International Applied Mechanics, Vol. 48, No. 4, July, 2012

MODELING THE MANEUVERING OF A VEHICLE

E. Ya. Antonyuk and A. T. Zabuga

A kinematic model of one- and two-link robotic vehicles with two or three steerable wheels is considered. A nonsmooth path in the form of an astroid enveloping the positions of the robot is planned. The motion of a two-link vehicle with such a trajectory is modeled. A numerical analysis of the dynamic of robots is performed determining the reactions of nonholonomic constraints Keywords: one- and two-link robots, planned path, astroid, steerable wheel Introduction. Problems of the remote control of mobile vehicles maneuvering in a space-limited environment have recently attracted great interest [4–12, 16, 17]. This is because of the necessity of supporting various industrial processes, performing handling operations at large warehouse terminals, providing production maintenance in hazardous conditions, etc. To solve practical problems, the range of types of wheeled robots is constantly extended. Among them are one- and multilink robots with one or several steerable wheels, which highly improve their mobility and usability. It is of scientific interest to examine the limiting capabilities of certain types of mobile robots on a flat surface. Here we will design a path for one- and two-link mobile robots with two and three steerable wheels in an L-shaped holding alley as a typical obstacle. As a mobile robot moves in such an alley, its longitudinal velocity vector turns by an angle of p/2, or nearly so. In motion planning, more complicated obstacles can be considered to consist of such alleys. Since the more there are steerable wheels, the higher the manoeuvrability of robots [5, 6, 10, 16], we will discuss some geometrical and kinematic grounds for planning a path of a one- and two-link robots with two and three steerable wheels, respectively. Dynamic models will be developed, and examples will be given. 1. Path Planning for a One-Link Robot. Let us consider one of the possible approaches to the path planning for a one-link one-dimensional robot (modeled by a straight-line segment) with two steerable wheels at the ends. Increasing the number of steerable wheels improves the maneuverability of the vehicle so that it can execute composite paths in one or two continuous maneuvers. Let the sensors of a one-link robot (as a nonholonomic system) measure the three generalized coordinates defining its position on a plane at every instant. Let also the sideways overturning of the robot be impossible. The frictional constraints between the wheels and the flat supporting sufrace are bilateral. Figure 1 shows (top view) a one-link robot AB in an L-shaped holding alley with legs I and II. The wheels A and B are steerable, i.e., can be turned about the vertical axis, and their size is neglected. Let us determine path 1 of the typical (moving) point K 0 of the robot A 0 B 0 that moves extremely close to the left corner (point O). Then, the point A 0 continuously moves along the wall OS, while the point B 0 moves along the wall PO. The current angle between the robot and the horizontal is denoted by q. During such motion, the family of intervals A 0 B 0 is described by the equations y = kx + b,

k = tan q,

b = -l sin q,

(1.1)

where l is the length of the robot; k and b are the parameters of straight lines depending on the angle q. The envelope of the family of straight lines A 0 B 0 F ( x, y, q ) = y - kx - b can be found by solving the system of equations

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterova St., Kyiv, Ukraine 03057, e-mail: [email protected]. Translated from Prikladnaya Mekhanika, Vol. 48, No. 4, pp. 105–117, July–August 2012. Original article submitted July 26, 2010. 1063-7095/12/4804-0447 ©2012 Springer Science+Business Media, Inc.

447

y O

-y 1

A0

x

ay

A

E2 K0

S

E3

M

E4

K

a2 y2

B0 1

q

T

C

l

B

2

II

E1 ax E0

a1 I

P

Fig. 1

F ( x, y, q ) = 0,

¶F ( x, y, q ) = 0, ¶q

(1.2)

y = -l sin 3 q.

(1.3)

i.e., after some transformations, we get x = l cos 3 q,

System (1.3) is the parametric equation of astroid 1 (Fig. 1) in the fourth quadrant. Eliminating the parameter q from (1.3), we obtain the equation of astroid 1 in Cartesian coordinates: x 2/ 3 + y 2/ 3 = l 2/ 3 .

