This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics Robust Tube-based Predictive Control for Visual Servoing of Constrained Differential-Drive Mobile Robots Fan Ke, and Zhijun Li, Senior Member, IEEE, and Chenguang Yang, Senior Member, IEEE Abstract?This work proposes a control strategy for the visionbased control problem of a noholonomic constrained differentialdrive mobile robots with bounded disturbance by using robust tube-based model predictive control (MPC) method. The proposed control strategy mainly consists of an ancillary state feedback controller and a MPC control for a nominal robotic system. Firstly, the states-error kinematics of the nominal system is converted into a chained form system, and then its MPC optimization can be computed to deal with a quadratic programming (QP) optimization problem by integrating a linear variable inequality-based primal-dual neural network (LVI-PDNN). Next, the gain scheduling of the ancillary state feedback can be obtained via solving robust pole assignment using LVI-PDNN. An optimal state trajectory can be generated for the nominal robotic system by the MPC without various uncertainties, the ancillary state feedback control forces the state variables to be constrained within an invariant designed tube. Finally, extensive experimental results on stabilization control of the noholonomic mobile robot are provided to verify the effectiveness of the proposed robust tube-based MPC. Index Terms?Mobile robots, Image-based servoing, Tuberbased model predictive control, Neural network optimization. I. I NTRODUCTION Vision-based control is one of main research fields for autonomous mobile robot, which can control the position and orientation of a robot via tightly combining the current visual feedback information obtained from cameras [1][3]. Until now, many previous approaches related to visionbased control of mobile robots have been investigated. For example, an adaptive tracking controller was proposed for a nonholonomic mobile robot with an un-calibrated camera [5]. A motion-estimation technique are proposed, which only needs THREE features to obtain a unique solution [4]. The method also can use the continuity of motion to eliminate the ambiguity problem even only two visible feature points. In [6], a robust stabilization control was proposed to steer the mobile robot to the desired posture without precise internal parameters of camera and the depth information. In order to achieve robust autonomous navigation based on vision servoing, various methods have been explored, such as specific Manuscript received February 22, 2017; revised August 06, 2017; and accepted August 30, 2017. This work is supported in part by National Natural Science Foundation of China grants (Nos. 61573147, 91520201, 61625303), and Guangzhou Research Collaborative Innovation Projects (No. 2014Y200507), Guangdong Science and Technology Research Collaborative Innovation Projects under Grant Nos. 2014B090901056, 2015B020214003, Guangdong Science and Technology Plan Project (Application Technology Research Foundation) No. 2015B020233006, National High-Tech Research and Development Program of China (863 Program) (Grant No. 2015AA042303). ? Corresponding author: Zhijun Li (zjli@ieee.org). F. Ke, Z. Li and. C. Yang are with College of Automation Science and Engineering, South China University of Technology, Guangzhou, China. Email: zjli@ieee.org. geometry features (line orientations and vanishing points) [7], image memory or planning of image trajectory [8], [9], catadioptric cameras, stereo cameras and omnidirectional [10][11], embedded velocity fields [12]. However, it should be noted that the above studies did not consider the problem of the internal constraints inside the robot, such as velocity limitation and actuator saturation [13]. Due to the ability to cope with complex nonlinear or linear systems with various constraints, MPC has received a considerable attention. In [14], under condition of the presence of both kinematic and dynamic constraints, a MPC method was proposed to stabilize an nonholonomic mobile robot(NMR). MPC also had been proposed for unmanned aerial vehicles [15], and humanoid robot [16]. However, it should be noted that the these works didn?t explicitly consider the problem of handling the external disturbance in the actual implementation and robust stability is dependent on the condition that the nominal system itself is robust. Unfortunately, the properties of inherently robust were not always present in the predictive controllers [17]. However, in this paper, the proposed Tubebased MPC can explicitly suppress the effect of the external disturbance by introducing a state feedback item in the designed control. Tube-based MPC is a double layer control strategy which is composed of an inner state feedback loop to robustify the system trajectory and an open-loop MPC with constraint condition [18], [19]. The first one is to design an auxiliary state feedback control law to keep all the possible trajectories of a uncertain system within an admissible sequential tube, which is determined by considering constraints satisfaction. Then, the second step is to solve a MPC problem under the constraint conditions that generates a nominal trajectory to the final goal. In tube-MPC context, actual trajectory obtained by state feedback control will generate a series of sequences in the state space, which is always located in a nominal trajectorycentric tube. Thus, tube-based MPC is attractive for practical implementation due to the relatively low computational complexity. On the other hand, real-time solving optimization