Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
WeP08-3
Identification and Control for Future Restraint Systems R.J. Hesseling*,**, M. Steinbuch*, F.E. Veldpaus*, T. Klisch** * Eindhoven University of Technology Faculty of Mechanical Engineering Control Systems Technology, 5600 MB, Eindhoven, The Netherlands
** BMW A.G. Advanced Safety Engineering 80788, Munich, Germany
Abstract— In this paper, an approach to design an ideal restraint system is discussed. The problem, normally solved by optimization approaches, is translated to a tracking problem, where a given reference trajectory has to be tracked. Before a controller can be designed, identification of the local dynamic input-output behavior is performed in several operating points using stepwise perturbations added to the input. Using the obtained model, a stabilizing controller is designed with loop shaping for performance. Results are shown for the design of the belt force, such that the maximum chest deceleration of the driver in a frontal crash test with 56 km/h crash velocity, is minimized. A reduction of 60% of the maximum chest acceleration is achieved.
I. I NTRODUCTION The last four decades, a lot of effort is put in the reduction of the number of fatalities and injuries in motor vehicle crashes. Widely known facilities for this purpose are the safety belt and the airbag, with the safety belt being the focus of this paper. Their objective is to smoothly absorb the kinetic energy of the occupant. In the U.S.A., safety belts have shown to reduce the risk of fatal injuries of front-seat passenger car occupants by 45 percent, and the airbag by 12 percent, according to [1]. Two phenomena are the stimulus for continuous research and development of restraint systems. First, although injuries are heavily influenced by characteristics of the occupant and the crash, the current restraint systems can not yet properly adapt to the relevant crash and occupant characteristics, [2]. Secondly, the restraint system itself is sometimes the cause of injuries, e.g., if the occupant is in the path of a deploying airbag, [3]. The developments in the field of restraint systems aim to actually adapt to the relevant characteristics. To determine the most relevant occupant characteristics during the crash, an abundance of methods exist, varying from strain gauges to ultrasonic sensors, e.g., [4]. In [5], an overview is given how the most relevant crash characteristics are determined in many of the modern vehicles. Focussing on the adaptivity of the safety belt system, most of the introduced devices aim to influence the force in the webbing. Examples are stitches in webbing, tearing at predefined force levels, [6], and a load limiter, to which the webbing is attached at one end, that can be switched, such that its stiffness characteristic changes, [7]. Corresponding author is R.J. Hesseling:
[email protected]
0-7803-7924-1/03/$17.00 ©2003 IEEE
To acquire insight in the occupant injuries, standard crash tests, representing real world crash test, are performed, e.g., [8], with real vehicles and with crash dummies instead of humans. The crash test used throughout this paper is the US-NCAP crash test, [9], a full-width, frontal crash with a velocity of 56 km/h into a rigid barrier. During such a crash test, variables as, for instance, accelerations and forces, are measured. Their translation to measures that reflect the occupant injuries, is the topic of ongoing research, e.g., [10]. Nevertheless, standardized injury measures have been formulated, and the maximum of the chest deceleration c¨ is one of the most used measures to assess the performance of the safety belt system. Crash tests are time and money consuming, and therefore, the restraint system design is supported by numerical simulation. A widely used package for that purpose is the commercially available MADYMO package, [11]. Results throughout this paper originate from simulations with MADYMO. The used numerical model, henceforth referred to as M , represents the US-NCAP crash test with a midsize passenger vehicle, Fig. 1. It includes a commercially available, numerical model of an average man, [11], and approximately 50 rigid bodies, 50 (non)-linear springs and dampers. To simulate a crash of 150 ms with a time step of 1 · 10−6 s, takes approximately one hour. D-ring
z x
y
Load limiter Chassis attachment Fig. 1. Graphical view of M , with the most essential components highlighted
The paper is organized as follows. The objective and problem formulation is discussed in Section II, the general
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x¨veh (t) rc¨ (t) + e(t) Fig. 2.
