Steering Behavior of a Four-Wheeled Road Vehicle with Automatic Course Control on a Fixed Path

This document is an English translation of the original German-language article “Führungsverhalten des vierrädrigen Straßenfahrzeugs bei Regelung des Kurses auf festgelegter Bahn” by W. Zimdahl, published in Regelungstechnik, Vol. 13, No. 5, 1965, pp. 221–268.

Abstract (bilingual in the original)

In recent years many papers have been published about the so-called “automatic highway”, where a vehicle, equipped with the required devices, is guided along a pre-established line at the center of the highway. In this paper the problems and their solutions — those proposed in the past as well as a new one suggested here — shall be discussed from the point of view of control theory. Of particular interest is the comparison of the actual vehicle path (trajectory) with the prescribed guide-line.

1. Introduction

In recent years many experimental investigations have been carried out on the motion of the four-wheeled road vehicle under automatic guidance. However, theoretical investigations on this problem are virtually absent or unusable. The reason for this is not so much an insufficient knowledge of the controlled plant, namely the vehicle. For example, Fonda in his 1957 paper [1] was able to set up differential equations, using a vehicle equivalent circuit, that describe the principal motion of the vehicle with sufficient accuracy. Attempts have also often been made to make use of the somewhat more advanced theory of automatically steered missiles, but in many cases false assumptions were made and inadmissible simplifications were adopted, so that the corresponding papers must be regarded as unusable. In this paper, the behavior of the vehicle is first examined on the basis of Fonda’s theory [1]. A second section discusses the road–vehicle connection via the guidance system and the sensing elements; in the final section, solutions to the problem are proposed and examined.

2. The Vehicle

a) Simplifications

First, some agreements on simplifications must be made. Corresponding to the six degrees of freedom that a body has in space, a road vehicle can perform six different motions (Fig. 1). Using the notation of Fig. 1, these are: three translational motions — in the direction of the x-axis: forward or longitudinal motion (u), in the direction of the y-axis: lateral or transverse motion (v), in the direction of the z-axis: [vertical]; and three rotational motions — about the x-axis: rolling (p), about the y-axis: pitching (r), about the z-axis: yawing (γ).

Setting up a system of equations for all six velocities simultaneously is very difficult. It is also often unnecessary, since, for example, pitching and motion in the z-direction are of no interest for considerations of lateral stability and exert only a very small influence on the motions in the other axes. In addition, the number of required equations can be reduced by one more by introducing the sideslip angle β with

tan β = v/u

β ≈ v/u

for small β. If necessary, the absolute magnitudes of the velocities v and u can be determined with the aid of the equations

v = V · sin β and u = V · cos β

The path speed V is thus given by

V² = u² + v² (1)

Kohr [2] found, as one of the results of his experiments, that rolling has a relatively weak influence on yawing and lateral motion; for this reason rolling will also be disregarded here.

b) Vehicle Equations

For the two remaining variables — yaw rate γ̇ and lateral velocity v — Fonda [1] gives the following two differential equations:

MV(γ̇ + β̇) + C(β − (a’/V)γ̇) = C_D δ (3a)

I_z γ̈ + N_1 γ̇ + χMVα = a’C_D δ (3b)

Here δ is the mean steer angle of the front wheels, hereafter called the steering angle, and M, C, C_D, I_z, N_1, χ, and a’ are constant quantities describing the vehicle. With I_z = k²M, these equations become:

MV(γ̇ + β̇) + C(β − (a’/V)γ̇) = C_D δ (5)

k²Mγ̈ + N_1γ̇ + χMV(γ̇ + β̇) = a’C_D δ (6)

Using the Laplace transform, the frequency responses are obtained from the two differential equations (5) and (6):

β/δ (p) and γ̇/δ (p) (7)

c) Transfer Functions and Pole–Zero Distributions

From the differential equations, the transfer functions are now derived. First — because of the importance of the case — this is done for a linearly increasing steering angle: δ = d·t with d = const.

