DYNAMICS OF A CONTROLLED ARTICULATED N-TRAILER WHEELED VEHICLE Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 1, pp. 39-48. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210104

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 74H40, 70F10

Dynamics of a Controlled Articulated n-trailer

Wheeled Vehicle

This article is devoted to the study of the dynamics of movement of an articulated n-trailer wheeled vehicle with a controlled leading car. Each link of the vehicle can rotate relative to its point of fixation. It is shown that, in the case of a controlled leading car, only nonholonomic constraint equations are sufficient to describe the dynamics of the system, which in turn form a closed system of differential equations. For a detailed analysis of the dynamics of the system, the cases of movement of a wheeled vehicle consisting of three symmetric links are considered, and the leading link (leading car) moves both uniformly along a circle and with a modulo variable velocity along a certain curved trajectory. The angular velocity remains constant in both cases. In the first case, the system is integrable and analytical solutions are obtained. In the second case, when the linear velocity is a periodic function, the solutions of the problem are also periodic. In numerical experiments with a large number of trailers, similar dynamics are observed.

Keywords: wheeled vehicle, nonholonomic constraint, mathematical model, dynamics of system

Introduction

Trends in modern industry are such that, in the production process, preference is increasingly given to robotic devices. In connection with the development of so-called "smart" production, there is a need for rational transportation of goods without human intervention. This problem is an integral part of modern production and is solved by the introduction of autonomous transport robots. They could automate the process of transporting goods and still make it safe even without human intervention.

Received December 08, 2020 Accepted March 16, 2021

Evgeniya A. Mikishanina [email protected]

Chuvash State University

Moskovskii prosp. 15, Cheboksary, 428015 Russia

E. A. Mikishanina

One of the first works on the dynamics of wheeled vehicles is that of Rocard [1]. Later, more extensive research in this area was carried out in the work of L. G. Lobas [2], who presented the results of research on the dynamical properties of mechanical rolling systems that model cars, motorcycles and intrafactory carts. However, the first works were characterized mainly by the complexity of the statements and calculations. Modern works on the dynamics of wheeled vehicles are initiated by the development of the robotic sphere [3, 4] and by the need to solve the problem of controlling n-trailer vehicles [5, 6].

The works [7-10] are devoted to the movement of wheeled vehicles on a plane. Wheeled vehicle designs can be very diverse. The issue of kinematics and dynamics of an articulated n-trailer vehicle is considered in the most meaningful way in [7]. In that paper, the authors provide a derivation of the equations of motion based on the method proposed in [11]. A vehicle with two wheel pairs is considered in [8] and a vehicle with two symmetric links is considered in [9]. A nonholonomic model of a snakeboard, consisting of three links and two wheel pairs that are attached to the platform, is presented in [10]. This model can be regarded as a generalization of the classical nonholonomic model of the Chaplygin sleigh [12]. A discussion of the kinematics of wheeled vehicles can also be found in [13-15].

In this paper, we consider another model of an articulated n-trailer wheeled vehicle consisting of an arbitrary number of links. The links hinged to each other contain one wheel pair and can rotate around their points of fixation with other parts, and the mass centers of the links coincide with the geometric centers of the links and the geometrical centers of the wheel pairs. This model is slightly different from the models that were considered in the above-mentioned works. However, a similar design is found in [16]. It is also assumed that the leading car moves in a prescribed manner, that is, the system is under control. The work on the dynamics of an uncontrollable wheeled vehicle [17] was published by the authors earlier. To describe the dynamics of a nonholonomic system, it is necessary to solve the equations of motion written in the Lagrange form with undetermined multipliers [18]. In the case of a controlled leading car, it is shown that only the use of nonholonomic constraint equations is sufficient to describe the dynamics of the entire vehicle.

This paper also considers special cases of movement of three-link and four-link wheeled vehicles with constant angular velocity. As shown by numerical experiments, with a larger number of trailers, similar dynamics is observed. Analytical solutions are obtained for the case of uniform movement of the leading car in a circle. Graphs of numerical solutions are plotted in the case of uneven movement of the leading car along a given curved path. Also, graphs of numerical solutions are presented.

1. Mathematical model of a mobile wheeled vehicle

Consider a system of (n + 1) articulated links (a leading car and trailers) moving on a plane. For convenience, we will call the leading link zero, the next link first, and so on. Each link contains one wheel pair. The links are pivotally attached to each other and can rotate around their attachment points. Assume that the centers of mass of each element of the structure are at their geometric centers and the geometric centers of the wheel pairs and lie on the same axis with the points of attachment of this element with neighboring elements. For convenience, we will consider the leading link to be zero, the next link will be considered the first, and so on.

