Sliding mode control and estimation theory has been used to estimate vehicle steering states for a steer-by-wire system. (Krishnaswami and Riozzoni, 1995).
UILU-ENG-98-7033
Model Recognition and Simulation of an E/H Steering Controller on Off-Road Equipment
D. Wu, Q. Zhang, J.F. Reid, H. Qiu and E.R. Benson Department of Agricultural Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801
[Abstract] An automated E/H steering controller is essential for autonomous off-road equipment. This paper presents a methodology for designing the E/H steering controller through a combination of testing and simulation. The steering kinematic model provides the steering linkage gain between the hydraulic actuator and the front wheel. The model makes it possible to close the steering control loop based on the signal of the hydraulic steering actuator rather than the front wheel. Test results were used to identify the non-linear and dynamic characteristics of the original E/H steering system. A Matlab model of E/H steering controller was developed based on the experimental test results. [Key words] steering controller, electrohydraulic, dynamic model, system identification, test and simulation, off-road equipment
1 Introduction Automation of off-road equipment (ORE) will lead to more precise maneuvering, more consistent performance, higher efficiency, and less labor costs in operation. Advancements in precision agriculture reinforce the need for automated off-road equipment. One of the key tasks in developing autonomous ORE is automated steering control. In addition, automated steering control on a semi-autonomous off-road equipment will also improve the safety of the human operator and equipment, simplify and speed the operation, and allow the operator to concentrate on handling other implementation functions. ORE steering controller design differs from that of onhighway vehicles due to its operating conditions. Off-road equipment often operates on unprepared, changing and unpredictable terrain, ranging from asphalt to spongy topsoil in the field. ORE steering controllers should be able to provide appropriate steering actions in response to the
variations in equipment operation states, travelling speed, tire cornering stiffness, ground conditions, and many other parameters influencing steering dynamics. Laine (1994) analyzed E/H control techniques for a parasitic steering valve. The steering controller design depended on many factors other than E/H steering elements and vehicle dynamics, including ground condition and vehicle speed. Erbach et al. (1991) stated that neither negligible nor constant friction could produce significant and unpredictable sideslip. Grovum and Zoerb (1970) developed an agricultural vehicle steering dynamic simulation model. Julian (1971) developed transfer function model for the turning (yaw) rate of a tractor. O’Connor et al. (1996) developed a steering controller based on a set of linear motion equations. Lee (1997) used a "model-following" control method to modify both the steady state and the transient lateral response characteristics of a small-size Variable Dynamic Testbed Vehicle (VDTV) for both compact-size and mid-size vehicles. Sliding mode control and estimation theory has been used to estimate vehicle steering states for a steer-by-wire system (Krishnaswami and Riozzoni, 1995). All previous research indicated that ORE requires a steering controller with stable and fast response. Due to the high degree of non-linearity and many unknown factors involved in ORE steering, it is difficult to design an appropriate steering controller from traditional design methods. This paper describes an effective method for designing ORE steering controller based on the investigation of E/H steering characteristics on an agricultural tractor. 2 Approach A 115-kw Case 7220 Magnum 2WD tractor was modified to serve as the research platform for this research. A pulse width
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modulated (PWM) electrohydraulic valve was installed in parallel to an existing steering handpump for actuating the automated steering. A data acquisition card was installed in the cab-mounted computer to convert voltage signal for steering control into PWM signal to drive the E/H valve. This research intended to develop a method for designing a digital steering controller D(z) based on experimental results. The digital controller should be able to provide optimal steering control with changing speed, load, soil conditions, and operating conditions. Understanding the characteristics of the electrohydraulic steering system and its system responses, and determining key factors affecting the steering control were critical in accomplishing the task. The non-linearity in the system could be identified via steering system characteristic tests. A mathematical model of steering system was developed based on both time and frequency domain test results.
F1 = P1 A1-P2 A2 F2 = kF1 F3 = F2 cosθ1 F4 = F3 cos ( 90-θ 7 )
where P1 and P2 are the pressures on either side of the piston; A1 and A2 are the head-end and rod-end areas of the steering cylinder; k is the geometrical gain of the steering linkage; θ1 and θ7 can be calculated from steering linkage geometry. From the force balance equations, the steering spindle angle (θ) and torque (T) relationships can be derived.
