Experimental Comparison of Linear and Nonlinear Controllers Applied ...
2014 IEEE International Conference on Control Applications (CCA) Part of 2014 IEEE Multi-conference on Systems and Control October 8-10, 2014. Antibes, France
Experimental comparison of linear and nonlinear controllers applied to an Antilock Braking System* Marcela Martinez-Gardea1 , Student Member, IEEE, Iordan J. Mares Guzm´an1 , Stefano Di Gennaro2 and Cuauhtemoc Acosta Lua1 , Member, IEEE The ABS control solutions can be grouped in two main categories, wheel acceleration and tire slip control. The first category approaches slip control indirectly, by controlling the deceleration/acceleration of the wheel through the brake pressure from the actuator. The second one is a direct slip control, and has many different approaches, [9]–[12] present P-I and P-D controllers, without developing of calculations, to obtain the control gains that help obtaining the optimal performance of this controllers. However, [12] proposed a fuzzy control solution for ABS control, in [13] and [14] a control strategy based on sliding mode analysis is proposed, and [15] proposed an optimal control of an ABS laboratory setup by INTECO. This paper focuses on the study of an ABS laboratory setup, manufactured by INTECO. A space-state mathematical model was obtained from this system. The first proposed control strategy is a PID controller, then a control law is proposed based on the Control Lyapunov Function technique. This control strategy is a generalization of the notion of Lyapunov function used in the stability analysis of systems. The fact that the Control Lyapunov Functions are, as a general rule, easier to obtain than the feedback law, will be helpful when obtaining nonlinear stabilization. Then the two techniques are compared to establish their differences in performance. The paper is organized as follows: Section II briefly presents the development of a quarter of a vehicle mathematical model. Section III is dedicated to the linearization of the system under given conditions. The different techniques used in the development of the PID controller are shown in Section IV. The demonstration of the asymptotic stability provided by the error feedback using Control Lyapunov Function techniques is done in Section V. Finally in the last Section, the results and comparison of the two controllers are presented, through charts.
Abstract— An experimental comparison of two control techniques applied to a quarter of a vehicle ABS, Antilock Braking System, laboratory setup is presented. The first technique is a basic PID controller and the second one is a nonlinear control method through error feedback using Control Lyapunov Function techniques.
I. INTRODUCTION The origin of the ABS, Antilock Braking System, dates back to the early 20th century when Bosch was granted a patent for an ”Apparatus for preventing lockbraking [sic] of wheels in a motor vehicle” in 1936 [1]. For several years there were not significant advances due to the fact that the available technology could not implement the antilock concept successfully. Between 1967 and 1970 MercedesBenz engineers changed the mechanical sensors for contactless sensors that operated under the induction principle [2]. Finally, with the arrival of electronic integrated circuits (that were small and robust enough) it was possible to record data from the wheel’s sensor. The latter enabled the use of more reliable actuators for adjusting the brakes’ hydraulic pressure. An important milestone in ABS history occurred in 1978 when Bosch and Mercedes-Benz, working together, launched the second generation of ABS (ABS2) installed in the Mercedes-Benz Class S (as an option) in mass production [2]. The ABS was developed to prevent the wheels from locking up while braking thus preventing the slippage of the wheels on the surface [3]–[5]. Modern ABS not only try to prevent the wheels from locking up, but also they aim to obtain maximum wheel grip on the surface while the car is braking [6], [7]. The ABS is designed to increase the braking efficiency and to maintain the car’s maneuverability [8]. Besides, it also functions as a control mechanism to adjust the brake fluid pressure level of each wheel to reduce the driving instability and decrease the braking distance. Several algorithms had been proposed for controlling the ABS with the aim of, obtaining a minimum braking time, and an increased driving capability of the vehicle.
II. SYSTEM MODEL As shown in Fig. 1, the ABS laboratory setup is comprised of two wheels; the lower wheel simulates the relative motion of the vehicle over the road or surface, and the upper wheel simulates the motion of the vehicle’s wheel. This is considered as a quarter of a vehicle system and it is introduced for practical analysis and design of the braking system. The proposed model was created by INTECO, which takes into consideration: the braking forces exerted on the wheels, the wheel motion and the vehicle motion.
*Research supported by CONACYT scholarships 513153 and 512691. 1 M. Martinez-Gardea, I. J. Mares Guzm´an and C. Acosta Lua are with Universidad de Guadalajara, Centro Universitario de la Ci´enega, Ocotl´an, Jalisco, 47820, M´exico, {marcela.martinezg, iordan.mares}
@alumno.udg.mx, [email protected] 2 S. Di Gennaro is with the Department of Information Engineering, Computer Science, and Mathematics, and also with the Center of Excellence DEWS, University of L’Aquila, 67100, L’Aquila, Italy,
[email protected] 978-1-4799-7408-5/14/$31.00 ©2014 IEEE
71
torque in the lower bearing m20 and the friction torque among the wheels µ(λ)fn . In Fig. 3 an auxiliary diagram of the model is shown. All the values for the different constants used in the following calculations are defined in Table I. During the deceleration process, the braking torque is applied on the upper wheel, consequently the velocity of the wheel decreases, the ABS dynamic can be described as [8]: J1 x˙ 1 = Ft r1 − d1 x1 + m10 + Tb (1) J2 x˙ 2 = − Ft r2 + d2 x2 + m20
For the implementation of the ABS, the upper wheel is equipped with a disk brake system which is driven by a small DC motor. The braking system includes two encoders that will provide the necessary measurements to estimate the wheel angular velocities [8].
where Ft is given by: Ft = µ(λ)fn
(2)
and fn is calculated as: fn = (d1 x1 +m10 +Tb +mg )/[l(sin(ϕ)−µ(λ) cos(ϕ))] (3) where l is the distance between the contact point of the wheels and the rotational axis of the balance lever, and ϕ is the angle between the normal in the contact point and the line l.
Fig. 1: Physical system. A. System diagram In Fig. 2 a physical diagram of the ABS experimental prototype is shown. The laboratory set up presents the angular motion of the wheel during braking. Although, the model is simple, it maintains the fundamental characteristics of the real system [8]. When the dynamic equations of the systems are derived, some assumptions are made. First, it only considers the longitudinal dynamic of the vehicle, therefore, the lateral and vertical motion of the wheels are neglected. Second, the rolling resistance force is omitted because its magnitude is very small in comparison to the braking force.