Using the known methods of the s-transform, the result in all three cases takes the form

f(t) = f_∞(t) + f_T(t) (13)

where the first term has the form

f_∞(t) = c₁·(t − T) (14)

and the second the form

f_T(t) = c₂·e^(−r₁t) + c₃·e^(−r₂t) or f_T(t) = c₀·e^(−σt)·sin(ωt + φ) (15a/b)

The expression f_∞ thus represents the vehicle’s response for t → ∞; f_T describes the transient. The solutions for a step steering angle δ = δ₀ at time t = 0 are completely analogous in structure, except that here the term f_∞ is a constant — the steady-state value.

Fig. 2 shows a possible qualitative progression of the four functions ψ, β, γ̇, and v for a steering step δ = δ₀ at t = 0.

At the end of this section, Fig. 3 shows a pole–zero distribution (as a function of path speed V).

3. The Road–Vehicle Connection

The path of the road is the reference variable for the control system road–vehicle. Under automatic course control on a fixed path, the connection between road and vehicle is established by the guidance system: the prescribed driving path, which at any instant represents the set value for the lateral position of the vehicle, is marked in some way on the roadway; sensing elements mounted on the vehicle scan this marking so as to derive from the set and actual positions the control deviation. Through a controller an intervention at the actuator — the steering — then attempts to reduce the control deviation to zero.

The following guidance systems can be considered:

a) Mechanical Guidance Systems

To guide a vehicle with mechanical scanning of the prescribed path, a groove or rail in or on the roadway can serve this purpose; likewise a rail or wire above the roadway can be used. There are few examples of this type of vehicle guidance; the system is also unsuitable for high speeds.

b) Optical Guidance Systems

In these systems a strip on the roadway, which has a different color from the surface, serves to guide the vehicle. The strip is scanned by two photocells; the difference in their output voltages forms the control deviation. This is fed to a control amplifier that drives the steering servo motor. Here too there exist only model installations at reduced scale and at low speeds.

c) Electrical Systems [3], [4], [5], [6], [7]

Of these, several designs have been implemented as model and experimental installations, both at reduced scale and at full size. Electrical systems have the best prospects for future practical use.

In most electrical systems a guidance cable (guidance cable) is used beneath the center of the road lane; this is fed with low-frequency alternating current and acts as an antenna with a small effective range. A voltage is induced in two coils in the vehicle that depends, among other things, on the distance of the coils from the guidance cable. The difference of these two voltages is the control deviation; it is zero when both coils are equidistant from the guidance cable. If the coils are symmetrically mounted with respect to the vehicle’s longitudinal axis, then the control deviation is zero when the longitudinal axis lies above the cable (longitudinal axis parallel to the cable). With optimal placement of the coils, the zero-crossing of the differential voltage — which represents the control deviation — is very steep. Depending on the design, the differential voltage varies almost linearly with the lateral displacement (lateral deviation) over a narrow or wide range.

4. Mathematical Treatment of the Control System

A discussion of longitudinal control for the purpose of maintaining a constant distance from the leading vehicle or a similar criterion is omitted here, as the problems are completely different from those of lateral control. The vehicle’s speed is regulated internally, with appropriate set values being set as required. Thus the forward speed V₀ enters the following calculation only as a parameter.

For the mathematical treatment of the problem, a suitably chosen coordinate system is needed first.

Cartesian coordinates have two serious disadvantages:

A function y(x) that is to represent explicitly an arbitrary path can be multi-valued (this is already the case in the simple example of circular travel).
In control engineering a system is often characterized by its step response. Here the input variable (for a closed loop, e.g. the reference variable) is changed in a step. Representing a driving path in Cartesian coordinates would therefore mean that e.g. with a continuous change of x, y changes in a step from y₁ to y₂ — but this is a physical and technical absurdity.

Because of these serious disadvantages, the following coordinates are chosen for the mathematical treatment: curvature κ and path length s. The driving path is thus represented by the curvature as a function of the distance traveled s (κ(s)), with the distance traveled measured from an arbitrary point on the path. The two disadvantages of Cartesian coordinates do not arise:

The function κ(s), which is to represent a path explicitly, is unambiguous since s is unambiguous. A further advantage arises: s changes continuously with time as the path is traveled; in the simple case of constant speed s is even proportional to time t.
To record step responses, κ can be changed in a step. In practice this means that the path radius changes in a step; a transition from κ = 0 to κ = κ₀ is equivalent to the transition from straight-ahead travel into circular travel.

