lizhijun2013iet biped

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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: [email protected]

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

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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]

Mrc 0

0 M

rc

q

+

0

C(q, q)q

+

G

0

+

Dr

Dq

=0

t +I

JT lG (2)

where Mrc [ R33 is the diagonal mass matrix for the COM

of the biped, M[ Rnn is the inertia matrix, C(q, q)q [ R

n

is the centrifugal and Coriolis term, andI[ R33 denotes theidentity matrix.

The rst part of (2) corresponding to the dynamics of theCOM is the simple linear dynamics

Mrcrc + G+Dr= lG (3)

which can be used to produce the desired forces from theground for dynamic balancing of the biped.

There are some useful properties for the dynamics of COMlisted as follows.

Property 1: MatrixMrc is symmetric and positive denite.

Fig. 1 Biped robot

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Property 2: There exist some nite unknown positiveconstants q1, q2, q3 such that q, q [ R

n, Mrc q1,G q2, and supDr q3.

The second part of (2) corresponding to the dynamics oftrunks is the non-linear dynamics

M(q)q + C(q, q)q +Dq = t+JTlG (4)

In the following section, we will eliminate the constraintsforce to obtain the reduced dynamics for (4), whichinvolves only the selected independent variables anddependent variables that are related through a Jacobianmatrix for the single-support (SS), impact, and double-support (DS).

3 Ground constraints

The following constraints need to be considered for neitherfoot can penetrate the ground, the knee joints cannot extend

beyond the fully straight position, and both feet areassumed not to slide when in contact with the ground.

3.1 Non-holonomic constraints

Consider no-slip between each foot and the ground. Thebiped is subjected to non-holonomic constraint with matrixJn. Assume that the l non-integrable and independentvelocity constraints can be

Jn(q)q = 0 (5)

where Jn(q) [ Rln.

Since Jn(q) [ Rln, it is always possible to nd an l

rank matrix S(q) [ Rn(nl) formed by a set of smoothand linearly independent vector elds spanning thenull space of Jn(q), that is, S

T(q)JTn(q) = 0. SinceS(q) = [s1(q), . . . ,snl(q)] is formed by a set of smoothand linearly independent vector spanning the null spaceof Jn(q), there exists dene an auxiliary time functionz(t) = [z1(t), . . . , znl(t)]

T[ R

nl, such that

q = S(q)z(t) = s1(q)z1(t) + +snl(q)znl(t) (6)

It is easy to have

q = S(q)z+ S(q)z (7)

Considering (6) and (7), we can rewrite (4) as

M(q)S(q)z+ [M(q)S(q) + C(q, q)S(q)]z+Dq

= t+JTn(q)ln +J

Th(q)lh (8)

Multiplying (8) by ST

(q), we have

M1 z+ C1 z+D1 = STt+ S

TJ

Thlh (9)

whereM1 = ST

(q)M(q)S(q),C1 = ST

(q)[M(q)S(q) + C(q, q)S(q)], andD1 = S

T(q)Dq.

The force multiplierln can be obtained by (8)

ln = Z1((M(q)S(q) + C(q, q)S(q))z+Dq tJ

T

hlh) (10)

whereZ1 = (Jn(q)M1(q)JTn(q))

1Jn(q)M1(q). Consider the

control input decoupled into the locomotion control ta and

the interactive force control tb as t= ta JTntb. Then, (9)

and (10) can be changed to

M1 z+ C1 z+D1 = STta + S

TJ

Thlh (11)

ln = Z1([M(q)S(q) + C(q, q)S(q)]z+Dq ta JThlh) + tb

(12)

3.2 Holonomic constraints

Assume that both feet are in contact with a certainconstrained surface (z) that is represented as ((z)) = 0,where ((z)) is a given scalar function, x(z) [ Rm

denotes the position vector of the end-effector in contactwith the environment.

Remark 1: Assume that the constraint surface is rigid andhas a continuous gradient. The Jacobian J =

x

z is of full

row rank m, such that the joint coordinate z can be

partitioned into z = [zh,zc]T where zh [ Rnlm andzc [ R

m, with zc = V(zh) with a non-linear mapping

function () from an open set Rnlm R Rm. Theterms V/zh,

2V/q

2h, V/t,

2V/t

2exist and are

bounded in the workspace.

