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Neural network control for automatic braking control system
Neural Networks, Vol. 7, No. 8, pp. 1303-|312, 1994 Copyright © 1994 ElsevierScienceLtd Printed in the USA. All rights reserved 0893-6080/94 $6.00 + .00
Pergamon 0893-6080( 94 ) E0001-2
CONTRIBUTED ARTICLE
Neural Network Control for Automatic Braking Control System HIROSHI OHNO, 1 TOSHIHIKO SUZUKI, 2 KEIJI AOKI, 2 ARATA TAKAHASI, l AND GUNJI SUGIMOTO 1 ~Toyota Central Research & Development Labs., Inc. and 2Toyota Motor Corporation
(Received 10 September 1993; revised and accepted 21 December 1993)
Abstract--We have developed an automatic braking control system for automobiles applying a three-layer neural network model. This system enables the vehicle to decelerate smoothly and to stop at the specified position behind the vehicle ahead. This paper focuses on the use of the three-layer neural network model in the automatic braking control system. Because vehicle dynamics is varied by the variations in road conditions or vehicle characteristics, it cannot be represented accurately by mathematical models. According to this reason, the conventional control methods, such as proportional integrative derivative ( PID ) control, cannot achieve satisfactory control performance. Therefore, we have constructed the neural network adaptive control system based on the feedback error learning method. This learning method enables the system to adapt to the changes in road grade and vehicle weight without using any specific sensors. Experimental results show satisfactory control performance and reveal that the neural network adaptive control system based on the feedback error learning method is available for the automatic braking control system.
Keywords--Feedback error learning, Neural networks, Feedforward control, Automatic braking control system, Vehicle control, Hysteresis characteristics. 1. INTRODUCTION
the variations in road conditions (e.g., road friction, road grade, etc.) and vehicle characteristics (e.g., vehicle weight, brake pad friction, etc.), and the automatic braking control system does not work well by constant PID gains. Moreover, if we use variable gains, PID gain tuning is very difficult adapting to the variations in road conditions and vehicle characteristics. Therefore, we decided to make a new control scheme. In recent years, some control applications of neural networks have been reported (Psaltis, Sideris, & Yamamura, 1988; Li & Slotine, 1989; Levin, Gewirtzman, & Inbar, 1991; Schiffmann & Geffers, 1993). These applications use multilayer neural networks as well as back propagation (Rumelhart, Hinton, & Williams, 1986) as its learning algorithm. Because it is clarified that a three-layer neural network can theoretically construct a universal approximator (Funahashi, 1989), multilayer neural networks have been used as controllers and identifiers for nonlinear controlled objects. Moreover, many control schemes using neural networks have been proposed. For example, Kawato (Kawato, Furukawa, & Suzuki, 1987; Miyamoto et al., 1988; Kawato, 1990) proposed the feedback error learning method from the inspiration in biological systems. The neural network trained by the feedback error learning
In view of vehicle safety, the automatic braking control system, or rear-end collision avoidance system, is being developed vigorously. We also have developed an automatic braking control system for the longitudinal speed control of a vehicle. The purpose of this system is to make a vehicle decelerate smoothly and stop at the position just behind the preceding stopped vehicle. To make the vehicle stop automatically, the vehicle speed must be controlled by the braking control system according to the distance between the vehicle controlled and the vehicle ahead (see Figure 1). Using a conventional control method such as PID, however, we cannot achieve satisfactory control performance with regard to keeping a safety interval at stopped position. This is because the vehicle dynamics changes are affected by Acknowledgements: We would like to thank Dr. M. Kawato and Mr. Y. Koike of ATR Human Information Processing Research Laboratories, Mr. Y. Kuroda of Toyota Motor Corporation, and Mr. T. Hongo of Toyota Central Research & Development Labs., Inc., for help in the preparation of this manuscript. Requests for reprints should be sent to Hiroshi Ohno, Toyota Central Research & Development Labs., Inc., 41-1, Asa Yokomichi, Oaza Nagakute, Nagakute-cho, Aichi-gun, Aichi-ken, 480-1 l, Japan.
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FIGURE 1. A situation of using the automatic braking control system. The aim of this system is to control vehicle velocity in
deceleration automatically to stop the vehicle accurately behind the preceding stopped vehicle.
method can represent the inverse dynamics model of the controlled object. The neural network control based on the feedback error learning method achieves high control performance for robot manipulator control. Jordan and Rumelhart (Jordan, 1988) proposed the forward and inverse modeling. In this modeling, the forward model of the controlled object is learned, then the inverse model of the controlled object is learned by the error signal that is back propagated through the forward model. We require a new scheme to control vehicle speed in the case of the dynamics of the vehicle that changes frequently in terms of the variations in road conditions and vehicle characteristics in real-time. The dynamics of the vehicle also has nonlinear characteristics. The feedback error learning method has three advantages: 1) implementation is rather simple and easy, 2) control and learning can be simultaneously done, and 3 ) special teaching signals are not required. Therefore, we expect that the neural network control based on the feedback error learning method has high control performance for our purpose. From these backgrounds, we adopted the feedback error learning method and a three-layer neural network model so that the automatic braking control system could adapt to the variations in road conditions and vehicle characteristics in real-time. Generally, under running conditions of the vehicle, the longitudinal dynamics of the vehicle changes more frequently rather than that of robot manipulator because of the variations in road conditions and vehicle characteristics. But, the learning ability of the neural network control is not well known for these frequent changes of the dynamics of the controlled object. We confirmed the effectiveness of the learning ability of the neural network control in our application. On the other hand, several applications of neural networks to vehicle control and modeling have been reported. Pomerleau (1993) developed the vision-based vehicle lateral control using a three-layer neural network. In the vehicle lateral control, Kornhauser ( 1991 ) discussed the convergence property of learning with two types of neural networks, a three-layer neural network, and adaptive resonance theory network, in simulations. Giacomin ( 1991 ) developed a vehicle suspension model using a four-layer neural network. But, in these applications, learning of the neural network is done off-line, such as pattern recognition problems. We confirmed
the effectiveness of the real-time learning of neural networks in real-time in our application. In the next section, we describe an overview of the automatic braking control system. The neural network adaptive controller that is realized by a digital signal processor (DSP)-based hardware system to compute the high-speed calculation of the neural network model is described in detail. Finally, experimental results are shown and discussed. 2. AUTOMATIC BRAKING CONTROL SYSTEM
2.1. System Construction The block diagram of the automatic braking control system and a photograph of the experimental car are shown in Figure 2. The automatic braking control system consists of a desired pattern generator that is realized by a eight-bit microcomputer and a neural network adaptive controller that is realized by a DSP-based hardware system. The desired pattern generator outputs the time series data, that is, the desired patterns, which consist of the velocity and the deceleration, to make a vehicle stop at the rear of another vehicle stopped before it. The brake actuator has the hysteresis characteristics of the solenoid current input/brake pressure output relationship. Therefore, the controlled object has the nonlinear characteristics. In the neural network adaptive controller, the input signals are the desired velocity, the desired deceleration, and the actual velocity. The output signal is the solenoid current value that drives the brake actuator. In the automatic braking control system, the neural network adaptive controller adapts to the vehicle dynamics changes in road grade and vehicle weight by using only speed sensor signal. The vehicle dynamics changes are detected from the difference between the actual velocity and the desired velocity.