Fig. 3: Auxiliary diagram. A new variable is proposed: S(λ) = µ(λ)/[l(sin(ϕ) − µ(λ) cos(ϕ))]
(4)
So (2) can be rewritten as: Ft = S(λ)(d1 x1 + m10 + Tb + mg )
(5)
Considering: µ(λ) = w4 λp /(a + λp ) + w3 λ3 + w2 λ2 + w1 λ
(6)
The relative difference among the velocities of both wheels can be represented in terms of a coefficient, and is defined as: λ = (r2 x2 − r1 x1 )/(r2 x2 ) (7)
Fig. 2: ABS schematic diagram.
The brake actuator is described by the following equation:
B. Mathematical model
T˙b = c3 1(b(u) − Tb )
There are three torques acting on the upper wheel: the braking torque Tb , the friction torque in the upper bearing m10 and the friction torque among the wheels µ(λ)fn . Model parameters and variables description are given in Table I. There are two torques acting on the lower wheel: the friction
where the function b(u) can be approximated by: ( b1 u + b2 u ≥ u0 b(u) = 0 u < u0 72
(8)
(9)
with u0 as the operating threshold of the brake driving system. The system described in (1) can be generalized as: x˙ = f (x) + gTb + d where x = ( x1 , x2 )
Substituting (13), (17) and (18) in (14) and (15) the linearized system is created and by simple mathematical procedures a matrix form is obtained: ∆λ˙ a11 a12 ∆λ b1 = + ∆Tb (19) ∆x˙ 2 a21 a22 ∆x2 b2 ∆λ y = (1 0) (20) ∆x2
(10)
T
and f (x) =
where:
S(λ) dJ1 r11 − Jd11 )x1 S(λ) Jr11 − J11 ; g = −S(λ) Jr22 −S(λ) dJ1 r22 x1 − Jd22 x2 S(λ) Jr11 (m10 + mg ) − mJ10 1 d= −S(λ) Jr22 (m10 + mg ) − mJ20 2
a11
meanwhile (7) can be represented as: y = h(x) = λ
(11) a12
Rewriting equation (10): x˙ 1 c11 S(λ) + c13 0 x1 = + x˙ 2 c21 S(λ) c23 x2 c15 S(λ) + c16 c12 S(λ) + c14 + Tb + c25 S(λ) c22 S(λ) + c24 (12) with r1 d1 J1 − mJ10 1 − rJ2 d21 − mJ20 2
c11 = c14 = c21 = c24 =
(m10 +mg )r1 J1 c15 = Jr11 (m +m )r c22 = − 10 J2 g 2 c25 = − Jr22
c12 =
a21
kr1 (c11 x10 + c12 + c15 Tb0 − ! c21 x10 + c22 + k − = −r2 x20 −r2 (1 − λ0 ) − +c25 Tb0 −r x S(λ )c /r 2 20 0 21 1 −r2 x20 [S(λ0 c11 + c13 ] (1 − λ0 ) S(λ0 )c11 + c13 + c23 − =− −r2 (1 − λ0 )S(λ0 )c21 /r1 x20 = k(c21 x10 + c22 + c25 Tb0 ) − r2 x20 S(λ0 )c21 /r1
a22 = r2 (1 − λ0 )S(λ0 )c21 /r1 + c23 b1 = −[r1 (c15 S(λ0 ) + c16 ) − r2 (1 − λ0 )c25 S(λ0 )]/r2 x20 b2 = c25 S(λ0 )
c16 = − J11
From (19) and (20) the following transfer function is obtained: (s − a22 )b1 + b2 a12 (21) G(s) = s2 + (−a22 − a11 )s − a12 a21 + a11 a22
c23 = − Jd22
and rearranging some terms:
c13 = − Jd11
G(s) =
III. S YSTEM LINEARIZATION To linearize the system some initial values for operation points are selected P0 = [x10 , x20 , λ0 , Tb0 ]. Linearizing (4), (7) and (12) developing Taylor series expansions [16] and using (6): ∆x˙ 1 = ∆x1 S(λ0 )c11 + c13 + + ∆S(λ)(c11 x10 + c12 + c15 Tb0 )+ + ∆Tb c15 S(λ0 ) + c16 ∆x˙ 2 = ∆x1 S(λ0 )c21 + ∆x2 c23 + + ∆S(λ) c21 x10 + c22 + c25 Tb0 + + ∆Tb c25 S(λ0 ) ∆λ˙ = −(∆x˙ 1 r1 − r2 (1 − λ0 )∆x˙ 2 )/(r2 x20 ) w4 λp p w4 λ2p 0 p 0 − λ0 (a+λ p 2+ ) ) λ0 (a+λp sin(ϕ) 0 0 +3w3 λ20 + 2w2 λ0 + w1 ∆λ ∆S(λ) = l(sin(ϕ) − µ(λ0 ) cos(ϕ))2
ks = b1 T1 = (a12 b2 − a22 b1 )/b1 q T2 = a22 + a11 − a222 + a211 + 4a12 a21 − 2a11 a22 /2 q T3 = a22 + a11 + a222 + a211 + 4a12 a21 − 2a11 a22 /2
(13)
It is possible to evaluate in operations points: λ0 ∈ {0.05, 0.14, 0.18, 0.19, 0.4, 0.8}; x20 ∈ {178, 89, 1}; Tb0 ∈ {9, 5, 1} [12]. In Table II some values are shown for these operation points. Analyzing the values, the system begins to loose stability from λ0 = 0.19, so a reference value of λ0 is proposed as λref = 0.14.
(14)
(15)
IV. CONTROL STRATEGIES The ABS laboratory setup, like most physical systems, has a nonlinear nature that makes the calculations and approximations of mathematical models harder to solve. Besides, if it is considered that the systems are subject to variations and perturbations, which are difficult to estimate, the model complexity increases. In this section two control strategies are introduced. A PID controller using the linearized system and a nonlinear control law through error feedback using Lyapunov function techniques.
