Application of Nonlinear Control to a Collision Avoidance System

Transcription

1 Application of Nonlinear Control to a Collision Avoiance System Pete Seiler Mechanical Eng. Dept. U. of California-Berkeley Berkeley, CA , USA Phone: Fax: pseiler@euler.me.berkeley.edu Bong-Sob Song Mechanical Eng. Dept. U. of California-Berkeley Berkeley, CA , USA Phone: Fax: bsong@vehicle.me.berkeley.edu J. Karl Herick Mechanical Eng. Dept. U. of California-Berkeley Berkeley, CA , USA Phone: Fax: kherick@euler.me.berkeley.edu SUMMARY The esign of a collision avoiance (CA) algorithm will be presente. Previous work relie on a simple CA algorithm which use on/off brake control. This work is extene by using nonlinear control techniques to improve brake control an minimize river iscomfort. First, a quarter car moel suitable for control esign will be escribe. Then, a multi-surface sliing controller with on/off hysteresis will be evelope. The on/off hysteresis will be moifie to inclue a switch with st orer ynamics in an attempt to minimize large initial accelerations. Finally, the simulation results of the CW/CA algorithms will be given. INTRODUCTION The potential for increase vehicle safety motivates the evelopment of collision avoiance systems. Half of the.5+ million rear-en crashes that occurre in 994 coul have been prevente by a collision avoiance system []. Collision avoiance systems can react to situations that humans can not (ue to slow response) or o not (ue to river error). Therefore, they are able to reuce the severity of accients. A commercially viable CA system must meet several (often conflicting) criteria. The CA system must balance the traeoff between performance an river interference. A river who is attempting an avoiance maneuver, such as steering, may be startle an possibly lose vehicle control if the system automatically applies the brakes [3]. Therefore, a CA system coul be esigne to be conservative an avoi any collision, but this system woul be more likely to apply the brakes uring normal riving maneuvers. The system must also account for iniviual riving styles [] an a variety of weather conitions. A previously esigne Binary CA Algorithm will now be briefly escribe [4]. When the vehicle-to-vehicle spacing rops below a pre-efine critical braking istance, br, the brakes are fully applie until the vehicle comes to a stop. There is a goo chance that a full-on application of the brakes will startle the river; thus small critical braking istances must be use with the Binary CA Algorithm to ensure that the brakes are only applie at the last possible moment of an impening collision. As a result, most extreme collisions will not be avoie; they will only have their impact velocity reuce. The objective of this paper is to exten the Binary Algorithm by using a slip-base controller. The brake controller will allow the brakes to be applie smoothly an to be moulate to prevent collisions. The benefit of brake moulation is that the critical

2 istance can be more conservative (i.e. larger braking critical istance) since smooth application of the brakes will be less likely to startle the river. A more conservative system leas to better collision avoiance performance. The system performance will be stuie in two funamentally ifferent scenarios. In the first scenario, the CA vehicle is following another vehicle with a negligible relative velocity when the lea car brakes at its maximum eceleration rate (8 m/s ). This will be referre to as the critical scenario. In this situation, the system shoul initially react slowly to give the river time to overrie the controller if esire. If the river oes not overrie the controller, then the system shoul react quickly to avoi the collision. In the secon scenario, the CA vehicle is again following another vehicle with no relative velocity when the lea car slows own gently from 7.8 m/s to 4 m/s over a 3.5 secon perio. This scenario will be referre to as a rift scenario. In this situation, the goal is to prevent the vehicle-to-vehicle spacing from ropping to an unsafe level. The brakes shoul be gently applie so that the river is aware that the CA controller has taken over, but is not surprise by this action. In the 96 s, Golman an von Gierke reporte that automotive ecelerations up to.5 m/s were comfortable to human passengers [8]. Thus, by gently applie, we mean that the vehicle eceleration shoul not excee this limit. CONTROL MODEL The problem of brake control for vehicle following has been stuie greatly in the last ecae. Applications range from Intelligent Cruise Control to Automate Highways [5,6]. However, most of the previous work on vehicle following assumes that the tire slip can be neglecte so that the relation between tire slip an longituinal force oes not enter the control problem. The CA scenario is funamentally ifferent in that the brakes are use to generate large longituinal tractive forces. Therefore, the no-slip assumption fails to hol. Thus, several assumptions nee to be given to arrive at a quarter car moel suitable for CA brake control esign. First, the prototype vehicle will be fitte with a brake actuator base on a esign evelope by the Partners for Avance Transit an Highways (PATH), escribe in [5],[6]. The esign relies on an intermeiate cyliner place between the vacuum booster an master cyliner. An electro-hyraulic servovalve is use to connect the intermeiate cyliner to a high pressure noe (a pump an accumulator) an force movement of the master cyliner. In this setup, the river is still able to increase the level of braking through the brake peal. Furthermore, the actuator response is fast enough that the master cyliner pressure, P mc, can be consiere the control input. A major rawback of this esign is that it oes not allow control of brake pressure at iniviual wheels. Since we can only control the aggregate brake pressure, P mc, we nee to assume similar behavior all on four wheels. Specifically, we will assume that all wheels are operating below the peak slip of the force versus slip curve. Furthermore, the force-slip curve is fairly linear up to its peak, so we will assume that tractive force is proportional to tire slip.

