CC BY
0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 681-707. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221204
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 37J60, 70Exx, 70Kxx
A Nonholonomic Model and Complete Controllability of a Three-Link Wheeled Snake Robot
E. M. Artemova, A. A. Kilin
This paper is concerned with the controlled motion of a three-link wheeled snake robot propelled by changing the angles between the central and lateral links. The limits on the applicability of the nonholonomic model for the problem of interest are revealed. It is shown that the system under consideration is completely controllable according to the Rashevsky - Chow theorem. Possible types of motion of the system under periodic snake-like controls are presented using Fourier expansions. The relation of the form of the trajectory in the space of controls to the type of motion involved is found. It is shown that, if the trajectory in the space of controls is centrally symmetric, the robot moves with nonzero constant average velocity in some direction.
Keywords: nonholonomic mechanics, wheeled vehicle, snake robot, controllability, periodic control
1. Introduction
One of the simplest and nevertheless most productive approaches to the study of the dynamics of wheeled systems is to investigate them using the nonholonomic model of rolling without slip-
Received September 15, 2022 Accepted November 16, 2022
The work of A. A. Kilin (Sections 1-3) was performed at the Ural Mathematical Center (Agreement No. 075-02-2022-889). The work of E. M. Artemova (Section 4, Appendices) was supported by the Russian Science Foundation (No. 21-71-10039) and was supported in part by the Moebius Contest Foundation for Young Scientists.
Elizaveta M. Artemova liz-artemova2014@yandex.ru
Ural Mathematical Center, Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
Alexander A. Kilin aka@rcd.ru
Ural Mathematical Center, Udmurt State University
ul. Universitetskaya 1, Izhevsk, 426034 Russia
Institute of Mathematics and Mechanics of the Ural Branch of RAS
ul. S. Kovalevskoi 16, Ekaterinburg, 620990 Russia
ping. This model is successfully applied in describing both simple model problems [1-3] and complex many-body systems [4, 5]. In spite of the simplicity this model provides a description of the main dynamical effects observed in the systems under consideration. Brief historical overviews of the main stages in the development of nonholonomic mechanics are presented in [6, 7].
The application of the formalism of quasi-holonomic constraints is particularly convenient for wheeled systems. This formalism was developed in [8] on the basis of the classical work of Hadamard and Hamel [9, 10]. Its application to wheeled vehicles has made it possible to prove that their dynamics is identical both kinematically and dynamically to the dynamics of sleighs (vehicles in which the wheels are replaced with skates). This made it possible to extend a number of classical and modern results on the dynamics of the simplest nonholonomic systems to the dynamics of wheeled vehicles. For example, in accordance with the above-mentioned analogy, the Chaplygin sleigh problem [11] describes the free dynamics of the simplest two-wheeled vehicle. We recall that by the Chaplygin sleigh one means a rigid body with a fastened skate in contact with a horizontal plane. The projection of the translational velocity of motion onto the normal vector to the plane of the wheel pair (knife edge) is zero.
We review in more detail publications concerned with the dynamics of various models of wheeled vehicles.
The simplest wheeled robot is a two-wheeled vehicle with a differential gear drive. In practice, the stability of such a two-wheeled vehicle is achieved by installing an additional spherical support or a castor wheel (their influence on the dynamics is usually neglected). The free dynamics of such a system has been well studied, but is mainly related to various versions of the problem of the Chaplygin sleigh rather than wheeled vehicles. The dynamics of the Chaplygin sleigh on various surfaces is analyzed in [11-13]. In [14, 15] the dynamics of a sleigh with periodically changing mass distribution is investigated. In particular, in [14] it is shown that small periodic changes in the mass distribution can lead to an unbounded acceleration of the sleigh. The problem of controlling the sleigh by periodically switching the nonholonomic constraints is addressed in [16]. The problem of a sleigh moving under the action of viscous dissipation and periodic rotation of a rotor is dealt with in [17], where it is shown that the system has a stable limit cycle. In [18] the planar motion of a rigid body with a fastened blade in an ideal fluid is examined. This problem is a hydrodynamical generalization of the classical Chaplygin sleigh problem. In particular, in [18] the equations of motion of a hydrodynamical sleigh are integrated in explicit form and the asymptotic nature of its motion is shown.
The controlled motion of two-wheeled vehicles has also been well studied. A large number of publications in this direction are concerned with practical problems of controlling modern robotic systems on two-wheeled platforms. Vacuum-cleaning robots may serve as examples of wide application of such a design. A detailed review of results along these lines can be found, for example, in [19, 20].