θ = f1 ( g ) T = f 2 ( g , F1 )
Figure 1 shows the block diagram of the steering controller. A linear potentiometer was installed on the steering cylinder to provide steering actuator linear displacement feedback. A signal processor was implemented to estimate the steering angle of the front wheels from the feedback signal. E/H actuator
θi D(z)
G(s)
( 2)
T M2
M1 M3
θ
g M4
Linkage
L
K(L)
F1
θο
m
K-1 (L)
Figure 1. Steering controller model. The design of steering controller was based on both time domain and frequency domain technical specifications for the developed system model, like rise time, settle time, overshoot, gain margin, and phase margin. Since a common PID controller could not meet these technical specifications, nonlinear controller, optimal controller, or fuzzy controller could be selected as feasible alternatives. 3 Analysis of Steering Linkage A feedback signal processor, which was the inverse function of the steering linkage gain, was been designed to convert steering cylinder displacement signal into front wheel angle signal. The design of the signal processor was based on the analysis, calibration and simulation of the steering linkage kinematics model (Figure 2). From this kinematics model, one can easily get the force balance equations.
θ7
F3
M6
M5 F4
θ1
F2
Figure 2. Sketch of a typical tractor steering linkage kinematics model. The steering model was developed using LabView (National Instruments Corp, Austin, TX). Simulation results indicated that the front wheel angle was roughly linearly related to steering cylinder displacement. However, the torque on the front wheels had a nonlinear relation to steering cylinder displacement. (Figure 3 a & b). torque on front wheel axis[lb-in]
Digital controller
(1)
4000
3000
2000
1000
0 6
10
14
18
rod travel length [in.]
figure 3(a) nonlinear torque to rod travel with
a certain piston drive force
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-250
-300
-350 6
10
14
18
rod travel length[in.]
figure 3(b) Steering linkage kinematics
simulation. 4 The Identification of Model and Parameters 4.1 Calibration of the steering feedback sensor Zero degree wheel angle was defined as the front wheel angle at which the tractor would travel in a straight line. To determine the "zero point", the indicated wheel angle from the potentiometer was logged continuously and the tractor was driven along a straight-line path. Additionally, the indicated wheel angle from the potentiometer was logged while the steering was turned from maximum right to maximum left. The feedback signal was V0=2.777 volts at the "zero point" for the system tested. 4.2 Identifying system non-linearities An electrohydraulic steering system on an agricultural tractor has several non-linear characteristics, including dead zone, saturation, asymmetry, and hysteresis. Dead zone will greatly affect the dynamic performance of an E/H steering control system and result in a lag and an unstable response. It is essential to carefully identify and compensate for the dead zone for satisfactory performance. Saturation limits the operational range of the steering control system. System design includes quantitative evaluation of non-linear elements and appropriate compensation of those non-linearities. Stombaugh (1997) investigated the non-linearity of the E/H steering valve and found that the system non-linearities were affected not only by the valve characteristics, but also by the characteristics of the cylinder, wheel-road interactions. To obtain the complete information, the system should be tested under both loaded and no-load situations. Static friction and dynamic steering torque change with variations in tractor speed and wheel-road interaction. The influence of unknown factors on the system was dealt as disturbances to the steering system.
