Mathematical Models for Motion of the RearEnds of Vehicles

G.E. Prince ∗

S.P. Dubois †

October 31, 2008

Abstract

A new treatment of the path of the rear wheels of single and mul-tiple axle vehicles produces simple first order differential equations forthe motion of the rear wheels relative to the front wheels. We showfor the first time, under the assumption of no slipping, that the pathof the rear wheels depends only on the path of the front wheels andnot on the vehicle speed. We examine motion along straight, circularand winding roads, giving closed form solutions in the first two cases.We also provide a straightforward algorithm for computing vehicleofftracking for any road geometry. We believe these results will be ofsignificant benefit to road designers.

∗Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria3086, Australia. E-Mail: G.Prince@latrobe.edu.au

†Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria3086, Australia. E-Mail: sdubois@kpmg.com.au

1

1 Introduction

The object of this paper is to provide a more direct and accessible descriptionof the motion of the rear ends of single and multiple chassis vehicles giventhe path of the front wheels. The principal assumptions are that there is nowheel slip and the road is flat (it may be uniformly sloped). This problem inits simplest form was posed by Baylis [3] and Bender [4] and finally Freedmanand Riemanschneider [7, 8] gave a master equation which yielded closed formsolutions in the constant speed cases of straight line and circular motion of asingle chassis vehicle. Alexander and Maddocks [2] gave a careful discussionof the kinematics of vehicle rolling and analysed the problem of circularmotion of the front wheels in order to approximate vehicle turning. Theyalso discuss offtracking of the rear wheels and optimal steering strategiesneeded to minimise it. The significant advantages of the our approach arethat there is no constant velocity restriction (in fact we prove that the motionis independent of the vehicle speed) and that the problem of fish–tailing canbe addressed (that is, the problem of the back wheels crossing the pathof the front wheels). In addition, the differential equations for the modelare straightforward to deal with symbolically and numerically for more orless any road geometry that the designer cares to specify. We also give acomputationally simple approach to offtracking. Extensive Maple programsrelated to this work can be found in [5] and are available on request to theauthors.

In section 2 we introduce and deal with the motion of a bus and cab-trailer. We prove some general theorems (velocity independence and fishtail-ing) about the motion of the rear wheels, and then produce explicit solutionsfor the cases of (entry to) straight roads and (traversal around) roundabouts.

In section 3 we introduce and deal with the motion of an articulatedtruck with multiple trailers. We identify the ordinary differential equationsfor the motion of the axles. In particular we look at straight line and circularmotion, then produce some qualitative results for general motion.

The reason for modelling the rear motion of a vehicle is to know how far itofftracks (deviates) from the roadway. A numerical measure of this deviationfor a given road geometry is what designers want. In section 4 we introducenew methods to calculate this deviation in an effective manner which weillustrate in the case of circular turning. We believe that the material inthis section should be accessible to engineers without recourse to some of themore technical parts of the paper.

1

Figure 1: The basic configuration of the bus

The results in this paper are based mainly on the unpublished report byPrince [9] and the thesis of Dubois [5]. We do not address the related issueof vehicle jackknifing, this was treated by Fossum and Lewis [6] and theirresults are recovered by Dubois in [5] using the methods developed here.

2 The bus and cab-trailer problems

2.1 The bus problem

The notation for the problem is fixed in figure 1.Here and throughout the paper we assume that there is no slip and that theroad is flat although it may be uniformly sloped. Our assumptions about thesteering gear are the same as those of the papers cited in the introduction andwe refer to [2] for a detailed discussion. In this no slip case the motion of themidpoint of the front axle P (t) determines the motion of the midpoint of therear axle Q(t) because Q must always follow P. So we have the differentialequation

2

Figure 2: The relative motion of the midpoint of the rear axle

Q(t) = γ(t)(P (t)−Q(t)) (1)

(an overdot indicates ddt

.) Given the motion of both P and Q the motion ofany other point on the chassis (e.g. a rear corner) is known. The undeter-mined scalar function γ(t) is the (signed) speed of Q.

