Adaptive robust controls of biped robots

www.ietdl.org 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 robots Zhijun Li1, Shuzhi Sam Ge2,3 1

The Key Lab of Autonomous System and Network Control, College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, People’s Republic of China 2 Robotics Institute, and School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China 3 Department 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 and posture 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 are constructed in the presence of parametric and functional dynamics uncertainties. The control computes a desired ground reaction force required to stabilise the posture with unknown dynamics of COM and then transforms these forces into fullbody joint torques even if the external disturbances exist. Based on Lyapunov synthesis, the proposed adaptive controls guarantee that the tracking errors of system converge to zero. The proposed controls are robust not only to system uncertainties such as mass variation but also to external disturbances. The verification of the proposed control is conducted using the extensive simulations.

1

Introduction

Recently, advances in both mechanical and software systems have promoted development of biped robots around the world [1–10]. Although, many works on dynamics and control of biped robot had been investigated in [3, 11, 12], the realisation of reliable autonomous biped robots is still limited by the current level of motion control strategies. For example, some control algorithms were proposed by introducing passive dynamics, linearised model [4], and reduced-order non-linear dynamic model for biped robots in the past two decades [13–16]. In [13], a control strategy based on feedforward compensation and optimal linear state feedback was derived for a seven-link, 12 degree-offreedom (DOF), biped robot in the double-support phase. In [17], sliding-mode robust control applied to the walking of a 9-link (8-DOF) biped robot was investigated. The biped robot 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 proposed to find stable as well as unstable hybrid limit cycles for a planar compass-like biped on a shallow slope without integrating the full set of differential equations and approximating the dynamics. In [18], the energy-based and passivity-based control laws were design for exploiting the existence of passive walking gaits to achieve walking on different ground slopes. Efforts were also made to build complete models to represent the whole periodic walking motion and three phases of the walking cycle (single-support, double-support, and transition phase) as an integrated model, such that the IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

performance and stability analysis of the whole closed-loop motion system could be improved, such as in [11, 15, 19]. In the recent, using the approximation property of the fuzzy systems and the neural networks, adaptive control have obtained many results [20–23]. Fuzzy neural networks (FNN) quadratic stabilisation output feedback control scheme was proposed for a biped robot in [24]. In [25], a design technique of a recurrent cerebellar model articulation controller (RCMAC)-based on fault-tolerant control system was investigated to rectify the non-linear faults of a biped robot. In [26], the impact dynamics of a five-link biped walking on level ground were studied and the results can be used to correlate the gait parameters with the contact event following impact. In [27], a systematic architecture and algorithm of gait control based on energy-efficiency optimisation was presented to reduce the high-energy consumption. In [3], an approach for the closed-loop control of a fully actuated biped robot that leverages on its natural dynamics when walking was presented, the input state-dependent torques were constructed from a combination of low-gain spring-damper couples. Most biped robots founded in the real world are composed of 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 difficult control problems in the category. Owing to the complexity of the multi-degrees-of-freedom (multi-DOF) mechanism of humanoid robots, an intuitive and efficient method for whole-body control is required. However, how to improve the tracking performance of biped robots through designed controls is still an challenging research topic that attracts 161

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www.ietdl.org great attention from robotic community. In this paper, considering both the dynamic balance and the posture position to be guaranteed, we decouple the dynamics of biped into the dynamics of centre of mass (COM) and the trunks, and then implement decoupled control structure because of the biped’s specific physical nature. Owing to the finite foot support area, pure position control is insufficient for executing bipedal locomotion trajectories. Therefore some approaches utilised force sensors in the feet for implementing an inner force or zero moment point (ZMP) control loop [28–31]. However, in this paper, we propose an approach that gives a desired applied force from the robot to the ground to stabilise the posture position and ensures the desired contact state between the robot and the ground, then distributes that force among predefined contact points and transforms it to the joint torques directly. The approach does not require contact force measurement or inverse kinematics or dynamics. Since, along the walk, toe and heel are independently characterised by non-penetration and no-slip constraint with the ground, in this paper, we consider the holonomic and non-holonomic constraints [24, 27] into the biped dynamics. The biped robot is firstly decoupled into the dynamics of COM and the trunks. Then, the adaptive robust control is constructed in the presence of parametric and functional dynamics uncertainties. The control computes a desired ground reaction 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, we develop the robust control based on the adaptive parameters mechanisms using on-line parameter estimation strategy in order to have an efficient approximation. The proposed control approach can ensure that the outputs of the system track the given bounded reference signals within a small neighbourhood of zero, and guarantee semi-global uniform boundedness of all the closed loop signals. Finally, simulation results are presented to verify the effectiveness of the proposed control.

2

Dynamics of biped robots

In general, the walking motion period of a biped robot is divided 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 configuration. The analysis of biped locomotion in both single-support and double-support phase is very important for improving the smoothness of the biped locomotion system, especially when the control becomes important for moving the centre of gravity and raising the heel. Consider a multi-DOF biped robot contacting with the ground, as shown in Fig. 1. Let r [ R3 be translational position coordinate (e.g. base position) and q [ Rn be the joint angles and attitude of the base. Using the generalised coordinates x = [rT , qT ]T [ R3+n , the exact non-linear dynamics of the biped with the holonomic constraints and non-holomic constraints (generated by the respective situations 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 + J T lG 162 & The Institution of Engineering and Technology 2013

(1)

