At large slip angles, the tire model will no longer be linear. This formulation can be used to obtain inertial acceleration values of a vehicle with yaw rate, roll and pitch rotational motion.
In this section, the formulation is used to obtain inertial acceleration along the lateral axis of a vehicle which has rotational yaw motion. Consider a rotating body, as shown in Figure , described in two coordinate systems : a coordinate system fixed in inertial space XYZ and a coordinate system fixed to the body xyz. At the time instant under consideration, assume that both coordinate systems have the same orientation.
Let the angular speed of the body be a. Inertial and body-fixed coordinate systems Consider a particle P with inertial coordinates [X Y z]T and body- T fixed coordinates [x y z ] located on the body.
Let 7 be the vector from the origin of the inertial coordinate system to the point P. Apply Eq. The lateral system in terms of rotating coordinates 2. Hence the lateral model developed in section 2. Consider a vehicle traveling with constant longitudinal velocity V,on a road of constant radius R. Define the rate of change of the desired orientation of the vehicle as The desired acceleration of the vehicle can then be written as Define el and e2 as follows Guldner, et. If the velocity were not constant, one would integrate Eq.
Hence the approach taken is to assume the longitudinal velocity is constant and obtain a LTI model. If the velocity varies, the LTI model is replaced with an LPV model in which longitudinal velocity is a time varying parameter see section 3.
Substituting from Eqs. Note that the lateral dynamics model shown above is a function of the longitudinal vehicle speed V, which has been assumed to be constant. If the influence of road bank angle is included, then Eq. The lateral dynamics of the vehicle is controlled by the front wheel steering angle 8. Single track model for vehicle lateral dynamics The body side slip angle can be related to el and e2 as follows. For small tire slip angles, the lateral tire forces can be approximated as a linear function of tire slip angle.
The front and rear tire forces and tire slip angles are defined as follows: where C and C, are the cornering stiffness of the front and rear tires respectively. Substituting 2. It is suitable for control system design, since a lane keeping controller must utilize body fixed measurements of position error with respect to road. To obtain a global picture of the trajectory traversed by the vehicle, however, the time history of the body-fixed coordinates must be converted into trajectories in inertial space.
From body fixed to global coordinates As shown in Figure , the lateral distance between the c. The position of the vehicle in global coordinates is therefore given by where X d e s , Ydes are the global coordinates of the point on the road centerline which lies on a line along the lateral axis of the vehicle. R Continuity of curvature is an important criterion that a road should satisfy in order to ensure that the lateral control system can track it.
Clothoid spirals are curves that are used to transition smoothly from one curvature value to another for example, in going from a straight road to a circular road. A clothoid is defined to be a spiral whose curvature is a linear function of its arc length and is mathematically defined in terms of Fresnel integrals Kiencke and Nielsen, Table 2.
Figure shows a clothoid spiral used to transition from a straight line segment to a circular arc. Clothoid spiral joining a straight line and a circle.
These models can be used to design steering control systems for lateral lane keeping. These models can also be extended for use in yaw stability control, rollover control and other vehicle control applications.
The major lateral models discussed in the chapter were 1. Kinematic vehicle model 2. Dynamic vehicle model in terms of inertial lateral position and yaw angle 3. Dynamic vehicle model in terms of road-error variables 4. It is a useful model for very low speed applications, for example vehicle control for automated parking. The dynamic models discussed in this chapter are useful for lane keeping applications and can also be extended for use in yaw stability control and rollover prevention applications.
The extension and use of these models for yaw stability control is discussed in Chapter 8. The transformation of coordinates from body-fixed to global axes was also presented. In addition road models were discussed and the use of clothoid spirals to transition smoothly from one road curvature to another was described. Donath, M. Hoffman, D. Jost, K. Kiencke, U. Leffler, H. Meriam, J. Peng, H. Rajamani, C. Zhu and L. Alexander , "Lateral control of a backward driven front- steering vehicle", Control Engineering Practice, Vol.
Sasipalli, V. ED, No. Taylor, C. Thorpe, C. Wang, D. This chapter discusses lateral control systems used to control a vehicle to stay in the center of its lane. The chapter is organized as follows. Control design by state feedback is discussed first in section 3. Steady state errors and the steady state steering angle required to negotiate a curved road are analyzed in sections 3.
The subsequent sections of the chapter concentrate on control design by output feedback sections 3. The following values of vehicle parameters will be used for all the simulations in this chapter. These values are representative of parameters for a passenger sedan.