A large amount of research is devoted to the search for the optimal control of two-wheeled vehicles. This problem is closely related to the solution of the sub-Riemannian problem on the group of motions of a plane and Euler's elastics. The optimal control of a wheeled vehicle consisting of one platform by changing the translational and angular velocities is considered in [21-23], where normal extreme trajectories of a wheeled vehicle are constructed and an optimal synthesis is performed. Results of experimental research on the motion of a wheeled robot along optimal trajectories are presented in [24]. The question of the controllability of one-link and two-link robots and that of optimal maneuvering are addressed in [25]. In [26] a robot control algorithm is proposed for constructing the minimal trajectory between two points in a bounded space.
More complicated wheeled vehicles include two-link vehicles consisting of two platforms with wheel pairs rigidly attached to them and connected to each other by a hinge. The classical motor-car design with one pivoted axle, which was studied in [27, 28], is a particular case of such a system. A more general nonsymmetric design of this type is usually called the roller-racer after sports equipment of the same name1. In the most general case the free dynamics of the roller-racer is considered in [5]. There it is shown that in the general case the roller-racer asymptotically tends to motion in a circle. The asymptotic stability of straight-line motions for the case where the center of mass of the system is at the point of attachment of the platform is investigated in [27].
Research on the control of the motion of a two-link vehicle can be divided into two main directions. The first is concerned with motion control by changing the angle of rotation of one platform relative to the other. One of the first publications analyzing the controllability and control of the motion of the roller-racer by periodically changing the angle between the platforms is [29]. In [30] it is shown that periodic changes in the angle can lead to an unbounded acceleration of the roller-racer. A theoretical and experimental study of the motion of the roller-racer under periodic controls is presented in [31]. In this paper we investigate an optimization of the parameters of periodic controls and mass-geometric parameters of a vehicle for a generally straight-line motion of the roller-racer. An optimization of control parameters in a given class of periodic functions for two- and three-link vehicles is also described in [32, 33]. In [33] an analytic and numeric investigation of the dynamics of the roller-racer under small and large amplitudes of the control angle is performed. In [34] the control of the rotation angle of the roller-racer is examined taking into account dry friction between the wheels and the ground.
The second direction of research on the controlled motion of wheeled systems of roller-racer type is concerned with the problems of controlling a car-like robot with trailer. In this setting, by controls one means the translational and angular velocities of the leading platform (car). For such systems one usually solves problems of construction of controls (including optimal control) used to perform some maneuvers of practical importance (turning, reparking etc.) or problems of stabilizing specific motions (for example, backing). Since these problems are of practical significance, there is a very extensive literature in this direction. Here we present only a few references to publications on these topics [35-39]. The general problem of finding optimal trajectories and of constructing an optimal synthesis for a two-link robot still remains open. We mention here only a few papers concerned with solving this problem in some restricted settings. A study of optimal trajectories for a two-link robot in the special case where the second platform is fastened to the center of mass of the wheel pair of the first (leading) platform is presented in [40]. In [41, 42] the method of nilpotent approximation is proposed for an approximate solution to the problem of the optimal control of a two-link vehicle. Since there is no general analytic solution to the problem of optimal control of the roller-racer at present, methods based on fuzzy logic and neural networks [43-46] are used to construct optimal maneuvers and control systems or to stabilize motion.
Another model of a wheeled vehicle with two wheel pairs is the snakeboard [47]. It is a platform to which two freely turning wheel pairs spaced at some distance from each other2 are fastened. Such a design is a particular case of the three-link vehicle whose wheel pairs are mounted only on the lateral links. In [48] the free dynamics of the snakeboard is considered and a topological and qualitative analysis of the trajectories of this system is made. The basic
1 https://www.youtube.com/watch?v=IqyfJvIp6ZY
2 https://www.youtube.com/watch?v=HE8jxG7MJTA
gaits (elementary maneuvers) for controlling the snakeboard by changing the turning angle of the platform are obtained and analyzed in [49]. The control by changing the angles and the problem of the controllability of such a system are considered in [47]. The kinematic controllability of the snakeboard is shown in [50], and the problem of its motion along a prescribed trajectory is addressed in [51].
Wheeled vehicles consisting of three and more links (platforms) are usually called snakelike systems. The first qualitative study of the motion of a snake robot was presented in [52], whereas the first operating snake robot was constructed almost three decades later, in 1972 [53]. At present, there exist many practical modifications of snake robots, including wheeled robots, for example [54]. The book [55] provides a detailed overview of the modeling and control of snake robots in two-dimensional and three-dimensional settings.
We note that the problem of controlling a multilink vehicle in a nonholonomic setting is mainly kinematic in nature. This is due to a large number of constraints imposed on the system. As a result, the constraints not only restrict the motion of the system, but also must be consistent with each other. As a consequence, even the planning of the trajectory of motion becomes a nontrivial problem, and most of the publications concerned with multilink systems study kinematic controls and special features related to them. Periodic controls under which the controls of all joints of a snake are described by the same function of time, but with different delays, are the most often used class of controls for snake robots.
