Internet-Based Teleoperation of Carts Considering Effects of Time Delay via Continuous Pole Placement

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1 Aerican Journal of Engineering and Applied Sciences Original Research Paper Internet-Based Teleoperation of Carts Considering Effects of Tie Delay via Continuous Pole Placeent Theophilus Okore-Hanson and Sun Yi Departent of Mechanical Engineering, North Carolina A & T State University, Greensboro, USA Article history Received: Revised: Accepted: Corresponding Author: Sun Yi Departent of Mechanical Engineering, North Carolina A & T State University, Greensboro, USA Eail: syi@ncat.edu Abstract: A new control design for teleoperation considering tie delay is presented. Unlike existing ethods for counication delays, the design presented here identifies and continuously oves the rightost eigenvalues to desired positions. Signal transission through the Internet using protocols (e.g., UDP) introduces tie delays into the syste. The tie delays cause instability as the syste is forced to ake use of past inforation rather than the present in deterining the output one agent sends to another agent. The proposed ethod controls rightost eigenvalue in infinite spectru such that the coand signal to the cart ensures a sustained stabilized syste. Experiental results are presented to validate the design ethod. Keywords: Counication Delays, Eigenvalues, Teleoperation, UDP, Stabilization Introduction This paper looks at the stabilization of teleoperated systes under tie delays. Systes react to external actions and the reaction is typically not instantaneous due to transport and propagation phenoena. Including inforation about the systes past in the odel, can lead to instability of tie delayed systes. Our ai in this study is to understand the effects of tie delays on the stability of teleoperation and design a control syste to reverse the adverse effects of the delay. Results are validated through experient. For coparison, stable PV controller is designed for the teleoperation syste without consideration of tie delay using classical ethod. The gains are applied to a tie delay syste to investigate perforance under tie delay. The PV controller is then odeled considering tie delay and the robustness is investigated. The design is validated with an experient using two cart systes and application of the calculated controller gains. Tie delays are inherent in various processes both natural and anade, in biology, edicine, cheistry, physics, engineering, econoics. Hence, odeling systes without delay can be approxiations of the real life odels (Kuang, 1993). For exaple, the presence of delay in an autoatic regulator syste ay cause the appearance of self-exciting oscillations, of increase of overregulation and even of instability of the syste (Norkin, 1972). Counicating between agents over the Internet also causes tie delay. A perfect controller designed without consideration of tie delay ay becoe unstable due to the effects of the tie delay. Tie delay turns a finite-diensional syste into an infinite-diensional syste. The controller design ethodology was developed based on eigenvalue based continuous pole placeent ethod (Michiels et al., 2002). The ai is to control the rightost characteristic roots in a quasi-continuous way at desired locations in the Left Half-Plane (LHP). The difficulty is controlling infinite nuber of characteristic roots with finite nuber of controller paraeters. In addition, nonlinearities in typical anipulators further coplicate the analysis (Anderson and Spong, 1989). Two carts are subsequently used to ipleent the controller to experientally validate the results. User Datagra Protocol (UDP), one of the transport layer protocols, is used to create process to process counication path for the cart syste. The two host coputers counicate through their IP addresses. Tie delay arises due to connectionless inforation transfer through the Internet. In transporting signals fro the aster agent to the slave agent and feedback signals fro the slave to the aster, tie delay arises. This delay causes undesired responses and instability Theophilus Okore-Hanson and Sun Yi. This open access article is distributed under a Creative Coons Attribution (CC-BY) 3.0 license.