It is easy to have matrix J(z) = JhS= V/z, which canbe partitioned as J(z) = [J1, J2] with J1 = V/zhand J2 = V/zc, and the Jacobian matrix J2 [ R

mm

never degenerates in the set. It is easy to have z = Hzhwith H = Inlm J1J

12

T, where H(q) is full column

rank if and only if J12 exists. There exists a matrix

JT

such that HTJ

T= 0. Consider the control input

ST(q)ta decoupled into ta1 and the force control ta2 asS

T(q)ta = ta1 J

Tta2, and z = Hzh, a reduced-order

model is obtained by taking the above constraints intoconsideration, one obtains

M2 zh + C2 zh +D2 = U (13)

lh = Z2[C1 z+D1 ta1] + ta2 (14)

where M2 = HTM1H, Z2 = (JM

11J

T)1JM11 , C2 = H

T

[M1 H+ C1H], D2 = H

TD1, U = HTta1.

From (12) and (14), it is easy to have

lh = Z2(q)H+T(q)M2(q)zh + ta2 (15)

ln = Z1(q)S+T(q)M1(q)z+ tb (16)

where H+(q) = H(q)(HT(q)H(q))1 is the pseudo-inverse ofH(q) and S+(q) = S(q)(S

T(q)S(q))

1is the pseudo-inverse

ofS(q).

Remark 2 [27]: Matrices H+(q) and S+(q) exist and arebounded for all q.

Property 3: MatrixM2is symmetric and positive denite andmatrix M

2

2C2

is skew-symmetric.

Property 4: There exists a unknown nite-positive vectorC = [c1,c2,c3,c4]

T with ci . 0, such that q, q [ Rn,

M2 c1, C2 c2 + c3q, supt0D2 c4.

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Property 5: All Jacobian matrices are uniformly boundedand uniformly continuous if q is uniformly bounded andcontinuous.

According to the denition of (13),zhjis denoted as the jthelement ofzh [ R

(nlm), andzh = [zh1,zh2, . . . ,zh(nlm)]T

,M2 = [mji](nlm)(nlm), C2 = [cji](nlm)(nlm), D2 =[dj](nlm)1, then we can obtain the jth local dynamics as

mjjzhj + cjj(q, q)zhj + dj +nlm

i=1,i=j

mji zhi

+nlm

i=1,i=j

cji(q, q)zhi = Uj (17)

Remark 3: As in the scheme for the biped, the localdynamics (17) consists of two parts: the rst part isthe local dynamics of each subsystem with parameteruncertainty and local disturbances, and the second part isthe interconnections among these subsystems. Since the

bound on the parameter uncertainty and disturbances ofeach subsystems depend on local variables and arerelatively easy to obtain, their effects can be compensatedseparately by designing a control for each of them to reduceconservativeness.

4 Impact model

The impact between the swing foot and the ground isassumed as a rigid collision. We make two assumptionson the impact model: (i) there is kinetic energy reductionat every impact and; (ii) the impact velocity becomes verysmall and the legs have no bounce. If we assume that theimpact occurs over an innitesimally small period of time,then (1) all velocities remain nite and; (2) there is nochange in position of the system. If t is the duration ofcollision and Fext is the impact force during collision,then the force impulse because of the impact at time isgiven by

M2(z+h z

h) = Fext(t,zh, zh) (18)

where z+h (z

h ) denotes the velocity just after (respectively,

before) an impact.The rst assumption about the kinetic energy reduction

at impact is given by K+ K = DK 0, where

K = 12 (zh)TM2(zh) and K+ = 12 (z+h)TM2(z+h) denote thepre-impact and post-impact kinetic energy, respectively. Thesecond assumption leads to F(zh) = 0. Then J = F/zh,then we have that

Jzh = 0 (19)

and Fext = JTlf where lf = [lft,lfn] with lft and lfn

corresponding to the tangential and normal forces at themoment of impact.

5 Control objective

In order to balance the biped, we should give the desiredposition rdc and velocity r

dc for the COM. Therefore the

rst control objective is to design a balancing control such

that the tracking error of rc and rc from their respectivedesired trajectories r

dc and r

dc to be within a small

neighbourhood of zero, that is, rc rdc 11, and

rc rdc 12. The desired reference trajectory z

dh is

assumed to be bounded and uniformly continuous, and hasbounded and uniformly continuous derivatives up to thesecond order.

The second control objective can be specied as designing

a controller that ensures the tracking error of zh fromtheir respective desired trajectories z

dh to be within a

small neighbourhood of zero, that is, zh(t) zdh e1,

zh(t) zdh e2 where e1 . 0 ande2 . 0. Ideally, e1 and

e2 should be the threshold of measurable noise.In order to avoid the slipping or slippage and tip-over, from

(3), rc rdc brings the ground applied constraints force

to a desired value ldG= [ldTn ,l

dTh ]

T; therefore the constraint

force errors and (lG ldG) should be to be within a small

neighbourhood of zero, that is, lG ldG 6, where > 0

is the threshold of measurable noise. For the impact phase,we should guarantee the system stability during thetransition phase.

