an adaptive unscented kalman filter based adaptive tracking control for wheeled mobile robots with control constrains in the presence of wheel slipping

Research Article An adaptive unscented Kalman filter-based adaptive tracking control for wheeled mobile robots with control constrains in the presence of wheel slipping International Journal of Advanced Robotic Systems September-October 2016: 1–15 ª The Author(s) 2016 DOI: 10.1177/1729881416666778 arx.sagepub.com Mingyue Cui, Hongzhao Liu, Wei Liu, Rongjie Huang, and Yi Qin Abstract A novel control approach is proposed for trajectory tracking of a wheeled mobile robot with unknown wheels’ slipping The longitudinal and lateral slipping are considered and processed as three time-varying parameters The adaptive unscented Kalman filter is then designed to estimate the slipping parameters online, an adaptive adjustment of the noise covariances in the estimation process is implemented using a technique of covariance matching in the adaptive unscented Kalman filter context Considering the practical physical constrains, a stable tracking control law for this robot system is proposed by the backstepping method Asymptotic stability is guaranteed by Lyapunov stability theory Control gains are determined online by applying pole placement method Simulation and real experiment results show the effectiveness and robustness of the proposed control method Keywords Adaptive unscented Kalman filter, unknown wheels’ slipping, pole placement method, tracking control, wheeled mobile robot Date received: 27 February 2016; accepted: August 2016 Topic: Mobile Robots and Multi-Robot Systems Topic Editor: Lino Marques Associate Editor: Euntai Kim Introduction Over the last several years, the control problem of the wheeled mobile robot (WMR) has been regarded as a fascinating problem because of the property of its nonholonomic constraints Many developed controllers have been designed for tracking and stabilization of nonholonomic mobile robots using several nonlinear control techniques such as sliding mode control,1–6 adaptive control,7–11 backstepping control12–14 and intelligent control based on neural networks,15–19 fuzzy control,20–23 and other intelligent control method.24,25 The previous papers1–24 assume nonholonomic constraints for the controlled WMR The nonholonomic constraints are generated by the assumption that the mobile robots are subject to a ‘‘pure rolling without slipping.’’ However, since the robotic wheels’ slipping can happen in various practical environments such as the on wet or icy roads, rough terrain, or the rapid cornering, the nonholonomic constraint can be disturbed in a few literatures.26–29 To deal with this case, control methods for mobile robots considering slipping were proposed in a few literatures.26–35 Wang and Low 32 proposed models of the WMR with wheels’ slipping and examined their controllability according to the outdoor maneuverability of the WMR Moreover, College of mechanic and electronic engineering, Nanyang Normal University, Nanyang Henan, China Corresponding author: Mingyue Cui, Nanyang Normal University, Wolong, Nanyang, Henan 473061, China Email: cuiminyue@sina.com Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 they presented control approaches for trajectory tracking of mobile robots considering skidding and slipping.31,32 However, the information of skidding and slipping is measured by the global positioning system and small initial conditions between the actual robot and the reference robot are required to design the controllers in 31,32 Additionally, these papers33,34 only considered the longitudinal slipping of the kinematic model for mobile robots without lateral slipping The study by Zhou et al.35 only considered the lateral slipping of the kinematic model for the WMR without longitudinal slipping The papers by Cui et al (2014), Tian and Sarkar (2014), and Yoo (2011)26–28 are excellent and distinguished works to deal with slipping perturbation; however, practical physical control constrains are not considered in the process of controller design Accordingly, the main theory contributions of our work are the design of an adaptive controller for tracking trajectory the WMR with unknown wheels’ slipping at the kinematics level More specifically, in the theoretical aspect of this article, considering practical physical constrains, we design a control law that guarantees tracking with bounded error for the WMR with unknown slipping First, the kinematic model of the WMR considering the slipping is induced from nonholonomic constraints Three slipping parameters are estimated online by the adaptive unscented Kalman filter (AUKF) An adaptive adjustment of the noise covariances in the estimation process is implemented using a technique of covariance matching in the AUKF context Meanwhile, a stable tracking control law for this nonholonomic system is proposed and the asymptotic stability is guaranteed by Lyapunov theory From the Lyapunov stability theorem, we prove that tracking errors of the controlled closed-loop system are uniformly bounded and the tracking errors can be made arbitrarily small by adjusting design parameters regardless of large initial tracking errors and unknown slipping Second, to simplify the complexity of the tuning parameters for the proposed