The control of wheeled multilink vehicles is examined, for example, in [55-57]. In [56], an analysis is made of the control of the trajectory of a snake robot raising its head (the leading platform) on the plane. An overview of analytic studies of such systems and studies concerned with the modeling of snake robots is presented in [58]. In [59], a method for generation of controls for flat nonholonomic (n + 1)-link robots (n ^ 3) controlled by changing the angles between all platforms is proposed. The motions generated by the proposed method satisfy nonholonomic restrictions. Equations for four- and five-link controlled robots are obtained in [60].
Much less attention has been given to the study of the free dynamics of multilink vehicles. The papers covering this topic mainly confine themselves either to a small number of links (up to three inclusive) or to investigating some special regimes of motion. For example, a special integrable case in the dynamics of a symmetric snake-like three-link robot is found in [61]. The motion of a prime mover with two trailers is dealt with in [62]. In [63], the modes of oscillations of various configurations of robots arising under the periodic control of only one part of the angles between the links are investigated. The dynamics of a wheeled vehicle consisting of three and more platforms is discussed in [64]. In this paper, reduced equations of motion are obtained for an articulated n-trailer motor car performing inertial motion on a plane. However, the free dynamics of an arbitrary n-link (n = 3, 4, ...) of a wheeled vehicle is still poorly understood.
We now consider in more detail the problem of controlling three-link snake robots. In [65] kinematic equations of motion are presented which describe the controlled motion of a multilink vehicle and, in particular, of a three-link vehicle, using the nonholonomic model. The problem of constructing gaits for a three-link vehicle was addressed in [66]. The gaits obtained were used to control a real robot. In [67], a control is constructed under which a three-link vehicle performs stable backing along a periodic trajectory. In [68] the method of nilpotent approximation is applied for motion planning for a three-link vehicle. As control actions one often considers changes in the angles between the platforms. However, in [69] the control of only one angle in the three-link vehicle is considered, whereas the second angle remains passive.
It is well known that, when one chooses the angles between the platforms as controls, "singular" configurations [65, 70, 71] appear in the multilink systems. When the robot passes through
these configurations, the constraint reactions begin to increase without bound, which results in the nonholonomic model becoming inapplicable. Various approaches have been proposed to overcome this problem. For example, in [72] a hybrid model is proposed for a symmetric robot. This model provides for switching between the nonholonomic model and the model with sliding obeying the law of Coulomb friction. Also, this paper presents a condition for passage through a singular configuration for a three-link robot without transition to hybrid dynamics.
Another hybrid model for overcoming singular configurations was proposed in [73]. Within the framework of this approach, it is proposed to use the kinematic nonholonomic model of motion until the robot approaches singular configurations. As the robot approaches a singularity, switching to a dynamical model occurs in which one of the joints becomes passive or blocked, which makes it possible to avoid the onset of unbounded constraint reaction forces.
In spite of the restrictions imposed on the control angles in the nonholonomic model, the robot consisting of three platforms is completely controllable. That is to say, it is possible to choose controls bringing the robot from any initial position and orientation to any final ones. A standard approach to proving the controllability is to apply the Rashevsky-Chow theorem [74]. A proof of the controllability of the three-link vehicle can be found in [75, 76]. We note that the problems of optimal control of a three-link vehicle in a general setting have apparently not been investigated. A partial solution to the problem of optimization in the class of given functions can be found in [32, 33].
In this paper we consider the control of a symmetric three-link robot by changing the angles between the central platform and the lateral platforms. Section 2 is devoted to deriving equations of motion and to defining singular configurations near which the nonholonomic model is inapplicable. Section 3 presents an analysis of the controllability of the system. Section 4 analyzes possible types of motion of the three-link robot under periodic (snake-like) controls. In this section it is shown that, if the trajectory in the space of controls is centrally symmetric, the robot moves with nonzero constant average velocity in some direction.
2. The Mathematical model
2.1. Equations of motion
In this paper we consider the controlled motion of a three-link wheeled snake robot on a plane without slipping. We assume that the robot consists of three identical platforms (links). The platforms are connected to each other by joints, and a wheel pair is rigidly fastened to the center of mass of each platform. By a wheel pair we mean two wheels rotating independently on the same axis.
We formulate the main assumptions concerning the system under consideration:
- when the wheeled vehicle rolls, there is no slipping of the wheels relative to the plane;
- the platforms can turn relative to the points of attachment (joints);
- the center of mass of each platform coincides with the center of mass of the wheel pair fastened to it.
We define two coordinate systems:
- a fixed coordinate system Oxy;
- a moving coordinate system Cx1x2 with origin C at the center of mass of the central platform (see Fig. 1) and with axes Cx1 and Cx2 directed along and perpendicular to the axis of the wheel pair, respectively.