Freedman and Riemanschneider analysed (1) through the function γ(t).They eliminated Q(t) obtaining a third order differential equation for the

integrating factor ξ(t) := exp(∫ t

0γ(s)ds

)before returning to an expression

for Q(t) to obtain their final solution when P (t) satisfied a second order,linear, constant–coefficient differential equation (a significant assumption).We will instead concentrate on the motion of Q relative to P and eliminateγ(t) at a very early stage. In this way we obtain a first order ordinarydifferential equation for the angle ψ shown in figure 2, without any restrictionon P .In terms of these variables Q(t) is given by

Q(t) = P (t) + B(t) = P (t) + L(cos(ψ), sin(ψ)) (2)

From (1) we obtain

B(t) = −γ(t)B(t)− P (t) (3)

=⇒ B ·B = −γ‖B‖2 − P ·B=⇒ 0 =

d

dt(‖B‖2) = −2γL2 − 2P ·B.

3

Hence,

γ = − P ·BL2

. (4)

Replacing γ(t) in (3) we have

B =

(P ·BL2

)B − P . (5)

In co-ordinates

(B1, B2) =P1B1 + P2B2

L2(B1, B2)− (P1, P2) (6)

In itself (5) is an improvement on (3) and may be solvable for certainforms of P (t). Indeed it shows that −B is the component of P perpendicularto B. However, we can do a lot better by producing an expression for dψ

dt. By

taking the dot product of (5) with B⊥ := (−B2, B1) we see that the angularspeed of P around Q is

Ldψ

dt= −B⊥ · P

L(7)

so thatdψ

dt=

sin(ψ)P1 − cos(ψ)P2

L(8)

This equation is valid for t ≥ 0, ψ ∈ R.In this way we find Q(t) through (2) without recourse to analysis of γ(t).Before we go on to illustrate the efficacy of this method by considering cir-cular and straight line motion, we will prove that the path of the rear ofthe vehicle is independent of the speed of P along its path. This new resultis important in both numerical and symbolic simulation because it meanswe can model the road with any parametrisation we like without having torealistically model the vehicle speed.

Theorem 2.1. The path of Q is independent of the speed of P along its path.(The path of the rear wheels is independent of the speed of the vehicle.)

Proof. Changing the speed of P along its path is equivalent to a regularre-parametrisation of the path (the change in speed is assumed smooth andwithout halts). If the new time parameter is s then we have s = h(t) whereh and its inverse are at least once differentiable on appropriate intervals, andwe can assume without loss that h(0) = 0.

4

In this parametrisation the derivation producing equation (8) gives in-stead

dψ

ds=

sin(ψ)dP1

ds− cos(ψ)

dP2

dsL

, ψ(0) = ψ0, (9)

where, by assumption, P1(s) = P1(h−1(t)) = P1(t), P2(s) = P2(h

−1(t)) =

P2(t) and ψ(s) is the angle−→PQ makes with the horizontal at “time” s (by

assumption ψ(0) = ψ0). We must show that ψ(s) = ψ(t), that is, ψ = ψ hor ψ = ψ h−1.

Nowd

dt=

ds

dt

d

ds= h′(t)

d

ds, so (8) becomes

h′(t)dψ(h−1(s))

ds=

(sin(ψ(h−1(s)))

dP1

ds− cos(ψ(h−1(s)))

dP2

ds

)h′(t)

L

=⇒ dψ(h−1(s))

ds=

sin(ψ(h−1(s)))dP1

ds− cos(ψ(h−1(s)))

dP2

dsL

since h′(t) 6= 0 by assumption, and ψ(h−1(0)) = ψ(0) = ψ0.Comparison with (9) and an appeal to the uniqueness and existence theoremfor these initial value problems shows that ψ h−1 = ψ as required.