controller, the controller gains are computed using pole placement method To the best of our knowledge, this is the first design of a tracking controller for the WMR with two independently driving wheels Finally, we provide simulation results to verify the effectiveness of the proposed controller This article is organized as follows In ‘‘Kinematic model of the WMR with wheels’ slipping’’ section, we present the kinematics model of WMRs with longitudinal and lateral slipping induced from nonholonomic constraints In ‘‘A scheme of the robotic slipping parameter estimation’’ section, an AUKF is employed to estimate three unknown sliding parameters In ‘‘A scheme of the robotic slipping parameter estimation’’ section, a tracking controller for mobile robots in the presence of unknown longitudinal and lateral slipping is designed, and the stability of the proposed control system is analyzed In ‘‘Adaptive adjustment of control parameters’’ section, a method of adjusting the control parameters online is proposed Simulation results International Journal of Advanced Robotic Systems yw ym xm xm v y om Ow ym x xw Figure WMR with two independent driving wheels WMR: wheeled mobile robot are discussed in ‘‘Simulations and experiment’’ section Finally, ‘‘Conclusions’’ section gives some conclusions Kinematic model of the WMR with wheels’ slipping A scheme of the WMR is shown as Figure The two front wheels are driven independently by two direct current servo motors, respectively The two rear wheels are supporting rollers, and they only play the roles of supporting car body, but no guidance To describe the motion features of tracked WMR simply and rigorously in the general plane motion, a fixed reference coordinate frame F1 ðxw ; yw Þ and a moving coordinate frame F2 ðxm ; ym Þ are defined which attach to the robot body with origin at the geometric center Om ! is the angular velocity of the WMR around the geometric center Om The linear velocities of left and right driving wheels of mobile robot without slipping are given as follows vL ¼ r!L vR ¼ r!R (1) where !L and !R are the angular velocities of the left and right wheels, respectively and r is radius of the wheels Longitudinal slipping ratio of the left and right wheels of the WMR are defined as follows33 r!L À vsL r!L r!R À vsR aR ¼ r!R aL ¼ (2) where vsL and vsR are the relative to the ground linear velocities of the left and right wheels of the WMR with wheels’ slipping, respectively Assumption The range of longitudinal slip ratio aL ; aR 1; 1ị [ 1; ỵ1ị Cui et al Remark If aR ¼ aL ¼ 1, from equation (2), we know that vsL ¼ vsR ¼ 0, it implies a complete slipping, that is, the wheels of the mobile robot are rotating, while its forward speed is zero, the mobile robot is uncontrollable, this case is not considered When vsL > r!L or vsR > r!R , that is, aR < or aL < 0, it indicates decelerated slipping (such as braking process) Lateral slipping ratio of the WMR is defined as35  ¼ tan (3) where is the lateral slipping angle of a mobile robot (see Figure 1), it is the angle between the velocity of the mobile robot v and the x axis of a local frame attached to the mobile robot Assumption The lateral slip angle lies in ð0; =2Þ Remark If ¼ =2, it implies that mobile robot is in a state of complete lateral slipping, the mobile robot is uncontrollable, this case is not considered That is, lateral slip ratio  is bounded From equation (2), the linear velocities of the left and right wheels of the mobile robot with wheels’ slipping are given as vsL ¼ r!L aL ị vsR ẳ r!R aR Þ (4) In coordinate frame F1 ðxw ; yw Þ, the kinematic mode of the WMR without wheels’ slipping is described as 3 ! x_ cos y_ ¼ sin v (5) ! _ In coordinate frame F2 ðxm ; ym Þ, a suitable model with slipping can be written as x_m ẳ r!L aL ị ỵ r!R aR ị y_m ẳ r!L aL ị ỵ r!R aR ị  (6) r!R ð1 À aR Þ À r!L aL ị _ ẳ b where b is the distance between two driving wheels As shown in Figure 1, the coordinate rotation transformation from F2 ðxm ; ym Þ to F1 ðxw ; yw Þ is given by ! ! ! x cos À sin xm ¼ (7) ym y sin cos From equations (6) and (7), in coordinate frame F1 ðxw ; yw Þ, the kinematic mode of the differential WMR with slipping is described as follows x_ ẳ r!L aL ị ỵ r!R aR ị cos ỵ  sinị y_ ẳ r!L aL ị ỵ r!R aR Þ ð sin À  cosÞ (8) r!R ð1 À aR Þ À r!L ð1 À aL Þ _ ¼ b where ½x; y; ŠT is posture vector of the mobile robot and  is heading angle of the WMR Assumption Slipping parameters aR , aL , and  are not measurable Define an auxiliary control input ½v; !ŠT , and then the relationship between auxiliary control input and real control input ½!L ; !R ŠT is regarded as r1 aL ị!L ỵ r1 aR Þ!R ! ! v ¼ T !L (9) ¼6 Àrð1 À aL ị!L ỵ r1 aR ị!R ! !R b À aL 6 where matrix T ẳ r6 a ị L b À aR 7 7, T is a nonsin1 À aR b gular matrix From equation (9), virtual control input ½!L ; !R ŠT can be obtained as follows b À ! ! ! 