problem is always important for MPC implementation. The target cost function of MPC is converted to a quadratic programming problem. Traditionally, the sequential quadratic programming (SQP) method requires repeated calculation of the Hessian matrix of the Lagrange to solve a QP, which brings extra computational complexity [20]. However, since the neural network have its inherent nature of parallel and distributed information processing, the emerging neural network optimization method has become a recognized efficient method to deal with complex computation. Primal-dual neural network (PDNN) can be treated as a promising computational tool for 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics Feature point Yw Ydc P zdc xdc Yc Oc ? xc zc w Ydr c vc xw w1 zdr xd yr r Oc w2 xr r Or Desired position Or zr w r ow vr zw Initial position Fig. 1. Coordinate systems relationship. solving quadratic programming (QP) problem [21], [14]. This paper describes a robust tube-MPC controller based on the visual servoing control model for nonholonomic mobile robots subject to bounded external disturbances. The proposed Tube-based MPC approach introduces the state feedback controller into the nonlinear system to compensate mismatch between real evolution of the system and the nominal system because of the effect of the bounded uncertainties, where the feedback gain can be solved online by using two linear variable inequality-based primal-dual neural network (LVI-PDNN) so that the state variables of the mobile robot can be constrained in a allowed bounded tube and a nominal MPC is applied to stabilize the nominal system which is limited to the tighter constraint sets. The main contributions of the paper are summarized as follows: 1) The proposed Tube-based MPC approach makes the state variables of the mobile robot constrained in an allowed bounded tube by introducing a state feedback controller into the closed-loop nonlinear system, where the feedback gain can be solved online by using two LVI-PDNN, to compensate the effect of the bounded uncertainties such as the external disturbance from light source and the vibration of camera. 2) Compared with the similar approaches where the boundary of tube can be obtained offline, the boundary of the feasible tube is enlarged and the time-varying parameter system reduces the conservativeness because of the proposed Tube-based MPC. 3) Considering the several feature points unavailable in the actual implementation, only one feature point is utilized to obtain the relative position of the robot. 4) The tube-based MPC can successfully achieve stabilizing the vision-based mobile robot under various constraints including velocity increment limit (acceleration limit), the velocity limits and the field-of-view limit of the onboard camera. II. P ROBLEM F ORMULATION A. Kinematic Model We firstly define the relationship of three sets of coordinate, namely, the robot coordinate, the camera coordinates and the world coordinate which are shown in Fig. 1, where Ow X w Y w Z w is the world coordinate frame attached on the ground, Or X r Y r Z r is the robot coordinate fixed to the mobile robot, while Oc X c Y c Z c is the camera coordinate rigidly attached to the camera, vr , wr and vc , wc are the linear velocity and angular velocity of the robot and camera, whose orientation of the linear velocity is the same as their Z-axis, respectively. Thus we can obtain the transformation matrix between the two coordinates as [22] [ r ] Rc drc Hcr = (1) 0 1 where drc = [dx , dy , dz ]T , Hcr ? R4�represents the homogeneous transformation, Rcr ? R3�is the rotational matrix and drc ? R3 is the position vector in the robot coordinate expressed in the camera coordinate, dx , dy , dz are the relative displacements between the robot coordinate and camera coordinate. In order to facilitate the experiment, we fixed the camera on the mobile robot base. Thus, the relationship between the robot velocity and the camera velocity from (1) can be described as [ ] Rcr s(drc )Rcr r T ?r = [0, 0, vr , 0, wr , 0] = ?cc (2) 03�Rcr where ?cc = [vx , vy , vz , wx , wy , wz ]T is the camera velocity vector relative to the camera coordinate, ?rr is the robot velocity vector relative to the robot coordinate, s(drc ) is the skew symmetric matrix of the vector drc , and wr and vr are the rotational and translational velocities of the robot on the ground. Then, we can obtain the wheel speeds as [ ] [ ]T [ ] w1 L/2 L/2 vr = (3) w2 L/l ?L/l wr where w1 and w2 are the speeds of the wheels, l is the distance between the two wheels and L is the wheel diameter. B. Camera Projection Model We assume that P is a feature point of the world coordinate, whose coordinate is (Xc , Yc , Zc ) relative to the camera frame. Correspondingly, we define that p is a projection point on the image plane coordinates of the feature points with (r? , c? ) relative to the image plane coordinate, whose pixel coordinate is (r, c). The coordinates of the original point is (r0 , c0 ) relative to the pixel coordinate and the focal length of camera is denoted by f . The transformation relation between these coordinates are given by { { ? c r = fx X r = sx (r ? r0 ) Zc + r0 (4) ? Yc c = sy (c ? c0 ) c = f y Zc + c 0 where sy and sx are the vertical and horizontal dimension of a pixel, respectively, and fx = f /sx and fy = f /sy . We can normalize image coordinates as [ x y 1 ]T [ = r?r0 fx c?c0 fy 1 ]T (5) 2 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics where x, y are the normalized image coordinates. Considering the derivative of (5), we can obtain [ ] [ ] [ X?c ?xZ?c ] x? r? Z = = (6) Y?c ?y Z?c y? c? Zc The moving velocity of p can be described as [ ] x? = L?cc = y? [ ] x 0 xy ?