2.2. Desired Pattern Generating Method From the view of a good comfortable ride, it is desirable that the desired patterns should be smooth with respect to time. Several mathematical criteria have been proposed to provide for the smoothness performance (Flash & Hogan, 1985; Uno, Kawato, & Suzuki, 1989). For the smooth deceleration, we adopted the minimum jerk criterion to generate the desired patterns because the model based on its criterion was easily computed. The minimum jerk criterion is to minimize the integral of the square of the jerk (rate of change in deceleration ). According to this criterion, the deceleration is expressed by time third polynominals and the velocity is expressed by time forth polynominals. Each coefficient in the polynominals is uniquely determined by the maximum deceleration, the initial velocity, and the travel distance
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(b) Photograph FIGURE 2. BlOCk diagram of automatic braking control system and photograph of experimental car. The automatic braking control system consists of a desired pattern generator and a neural network adaptive controller. The brake actuator has the hysteresis charactedstics of the solenoid current input/the brake pressure ouput relationship.
until full stop at the specified position just behind the vehicle stopped ahead. Further details of deriving the desired patterns are provided in the Appendix. Figure 3 shows an example of desired patterns, the desired velocity, and the desired deceleration generated by the minimum jerk criterion. Here, the maximum deceleration was set to 0.2G ( 1G = 9.8 m/s2), the initial velocity was set to 30 km/h, and the travel distance was set to 40 m. Except in an emergency, 0.2G is usually adopted as the maximum deceleration, at which deceleration no one felt uncomfortable in our other experiments. The vehicle deceleration can be smoothly achieved by these desired patterns.
2.3. Neural Network Adaptive Controller The block diagram of the neural network adaptive controller is shown in Figure 4. The neural network adaptive controller consists of a feedforward controller using a three-layer neural network model and a proportional feedback controller. The output of the neural network adaptive controller, the solenoid current Is, is expressed by: Is = Kp.(V= - Vd) + F,(Vd, Gd, W).
(1)
W is the weight vector of the neural network model. F, is the function corresponding to the output that the neural network model generates from input signals. Vv is the actual velocity from the speed sensor, Vd is the desired velocity, Gd is the desired deceleration, and Kp is a proportional gain. The learning rule of the feedback error learning method is represented as: dW .(OF, It dt =*1 \OW] "Ifb here Ifb=Kp.(Vv-Va),
(2)
where ~ is the learning coefficient. As the learning of the neural network model proceeds, the feedforward control gradually achieves taking the place of the feedback control. Once the proportional feedback controller output becomes zero, that is, Vv = Va, the inverse dynamics model of the controlled object is acquired in the feedforward controller. The structure of the neural network model is shown in Figure 5. The desired velocity and the desired deceleration are given to the input layer. These input signals are normalized in a range between 0 and 1 in the forward calculation of the neural network. The number of the hidden units is 12. The input/output function
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FIGURE 3. An example of desired patterns (velocity and deceleration). These patterns are generated by the minimum jerk criterion. The maximum deceleration was set to 0.2G. Initial velocity was set to 30 km/h. The travel distance was set to 40 m.
of the hidden unit is a sigmoid function with asymptotes at 0 and 1. Calculation speed, which is provided by hardware systems, is an important factor in the design and implementation of neural networks, because it affects the control interval and the learning interval necessary for assuring stable learning. Therefore, we developed the DSP-based hardware system for the neural network adaptive controller that enables the high-speed calculation of the neural network model. Figure 6a shows the photograph of the DSP-based hardware system and Figure 6b shows the block diagram. Calculations of the neural network model are done in software. A 32-bit floating point DSP with a clock rate of 33.33 MHz was used to satisfy the need for high-speed floating point calculation capabilities. The I / O signals are received from and transmitted to external devices as 0 ~ 5 volt analog values. The proportional control is performed by an analog circuit.
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FIGURE 4. Block diagram of neural network adapt/re controller. The neural network adaptive controller consists of a proportional feedback controller and a feedforward controller using a three-layer neural network model. The learning of the feedforward controller is based on the feedback error leaming method. G~ is the desired deceleration, V~is the desired velocity, V, is the actual velocity, and I, is the solenoid current.
Initial weight values of the neural network model are transmitted from outside using a personal computer and are stored in Random access memory (RAM, 32KB). The number of hidden units and the learning coefficient are also stored in RAM. Calculation time contained both a feedforward and a backward calculation of the neural network model and can be varied from 1-10 ms. 3. LEARNING AND ADAlYI'IVE C O N T R O L RESULTS The learning of the neural network model is done with the following two steps: • Step 1: A specific pattern for deceleration from a steady initial velocity of 30 k m / h to 5 k m / h is learned. Other parameters are set at one occupant and a flat road. During the learning, the engine torque necessary for a initial velocity of 30 k m / h is maintained. • Step 2: The pattern learned in Step 1 is extended to a full stop, but the engine torque is reduced to 0 by disengaging the clutch. The reason for taking these two steps is that velocity control in deceleration by using brakes alone is nonlinear. For example, although positive brake pressure decelerates the vehicle, negative brake pressure, which means brakes are released, cannot accelerate the vehicle. This effect causes unstable learning when errors are large, because the learning process sometimes causes the feedforward controller to generate a meaningless output of negative brake pressure. However, if the engine torque is maintained, the vehicle will accelerate
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when the brakes are released. Step 1 provides the bias necessary for initial learning convergence. The control interval is 100 ms. Within 100 ms, the learning of the feedforward controller is performed and the output value of the neural network adaptive controller is calculated by the DSP-based hardware system.