(1.4)

An astroid is a hypocycloid (a roulette generated by the trace of a fixed point on a small circle rolling within a larger circle, the ratio of their radii being R / r = 4). The tangent to an astroid at a point with angular coordinate q is characterized by dy = ( l 2 / 3 - x 2 / 3 )1/ 2 x -1/ 3 = tan q. dx

(1.5)

The limiting (no-go) length (lmax ) of a one-link robot in an L-shaped alley with leg lengths a1 and a 2 (Fig. 1) is defined by lmax =

a1 cos q m

+

a2 sin q m

,

(1.6)

where q m is the angle at which the robot touches the walls PO and SO and corner C of the alley with side A 0 B 0 (of maximum possible length) including points A 0 and B 0 . After appropriate transformations, we obtain lmax = ( a12 / 3 + a 22 / 3 ) 3 / 2 ,

tanq m = 3

a2 a1

=3 -

yc xc

.

(1.7)

The envelope of positions of a one-link robot of length l < lmax can be defined by some astroid 2 (Fig. 1) located between astroid 1 (1.4) and the corner point C. It is well if this astroid comes through the middle point T of the segment CM

448

located on the normal drawn through the corner point C to astroid 1 and the point M of intersection between this normal and astroid 1. This distance is defined by lCM = l 2 (sin 6 q c + cos 6 q c ) + 2l( y c sin 3 q c - x c cos 3 q c ) + x c2 + y c2 .

(1.8)

The angle q c can be determined from the implicit equation l cos( q c - j ) = cos 2q c , r

j = arctan

yc xc

r = x c2 + y c2 ,

,

l £ r,

(1.9)

where x c and y c are the coordinates of the corner point C. The quarter of “middle” astroid 2 (Fig. 1) located in the fourth quadrant is described by the equation x = l cos 3 q + a x ,

y = -l sin 3 q + a y .

(1.10)

Eliminating the parameter q, we obtain

[

y = a y - l 2/ 3 - (x - ax )2/ 3

]

3/ 2

,

(1.11)

where a x = x c - 0.5lCM sin q c - l cos 3 q c ,

a y = y c + 0.5lCM cos q c + l sin 3 q c .

(1.12)

Astroid (1.10), (1.11) has singular points associated with infinitely great curvature. For (1.11), one singular point has coordinates x and y equal to a x and a y - l (parameter q = p / 2), and the coordinates of the other point are equal to a x + l and a y (parameter q = 0). The curvature k of the astroid defined by k=

x¢ y¢¢ - y¢ x¢¢

[

( x¢ ) 2

]

(1.13)

3 + ( y¢ ) 2

satisfies the equality k=

2 , 3l sin 2q

(1.14)

where x¢ and y¢ are the first-order derivatives of the coordinates x and y with respect to the parameter q; x¢¢ and y¢¢ are the second-order derivatives with respect to the same parameter. If q = 0 or q = p / 2, then k = ¥, and the radius of curvature R = 1/ k = 1.5l sin 2q = 0. The astroids remain continuous at those points. In kinematic and dynamic analyses, however, these singularities have to be treated appropriately. The point K at which astroid (1.11) touches the robot AB moves along the astroid with velocity v k = 1.5lq& sin 2q so that the arc length S of the astroid between 0 and q = p/2 is equal to 1.5l, i.e., is less than a quarter the perimeter of a circle of radius l. If the turn of the alley in which the robot moves is p / 2+ b rather than p / 2 (b is an algebraic value), then, according to (1.2), the envelope A 0 B 0 is defined not by (1.3), but by x = l cos 3 q,

y = -l sin 3 q + l cos 3 qtanb.

(1.15)

The tangent of the angle between the tangent line and envelope (1.15) is given by dy = tan q + tan b. dx

(1.16)

449

The path of the point A of the one-link robot AB before and after the turn in the L-shaped alley can be planned as a line E 0 E1 E 2 E 3 E 4 shown in Fig. 1. The first segment is vertical (E 0 E 2 , x = a x ). Moving along it, the points A and B reach the points E 2 and E1 , respectively. As the point A moves along the segment E 2 E 3 and the point B along the segment E1 E 2 , the point K follows astroid 2, according to (1.10). As the one-link robot moves along leg II, the points A and B move along the third segment E 3 E 4 , which is horizontal ( y = a y ). The control angles y 1 = y 2 = 0 on the first segment, y 1 = -q, y 2 = p / 2 - q on the second segment (the point K moves along the astroid E1TE 3 ), and y 1 = y 2 = 0 on the third segment (the points A and B move along the lines E 3 E 4 and E 2 E 3 ). If the planned path is strictly followed, then on the segment where the astroid forms, the velocity of the point A will be directed horizontally, and the velocity of the point B vertically. During this motion, the robot will touch the imaginary astroid at the point K, revolving and sliding around it. Thus, the planned angles between the robot and the planes of front and rear wheels are y 1 = -q,

y 2 = p / 2 - q.