The steering dead zone also reflects the overlap in the E/H valve, the static friction between steering linkage and hydraulic cylinder, as well as the interaction between the wheel and the road. Hydraulic fluid temperature and other factors can also cause inconsistency in the dead zone. It is difficult to develop an exact multi-variable function to account for the contribution from each of the factors. Our test results showed that the dead zone for right-turn was almost constant (Table 1), but the left-turn dead zone varied with operating conditions. Changing the travelling surface from concrete to soil had limited effect on the dead zone when making right turn. However, it resulted in significant variation when making left turn. The dead zone testing method consisted of gradually ramping up the input signal until a change was observed in the system. The dead zone is just the input signal at which the system begins to react. Table 1. Dead zone Level variations with different wheelroad interaction
dead zone
turn left
free load Average -0.157 Std. dev. 0.014
turn right
con- sponge free con- sponge crete topoil load crete topoil -0.163 -0.234 0.245 0.240 0.240 0.020 0.029 0.000 0.001 0.008
The response range of the E/H valve and the controller need to be properly matched to avoid driving one element into saturation. The wheel angle sensor had the capacity to provide a voltage output of 0 to 5 V with a center voltage position representing zero-angle (straightforward) steering. Actual full-scale output range was 0.65V for full right turn and 4.6V for full left turn. 4.3 Developing a model of the unmodified E/H plant Test data for the system gain (Figure 4) shows that for a 40 o]
front wheel angle [
o
front wheel angle[ ]
-200
Input=1.25v
20
Input=-0.75v
0
Input=-1.25v Input=0.75v
-20
-40 0.0
0.5
1.0
1.5
2.0
2.5
tiome[s]
Figure 4. E/H steering system step response for varying input voltages
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20
2
Gain (dB)
0
-20
0
-40
-60
-2 0
input voltage[v]
(b)
x
(c)
x
(d)
x
1 s(τ 1 s +1)
y
ω c2 s(s + 2ξ cω c s +ω c2 )
NonLinearity
y
2
Figure 6. Four E/H steering models four signal transmission models of the original steering system. The potentiometer was assumed to be linear. The output (y) represents the travel of cylinder rod. A time constant is used in model (b) and (c) to roughly represent the E/H steering system. Model (d) includes the non-linear
-45
-90
-135
-180
-225
0
1
100
The E/H steering system was modeled with a frequency response approach. An inverted function of Fnl (vi) for the digital controller was inserted in the system to reduce the effect of non-linearity on the transfer function. The sampling and control rate (50 1/s) was used to counteract the nonlinearity of physical system with the digital inverted function. 10
0.6 simulation test input
o
angular velocity [ /s]
0.032v/s
0.030 0.025 0.020 0.015
0.01177v/s
0.010
0
0
-10
-0.6
-20
T=1/a=0.11s
-1.2 0
1
0
0.000
time(s)
Figure 7.
10
frequency (rad/sec)
A zero-input response was used to estimate the overall time constant τ0, and investigate the behavior of the output signal from a finite rate to zero. Figure 7 shows cylinder velocity curve differentiated from the potentiometer reading. The time constant, τ0 = 0.11s, was estimated from the curve.
0.035
0.005
100
Figure 8(b) Phase Frequency Characteristics elements, and a hydraulic cylinder. A more realistic model (Figure 12) would include E/H valve and noise source to simulate pressure wave effects.
y
y 1 s (τ 0 s +1) NonLinearity
10
Zero-input response of output rate.
command signal [v]
(a)
E/H steering system
1
Frequency (rad/sec)
Figure 8(a) Amplitude Frequency Characteristics
Figure 5. E/H steering system’s steady state characteristics certain input signal, the output curve almost has a constant slope. The curves on the left side of Figure 4 represent the transient period rather than the steady state characteristics. The test results indicate that a certain input voltage corresponds to a certain rod translation velocity in the original E/H steering system. Corrected for nonlinear elements, it includes a single integrator rather than a double integrator. The steady-state characteristics of the E/H steering system (Figure 5) indicate that there are the saturation and the dead zone. Figure 6 shows x
0.1
5
Phase (degree)
-5
0.2
0.4
0.6
time[s]
Figure 9. Simulation and test results for the zero-input response.
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5. Simulation and controller angle of front wheel[o]
8
A model of the original E/H steering system model was developed based on tractor test data (Figure 11). This model has taken consideration of the non-linearities, static and dynamic frictions. The test and simulation curves are compared in Fig 9 and 10. The test and simulation curves indicate that there is good consistence between the time domain step. Sine responses comparative analysis of results from tests with and without the inverted function indicated that there were dynamic asymmetry and non-linearity in the system.
open-loop frequency response
6
simulation
4 2
test in field 0
test on road -2 0
2
4
time[s]
6
8
Figure 10. The simulation and test results with 0.5v and 0.5Hz sine input.
In addition, precautions were taken to avoid driving the output into saturation. The first order time constant was estimated as:
θι
1 = 0.175 s ω1
+
+ Inverted linkage
The inverted function in the forward path could not entirely eliminate the non-linear factors. The dynamic asymmetry that remained could be imitated with a balancing coefficient (ρ=0.95).