A further advantage is that — with ideal sensing element and controller — a step change in κ also results in a step change in steering angle, so that — from the control engineer’s point of view — the quantities κ and δ are linked only by a constant, not by a nonlinearity or a time-dependent function.

a) Controller Input is Lateral Deviation

With the above assumptions one can now proceed to the mathematical treatment of the problem.

In practice, the sensing elements in installations built so far measure the distance of the guidance cable from the vehicle’s longitudinal axis, i.e. the ordinate y_F₀ of the nearest guidance cable point in a vehicle-fixed Cartesian coordinate system designated (x_F, y_F) here (see also Fig. 1). Since in the calculation the set and actual paths are to be represented by κ₀(s₀) and κ₁(s₁) respectively, an extensive mathematical apparatus is needed to compute the control deviation Δy_F₀ from κ₀(s₀) and κ₁(s₁). In the block diagram of the control loop shown in Fig. 4, the dashed box is intended to indicate this conversion.

The result of the calculation of Δy_F₀ from Δκ and ψ is a system of two nonlinear, inhomogeneous differential equations in which expressions of the form f₁(t)·f₂(t) appear; in the general case it is not solvable.

To arrive at least at an estimate of the control loop behavior within the scope of this paper, the following agreements are made:

The forward speed V₀ is constant; terms with (Δκ)² can be neglected.
The sideslip angle β and its time derivatives shall be zero.
Furthermore, one restricts to very small deviations of the actual path from the set path.

With these simplifications, the originally very complicated differential equations can be considerably simplified; the result is:

Δy_F₀ = −V₀²·Δκ/κ with Δκ = κ₀ − var. (16)

or in the frequency response representation:

Δy_F₀/Δκ = −V₀²/p² (17)

By means of this estimation — which is very rough, and that must be emphatically stated once more — the loop is thus linearized. With the aid of Eq. (17) one arrives at the block diagram of the loop shown in Fig. 5.

Already upon looking at this block diagram the assumption is close at hand that the loop will be hard to stabilize without the addition of auxiliary control variables, that it may even become structurally unstable due to the 1/p² element (designated F_g in Fig. 5). Oppelt points out in his “Kleines Handbuch” [8] on page 164 that this possibility is generally given in vehicle control systems and illustrates it with an example (see [8], p. 467).

For simplicity, pure P-behavior is assumed for the controller; it is obvious that adding an I-component to the controller, or even a too-large time constant, could significantly worsen stability.

The frequency response of the vehicle has the following form:

F_F(p) = [numerator terms] / [denominator terms] (18)

Through more detailed calculations using data from real vehicles (these data were taken from the papers by Mitschke [9] and Gauß-Wolf [10]), it could be shown that for approximately V > 20 km/h the zeros of the frequency response are conjugate complex (σ₂ real part, ω₁ imaginary part of the zero). Furthermore, an oversteering vehicle is assumed that exhibits aperiodic transient behavior with time constants T₁, T₂.

Thus the frequency response F₀ of the open loop becomes:

F₀ = K_R · K_F · [expression with p terms] (19)

The characteristic equation of the control loop is:

1 − F₀ = 0: [polynomial in p with K_R terms] (20)

Although all coefficients are present in the characteristic equation — so the loop is not structurally unstable — it has a tendency in that direction, since with increasing speed V₀ the coefficient of p becomes smaller (because σ₂ becomes smaller and ω₁² larger, see Fig. 3), while all other coefficients grow or remain approximately constant.

The fully nonlinear control loop, without the simplifications of Eqs. (16) and (17), was simulated on the analog computer. Numerical values for the vehicle quantities were taken from the already cited papers [9] and [10]. Fig. 6 shows as a result of this simulation the step responses of the control loop for a set-value step from κ₀ = 0 to κ₀ = κ₀₀ (transition from straight-ahead travel into a circular arc). From this one can see that the guidance behavior of the system is poor, since the step responses show multiple strong overshoots.