Fig. 1 Biped robot




 Mr Mrq where M(x) = [ R(n+3)×(n+3) is the inertia Mqr Mq
 Cr Crq matrix; C(x, x˙ ) = is the [ R(n+3)×(n+3) Cqr Cq centrifugal and Coriolis force term; G [ R(n+3) is the gravitational torque vector; D [ R(n+3) is the external disturbance vector; u = [03×1 , tTn×1 ]T [ R(n+3) is the control input vector; J = [JnT , JhT ]T [ R3×(n+3) and lG = [lTn , lTh ]T [ R3 are Jacobian matrix and Lagrangian multiplier corresponding to the non-holonomic and holonomic constraints. Let rc = [xc , yc , zc ]T [ R3 be the position vector of the COM coordinate, and rp = [xp , yp , zp ]T [ R3 be the position vector from COM to the contact point. The contact point does not move on the ground surface. The constraint forces lG = [lTn , lTh ]T and a ground reaction force fR satisfy lG + fR = 0. If we replace r by rc , we can rewrite the dynamics (1) as the decoupled dynamics [32]


Mrc

0

0

M






   Dr 0 r¨ c G + + + Dq q¨ C(q, q˙ )˙q 0

 I 0 + T lG (2) = t J

where Mrc [ R3×3 is the diagonal mass matrix for the COM of the biped, M [ Rn×n is the inertia matrix, C(q, q˙ )˙q [ Rn is the centrifugal and Coriolis term, and I [ R3×3 denotes the identity matrix. The first part of (2) corresponding to the dynamics of the COM is the simple linear dynamics Mrc r¨ c + G + Dr = lG

(3)

which can be used to produce the desired forces from the ground for dynamic balancing of the biped. There are some useful properties for the dynamics of COM listed as follows. Property 1: Matrix Mrc is symmetric and positive definite. IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

www.ietdl.org Property 2: There exist some finite unknown positive constants q1 , q2 , q3 such that ∀q, q˙ [ Rn , Mrc  ≤ q1 , G ≤ q2 , and supDr  ≤ q3 . The second part of (2) corresponding to the dynamics of trunks is the non-linear dynamics M (q)¨q + C(q, q˙ )˙q + Dq = t + J T lG

(4)

In the following section, we will eliminate the constraints force to obtain the reduced dynamics for (4), which involves only the selected independent variables and dependent variables that are related through a Jacobian matrix for the single-support (SS), impact, and doublesupport (DS).

3

Ground constraints

The following constraints need to be considered for neither foot can penetrate the ground, the knee joints cannot extend beyond the fully straight position, and both feet are assumed not to slide when in contact with the ground. 3.1

Non-holonomic constraints

Consider no-slip between each foot and the ground. The biped is subjected to non-holonomic constraint with matrix Jn . Assume that the l non-integrable and independent velocity constraints can be Jn (q)˙q = 0

(5)

where Jn (q) [ Rl×n . Since Jn (q) [ Rl×n , it is always possible to find an l rank matrix S(q) [ Rn×(n−l) formed by a set of smooth and linearly independent vector fields spanning the null space of Jn (q), that is, S T (q)JnT (q) = 0. Since S(q) = [s1 (q), . . . , sn−l (q)] is formed by a set of smooth and linearly independent vector spanning the null space of Jn (q), there exists define an auxiliary time function z˙(t) = [˙z1 (t), . . . , z˙n−l (t)]T [ Rn−l , such that q˙ = S(q)˙z(t) = s1 (q)˙z1 (t) + · · · + sn−l (q)˙zn−l (t)

(6)

It is easy to have ˙ z + S(q)¨z q¨ = S(q)˙

(7)

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

˙ + C(q, q˙ )S(q)]˙z + Dq − ta − JhT lh ) + tb ln = Z1 ([M (q)S(q) (12)

3.2

Holonomic constraints

Assume that both feet are in contact with a certain constrained surface Φ(z) that is represented as Φ(χ(z)) = 0, where Φ(χ(z)) is a given scalar function, x(z) [ Rm denotes the position vector of the end-effector in contact with the environment. Remark 1: Assume that the constraint surface is rigid and has a continuous gradient. The Jacobian J = ∂∂zx is of full row rank m, such that the joint coordinate z can be partitioned into z = [zh , zc ]T where zh [ Rn−l−m and zc [ Rm , with zc = V(zh ) with a non-linear mapping function Ω(·) from an open set Rn−l−m × R  Rm . The terms ∂V/∂zh , ∂2 V/∂q2h , ∂V/∂t, ∂2 V/∂t 2 exist and are bounded in the workspace. It is easy to have matrix J(z) = Jh S = ∂V/∂z, which can be partitioned as J(z) = [J1 , J2 ] with J1 = ∂V/∂zh and J2 = ∂V/∂zc , and the Jacobian matrix J2 [ Rm×m never degenerates in the set Ω. It is easy to have z˙ = H z˙h  T with H = In−l−m −J1 J−1 , where H(q) is full column 2 −1 rank if and only if J2 exists. There exists a matrix JT such that H T JT = 0. Consider the control input S T (q)ta decoupled into ta1 and the force control ta2 as S T (q)ta = ta1 − JT ta2 , and z˙ = H z˙h , a reduced-order model is obtained by taking the above constraints into consideration, one obtains M2 z¨h + C2 z˙h + D2 = U

(13)

lh = Z2 [C1 z˙ + D1 − ta1 ] + ta2

(14)

where M2 = H T M1 H, Z2 = (JM11 JT )−1 JM1−1 , C2 = H T [M1 H˙ + C1 H], D2 = H T D1 , U = H T ta1 . From (12) and (14), it is easy to have

(8)

ln = Z1 (q)S +T (q)M1 (q)¨z + tb

(16)

(9)

where H + (q) = H(q)(H T (q)H(q))−1 is the pseudo-inverse of H(q) and S + (q) = S(q)(S T (q)S(q))−1 is the pseudo-inverse of S(q).