(16)
(17)
considering that w4 , w3 , w2 , w1 , λ0 , p, ϕ and a are constants defined in Tables I and II. Next, a variable change is proposed: r1 ∆x1 = r2 (1 − λ0 )∆x2 − r2 x20 ∆λ
(22)
where:
Eq. (16) can be rewritten as: ∆S(λ) = k∆λ
ks (s − T1 ) (s + T2 )(s + T3 )
(18) 73
The goal of the ABS controller is to track a predeterminated slip set point λref . For this reason, in this article we must be a change the variable x = (x1 , x2 )T to xn = (λ, x2 )T in the both control strategies.
stable system the Lyapunov derivative must be negative definite. [18]. This control technique use the information of Lyapunov function for the construction of the control law kl from V (x) and the vector fields defining the system. An advantage of this technique is to extreme simplicity and ease of implementation moreover is that provides automatically an analytic feedback law [19]. In this section, a variable change will be used, x = (x1 , x2 )T to xn = (λ, x2 )T , which will generate the following transformed system: λ˙ = r1 m10 − fn µ(λ)r1 /x2 r2 J1 + d1 (1 − λ)/J1 −
A. PID controller The PID controller is an attractive option, because it is widely used in the industry [17], you can use it even if the plant is unknown and it is easy to tune. With the ABS laboratory setup there is the advantage that you can test it and adjust the parameters as you wish. There are several techniques used to tune the PID: the Ziegler-Nichols rule [17], the root locus, the Routh-Hurwitz criterion, the pole-zero cancellation or the empirical method. Considering the transfer function: 0.6142s + 0.005378 (23) G(s) = (s + 1.291101)(s + 0.009898)
− (1 − λ)(fn µ(λ)r2 + d2 x2 + m20 )/x2 J2 + + r1 Tb /x2 r2 J1 x˙ 2 = − Ft r2 + d2 x2 + m20 /J2 (24) To properly apply this technique [19], λ must track a constant reference λref , so an error function is defined:
and applying the Routh-Hurwitz criterion on the transfer function closed loop:
e = λ − λref
(25)
e˙ = λ˙
(26)
and its derivative:
Y (s) kp (0.6142s + 0.005378) = 2 R(s) s + (1.301 + 0.614kp )s + 0.01278 + 0.005378kp
Therefore, the following lemma is proposed: Lemma 1: The control law that ensures asymptotic stability in the sense of Lyapunov for the ABS system is: r1 (fn µ(λ)r1 − m10 )/(x2 r2 J1 )+ x2 r2 J1 (1−λ) Tb = + x2 J2 (fn µ(λ)r2 + d2 x2 + m20 )− r1 −d1 (1 − λ)/J1 − kl e (27) Proof: The following Lyapunov candidate function is proposed:
From this analysis no useful information was obtained, therefore the Ziegler-Nichols rule was kept in use; however, the system does not meet the minimum requirements for this technique to be applied. The next method is the root locus. With this method one or more parameters from the transfer function vary and the geometric place of the closed loop transfer function poles are located. First we place the poles and the zeroes on the plane (open loop). The branches always begin in the poles and finish in the zeroes. Then, the asymptotes are calculated. Finally we draw the trajectories between poles and zeroes (just the odd) and the behavior of kp from zero to infinity. With (23), the dynamic analysis of the root locus indicates that the pole (-1.291101) will track a trajectory over the x-axis with -∞ direction. Meanwhile, the zero (-0.008756) attracts the pole (0.009898). All the trajectories are over the left side of the plane, so no matter how much kp grows it is still on the left part of the plane. So, for the PID controller design, an empirical approach was used, taking into consideration that the ABS laboratory Setup is available for tuning the parameters. The following values are proposed: kp = 2.8, ki = 3.25, kd = 0.3.
V (e) = e2 /2
(28)
V˙ (e) = ee˙
(29)
and its derivative: Substituting (26) and (24) in (29): r1 (m10 − fn µ(λ)r1 )/x2 r2 J1 − V˙ (e) = e − (1−λ) x2 J2 (fn µ(λ)r2 + d2 x2 + m20 )+ +d1 (1 − λ)/J1 + r1 Tb /x2 r2 J1
(30)
Applying the control law (27), the equation is reduced to: V˙ (e) = e(−kl e) = −kl e2
(31)
which proves that V˙ (e) < 0 ↔ kl > 0, ensuring that the system is asymptotically stable [18].
B. Nonlinear control law through of Control Lyapunov function techniques
V. R ESULTS
Stability is an important concern in systems theory. In the following control strategy proposal, the main concern is the stability of equilibrium points, which are usually characterized in the sense of Lyapunov. A simple definition for a Lyapunov function for a system is a positive definite function monotonically decreasing along the system trajectories, this means that the Lyapunov function are V (0) = 0, V (x) > 0 and V (x) → ∞ as |x| → ∞. This can be proven just by means of the Lyapunov derivative, even if the system trajectories are unknown. For an asymptotically
Experimentation is performed on the ABS laboratory setup INTECO. The initial conditions were x1 (0) = x2 (0) = 1700 RPM (178.02 rad/s). During the runs, a control signal is sent to the actuator in order to begin with the braking process. In the following, a comparison between the two controllers is shown. Fig. 4 shows the performance of both controllers, it is evident that the Lyapunov controller accomplishes the braking process in less time and distance. In Fig. 5 it is 74
Fig. 4: Braking distance vs. time. PID controller (dash), Lyapunov controller (solid). Fig. 7: Wheel velocity vs. time. PID controller (dash), Lyapunov controller (solid).
Fig. 5: Slip vs. time. PID controller (dash), Lyapunov controller (solid).
Fig. 8: Vehicle velocity vs. time. PID controller (dash), Lyapunov controller (solid).
shown that tracking the reference slip is a difficult task for both controllers, but from 8.5 seconds to 9 seconds in the braking process the controllers manage to properly track the reference. Fig. 6 exhibits the smoothness of the control signal attained by the Lyapunov controller in comparison with the PID. This represents less wear of the braking actuator by the Lyapunov controller. Fig. 7–8 show that the PID controller maintains a constant deceleration, however the Lyapunov controller generates the same velocity change in both wheels (x1 and x2 ) and this can be accomplished because the Lyapunov controller follows the reference no matter how drastic the change of the slip is. Fig. 6: Control variable (u) vs. time. PID controller (dash), Lyapunov controller (solid).
VI. CONCLUSIONS In this paper, two control strategies were designed and tested. The first approach was to work with a linearized version of the ABS laboratory setup mathematical model 75
through a PID controller. The second approach was to work directly with the nonlinear model with a control law designed by error feedback through a Lyapunov function. The Lyapunov controller achieves a shorter braking distance in less time than the PID controller, furthermore, the Lyapunov controller provides smooth signals to the actuators, maintaining their durability. The Lyapunov controller prevents braking instability by following an established reference, thus increasing passenger safety.
