Automatica 44 (2008) 776 – 784 www.elsevier.com/locate/automatica
Brief paper
Robust adaptive control of uncertain force/motion constrained nonholonomic mobile manipulators夡 Z. Li a,b , S.S. Ge a,∗ , M. Adams b , W.S. Wijesoma b a Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore b School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore 639798, Singapore
Received 28 May 2006; received in revised form 28 May 2007; accepted 9 July 2007
Abstract In this paper, force/motion tracking control is investigated for nonholonomic mobile manipulators with unknown parameters and disturbances under uncertain holonomic constraints. The nonholonomic mobile manipulator is transformed into a reduced chained form, and then, robust adaptive force/motion control with hybrid variable signals is proposed to compensate for parametric uncertainties and suppress bounded disturbances. The control scheme guarantees that the outputs of the dynamic system track some bounded auxiliary signals, which subsequently drive the kinematic system to the desired trajectory/force. Simulation studies on the control of a wheeled mobile manipulator are used to show the effectiveness of the proposed scheme. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Mobile manipulator; Trajectory/force control; Nonholonomic/holonomic constraint
1. Introduction Mobile manipulators have received increasing attention over the last 15 years, because of their industrial relevance and academic interest (Dong, 2002; Liu & Lewis, 1990; Tan, Xi, & Wang, 2003; Watanabe, Sato, Izumi, & Kunitake, 2000). However, the motion planning and control of mobile manipulators cannot be addressed by traditional methods due to the nonholonomic nature of the systems which are imposed by the wheeled platforms. Due to Brockett’s theorem (1983), it is well known that nonholonomic systems with restricted mobility cannot be stabilized to a desired configuration nor posture-via differentiable, or even continuous, pure state feedback. Therefore, the control design for these systems is a challenging problem, and has attracted much attention in the robotics and control community (Ge, Wang, Lee, & Zhou, 2001, 2003; Oya & Su, 2003; Wang, Ge, & Lee, 2004). 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Toshiharu Sugie under the direction of Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +65 6516 6821; fax: +65 67791103. E-mail address:
[email protected] (S.S. Ge).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.07.012
With the assumption of known dynamics, much research has been carried out to control mobile manipulators including input–output feedback linearization (Tan et al., 2003), nonlinear feedback control (Yamamoto & Yun, 1996). Because of dynamic uncertainty, adaptive neural network controls (Lin & Goldenberg, 2001) have been proposed for motion control of mobile manipulators. However, for practical applications, not only the motion but also the force of the end-effector of the arm should be considered. In Dong (2002), adaptive control was proposed for trajectory/force control of mobile manipulators subjected to holonomic and nonholonomic constraints with unknown inertia parameters. Most control approaches for mobile manipulator deal with uncertainty in system dynamics and assume that the exact holonomic constraints of the interconnected system are known (Ge & Lewis, 2006). However, in practical applications, environmental uncertainties arise in mobile manipulator applications which can affect the system stability and performance. In this paper, under holonomic uncertainty, we consider the trajectory and force tracking for general dynamic nonholonomic mobile manipulator systems in which the system is transformed into a chained form. Since the general motion/force decomposition control is not valid with constraint uncertainties, we develop a low-pass force filter to assure the
Z. Li et al. / Automatica 44 (2008) 776 – 784
777
boundedness of force error and simultaneously estimate the uncertain constraint Jacobian matrix. A main concern in our design is the magnitude of the Jacobian error as it influences the residual force error. By investigating the dynamics error, we apply a reduction procedure and define a mixed tracking error of trajectory and force. Then, robust adaptive force/motion control is presented in which adaptive controls are used to compensate for parametric uncertainties, suppress constraint uncertainties and bounded disturbances. The control guarantees the outputs of the dynamic system to track some bounded hybrid signals which subsequently drive the kinematic system to the desired trajectory/force.
where H = diag[Hv (qv ), Ia ], M1 (q) = H T (q)M(q)H (q), ˙ = H T (q)[M(q)H˙ (q) + C(q, q)H ˙ (q)], G1 (q) = C1 (q, q) T H (q)G(q), d1 = H T (q)d(t), B1 (q) = H T (q)B(q), v = [vb , q˙a ]T , and f1 = J1 (q)T h .
2. System description
with T2 (q) = diag[T21 (qv ), I ] and u = [ub , ua ]T , then the kinematic system (4) could be locally or globally converted to the chained form (Walsh & Bushnell, 1995)
2.1. Dynamics of mobile manipulators Consider an n DOF manipulator mounted on a nonholonomic mobile platform (Li, Ge, & Ming, 2007) M(q)q¨ + C(q, q) ˙ q˙ + G(q) + d(t) = B(q) + f ,
(1)
where q = [q1 , . . . , qn ]T ∈ R n denote the generalized coordinates; M(q) ∈ R n×n is the symmetric bounded positive definite inertia matrix; C(q, q) ˙ q˙ ∈ R n denotes the Centripetal and Coriolis torques; G(q) ∈ R n is the gravitational torque vector; d(t) ∈ R n denotes the external disturbances; ∈ R p are the control inputs; B(q) ∈ R n×p is a known input transformation matrix; and f = J T (q) ∈ R n denotes the generalized constraint forces. The generalized coordinates may be separated into two sets as q = [qv , qa ]T with qv ∈ R nv denoting the generalized coordinates for the vehicle and qa ∈ R na denoting the coordinates of the arm.