One can also see that the loop is strongly nonlinear: the oscillations first grow, then decay.

One might try to improve the guidance behavior by feeding in auxiliary variables, e.g. by adding the differential angle Δψ between the guidance cable and the vehicle longitudinal axis, but this would not bring a fundamental improvement either. The reason for this is: both the control deviation y_F₀ and the differential angle Δψ grow with a step in κ₀ in the first approximation quadratically with the path length (or with time at V₀ = const.). But the controller output — which is to influence the steering angle — would also need to rise approximately in a step in order to prevent y_F₀ from growing inadmissibly in the first place.

b) Controller Input is Curvature Difference

In order to solve the above-mentioned problem, the structure of the control loop must be fundamentally changed. Since none of the quantities measurable from the vehicle changes at the same rate in time as κ, the set-path curvature κ₀ must be fed directly into the vehicle. How this can be technically realized will be shown in Section 5. The actual-path curvature κ₁ can be measured indirectly via the yaw rate γ̇ and the lateral acceleration v̇; at constant forward speed V₀ the following simple linear operation yields κ₁:

κ₁ = (1/h)·(γ̇₁/V₀ + v̈/V₀²) (21)

The Δκ formed in a comparison element is now fed directly to the controller; the main control variable is thus the curvature. The decisive advantage of this arrangement results from the elimination of the geometry-conditioned element 1/p² in the control loop (designated F_g in Fig. 5). The main control loop thus simplifies to a follow-up system.

However, one cannot do without a feed-forward of the lateral deviation y_F₀ — which now serves as an auxiliary variable — since as already mentioned the representation κ(s) does not fix the path in absolute terms. Even with perfectly regulated curvature, a permanent y_F₀ will result from the initial deviations; a supplementary branch must therefore be present in the controller for this.

This gives a control loop according to Fig. 7. In it the controller consists of two channels. The first channel is

F_R1 = c₁ + c₂/p

and receives the curvature difference Δκ as input; the second is F_R2 = c₃ and receives the lateral deviation y_F₀ as input. The overall gain of the controller is set by K; the individual components are adjusted by the c_i.

The frequency response of the controller thus becomes:

F_R = K · {F_R1 + (1/p²) · F_R2} = K · (c₁ + c₂p·σ₂ + c₃p²) / p²

F_R · p = K’ · c₁V₀² + c₂p + c₃p² (22)

By adding the component F_R1, two zeros are moved from infinity into the finite region of the p-plane. For the case c₂ = 0 these zeros become either purely imaginary, or one of them has a positive real part: the latter case means (monotonic) structural instability of the closed loop; the former case gives — depending on the position of the other poles and zeros — at minimum very poor stability, if not outright instability. Hence when c₂ is present, c₁ must also be present. One tries to make the zeros real, since then the roots running into them at high loop gains also become real. Three cases can be distinguished:

The zeros p_{z1,2} of the controller lie outside the vehicle poles in the p-plane, i.e. the absolute value of the zeros is greater than that of the poles.
At least one of the zeros lies between the vehicle poles.
Both zeros lie between the absolutely smallest vehicle pole and the double pole p_{R,0} of the controller at the origin.

The root loci of the control loop then have the following principal progressions:

For Case 1: The poles p₁,₂ of the vehicle first remain real with increasing K, then become complex (damping depending on the position of the controller zeros p_{z1,2}), at high K values become real again and run toward p_{z1,2}. The pole pair p₃,₄ of the controller is initially complex with positive real part, but with larger K the real part becomes negative. The damping of this natural oscillation is in general not greater than the value given by the vehicle zeros p_{z1,2}.

For Case 2: The poles p₁,₂ are always real; the principal progression of p₃,₄ does not change.

