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Institutionen för systemteknik Department of Electrical Engineering Examensarbete AFS-Assisted Trailer Reversing Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping av Olof Enqvist LITH-ISY-EX--06/3752--SE

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Institutionen för systemteknik Department of Electrical Engineering Examensarbete AFS-Assisted Trailer Reversing Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping av Olof Enqvist LITH-ISY-EX--06/3752--SE Linköping 2006 Department of Electrical Engineering Linköpings universitet SE Linköping, Sweden Linköpings tekniska högskola Linköpings universitet Linköping AFS-Assisted Trailer Reversing Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping av Olof Enqvist LITH-ISY-EX--06/3752--SE Handledare: Examinator: M.Sc. Christian Lundquist ZF Lenksysteme Lic. Gustaf Hendeby isy, Linköpings universitet Dr. Wolfgang Reinelt ZF Lenksysteme Dr. Jacob Roll isy, Linköpings universitet Linköping, 27 January, 2006 Avdelning, Institution Division, Department Division of Automatic Control Department of Electrical Engineering Linköpings universitet S Linköping, Sweden Datum Date Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LITH-ISY-EX--06/3752--SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version Titel Title Aktiv styrning vid backning med släp AFS-Assisted Trailer Reversing Författare Author Olof Enqvist Sammanfattning Abstract Reversing with a trailer is very difficult and many drivers hesitate to even try it. This thesis examines if active steering, particularly AFS (Active Front Steering), can be used to provide assistance. For analysis and controller design a simple geometric model of car and trailer is used. The model seems to be accurate enough at the low speeds relevant for trailer reversing. It is shown that the only trailer dependent model parameter can be estimated while driving. This enables use with different trailers. Different schemes to control the system are tested. The main approach is to use the steering wheel as reference for some appropriate output signal, for example the angle between car and trailer. This makes reversing with a trailer more like reversing without a trailer. To turn left, the driver simply turns the steering wheel left and drives. Test driving, as well as theoretical analysis, shows that the resulting system is stable. Of the eight drivers that have tested this type of control, five found it to be a great advantage while two considered it more confusing than helpful. A major problem with this control approach has to do with the way AFS is constructed. With AFS, the torque required to turn the front wheels results in a reaction torque in the steering wheel. Together with the reference tracking controllers, this makes the steering wheel unstable. Theoretical analysis implies that this problem has to be solved mechanically. One solution would be to combine AFS with electric power steering. This thesis also presents a trajectory tracking scheme to autonomously reverse with a trailer. Starting from the current trailer position and the desired trajectory an appropriate turning radius for the trailer is decided. Within certain limits, this will stabilize the car as well. The desired trajectory can be programmed beforehand, but it can also be saved while driving forward. Both variants have been tested with good results. Nyckelord Keywords Automotive Control, Trajectory Tracking, Active Steering Abstract Reversing with a trailer is very difficult and many drivers hesitate to even try it. This thesis examines if active steering, particularly AFS (Active Front Steering), can be used to provide assistance. For analysis and controller design a simple geometric model of car and trailer is used. The model seems to be accurate enough at the low speeds relevant for trailer reversing. It is shown that the only trailer dependent model parameter can be estimated while driving. This enables use with different trailers. Different schemes to control the system are tested. The main approach is to use the steering wheel as reference for some appropriate output signal, for example the angle between car and trailer. This makes reversing with a trailer more like reversing without a trailer. To turn left, the driver simply turns the steering wheel left and drives. Test driving, as well as theoretical analysis, shows that the resulting system is stable. Of the eight drivers that have tested this type of control, five found it to be a great advantage while two considered it more confusing than helpful. A major problem with this control approach has to do with the way AFS is constructed. With AFS, the torque required to turn the front wheels results in a reaction torque in the steering wheel. Together with the reference tracking controllers, this makes the steering wheel unstable. Theoretical analysis implies that this problem has to be solved mechanically. One solution would be to combine AFS with