280 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 A New Performance Index for the Repetitive Motion of Mobile Manipulators Lin Xiao and Yunong Zhang, Member, IEEE Abstract—A mobile manipulator is a robotic device composed of a mobile platform and a stationary manipulator ﬁxed to the platform. To achieve the repetitive motion control of mobile manipulators, the mobile platform and the manipulator have to realize the repetitive motion simultaneously. To do so, a novel quadratic performance index is, for the ﬁrst time, designed and presented in this paper, of which the effectiveness is analyzed by following a neural dynamics method. Then, a repetitive motion scheme is proposed by combining the criterion, physical constraints, and integrated kinematical equations of mobile manipulators, which is further reformulated as a quadratic programming (QP) subject to equality and bound constraints. In addition, two important Bridge theorems are established to prove that such a QP can be converted equivalently into a linear variational inequality, and then equivalently into a piecewiselinear projection equation (PLPE). A real-time numerical algorithm based on PLPE is thus developed and applied for the online solution of the resultant QP. Two tracking-path tasks demonstrate the effectiveness and accuracy of the repetitive motion scheme. In addition, comparisons between the nonrepetitive and repetitive motion further validate the superiority and novelty of the proposed scheme. Index Terms—Mobile manipulators, neural dynamics, quadratic programming (QP), real-time numerical algorithm, repetitive motion scheme. I. Introduction A ROBOT manipulator is said to be redundant when more degrees of freedom (DOF) are available than the minimum number that is required to perform a speciﬁc endeffector primary task. The potential application of a redundant manipulator is determined, in some sense, by the DOF number, as well as by the manipulator’s structure and its control scheme. The end-effector’s motion will not be accurate or even cannot be fulﬁlled if a manipulator does not have sufﬁcient DOF number. The use of redundant manipulators is expected to increase dramatically in the future because of their ability Manuscript received September 26, 2012; revised February 24, 2013; accepted March 14, 2013. Date of publication May 7, 2013; date of current version January 13, 2014. This work was supported in part by the National Natural Science Foundation of China, under Grants 61075121 and 60935001, and the Specialized Research Fund for the Doctoral Program of Institutions of Higher Education of China with Project 20100171110045. This paper was recommended by Associate Editor H. M. Schwartz. The authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, China (e-mail: xiaolin860728@163.com; zhynong@mail.sysu.edu.cn). Color versions of one or more of the ﬁgures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identiﬁer 10.1109/TCYB.2013.2253461 to avoid the internal singularity conﬁgurations and obstacles, and to optimize dynamic performance. Fixed-based redundant manipulators have long been studied and widely applied in factory automation [1]–[6], such as working in hazardous or cluttered environments, carrying radioactive materials, doing repetitive and dull work within a limited workspace. In addition, many techniques have been developed for solving the inverse kinematics of ﬁxed-based redundant manipulators. The most popular method is to apply the pseudoinverse formulation for obtaining a general solution at the joint-velocity level, which contains a minimumnorm particular solution and a homogeneous solution [7], [8]. However, even though the joint velocities are instantaneously minimized, there is no guarantee that the kinematic singularities are avoided. Moreover, this method has the generally undesirable property that repetitive end-effector motions do not necessarily yield repetitive joint motions. Thus, the manipulator’s behavior is difﬁcult to predict when the endeffector traces a closed path in its workspace. In addition, it is less efﬁcient to readjust the manipulator’s conﬁguration after every cycle via self-motion such that the joint angles return to their initial values. Klein and Huang [9] were the ﬁrst to observe this phenomenon for the case of the pseudoinverse control to a planar three-link manipulator. A large number of research on this topic have been produced in the last three decades. For example, Brockett [10] and Shamir and Yomdin [11] provided differential geometric interpretations along with necessary and sufﬁcient conditions for repeatability. In [12], an extended Jacobian matrix method was for the ﬁrst time presented to achieve repetitive motion control of ﬁxed-based redundant manipulators. Roberts and Maciejewski [13] were devoted to the optimal synthesis of repeatable extended Jacobian inverse kinematics algorithms with prescribed properties. Marcos et al. [14], [15] provided a technique that combines the closed-loop pseudoinverse method with genetic algorithms, leading to an optimization criterion for repeatable control of redundant manipulators, and avoiding the joint-angle drift problem. Zhang et al. [16]–[18] proposed a cyclic-motion generation scheme, which can remedy effectively the jointangle drift phenomenon of ﬁxed-base redundant manipulators and make the joint variables kept within their limits. With the evolution of the complex technological society and the introduction of new notions and innovative theoretical tools in the ﬁeld of intelligent systems, mobile manipulators are attracting signiﬁcant interest in the industrial, military, and public service communities [19]–[21], because of their large- 2168-2267 c 2013 IEEE XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS 281 scale mobility and manipulation abilities, as compared to these of the ﬁxed-base manipulators. However, the integration of a redundant manipulator and a mobile platform gives rise to many new difﬁculties; for example, how to coordinate a given task into ﬁne motions to be carried out by the manipulator and the gross motions to be achieved by the mobile platform; and how to deﬁne the repeatability of an inverse-kinematic scheme for a mobile manipulator. It is well known that, for a stationary manipulator, an inverse-kinematic scheme is called repetitive, if it maps closed paths in the task space to closed trajectories in the