(1 + x2 ) y ? Z1c Zc ?cc 0 ? Z1c Zyc 1 + y 2 ?xy ?x (7) where ?cc has been defined in (2), the Zc in the interactionmatrix (7) denotes the depth information. Since the robot motion is in the two-dimensional plane, the linear and angular velocities of camera are consistent with the Z-axis direction and Y-axis direction of coordinates, respectively. Thus, we can formulate the image Jacobian matrix L as: [ x ] ?(1 + x2 ) L = Zyc (8) ?xy Zc C. Error Model and e0 = ? ? ? ? (13) where ? ? R represents the rotation angle of the robot, ?? is the desired angle. Considering (12), (13) and (11), we can obtain the error system as e?0 = wc e?1 = ?e2 wc e?2 = ? Y1c vc + e1 wc (14) To facilitate the nominal MPC design, let us define u1 = ?c , u2 = ? Y1c vc + e1 ?c , then the chained form system is written as e?0 = u1 e?1 = ?e2 u1 e?2 = u2 (15) Then, we transform (15) into two subsystems The underlying concept of vision-based control can be described as that the controller continuously adjusts the velocities of the mobile robot through the position errors of the feature point so that the mobile robot can continuously move from the initial position to the desired position. The following two state variables are introduced as x 1 r ? r 0 fy fy ?1 = = , ?2 = = (9) y c ? c0 fx y c ? c0 It is clear that the equation (9) is singularity when y = 0. So we set the camera optical center higher or lower than the feature point, i.e. y < 0 or y > 0. By utilizing (7), (8) and (9), we can obtain the kinematics model as [4]: ??1 = ??2 ?c robot, P osi is the total number of gratings of the encoder of left wheel or right wheel in ?t time, Red is precision ratio of the motor and Cnt is the resolution of the encoder so that the location of the robot can be unique determined. The error signals can be presented as [ ] [ ] [ ][ ? ] e1 ?1 cos ? ? sin ? ?1 = ? (12) e2 ?2 sin ? cos ? ?2 ? ??2 = ? Z1c ?2 vc + ?1 ?c (10) 1 c Using (4) and (9), it is clear that Z1c ?2 = Z1c Z Yc = Yc , where Yc is the height of the camera coordinate origin to the feature points and can be obtained by measurement. Thus, we can rewrite (10) as ??1 = ??2 ?c ??2 = ? Y1c vc + ?1 ?c (11) by using (5), we can? define the new desired variables in fy ?r0 fy term of (9) : ?1 ? = rc? ?c , ?2 ? = c? ?c , where (r? , c? ) 0 fx 0 represents the desired coordinate of the feature point in the image plane coordinate. Then, the control objective turns into the construction of appropriate velocities wc and vc to guarantee that ?1 ? ?1 ? , ?2 ? ?2 ? . Assume that the state variables are only the coordinate of the feature point, the control model will be singularity. And we only obtain the position of the robot relative to the feature point and can?t make sure the unique position of the robot in the world coordinate. Thus, a new global state variable ? is introduced where the relative angle ?(k + 1) = ?(k) + ??(k), ?(k) represents the relative angle of the time k, ??(k) = osi 1 ?L2 , Li = 2?識ad譖 arctan L2Dis 4Red證nt , i = 1 or 2, Dis is the distance of the two wheel, Rad is the wheel radius of the e?0 = u1 [ ] ?e2 u1 e? = u2 (16) (17) where e = [e1 , e2 ]T is the state of the system (17). We can rewrite two sub-systems (16) and (17) as e?0 = f1 (e0 ) + g1 (e0 )u1 (18) e? = f2 (e, u1 ) + g2 (e)u2 (19) where f1 (e0 ) = 0, g1 (e0 ) = 1, f2 (e, u1 ) ? R2 and g2 (e) ? R2 are defined below: f2 (e, u1 ) = [?u1 e2 , 0]T , g2 (e) = [0, 1]T . It should be noted that this system of the second subsystem (19) is not necessarily controllable when the input of the system (18) is on singular manifold. i.e., the control input u1 (0) = 0 will make the state transition matrix of the system (19) become a zero matrix so that the system becomes uncontrollable. Hence, we can prevent the uncontrollable system by making e0 (t) out of singular manifold. Integrating an exponential decaying term, the control input can be written as u1 = u?1 + ?e??t . (20) where the notation ? ? R is a positive constant and ? ? R is a nonzero constant, u?1 is the optimal input for (18). The convergence and amplitude of u1 are mainly dependent on u?1 , because ?e??t is the exponential decaying term. Obviously, if t ? ?, u1 ? u?1 . Noted that the decay term can make the speed of the input u1 to 0 slow, so that the subsystem (19) keeps its controllability until the state of system reaches the desired state. For the ancillary state feedback gains design, if we treat wc as a time-varying parameter which is obtained at real time, we can transform the model (19) into an linear parameter-varying (LPV) model : e(k + 1) = A(w(k))e(k) + Bu(k) (21) 3 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics xr + e MPC u x Nominal system ? x + + u where u? ? Rm is the nominal control, x? ? Rn is the nominal state. And, we assume the solution of (23) is ??(k; x?, u?) at time k when the control sequence is u? = {u(0), u(1), � � � , u(k)} and the initial state is x?(0). The nominal trajectory is a feasible trajectory for the nominal system to enable the state variables satisfy these constraints by the ancillary state feedback controller. The nominal optimal control problem V?N (�) for the nominal robotic system (23) is described as ? v? K + Actual system x min V?N (x?(0), xd ) = min u? Fig. 2. s.t. Control scheme of the system. u? k+N ??1 ?(x?(k) ? xd , u?(k)) j=k (24) u?(k) ? U?, x?