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    Published in IET Control Theory and Applications

    Received on 23rd February 2012

    Revised on 30th September 2012

    Accepted on 9th November 2012

    doi: 10.1049/iet-cta.2012.0066

    ISSN 1751-8644

    Adaptive robust controls of biped robotsZhijun Li1, Shuzhi Sam Ge2,3

    1The Key Lab of Autonomous System and Network Control, College of Automation Science and Engineering, South China

    University of Technology, Guangzhou 510641, Peoples Republic of China2Robotics Institute, and School of Computer Science and Engineering, University of Electronic Science and Technology of

    China, Chengdu 610054, Peoples Republic of China3Department of Electrical and Computer Engineering, The National University of Singapore, Singapore 117576, Singapore

    E-mail: zjli@ieee.org

    Abstract: This paper presents a structure of robust adaptive control for biped robots, which includes balancing andposture control for regulating the centre-of-mass (COM) position and trunk orientation of bipedal robots in a compliant way.First, the biped robot is decoupled into the dynamics of COM and the trunks. Then, the adaptive robust controls areconstructed in the presence of parametric and functional dynamics uncertainties. The control computes a desired groundreaction force required to stabilise the posture with unknown dynamics of COM and then transforms these forces into full-

    body joint torques even if the external disturbances exist. Based on Lyapunov synthesis, the proposed adaptive controlsguarantee that the tracking errors of system converge to zero. The proposed controls are robust not only to systemuncertainties such as mass variation but also to external disturbances. The verication of the proposed control is conductedusing the extensive simulations.

    1 IntroductionRecently, advances in both mechanical and software systemshave promoted development of biped robots around theworld [110]. Although, many works on dynamics andcontrol of biped robot had been investigated in [3, 11, 12],the realisation of reliable autonomous biped robots is stilllimited by the current level of motion control strategies. Forexample, some control algorithms were proposed byintroducing passive dynamics, linearised model [4], andreduced-order non-linear dynamic model for biped robots inthe past two decades [1316]. In [13], a control strategy

    based on feedforward compensation and optimal linear statefeedback was derived for a seven-link, 12 degree-of-

    freedom (DOF), biped robot in the double-support phase. In[17], sliding-mode robust control applied to the walking ofa 9-link (8-DOF) biped robot was investigated. The bipedrobot is assumed to involve large parametric uncertainty,while its locomotion is constrained to be on the sagittal

    plane. In [16], an name of this approach was proposedto nd stable as well as unstable hybrid limit cycles fora planar compass-like biped on a shallow slope withoutintegrating the full set of differential equations andapproximating the dynamics. In [18], the energy-based and

    passivity-based control laws were design for exploiting theexistence of passive walking gaits to achieve walking ondifferent ground slopes.

    Efforts were also made to build complete models torepresent the whole periodic walking motion and threephases of the walking cycle (single-support, double-support,and transition phase) as an integrated model, such that the

    performance and stability analysis of the whole closed-loopmotion system could be improved, such as in [11, 15, 19].In the recent, using the approximation property of the fuzzysystems and the neural networks, adaptive control haveobtained many results [2023]. Fuzzy neural networks(FNN) quadratic stabilisation output feedback controlscheme was proposed for a biped robot in [24]. In [25], adesign technique of a recurrent cerebellar model articulationcontroller (RCMAC)-based on fault-tolerant control systemwas investigated to rectify the non-linear faults of a bipedrobot. In [26], the impact dynamics of a ve-link bipedwalking on level ground were studied and the results can

    be used to correlate the gait parameters with the contactevent following impact. In [27], a systematic architecture

    and algorithm of gait control based on energy-efciencyoptimisation was presented to reduce the high-energyconsumption. In [3], an approach for the closed-loopcontrol of a fully actuated biped robot that leverages onits natural dynamics when walking was presented, theinput state-dependent torques were constructed from acombination of low-gain spring-damper couples.