(1.17)

The absolute velocities v i of the points B and K of the robot and its angular velocity q& are defined by vB =

x& A tan q

,

v K = x& K2 + &y K2 = -3lq& sin q cos q,

q& = -x& A / ( l sin q ),

(1.18)

where x& K and &y K are found by differentiating (1.10) with respect to time; the velocity v K is directed at an angle q to the Ox-axis. The current length of the robot from the point B to the point K (contact with astroid 2) is calculated from the formula lBK = l cos 2 q,

(1.19)

whence it follows that the velocity v Kr of relative motion of the point K along the interval BA is twice the velocity v Ke of the frame of reference (v Kr = l&BK = -2lq& cos q sin q, v Ke = -lq& cos q sin q). The velocities v Ke and v Kr are codirectional vectors. A quarter of the length of the astroid S a = 1.5l, i.e., is by 4.7% less than a quarter of the perimeter of a circle of radius l. As the robot follows the planned path, the point K describes the astroid x K = x A - l sin 2 q cos q,

y K = y A - l sin 3 q,

(1.20)

i.e., (1.10). If the robot has more or less large transverse dimension (i.e., is formed by, for example, a rectangle of length l and width d), the envelopes of the left and right sides are described by the equations of astroid x = l cos 3 q,

y = -l sin 3 q

(1.21)

y = -l sin 3 q - d cos q.

(1.22)

and elongated astroid x = l cos 3 q + d sin q,

Curve (1.22) is equidistant to curve (1.21). For the given angle q between the robot and the Ox-axis, both envelopes (1.21) and (1.22) have identical derivatives dy / dx = tan q. Thus, the normal to one of the envelopes is normal to the other envelope. The normal distance L between the points at which it intersects curves (1.21) and (1.22) is always constant and equal to the width d of a rectangular vehicle for any angles q, i.e., L = d. The singular points of the elongated astroid (1.22) at which the radius of curvature equals zero are defined by the following formulas in the fourth quadrant: 1 2d q1 = arcsin , 2 3l

450

q 2 = p / 2 - q1 .

(1.23)

y

1 l1

2

B

l2

y2

B

q2

C2

-y 1 q1

C1

y3 D

A0 A

p / 2- y2

S

x D2

O

l1

l2

l1

O1

3

A1

l2

p / 2 + y1

2

A2

a2

II

T 1 C l2

D1 B0

-y 1 + y 2

p / 2- y3

ay

M B2

ax p / 2 - q2 + q1 + y 2

I O1

q2 - q1 - y 2 + y 3

D0 O2

a1

P

Fig. 2

Fig. 3

The motion of a one-link robot represented by the straight-line segment AB (Fig. 1) with steerable wheels at the ends is kinematically described by the following equations [8, 9]: é ù é x& A ù ê cos( q + y 1 ) ú ê &y ú = vê sin( q + y ) ú, ú 1 ê Aú ê êë q& úû ê- sin( y 2 - y 1 ) ú ê ú l cos y 2 û ë

(1.24)

where x A and y A are the Cartesian coordinates of the point A; q is the angle between the robot and the Ox-axis; y 1 and y 2 are the angles between the robot and the vertical planes of the wheels at the points A and B, which depend on where the robots is moving; v is the set velocity. It is natural that the integration of (1.24) at given controls y 1 and y 2 produces the above-mentioned planned path of the robot AB, i.e., the broken line E 0 E1 E 2 E 3 E 4 and the enveloping astroid E1TE 3 (Fig. 1). In the kinematic approximation, it is possible [11, 12] to eliminate time t and velocity v and to change over to the variables y A = y A ( x A ), q = q ( x A ), where x A is an independent variable. In this case, the equations of motion (1.24) become: tan( q + y 1 ) é ù é y¢A ù ê ú, sin( y y ) 2 1 êq¢ ú = êë A û ê l cos( q + y )cos y úú ë 1 2û

y¢A =

dy A dx A

,

q¢A =

dq . dx A

(1.25)