The controller design will consist of two stages: (i) a virtualcontrol input l

dG is designed, so that the subsystems (3)

converge to the desired trajectory, and (ii) the actual controlinput is designed in such a way thatzh z

dh andlG l

dG

to be stabilised to the origin.

Lemma 1: Forx > 0 and 1, we have ln(cosh(x)) + d x.

Proof: Ifx 0, we have

x0

2

e2s + 1ds ,

x0

2

e2sds = 1 e

2x, 1

Therefore ln(cosh(x)) + d ln(cosh(x)) +x

02

e2s+1ds with

1. Let

f(x) = ln(cosh(x)) +

x0

2

e2s + 1ds x

we have

f(x) = tanh(x) + 2

e2x + 1 1

=e

x e

x

ex + ex + 2

e2x + 1 1 = 0

From the mean value theorem, we have

f(x) f(0) = f(x)(x 0)

Since f(0) = 0, we have

f(x) = 0

that is, ln(cosh(x)) +x

0 2e2s+1 ds = x, then, we haveln(cosh(x)) + d x. This completes the proof.

Remark 4: Lemma 1 is used to facilitate the control design.

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Assumption 1: Time-varying positive function f(t) convergesto zero as t and satises

limt1

t0

f(v) dv= @, 1

with a nite constant.

6 Adaptive robust control

6.1 Balancing control

For subsystem (3), we can dene

ec = rc rdc (20)

rrc = rdc Lcec (21)

s = ec + Lcec (22)

with Lc being diagonal constant matrix.Considering (21) and (22), we can rewrite (3) as

Mrs = lG D (23)

D = Mrrrc + G+Dr (24)

where Mr is diagonal andlG[ R3.

Lemma 2: Consider Property 2, the upper bound of kthsub-vectorDk ofsatises

Dk ln (cosh(Ck)) + d (25)

where 1 is a small function, Ck = gTkwk with wk =

[1, sup sk]T

, and gk= [gk1,gk2]T

is a vector of positiveconstants dened below.

Proof:According to Property 2, the upper bound ofDksatises

Dk q1rdck Leck + q2 + q3 q1r

dck

+ q1Leck + q2 + q3 (26)

Consider the linear system dened by eck = Leck+ sk,eck(0) = e0. Since the matrix is Hurwitz, there exist

constants b1, b2, b3 and b4 such that eck(t) b1eck0+b2sup sk and eck(t) b3eck0 + b4sup sk.Substituting the later equation into (26), we could nallyobtain Dk gk1 + gk2sup sk.

For thekth vectorlGk, we can design the desired producingconstrain force lGk as

lGk = Yksk ln(cosh(Ck))sgn(sk) dsgn(sk)

Ck = gTkwk

(27)

gk = hgk+ kwksk (28)

where the designed constant Yk. 0, k. 0, if sk 0,sgn(sk) = 1, else sgn(sk) = 1; 1 and in the simulation,we choose d= 1 + 1

(1+t)2; and satises Assumption 1, that

is, limt1 h(t) = 0 and limt1t

0h(v)dv= @h, 1 with

the nite constant@h, that is, can be chosen as 1

(1+t)2.

Remark 5: A control block diagram that summarises thecontrol is shown in Fig. 2. The proposed approach is

based generally on the idea of duality of control withrespect to position and force, as well as to theorthogonality of the subspaces of possible displacementsand reaction forces of a robot in contact with theenvironment. Consider the position of the base (COM) isstabilised by the force control, the required force isrealised by the ground reaction force. While the postureand the joint angle are used to project dynamics on thereduced motion subspace of the generalised coordinates. Acomputed torque method can be established to linearisethe motion subproblem. Once this is done, a force controlwill follow as in any other hybrid control scheme. Thus,the proposed control (26) and (41)(43) constitute one of

the main contributions of this paper, as seen in Fig. 2,achieves global tracking convergence. Once we havedecouple linearised the position and force patterns, we will

proceed to synthesise respectively a full-order feedbackcontroller for the position loop and an proportional actionto regulate the force loop.

Theorem 1:Consider the dynamics of COM described by (3),using the control law (27) and the adaptive law (28), thefollowing hold for any (rc(0), rc(0)):

(i) rc = [rc1,rc2,rc3]T converges to the desired trajectory

rdc = [rdc1,r

dc2,r

dc3]

T as t;(ii) eckandeckconverge to 0 ast, andlGis bounded fort 0.