When the system is subjected to nonholonomic constraints, assume that the nv − m nonintegrable and independent velocity constraints can be expressed as A(qv )q˙v = 0. Since A(qv ) ∈ R (nv −m)×nv , it is always possible to find an m rank matrix Hv (qv ) ∈ R nv ×m formed by a set of smooth and linearly independent vector fields spanning the null space of A(qv ), i.e., HvT (qv )AT (qv ) = 0. Since Hv (qv ) = [Hv1 (qv ), . . . , Hvm (qv )] is formed by a set of smooth and linearly independent vectors spanning the null space of A(qv ), define an auxiliary function vb = [vb1 , . . . , vbm ]T ∈ R m such that (2)
which is the so-called kinematic model of nonholonomic system. Differentiating (2) yields q¨v = H˙ v (qv )vb + Hv (qv )v˙b .
There exist a coordinate transformation T1 (q) and a state feedback T2 (q), such that = [, qa ]T = T1 (q) = [T11 (qv ), qa ]T ,
(6)
v = [vb , q˙a ]T = T2 (q)u = [T21 (qv )ub , ua ]T
(7)
˙ 1 = u1 ,
˙ i = u1 i+1 (2 i nv − 1),
˙ nv = u2 ,
q˙a = ua ,
(8)
where ua = [ua1 , . . . , uana ]T . Remark 2.1. In Murray and Sastry (1993), a necessary and sufficient condition for the transformation of the kinematic system (4) of a differential drive wheeled mobile platform into single chain was given. In Walsh and Bushnell (1995), the existence condition of the transformation (multi-chain case) was discussed for the other types of mobile platform. Consider the above transformations, the dynamics (5) is converted as ˙ + G2 () + d2 = B2 + f2 , M2 ()u˙ + C2 (, )u
2.2. Nonholonomic constraints
q˙v = Hv (qv )vb = Hv1 (qv )vb1 + · · · + Hvm (qv )vbm
2.3. Reduced model and state transformation
(9)
˙ = T T (q) where M2 () = T2T (q)M1 (q)T2 (q)|q=T −1 () , C2 (, ) 2 1 [M1 (q)T˙2 (q) + C1 (q, q)T ˙ 2 (q)]| , G2 () = T T (q) −1 q=T1 ()
2
G1 (q)|q=T −1 () , B2 = T2T (q)B1 (q)|q=T −1 () , d2 = T2T (q) 1
1
d1 |q=T −1 () , f2 = T2T (q)f1 = J2T (q)h |q=T −1 () . 1
1
Property 2.1. The matrix M2 is symmetric and positive definite, and the matrix M˙ 2 −2C2 is skew-symmetric (Dong, 2002). Property 2.2. There exist some finite non-negative constants ci 0 (i =1, . . . , 5) such that ∀ ∈ R n , ∀˙ ∈ R n , M2 ()c1 , ˙ c2 + c3 , ˙ G2 () c4 , and sup C2 (, ) t 0 d2 (t) c5 (Wang et al., 2004). 2.4. Uncertain holonomic constraints
(3)
Therefore, the dynamics (1), after eliminating the nonholonomic constraint AT n , can be reformulated as q˙ = H (q)v,
(4)
˙ + G1 (q) + d1 = B1 (q) + f1 , M1 (q)v˙ + C1 (q, q)v
(5)
Assume that the r-rigid-link manipulator is in contact with a certain constrained surface (q) can be represented as: ((q)) = 0, where ((q)) is a given scalar function, (q) ∈ R l denotes the position vector of the end-effector in contact with the environment. If the constraint surface is rigid and has a continuous gradient, the contact force in (9) is then
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Z. Li et al. / Automatica 44 (2008) 776 – 784
given by f2 = J2T (q)h , where h is the magnitude of the contact force. However, when the robot end-effector is rigidly in contact with the uncertain surface, the environmental constraint could be expressed as an algebraic equation of the coordinate in the task space. Without loss of ¯ generality, the uncertain surface constraint ((q)) can be decomposed into a nominal part ((q)) and an unknown constraint error part ((q)) in an additive manner as follows:
Integrating (14) into (11) and considering (14) and letting h = j((q))/jt, we have
¯ ((q)) = ((q)) + ((q)),
where
(10)
¯ where ((q)) is the constrained surface. ¯ Denoting J¯2 and J2 as the Jacobian matrix of ((q)) and ((q)) with respect to q, and since q˙v = Hv (qv )T21 (qv )ub , J¯2 (q) = J¯ [J2v Hv (qv )T21 (qv ), J2a ], J2 (q) = J [J2v Hv (qv )T21 ¯ (qv ), J2a ], J¯ = ((q))/j, J = j((q))/j, J2v = j/jqv and J2a = j/jqa . Integrating (4) and (7), we have J2 (q)u +
j((q)) = 0. jt
(11)
⎤
ub
⎥ ⎢ q˙a1 ⎥ ⎢ u= ⎢ ⎥ −1 ⎣ −J2a2 [J1v Hv (qv )T21 (qv )ub + J1a1 q˙a1 ] ⎦ −1 −1 −J1a2 J h
= Lu1 + ε, L = [Lv La ]T ⎡
(15)
Iv
0
0
Ia1
−1 −J2a2 J2v Hv (qv )T21 (qv )
−1 −J2a2 J2a1
⎢ =⎣
⎤ ⎥ ⎦,
(16)
u1 = [ub q˙a1 ]T ,
(17)
ε = J h ,
(18)
−1 −1 T J = [0 0 − J2a2 J ] .
(19)
It is easy to have LT J2T (q) = 0.