3.1. Learning Results In the feedback error learning method, determination of the optimal feedback gain in the feedback controller is important for the accuracy of inverse dynamics model (Kawato, 1990). In our experiments, however, the optimal proportional gain could not be selected because of unknown vehicle dynamics. After some trials by the proportional feedback controller, the proportional gain was set at Kp = 1.0. Figure 7 shows the desired velocity and the desired deceleration pattern output from the desired pattern generator. The learning of the neural network was started at 0 s and stopped at 6 s. Twenty trial decelerations from 30 km/h to 5 km/h (Step 1 ) were performed followed by the decelerations to zero (i.e., full stop) ( Step 2 ). Figure 8 shows the brake pressure, the vehicle velocity, and the velocity error at the first, 10th, and 20th Step 1 trials and a Step 2 trial. All the weight values of the neural network model were initialized at random values in the range [-1, +1] for the initial learning stage. The learning coefficient of the back-propagation algorithm was 0.001 in Step 1 and 0.005 in Step 2. By the 20th trial of Step 1, the learning had converged. Although the learning was not done for taking the vehicle velocity from 5 k m / h to dead stop in Step 1, it always completed correctly on the first trial of Step 2. At the first trial of Step 1, the brake pressure had a peak at about 0.7 s. This was caused by the random values of the weights. As the trial proceeded, the brake pressure became smoother in shape, and the velocity error also converged to zero at the 20th trial. This indicates that the feedforward controller alone controls the vehicle velocity. Consequently, the inverse longitudinal dynamics model of the vehicle is aquired by repeating the trials. In addition, by keeping the weights of the neural network model intact after Step 2 and turning the
learning off, the generalization capability of the neural network model was evaluated. The evaluation was done under the following two situations: 1. Initial velocity at the commencement of deceleration is 20 km/h; maximum deceleration is 0.2G. 2. Initial velocity at the commencement of deceleration is 30 km/h; maximum deceleration is 0.15G. The results are shown in Figure 9. Both the learning of a single deceleration and the feedforward quantities learned during the learning process are effective, even under the conditions that the velocity at the commencement of deceleration and maximum deceleration values have been slightly changed. In Figure 9a, large errors are observed at the beginning of situation 1 because of tire friction. Therefore, after 5.5 s, the neural network adaptive controller was turned on, but learning turned off. So, the velocity error becomes zero. As a result, once the feedforward controller learned a deceleration trial, it could control the vehicle speed even under different situations of deceleration trials because of its generalization ability.
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to changes in the number of passengers or the degree of road gradient. The experimental conditions are as follows: 1. Weight values from Step 2 are maintained, learning is turned on, and vehicle weight increases by 200 kg. 2. All the conditions are maintained except that the deceleration experiment is conducted on a 3% downgrade. Figures 10 and 11 show the results of the control with learning on in real-time and off, respectively. In Figure 10, the vehicle stopping distance with learning offwas set five times longer than that with learning on. In Figure 11, stopping distance increased by about 1 m when the learning was turned on. However, with learning turned off, the vehicle could not come to a complete stop. Because the proportional feedback controller cannot act at a velocity of zero, the vehicle continues to roll slowly down the grade at about 3 km/h. Each result shows that the learning of feedforward quantities in real-time improved the velocity control performance. From these results, it would be possible to adapt the neural network adaptive controller to the changes in
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Fb is the braking force, which has certain nonlinear characteristics and is affected by the change in brake pad friction factor p. Is is the solenoid current, P is the brake cylinder pressure, which is a nonlinear function o f / , and Kis the constant. Fc is the travelling resistance. Fg is the component of gravity in the longitudinal direction due to the road gradient f~. my is the weight of the vehicle, mp is the weight of passengers and packages in the vehicle, and g is the gravity. Here, the variations that must be considered are changes in brake pad friction factor (p), gross vehicle weight (my + mp), and road grade (fl). All of these variations must be c.m-
In this section, we now discuss the inverse longitudinal dynamics model of the vehicle with the hysteresis characteristics of the brake actuator and consider the contribution with the input signals of the neural network model to control. Figure 12 shows the relationship between deceleration and forces acting upon the vehicle in the longitudinal dynamics model (e.g., Hauksdottir & Fenton, 1985; Frank, Liu, & Liang, 1989; Maretzke et al., 1990). The following equations are given to represent this relationship;
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pensated for when the solenoid current of the brake actuator is regulated. Now suppose that the road gradient 9 and the brake pad friction factor ~ are constant. From eqns ( 3 )- ( 6 ), we obtain, I/v :
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After learning, that is, Vv = Va, the inverse dynamics model of the vehicle is expressed as follows:
The inverse dynamics model is a function of only the desired deceleration Gal. Figure 13 shows the hysteresis characteristics of the solenoid current input/brake pressure output relationship. To compensate the hysteresis characteristics, the output of the neural network model must be a different value corresponding to a Gd value at two different times. Because the neural network model does not have an
inner state memory, its output cannot compensate for the hysteresis characteristics by only the Gd value. Therefore, the desired velocity, Vd, which is another input signal for the neural network model, would act effectively to compensate for the hysteresis characteristics. 5. CONCLUSION We have developed an automatic braking control system based on a three-layer neural network model using the feedback error learning method. The DSP-based hardware system was developed for the neural network adaptive controller to achieve the high-speed calculation of the neural network model. The experimental results indicate that the neural network adaptive controller can adapt to the vehicle dynamics changes in road grade and vehicle weight in real-time. The feedforward controller using a three-layer neural network model could learn the inverse dynamics model of the vehicle in spite of the hysteresis characteristics of the brake actuator. Because the input signals for the neural network model could be combined effectively by learning, the feedforward controller output (i.e., the neural network output) could compensate the hysteresis characteristics of the brake actuator. Although the optimal proportional gain
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neural network controller. IEEE Control Systems Magazine, pp. 17-21. Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323, 533536. Schiffmann, W. H., & Geffers, H. W. (1993). Adaptive control of dynamic systems by back propagation networks. Neural Networks, 6, 517-524. Uno, Y., Kawato, M., & Suzuki, R. (1989). Formation and control of optimal trajectory in human arm movement-minimum torquechange model. Biological Cybernetics, 61, 89-101.
SOLENOID CURRENT[A] FIGURE 13. Hysteresis characteristics of brake actuator.
was not chosen, the feedback error learning method was effective for the learning of the feedforward controller. We presented the effectiveness of the learning ability of the neural network model for the real-time learning control of the vehicle.
APPENDIX We derive expressions of the desired patterns as functions of time t. Let us consider three periods of time as shown in Figure A I. In the period t _< To, the desired deceleration Ao(t) is zero. Therefore, the desired velocity Vo(t) is equal to the initial velocity//in. In the period To -< t ~ To + T,, according to the minimum jerk criterion, the desired deceleration A~(t) can be expressed as:
At(t) = aat 3 + a2t 2 + art + ao,
(Al)
where ai(i = 0, 1, 2, 3) could be determined by the boundary conditions as follows:
REFERENCES Flash, T., & Hogan, N. ( 1985 ). The coordination of arm movements: An experimentally confirmed mathematical model. The Journal of Neuroscience, 5(7), 1688-1703. Frank, A. A., Liu, S. J., & Liang, S. C. (1989). Longitudinal control concepts for automated automobiles and trucks operating on a cooperative highway. SAE Technical Paper Series, No. 891708. Funahashi, K. (1989). On the approximate realization of continuous mapping by neural networks. Neural Networks, 2, 183-192. Giacomin, J. (1991). Neural network simulation of an automotive shock absorber. Engng Applic. Artif Intell., 4, 1, 59-64. Hauksdottir, A. S., & Fenton, R. E ( 1985 ). On the design of a vehicle longitudinal controller. IEEE Transactions on Vehicle Technology VT-34(4), 182-187. Jordan, M. I. (1988). Supervised learning and systems with ~:~'cess degrees ~dTkeedom (COINS Tech. Rep. 88, 27, 1-41), Amherst, MA: University of Massachusetts, Computer and Information Sciences. Kawato, M. (1990). Feedback-error-learning in neural networks for supervised motor learning. In R. Eckmiller (Ed.), Advanced neural computers (pp. 365-372). Amsterdam: Elsevier. Kawato, M., Furukawa, K., & Suzuki, R. (1987). A hierarchical neural network model for control and learning of voluntary movement. Biological Cybernetics, 57, 169-185. Kornhauser, A. L. ( 1991 ). Neural network approaches for lateral control of autonomous highway vehicles. SAE Technical Paper Series, No. 912871. Levin, E., Gewirtzman, R., & lnbar, G. E (1991). Neural-network architecture for adaptive system modeling and control. Neural Networks, 4, 185-191. Li, W., & Slotine, J. E. (1989). Neural network control of unknown nonlinear systems. In Proceedings of the American Control Conference, Pittsburgh, PA, June. Maretzke, J., Dreyer, W., Hoppe, P., & Jacob, U. (1990). Systems for driver support in the longitudinal and lateral control of motor vehicles. SAE Technical Paper Series, No. 905165. Miyamoto, H., Kawato, M., Setoyama, T., & Suzuki, R. (1988). Feedback-error-learning neural network for trajectory control of a robotic manipulator. Neural Networks, 1, 251-265. Pomerleau, D. A. ( 1993 ). Neural network perception for mobile robot guidance. London: Kluwer. Psaltis, D., Sideris, A., & Yamamura, A. ( 1988, April). A multilayered
At(To) = 0
(A2)
A,(To + T,) = Amax
(A3)
A,(T0) = -4,(T0 + T,) = 0,
(A4)
where the dot denotes the derivation with respect to time t. Therefore, we obtain: A,(/) = -
2Am~x 3A,~ T--'T"I(t - To) 3 + T--T'-~(t - To) 2,
(A5)
where A~,~ is the maximum deceleration, which is given beforehand. The desired velocity Vt (t) is derived by integrating A, (t) with respect to time as follows:
Vl(t) = V i , -
A,(r)dT.