The coordinates of the center of mass of each link are denoted by (xi,yi). The coordinates of the points Oi of the attachment of the (i — 1)th and the ith element are denoted (Xi, Yt).

Fig. 1. Example of a four-link wheeled vehicle.

The vector connecting the center of mass of the leading element to the point O1 is denoted by r, the vector connecting the point Oi to the center of mass of the kth link is denoted by pi X and the vector connecting the center of mass of the ith element to the point of attachment of the elements Oi+1 is denoted by pi 2. The identity pi = pi X + pit2 is also valid.

Let us define the lengths of the introduced vectors:

\r\ = a \pi,l\ = ai,1, \pi,2\ = ai,2, \pi \ = ai-

Let us denote the angle between the axis of the leading link and axis Ox by 0, which is the angle between the vector —r and the positive direction of the axis Ox, the angle between the axis of the ith link (vector —p^^) and the positive direction of the axis Ox by 0i, and the angle between the axes of the (i — 1)th element and the ith element (the angle of rotation from vector —pi-1 to vector pi) by ai and ai,di £ [—n,n). Then the angular velocity of the zero (leading) link is 0 = w. For the other links the following applies:

9i = 0 + Y, ak, 0

i = wi-

k=1

Then the coordinates of the centers of mass (xi,yi) of each element can be calculated using the formula

i-1

x0 — a cos 0 — ak cos 0k — aiy1 cos 0i,

k=1 (1.1)

i1

xi

yi = yo — a sin 0 — ak sin 0k — a^ sin 0i

k=1

where (x0,y0) are the coordinates of the center of mass of the leading car. We assume that

Ek=1 fk = 0.

The kinetic energy of the leading car has the form

T0 = i (I0w2 + m0 (¿o + Vo))>

Тг = -2{1гв2г+ГПг (.Й ■ fjf)).

where I0 is the moment of inertia of the first link relative to the center of mass and m0 is the hollow mass of the first link. The kinetic energy of the ith link has the form

2

where Ii is the moment of inertia of the ith link relative to the center of mass and mi is the hollow mass of the ith link. The kinetic energy of the system will take the form

1 / n n \

T = - low2 + mo (±1 +il2o)+Y. + I] № + Vi) • (1-2)

V i=1 i=1 J

Calculate the derivatives in equations (1.1)

i-1

Xi = X0 + aw sin 9 + ^ akwk sin 9k + ai,1wi sin 9i, k=1 i-1

yi = y0 — aw cos 9 — ^ akwk cos dk — ai,1wi cos 9i

(1.3)

k=i

Let us introduce the following assumption for convenience: riii = m, i = 0,n. Calculate the following sum:

n

y^ (X2 + y2) = n{±2 + y0 + a2w2) + 2anw (X0 sin 9 — y0 cos 9) +

i=1

nn

+ Yl(n — k)ak wk + Y1 ak,1wk + k=1 k=1

nn

+ 2 ^^ (n — k)akWk (Xo sin 9k — yo cos dk) + 2 ak,1Wk (Xo sin 9k — yo cos 9k) + k=1 k=1

nn

+ 2aw ^ (n — k)akwk cos (9k — 9) + 2aw ^ ak}1wk cos (9k — 9) + k=1 k=1 n k-1 n k-1

+ 2 ^ ^ aiak,1WiWk cos (9k — 9i) + 2 ^ ^ (n — k)aiakWiWk cos (9k — 9i). k=2i=1 k=2i=1

-.2 , „2

After the substitution Ak = (n — k)ak + ak>1, 5k = (n — k)ak + ak 1, the kinetic energy can be represented as

1

T = - ^(/o + mna2) w2 + т(?г + 1) (v2 + г>|) + 2тпаиз (vi sin в — v-2 cos в) +

n n

+ + mBk) w2^ + Akшk (vi sin dk - V2 cos dk) +

k=i k=i

n n k-1

+ AkUk cos(6>k - &) + ^ ^ Akaww cos(6k -

k=1 k=2г=1

where ¿0 = v1, y0 = v2 are the components of the linear velocity. For further modeling, it is necessary to formulate equations of motion in Lagrangian form and equations with a nonholonomic constraint.