(k) ? X? , k = 0, 1, . . . , N The proposed control can be described by the combination of a nominal MPC and an ancillary feedback control law. The nominal MPC produces a feasible trajectory without external environmental disturbance for the future predicted states and inputs. Meanwhile, the proposed ancillary feedback control is to maintain the state variables close to the reference (nominal) trajectory. Fig. 2 demonstrates the control architecture. The control input can be presented as u = u?+e v , where ve = K(x? x?) is the control input generated by the ancillary feedback controller and u? is the control input generated by the nominal MPC and K ? Rm譶 is a constant state feedback gain matrix. where x?(k) = ??(k; x?(0), u?), x?(0) is the initial state, xd is a desire states for the nominal system satisfying, U? and X? are nominal input and state constraints, respectively; the function 2 2 ?(x, u) = ?x?Q + ?u?R , where Q ? Rn譶 and R ? Rm譵 are weighting matrices being symmetric and positive definite. Definition 3.2: Assume a control invariant set ? for (22), if for any state variable x(k) ? ?, there exists a feasible control input u(k) ? U so that x(k + 1) ? ? is established for any d(k) ? D and k ? 0. Definition 3.3: Assume a set of sequences {?k } is an invariant tube for (22), if for any state variable x(k) satisfying with x(k) ? ?k , there exists a control input u(k) ? U so that all x(k + 1) ? ?k+1 is established for any d(k) ? D. In the sampling time k, we can define the optimal control sequence and the nominal state trajectory as follows: x?? = {x?(k), x?(k + 1), . . . , x?(k + N ? 1)}, u?? = {u?(k), u?(k + 1), . . . , u?(k + Nu ? 1)}, where u?? is the optimal control sequence obtained by solving (24), x?? is the nominal state trajectory by applying u?? . A. The Nominal Model Predictive Controller B. Gain Scheduling To illustrate the operation and inclusion relations between sets and the convenience of later discussions about robust tubebased MPC scheme, we will introduce some basic concepts. Definition 3.1: Given two sets A, B ? Rn , A ? B := {a + b|a ? A, b ? B} and A ? B := {a|a ? B ? A} are defined as the Minkowski set addition and Pontryagin set subtraction, respectively. The nonlinear uncertain system model in discrete time form to be controlled is presented as By integrating (22) into a linear time-variant model x(k + 1) = Ax(k) + Bu(k) + d(k), then we can design a robust feedback control as [ ] [ ] 0 1 ?T wc (k) , ,B = ? YTc T wc (k) 1 T represents the sampling time, wc is the angular velocity around the Y-axis and u = vc is the linear velocity of camera. where A(w(k)) = III. T UBE - BASED MPC x(k + 1) = f (x(k), u(k)) + d(k) (22) where x ? Rn is the state variable, u ? Rm the control input of system and d ? Rn is a bounded white noise disturbance; f (�) is a nonlinear function which is twice continuously differentiable for all (x, u) with f (0, 0) = 0. We assume all state variables are available. The state and control constraints of the system are defined as x ? X ? Rn , u ? U ? Rm , where U and X are compact set with containing the origin in its interior. The disturbance d is defined to be bounded d ? D with a compact convex set D containing the origin in its interior. We define the nominal(ref erence) system corresponding to (22) as follows. x?(k + 1) = f (x?(k), u?(k)) (23) u = u?? + K(x ? x?? ) ? (25) ? where the notations x, x? , K, u, u? have been defined above. Combining with Definitions 3.1 and (25), we can have a proposition as follows : Proposition 3.1: Assume S is a disturbance invariant set ? for x(k + 1) = AK x(k) + d(k), where AK = A + BK. If any x ? x??S and u = u?+K(x? x?), then x(k +1) ? x?(k +1)?S for all d ? D where x(k + 1) = Ax(k) + Bu(k) + d(k) and x?(k + 1) = Ax?(k) + B u?(k). Proof: Replacing the u of (22) with the formula of (25), we can obtain x(k + 1) ? x?(k + 1) = (A + BK)(x(k) ? x?(k)) + d(k) (26) ? Let AK = A + BK and a set S is a disturbance invariant set for the uncertain system (22), namely x ? x? ? S, satisfying, therefore AK S ? D ? S according to the Definitions 3.1. It is desirable that S is as small as possible (the minimal k disturbance invariant set is ?? k=0 AK D) to reduce conservativeness. So we can obtain x(k + 1) ? x?(k + 1) ? S, i.e., 4 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics x(k+1) ? x?(k+1)?S where the set S is computed as a outer invariant approximation, disturbance invariant and polynomial of the minimal disturbance set [18]. Proposition 3.1 illustrates the states x(k) of the system (22) can be close to the states x?(k) of the nominal system (23) (for all u?, x(k) ? x?(k) ? S) by applying the state feedback control policy u(k) = u?(k) + K(k)(x(k) ? x?(k)) to compensate the effect of the bounded uncertainties. According to Definitions 3.2, 3.3 and Proposition 3.1, we can obtain an invariant tube by {?k } = x?? ? S where S is a robust invariant set for the control system and the set S is robustly exponentially stable for the controlled uncertain system (22) where d ? D [18] so that the state trajectory of the uncertain system (22) robustly converges to the invariant tube {?k }. Considering the above problems, we adopt state-feedback gain scheduling technique in the control design, i.e., the timevarying state feedback control [24] is presented as u(k) = u?? (k) + ve(k) (27) where ve(k) = K(k)(x(k) ? x?? (k)), K(k) is a feedback gain matrix with its value constantly varying with the instant k. Designing feedback gain K(k) is desirable to achieve robustness against parametric perturbations, since we can hardly obtain model accurate parameters A(w(k)) and B(w(k)) in practice. In order to obtain robust gains in real time, robust pole assignment is an effective approach by using neural network optimization [25]. A time-varying LPV model (21) can be translated into a time-discrete LPV model during each sampling time interval T , i.e., e?