2 Therefore, there is need to design a control syste capable of stabilizing the plants considering the effects of the tie delay. First a stable controller is designed without consideration of tie delay and tested with tie delay. The DDE-BIFTOOL is used to copute branches of steady-state solution. The DDE-BIFTOOL is a collection of routines written in MATLAB (Engelborghs et al., 2000) ostly used in scientific coputing environents. Delay Differential Equations (DDEs) differ fro Ordinary Differential Equations (ODEs) in that the derivative at any tie depends on the solution at prior ties. The package provides a tool for nuerical bifurcation analysis of steady state solutions and periodic solutions of delay differential equations with ultiple fixed, discrete delays. The DDEBIFTOOL is used to find the critical eigenvalues of the syste. Given an equilibriu, it allows to approxiate the rightost, stability deterining roots of the infinite diensional characteristic equation. The results fro the DDE-BIFTOOL are copare to siulation results and finally ipleented on the twocarts syste for validation. The paper is organized as follows; Section 2 presents an overview of the experiental setup used for the study. In section 3, odeling of a teleoperation syste with tie delay is presented. Stability analysis is presented in section 4. Siulation and experiental results are presented in 5 and lastly, conclusions fro the study are presented in section 6. Experiental Setup Two carts ade by Quanser with two clients and Quarc UDP Server are used for the experient. Multiagent synchronization without the effects of tie delay was presented in (Okore-Hanson and Yi, 2016). This paper looks at synchronization of two cart systes under the effects of tie delay. Tie delay in transferring the signal fro the aster agent to the slave agent can be noticed by the otion of the slave. The setup is shown in Fig. 1. The setup consists of the following: Two Carts with DC otors and encoder sensors QPID data acquisition device by Quanser 2 voltage aplifiers 2 coputers and hosts for ipleenting UDP counication using Matlab/Siulink The UDP transport protocol is responsible for creating a process to process connection and ensuring that the packet data gets to the correct process. If a strea of data is produced, the transport layer takes this strea of data fro the application layer and break the into packets. Since UDP does not need to retransit lost packets it is advantageous especially in situations where delay is of concern. The data is sapled at a rate of Hz. Fig. 1. Scheatic diagra of experiental tele-robotic syste 328

3 Modeling and Control Design Generalized Delay Differential Equations (DDEs) can be represented in a atrix-vector for as: xɺ ( t) = Ax( t) + A x( t h) (1) d where, A and A d are nxn coefficient atrices and x(t) is an nx1 state vector in the tie doain. The cart is operated by a Faulhaber coreless DC otor. The transfer function without tie delay is: K X( s) P( s) = = s( τs+ 1) V ( s) (2) where,τ = [s] is the otor tie constant, K = rad 1 Vs is the otor steady state gain, X(s) is the state vector in the Laplace doain and V (s) is the input voltage. The proportional-velocity controller is used to yield a stable control signal to regulate the otor position. It has the following structure for otors 1 and 2 respectively: ( ) ( ) ( t h ) V ( s) = K x x + K x x K sx 1 p1 d c1 u1 c2 c1 v1 c1 V ( s) = K x K x K sx 2 u2 c1( t h) p2 c2 v2 c2 (3) where, K u1 and K u2 are proportional gains for otor 1 and otor 2 delayed signals respectively. K p1 and K p2 are proportional control gains and K v1 and K v2 are velocity control gains. The displaceents x c1 and x c2 are the easured positions of carts 1 and 2, respectively and x d is the desired reference. Finally, the Delay Differential Equation (DDE) for the PV control is written in the atrix vector-for as: xɺ 1 KK 1 ( 1+ KK p v1) 0 0 x ɺ τ τ = xɺ xɺ 4 KKp2 0 0 τ τ x1 ( t h) 0 KKu2 KK u2 0 0 x Kk p τ τ 2( t h) + + τ x x d 3( t h) 0 KKu x 4( t h) 0 τ ( 1+ KKv2 ) x1 x x x 4 (4) T T where,[ x1 x2 x3 x 4] = [ xc 1 xɺ c1 xc2 xɺ c2]. The DDE-BIFTOOL is used to copute the rightost, stability deterining roots of the characteristic equation. Results fro DDE-BIFTOOL and closed-loop control siulation is presented in subsequent sections. The block diagra for the PV control is shown in Fig. 2. Stability Analysis using Eigenvalues Stability criteria used to deterine asyptotic stability of the teleoperation syste is presented. The theoretical explanation is followed by siulation and experiental results using the two-cart syste. Due to the infinite spectru, it is difficult to deterine the rightost eigenvalues. The DDE-BIFTOOL is used to deterine the stability of the syste by allowing to calculate the rightost, stability deterining root of the characteristic Equation 4. The general DDE syste is given by d x ( t ) = f ( x ( t ), x ( t h 1), dt x t h,.., ( 2) ( ) x t h, η) n ( 1) where, ( ), : n + p n x t R f R * R R is a nonlinear sooth function depending on a nuber of paraeters η R p and delay h i >0, i = 1,,. Derivative xɺ ( t) depends on the solution at previous tie(s), it is therefore necessary to provide an initial history function to specify the value of the solution before tie t = 0. The linearization of (5) around a solution x * (t) is the variational equation, given by, d y t A t y t A t y t h (5) ( ) = 0( ) ( ) + i( ) ( i) (6) dt = i 1 where, using f f(x 0,x 1,,x,η): f Ai( t) =, i= 0,1,, x i * * * x ( t), x ( t h ),, x ( t h 1 ), η) If x * (t) corresponds to steady state solution: n ( ) R ( ) =,,,, = 0 * * * * x t x with f x x η (7) Then the atrices A i (t) are constants A i (t) A i and the corresponding variational Equation 6 leads to a characteristic equation. Define the n n - diensional atrix as: ( ) = 0 λ λi A Ae λ i i= 1 hi 329

