Stable control of networked robot subject to communication delay, packet loss, and out-Of-order delivery

Volume 36 Number 3 3 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 3 (2014), pp. 215 – 233 STABLE CONTROL OF NETWORKED ROBOT SUBJECT TO COMMUNICATION DELAY, PACKET LOSS, AND OUT-OF-ORDER DELIVERY Manh Duong Phung∗, Thuan Hoang Tran, Quang Vinh Tran University of Engineering and Technology, VNU, Hanoi, Vietnam ∗E-mail: duongpm@vnu.edu.vn Received July 02, 2013 Abstract. Stabilization control of networked robot system faces uncertain factors caused by the network. Our approach for this problem consists of two steps. First, the Lyapunov stability theory is employed to derive control laws that stabilize the non-networked robot system. Those control laws are then extended to the networked robot system by imple- menting a predictive filter between the sensor and controller. The filter compensates influences of the network to acquire accurate estimate of the system state and conse- quently ensures the convergence of the control laws. The optimality of the filter in term of minimizing the mean square error is theoretically proven. Many simulations and exper- iments have been conducted. The result confirmed the validity of the proposed approach. Keywords: Networked robot, stabilization control, optimal filter. 1. INTRODUCTION The stable movement from one point to another is essential for the efficient operation of a control system and is basic for the development of real-world applications. In non- networked robot system, a number of researches have been introduced and the problem of stabilization control has been solved in both theoretic and experimental aspects [1–3]. Networked robot systems (NRSs) however have differences. The occurrence of network delay, packet loss and out-of-order delivery influences the accuracy of state estimation and control so that directly applying previous control methods is no longer practical. Several new approaches have been proposed. Wargui et al. developed a stable controller for NRSs with nonholonomic constraints [4]. Control laws that stabilize the system in delay-free scenario are first derived. They are then extended to work with the time delay by adding a state estimator between the system output and the controller. The estimator uses a multistep prediction mechanism to predict the state that will be used for the controller. In [5], the maximum time delay allowed at the control input without the loss of system stability is estimated. Based on it, a single layered neural network is designed for the controller to control the robot with unknown dynamics. In [6], Luck uses a time buffer which was longer than the worst delay 216 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran to make the system to be time-invariant and then applied the classic control theory. In [7], Xi and Tarn propose an event-based (non-time) control scheme to reduce the impact of time delay on the system operation. This idea is expanded by Wang and Liu [8] in which they introduced the telecommanding concept. In telecommanding, each command is designed for an independent task and is defined with multiple events so that the robot can deliberately respond to expected events while actively respond to unexpected events. The main limitation of the proposed works is that they mainly focus on dealing with the time delay, hardly address other issues such as the out-of-order delivery. In this study, we address the problem of stabilization of NRS under the influence of three inevitable network induced problems: the communication delay, packet loss, and out-of-order delivery. The core of the scheme is the development of a new filter that is able to predict the robot’s pose based on past observations so that control laws designed for non-networked robot system can be used to stabilize the NRS. The filter is optimal in sense that it minimizes the mean of the squared error. In addition, expansion of the filter to nonlinear systems is also derived. The paper is organized as follows. Section 2 introduces the system model and formulates the control problem. Section 3 describes the stabilization of non-networked robot system. The stabilization of NRS is introduced in section 4. Section 5 presents the simulations, experiments, and discussions. Conclusions are drawn in section 4. 