vehicle velocity (m/s) relative velocity (m/s) 22 relative velocity (m/s) 1.5 vehicle velocity (m/s) time (s)

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1 Proceedings of the 4th IEEE Conference on Decision and Control, New Orleans, LA, December 99, pp. 477{48. Variable Time Headway for String Stability of Automated HeavyDuty Vehicles Diana Yanakiev and Ioannis Kanellakopoulos UCLA Electrical Engineering Los Angeles, CA 9994 Abstract We present adaptive nonlinear schemes for longitudinal control of automated heavy duty vehicles. An important control objective is string stability, which ensures that errors decrease as they propagate downstream through the platoon. It is well known that string stability requires intervehicle communication if a constant spacing policy is adopted. When vehicles operate autonomously, string stability can be achieved if speeddependent spacing with constant time headway is used. This, however, results in larger steadystate spacings, which increase the platoon length hence decreasing trac throughput. In this paper we propose a new spacing policy in which the time headway varies linearly with the velocity error. Our simulation results demonstrate that this modication signicantly reduces the transient errors and allows us to use much smaller spacings in the autonomous mode of platoon operation. Introduction A widely proposed strategy for eectively increasing traf c throughput on existing highways is to group automatically controlled vehicles in platoons [,, 4, ]. These are formations which require small intervehicle spacing and many vehicles in each platoon in order to provide signicantly higher trac throughput. Therefore, the control design for platoons of automated vehicles has to guarantee not only the desired performance of any individual vehicle but also of the whole formation. A key property is string stability, which ensures that errors decrease as they propagate downstream the platoon. Sheikholeslam and Desoer [4] showed that string stability cannot be achieved for platoons with constant intervehicle spacing under autonomous operation, and proposed a scheme which guarantees string stability assuming that the lead vehicle is transmitting its velocity and acceleration information to all other vehicles in the platoon. This approach yields stable platoons with small intervehicle spacings at the cost of introducing and maintaining continuous intervehicle communication. Chien and Ioannou [], on the other hand, proved that string stability can be recovered in autonomous operation if a speeddependent spacing policy is adopted, which incorporates a xed time headway term in addition to the constant distance. Their approach avoids the communication overhead, but results in larger spacings between adjacent vehicles, thereby reducing trac throughput. This disadvantage becomes more pronounced in the case of heavyduty vehicles, since the xed time headway required to achieve string stability is much larger than for passenger cars. However, somewhat surprisingly, our results now demonstrate that signicant benets are obintervehicle communication s : h : x r : s d = s + hv f : v l : v f : v r = v l? v f : = x r? s d : v f x r v l hv f s s d minimum distance between vehicles time headway (for speeddependent spacing) vehicle separation desired vehicle separation velocity of leading vehicle velocity of following vehicle relative vehicle velocity separation error Figure : Parameters of a truck platoon. tained if the time headway decreases slowly as the relative velocity increases. Comparative simulations between xed and variable timeheadway policies show that the improvement becomes more dramatic as the number of heavyduty vehicles in the platoon increases. The control objective specic to the platoon scenario is formulated in Section. A simplied version of the adaptive nonlinear controller presented in [7] is given in Section. The new variable time headway spacing policy is presented in Section 4, where we also present simulation results comparing the performance of xed and variable time headway policies. Control Objective The parameters relevant to any two adjacent vehicles in a platoon are illustrated in Fig.. In the platoon scenario, the controller has to drive to zero both the relative velocity v r and the separation error, v r = v l? v f ; = x r? s d ; (.) where v l and v f are the velocities of the leading and following vehicle, while x r is the actual and s d the desired separation between vehicles. The desired separation may be a function of the following vehicle's velocity: s d = s + hv f (.) as shown in Fig. or constant, i.e., h =. The parameter h is called time headway and its eect is introducing more spacing at higher velocity in addition to the constant headway s. The tasks of providing zero relative velocity and zero separation error can be combined into trying to maintain 477