PID +
Nonlinearity Compensation
E/H steering system
θο linkage
Figure 12. E/H steering actuator and controller 6 open-loop frequency response o
front wheel angle [ ]
The difference between the time constant estimations for the zero-state (0.11 s) and frequency response (0.175 s) was due to the inverted digital controller during the frequency response test. Generally speaking, the natural frequency of the E/H valve (ωs) and the cylinder chamber pressure wave (ωpw) are higher than the natural frequency of the cylinder (ωc). Model (d) in Figure 6 was used to simulate the E/H steering system. From Figure 8(b), the cylinder phase crossover frequency was 15 rad/s (2.4 Hz). The higher frequency components (ωs and ωpw) are difficult to distinguish. Pressure waves within the cylinder chamber showed up as high frequency oscillations on the fundamental feedback signal from the potentiometer
Filter
3 command angle 0
-3 real angle
-6 0
2
4
time[s]
6
8
10
Figure 13(a). Angle control of tractor front wheel (5osin0.1Hz) 6 real angle
front wheel angle [ o ]
τ1 =
Differ -ential
Filter2
3 0 command angle
-3
t time
Clock
1
In1 Out1
s
sin
non-linearity Integrator
pressure noise 910
-6 0
s2 +36s+910 solenoid 5Hz
0.067 pc/wc net force force v
Sum3
Out1 Out2
startstop
friction control
2
4
time[s]
6
8
10
Figure 13(b). Angle control of tractor front wheel (5osin0.5Hz)
1
v
dynamic system
s integ3 Sum5
Out1 In1
2.57
dynamic friction control
yo
Figure 11 Simulation model of E/H steering actuator
The purpose of the simulation was to design an E/H controller. The system gain and non-linearity, which was estimated and validated, was used to design the compensation function (figure 12). Figure 13(a) and 13(b) are the front wheel angle test curves with a two-second moving average controller. The wheel angle error could be controlled within 1o for 0.1 ~0.5Hz sine angle input. The system had a large amount of noise for both the guidance and E/H steering system (>0.2 V).
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Considering the noise inherent to the test system, this error level was acceptable. 6.
Grovum, M.A., and G.C. Zoerb, 1970. An automatic guidance system for off-road vehicles, Transactions of the ASAE 13(5):565-573.
Conclusion
This paper presents a method for designing an E/H steering controller for off-road equipment. It is necessary to compensate the main non-linear factors, including dead zone and asymmetry to obtain satisfactory steering control performance. The effect of wheel-road interaction can be modeled either as changing model parameters or as an input disturbance. Test results on an agricultural tractor validated this methodology.
Julian, A.P., 1971. Design and performance of a steering control system for agricultural tractors. Journal of Agricultural Engineering Research 16(3):324-336. Krishnaswami, V. and G. Riozzoni, 1995. Vehicle steering system state estimation using sliding mode observers, Proceedings of the 34th Conference on Decision & Control, New Orleans, LA, vol. 4, p3391-3396.
Acknowledgments
Laine, P., 1994. Development methods of controller used in automatic guidance system, In Proceeding Of XII World Cong. On Agricultural Engineering 2:1159-1166.
This research was supported by the University of Illinois at Urbana-Champaign Research Board, Illinois Council on Food and Agricultural Research, and Case Corporation. Mr. Jeffrey Will assisted in preparing the tests. All of the mentioned support is gratefully acknowledged.
Lee, A.Y., 1997. Matching vehicle response using the modelfollowing control method, Vehicle Dynamics and Simulation, SAE Inc., 970561, P57-69.
Reference Erbach T.C., C.H. Choi, and K. Noh. 1991. Automated guidance for agricultural tractors. In: Proceeding automated agriculture for the 21st century, ASAE Publication No. 11-91: 182-191.
O’Connor, M., T. Bell, G. Elkaim, and B. Parkinson, 1996. Automatic steering of farm vehicles using GPS. Paper presented at the 3rd international conference on precision agriculture. Minneapolis, MN, June 23-26. Stombaugh, T.S., 1997. Automatic Guidance of Agricultural Vehicles at Higher Speeds, Ph.D. dissertation. Dept. of Agriculture Engineering, UIUC.