ACKNOWLEDGMENT The authors would like to thank Universidad de Guadalajara – Centro Universitario de la Ci´enega, especially to the Master and Doctoral Departments. R EFERENCES [1] Robert Bosch GmbH. (2003). ABS a success story. [Online]. Available: http://www.bosch.com/assets/en/company/innovation/theme03.htm [2] Daimler AG. (2008, July 1). Mercedes-Benz and the invention of the anti-lock braking system: ABS, ready for production in 1978. [Online]. Available: http://media.daimler.com/dcmedia/0-921-6574861-803841-1-0-0-0-0-0-11701-614318-0-1-0-0-0-0-0.html [3] M. Petrov, V. Balankin and O. Naruzhnyi, Study of automobiles brakes and pneumatic tires work model of the work process of antiblock brake system, Novosibirsk, NISI, Tech. Rep. 1977. [4] N. Rittmannsberger, Antilock braking system and traction control in Proc. Int. Congr. Transportation Electronics Convergence 88, 1988, pp. 195-202. [5] R. Emig, H. Goebels and J. Schramm, Antilock braking systems (ABS) for commercial vehicles-status 1990 and future prospects, in Proc. Int. Transportation Electronics Congr. Vehicle Electronics in the 90, 1990, pp. 515-523. [6] U. Kiencke and L. Nielsen, Automotive control systems: For engine, driveline, and vehicle. 2nd ed. 2005 Springer, pp. 11. [7] R. Rajamani, Vehicle dynamics and control (mechanical engineering series), softcover reprint of hardcover 1st ed. 2006 ed. Springer, pp.2. [8] Intenco Ltd, The laboratory antilock braking system controlled from pc, User manual, Poland, 2006. [9] M. H. Al-mola, M. Mailah, P. M. Samin, Performance Comparison between Sliding Mode Control and Active Force Control for A Nonlinear Anti-Lock Brake System WSEAS Trans. on Syst. and Control, Volume 9, 2014. [10] S. John, and J. O. Pedro, A Comparative Study of Two Control Schemes for Anti-Lock Braking Systems 2009 Control Conference (ASCC), 9th Asian, Istambul 23-26 June 2009. [11] M. B. Radac, R. E. Precup, S. Preitl, J. K. Tar, E. M. Petriu. Gain-Scheduling and Iterative Feedback Tuning of PI Controllers for Longitudinal Slip Control, IEEE 6th International Conference on Computational Cybernetics, pp. 183-188 Stara Lesna, Slovakia, November 2008. [12] M. B. Radac, R. E. Precup, S. Preitl, J. K. Tar, E. M. Petriu.”Linear and fuzzy control solutions for a laboratory Anti-lock Braking System”, SISY 2008, 6th Int. Symp. on Intelligent Syst. and Informatics, vol. 1, no.6, pp. 26-27 Sept. 2008. doi: 10.1109/SISY.2008.4664947. [13] Y. Oniz, E. Kayacan and O. Kaynak, A Dynamic Method to Forecast the Wheel Slip for Antilock Braking System and Its Experimental Evaluation IEEE Transaction on System, Man and Cybernetics–Part B: Cybernetics, Vol. 39, Number 2, April 2009. [14] L. Li, F.-Y. Wang and Q. Zhou, Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control, IEEE Trans. Intell. Transp. Syst., vol. 7, no. 1, pp. 1-19, 2006. [15] P. Bania, A. Korytowski, M. Szymkat and P. Gorczyca Optimal control of a laboratory anti-lock brake system in Proceedings of the 16th IFAC World Congress, Prague, pp. 1-6., 2005. [16] B. C. Kuo, Sistemas Autom´aticos de Control, 2nd ed. CECSA, 1991. [17] K. Ogata, Ingenier´ıa de Control moderna, 4th ed. Prentice Hall, 2003. [18] H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2002. [19] E. D. Sontag. ”A Universal Construction of Artstein’s Theorem on Nonlinear Stabilization”, Report SYCON-98-03, Rutgers Center for System and Control, February 1989.
APPENDIX TABLE I: SYSTEM PARAMETERS Symbol x1 x2 Tb r1 r2 J1 J2 d1 d2 µ(λ) λ fn m10 m20 mg l ϕ u Ft a bu1 bu2 c31 p u0 w1 w2 w3 w4
Description Angular velocity of the upper wheel Angular velocity of the lower wheel Braking torque Radius of the upper wheel Radius of the lower wheel Moment of inertia of the upper wheel Moment of inertia of the lower wheel Viscous friction coefficient of the upper wheel Viscous friction coefficient of the lower wheel Friction coefficient between wheels Slip (relative difference of the wheel velocities) Normal force Static friction of the upper wheel Static friction of the lower wheel Gravitational and shock absorber torques acting on the balance lever Distance between the contact point of the wheels and the rotational axis of the balance lever Angle between the normal in the contact point and the line L Control of the brake Total force generated by the upper wheel and pressing on the lower wheel Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant
Value/Unit rad/s rad/s Nm 0.0995m 0.099m 7.53x10-3 kgm2 25.6x10-3 kgm2 118.74x10-6 kgm2 /s 214.68x10-6 kgm2 /s
58.214 N 0.0032 Nm 0.0925 Nm 19.62 Nm 0.370 m 65.61°
N 0.00025724 15.24 -6.21 20.37 2.09945271 0.415 -0.04240011 2.9375x10-10 0.03508217 0.40662691
TABLE II: SYSTEM BEHAVIOR UNDER DIFFERENT INITIAL CONDITIONS λ0 0.02 0.02 0.02 0.4 0.4 0.4 0.8 0.8 0.8
x20 178 178 178 89 89 89 1 1 1
x10 174.4 174.4 174.4 53.4 53.4 53.4 0.2 0.2 0.2
Tb0 9 5 1 9 5 1 9 5 1
ks 0.0058 0.0058 0.0058 0.0108 0.0108 0.0108 0.9584 0.9584 0.9584
T1 116.7632 116.7632 116.7632 115.8296 115.8296 115.8296 118.1109 118.1109 118.1109
T2 104.7154 90.0932 75.4710 0.0095 0.0095 0.0095 44.3511 38.1546 31.9581
T3 0.0100 0.0100 0.0100 -0.4635 -0.3970 -0.3305 0.0088 0.0088 0.0088
76
Experimental comparison of linear and nonlinear controllers applied to an Antilock Braking System* Marcela Martinez-Gardea1 , Student Member, IEEE, Iordan J. Mares Guzm´an1 , Stefano Di Gennaro2 and Cuauhtemoc Acosta Lua1 , Member, IEEE The ABS control solutions can be grouped in two main categories, wheel acceleration and tire slip control. The first category approaches slip control indirectly, by controlling the deceleration/acceleration of the wheel through the brake pressure from the actuator. The second one is a direct slip control, and has many different approaches, [9]–[12] present P-I and P-D controllers, without developing of calculations, to obtain the control gains that help obtaining the optimal performance of this controllers. However, [12] proposed a fuzzy control solution for ABS control, in [13] and [14] a control strategy based on sliding mode analysis is proposed, and [15] proposed an optimal control of an ABS laboratory setup by INTECO. This paper focuses on the study of an ABS laboratory setup, manufactured by INTECO. A space-state mathematical model was obtained from this system. The first proposed control strategy is a PID controller, then a control law is proposed based on the Control Lyapunov Function technique. This control strategy is a generalization of the notion of Lyapunov function used in the stability analysis of systems. The fact that the Control Lyapunov Functions are, as a general rule, easier to obtain than the feedback law, will be helpful when obtaining nonlinear stabilization. Then the two techniques are compared to establish their differences in performance. The paper is organized as follows: Section II briefly presents the development of a quarter of a vehicle mathematical model. Section III is dedicated to the linearization of the system under given conditions. The different techniques used in the development of the PID controller are shown in Section IV. The demonstration of the asymptotic stability provided by the error feedback using Control Lyapunov Function techniques is done in Section V. Finally in the last Section, the results and comparison of the two controllers are presented, through charts.
