Adaptive Self-triggered Control ofa Remotely Operated Robot

Carlos Santos1, Manuel Mazo Jr.23, and Felipe Espinosa1

1 Electronics Department, University of Alcala (Spain).2 INCAS3, Assen (The Netherlands), http:www.incas3.eu

3 Faculty of Mathematics and Natural Sciences,University of Groningen (The Netherlands)

Abstract. We consider the problem of remotely operating an autonomousrobot through a wireless communication channel. Our goal is to achievea satisfactory tracking performance while reducing network usage. Toattain this objective we implement a self-triggered strategy that adjuststhe triggering condition to the observed tracking error. After the theo-retical justification we present experimental results from the applicationof this adaptive self-triggered approach on a P3-DX mobile robot re-motely controlled. The experiments show a relevant reduction on thegenerated network traffic compared to a periodic implementation and toa non-adaptive self-triggered approach, while the tracking performanceis barely degraded.

1 Introduction

Network Control Systems (NCSs) are spatially distributed systems in whichthe communication between plants (including sensors and actuators) and con-trollers occurs through a shared digital network with limited bandwidth [10]. Thereal-time implementation of these systems using periodic control laws presentstwo main limitations: the inefficient use of the electronic resources available;and the overload of the communication channel. This has motivated the ap-pearance of many techniques based on the idea of sampling only when neces-sary [12], [13], [3], [18], [19], [21]. These techniques use feedback from the state ofthe plant to decide when the control signal needs to be recomputed with recentmeasurements, two of the most important are event-triggered and self-triggeredtechniques. In event-triggered control implementations [18], [3], the current stateof the plant is measured constantly in order to decide when the control execu-tion must be triggered. Consequently, these implementations require dedicatedresources to continuously supervise the state of the plant. The main idea ofself-triggered control is to emulate the event-triggered implementation, insteadof measuring continuously the state of the plant [12], [13], and determining thenext update times from the last measurement acquired. In this way no contin-uous monitoring of the state of the plant is required. We propose a practicalimprovement on the self-triggering control technique on [12], [13]. This modifi-cation consists on adapting the triggering condition to the tracking error, thus

establishing a trade-off between the number of accesses to the wireless channeland the control performance.

In the robotics context, NCS are especially interesting for remote control andcooperative guidance. Examples of remote control can be found in [17] where aDC motor is periodically controlled using a TCP/IP network communication;in [15] where a fixed-time sampling PID is remotely implemented to control athermal system; and in [20] where a robotic vehicle is bilaterally tele-operatedby means of a power-based passivity controller implemented with a periodicsampling time. Much work has been devoted to the cooperative guidance ofrobots, for example the string stability of a convoy is studied in [4] and [22].

Some of the authors of this paper have also designed control solutions forthe remote control of a robotic unit [6] and for the guidance of robots platoon-ing [2], [7], [16], [8], but always implemented with periodic sampling. However, inthe context of robotics formation or applications based on robotics tele-operationis of foremost importance to minimize the computational load of the on-boardelectronics system of the mobile unit [9], [14], as well as to reduce the com-munications burden. This justifies the objective of this paper: the design andimplementation of NCS for robotic applications minimizing the wireless commu-nications load enabling other applications to share the channel bandwidth andthe on-board electronic resources. As a first approach we consider a scenariowhere a Pioneer P3-DX robot is remotely guided and controlled from a PCworking as remote center. The remote center (RC) runs a self-triggered veloc-ity servo-controller with an adaptive triggering condition. The adaptive solutiongoal is to relax the performance requirements of the controller when the systemis close to the equilibrium point. In this way the wireless network traffic is greatlyreduced while the tracking performance is not degraded significantly.

2 Theory

In this section we introduce the notions of self-triggered control for time-invariantlinear systems on the continuous time domain, the specific problem statementand the interest of applying an adaptive triggering condition to solve it.

2.1 Preliminaries

We denote by R+ the positive real number, by R+0 = R+ ∪ {0} and by N the

natural numbers. The usual Euclidean (l2) vector norm is represented by | · |.We remind the readers of a classic result on stability theory for linear systems

in the following theorem, see e.g. [1].