C
F(t)
M
A. Identification for control design c(t) ¨
Closed loop with M
approach is outlined in Section III. Section IV discusses the identification of the relevant transfer in M , required for the control design in Section V. Finally, some conclusions are presented in Section VI. II. O BJECTIVE AND PROBLEM FORMULATION The objective is the development of an easy to use approach for the design of a controller for the belt force F, such that the maximum of the chest deceleration c¨ is minimized. This objective requires the measurement of occupant state variables, and the regulation of the belt force F. For a real world implementation, such sophisticated devices do not (yet) exist, but on the level of numerical simulation, the requirements can be easily fulfilled. To prevent from severe injuries: • •
the chest may not have contact with inner vehicle components like the steering wheel, and the chest velocity at the end of the crash has to be equal to or lower than the vehicle velocity at that time instant.
Since these bounds reflect the motion of the chest, the objective can be translated into a tracking problem, where following rc¨ does not result in violation of the bounds. In Fig. 2, a general closed loop with M is shown, where e(t) = rc¨ (t) − c(t) ¨ is the error, x¨veh (t) is the deceleration of the vehicle, and C is the to be designed controller. The problem is now precisely formulated as: a controller design approach for a controller C, stabilizing the closed loop of M such that the error e(t) is small. In this paper, it is assumed that the vehicle deceleration over the full length of the crash is known. Furthermore, the focus is on the approach, and therefore, only one combination of a crash, dummy, and vehicle is considered. Nevertheless, it has to be possible to apply the approach to another combination of the dummy, vehicle, and crash. III. A PPROACH The model M is not suitable for control design, due to its complexity and the time needed to simulate. Nevertheless, knowledge about the relevant dynamic behavior of the occupant, the vehicle, the restraint system, and their relevant interactions during the considered crash is needed for proper control design. In Section III-A, the approach to identify the relevant dynamic behavior is elucidated, whereas in Section III-B, the control design is discussed.
The MADYMO package does not offer the facility to output the differential equations of M , either non-linear or linearized. Because of the complexity of M , the only alternative left is to estimate one or more linear models using measurements on M . The interesting transfer, namely from F to c, ¨ is linearized using perturbations δF added to a known input trajectory F0 . Such perturbed output responses reflect local dynamic behavior, which is easier to identify, due to the reduced influence of phenomena. Besides that, the identification of global dynamic behavior may lead to complex models without any benefit. Note that the perturbed output responses do not reflect perturbations in a stationary operating point, but cover more than one operating point, and one has to be very careful when interpreting the results. To analyze whether it is justified to use linear model(s), different perturbations are used, and to investigate whether it is possible to use the obtained model(s) independently from the operating point, perturbations are applied at different operating points. Suppose that it is justified to use one or more linear models, then a variety of estimation techniques exists to arrive at a linear model starting from the measurements, [12]. Here, a straightforward technique is desired that does not require detailed knowledge of the underlying behavior, like the order of the investigated dynamic behavior. Furthermore, the technique has to be able to deal with step responses that do not cover the full time, needed to attain a steady state. The approximate realization algorithm, as extended in [13], [14] meets these requirements and leads to a discrete-time, finite order model. B. Control design Clearly, model errors are introduced during identification. They play an important role in the closed loop behavior. To reduce the associated consequences in advance and to enable the use of C for slightly different crash situations, the concept of feedback is used. Besides that, feed forward cannot be used with sufficient accuracy considering the model of local dynamic behavior. A typical control design procedure for this type of tracking problems is loop shaping. It is widely accepted as a starting point for control design and as a benchmark technique. Its results can be intuitively explained, the resulting controller structure (PID or a-like) is widely known, and the controller can be designed such that is capable to deal with model errors. The to be designed controller has to satisfy criteria on stability, robustness, and performance. The desired stability and robustness is formulated by the gain margin GM and the phase margin PM. The closed loop performance is characterized by a maximum allowable error emax and a desired 5 % settling time of the closed loop system, t5 % .
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The desired bandwidth ωBW , defined as the 0 dB crossover frequency of the open loop transfer function, is an important characteristic of the closed loop behavior, but is very complex to determine accurately. Generally, the bandwidth is chosen in a range based on characteristics of the closed loop components. Here, the lower bound is based on the total time of a crash, and the upper bound on the computational noise in the signals from MADYMO. Within this range, the lowest possible bandwidth that still satisfies the control design criteria is the desired bandwidth, since this bandwidth means less expensive implementation. IV. I DENTIFICATION FOR CONTROL DESIGN The approach is applied to minimize the maximum chest deceleration by manipulation of the belt force F. The original belt force Fo (t) and the original chest deceleration c¨o (t), with the original load limiter, are shown in Fig. 3 and Fig. 4. The shape of Fo (t) seems peculiar, but reflects the similarly shaped non-linear stiffness of the load limiter in the original belt system, [7].
F0(t)
4
therefore, a body with infinitesimal mass has to be attached to the belt, and the force F(t) is applied to this mass instead. To perturb the dynamic behavior locally, small amplitudes ∆F are chosen, whereas more than one amplitude is chosen to investigate whether the relevant dynamic behavior may be described by one linear model. Five amplitudes up to 50 N are chosen, i.e., ∆F1 = 10 N, ∆F2 = 20 N, ∆F3 = 30 N, ∆F4 = 40 N, and ∆F5 = 50 N. The operating points τ are chosen in the time span (20 ms, 35 ms), namely τ1 = 20 ms, and τ2 = 25 ms. Before 20 ms, settling phenomena, due to the tightening of the webbing and settling phenomena in the crash dummy, are present, and after 35 ms, contact with the airbag normally is established, such that identification for t > 35 ms does not represent normal operating points. So, ten experiments with the open loop system with M ˜ = Fo (t) + ∆F · are performed, with the perturbed input F(t) ˜¨ = c¨o (t) + δc(t) ε(t − τ) as the input and c(t) ¨ as the output. c(t) ¨ The normalized responses, defined as δ∆F , for τ1 =20 ms, are shown in Fig. 5. Analysis of M learns that they are disturbed by forces, due to a contact of the right femur of the dummy with the seat cushion at 37 ms. Nevertheless, it may be concluded that for t < 37 ms, the underlying dynamic behavior is linear.
0.08
δc(t) ¨ −2 −1 ∆F [ms N ]
F(t) [kN]
3 2 1
0.04 0
δc¨ (t)/∆F1 δc¨ (t)/∆F2 δc¨ (t)/∆F3 δc¨ (t)/∆F4 δc¨ (t)/∆F5
−0.04
0
0
0.025
0.05
0.075 t [s]
0.1
0.125
−0.08 0.02
Fig. 3.
The original belt force Fo (t).
0
c¨ (t) [ms−2]
−100 −200 −300 −400
c¨o (t) 0
Fig. 4.
0.025
0.05
0.075 t [s]
0.1
0.04
0.05
t [s]
To prescribe a belt force F(t), the stiffness of the load limiter in M is replaced by an actuator. Furthermore, it is not possible to apply a force directly to the webbing, and
−500
0.03
0.125
The original chest deceleration c¨o (t).
Fig. 5.
Normalized output response
δc(t) ¨ ∆F
for τ1 =20 ms.
The idea is now to try whether the smooth part of the output responses can be used to accurately identify the relevant dynamic behavior. For that purpose, the averaged, normalized output responses are computed to investigate whether a constant model can be used for both operating points. From Fig. 6, it can be concluded that the relevant transfer, from δF to δc, ¨ depends from the operating point. Therefore, a model for each operating point will be realized. The concept of extended approximation realization, based on a so-called Hankel matrix, defined as G p+q−1 for p + q < n + 1, (1) Tp,q = 0 for p + q ≥ n + 1, with Gi , the measured impulse response at t = i, and n + 1 the length of the measured impulse response. Here, the
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0
−20 M(ω) [dB]
δc(t) ¨ −2 −1 ∆F [ms N ]
averaged δc¨ (t)/∆F for τ1 averaged δc¨ (t)/∆F for τ2
−0.02
−40
2
−80
2
3
10
10
45
0
0.005
0.01 t − τ [s]
0.015
Φ(ω) [o]
−0.06
1
10
−0.04
0.02
0 −45 −90
Fig. 7.
Hankel matrix is constructed from the impulse response. One extension, [13], enables the use of step responses instead of impulse responses by transforming the Hankel matrix with elementary matrix operations. Another extension of the algorithm, [14], enables it to get insight in the expected order of the underlying system by investigation of the singular values of the transformed Hankel matrix. The largest five singular values of the two transformed Hankel matrices, T1 and T2 , constructed from the averaged, normalized output responses for τ1 =20 ms and τ2 =25 ms, are given in Table I.
T HE LARGEST FIVE SINGULAR VALUES OF T1 AND T2 σi of T1 1.4620 0.2955 0.0350 0.0118 0.0100
σi of T2 1.0945 0.1206 0.0117 0.0054 0.0047
TABLE II S YSTEM PARAMETERS OF H1 (s) AND H2 (s). ωn 2π
[Hz] 40.9 42.3
ζ [-] 0.564 0.592
The Bode diagrams of H1 (s) and H2 (s).
c(t) chest
seat
3
10
steering wheel `o
vehicle
xveh (t)
V. C ONTROL DESIGN
It seems justified to neglect the third and higher singular values, so that the obtained linear models for τ=20 ms and τ2 =25 ms are second order models. The obtained linear models are transformed into continuous-time using zero order hold conversion, resulting in H1 (s) and H2 (s), with s the Laplace transform. Their Bode diagram is shown in Fig. 7, and their static gain Kst , eigenfrequency ωn , dimensionless damping ζ, and their system zero z are given in Table II.
Kst [ms−2 N−1 ] 0.0266 0.0246
2
10 ω [Hz]
Fig. 8. The simplified one-dimensional model, where the chest and the vehicle are modelled by a single body having initial velocity x˙v (t0 ).
TABLE I
i 1 2 3 4 5
1
10
Fig. 6. Averaged, normalized output responses for τ1 =20 ms, and τ2 =25 ms.
H1 (s) H2 (s)
H (s) 1 H (s)
−60
z 2π
[Hz] 8.1 4.8
The influence of the operating point on the characteristics of the obtained models is relatively small, for instance 8% for Kst .
For control design, first an appropriate reference trajectory is determined, then the control design criteria are formulated and, if needed, translated to the frequency domain, and finally, the controller C is designed and implemented. Based on observations of the occupant motion during standardized frontal crash tests, [15], the trajectory of the chest may be approximated by a straight line. Then, the essential characteristics of the chest motion can be approximated using Fig. 8. Using the equation of motions of the simplified model and the a priori known vehicle deceleration, a trajectory for rc¨ can be determined for which the maximum deceleration of the chest is minimal and the crash dummy bounds, exactly formulated by 0 ≤ c(t) − xveh (t) ≤ `0 and c(t ˙ e ) ≤ x˙veh (te ) using Fig. 8, are not violated. x¨veh (t) , 0 ≤ t ≤ t 0 x¨veh (t 0 ) ,t 0 ≤ t ≤ te (2) rc¨ (t) = 0 ,t ≥ te Time point t 0 is iteratively found, and te typically is 100 ms for the considered crash, Fig. 9. A desired gain and phase margin of 3 for GM and 45o for PM is chosen. Relatively large values are chosen, to account for the lack of accurate knowledge about model errors.
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t0
te
Considering the Bode diagram of H1 (s) and H2 (s), a proportional controller, denoted be P, gives sufficiently stable closed loop behavior. However, the desired performance reflected by the sensitivity bound of (100 Hz, -20 dB) will not be met. Therefore, an integral controller, denoted by τI , is added to increase the performance for low frequencies:
0
−200 −300
−500
Fig. 9.