Abstract— An experimental comparison of two control techniques applied to a quarter of a vehicle ABS, Antilock Braking System, laboratory setup is presented. The first technique is a basic PID controller and the second one is a nonlinear control method through error feedback using Control Lyapunov Function techniques.
I. INTRODUCTION The origin of the ABS, Antilock Braking System, dates back to the early 20th century when Bosch was granted a patent for an ”Apparatus for preventing lockbraking [sic] of wheels in a motor vehicle” in 1936 [1]. For several years there were not significant advances due to the fact that the available technology could not implement the antilock concept successfully. Between 1967 and 1970 MercedesBenz engineers changed the mechanical sensors for contactless sensors that operated under the induction principle [2]. Finally, with the arrival of electronic integrated circuits (that were small and robust enough) it was possible to record data from the wheel’s sensor. The latter enabled the use of more reliable actuators for adjusting the brakes’ hydraulic pressure. An important milestone in ABS history occurred in 1978 when Bosch and Mercedes-Benz, working together, launched the second generation of ABS (ABS2) installed in the Mercedes-Benz Class S (as an option) in mass production [2]. The ABS was developed to prevent the wheels from locking up while braking thus preventing the slippage of the wheels on the surface [3]–[5]. Modern ABS not only try to prevent the wheels from locking up, but also they aim to obtain maximum wheel grip on the surface while the car is braking [6], [7]. The ABS is designed to increase the braking efficiency and to maintain the car’s maneuverability [8]. Besides, it also functions as a control mechanism to adjust the brake fluid pressure level of each wheel to reduce the driving instability and decrease the braking distance. Several algorithms had been proposed for controlling the ABS with the aim of, obtaining a minimum braking time, and an increased driving capability of the vehicle.
II. SYSTEM MODEL As shown in Fig. 1, the ABS laboratory setup is comprised of two wheels; the lower wheel simulates the relative motion of the vehicle over the road or surface, and the upper wheel simulates the motion of the vehicle’s wheel. This is considered as a quarter of a vehicle system and it is introduced for practical analysis and design of the braking system. The proposed model was created by INTECO, which takes into consideration: the braking forces exerted on the wheels, the wheel motion and the vehicle motion.
*Research supported by CONACYT scholarships 513153 and 512691. 1 M. Martinez-Gardea, I. J. Mares Guzm´an and C. Acosta Lua are with Universidad de Guadalajara, Centro Universitario de la Ci´enega, Ocotl´an, Jalisco, 47820, M´exico, {marcela.martinezg, iordan.mares}
@alumno.udg.mx, [email protected] 2 S. Di Gennaro is with the Department of Information Engineering, Computer Science, and Mathematics, and also with the Center of Excellence DEWS, University of L’Aquila, 67100, L’Aquila, Italy,
[email protected] 978-1-4799-7408-5/14/$31.00 ©2014 IEEE
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torque in the lower bearing m20 and the friction torque among the wheels µ(λ)fn . In Fig. 3 an auxiliary diagram of the model is shown. All the values for the different constants used in the following calculations are defined in Table I. During the deceleration process, the braking torque is applied on the upper wheel, consequently the velocity of the wheel decreases, the ABS dynamic can be described as [8]: J1 x˙ 1 = Ft r1 − d1 x1 + m10 + Tb (1) J2 x˙ 2 = − Ft r2 + d2 x2 + m20
For the implementation of the ABS, the upper wheel is equipped with a disk brake system which is driven by a small DC motor. The braking system includes two encoders that will provide the necessary measurements to estimate the wheel angular velocities [8].
where Ft is given by: Ft = µ(λ)fn
(2)
and fn is calculated as: fn = (d1 x1 +m10 +Tb +mg )/[l(sin(ϕ)−µ(λ) cos(ϕ))] (3) where l is the distance between the contact point of the wheels and the rotational axis of the balance lever, and ϕ is the angle between the normal in the contact point and the line l.
Fig. 1: Physical system. A. System diagram In Fig. 2 a physical diagram of the ABS experimental prototype is shown. The laboratory set up presents the angular motion of the wheel during braking. Although, the model is simple, it maintains the fundamental characteristics of the real system [8]. When the dynamic equations of the systems are derived, some assumptions are made. First, it only considers the longitudinal dynamic of the vehicle, therefore, the lateral and vertical motion of the wheels are neglected. Second, the rolling resistance force is omitted because its magnitude is very small in comparison to the braking force.