2ð1 À aL Þ 16 !L À aL v À1 v ¼ ¼T b ! !R ! r4 À aR 2ð1 À aR Þ (10) Then, equation (8) can be rewritten as 3 ! x_ cos þ  sin y_ ¼ sin À  cos v ! _ (11) We can see from equation (8), to handle tracking control problem of the WMR with unknown slipping parameters aL , aR , and , it is the top priority to estimate time-varying slipping parameters online, and then to design tracking controller on the basis of the estimation of the slipping parameters A scheme of the robotic slipping parameter estimation Because three slipping parameters aR , aL , and  in equation (8) cannot be measured directly, it is necessary to estimate slipping parameters in order to design tracking controller 4 International Journal of Advanced Robotic Systems To estimate the states and slipping parameters, joint estimation technique can be used, that is, states and parameters are estimated simultaneously using a same filter.36 It is often used to solve the state feedback control with uncertain parameters, or the modeling of the parameters with noise and states that can’t be measured directly Because of the incorporation of the states and the parameters, more accurate results may be made using this approach In the localization of the WMR with slipping, the pose and the slipping parameters should be estimated at the same time A new state vector P ẳ ẵx; y; ; aR ; aL ; ŠT is defined as a combination of the old states and parametric vector In this augmented state, the dynamic of the slipping parameters are often unknown In discrete time domain, it can be rewritten as following Pkỵ1 ẳ Pk ỵ wp;k ; k ẳ 0; 1; 2; (12) where Pk Rp is the discrete parametric vector and wp;k Rp is the additive process noise which drives the model The UKF is introduced to estimate jointly the state and slipping parameters Unlike extended Kalman filter (EKF),37 the UKF is able to approximate the nonlinear process and observation models.38 Instead, it uses the true nonlinear models and approximates the distribution of the state random variable The UKF, which does not need to compute the Jacobian, the so-called unscented transform and sigma points are used to propagate all of them through models As a result, the UKF often leads to more accurate estimations than the EKF Given the following general nonlinear system ( xkỵ1 ẳ f xk ; uk ị ỵ wk (13) ykỵ1 ẳ hxk ị ỵ vk where f xk ; uk ị and h ðxk Þ are the nonlinear process and measurement models of the robot, respectively The immeasurable state vector is represented by xk ẳ ẵx; y; ; aR ; aL ; ŠT , uk is known as the control input vector, and _ T is the observed output wk and vk are process _ y; _  yk ẳ ẵx; and measurement noise sequences with covariance Qk and Rk , respectively The initial state vector is defined as x The UKF algorithm is given as follows:38 (2) For K=1,2, ,1 (a) Calculate sigma points pffiffiffiffiffi p Xk ẳ ẵ x^ak x^ak ỵ Pak x^ak Pak (b) (16) The prediction step Xkỵ1;k ¼ f ðXk ; uk Þ > > > > > 2n > X > > > Wim Xkỵ1;k iị x^akỵ1;k ẳ > > > > iẳ0 > > > > 2n > h ih iT X > > < Pa Wic Xkỵ1;k iị x^akỵ1;k Xkỵ1;k iị x^akỵ1;k ỵ Qk k;kỵ1 ẳ iẳ0 > > a p > a a > > > Xkỵ1;k ẳ x^kỵ1;k x^kỵ1;k ỵ Pk;kỵ1 > > > > Ykỵ1;k ẳ hXkỵ1;k ị > > > > > > 2n X > > > > y^kỵ1;k ẳ Wim Ykỵ1;k iị : x^akỵ1;k p Pak;kỵ1 iẳ0 (17) where Qk is the process noise covariance matrix, and the weights Wim and Wic are defined as follows  > m > > Wi ẳ n ỵ  ; i ẳ > > > > > <  ỵ ỵ ị; i ẳ Wic ẳ (18) n ỵ  > > > > > > > ; i ¼ 1; 2; ; 2n > Wim ẳ Wic ẳ : 2n ỵ ị where n is the dimension of the augmented states; controls the size of the sigma point distribution and should be ideally a small number to avoid sampling nonlocal effects when the nonlinearities are strong; and is a nonnegative weighting term, which can be used to acknowledge the information of the higher order moments of the distribution For a Gaussian prior the optimal choice is ¼ 2, and to guarantee positive semi-definiteness of the covariance matrix tuning, parameter  ! is chosen The rest parameters are defined as (  ẳ n ỵ ị n (19) p ẳ nỵ (c) The update step Standard UKF (1) Initization at k ẳ ( x0 ẳ Eẵx Pyy ẳ T P0 ẳ Eẵx x0 ịx x0 ị iẳ0 (14) where x0 is the expected value of the initial state and P0 is initial covariance The augmented state including original states, parameters, and process noises are defined as ( x^a0 ẳ ẵ xT0 0ŠT (15) Pa0 ¼ diagðP0 ; Q0 ; R ị 2n X Wic ẵYkỵ1;k iị y^kỵ1;k ẵYkỵ1;k iị y^kỵ1;k T ỵ Rk Pxy ẳ 2n X Wic ẵXkỵ1;k iị x^akỵ1;k ẵYkỵ1;k iị y^kỵ1;k T iẳ0 ^akỵ1 ẳ x^akỵ1;k ỵ Kk ykỵ1 y^kỵ1;k ị Kk ẳ Pxy P1 yy ; x Pakỵ1 ẳ Pak;kỵ1 Kk Pyy KkT (20) where Rk is the measurement noise covariance matrix Cui et al AUKF u Low-pass filter y y e3 r yr Figure Filtering processing of the estimation signal y x xr x Figure Robotic pose error coordinates scheme Qk ¼ Kk Ck ðKk ịT Rk ẳ Ck ỵ e2 e2 Adaptive UKF In order to further improve the estimation precision, an adaptive adjustment of the noise covariances in the estimation process is implemented using a technique of covariance matching in the UKF context More specifically, the adaptive estimation of the process noise covariance Qk and measurement noise Rk on the basis of the pose sequence of the mobile robot will be considered Therefore, Qk and Rk are estimated and updated iteratively from the following39 e1 2n X Wic ẵYkỵ1;k iị y^kỵ1;k ẵYkỵ1;k iị y^kỵ1;k T iẳ0 (21) where y^kỵ1;k is measured pose of the mobile robot and Ck is defined as Ck ẳ k X Ei Ei ịT (22) iẳkLỵ1 where E ẳ ẵ x x^ y y^  À ^ ŠT are the pose estimation errors of the mobile robot at time step k, Ck is an approximation to the covariance of the voltage residual at time step k, and L is window size for covariance matching More details can be found in the article by Cui et al (2005)37 Moreover, in order to reduce the chattering of AUKF, a low-pass filter (LPF) is applied to process the estimation