Assume that J¯2 (q) = J2 (q) + J2 (q)
⎡
(12)
(20)
Differentiating (15) and substituting it into (13), we have M3 ()u˙ 1 + C3 u1 + G3 () + d3
with J2 (q) defined later. Since the uncertain constraint error (10) is introduced, integrating (12) into (9) yields
(13)
Assumption 2.1. The Jacobian matrices J¯2 (q) is uniformly bounded and uniformly continuous if q is uniformly bounded and continuous, respectively. Assumption 2.2. The manipulator is operating away from any singularity. Remark 2.2. Under Assumption 2.2, the Jacobian J2a =j/jqa is of full row rank l, such that the joint coordinate qa can be parT , q T ]T where q ∈ R na −l and q ∈ R l , titioned into qa = [qa1 a1 a2 a2 with qa2 =(qa1 ) with a nonlinear mapping function (·) from 2 , an open set R na −l × R → R l . The terms j/jqa1 , j2 /jqa1 2 2 j/jt, j /jt exist and are bounded in the workspace. Since the dimension of the constraint is l, the configuration space of the manipulator is left with na − l degrees of freedom. Based on the full row rank for Ja , the existence of (qa1 ) (You & Chen, 1993; Yuan, 1997), it is easy to obtain J2 (q) = J [J2v Hv (qv )T21 (qv ), J2a1 , J2a2 ].
(21)
˙ ˙ ˙ where M3 ()=M2 ()L, C3 (, )=M 2 ()L+C 2 (, )L, G3 ()= ˙ G2 (), B3 = B2 , d3 = M2 ()˙ε + C2 (, )ε + d2 .
˙ + G2 () + d2 M2 ()u˙ + C2 (, )u = B2 + (J2 (T1−1 ()) + J2 (T1−1 ()))T h .
= B3 + (J2 (T1−1 ()) + J2 (T1−1 ()))T h ,
(14)
Assumption 2.3. The set of the constrained surface reachable by the end-effector of mobile manipulator, defined by ¯ S := { : (,
) = 0, ∈ R l1 }
(22)
is bounded and belong to a class of continuously differentiable ¯ manifolds (,
) = f (1 , 2 , . . . , l1 ) + g(l1 +1 , l1 +2 , . . . , l ) + with l1 l n and ∈ R n , where = [ 1 , . . . , l1 , 0, . . . , 0]T ∈ R l and = [0, . . . , 0, 1, . . . , 1]T ∈ R l are constant vectors, f (∗) = [f1 , . . . , fl1 , 0, . . . , 0] ∈ R 1×l and g(∗) = [0, . . . , 0, g1 , . . . , gl ] ∈ R 1×l are bounded and uniformly continuous differentiable vectors, and is a constant scalar. Considering Assumption 2.3, the uncertainty h could be expressed with h = J2 (T1−1 ())
d −1 (T ()), dt 1
J2T = JT (S ()W + C ),
(23) (24)
where S = [jf/j1 , jf/j2 , . . . , jf/jl1 , 0, . . . , 0]T ∈ R l×l , C = [0, . . . , 0, jg/jl1 +1 , jg/jl1 +2 , . . . , jg/jl ]T ∈ R l×l ,
=1/S ()W +C , and J =j/j. From Assumption 2.3,
Z. Li et al. / Automatica 44 (2008) 776 – 784
the weight vector W = [W1 , . . . , Wl1 , 0, . . . , 0]T ∈ R l is unknown positive, and = [0, . . . , 0, 1, . . . , 1]T ∈ R l . Property 2.3. S C = 0 and W T = 0.
the reference signals 1T T u1d = [u1T bd , uad ] , ⎡
Define the estimated Jacobian Jˆ2 by Jˆ2T
u1bd
= J (S ˆ Wˆ + C ) T
(25)
with ˆ = 1/S ()Wˆ + C . Consider Property 2.3, the error in Jacobian matrix J˜2 = Jˆ2 − J2 can be expressed by J˜2T = JT (S ( ˆ Wˆ − W ) + C ˜ ),
(26)
where ˜ = ˆ − . Consider (26), the force error can be expressed as ef = J+ J2T h − F = J+ J˜2T h , T
T
(27)
where J+ = J (JT J )−1 and the force error ef can be calcuT
lated from J+ J2T h and F from force sensor. T
Assumption 2.4. There exist some finite non-negative known constants b 1 and b 2 , such that, ∀ ∈ , h b 1 (d/dt) T −1 (), ˙ h b 1 (d2 /dt 2 )T −1 () + b 2 (d/dt)T −1 (). 1
1
1
3. Adaptive control
779
⎥ ⎢ ⎢ ud2 − snv −1 ud1 − knv snv ⎥ ⎥ ⎢ ⎢ n −3 j(env − snv ) (i+1) ⎥ v ⎢ = ⎢ + i=0 ud1 ⎥ ⎥, (i) ⎥ ⎢ jud1 ⎥ ⎢ ⎦ ⎣ nv −1 j(env − snv ) + i=2 ei+1 jei
u1ad = q˙a1d − Ka (qa1 − qa1d ), where ⎡
Assumption 3.1. The desired reference trajectory d (t) is assumed to be bounded and uniformly continuous, and has bounded and uniformly continuous derivatives up to the (n − 1)th order. The desired Lagrangian multiplier dh is also bounded and uniformly continuous. Assumption 3.2. Time varying positive functions t i and hi converge to zero as t → ∞ and satisfy limt→∞ 0 i () d = t ai < ∞, limt→∞ 0 hi () d = bi < ∞ with finite constants ai and bi , i = 1, . . . , 5. There are many choices for i and hi that satisfy the Assumption 3.2, for example, i = hi = 1/(1 + t)2 . Denote the tracking errors as e = − d and e = h − dh , and define