2. SYSTEM MODEL AND PROBLEM FORMULATION 2.1. System Model The robot considered in this study is the two-wheeled, differential-drive mobile robot with non-slipping and pure rolling. The pose or state of the robot includes the position of the wheel axis center (x, y) and the chassis orientation θ with respect to the x axis (Fig. 1). XG X R YR YG (t) v(t) R L 0 Fig. 1. The robot’s pose and parameters Actuator Sensors Controller u ca u k+n+1 z sc z Network k k+n k+n+m+1 Process x k+n+1 k k+n+1 (Delay, Loss, Out of Order) Fig. 2. Model of a networked robot system The kinematic models of the robot in continuous and discrete time domains in case of no noise affected are given by x˙ = v cos θ, y˙ = v sin θ, θ˙ = ω, (1) xk+1 = xk + Tsvk cos θk, yk+1 = yk + Tsvk sin θk, θk+1 = θk + Tsωk, (2) Stable control of networked robot subject to communication delay, packet loss, . . . 217 where Ts is the sampling period and v and ω are the tangent and angular velocities of the robot, respectively. In practice, there always exist two kinds of noises in a robotic system including the input and measurement noises. In this study, these noises are assumed to be independent, zero-mean, and white-noise processes with normal probability distributions: wk ∼ N(0, Qk), vk ∼ N(0, Rk), E(wivTj ) = 0. This assumption is sufficient and widely adopted due to the central limit theorem [9–12]. With the existence of noises, the robot system can be described in state-space representation as follows. Let x = [x y θ]T be the state vector. This state can be observed by a measurement z. This measurement is described by a function h of the state vector and measurement noise v. Denoting function (2) as f , with an input vector u = [v ω] and input disturbance w, the representation of the robot in state space is given by xk+1 = f(xk,uk,wk), zk = h(xk,vk). (3) When distributing over communication networks, the robot becomes decentralized and its functioning operation depends on a number of network parameters. The networks are in general very complex and can greatly differ in their architecture and implementation depending on the medium used, and on the applications they are meant to serve. In this work, a network is modeled as a module between the process and controller which delivers input signals and observation measurements with possible delay, loss, and out-of-order (Fig. 2). The delay is assumed to be random but measurable at each sampling time. The packet loss is modeled as a binary random variable λk defined as follows λk = { 1, if a packet arrives at time k 0, otherwise (4) The out-of-order packet with sequence number i arrived at time k(i < k) is modeled as a delayed packet with the time delay ∆ti = ∆tk + (j − i)Ts, (5) where ∆tk is the transfer time at time k and j is the sequence number of the last received chronological packet. Let n be the network delay (in number of sampling periods) between the controller and the actuator,m be the network delay between the sensor and the controller, λcak be the binary random variable described the arrival of inputs from the controller to the actuator, λsck be the binary random variable described the arrival of measurements from the sensor to the controller. The system (3) becomes time-varying and can be rewritten as follows xk = f(xk−1, λcak−n−1uk−n−1,wk−1), z˜k = λsck−mzk−m = λ sc k−mh(xk−m,vk−m). (6) 2.2. Problem Formulation Consider the control scenario shown in Fig. 3, with an arbitrary position and ori- entation of the robot and a predefined goal position and orientation. Let the difference between the present pose (x, y, θ) and the final pose (x2, y2, θ2) given in the robot refer- ence frame {XR, YR, θR} be the error vector, e = (x2 − x, y2 − y, θ2 − θ)T . The task of 218 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran the controller layout is to find a control constraint, if it exists, of the tangent and angular velocities such that the error e is driven toward zero: lim t→∞ e(t) = 0. XG XR YR YG 0 start goal v(t)(t) Fig. 3. The goal of the controller O X Y Y 2 X2 O2 2 OG XG YG Fig. 4. The robot poses and navigation variables Our approach for this problem consists of two steps. First, control laws that stabilize the non-networked robot system are derived. Then, a predictive filter is introduced to extend those control laws to the NRS. 3. STABILIZATION OF NON-NETWORKED ROBOT SYSTEM In (1), it can be seen that one degree of freedom of the system is constrained as x˙ sin θ − y˙ cos θ = 0. (7) Thus, the system is nonholonomic. According to a work of Brockett [13], Cartesian state-space representations of the robot cannot be asymptotically stabilized via smooth and time invariant feedback laws. We therefore define a new coordinate system as shown in Fig. 4. The new coordinate system consists of three parameters (ρ, α, φ) called navigation variables. Let OXY and O2X2Y2 be the local coordinate frames attached to the present and final poses of the robot, respectively, ρ is then defined as the distance between O and O2, φ is the angular made by the vector connecting O and O2 and the vector connecting O2 and x2, and α is the angular made by the vector connecting O and O2 and the vector connecting O and x ρ = √ (x2 − x)2 + (y2 − y)2, ϕ = a tan 2 (y2 − y, x2 − x)− θ2, α = a tan 2 (y2 − y, y2 − x)− θ. (8) Without lost of generality, we assume that the goal desired pose of the robot is (x2, y2, θ2) = (0, 0, 0) which can also be expressed by (ρ2, α2, ϕ2) = (0, 0, 0) (This can be done by setting the origin of the global coordinate frame at the final position). Let wv and wω be the input disturbances of the tangent and angular velocities, respectively. With the existing of input disturbances, the kinematic model (1) is rewritten in navigation variable Stable control of networked robot subject to communication delay, packet loss, . . . 219 domain (ρ, α, φ) as follows ρ˙ = −(v + wv) cosα, α˙ = −(ω + wω) + (v + wv) sinαρ , ϕ˙ = (v + wω) sinαρ . (9) The goal of the controller now is to establish smooth control laws that drive the navigation variables (ρ, α, φ) toward zero. By using Lyapunov theory, it is shown that (9) can be asymptotically stabilized using the following control laws v = (γ cosα)e, ω = λα+ γ cosα sinαα (α+ hθ). (10) where γ, λ and h are the positive parameters [1, 14]. In discrete time domain, the control laws are given by vk = (γ cosαk)ek, ωk = λαk + γ cosαk sinαkαk (αk + hθk). (11) Those control laws combined with a predictive filter are capable to stabilize the NRS as presented in the next section. 4. STABILIZATION OF NETWORKED ROBOT SYSTEM As shown in (6), the NRS is time-varying in which the control input at time k would not reach the actuator until time k+ n while the measurement at time k actually reflects the system state at time k −m. Thus, in order to ensure the stabilization of control laws (11) for NRS, the system state at time k + n need be estimated based on measurements taken at time k−m, xˆ(k+ n|k−m) (Fig. 5). This estimate will be used at the controller to generate the control signal uk+n so that it will arrive to the actuator at time k + n as expected. This approach is similar to [4] but we have developed a new state estimator which is able to handle not only the network delay but also the packet loss and out-of-order delivery. The filter called past observation-based predictive filter (PO-PF) is derived based on the Kalman filter’s theory [9]. Actuator Sensors Controller State Estimator uk+n k-m zk-m Network k k+n k k Process x k-m xk+n ^ uk+n zk-m Fig. 5. NRS with the presence of a state estimator 220 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran In this section, the standard Kalman filter is first briefly described. The derivation of the PO-PF for linear systems is then introduced. Finally, the expansion of the PO-PF for nonlinear systems is presented. 4.1. The standard Kalman filter Consider the following discrete time linear stochastic system xk = Ak−1xk−1 +Bk−1uk−1 +wk−1, zk = Hkxk + vk, (12) where k ∈ N, x and w ∈ Rn, z and v ∈ Rm, u ∈ Rl, A ∈ Rn×n, B ∈ Rn× l, H ∈ Rm×n, (x0,w,v) are Gaussian, uncorrelated, white, with mean (x¯, 0, 0) and covariance (P0, Q,R) respectively. The steps to calculate the Kalman filter can be summarized as follows [9]: The time update equations (prediction phase) xˆ−k = Ak−1xˆ + k−1 +Bk−1uk−1, (13) P−k = Ak−1P + k−1A T k−1 +Qk−1, (14) where xˆ−k ∈ Rn is the priori state estimate at step k given knowledge of the process prior to step k, and P−k denotes the covariance matrix of the priori estimate error. The data update equations (correction phase) Kk = P−k H T k [HkP − k H T k +Rk] −1, (15) xˆ+k = xˆ − k +Kk[zk −Hkxˆ−k ], (16) P+k = [I −KkHk]P−k , (17) where xˆ+k ∈ Rn is the posteriori state estimate at step k given measurement zk, Kk is the Kalman gain, and P+k is the covariance matrix of the posteriori estimate error. 