˙ + C(q, q˙ ) where M1 = S T (q)M (q)S(q), C1 = S T (q)[M (q)S(q) S(q)], and D1 = S T (q)Dq . The force multiplier ln can be obtained by (8) ˙ + C(q, q˙ )S(q))˙z + Dq − t − JhT lh ) (10) ln = Z1 ((M (q)S(q) where Z1 = (Jn (q)M −1 (q)JnT (q))−1 Jn (q)M −1 (q). Consider the control input τ decoupled into the locomotion control ta and IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

(11)

(15)

Multiplying (8) by S T (q), we have M1 z¨ + C1 z˙ + D1 = S T t + S T JhT lh

M1 z¨ + C1 z˙ + D1 = S T ta + S T JhT lh

lh = Z2 (q)H +T (q)M2 (q)¨zh + ta2

˙ + C(q, q˙ )S(q)]˙z + Dq M (q)S(q)¨z + [M (q)S(q) = t + JnT (q)ln + JhT (q)lh

the interactive force control tb as t = ta − JnT tb . Then, (9) and (10) can be changed to

Remark 2 [27]: Matrices H + (q) and S + (q) exist and are bounded for all q. Property 3: Matrix M2 is symmetric and positive definite and ˙ 2 − 2C2 is skew-symmetric. matrix M Property 4: There exists a unknown finite-positive vector C = [c1 , c2 , c3 , c4 ]T with ci . 0, such that ∀q, q˙ [ Rn , M2  ≤ c1 , C2  ≤ c2 + c3 ˙q, supt≥0 D2  ≤ c4 . 163

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www.ietdl.org Property 5: All Jacobian matrices are uniformly bounded and uniformly continuous if q is uniformly bounded and continuous. According to the definition of (13), zhj is denoted as the jth element of zh [ R(n−l−m) , and zh = [zh1 , zh2 , . . . , zh(n−l−m) ]T , M2 = [m ji ](n−l−m)×(n−l−m) , C2 = [c ji ](n−l−m)×(n−l−m) , D2 = [d j ](n−l−m)×1 , then we can obtain the jth local dynamics as m jj z¨hj + c jj (q, q˙ )˙zhj + d j +

n−l−m 

m ji z¨hi

i=1,i=j

+

n−l−m 

c ji (q, q˙ )˙zhi = U j

(17)

i=1,i=j

Remark 3: As in the scheme for the biped, the local dynamics (17) consists of two parts: the first part is the local dynamics of each subsystem with parameter uncertainty and local disturbances, and the second part is the interconnections among these subsystems. Since the bound on the parameter uncertainty and disturbances of each subsystems depend on local variables and are relatively easy to obtain, their effects can be compensated separately by designing a control for each of them to reduce conservativeness.

that the tracking error of rc and r˙ c from their respective desired trajectories rcd and r˙ dc to be within a small neighbourhood of zero, that is, rc − rcd  ≤ 11 , and ˙rc − r˙ dc  ≤ 12 . The desired reference trajectory zdh is assumed to be bounded and uniformly continuous, and has bounded and uniformly continuous derivatives up to the second order. The second control objective can be specified as designing a controller that ensures the tracking error of zh from their respective desired trajectories zdh to be within a small neighbourhood of zero, that is, zh (t) − zdh  ≤ e1 , ˙zh (t) − z˙dh  ≤ e2 where e1 . 0 and e2 . 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  rcd brings the ground applied constraints force dT T to a desired value ldG = [ldT n , lh ] ; therefore the constraint d force errors and (lG − lG ) should be to be within a small neighbourhood of zero, that is, lG − ldG  ≤ 6, where ς > 0 is the threshold of measurable noise. For the impact phase, we should guarantee the system stability during the transition phase. The controller design will consist of two stages: (i) a virtual control input ldG is designed, so that the subsystems (3) converge to the desired trajectory, and (ii) the actual control input τ is designed in such a way that zh  zdh and lG − ldG to be stabilised to the origin. Lemma 1: For x > 0 and δ ≥ 1, we have ln(cosh(x)) + d ≥ x.

4

Impact model

Proof: If x ≥ 0, we have

The impact between the swing foot and the ground is assumed as a rigid collision. We make two assumptions on the impact model: (i) there is kinetic energy reduction at every impact and; (ii) the impact velocity becomes very small and the legs have no bounce. If we assume that the impact occurs over an infinitesimally small period of time, then (1) all velocities remain finite and; (2) there is no change in position of the system. If t is the duration of collision and Fext is the impact force during collision, then the force impulse because of the impact at time is given by M2 (˙z+ ˙− ˙h ) h −z h ) = Fext (t, zh , z

(18)

z− where z˙+ h (˙ h ) denotes the velocity just after (respectively, before) an impact. The first assumption about the kinetic energy reduction at impact is given by K+ − K− = DK ≤ 0, where T + 1 + T K− = 12 (˙z− z− zh ) M2 (˙z+ h ) M2 (˙ h ) and K = 2 (˙ h ) denote the pre-impact and post-impact kinetic energy, respectively. The second assumption leads to F(zh ) = 0. Then J = ∂F/∂zh , then we have that J z˙h = 0

(19)

and Fext = J T lf where lf = [l ft , l fn ] with l ft and l fn corresponding to the tangential and normal forces at the moment of impact.

x

2 ds , 2s 0e +1

x

2 ds = 1 − e−2x , 1 2s 0e

Therefore ln(cosh(x)) + d ≥ ln(cosh(x)) + δ ≥ 1. Let

x

2 ds 0 e2s +1

with

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

2 ds − x 2s + 1 e 0

we have f˙ (x) = tanh(x) + =

2 −1 e2x + 1

ex − e−x 2 −1=0 + 2x x −x e +e e +1

From the mean value theorem, we have f (x) − f (0) = f˙ (x)(x − 0) Since f(0) = 0, we have f (x) = 0

5

Control objective

In order to balance the biped, we should give the desired position rcd and velocity r˙ dc for the COM. Therefore the first control objective is to design a balancing control such 164 & The Institution of Engineering and Technology 2013

x that is, ln(cosh(x)) + 0 e2s2+1 ds = x, then, ln(cosh(x)) + d ≥ x. This completes the proof.

we

have □

Remark 4: Lemma 1 is used to facilitate the control design. IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

www.ietdl.org Assumption 1: Time-varying positive function f (t) converges to zero as t → ∞ and satisfies t f (v ) d v = @ , 1

lim

t1

0

with a finite constant ϱ.

