin t t max 3, 4 3 q = q ε an q = Cte The actual control u is given by combining (4), (6) an (7), after some algebra as: u = M± v+ Nv + h % + % φ (8) The non collocate linearization approach transfers the system () into two subsystems written as follows: & η = Aη (9) z& = s( η, z, t) T T T T 0 Whereη = ( η, η) = ( q q, q& q& ), z = ( z, z) = ( q, q& ), A = kp k an z s( η, z, t) = (0) M( h + φ ) MM( q& kpη kη) We refer to the linear subsystem as η -subsystem. Accoringly the non linear subsystem in (9) is enote as z -subsystem where s(0, z, t ) efines the zero ynamics relative to the outputη. 3.3 Fining perio one gait cycles an step perio After stabilizing the torso an the knees we procee to simulate the motion of the bipe robot ownhill a slope. The walker s motion can exhibit perioic behaviour. Limit cycles are often create in this way. At the start of each step we nee to specify initial conitions ( qq&, ) such that after T secons (T is the minimum perio of the limit cycle) the system returns to the same initial conitions at the start. A general proceure to stuy the bipe robot moel is base on interpreting a step as a Poincaré map. Limit cycles are fixe points of this function. A Poincaré map samples the flowφ x of a perioic system once every perio [].The concept is illustrate in figure (). The limit cycle Γ is stable if oscillations approach the limit cycle over time. The samples provie by the corresponing Poincaré map approach a fixe point x. A non stable limit cycle results in ivergent oscillations, for such a case the samples of the Poincaré map iverge. Figure () Poincaré map 3.4 Gait cycle stability Stability of the Poincaré map (0) is etermine by linearizing P aroun the fixe point x, leaing a iscrete evolution equation: xk+ = DP( x ) xk () The matrix A must be Hurwitz.

6 The major issue is how to obtain DP( x- ) The Jacobean matrix- while the bipe ynamics is rather complicate; a close form solution for the linearize map is ifficult to obtain. But one can be obtaine by the use of a recent generalization of trajectory sensitivity analysis [0] [] [4] [5]. 3.5 Numerical proceure A numerical proceure [0] [] [4] [5] is use to test the walking cycle via the Poincaré map, it s resume as follows:. With an initial guess we use the multiimensional Newton-Raphson metho to etermine the fixe point x of P + (immeiately prior the switching event).. Base on this choice of x, we evaluate the eigenvalues of the Poincaré map after one perio by the use of the trajectory sensitivity. Figure (3) Algorithm for the numerical analysis 3.6 Simulation results Consier the moel (), with the following values: Paramétres Tronc Fémur Tibia Masse kg M T 0 5 M M 4 3. Longueurs m L T 0. 5 L L Position es centres e masses m Z T 0. Z Z Premiers moments inerties kg. m MY 0.0 MZ 4 MZ3. MZ Moments inerties orre kg. m XX. XX3.08 XX4 0.93

7 We choose the hyper plane Σ as the event plane. We let the bipe robot on a ownhill slope with the control scheme (8). Starting with a suitable initial guess we obtain the following result: Figure (4) Nearly passive limit cycles for the 7 of bipe robot γ = 0.075ra, q = 0.ra, Kp = 400, Kv = 0, Kp = 500, Kv = 00, Kp3 = 500, Kv3 = 00 we obtain x = ;3.0483; ;.85776; ; ;0.6996; ; ; [ ] 5 CONCLUSION In this paper we present that by the use of a simple nonlinear feeback control with a numerical optimization, we can obtain a semi passive walking gait. The gaits correspon to limit cycles or region of attraction efine with the energy of the almost passive walking on the ownhill slope. The approach is applie to a 7 DOF bipe robot. Simulation results emphasize performance an efficiency of the propose methoology. REFERENCES: [] N.K.M Siri, N.Manamani an D.Elghanami, Control approach for legge robots with fast gaits: Controlle Limit Cycles, Journal of intelligent an robotic systems, Vol. 7, n 4, pp 3-34, 000 [] N.K.M Siri, N.Manamani an N.Najar- Gauthier Methoology base on CLC for control of fast legge robots, Intl. Conference on intelligent robots an systems, proceeings of IEEE/RSJ, October 998. [3] N.K.M Siri, D.Elghanami, T.Boukhobza an N.khraief Gait stabilization for legge robots using energetic regulation, accepte for IEEE/ICAR 00.