2. Path Planning for a Two-Link Robot. Figure 2 schematizes a robot with two links 1 and 2 (AB and BD) of lengths l1 and l2 (l1 > l2 ) connected by a hinge at the point B and three steerable wheels. In planning the path, we assume that each link, when turning the corner as in Sec. 1, has an envelope in the form of astroid (1.10) with parameter l1 or l2 , respectively. Let us examine two most important stages of motion (Fig. 3). At the first stage, the robot moves from the starting position A 0 B 0 D0 to the position A1 B1 D1 . The angle q1 = [ p / 2, 0], q 2 = 0. The planned angles y 1 = -q1 , y 2 = p / 2 - q1 , y 3 = 0. The envelope of positions of the first link (AB) is astroid 1. The second stage (motion from the position A1 B1 D1 to A 2 B 2 D2 ) begins after the robot stops and the wheel B turns by an angle of -p / 2(the angle y 2 becomes equal to zero). Then the link BD follows astroid 2 at control angles y 1 = 0, y 2 = -q 2 , y 3 = p / 2 - q 2 with q1 º 0, q 2 = [ p / 2, 0]. For the straight-lint motion of the robot to the right along leg II (Fig. 3) from the position A 2 B 2 D2 , the angle y 3 has to be changed stepwise from p / 2 to zero, which can also be accomplished during a short stop.

451

During these two stages, the robot travels distances l1 and l2 , respectively. If astroid 1 with greater l (i.e., l1 ) is chosen to be a middle one, then astroid 2 will be in the domain bounded by astroid 1 (Fig. 3) and the segments B 0 O1 and O1 A1 . During planned motion in which the initial position of the point A of link 1 coincides with the point A 0 , the point A moves horizontally to the right to the point A1 , traveling distance l1 . The point B coinciding with B 0 in the starting position and moving vertically will occupy the position B1 , while the point D0 , the position D1 . During this motion, the link AB follows (Fig. 3) astroid 1. After the robot stops and the wheel B turns by an angle of -p / 2, the point A1 moves to the position A 2 , traveling distance l2 . The point B1 appears in the position B 2 , while the point D1 , in the position D2 . The link BD follows astroid 2. Two-link robots with two steerable wheels are less maneuverable. If both steerable wheels (A and B) are mounted on driving link 1, then the motion of this link will be the same as the motion of the one-link robot in Sec. 1 with formation of an enveloping astroid. The path of the point D of driven link 2 for x A ³ a X + l1 can be identified from the equation dx B = -l2

dq 2 sin q 2

,

(2.1)

integrating which yields q 2 = 2arctan exp[( x B - a X ) / l2 ], where q 2 is the angle between link 2 and the Ox-axis, x B = x A - l1 . The coordinates of the path of the point D are defined by x D = x B - l2 cos q 2 and y D = a y - l2 sin q 2 . For link 2 to turn the corner, it is necessary that x B < x c + l2 cos q cm , where q cm = arcsin[( aY - y c ) / l2 ]. If a two-link mobile robot has two steerable wheels at the points A and D, then it is still expedient that the planned path of the wheel A of the driving link be the straight line y = a y with the initial position at the point x A ( 0) = a x . The point B of the second (non-steerable) wheel of the first link follows the path q1 = 2arctan exp[( x A - a X ) / l1 ] , x B = x A - l1 cos q1 ,

y B = a y - l1 sin q1 .

(2.2)

3. Dynamic Model. The equations of motion of one- and multi-link vehicles as nonholonomic systems are presented in [1–3, 5, 8, 12]. The inertial two-link robot ABD with three steerable wheels at the points A, B, D is schematized in Fig. 2 (q1 and q 2 are the angles between the horizontal and links 1 and 2 represented by the straight-line segments AB and BD; y 1 , y 2 , y 3 are the angles of turn of the steerable wheels A, B, D around the vertical axes reckoned from the link AB for y 1 and y 2 and from the link BD for y 3 ; C1 and C 2 are the centers of mass of links 1 and 2; l1 and l 2 are the distances lBC and lBC from the point B to 1 2 the centers of mass C i ; O1 and O2 are the instantaneous velocity centers of links 1 and 2). The angles are measured counterclockwise. The nonholonomic constraints at the points A, B, D between the supporting surface and the wheels are imposed so that the horizontal vectors of linear velocities of the points A, B, D are in the planes of the wheels and equal to zero in the other (including perpendicular) directions. The instantaneous velocity centers O1 and O2 of links 1 and 2 are at the intersection of the perpendiculars to the wheel planes at the points A, B and B, D. We will derive the equation of planar motion of the system in Fig. 2 in vector form from the general dynamic (d’Alembert–Lagrange) equation for a system with ideal constraints [3]: N