Theorem 1 A linear system x = Ax is exponentially stable, i.e. ∃ M,λ ∈R+ such that |x(t)| ≤ Me−λt|x(0)|, if and only if there exists positive definitematrices P,Q such that:

ATP + PA ≤ −Q. (1)

If the linear system is stable then, the function V (t) = x(t)TPx(t) is said tobe a Lyapunov function for the system.

2.2 Self-triggered Control Review

In this section, we briefly review the self-triggering policy presented in [12]. Thesystem under consideration is

x(t) = Ax(t) +Bu(t)y(t) = Cx(t)

where A ∈ Rnxn, B ∈ Rnxr, C ∈ Rmxn are the characteristic matrices and x(t) ∈Rn, u(t) ∈ Rr and y(t) ∈ Rm the state, input and output vectors respectively. Ifthe pair (A,B) is stabilizable we find a linear feedback controller rendering theclosed loop asymptotically stable:

u(t) = Kx(t)

The resulting closed loop system is thus described by the equation:

x(t) = (A+BK)x(t)

If the closed loop system is asymptotically stable, there exists a Lyapunovfunction of the form

V (t) = x(t)TPx(t)

where function V (t) is considered to be positive-definite.Given a symmetric and positive-definite matrix Q, P is the solution to the

Lyapunov equation (1).Now we define the state measurement error as

e(t) = x(tk)− x(t), t ∈ [tk, tk+1[

where tk is the latest actuation update instant. At the sampling instant tk, thestate variable vector x(tk) is available through measurement and provides thecurrent plant information. That is, at times tk the controller is recomputed withfresh measurements and the input kept constant until a new measurement isreceived, i.e.:

u(t) = kx(tk), t ∈ [tk, tk+1[

The closed loop dynamics under this sample and hold implementation of thecontroller is: [

x(t)e(t)

]=

[A+BK BK−A−BK −BK

] [x(t)e(t)

]t ∈ [tk, tk+1[ (2)

e(tk) = 0

The objective of the self-triggered control strategy is finding a sequence ofupdate times {tk} such that the number of updates is minimized while sta-bility is preserved. This sequence will be implicitly defined as the times whensome triggering condition is violated. To guarantee the stability of the closedloop implementation we introduce a performance function S : R+

0 ×Rn → R+0 .

This function is forced to upper-bound the evolution of V, thus determining thedesired performance of the implementation. Hence, the update times {tk} aredetermined by the time instants at which:

V (t, xt0) ≤ S(t, xt0), t ≥ t0is violated.

The inter-executions times (tk+1 − tk) should be lower bounded by somepositive quantity τmin in order to avoid Zeno executions4 [23] of the hybridsystem (2).

In order to guarantee inter-execution times greater than zero it is sufficientto design S satisfying V (tk) < S(tk) at the execution times tk. In [12] it issuggested the use of a function S(t) given by:

S(t) := xs(t)TPxs(t)

xs(t) = Asxs(tk), t ∈ [tk, tk+1[

xs(tk) = x(tk)

where As is a Hurwitz matrix5 satisfying the Lyapunov equation:

ATs P + PAs = −R

with 0 < R < Q, which guarantees that V (tk) < S(tk). The matrix R describesthe stability requirements for the implementation as it defines As, which in turndetermines S(t).

Finally, how can one compute the next time the controller needs to be re-freshed with new measurements? Following the technique in [13], one can predictfrom a measurement at time tk and the dynamics of the system, the evolutionof the state x(tk + τ), τ ∈ R+. Thus, one can compute ahead of time V and Sat times separated ∆ units of time, and check if

V (tk + p∆, xtk) ≤ S(tk + p∆, xtk), p ∈ [1, 2, . . . , N ],

for N ∈ N some pre-specified horizon. Then, one can compute tk+1 = tk + p∆such that: either p = N or

V (tk + (p+ 1)∆,xtk) > S(tk + (p+ 1)∆,xtk).

Note that the discretization time ∆ selected for the implementation should begreater than τmin.

In our application we want to perform tracking of piecewise constant ref-erence signals. This means that for non-zero references the equilibrium of thesystem xeq is different than the origin, while all the above discussion is performedassuming the equilibrium of the system to be the origin. Nonetheless, this posesno problem, as the new equilibrium point is easily computed from the reference,and a simple change of coordinates: x = x− xeq, brings the equilibrium back tothe origin.