4 Fig. 2. Block diagra for PV control of telerobotic syste Then the characteristic equation reads: ( ( λ I) ) = (8) det 0 Equation 8 has an infinite nuber of roots λ C, which deterine the stability of the steady-state solution x *. The steady state solution is (asyptotically) stable provided all roots of the characteristic equation in (8) have negative real parts (Engelborghs et al., 2000). Fro (Forde and Nelson, 2004) the transcendental equation of delay differential equation, at steady state deterined for h = 0, will have the for: ( ) ( ) ( ) P λ, h P λ + P λ e λh = 0 (9) 1 2 where, h is the length of discrete tie delay and P 1 and P 2 are polynoials in λ. We can rewrite Equation 9 as: N M j λτ jλ j= 0 j= 0 j a + e bjλ = 0 (10) and assue that the steady state about which we have linearized is stable in the absence of delay. Therefore, for h = 0 all the roots of the polynoial have negative real part. As h varies, the roots change. We are interested in any critical value of h, such that the characteristic equation has a purely iaginary root (Norkin, 1972). That is to deterine whether bifurcation can occur as a result of the introduction of delay. After finding a critical delay h * and the point λ = iv * at which the roots of a characteristic equation hit the iaginary axis, it is necessary to confir that the root continues into the positive half-plane as h increases past h *. Siulation and Experient The cart is operated by a DC otor with the following characteristics: Tie constant τ = [s] and the otor steady-state gain K = rad 1 Vs. To estiate the tie delay in the UDP counication setup through the internet, the responses of the aster and slave were plotted on the sae axis and the tie delay was estiated fro the difference between the two responses. The tie delay for the UDP cart syste was estiated to be h = 1.62[secs]. Figure 3 shows the response fro gains calculated for a tie delay h = 0 in Table 1. Without tie delay the above calculated gains yield a stable response as shown in Fig. 3, however with tie delay the syste becoes unstable as shown in Fig. 4. It is therefore iportant to consider tie delay when designing systes with delayed signals. The controller gains calculated at h = 0 were applied to Siulink block diagra with tie delay h = 1.62[secs], the unstable output is presented in Fig. 4. Using the DDEBIFTOOL with a tie delay h = 1.62[secs] and the paraeters in Table 1 obtained at h = 0, instability of the syste with the rightost eigenvalue R(λ) = is confired. Result of the DDEBIFTOOL is shown in Fig. 5. The stable eigenvalues calculated without considering delay are shown in blue circles. The result agrees with the earlier results obtained in Fig. 4. Tie delay shifts the rightost eigenvalues to the unstable region. In order to verify the obtained results, the paraeters calculated at tie delay h = 0 are applied to the Carts syste and the result is displayed in Fig. 6. Figure 7 shows poles obtained with the DDEBIFTOOL fro gains calculated at h = 1.62 [secs]. The obtained results prove that tie delay can liit and degrade the achievable perforance of controlled systes and even induce instability (Yi et al., 2010). It is therefore iperative to consider tie delay when designing control systes. 330