Fig. 3: Auxiliary diagram. A new variable is proposed: S(λ) = µ(λ)/[l(sin(ϕ) − µ(λ) cos(ϕ))]
(4)
So (2) can be rewritten as: Ft = S(λ)(d1 x1 + m10 + Tb + mg )
(5)
Considering: µ(λ) = w4 λp /(a + λp ) + w3 λ3 + w2 λ2 + w1 λ
(6)
The relative difference among the velocities of both wheels can be represented in terms of a coefficient, and is defined as: λ = (r2 x2 − r1 x1 )/(r2 x2 ) (7)
Fig. 2: ABS schematic diagram.
The brake actuator is described by the following equation:
B. Mathematical model
T˙b = c3 1(b(u) − Tb )
There are three torques acting on the upper wheel: the braking torque Tb , the friction torque in the upper bearing m10 and the friction torque among the wheels µ(λ)fn . Model parameters and variables description are given in Table I. There are two torques acting on the lower wheel: the friction
where the function b(u) can be approximated by: ( b1 u + b2 u ≥ u0 b(u) = 0 u < u0 72
(8)
(9)
with u0 as the operating threshold of the brake driving system. The system described in (1) can be generalized as: x˙ = f (x) + gTb + d where x = ( x1 , x2 )
Substituting (13), (17) and (18) in (14) and (15) the linearized system is created and by simple mathematical procedures a matrix form is obtained: ∆λ˙ a11 a12 ∆λ b1 = + ∆Tb (19) ∆x˙ 2 a21 a22 ∆x2 b2 ∆λ y = (1 0) (20) ∆x2
(10)
T
and f (x) =
where:
S(λ) dJ1 r11 − Jd11 )x1 S(λ) Jr11 − J11 ; g = −S(λ) Jr22 −S(λ) dJ1 r22 x1 − Jd22 x2 S(λ) Jr11 (m10 + mg ) − mJ10 1 d= −S(λ) Jr22 (m10 + mg ) − mJ20 2
a11
meanwhile (7) can be represented as: y = h(x) = λ
(11) a12
Rewriting equation (10): x˙ 1 c11 S(λ) + c13 0 x1 = + x˙ 2 c21 S(λ) c23 x2 c15 S(λ) + c16 c12 S(λ) + c14 + Tb + c25 S(λ) c22 S(λ) + c24 (12) with r1 d1 J1 − mJ10 1 − rJ2 d21 − mJ20 2
c11 = c14 = c21 = c24 =
(m10 +mg )r1 J1 c15 = Jr11 (m +m )r c22 = − 10 J2 g 2 c25 = − Jr22
c12 =
a21
kr1 (c11 x10 + c12 + c15 Tb0 − ! c21 x10 + c22 + k − = −r2 x20 −r2 (1 − λ0 ) − +c25 Tb0 −r x S(λ )c /r 2 20 0 21 1 −r2 x20 [S(λ0 c11 + c13 ] (1 − λ0 ) S(λ0 )c11 + c13 + c23 − =− −r2 (1 − λ0 )S(λ0 )c21 /r1 x20 = k(c21 x10 + c22 + c25 Tb0 ) − r2 x20 S(λ0 )c21 /r1
a22 = r2 (1 − λ0 )S(λ0 )c21 /r1 + c23 b1 = −[r1 (c15 S(λ0 ) + c16 ) − r2 (1 − λ0 )c25 S(λ0 )]/r2 x20 b2 = c25 S(λ0 )
c16 = − J11
From (19) and (20) the following transfer function is obtained: (s − a22 )b1 + b2 a12 (21) G(s) = s2 + (−a22 − a11 )s − a12 a21 + a11 a22
c23 = − Jd22
and rearranging some terms:
c13 = − Jd11
G(s) =
III. S YSTEM LINEARIZATION To linearize the system some initial values for operation points are selected P0 = [x10 , x20 , λ0 , Tb0 ]. Linearizing (4), (7) and (12) developing Taylor series expansions [16] and using (6): ∆x˙ 1 = ∆x1 S(λ0 )c11 + c13 + + ∆S(λ)(c11 x10 + c12 + c15 Tb0 )+ + ∆Tb c15 S(λ0 ) + c16 ∆x˙ 2 = ∆x1 S(λ0 )c21 + ∆x2 c23 + + ∆S(λ) c21 x10 + c22 + c25 Tb0 + + ∆Tb c25 S(λ0 ) ∆λ˙ = −(∆x˙ 1 r1 − r2 (1 − λ0 )∆x˙ 2 )/(r2 x20 ) w4 λp p w4 λ2p 0 p 0 − λ0 (a+λ p 2+ ) ) λ0 (a+λp sin(ϕ) 0 0 +3w3 λ20 + 2w2 λ0 + w1 ∆λ ∆S(λ) = l(sin(ϕ) − µ(λ0 ) cos(ϕ))2
ks = b1 T1 = (a12 b2 − a22 b1 )/b1 q T2 = a22 + a11 − a222 + a211 + 4a12 a21 − 2a11 a22 /2 q T3 = a22 + a11 + a222 + a211 + 4a12 a21 − 2a11 a22 /2
(13)
It is possible to evaluate in operations points: λ0 ∈ {0.05, 0.14, 0.18, 0.19, 0.4, 0.8}; x20 ∈ {178, 89, 1}; Tb0 ∈ {9, 5, 1} [12]. In Table II some values are shown for these operation points. Analyzing the values, the system begins to loose stability from λ0 = 0.19, so a reference value of λ0 is proposed as λref = 0.14.
(14)
(15)
IV. CONTROL STRATEGIES The ABS laboratory setup, like most physical systems, has a nonlinear nature that makes the calculations and approximations of mathematical models harder to solve. Besides, if it is considered that the systems are subject to variations and perturbations, which are difficult to estimate, the model complexity increases. In this section two control strategies are introduced. A PID controller using the linearized system and a nonlinear control law through error feedback using Lyapunov function techniques.