conﬁguration space. However, for a mobile manipulator, if the mobile platform does not return to the initial position, a repetitive inverse-kinematic scheme for a stationary manipulator is no longer ﬁt for a mobile manipulator. Note that repetitive motion control of mobile manipulators starts to play a more and more important role in practical applications, which urgently requires an effective scheme for solving the nonrepetitive problem of mobile manipulators. In [22], Tchon´ was the ﬁrst to introduce the concept of repeatability of inverse kinematics algorithms for mobile manipulators by exploiting the endogenous conﬁguration space approach; and further presented repeatable inverse kinematics algorithms [23]–[25], which provides an insight into the mechanism of repeatability. However, among these inverse kinematics algorithms for repeatability of mobile manipulators, the physical constraints (e.g., joint-angle limits and joint-velocity limits) are usually not taken into account. If these physical limits are not considered, a saturation may occur in some cases. Thus, these schemes may be less effective to control mobile manipulators for generating cyclic motion. To achieve the repetitive motion control of mobile manipulators (with physical constraints considered), a novel quadratic performance index is for the ﬁrst time designed and presented in this paper, of which the effectiveness is analyzed by following Zhang et al.’s neural dynamics method. Then, a repetitive motion scheme is proposed by combining the criterion, physical constraints, and integrated kinematical equations of mobile manipulators, and further reformulated as a quadratic programming (QP) subject to equality and bound constraints. In addition, two important Bridge theorems are established, which prove that such a QP can be converted equivalently into a linear variational inequality (LVI) and then equivalently into a piecewise-linear projection equation (PLPE). A real-time numerical algorithm based on PLPE (or, say, discrete-time QP solver) is thus developed and applied for the online solution of the resultant QP. As an illustrative example, a wheeled mobile manipulator is studied, which is composed of a mobile platform driven by two independent wheels, and a six-DOF spatial manipulator mounted on the platform. Tracking-path tasks based on the wheeled mobile manipulator are performed, which further verify the effectiveness and accuracy of the proposed scheme, as well as the discrete-time QP solver. The remainder of this paper is organized into ﬁve sections. The kinematics model of the mobile manipulator is established in Section II. In Section III, a repetitive motion scheme is proposed and analyzed. A real-time numerical algorithm is developed in Section IV for the solution of the resultant QP. Section V illustrates tracking-path results based on a wheeled Fig. 1. Computer-aided design model of the mobile manipulator. mobile manipulator. Section VI concludes this paper with ﬁnal remarks. Before ending this section, the main contributions of this paper are listed as follows. 1) This paper proposes and analyzes a new performance index for the repetitive motion of mobile manipulators by deﬁning three different error functions and exploiting an exponent-type design formula. To the best of the authors’ knowledge, it is the ﬁrst time for researchers to design such a performance criterion, which may give new insights into the repetitive motion of mobile manipulators. 2) A repetitive motion scheme is proposed and presented in this paper, which can coordinate simultaneously the mobile platform and the manipulator to complete the endeffector task, and achieve the repetitive motion control. 3) Two important Bridge theorems are established to connect QP, LVI, and PLPE. Thus, a real-time numerical algorithm based on PLPE is ﬁnally developed and applied for the online solution of the resultant QP. In addition, its convergence analysis is provided for the completeness of this paper. 4) Two tracking-path tasks are conducted to demonstrate the effectiveness and accuracy of the repetitive motion scheme. In addition, this paper also makes a comparison between the nonrepetitive and repetitive motions of the wheeled mobile manipulator. II. Kinematics Modeling of Mobile Manipulators In this section, a wheeled mobile manipulator is developed to demonstrate the effectiveness of the repetitive motion scheme [26]–[29]. The computer-aided design model of the mobile manipulator is shown in Fig. 1, which is composed of a wheeled mobile platform and a six-DOF spatial manipulator. The mobile platform includes two independent drive wheels and two omni-directional passive supporting wheels. In this paper, only the end-effector position is considered, and thus the manipulator becomes a functionally redundant manipulator. In the following subsections, the kinematics equation of the mobile platform is ﬁrst analyzed and presented with respect to the world coordinate system. Then, the forward kinematics equation of the manipulator is given directly with respect to the base coordinate system (termed XCYCZC coordinate system). Finally, the velocity-level integrated kinematics model of the 282 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 TABLE I D-H Parameters of the Six-DOF Manipulator # θi (rad) di (m) ai (m) αi (rad) 1 θ1 0.698 0 0 2 θ2 0.065 0 π/2 3 θ3 0 0.555 0 4 θ4 0.190 0 π/2 5 θ5 0.130 0 π/2 6 θ6 0.082 0 −π/2 Fig. 2. Kinematical model of the mobile platform. wheeled mobile manipulator is obtained by combining the equations of the manipulator and the mobile platform. A. Kinematics of Mobile Platform In this subsection, the mathematical model of the mobile platform is ﬁrst established under the nonholonomic constraint condition. For a better understanding, the simpliﬁed model of the mobile platform is shown in Fig. 2 with the following notations used in the process of modeling. 