6 6.1

Adaptive robust control Balancing control

For subsystem (3), we can define

Fig. 2 Control structure

ec = rc − rcd

(20)

r˙ rc = r˙ dc − Lc ec

(21)

s = e˙ c + Lc ec

(22)

with Lc being diagonal constant matrix. Considering (21) and (22), we can rewrite (3) as Mr s˙ = lG − D

(23)

D = Mr r¨ rc + G + Dr

(24)

where Mr is diagonal and lG [ R3 . Lemma 2: Consider Property 2, the upper bound of kth sub-vector Dk of Δ satisfies Dk  ≤ ln (cosh(Ck )) + d

(25)

where δ ≥ 1 is a small function, Ck = gTk wk with wk = [1, sup sk ]T , and gk = [gk1 , gk2 ]T is a vector of positive constants defined below. Proof: According to Property 2, the upper bound of Dk satisfies Dk  ≤ q1 ¨rdck − L˙eck  + q2 + q3 ≤ q1 ¨rdck  + q1 L˙eck  + q2 + q3

(26)

Consider the linear system defined by e˙ ck = −Leck + sk , eck (0) = e0 . Since the matrix −Λ is Hurwitz, there exist constants b1 , b2 , b3 and b4 such that eck (t) ≤ b1 eck0 + b2 sup sk  and ˙eck (t) ≤ b3 eck0  + b4 sup sk . Substituting the later equation into (26), we could finally obtain Dk  ≤ gk1 + gk2 sup sk . □ For the kth vector lGk , we can design the desired producing constrain force lGk as ˆ ))sgn(s ) − d sgn(s ) lGk = −Yk sk − ln(cosh(C k k k ˆ = gˆ T w C k k k

g˙ˆ k = −hgˆ k + kwk sk 

(27) (28)

where the designed constant Yk . 0, k . 0, if sk ≥ 0, sgn(sk ) = 1, else sgn(sk ) = −1; δ ≥ 1 and in the simulation, 1 we choose d = 1 + (1+t) 2 ; and η satisfies Assumption 1, that IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

t is, limt1 h(t) = 0 and limt1 0 h(v)dv = @h , 1 with 1 the finite constant @h , that is, η can be chosen as (1+t) 2. Remark 5: A control block diagram that summarises the control is shown in Fig. 2. The proposed approach is based generally on the idea of duality of control with respect to position and force, as well as to the orthogonality of the subspaces of possible displacements and reaction forces of a robot in contact with the environment. Consider the position of the base (COM) is stabilised by the force control, the required force is realised by the ground reaction force. While the posture and the joint angle are used to project dynamics on the reduced motion subspace of the generalised coordinates. A computed torque method can be established to linearise the motion subproblem. Once this is done, a force control will 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 have decouple linearised the position and force patterns, we will proceed to synthesise respectively a full-order feedback controller for the position loop and an proportional action to regulate the force loop. Theorem 1: Consider the dynamics of COM described by (3), using the control law (27) and the adaptive law (28), the following hold for any (rc (0), r˙ c (0)): (i) rc = [rc1 , rc2 , rc3 ]T converges to the desired trajectory d d d T , rc2 , rc3 ] as t → ∞; rcd = [rc1 (ii) eck and e˙ ck converge to 0 as t → ∞, and lG is bounded for t ≥ 0. Proof: To facilitate the control design, consider the following ˜ = (·) − (·) ˆ as Lyapunov function candidate with (·) 3  2  1 1 V = sT Mrc s + g˜ g˜ 2 2k k i k i k=1 i=1

(29)

The derivative of V along (23) is given by V˙ = sT Mrc s˙ +

3  2  1 k=1 i=1

=

3  k=1

k

sk [lGk − Dk ] +

g˜ k i g˙˜ k i 3  2  1 k=1 i=1

k

g˜ k i g˙˜ k i

(30)

165

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

Integrating (27) into (30), we have V˙ ≤

3 

sk [lGk − Dk ] +

=

g˜ k i g˙˜ k i

k

k=1 i=1

k=1 3 

3  2  1

6.2

ˆ ))sgn(s ) sk [−Yk sk − ln(cosh(C k k

Since the dynamics uncertainties of the system, such as dynamics parameters and disturbances in the system, are usually hard to measure and construct, we need to estimate those uncertainties in this paper, we develop robust control combing on-line parameters identification. Let

k=1

− dsgn(sk ) − Dk ] +

3  2  1 k=1 i=1

=−

3 

3 

Yk s2k −

k=1

−

3 

dsk  +

k

g˜ k i g˙˜ k i

3  2 

3 

ˆ ))sgn(s ) − sk ln(cosh(C k k

3 

Yk s2k −

3 

sk D k

k=1

ˆ )) + sk  ln(cosh(C k

3 

k=1 i=1

k

g˜ k i gˆ k i −

3  2 

J=

g˜ k i wk i sk 

(31)

k=1 i=1

Considering Lemmas 1 and 2 and using g˜ k i gˆ k i = gˆ k i (gk i − gˆ k i ) = (1/4)g2k i − ((1/2)gk i − gˆ k i )2 ≤ (1/4)g2k i , we have V˙ ≤ −