3 If the brake controller attempts to track a tire slip which causes any tire to lock up, then the ABS will release pressure an cause the tire to stay at the peak slip. Geres [5] an Maciuca [6] extensively moele the brake system with this actuator setup. Using P mc as a control input, they evelope a nonlinear sliing controller to track a esire wheel pressure, P w,es, which was measure with a pressure sensor on one of the front wheels. This paper will use this controller an assume that it can track esire wheel pressures fast enough to assume that wheel pressure can be use as a control input. Using these assumptions, the vehicle ynamics reuce to the following two state quartercar moel: v& = m & ( λ) λ = m v [ c v F k λ] r λ r kλ r kb [ c v Fr k λ] λ + Pw λ J v J v () where m = vehicle mass, c = aeroynamic rag coefficient, F r = rolling resistance, k λ = lumpe tractive force gain, λ = ( v r ω )/ v = tire slip, v = velocity, r = tire raius, J = tire rotational inertia, k b = lumpe coefficient relating wheel pressure an brake torque. Finally, it will be assume that a reasonable slip estimate can be accurately calculate using only vehicle velocity an wheel angular velocity information. The wheel angular velocity can be obtaine from a wheel spee sensor. The absolute velocity can be obtaine using the wheel spee sensor an accelerometer measurements. Just before the braking maneuver, the unriven wheel on an actual vehicle has virtually no slip. Therefore, the wheel spee measurement on this wheel can be use to calculate the true vehicle velocity. Whenever a collision avoiance maneuver is being performe, the accelerometer measurements can be integrate (using the last unriven wheel spee measurement as an initial conition) to obtain true vehicle velocity. Collision avoiance maneuvers will typically have short time spans so that integrator error builup shoul not be a problem. Once the maneuver ens, the vehicle, if not stoppe, will return to a steay cruising spee an true velocity can again be calculate from unriven wheel spee measurements. CONTROL ALGORITHM DEVELOPMENT A two-surface sliing controller will be use to control the velocity of a quarter-car vehicle. The upper level (the velocity controller) uses the tire slip as a synthetic input for velocity control. This means that the tire slip is treate as a control input which can be use to track a esire velocity. In reality, the upper level commans a esire tire slip which will result in the necessary velocity tracking. It is the job of the lower level (slip controller) to track this esire tire slip. It is assume that the wheel angular ynamics are much faster than the vehicle velocity ynamics. 3