(A6)
From eqn (A6), we obtain: Am~ V,(t)= V i . + ~ 1 3 ( t -
Amp T0)4---~-~-t2 ( t - To) 3.
(A7)
In the period To + Tl < t ~ To + 2T,, the desired deceleration A2(t) can also be expressed as: AE(t) = a3t3 + a2t 2 + ad + ao.
(A8)
We also assume the boundary conditions as follows: A2(To + Tl) = A,,~,
(A9)
Amax
0
f~
TO
~
T1
~
TIME t
T1
FIGURE A1. Desired deceleration.
1312
H. Ohno et al. A2(To + 2T)) = 0
(AI0)
A2(To + T,) = A2(T o + 2 T , ) = 0.
(All)
Using eqns (A5) and ( A I 2 ) , we obtain: [/in
T~ =
.
(AI6)
Amax
Using these conditions, we obtain: Next, we determine To using Ama~ and 1/].. We consider the travel distance l~r, in the period To + T~ <_ t <- To + 2T.. The travel distance l~a~ can be calculated as follows:
A2(t) = 2A=~ T--T- (t - T O - TI) 3
3 A m ~ ( t - T o - T~) 2 + Am~.
(AI2)
l~n =
~T~0+Tt Vt ( r ) d r
~TS)+27"1l,~(r)dr.
+
(AI7)
+TI
Using eqns (A7) and ( A I 4 ) , we obtain: Then, the desired velocity V2(t) is also calculated as follows: W 2
V2(t) ~ Vt(To + Ti) -
£
l~rt
A2(r)dr
Am.x -(t-
2T;
Amax
+-'-~(t-
Therefore, To can be calculated as follows:
T o - Tj) 4
To
Ls
/part L~ 1/in
T o - T ~ ) x - a ~ x ( t - T o - T~).
(AI4)
l f A ~ and ~ . are known, because at the time To + 2T~ the velocity becomes zero, T, can be calculated using eqns (A5) and (A 12) as in the following equation: Vi. -
(AI8)
(AI3)
+T I
_ V~. 2
Amax"
fT~IA l ( r ) d r - fTT°+2TIA 2 ( r ) d r I
= 0.
(AI5)
Vm
V~,
(AI9)
Amax'
where L, is the travel distance until the specified position, which is given beforehand. Here, because To >- 0, A ~ , must satisfy the following condition:
v~
,1max > - - • L,
(A20)
Thus, from eqns (A5), (A7), ( A I 2 ) , (A14), ( A I 6 ) , and ( A I 9 ) , we can calculate the desired patterns using A . . . . V~,, and L~.
Pergamon 0893-6080( 94 ) E0001-2
CONTRIBUTED ARTICLE
Neural Network Control for Automatic Braking Control System HIROSHI OHNO, 1 TOSHIHIKO SUZUKI, 2 KEIJI AOKI, 2 ARATA TAKAHASI, l AND GUNJI SUGIMOTO 1 ~Toyota Central Research & Development Labs., Inc. and 2Toyota Motor Corporation
(Received 10 September 1993; revised and accepted 21 December 1993)
Abstract--We have developed an automatic braking control system for automobiles applying a three-layer neural network model. This system enables the vehicle to decelerate smoothly and to stop at the specified position behind the vehicle ahead. This paper focuses on the use of the three-layer neural network model in the automatic braking control system. Because vehicle dynamics is varied by the variations in road conditions or vehicle characteristics, it cannot be represented accurately by mathematical models. According to this reason, the conventional control methods, such as proportional integrative derivative ( PID ) control, cannot achieve satisfactory control performance. Therefore, we have constructed the neural network adaptive control system based on the feedback error learning method. This learning method enables the system to adapt to the changes in road grade and vehicle weight without using any specific sensors. Experimental results show satisfactory control performance and reveal that the neural network adaptive control system based on the feedback error learning method is available for the automatic braking control system.
Keywords--Feedback error learning, Neural networks, Feedforward control, Automatic braking control system, Vehicle control, Hysteresis characteristics. 1. INTRODUCTION
the variations in road conditions (e.g., road friction, road grade, etc.) and vehicle characteristics (e.g., vehicle weight, brake pad friction, etc.), and the automatic braking control system does not work well by constant PID gains. Moreover, if we use variable gains, PID gain tuning is very difficult adapting to the variations in road conditions and vehicle characteristics. Therefore, we decided to make a new control scheme. In recent years, some control applications of neural networks have been reported (Psaltis, Sideris, & Yamamura, 1988; Li & Slotine, 1989; Levin, Gewirtzman, & Inbar, 1991; Schiffmann & Geffers, 1993). These applications use multilayer neural networks as well as back propagation (Rumelhart, Hinton, & Williams, 1986) as its learning algorithm. Because it is clarified that a three-layer neural network can theoretically construct a universal approximator (Funahashi, 1989), multilayer neural networks have been used as controllers and identifiers for nonlinear controlled objects. Moreover, many control schemes using neural networks have been proposed. For example, Kawato (Kawato, Furukawa, & Suzuki, 1987; Miyamoto et al., 1988; Kawato, 1990) proposed the feedback error learning method from the inspiration in biological systems. The neural network trained by the feedback error learning
In view of vehicle safety, the automatic braking control system, or rear-end collision avoidance system, is being developed vigorously. We also have developed an automatic braking control system for the longitudinal speed control of a vehicle. The purpose of this system is to make a vehicle decelerate smoothly and stop at the position just behind the preceding stopped vehicle. To make the vehicle stop automatically, the vehicle speed must be controlled by the braking control system according to the distance between the vehicle controlled and the vehicle ahead (see Figure 1). Using a conventional control method such as PID, however, we cannot achieve satisfactory control performance with regard to keeping a safety interval at stopped position. This is because the vehicle dynamics changes are affected by Acknowledgements: We would like to thank Dr. M. Kawato and Mr. Y. Koike of ATR Human Information Processing Research Laboratories, Mr. Y. Kuroda of Toyota Motor Corporation, and Mr. T. Hongo of Toyota Central Research & Development Labs., Inc., for help in the preparation of this manuscript. Requests for reprints should be sent to Hiroshi Ohno, Toyota Central Research & Development Labs., Inc., 41-1, Asa Yokomichi, Oaza Nagakute, Nagakute-cho, Aichi-gun, Aichi-ken, 480-1 l, Japan.