5 Fig. 3. PV Control siulated cart response at delay h = 0 and R(λ) = Fig. 4. PV control siulated response at tie delay h = 1.62[secs] Fig. 5. Rightost eigenvalue R(λ) = for delay T d = 1.62[secs] Fig. 6. Cart position fro actual syste with control paraeters calculated at delay h = 0 ( ) Fig. 7. Rightost eigenvalue R(λ) = delay h = 1.62[secs] 331

6 Fig. 8. Siulated cart response tie delay h = 1.62[secs] rightost eigenvalue R(λ) = Fig. 9. Continuous shift of rightost eigenvalue as tie-delay changes Fig. 10. Experiental cart response PV-controller Table 1. Controller gains calculated with tie delay, h = 0 K p K p1 = 50, K p2 = 80 K u1 = 5, K u2 = 80.0 Table 2. Controller gains with delay h = 1.62 s and R(λ) = K p K p1 = 50, K p2 = 35 K v1 = K v2 = 0.5 Table 3. Controller gains for h = 1.62s and λ = K p K p1 = 80, K p2 = 35 K v1 = K v2 = 0.5 In the continuous pole placeent ethod used in this study, unstable eigenvalues are shifted to the stable region by adjusting the sensitivity of the rightost eigenvalues with respect to the feedback gains. The rightost eigenvalues are first coputed using the DDE- BIFTOOL. The coputed eigenvalues are oved in the direction of the left half-plane by applying sall changes to the feedback gains using their sensitivities. K u K v K v Response fro a PV controller with controller gains in Table 2 obtained by sensitivity analysis explained above considering tie delay h = 1.62[secs] are shown. Siulation of the response with the controller gains obtained by the continuous pole placeent ethod is shown in Fig. 8 with stabilized output. Using the continuous pole placeent ethod, the unstable eigenvalues can be shifted to the left half plane in a quasi-continuous way by applying sall changes to the feedback gain, while onitoring the eigenvalues with large real parts (Michiels et al., 2002). Figure 9 shows continuous variation of the eigenvalues with change in tie delay. The continuous pole placeent is used to obtain the controller gains in Table 3. The controller gains are applied to the cart syste through the internet and the response is displayed in Fig. 10. Conclusion A stabilization control ethod in teleoperation was developed and ipleented. The paper showed that a 332

7 stabilized control syste without consideration of the effect of tie delay ay yield instability. The designed controller considering the effect of tie delay effectively stabilized the teleoperated syste using the Cart systes. The DDE-BIFTOOL in Matlab provided an efficient tool in deterining the rightost eigenvalues. This ethod is very efficient especially when the diension of the delay differential equation syste is large. Author s Contributions All authors contributed to the writing of the anuscript. Ethics No part of this article ay be reproduced without written perission fro the publisher or authors. References Anderson, R.J. and M.W. Spong, Bilateral control of teleoperators with tie delay. IEEE Trans. Autoatic Control, 34: DOI: / Engelborghs, K., T. Luzyanina and G. Saaey, DDE-BIFTOOL: A Matlab package for bifurcation analysis of delay differential equations. TW Report. Forde, J. and P. Nelson, Applications of Stur sequences to bifurcation analysis of delay differential equation odels. J. Math. Anal. Applic., 300: DOI: /j.jaa Kuang, Y., Delay Differential Equations: With Applications in Population Dynaics. 1st Edn., Acadeic Press, Boston, ISBN-10: , pp: 398. Michiels, W., K. Engelborghs, P. Vansevenant and D. Roose, Continuous pole placeent for delay equations. Autoatica, 38: DOI: /S (01) Norkin, S., Introduction to the theory and Application of Differential Equations with Deviating Arguents. 1st Edn., Acadeic Press. Okore-Hanson, T. and S. Yi, Adaptive synchronization of networked ulti-agent systes considering transient responses and disturbances. Proceedings of the International Syposiu on Flexible Autoation, Aug. 1-3, IEEE Xplore Press, pp: DOI: /ISFA Yi, S., P.W. Nelson and A.G. Ulsoy, Tie-Delay Systes: Analysis and Control using the Labert W Function. 1st Edn., World Scientific, ISBN-10: , pp: