Adaptive Self-triggered Control of a Remotely Operated Robot

Transcription

1 Adaptive Self-triggered Control of a Remotely Operated Robot Carlos Santos 1, Manuel Mazo Jr. 23, and Felipe Espinosa 1 1 Electronics Department, University of Alcala (Spain). 2 INCAS3, Assen (The Netherlands), 3 Faculty of Mathematics and Natural Sciences, University of Groningen (The Netherlands) Abstract. We consider the problem of remotely operating an autonomous robot through a wireless communication channel. Our goal is to achieve a satisfactory tracking performance while reducing network usage. To attain this objective we implement a self-triggered strategy that adjusts the triggering condition to the observed tracking error. After the theoretical justification we present experimental results from the application of this adaptive self-triggered approach on a P3-DX mobile robot remotely controlled. The experiments show a relevant reduction on the generated network traffic compared to a periodic implementation and to a non-adaptive self-triggered approach, while the tracking performance is barely degraded. 1 Introduction Network Control Systems (NCSs) are spatially distributed systems in which the communication between plants (including sensors and actuators) and controllers occurs through a shared digital network with limited bandwidth [10]. The real-time implementation of these systems using periodic control laws presents two main limitations: the inefficient use of the electronic resources available; and the overload of the communication channel. This has motivated the appearance of many techniques based on the idea of sampling only when necessary [12], [13], [3], [18], [19], [21]. These techniques use feedback from the state of the plant to decide when the control signal needs to be recomputed with recent measurements, two of the most important are event-triggered and self-triggered techniques. In event-triggered control implementations [18], [3], the current state of the plant is measured constantly in order to decide when the control execution must be triggered. Consequently, these implementations require dedicated resources to continuously supervise the state of the plant. The main idea of self-triggered control is to emulate the event-triggered implementation, instead of measuring continuously the state of the plant [12], [13], and determining the next update times from the last measurement acquired. In this way no continuous monitoring of the state of the plant is required. We propose a practical improvement on the self-triggering control technique on [12], [13]. This modification consists on adapting the triggering condition to the tracking error, thus

2 establishing a trade-off between the number of accesses to the wireless channel and the control performance. In the robotics context, NCS are especially interesting for remote control and cooperative guidance. Examples of remote control can be found in [17] where a DC motor is periodically controlled using a TCP/IP network communication; in [15] where a fixed-time sampling PID is remotely implemented to control a thermal system; and in [20] where a robotic vehicle is bilaterally tele-operated by means of a power-based passivity controller implemented with a periodic sampling time. Much work has been devoted to the cooperative guidance of robots, 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 for the remote control of a robotic unit [6] and for the guidance of robots platooning [2], [7], [16], [8], but always implemented with periodic sampling. However, in the context of robotics formation or applications based on robotics tele-operation is of foremost importance to minimize the computational load of the on-board electronics system of the mobile unit [9], [14], as well as to reduce the communications burden. This justifies the objective of this paper: the design and implementation of NCS for robotic applications minimizing the wireless communications load enabling other applications to share the channel bandwidth and the on-board electronic resources. As a first approach we consider a scenario where a Pioneer P3-DX robot is remotely guided and controlled from a PC working as remote center. The remote center (RC) runs a self-triggered velocity servo-controller with an adaptive triggering condition. The adaptive solution goal is to relax the performance requirements of the controller when the system is close to the equilibrium point. In this way the wireless network traffic is greatly reduced while the tracking performance is not degraded significantly. 2 Theory In this section we introduce the notions of self-triggered control for time-invariant linear systems on the continuous time domain, the specific problem statement and 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 (l 2 ) 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 ẋ = Ax is exponentially stable, i.e. M, λ R + such that x(t) Me λt x(0), if and only if there exists positive definite matrices P, Q such that: A T P + P A Q. (1) If the linear system is stable then, the function V (t) = x(t) T P x(t) is said to be a Lyapunov function for the system.

3 2.2 Self-triggered Control Review In this section, we briefly review the self-triggering policy presented in [12]. The system under consideration is ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) where A R nxn, B R nxr, C R mxn are the characteristic matrices and x(t) R n, u(t) R r and y(t) R m the state, input and output vectors respectively. If the pair (A, B) is stabilizable we find a linear feedback controller rendering the closed loop asymptotically stable: u(t) = Kx(t) The resulting closed loop system is thus described by the equation: ẋ(t) = (A + BK)x(t) If the closed loop system is asymptotically stable, there exists a Lyapunov function of the form V (t) = x(t) T P x(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(t k ) x(t), t [t k, t k+1 [ where t k is the latest actuation update instant. At the sampling instant t k, the state variable vector x(t k ) is available through measurement and provides the current plant information. That is, at times t k the controller is recomputed with fresh measurements and the input kept constant until a new measurement is received, i.e.: u(t) = kx(t k ), t [t k, t k+1 [ The closed loop dynamics under this sample and hold implementation of the controller is: [ẋ(t) ] [ ] [ ] A + BK BK x(t) = t [t ė(t) A BK BK e(t) k, t k+1 [ (2) e(t k ) = 0 The objective of the self-triggered control strategy is finding a sequence of update times {t k } such that the number of updates is minimized while stability is preserved. This sequence will be implicitly defined as the times when some triggering condition is violated. To guarantee the stability of the closed loop implementation we introduce a performance function S : R + 0 Rn R + 0.

4 This function is forced to upper-bound the evolution of V, thus determining the desired performance of the implementation. Hence, the update times {t k } are determined by the time instants at which: V (t, x t0 ) S(t, x t0 ), t t 0 is violated. The inter-executions times (t k+1 t k ) should be lower bounded by some positive quantity τ min in order to avoid Zeno executions 4 [23] of the hybrid system (2). In order to guarantee inter-execution times greater than zero it is sufficient to design S satisfying V (t k ) < Ṡ(t k) at the execution times t k. In [12] it is suggested the use of a function S(t) given by: S(t) := x s (t) T P x s (t) x s (t) = A s x s (t k ), t [t k, t k+1 [ x s (t k ) = x(t k ) where A s is a Hurwitz matrix 5 satisfying the Lyapunov equation: A T s P + P A s = R with 0 < R < Q, which guarantees that V (t k ) < Ṡ(t k). The matrix R describes the stability requirements for the implementation as it defines A s, which in turn determines S(t). Finally, how can one compute the next time the controller needs to be refreshed with new measurements? Following the technique in [13], one can predict from a measurement at time t k and the dynamics of the system, the evolution of the state x(t k + τ), τ R +. Thus, one can compute ahead of time V and S at times separated units of time, and check if V (t k + p, x tk ) S(t k + p, x tk ), p [1, 2,..., N], for N N some pre-specified horizon. Then, one can compute t k+1 = t k + p such that: either p = N or V (t k + (p + 1), x tk ) > S(t k + (p + 1), x tk ). Note that the discretization time selected for the implementation should be greater than τ min. In our application we want to perform tracking of piecewise constant reference signals. This means that for non-zero references the equilibrium of the system x eq is different than the origin, while all the above discussion is performed assuming the equilibrium of the system to be the origin. Nonetheless, this poses no problem, as the new equilibrium point is easily computed from the reference, and a simple change of coordinates: x = x x eq, brings the equilibrium back to the origin. 4 A Zeno execution, in our context, is an execution of the system in which an infinite number 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 negative real part.

5 2.3 Adaptive Self-triggering Condition One of the contributions of this work, besides the practical demonstration on a robotics application, is the proposal of an adaptive self-triggering condition. To assure the condition V (t k ) < Ṡ(t k), we pick R = σq where 0 < σ < 1. The choice of σ provides a trade-off between the number of updates and the stability requirements. In a qualitative way, it can be said that if σ 0 we achieve a significant reduction on the controller updates and a corresponding degradation of the performance. On the other hand, when σ 1 one obtains a better performance at the cost of an increase in the number of updates. To evaluate the performance of the control system the Integral of the Squared Error (ISE) [5] Index is applied to the output tracking: ISE = ˆ 0 y(t) y ref (t) 2 dt. In our experiments we compute the ISE from sampled measurements, thus instead of the formula above we use: ISE = y(k ) y ref (k ) 2. k=0 The key idea of the adaptive triggering condition is to take advantage of the benefits of executions reduction (σ 0) without losing performance (σ 1). In order to achieve this, we select the value of σ depending on the deviation of the state vector of its equilibrium point, that means x(t) x eq. When x(t) x eq 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) x eq is small enough we work with the lowest range (σ 0) to obtain a significant reduction of the controller updates. The threshold values delimiting the mentioned ranges of x(t) x eq are selected ad-hoc after experimental tests on the under study as it is explained in next section. 3 Experimental Set-up and Results The objective of this work is the evaluation of a remote adaptive self-triggered servo-controller of a minimally instrumented robotic unit. The robotic unit only incorporates the lowest control level associated to the active wheels and a digital observer to recover the full state of the robot from odometric information (providing linear and angular velocities of the mobile). The remote center, a PC in the same wireless network as the robot, deals with three main tasks: generation of the velocity reference vector, calculation of the robotic control vector and execution of the self-triggered scheduler. The last one is responsible for deciding when the control action has to be applied and, at the same time, when the plant

6 state vector estimation has to be updated. It is clear that the highest the interval inter-executions the lowest the load of the wireless channel. Figure 1 shows the global structure of the self-triggered control remotely implemented and Figure 2 presents 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 from the remote center side, that means including the wireless channel as part of the plant to be controlled. For this matter, we consider an idealized channel without disturbances and/or packet dropouts, and thus the delay exhibited by the channel can be considered constant. The plant model has been obtained with standard system identification techniques and validated through experimental trials. Linear and angular velocity references (components of input vector u) were sent to the robot (including its embedded control loops) and the open-loop robotic response (linear and angular velocities as output vector y) was registered. A constant channel delay of L seconds is included as part of the plant model, where this non-linear element is approximated through a P ade (1, 1) approximation [5], [11]. A P ade (p, q) method approximates in the Laplace domain a delay by a rational model, being p the

7 numerator degree and q the denominator degree. In our case, the channel delay L is approximated as: e Ls 1 L 2 s 1 + L 2 s. The resulting continuous state-space model of the P3-DX robotic unit is: where: ṙ(t) = A d r(t) + B d u(t) = r(t) u(t) [ ] y(t) = C d r(t) = r(t) r(t) R 4 is the plant state vector; the first two components correspond with the output plant and the other ones are related to the wireless channel delays. u(t) R 2 is the input state vector (linear and angular velocities sent to the robot); y(t) R 2 is the measurement vector (linear and angular velocities obtained from the odometric system). 3.2 Servocontroller Design for Robotic Guidance To properly track the linear and angular velocities references we designed a servo-system. The structure of the servo-system, shown in Figure 3, assures null tracking error in steady state. Because the full state of the system r is not accessible from measurements, an observer is included in the robot to provide, based on the output measurements y, estimates of the state. Nevertheless, we design our controller relying on the separation principle [5] (between estimation and control) and apply an LQR design technique to the original dynamics extended with the integrator state n. In doing this, we are assuming the full state vector is available, i.e. we design the controller constants K I and K R for the following system: [ ] [ ] [ ] [ ] ṙ(t) Ad 0 r(t) Bd [ ] [ ] [ ] r(t) 0 = + KR K ṅ(t) C d 0 n(t) 0 I + y n(t) I ref (t). The weighting matrices used in the LQR design are: [ ] 0.1I2x2 0 Q LQR = 2x4 ; R 0 4x2 I LQR = I 2x2 4x4 and the resulting constants of the controller K R and K I are: [ ] [ K R = K I = ].

8 Fig. 3. Block diagram of the remote servo control for the robot guidance. Wireless links are represented by dotted arrows. The matrix P used in the self-triggered implementation is the one resulting from solving equation (1) with: [ ] [ ] Ad 0 Bd [ ] A = + KR K C d 0 0 I ; Q = I. (3) For a maximum value of σ = 0.9, with the provided design, the resulting minimum inter-transmission times is 35ms., and therefore we selected a sampling time of 10ms. for the sensors of the robot. This is also the time-step used in the self-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 Luenberger observer [11] (for the discretized system with sampling time ). The designed observer gain in discrete time is: L = The observer is periodically executed at every discretization step ( = 10ms) at the robot, which provides with the same granularity at which the self-triggered implementation runs, thus avoiding any issues when using the estimates in the self-triggered controller implementation Adaptive Self-triggered Implementation As previously stated, the triggering condition is chosen depending on the deviation from the equilibrium state x(t) x eq, being x(t) = [r(t) T n(t) T ] T and x eq = A 1 [0, y eq ] 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 updates of the remote control and the system performance level, as described in Section 2.3.