4 A Zeno execution, in our context, is an execution of the system in which an infinitenumber of discrete events (transmissions) happen within a finite time interval.

5 A square matrix A is said to be Hurwitz if all of its eigenvalues have strictly negativereal part.

2.3 Adaptive Self-triggering Condition

One of the contributions of this work, besides the practical demonstration ona robotics application, is the proposal of an adaptive self-triggering condition.To assure the condition V (tk) < S(tk), we pick R = σQ where 0 < σ < 1.The choice of σ provides a trade-off between the number of updates and thestability requirements. In a qualitative way, it can be said that if σ → 0 weachieve a significant reduction on the controller updates and a correspondingdegradation of the performance. On the other hand, when σ → 1 one obtains abetter performance at the cost of an increase in the number of updates.

To evaluate the performance of the control system the Integral of the SquaredError (ISE) [5] Index is applied to the output tracking:

ISE =

∞

0

|y(t)− yref (t)|2dt.

In our experiments we compute the ISE from sampled measurements, thus in-stead of the formula above we use:

ISE =

∞∑k=0

|y(k∆)− yref (k∆)|2∆.

The key idea of the adaptive triggering condition is to take advantage of thebenefits of executions reduction (σ → 0) without losing performance (σ → 1). Inorder to achieve this, we select the value of σ depending on the deviation of thestate vector of its equilibrium point, that means |x(t)− xeq|. When |x(t)− xeq|is significant we work with the highest range of the triggering condition (σ →1) to achieve a fast response of the control system. On the other hand, when|x(t)− xeq| is small enough we work with the lowest range (σ → 0) to obtain asignificant reduction of the controller updates.

The threshold values delimiting the mentioned ranges of |x(t)− xeq| are se-lected ad-hoc after experimental tests on the under study as it is explained innext section.

3 Experimental Set-up and Results

The objective of this work is the evaluation of a remote adaptive self-triggeredservo-controller of a minimally instrumented robotic unit. The robotic unit onlyincorporates the lowest control level associated to the active wheels and a digitalobserver to recover the full state of the robot from odometric information (pro-viding linear and angular velocities of the mobile). The remote center, a PC inthe same wireless network as the robot, deals with three main tasks: generationof the velocity reference vector, calculation of the robotic control vector andexecution of the self-triggered scheduler. The last one is responsible for decidingwhen the control action has to be applied and, at the same time, when the plant

state vector estimation has to be updated. It is clear that the highest the intervalinter-executions the lowest the load of the wireless channel. Figure 1 shows theglobal structure of the self-triggered control remotely implemented and Figure 2presents a picture of the actual setup.

Fig. 1. Structure of the implemented self-triggered servo control

Fig. 2. P3-DX robot tele-operated from a remote center.

3.1 Plant Model

The first stage for the experimental set-up is modeling the P3-DX robot fromthe remote center side, that means including the wireless channel as part ofthe plant to be controlled. For this matter, we consider an idealized channelwithout disturbances and/or packet dropouts, and thus the delay exhibited bythe channel can be considered constant.

The plant model has been obtained with standard system identification tech-niques and validated through experimental trials. Linear and angular velocityreferences (components of input vector u) were sent to the robot (including itsembedded control loops) and the open-loop robotic response (linear and angu-lar velocities as output vector y) was registered. A constant channel delay of Lseconds is included as part of the plant model, where this non-linear element isapproximated through a Pade(1, 1) approximation [5], [11]. A Pade(p, q) methodapproximates in the Laplace domain a delay by a rational model, being p the

numerator degree and q the denominator degree . In our case, the channel delayL is approximated as:

e−Ls '1− L

2 s

1 + L2 s.

The resulting continuous state-space model of the P3-DX robotic unit is:

r(t) = Adr(t) +Bdu(t)

=

−4.094 −0.015 1664 0.7227−0.008 −5.042 0.326 2023

0 0 −200 00 0 0 −200

r(t) +

−4.159 −0.002−0.001 −5.057

1 00 1

u(t)

y(t) = Cdr(t) =

[1 0 0 00 1 0 0

]r(t)

where:

– r(t) ∈ R4 is the plant state vector; the first two components correspondwith the output plant and the other ones are related to the wireless channeldelays.