(k + 1) = Ae?(k) + B u?(k) x(k + 1) = f (x(k)) + g(x(k))u(k) (29) where the real pseudo-diagonal matrix ? ? Rn譶 is defined as the desired eigenvalues matrix, whose eigenvalues all have negative real parts. Then, calculating the following optimization problem, we can obtain a robust feedback gain matrix K as min ?22 (Z) (30) s.t : Z? ? AZ = BG where Z ? Rn譶 is the control variable matrix with respect to the eigne-system and G ? Rm譶 is also a variable, then, we can calculate GZ ?1 , ?2 (Z) = ?1 the state feedback / gain by TK =1/2 T ?Z?2 Z 2 = (?max (Z Z) ?min (Z Z)) is the spectral condition number, where ?min and ?max are the smallest and largest eigenvalues, respectively. It is shown that if (28) is controllable, ? and A don?t have the same eigenvalue, then the solution Z of (30) is generally non-singular matrix relative to the parameter G. In conclusion, we can obtain the robust feedback control gain in (27) by repeatedly solving (30) at each sampling time. (31) subject to x(k) ? X , k = 1, 2, . . . , N , u(k) ? U , k = 1, 2, . . . , Nu , where x denotes the state variable vector, u denotes the control input; f (�) ? Rm and g(�) ? Rm are continuous nonlinear functions with f (0) = 0. Nu denotes the control horizon and N denotes the prediction horizon, 1 ? N and 0 ? Nu ? N . In order to steer the state variables converge to the origin for the system (31) by the proposed control, we can define the following cost function as ?(k) = N ? ||x (k + T j|k)||2Q j=1 + N? u ?1 ||?uT (k + j|k)||2R . (32) j=0 Nu 譔u In the quadratic form, R ? R and Q ? RmN 譵N represent appropriate weighting matrices which are symmetric and positive, where m = 1 or 2, ? � ? denotes the Euclidean norm, the x(k +j|k) denotes the predicted future horizon state and ?u(k+j|k) denotes the increment of system input. In term of control theoretical, it will guarantee stability if the finite control horizon and prediction horizon are large enough in stage cost. For the system (31), the quadratic program problem can be obtained by using the cost function (32). The two subsystems (18) and (19) can be rewritten in the discrete form as e0 (k + 1) = = e(k + 1) = (28) where the matrices of A and B are the time-varying parameters obtained by the matrices of A(w(k)) and B(w(k)) at the sampling time k, and then the state feedback control u?(k) = K e?(k) can be applied, if (A, B) is controllable and B is of full column rank. Thus, the closed-loop system can be described as: e?(k + 1) = (A + BK)e?(k) C. Tube-based Model Predictive Control The continuous time model (15) could be written as in the discrete time form as = e0 (k) + T u1 (k) f1 (e0 (k)) + g1 (e0 (k), u1 (k)) (33) ] [ ] [ ] [ 0 e1 ?T u1 e2 u2 + + ? YTc e2 T u1 e1 f2 (e(k), u1 (k)) + g2 (e(k), u1 (k)) (34) subject to constraints u1min 6 u1 (k) 6 u1max , u2min 6 u2 (k) 6 u2max , ?u1min 6 ?u1 (k) 6 ?u1max , ?u2min 6 ?u2 (k) 6 ?u2max , e0min 6 e0 (k) 6 e0max , emin 6 e(k) 6 emax , where e = [e1 , e2 ]T denotes the state variable vector of the subsystem (34), ?u(k) denotes the increment of the control input at the instant of k, umin , ?umin , emin and umax , ?umax , emax are the lower/upper bound of signals and states, respectively. The following vectors are defined: e?0 (k) = [e0 (k + 1|k), . . . , e0 (k + N |k)]T ? RN , e?(k) = [e(k + 1|k), . . . , e(k + N |k)]T ? R2N , u?(k) = T [u(k|k), . . . , u(k + Nu ? 1|k)] ? RNu , ?u?(k) = [?u(k|k), . . . , ?u(k + Nu ? 1|k)]T ? RNu . We can denote a vector [x1 , x2 ]T = [e?0 , e?]T , then, the predicted output is described as x?i (k) = Gi ?u?i (k) + f?i + g?i (35) where i = 1 or 2, Gi = ? gi (xi (k|k ? 1)) ? g (x i i (k + 1|k ? 1)) ? ? .. ? . ... ... .. . gi (xi (k + N ? 1)|k ? 1)) . . . 0 0 .. . ? ? ? ? ? gi (xi (k + N ? 1)|k ? 1)) 5 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics + u d x f (x, u) + F1 (? ) x+ F2 (? ) + + u PDNN1 PDNN 2 Fig. 3. ? ? f?i = ? ? ? ? ? g?i = ? ? P? + ? ? ? ?1 ? ? ?2 ? ? ? xd ?1 ?2 ? ? Fk (? ) K P? + ? ? ?k ? ? ?k Ck (? ) Block diagram of the robust MPC scheme ? ? C2 (? ) PDNN3 ? v? = K (x ? x) + C1 (? ) x ? LPV model P? Fig. 4. fi (xi (k|k ? 1)) fi (xi (k + 1|k ? 1)) .. . fi (xi (k + N ? 1|k ? 1)) gi (xi (k|k ? 1))ui (k ? 1) gi (xi (k + 1|k ? 1))ui (k ? 1) .. . ? network ? and its bounds ?� can be expressed as [ ] [ ] [ ] ?u? ?u?max ?u?min + ? ? := , ? := , ? := (40) ? +? + ?? ? ? ? ? ? ? ? ? ? ? gi (xi (k + N ? 1|k ? 1))ui (k ? 1) Thus, the original optimization objective (32) can be reexpressed as min ?Gi ?u?i (k) + f?i + g?i ?2Q + ??u?i (k)?2R (36) subject to ?u?min 6 ?u?i (k) 6 ?u?max , u?min 6 u?i (k ? 1) 6 ? i (k) 6 u?max , x?min 6 f?i + u?max , u?min 6 u?i (k ? 1) + I?u? ? ? I 0 贩� 0 ? I I 贩� 0 ? ? ? g?i + Gi ?u?i (k) 6 x?imax , where I? = ? . . . ? . . ... ? ? .. .. ? I I 贩� The structure frame of primal-dual neural network I where ?+ is the upper bound and ?? is the lower bound, for any i, the elements ?i + ? 0 in ? + are sufficiently positive to represent +?. Thus, we can define ? as the convex set which is expressed as ? = ?? ? ? ? ?+ . We can define the matrix M and the vector p as [ ] [ ] c1i W1i ?E1i T p= ,M = (41) ?b1i E1i 0 Then, the following lemma for the optimization of (37) can be presented as : Lemma 4.1: [21] The quadratic program problem (37) with the corresponding constrains of (38) and (39) is the same effect as, i.e.,to find a vector ?? ? ? satisfy the following linear variational inequalities as : (? ? ?? )T (M ?? + p) ? 