9 From all the possibilities for the triggering condition, σ ]0, 1[, we consider only three values, each one applied to the case of: lowest magnitude (σ 1 ), middle magnitude (σ 2 ) and highest magnitude (σ 3 ) of the tracking error. 1. σ 1 = 0.1, for values of x(t) x eq 0.01, that means state variables near to equilibrium point; 2. σ 2 = 0.5, for values of 0.01 < x(t) x eq 0.1, for half-way situations; 3. σ 3 = 0.9, for values of x(t) x eq > 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 remote guidance of the robotic unit. A combination of linear and angular velocities have been chosen as references and the servo-control is implemented in the remote center according to the block diagram shown in Figure 3. Every time the triggering condition is violated the control action is sent to the P3-DX robotic unit and the remote center receives information of the plant state variables. We considered for comparison purposes three different implementations of the controller: 1. A periodic implementation with constant sampling period equal to the discretization step = 10ms. 2. A conventional self triggered implementation presented in [12] with two fixed triggered 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 condition described 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 implementations. The left top figure corresponds to a fixed sampling time (10ms), and from it a good tracking solution is appreciated. The top right figure shows a high-performance self-triggered implementation (σ = 0.9), the bottom-left figure shows a low-performance self-triggered implementation (σ = 0.1), and the bottom-right figure shows our adaptive self-triggered solution. It can be appreciated that the highest the value of σ the best the servo-control performance. Nevertheless, the adaptive self-triggering solution presents a balanced solution with lower number of channel access and a good control performance. A similar behavior was observed for the tracking of the angular velocity (second component of 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 the ISE to measure the quality of the tracking, and the number of transmissions for the network load. Table 1 summarizes these performance measurements allowing to compare the previously mentioned experiments and confirming the benefits of the authors proposal. The adaptive solution presents, without a significant system performance degradation, a number of updates (wireless transmissions)

10 Fig. 4. Linear velocity registered (red line) when a reference (blue line) is applied to the robot. Results from different implementations are shown: periodic sampling (upper left corner), fixed high value of the sigma parameter (upper right corner), fixed low value of the sigma parameter (lower left corner) and adaptive solution proposed by the authors (lower right corner). σ = 0.1 σ = 0.9 σadaptive Periodic 10 ms Updates(Wifi Tx) Average time between updates (ms) ISE performance index Table 1. Key parameters for comparison of the experimental results concerning the different control strategies

11 significantly lower than the periodic case and the non-adaptive self-triggered solution with σ = 0.9. Finally, we want to note how in the self-triggered implementations there is a certain delay in reacting to changes on the commands. This is due to the fact that transmissions are scheduled before hand when a measurement was received. Thus, if the reference is updated before the next scheduled measurements the system does not react to it until the next measurement arrives. This can easily be addressed by requesting from the RC forced measurements whenever the reference 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 of a tracking controller in self-triggered form can greatly reduce the amount of changes in the control signals applied to the robot. This in turn helps reducing the network traffic when applied in a tele-robotics context. In this way, communications resources are liberated for other applications, e.g. video streams, sharing the channel bandwidth. One critical shortcoming of the self-triggered technique employed is the necessity of state-feedback, while, as illustrated in our application, it is quite common to not have access to the full state from measurements. Inspired by the classic separation principle we designed an observer running at the robot to obtain that way state estimates that could be used by the control scheme. A more detailed and careful theoretical study might be required to have a more solid theoretical justification of the efficacy of this solution. In the current implementation we also assumed constant delays, which given our single robot set-up was a reasonable assumption. Nevertheless, our goal is to have a set of robots moving in convoy and on real (not laboratory) conditions. This more realistic setup will present variable delays in the communications, which need also 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., Valdés, F.: Compensation of channel packet dropout based on TVKF optimal estimator for robotics teleoperation. In: International 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 (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), (march 2012) 5. Duton, K., Thompson, S., Barraclough, B.: The art of control engineering. Addison- Wesley (1997) 6. Espinosa, F., Salazar, M., Pizarro, D., Valdés, F.: Electronics proposal for telerobotics operation of P3-DX units. In: INTECH (ed.) Remote and Telerobotics, pp (2010)

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