8 [4] Mochon, S. an McMahon, Ballistic walking: an improve moel, Mathematical Biosciences, 980, 5: [5] T.McGeer, Passive Dynamic Walking, the international journal of robotics research, 9(), 6-8, April 990 [6] Goswami, A., Tuillot, B., an Espiau, B., Compass like bipeal robot part I: Stability an bifurcation of passive gaits INRIA Research report, N 996, 996. [7] Mariano, Aninya, C., Ruina, A., an Coleman, M. The simplest walking moel: Stability, complexity an scaling, 998 [8] Mark.W.Spong, Passivity base control of the compass gait bipe In: IFAC, worl congress, 999 [9] Choavarapu, P.A., an Spong, M.W., ``On Noncollocate Control of a Single Flexible Link,'' IEEE Intl. Conf. on Robotics an Automation, Minneapolis, MN, April, 996. [0] Spong, M.W., ``The Control of Uneractuate Mechanical System'', Plenary Aress at The First International Conference on Mechatronics, Mexico City, Jan. 6--9, 994. [] Spong, M.W., ``Partial Feeback Linearization of Uneractuate Mechanical Systems'', IROS'94, Munich, Germany, pp. 34-3, Sept [] Asano, F., Yamakita, M, an Furuta, K., Virtual passive ynamic walking an energy base control laws Proc on int conf on intelligent robots an systems, vol pp 49-54, 000 [3] Haruna, M., Ogino, M, Hosoa, K. an Minour, A., Yet another humanoi walking - Passive ynamic walking with torso uner a simple control- Int, Conf on intelligent robots an system, 00 [4] Nahla.khraief, N.K.M Siri, M.W.Spong Nearly passive ynamic walking of a bipe robot, International conference Signals, Systems, Decision & Information Technology, Mars 003. [5] N.Khraief, N.K.M Siri, M.W.Spong An almost passive walking of a kneeless bipe robot with torso, European Control conference 003 (ECC 03) University of Cambrige, UK: - 4 September 003. [6] J.W.Grizzle, Gabriel Abba an Franck Plestan, Asymptotically stable walking for bipe robots :Analysis via systems with impulse effects, IEEE TAC, 999 [7] J.W.Grizzle, Gabriel Abba an Franck Plestan, Proving asymptotic stability of a walking cycle for a five D.OF bipe robot moel. CLAWAR-99 [8] Hurmuzlu, Y., an Marghilu, D.B, Rigi boy collisions of palanar kinematic chains with multiple contact points, the international journal of robotics research, 3() : 8-9, 994 [9] Ian A.Hiskens, Stability of hybri system limit cycle : Application to the compass gait bipe robot,conference on ecision an control, Floria USA, December 00. [0] Ian A.Hiskens, M.A.Pai, Trajectory sensitivity analysis of hybri systems, IEEE transactions on circuit an systems, 000 [] T.S parker an L.O Chua, Practical Numerical Algorithms for chaotic systems, Springer Verlag, New York, NY, 989. [] N.Khraief an N.K.M'Siri. Energy base control for the walking of a 7-DOF bipe robot. To appear in a special issue of Acta Electronica en 004 'Avance Electromechanical motion systems'. [3] N. Khraief, L.Laval, N.K.M'Siri. From almost-passive to active ynamic walking For a kneeless bipe robot with torso. 6th International Conference on Climbing an Walking Robots (CLAWAR) - Italy September 7-9, 003. [4] N.K. M Siri, G. Beurier, A Benallegue an N. Manamani, Passive fee-back control for robots with close kinematic chains. IEEE-Syst. Man an Cyber. IMACS, CESA98, Hammamet, Tunisia, Avril 998.