Proof:To facilitate the control design, consider the followingLyapunov function candidate with () = () () as

V =1

2s

TMrcs +

3k=1

2i=1

1

2kgkigki (29)

The derivative ofValong (23) is given by

V = sTMrcs +

3

k=1 2

i=1

1

kgki

gki

=3k=1

sk[lGk Dk] +3k=1

2i=1

1

kgki

gki (30)

Fig. 2 Control structure

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Integrating (27) into (30), we have

V 3k=1

sk[lGk Dk] +3k=1

2i=1

1

kgki

gki

=

3

k=1

sk[Yksk ln(cosh(Ck))sgn(sk)

dsgn(sk) Dk] +3k=1

2i=1

1

kgki

gki

= 3k=1

Yks2k

3k=1

dsk +3k=1

2i=1

gkigki

3k=1

skln(cosh(Ck))sgn(sk)

3k=1

skDk

3k=1

Yks2k

3k=1

dsk

3k=1

sk ln(cosh(Ck)) +

3k=1

skDk

+3k=1

2i=1

h

kgkigki

3k=1

2i=1

gkiwkisk (31)

Considering Lemmas 1 and 2 and using gkigki= gki(gki

gki) = (1/4)g2ki ((1/2)gki gki)

2 (1/4)g

2ki, we have

V

3

k=1

Yks2k

3

k=1

sk ln(cosh(Ck))

+3k=1

sk ln(cosh(Ck) 3k=1

2i=1

gkiwkisk

+3k=1

2i=1

h

kgkigki

3k=1

Yks2k+

3k=1

2i=1

h

kgkigki

3k=1

Yks2k+

1

4

3k=1

2i=1

h

kg

2ki (32)

Since (1/4)3

k=1

2i=1

hkg2ki is bounded and converges

to zero as t by noting limt1 h= 0, there exists

t. t1, (1/4)3

k=1

2i=1

hkg2ki @2, when |sk|

@2Ymin

,

with Ymin = min(Y1, Y2, . . . , Ynlm),V 0, from above

all, sk converges to a small set containing the origin ast.Integrating both sides of the above equation gives

V(t) V(0) ,

t0

3k=1

Yks2kds +

1

4

3k=1

2i=1

@h

kg2ki (33)

by noting limt1 h= 0, and limt1 t0 h(v)dv= @h, 1.Thus V is bounded, which implies that s [ L1. From

s = ec + Lec, it can be obtained that ec, ec [ L1. As wehave established ec, ec [ L1, we conclude that rc, rc,rc [ L1.

Therefore all the signals on the right-hand side of (3) arebounded, it is easy to conclude that lG is bounded from(27).

6.2 Posture control

Since the dynamics uncertainties of the system, such asdynamics parameters and disturbances in the system, are

usually hard to measure and construct, we need to estimatethose uncertainties in this paper, we develop robust controlcombing on-line parameters identication.

Let

e = zh zdh (34)

zrh = zdh Lhe (35)

r= e + Lhe (36)

with Lh being diagonal positive-denite constant matrix.Considering (35) and (36), we can rewrite (13) as

M2r+ C2r = U J (37)

J = M2 zrh + C2 z

rh +D2 (38)

According to the denition of J [ R(nlm), we denote

Jk, k= 1,2, . . . ,(n l m) as the kth elements of J,which corresponds to the kth equation in the dynamicsof the jth sub-system. Similarly, we denote rk as the kthelement of r[ R

(nlm), and in addition, denote r=[r1,r2, . . . ,rnlm]

T.

We dene thekth component of trunk dynamics in (37) as

nlmj=1 m

kjrj +

nlmj=1 c

kj(q, q)rj= Uk

Jk (39)

Lemma 3:Consider Property 4, the upper bound ofJ satises

Jk ln( cosh (Fk)) + d (40)

where is a small function, Fj= aTkw with w=

[1, supr, supr2

]T

, and ak= [ak1,ak2,ak3]T

is avector of positive constants dened below.

Proof: According to Property 4, the upper bound of Jksatises

Jk c1zdhk Lek + (c2 + c3z

dhk+ ek)z

dhk Lek

+ c4 + c5

c1zdhk + c1Lek + c2z

dhk + c2Lek

+ c3zdhk

2+ c3z

dhkLek + c3ekz

dhk

+ c3Lekek + c4 + c5

c1zdhk + c2z

dhk + c3z

dhk

2+ c4 + c5

+ (c2L + c3zdhkL)ek + c3Lekek

+ (c1L

+ c3

z

d

hk)ek

b1 + b2ek + b3ek + b4ek2

+ b5ek2

(41)

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where b1 = c1zdhk + c2z

dhk + c3z

dhk

2+ c4 + c5, b2 =

c2L + c3zdhkL, b3 = c1L + c3z

dhk, b4 =

12

c3L,b5 = b4.