electric power steering. This thesis also presents a trajectory tracking scheme to autonomously reverse with a trailer. Starting from the current trailer position and the desired trajectory an appropriate turning radius for the trailer is decided. Within certain limits, this will stabilize the car as well. The desired trajectory can be programmed beforehand, but it can also be saved while driving forward. Both variants have been tested with good results. v Acknowledgements A lot of people has helped me with this thesis. Christian Lundquist worked together with me all the way, providing valuable feedback and ideas. Wolfgang Reinelt invited me to do the thesis and shared his experience in control. I want to thank them both. I am also very grateful to Gustaf Hendeby and Jacob Roll, for their patience and for good advice concerning the thesis. Further, my family and friends, Camorasan, Alexander, Christine, Frank, Gerd, Ralf, Reinhard, Samuel, Schusterle, Thomas and Thorsten are not forgotten... och Christian, se nu till att du kommer ut och får sladda lite ibland också. vii Contents 1 Introduction Active Front Steering This Thesis Modelling Basics Coordinates Geometric Constraints Kinematic Constraints Differential Equations Model Validation Trailer Length Estimation System Characteristics Equilibria Left-Right Steering Jackknifing Stability Reference Tracking Controlling γ Input-Output Linearization Optimal Feedback Comparing the γ-controllers Controlling the Turning Radius Anti-Jackknifing Test Driving Alternatives Error Correction Modified Steering Characteristics Assisting Steering Wheel Torque ix 5 Steering Wheel as Reference Linear Analysis Stabilizing the Steering Wheel No Steering Shaft Friction With Steering Shaft Friction Limiting Instability Autonomous Steering Tracking Positioning Saving a Trajectory Special Trajectories Conclusions and Future Work Modelling Reference Tracking Controllers Trajectory Tracking A Derivations 37 A.1 Deriving the Differential Equations A.2 Deriving the Trailer Velocity B Model with Lateral Slip 39 B.1 Slip Angles B.2 Dynamics Bibliography 43 Chapter 1 Introduction When reversing, a car with an attached trailer constitutes an unstable system. Thus, the trailer will tend fold up to the car like a jackknife. To avoid this the driver has to compensate for the movement of the trailer using the steering wheel. This is especially difficult due to another characteristic of the system: To get the trailer more to the left, you need to turn the steering wheel more to the right, which can be quite confusing. All of this makes trailer reversing very difficult and many drivers hesitate to even try it. This thesis examines if active steering, particularly AFS (Active Front Steering), can be used to provide assistance. We will soon discuss the aim and contents of the thesis more thoroughly, but first a short introduction to active steering might be in place. 1.1 Active Front Steering Normally, the front wheel angles of a car depend solely on the steering wheel angle. With Active Front Steering (AFS) it is possible to superpose an additional, electronically controlled angle (Figure 1.1). Thus, the steering characteristics can be adjusted according to the driving situation. The superposition angle is provided by an electric motor connected to a planetary gear. This way, the mechanical connection between steering wheel and front road wheels is maintained. In case of malfunction, the planetary gear will be mechanically locked, allowing the driver to steer normally. AFS was developed by ZF Lenksysteme. It is optional equipment for some BMW series. There it is used to adjust the steering ratio depending on vehicle speed. At low and medium speeds the steering becomes more direct, requiring less steering effort in, for example, a parking manuevre. At higher speeds, the steering is less direct to enhance directional stability. BMW also uses AFS to stabilize the car in critical driving situations. 1 2 Introduction steering wheel angle superposition angle Figure 1.1. With AFS, an angle is superposed to the steering wheel angle. 1.2 This Thesis The work behind this thesis was conducted at ZF Lenksysteme. They were looking for new applications for AFS. Consequently, the aim of this thesis is to explore the potential of using AFS for trailer reversing. To do so, different assistance functions are developed and evaluated. Though they are primarily intended for presentations, an aim is to make them suitable for serial production as well. Thus, they use existing sensors, with the sole addition of an angle sensor to measure the trailer position. In serial production such a sensor could be placed in the towing hook of the car. For analysis and controller design a model of the car-trailer system is required. Such a model is derived and validated in Chapter 2. To enable use with different trailers, we need a method to estimate the trailer length while driving. Such a method is presented in Section 2.7. Chapter 3 discusses