(28) ⎤
e1
⎢ ⎥ e2 ⎢ ⎥ ⎢ ⎥ 2p−1 ⎢ ⎥ e3 + k2 e2 ud1 ⎢ ⎥ ⎢ ⎥ 2p−1 ⎢ ⎥ e4 + s2 + k3 s3 ud1 ⎢ ⎥ ⎢ ⎥ ⎢ 1 0 j(e3 − s3 ) (i+1) 2 j(e3 − s3 ) ⎥ ⎢− ud1 − i=2 ei+1 ⎥ ⎢ ud1 i=0 ⎥ (i) jei jud1 s=⎢ ⎥, ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ 2p−1 ⎢ ⎥ env + snv −2 + knv −1 snv −1 ud1 ⎢ ⎥ ⎢ ⎥ 1 nv −4 j(env −1 −snv −1 ) (i+1) ⎢ ⎥ − ud1 i=0 ud1 i ⎢ ⎥ jud1 ⎣ ⎦ nv −2 j(env −1 −snv −1 ) − i=2 ei+1 jei ˙ = −k0 − k1 s1 −
n
v −1 i=2
Given a desired motion trajectory qd and a desired constraint force, or, equivalently, a desired multiplier dh (t) should satisfy the constrained equations. Since the desired trajectory qd should satisfy Eq. (6), we can have the desired d . After the transformation for the chained form through d = T1 (qd ) and vd = T2 (qd )ud , we can obtain d and ud and, equivalently, u1d . The trajectory and force tracking control can be restated as seeking a strategy for specifying a control law subjected ˙ → to the uncertain holonomic constraint, such that {h , , } {dh , d , ˙ d }.
⎤
ud1 +
si i+1 +
nv
j =3
sj
j −1
j(ej − sj ) i+1 , jei i=2
(i)
p = nv − 2, ud1 is the ith derivative of ud1 with respect to t, and ki is positive constant, Ka ∈ R (na −l)×(na −l) is diagonal positive. Define new variables to handle the force control ϑ˙ = −Kϑ ϑ − Kϑ J2T e ,
(29)
where ϑ = [0, ϑ1 ] with ϑ1 ∈ R na , e = h − dh , and Kϑ = diag[0, kϑi ] > 0. Defining the following auxiliary signals as u˜ 1 = u1 − u1d = [u˜ 1 , u˜ 2 , u˜ a1 ]T , ˙ = u˜ 1 and ur = u1d − Ku , we have r = ˙ + Ku ,
(30)
= Lr + ϑ,
(31)
= Lur − ϑ,
(32)
where Ku = diag[0, Ku1 ] > 0 and Ku1 ∈ R (na −l)×(na −l) . From (30) and the definitions of u˜ 1 , ur and ˙ above, we have u1 = ur + r.
(33)
The time derivatives of and are given by ˙ 1 + Lu˙ 1 − ˙ , ˙ = Lu
(34)
˙ ˙ r + Lu˙ r − ϑ. ˙ = Lu
(35)
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Z. Li et al. / Automatica 44 (2008) 776 – 784
From (31) to (33) we have + = Lu1 .
(36)
From the dynamics (21) together with (34)–(36), we have
s˙1 = + u˜ 1 ,
˙ + M2 ()˙ + C2 (, ) ˙ M2 ()˙ + C2 (, ) + G2 () + d3 (t) = B2 () + (J2T
+ J2T )h .
2p
(37)
5
cˆi 2i 2 Yi2 (, ˙ ) B2 = − − K − i Yi (, ˙ ) + i
.. .
(46)
(38)
s˙nv −1 = ( + u˜ 1 ) nv −
nv −1
s˙nv = ( + u˜ 1 )
(47)
j(env − snv ) i+1 − knv snv jei
− snv −1 ud1 + u˜ 2 ,
(48)
˙ = −k0 − 1 , M2 ˙ = − C2 − − K −
(49) 5
cˆi 2i 2 Yi2 (, ˙ ) i Yi (, ˙ ) + i i=1
− J2T (dh − K e ) + J2T h
and the adaptive laws
− J˜2T h + Jˆ2T (K + I )e − h − ,
˙ˆ = T W ˆ h S J , 1 J S 2 , 2C (i = 1, . . . , 5),
where
d
d −1
Y1 (, ˙ ) = ˙ + b 1 J + b 2 J T1 ()
dt dt
2
d −1
+ b 1 J 2 T1 ()
, dt
d −1
Y2 (, ˙ ) = + b 1 J T1 ()
, dt
˙ + b 1 J d T −1 () , Y3 (, ˙ ) =
dt 1
Y4 (, ˙ ) = Y5 (, ˙ ) = 1.
n
v −2 i=2
J + sgn()ef 2 + (K + I )Jˆ2T (h − dh ), (40) 2C 1 if 0, sgn() = −1 if < 0
i 2i Yi2 2 i Yi (, ˙ ) + i
2p
J ˆ + C 2 ) sgn()(S 2 2C
c˙ˆi = −hi cˆi +
j(env −1 − snv −1 ) i+1 jei
+ snv ud1 − snv −2 ud1 − knv −1 snv −1 ud1 ,
⎤
k1 s1 + i=2 si i+1 ⎥ ⎢ j −1 j(ej − sj ) nv 1 = ⎢ i+1 ⎥ ⎦, ⎣ − j =3 sj i=2 jei snv
˙ˆ =
n
v −2 i=2
(39)
⎡
h =
j(e3 − s3 ) 3 s˙3 = ( + u˜ 1 ) 4 − je2 2p
− (J2T + Jˆ2T )dh + (J2T + Jˆ2T )K (h − dh )
= [1 0] ,
(45)
+ s4 ud1 − s2 ud1 − k3 s3 ud1
i=1
T
(44)
s˙2 = ( + u˜ 1 )3 + s3 ud1 − k2 s2 ud1 ,
Consider the control law given by
− h ,
K and K are positive definite matrices, i > 0 and i > 0 are constant. From the dynamic equation (37) together with (38), the closed-loop system dynamics can be written as
(50)
(41)
where Υ = M2 ˙ + C2 + G2 + d3 .