4 Therefore, the slip controller shoul force λ λ very quickly from the view of the velocity controller. In fact, it is hope that this convergence will happen so quickly that the velocity controller can be esigne using the tire slip as the control input, hence the term synthetic input. BRAKING CRITICAL DISTANCE The range at which the brakes are applie by the CA system, br, can be efine in many ways. Typically, br is a function of vehicle velocity an relative velocity between vehicles. For this analysis, br will be erive from the kinematics of two vehicles braking at their maximum ecelerations until they come to a stop. The efinition will be: v v p br = + v τ + o () α α where α = eceleration rate of both vehicles = 6 m/s, τ = system (. sec) + river elays (. sec) =. sec, an o = safety offset = 5m. This critical istance efinition is very conservative, meaning it results in large following istances. In actual implementation, the river will be allowe to scale br (within bouns) to obtain a critical istance suitable to their riving style. VELOCITY CONTROLLER The velocity controller escribe in this section follows the usual sliing controller evelopment given in [7]. First, efine S to be: S v v + Λ x x = v v v ) + Λ x ( x ) (3) ( ) ( ) ( ) ( ) ( p rel, es p br r x v p rel The first sliing surface will be given by S =. Define = x an = v v an rewrite S as: ( v v ) + Λ ( r) S = rel, es rel br (4) Furthermore, we can etermine v rel,es by inverting the critical braking istance relation, br = f(v,v rel,es ). Thus, v rel,es = f - (r,v). Differentiating S, using Equation an assuming v& = yiels: [ c v F k λ] + v& + Λ ( & r& ) = b h ( v, &, r& ) S& = r λ rel, es br λ + br (5) m where b an h have the implie efinitions. If we knew b an h exactly, we coul cancel out these nonlinear ynamics an replace them with more esirable ynamics. Since both of these terms have uncertainty, we can only use our best estimate to cancel these terms out: p [ hˆ K S] λ = (6) bˆ The hat is use to enote the best estimate of the uncertain quantities. The overbar on the esire slip implies that this esire slip will be filtere before its use by the slip controller. The necessary filtering is escribe in the next section. p 4

5 For the moment, assume that the filtering ynamics are fast an the slip controller causes fast convergence so that λ λ λ. Then, the first term of Equation 7 cancels out the unwante nonlinear ynamics with the best moel estimate while the secon term attempts to overcome any errors between the moele an true values. For example, if λ = λ, Equation 6 becomes: b S& = [ hˆ K S] + h ˆ = β K S + ( β) h + h (7) bˆ where β = b bˆ > an h = h ĥ. The assumption that β > means that we must know the sign of b. Finally, if λ = λ, the gain neee to guarantee convergence to a bounary layer of the Φ surface, S, is: ( β ˆ,min ) h + max( h ) β,min Φ < K (8) DESIRED SLIP FILTER In the evelopment of the velocity controller, br an v rel,es were ifferentiate. Similarly, the esire slip will nee to be ifferentiate uring the slip controller esign. Notice that the erivative of λ (Equation 6) results in many terms. To avoi this explosion of terms, the following filter is use to obtain λ : τ & λ + λ = λ with λ () = λ () (9) f The use of this filter allows & λ to be easily compute. It shoul be note that br an v rel,es are also filtere for the same reason before their use by the velocity controller. SLIP CONTROLLER Following the velocity controller evelopment, efine: S = λ λ () Differentiating S an using Equation gives: ( ) r k r kb S & λ λ = [ c v Fr kλ λ] λ + Pw & λ = b Pw h ( v, λ, & λ ) m v J v J v + () To cancel the unwante nonlinear ynamics, efine the esire wheel brake pressure to be: P [ ˆ w, es = h K S ] () bˆ Again, P w, es will be filtere before being use by the brake controller esigne by Maciuca an Geres. Similar to above, the gain neee to converge to S Φ if P w = P w, es is: 5

6 ( β ˆ,min ) h + max( h ) β,min Φ < K (3) ON/OFF HYSTERESIS The CA controller will switch on whenever r br an it will try to force r br. Switching logic with hysteresis is necessary to prevent the controller from turning on an off when r. The first iteration switching logic turne on the controller when br r br an turne off the controller when r > br + 5. Unfortunately, this logic cause the brakes to be applie harshly when the controller turne on resulting in large initial accelerations. We wante to eliminate these large initial accelerations without hinering controller performance over the remainer of the collision avoiance maneuver. Instea of turning on the controller abruptly base on the above logic, the esire wheel pressure was scale by the following first orer switch whenever the controller turne on: if ( tca < ) ( SWITCH)/t =. ( SWITCH + ) (4) else ( SWITCH)/t =. ( SWITCH + ) For the first secon after the controller turns on, the SWITCH rises slowly. This scaling factor prevents the esire wheel pressure from rising too rapily an startling the river. After the first secon of the CA maneuver, the SWITCH rises rapily an until its scaling effect is negligible. This allows high gains to be use in the controller without the consequence of large initial accelerations. RESULTS The critical an rift scenarios escribe above were simulate on a half car vehicle moel. The vehicle parameters use by the controller (m, J, r, c, k b, an F r ) were given % error. The tire slope, k λ, is ifficult to etermine an was given 5% error. Furthermore, the front tire slip (%) was use for feeback. To simulate the use of a slip estimate, noise an a large bias were ae to the front tire slip. Figure shows the controller performance in the critical scenario without the first orer hysteresis switch. When the range rops below the critical braking istance (upper right plot), the controller switches on (upper left). This causes the slip controller to comman a large step increase in wheel pressure, which is then tracke by the brake controller (lower right). As a result, the passenger experiences a large initial acceleration (upper left). After this large initial acceleration, the controller performs well, forcing the relative velocity to zero an the range to the critical braking istance. Also notice that the brake controller faithfully tracks the filtere esire wheel pressure prouce by the slip controller, so that the assumption for the multi-surface controller hols. Finally, notice that the estimate slip use by the slip controller is noisy an biase as escribe above. The slip controller causes the estimate slip to converge to the filtere esire slip, but the true slip oes not track the esire value. Yet, the velocity controller is able to overcome this error an still perform well. 6