The open-loop matrix A has two eigenvalues at the origin and is unstable. The system has to be stabilized by feedback. Calculations show that the pair A, B I is controllable. Hence, using the state feedback law the eigenvalues of the closed-loop matrix A - B K can be placed at any desired locations. The closed-loop system using this state feedback controller is The following Matlab command can be used to place the eigenvalues of the closed-loop system. The road is initially straight and then becomes circular with a radius of meters starting at a time of 1 second.
The desired yaw R rate is shown in Figure and is a step input from 0 to 1. The time histories of the lateral error el and yaw angle error e2 are shown in Figure and Figure respectively.
Due to the presence of the B2Pdesterm in equation 3. The steady state values of el and e2 are non-zero because the input due to road curvature Pdes is non-zero. A physical interpretation of these steady state errors is provided in sections 3. Lateral position error using state feedback Steering Control for Automated Lane Keeping 3.
In this section, we will investigate whether the use of a feedforward term in addition to state feedback can ensure zero steady state errors on a curve. Assume that the steering controller is obtained by state feedback plus a feedforward term that attempts to compensate for the road curvature : Then, the closed-loop system is given by Taking Laplace transforms, assuming zero initial conditions, we find where L Q and L v d e S are Laplace transforms of aff and pdes respectively.
However, cannot influence the steady state yaw error, as seen from equation 3. Steering Control for Automated Lane Keeping has a steady state term that cannot be corrected, no matter how the feedforward steering angle is chosen. The steady state yaw-angle error is The steady state lateral position error can be made zero if the feedforward steering angle is chosen as which upon close]r inspection is seen to be where Kv - 1 rm jfm is called the understeer -5qj-jZJ-m vx2.
Steering Control for Automated Lane Keeping 59 In conclusion, the lateral position error el can be made zero at steady state by appropriate choice of the feedforward input 6 f l. As expected, the geometric analysis provides the same answer as the feedforward system analysis of the previous section.
However, a better physical understanding of the lateral tire force requirements is obtained from the geometric analysis. As discussed in the previous chapter, the slip angle at each wheel is the angle between the orientation of the wheel and the orientation of its velocity vector. Let the slip angle at the front wheel be denoted by af and that at the rear wheel be denoted by a,. The instantaneous turn center 0 of the vehicle is the point at which the two lines perpendicular to the velocities of the two wheels meet.
Steering angle for high speed cornering Hence the steady state steering angle is given by v The steady state slip angles afand a, are related to the road radius as follows. In words, the lateral force developed at the rear axle is m, times the lateral acceleration. Assume that the slip angles are small so that the lateral tire force at each wheel is proportional to its slip angle.
This is the same as equation 3. Depending on the relative values of the front and rear cornering stiffness and mass distribution values, three possibilities exist for the value of Kv : 1. Neutral steer In this case the understeer gradient Kv is zero due to equal slip angles at the rear and front tires. Steering Control for Automated Lane Keeping In the case of neutral steer, on a constant radius turn, no change in the steering angle is required as speed is varied.
The steering angle depends only on the curve radius and the wheelbase. In the case of understeer, on a constant radius turn, the steering angle will have to increase with speed in proportion to KV times the lateral acceleration. In the case of oversteer, on a constant radius turn, the steer angle will have to decrease as the speed is increased. The steering angle as a function of vehicle longitudinal speed is shown in Figure for the three cases of neutral steer, understeer and oversteer.
Note that in the case of oversteer, the steering angle decreases with speed and could eventually reach zero at a speed called critical speed. Chapter 3 understeer neutral steer critiial speed Figure Steering angle variation with speed 3. If the parameters of the vehicle and the vehicle speed were such that then the steady state yaw error of equation 3.
This happens at one particular speed VXatwhich equation 3. The physical interpretation of equation 3. The right hand side of the equation, as we have seen during the geometric analysis, is the slip angle at the rear tire. The left hand side of the equation is the angle y subtended by the rear portion of the vehicle at the center of the circular path, as shown in Figure below.
Steady state yaw angle error Since the vehicle has a finite length, both its lateral position error and its yaw-angle error cannot always be made simultaneously zero. If the steady- state lateral position error is zero, then the steady state yaw-angle error can be zero only if the slip angle at the rear is the same as the angle y subtended by the vehicle at the center of the circular path.
This happens at one particular speed V,at which equation 3. Is non-zero yaw angle error a concern? The above geometric analysis shows that no matter which control law is used, the yaw angle error e2 will have a steady state value. This is because the slip angles at the rear and front wheels are completely determined, once the radius of the road and the vehicle speed V, are fixed. Hence the slip angle of the vehicle ,8 is automatically determined. Since the steady state error in e2 is equal to P , from equation 3.
A constant state feedback matrix K can be used to obtain stability for varying velocity by exploiting the convex nature of the lateral dynamic system. The following Theorem summarizes the design result that can be used for full state feedback control system design.