signal A first-order LPF is described in Figure The relationship between input u and output y of LPF is given as y_ ỵ y ẳ u; y0ị ẳ u0ị (23) where u ẳ ẵ^ aL ; a^R ; ^ŠT is estimation output of the AUKF, ¼ diag ð ; ; Þ; ; ; > are the filter parameters, y is the output, and is the filter time constant and a very small value that lies i 1; i ¼ 1; 2; Remark Note that we employ the first-order LPF (23) to reduce the chattering problem and guarantees smoothness of the estimation signal After the estimation of the slipping parameters is processed by the LPF, their first-order deri_ vative values a^_ L , a^_ R , and ^ are all bounded equation (8), so actual posture of the WMR is decided by equation (8) The desired posture is defined as pr ẳ ẵxr ; yr ; r T , the desired posture satisfies the following equations 3 ! x_r cos r y_r ¼ sin r vr (24) !r _ r where vr and !r are reference linear velocity and angular velocity of the mobile robot, respectively Assumption The reference linear velocity vr and !r as well as their derivatives are all bounded Our control objective is to design a trajectory tracking controller, when the wheels’ slipping will occur between the WMR and the ground, which making the actual and desired posture of the WMR satisfy as following lim p pr ị ẳ t!1 (25) In the coordinate frame F1 ðx; yÞ, the tracking error dynamics of the WMR are described as follows 32 cos  ỵ  sin  sin  À  cos  xr À x e1 e ẳ sin  ỵ  cos  cos  ỵ  sin  54 yr À y 0 r À  e3 (26) Design of the tracking control law where e , e , and e are state-tracking errors, they are expressed in the frame of real robot, as shown in Figure Tracking errors vector of the WMR is defined as e ẳ ẵe ; e ; e ŠT , the error dynamics of the WMR is obtained as follows11 3 e_ !e ỵ ỵ  ịvr cos e ỵ  ịv e_ ẳ (27) !e ỵ ỵ  ịvr sin e e_ !r À ! Assume there is a superposition between center of mass and movement geometric center of the WMR The kinematic model of the WMR with the slipping can be described by In the presence of the wheels’ slipping, applying backstepping method, the auxiliary control input is designed as follows7 Design of the tracking controller International Journal of Advanced Robotic Systems ! ! Remark In equation (32), the nominal velocity control v vr cose ỵ k ðvr ; !r Þe commands !L and !R are computed from equation (30) ẳ ! !r ỵ ỵ  ịk vr e ỵ ỵ  Þk ðvr ; !r Þvr sin e (28) where k is a positive design constant, and k ðvr ; !r Þ and k ðvr ; !r Þ are bounded continuous functions with bounded first derivatives, strictly positive on R  R À ð0; 0Þ Now, if the slipping parameters aR , aL , and  that appear in equation (8) are unknown, we cannot choose directly the auxiliary control input as given by equation (28) Hence, we will employ an AUKF with the LPF to attain the estimation of the slipping parameters If a^R , a^L , and ^ denote the estimatiotions of aR , aL , and , respectively, from equations (10) and (28), we can obtain auxiliary control input as follows ! ! vr cose ỵ k e v ẳ 2 ! !r ỵ ỵ ^ ị k vr e ỵ ỵ ^ Þ k vr sin e (29) where k ẳ k vr ; !r ị and k ẳ k vr ; !r ị are defined in equation (28) Actual control input !L and !R can also be obtained as follows b À ! ! 2ð1 À a^L Þ 16 !L À a^L v (30) ¼ b !R ! r4 À a^R 2ð1 À a^R Þ Remark The estimation values of the slipping parameters are possible to be ones when we use AUKF Once a^L ¼ or a^R ¼ 1, from equation (30), we know that actual control input !L or !R will go to infinity This can not be implemented by controller One way to avoid this is by letting a^L ¼ or a^R ¼ be replaced by a^L À " or a^R À "(" is an arbitrarily small positive number) Control constraints Because of the bounded velocity capability of the motors, each wheel can achieve a maximum angular velocity ! max With the saturation, it is necessary to perform a suitable velocity scaling so as to preserve the curvature radius corresponding to the nominal velocities !L and !R The actual commands !cL and !cL are then computed by defining   j!L j j!R j ; ;1 (31) c ¼ max ! max ! max From equation (27), the actual commands computed as follows !cL and !cL ¼ !L ; !cR ¼ !R ; c ¼1 !L j!R j c c !L ¼ ; !R ¼ ! max sgnð!R Þ; c ¼ ! max c !R ; else !cL ẳ ! max sgn!L ị; !cR ¼ c !cR are (32) Stability analysis of control system Theorem The trajectory tracking errors e ẳ ẵe ; e ; e ŠT in the tracking error dynamics equation (27) of the WMR with unknown slipping parameters will asymptotically converge to zero vector, if the control law equation (29) and AUKF are applied Proof We choose the following Lyapunov function candidate as 1 À cos e V tị ẳ e 21 ỵ e 22 ị ỵ k2 (33) The first-order derivative of the Lyapunov function V ðtÞ can be obtained as e_ sin e V_ tị ẳ e e_ þ e e_ þ k2 (34) The error dynamics equation (27) is substituted into equation (34), V_ ðtÞ can be obtain as follows h     i 2 V_ tị ẳ e !e þ þ ^ vr cos e À þ ^ v þ h   i ð! À !