some important characteristics of the car-trailer system. The part on control is divided into two parts. Chapter 4 concerns reference tracking controllers, where the driver gives the reference value with the steering wheel. In Chapter 6 a couple of autonomous steering functions are constructed. Using the steering wheel to provide the reference value causes some special problems with AFS. They are analysed in Chapter 5. Chapter 2 Modelling The design and analysis of a controller requires an appropriate model of the system that we want to control. In this chapter, a model for the car-trailer system is derived. The modelling is made in a structured manner to enable expansions. Sections discuss the basic assumptions and formulate them as algebraic and differential constraint equations. From these equations a system of differential equations is derived. The derivations can be found in Appendix A and the resulting differential equations in Section 2.5. The validity of the model is examined in Section 2.6 and Section 2.7 presents a method to estimate the trailer length while driving. 2.1 Basics An important part of modelling, is to choose what kind of model to use. Which effects are essential and which can be disregarded? Choosing the most complex model is not always a good idea. Though such a model is theoretically more accurate, it tends to be more sensitive to variations and it can be difficult to estimate all the parameters. A simpler model is also easier to analyse. One choice we have to make is whether to include lateral slip in our model. Lateral slip is an effect of cornering. To turn, a car has to be affected by a lateral force. This force is provided by friction when the tyres slip sideways. A few facts can guide us when we decide whether to include slip in our model. One is that the weight of a specific trailer varies, and thus tyre friction and dynamic properties. If the model depends on these parameters, they would have to be estimated each time the car starts. Moreover, car owners tend to use their car with different trailers. Therefore, we want controllers that can adapt to a new trailer, and thus all trailer dependent parameters have to be estimated while driving. Since slip is linked to the lateral forces, a model with slip would depend on the dynamic properties of both the car and the trailer. A model without slip, on the other hand, only depends on the geometry of the car and the trailer. Besides, trailer reversing mostly takes place at low speeds, where side forces and lateral slip are small. 3 4 Modelling It seems a model without slip is more suitable for our purposes. (In cases when slip cannot be disregarded the model in Appendix B might be a starting point.) We will now look a little closer at the geometry of the car. To allow all wheels of the car to roll without lateral slip, the inner front wheel needs to turn more than the outer. The ideal geometry, often called Ackermann steering geometry, is shown in Figure 2.1. Here, all wheels are aligned to move in circles around a common central point, M. The relation between the left front wheel angle, δ F L, and the right, δ F R, is tan δ F L tan δ F R = w 1 l 1 tan δ F L tan δ F R, where w 1 and l 1 are the distances between the wheels as can be seen in Figure 2.1. Note that the wheel angles are positive when turning left. M δf L δ F R δ F. v F P v R l 1 P w 1. Figure 2.1. The Ackermann steering geometry. For our model, we will assume that the car has Ackermann steering geometry and that all wheels roll without lateral slip. Thus, all points on the car will move on circles around a common point (Figure 2.1). We define the points P 0 and P 1 as in Figure 2.1 and v F and v R as their velocity vectors. The movement of the car can be specified by the signed speed v R = ± v R, and the angle δ F between v F and the central axis of the car. In practice δ F is computed from measurements of the front wheel angles, using tan δ F = tan δ F L tan δ F R =. 1 + w1 2 l 1 tan δ F L 1 w1 2 l 1 tan δ F R From now on we will call δ F the front wheel angle of the car, though that is not entirely true. 2.2 Coordinates Coordinates We introduce a (global) inertial frame R with coordinates x R and y R as well as local frames L i with coordinates x Li and y Li (Figure 2.2). The car-fixed frame, L 1, has its origin in the point P 1 between the rear wheels, and its x-axis in the forward direction of the car. Frame L 2 is trailer-fixed with origin between the wheels of the trailer, and x-axis in the forward direction of the trailer. We also define a frame L 0 with origin in P 0, between the front car wheels, and x-axis coinciding with the velocity vector, v F (see Section 2.1). y R. L 0 x R... R L L Figure 2.2. The different coordinate frames. The position of body i in the plane is specified by the global coordinates of the origin P i of the body-fixed frame L i, and the orientation of body i is specified by ψ i = ψ LiR, the angle of rotation of L i with respect to R. For convenience, we also introduce the displacement vector r R P io = (xr P io, yr P io )T (see Figure 2.3). r R... P io R ψ i... L i... Figure 2.3. Displacement vector and angle of rotation. 