Theorem 3. Sensor systems used for measurement of lateral position include differential GPS Donath, et. Look ahead lateral position measurement with respect to road If we assume that the yaw angle error e2 is small so that chord lengths can be approximated by arc lengths, then the measurement equation that relates the output to the states is as follows: where d, is the longitudinal distance of the point ahead of the vehicle c.
Chapter 3 3. Here P s is the plant transfer function between the steering angle input for the vehicle and the lateral position measurement output described in section 3.
C s represents the transfer function for the controller to be determined later. The road-determined desired yaw rate pdes affects the system dynamics through a transfer function denoted in Figure as G s. The signal n t is the sensor noise that affects the system. P s has two poles at the origin, a pair of complex conjugate poles and a pair of complex conjugate zeros. Note that the zeros in Figure are much better damped than the zeros in Figure As d , is increased, the damping increases for the complex conjugate pair of zeros.
A longitudinal velocity of 2 5 d s has been used in the model. The open-loop transfer function P s has two poles at the origin, an additional pair of complex conjugate poles and a pair of complex conjugate zeros.
Steering Controlfor Automated Lane Keeping The contour T, that s traverses in the complex plane for purposes of plotting the Nyquist plot must not pass through any poles or zeros of the open loop transfer function PC s.
Hence it must not pass through the origin. Hence the following contour Tsas shown below in Figure was used for the Nyquist plot. The l? The contour rpC must be drawn for all values of s that s takes from the rs contour. It is important to draw the rPC contour for sections 1, 2, 4 and 5 of rs see Figure and determine how many times this contour encircles the -1 point.
The rPC contour for sections 1 and 2 is shown in the Nyquist plot in Figure The solid line in this figure corresponds to sections 1 and 2 of r, while the dashed line corresponds to sections 4 and 5 r,.
To determine how many times the above rPC contour encircles the -1 point, it is necessary to zoom into the region near the -1 point, as is being done in Figure and Figure In this case, the rpC contour encircles the -1 point twice: once clockwise and once counterclockwise.
The clockwise encirclement can be easily seen in the big picture Nyquist plot of Figure In the zoomed section of Figure , a counter clockwise encirclement can be seen. Figure 8 shows the root locus plot for varying feedback gain with the proportional controller. Again it can be seen that for small proportional gain, there is a pair of complex conjugate poles that are unstable. As the proportional gain is increased, these poles become stable. It is important to note that with adequately large proportional gain, although the closed loop system gets stabilized, it still has poor phase margin.
This can be seen from the Nyquist plots as well as the Bode plot showing the gain and phase margins in Figure In Figure , with a proportional gain of 1, a phase margin of 18 degrees is obtained. It can be deduced from the plot that this is close to the best phase margin that can be obtained for this system. With a smaller gain of 0. With a higher proportional gain of 10, the system only has a phase margin of 8 degrees. Phase uncertainty can therefore easily change the number of encirclements of the -1 point for this system.
Frequency radlsec Figure Steering Controlfor Automated Lane Keeping 3. Hence a lead compensator is suggested. The following transfer function can be used for the controller compensator Values for Tn and Td can be chosen so as to design the closed-loop system to have any desired value of phase margin. In the plots shown in the next few pages, the above arbitrary values of Tn and Td are used just to show that this compensator will increase the phase margin of the system.
Figure shows the Bode plot for PC s using the above lead compensator. It is clear that with the lead compensator phase has been added at the low frequencies to improve phase margin. Figures and show the Nyquist plot for PC s. Figure shows the Nyquist plot corresponding to sections 1 and 2 of I?
It is clear that the Nyquist curve does not encircle the -1 point and the closed-loop system is stable for all values of the compensator gain K. Figure shows the root locus plot for the system with lead compensator. Again, it is clear that the closed-loop system is stable for all values of the compensator gain K. Chapter 3 Figure The plots show that the transfer function has less damping at higher speeds. It can be seen that the closed-loop system also is better damped at lower speeds and has less damping at higher speeds.
Chapter 3 mIs As seen in Figure and Figure , as the variable d, is increased, the system is better damped. This is also observed in the time response plots shown in Figure , where the higher values of d, gives a better damped step response. Large values of d, correspond to "look-ahead measurement in which the lateral position error with respect to road is measured at a distance significantly ahead of the vehicle.
Look ahead measurement is typical when a vision system is used for lateral position measurement. Steering Controlfor Automated Lane Keeping 89 angle measurement so as to extrapolate the lateral position error to a look- ahead point.