Þ sin e r e À!e þ þ ^ vr sin e þ k2 (35) Substituting equation (29) into equation (35), V_ ðtÞ can be obtained as    2 k3  ỵ ^ vr sin2 e (36) V_ tị ẳ k 1 ỵ ^ e 21 k2 The domain D is defined by D ¼ fe R j À  < e < g, then the Lyapunov function given in equation (33) is positive definite in D ¼ fe R j À  < e < ; e 6¼ 0g with V_ ðtÞ in domain D, so we have   k   2 þ ^ vr sin2 e (37) V_ ðtÞ ẳ k e 21 ỵ ^ k2 As t ẵ0; 1ị, V tị is a nonincreasing function, we can obtain as V ðtÞ V ð0Þ; 8t ! (38) This implies that V ðtÞ is bounded as t ẵ0; 1ị From equaion (33), we know e and e are all bounded as t ẵ0; 1ị Since desired velocity vr and !r are assumed to be bounded, from equation (28), we know the auxiliary control inputs v and ! are bounded From equation (27), we can know e_ ẳ ẵe_ ; e_ ; e_ ŠT is bounded The Lyapunov function V ðtÞ is taken the second-order derivative is given as Cui et al     2 _ Vtị ẳ k_1 e 21 ỵ ^ 2k 1 ỵ ^ e e_ 2k e21 ^^ À  2k ^^_ k_3  vr ỵ ^ sin e À vr  sin e k2 k2 À k3 2k vr ỵ  ịe_ sin e cos e v_r ỵ  Þ sin e À k2 k2 (39) From remark 3, assumption 4, and equation (39), we can know V€ðtÞ is bounded, so V_ ðtÞ is uniformly continuous, Barbalat’s Lemma40 shows that V_ ðtÞ ! as t ! From equation (37), we know e ! and e ! as t ! From equations (27) and (29), we have     2 (40) e_ ¼ Àk vr þ ^ e À k vr þ ^ sin e Thus     2 _ e3 ẳ k v_r ỵ ^ e À 2k vr ^^e À k vr ỵ ^ e_     2 k_3 vr ỵ ^ sin e k v_r ỵ ^ sin e   _ À 2k vr ^^ sin e k vr ỵ ^ e_ cos e (41) _^ Because e_ , e_ , and  are all bounded, the reference linear velocity vr and its derivative are finite value, k_3 is bounded, thus e€3 is bounded, that is e€3 L1 , so e_ is uniformly continuous From Barbalat’s Lemma,40 we know e_ ! as t ! Since e ; e_ ! as t ! 1, from equation (40), we have k vr ỵ ^ ịe ! as t ! If the reference linear velocity vr 6¼ as t ! 1, then e ! as t ! In conclusion, the equilibrium point e ẳ ẵe ; e ; e T ẳ ẵ0; 0; 0T is uniform and asymptotically stable This implies the WMR can converge asymptotically to the reference posture pr ẳ ẵxr ; yr ; r ŠT from the arbitrary initial posture Thus, the control objective is implemented accordingly Adaptive adjustment of control parameters Since the location of the poles determines the damping ratio  of transient response near the reference trajectory, so the transient response of closed-loop robot system can be analyzed from the location of the poles in a closed-loop system The theory of the location of poles shows that the poles are always in the area not exceeding +45 in the lefthalf s-plane That is to say, cos45  is guaranteed It implies the transient response of trajectory tracking always converges without oscillation regardless of gain parameters Therefore, we need not choose carefully the gain parameters to reduce oscillation or overshoot The tracking error vector of the WMR is defined as e ẳ ẵe ; e ; e ŠT , after substituting equation (28) into equation (27), the tracking error differential equations of closedloop robotic system can be obtained as h       i 2 2 k1 1ỵ ^ e1 ỵ !r ỵk2 1ỵ ^ vr e2 ỵk3 1ỵ ^ sine3 e2 e_ h     i   7 ^ vr e2 ỵk3 1ỵ ^ sine3 e1 ỵvr 1ỵ ^ sine3 ! ỵk 1ỵ e_ ¼ r     e_ 2 Àk2 1ỵ ^ vr e2 k3 1ỵ ^ sine3 (42) After linearizing the equation (42) around the equilibrium point ½e ; e ; e T ẳ ẵ0; 0; 0ŠT , we have   2 3 k1 1ỵ ^ !r e_   e1 e_ ẳ 1ỵ ^ vr 74 e2 À!r    5 e e_ 2 k2 1ỵ ^ vr k3 1ỵ ^ ¼ Aq e (43) Remark Note that the mismatch between the linearized model and the nonlinear system grows for values of e far from zero It will be shown that, in practice, after the transient, e remains very close to zero Then, the problem could be present at the first instants, due to the initial condition For that reason, we assume that, in practice, the real robot and the reference virtual robot start close In this case, the linearization is successful Generally, the trial-and-error method is used to determine controller gains If so, not only the accuracy of the controlled robotic system can not be guaranteed but also its adaptation is bad Based on the linearization model, an online computing method of control gains is proposed by the pole placement strategy The controller gains can be determined by comparing the actual and desired characteristic polynomial equations Here, pole placement methodology is employed to decide control gains of robotic tracking controller The desired poles are chosen as s ¼ À2!n and s ; s ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi À!n + j!n À  ( is damping ratio of controlled mobile robot system and !n is the characteristic frequency of controlled WMR) So the desired characteristic polynomial of the closed-loop robotic system takes the following form as:   s ỵ 2!n ị s þ 2!n s þ !n2 (44) where  is desired damping coefficient of closed-loop robotic system, !n is the characteristic frequency, both of them are positive constants, and s is Laplace operator Equation (43), the characteristic polynomial of the system matrix Aq can be obtained as International Journal of Advanced Robotic Systems vr , T r Pole placement k1 , k2 , k3 Reference Trajector xr , yr , T r Posture e1 , e2 , e3 T Auxiliary contro l errors input v, T L Control , T R input aˆ L , aˆ R δˆ T aL , aR , T Mobile xr , yr , T r robot T Adaptive unscented Kalman filter Figure A scheme of the trajectory tracking control principle for the WMR WMR: wheeled mobile robot   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 detsI Aq ị ẳ s þ ðk þ k Þ þ ^ s 2 2 !r ỵ d ỵ ^ vr2 h     i ; k2 ¼ d (49) k1 ¼ k3 ¼ 2 2 ỵ k k ỵ ^ ỵ k ỵ ^ vr2 ỵ !r2 s ỵ ^     ỵ k k ỵ ^ vr2 ỵ k !r2 ỵ ^ Remark In equation (49), the slipping parameter  is (45) estimated by AUKF online (see ‘‘Scheme of the robotic slipping parameter estimation’’ section) The robotic closed-loop poles are now equal to the roots of the characteristic