1) P0: the middle point of the two-drive wheel axis; its coordinate in the world coordinate system is (x0, y0, z0); 2) PC: the point connecting the manipulator and the mobile platform; its coordinate in the world coordinate system is (xC, yC, zC) with zC = 0; 3) d: the distance between point P0 and point PC; 4) b: the distance between the drive wheels and point P0; 5) r: the radius of the drive wheels; 6) φ: the heading angle of the mobile platform; its time derivative is the heading velocity φ˙ ; 7) Q: a hypothetical point around which the mobile plat- form rotates; 8) R: the distance between point Q and the left drive wheel; 9) ω: the rotational velocity with which the mobile platform rotates around point Q; numerically ω = φ˙ ; 10) ϕ˙ l, ϕ˙ r: the rotational velocities of the left and right wheels. It is worth pointing out that the mobile platform and each link of the manipulator are rigid, and the motion of the mobile platform is only in the XOY-plane. Under the assumption that no-slipping and no-sideways sliding occur to the platform (i.e., rolling without slipping), the velocities of left and right wheels are both strictly perpendicular to the drive wheel axle. Thus, according to the geometrical relationship of Fig. 2, we have φ˙ = rϕ˙ l = x˙0 cos φ + y˙0 sin φ = rϕ˙ r (1) R R+b R + 2b where x˙0 cos φ + y˙0 sin φ = x˙C cos φ + y˙C sin φ in view of the velocity projection theorem that the projections of the velocities of PC and P0 in the XC-axis are equivalent to each other. Note that (1) describes the velocity constraint of the mobile platform, which is perpendicular to the drive wheel axle (i.e., the XC-axis). In the direction along the drive wheel axle (i.e., the YC-axis), the velocity of the mobile platform has to equal zero in view of the assumption of rolling without slipping. So, the velocity of the mobile platform in YC-axis can be given as x˙C sin φ − y˙C cos φ + dφ˙ = 0. (2) Combining (1) and (2), one can obtain the following important equations: φ˙ = r 2b (ϕ˙ r − ϕ˙ l) x˙C = r dr cos φ + sin φ 2 2b ϕ˙ l + r cos φ − dr sin φ 2 2b ϕ˙ r y˙C = r sin φ − dr cos φ 2 2b ϕ˙ l + r dr sin φ + cos φ 2 2b ϕ˙ r where x˙C and y˙C denote the time derivatives of xC and yC, respectively. Therefore, the mathematical models of the mobile platform are presented as follows by rewriting the above equations in a compact matrix form: Aϕ˙ = φ˙ (3) Bϕ˙ = p˙ C (4) where ϕ˙ = [ϕ˙ l, ϕ˙ r]T, and p˙ C = [x˙C, y˙C]T. In addition, matrices A and B are deﬁned as follows: r A= 2b −1 1 T ,B = r 2 cos φ sin φ − sin φ cos φ 1 −d/b 1 d/b . Note that r = 0.1025 m, b = 0.32 m, and d = 0.1 m for the mobile platform shown in this paper. B. Kinematics of Six-DOF Manipulator It is well known that, for a stationary manipulator, its forward kinematical equation (i.e., the transformation of jointspace vector θ ∈ Rn to end-effector position-and-orientation vector rb ∈ Rm) can be established directly with respect to the base coordinate system (i.e., XCYCZC coordinate system) f (θ) = rb where f (·) is a differentiable nonlinear function with a known structure and parameters for a given manipulator. For example, XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS 283 the speciﬁc kinematics equation of the manipulator investigated in this paper is (i.e., n = 6 and m = 3 with D-H parameters shown in Table I [30], [31]) rb =⎡ f (θ) ∈ Rm ⎤ l65(c5s32c1 − s5c4c32c1 + s5s4s1) + l43s32c1 + l2s2c1 = ⎣l65(c5s32s1 − s5c4c32s1 − s5s4c1) + l43s32s1 + l2s2s1⎦ (5) l65(s5c4s32 + c5c32) + l43c32 + l2c2 + l1 + l0 where I denotes the identity matrix and the augmented matrix K is deﬁned as follows: K= B 0 0 0 + J(θ, φ) A 0 0 I ∈ Rm×(n+2). The velocity-level integrated kinematics model (9) of the wheeled mobile manipulator is thus obtained. where li denotes the length of the ith link, ci := cos θi and si := sin θi with i = 1, 2, · · · , 6. In addition, l0 denotes the height of the ﬁrst joint from the bottom of the mobile platform, l65 := l6 + l5, l43 := l4 + l3, s32 := sin(θ3 + θ2) and c32 := cos(θ3 + θ2). In this paper, parameters l0 = 0.698 m, l1 = 0.065 m, l2 = 0.555 m, l3 = 0.190 m, l4 = 0.130 m, l5 = 0.082 m, and l6 = 0.133 m. Note that such a manipulator is redundant (i.e., m < n), so (5) is under-determined and generally admits an inﬁnite number of solutions in terms of inverse kinematics. C. Integrated Kinematics of Mobile Manipulator Based on the kinematics equations of the manipulator and the mobile platform, the integrated mathematical model of the mobile manipulator is developed in this subsection. First, for a mobile manipulator, the motion of the endeffector is relative to the world coordinate system instead of the base coordinate system. Thus, by adopting a transformation matrix from the base coordinate system to the world coordinate system, a new integrated kinematical equation with respect to the world coordinate system can be obtained for the mobile manipulator ⎡ ⎤ ⎡⎤ cos φ rw = ⎣sin φ 0 − sin φ cos φ 0 0 0 1 xC yC⎦ 0 f (θ) 1 xC := ⎣yC⎦ + g(θ, φ). 0 (6) Note that the above equation is the kinematics model of the mobile manipulator with respect to the world coordinate system at the position level. By differentiating (6) with respect to time t, the integrated kinematics of the mobile manipulator at the velocity level is obtained in the following form: r˙w = p˙ C 0 + J(θ, φ) φ˙ θ˙ (7) III. Design of Repetitive Motion Scheme As discussed previously, for mobile manipulators, we have to consider the effect of the mobile platform on repeatability of an inverse kinematics algorithm. Thus, the aforementioned repetitive motion scheme for stationary manipulators may not be effective for the repetitive motion control of mobile manipulators. In addition, the physical constraints of mobile manipulators have to be considered. Motivated by these points, we design and propose a novel repetitive motion scheme for mobile manipulators. Such a scheme can not only solve the joint-angle drift problem, but also make the mobile platform return to the initial position. That is to say, the ﬁnal state of the mobile manipulator is repetitive with its initial state after completing a closed-path task. Repetitive Motion Scheme: For mobile manipulators with physical constraints considered, the repetitive motion scheme is designed as minimize (Dq˙ + z)T(Dq˙ + z)/2 (10) subject to Kq˙ = r˙w (11) q− q q+ (12) q˙− q˙ q˙+ (13) where q− and q+ denote, respectively, the lower and upper limits of the combined angle vector, q˙− and q˙+ denote, respectively, the lower and upper limits of the combined velocity vector, and, matrix D and vector z are deﬁned as D= C 0 0 I ∈ R(n+m)×(n+2), z = a c ∈ Rn+m with a = [λ3 (pC − pC(0)) ; λ2(sin φ − sin φ(0))] ∈ Rm, C = [B; A cos φ] ∈ Rm×2 and c = λ1[θ − θ(0)] ∈ Rn. Note that the physical limits (i.e., q−, q+, q˙− and q˙+) used in this wheeled mobile manipulator can be seen in Table II. where Jacobian matrix J(θ, φ) is deﬁned as J(θ, φ) = ∂g(θ, φ)/∂w ∈ Rm×(n+1) with