(16)
(17)
considering that w4 , w3 , w2 , w1 , λ0 , p, ϕ and a are constants defined in Tables I and II. Next, a variable change is proposed: r1 ∆x1 = r2 (1 − λ0 )∆x2 − r2 x20 ∆λ
(22)
where:
Eq. (16) can be rewritten as: ∆S(λ) = k∆λ
ks (s − T1 ) (s + T2 )(s + T3 )
(18) 73
The goal of the ABS controller is to track a predeterminated slip set point λref . For this reason, in this article we must be a change the variable x = (x1 , x2 )T to xn = (λ, x2 )T in the both control strategies.
stable system the Lyapunov derivative must be negative definite. [18]. This control technique use the information of Lyapunov function for the construction of the control law kl from V (x) and the vector fields defining the system. An advantage of this technique is to extreme simplicity and ease of implementation moreover is that provides automatically an analytic feedback law [19]. In this section, a variable change will be used, x = (x1 , x2 )T to xn = (λ, x2 )T , which will generate the following transformed system: λ˙ = r1 m10 − fn µ(λ)r1 /x2 r2 J1 + d1 (1 − λ)/J1 −
A. PID controller The PID controller is an attractive option, because it is widely used in the industry [17], you can use it even if the plant is unknown and it is easy to tune. With the ABS laboratory setup there is the advantage that you can test it and adjust the parameters as you wish. There are several techniques used to tune the PID: the Ziegler-Nichols rule [17], the root locus, the Routh-Hurwitz criterion, the pole-zero cancellation or the empirical method. Considering the transfer function: 0.6142s + 0.005378 (23) G(s) = (s + 1.291101)(s + 0.009898)
− (1 − λ)(fn µ(λ)r2 + d2 x2 + m20 )/x2 J2 + + r1 Tb /x2 r2 J1 x˙ 2 = − Ft r2 + d2 x2 + m20 /J2 (24) To properly apply this technique [19], λ must track a constant reference λref , so an error function is defined:
and applying the Routh-Hurwitz criterion on the transfer function closed loop:
e = λ − λref
(25)
e˙ = λ˙
(26)
and its derivative:
Y (s) kp (0.6142s + 0.005378) = 2 R(s) s + (1.301 + 0.614kp )s + 0.01278 + 0.005378kp
Therefore, the following lemma is proposed: Lemma 1: The control law that ensures asymptotic stability in the sense of Lyapunov for the ABS system is: r1 (fn µ(λ)r1 − m10 )/(x2 r2 J1 )+ x2 r2 J1 (1−λ) Tb = + x2 J2 (fn µ(λ)r2 + d2 x2 + m20 )− r1 −d1 (1 − λ)/J1 − kl e (27) Proof: The following Lyapunov candidate function is proposed:
From this analysis no useful information was obtained, therefore the Ziegler-Nichols rule was kept in use; however, the system does not meet the minimum requirements for this technique to be applied. The next method is the root locus. With this method one or more parameters from the transfer function vary and the geometric place of the closed loop transfer function poles are located. First we place the poles and the zeroes on the plane (open loop). The branches always begin in the poles and finish in the zeroes. Then, the asymptotes are calculated. Finally we draw the trajectories between poles and zeroes (just the odd) and the behavior of kp from zero to infinity. With (23), the dynamic analysis of the root locus indicates that the pole (-1.291101) will track a trajectory over the x-axis with -∞ direction. Meanwhile, the zero (-0.008756) attracts the pole (0.009898). All the trajectories are over the left side of the plane, so no matter how much kp grows it is still on the left part of the plane. So, for the PID controller design, an empirical approach was used, taking into consideration that the ABS laboratory Setup is available for tuning the parameters. The following values are proposed: kp = 2.8, ki = 3.25, kd = 0.3.
V (e) = e2 /2
(28)
V˙ (e) = ee˙
(29)
and its derivative: Substituting (26) and (24) in (29): r1 (m10 − fn µ(λ)r1 )/x2 r2 J1 − V˙ (e) = e − (1−λ) x2 J2 (fn µ(λ)r2 + d2 x2 + m20 )+ +d1 (1 − λ)/J1 + r1 Tb /x2 r2 J1
(30)
Applying the control law (27), the equation is reduced to: V˙ (e) = e(−kl e) = −kl e2
(31)
which proves that V˙ (e) < 0 ↔ kl > 0, ensuring that the system is asymptotically stable [18].
B. Nonlinear control law through of Control Lyapunov function techniques
V. R ESULTS
Stability is an important concern in systems theory. In the following control strategy proposal, the main concern is the stability of equilibrium points, which are usually characterized in the sense of Lyapunov. A simple definition for a Lyapunov function for a system is a positive definite function monotonically decreasing along the system trajectories, this means that the Lyapunov function are V (0) = 0, V (x) > 0 and V (x) → ∞ as |x| → ∞. This can be proven just by means of the Lyapunov derivative, even if the system trajectories are unknown. For an asymptotically
Experimentation is performed on the ABS laboratory setup INTECO. The initial conditions were x1 (0) = x2 (0) = 1700 RPM (178.02 rad/s). During the runs, a control signal is sent to the actuator in order to begin with the braking process. In the following, a comparison between the two controllers is shown. Fig. 4 shows the performance of both controllers, it is evident that the Lyapunov controller accomplishes the braking process in less time and distance. In Fig. 5 it is 74
Fig. 4: Braking distance vs. time. PID controller (dash), Lyapunov controller (solid). Fig. 7: Wheel velocity vs. time. PID controller (dash), Lyapunov controller (solid).
Fig. 5: Slip vs. time. PID controller (dash), Lyapunov controller (solid).
Fig. 8: Vehicle velocity vs. time. PID controller (dash), Lyapunov controller (solid).
shown that tracking the reference slip is a difficult task for both controllers, but from 8.5 seconds to 9 seconds in the braking process the controllers manage to properly track the reference. Fig. 6 exhibits the smoothness of the control signal attained by the Lyapunov controller in comparison with the PID. This represents less wear of the braking actuator by the Lyapunov controller. Fig. 7–8 show that the PID controller maintains a constant deceleration, however the Lyapunov controller generates the same velocity change in both wheels (x1 and x2 ) and this can be accomplished because the Lyapunov controller follows the reference no matter how drastic the change of the slip is. Fig. 6: Control variable (u) vs. time. PID controller (dash), Lyapunov controller (solid).
VI. CONCLUSIONS In this paper, two control strategies were designed and tested. The first approach was to work with a linearized version of the ABS laboratory setup mathematical model 75
through a PID controller. The second approach was to work directly with the nonlinear model with a control law designed by error feedback through a Lyapunov function. The Lyapunov controller achieves a shorter braking distance in less time than the PID controller, furthermore, the Lyapunov controller provides smooth signals to the actuators, maintaining their durability. The Lyapunov controller prevents braking instability by following an established reference, thus increasing passenger safety.
ACKNOWLEDGMENT The authors would like to thank Universidad de Guadalajara – Centro Universitario de la Ci´enega, especially to the Master and Doctoral Departments. R EFERENCES [1] Robert Bosch GmbH. (2003). ABS a success story. [Online]. Available: http://www.bosch.com/assets/en/company/innovation/theme03.htm [2] Daimler AG. (2008, July 1). Mercedes-Benz and the invention of the anti-lock braking system: ABS, ready for production in 1978. [Online]. Available: http://media.daimler.com/dcmedia/0-921-6574861-803841-1-0-0-0-0-0-11701-614318-0-1-0-0-0-0-0.html [3] M. Petrov, V. Balankin and O. Naruzhnyi, Study of automobiles brakes and pneumatic tires work model of the work process of antiblock brake system, Novosibirsk, NISI, Tech. Rep. 1977. [4] N. Rittmannsberger, Antilock braking system and traction control in Proc. Int. Congr. Transportation Electronics Convergence 88, 1988, pp. 195-202. [5] R. Emig, H. Goebels and J. Schramm, Antilock braking systems (ABS) for commercial vehicles-status 1990 and future prospects, in Proc. Int. Transportation Electronics Congr. Vehicle Electronics in the 90, 1990, pp. 515-523. [6] U. Kiencke and L. Nielsen, Automotive control systems: For engine, driveline, and vehicle. 2nd ed. 2005 Springer, pp. 11. [7] R. Rajamani, Vehicle dynamics and control (mechanical engineering series), softcover reprint of hardcover 1st ed. 2006 ed. Springer, pp.2. [8] Intenco Ltd, The laboratory antilock braking system controlled from pc, User manual, Poland, 2006. [9] M. H. Al-mola, M. Mailah, P. M. Samin, Performance Comparison between Sliding Mode Control and Active Force Control for A Nonlinear Anti-Lock Brake System WSEAS Trans. on Syst. and Control, Volume 9, 2014. [10] S. John, and J. O. Pedro, A Comparative Study of Two Control Schemes for Anti-Lock Braking Systems 2009 Control Conference (ASCC), 9th Asian, Istambul 23-26 June 2009. [11] M. B. Radac, R. E. Precup, S. Preitl, J. K. Tar, E. M. Petriu. Gain-Scheduling and Iterative Feedback Tuning of PI Controllers for Longitudinal Slip Control, IEEE 6th International Conference on Computational Cybernetics, pp. 183-188 Stara Lesna, Slovakia, November 2008. [12] M. B. Radac, R. E. Precup, S. Preitl, J. K. Tar, E. M. Petriu.”Linear and fuzzy control solutions for a laboratory Anti-lock Braking System”, SISY 2008, 6th Int. Symp. on Intelligent Syst. and Informatics, vol. 1, no.6, pp. 26-27 Sept. 2008. doi: 10.1109/SISY.2008.4664947. [13] Y. Oniz, E. Kayacan and O. Kaynak, A Dynamic Method to Forecast the Wheel Slip for Antilock Braking System and Its Experimental Evaluation IEEE Transaction on System, Man and Cybernetics–Part B: Cybernetics, Vol. 39, Number 2, April 2009. [14] L. Li, F.-Y. Wang and Q. Zhou, Integrated longitudinal and lateral tire/road friction modeling and monitoring for vehicle motion control, IEEE Trans. Intell. Transp. Syst., vol. 7, no. 1, pp. 1-19, 2006. [15] P. Bania, A. Korytowski, M. Szymkat and P. Gorczyca Optimal control of a laboratory anti-lock brake system in Proceedings of the 16th IFAC World Congress, Prague, pp. 1-6., 2005. [16] B. C. Kuo, Sistemas Autom´aticos de Control, 2nd ed. CECSA, 1991. [17] K. Ogata, Ingenier´ıa de Control moderna, 4th ed. Prentice Hall, 2003. [18] H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2002. [19] E. D. Sontag. ”A Universal Construction of Artstein’s Theorem on Nonlinear Stabilization”, Report SYCON-98-03, Rutgers Center for System and Control, February 1989.
APPENDIX TABLE I: SYSTEM PARAMETERS Symbol x1 x2 Tb r1 r2 J1 J2 d1 d2 µ(λ) λ fn m10 m20 mg l ϕ u Ft a bu1 bu2 c31 p u0 w1 w2 w3 w4
Description Angular velocity of the upper wheel Angular velocity of the lower wheel Braking torque Radius of the upper wheel Radius of the lower wheel Moment of inertia of the upper wheel Moment of inertia of the lower wheel Viscous friction coefficient of the upper wheel Viscous friction coefficient of the lower wheel Friction coefficient between wheels Slip (relative difference of the wheel velocities) Normal force Static friction of the upper wheel Static friction of the lower wheel Gravitational and shock absorber torques acting on the balance lever Distance between the contact point of the wheels and the rotational axis of the balance lever Angle between the normal in the contact point and the line L Control of the brake Total force generated by the upper wheel and pressing on the lower wheel Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant
Value/Unit rad/s rad/s Nm 0.0995m 0.099m 7.53x10-3 kgm2 25.6x10-3 kgm2 118.74x10-6 kgm2 /s 214.68x10-6 kgm2 /s
58.214 N 0.0032 Nm 0.0925 Nm 19.62 Nm 0.370 m 65.61°
N 0.00025724 15.24 -6.21 20.37 2.09945271 0.415 -0.04240011 2.9375x10-10 0.03508217 0.40662691
TABLE II: SYSTEM BEHAVIOR UNDER DIFFERENT INITIAL CONDITIONS λ0 0.02 0.02 0.02 0.4 0.4 0.4 0.8 0.8 0.8
x20 178 178 178 89 89 89 1 1 1
x10 174.4 174.4 174.4 53.4 53.4 53.4 0.2 0.2 0.2
Tb0 9 5 1 9 5 1 9 5 1
ks 0.0058 0.0058 0.0058 0.0108 0.0108 0.0108 0.9584 0.9584 0.9584
T1 116.7632 116.7632 116.7632 115.8296 115.8296 115.8296 118.1109 118.1109 118.1109
T2 104.7154 90.0932 75.4710 0.0095 0.0095 0.0095 44.3511 38.1546 31.9581
T3 0.0100 0.0100 0.0100 -0.4635 -0.3970 -0.3305 0.0088 0.0088 0.0088
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