6 Modelling 2.3 Geometric Constraints In this section we set up the purely algebraic constraint equations that originate from the geometry of the car-trailer system. The angles between the front wheels and the car are controlled by the driver together with the controller. These angles are specified by the single angle δ F (see Section 2.1). Thus, with ψ 0 and ψ 1 as defined in the previous section, ψ 0 ψ 1 δ F = 0. (2.1). P P 1 l 1 P... 2 Q 2 Q l 12.. l 2 Figure 2.4. Points in the different frames. By definition the origin P 0 of frame 0 is fixed in the L 1 frame r R P 0O A RL1 r L1 P 0P 1 r R P 1O = 0, where A RL1 = ( ) cos ψ1 sin ψ 1 sin ψ 1 cos ψ 1 and r L1 P 0P 1 = ( ) l1. 0 We get the constraint equations x R P 0O l 1 cos ψ 1 x R P 1O = 0 y R P 0O l 1 sin ψ 1 y R P 1O = 0. (2.2a) (2.2b) 2.4 Kinematic Constraints 7 We define a point Q 1 for the towing hook of the car and a point Q 2 for the coupling of the trailer (see Figure 2.4). Since the trailer is attached to the car, these points coincide, r R P 1O + r R Q 1P 1 (r R P 2O + r R Q 2P 2 ) = 0. where We get ( ) r R l12 Q 1P 1 = A RL1 0 and r R Q 2P 2 = A RL2 ( l2 0 ). x R P 1O l 12 cos ψ 1 x R P 2O l 2 cos ψ 2 = 0 y R P 1O l 12 sin ψ 1 y R P 2O l 2 sin ψ 2 = 0. (2.3a) (2.3b) 2.4 Kinematic Constraints So far, we have only considered the geometry of the car and trailer system. To get further we use the properties discussed in Section 2.1. By definition, P 0 moves only along the x-axis of the L 0 frame and as an effect of the no lateral slip assumption P 1 only move along the x-axis of L 1. With a similar argument we assume that the trailer has zero lateral slip. Thus the point P 2 will only move along the x-axis of L 2. We have ( A LiR ṙ R P io =, i = 0, 1, 2 0) which yields the constraint equations sin ψ 0 ẋ R P 0O + cos ψ 0 ẏ R P 0O = 0 sin ψ 1 ẋ R P 1O + cos ψ 1 ẏ R P 1O = 0 sin ψ 2 ẋ R P 2O + cos ψ 2 ẏ R P 2O = 0. (2.4a) (2.4b) (2.4c) Finally, the speed of the car can be measured. Let us assume that the speed of the rear wheels is measured. Since there is no slip, ( ) A L1R ṙ R vr P 1O =, 0 which yields cos ψ 1 ẋ R P 1O + sin ψ 1 ẏ R P 1O = v R. (2.5) Assuming that initial conditions are known we now have enough equations to decide the behaviour of the system as a result of the steering angle, δ F, and rear wheel speed, v R. 8 Modelling 2.5 Differential Equations To make our model easier to handle we rewrite it as a system of differential equations. The derivations can be found in Appendix A. Since we have four nonalgebraic constraint equations, (2.4) and (2.5), we get four independent variables in our equations. We choose these to be x 1 = x R P 1O, y 1 = y R P 1O, ψ 1 and γ = ψ 1 ψ 2. Note that γ is the angle between car and trailer, defined as positive in a left curve. The resulting equations are ẋ 1 = v R cos ψ 1 ẏ 1 = v R sin ψ 1 (2.6a) (2.6b) ψ 1 = v R tan δ F (2.6c) l ( 1 vr γ = + v ) R l 12 cos γ tan δ F v R sin γ. (2.6d) l 1 l 1 l 2 l 2 For obvious reasons the input δ F is bounded. We have δ F δ bd F. Sometimes it is more appropriate with a model that uses traveled distance, rather than time, as independent variable. We introduce σ as the distance travelled backwards by the rear wheels. Using the chain rule we get γ(t) = γ (σ) σ(t) = v R γ (σ) and ( 1 γ (σ) = + l ) 12 cos γ tan δ F + 1 sin γ. (2.7) l 1 l 1 l 2 l 2 Naturally, the same substitution could be performed in the first three equations, but that will not be necessary. 2.6 Model Validation To validate the model, we use measurements of rear speed, v R, front wheel angle, δ F, and car-trailer angle, γ, from test drives. By solving the model equations with measured inputs v R and δ F, we get a simulated output γ. Comparing measured and simulated γ gives an idea of the accuracy of the model. Figure 2.5 shows this comparison for a typical test drive. It should be mentioned that the agreement between measured and simulated γ is bad when reversing. The reason is that since the system is unstable, a small initial deviation will tend to increase. For this reason, we use measurements from forward driving, trusting that our assumptions are just as valid for reversing. 2.7 Trailer Length Estimation γ [rad] Time [s] Figure 2.5. Measured (solid) and simulated (dashed) γ, when driving forward at low speed. 2.7 Trailer Length Estimation To allow driving with trailers of different lengths we need some way to estimate the trailer length while driving. The standard solution is to use a prediction error method, choosing the parameter value that minimizes the quadratic sum of the prediction errors. The basics of parameter estimation can be found in [6]. The basis of our estimation is measurements of γ, δ F and v R. We use v R to get sequences γ(k) and δ F (k) that are equidistant in space rather than

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