2. Equations of motion

Let the leading car be under control, i.e., let the components of the velocity v1 (t), v2(t) be known. Then one can determine the value of 0 from the equation

v1 sin 0 — v2 cos 0 = 0 (2.1)

using the standard ATAN2 function

0(t) = atan2(v2(t),v1(t)), 0 £ [—n,n). Then the expression for the angular velocity takes the form

v2v1 — vv W(i) = vj+vi •

The equations of motion for a controlled n-trailer wheeled vehicle have the form

dt\du)kJ d9k ^ dujk ^ ''' duok' ^

where the equations

fi = ±i sin 9i - in cos 0i = 0, i = l,n, (2.3)

are nonholonomic constraints. Conditions (2.1) and (2.3) imply that the velocity vector of the center of mass of the link is always aligned with the longitudinal axis of the link.

Let us rewrite the constraint equations in terms of angular velocities and rotation angles

i- 1

fi = v1 sin 0i — v2 cos 0i + aw cos (0i — 0) + ^ akwk cos (0i — 0k) + ai,1wi = 0. (2.4)

k=1

To determine the constants A1,..., Xn, it is necessary to jointly solve the system of equations (2.2) and the time derivatives of the constraints (2.4). Equations (2.2) define the phase flow on a 2n-dimensional space M2n = {01,...,0n,w1,...,wn}, with the constraints (2.4) being its integrals. The equations of motion of the system under consideration lie on the zero level set of the integrals (2.4).

If the law of motion of the leading car is prescribed, one can simplify the problem by dropping the equations of motion. It is possible to determine the desired components of the vector (01,...,0n,w1,..., wn) by alternately solving the constraint equations (2.4) without using the equations of motion.

3. Motion with constant angular velocity

Let the velocity of the components of the center of mass of the leading car change according to the law

v1 = u(t)cos(Qt), v2 = u(t)sin(Qt). Then the angles of rotation and the angular velocity of the leading car have the form

0 = Qt, w = Q,

and Eqs. (2.4) take the form

i-1

fi = u(t) sin (0i — Qt) + aw cos (0i — 0) + ^ akwk cos (0i — 0k) + ai,1wi = 0. (3.1)

Thus, we consider here motion with a constant angular velocity.

For convenience, we can go to the angles between the axes of the links a1 = 01 — 0, a2 = 02 — 01..., ai = 0i — 0i-1,____Then the system (3.1) takes the form

fi = u(t) sin PT am\ + aQcos ^ am\ + ^ ak Wk cos ^ am\ + ai^Wi = 0. (3.2) \m=1 J \m=1 J k=1 W=k+1 )

For convenience, we further assume that ai}1 = ai}2 = 1. Equations (3.2) can be solved with respect to angular velocities:

Wi = (—1)i ( u(t) sin ( jr (—1)m+1a J + Qcos ( ^T (—1)m+1 a J ). (3.3)

V \m=1 J \m=1 J )

Using Eqs. (3.3) and considering that the condition

ai = Wi — Wi-1

is satisfied, we can obtain a closed system with respect to variables a1,a2,..., an: a1 = —Q — u(t) sina1 — Qcosa1, da = (-1^2cos (y) (u(t) sinf3i + Qcos fr), i > 1,

(3.4)

where fa = E™ii("l)m+1am + (-1 )mf •

Since in general the solution of the resulting system of equations is quite time-consuming, let us consider special cases.

3.1. Uniform movement in a circle

Let the leading car move in a circle at a constant velocity u = const. Then, after simplification, in the case of three links including the leading car (n = 2), Eqs. (3.4) become

à\ = —Q — u sin a\ — Qcos ai,

1 (3.5)

a2 = u (sin(a1 — a2) + sin a1 ) + Q (cos(a1 — a2) + cos a1)

with the specified initial conditions a1 (0) = c1, a2 (0) = c2. This system is autonomous and integrable. The analytical solution has the form

( B Q

ai = 2 arct.an e---

uu

iAe-ut ((B — Q)2+u2) + (u2 + —Q + Qe-ut+2Bute-ut) + B2Q {e-2ut — e-ut)

«2 = 2 arct.an ------r-

y u(u2 + (Q — Be-ut)2)

(3.6)

where B = utan(c1/2) + Q, A = utan(c2/2).

Figures 2 and 3 show graphs of the desired functions for different parameter values.

1.5-

0.5-

ai\

,«2

0.2

0.4 0.6

0.8

Fig. 2. Graphs of functions ai(t), a2(t) for u = 20, Q = 0.5, ci = 2, c2 = 1.

2l

-1J

0.4 0.6

t

0.8

Fig. 3. Graphs of functions ai(t), a2(t) for u = 20, Q = -0.3, ci = 2, c2 = -1.

The desired functions are smooth and the condition ai(t) ^ 2arctan(-Q/u), a2(t) ^ 2arctan(-Q/u) for t ^ to is true. Therefore, the larger the radius R = u/Q of the circle along which the leading car moves, the closer to zero the values of the angles a1, a2. A similar solution will be found with a larger number of links.