A lag compensator would be able to adequately perform this task. First, the use of full information in the form of state feedback was presented. The lateral system is controllable and can be stabilized by state feedback. On a straight road, with the use of a state feedback controller, all position and yaw errors were shown to converge to zero. On a circular road, however, these errors do not converge to zero with state feedback.
The use of a feedforward term in the control system enables the position error to converge to zero. However, the yaw angle error will always have a steady state value, resulting in a steady state vehicle slip angle. Equations for the feedforward term and for the steady state slip angle were presented. Next, control system design using output feedback was discussed. The output measurement was assumed to be lateral position measurement with respect to road center at a look-ahead point.
Such a measurement is available from vision cameras and can also be obtained from other types of lateral position measurement systems. Nyquist plots were used to design a control system. It was shown that a proportional controller could stabilize the system if adequately large gains could be used. However, it would still suffer from poor phase margin. The use of a lead compensator together with proportional feedback ensures both adequate phase and gain margins and good performance. Another important result presented in the chapter was that by increasing the look-ahead distance at which lateral position measurement is made, a simple lag compensator would be adequate at providing good performance and robustness.
Steering Control for Automated Lane Keeping steering wheel angle turn radius of vehicle or radius of road feedback gain matrix for state feedback controller feedforward steering angle steady state steering angle steady state yaw angle error understeer gradient steady state tracking errors on a curve lateral tire force lateral tire force on front tires lateral tire force on rear tires longitudinal velocity at c. Steinhauser, R. Chen, C. Gillespie, T. Guldner, J. Malik, J.
Patwardhan, S. S and Guldner, J. Kosecka, J. Rajamani, H. Tan, B. Law and W. Tan, H. Weber, J. Wong, J. Zhang and R. Common systems involving longitudinal control available on today's passenger cars include cruise control, anti-lock brake systems and traction control systems. Other advanced longitudinal control systems that have been the topic of intense research include radar-based collision avoidance systems, adaptive cruise control systems, individual wheel torque control with active differentials and longitudinal control systems for the operation of vehicles in platoons on automated highway systems.
This chapter presents dynamic models for the longitudinal motion of the vehicle. The two major elements of the longitudinal vehicle model are the vehicle dynamics and the powertrain dynamics. The vehicle dynamics are influenced by longitudinal tire forces, aerodynamic drag forces, rolling resistance forces and gravitational forces.
Models for these forces are discussed in section 4. The longitudinal powertrain system of the vehicle consists of the internal combustion engine, the torque converter, the transmission and the wheels. Models for these components are discussed in section 4.
The external longitudinal forces acting on the vehicle include aerodynamic drag forces, gravitational forces, longitudinal tire forces and rolling resistance forces. These forces are described in detail in the sub-sections that follow. Longitudinal forces acting on a vehicle moving on an inclined road A force balance along the vehicle longitudinal axis yields where Fxf is the longitudinal tire force at the front tires Fx, is the longitudinal tire force at the rear tires Fa,, is the equivalent longitudinal aerodynamic drag force Rd is the force due to rolling resistance at the front tires Rxr is the force due to rolling resistance at the rear tires m is the mass of the vehicle g is the acceleration due to gravity 6 is the angle of inclination of the road on which the vehicle is traveling The angle 6 is defined to be positive clockwise when the longitudinal direction of motion x is towards the left as in Figure It is defined to be positive counter clockwise when the longitudinal direction of motion x is towards the right.
Longitudinal Vehicle Dynamics 4. Atmospheric conditions affect air density p and hence can significantly affect aerodynamic drag. The commonly used standard set of conditions to which all aerodynamic test data are referred to are a temperature of 15 0 C and a barometric pressure of The corresponding mass density of air p may be taken as 1.
According to Wang, , the following relationship between vehicle mass and frontal area can be used for passenger cars with mass in the range of kg: The aerodynamic drag coefficient Cd can be roughly determined from a coast-down test White and Korst, In a coast down test, the throttle angle is kept at zero and the vehicle is allowed to slow under the effects of aerodynamic drag and rolling resistance.
Under these conditions, the longitudinal dynamics equation can be re- written as Integrating equation 4. Longitudinal Vehicle Dynamics In equation 4. Equation 4. From such a plot, the value of P for a particular T vehicle can be obtained.
Once p has been obtained from equation 4. Experimental results have established that the longitudinal tire force generated by each tire depends on a the slip ratio defined below , b the normal load on the tire and c the friction coefficient of the tire-road interface.
The vertical force on a tire is called the tire normal load. Section 4. Jetzt verschenken. In den Warenkorb. Sie sind bereits eingeloggt. Klicken Sie auf 2. Lateral Vehicle Dynamics. Longitudinal Vehicle Dynamics. Only valid for books with an ebook version.
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