HC = P(
x¨veh (t) rc¨ (t)
−400
0
0.025
0.05
0.075 t [s]
0.1
0.125
The vehicle deceleration x¨veh (t) and the reference trajectory rc¨ (t).
A settling time t5% of 10 ms is desired, and is, for a second order model, defined by, [16]: t5 % ≈
3 . ζ ωBW
(3)
100 95
100
Fig. 10.
S (s) 1 S2(s)
−20
1
2
10
Fig. 11.
3
10 ω [Hz]
10
The sensitivity function for S1 (s) and S2 (s).
Evaluation of the designed controller C is performed in the closed loop with M . The controlled chest deceleration and the required belt force are shown in Fig. 13 and Fig. 14. The smooth closed loop results indicate a stable closed loop, and show that the maximum chest deceleration is reduced with 60% compared with the original chest deceleration.
GM1 = 3.8, GM2 = 3.2
(100 Hz, 95%) 90 0 0
1000
250 2000 3000 ω [Hz]
0
500 4000
(4)
(100 Hz, −20 dB)
95
75
50
0
−40
Imaginary axis
cum. power spectral density [%]
A maximum error emax of 10% is allowed, whereas the now available input and output signals reflect open loop behavior. However, the closed loop disturbance rejection is related with the open loop disturbance by the sensitivity function, S(s), meaning that desired closed loop performance can be formulated as bounds on the sensitivity transfer function S(s). To determine the expected frequencies in e(t), an analysis of c¨o (t) is done using spectral density, [12]. The cumulative spectral density of c¨o (t) in Fig. 10 shows that frequencies up to 100 Hz cover approximately 95% of the frequencies in c¨o (t). So, to aim at a maximum error of 10% or -20 dB, covering 95% of the expected frequency content, the sensitivity function HS is bounded by (100 Hz, -20 dB).
τI s + 1 ω2LP )( 2 ), τI s s + 2βLP ωLP s + ω2LP
where (ωLP , βLP ) denotes an underdamped low-pass filter to suppress computational noise. Next, the bandwidth is iteratively increased, until a PI-controller can be designed, satisfying all control design criteria. The result is a bandwidth of 410 Hz for H1 (s) and 480 Hz BW , ωLP = for H2 (s), with rules of thumb being τI = 0.4 · ω2π 0.7 3 · ωBW , and βLP = 2π , and P is computed such that the 0 dB crossover frequency function equals the desired bandwidth. The modulus of S1 (s) and S2 (s) and the Nyquist plot are shown in Fig. 11 and Fig. 12.
|M(ω)| [dB]
c¨ (t) [m s−2]
−100
5000
The cumulative spectral density plot of c¨o (t).
−0.5
PM = 45 o 1 o PM2 = 43
2
ω
BW, 1
−1
Next, the desired bandwidth is determined. For the lower and upper bound of ωBW , a frequency of t1e = 10 Hz respectively 1 kHz is chosen respectively experimentally obtained. Furthermore, using Eq. 3 with ζ1 and ζ2 , the first estimate of the desired bandwidth is 85 Hz and 80 Hz.
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|H1(s) C(s)| |H (s) C(s)|
= 410 Hz
ωBW, 2 = 481 Hz −1
Fig. 12.
−0.5 Real axis
The Nyquist plot.
0
t0
te
0
[2]
c¨ (t) [m s−2]
−100 −200
[3]
−300
c¨o (t) c¨ (t) rc¨ (t)
−400 −500
0
Fig. 13.
0.025
0.05
0.075 t [s]
0.1
[4]
0.125
[5]
The trajectories of c¨o (t), c(t), ¨ and rc¨ (t)
15
[6]
F0(t) F(t) F(t) [kN]
10
[7] 5
0
0
Fig. 14.