4.2. The past observation-based predictive filter for linear NRSs Consider the NRS described in (6). If functions f and h are linear, the system is rewritten as xk = Ak−1xk−1 + λcak−n−1Bk−1uk−n−1 +wk−1 = Ak−1xk−1 +Bk−1u˜k−n−1 +wk−1, (18) z˜ik = λ sc k−mzk−m = λsck−mHk−mxk−m + λ sc k−mvk−m = H˜ixi + v˜i, (19) where u˜k−n−1, z˜ik, H˜k−m, v˜k−m, and i are defined by the above equations. We can derive the optimal filter to estimate the state of the stochastic linear system given in (18) and (19) as follows. Stable control of networked robot subject to communication delay, packet loss, . . . 221 Priori State Estimate at prediction phase The priori estimate, xˆ−k , is defined as the expectation of the state xk given all measurements up to and including the last step k − 1. From (18), it is given by xˆ−k = E(xk) = Ak−1E(xk−1) +Bk−1E(u˜k−n−1) + E(wk−1). (20) As E(xk−1) is the posteriori state estimate at time k− 1, u˜k−n−1 is a known input, and wk−1 is zero-mean, (20) becomes xˆ−k = Ak−1xˆ + k−1 +Bk−1u˜k−n−1. (21) At time k, the controller sends the input signal u˜k+n, (21) becomes xˆ−k = Ak−1xˆ + k−1 +Bk−1u˜k−1. (22) Priori Error Covariance Let e−k and e + k be the priori and posteriori estimate errors, respectively e−k = xk − xˆ−k , (23) e+k = xk − xˆ+k . (24) From (18) and (22), the covariance of the priori estimate error is given by P−k = E(e − k e −T k ) = E(Ak−1e+k−1e +T k−1A T k−1 +wk−1w T k−1 +Ak−1e+k−1w T k−1 +wk−1e +T k−1A T k−1) = Ak−1P+k−1A T k−1 +Qk−1. (25) Posteriori State Estimate at correction phase From (19) the measurement z˜ik received at time k would update the system state at a previous time i rather than the present time k. Due to the network, this measurement could not reach the estimator until time k. We therefore construct the data update equation of the form xˆ+k = xˆ − k +Kk(z˜ i k − H˜ixˆ−i ), (26) and recompute the Kalman gain and error covariance to ensure the optimality of the new data update equation. Kalman gain and Posteriori Error Covariance Assume that the measurement is fused using (26) with an arbitrary gain Kk. The covariance of the posteriori estimate error, P+k , is determined as P+k = E(e + k e +T k ) = E[e−k e −T k − e−k e−Ti (KkH˜i)T − e−k v˜Ti KTk −KkH˜ie−i e−Tk +KkH˜ie−i e −T i (KkH˜i) T +KkH˜ie−i v˜ T i K T k −Kkv˜ie−Tk +Kkv˜ie−Ti (KkH˜i) T +Kkv˜iv˜Ti K T k ]. (27) 222 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran Due to the independence between e− and v˜, (27) can be simplified to P+k = E(e − k e −T k )− E(e−k e−Ti )(KkH˜i)T −KkH˜iE(e−i e−Tk ) +KkH˜iE(e−i e −T i )(KkH˜i) T +KkE(v˜iv˜Ti )K T k ] = P−k − LT H˜Ti KTk −KkH˜iL+KkH˜iP−i H˜Ti KTk +KkR˜iKTk , (28) where L = E(e−i e −T k ). As the matrix Kk is chosen to be the gain or blending factor that minimizes the posteriori error covariance, this minimization is accomplished by taking the derivative of the trace of the posteriori error covariance with respect to Kk, setting that result equal to zero, and then solving for Kk. Applying this process to (28) obtains ∂tr(P+k ) ∂Kk = 2(−LT H˜Ti +KkH˜iPiH˜Ti +KkR˜i) = 0, (29) ⇔ Kk = LT H˜Ti [H˜iP−i H˜Ti + R˜i]−1. (30) Inserting (30) in (28) leads to a simpler form of P+k P+k = P − k −KkH˜iL. (31) In order to compute L, the priori state estimate at time k needs determining from the estimate at time i. Through the time update (21) and the data update (26), e− becomes e−k = xk − xˆ−k = Ak−1e+k−1 −wk−1 = Ak−1[(I −Kk−1H˜k−1)e−k−1 +Kk−1v˜k−1]−wk−1. (32) After m updating steps, the estimation error becomes e−k = Fe − i + ξ1(wi, . . . ,wk−1) + ξ2(v˜i, . . . , v˜k−1), (33) where F = m∏ j=1 Ak−j(I −Kk−jH˜k−j), (34) and ξ1 and ξ2 are the functions of noise sequencesw and v˜. From (33) and the independence between e− and noise sequences, the covariance L becomes L = E(e−i e −T k ) = P − i F T . (35) Substituting (35) in (31) and (30) yields P+k = P − k −KkH˜iP−i F T , (36) and Kk = FP−i H˜ T i [H˜iP − i H˜ T i + R˜i] −1. (37) Stable control of networked robot subject to communication delay, packet loss, . . . 