1303
1304 Stopped v e h i c l e
H. Ohno et al. <
Oi stance
>
Stop vehicle automatical ly
FIGURE 1. A situation of using the automatic braking control system. The aim of this system is to control vehicle velocity in
deceleration automatically to stop the vehicle accurately behind the preceding stopped vehicle.
method can represent the inverse dynamics model of the controlled object. The neural network control based on the feedback error learning method achieves high control performance for robot manipulator control. Jordan and Rumelhart (Jordan, 1988) proposed the forward and inverse modeling. In this modeling, the forward model of the controlled object is learned, then the inverse model of the controlled object is learned by the error signal that is back propagated through the forward model. We require a new scheme to control vehicle speed in the case of the dynamics of the vehicle that changes frequently in terms of the variations in road conditions and vehicle characteristics in real-time. The dynamics of the vehicle also has nonlinear characteristics. The feedback error learning method has three advantages: 1) implementation is rather simple and easy, 2) control and learning can be simultaneously done, and 3 ) special teaching signals are not required. Therefore, we expect that the neural network control based on the feedback error learning method has high control performance for our purpose. From these backgrounds, we adopted the feedback error learning method and a three-layer neural network model so that the automatic braking control system could adapt to the variations in road conditions and vehicle characteristics in real-time. Generally, under running conditions of the vehicle, the longitudinal dynamics of the vehicle changes more frequently rather than that of robot manipulator because of the variations in road conditions and vehicle characteristics. But, the learning ability of the neural network control is not well known for these frequent changes of the dynamics of the controlled object. We confirmed the effectiveness of the learning ability of the neural network control in our application. On the other hand, several applications of neural networks to vehicle control and modeling have been reported. Pomerleau (1993) developed the vision-based vehicle lateral control using a three-layer neural network. In the vehicle lateral control, Kornhauser ( 1991 ) discussed the convergence property of learning with two types of neural networks, a three-layer neural network, and adaptive resonance theory network, in simulations. Giacomin ( 1991 ) developed a vehicle suspension model using a four-layer neural network. But, in these applications, learning of the neural network is done off-line, such as pattern recognition problems. We confirmed
the effectiveness of the real-time learning of neural networks in real-time in our application. In the next section, we describe an overview of the automatic braking control system. The neural network adaptive controller that is realized by a digital signal processor (DSP)-based hardware system to compute the high-speed calculation of the neural network model is described in detail. Finally, experimental results are shown and discussed. 2. AUTOMATIC BRAKING CONTROL SYSTEM
2.1. System Construction The block diagram of the automatic braking control system and a photograph of the experimental car are shown in Figure 2. The automatic braking control system consists of a desired pattern generator that is realized by a eight-bit microcomputer and a neural network adaptive controller that is realized by a DSP-based hardware system. The desired pattern generator outputs the time series data, that is, the desired patterns, which consist of the velocity and the deceleration, to make a vehicle stop at the rear of another vehicle stopped before it. The brake actuator has the hysteresis characteristics of the solenoid current input/brake pressure output relationship. Therefore, the controlled object has the nonlinear characteristics. In the neural network adaptive controller, the input signals are the desired velocity, the desired deceleration, and the actual velocity. The output signal is the solenoid current value that drives the brake actuator. In the automatic braking control system, the neural network adaptive controller adapts to the vehicle dynamics changes in road grade and vehicle weight by using only speed sensor signal. The vehicle dynamics changes are detected from the difference between the actual velocity and the desired velocity.
2.2. Desired Pattern Generating Method From the view of a good comfortable ride, it is desirable that the desired patterns should be smooth with respect to time. Several mathematical criteria have been proposed to provide for the smoothness performance (Flash & Hogan, 1985; Uno, Kawato, & Suzuki, 1989). For the smooth deceleration, we adopted the minimum jerk criterion to generate the desired patterns because the model based on its criterion was easily computed. The minimum jerk criterion is to minimize the integral of the square of the jerk (rate of change in deceleration ). According to this criterion, the deceleration is expressed by time third polynominals and the velocity is expressed by time forth polynominals. Each coefficient in the polynominals is uniquely determined by the maximum deceleration, the initial velocity, and the travel distance
Neural Network Control
1305 ...." Automatic braking control system
,." :."
Desired patterns (velocity, deceleration)
Solenoid current
De;iredpattern~Neural network p=.i-00o. o .,I~ e Actual velocity .........]
Brake pressure
a'~tuat~ /
~..~
~
(a) BIock diagram
(b) Photograph FIGURE 2. BlOCk diagram of automatic braking control system and photograph of experimental car. The automatic braking control system consists of a desired pattern generator and a neural network adaptive controller. The brake actuator has the hysteresis charactedstics of the solenoid current input/the brake pressure ouput relationship.
until full stop at the specified position just behind the vehicle stopped ahead. Further details of deriving the desired patterns are provided in the Appendix. Figure 3 shows an example of desired patterns, the desired velocity, and the desired deceleration generated by the minimum jerk criterion. Here, the maximum deceleration was set to 0.2G ( 1G = 9.8 m/s2), the initial velocity was set to 30 km/h, and the travel distance was set to 40 m. Except in an emergency, 0.2G is usually adopted as the maximum deceleration, at which deceleration no one felt uncomfortable in our other experiments. The vehicle deceleration can be smoothly achieved by these desired patterns.
2.3. Neural Network Adaptive Controller The block diagram of the neural network adaptive controller is shown in Figure 4. The neural network adaptive controller consists of a feedforward controller using a three-layer neural network model and a proportional feedback controller. The output of the neural network adaptive controller, the solenoid current Is, is expressed by: Is = Kp.(V= - Vd) + F,(Vd, Gd, W).