3 

Yk s2k −

k=1

+

3 

3 

ˆ )) sk  ln(cosh(C k

3  2  h k=1 i=1

≤−

3 

3  k=1

3  2 

j=1

g˜ k i wk i sk 

r = e˙ + Lh e

(36)

M2 z¨rh

+

C2 z˙rh

+ D2

(37) (38)

mkj r˙ j +

n−l−m 

ckj (q, q˙ )r j = U k − Jk

(39)

j=1

k

Lemma 3: Consider Property 4, the upper bound of J satisfies

g˜ k i gˆ k i

Yk s2k + Yk s2k +

Jk  ≤ ln( cosh (Fk )) + d

3  2  h k=1 i=1

k=1

≤−

(35)

k=1 i=1

k=1

+

z˙rh = z˙dh − Lh e

According to the definition of J [ R(n−l−m) , we denote Jk , k = 1, 2, . . . , (n − l − m) as the kth elements of J, which corresponds to the kth equation in the dynamics of the jth sub-system. Similarly, we denote rk as the kth element of r [ R(n−l−m) , and in addition, denote r = [r1 , r2 , . . . , rn−l−m ]T . We define the kth component of trunk dynamics in (37) as n−l−m 

k=1

sk  ln(cosh(Ck ) −

(34)

M2 r˙ + C2 r = U − J

sk Dk 

k=1

3  2  h

e = zh − zdh

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

dsk 

k=1

+

3  k=1

k=1

−

g˜ k i g˙˜ k i

k=1 i=1

k=1

k=1

≤−

Posture control

k

where δ is a small function, Fj = aTk w with w = [1, supr, supr2 ]T , and ak = [ak1 , ak2 , ak3 ]T is a vector of positive constants defined below.

g˜ k i gˆ k i

3  2 1 h 2 g 4 k=1 i=1 k k i

(32)

  Since (1/4) 3k=1 2i=1 hk g2k i is bounded and converges to zero as t → ∞ by noting limt1 h = 0, there exists 3 2 h 2 t . t1 , (1/4) k=1 i=1 k gk i ≤ @2 , when |sk | ≥ Y@2 , min with Ymin = min(Y1 , Y2 , . . . , Yn−l−m ), V˙ ≤ 0, from above all, sk converges to a small set containing the origin as t → ∞.Integrating both sides of the above equation gives t  3

3  2 @h 2 1 g Yk s2k ds + V (t) − V (0) , − 4 k=1 i=1 k k i 0 k=1

t

(33)

by noting limt1 h = 0, and limt1 0 h(v)dv = @h , 1. Thus V is bounded, which implies that s [ L1 . From s = e˙ c + Lec , it can be obtained that ec , e˙ c [ L1 . As we have established ec , e˙ c [ L1 , we conclude that rc , r˙ c , r¨ c [ L1 . 166 & The Institution of Engineering and Technology 2013

(40)

Proof: According to Property 4, the upper bound of Jk satisfies Jk  ≤ c1 ¨zdhk − L˙ek  + (c2 + c3 ˙zdhk + e˙ k )˙zdhk − Lek  + c4 + c5 ≤ c1 ¨zdhk  + c1 L˙ek  + c2 ˙zdhk  + c2 Lek  + c3 ˙zdhk 2 + c3 ˙zdhk Lek  + c3 ˙ek ˙zdhk  + c3 Lek ˙ek  + c4 + c5 ≤ c1 ¨zdhk  + c2 ˙zdhk  + c3 ˙zdhk 2 + c4 + c5 + (c2 L + c3 ˙zdhk L)ek  + c3 Lek ˙ek  + (c1 L + c3 ˙zdhk )˙ek  ≤ b1 + b2 ek  + b3 ˙ek  + b4 ek 2 + b5 ˙ek 2 (41) IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

www.ietdl.org where b1 = c1 ¨zdhk  + c2 ˙zdhk  + c3 ˙zdhk 2 + c4 + c5 , b2 = c2 L + c3 ˙zdhk L, b3 = c1 L + c3 ˙zdhk , b4 = 12 c3 L, b5 = b4 .

Considering Property 3, and integrating (42) into (47), we have V˙ ≤

Consider the linear system defined by e˙ k = −Lek + rk , ek (0) = e0 . Since matrix −Λ is Hurwitz, there exist constants a1 , a2 , a3 and a4 , such that ek (t) ≤ a1 ek0  + a2 suprk  and ˙ek (t) ≤ a3 ek0  + a4 suprk . Substituting these two equations into (41), we could finally obtain Jk  ≤ ak1 + ak2 suprk  + ak3 suprk 2 . □

ta2 =

− Kh (lh −

ldh )

=

tb =

− Kn (ln −

ldn )

ˆ ))sgn(r ) rk [ − kk rk − ln(cosh (F k k

− dsgn(rk ) − Jk ] n−l−m 3  

+ (42)

a˜ k i G−1 a˙˜ k i

i=1

k=1 n−l−m 

kk rk2 −

n−l−m 

k=1

(43)

(44)

(45) −

ˆ ))sgn(r ) − rk ln(cosh(F k k

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), z˙h (0)):

+

kk rk2 −

n−l−m 

drk 

k=1

n−l−m 

ˆ )) + rk  ln(cosh (F k

n−l−m 

V˙ ≤ −

k=1

i=1

n−l−m 3  

i=1

n−l−m 

kk rk2 −

k=1

+

n−l−m 

ˆ )) rk  ln(cosh (F k

rk  ln(cosh (Fk ) −

n−l−m 3   k=1

n−l−m 3  

i=1

k=1

≤−

(46)