polynomial equation (45) Comparing Remark From assumption and equation (49), we know equation (44) with equation (45), the following equations that ki 0; 1ị; i ẳ 1; 2; 3Þ are realizable in physics can be obtained as Since the dynamic performance of closed-loop system   is determined by the damping ratio , thus we can prede2 k ỵ k ị ỵ ^ ¼ 4!n termine the damping ratio  in accordance with the desired     performance requirements Therefore, controller gains k , 2 2 k k ỵ ^ ỵ k ỵ ^ vr2 ỵ !r2 ẳ 4 !n2 ỵ !n2 k , and k can be determined only by adjusting parameter     d, and then the control parameter adjusting process is k k ỵ ^ vr2 ỵ k !r2 ỵ ^ ẳ 2!n3 (46) simplified greatly Since the reference velocity vr and !r are time varying, then the control gains k and k are also Equation (46), we can obtained as time varying Accordingly, they can be adjusted online in 2 accordance with equation (49) Hence, the real-time prop2!n ! À! k1 ¼ k3 ¼ ; k ¼  n r2 (47) erty and flexibility of the controlled robot system are 2 ỵ ^ ỵ ^ vr2 improved greatly From the above analysis (see ‘‘A scheme of the robotic where !n should be larger than the maximum-allowed slipping parameter estimation,’’ ‘‘Design of the tracking robotic angular velocity, !n > !r max , !r max is the controller,’’ and ‘‘Adaptive adjustment of control paramaximum-allowed robot angular velocity !r max can be meters’’ sections), tracking close-loop control principle obtained as follows of the WMR can be described by the scheme shown in Figure 4r! max (48) !r max ¼ b where ! max is robot’s wheel, which can achieve the maximum angular velocity, r is radius of the wheels, and b is the distance between two driving wheels In equation (47), vr ! and k ! One way to avoid this is by allowing the closed-loop poles only depend on the values of vr and !r Consequently, a gain scheduling should be chosen for rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 ^ !n ẳ !r ỵ d ỵ  vr2 with a positive constant d Then, the control gains can become Simulations and experiment Simulations In this section, to verify the effectiveness of the proposed tracking control scheme, some simulations are performed on the kinematic model of the tracked WMR with slipping In the simulations, the angular velocities of the two driving wheels are considered as control input variables To observe and compare the simulation results more easily, Cui et al two kinds of reference trajectories are chosen as follows: one is a straight-line trajectory, and the other is a circle one The parameters of the WMR are chosen as follows: b ¼ m, r ¼ 0:125 m,  ¼ 0:707 (both to ensure that convergence speed of the tracking errors and ensure that as far as possible little overshoot, the best damping ratio  ¼ 0:707 is chosen), robot’s wheels can achieve a maximum angular velocity !max ¼ 8.5rad/s, k2 ¼ d ¼ 60; the parameters of the AUKF should be determined carefully For the AUKF, the three constant filter parameters are chosen as follows: ¼ 1, ¼ and ¼ 0, L ¼ 100 The parameters of the LPF are chosen as follows: ¼ 0.6, ¼ 0.2, ¼ 0.8 Let us suppose the three states about the robot’s poses, which can be measured directly Note that the kinematics and the dynamics of the robot are described by the continuous-time equations and (26) On the other hand, the AUKF is a discrete-time algorithm Thus, to perform the computer simulation, the continuoustime equations (8) and (26) are discretized using Euler’s forward difference scheme with a sampling period of Ts ¼ 0:02 s adaptive tracking controller has excellent ability to overcome the wheel’s slipping perturbation Meanwhile, we can see that the AUKF can estimate the time-varying slipping parameters rather precisely The estimations of the slipping parameters have a rather light oscillation due to LPF is applied The circle reference trajectory tracking The equation of the circle reference trajectory is given as & xr ¼ 2cost yr ¼ sint where t is simulation time The initial posture of the reference trajectory is set at h iT  ẵxr 0ị; yr 0ị; r 0ị T ẳ m; m; rad The actual initial posture of the WMR is h iT  ẵx0ị; y0ị; 0ịT ẳ m; m; rad The actual initial posture errors of the WMR is The straight-line reference trajectory tracking In this case, a straight-line reference trajectory is considered The initial posture of the reference trajectory is set at h iT  ẵxr 0ị; yr 0ị; r 0ịT ẳ m; m; rad The actual initial posture of the WMR is given as h iT  ẵx0ị; y0ị; 0ịT ẳ m; m; rad The actual initial posture errors of the WMR is h p iT ẵe 0ị; e 0ị; e 0ịT ẳ 2 m; m;0 rad Reference velocities are given as vr ¼ m=s, !r ¼ rad=s, and the reference orientation angle r ¼  rad=4 In order to demonstrate the tracking performance, abrupt changes are simulated to occur in the three slipping parameters at time t ¼ 15 s, they are given as follows: ( 0:15 sin 0:2ðt À 15Þ; t ! 15 s aL ¼ 0; else ( À0:15 sin 0:2ðt 15ị; t ! 15 s aR ẳ 0; else ( 0:12 sin 0:2ðt À 15Þ; t ! 