N

i =1

i =1

å mi Wi × dr i = å F i × dr i ,

(3.1)

whence for the discrete system consisting of two hingedly connected rigid links (Fig. 2) and undergoing planar motion, we have -m1V&C × dS C - m2V&C × dS C - J C &&q1 × dq1 - J C &&q 2 dq 2 + FA × dS A + FB × dS B + FD × dS D = 0, 1

1

2

2

1

(3.2)

2

where mi is the mass of the ith link (i = 2); Wi is the acceleration of the center of mass of the ith link; dri is the virtual displacement; Fi are the active external forces acting on the respective points of the system; FA , FB , FD are the forces developed by the wheel drive motors or brake mechanisms and referred to the horizontal axes of rotation of the wheels A, B, D; V& ,V& and dS , dS are the accelerations of the centers of mass of links 1 and 2 and their virtual displacements; dq , dq are C1

452

C2

C1

C2

1

2

the virtual angular displacements of links 1 and 2 about the vertical axes; dS A , dS B , dS D are the virtual displacements of the points A, B, D; J C and J C are the central moments of inertia of links 1 and 2 about the vertical axes. 1 2 The nonholonomic system (3.2) in Fig. 2 has one degree of freedom (one independent variation of the generalized coordinates) and its position on a plane is described by three generalized coordinates x A , q1 , q 2 . Let the independent coordinate be q1 . In addition to the variables mentioned above (Sec. 3), the model uses the variable distances lO C , lO C from the 1 1 2 2 instantaneous centers of rotation (O1 , O2 ) to the points C i . At the first stage, link 1 has planar motion (the point A moves along the straight line O1 A1 , the point B along B 0 O1 not reaching the point O1 (Fig. 3)). Link 2 translates along D0 O1 . Since the controls y 1 , y 2 , y 3 are set as functions of the generalized coordinates (at the first stage y 1 = -q1 , y 2 = 0.5p - q1 , y 3 = 0), the system is actually holonomic, and Lagrange equations of the second kind can be used to describe its dynamics. For simplicity, we set l1 = 0.5l1 , l 2 = 0.5l2 . Then the equation of motion at the first stage has the following form according to (3.2): &&q = 1 ( q& 2 m l 2 sin q cos q - F l sin q + F l cos q + F l cos q ), 1 1 1 A 1 1 B 1 1 D 1 1 J 1r 1 2 1

(3.3)

where FA is the force developed by the engine; FB and FD are the forces developed by the brake mechanisms acting on the wheels B and D; the reduced moment of inertia J 1r = J C + 0.25m1 l12 + m2 l12 cos 2 q1 . 1

It is convenient to resolve the equation of motion at the second stage for &&q 2 :

[

& & &&q = 1 -q& u& ( J u + m u l 2 ) - m q& u 2 &l 2 2 12 C1 12 1 12 O1C1 1 2 12 O1C1 lO1C1 - m2 q 2 lO2C 2 lO2C 2 J 2r +1/ cos 2q1 ( -FA l2 sin q 2 + FB l2 sin q 2 + FD l2 cos( 2q1 - q 2 ))] , 2 2 l2 2 J 2 r = J C u12 + J C + m1 u12 O C + m2 lO 1

2

1 1

2C 2

.

(3.4)

Equation (3.4) describes the case where after the first stage, i.e., stop of the two-link vehicle ABD, the angle q1 = q1* > 0 and the controls y 1 = -q1 , y 2 = q1 , y 3 = p / 2 - q 2 . The boundary conditions are q 2 ( 0) = p / 2, q& 2 ( 0) = 0. In (3.4), the following equalities hold for the second stage: u12 =

q& 1 2l2 sin q1 sin q 2 , = l cos 2q q& 2

lO

2C 2

1

1

lO C = 1 1

l1 2 tan q1

,

= l2 0.25 + cos 2 ( 2q1 - q 2 ) / cos 2 2q1 - cos( 2q1 - q 2 )cos q 2 / cos 2q1 , x B = a X + l2 cos q 2 ,

y B = a y - l1 sin q1 .