(42)
Theorem 3.1. Consider the mechanical system described by (1), (2), and (10). Under Assumption 3.1, the control law ˙ h converge to d , ˙ d , d at t → ∞; (38) leads to: (i) , , h and (ii) all the signals in the closed-loop are bounded for all t 0.
(43)
Proof. Consider the Lyapunov function candidate: V (t) =
v
1 1
1 1 1 si2 + k1 s12 + 2 + T M2 + c˜2 2 2 2 2 2i i
n
5
i=2
i=1
1 1 + W˜ T W˜ + ϑT (I + K )Kϑ−1 ϑ 2 2 1 + (W 2 − ) ˆ 2, 2
(51)
˙ˆ , c˙˜ = c˙ˆ . ˙˜ = W where W˜ = Wˆ − W , c˜i = cˆi − ci , we have W i i Considering Property 2.1 and integrating (43) and (50) into the
Z. Li et al. / Automatica 44 (2008) 776 – 784
derivative of V yield V˙ −
n
v −1
From (41), (57)–(59), it can be shown that V˙ − k0 2 − knv snv −
2 2 ki si2 u2l ˜ 1T − T K d1 − knv snv − k0 + u
i=2
−
cˆi 2i 2 Yi2 (, ˙ )
i=1
i Yi (, ˙ ) + i
+ Υ
+
5
hi cˆi c˜i
i
+
(52)
T ef = J+ J˜2T h
˜ h . = J+ JT S ( ˆ Wˆ − W )h + J+ JT C T
(53)
Considering Property 2.3, we have ef = J J S ( ˆ Wˆ − W ) h +T T 2
2
+T T 2
2
(54)
ˆT
W˜ + J (k + I )e T
(56)
where = Jˆ2T (K + I )e . From (55) and (56), we have − T JT h (S ˜ W + C ) J (ef 2 + S 2 W 2 + C 2 ) 2C
(57)
by noting Property 2.3. Moreover 5 5
cˆi 2i 2 Yi2 (, ˙ ) hi cˆi c˜i Υ − − i Yi (, ˙ ) + i i i=1
+
i=1
V˙ − k0 2 − knv snv −
(55)
= − J (S ˜ W + C )h + T ,
5
(61)
Integrating (61), one obtains
Considering (26), (53) and using the adaptive parameter law (41), rewriting the 10th, the 11th and the 16th right-hand terms in (52), we have
From (16) and (30), we have Lv =[Iv , 0], and Ku =diag[0, Ku1 ], r T [LTv LTa ][1 0]T = u˜ 1T 1 , subsequently we obtain
n
v −1
T ki si2 u2l d1 − K
i=2
˜ h ef /C .
−
(60)
u˜ 1T − T = u˜ 1T − (r T LT + ϑT ) 1 1 1 = u˜ 1T − r T [LTv LTa ] − [0 ϑ1 ] . 0 0 0
u˜ 1T − T = 0.
Therefore, we obtain
T
J (S 2 W 2 + C 2 ) + T − T h 2C
2
+ J J C ˜ h .
˙˜ J˜2T h + W T T
J hi 2 ci + ci i − ϑT (K + I )ϑ + ef 2 4i 2C
From (16), (29), and (39), considering the fourth and the 12th right-hand terms in (60), we have
i=1
From (26) and (27), we can obtain
T
ki si2 u2l ˜ 1T − T K d1 + u
˙ˆ − ϑT (I + K )ϑ. − T − (W 2 − ) ˆ
5
c˜i 2i 2 Yi2 (, ˙ ) ˙˜ T W˜ + +W i Yi (, ˙ ) + i
˙ˆ ˆ . + ϑT (I + K )Kϑ−1 ϑ˙ − (W 2 − )
2
5
i=1
− T J˜2T h + T Jˆ2T (K + I )e − T h − T
i=1
n
v −1 i=2
5
− T J2T (dh − K e ) + T J2T h
−
781
c˜i 2i 2 Yi2 (, ˙ ) i Yi (, ˙ ) + i
i=1
5
i=1
hi 2 c i + c i i . 4i
(58)
In addition, from (29) and (31), T =r T LT +ϑT . Thus, we have − T J2T (dh − K e ) + T J2T h + ϑT (I + K )Kϑ−1 ϑ˙ = −ϑT (I + K )ϑ + r T LT J2T (I + K )e by noting LT J2T = 0 from (20).