In hindsight the result seems physically obvious given the no slip con-dition, but we don’t have a completely watertight kinematic proof. It ishowever possible to employ the same proof technique to equations (1) and(4) but this is more tedious.

2.2 Straight line motion

The straight line motion of P can be used to model entry onto a straight roador part of a lane changing manoeuvre. We will examine the former situation,in part because we can model the latter situation more comprehensively withan appropriate polynomial P (t). We suppose that the straight road is at anangle φ to the horizontal and that the chassis is initially at an angle ψ0 tothe horizontal.Let

P (t) := f(t)α + β.

5

Here f(0) = 0, α, β are constant vectors with α := (cos(φ), sin(φ)) and f(t)is assumed C1(R). As a result of theorem 2.1 we could assume that f(t) = tfor forward motion, however the problem presents no difficulties as it stands.Equation (8) gives

dψ

dt=

1

Lsin(ψ − φ)f(t), ψ(0) = ψ0. (10)

The solution of this initial value problem is given implicitly by

tan

(ψ − φ

2

)= tan

(ψ0 − φ

2

)e

1L

f(t).

We can suppose that f(t) is an increasing function so it follows directly

from the solution above that limt→∞

(ψ − φ

2

)=

π

2, which implies lim

t→∞ψ = π+φ

as expected. The physical consequence of this is that the rear of the bus doesnot swing across the line of motion of P (i.e. there is no fish–tailing; see alsoproposition 2.2 and section 4).

2.3 Circular motion

Now we model motion on a roundabout of radius R, or equivalently, thecircular turning with radius of curvature R, of a vehicle around a corner.(But again, we can model the corner turning more comprehensively with adesigned function P (t).) Let

P (t) := R(cos(Ω(t)), sin(Ω(t)))

with Ω(0) = Ω0, Ω(t) is C1(R) and R is a positive constant. Equation (8)yields the separable equation

d(ψ − Ω)

dt= −Ω

(R

Lcos(ψ − Ω) + 1

), ψ(0) = ψ0, Ω(0) = Ω0 (11)

whose solution is available from standard integrals in each of the 3 cases ofthe ratio R/L.

This circular case is much more interesting than straight line motion. Toproceed with the analysis we assume the following standard configuration:Ω0 = 0 and ψ(0) = 0 (so that the bus enters the roundabout on the right

6

and at right angles to it) and Ω is an increasing function (so that the busproceeds anti-clockwise around the roundabout: see figure 3).

When L < R, equation (11) (without the accompanying initial condi-tions) has constant solutions given by

cos(ψ − Ω) = −L

R. (12)

In equation (2) this produces a circle of radius√

R2 − L2 centred on the ori-gin. This is a limit cycle and the rear wheels of a bus entering the roundaboutin our standard configuration in the case L < R spiral onto this circle.

When L = R this limit cycle degenerates to a fixed point at the originand the bus eventually performs circular motion about the point Q at theorigin! (Of course the steering gear won’t allow this.)

When L > R there are no constant solutions of equation (11) and ψ −Ωis a decreasing function. Our standard solution has points of closest andfurthest approach lying on circles of radius

√L−R and

√L + R respectively

derived fromd‖Q‖2

dt= −2RL

d(ψ − Ω)

dtsin(ψ − Ω) (13)

obtained from equation (2).The problem of fish-tailing is easily analyzed in the circular case: Q

crosses the roundabout when B forms a chord on the circle of radius R. Alittle geometry shows that this occurs precisely when

cos(ψ − Ω) = − L

2R. (14)

For L ≤ R this occurs precisely once for our standard solution. When R <L ≤ 2R it occurs repeatedly (the closed form solution of equation (11) isrequired for the analysis). When L > 2R it doesn’t occur at all and Qlies outside the roundabout at all times. Again note that we could assumeΩ(t) = t or Ω(t) = ωt by virtue of theorem 2.1.