Consider the linear system dened by ek= Lek+ rk,ek(0) = e0. Since matrix is Hurwitz, there existconstants a1, a2, a3 and a4, such that ek(t) a1ek0 + a2suprk and ek(t) a3ek0 + a4suprk.

Substituting these two equations into (41), we could nallyobtain Jk ak1 + ak2suprk + ak3suprk

2.

We propose the following control for the biped

Uk = kkrk ln(cosh(Fk))sgn(rk) dsgn(rk) (42)

ta2 = ldh Kh(lh l

dh) (43)

tb = ldn Kn(ln l

dn ) (44)

Fk = aTkwk

ak = Sak+ Gwkrk(45)

where kk. 0, > 1, if rk 0, sgn(rk) = 1, elsesgn(rk) = 1, and > 0, satises Assumption 1, such as,limt1 S = 0, and limt1

t0S(v)dv= rS , 1 with the

nite constantrS, that is, S = 1

(1+t)2. It is observed that the

controller (42) only adopt the local feedback information.

Theorem 2: Consider the mechanical system described by

(13) and its dynamics model (39), using the control law(42) and (45), the following hold for any (zh(0), zh(0)):

1. rkconverges to a set containing the origin as t;2. ekandekconverge to 0 as t; and are bounded for allt 0; and3. lG l

dG= [e

Th ,e

Tn ]

T= [(lh l

dh )

T,(ln l

dn)

T]

Tis

bounded and can be made arbitrarily small.

Proof:To facilitate the control design, consider the followingLyapunov function with ak = ak ak as

V =1

2r

TM2r+

n1mk=1

3i=1

1

2Gakiaki (46)

The derivative ofValong (39) is given by (see (47))

Considering Property 3, and integrating (42) into (47), wehave

V nlmk=1

rk[Uk Jk] +nlmk=1

3i=1

1

Gaki

aki

= nlm

k=1

rk[ kkrk ln(cosh (Fk))sgn(rk)

dsgn(rk) Jk]

+nlmk=1

3i=1

akiG1 aki

= nlmk=1

kkr2k

nlmk=1

drk +nlmk=1

3i=1

akiG1 aiki

nlmk=1

rkln(cosh(Fk))sgn(rk)

nlmk=1

rkJk

n

l

m

k=1

kkr2k

n

l

m

k=1

drk

nlmk=1

rk ln(cosh (Fk)) +

nlmk=1

rkJk

+nlmk=1

3i=1

S

Gakiaki

nlmk=1

3i=1

akiwkirk

(48)

Considering Lemmas 1 and 3 and using akiaki= aki(aki

a ki) = (1/4)a2ki ((1/2)aki aki)

2 (1/4)a

2ki, we have

V nlmk=1

kkr2k

nlmk=1

rk ln(cosh (Fk))

+nlmk=1

rk ln(cosh (Fk) nlmk=1

3i=1

akiwkirk

+nlmk=1

3i=1

S

Gakia ki

nlm

k=1

kkr2k+

nlm

k=1 3

i=1

S

Gakia ki

nlmk=1

kkr2k+

1

4

nlmk=1

3i=1

S

Ga

2ki (49)

Since (1/4)nlm

k=1

3i=1

SGa

2ki is bounded and converges to

V =1

2[r

T M2r+ rTM2r+ r

TM2r] +

nlmk=1

3i=1

1

Gaki

aki

=1

2 nlm

k=1 nlm

j=1

rkmkjrj + nlm

k=1 nlm

j=1

rkmkjrj + nlm

k=1 3

i=1

1

Gaki

aki

=nlmk=1

rk

nlmj=1

1

2mkjrj

nlmj=1

ckjrj + Uk Jk

+

nlmk=1

3i=1

1

Gaki

aki (47)

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zero as t by noting limt1 G = 0, there exists t. t1,14

nlmk=1

3i=1

SGa

2ki r3 with a nite small constant r3,

when |rk| r3

kmin

with kmin = min(k1,k2, . . . ,knlm),

V 0, from above all, rk converges to a small setcontaining the origin as t.