    Most biped robots founded in the real world are composedof a lot of interconnected joints, and the dynamic balance and

    posture need to be considered simultaneously. As such,non-linear biped systems are one of the most difcultcontrol problems in the category. Owing to the complexityof the multi-degrees-of-freedom (multi-DOF) mechanism of

    humanoid robots, an intuitive and ef

    cient method forwhole-body control is required. However, how to improvethe tracking performance of biped robots through designedcontrols is still an challenging research topic that attracts

    www.ietdl.org

    IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161175 161

    doi: 10.1049/iet-cta.2012.0066 & The Institution of Engineering and Technology 2013

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    great attention from robotic community. In this paper,considering both the dynamic balance and the posture

    position to be guaranteed, we decouple the dynamics ofbiped into the dynamics of centre of mass (COM) and thetrunks, and then implement decoupled control structure

    because of the bipeds specic physical nature.Owing to the nite foot support area, pure position control

    is insufcient for executing bipedal locomotion trajectories.

    Therefore some approaches utilised force sensors in the feetfor implementing an inner force or zero moment point(ZMP) control loop [2831]. However, in this paper, we

    propose an approach that gives a desired applied force fromthe robot to the ground to stabilise the posture position andensures the desired contact state between the robot and theground, then distributes that force among predened contact

    points and transforms it to the joint torques directly. Theapproach does not require contact force measurement orinverse kinematics or dynamics.

    Since, along the walk, toe and heel are independentlycharacterised by non-penetration and no-slip constraintwith the ground, in this paper, we consider the holonomicand non-holonomic constraints [24, 27] into the bipeddynamics. The biped robot is rstly decoupled into thedynamics of COM and the trunks. Then, the adaptive robustcontrol is constructed in the presence of parametric andfunctional dynamics uncertainties. The control computesa desired ground reaction force required to stabilise the

    posture with unknown dynamics of COM and thentransforms these forces into full-body joint torques evenif the external disturbances exist. Based on Lyapunovsynthesis, we develop the robust control based on theadaptive parameters mechanisms using on-line parameterestimation strategy in order to have an efcient approximation.The proposed control approach can ensure that the outputsof the system track the given bounded reference signals

    within a small neighbourhood of zero, and guaranteesemi-global uniform boundedness of all the closed loopsignals. Finally, simulation results are presented to verifythe effectiveness of the proposed control.

    2 Dynamics of biped robots

    In general, the walking motion period of a biped robot isdivided into the single-support phase, the double-support

    phase, and the transition phase. In biped locomotion,the double-support and single-support phases alternate.The biped robot usually starts and stops motion at the

    double-support con

    guration. The analysis of bipedlocomotion in both single-support and double-support phaseis very important for improving the smoothness of the

    biped locomotion system, especially when the controlbecomes important for moving the centre of gravity andraising the heel.

    Consider a multi-DOF biped robot contacting with theground, as shown in Fig. 1. Let r[ R

    3be translational

    position coordinate (e.g. base position) and q [ Rn

    be thejoint angles and attitude of the base. Using the generalisedcoordinates x = [r

    T,q

    T]

    T[ R

    3+n, the exact non-linear

    dynamics of the biped with the holonomic constraints andnon-holomic constraints (generated by the respectivesituations of one or both feet grounded with no-slip) can be

    derived using a standard Lagrangian formulation

    M(x)x + C(x, x)x + G + D = u +JTlG (1)

    where M(x) = Mr MrqMqr Mq

    [ R(n+3)(n+3) is the inertia

    matrix; C(x, x) = Cr Crq

    Cqr Cq

    [ R

    (n+3)(n+3) is the

    centrifugal and Coriolis force term; G [ R(n+3) is the

    gravitational torque vector; D [ R(n+3) is the external

    disturbance vector; u = [031,tTn1]

    T[ R

    (n+3) is the

    control input vector; J = [JTn, J

    Th]

    T[ R

    3(n+3) and

    lG= [lTn ,l

    Th ]

    T[ R

    3 are Jacobian matrix and Lagrangianmultiplier corresponding to the non-holonomic andholonomic constraints.

    Let rc = [xc,yc,zc]T[ R

    3 be the position vector of

    the COM coordinate, and rp=

    [xp,yp,zp]

    T[ R

    3

    be theposition vector from COM to the contact point. The contactpoint does not move on the ground surface. The constraintforces lG= [l

    Tn ,l

    Th ]

    T and a ground reaction force fR satisfylG +fR = 0.

    If we replacerbyrc, we can rewrite the dynamics (1) as thedecoupled dynamics [32]

    M