(3.5)

At the second stage, the angles q1 and q 2 are related by ì l ï q 2 = arccos í 1 ïî 2l2

é ½tan 0.5q *½ ù üï 1 * êln½ ½ + 2(cos q1 - cos q1 )ú ý , êë ½tan 0.5 q1½ úû ïþ

q1 > 0.

4. Numerical Example. Let us analyze the motion of a two-link robot with three steerable wheels (Fig. 2) with the following parameters: m1 = 100 kg, m2 = 10 kg, J C = 5 kgm2, J C = 2 kgm2, l1 = 1.5 m, l2 = 1.2 m, l1 = 0.75 m, l 2 = 0.6 m. 1 2 Since the planned path at the first stage is such that the motion of links 1 and 2 is planar and translational, respectively, some of the variables (lij , l&ij ) in (3.3), (3.4) may become unbounded. For example, if q1 ® 0 and x& A ¹ 0, then &y B = v B ® ¥ according to (1.18). In the inertial system of rigid links, such a dynamic situation corresponds to an abrupt stop followed by impact and inevitable breaking of the frictional constraints between the supporting surface and the wheels. To eliminate this

453

q&1, sec–1

q1, rad

x& A , x&B , m/sec

2

1.4

1

–0.04 1.0

0.2

–0.08

1

2

–0.12

0.6

0.1 –0.16

0.2 0

2

4

6

8

–0.2 10 t, sec 0

2

4

6

a

8

10 t, sec

0

2

4

b

FA , FD , N

6

8

10 t, sec

8

10 t, sec

c

N A, N

N B, N

1 4 3

1

0

3

2 2

–1

2

1

1 –2 0 0

2

4

6

8

10 t, sec

0

d

2

4

6

8

10 t, sec

0

2

4

e Fig. 4

6

f

singularity, it is necessary to take into account the elasticity of elements (tires, chassis drive links, etc.). It is more reasonable to stop the vehicle in advance at the end of the first stage (t = t 1* ) with, for example, a dry-friction brake mechanism. Numerically, this time point corresponds to the change of the sign of the angular velocity q& from minus to plus. It is also necessary to provide 1

that q1 ( t 1* ) > 0, which is possible to do by choosing the appropriate braking force. The drive motor of the wheel A and the brake motor D operate at the first stage. The forces acting on the system are defined by ì k1 ( x A - a x ), ïq , ï FA = í a ï k1 ( -x A + a x + x a ), ïî 0,

FD

a x £ x A < a x + x1 , a x + x1 £ x A < a x + x a - x1 , a x + x a - x1 £ x A < a x + x a , a x + xa £ xA ,

ì 0, y D < a y - l2 - x b , ï = í q b , a y - l2 - x b £ y D < a y - l2 , ï 0, a y - l2 £ y D , î

x A = a X + l1 cos q1 ,

y B = a y - l1 sin q1 ,

y D = a y - l1 sin q1 - l2

(k1 = 100 N/m, x1 = 0.05 m, q a = 5 N, q b = 3 N, x a = 0.3 m, x b = 0.5 m). 454

q& i , sec–1

qi , rad

1

2

1.2

N A, N 0.4

–0.04 2

0.3

–0.08

0.8

0.2 0.1

–0.12

0.4

0 1

0

–0.16 5

10

t, sec

0

5

a

10

t, sec

0

5

b

N B, N

10

t, sec

10

t, sec

c

N D, N

RA , N 5

40 1.0 30

3

0.6 20 10

0

0.2

5

10

t, sec

0

1

5

d

10

t, sec

0

5

e Fig. 5

f

The initial conditions: q1 ( 0) = p / 2, q& 1 ( 0) = 0. The motor A and the brake motor B operate at the second stage. The forces FA and FB acting on the system are defined by ì 0, ï * ïï k1 ( x A - a X - l1 cos q1 ), FA = í q a , ï k ( -x + a + l cos q * + x ), A X a 1 1 ï 1 ïî 0,

x A - a X £ l1 cos q1* ,

l1 cos q1* < x A - a X £ l1 cos q1* + x1 ,

l1 cos q1* + x1 £ x A - a X < l1 cos q1* + x a - x1 ,

l1 cos q1* + x a - x1 £ x A - a X < l1 cos q1* + x a ,

l1 cos q1* + x a £ x A - a X ,

ì 0, x B - a x £ l2 cos q *2 - x b , ï FB = í q b , l2 cos q *2 - x b < x B - a x £ l2 cos q *2 , ï 0, l2 cos q 2* < x B - a x . î

FD = 0.