15 s ¼ 0; else From Figure 5, we can see that proposed control method can track the desired straight-line trajectory rather quickly Furthermore, when the wheel’s slipping are introduced into the robot’s system, we can observe that the proposed ½e 0ị; e 0ị; e 0ịT ẳ ẵ3 m; m;0 radŠT To facilitate comparison of the simulation results, the control parameters, the parameters of the AUKF, and the changes of the slipping parameters are all the same as in the previous simulation In this case, the reference line velocity vr tị ẳ rad=s, and the tangential angle velocity of each point on the reference trajectory is given as follow: !r tị ẳ rad=s From Figure 6, we can see that the proposed control approach has the good tracking control performance for the curved path in spite of the effects of the unknown wheels’ slipping That is to say, the proposed control method can effectively conquer the slipping effect for the given curved trajectory tracking of the WMR Meanwhile, the AUKF can estimate three time-varying slipping parameters accurately, even when sudden changes happen Furthermore, from Figures and 6, we can futher find that the proposed control method can effectively overcome the slipping for the given trajectory tracking of the WMR, this is mainly because that the designed tracking controller have an adaptive ability, whose some control parameters k and k can be adaptively modifying in real time Moreover, even if the wheels’ slipping parameters change suddenly, the AUKF can still exactly estimate slipping parameters in real time to satisfy the demands of the robot in the actual working environment Consequently, the adaptive tracking control algorithm has good robustness and adaptive ability to confront slipping parameter perturbations of the WMR Further, by comparing Figure with Figure 6, we can also find that the tracking errors of circle trajectory is bigger than straight-line trajectory, this is mainly because that 10 International Journal of Advanced Robotic Systems 25 Desired trajectory Real trajectory 20 2.5 x error e1 (m ) y (m) 15 10 1.5 0.5 0 10 x (m) 15 20 –0.5 25 10 0.8 0.4 0.7 0.3 0.6 Angle error e (rad) 0.5 0.1 –0.1 –0.2 0.2 0.1 –0.4 –0.1 15 20 25 30 35 –0.2 40 10 15 20 25 30 Time (s) (c) y orientation error e (d) Heading angle error e3 35 40 0.2 Real value Estimation value 0.15 Real value Estimation value 0.15 0.1 Slipping parameter aR Slipping parameter aL Time (s) 0.2 0.05 0.1 0.05 –0.05 –0.05 –0.1 –0.15 –0.2 40 0.3 10 35 0.4 –0.3 30 0.5 y error e2 (m ) 0.2 20 25 Time (s) (b) x orientation error e1 (a) Straightline trajectory tracking result –0.5 15 –0.1 –0.15 10 15 20 25 30 35 40 –0.2 10 15 20 25 30 35 Time (s) Time (s) (e) Slipping parameter aL estimation (f) Slipping parameter aR estimation 40 Figure Straight-line trajectory tracking (a) Straight-line trajectory tracking result; (b) x orientation error e1 ; (c) y orientation error e ; (d) heading angle error e ; (e) slipping parameter aL estimation; (f) slipping parameter aR estimation; (g) slipping parameter  estimation; (h) control input !L ; (i) control input !R ; (j) control gains (k ¼ k ) 11 0.1 (ra d ) 0.15 L 0.05 –0.05 C o n tro l in p u t Slipping parameter δ Cui et al Real value Estimation value –0.1 –0.15 –0.2 5 10 15 20 25 30 35 40 10 15 Time (s) (g) Slipping parameter δ estimation 30 35 40 30 35 40 L 25 20 R (ra d ) 25 (h) Control input 15 k1, k C o n tro l in p u t 20 Time (s) 10 5 10 15 20 25 30 35 40 10 15 20 25 Time (s) Time (s) (i) Control input R (j) Control gains ( k1 = k ) Figure (Continued) circle path have a time-varying reference angular velocity r while reference angular velocity of the straight-line reference path is constant Compared Figure 5(j) with Figure 6(j), we can see that k and k are all constants in straight-line trajectory environment, from equation (47), pffiffiffi we know k ¼ k ¼ 2vr d ¼ 21:9056 However, in circular reference trajectory environment, due to reference angular velocity r is time-varying, control gains k and k must also be time varying to meet better circular reference trajectory Real experiment In order to demonstrate the effectiveness and applicability of the proposed method, a real-time control system is implemented for the mobile robot In the experiment, a mobile robot with one vision navigation system fixed on the top moves along the marking line Figure shows the picture of the robot which is used in the experiment It has the same structure as Figure 8, with two driving wheels and two passive wheels The diameter of the robot is 50 cm and the radius of driving wheel is 12.5 cm The driving wheels are driven by motors with the maximum permissible speed of 3900 n/min The motor and the driving wheel are connected by a reduction gearbox For the convenience of comparing, the control parameters are all same as the simulations The control board of the mobile robot consists of the main controller and motor controller The main controller of the robot is dsPIC30F6014, which is running at 32 MHz It is used to communicate with host computer and motor controller It receives the voltage instruction from the host computer and calculates the voltage distribution on the right and left motors, respectively, and then sends the data through SPI communication to the auxiliary motor controller, dsPIC4012 The motor controller generates pulse-width modulation (PWM) signal with different duty cycles according to the voltage instruction 12 International Journal of Advanced Robotic Systems 2.5 Desired tajectory Real tajectory 1.5 2.5 x error e1 (m ) Y (m) 0.5 0.02 1.5 –0.5 –0.02 –0.04 –1 15 15.2 15.4 15.6 15.8 0.5 –1.5 –2 –2.5 –2.5 –2 –1.5 –1 –0.5 0.5 1.5 2.5 X (m) 10 15 20 25 30 35 40 35 40 35 40 Time (s) (b) x orientation error e1 (a) Circle trajectory