For Case 3: The first pole pair p₁,₂ is real at small K values, then complex with negative real part, running toward p_{z1,2}; the minimum damping of this natural oscillation is given by the position of p_{z1,2}. The second pole pair p₃,₄ is initially complex with negative real part; the damping of this natural oscillation increases steadily; at a medium K value D > 1, i.e. the poles p₃,₄ become real; the decay constant remains small.

The only practically useful solution — Case 3 — makes the system structurally stable; a K value can be found at which a compromise between the dampings of the two pole pairs is possible. The disadvantage lies in the very small natural frequency of the second oscillation; but in the Case 3 structure this is fundamentally caused by the vehicle poles, whose position can only be substantially influenced by a completely different vehicle design.

As the controller component for the lateral deviation y_F₀ only a P-controller appears; consequently a steady-state deviation must appear here. Although the component c₃ responsible for this must be very small because of the just-required position of p_{z1,2}, useful and reasonable values for the steady-state deviations in y_F₀ result — at medium gain factor K of the controller. The steady-state deviation is, incidentally, dependent on the forward speed V₀ only insofar as the transmission factor of the vehicle also depends on it, which is equivalent to the strength of the over- or understeer tendency of the vehicle.

5. Input of the Set Curvature (Set Value) into the Controller

Finally, a constructive proposal shall be made for the input of the set curvature κ₀ into the controller. The following prerequisite must be made: the curvature κ₀ is either constant (including κ₀ = 0) or linearly dependent on the path (κ₀ = k·s), which is almost exclusively the case in modern road construction.

It was already stated that the guidance cable fed with low-frequency current represents an antenna with a small effective range. The information about the magnitude of the curvature or its rate of change can be impressed on the supply voltage by frequency modulation (FM). It is practical to work with two carrier frequencies, since it is technically difficult to vary the frequency of the supply voltage continuously along the cable: one carrier is active on sections where the curvature is constant and contains there the information proportional to the curvature; the other carrier is active where the curvature changes linearly and contains there the information proportional to the rate of change of curvature. In a demodulator built into the vehicle, these pieces of information are converted in a suitable manner into an analog voltage (Fig. 8) that serves as the input variable for the control loop. As the antenna in the vehicle, one of the two receiving coils needed for measurement of the lateral deviation can readily be used.

References

[1] Fonda, A. G.: Theory of a practical lateral simulator for the automobile, and derived concepts of vehicle behaviour. Cornell Aeronautical Laboratory of Cornell University, Buffalo, N.Y., Report No. YA-804-F-3, 1957.

[2] Kohr, R. H.: Application of computers to automobile control and stability problems. Proc. Eastern Joint Computer Conference, 1957.

[3] The RCA electronic highway system. RCA Laboratories, David Sarnoff Research Center, Princeton, N.J.

[4] Bidwell, J. B., Welch, A. F. and Hanysz, E. J.: Electronic highways. General Motors Research Laboratories, GMR-245, 1960.

[5] Paper presented by Road Research Laboratory of DSIR, 1961, Crowthorne, Berks., GB.

[6] The Robotug, The EMI Control System for driverless factory vehicles. Document Ref. No. CP 180, 1958, EMI Electronics Ltd., Hayes, Middlesex, England.

[7] Hoops, J.: Modell eines auf festgelegte Fahrbahn elektronisch gelenkten Fahrzeugs. Elektronische Rundschau 11 (1957), pp. 277–278.

[8] Oppelt, W.: Kleines Handbuch technischer Regelvorgänge. 4th ed. Verlag Chemie, Weinheim/Bergstr., 1964.

[9] Mitschke, M.: Fahrtrichtungshaltung und Fahrstabilität von vierrädrigen Kraftfahrzeugen. Deutsche Kraftfahrtforschung und Straßenverkehrstechnik, H. 135, 1960, VDI-Verlag, Düsseldorf.

[10] Gauß, F. and Wolf, H.: Über die Seitenführungskraft von Personenwagenreifen. Deutsche Kraftfahrtforschung und Straßenverkehrstechnik, H. 133, 1959, VDI-Verlag, Düsseldorf.

[Translation covers all 6 pages of this article; the original is complete within this document.]