0, ?? ? ? (42) RNu 譔u . Then, the optimization objective (36) of the nominal system will be translate into QP problem. We assume the dimension parameter for the ith subsystem where i = 1, 2 is defined as the integer h = 1 or 2. The QP problems can be expressed as: where ?, M , p are defined in (40) and (41), respectively. Moreover, we can obtain linear variational inequality (42) transformed to the following system of piecewise linear equation: P? (? ? (M ? + p)) ? ? = 0 (43) 1 ?u?i (k)T W1i ?u?i (k) + cT1i ?u?i (k) 2 where P? (�) is the projection operator onto ? and defined as P? (?) = [P? (?1 ), � � � ? , P? (?6Nu +4N )]T , ?i ? {1, � � � , 6Nu + ? ?i ? , ?i < ?i ? ?i , ?i ? ? ?i ? ?i + . 4N }, with P? (?i ) = ? + ?i , ?i > ?i + To solve the linear projection equation (43), we may build a dynamical system by using the dual dynamical system design approach. However, the matrix M is asymmetric so that leads to such a unstable system. Thus, motivated by the design experience, we can propose the following modified dynamical neural network to solve (43) min (37) subject to E1i ?u?i 6 b1i ?u?min 6 ?u?i 6 ?u?max (38) (39) where the coefficients are W1i = 2(GTi Qi Gi + Ri ) ? RNu 譔u , c1i = 2GTi Qi (g?i + f?i ) ? RNu , E1i = ? I, ? ?Gi , Gi ]T ? R(2Nu +2hN )譔u , b1i = [?u?min + u?i (k? [?I, 1), u?max + u?i (k ? 1), ?x?imin + f?i + g?i , x?imax ? f?i ? g?i ]T ? R2Nu +2hN . ?? = ?(I + M T ){P? (? ? (M ? + p)) ? ?} IV. N EURODYNAMIC A PPROACH A. First PDNN Optimization For the constraints (38) and (39), we can defined ? ? R4Nu +4N . Hence, the decision vector of the primal-dual neural (44) where ? is a designed positive parameter, by adjusting which the convergence rate of the system can be tuned. If we set ? = I + M T , S(?) = ? ? (M ? + p) and C(?) = ?, (44) can be simplified as ?? = ??(P? (S(?)) ? C(?)). 6 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics Remark 4.1: MPC approach is iteratively transformed as a constrained quadratic programming (QP) problem with taking into account the constraints, where the resulting convex optimization problem is solved in real time by applying a primal-dual neural network (PDNN). The neural network structure is shown in Fig.4, where ?i represents the ith row of the scaling matrix ?. When the dimensions of input ? is k = 6Nu + 4N , the circuit for PDNN (44) consists of k integrators, k limiters (being the piecewise-linear activation functions), 2k 2 multipliers, and 5k summers, so the proposed PDNN methods contain O(7(6Nu + 4N ) + 2(6Nu + 4N )2 ) operations. To solve QP (37)?(39), a traditional sequential quadratic programming (SQP) using gradient descent methods is usually adopted, where the computational complexity requires repeated calculation of the Hessian matrix to solve a quadratic program. Traditional QP solution needs O(N 4 + N + (6Nu + 4N ) ? Nu2 + (7Nu + 4N )3 ) operations; it is impossible to be conducted online for mobile robot systems, owing to inefficient numerical algorithm. When compared with the computational complexity, the proposed PDNN approach can reduce the computational cost. Thus the MPC solved by the PDNN has a much smaller computational complexity than SQP method. B. Second PDNN Optimization According to the understand for vectorization techniques and Kronecker product, we can rewrite the equation AZ ? Z? + BG = 0 as Hz = 0 (45) where z = [vec(Z)T , vec(G)T ]T ? Rn(n+m) and H = [In ? A ? ?T ? In |In ? B] ? Rnn譶(n+m) . Hence, according to the standard quadratic form, the optimization problem (30) can be written in a standard form : min ?2 (Z) = 12 ?T (Z)W ?(Z) (46) s.t Hz = 0 (47) / where ?(Z) = (?max (Z T Z) ?min (Z T Z))1/2 , W = 2 represents appropriate weighting matrices, T T ] and zi is the ith z = [z1T , z2T , . . . , znT , g1T , . . . , gm column of Z, gi is the ith column of G. Notice that Z T Z is symmetric and positive definite. For the constraints (47), the corresponding dual decision vector is n(n+m) defined , the primal-dual decision vector [ as ? ? ]RT ?2 = ?(Z) ? ? Rn(n+m)+1 . It is suitable to apply the following piecewise linear equation for solving (46) for its pseudo-convexity ??2 = ?2 (I + M2 T ){P? (?2 ? (M2 ?2 + p2 )) ? ?2 } (48) [ ] W ?H T where the coefficient matrix M2 = ? H 0 n(n+m)+1譶(n+m)+1 R , and the vector p2 = 0, ?2 is a positive proportional constant, and P? is a discontinuous vector function with its components have been defined in (43). C. Third PDNN Optimization To obtain the eigenvalues and eigenvectors of real symmetric matrices Z T Z, we assume that u is the eigenvector of the matrix Z T Z, and correspondingly, the eigenvalue is ?. Then, we can obtain Z T Zu = ?u (49) and, the both sides of equation (49) are multiplied by uT : ?uT u = uT Z T Zu T (50) T Further, due to ? = u u and Z Z is symmetric and positive definite, we can get the final object function: ?2 = uT Z T Zu (51) The eigenvalue problem for the matrix A = Z T Z can be rewritten as the following optimization problem: min subject to 1 T 2 u W3 u umin ? u ? umax E3 u ? b3 (52) (53) where W3 = 2A ? Rn譶 , E3 = [?I, I]T ? R2n譶 , b3 = [?umin , umax ]T ? R2n . According to [27], if A = Z T Z, the equilibrium vector umin is the smallest eigenvector of the matrix A, and correspondingly, the smallest eigenvalue is ?min = uTmin umin ; If A = Z T Z ? ?min I, the equilibrium vector umax is the largest eigenvector of the matrix A, and correspondingly, the largest eigenvalue is ?max = uTmax umax + ?min . For the constraints (53), the corresponding dual decision vector is Rn , the primal-dual decision vector [ defined ]T as ? ? 