Integrating both sides of the above equation gives

V(t) V(0) ,

t0

nlmk=1

kkr2kds +

1

4

nlmk=1

3i=1

r

Ga2ki (50)

Table 1 Range of each joint for the biped robot

Range of jointangel for human

Range of jointangle for our robot

leg

hip roll 4520 4020

pitch 12515 13040

yaw 4545 4545

knee pitch 0130 0150

ankle roll 2030 3040

pitch 2045 5045

toe pitch 4530 300

Fig. 3 Video snapshots of walking

12.09.06.03.00.0

0.0

1.0

0.5

0.0

-0.5

-1.0

-1.5

Analysis: LastRun_Adaptive 2012-05-28 13:04:47

Joint Angles of Left Lower Limb

Time (sec)

Angle(rad)

Actual_Left_Ankle_PitchDesired_Left_Ankle_PitchActual_Left_Ankle_RollDesired_Left_Ankle_RollActual_Left_Hip_PitchDesired_Left_Hip_PitchActual_Left_Hip_RollDesired_Left_Hip_RollActual_Left_Knee_PitchDesired_Left_Knee_Pitch

Fig. 4 Trajectories of left leg (unit: rad) under adaptive control

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by noting limt1 S = 0, and limt1t

0S(v)dv= rS , 1.

Thus V is bounded, which implies that r[ L1

. Fromr= e + Le, it can be obtained that e, e [ L

1. As we have

establishede, e [ L1

, we conclude thatzh, zh, zh [ L1.

Therefore all the signals on the right-hand side of (39) arebounded, it is easy to conclude thatUkis bounded from (42).

We substitute the control ta2 = ldh Kh(lh l

dh ) and

tb = ldn Kn(ln l

dn) with the constant matrices of

proportional control feedback gains Kh and Kn into thereduced order dynamics (15) and (16) yielding (Kh + 1)(l

dh lh) = Z2(q)H

+T(q)M2(q)zh, (Kn + 1)(ldn ln) =

Z1S+T(q)M1(q)z. Since zh z

dh , zh z

dh, zh z

dh, z z

d,

z zd

, z zd

; therefore Z2(q)H+T

(q)M2(q) and Z1S

+T(q)M1(q) are bounded; therefore the size of (lh ldh )

and (ln ldn) are bounded and can be regulated by

choosing suitableKn andKh to arbitrary small.

12.09.06.03.00.0

0.0

4.5E+005

4.0E+005

3.5E+005

3.0E+005

2.5E+005

2.0E+005

1.5E+005

1.0E+005

50000.0

0.0

-50000.0


Joint Torques of Left Lower Limb

Time (sec)

Torque(N*mm)

torque_LAPtorque_LARtorque_LHPtorque_LHRtorque_RAP

Fig. 5 Torques of left leg (unit: Nmm) under adaptive control

12.09.06.03.00.0

0.0

1.5

1.0

0.5

0.0

-0.5

-1.0


Joint Angles of Right Lower Limb

Time (sec)

Angle(rad)

Actual_Right_Ankle_PitchDesired_Right_Ankle_PitchActual_Right_Ankle_RollDesired_Right_Ankle_RollActual_Right_Hip_PitchDesired_Right_Hip_Pitch

Actual_Right_Hip_RollDesired_Right_Hip_RollActual_Right_Knee_PitchDesired_Right_Knee_Pitch

Fig. 6 Trajectories of right leg (unit: rad) under adaptive control

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Remark 6: From the designed controller (25), (40), (41) and(42), the control does not use the dynamics information,that is, the dynamics are unknown for the controller.Although the unmodelled dynamics exists, and it is

suppressed by the proposed control and the systemstability is achieved, the following simulation veries theeffectiveness of the proposed control.

7 Switching stability

For the system switching stability between the single supportand double support, we give the following theorem as follows:

Theorem 3: Consider system (13) with single support phaseand the double support phase, if the system is both stable

12.09.06.03.00.0

0.0

5.0E+005

4.0E+005

3.0E+005

2.0E+005

1.0E+005

0.0

-1.0E+005


Joint Torques of Right Lower Limb

Time (sec)

Torque(N*mm)

torque_RAPtorque_RARtorque_RHPtorque_RHRtorque_RKN

Fig. 7 Torques of right leg (unit: Nmm) under adaptive control

12.09.06.03.00.0

0.0

3000.0

2500.0

2000.0

1500.0

1000.0

500.0

0.0

-500.0

2012-05-27 23:50:13

Centroid

Time (sec)

Length(mm)

.body.CM_Position.X

.body.CM_Position.Y

.body.CM_Position.Z

Fig. 8 Position of COM under adaptive control, Z-axis is the forward direction

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before and after the switching phase using the control law(42), even if there exist external impacts during theswitching, the system is also stable during the switching

phase.