Using D’Alembert’ principle, i.e., applying the inertial forces and moments depending on q i , q& i , &&q i ( i = 1, 2) to the links and methods of kinetostatics, it is possible to find the vectors of current dynamic reactions between the wheels A, B, D and the supporting surface. Comparing these reactions with the limiting forces of friction, we can ascertain whether the frictional constraints are broken. If they are, the wheels slip, which requires decreasing the velocity. Another method to determine the constraint forces of a two-link wheeled robot was addressed in [13, 17]. 455

The normal reaction forces N i (perpendicular to the plane of the ith wheel) exerted by the supporting surface on the wheels are expressed as N A = l1 ( 0.5m1 + m2 )[ f 0 ( q1 )cos q1 - q& 12 sin q1 ] - FB - FD , N B = -0.5m1 l1 [ f 0 ( q1 )sin q1 + q& 12 cos q1 ] - FA ,

ND =0

for the first stage and NA =

(

) f (q

1 é -J + 0.25m l 2 cos 2q C1 11 1 l1 cos q1 êë

1

2

) - 0.25m1 l12 q& 12 sin 2q

]

-0.5m1 l1 l2 sin q1 sin q 2 f 2 ( q 2 ) - 0.5m1 l1 l2 sin q1 cos q 2 q& 22 - FA l1 sin q1 , ND =

(

1 é J + 0.25m l 2 cos 2q 2 2 2 l2 sin q 2 êë C 2

)f

2 (q 2

) - 0.25m2 l22 q& 22 sin 2q 2

]

+ 0.5m2 l1 l2 cos q1 cos q 2 f1 ( q 2 ) - 0.5m2 l1 l2 sin q1 cos q 2 q& 12 - FD l2 cos q 2 , NB =

[

1 l cos q1 ( 0.5m1 + m2 ) f1 ( q 2 ) - l1 sin q1 ( 0.5m1 + m2 )q& 12 + 0.5m2 l2 cos q 2 f 2 ( q 2 ) cos 2q1 1 - 0.5m2 l2 sin q 2 q& 22 - N A - FB sin 2q1 - FD

]

for the second stage, where f 0 ( q1 ) and f 2 ( q 2 ) are the right-hand sides of the differential equations (3.3) and (3.4); f1 ( q 2 ) = f 2 ( q 2 ) u12 + q& 2 u& 12 . The total reactions: R i = Fi2 + N i2 , i = A , B , D. Figure 4 shows the variables characterizing the first stage of motion. The curves have been plotted by numerically integrating Eqs. (3.3) (the angles q1 and q 2 are denoted by 1 and 2). The angular velocity q& 1 of link 1 is represented in Fig. 4b ( q& 2 º 0). Curves 1 and 2 of the linear velocities x& A and &y B of the points A and B of the two-link vehicle in Fig. 4c reflect the above-mentioned singularity of the system associated with a “soft” impact: the angle between the tangent at the end point and &y B changes stepwise to zero, whereas the velocity x& A of the point A has no such discontinuities. The forces FA and FD are given in Fig. 4d as functions of time t. The normal reaction forces N A and N B exerted by the supporting surface on the wheels A and B are shown in Fig. 4e and Fig. 4f. The reaction N A is strong at the beginning of motion. To decrease N A , it is reasonable to start the motors A and D simultaneously at the beginning of motion. The curves representing the second stage of motion are shown in Fig. 5a–e. Curves 1 and 2 represent the angles q1 and q 2 in Fig. 5a and the angular velocities q& 1 and q& 2 in Fig. 5b. The normal reactions N A , N B , N D and the total reaction R A acting on the wheel A are shown in Fig. 5c–f. It can be seen that the normal reaction N B is much stronger than the other two (N A and N D ). In an L-shaped holding alley, the two-link robot has the same maneuverability as the one-link robot according to the method of Sec. 2 and has somewhat underestimated maneuverability according to the method of Sec. 4 (in both cases, it is assumed that the length of the one-link robot is equal to the length of the longest link of the two-link robot). Conclusions. We have analyzed kinematic models of one- and two-link robots with two and three steerable wheels. A nonsmooth path in the form of an asteroid enveloping the positions of the robot has been planned. A dynamic model of a two-link vehicle with such a path has been set up. The dynamics of the robots has been analyzed numerically, and the reactions of nonholonomic constraints have been determined.

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