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22nd IEEE International Symposium on Intelligent Control Part of IEEE Multi-conference on Systems and Control Singapore, 1-3 October 2007 TuA08.3 Robust Adaptive Control of Cooperating Mobile Manipulators with Relative Motion Zhijun Li, Shuzhi Sam Ge∗, Martin Adams, and Wijerupage Sardha Wijesoma Abstract— In this paper, coupled dynamics are presented for two cooperating mobile robotic manipulators manipulating an object with relative motion in the presence of system dynamics uncertainty and external disturbances. Centralized robust adaptive controls are introduced to guarantee the motion and force trajectories of the constrained object converge to the desired manifolds with prescribed performance. The stability of the closed-loop system and the boundedness of tracking errors are proved using Lyapunov stability synthesis. The tracking of the constraint trajectory/force up to an ultimately bounded error is achieved. The proposed adaptive controls are robust against relative motion disturbances and parametric uncertainties and validated by simulation studies. I. INTRODUCTION The controls of multiple mobile manipulators present a significant increase in complexity over the single mobile manipulator case [1], [2], [3], [4], [5]. The difficulties of the control problem lie in the fact that, when multiple mobile manipulators coordinate each other, they form a closed kinematic chain mechanism. This will impose a set of kinematic and dynamic constraints on the position and velocity of coordinated mobile manipulators. As a result, the degrees of freedom of the whole system decrease, and internal forces are generated which need to be controlled. Thus far, there are two categories of coordination schemes for multiple mobile manipulators reported in the literature. These methods can be classified into two categories as: (i) hybrid position force control by decentralized/centralized scheme, where the position of the object is controlled in a certain direction of the workspace, and the internal force of the object is controlled in a small range of the origin [1], [4]; (ii) leader-follower control for mobile manipulator, where one or a group of mobile manipulators or robotic manipulator play the role of the leader or master, which track a preplanned trajectory, and the rest of the mobile manipulators form the follow group which are moved in conjunction with the leader mobile manipulators [2], [6]. However, in the hybrid position force control of constrained coordinated multiple mobile manipulator, such as [1], [4], although the constraint object is moving, it is usually ∗To whom all correspondences should be addressed. Tel. (+65) 65166821, Fax (+65) 6779-1103, E-mail: elegesz@nus.edu.sg S. S. Ge is with the Department of Electrical and Computer Engineering, National University of Singapore,Singapore 117576, Email: elegesz@nus.edu.sg. M. Adams and W. S. Wijesomais are with School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore 639798. Z. Li was with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, and is currently with Department of Automation, Shanghai Jiaotong University, Shanghai, 