Let V

=12 (z

h zh)

T

M2(zh zh) and V

+

=12 (z

+h zh)

T

M2(z+h zh) denote the Lyapunov function candidate before

and after the switching, and z+h and zh represent the post-

and pre-switch velocities, and zh denotes the single-supportor double support velocity, respectively. The Lyapunovfunction change during the switching can be simplied asfollows: DV=V+ V =K+ K [z+

Th M2 zh z

Th M2 zh]=

DK zThM2(z

+h z

h). Because the foot cannot be penetrated

into the ground; therefore zh should be on the tangentialplane of the ground, while Dzh should be in the verticalplane of the ground. Considering (18) and (19), we have

12.09.06.03.00.0

0.0

1.0

0.375

-0.25

-0.875

-1.5

Analysis: LastRun_PD_002 2012-05-27 17:25:10

Joint Angles of Left Lower Limb

Time (sec)

Angle(rad)

Actual_Left_Ankle_PitchDesired_Left_Ankle_PitchActual_Left_Ankle_RollDesired_Left_Ankle_RollActual_Left_Hip_PitchDesired_Left_Hip_PitchActual_Left_Hip_RollDesired_Left_Hip_RollActual_Left_Knee_PitchDesired_Left_Knee_Pitch

Fig. 9 Trajectories of left leg (unit: rad) under PD control

12.09.06.03.00.0

0.0

4.5E+005

4.0E+005

3.5E+005

3.0E+005

2.5E+005

2.0E+005

1.5E+005

100000.0

50000.0

0.0


Joint Torques of Left Lower Limb

Time (sec)

Torque(N*mm)

torque_LAPtorque_LARtorque_LHPtorque_LHRtorque_LKN

Fig. 10 Torques of left leg (unit: Nmm) under PD control

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DV= DK zThFext= DK z

Th J

Tlf = DK , 0. Therefore

the Lyapunov function is decreasing during impact, themotion of the system is also stable.

8 Simulations

Consider a 12-DOF biped robot shown in Fig. 1 modellingusing ADAMS, which consists of a torso, and a pair of legs

composed of six links. The left and right legs are numberedLegs 1 and 2, respectively. The height of the biped is1.2 m, the lower limbs are 460 mm, and the height of footis 90 mm, the weight is 22 kg. The range of each joint for

the biped in ADAMS is shown in Table 1, and inertiaparameters of the biped are listed in Table 2.

In this study, a cycloidal prole is used for the trajectoriesof the hip and ankle joints of the swinging leg, which can be

12.09.06.03.00.0

0.0

1.5

1.0

0.5

0.0

-0.5

-1.0


Joint Angles of Right Lower Limb

Time (sec)

Angle(rad)

Actual_Right_Ankle_PitchDesired_Right_Ankle_PitchActual_Right_Ankle_RollDesired_Right_Ankle_RollActual_Right_Hip_PitchDesired_Right_Hip_PitchActual_Right_Hip_RollDesired_Right_Hip_RollActual_Right_Knee_PitchDesired_Right_Knee_Pitch

Fig. 11 Trajectories of right leg (unit: rad) under PD control

12.09.06.03.00.0

0.0

4.0E+005

3.0E+005

2.0E+005

1.0E+005

0.0


Joint Torques of Right Lower Limb

Time (sec)

Torque(N*mm)

torque_RAPtorque_RARtorque_RHPtorque_RHRtorque_RKN

Fig. 12 Torques of right leg (unit: Nmm) under PD control

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found in [33]. This prole is used because it shows a similarpattern to a humans ankle trajectory in normal walking andit describes a simple function, which can be easily changedfor different walking patterns. The equations are given asfollows: xa(i) =

ap

[ 2pk

i sin( 2pk

i)], za(i) =d2

[1 cos( 2pk

i)],xhip(i) =

12xa(i) +

a2, zhip(i) =

12za(i) + l1 + l2

d2, where xhip

and zhip denote the positions of the hip, and xa and za

denote the positions of the swinging angle, a is the step

length, d is the height of the swinging ankle, is the totalsampling number of a step, andi is the sampling index, andli is the length of link i. In order to avoid the tumbling, wedesign the lateral trajectory as yh(i) = 102.5 sin(

pk

i), whereyh is the projection of COM on the ground such that theposition of COM is in the foot support area, n is the totalsampling number of a step, and i is the sampling index,and 102.5 mm is the distance between the COM andsupport leg. In the simulation, we choose the parametersas a = 200 mm, d= 120 mm, and l1 = 235.5 mm,l2 = 233.5 mm, = 200. Therefore we can obtain the every

joint in the working space. For the support leg, q1 and q2,the constraint equation is given by l2cos(q2) + l3cos(q3) =

zhip

l4

with the angle height l4

. Therefore, q2

isindependent coordinate, and q3 = F(q2) =cos1

zhipl4l1cos (q2)

l3we design the H = [1,0,0,0; F

q1,0,0,

0;0,1,0,0;0,0,1,0;0,0,0,1], Jn = [0,0,0,1,0,0,0,0,1,0;0,0,0,0,1,0,0,0,0,1], S= [133,033;023,023;033,133,023,033], l=4,m=2.