– u(t) ∈ R2 is the input state vector (linear and angular velocities sent to therobot);

– y(t) ∈ R2 is the measurement vector (linear and angular velocities obtainedfrom the odometric system).

3.2 Servocontroller Design for Robotic Guidance

To properly track the linear and angular velocities references we designed aservo-system. The structure of the servo-system, shown in Figure 3, assures nulltracking error in steady state. Because the full state of the system r is not acces-sible from measurements, an observer is included in the robot to provide, basedon the output measurements y, estimates of the state. Nevertheless, we designour controller relying on the separation principle [5] (between estimation andcontrol) and apply an LQR design technique to the original dynamics extendedwith the integrator state n. In doing this, we are assuming the full state vectoris available, i.e. we design the controller constants KI and KR for the followingsystem:[

r(t)n(t)

]=

[Ad 0−Cd 0

] [r(t)n(t)

]+

[Bd0

] [KR KI

] [ r(t)n(t)

]+

[0I

]yref (t).

The weighting matrices used in the LQR design are:

QLQR =

[0.1I2x2 02x4

04x2 I4x4

]; RLQR = I2x2

and the resulting constants of the controller KR and KI are:

KR =

[0.26 0 2.167 −0.004−0.001 0.222 −0.004 2.256

]KI =

[1 0.001

−0.001 1

].

Fig. 3. Block diagram of the remote servo control for the robot guidance. Wirelesslinks are represented by dotted arrows.

The matrix P used in the self-triggered implementation is the one resulting fromsolving equation (1) with:

A =

[Ad 0−Cd 0

]+

[Bd0

] [KR KI

]; Q = I. (3)

For a maximum value of σ = 0.9, with the provided design, the resulting mini-mum inter-transmission times is 35ms., and therefore we selected a sampling timeof 10ms. for the sensors of the robot. This is also the time-step ∆ used in theself-triggered implementation to predict the next measurements’ transmission,as described in [12] and reviewed in Section 2.2.

Independently of the controller design, we construct a discrete time Luen-berger observer [11] (for the discretized system with sampling time ∆). Thedesigned observer gain in discrete time is:

L =

0.01 0

0 0.010 00 0

.The observer is periodically executed at every discretization step (∆ = 10ms) atthe robot, which provides with the same granularity at which the self-triggeredimplementation runs, thus avoiding any issues when using the estimates in theself-triggered controller implementation.

3.3 Adaptive Self-triggered Implementation

As previously stated, the triggering condition is chosen depending on the de-viation from the equilibrium state |x(t)− xeq|, being x(t) = [r(t)Tn(t)T ]

Tand

xeq = −A−1[0, yeq]T , where A is defined as in (3). In other words, choosing the

right σ values we achieve a balance between reduction on the number of updatesof the remote control and the system performance level, as described in Section2.3.

From all the possibilities for the triggering condition, σ ∈]0, 1[, we consideronly three values, each one applied to the case of: lowest magnitude (σ1), middlemagnitude (σ2) and highest magnitude (σ3) of the tracking error.

1. σ1 = 0.1, for values of |x(t)− xeq| ≤ 0.01, that means state variables nearto equilibrium point;

2. σ2 = 0.5, for values of 0.01 < |x(t)− xeq| ≤ 0.1, for half-way situations;3. σ3 = 0.9, for values of |x(t)− xeq| > 0.1, when a change in the reference is

applied and a quick answer of the servo-control is required.

3.4 Experimental results

In order to evaluate the proposal we carried out three experiments on the remoteguidance of the robotic unit. A combination of linear and angular velocities havebeen chosen as references and the servo-control is implemented in the remotecenter according to the block diagram shown in Figure 3. Every time the trig-gering condition is violated the control action is sent to the P3-DX robotic unitand the remote center receives information of the plant state variables.

We considered for comparison purposes three different implementations ofthe controller:

1. A periodic implementation with constant sampling period equal to the dis-cretization step ∆ = 10ms.

2. A conventional self triggered implementation presented in [12] with two fixedtriggered conditions: one close to 0 (σ = 0.1) and other close to 1 (σ = 0.9).