1 ˙ 2 r + r˙ T M2 r + rT M2 r˙ ] + V˙ = [rT M 2

n−l−m  k=1

n−l−m 

≤−

kk rk2 + kk rk2 +

k=1

Since (1/4) n−l−m 3   k=1

i=1

n−l−m 3   k=1

n−l−m 

a˜ k i wk i rk 

i=1

S a˜ aˆ G ki ki

k=1

The derivative of V along (39) is given by (see (47))

n−l−m 3   S a˜ k i aˆ k i − a˜ k i wk i rk  G k=1 i=1

k=1

Proof: To facilitate the control design, consider the following Lyapunov function with a˜ k = ak − aˆ k as 1 a˜ a˜ 2G k i k i

rk Jk 

k=1

Considering Lemmas 1 and 3 and using a˜ k i aˆ k i = aˆ k i (ak i − aˆ k i ) = (1/4)a2k i − ((1/2)ak i − aˆ k i )2 ≤ (1/4)a2k i , we have

+

1 V = r T M2 r + 2

rk J k

(48)

1. rk converges to a set containing the origin as t→∞; 2. ek and e˙ k converge to 0 as t→∞; and τ are bounded for all t ≥ 0; and 3. lG − ldG = [eTh , eTn ]T = [(lh − ldh )T , (ln − ldn )T ]T is bounded and can be made arbitrarily small.

n−1−m 3  

n−l−m  k=1

n−l−m 

k=1

a˜ k i G−1 a˙˜ ik i

i=1

k=1

k=1

where kk . 0, δ > 1, if rk ≥ 0, sgn(rk ) = 1, else sgn(rk ) = −1, and Γ > 0, Σ satisfies Assumption 1, such as,

t limt1 S = 0, and limt1 0 S(v)dv = rS , 1 with the 1 finite constant rS , that is, S = (1+t) 2 . It is observed that the controller (42) only adopt the local feedback information.

n−l−m 3  

k=1

k=1

a˙ˆ k = −Saˆ k + Gwk rk 

drk  +

k=1

n−l−m 

≤−

ˆ = aˆ T w F k k k

i=1

1 a˜ a˙˜ G ki ki

k=1

−

ldn

n−l−m 3   k=1

n−l−m 

=−

ldh

rk [U k − Jk ] +

k=1

We propose the following control for the biped ˆ ))sgn(r ) − dsgn(r ) Uk = −kk rk − ln(cosh(F k k k

n−l−m 

3   1 n−l−m S 2 a 4 k=1 i=1 G k i

n−l−m 3 k=1

i=1

S a˜ aˆ G ki ki

S 2 i=1 G ak i

(49)

is bounded and converges to

1 a˜ a˙˜ G ki ki

n−l−m n−l−m 3  n−l−m      n−l−m 1 n−l−m 1 ˙ kj r j + rk m rk mkj r˙ j + a˜ k i a˙˜ k i 2 k=1 j=1 G j=1 k=1 k=1 i=1   n−l−m n−l−m n−l−m n−l−m 3      1 1 ˙ kj rj − m = rk ckj r j + U k − Jk + a˜ k i a˙˜ k i 2 G j=1 j=1 k=1 k=1 i=1

=

IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

(47)

167

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www.ietdl.org Table 1 Range of each joint for the biped robot

hip leg

knee ankle toe

roll pitch yaw pitch roll pitch pitch

Range of joint angel for human

Range of joint angle for our robot

−45°−20° −125°∼15° −45°−45° 0°–130° −20°–30° −20°–45° −45°–30°

−40°−20° −130°−40° −45°–45° 0°–150° −30°–40° −50°–45° −30°–0°

zero as t→∞ by noting limt1 G = 0, there exists t . t1 , n−l−m 3 S 2 1 ak i ≤ r3 with a finite small constant r3 , i=1 k=1 4 G when |rk | ≥ kr3 with kmin = min(k1 , k2 , . . . , kn−l−m ), min V˙ ≤ 0, from above all, rk converges to a small set containing the origin as t→∞. Integrating both sides of the above equation gives V(t) − V(0) , −

t n−l−m 

kk rk2 ds +

0 k=1

3   1 n−l−m r 2 a 4 k=1 i=1 G k i

(50)

Fig. 3 Video snapshots of walking

Joint Angles of Left Lower Limb

1.0

0.5

Angle (rad)

0.0

-0.5

Actual_Left_Ankle_Pitch Desired_Left_Ankle_Pitch Actual_Left_Ankle_Roll Desired_Left_Ankle_Roll Actual_Left_Hip_Pitch Desired_Left_Hip_Pitch Actual_Left_Hip_Roll Desired_Left_Hip_Roll Actual_Left_Knee_Pitch Desired_Left_Knee_Pitch

-1.0

-1.5 0.0 Analysis: LastRun_Adaptive

3.0

6.0 Time (sec)

9.0

12.0 2012-05-28 13:04:47

Fig. 4 Trajectories of left leg (unit: rad) under adaptive control 168 & The Institution of Engineering and Technology 2013

IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

www.ietdl.org Joint Torques of Left Lower Limb

4.5E+005

torque_LAP torque_LAR torque_LHP torque_LHR torque_RAP

4.0E+005

3.5E+005

Torque (N*mm)

3.0E+005

2.5E+005

2.0E+005

1.5E+005

1.0E+005

50000.0

0.0

-50000.0 0.0 Analysis: LastRun_Adaptive

3.0

6.0 Time (sec)

9.0

12.0 2012-05-28 13:04:47

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

Joint Angles of Right Lower Limb

1.5

1.0

Angle (rad)