The parameters in the adaptive control are set asq = [0.0,0.0,0.0]

T, a = [0.0, , . . ., 0.0]

T, and k= G =

diag[1.0], h= S = diag[ 1(1+t)2

], d= 1 + 1(1+t)2

, the balancecontrol gain are choose as Lc = diag[10] andY = [20000,20 000,20 000]

T. The posture control gain is

listed in Table 4. For comparison, we implement the PDcontrol in the biped robot, and the control is set asti = Piei Diei with ei = zh z

dh, and the control gain is

listed in Table3.The video snapshots are shown in Fig. 3. The positions

tracking for each joint proles of the left and right legs areshown in Figs. 4 and6. Similarly, the input torques for the

joints of the left and right leg are shown in Figs. 5 and 7.The position of COM is shown in Fig. 8. For comparisonwith the traditional PD control, Figs. 9 and11are the joint

positions using PD control, the corresponding inputtorques are listed in Figs. 10 and 12. The summary of

Fig. 13 Sum of all joint position errors using adaptive robust control (red) and PD control (blue)

Table 2 Inertia parameters of the bipedName Mass, kg

body 40.410right hip yaw harmonic driver 0.981left hip yaw harmonic driver 0.981right hip 1.066left hip 1.066right hip pitch harmonic driver 0.979left hip pitch harmonic driver 0.979right thigh 1.006left thigh 1.006right knee motor 1 0.480left knee motor 1 0.480right hip pitch motor 0.480left hip pitch motor 0.480

right knee motor 2 0.480left knee motor 2 0.480right knee harmonic driver 1.500left knee harmonic driver 1.500right knee bearing 4.624E-002left knee bearing 4.624E-002right shank 0.718left shank 0.718right ankle pitch motor 0.342left ankle pitch motor 0.342right ankle 1.943left ankle 1.943right foot 1.205left foot 1.205

Table 3 Parameters in the PD control

Parameters Ankle pitch Ankle roll Hip pitch Hip roll Knee

Pi 400 000 29 998 500 2 500 000 35 500 000 900 000Di 40 000 3500 20 000 250 000 20 000

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all joint tracking errors and all joint input torques areshown in Fig. 13 using both PD control and adaptiverobust control, respectively, and we can see the tackingerrors using the proposed adaptive robust control are

bounded and especially smaller than PD control, that is,the walking locomotion using adaptive control is morestable since the robot could adaptively update the control

parameters online, while the PD control is without thecapability. Since the initial values of the dynamics areassumed to be unknown for the controls in the

simulation, from these gures, even if the nominalparameters of the system are uncertain, and the initialdisturbances boundedness from the environment areunknown, we can obtain satisfactory performance by the

proposed control, which is veried by the ADAMSenvironment.

9 Conclusions

In this paper, a structure of adaptive robust control has beenpresented for a biped robot, which includes balancingcontrol and posture control for regulating the COM positionand trunk orientation of bipedal robots in a compliant way.

The biped robot can be decoupled into the decoupleddynamics of COM. The trunks and the adaptive robustcontrol are constructed in the presence of parametric andfunctional dynamics uncertainties. The controller computesa desired ground reaction force required to stabilise the

posture with unknown dynamics of COM and thentransforms these forces into full-body joint torques even ifthe external disturbances exist. The verication of the

proposed control has been conducted by using the extensivesimulations.

10 Acknowledgments

This work was supported by the Natural Science Foundationof China under Grants 61174045, 61111130208, 60935001,the International Science and Technology CooperationProgramme of China under 0102011DFA10950, and theFundamental Research Funds for the Central Universities(Grant no. 2011ZZ0104), National High TechnologyResearch and Development Programme of China (863,2011AA040701), and the Program for New CenturyExcellent Talents in University.

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Table 4 Parameters in the adaptive control

Joints DOF h kk

hip roll 20 250 000hip pitch 125 20 000knee pitch 45 20 000angle roll 140 3500angle pitch 10 40 000

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