7 5-5 SWITCH* Acceleration (m/s ) Relative Vel. (m/s) Spacing (m) 6 4 br (m) r (m) Slip es,filt Pressure (kpa) 4 3 Pw es,filt Pw Slip (%) 5 5 Slip f,est Slip f Figure : Critical Scenario On/Off Switch Figure shows the rift scenario without the first orer switch. As soon as the range rops below the critical braking istance, the controller harshly applies the brakes. This large initial acceleration woul not be acceptable to most rivers consiering the milness of the situation. This first orer switch was then implemente to reuce this initial harshness. The results (Figure 3) show that the switch turns on very slowly for the first secon after the controller turns on. As a result, the initial accelerations are mil. After the first secon, the switch rises rapily to a gain, allowing the controller performance to improve as the error ecreases. One concern about using the switch is that it may hiner the vehicle performance in critical situations. The results of the critical scenario with the first orer switch (Figure 4) show the controller is still able to prevent the collision. Furthermore, the controller oes not slam on the brakes, making the results more esirable from a human factors point of view. ACKNOWLEGEMENTS The authors woul like to acknowlege the financial support of the KIA Motor Corporation an the comments of Professor Kyongsu Yi of Hangyang University. 7

8 - -4 SWITCH* Acceleration (m/s ) Relative Vel. (m/s) Spacing (m) br (m) r (m) Figure : Drift Scenario with On/Off Switch - SWITCH* Acceleration (m/s ) Relative Vel. (m/s) Spacing (m) br (m) r (m) Figure 3: Drift Scenario with st Orer Switch 8

9 -5 - SWITCH* Acceleration (m/s ) Relative Vel. (m/s) Spacing (m) br (m) r (m) Figure 4: Critical Scenario with st Orer Switch REFERENCES. NHTSA. Report to Congress on the National Highway Traffic Safety Aministration ITS Program. Technical Report, U.S. Department of Transportation, January A.D. Horowitz an T.A. Dingus. Warning Signal Design: A Key Human Factors Issue in an In-Vehicle Front-to-Rear Collision Warning System Proceeings of the Human Factors Society, vol., p. -3, Atlanta, Georgia, October Y. Fujita, K. Akuzawa, an M. Sato. Raar Brake System 995 Annual Meeting of ITS America, vol., p. 95-, Washington, D. C., March P. Seiler, B. Song an J. K. Herick. Development of a Collision Avoiance System SAE Conference, p. 97-3, Detroit, Michigan, February J.C. Geres. Decouple Design of Robust Controllers for Nonlinear Systems: As Motivate by an Applie to Coorinate Throttle an Brake Control for Automate Highways Ph.D. Thesis, University of California at Berkeley, Berkeley, CA, D.B. Maciuca. Nonlinear Robust Control with Application to Brake Control for Automate Highway Systems Ph.D. Thesis, University of California at Berkeley, Berkeley, CA, J.J.E. Slotine an W. Li. Applie Nonlinear Control, Prentice Hall, D.E. Golman an H.E. von Gierke. Chapter 44 in Shock an Vibration Hanbook, Cyril M. Harris (e.), McGrw-Hill Book Company, 3 r eition,