2 v r +k =, where k is a constant coecient which can be tuned depending on the performance requirements. As we have shown in [7] for constant time headways, when the leading vehicle's velocity is constant, v r + k implies that v r! and!. In general, for variable h the above is not necessarily true and has to be shown for the particular form of h. Control Design The idea of using the same controller for fuel as well as for brake control is suggested by the fact that the fuel and the brake command are mutually exclusive in nature. This approach is appealing because it eliminates the undesirable overhead associated with switching between controllers thus providing a less complicated and more reliable design. When the output of the controller is positive a fuel command has to be issued. A negative output should activate the brakes. In order to avoid undesirable switching between fuel and brake command in the region around zero, a hysteresis element is employed in the switching algorithm. The lower actuationtoweight ratio of heavyduty vehicles requires a more aggressive controller without undesirable overshoot. The platoon scenario is even more demanding in terms of fast response to accelerating and braking commands as well as disturbance rejection. We thus use an adaptive scheme which consists of a proportional, an integral, and a nonlinear term. We call the nonlinear term a Q term because it has a signedquadratic form. This term was used successfully in the speed control problem of heavyduty vehicles in [7], where it proved to be more ecient in avoiding overshoot than an antiwindup term and provided faster attenuation of errors compared to linear controllers. In this paper we use a simplied version of the nonlinear adaptive controller for vehicle following presented in [7]. A rst order linearized vehicle model is appropriate for the design of our longitudinal control scheme (in our simulations, however, we use the full nonlinear vehicle model). Since the control objective is to maintain v r + k =, we linearize the model around the corresponding trajectory and obtain: _v f = a(v r + k) + bu + d ; (.) where d incorporates external disturbances as well as modeling errors. We propose the adaptive PIQ control law: u = ^k (v r + k) + ^k + ^k (v r + k) jv r + kj ; (.) where ^k, ^k, ^k are timevarying parameters which are being updated by an adaptive law. Substituting equation (.) into equation (.) yields: _v f = (a + b^k )(v r + k) + b^k + b^k (v r + k) jv r + kj + d : (.) To design update laws for the parameter estimates, we consider a nonlinear reference model: _v m = a m (v l? v m + k) + q m (v l? v m + k)jv r + kj : (.4) If a, b, and d were known, the coecients of the controller could be chosen so that the plant and the reference model were responding identically to the same input signal. The corresponding values k, k, and k are computed from the following equations: a + bk = a m bk =? d (.) bk = q m : However, the parameters of the plant are unknown and timevarying. Therefore, in the control law we use estimates of the parameters which are being updated by an adaptive law. The latter is based on the tracking error e r = v f? v m. We compute _e r from equations (.), (.4), and (.): _e r = _v f? _v m (.6) =?a m e r? q m e r jv r + kj?b[ ~ k (v r + k) + ~ k + ~ k (v r + k)jv r + kj] : We use directly the absolute error e r in the update law instead of the normalized estimated error introduced in [7] to improve the robustness of the adaptive controller. The performance of this simplied version is only slightly inferior to the one using. However, the reduced complexity of the controller is essential in this case when we introduce variable time headway. The update law is obtained via the partial Lyapunov function: V = e r + k ~ b ~k ~k + b + b ; (.7) where,, are positive design constants and b is unknown but positive. With the choices: we obtain for _ V : _^k =? e r (v r + k) _^k =? e r (.8) _^k =? e r (v r + k)jv r + kj ; _V =?a m e r? q m jv r + kje r : (.9) This guarantees the boundedness of e r, ^k, ^k, ^k and the regulation of e r. The presented adaptive PIQ control law guarantees individual stability of the vehicles in the platoon. Now we also need to address the string stability issue, i.e., to ensure attenuation of errors as they propagate downstream. In the case of autonomous operation, the information available to each vehicle is its own velocity and the relative velocity and separation from the preceding one. If the desired intervehicle spacing is constant, i.e. h =, string stability cannot be achieved. This result is not specic to truck platoons. Similar results are available for passenger cars [,,, 4] which is expected because this phenomenon is caused by the nature of propagating information in the platoon rather than the particular vehicle dynamics. A simple way to circumvent this problem without providing any additional information to the vehicles is to 478