Figures 3, 4 and 5 illustrate the motion (with Ω(t) = t and L = 1) forthe 3 cases L < R, L = R and L > R respectively. These diagrams wereproduced from exact solutions of equation (11) obtained with Maple and arethe first frames of animations of the motion. In each diagram the circularpath (the roundabout) is that of the midpoint of the front axle and theother path is that of the midpoint of the rear axle. The solid line segment

7

–2

–1

0

1

2

–2 –1 1 2 3

Figure 3: Circular motion of bus with L < R

on the horizontal axis represents the vehicle. In the last two figures theroundabout has to be traversed multiple times for the rear axle to traversethe entire red curve, hence the very unphysical appearance. Because in realitya vehicle only traverses part of the roundabout, the motion is physical andthe diagrams indicate the potentially disastrous effect of a vehicle too longfor the roundabout.

8

–1

–0.5

0.5

1

–1 –0.5 0.5 1 1.5 2

Figure 4: Circular motion of bus with L = R

–1.6

–1

–0.6

0.6

1

1.6

–1.6 –1 –0.6 0.6 1 1.6

Figure 5: Circular motion of bus with L > R

9

There are two fishtailing problems: does Q cross lanes on entry to aroadway, and does Q cross lanes on a road with changing curvature? We cananswer the first fishtailing problem here: if P travels on a concave path witha positive slope with the following proposition which we give without proof(see [5]).

Proposition 2.2. Suppose that P travels on a concave curve with positiveslope (see figure 6). Assume Q is initially under the path of P with ψ0 ∈(−π,−π

2), then Q travels in a concave curve with a positive slope and the

path of Q never crosses the path of P. In the limit t →∞ Q approaches thepath of P.

The second fishtailing problem will be discussed in section 4.

¯¯¯¯¯¯ P0

Q0

Figure 6: Q0 is under the curve of P.

2.4 The cab-trailer problem

Our cab-trailer has a comparatively small cab, length l, compared to itstrailer and only 2 axles. One is on the cab (which we have assumed to be inthe middle of the cab) and the other set is at the rear of the trailer. Assumealso that the hitching point for the trailer is directly over the cab’s axle.(These are not mathematical simplifications, nothing is changed by addingmore axles.) This is shown in figure 7.

The motion of the rear of a cab-trailer and the motion of the rear of abus are essentially the same, thus the basic configuration of the cab-traileris the same as the bus (as seen in figure 1). The difference appears only insimulation because the cab will not necessarily face along the same line as thetrailer, since it must face along the tangent line to its path (the road) at any

10

Figure 7: The basic configuration of the cab-trailer.

particular time. This means separate equations must be used in modellingthe cab and the trailer. The trailer equations are the ones that match thoseof the bus, the only difference is the range that they are calculated for, as thebody of the trailer does not extend from axle to axle as shown in figure 7.

Because the cab is to be represented by a line segment disjoint from thatof the trailer in the Maple simulations we create two equations for the cab:one for the horizontal coordinate and the other for the vertical coordinate,denoted cab1 and cab2 respectively (see figure 8). Both have time and sdependence, where the range of s defines the length of the cab and timedefines the position of P. The equations for the trailer are the same used forthe bus, the only difference is the range of s used in the calculation of trailerlength.

Since the cab must, at any specific time, be pointing along the tangentto the curve, have length l and the midpoint of the axle is at the point P (t),then we have equations for the line segment representing the cab (see figure8)

11

cab1(t, s) := P1(t) + sP1(t), cab2(t, s) := P2(t) + sP2(t), (15)

where s ∈ (− l2, l

2).

´´

´´

´´

´´

´3S

SS

SSSo

SS

¶¶

¶¶

¶¶

¶¶

¶¶7

£££££££££££££±cab

(t, l

2

): front of cab

P

P

cab(t,− l

2

): rear of cab

O

Figure 8: The basic configuration of the cab.

2.5 Straight line motion

As before letP (t) := f(t)α + β

with f(0) = 0, α, β constant vectors with α := (cos(φ), sin(φ)). f(t) isassumed C1(R).