3. An adaptive self-triggered implementation applying the triggering conditiondescribed in section 3.3.

In Figure 4, the linear velocity (first component of the output vector y(t))captured from the P3-DX through the odometric system with different imple-mentations. The left top figure corresponds to a fixed sampling time (10ms),and from it a good tracking solution is appreciated. The top right figure showsa high-performance self-triggered implementation (σ = 0.9), the bottom-left fig-ure shows a low-performance self-triggered implementation (σ = 0.1), and thebottom-right figure shows our adaptive self-triggered solution. It can be appre-ciated that the highest the value of σ the best the servo-control performance.Nevertheless, the adaptive self-triggering solution presents a balanced solutionwith lower number of channel access and a good control performance. A similarbehavior was observed for the tracking of the angular velocity (second componentof the output vector y(t)), which for the sake of space are omitted from the paper.These observed behavior is confirmed employing a performance measure as theISE to measure the quality of the tracking, and the number of transmissions forthe network load. Table 1 summarizes these performance measurements allowingto compare the previously mentioned experiments and confirming the benefitsof the authors’ proposal. The adaptive solution presents, without a significantsystem performance degradation, a number of updates (wireless transmissions)

Fig. 4. Linear velocity registered (red line) when a reference (blue line) is applied tothe robot. Results from different implementations are shown: periodic sampling (upperleft corner), fixed high value of the sigma parameter (upper right corner), fixed lowvalue of the sigma parameter (lower left corner) and adaptive solution proposed by theauthors (lower right corner).

σ = 0.1 σ = 0.9 σAdaptive Periodic 10 ms

Updates(Wifi Tx) 66 1289 127 10000

Average time between updates (ms) 1515 78 787 10

ISE performance index 1.697 0.410 0.419 0.271

Table 1. Key parameters for comparison of the experimental results concerning thedifferent control strategies

significantly lower than the periodic case and the non-adaptive self-triggeredsolution with σ = 0.9.

Finally, we want to note how in the self-triggered implementations there isa certain delay in reacting to changes on the commands. This is due to the factthat transmissions are scheduled before hand when a measurement was received.Thus, if the reference is updated before the next scheduled measurements thesystem does not react to it until the next measurement arrives. This can easilybe addressed by requesting from the RC forced measurements whenever thereference changes and it will be included in our future developments.

4 Conclusions and future work

We have shown in a real mobile robot how a rather simple implementation ofa tracking controller in self-triggered form can greatly reduce the amount ofchanges in the control signals applied to the robot. This in turn helps reducingthe network traffic when applied in a tele-robotics context. In this way, com-munications resources are liberated for other applications, e.g. video streams,sharing the channel bandwidth. One critical shortcoming of the self-triggeredtechnique employed is the necessity of state-feedback, while, as illustrated in ourapplication, it is quite common to not have access to the full state from mea-surements. Inspired by the classic separation principle we designed an observerrunning at the robot to obtain that way state estimates that could be used by thecontrol scheme. A more detailed and careful theoretical study might be requiredto have a more solid theoretical justification of the efficacy of this solution. In thecurrent implementation we also assumed constant delays, which given our singlerobot set-up was a reasonable assumption. Nevertheless, our goal is to have a setof robots moving in convoy and on real (not laboratory) conditions. This morerealistic setup will present variable delays in the communications, which needalso to be handled in a more elaborate way.

References

1. Antsaklis, P., Michel, A.N.: Linear Systems. McGraw-Hill (1997)2. Bocos, A., Espinosa, F., Salazar, M., Valdes, F.: Compensation of channel packet

dropout based on TVKF optimal estimator for robotics teleoperation. In: Interna-tional Conference on Robotics and Automation (ICRA) (2008)

3. Cogill, R.: Event-based control using quadratic approximate value functions. In:Proceedings of the 48th IEEE Conference on Decision and Control. pp. 5883 –5888(Dec 2009)

4. Dunbar, W., Caveney, D.: Distributed receding horizon control of vehicle platoons:Stability and string stability. IEEE Transactions on Automatic Control 57(3), 620–633 (march 2012)