223 Predictive state estimate at extrapolated phase The extrapolated phase is added to predict the system state at time k+n from time k. As there is no additional measurement, this phase is an open-loop propagation xˆ−k+n = Ak+n−1xˆ + k+n−1 +Bk+n−1u˜k+n−1. (38) • Remark 1: When the delay n and m is zero, F = I and the Kalman gain (37) reduces to the standard form (15) and the error covariance (36) reduces to (17) • Remark 2: Eq. (37) can be rewritten as Kk = FK∗i , (39) where K∗i is the Kalman gain at time i of the standard Kalman filter (15). It turns out that the past residual (z˜−k H˜ixˆ − i ) in Eq. (26) can be normally updated to the posteriori estimate at time k as at time i but the Kalman gain needs to be changed by the factor F . This factor describes the relevant of the measurement updated at time i to the state at time k. • Remark 3: Eq. (26) can be rewritten as xˆ+k = xˆ − k + λiKk(z i k −Hixˆ−i ). (40) It implies that if a measurement is not arrived (λi = 0), the estimator does not use any information of the “dummy” observation to the estimate. It simply sets the posteriori estimate to the value of the priori estimate. 4.3. The past observation-based predictive filter for nonlinear NRSs Though the filter derived in previous section is capable for NRSs, the system has to be linear. As practical robot systems are often nonlinear, further modification needs to be accomplished. The main idea is the linearization of a nonlinear system around its previous estimated states. Performing a Taylor series expansion of the state equation around (xˆ+k−1, u˜k−1, 0) gives xk = f(xˆ+k−1, u˜k−1, 0) + ∂f ∂x ∣∣∣ (xˆ+k−1,uk−1,0) (xk−1 − xˆ+k−1) + ∂f∂w ∣∣∣ (xˆ+k−1,uk−1,0) wk−1 = f(xˆ+k−1, u˜k−1, 0) +Ak−1(xk−1 − xˆ+k−1) +Wk−1wk−1 = Ak−1xk−1 + [f(xˆ+k−1, u˜k−1, 0)−Ak−1xˆ+k−1] +Wk−1wk−1 = Ak−1xk−1 + u˜∗k−1 +w ∗ k−1, (41) where Ak−1, Wk−1, u˜∗k−1, w˜ ∗ k−1 are defined by the above equation. Similarly, the measure- ment equation is linearized around (xˆ−i , 0) to obtain z˜ik = λi[h(xˆ − i , 0) + ∂h ∂x ∣∣ (xˆ−i ,0) (xi − xˆ−i ) + ∂h∂v ∣∣ (xˆ−i ,0) vi] = h˜(xˆ−i , 0) + H˜i(xi − xˆ−i ) + V˜ivi = H˜ixi + [h˜(xˆ−i , 0)− H˜ixˆ−i ] + V˜ivi = H˜ixi + ε∗i + v˜ ∗ i , (42) 224 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran where h˜i, H˜i, V˜i , ε∗i , v˜ ∗ i are defined by the above equation. The system (41) and the measurement (42) are now linear and the proposed filter can be applied to obtain the optimal filter for nonlinear NRSs as follows: • The time update equations at prediction phase xˆ−k = fk−1(xˆ + k−1, u˜k−1,0), P−k = Ak−1P + k−1A T k−1 +Wk−1Qk−1W T k−1. (43) • The data update equations at correction phase F = m∏ j=1 Ak−j(I −Kk−jH˜k−j), Kk = FP−i H˜ T i (H˜iP − i H˜ T i + V˜iR˜iV˜ T i ) −1, xˆ+k = xˆ − k +Kk[z˜ i k − h˜(xˆ−i ,0)], P+k = P − k −KkH˜iP−i F T . (44) • The predictive equation at extrapolated phase xˆ−k+n = fk+n−1(xˆ − k+n−1, u˜k+n−1,0). (45) 5. SIMULATIONS AND EXPERIMENTS 5.1. Simulations In order to evaluate the validity of the proposed control algorithm, simulations were conducted in MATLAB [14]. Parameters for simulations are extracted from the real NRS described in next section. The process noise is modeled as being proportional to the angular speed ωL(k) and ωR(k) of the left and right wheels, respectively. The variances equal to δω2L and δω 2 R, where δ is a constant determined by experiments as follows: the deviations between the true robot position and the position estimated by the kinematic model when driving the robot straight forwards several times (from the minimum to the maximum tangential speed of the robot) is determined. The deviations between the true robot orientation and the orientation estimated by the kinematic model when rotating the robot around its own axis several times (from the minimum to the maximum angular speed of the robot) is also determined. Based on those errors in position and orientation, the parameter δ is calculated. In our system, the δ is estimated to be 0.01. The input-noise covariance matrix Qk is then given as Qk = [ δω2R(k) 0 0 δω2L(k) ] . (46) The measurement is accomplished by two kinds of sensor: the optical encoder for the position (x, y) and the compass sensor for the orientation θ. The determination of the measurement-noise covariance matrix Rk is carried out similarly to the determination of the constant δ in which the measurement values are compared with the true values under different configurations in many times to estimate the expectation and variance of the Stable control of networked robot subject to communication delay, packet loss, . . . 225 noise. In our system, the measurement-noise covariance matrix is identified as Rk =  0.01 0 00 0.01 0 0 0 0.018  . (47) Remaining parameters are retrieved from the kinematic model of the robot (2) as follows Ak+1 = ∂fk ∂x ∣∣∣∣ (xˆ+k ,uk,0) =  1 0 −Tsvk sin θˆ+k0 1 Tsvk cos θˆ+k 0 0 1  , (48) Wk+1 = ∂fk ∂w ∣∣∣∣ (xˆ+k ,uk,0) = Ts R 2  cos θˆ+k cos θˆ+ksin θˆ+k sin θˆ+k 2 L 2 L  , (49) Hk = Vk = I. (50) Parameters for the controller are chosen as follows: λ = 6, h = 1, γ = 3. The first simulation evaluates the performance of the predictive filter PO-PF by comparing it with two popular localization algorithms including the extended Kalman filter (EKF) [9] and the optimal filter proposed in [15]. The comparison does not include the predictive phase as both the EKF and LEKF do not have this phase. The EKF is implemented with the assumption that it does not know if a measurement is delayed or not. It incorporates every measurement it receives as there is no delay. On the other hand, the filter proposed in [15] is specifically designed for the system subject to random delay and packet drop. It is called Lucas-Extended Kalman filter (LEKF) in this study. The LEKF uses an infinite buffer to store and rearrange the delayed and out-of-order measurements. Loss measurements are stored as zero values. At each estimation step, the filter iteratively executes the Kalman equations from the initial to the being estimated state. This method was proven to be optimal. Fig. 6. The EKF, LEKF, PO-PF, and true trajectories in the motion plane Fig. 7. RMSE of the EKF, LEKF, PO-PF esti- mation and the true trajectories in X direction 226 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran Fig. 8. RMSE between the EKF, LEKF, PO-PF and the true trajectories in Y direction Fig. 9. RMSE between the EKF, LEKF, PO-PF and the true trajectories in orientation In the simulation, the network parameters are set as an extreme scenario as follows: the time delay is between 800ms and 1500ms, the out-of-order rate is 15%, and the loss rate is 10%. Figure show the estimate and true trajectories in the motion plane. The overlap between the trajectories implies the good estimation. In order to realize the difference between algorithms, the root mean square error (RMSE) was calculated. Figs. 6-9 give the comparative curves in X, Y and θ directions simulated by 100-time Monte Carlo tests. We see that EKF has the largest error while the PO-PF and LEKF introduce equivalent small error. In addition to the accuracy, the computational burden is also considered. Standard MATLAB functions such as tic toc and flops [16] were used to calculate the amount of floating point operations and execution time of the filters. Tab. 1 shows the result scaled with respect to the EKF. It shows that the PO-PF is around two times higher than the EKF, but many times smaller than the LEKF. Table 1. Normalized computational burden of filters Parameter EKF LEKF PO-EKF Floating point operations 1.0 36.5 4.7 Execution time 1.0 33.7 2.4 The next simulation verifies the asymptotic stabilization of the control laws (11) for non-networked robot system. Figs. 10-11 present the results with the initial pose (-4, -3, 900) and the final pose (0, 0, 00). The (x, y) coordinate and θ orientation go toward the final state (0, 0, 00) while the tangent and angular velocities also go toward zeros. Thus, the system is asymptotically stable. The third simulation applies the same stable control laws to the NRS. Simulation results with 500ms network delay, 5% out-of-order, and 1% message loss are shown in Figs. 12-13. The robot’s pose (x, y, θ) cannot reach (0, 0, 00) and the angular speed does not go to zero as expected. The system is therefore not asymptotically stable. Stable control of networked robot subject to communication delay, packet loss, . . . 227 -4 -3 -2 -1 0 -3 -2.5 -2 -1.5 -1 -0.5 0 X (m) Y (m ) Y (m ) (a) 0 5 10 15 20 25 0 20 40 60 80 100 Time (s) T h e t a (d e g r e e ) T h e t a (d e g r e e ) (b) Fig. 10. Stabilization control of the non-networked robot: (a) Trajectory of the robot in the motion plane; (b) Variation of the direction of the robot 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) u ( m /s ) u ( m /s ) (a) 0 5 10 15 20 25 -0.5 -0.4 -0.3 -0.2 -0.1 0 Time (s) w (r a d / s ) w (r a d / s ) (b) Fig. 11. Robot’s velocities during the stabilization control of non-networked robot system: (a) Variation of the tangent velocity; (b) Variation of the angular velocity -4 -3 -2 -1 0 1 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 X (m) Y ( m ) Y ( m ) (a) 0 5 10 15 20 25 -150 -100 -50 0 50 100 Time (s) T h e