Now using equations (15) we get for the cab

cab1(t, s) = f(t) cos(φ) + β + s(f(t) cos(φ)),

cab2(t, s) = f(t) sin(φ) + β + s(f(t) sin(φ)).

The motion of the trailer is given by (10).

12

2.6 Circular motion

As before let

P (t) := R(cos(Ω(t)), sin(Ω(t)))

with Ω(0) = Ω0, Ω(t) is C1(R) and R is a positive constant.

–2

–1

1

2

–2 –1 1 2 3

Figure 9: Cab-trailer travelling along circle R > L

Using (15) we get

cab1(t, s) = R cos(Ω(t))− sR Ω(t) sin(Ω(t)), (16)

cab2(t, s) = R sin(Ω(t)) + sR Ω(t) cos(Ω(t)). (17)

The motion of the trailer is given by (11). Figure 9 illustrates the motion(with Ω(t) = t and L = 1 for the case L < R). It is obtained with Mapleand is one of the frames of animation of the motion and was produced fromexact solutions of equation (11) along with (16).

13

2.7 Motion on an S bend

A toy model of the motion of an articulated vehicle on an S bend can beobtained with

P (t) := (t, t(t− 1)(t− 2)).

Figure 10: Cab-trailer on an S bend

We used Maple to numerically integrate (8) and, along with (2) and (15),to produce figure 10 in which P is assumed to travel along the centre of theleft-hand lane. This diagram is the first frame of an animation showing thepaths of Q and the rear corners of the trailer. (The road and its centrelineare dark and the paths of the rear of the vehicle are lighter.) Of course, moresophisticated polynomials need to be used to fit actual lane geometries, butfrom a computational point of view we can always parametrise the path ofP by the horizontal co-ordinate as we did here by virtue of theorem 2.1.

14

3 The articulated truck problem

The object of this section is to provide a direct and accessible descriptionof the motion of the rear end of an articulated truck, given the path of thefront wheels. Our articulated truck is a comparatively small cab (length l)with multiple trailers (n) attached and having n + 1 axles. One of the axlesis on the cab (which we have assumed to be in the middle of the cab) andthe others are at the rear of each trailer. Since all axles on the trailers arefixed, it wouldn’t make a difference if there were more than one axle on atrailer, L would be taken to be the distance between the front axle and thenearest rear axle.

We begin by examining the motion of the 2 trailer articulated truck shownin figure 11.

Figure 11: The basic configuration of the 2 trailer articulated truck.

15

Assuming there is no slip and a flat road, and following the approach ofsection 2 we obtain

Q(t) = γ1(t)(P (t)−Q(t)), (18)

M(t) = γ2(t)(Q(t)−M(t)). (19)

Given the motion of P , Q and M, the motion of any other point onthe vehicle (e.g. a rear corner) is known. We concentrate on the motionof Q relative to P and M relative to Q. In this way we obtain two firstorder ordinary differential equations for the angles ψ1 and ψ2 shown in figurebelow, without any restriction on P or Q.

Figure 12: The relative motion of the midpoints of axles of the trailers.

In terms of these variables, Q(t) and M(t) is given by

Q(t) = P (t) + B1(t) = P (t) + L1(cos(ψ1), sin(ψ1)), (20)

M(t) = Q(t) + B2(t) = Q(t) + L2(cos(ψ2), sin(ψ2)). (21)

Following the derivation of (7) we obtain

L1dψ1

dt= −B⊥

1 · PL1

, (22)

L2dψ2

dt= −B⊥

2 · QL2

. (23)

Thus we have a pair of equations that show us the angular speed ofmidpoint of the front axle of each of the trailers around the midpoint of the

16

rear axle of the corresponding trailer. In this way we find Q(t) and M(t)without recourse to analysis of γ1(t) or γ2(t). Because of these equationsare coupled it is unlikely that any closed form solutions will be able to befound even for straight line and circular motion and the ordinary differentialequations will need to be solved numerically. This is a routine matter.