2n u ? ?3 = ? R . The neurodynamic optimization model for solving (52) can be defined as follow ??3 = ?3 (I + M3 T ){P? (?3 ? (M3 ?3 + p3 )) ? ?3 } (54) [ ] [ ] W3 ?E3 0 where M3 = , p3 = . E3 0 ?b3 From Fig. 3, we can know that the proposed robust MPC scheme is based on a LIV-PDNN and two interactive PDNNs. In every sample period, the nominal MPC problem transformed into a QP problem can be calculated by the neural network (44), and the ancillary feedback control gain K can also be obtained by combining the neural networks (48) and (54) to solve the robust pole assignment problem. V. E XPERIMENT V ERIFICATION A. Robotic System Description The proposed tube-based MPC method has been implemented on the developed differential driven robot in Fig. 5. The robot has two differential driving wheels, two support wheels for balance purpose, 24-V battery and a Microsoft Kinect camera, which equipped with a normal RGBD camera, but we only use a RGB camera because the visual stabilization of robot is based on image-based visual servoing(IBVS). The wheels are mounted on a chassis of length 45 cm and have a radius 6.4 cm. The wheels are driven by the motors with its rated torque 72.1 mNm/A at 5200 rpm. And, in order to 7 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics Envolution of the pixel coordinate r and c 350 300 vc 1 wc 0.8 250 input value pixel value(pixels) The control inputs w amd v of camera r r? c c? 200 150 0.6 0.4 0.2 0 100 ?0.2 50 ?0.4 0 0 10 20 30 40 50 60 70 0 10 time(s) The system state errors e1 and e2 3.5 2.5 30 40 50 60 70 time(s) The rotation angle of camera ? e1 e?1 e2 e?2 3 error value(unitless) 0.1 0 2 ?(rad) Fig. 5. The developed mobile robot in experiments and the different experimental road surface condition. a) smooth ceramic tile ground; b) rough concrete ground;c) epoxy resin ground; d) flat marble floor. 20 1.5 1 ?0.1 ?0.2 0.5 ?0.3 0 ?0.5 ?0.4 ?1 0 10 20 30 40 50 60 ?0.5 0 70 10 20 time(s) 30 40 50 60 70 time(s) Fig. 8. Experiment environmental condition is rough concrete floor, ? = ?0.65, ? = 1.05. Envolution of the pixel coordinate r and c accurately control the motor, each motor must be equipped with a drive gear whose reduction ratio is 85.33 and a incremental encoders counting 1024 pulses/turn. The instruction of controlling the wheels are sent out mainly through CAN bus by Elmo drives. B. Experiment Results In each experiment, the parameters of the proposed model predictive control are chosen as R1 = 0.1I, Q1 = 0.1I, R2 = 100I, Q2 = 10I, Nu = 2, N = 3. We choose the sampling time as T = 0.2s according to program running time. In each Envolution of the pixel coordinate r and c input value 0.2 0 ?0.2 10 20 30 40 50 60 ?0.4 0 70 10 time(s) The system state errors e1 and e2 20 30 40 50 60 2 ? 0.1 1 0 0 e1 e?1 e2 e?2 ?1 ?2 ?3 ?0.1 ?0.2 ?0.3 ?0.4 ?4 ?5 0 10 20 30 40 50 60 ?0.5 0 70 10 20 time(s) 30 40 50 60 The control inputs w amd v of camera 100 vc 0.5 0 1 150 100 30 40 50 60 70 time(s) The system state errors e and e 1 2.5 30 40 50 60 70 0.4 ?(rad) 1 0.3 0.2 0.1 0.5 0 0 ?0.1 30 40 time(s) 50 60 40 50 60 70 ?0.2 0 3 20 30 40 50 60 70 time(s) Fig. 7. Experiment environmental condition is epoxy resin ground, ? = 0.66, ? = 0.60. 10 20 30 40 50 60 70 time(s) The rotation angle of camera ? 0.3 0.25 0.2 2 0.15 0.1 0.05 1 0 ?0.05 0 10 ?0.2 0 70 e1 e?1 e2 e?2 4 0.5 1.5 20 30 time(s) The system state errors e1 and e2 0.6 2 10 20 ? e1 e?1 e2 e?2 3 20 time(s) The rotation angle of camera 2 3.5 10 0.4 0 10 ?(rad) 20 0 error value(unitless) 10 0.6 0.2 50 ?0.5 0 wc 0.8 r r? c c? input value r r? c c? 150 70 time(s) Experiment environmental condition is flat marble floor, ? = Fig. 9. ?0.3, ? = 0.5. pixel value(pixels) 200 70 time(s) The rotation angle of camera Envolution of the pixel coordinate r and c wc input value pixel value(pixels) 150 200 50 error value(unitless) wc 1 250 ?0.5 0 200 100 0 vc 0 0 250 The control inputs w amd v of camera 300 vc 0.4 ?(rad) The different result of the four experiments. The control inputs w amd v of camera 0.6 r r? c c? 300 error value(unitless) Fig. 6. pixel value(pixels) 350 0 10 20 30 40 time(s) 50 60 70 ?0.1 0 10 20 30 40 50 60 70 time(s) Fig. 10. Experiment environmental condition is smooth tile floor, ? = 0.47, ? = 0.95. 8 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics experiment, the parameter ? and ? of the exponential decaying term need to be adjust constantly until the state error is zero when the initial state or target state changes. The boundaries of the input are chosen as u1max = [0.5, � � � , 0.5]T , u1min = ?u1max , u2max = [1, � � � , 1]T , u2min = ?u2max , ?u1max = [0.5, � � � , 0.5]T , ?u1min = ??u1max , ?u2max = [1, � � � , 1]T , ?u2min = ??u2max . The boundaries of the state variable are [e0max , emax ]T = [1, 6, 6, � � � , 1, 6, 6] ? R3N , [e0min , emin ]T = [?1, ?6, ?6, � � � , ?1, ?6, ?6] ? R3N Since the road surface condition can affect the dynamic conditions, we have conducted the experiments using different road conditions such as rough concrete ground, smooth ceramic tile ground, epoxy resin ground and flat marble ground as shown in Fig. 5 for comparative experiment. In each experiment, the initial states are different, but the initial input vector of the robot is same. We set the initial input vector as [wr (0), vr (0)]T = [0, 0]T . Figs. 7?10 show the four experimental results which include envolution of the pixel coordinate, control inputs, the states error and the angle of camera. In each experiment, we chose control parameters properly for individual case for different dynamic condition and different initial states. From these figures we can see that the error state e2 of the uncertain system can track well the error state of the nominal system, and correspondingly, the pixel coordination c also shows good tracking performance. However, the error state e1 of the uncertain system is not well tracking