w = [φ; θ] ∈ Rn+1. To further simplify (7), one can eliminate the intermediate variable φ˙ in view of (3) and (4). Thus, the above equation can be rewritten as r˙w = Bϕ˙ 0 + J(θ, φ) Aϕ˙ θ˙ . (8) Let q = [ϕ; θ] ∈ Rn+2 denote the combined angle vector, and q˙ = [ϕ˙ ; θ˙] denote the time derivative of q (i.e., the combined velocity vector). Equation (8) can be further rewritten in a compact matrix form as r˙w = B 0 0 0 ϕ˙ θ˙ + J(θ, φ) A 0 0 I ϕ˙ θ˙ := Kq˙ (9) A. Theoretical Analysis In order to achieve the repetitive motion control of mobile manipulators, let us consider three factors of mobile manipulators, i.e., the joint angle of the manipulator θ, the rotational angle of the platform φ, and the location of the manipulator on the mobile platform pC = [xC, yC]T. Evidently, the mobile manipulator can return to the original state if and only if these three variables return to their initial positions, when the endeffector of mobile manipulators performs a closed trajectory. That is to say, the repetitive motion of mobile manipulators is equivalent to the repetitive motion of variables θ, φ, and pC. Thus, by following Zhang et al.’s design method [32], [33], the repetitive motion of θ, φ, and pC can be achieved through the following design steps. 284 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 TABLE II Physical Limits Used in the Wheeled Mobile Manipulator i qi− (rad) qi+ (rad) q˙i+ (rad/s) q˙i− (rad/s) 1 −∞ +∞ +30.0000 −30.0000 2 −∞ +∞ +30.0000 −30.0000 3 −0.5236 +0.5236 +3.0000 −3.0000 4 −0.1745 +2.3562 +3.0000 −3.0000 5 −6.2832 +6.2832 +3.0000 −3.0000 6 −6.2832 +6.2832 +3.0000 −3.0000 7 −6.2832 +6.2832 +3.0000 −3.0000 8 −6.2832 +6.2832 +3.0000 −3.0000 Step 1. We can deﬁne an indeﬁnite vector/sclar-valued error function E(t) according to the corresponding problem solving. Step 2. We can choose its time derivative E˙ (t) via the following exponent-type design formula so that E(t) exponentially converges to zero (with λ denoting a scalar-valued positive design parameter): E˙ (t) = −λE(t). (14) Step 3. We can expand the above design formula in terms of the speciﬁc error function, and thus obtain the desired dynamics equations. Speciﬁcally, in this paper, we deﬁne three different error functions as follows: E(t) := θ − θ(0) (15) E(t) := sin φ − sin φ(0) (16) E(t) := pC − pC(0) (17) where θ(0), φ(0), and pC(0) denote the initial states of θ, φ, and pC, respectively. It is worth pointing out that, when the mobile platform moves circularly (i.e., the heading angle φ makes 360-degree turns), the heading angle can return to the initial position. In this situation, the value of the heading angle is φ = φ(0) + 2kπ (k = ±1, ±2, · · · ). If φ − φ(0) is used [instead of sin φ − sin φ(0)], the restrictions become harsh and the resultant repetitive motion scheme is difﬁcult to realize. Thus, the sine function is exploited, and the simulation and modeling of the scheme become easy. Then, by combining the design formula (14) and the above three error functions respectively, three corresponding differential equations can be obtained θ˙ + λ1[θ − θ(0)] = 0 (18) φ˙ cos φ + λ2[sin φ − sin φ(0)] = 0 (19) p˙ C + λ3[pC − pC(0)] = 0 (20) where λ1 > 0, λ2 > 0, and λ3 > 0 denote scalar-valued positive design parameters used for achieving the repetitive motion. In addition, one can prove theoretically that, in (18)– (20), as t → ∞, θ, sin φ and pC can converge to their initial states globally and exponentially. In addition, the variables of the integrated kinematics at the velocity level for the mobile manipulator only include ϕ˙ and θ˙, so (18)–(20) have to be converted into a [ϕ˙ ; θ˙]-based matrix–vector equation for meeting the needs of the simulation modeling. In view of (3) Aϕ˙ = φ˙ and (4) Bϕ˙ = p˙ c, (19) and (20) can be merged into an equation in the form of B A cos φ ϕ˙ + λ3(pC − pC(0)) λ2(sin φ − sin φ(0)) = 0. (21) Then, let a = [λ3(pC − pC(0)), λ2(sin φ − sin φ(0))]T, C = [B, A cos φ]T, and c = λ1[θ − θ(0)], (18) and (21) can be combined into a uniﬁed matrix–vector equation as follows: Dq˙ + z = 0 (22) where the augmented matrix D and vector z are just what we have deﬁned before. Therefore, from the above derivation, one knows that solving (18)–(20) is equivalent to solving (22). In addition, when (22) is solved, the resultant solutions can achieve the repetitive motion control of mobile manipulators. However, because the end-effector motion-trajectory requirement has to be considered and physical constraints always exist in mobile manipulators, it is better to minimize Dq˙ + z 22/2 (with · 2 denoting the two-norm of a vector), rather than using Dq˙ + z = 0 directly. Thus, we can obtain the novel quadratic performance criterion (10) for achieving the repetitive motion planning and control of mobile manipulators. Based on the above analysis, performance criterion (10) is effective for repetitive motion control of mobile manipulators. Furthermore, the integrated kinematics equation and the physical limits of mobile manipulators are incorporated as constraints to guarantee that the primary task will be completed successfully with all variables being kept within their limits. Therefore, we can conclude that the repetitive motion scheme (10)–(13) is effective. B. Scheme Reformulation As the repetitive motion scheme (10)–(13) is resolved at the combined-vector velocity level [i.e., in terms of q˙], the limited combined-angle vector range [q−, q+] has to be converted into a q˙-based expression. The following conversion technique is adopted: μ(q− − q) q˙ μ(q+ − q) (23) where μ > 0 is used to scale the feasible region of q˙. Note that large values of μ may cause sudden deceleration for joints and wheels when the mobile manipulator approaches their physical limits, and that, in mathematically, μ qi−), −q˙i−/(qi+ − qi−)}. Therefore, the 2max1 following i n+2{q˙i+/(qi+− new combined bound constraints can be used to replace (12)–(13): ξ− q˙ ξ+ (24) XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS 285 where the ith elements of ξ− and ξ+ are, respectively, deﬁned as (with i = 1, 2, 3, · · · , n + 2) ξi− = max{μ(qi− − qi), q˙i−} and ξi+ = min{μ(qi+ − qi), q˙i+}. By summarizing the above analysis and deﬁning the decision variable vector x = q˙ ∈ Rn+2, the repetitive motion scheme (10)–(13) for mobile manipulators can be reformulated as the following standard QP, in view of (Dx + z)T(Dx + z)/2 = xTDTDx/2 + zTDx + zTz/2: minimize 1 xTWx + hTx (25) 2 subject to Kx = r˙w (26) ξ− x ξ+ (27) where W = DTD ∈ R(n+2)×(n+2) and h = DTz ∈ Rn+2. The other parameters (i.e., K, r˙w, and ξ±) are deﬁned the same as before. The performance criterion (25) is for the repetitive motion control and results from the simpliﬁcation of (10). The equality constraint (26) describes the integrated kinematics relationship of mobile manipulators at the combinedvector velocity level. The inequality constraint (27) is used to equivalently replace constraints (12) and (13) by exploiting a conversion technique. Note that almost all coefﬁcients in (25)– (27) are time-varying and state-dependent, which requires an efﬁcient solver for the above time-varying QP (25)–(27). IV. Bridge Theorems and Numerical Solver In the above section, the repetitive motion scheme is for the ﬁrst time designed and reformulated as a standard QP for repetitive motion control of mobile manipulators. In order to solve the above QP (25)–(27) rapidly, an efﬁcient discretetime solver based on PLPE is developed and investigated in this section. To do this, two Bridge theorems are developed to connect QP, LVI, and PLPE. Theorem 1 (Bridge Between QP and LVI): Assume the existence of the optimal solution x∗ to QP (25)–(27). QP (25)–(27) is equivalent to the LVI that is to ﬁnd a vector u∗ ∈ := {u|u− u u+} ⊂ RN such that ∀u ∈ (with N = n + m + 2) (u − u∗)T(Mu∗ + p) 0 (28) where the primal-dual decision variable vector u ∈ RN and its upper and lower bounds u± ∈ RN are deﬁned, respectively, as u= x y , u+ = ξ+ 1v , u− = ξ− − 1v where y ∈ Rm is the dual decision vector deﬁned corresponding to equality constraint (26); 1v := [1, · · · , 1]T denotes an appropriately-dimensioned vector composed of ones, constant = 1010 ∈ R is deﬁned to replace +∞ numerically, and, the augmented matrix M and vector p are deﬁned as M= W K −KT 0 ∈ RN×N , p= h −r˙w ∈ RN . Proof: It can be generalized from [16] and references therein by using the Lagrangian multipliers. ٗ Theorem 2 (Bridge Between LVI and PLPE): Assume the existence of the optimal solution x∗ to the QP (25)–(27). Solving the LVI (28) is equivalent to solving the following PLPE: P (u − (Mu + p)) − u = 0 (29) where P (·) : RN → is a piecewise-linear projection operator deﬁned from space RN onto set , and the ith element of P (u) is deﬁned as ⎧ ⎪⎨u−i , if ui u−i [P (u)]i = ⎪⎩uui+i,, if u−i < ui < u+i ∀i ∈ {1, 2, · · · , N} if ui u+i . Proof: See Appendix A. ٗ Therefore, via two important Bridge theorems, solving QP (25)–(27) can be equivalent to solving PLPE (29). Next, in order to solve PLPE (29), as well as QP (25)–(27), guided by [34]–[37], we ﬁrst deﬁne the following vector-valued error function: e(u) = u − P (u − (Mu + p)). (30) Obviously, the solution of (29) is equivalent to ﬁnding a zero point of (30). Let ∗ = {u∗|u∗ be a solution of (29)}. When the initial value of primal-dual decision variable vector u0 ∈ RN is given, for iteration index k = 0, 1, 2, · · · , if uk ∈/ ∗, we have the following iteration formula for solving (29), as well as the resultant QP (25)–(27): uk+1 = uk − e(uk ) 2 2 (MT + I)e(uk) 2 (MT + I)e(uk). 2 (31) An important criterion of measuring the performance of numerical algorithms is their computational complexity. As seen from the above iteration formula (31), within one iteration, it only contains 2N2 + 3N + 1 multiplications and 2N2 + 5N − 2 additions. Therefore, the discrete-time QP solver (31) has a low computational complexity, i.e., O(N2). In addition, we usually choose the solution obtained in the previous time step as the initial value of the new time step, as such, the discretetime QP solver (31) has better real-time performance. Besides, by following [34]–[37], an important theorem about the global convergence of the discrete-time QP solver (31) is presented. Theorem 3: The sequence {uk} generated by the discretetime solver (31) for PLPE (29), as well as for QP (25)–(27), satisﬁes uk+1 − u∗ 2 2 uk − u∗ 2 2 − e(uk ) 22/ MT + I 2 F for all u∗ ∈ ∗, where · F denotes the Frobenius norm of a matrix. In addition, the sequence {uk} globally converges to a solution u∗, of which the ﬁrst n + 2 elements constitute the optimal solution x∗ to QP (25)–(27). Proof: See Appendix B.ٗ V. Verification of Repetitive Motion Scheme In this section, computer simulations are conducted on the wheeled mobile manipulator to demonstrate the effectiveness 286 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 Fig. 3. Simulation results when the mobile manipulator tracks the given circular path synthesized by the nonrepetitive motion scheme. (a) Motion trajectories of the mobile manipulator. (b) Top view of motion trajectories of the mobile manipulator. (c) Motion trajectories of the mobile platform. Fig. 4. Proﬁles of point of junction [xC, yC]T, heading angle φ, joint angle θ, left, and right wheels when the mobile manipulator tracks the given circular path synthesized by the nonrepetitive motion scheme. (a) Point of junction and heading angle proﬁles. (b) Joint-angle proﬁles. (c) Left and right wheel proﬁles. of repetitive motion scheme (25)–(27) and the corresponding discrete-time QP solver. Note that the ﬁnal error tolerance of e(uk) 2 is set to be 1.0 × 10−6 for discrete-time QP solver (31). In the ﬁrst example, the end-effector of the mobile manipulator is expected to track a circular path, and in the second example the end-effector is to follow a Lissajous path. Without loss of generality, we set μ = 2, initial state θ(0) = [π/12, π/3, π/3, π/3, π/3, π/3]T rad, and φ(0) = xC(0) = yC(0) = 0 rad, for the wheeled mobile manipulator. A. Circular Path Tracking In this subsection, the mobile manipulator’s end-effector is expected to track a circular path with radius ψ being 0.3 m. The X-axis, Y -axis, and Z-axis functions of the desired circular path are ⎧ ⎪⎨r˙wX = − 2π2 T ψ sin(2π sin2( πt 2T ) + π/6) sin( πt 2T ) cos( πt 2T ) ⎪⎩rr˙˙wwYZ = = 2π2 ψ T 0 cos(2π sin2( πt 2T ) + π/6) sin( πt 2T ) cos( πt 2T ) where task duration T = 5 s. 1) Nonrepetitive Motion: First, we show the nonrepetitive motion of the mobile manipulator. Since λ1, λ2, and λ3 are greater than zero, a nonrepetitive motion scheme is produced when λ1 = λ2 = λ3 = 0. In this situation, the circular-path tracking results are shown in Fig. 3. Fig. 3(a) shows the whole tracking process of the mobile manipulator. It follows from Fig. 3(a) that the ﬁnal state of the mobile manipulator does not return to the initial one, i.e., the solution in this situation is not repetitive. For a better understanding, Fig. 3(b) shows the top view of motion trajectories of the mobile manipulator, and Fig. 3(c) shows the motion trajectories of the mobile platform. It follows from Fig. 3(b) and (c) that the mobile platform is not repetitive after the end-effector completing the circularpath tracking task. Besides, Fig. 4 shows proﬁles of point of junction [xC, yC]T, heading angle φ, joint angle θ, left and right wheels synthesized by the non-repetitive motion scheme when the mobile manipulator tracks the given circular path. From Fig. 4, we can see that these variables do not return to their initial values. In engineering applications, this situation may induce a problem wherein the behavior of the mobile manipulator is difﬁcult to predict. Readjusting the conﬁguration of the manipulator through self-motion is inefﬁcient. 