In general, the angular speed of the midpoint, Qk, of the rear axle of thekth trailer around the midpoint of the rear axle of the (k + 1)tst trailer isgiven by (where Q0 := P )

Lk+1dψk+1

dt= −B⊥

k+1 · Qk

Lk+1

. (24)

Finally, we can state the analogue of theorem 2.1:

Theorem 3.1. The path of Qk, k = 1 . . . n is independent of the speed of Palong its path. (The paths of the rear wheels of the trailers is independent ofthe speed of the vehicle.)

The reader is referred to the paper by Altafini [1] for a control theoreticapproach to the n trailer problem.

3.1 Straight line motion

Despite not being able to decouple the two ODEs (22) and (23), we are stillable to produce some qualitative analysis of articulated trucks. Since weare concerned about the path of the rear of the final trailer we can ask thefollowing question: for an articulated truck with n trailers entering a straightroad, does the rear of the final trailer swing across the line of motion of P (i.e.is there any fish-tailing)? Using results from section 2 we have the followingresult (see [5]):

Theorem 3.2. Suppose that the path of P is a straight line and assume thatinitially all midpoints of trailer axles lie on the same side of the straight linewith ψi < ψj, i < j, ψk ∈ (−π + φ,−π

2), k = 1, ..., n initially.

Then the rear of a trailer does not cross the path of the trailer precedingit and the path of each trailer is a concave curve with positive slope.

3.2 Circular motion

Since we are concerned about the path of the rear of the final trailer wecan ask the following question for circular motion: for an articulated truck

17

with n trailers travelling around a roundabout, when does the final trailerhave spiral motion. Using the facts that we know about a vehicle travellingin circular motion from the bus and semitrailer cases we have the followingtheorem (see [5] for a proof):

Theorem 3.3. For an articulated truck with n trailers of equal length trav-elling in circular motion, when R =

√nL, the final trailer has spiral motion

and the path of the n− 1 other trailers produce circles.

Similarly, if R >√

nL then the rear of all the trailers traverse limitcycles. If R <

√nL then jackknifing will occur when multiple traversal of a

roundabout is attempted.Figure 13 illustrates the motion of a 2 trailer articulated truck (with

Ω(t) = t and L1 = L2 = 1) in the second of the above cases.

–1

–0.5

0.5

1

–1 1 2 3

Figure 13: Circular motion of an articulated truck with R =√

2L

18

4 Calculation of offtracking

The professionals in truck and road design are concerned with “offtracking”,the amount the path of a trailer or the rear of a bus diverges from the path ofthe front. Such people are not interested in exposing the mathematical aspectsof the problem, but in quick, reliable rules, usually with a margin of error,for design work.

J.C. Alexander and J.H. Maddocks [2]

From our simulations (eg figure 10) we can visualize the deviation of therear of the vehicle from the road, now we introduce a new angle ξ from whichwe will be able to explicitly calculate the three measures of offtracking shownin figure 15: the perpendicular deviation PD(t), chord deviation C(t) andarc deviation A(t).

Assume that we are dealing with the motion of a bus or cab-trailer. Torecap, the position of the midpoint of the front axle P (t) is given and themidpoint of the rear axle is at Q(t). The distance between Q(t) and P (t) isL. Now if we have appropriate restrictions on P, there exists a point P (t′) onthe path of P , such that the distance from P (t′) to P (t) is also L, that is,

P (t′)− P (t) = L(cos(ζ), sin(ζ)),

where t′ < t.Now ξ is the angle between

−−−−−−→P (t)P (t′) and

−−−−−→P (t)Q(t). The relationship between

ζ, ξ and ψ is shown in figure 14.So to find ξ there are two steps:

Step 1. Find the largest t′ < t such that

‖ P (t′)− P (t) ‖= L. (25)

Step 2. Solve the following equation for ξ(t):

[P (t′)− P (t)] · [Q(t)− P (t)] = L2 cos(ξ(t)).