0.025
0.05
0.075 t [s]
0.1
[8] [9]
0.125
The original and the required belt force.
[10] VI. C ONCLUSION An approach is presented to design the restraint system variables, such that the occupant injuries are reduced, consisting of two steps, being the identification of the relevant dynamic behavior and control design. For the identification of the relevant transfer in M , a time span of 17 ms and 12 ms shows to be sufficient for control design purposes. Furthermore, the identified transfer from δF to δc¨ is second order in the considered operating points, and can be reasonably described by two constant linear, models of order two. The fact that the designed controller stabilizes the closed loop of M , meeting the a priori defined performance criteria, shows that the obtained models make sense. Research has been started to set up a strategy accounting for constraints on the belt force. Preliminary results show that classical predictive control concept is very effective. Future research emphasizes the manipulation of the airbag, and the control of the chest deflection. After that, the topic of interaction between the airbag and the belt will be explored.
[11]
[12]
[13]
[14]
[15]
VII. REFERENCES [16] [1] National Highway Traffic Safety Administration (NHTSA), “Traffic safety facts 2001 - occupant
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protection,” U.S. Department of Transportation, Tech. Rep. DOT HS 809 474, 2001. M. Mackay, S. Parkin, and A. Scott, “Intelligent restraint systems - what characteristics should they have?” in Advances in Occupant Restraint Technologies, Lyon, France, September 1994, pp. 113 – 126. C. R. Bass, J. R. Crandall, and J. R. Bolton, “Deployment of air bags into the thorax of an out-ofposition dummy,” SAE Technical Paper, Nr. 1999-010764, 1999. D. S. Breed, “Smart airbag system,” World Patent, May 1998, WO 98/54638. C. Y. Chan, “A treatise on crash sensing for automotive air bag systems,” IEEE Transactions on Mechatronics, vol. 7, no. 2, pp. 220 – 234, June 2002. F. Bendjellal, G. Walfisch, and C. Steyer, “The combination of a new air bag technology with a belt load limiter,” in Proc. of the 16th Int. Technical Conf. on the Enhanced Safety of Vehicles, Windsor, Canada, May, 1998, paper Nr. 98-S5-O-14. G. Clute, “Potentials of adaptive load limitation,” in Proc. of the 17th Int. Technical Conf. on the Enhanced Safety of Vehicles, Amsterdam, the Netherlands, June 2001, 134. — , “http://www.crashtest.com,” 2003. National Highway Traffic Safety Administration (NHTSA), “Laboratory test procedure frontal impact testing,” U.S. Department of Transportation, Tech. Rep. NHTSA-99-4962-37, 1999. Int. Research Council on Biokinetics of Impacts (IRCOBI), “http://www.ircobi.org/,” 2003. TNO Automotive, MADYMO Manuals, 5th ed., TNO Road-Vehicles Research Institute, Delft, The Netherlands, November 1999. L. Ljung, System Identification - Theory for the User, 2nd ed., Upper Saddle River, USA: PTR Prentice Hall, 1999, ISBN 0-13-656695-2. J. B. van Helmont, A. J. J. van der Weiden, and H. Anneveld, “Design of optimal controllers for a coal fired benson boiler based on a modified approximate realization algorithm,” in Proc. of Application of Multivariable System Techniques. London, UK: Elsevier, 1990, pp. 313 – 320. S. Y. Kung, “A new identification and model reduction algorithm via singular value decomposition,” in Proc. of the 12th Asilomar Conf. on Circuits, Systems and Computers, California, USA, 1978, pp. 705 – 714. M. Mackay, A. M. Hassan, and J. R. Hill, “Observational studies of car occupants’ positions,” in Proc. of the 16th Int. Technical Conf. on the Enhanced Safety of Vehicles, Windsor, Canada, May 1998, 98-S6-W42. M. Driels, Linear control systems engineering, 1st ed., ser. McGraw-Hill series in mechanical engineering. Singapore: McGraw-Hill, 1996, ISBN 0-07-113997-4.