ta (d e g re e ) T h e ta (d e g re e ) (b) Fig. 12. Control of the NRS without using the PO-PF: (a) Trajectory of the robot in the motion plane; (b) Variation of the direction of the robot Figs. 14-15 present the control results with the use of the PO-PF. It is able to see that the robot’s pose and velocities go to zero again. The system is asymptotically stable. It confirms the success of the predictive filter in ensuring the stability of the NRS. 228 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Time (s) u ( m /s ) u ( m /s ) (a) 0 5 10 15 20 25 -0.5 -0.4 -0.3 -0.2 -0.1 0 Time (s) w (r a d / s ) w (r a d / s ) (b) Fig. 13. Robot’s velocities during the control of NRS without using the PO-PF: (a) Variation of the tangent velocity; (b) Variation of the angular velocity 0 5 10 15 20 25 0 20 40 60 80 100 Time (s) T h e ta (d e g re e ) T h e ta (d e g re e ) (b) 0 5 10 15 20 25 0 20 40 60 80 100 Time (s) T h e ta (d e g re e ) T h e ta (d e g re e ) Fig. 14. Control of NRS with the use of the PO-PF: (a) Trajectory of the robot in the motion plane; (b) Variation of the direction of the robot 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Time (s) u ( m /s ) u ( m /s ) (a) 0 5 10 15 20 25 -0.4 -0.3 -0.2 -0.1 0 0.1 Time (s) w (r a d / s ) w (r a d / s ) (b) Fig. 15. Robot’s velocities during the control of NRS with the use of the PO-PF: (a) Tangent velocity; (b) Angular velocity 5.2. Experiments Experiments were conducted in a real NRS. The robot is a Multi-Sensor Smart Robot (MSSR) developed at our laboratory. It contains basic components for motion control, sensing, navigation (Figs. 16-17). The communication environment is the Internet. The remote controller is written in Visual C++ and implemented in a laptop computer. More details of the system can be referred from our previous work [17]. Stable control of networked robot subject to communication delay, packet loss, . . . 229 Fig. 16. Communication configuration of the networked robot system Fig. 17. The hardware configuration of the robot -4 -3 -2 -1 0 -4 -3 -2 -1 0 X (m) Y (m ) Y (m ) 0 degree 45 degree 90 degree (a) 0 20 40 60 80 0 20 40 60 80 100 Time (s) O ri e n t a ti o n (d e g re e ) O ri e n t a ti o n (d e g re e ) 0 degree 45 degree 90 degree (b) Fig. 18. Stable control of NRS with three different initial poses (−4,−4, 00), (−4,−4, 450), and (−4,−4, 900): (a) Trajectories of the robot in the motion plane; (b) Variation of the direction of the robot 0 20 40 60 0 0.1 0.2 0.3 Time (s) T a n g e n t v e lo c it y (m /s ) T a n g e n t v e lo c it y (m /s ) 0 degree 45 degree 90 degree (a) 0 20 40 60 -0.2 0 0.2 0.4 0.6 0.8 Time (s) A n g u la r v e lo c it y (r a d /s ) A n g u la r v e lo c it y (r a d /s ) 0 degree 45 degree 90 degree (b) Fig. 19. Robot’s velocities during stable control experiments with three different initial poses (-4,-4,00), (-4,-4,450), and (-4,-4,900): (a) Tangent velocity; (b) Angular velocity In experiments, the goal is to navigate the mobile robot from starting points (−4,−4, 0), (−4,−4, 45), and (−4,−4, 90) to reach the destination (0, 0, 00). The param- eters of the controller were chosen as: λ = 200, h = 3, γ = 100. The network parameters 230 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran were measured as follows: the time delay was between 300ms and 500ms; the out-of-order rate was 2.4%; and the loss rate was 1.3%. Fig. 18 describes the trajectories of the robot. Fig. 19 shows the tangent and angular velocities of the robot during the operation. They proved the success of robot in reaching the destinations and the convergence to zero of velocities at those positions. In another word, the system is asymptotically stable. 5.3. Discussion We have carried out many experiments with different initial positions, network con- figurations, and control parameters (λ, h, γ). At all time, the robot succeeded in reaching the destinations. However, we also recognize that the control parameters (λ, h, γ) play an important role in determining the way the robot reaches the destination. For example, the high values of λ and γ speed up the goal reaching process but introduce more fluctuation to the trajectory while the high value of h implies faster change in robot’s direction. Fig. 20 compares trajectories of the robot during the stable control from point (−4,−4, 0) to point (0, 0, 0) with different control parameters. It is recognizable that the configuration with λ = 200, h = 3, γ = 100 gives the best result in term of traveling distance and time response. It suggests that tuning these parameters need be carefully taken into ac- count when implementing practical systems as they can significantly enhance the control performance. -5 -4 -3 -2 -1 0 -4 -3 -2 -1 0 1 Time (s) A n g u la r v e lo c it y (r a d /s )  =50, h=15, =50  =100, h=8, =100  =200, h=3, =100 A n g u la r v e lo c it y (r a d /s ) Fig. 20. Trajectories of the robot during the stable control with different control parameters We have also evaluated the performance of the PO-PF with non-Gaussian and non- zero mean noises. Fig. 21 shows the estimation result with the uniform distribution noise. It shows that the accuracy is similar to the case with Gaussian noise (Figs. 6-9). Fig. 22 shows the estimation result with the noise having the mean of 20 cm. In this case, the accuracy is reduced comparing to the estimate with zero-mean noise (Figs. 6-9). It is concluded that the PO-PF still works with different types of noise but the accuracy may be slightly affected. The next discussion is the acceptable error of the system at the destination position. More specifically, we stated from the problem formulation that the goal of the controller is to drive the error e between the present and final poses toward zero as the time goes Stable control of networked robot subject to communication delay, packet loss, . . . 231 (a) (b) Fig. 21. Estimate by the PO-PF with uniform distribution noise: (a) RMSE in X and Y directions; (b) RMSE in orientation (a) (b) Fig. 22. Estimate by the PO-PF with non-zero mean noise: (a) RMSE in X and Y directions; (b) RMSE in orientation (a) Fig. 23. RMSE of the predictive filter with the time-out value set to 5 seconds: (a) RMSE in X and Y directions; (b) RMSE in orientation infinite: lim t→∞ e(t) = 0. It means that it may take a very long time for the robot to exactly reach the destination. In practice, this condition can be relaxed as each control task often tolerates certain error. For example, the time for our robot to reach the destination (0, 0, 0) from point (−4,−4, 0) with the error of 1 cm is 75 seconds. If the acceptable error is 5 cm, 232 Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran the goal reaching time reduces to 15 seconds. In conclusion, there is a trade off between the time response and the accuracy depending on control requirements. Finally, it is noted that if the estimation is 100% accurate, the robot’s pose will converge to zero as proven in section 3. In practice, error in estimate is unavoidable. If the error is sufficiently small, it can be considered as the measurement noise and the system is still asymptotically stable. However, as the time delay grows, the error is accumulated and in the worst case it can break the system stability. To avoid this, a time-out value is set. It is the maximum delay at which the estimate error still meets the system requirement. The requirement comes from the acceptable error discussed above. In our system, the time-out value is set to 5 seconds. This value ensures that the estimation error is within 30 cm which is the range required for the functioning operation of the system (Fig. 23). It is also noted that a delay longer than 5 seconds over the Internet would indicate a network congestion or interruption which stops the NRS from functioning operation. 6. CONCLUSION This paper addresses the stabilization control problem of NRS subject to commu- nication delay, loss, and out-of-order. The main contribution is the derivation of a new state estimation filter namely past observation-based predictive filter (PO-PF). This filter enables the prediction of system state from past measurements. When combined with sta- ble control laws of non-networked robot, it ensures the asymptotic stability of the NRS. Simulations and experiments have been conducted. 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Dao Van Dung, Vu Hoai Nam, An analytical approach to analyze nonlin- ear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part 1: Governing equations establishment. 201 5. Manh Duong Phung, Thuan Hoang Tran, Quang Vinh Tran, Stable control of networked robot subject to communication delay, packet loss, and out-of- order delivery. 215 6. Phan Anh Tuan, Vu Duy Quang, Estimation of car air resistance by CFD method. 235