the error state of nominal system for the existence of the robot?s speed limitation and disturbance, but the errors of uncertain system are finally tracked to the nominal system, which show the feedback control law can force the actual system states to track the nominal system state with considering the disturbances. And, the greater the friction coefficient on the ground, the greater the impact of disturbance on robot so that results in a greater amplitude of oscillation. But the state errors of the actual system are able to approach to zero gradually under an exponential convergent rate so as to the current image would approach to the desired one, which indicates that the nominal MPC can force the nominal system states to desired values. Finally, Fig. 6 shows the different result of the four experiments according to the rolling friction coefficient of the four ground materials. We can clearly know that the larger the friction coefficient, the longer the response time of the system and the greater the oscillation amplitude of linear velocity of the robot. VI. C ONCLUSIONS In the paper, a robust MPC method has been proposed for vision-based mobile robots. Firstly, the kinematics is transformed into a chained form and partitioned into two subsystems. Then, a quadratic programming (QP) optimization problem through MPC framework can be formulated and is computed by LVI-PDNN. Moreover, the gain scheduling of the ancillary state feedback can be obtained via solving robust pole assignment using LVI-PDNN. Finally, extensive experimental results are provided to verify the effectiveness of the proposed tubed-based MPC approach on the actual mobile robotic system. R EFERENCES [1] S. Hutchinson, G. D. Hager, and P. I. Corke, ?A tutorial on visual servo control,? IEEE Trans. Robot. Autom., vol. 12, no. 5, pp. 651?670, Oct. 1996. [2] F. Chaumette and S. Hutchinson, ?Visual servo control part I: basic approaches,? IEEE Robot. Autom. Mag., vol. 13, no. 4, pp. 82?90, Dec. 2006. [3] F. Chaumette and S. Hutchinson, ?Visual servo control part II: advanced approaches,? IEEE Robot. Autom.Mag., vol. 14, no. 1, pp. 109?118, Mar. 2007. [4] X. Zhang, Y. Fang, and X. Liu, ?Motion-estimation-based visual servoing of nonholonomic mobile robots,? IEEE Trans. Robot., vol. 27, no. 6, pp. 1167?1175, Dec. 2011. [5] W. E. Dixon, D. M. Dawson, E. Zergeroglu, and A. 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Silva, ?A hybird visual servo controller for robust grasping by wheeled mobile robots?, IEEE/ASME Transactions on Mechatronics, vol 15, no 5, pp 2323?2334, Oct, 2010. 9 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2756595, IEEE Transactions on Industrial Electronics [23] M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot modeling and control. New York: Wiley, 2006. [24] D. Mayne, E. Kerrigan, E. Wyk, and P. Falugi, ?Tube-based robust nonlinear model predictive control,? Int. J. Robust Nonlinear Control, vol. 21, pp. 1341?1353, 2011. [25] X. Le and J. Wang, ?Robust pole assignment for synthesizing feedback control systems using recurrent neural networks,? IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 2, pp. 383?393, Feb. 2014. [26] Y. Xia and J. Wang, ?A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints,? IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 7, pp. 1385?1394, Jul. 2004. [27] Y. Liu, Z. You, and L. Cao, ?A simple functional neural network for computing the largest and smallest eigenvalues and corresponding eigenvectors of a real symmetric matrix,? Neurocomputing, vol. 67, pp. 369?383, Aug. 2005. Fan Ke received the B.Eng. degree in College of Automation Science and Engineering, Lanzhou University of Technology, Lanzhou, China, in 2015. He is working toward Master Degree in College of Automation Science and Engineering, South China Univ. of Technology. His research interests are model predictive control, mobile robot, neural network control and optimization. Zhijun Li (M?07-SM?09) received the Ph.D. degree in mechatronics, Shanghai Jiao Tong University, P. R. China, in 2002. 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. Since 2012, he is a Professor in College of Automation Science and Engineering, South China university of Technology, Guangzhou, China. From 2016, he has been the Co-Chairs of Technical Committee on Biomechatronics and Biorobotics Systems (B 2 S), IEEE Systems, Man and Cybernetics Society, and Technical Committee on Neuro-Robotics Systems, IEEE Robotics and Automation Society. He is serving as an Editor-at-large of Journal of Intelligent & Robotic Systems, and Associate Editors of several IEEE Transactions. He has been the General Chair and Program Chair of 2016 and 2017 IEEE Conference on Advanced Robotics and Mechatronics (IEEEARM), respectively. Dr. Li?s current research interests include service robotics, tele-operation systems, nonlinear control, neural network optimization, etc. Chenguang Yang (M?10-SM?16) received the B.Eng. degree in measurement and control from Northwestern Polytechnical University, Xi?an, China, in 2005, and the Ph.D. degree in control engineering from the National University of Singapore, Singapore, in 2010. He is a Senior Lecturer with Zienkiewicz Centre for Computational Engineering, Swansea University, UK. He received postdoctoral training in the Department of Bioengineering at Imperial College London, UK. His major research interests lie in robotics, automation and computational intelligence. 10 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 0278-0046 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

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