Using (2) this becomes

[P (t′)− P (t)] · (cos(ψ), sin(ψ)) = L cos(ξ(t)). (26)

19

Figure 14: Relationship between ζ, ξ and ψ.

From a computational perspective finding ξ(t) from (26) involves, for eacht, solving (25) for t′ for the given (polynomial say) road geometry P (t). Thenthese t′ values, along with the numerical solution to (8), are used in (26) tocompute ξ(t). Lane crossing occurs when ξ(t) = 0 and Q(t) is not parallel toP (t′).

The formulae for the three measure of offtracking are

A(t) := Lξ(t) (27)

C(t) := 2L sin

(ξ(t)

2

)(28)

PD(t) := L sin(ξ(t)) (29)

The Taylor expansions for the chord and perpendicular deviations relatingthem to A are

PD = A

(1− 1

3!

(A

L

)2

+1

5!

(A

L

)4

+ O(A6)

),

C = A

(1− 1

4.3!

(A

L

)2

+1

16.5!

(A

L

)4

+ O(A6)

).

20

Figure 15: Visual description of the three measures of deviation.

As expected A is a closer approximation to C than it is to PD when(

AL

) ¿1. In any event computation of either C(t) or PD(t), with (28) and (29)respectively, poses no difficulty.

4.1 Straight-line motion

This time we will take advantage of theorem 2.1 and take

P (t) := t(1, 0) + β

with β a constant vector giving the initial position of the vehicle. Clearly inthis case ξ = π + ψ, and hence

tan

(ξ − π

2

)= tan

(ψ0

2

)e

tL .

The deviation definitions (27), (28) and (29) all demonstrate that the off-tracking is consistent with the results in section 2.2.

4.2 Circular motion

LetP (t) := R(cos(Ω(t)), sin(Ω(t))).

21

Without loss we take Ω(t) = t. Given the definition of ξ we see that we canconsider only the case L ≤ 2R since the maximum chord length on the pathof P is 2R. Notice also that in this circular case PD(t) = |R− ‖Q(t)‖|.To obtain ξ we first solve equation (25), giving

t′ = t− 2 arcsin(L

2R) =: t− T.

Equation (26) gives

L cos(ξ) = 2R sin(t− ψ − T

2) sin(

ψ

2).

Figure 16 shows the arc deviation for the case L = R computed withMaple, compare with vehicle path shown in figure 4.

Figure 16: Arc deviation from roundabout to Q with L = R

22

5 Conclusion

We believe that we have overcome a number of theoretical and computationaldifficulties in modelling the motion of long, single and multiple axle vehicles.With the tools described here the road designer can explicitly compute thefootprint of a vehicle as it traverses a given road geometry and producegraphs of offtracking. Such tools should take some of the guesswork andrules of thumb reckoning out of the existing design tools.

References

[1] Claudio Altafini, Some Properties of the General n-Trailer, Internat.J. Control, 74 (2001), 409–424.

[2] James C. Alexander and J. H. Maddocks, On the maneuvering ofvehicles, SIAM J. Appl. Math., 48 (1988), 38–51.

[3] John Baylis, The mathematics of a driving hazard, Math. Gaz., 57(1973), 22–26.

[4] Edward A. Bender, A driving hazard revisited, SIAM Rev., 21 (1979),136–138.

[5] S. Dubois, Mathematical Models for Motion of the Rear Ends of Vehi-cles, Honours thesis, Department of Mathematics, La Trobe University.(December 2002).

[6] T. V. Fossum and G. N. Lewis,A mathematical model for trailer-truckjack knifing SIAM Rev.,23 (1981), 95–99.

[7] H. I. Freedman and S. D. Riemenschneider, Determining the pathof the rear wheels of a bus, SIAM Rev., 25 (1983), 561–567.

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