tracking result 0.8 0.7 Angle error e (ra d ) 0.6 y error e2 (m ) 0.5 0.4 0.3 0.8 0.6 0.4 0.2 –0.2 –0.4 –0.6 –0.8 –1 0.2 0.1 –0.1 –0.2 10 15 20 25 30 35 40 10 15 Time(s) (c) y orientation error e2 25 30 (d) Heading angle error e3 Real value Estimation value Slipping parameter aR Slipping parameter aL 0.25 0.2 20 Time (s) 0.15 0.1 0.05 –0.05 0.25 0.2 0.15 0.1 0.05 Real value Estimation value –0.05 –0.1 –0.15 –0.2 –0.25 –0.1 –0.15 –0.2 –0.25 10 15 20 25 30 35 40 10 15 20 25 30 Time (s) Time (s) (e) Slipping parameter aL estimation (f) Slipping parameter aR estimation Figure Circle trajectory tracking (a) Circle trajectory tracking result; (b) x orientation error e ; (c) y orientation error e ; (d) heading angle error e ; (e) slipping parameter aL estimation; (f) slipping parameter aR estimation; (g) slipping parameter  estimation; (h) control input !L ; (i) control input !R ; (j) control gains (k1 ¼ k ) Cui et al 13 0.2 Real value Estimation value (ra d ) 0.05 –0.05 –0.1 –0.15 –0.2 L 0.1 C o n tro l in p u t Slipping parameter δ 0.15 10 15 20 25 30 35 40 10 15 (g) Slipping parameter δ estimation 30 35 40 L 21.9512 21.951 21.9508 21.9506 k1, k R (ra d ) 25 (h) Control input C o n tro l in p u t 20 Time (s) Time (s) 21.9504 21.9502 21.95 21.9498 10 15 20 25 30 35 40 21.9496 10 (i) Control input 15 20 25 30 35 40 Time (s) Time (s) (j) Control gains ( k1 = k ) R Figure (Continued) r e [e1 , e2 , e3 ] Reference tracjectory + [ Proposed controller _ L , R ] dsPIC controller PWM x, y , Camera navigation Mobile robot Motor Figure Schematic diagram of the experiment control system Figure Mobile robot in real experiment Figure shows the whole schematic diagram of the trajectory tracking system for the mobile robot Because of the complexity of the calculation process, the proposed adaptive tracking controller based on AUKF is carried out in the main computer running at the frequency of 1.86 MHz 14 International Journal of Advanced Robotic Systems Tracking errors e1 (m ), e2 (m ), e (rad) 0.8 e1 0.6 e2 0.4 e3 0.2 –0.2 –0.4 –0.6 –0.8 10 15 20 25 30 35 40 Time (s) Figure Experimental results for the straight-line tracking errors with initial error (0 m, m, rad) The software for implementing the algorithm is developed in Visual Cỵỵ6.0 After the reference trajectory has been set up, the proposed adaptive tracking controller generates the real voltage instruction The dsPIC controller can generate the PWM signal to control the velocity of the mobile robot so that the mobile robot moves according to the instruction The vision navigation system evaluates the posture of the robot and feedback the information to the host computer until the posture error is minimized In order to validate the applicability of the proposed control scheme, the mobile robot was required to track reference trajectories The real position of the mobile robot is feedback to the mobile robot every 0.2 s by camera navigation system In order to prove the effectiveness of the proposed adaptive tracking controller based on AUKF, real experiment is implemented for the desired trajectory of a straight line Reference velocities are given as vr ¼ m=s and !r ¼ rad=s, The robot started tracking with initial errors e ¼ m, e ¼ m, and e ¼ rad At 25 s, the arbitrary external wheels’ slipping disturbance is fed in robotic system by laying sand on the ground The experimental results are shown in Figure From Figure 9, we can see that the tracking errors of the proposed adaptive control algorithm are asymptotically convergent It implies that the mobile robot eventually approaches the reference trajectory with asymptotic stability within s to 0.32% error bound by the proposed adaptive controller This fact demonstrates the effectiveness of the proposed adaptive control algorithm Conclusions The wheels’ slipping effects appear generally in practice and are really significant problems for the control of the WMR In this article, the wheels’ slipping is taken into account and modeled as three time-varying parameters The kinematic model containing the slipping parameters is proposed and AUKF is designed to estimate unknown slipping The estimated results are introduced into a uniformly asymptotically stable tracking controller to implement the path tracking objective Furthermore, based on pole placement strategy, the controller parameters are adjusted online in real time in accordance with the practial reference trajectory The simulation and real experiment results are given to validate the performance of the proposed tracking control approach More works should be done to research the physical mechanism of slipping and the dynamic tracking controller design will be investigated in future Acknowledgement Authors Mingyue Cui, Hongzhao Liu, Wei Liu and Rongjie Huang are also affiliated with Oil Equipment Intelligent Control Engineering Laboratory of Henan province, Nanyang, Henan, 473061, China Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation (no.U1404614), the Henan Province Education Department Foundation (no.14B120003 and no.17A413002), and the Henan Province Scientific and Technological Foundation (no 142300410455) References Yang JM and Kim JH Sliding mode control of trajectory tracking of nonholonomic wheeled mobile robots IEEE Trans Robot Autom 1999; 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