0.5

0.0

Actual_Right_Ankle_Pitch Desired_Right_Ankle_Pitch Actual_Right_Ankle_Roll Desired_Right_Ankle_Roll Actual_Right_Hip_Pitch Desired_Right_Hip_Pitch Actual_Right_Hip_Roll Desired_Right_Hip_Roll Actual_Right_Knee_Pitch Desired_Right_Knee_Pitch

-0.5

-1.0 0.0 Analysis: LastRun_Adaptive

3.0

6.0 Time (sec)

9.0

12.0 2012-05-28 13:04:47

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

t by noting limt1 S = 0, and limt1 0 S(v)dv = rS , 1. Thus V is bounded, which implies that r [ L1 . From r = e˙ + Le, it can be obtained that e, e˙ [ L1 . As we have established e, e˙ [ L1 , we conclude that zh , z˙h , z¨h [ L1 . Therefore all the signals on the right-hand side of (39) are bounded, it is easy to conclude that Uk is bounded from (42). We substitute the control ta2 = ldh − Kh (lh − ldh ) and tb = ldn − Kn (ln − ldn ) with the constant matrices of IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

proportional control feedback gains Kh and Kn into the reduced order dynamics (15) and (16) yielding (Kh + 1) (ldh − lh ) = Z2 (q)H +T (q)M2 (q)¨zh , (Kn + 1)(ldn − ln ) = +T d Z1 S (q)M1 (q)¨z. Since zh  zh , z˙h  z˙dh , z¨h  z¨dh , z  zd , z˙  z˙d , z¨  z¨d ; therefore Z2 (q)H +T (q)M2 (q) and Z1 S +T (q)M1 (q) are bounded; therefore the size of (lh − ldh ) and (ln − ldn ) are bounded and can be regulated by choosing suitable Kn and Kh to arbitrary small. □ 169

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www.ietdl.org Joint Torques of Right Lower Limb

5.0E+005

torque_RAP torque_RAR torque_RHP torque_RHR torque_RKN

4.0E+005

Torque (N*mm)

3.0E+005

2.0E+005

1.0E+005

0.0

-1.0E+005 0.0 Analysis: LastRun_Adaptive

3.0

6.0 Time (sec)

9.0

12.0 2012-05-28 13:04:47

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

Centroid

3000.0 .body.CM_Position.X .body.CM_Position.Y .body.CM_Position.Z 2500.0

Length (mm)

2000.0

1500.0

1000.0

500.0

0.0

-500.0 0.0

3.0

6.0 Time (sec)

9.0

12.0 2012-05-27 23:50:13

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

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 system stability is achieved, the following simulation verifies the effectiveness of the proposed control.

170 & The Institution of Engineering and Technology 2013

7

Switching stability

For the system switching stability between the single support and double support, we give the following theorem as follows: Theorem 3: Consider system (13) with single support phase and the double support phase, if the system is both stable

IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

www.ietdl.org Joint Angles of Left Lower Limb

1.0

0.375

Angle (rad)

0.0

-0.25

Actual_Left_Ankle_Pitch Desired_Left_Ankle_Pitch Actual_Left_Ankle_Roll Desired_Left_Ankle_Roll Actual_Left_Hip_Pitch Desired_Left_Hip_Pitch Actual_Left_Hip_Roll Desired_Left_Hip_Roll Actual_Left_Knee_Pitch Desired_Left_Knee_Pitch

-0.875

-1.5 0.0 Analysis: LastRun_PD_002

3.0

6.0 Time (sec)

9.0

12.0 2012-05-27 17:25:10

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

Joint Torques of Left Lower Limb

4.5E+005

torque_LAP torque_LAR torque_LHP torque_LHR torque_LKN

4.0E+005

3.5E+005

Torque (N*mm)

3.0E+005

2.5E+005

2.0E+005

1.5E+005

100000.0

50000.0

0.0 0.0 Analysis: LastRun_PD_002

3.0

6.0 Time (sec)

9.0

12.0 2012-05-27 17:25:10

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

before and after the switching phase using the control law (42), even if there exist external impacts during the switching, the system is also stable during the switching phase. Let V− = 12 (˙z− ˙h )T M2 (˙z− ˙h ) and V+ = 12 (˙z+ ˙ h )T h −z h −z h −z + M2 (˙zh − z˙h ) denote the Lyapunov function candidate before and after the switching, and z˙+ ˙− h and z h represent the post-

IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

and pre-switch velocities, and z˙h denotes the single-support or double support velocity, respectively. The Lyapunov function change during the switching can be simplified as follows: DV=V+ −V− =K+ −K− −[˙z+T ˙h −˙z−T ˙h ]= h M2 z h M2 z T + − DK−˙zh M2 (˙zh −˙zh ). Because the foot cannot be penetrated into the ground; therefore z˙h should be on the tangential plane of the ground, while D˙zh should be in the vertical plane of the ground. Considering (18) and (19), we have

171

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www.ietdl.org Joint Angles of Right Lower Limb

1.5

1.0

Angle (rad)

0.5

0.0

Actual_Right_Ankle_Pitch Desired_Right_Ankle_Pitch Actual_Right_Ankle_Roll Desired_Right_Ankle_Roll Actual_Right_Hip_Pitch Desired_Right_Hip_Pitch Actual_Right_Hip_Roll Desired_Right_Hip_Roll Actual_Right_Knee_Pitch Desired_Right_Knee_Pitch

-0.5

-1.0 0.0 Analysis: LastRun_PD_002

3.0

6.0 Time (sec)

9.0

12.0 2012-05-27 17:25:10

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

Joint Torques of Right Lower Limb

4.0E+005

torque_RAP torque_RAR torque_RHP torque_RHR torque_RKN

Torque (N*mm)

3.0E+005

2.0E+005

1.0E+005

0.0 0.0 Analysis: LastRun_PD_002

3.0

6.0 Time (sec)

9.0

12.0 2012-05-27 17:25:10

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

DV = DK − z˙Th Fext = DK − z˙Th J T lf = DK , 0. Therefore the Lyapunov function is decreasing during impact, the motion of the system is also stable.

8

Simulations

Consider a 12-DOF biped robot shown in Fig. 1 modelling using ADAMS, which consists of a torso, and a pair of legs 172 & The Institution of Engineering and Technology 2013

composed of six links. The left and right legs are numbered Legs 1 and 2, respectively. The height of the biped is 1.2 m, the lower limbs are 460 mm, and the height of foot is 90 mm, the weight is 22 kg. The range of each joint for the biped in ADAMS is shown in Table 1, and inertia parameters of the biped are listed in Table 2. In this study, a cycloidal profile is used for the trajectories of the hip and ankle joints of the swinging leg, which can be IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

www.ietdl.org

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

Table 2 Inertia parameters of the biped Name

Mass, kg

body right hip yaw harmonic driver left hip yaw harmonic driver right hip left hip right hip pitch harmonic driver left hip pitch harmonic driver right thigh left thigh right knee motor 1 left knee motor 1 right hip pitch motor left hip pitch motor right knee motor 2 left knee motor 2 right knee harmonic driver left knee harmonic driver right knee bearing left knee bearing right shank left shank right ankle pitch motor left ankle pitch motor right ankle left ankle right foot left foot

40.410 0.981 0.981 1.066 1.066 0.979 0.979 1.006 1.006 0.480 0.480 0.480 0.480 0.480 0.480 1.500 1.500 4.624E-002 4.624E-002 0.718 0.718 0.342 0.342 1.943 1.943 1.205 1.205

found in [33]. This profile is used because it shows a similar pattern to a human’s ankle trajectory in normal walking and it describes a simple function, which can be easily changed for different walking patterns. The equations are given as follows: xa (i) = pa [ 2kp i − sin( 2kp i)], za (i) = d2 [1 − cos( 2kp i)], xhip (i) = 12 xa (i) + a2, zhip (i) = 12 za (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, κ is the total sampling number of a step, and i is the sampling index, and li is the length of link i. In order to avoid the tumbling, we design the lateral trajectory as yh (i) = 102.5 sin( pk i), where yh is the projection of COM on the ground such that the position of COM is in the foot support area, n is the total sampling number of a step, and i is the sampling index, and 102.5 mm is the distance between the COM and support leg. In the simulation, we choose the parameters as a = 200 mm, d = 120 mm, and l1 = 235.5 mm, l2 = 233.5 mm, κ = 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 l2 cos(q2 ) + l3 cos(q3 ) = zhip − l4 with the angle height l4 . Therefore, q2 is independent coordinate, and q3 = F(q2 ) = ∂F cos−1 zh ip−l4 −ll 1 cos (q2 ) we design the H = [1, 0, 0, 0; ∂q , 0, 0, 3 1 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 = [13×3 , 03×3 ; 02×3 , 02×3 ; 03×3 , 13×3 , 02×3 , 03×3 ], l=4, m=2. The parameters in the adaptive control are set as qˆ = [0.0, 0.0, 0.0]T , aˆ = [0.0, , . . . , 0.0]T , and k = G = 1 1 diag[1.0], h = S = diag[ (1+t) 2 ], d = 1 + (1+t)2 , the balance control gain are choose as Lc = diag[10] and Y = [20 000, 20 000, 20 000]T . The posture control gain is listed in Table 4. For comparison, we implement the PD control in the biped robot, and the control is set as ti = −Pi ei − Di e˙ i with ei = zh − zdh , and the control gain is listed in Table 3. The video snapshots are shown in Fig. 3. The positions tracking for each joint profiles of the left and right legs are shown in Figs. 4 and 6. 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 comparison with the traditional PD control, Figs. 9 and 11 are the joint positions using PD control, the corresponding input torques are listed in Figs. 10 and 12. The summary of

Table 3 Parameters in the PD control Parameters Pi Di

Ankle pitch

Ankle roll

Hip pitch

Hip roll

Knee

400 000 40 000

29 998 500 3500

2 500 000 20 000

35 500 000 250 000

900 000 20 000

IET Control Theory Appl., 2013, Vol. 7, Iss. 2, pp. 161–175 doi: 10.1049/iet-cta.2012.0066

173

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www.ietdl.org Table 4 Parameters in the adaptive control Joints

DOF

Λh

kk

hip hip knee angle angle

roll pitch pitch roll pitch

20 125 45 140 10

250 000 20 000 20 000 3500 40 000

all joint tracking errors and all joint input torques are shown in Fig. 13 using both PD control and adaptive robust control, respectively, and we can see the tacking errors using the proposed adaptive robust control are bounded and especially smaller than PD control, that is, the walking locomotion using adaptive control is more stable since the robot could adaptively update the control parameters online, while the PD control is without the capability. Since the initial values of the dynamics are assumed to be unknown for the controls in the simulation, from these figures, even if the nominal parameters of the system are uncertain, and the initial disturbances boundedness from the environment are unknown, we can obtain satisfactory performance by the proposed control, which is verified by the ADAMS environment.

9

Conclusions

In this paper, a structure of adaptive robust control has been presented for a biped robot, which includes balancing control and posture control for regulating the COM position and trunk orientation of bipedal robots in a compliant way. The biped robot can be decoupled into the decoupled dynamics of COM. The trunks and the adaptive robust control are constructed in the presence of parametric and functional dynamics uncertainties. The controller computes a desired ground reaction 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. The verification of the proposed control has been conducted by using the extensive simulations.

10

Acknowledgments

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

11

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