(59)
+
5
hi 2 ci + ci i − ϑT (I + K )ϑ 4i i=1
+
J S 2 ˙ˆ ˆ − (W 2 − ) ˆ . (W 2 − ) 2C
(62)
Considering the parameter ˆ update law (42), it results that v −1 2 2l V˙ − k0 2 − knv snv − ni=2 ki si ud1 − T K + 5i=1 ((hi /4i )ci2 + ci i ) − ϑT (I + K )ϑ. Noting Assumption 3.2, we have 5i=1 ((hi /4i )ci2 + ci i ) → 0 as t → ∞. Integrating both of the above equation gives V (t) − V (0) < − t sides nv −1 2 2l T 2 + k s T (k 0 nv nv + i=2 ki si ud1 + K + ϑ (I + 0 5 K )ϑ) ds+ i=1 ((ai /4i )ci2 +ci bi ) < ∞. Thus, V (t) < V (0)+ 5 2 i=1 ((ai /4i )ci + ci bi ), therefore V (t) is bounded, which ˆ are bounded. From the defiimplies that , si , , cˆi , Wˆ , ϑ and nition of si in (29), it can be concluded that [e1 , e2 , . . . , env ]T is bounded, which follows that is bounded. Since is bounded, we can obtain r, u˜ 1 ∈ Ln−l from (31), therefore, qa1 − qa1d 2 and q˙a1 − q˙a1d are bounded, which follows that qa1 is bounded. Since ϑ is bounded, from (29), we have e is bounded. Therefore, it is concluded that si ud1 , snv , ∈ L2 , it can be shown that si ud1 → 0, snv → 0, → 0 as t → 0, respectively. It is further concluded that si → 0 as t → 0. Differentiating p p p p−1 p p ud1 yields (d/dt)ud1 = −k1 ud1 s1 + lud1 u˙ d1 − k0 ud1 − j −1 n n −1 v v uld1 { i=2 si i+1 − j =3 sj i=2 (j(ej − sj )/jei )i+1 }, where the first term is uniformly continuous and the other terms p converges to zero. Since (d/dt)ud1 converge to zero, s and s˙
Z. Li et al. / Automatica 44 (2008) 776 – 784
also tend to zero. It is obvious that si = 0, yields that i → di ˙ d3 , J˜2 , e and h are all and ˙ i → ˙ di as t → ∞. Since , , bounded, it can be concluded that is bounded from (38). 4. Simulation studies Consider a 3-DOF robotic manipulator with two revolute joints and one prismatic joint mounted on two wheeled mobile platform shown in Fig. 1, which is subjected to the following constraints: x˙ cos + y˙ sin = 0. Using the Lagrangian approach, we can obtain the standard form (1) with qv = [x, y, ]T , qa = [1 , 2 , 3 ]T , 2 = /2, q = [qv , qa ]T , and Av = [cos , sin , 0]T , and Mv11 Mv12 Cv11 Cv12 , Cv = , Mv = Mv21 Mv22 Cv21 Cv22 sin /r − cos /r −l/r T , Bv = − sin /r cos /r l/r Mv12 = [m123 d cos + m23 cos( + 1 ), m123 d sin + m23 sin(+1 )]T , Mv11 =diag[mp123 ], m23 =m2 l2 +m3 L3 , L3 = 2l2 +l3 +3 , Mv22 =Ip +I123 +m123 d 2 +m2 (l22 +2dl 2 cos 1 )+ m3 (L23 + 2dL3 cos 1 ), Mva = [Mva1 , Mva2 , Mva3 ], Mva1 = [m23 cos(+1 ), m23 sin(+1 ), I123 +m2 (l22 +2dl 2 cos 1 )+ m3 (L23 + 2dL3 cos 1 )]T , Mva2 = 0, Mva3 = [sin( + 1 ), − cos(+1 ), 0]T , Ba =diag[1.0], Ma =diag[I123 , I23 , m3 ], =[l , r , 1 , 2 , 3 ]T , Gv =[0.0, 0.0, 0.0]T , mp123 =mp +m123 , m123 = m1 + m2 + m3 , I123 = I1 + I2 + I3 + m3 L23 , T , C I23 = I2 + I3 + m3 L23 , Cv11 = 0, Cv12 = Cv21 v22 = ˙ ˙ ˙ 0], −2m23 d sin 1 1 , Ca =diag[−m23 d sin 1 , −m23 d sin 1 , ˙ ˙ ˙ ˙ Cv12 =[−m123 d sin −m23 sin(+1 )(+1 ), m123 d cos + m23 cos( + 1 )(˙ + ˙ 1 )]T , Ga = [0.0, m2 gl 2 , m3 gL3 ]T , ˙ Cva =[Cva1 , Cva2 , Cva3 ], Cva1 =Cva2 =[−m23 sin(+1 )(+ ˙1 ), −m23 sin cos( + 1 )(˙ + ˙ 1 ), 0]T , Cva3 = [−m3 cos( + T , 1 )(˙ + ˙ 1 ), −m3 sin cos( + 1 )(˙ + ˙ 1 ), 0]T , Cav1 = Cva1 T Cav2 = Cva2 , Cav3 = [m3 cos( + 1 )(˙ + ˙ 1 ), m3 sin( + 1 )(˙ + ˙ 1 ), m3 d sin ˙ 1 ].