(1)
W is the weight vector of the neural network model. F, is the function corresponding to the output that the neural network model generates from input signals. Vv is the actual velocity from the speed sensor, Vd is the desired velocity, Gd is the desired deceleration, and Kp is a proportional gain. The learning rule of the feedback error learning method is represented as: dW .(OF, It dt =*1 \OW] "Ifb here Ifb=Kp.(Vv-Va),
(2)
where ~ is the learning coefficient. As the learning of the neural network model proceeds, the feedforward control gradually achieves taking the place of the feedback control. Once the proportional feedback controller output becomes zero, that is, Vv = Va, the inverse dynamics model of the controlled object is acquired in the feedforward controller. The structure of the neural network model is shown in Figure 5. The desired velocity and the desired deceleration are given to the input layer. These input signals are normalized in a range between 0 and 1 in the forward calculation of the neural network. The number of the hidden units is 12. The input/output function
1306
H. Ohno et al.
30km/h
0.26 2.00
8.00
-
7.00
-
6.00
-
\ Z O
-
1.80
-
1.60
-
1,40
-
1.20
-
I----
>P"
5.00
-
4.00
-
l.IJ o~
~J >
¢~
E ~
3.00
-
2.00
-
1.00 - -
0.00
=<
1.00
--
u.=
0.80
--
0.60
-
0.40
--
0.20
--
0.00
-
E
/
I
-
0.O0
2.00
4.00
6.00
I 6.00
10.00
0.00
2.00
4,00
6.00
8,00
10.00
TIME
TIME
[sac]
[sec] (b) Desired deceleration
(a) Desired velocity
FIGURE 3. An example of desired patterns (velocity and deceleration). These patterns are generated by the minimum jerk criterion. The maximum deceleration was set to 0.2G. Initial velocity was set to 30 km/h. The travel distance was set to 40 m.
of the hidden unit is a sigmoid function with asymptotes at 0 and 1. Calculation speed, which is provided by hardware systems, is an important factor in the design and implementation of neural networks, because it affects the control interval and the learning interval necessary for assuring stable learning. Therefore, we developed the DSP-based hardware system for the neural network adaptive controller that enables the high-speed calculation of the neural network model. Figure 6a shows the photograph of the DSP-based hardware system and Figure 6b shows the block diagram. Calculations of the neural network model are done in software. A 32-bit floating point DSP with a clock rate of 33.33 MHz was used to satisfy the need for high-speed floating point calculation capabilities. The I / O signals are received from and transmitted to external devices as 0 ~ 5 volt analog values. The proportional control is performed by an analog circuit.
_1 >t Fe.d~..... d co~tro~ ~er
Gd Vd
VV
-
''J.,7 ( t h r e e - l a y e r
~Proportiona[
neural
feedback
/
network) ~[ I
control
>
I S
FIGURE 4. Block diagram of neural network adapt/re controller. The neural network adaptive controller consists of a proportional feedback controller and a feedforward controller using a three-layer neural network model. The learning of the feedforward controller is based on the feedback error leaming method. G~ is the desired deceleration, V~is the desired velocity, V, is the actual velocity, and I, is the solenoid current.
Initial weight values of the neural network model are transmitted from outside using a personal computer and are stored in Random access memory (RAM, 32KB). The number of hidden units and the learning coefficient are also stored in RAM. Calculation time contained both a feedforward and a backward calculation of the neural network model and can be varied from 1-10 ms. 3. LEARNING AND ADAlYI'IVE C O N T R O L RESULTS The learning of the neural network model is done with the following two steps: • Step 1: A specific pattern for deceleration from a steady initial velocity of 30 k m / h to 5 k m / h is learned. Other parameters are set at one occupant and a flat road. During the learning, the engine torque necessary for a initial velocity of 30 k m / h is maintained. • Step 2: The pattern learned in Step 1 is extended to a full stop, but the engine torque is reduced to 0 by disengaging the clutch. The reason for taking these two steps is that velocity control in deceleration by using brakes alone is nonlinear. For example, although positive brake pressure decelerates the vehicle, negative brake pressure, which means brakes are released, cannot accelerate the vehicle. This effect causes unstable learning when errors are large, because the learning process sometimes causes the feedforward controller to generate a meaningless output of negative brake pressure. However, if the engine torque is maintained, the vehicle will accelerate
1307
Neural Network Control
InputlaY~er/~
layer
Hidden layer FIGURE 5. Structure of a neural network model.
when the brakes are released. Step 1 provides the bias necessary for initial learning convergence. The control interval is 100 ms. Within 100 ms, the learning of the feedforward controller is performed and the output value of the neural network adaptive controller is calculated by the DSP-based hardware system.
3.1. Learning Results In the feedback error learning method, determination of the optimal feedback gain in the feedback controller is important for the accuracy of inverse dynamics model (Kawato, 1990). In our experiments, however, the optimal proportional gain could not be selected because of unknown vehicle dynamics. After some trials by the proportional feedback controller, the proportional gain was set at Kp = 1.0. Figure 7 shows the desired velocity and the desired deceleration pattern output from the desired pattern generator. The learning of the neural network was started at 0 s and stopped at 6 s. Twenty trial decelerations from 30 km/h to 5 km/h (Step 1 ) were performed followed by the decelerations to zero (i.e., full stop) ( Step 2 ). Figure 8 shows the brake pressure, the vehicle velocity, and the velocity error at the first, 10th, and 20th Step 1 trials and a Step 2 trial. All the weight values of the neural network model were initialized at random values in the range [-1, +1] for the initial learning stage. The learning coefficient of the back-propagation algorithm was 0.001 in Step 1 and 0.005 in Step 2. By the 20th trial of Step 1, the learning had converged. Although the learning was not done for taking the vehicle velocity from 5 k m / h to dead stop in Step 1, it always completed correctly on the first trial of Step 2. At the first trial of Step 1, the brake pressure had a peak at about 0.7 s. This was caused by the random values of the weights. As the trial proceeded, the brake pressure became smoother in shape, and the velocity error also converged to zero at the 20th trial. This indicates that the feedforward controller alone controls the vehicle velocity. Consequently, the inverse longitudinal dynamics model of the vehicle is aquired by repeating the trials. In addition, by keeping the weights of the neural network model intact after Step 2 and turning the
learning off, the generalization capability of the neural network model was evaluated. The evaluation was done under the following two situations: 1. Initial velocity at the commencement of deceleration is 20 km/h; maximum deceleration is 0.2G. 2. Initial velocity at the commencement of deceleration is 30 km/h; maximum deceleration is 0.15G. The results are shown in Figure 9. Both the learning of a single deceleration and the feedforward quantities learned during the learning process are effective, even under the conditions that the velocity at the commencement of deceleration and maximum deceleration values have been slightly changed. In Figure 9a, large errors are observed at the beginning of situation 1 because of tire friction. Therefore, after 5.5 s, the neural network adaptive controller was turned on, but learning turned off. So, the velocity error becomes zero. As a result, once the feedforward controller learned a deceleration trial, it could control the vehicle speed even under different situations of deceleration trials because of its generalization ability.
(a) P h o t o g r a p h
A/D converter Desired
(Gd)
inputO
D/A converter /
~DSP r~ ~ (TMS320C30) I ~RAM32KB _
Desired (Vd) input C) ~ Measured inputO (Vv)
~pl
i~
Output t~Pr°p°rti°nal Voutput
(b) BI0ck diagram FIGURE 6. Photograph and block diagram of DSP-based hardware system for the neural network adaptive controller. Calculations of the neural network model are done by DSP in software. Calculation time can be varied from 1-10 ms.