3 introduce a xed time headway in addition to the constant spacing, i.e., h >. This strategy is successful in achieving string stability but due to the lower actuationtoweight ratio of the heavyduty compared to the passenger vehicles, the necessary minimum value of h is signicantly higher. Most maneuvers require h :7s, compared to.s for passenger cars. Hence, the intervehicle spacings become quite large at higher speeds. If the resulting reduction in trac throughput is not acceptable, the only available alternative is the introduction of intervehicle communication. In [7] we showed that if the lead vehicle transmits its desired speed to all following vehicles in the platoon, string stability can be achieved even with h =, thus yielding much smaller intervehicle spacings. 4 Variable Time Headway Choosing between the two approaches for providing platoon stability discussed in the previous section is a tradeo between cost and performance. Introducing even simple intervehicle communication can guarantee string stability with small spacing between vehicles. However, if we want to avoid the inevitable overhead associated with establishing and maintaining reliable communication, and turn to the speeddependent spacing scheme, we must come to terms with the disadvantage of signicantly increased vehicle separation. This undesirable eect of constant time headway policies can be overcome by using a variable h. During steady state operation, when the velocity and separation errors are zero, the time headway component hv f of the desired separation in equation (.) should be small compared to the constant component s. During acceleration and deceleration maneuvers, the value of h should be allowed to vary with the speed error v r. Our initial intuition suggested that h should increase during acceleration and decrease during deceleration maneuvers. However, this modication resulted in increased errors and did not help achieve string stability. In contrast, the opposite modication turned out to be very helpful. Our choice for h is a function of the relative velocity v r : h = h? c v r ; (4.) where h > ; c > are constant. Since the time headway has to be h and very large headways are undesirable as seen in [7], we limit the headway in the interval [; ]: 8 < : if h? c v r, h = sat(h? c v r ) = h? c v r if < h? c v r <, otherwise. (4.) This choice of h is illustrated in the following diagram. h = sat(h? c v r ) h 6 As mentioned in Section, we need to show that v r + k implies v r! and! for the new choice of v r h. With the velocity of the lead vehicle being constant ( _v l = ), we have = x r? hv f? s ) _ = v r? h _v f? hv _ f (4.) v r = v l? v f ) _v r =? _v f : (4.4) But v r + k implies that _v r + k _. Combining this with (4.), (4.4), and (4.) we obtain (+kh+(h)kc v f ) _v r +kv r ; (h) = < h < ; otherwise, (4.) which shows that v r and hence converge to zero. In order to compare the performance of the developed control schemes, we simulated platoons consisting of 4 and trucks. The commanded prole of the simulation results presented here consists of a 4 m=s step increase of the velocity at t = s followed by an 8 m=s step decrease at t = 7 s. In all our simulation plots, the thickest black lines represent the lead vehicle, while following vehicles are represented by lines of decreasing thickness as their platoon position number increases. When examining the simulation results, we should keep in mind that, from an implementation viewpoint, the \vehicle separation" plot is the more interesting indicator of string stability. The \separation error" represents the variable which, when h >, is velocity dependent. The dierence between the actual vehicle separation x r and is best illustrated in Fig., where h = :7s and x r shows a stable trend in contrast to which grows slightly as it is propagated downstream in the platoon. From the \vehicle separation" plots we see that achieving string stability through time headway comes at the cost of signicantly increasing the intervehicle spacing. With h = :s (Fig. ) the spacing is small, but the errors increase downstream. The smallest headway for which x r is stable is h = :7s (Fig. ). This value is much larger than the h = :s often used for passenger cars, due to the lower actuationtoweight ratio of heavyduty vehicles. Simulation results using h from equation (4.) with h = :s and c = :s /m are shown in Fig. 4. Our choice of variable time headway does not guarantee string stability in the strict sense, but it results in signicantly improved performance compared to the constant h case observed in Fig. with much tighter vehicle separation compared to the constant h case in Fig. at steady state and even during maneuvers. This improvement is even more dramatic in a platoon of vehicles, shown in Fig. for constant h = :s and in Fig. 6 for variable h with h = :s and c = :s /m. Acknowledgement This work is supported by the California Department of Transportation (CalTrans) under PATH MOU 4. The contents of this paper reect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reect the ocial views or policies of the State of California. This paper does not constitute a standard, specication or regulation. 479

4 Figure : 4 autonomous vehicles, constant h = :s.. Figure 4: 4 autonomous vehicles, h = sat(:? :vr) References [] C. Chien and P. Ioannou, \Automatic vehicle following," Proc. 99 ACC, Chicago, IL, pp. 748{7. [] J. K. Hedrick, D. H. McMahon, V. K. Narendran, and D. Swaroop, \Longitudinal vehicle controller design for IVHS systems," Proc. 99 ACC, Boston, MA, pp. 7{. [] P. Ioannou and Z. Xu, \Throttle and brake control systems for automatic vehicle following," IVHS Journal, vol., pp. 4{77, 994. [4] S. Sheikholeslam and C. A. Desoer, \Longitudinal control of a platoon of vehicles," Proc. 99 ACC, San Diego, CA, pp. 9{97. [] P. Varaiya, \Smart cars on smart roads: problems of control," IEEE Trans. Automat. Control, vol. 8, pp. 9{7, 99. [6] Z. Xu and P. Ioannou, \Adaptive throttle control for speed tracking," Vehicle System Dynamics, vol., pp. 9{6, 994. [7] D. Yanakiev and I. Kanellakopoulos, \Longitudinal control of heavyduty vehicles for automated highway systems," Proc. 99 ACC, Seattle, WA, pp. 96{.. Figure : 4 autonomous vehicles, constant h = :7s. 48

5 Figure : autonomous vehicles, constant h = :s.. Figure 6: autonomous vehicles, h = sat(:? :vr). 48