Remark 4.1. In such case that 3-DOF mobile manipulator consists of two revolute joints and one prismatic joint, ∀ ∈ R n , ∀˙ ∈ R n , M2 () km1 +km2 6 2 with km1 and km2 > 0, G2 () kg1 + kg2 6 with kg1 and kg2 > 0, where 6 = 3 , if the boundedness of 6 is known, there still exist some finite non-negative constants ci 0 (i = 1, . . . , 4), therefore Property 2.2 holds. Remark 4.2. The existence of sign-function in the controller (46) may inevitably lead to chattering in control torques. To avoid chattering, a sat-function can be used to replace the signfunction (Slotine & Li, 1991). Given the desired trajectory qd = [xd , yd , d , 1d , 2d ]T with xd = 2.0 cos(t), yd = 2.0 sin(t), d = t, 1d = /2 rad, 2d = /2 rad and the geometric constraint which the end-effector subjected to: = (x 2 + y 2 ) + z − c = 0 with c = 2.25 m, and d = 10.0 N, the desired value of the parameter is 1.0, and the joint 3 is redundant prismatic joint used to compensate the position errors caused by uncertain holonomic constraints. Assume that 3 ∈ [0.0 m, 0.3 m]. The transformation T1 (q) is defined as 1 = , 2 = x cos + y sin , 3 = −x sin + y cos , 4 = 1 , 5 = 2 , 6 = 3 , u1 = vb2 , u2 = vb1 − (x cos + y sin )vb2 , u3 = ˙ 3 , u4 = ˙ 4 , u5 = ˙ 5 , one can obtain the kinematic system in the chained form as ˙ 1 = u1 , ˙ 2 = 3 u1 , ˙ 3 = u2 , ˙ 4 = u3 , ˙ 5 = u4 , ˙ 6 = u5 . Using the above diffeomorphism
12 10 Positions (m, rad)
782
8 6
d
4
2
1d
0 -2
0
x
2
2l2
2l1
3 3d
10
Fig. 2. Positions tracking.
40
3
m1
1
O
r
x driving wheel
1
0 Torques (Nm)
l
l
20
l d
8
m3
z y
6
Time (s)
2l3
m2
y yd
xd
-4 2
4
2 2
0 -20
r
4
6 Time (s)
-40 -60 -80
2r Fig. 1. 3-DOF robotic manipulator mounted on a mobile platform.
-100 Fig. 3. Torques of joints.
8
3
10
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References
12 Constact force (N) h
10 8 6 4
Estimated uncertainty
2 0 0
2
4
6
8
10
Time (s)
Fig. 4. The contact force (N) and estimated .
transformation, we can obtain d1 = t, d2 = 2.0, d3 = 0.0, d4 = /2, d5 = /2, dh = 10.0 N with ud1 = 1.0, ud2 = 0.0, ud3 =0.0, ud4 =0.0. In the simulation, mp =5.0 kg, m1 =1.0 kg, m2 =m3 =0.5 kg, Ip =2.5 kg m2 , I1 =1.0 kg m2 ,I2 =0.5 kg m2 , I3 = 0.5 + m3 23 kg m2 , d = l = r = 0.5 m, 2l1 = 1.0 m, 2l2 = 0.3 m. The initial condition select x(0) = 2.0 m, y(0) = 0.0 m, (0) = /2 rad, 1 (0) = /2 rad, 2 (0) = /2 rad, 3 (0) = 0.1 m, ˙ (0)=0.0 N and x(0)=0.5 ˙ m/s, y(0)= ˙ (0)= ˙ 1 (0)= ˙ 2 (0)= ˙ 3 (0)=0.0, (0)=0.1, cˆi (0)=1.0, i =1, . . . , 5. In the simulation, the design parameters are set as b 1 = b 2 = 1.0, k0 = 30.0, k1 = 200.0, k2 = 1.0, k3 = 1.0,Kϑ = diag[0, 0.01], K = 0.5, (0) = 0, K = diag[1.0], Ka = diag[1.0], the adaptive gains in the adaptive laws are chosen as i = 0.1, i = 1.0, hi = i = 1/(1+t)2 . The disturbances on the mobile base are set to a time varying form as 0.5 sin(t) and 0.5 cos(t). The control results are shown in Figs. 2–4. Fig. 2 show the trajectory tracking (q − qd ) with the disturbances, and the corresponding torques are shown in Fig. 3. Fig. 4 shows the contact force tracking h − dh and the evolution of . The simulation results show that the position tracking error converges to zero, the estimated uncertainty converges and the contact force error converges to the desired contact force in Fig. 4. 5. Conclusion In this paper, the trajectory and force tracking controls of nonholonomic mobile manipulators have been investigated with unknown inertia parameters, constraints and disturbances. The controls ensure that the output of the dynamic system tracks hybrid variable signals and makes the whole system stable with respect to the desired force/motion. Throughout this paper, feedback control design and stability analysis are performed via explicit Lyapunov techniques. Simulation studies on the control of a two wheels driven mobile manipulator illustrate the effectiveness of the proposed control. Acknowledgment The authors enjoyed the discussions with Pey Yuen Tao.
Brockett, R. (1983). Asymptotic stability and feedback stabilization. Differential Geometry Control Theory, 181–208. Dong, W. (2002). On trajectory and force tracking control of constrained mobile manipulators with parameter uncertainty. Automatica, 38(8), 1475–1484. Ge, S. S., & Lewis, F. L. (Eds.). (2006). Autonomous mobile robots: Sensing, control, decision-making, and applications. Boca Raton, FL: CRC Press, Taylor & Francis Group. Ge, S. S., Wang, J., Lee, T. H., & Zhou, G. Y. (2001). Adaptive robust stabilization of dynamic nonholonomic chained systems. Journal of Robotic System, 18(3), 119–133. Ge, S. S., Wang, Z., & Lee, T. H. (2003). Adaptive stabilization of uncertain nonholonomic systems by state and output feedback. Automatica, 39(8), 1451–1460. Li, Z., Ge, S. S., & Ming, A. (2007). Adaptive robust motion/force control of holonomic-constrained nonholonomic mobile manipulator. IEEE Transactions on System, Man, and Cybernetics, Part B: Cybernetics, 37(3), 607–617. Lin, S., & Goldenberg, A. A. (2001). Neural-network control of mobile manipulators. IEEE Transactions on Neural Networks, 12(5), 1121–1133. Liu, K., & Lewis, F. (1990). Decentralized continuous robust controller for mobile robots. In Proceedings of IEEE international conference robotics and automation (1822–1827), Cincinnati, OH. Murray, R. M., & Sastry, S. S. (1993). Nonholonomic motion planning: Steering using sinusoids. IEEE Transactions on Automatic Control, 38, 700–716. Oya, M., & Su, C. (2003). Robust adaptive motion/force tracking control of uncertain nonholonomic mechanical systems. IEEE Transactions on Robotics and Automation, 19(1), 175–181. Slotine, J. E., & Li, W. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Tan, J., Xi, N., & Wang, Y. (2003). Integrated task planning and control for mobile manipulators. International Journal of Robotics Research, 22(5), 337–354. Walsh, G. C., & Bushnell, L. G. (1995). Stabilization of multiple input chained form control systems. System Control Letter, 25, 227–234. Wang, Z., Ge, S. S., & Lee, T. H. (2004). Robust motion/force control of uncertain holonomic/nonholonomic mechanical systems. IEEE/ASME Transactions on Mechatronics, 9(1), 118–123. Watanabe, K., Sato, K., Izumi, K., & Kunitake, Y. (2000). Analysis and control for an omnidirectional mobile manipulator. Journal of Intelligent and Robotic Systems, 27(1), 3–20. Yamamoto, Y., & Yun, X. (1996). Effect of the dynamic interaction on coordinated control of mobile manipulators. IEEE Transactions on Robotics and Automation, 12(5), 816–824. You, L. S., & Chen, B. S. (1993). Tracking control design for both holonomic and non-holonomic constrained mechanical systems: A unified viewpoint. International Journal of Control, 58(3), 587–612. Yuan, J. (1997). Adaptive control of a constrained robot ensuring zero tracking and zero force errors. IEEE Transactions on Automatic Control, 42(12), 1709–1714. Zhijun Li received the B.E. and M.E. degrees in mechanical engineering, and the Dr. Eng. degree in mechatronics from Wuhan Institute of Technology (currently, Wuhan University of Technology), PR China and Shanghai Jiao Tong University, PR China, in 1996, 1999 and 2002, respectively. From 2003 to 2005, he was a postdoctoral fellow in Department of Mechanical Engineering and Intelligent systems, The University of Electro-Communications, Tokyo, Japan. From 2005 to 2006, he was a research fellow in the Department of Electrical and Computer Engineering, National University of Singapore and Nanyang Technological University, Singapore. Currently, he is with Department of Automation, Shanghai Jiao Tong University as a Lecturer. Dr. Li’s current research
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interests are in the adaptive/robust control, friendly manipulator, nonholonomic system, etc. Shuzhi Sam Ge, IEEE Fellow, P. Eng, is the Director of Social Robotics Lab, Interactive Digital Media Institute, and Professor at Department of Electrical and Computer Engineering, the National University of Singapore. He received his B.Sc. degree from Beijing University of Aeronautics and Astronautics (BUAA), and the Ph.D. degree and the Diploma of Imperial College (DIC) from Imperial College of Science, Technology and Medicine, University of London. He has (co)-authored three books: Adaptive Neural Network Control of Robotic Manipulators (World Scientific, 1998), Stable Adaptive Neural Network Control (Kluwer, 2001) and Switched Linear Systems: Control and Design (Springer, 2005), and over 300 international journal and conference papers. He has been serving as Associate Editors for IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IEEE Transactions on Neural Networks, Automatica, and Editor for International Journal of Control, Automation & Systems, and Taylor & Francis Automation and Control Engineering Series. Dr. Ge is the recipient of the 1999 National Technology Award, 2001 University Young Research Award, 2002 Temasek Young Investigator Award, Singapore, and 2004 Outstanding Overseas Young Researcher Award from National Science Foundation, China. His current research interests include social robotics, intelligent control, hybrid systems, and intelligent multimedia fusion. Martin Adams is an Associate Professor at the School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore. He obtained his first degree in Engineering Science at the University of Oxford, UK, in 1988 and continued to study for a D.Phil. at the Robotics Research Group, University of Oxford, which he received in 1992. He continued his research in autonomous robot navigation as a project leader and part time lecturer at
the Institute of Robotics, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. He was employed as a guest professor and taught control theory in St. Gallen (Switzerland) from 1994 to 1995. From 1996 to 2000, he served as a senior research scientist in robotics and control, in the field of semiconductor assembly automation, at the European Semiconductor Equipment Centre (ESEC), Switzerland. His research work focuses on autonomous robot navigation, sensing, sensor data interpretation and control, and he has published many technical papers in these fields. He has been the principle investigator and leader of many robotics projects, coordinating researchers from local industries and local and overseas universities and is an associate editor of the International Journal of Systems Science.
Wijerupage Sardha Wijesoma received the B.Sc. Engineering Hons. degree in electronics and telecommunication engineering from the University of Moratuwa, Sri Lanka, in 1983, and the Ph.D. degree in robotics from Cambridge University, Cambridge, UK, in 1990. He is an Associate Professor of the School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore. He is also the Program Director for Mobile Robotics of the Center for Intelligent Machines, NTU. His research interests are in autonomous land and underwater vehicles, with emphasis on problems related to navigation and perception.