1308
H. Ohno et al.
~-~" 0 r l ' ' ' l ' ' - l ' ' ' l ' ' - l ' ' ' l ' ' ' i ' 0 2 4 6
,
to changes in the number of passengers or the degree of road gradient. The experimental conditions are as follows: 1. Weight values from Step 2 are maintained, learning is turned on, and vehicle weight increases by 200 kg. 2. All the conditions are maintained except that the deceleration experiment is conducted on a 3% downgrade. Figures 10 and 11 show the results of the control with learning on in real-time and off, respectively. In Figure 10, the vehicle stopping distance with learning offwas set five times longer than that with learning on. In Figure 11, stopping distance increased by about 1 m when the learning was turned on. However, with learning turned off, the vehicle could not come to a complete stop. Because the proportional feedback controller cannot act at a velocity of zero, the vehicle continues to roll slowly down the grade at about 3 km/h. Each result shows that the learning of feedforward quantities in real-time improved the velocity control performance. From these results, it would be possible to adapt the neural network adaptive controller to the changes in
,
8
]0 TIME
[sec]
(a) Oesi red v e l o c i t y
-~ 0 s
,
0
i " - " l - " ' l - - - 1 - - - i - ' " l - ' - r" " ' r " "'r'l 2 4 s 8 10
TIME
[sec]
(b) Desired deceleration FIGURE 7. Desired velocity and deceleration patterns. The maximum deceleration was set to 0.2G.
3.2. Adaptive Control Results Further experiments were conducted to ascertain the adaptability of the neural network adaptive controller 1st t r i a l
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on Stepl
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lnO TIME [sec]
FIGURE 8. Expedmental results of leaming process. The leaming of the feedforward controller is started at 0 s and is stopped at 6 s. Brake pressure, vehicle velocity, and velocity error profiles at the first, 10th, and 20th Step I trials and one Step 2 tdaL
Neural Network Control
1309
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(b) Changed max G from 0.26 to O. 15G
to 20[km/h]
FIGURE 9. Experimental results in Step 2 of leaming process. (a) Changed initial velocity from 30 km/h to 20 km/h; (b) changed maximum deceleration from O.2G to 0.15G.
road grade and vehicle weight in real-time. Therefore, the effectiveness of the learning ability of the neural network model for the frequent changes of the longitudinal dynamics of the vehicle is presented.
4.
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100
(3)
Fb is the braking force, which has certain nonlinear characteristics and is affected by the change in brake pad friction factor p. Is is the solenoid current, P is the brake cylinder pressure, which is a nonlinear function o f / , and Kis the constant. Fc is the travelling resistance. Fg is the component of gravity in the longitudinal direction due to the road gradient f~. my is the weight of the vehicle, mp is the weight of passengers and packages in the vehicle, and g is the gravity. Here, the variations that must be considered are changes in brake pad friction factor (p), gross vehicle weight (my + mp), and road grade (fl). All of these variations must be c.m-
In this section, we now discuss the inverse longitudinal dynamics model of the vehicle with the hysteresis characteristics of the brake actuator and consider the contribution with the input signals of the neural network model to control. Figure 12 shows the relationship between deceleration and forces acting upon the vehicle in the longitudinal dynamics model (e.g., Hauksdottir & Fenton, 1985; Frank, Liu, & Liang, 1989; Maretzke et al., 1990). The following equations are given to represent this relationship;
§
l ~ . ( F b + Fc+ Fg) m~ + mp
i I
I
I
I
I
2
I
I
I
I
4
I
I
I
I
6
I
I
I
]
8
I
I
I
I
i
i
]
I
1
I
4
I
I
I
I
I
I
6
I
I
I
I
I
I
8
o
§~-lz
"1
10 I
I
]
I
2
4
6
8
i
]0
I
J|
,1
lO
100
_ ~ so I
i
2
,
i
4
i
,
6
,-
]
8
I
I
10 TIME [sec]
(a) Learning ON: Stopping distance prolonged O. J[m]
~
o 0
2
4
6
8
10 TIME [sec] (b) Learning OFF: Stopping distance prolonged 0.5[m]
FIGURE 10. Experimental results. (a) Leaming on; (b) leaming off. Vehicle weight is increased by 200 kg.
1310
H. Ohno et al.
60 f . . . . . . . . %, 30
{~
o ~---~---~--0
"
I
2
4
I
J l
~ I
6
i
1
I 8 I
I 10 I
'
0
°
2
4
I
'
'
'
,
,
i
t
2' "
T ' - - r ' - ' r ' - ' r ' - 6T - 4
' "
i'oI
8
O----
P~-12-1 ~-~ 0
[
2
4
I
6
I
,-",-','--,---,'-','-, 2 4 6
0
8
"'-r
I
l0 TIME
Csec] (a) Learning ON: Stopping distance prolonged l[m]
§~-12
~
o
100 F I
~
0
L
O~
2
,
,
i
i
4
t
l
i
l
e
,
!'0 I
!
I~
'
'1
10 TIME
[s~]
(b) Learning OFF: No stopping
FIGURE 11. Experimental results with 3% down grade. (a) Learning on; (b) learning off.
pensated for when the solenoid current of the brake actuator is regulated. Now suppose that the road gradient 9 and the brake pad friction factor ~ are constant. From eqns ( 3 )- ( 6 ), we obtain, I/v :
Cl ° P(Is) + C2,
(7)
where Cl and C2 are constant. Assume that P is invertible, that is, p-l, then we obtain,
i~:
p -\--~--, l( v- c2/).
(8)
After learning, that is, Vv = Va, the inverse dynamics model of the vehicle is expressed as follows:
The inverse dynamics model is a function of only the desired deceleration Gal. Figure 13 shows the hysteresis characteristics of the solenoid current input/brake pressure output relationship. To compensate the hysteresis characteristics, the output of the neural network model must be a different value corresponding to a Gd value at two different times. Because the neural network model does not have an
inner state memory, its output cannot compensate for the hysteresis characteristics by only the Gd value. Therefore, the desired velocity, Vd, which is another input signal for the neural network model, would act effectively to compensate for the hysteresis characteristics. 5. CONCLUSION We have developed an automatic braking control system based on a three-layer neural network model using the feedback error learning method. The DSP-based hardware system was developed for the neural network adaptive controller to achieve the high-speed calculation of the neural network model. The experimental results indicate that the neural network adaptive controller can adapt to the vehicle dynamics changes in road grade and vehicle weight in real-time. The feedforward controller using a three-layer neural network model could learn the inverse dynamics model of the vehicle in spite of the hysteresis characteristics of the brake actuator. Because the input signals for the neural network model could be combined effectively by learning, the feedforward controller output (i.e., the neural network output) could compensate the hysteresis characteristics of the brake actuator. Although the optimal proportional gain
Neural NetworkModel
FIGURE 12. Block d i a g r a m of control s y s t e m with the neural n e t w o r k a d a p t i v e controller b a a e d on the f e e d b a c k error learning method.