3.2. Curvilinear motion with constant angular velocity

Let us consider another case when the leading car moves along a certain curve also with a constant angular velocity.

We define the components of the variable velocity

vi = A (2 + sin(Qt)) cos (Qt), = A (2 + sin(Qt)) sin (Qt).

2 A

The trajectory of the leading car for the initial conditions a?o(0) = 0, yo(0) = giyen by

the following equations:

A sin (Qt) (4 + sin (Qt)) A (Qt - 4 cos (Qt) - sin (Qt) cos (Qt))

-2Q-' V°--2Q-*

The trajectory is shown in Fig. 4.

The dependence of the velocity on time is shown in Fig. 5.

In the case of three links including the leading car (n = 2), the system of differential equations with respect to angles a1, a2 takes the form (3.5) for u = A(2 + sin(Qt)).

In the case of four links including the leading car (n = 3), the system of differential equations with respect to angles a1, a2, a3 takes the form

a 1 = -Q - u sin a1 - Qcos a1,

a2 = u (sin(a1 - a2) + sin a1) + Q (cos(a1 - a2) + cos a1), (3.7)

a3 = -u (sin(a1 - a2 + a3) + sin (a1 - a2)) - Q (cos(a1 - a2 + a3) + cos (a1 - a2)),

where u = A(2 + sin(Qt)).

This system is nonautonomous. The periodic solutions may exist in this system.

Fig. 4. The trajectory of the leading link for A = 10, Fig. 5. Speed graph u for A = 10, Q = 0.5. Q = 0.5.

Let us construct a solution of the system for the given numerical parameters. For example, for A = 10, Q = —0.5 and the initial conditions a1(0) = 2, a2(0) = 1, a3(0) = —1, graphs of the unknown functions are shown in Fig. 6.

Clear differences in the behavior of the angle functions are observed at the beginning of the movement. Over time, these functions become periodic. Moreover, over time, the dynamics of the angles does not depend on the initial conditions. But as the link sequence number increases, changes in the behavior of the angle function become more noticeable. For example, for a vehicle consisting of 6 links (including the leading car), the graph of the function a5 is shown in Fig. 7.

For a more detailed analysis of this system (for u = A (2 + sin (Qt))), three-dimensional

maps for the period ^ in space ai, a-2, «3 were constructed. Numerical experiments showed

that, for different values of constants A and Q, the mappings contain only fixed points. Thus, for different initial conditions, the trajectories are attracted to a certain limit cycle.

Numerical experiments have shown that, with the controlled leading car moving uniformly in a circle, subsequent links occupy a certain limit position over time.

4. Conclusion

Two separate problems can be singled out for modeling the dynamics of articulated n-trailer wheeled vehicles: the movement of the vehicle when the leading car is under control, and the uncontrolled movement of the vehicle. To describe the dynamics of motion of an uncontrolled wheeled vehicle, it is necessary to jointly solve the equations of motion written in Lagrangian form and nonholonomic constraint equations. And to describe the dynamics of a controlled wheeled vehicle, that is, when the law of motion of the leading car is prescribed, nonholonomic constraint equations are sufficient.

In this work, equations describing the dynamics of a controlled symmetric wheeled vehicle are formulated. Numerical experiments for three-link and four-link wheeled vehicles are constructed. Analytical solutions are obtained for a symmetric design (ai;1 = ai>2) for the case of uniform movement of the leading car in a circle. In the case of curved uneven movement, when the linear

-0.5tt-

0 0.25 0.5 0.75 1

a3

0.5tt-

-0.5tt-

0 0.25 0.5 0.75 1

0.25

0 25 50 75 100

"2

0.1875-

0.125-

0.0625

0.25

0.1875-

0 25 50 75 100

C*3

0 25 50 75 100

Fig. 6. Graphs of the functions ai(t), a2(t), a3(t).

0.5tr

-0.5tt

a 5

0.25 0.1875 0.125 0.0625 0

0 0.5 1 1.5 2 0 25 50 75 100

Fig. 7. Graphs of the functions a5 (t).

velocity of the leading car changes periodically, the trajectories are attracted to the limit cycles. In the case under consideration, increasing the number of links does not significantly affect the dynamics of the vehicle. The question of the possibility of the existence of more complex dynamics for an arbitrarily given velocity function remains open. In the cases considered, no chaotic oscillations have been observed.

Acknowledgments

The author expresses her gratitude to A. V. Borisov for guidance and discussion of the results when working on the article.

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