2) Repetitive Motion: To ﬁnish the circular-path tracking task in a repetitive manner, the repetitive motion scheme (25)–(27) with λ1 = λ2 = λ3 = 105 and the discrete-time QP solver are applied to the control of the mobile manipulator. The corresponding simulation results are shown in Figs. 5–8. Fig. 5(a) shows the motion trajectories of the mobile manipulator during the whole tracking process. As seen from Fig. 5(a), the repetitive motion scheme (25)–(27) not only coordinates simultaneously the mobile platform and the manipulator to complete the given end-effector task, but also makes the mobile manipulator return to the initial state. In XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS 287 Fig. 5. Simulation results when the mobile manipulator tracks the given circular path synthesized by repetitive motion scheme (25)–(27). (a) Motion trajectories of the mobile manipulator. (b) Desired circular path and actual end-effector trajectory. (c) Corresponding tracking position error proﬁles. Fig. 6. Simulation results when the mobile manipulator tracks the given circular path synthesized by repetitive motion scheme (25)–(27). (a) Top view of motion trajectories of the mobile manipulator. (b) Motion trajectories of the mobile platform. (c) Corresponding left and right wheel proﬁles. Fig. 7. Proﬁles of point of junction [xC, yC]T, heading angle φ, and joint angle θ when the mobile manipulator tracks the given circular path synthesized by repetitive motion scheme (25)–(27). (a) Point of junction and heading angle proﬁles. (b) Joint-angle proﬁles. 288 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 Fig. 8. Proﬁles of [x˙C, y˙C]T, heading velocity φ˙ , joint velocity θ˙, and the corresponding tracking velocity error when the mobile manipulator tracks the given circular path synthesized by repetitive motion scheme (25)–(27). (a) Combined velocity vector proﬁles. (b) Tracking velocity error proﬁles. Fig. 9. Simulation results when the mobile manipulator tracks the given Lissajous path synthesized by repetitive motion scheme (25)–(27). (a) Motion trajectories of the mobile manipulator. (b) Desired Lissajous path and actual end-effector trajectory. (c) Corresponding tracking position error proﬁles. addition, from Fig. 5(b), we can see that the actual end-effector motion trajectory is close enough to the desired circular path. Fig. 5(c) shows the corresponding tracking position errors ε := rw − g(θ, φ) − [xC, yC, 0]T, where εX, εY and εZ denote, respectively, the X-axis, Y -axis, and Z-axis components of ε. As seen from Fig. 5(c), the corresponding X-axis, Y -axis, and Z-axis components of the tracking position error are less than 5 × 10−6 m. These results demonstrate that the mobile manipulator can ﬁnish the circular path tracking task well, as synthesized by the repetitive motion scheme (25)–(27). To further illustrate and verify the effectiveness of the repetitive motion scheme (25)–(27), in Fig. 6, we show the movements of the mobile platform. Speciﬁcally, Fig. 6(a) shows the top view of motion trajectories of the mobile manipulator, Fig. 6(b) shows the motion trajectories of the mobile platform, and Fig. 6(c) shows the corresponding left and right wheel proﬁles. From Fig. 6, we can conclude that the mobile platform is repetitive after the end-effector completing the circular-path tracking task. In addition, what we are more interested in are some important variables, such as point of junction [xC, yC]T, heading angle φ, and joint angle θ. This reason lies in that if and only if these variables return to their initial states, the mobile manipulator can achieve the repetitive motion. Fig. 7 shows the proﬁles of [xC, yC]T, φ and θ when the mobile manipulator tracks the given circular path. As seen from Fig. 7(a), xC, yC, and φ return to their initial states. In addition, Fig. 7(b) shows that θ also returns to its initial state θ(0). That is to say, these variables all return to their initial states so that the mobile manipulator can achieve the repetitive motion. Fig. 8 shows the proﬁles of [x˙C, y˙C]T, heading velocity φ˙ , joint velocity θ˙ and the corresponding tracking velocity error when the mobile manipulator tracks the given circular path. From Fig. 8, we can see that the ﬁnal states of the combined velocity equal zero, and that the corresponding X-axis, Y -axis, and Z-axis components of the tracking velocity error are less than 1×10−6 m/s. The results further demonstrate the high accuracy of the repetitive motion scheme (25)–(27) and discrete-time QP solver (31). It is worth pointing out that, in Fig. 8(a), θ˙5 reaches its upper bound θ˙5+, but never exceeds the upper bound, which demonstrates that the bound constraint (27) is activated and effective. B. Lissajous Path Tracking In this subsection, the mobile manipulator’ end-effector is expected to track a Lissajous path with parameter γ being 0.45 m. The X-axis, Y -axis, and Z-axis velocity functions of XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS 289 Fig. 10. Simulation results when the mobile manipulator tracks the given Lissajous path synthesized by repetitive motion scheme (25)–(27). (a) Top view of motion trajectories of the mobile manipulator. (b) Motion trajectories of the mobile platform. (c) Corresponding left and right wheel proﬁles. Fig. 11. Proﬁles of point of junction [xC, yC]T, heading angle φ, and joint angle θ when the mobile manipulator tracks the given Lissajous path synthesized by repetitive motion scheme (25)–(27). (a) Point of junction and heading