Neural Network Control 0.1Hz
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E
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neural network controller. IEEE Control Systems Magazine, pp. 17-21. Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323, 533536. Schiffmann, W. H., & Geffers, H. W. (1993). Adaptive control of dynamic systems by back propagation networks. Neural Networks, 6, 517-524. Uno, Y., Kawato, M., & Suzuki, R. (1989). Formation and control of optimal trajectory in human arm movement-minimum torquechange model. Biological Cybernetics, 61, 89-101.
SOLENOID CURRENT[A] FIGURE 13. Hysteresis characteristics of brake actuator.
was not chosen, the feedback error learning method was effective for the learning of the feedforward controller. We presented the effectiveness of the learning ability of the neural network model for the real-time learning control of the vehicle.
APPENDIX We derive expressions of the desired patterns as functions of time t. Let us consider three periods of time as shown in Figure A I. In the period t _< To, the desired deceleration Ao(t) is zero. Therefore, the desired velocity Vo(t) is equal to the initial velocity//in. In the period To -< t ~ To + T,, according to the minimum jerk criterion, the desired deceleration A~(t) can be expressed as:
At(t) = aat 3 + a2t 2 + art + ao,
(Al)
where ai(i = 0, 1, 2, 3) could be determined by the boundary conditions as follows:
REFERENCES Flash, T., & Hogan, N. ( 1985 ). The coordination of arm movements: An experimentally confirmed mathematical model. The Journal of Neuroscience, 5(7), 1688-1703. Frank, A. A., Liu, S. J., & Liang, S. C. (1989). Longitudinal control concepts for automated automobiles and trucks operating on a cooperative highway. SAE Technical Paper Series, No. 891708. Funahashi, K. (1989). On the approximate realization of continuous mapping by neural networks. Neural Networks, 2, 183-192. Giacomin, J. (1991). Neural network simulation of an automotive shock absorber. Engng Applic. Artif Intell., 4, 1, 59-64. Hauksdottir, A. S., & Fenton, R. E ( 1985 ). On the design of a vehicle longitudinal controller. IEEE Transactions on Vehicle Technology VT-34(4), 182-187. Jordan, M. I. (1988). Supervised learning and systems with ~:~'cess degrees ~dTkeedom (COINS Tech. Rep. 88, 27, 1-41), Amherst, MA: University of Massachusetts, Computer and Information Sciences. Kawato, M. (1990). Feedback-error-learning in neural networks for supervised motor learning. In R. Eckmiller (Ed.), Advanced neural computers (pp. 365-372). Amsterdam: Elsevier. Kawato, M., Furukawa, K., & Suzuki, R. (1987). A hierarchical neural network model for control and learning of voluntary movement. Biological Cybernetics, 57, 169-185. Kornhauser, A. L. ( 1991 ). Neural network approaches for lateral control of autonomous highway vehicles. SAE Technical Paper Series, No. 912871. Levin, E., Gewirtzman, R., & lnbar, G. E (1991). Neural-network architecture for adaptive system modeling and control. Neural Networks, 4, 185-191. Li, W., & Slotine, J. E. (1989). Neural network control of unknown nonlinear systems. In Proceedings of the American Control Conference, Pittsburgh, PA, June. Maretzke, J., Dreyer, W., Hoppe, P., & Jacob, U. (1990). Systems for driver support in the longitudinal and lateral control of motor vehicles. SAE Technical Paper Series, No. 905165. Miyamoto, H., Kawato, M., Setoyama, T., & Suzuki, R. (1988). Feedback-error-learning neural network for trajectory control of a robotic manipulator. Neural Networks, 1, 251-265. Pomerleau, D. A. ( 1993 ). Neural network perception for mobile robot guidance. London: Kluwer. Psaltis, D., Sideris, A., & Yamamura, A. ( 1988, April). A multilayered
At(To) = 0
(A2)
A,(To + T,) = Amax
(A3)
A,(T0) = -4,(T0 + T,) = 0,
(A4)
where the dot denotes the derivation with respect to time t. Therefore, we obtain: A,(/) = -
2Am~x 3A,~ T--'T"I(t - To) 3 + T--T'-~(t - To) 2,
(A5)
where A~,~ is the maximum deceleration, which is given beforehand. The desired velocity Vt (t) is derived by integrating A, (t) with respect to time as follows:
Vl(t) = V i , -
A,(r)dT.
(A6)
From eqn (A6), we obtain: Am~ V,(t)= V i . + ~ 1 3 ( t -
Amp T0)4---~-~-t2 ( t - To) 3.
(A7)
In the period To + Tl < t ~ To + 2T,, the desired deceleration A2(t) can also be expressed as: AE(t) = a3t3 + a2t 2 + ad + ao.
(A8)
We also assume the boundary conditions as follows: A2(To + Tl) = A,,~,
(A9)
Amax
0
f~
TO
~
T1
~
TIME t
T1
FIGURE A1. Desired deceleration.
1312
H. Ohno et al. A2(To + 2T)) = 0
(AI0)
A2(To + T,) = A2(T o + 2 T , ) = 0.
(All)
Using eqns (A5) and ( A I 2 ) , we obtain: [/in
T~ =
.
(AI6)
Amax
Using these conditions, we obtain: Next, we determine To using Ama~ and 1/].. We consider the travel distance l~r, in the period To + T~ <_ t <- To + 2T.. The travel distance l~a~ can be calculated as follows:
A2(t) = 2A=~ T--T- (t - T O - TI) 3
3 A m ~ ( t - T o - T~) 2 + Am~.
(AI2)
l~n =
~T~0+Tt Vt ( r ) d r
~TS)+27"1l,~(r)dr.
+
(AI7)
+TI
Using eqns (A7) and ( A I 4 ) , we obtain: Then, the desired velocity V2(t) is also calculated as follows: W 2
V2(t) ~ Vt(To + Ti) -
£
l~rt
A2(r)dr
Am.x -(t-
2T;
Amax
+-'-~(t-
Therefore, To can be calculated as follows:
T o - Tj) 4
To
Ls
/part L~ 1/in
T o - T ~ ) x - a ~ x ( t - T o - T~).
(AI4)
l f A ~ and ~ . are known, because at the time To + 2T~ the velocity becomes zero, T, can be calculated using eqns (A5) and (A 12) as in the following equation: Vi. -
(AI8)
(AI3)
+T I
_ V~. 2
Amax"
fT~IA l ( r ) d r - fTT°+2TIA 2 ( r ) d r I
= 0.
(AI5)
Vm
V~,
(AI9)
Amax'
where L, is the travel distance until the specified position, which is given beforehand. Here, because To >- 0, A ~ , must satisfy the following condition:
v~
,1max > - - • L,
(A20)
Thus, from eqns (A5), (A7), ( A I 2 ) , (A14), ( A I 6 ) , and ( A I 9 ) , we can calculate the desired patterns using A . . . . V~,, and L~.