5. Duton, K., Thompson, S., Barraclough, B.: The art of control engineering. Addison-Wesley (1997)

6. Espinosa, F., Salazar, M., Pizarro, D., Valdes, F.: Electronics proposal for teler-obotics operation of P3-DX units. In: INTECH (ed.) Remote and Telerobotics, pp.1–16 (2010)

7. Espinosa, F., Salazar, M., Valdes, F., Bocos, A.: Communication architecture basedon player/stage and sockets for cooperative guidance of robotic units. In: 16thMediterranean Conference on Control and Automation. pp. 1423 –1428 (June 2008)

8. Espinosa, F., Santos, C., Marron-Romera, M., Pizarro, D., Valdes, F.,Dongil, J.: Odometry and laser scanner fusion based on a discrete extendedkalman filter for robotic platooning guidance. Sensors 11(9), 8339–8357 (2011),http://www.mdpi.com/1424-8220/11/9/8339/

9. Hashimoto, H.: Present state and future of intelligent spaces - discussion on theimplementation of rt in our enviorement. In: Artificial Life and Robotics, vol. 11,pp. 1–7 (2007)

10. Hespanha, J., Naghshtabrizi, P., Xu, Y.: A survey of recent results in networkedcontrol systems. Proceedings of the IEEE 95(1), 138 –162 (jan 2007)

11. Levine, W.: The control handbook. IEEE-Press (1996)12. Mazo Jr, M., Anta, A., Tabuada, P.: On self-triggered control for linear systems:

Guarantees and complexity. In: European Control Conference (2009)13. Mazo Jr., M., Anta, A., Tabuada, P.: An ISS self-triggered implementation of linear

controller. Automatica 46(8), 1310–1314 (Aug 2010)14. Monekosso, D., Remagnino, P., Kuno, Y.: Intelligent environments: methods, al-

gorithms and applications. Springer (2009)15. Santana, I., Ferre, M., Izaguirre, E., Aracil, R., Hernandez, L.: Remote labora-

tories for education and research purposes in automatic control systems. IEEETransactions on Industrial Informatics PP(99), 1 (2012)

16. Santos, C., Espinosa, F., Pizarro, D., Valdes, F., Santiso, E., Dıaz, I.: Fuzzy de-centralized control for guidance of a convoy of robots in non-linear trajectories.In: IEEE Conference on Emerging Technologies and Factory Automation (ETFA).pp. 1 –8 (Sept 2010)

17. Stefan, O., Codrean, A., Dragomir, T., Silea, I.: Time delay and information losscompensation in a network control system for a dc motor. In: 6th IEEE Interna-tional Symposium on Applied Computational Intelligence and Informatics (SACI).pp. 131 –135 (May 2011)

18. Tabuada, P.: Event-triggered real-time scheduling of stabilizing control tasks. IEEETransactions on Automatic Control 52(9), 1680 –1685 (Sept 2007)

19. Wang, X., Lemmon, M.: Event design in event-triggered feedback control systems.In: 47th IEEE Conference on Decision and Control. pp. 2105 –2110 (Dec 2008)

20. Ware, J., Pan, Y.J.: Realisation of a bilaterally teleoperated robotic vehicle plat-form with passivity control. Control Theory Applications, IET 5(8), 952 –962 (may2011)

21. Xue, Y., Liu, K.: Controller design for variable-sampling networked control systemswith dynamic output feedback. In: 7th World Congress on Intelligent Control andAutomation, WCICA. pp. 6391 –6396 (June 2008)

22. Yazbeck, J., Scheuer, A., Simonin, O., Charpillet, F.: Improving near-to-near lat-eral control of platoons without communication. In: IEEE/RSJ International Con-ference on Intelligent Robots and Systems (IROS). pp. 4103 –4108 (Sept 2011)

23. Zhang, J., Johansson, K.H., Lygeros, J., Sastry, S.: Dynamical systems revisited:Hybrid systems with Zeno executions. In: Krogh, B., Lynch, N. (eds.) HybridSystems: Computation and Control, Lecture Notes in Computer Science, vol. 1790.Springer-Verlag, New York (2000)