angle proﬁles. (b) Joint-angle proﬁles. the desired Lissajous path are ⎧ ⎪⎨r˙wX(t) = − 4π2 T γ sin(4π sin2( πt 2T ) + π/6) sin( πt 2T ) cos( πt 2T ), ⎪⎩rr˙˙wwYZ((tt)) = = 2π2 γ T 0. cos(2π sin2 ( πt 2T ) + π/6) sin( πt 2T ) cos( πt 2T ), In order to investigate the effectiveness of repetitive motion scheme (25)–(27) for different values of parameters, in this example, we set task duration T = 10 s, λ1 = λ2 = 103, and λ3 = 102. Thus, the corresponding simulation results, synthesized by repetitive motion scheme (25)–(27) and discrete-time QP solver (31), are shown in Figs. 9–12. Fig. 9(a) shows the whole tracking process of the mobile manipulator. As seen from Fig. 9(a), the repetitive motion of the mobile manipulator is achieved. Speciﬁcally, Fig. 9(b) shows the desired Lissajous path and the actual end-effector trajectory, and Fig. 9(c) shows the corresponding tracking position errors. As observed from Fig. 9(b), the actual motion trajectory of the mobile manipulator’ end-effector is sufﬁciently close to the desired Lissajous path. The corresponding X-axis, Y -axis, and Z-axis components of the tracking position error shown in Fig. 9(c) are less than 1 × 10−5 m. These demonstrate that the given Lissajous-path tracking task is performed well via the repetitive motion scheme (25)–(27). To more clearly see the repetitive motion of the mobile manipulator, its mobile platform is visualized in Fig. 10. Speciﬁcally, Fig. 10(a) shows the top view of motion trajectories of the mobile manipulator, Fig. 10(b) shows the motion trajectories of the mobile platform, and Fig. 10(c) shows the corresponding left and right wheel proﬁles. It follows from Fig. 10 that the mobile platform is repetitive after the endeffector completing the Lissajous-path tracking task. Fig. 11 shows the proﬁles of [xC, yC]T, φ and θ when the mobile manipulator tracks the given Lissajous path. As seen from Fig. 11(a), xC, yC, and φ go back to their initial states. In addition, Fig. 11(b) shows that θ also returns to its initial state θ(0). These illustrate and verify again the effectiveness of such a repetitive motion scheme, and the nonrepetitive problem of the mobile manipulator has been solved by using the repetitive motion scheme. Fig. 12 further shows the proﬁles of [x˙C, y˙C]T, heading velocity φ˙ , joint velocity θ˙, and the corresponding tracking velocity error when the mobile manipulator tracks the given Lissajous path. From Fig. 12(a), we can see that the ﬁnal states of the combined velocity equal zero. Note that, if the ﬁnal states of combined velocity are not zero, the mobile manipulator will not stop immediately at the end of the task duration, and thus, the nonrepetitive problem may happen. From Fig. 12(b), one can see that the corresponding X-axis, 290 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 Fig. 12. Proﬁles of [x˙C, y˙C]T, heading velocity φ˙ , joint velocity θ˙ and the corresponding tracking velocity error when the mobile manipulator tracks the given Lissajous path synthesized by repetitive motion scheme (25)–(27). (a) Combined velocity vector proﬁles. (b) Tracking velocity error proﬁles. Y -axis, and Z-axis components of the tracking velocity error are less than 1 × 10−6 m/s, where ε˙X, ε˙Y and ε˙Z denote the time derivatives of εX, εY and εZ, respectively. The results further demonstrate the high accuracy of the repetitive motion scheme (25)–(27) and discrete-time QP solver (31). It is also worth noting that all the variables (e.g., θ, θ˙, φ and ϕ) in simulations are kept within their limits due to consideration of physical constraints of mobile manipulators. Thus, the given end-effector task can be completed successfully. In summary, the presented two examples performed on the mobile manipulator, i.e., tracking a circular path and a Lissajous path, have both demonstrated the efﬁcacy of the proposed repetitive motion scheme and the corresponding discrete-time QP solver, which can solve the non-repetitive problem well. Furthermore, the tracking position and velocity errors shown in Figs. 5(c), 8(b), 9(c) and 12(b), have validated well the high accuracy of such a repetitive scheme. VI. Conclusion In this paper, a repetitive motion scheme was proposed and investigated for achieving the repetitive motion control of mobile manipulators with physical limits considered. As illustrated via computer-simulation results based on a wheeled mobile manipulator, the nonrepetitive problem was solved effectively. To do so, the scheme was reformulated as a standard QP subject to an equality constraint and a bound constraint. In addition, two important Bridge theorems were established, which proved that such a QP can be converted equivalently into a LVI and then equivalently into a PLPE. Then, a real-time numerical algorithm based on PLPE was developed and applied for the online solution of the resultant QP. Finally, computer-simulation results based on such a mobile manipulator further veriﬁed the effectiveness, accuracy, and superiority of the repetitive motion scheme and the corresponding discrete-time QP solver. Acknowledgment The authors would like to thank the editors and anonymous reviewers for their valuable suggestions and constructive comments that have really helped to greatly improve the presentation and quality of this paper. Appendix A. Proof of Theorem 2 Let us divide two parts to prove the equivalence of LVI (28) and PLPE (29), i.e., the proof of sufﬁciency and the proof of necessity. Part one (sufﬁciency, LVI → PLPE): As u∗ ∈ and u ∈ are guaranteed in LVI (28), there are three situations discussed as follows [where [P (·)]i denotes the ith element of P (·), with i = 1, 2, · · · , N]. Situation 1: If u∗i = u+i , due to ui − u∗i = ui − u+i 0 and (ui − u∗i )(Mu∗ + p)i 0, we have (Mu∗ + p)i 0. Then [P (u∗ − (Mu∗ + p))]i = u+i = u∗i . Situation 2: If u∗i = u−i , due to ui − u∗i = ui − u−i 0 and (ui − u∗i )(Mu∗ + p)i [P (u∗ − (Mu∗ + p))]i = 0, we have u−i = u∗i . (Mu∗ + p)i 0. Then Situation 3: If u−i < u∗i < u+i , then ui − u∗i is indeﬁnite; while (ui−u∗i )(Mu∗+p)i 0 should be always true. Therefore, (Mu∗ + p)i = 0, and we have [P (u∗ − (Mu∗ + p))]i = u∗i . So, in view of the above three situations, we always have [P (u∗ − (Mu∗ + p))]i = u∗i for any i, i.e., we have P (u∗ − (Mu∗ + p)) = u∗. This implies that u∗ is a solution of PLPE (29). The proof from LVI to PLPE is thus completed. Part two (necessity, PLPE → LVI): Consider that u∗ is a solution of PLPE (29). That is, u∗ − P (u∗ − (Mu∗ + p)) = 0. Since P (u∗ − (Mu∗ + p)) ∈ = {u|u− u u+}, we have u∗ ∈ . Thus, one can discuss three situations as follows. Situation 1: If u∗i = u+i , due to [P (u∗ − (Mu∗ + p))]i = u∗i = u+i , For we know any u ∈ u+iw−ith(Mu−iu∗ + p)i ui u+i . Thus, (Mu∗ + p)i 0. u+i , we have ui − u+i 0 and (ui − u+i )(Mu∗ + p)i 0. u∗i Situation 2: = u−i , we If u∗i know (Mu∗ + p)i 0. For = u−i , due u−i − (Mu∗ any u ∈ to [P (u∗ − + p)i u−i . with u−i ui (Mu∗ + p))]i = It follows that u+i , we have ui − u−i 0 and (ui − u−i )(Mu∗ + p)i 0. XIAO AND ZHANG: NEW PERFORMANCE INDEX FOR MOBILE MANIPULATORS 291 Situation 3: If u−i < u∗i < u+i , due to [P (u∗ −(Mu∗ +p))]i = aun∗i y, wue∈havewuit∗ih−u(−iMu∗u+ip)i = u+i u∗i ; i.e., (Mu∗ +p)i = 0. , we have (ui−u∗i )(Mu∗ Thus, for +p)i = 0, which makes the inequality (ui − u∗i )(Mu∗ + p)i 0 hold true (by taking the equal sign). So, in view of the above three situations, we always have (ui − u∗i )(Mu∗ + p)i 0 for any i, i.e., we have (u − u∗)T(Mu∗ + p) 0. That is, the solution u∗ of the PLPE (29) also satiﬁes LVI (28). The proof from PLPE to LVI is thus completed. Therefore, as seen from the above analysis, it can be summarized that solving the LVI (28) is equivalent to solving the PLPE (29). The proof is thus completed.ٗ Appendix B. Proof of Theorem 3 Let ρ(uk) = e(uk ) 22/ d (uk ) 2 2 with d (uk ) being (MT + I)e(uk). According to (31), we obtain uk+1 − u∗ 2 2 = uk − u∗ − ρ(uk)d(uk) 2 2 = uk − u∗ 2 2 − 2ρ(uk )(uk − u∗ )T d (uk ) + ρ2 (uk ) d (uk ) 2 2 = uk − u∗ 2 2 − ρ(uk )[2(uk − u∗ )T d (uk ) − e(uk ) 2 2 ]. (32) Then, based on the well-known projection property, for any v ∈ RN and any s ∈ , the following inequality holds true: [s − P (v)]T[v − P (v)] 0. (33) Setting v = u − (Mu + p) and s = u∗, in view of e(u) = u − P (u − (Mu + p)), we obtain [u∗ − P (u − (Mu + p))]T[e(u) − (Mu + p)] 0. It follows from the above equation that [P (u − (Mu + p)) − u∗]Te(u) [P (u − (Mu + p)) − u∗]T(Mu + p) and then [due to e(u) = u − P (u − (Mu + p))] (u − u∗)Te(u) e(u) 2 2 + [P (u − (Mu + p)) − u∗]T(Mu + p).(34) Besides, note that (u − u∗)TMTe(u) = e(u)T(Mu + p) − e(u)T(Mu∗ + p). (35) Adding (34) and (35), and knowing M is positive deﬁnite, we have (u − u∗)T(MT + I)e(u) (u − u∗)T(Mu + p) − e(u)T(Mu∗ + p) + e(u) 2 2 = e(u) 2 2 + (u − u∗)T(Mu + p) − (u − u∗)T(Mu∗ + p) − [u∗ − P (u − (Mu + p))]T(Mu∗ + p) = e(u) 2 2 + [P (u − (Mu + p)) − u∗]T(Mu∗ + p) + (u − u∗)TM(u − u∗) e(u) 2 2 + [P (u − (Mu + p)) − u∗]T(Mu∗ + p). (36) In addition, using the projection property (33) by setting s = P (u − (Mu + p)) and v = u∗ − (Mu∗ + p), in view of u∗ = P (u∗ − (Mu∗ + p)), we obtain [P (u − (Mu + p)) − u∗]T(Mu∗ + p) 0 and thus (36) yields (u − u∗)T(MT + I)e(u) e(u) 22. With d(u) = (MT + I)e(u), we can get 2(uk − u∗)Td(uk) 2 e(uk ) 2 2 0. So, it follows from (32) that uk+1 − u∗ 2 2 uk − u∗ 2 2 − ρ(uk ) e(uk ) 2 2 . (37) Since (MT + I)e(uk) 2 2 (MT + I) 2 F e(uk ) 22, we have ρ(uk) = e(uk ) 22/ (MT + I)e(uk) 2 2 1/ MT + I 2 F . Substituting the above inequality into (37), we obtain uk+1 − u∗ 2 2 uk − u∗ 2 2 − e(uk ) 2 2 / MT + I 2F. (38) Evidently, MT + I is not a zero matrix in this paper, and thus MT+I 2 F > 0. Therefore, it follows from (38) that, if uk ∈/ ∗, then e(uk ) 2 2 >0 and uk+1 − u∗ 2 2 < uk − u∗ 22, k = 0, 1, 2, · · · . Moreover, also from (38), we obtain e(uk ) 2 2 / MT + I 2 F uk − u∗ 2 2 − uk+1 − u∗ 2 2 . Thus, we further have +∞ e(uk ) 2 2 / MT + I 2 F k=0 u0 − u∗ 2 2 − lim j→+∞ uj+1 − u∗ 2 2 u0 − u∗ 22. That is, +∞ k=0 e(uk ) 2 2 u0 − u∗ 2 2 MT + I 2 F < +∞. So, there exists a positive constant δ ∈ [0, u0 − u∗ 2 2 MT + I 2 F ], which satisﬁes j lim j→+∞ e(uk ) 2 2 = δ. k=0 According to the deﬁnition of series convergence, the sum- mation sequence { j k=0 e(uk ) 2 2 } is convergent, and thus limk→+∞ e(uk) = 0. Because u∗ ∈ ∗ is a solution of (29) [as well as LVI (28) and QP (25)–(27)], we have e(u∗) = 0, and limk→+∞ uk = u∗ in view of limk→+∞ e(uk) = 0. Then, according to [38, Theorem 3.4.2], the solution of QP (25)– (27) is unique, which would guarantee the global convergence of the sequence {uk}. The proof is thus completed.ٗ 292 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 2, FEBRUARY 2014 References [1] T. A. Lasky and B. Ravani, “Sensor-based path planning and motion control for a robotic system for roadway crack sealing,” IEEE Trans. Contr. Syst. Technol., vol. 8, no. 4, pp. 609–622, Jul. 2000. [2] Y. Xia and J. Wang, “A dual neural network for kinematic control of redundant robot manipulators,” IEEE Trans. Syst., Man, Cybern. 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He, “Inexact implicit methods for monotone general variational inequalities,” Math. Program., vol. 86, no. 1, pp. 199–217, Sep. 1999. [36] B. He, “A new method for a class of linear variational inequalities,” Math. Program., vol. 66, nos. 1–3, pp. 137–144, Aug. 1994. [37] B. He and L. Liao, “Improvements of some projection methods for monotone nonlinear variational inequalities,” J. Optimiz. Theory Appl., vol. 1, no. 1, pp. 111–128, Jan. 2002. [38] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms. New York, NY, USA: Wiley, 1993. Lin Xiao received the B.S. degree in electronic information science and technology from Hengyang Normal University, Hengyang, China, in 2009. He is currently pursuing the Ph.D. degree in communication and information systems at the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou, China. His current research interests include robotics, neural networks, intelligent information processing, and related areas. Yunong Zhang (S’02–M’03) received the B.S. degree from the Huazhong University of Science and Technology, Wuhan, China, in 1996, the M.S. degree from the South China University of Technology, Guangzhou, China, in 1999, and the Ph.D. degree from the Chinese University of Hong Kong, Shatin, Hong Kong, in 2003. He is currently a Professor with the School of Information Science and Technology, Sun YatSen University (SYSU), Guangzhou. Before joining SYSU in 2006, he was with the National University of Ireland, Maynooth, the University of Strathclyde, Glasgow, U.K., and the National University of Singapore, Singapore, since 2003. His webpage is now available at http://sist.sysu.edu.cn/∼zhynong. His current research interests include robotics, neural networks, computation, and optimization.

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