200240. assumed, for the ease of analysis, to be held tightly and thus has no relative motion with respect to the end effectors of the mobile manipulators. These works have focused on dynamics based on pre-defined, fixed constraints among them. The assumption of these works are not applicable to some applications which require both the motion of the object and its relative motion with respect to the end effectors of the manipulators, such as sweeping tasks and cooperating assembly tasks by two or multiple mobile manipulators. The object/tool is required to move with respect to not only the world coordinates but also the end effectors of the mobile manipulators. The motion of the object with respect to the mobile manipulators can also be utilized to cope with the limited operational space and to increase task efficiency. Such tasks need simultaneously control of position and force in the given direction, therefore, the impedance control like [2], [6] may not be applicable. Model-based and neural-network-based adaptive control were developed for a class of constrained robots where one robotic manipulator (manipulator I) performs constrained motion on the surface of an object which is held tightly by another robotic manipulator (manipulator II) [7]. Motivated by the results, we investigate, in this paper, a similar class of system where manipulators I and II are mobile manipulators. Mobile manipulator II is to be controlled such that the constraint object follows the planned trajectory, while mobile manipulator I is to be controlled such that its end effector follows a planned trajectory on the surface with the desired contact force. We first present the dynamics of two mobile robotic manipulators manipulating an object with relative motion. This will be followed by centralized robust adaptive control to guarantee the converge of of the motion/force trajectories tracking of the constraint object under the parameters uncertainties and the external disturbances. II. DESCRIPTION OF INTERCONNECTED SYSTEM The system under study is schematically shown in Fig. 1. The object is held tightly by the end effector of mobile manipulator II and can be moved as required in space. The end effector of mobile manipulator I follows a trajectory on the surface of the object, and at the same time exerts a certain desired force on the object. The dynamics of the constrained mobile manipulators can be described as M1(q1)q¨1 + C1(q1, q˙1)q˙1 + G1(q1) + d1(t) = B1τ1 +J1T λ1 (1) M2(q2)q¨2 + C2(q2, q˙2)q˙2 + G2(q2) + d2(t) = B2τ2 +J2T λ2 (2) 1-4244-0441-X/07/$20.00 ©2007 IEEE. 351 TuA08.3 where Mi(qi) = Mib Miab Miva Mia , Ci(q˙i, qi) = Cib Ciab Ciba Cia , Gi(qi) = Gib Gia , di(t) = dib(t) dia(t) (i = 1, 2), λ1 = λ1n λc , λ2 = λ2n λc , Mi(qi) ∈ Rni×ni is the symmetric bounded positive definite inertia matrix, Ci(q˙, q)q˙ ∈ Rni denote the Centripetal and Coriolis forces; Gi(q) ∈ Rni are the gravitational forces; τi ∈ Rpi is the vector of control inputs; Bi ∈ Rni×pi is a full rank input transformation matrix and is assumed to be known because it is a function of the fixed geometry of the system; di(t) ∈ Rni is the disturbance vector; qi = [qib, qia]T ∈ Rni , and qib ∈ Rniv describe the generalized coordinates for the mobile platform; qia ∈ Rnia are the coordinates of the manipulator, and ni = niv + nia; Fi = JiT λi ∈ Rni denotes the vector of constraint forces; ATi λin = 0 satisfies the nonholonomic constraint; λi = [λin, λc]T ∈ Rpi with λin being the Lagrangian multipliers with the nonholonomic constraints. Assumption 2.1: The mobile manipulator is subject to known nonholonomic constraints. Remark 2.1: In actual implementation, we can adopt the methods of producing enough friction between the wheels of the mobile platform and the ground such that this assumption holds. Since Ai ∈ R(niv−m)×nv , it is always realizable to find an m rank matrix Si ∈ Rniv×m formed by a set of smooth and linearly independent vector fields spanning the null space of Ai, i.e. SiT ATi = 0. Since Si = [si1, · · · , sim] is formed by a set of smooth and linearly independent vector fields spanning the null space of Ai, define an auxiliary time function vib = [vib1, · · · , vibm]T ∈ Rm such that q˙ib = Sivib = si1vib1 + · · · + simvibm, which is the so-called the kinematics of nonholonomic system. Let via = q˙ia. One can obtain q˙i = Ri(qi)vi (3) where vi = [vib, via]T , and Ri(qi) = diag[Si, Inia×nia ]. Differentiating equation (3) yields q¨i = R˙i(qi)vi + Ri(qi)v˙i (4) Substituting (4) into (1) and (2), and multiplying both sides with RiT (qi) to eliminate λin, yields: Mi1(qi)v˙i + Ci1(qi, q˙i)vi + Gi1(qi) + di1(t) = Bi1(qi)τ +JiT1λi (5) where Mi1(qi) = RiT (qi)Mi(qi)R˙ i(qi) + RiT Mi(qi)Ri, Ci1(qi, q˙i) RiT Ci(qi, q˙i)Ri(qi), Gi1(qi) = = RiT (qi)Gi(qi), di1(t) = RiT (qi)di(t), Bi1 = Ri1(qi)Bi(qi), JiT1 = RiT (qi)JiT , λi = λc. There exist a coordinate transformation and a state feed- back ζi = [ζib, ζia]T = T1(qi) = [T11(qib), qi1a, T12(qib, qi1a)]T (6) vi = [vib, via]T = T2(qi)ui = [T21(qib)uib, uia]T (7) with T2(q) = diag[T21(qiv), I] and u = [uib, uia]T , uia = q˙ia, so that the kinematic system (3) could be locally or globally converted to the chained form [8], [9]     ζ˙ib1 = ui1 ζ˙ibj = ui1ζib(j+1)(2 ≤ j ≤ nv − 1)    ζ˙ibnv = ui2 ζ˙ia = q˙ia = uia (8) Consider the above transformations, the dynamic system (1) and (2) could be correspondingly converted into the following canonical transformation Mi2(ζi)u˙ i + Ci2(ζi, ζ˙i)ui + Gi2(ζi) + di2(t) = Bi2τi + JiT2λi (9) JTGTMi22TTTi2iA22(((=s(qqζsζiiiu))iT))ms[M2Tip1(=(tiqi1toi)(=)n|qqJTii2)i2T=T1T.˙2|T(2q:1q−(=Tiq1TT)2Ti(M1)ζ− h(ie1q)i(,1iζ)(JiB+Gqa).c)iiTo21b(2q(i=Caiq)nii|)1qT|(imq2=qTii=aT(,tq1T−rq˙i1i− i1)x)(1BTζ(iJζi2)1i(i,)(2q,qCii)i)id]s2||iqq(2iiuζ(==nit,TT)ifζ11− − ˙oi11)r((mζζiil==))y,, bounded and uniformly continuous, if qi is uniformly bounded and uniformly continuous. Assumption 2.3: Each manipulator is redundant and op- erating away from any singularity. Remark 2.2: Under Assumptions 2.2 and 2.3, the Jaco- bian Ji2 is of full rank. The vector qia ∈ Rnia can always be properly rearranged and partitioned into qia = [qi1a, qi2a]T , qi1a the = [qi1a1, . . . manipulator ,aqni1daκqi ]i2ade∈scRribnieas−tκhie constrained denotes the motion of remaining joints variables which make the arm redundant such that the possible breakage of contact could be compensated. Therefore, we have Ji2(qi) = [Ji2b, Ji12a, Ji22a]. Con- sidering the object trajectory and relative motion trajectory as holonomic constraints, we could obtain q˙i2a = −(Ji22a)−1[Ji2bq˙ib + Ji12aq˙i1a]. From (6) and (7), q˙ib = Si(qib)T21(qib)uib, we have  uib  ui =  q˙i1a  −(Ji22a)−1[Ji2bSi(qib)T21(qib)uib + Ji12aq˙i1a] = Liu1i (10) where Li  I 0  = 0 I  −(Ji22a)−1Ji2bSi(qib)T21(qib) −(Ji22a)−1Ji12a u1i = uib q˙i1a T It is easy to know that LTi Ji2T = 0. Combining (9) and (10), we can obtain the following compact dynamics: M u˙ 1 + Cu1 + G + d = Bτ + J T λ (11) where M = M12L1 0 0 M22L2 ,L= L1 0 0 L2 ,C= M12L˙ 1 + C12L1 0 0 M22L˙ 2 + C22L2 ,G = G12 G22 , 352 TuA08.3 d= B12 0 0 B22 ,d= d12(t) d22(t) ,τ = τ1 τ2 , JT = J1T2 J2T2 , λ = λc. Property 2.1: Matrices M = LT M , G = LT G are uni- formly bounded and uniformly continuous if ζ = [ζ1, ζ2]T is uniformly bounded and continuous, respectively. Matrix C = LT C is uniformly bounded and uniformly continuous if ζ˙ = [ζ˙1, ζ˙2]T is uniformly bounded and continuous. Property 2.2: ∀ζ ∈ R2n, 0 < λmin(M)I ≤ M(ζ) ≤ βI where λmin is the minimal eigenvalue of M and β > 0. III. ROBUST ADAPTIVE CONTROLS DESIGN Let rod(t) be the desired trajectory of the object, rcdo(t) be the desired trajectory on the object and λdc (t) be the desired constraint force. The first control objective is to drive the mobile manipulators such that ro(t) and rco(t) track their desired trajectories rod(t) and rcdo(t) respectively. The second objective is to make λc(t) its desired trajectory λdc (t). The centralized control is used for two mobile manipulators. Assumption 3.1: Time varying positive functions δk and ας converge to zero as t → ∞ and satisfy limt→∞ t 0 δk (ω )dω = ak < ∞, and limt→∞ t 0 ας (ω)dω = bς < ∞ with finite constants ak with k = 1, . . . , 6, and bς with ς = 1, . . . , 5 [11]. For the given ζd, the tracking errors are denoted as e = ζ − ζd = [e1, e2]T , ei = ζi − ζid and eλ = λc − λdc . Define the reference signals z = [z1, z2]T and zi = [zib, zia]T as  uid1 + ηi   uid2 − si(niv−1)uid1 − kniv siniv  zib =  +   + u niv −3 ∂(einiv −siniv ) (j+1) j=0 ∂ ui(dj )1 id1 e niv −1 ∂(einiv −siniv ) j=2 ∂ eij i(j+1)     (12) zia = q˙i1ad − K1a(qi1a − qi1ad) (13)  ei1   ei2     ...    si =       einiv + si(niv−2) + kniv−1si(niv−1)ui2dl−1 1 −1 uid1 u niv −4 ∂(ei(niv −1)−si(niv −1)) (j+1) j=0 ∂ ui(dj )1 id1 − e niv −2 ∂(ei(niv −1)−si(niv −1)) j=2 ∂ eij i(j+1)       (14) We could rewrite (11) as M σ˙ + Cσ + M ν˙ + Cν + G + d = Bτ + JT λ (16) If the system is certain, we could choose the control law given by Bτ = M (ν˙ − Kσσ) + C(ν + σ) + G + d − JT λ(h17) with diagonal matrix Kσ > 0. The force control input λh as λh = λd − Kλλ˜ − KI t 0 λ˜dt, where λ˜ = λc − λdc , Kλ is a constant matrix of proportional control feedback gains, and KI is a constant matrix of integral control feedback gains. However, since M(ζ), C(ζ, ζ˙), G(ζ) are uncertain, to facilitate the control formulation, the following assumption is required. Assumption 3.2: There exist some finite positive constants b, cς > 0 (1 ≤ ς ≤ 4), and finite non-negative constant cς ≥ 0 (ς = 5) such that ∀ζ ∈ R2n, ∀ζ˙ ∈ R2n, ∆M = M−M0 ≤ c1, ∆C = C −C0 ≤ c2 +c3 ζ˙ , ∆G = G −G0 ≤ c4, and supt≥0||dL(t)|| ≤ c5, where M0, C0 and G0 are nominal parameters of the system. Let B = LT B, the proposed control for the system is given as Bτ = U1 + U2 (18) where U1 is the nominal control, U1 = M0(ν˙ − Kσσ) + C0(ν + σ) + G0, and U2 is designed to compensate for the parameter errors arising from estimating the unknown function M, C and G and the disturbance, respectively. U2 = U21 + U22 + U23 + U24 + U25 + U26 (19) U21 = − β λmin cˆ1 cˆ21 Kσσ − ν˙ 2σ Kσσ − ν˙ σ + δ1 U22 = − β λmin cˆ2 cˆ22 σ + ν 2σ σ + ν σ + δ2 U23 = − β λmin cˆ3 cˆ23 ζ˙ ζ˙ 2 σ + ν σ+ν σ 2σ + δ3 U24 = − β λmin cˆ4 cˆ24σ σ + δ4 U25 = − β λmin cˆ5 cˆ25 L L 2σ σ + δ5 U26 = −β u˜b Λ 2σ Λ σ 2 + δ6 niv −1 η˙i = −k0ηi − k1si1 − sij ζi(j+1) j=2 + niv k=3 sik k−1 j=2 ∂(eik − sik) ∂eik ζi(k+1) (15) and l = niv −2, u(idl)1 is the lth derivative of uid1 with respect to t, and kj is positive constants, Kia is diagonal positive. Denote u˜ = [u˜b, u˜a]T = u − z and define a filter tracking error Ku1 σ ∈ R=(nu˜ia+−Kκiu)×(0nt iu˜ad−sκiw).itWh eKcuo=ulddioabgt[a0imn ×σ˙m=, Ku˜˙u+1] > 0, Kuu˜ and u = ν + σ with ν = z − Ku t 0 u˜ds. where δk (k = 1, . . . , 6) satisfying Assumption 3.1, and cˆς denoting the estimate cς , which are adaptively tuned according to cˆ˙ 1 = −α1cˆ1 + γ1 λmin σ Kσσ − ν˙ , cˆ1(0) > 0 (20) cˆ˙ 2 = −α2cˆ2 + γ2 λmin σ σ + ν , cˆ2(0) > 0 (21) cˆ˙ 3 = −α3cˆ3 + γ3 λmin σ ζ˙ σ + ν , cˆ3(0) > 0 (22) cˆ˙ 4 cˆ˙ 5 = = −α4cˆ4 −α5cˆ5 + + γ4 λγm5in λmin σ , cˆ4(0) > 0 L σ , cˆ5(0) > 0 (23) (24) 353 TuA08.3 where ας > 0 satisfies Assumption 3.1 and γς > 0, (ς = 1, . . . , 5). Theorem 3.1: Consider the mechanical system described by (9), under Assumption 2.4, using the control law (18), the following can achieved: (i) ζ, ζ˙, λc converge to ζd, ζ˙d, λdc as t → ∞; and (ii) all the signals in the closed-loop are bounded for all t ≥ 0. Proof : Integrating the dynamic equation (16) together with (14), (15) and (18), the close-loop system dynamics can be written as s˙i1 = ηi + u˜i1 (25) s˙i2 = (ηi + u˜i1)ζi3 + si3uid1 − k2si2u2idl1 (26) ... s˙iniv = (ηi + u˜i1) niv −2 j=2 ∂(einiv − ∂eij siniv ) ζi(j+1) −kniv siniv − si(niv−1)uid1 + u˜i2 η˙i = −k0ηi − Λi1 (27) (28) M σ˙ = −M ν˙ − C(ν + σ) − G − d + Bτ +J T λ (29) Let D = LT d. Multiplying LT on both sides of (29), using (18), one can obtain σ˙ = −Kσσ + M−1∆M (Kσσ − ν˙ ) − M−1∆C(ν + σ) 6 −M−1∆G − M−1D + M−1 U2i (30) i=1 Let c˜ς = cˆς − cς , Λ = Λ1 Λ2 T , and Λi =  k1si1 + nv −1 j=2 sij ζi(j+1)  −  nv j=3 sj ζ j−1 ∂(eik−sik) k=2 ∂eik i(k+1) .  sinv  0 Consider the following positive definite functions: V = V1 + V2 (31) V1 = 1 2 2 niv s2ij + 1 2 2 ki1s2i1 + 1 2 2 ηi2 i=1 j=2 i=1 i=1 V2 = 1 2 σT σ + 5 ς =1 1 2γς c˜2ς Taking the time derivative of V1 with (25) - (28) results V˙1 = 2 niv −1 2 2 sij s˙ij + ki1si1s˙i1 + ηiη˙i i=1 j=2 i=1 i=1 2 niv −1 2 =− kij s2ij u2idl1 − kiniv s2iniv i=1 j=2 i=1 2 − k0ηi2 + u˜Tb Λ (32) i=1 Taking the time derivative of V2 and integrating (30) result V˙2 = −σT Kσσ + σT M−1∆M (Kσσ − ν˙ ) + σT M−1 U21 + 1 γ1 c˜1 cˆ˙ 1 + −σT M−1 ∆C (σ + ν) + σT M−1 U22 + 1 γ2 c˜2 cˆ˙ 2 + σT M−1U23 + 1 γ3 c˜3 cˆ˙ 3 + −σT M−1∆G + σT M−1U24 + 1 γ4 c˜4 cˆ˙ 4 + −σT M−1D + σT M−1 U25 + 1 γ5 c˜5 cˆ˙ 5 +σT M−1U26 (33) The second right-hand term of (33) is bounded by σT M−1∆M (Kσσ − ν˙ ) + σT M−1u21 + 1 γ1 c˜1 cˆ˙ 1 ≤ 1 λmin δ1 − α1 γ1 (cˆ1 − 1 2 c1 )2 + α1 4γ1 c21 (34) Similarly, the third right-hand term of (33) is bounded by −σT M −1∆C(σ + ν) + σT M −1u22 + σT M −1u23 + 1 γ2 c˜2 cˆ˙ 2 + 1 γ3 c˜3 cˆ˙ 3 ≤ 1 λmin δ2 − α2 γ2 (cˆ2 − 1 2 c2 )2 + α2 4γ2 c22 + 1 λmin δ3 − α3 γ3 (cˆ3 − 1 2 c3 )2 + α3 4γ3 c23 (35) Similarly, the fourth right-hand term of (33) is bounded by σT M −1∆G + σT M −1u24 + 1 γ4 c˜4 cˆ˙ 4 ≤ 1 λmin δ4 − α4 γ4 (cˆ4 − 1 2 c4)2 + α4 4γ4 c24 (36) Similarly, the fifth right-hand term of (33) is bounded by σT M−1D + σT M−1u25 + 1 γ5 c˜5 cˆ˙ 5 ≤ 1 λmin δ5 − α5 γ5 (cˆ5 − 1 2 c5 )2 + α5 4γ5 c25 (37) Integrating (32) and (33), it is easily to obtain that 2 niv −1 2 V˙ ≤ − kij s2ij u2idl1 − kiniv s2iniv i=1 j=2 i=1 − 2 i=1 k0ηi2 + u˜Tb Λ − σT Kσσ − 5 ς =1 ας γς (cˆς − 1 2 cς )2 + 1 λmin 5 k=1 δk + 5 ς =1 ας 4γς c2ς + σT M−1 U26 (38) The fourth and ninth right-hand term of (38) is bounded by u˜Tb Λ + σT M−1U26 ≤ u˜b Λ − u˜b Λ Λ2σ2 σ 2 + δ6 ≤ δ6 (39) 354 TuA08.3 Therefore, we can rewrite (38) as 2 niv −1 2 V˙ ≤ − kij s2ij u2idl1 − kinv s2iniv i=1 j=2 i=1 − 2 i=1 k0ηi2 − σT Kσ σ − 5 ς =1 ας γς (cˆς − 1 2 cς )2 + 1 λmin 5 k=1 δk + δ6 + 5 ς =1 ας 4γς c2ς (40) 1 2 Noting Assumption 3.1, we have B 5 ς =1 Let ας 4γς c2ς A + δ6 = → 0 as t → ∞. 2 i=1 k0ηi2 + 2 i=1 niv −1 j=2 kij s2ij u2idl1 + λmin (Kσ ) cς )2, and it is easy to know A > 0. =1 λmin 5 k=1 δk σ 2 i=1 2+ kin5ς =iv1s2iαγnςςiv(cˆς + + − Integrating both sides of (40) gives V (t) − V (0) ≤ − 5k0=t 1Aλdm askin++ t 0 Assumption 3.1, Bds < 5 ς =1 bς 4γς c2ς we have V − t 0 Ads + C, + a6 is a finite (t) < V (0) − t 0 V is bounded and V (t) is no increasing and where C = constant from Ads + C. Thus V˙ (t) < 0. Substituting the control (18) into the reduced order dy- namics ν˙ ) + C (11) (ν + yields JT σ) + G + [(Kλ + 1)eλ + KI d − L(LT L)−1(U1 t 0 + eλdt] = M (σ˙ U2). Since σ˙ , + σ, ν˙ , ν, ci, αi, ζ˙, γi, Λ, δi are all bounded, the right hand side of (III) is bounded, i.e., JT [(Kλ + 1)eλ + KI Γ(σ˙ , σ, ν˙ , ν, ci, αi, ζ˙, γi, Λ, δi), Γ(∗) ∈ L∞. t 0 eλdt] = Let t 0 eλdt = Eλ, then E˙ λ = eλ. By appropriate choos- ing Kλ = diag[Kλ,i], Kλ,i > −1 and KI = diag[KI,i], KI ,i > 0 to make Ei(p) = , p 1 (Kλ,i +1)p+KI ,i = d/dt a strictly proper exponential stable transfer function, it can be concluded that t 0 eλdt ∈ L∞, eλ ∈ L∞, and the size of eλ can be adjusted by choosing the proper gain matrices Kλ and KI . Since σ˙ , σ, ν˙ , ν, ci, αi, ζ˙, γi, Λ, δi, eλ and t 0 eλdt are all bounded, it is easy to conclude that τ is bounded. IV. SIMULATION STUDIES Consider the two 3-DOF mobile manipulators systems shown in Fig. 2. The parameters are set as as mp = 5kg, m1 = 1.0kg, m2 = m3 = 0.5kg, Iw = 1.0kgm2, Ip = 2.5kgm2, I1 = 1.0kgm2, I2 = 0.5kgm2, I3 = 0.5kgm2, d = l = r = 0.5m, 2l1 = 1.0m, 2l2 = 0.5m, 2l3 = 0.5m. The mass of the object is mobj = 0.5kg. Let us set the desired trajectories of the object as rod = [xod, yod, zod]T , xod = 1.7 cos(t), yod = 1.7 sin(t), zod = 2l1. Therefore, we could obtain the corresponding desired trajectory of mobile manipulator II as q2d = [x2d, y2d, θ2d, θ21d, θ22d]T with xd = 1.0 cos(t), yd = 1.0 sin(t), θd = t, θ21d = π/2rad, θi2, θi3 are redundant joints to control force and compensate the task space errors. The end-effector hold tightly on the top point of surface. The constraint relative motion by the mobile manipulator I is an arc with the center on the joint 2 of the mobile manipulator I, angle= π/2 − π/6 cos(t), R = 0.7m and zcd = 1.0m, and the constraint force is set as λdc = 15.0N . Therefore, from the constraint relative motion, we can obtain the desired trajectory of mobile manipulator I is q1d = [x1d, y1d, θ1d, θ11d, θ12d]T with the corresponding trajectories x1d = 2.4 cos(t), y1d = 4.0 sin(t), θ1d = t, θ11d = π/2−π/6 cos(t), and θ22, θ23 are used to compensate the position errors of the mobile platform. The initial conditions selected for mobile manipulator I are x1(0) = 2.5m, y1(0) = 0.0m, θ1(0) = 0.0rad, θ11(0) = 0.5253rad, θ12(0) = −0.9273rad, θ13(0) = 1.8546rad, λ(0) = 0.0N and x˙ 1(0) = 0.5m/s, y˙1(0) = θ˙1(0) = θ˙11(0) = θ˙12(0) = θ˙13(0) = 0.0, and the initial conditions selected for mobile manipulator II are x2(0) = 1.2m, y2(0) = 0m, θ2(0) = 0.0rad, θ21(0) = 1.57rad, θ22(0) = 2.62rad, θ23(0) = 1.05rad, and x˙ 2(0) = y˙2(0) = θ˙2(0) = θ˙12(0) = θ˙22(0) = θ˙23(0) = 0.0. In the simulation, the design parameters for each mobile manipulator are set as k0 = 5.0, k1 = 180.0, k2 = 5.0, k3 = 5.0, η(0) = 0.0, Ka1 = diag[2.0], Kλ = 0.3, KI = 1.5, Kσ = diag[0.5], Ku = diag[1.0]. The design parameters in u2 of (19) are γi = 0.1, αi = δi = 1/(1 + t)2, and cˆi(0) = 1.0. The disturbances on each joint of each mobile manipulator are set to a time varying form as 0.5 sin(t), 0.5 sin(t), 0.1 sin(t) and 0.1 sin(t), respectively. If using the control law (18), we can obtain Fig. 3 to describe the trajectory of the mobile platforms of both mobile manipulators. Fig. 4, Fig. 7, Fig. 5 and Fig. 8 show the trajectory tracking (ζ − ζd) of the joints with the disturbances for both mobile manipulators. Fig. 10 shows the contact force tracking λc − λdc . Therefore, the validity of the proposed controls are confirmed by these simulation results. V. CONCLUSION In this paper, dynamics and control of two mobile robotic manipulators manipulating a constrained object have been investigated. In addition to the motion of the object with respect to the world coordinates, its relative motion with respect to the mobile manipulators is also taken into consideration. Robust adaptive controls have been developed which can guarantee the convergence of positions and boundedness of the constraint force. REFERENCES [1] O. Khatib, K. Yokoi, K. Chang, D. Ruspini, R. Holmberg and A. Casal , “Coordination and decentralized cooperation of multiple mobile manipulators,” Journal of Robotic Systems, vol. 13, no. 11 , pp. 755764, 1996 [2] T. G. Sugar and V. Kumar, “Control of cooperating mobile manipulators,” IEEE Trans. Robotics and Automation, vol. 18, pp.94-103, 2002 [3] H. G. Tanner, K. J. Kyriakopoulos, and N. J. Krikelis “Modeling of multiple mobile manipulators handling a common deformable object,” Journal of Robotic Systems, vol. 15, pp. 599-623, 1998 [4] H. G. Tanner, S. Loizou, and K. J. Kyriakopoulos, “Nonholonomic navigation and control of cooperating mobile manipulators,” IEEE Trans. Robotics anf Automation, vol.19, pp. 53-64, 2003 [5] T. L. Huntsberger , A. Trebi-Ollennu, H. Aghazarian, P. S. Schenker, P. Pirjanian and H. D. Nayar “Distributed control of multi-robot systems engaged in tightly coupled tasks,” Autonomous Robots, vol. 17, no. 1, pp. 929-5593, 2004 [6] R. Fierro, L. Chaimowicz, and V. Kumar,“Multi-robot cooperation”, Autonomous mobile robots: sensing, control, decision making and application, S. S. Ge and F. L. Lewis Ed., Boca Raton: Taylor & Francis Group, pp. 417-459, 2006 355 Xh Yh Zo rho Zh Oh Oo Xo Yo rco Zc Yc Oc Xc ro Trajectory rh rc Φ(rco ) = 0 Z Mobile Manipulator II X O Y Mobile Manipulator I Fig. 1. Coordinated operation of two robots Trajectory θ 2 θ 2l2 3 m2 2l3 m3 Φ(rco ) = 0 Zo Yo Xo l l 2l1 m1 d d θ r θ 1 rh Oo rco m3 2l2 m2 θ 2 m1 2l1 driving wheel ro θ 1 2r Mobile Manipulator II Z θ r X OY d driving wheel Mobile Manipulator I Fig. 2. Cooperating 2-DOF mobile manipulators TuA08.3 Arm joint angle (rad) 2.5 2 θ 3 1.5 θ 3d 1 0.5 θ 1 0 θ 1d 0 1 2 3 4 5 6 7 8 9 10 -0.5 θ 2d -1 θ 2 -1.5 -2 Time (s) Fig. 5. Tracking of arm joint angles of mobile manipulator I 100 80 τ l 60 τ r Torque (Nm) 40 20 τ 3 0 τ 0 1 2 τ3 4 5 6 7 8 19 10 -20 2 -40 Time (s) Fig. 6. Input torques for mobile manipulator I [7] S. S. Ge and L. Huang and T. H. Lee, “Model-based and neuralnetwork-based adaptive control of two robotic arms manipulating an object with relative motion,” Int. J. Sys. Sci., vol. 32, no. 1, pp. 9-23, 2001 [8] G. C. Walsh and L. G. Bushnell, “Stabilization of multiple input chained form control systems,” System Control Letter, vol. 25, pp. 227234, 1995 [9] C. Samson, “Control of chained systems: Application to path following and time- varying point-stabilization of mobile robots,” IEEE Trans. Automatic Control, vol. 40, pp. 64-77, 1995. [10] R. M. Murray and S. S. Sastry, “Nonholonomic motion planning: Steering using sinusoids,” IEEE Trans. Automat. Contr., vol. 38, pp. 700-716, May 1993. [11] Z. P. Wang, S. S. Ge, and T. H. Lee, “Robust motion/force control of uncertain holonomic/nonholonomic mechanical systems”, IEEE/ASME Transactions on Mechatronics, vol. 9, no.1, pp.118-123, 2004. 12 10 8 (rad) 6 4 2θ d θ 0         Time (s)  Fig. 7. Tracking of θ for mobile manipulator II Arm joint angle (rad) 3 θ 2 2.5 θ 2d 2 θ 1 θ 1.5 1d θ 3 1 θ 3d 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Fig. 8. Tracking of arm joint angles of mobile manipulator II 3 Mobile manipulator I 2 Mobile manipulator II Actual position 1 Actual position Y-axis (m) 0 -3 -2 -1 0 1 2 3 -1 Desired position -2 Desired position -3 X-axis (m) Fig. 3. Tracking trajectories of both mobile platforms 12 10 8 (rad) 6 4 θ 2 d θ 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Fig. 4. Tracking of θ for mobile manipulator I 356 Torque (Nm) 120 τ 100 l 80 60 τ r 40 τ 20 1 τ τ3 0 2 0 1 2 3 4 5 6 7 8 9 10 -20 -40 Time (s) Fig. 9. Torques of mobile manipulator II Contact force (N) 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 Time (s) Fig. 10. Contact force of relative motion

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