Fig. 1. A three-link wheeled snake robot on a plane
The position of the central platform relative to the fixed coordinate system will be defined by the radius vector r = (x,y) of its center of mass (point C in Fig. 1), and the orientation, by the ^ angle between the axes Ox and Cx2. Let §1 and denote the turning angles of the lateral platforms relative to the central platform. We note that the angles §1 and $2 are measured from the axis Cx2 (see Fig. 1). In what follows we will assume that the angles and $2 are given functions of time.
Table 1 presents the mass-geometric characteristics of the wheeled robot. Here and below (unless otherwise specified) we assume that all vectors are referred to the moving coordinate system Cx1x2.
Table 1. Nomenclature of the mass-geometric characteristics of the wheeled robot
Symbols Description
m the total mass of each platform
Jo the total moment of inertia of the central platform relative to the vertical axis passing through point o
ok the point of attachment of the fcth platform to the central platform
j the total moment of inertia of the fcth platform relative to the vertical axis passing through point Ok
rk the radius vector of point Ok
Ck the center of mass of the fcth platform (coincides with the center of the wheel pair)
Pk the radius vector from the point of attachment of the kth platform to its center of mass Ck
nk the normal unit vector to the wheels' plane
Tk the tangent unit vector to the wheels' plane
The vectors pk, nk and tk depend on the turning angle §k. Referred to the moving coordinate system, they can be represented as
nk = (cosrdk, sinrdk), tk = (— sinrdk, cosrdk),
^ o ^ /cos ■dk — sin <dk\
Pk = QkPk> Qk = . Q q > Vsin §k cos vk I
where p0 are the constant radius vectors of the center of mass of the platform, referred to the coordinate system Cknktk. For the symmetric wheeled robot we consider here, these vectors can be represented as
and the vectors rk as
P0 = (0, a), = (0, -a),
ri = (0, a), r2 = (0, -a),
where a is the distance from the point of attachment to the center of mass of the platform.
As shown in [8], the dynamics of wheeled systems are identical to those of systems in which skates are used instead of wheel pairs. Therefore, we next consider a system of three coupled platforms to whose center of mass skates are fastened. If necessary, the time dependence of angles of rotation of the wheels can be recovered using straightforward quadratures [8].
Remark 1. Without loss of generality we assume that m =1, a = 1 (in this case j = j0 + 1).
The no-slip condition for the skates imposes the following three time-dependent nonholonomic constraints on the system:
fi = v1 = 0,
f2 = v1 cos $ + v2 sin $ — w cos $ — (w + $1) = 0, f3 = v1 cos $2 + v2 sin $2 + w cos $2 + (w + $2) = 0,
(2.1)
where v = (v1, v2) are the components of the linear velocity of the point C (see Fig. 1), w is the angular velocity of the central platform, and •&i are, as stated above, given functions of time. Without loss of generality we will assume that §i G (—n, n].
The equations of motion for such systems can be written in the form of Lagrange - Euler equations in quasi-velocities [8]. These equations are a closed system of three nonautonomous differential first-order equations in the variables w = (v1, v2, w). By construction, these equations admit three integrals f = (/1, f2, f3) and are defined on the zero level set f = 0.
We note that in the case at hand the dimension of the velocity space w is the same as the number of nonholonomic constraints. As a result, the Lagrange - Euler equations are identical to the time derivatives of the constraints (2.1). And the time dependences of velocities (v1, v2, w) are defined by the constraint equations. However, the Lagrange - Euler equations are necessary to calculate the constraint reaction forces Ni. An explicit form of the reaction forces Ni is presented in Appendix A.
Supplementing the constraint equations (2.1) depending on (v1, v2, w) with the kinematic relations
x = v2 cos ^, y = v2 sin ^, ip = w, (2.2)
we obtain a complete system of equations describing the dynamics of the system.
Solving the constraint equations (2.1) for velocities, we obtain
cos
v1 = 0,
v2 =
cos
2 sin
2
cos
$ 1 -
2 sin
2
cos
Sin ^f
w=
sin —o
-$ 1 -
(2.3)
2.
ß
■o
2
1
2
o
o
1
2
2
2
o
o
Substituting these expressions for velocities into the kinematic equations
fx\
y
w
Vl +
)cos
V
) cos ^
2 sin ( ^ 2 ^ cos^ ) sin ^
2sinfir2 w
sin 1 - w
2 sin ( ^ 2 cos
V 2
(2.4)
yields a system of differential equations describing the controlled motion of a symmetric three-link robot. In what follows, we will investigate the resulting system (2.4).
2.2. Degeneracy of constraints
Before we turn to analysis of controllability, we consider singular values of and at which the homogeneous (in velocities) parts of the nonholonomic constraints (2.1) become linearly dependent. It is easy to show that the degeneracy occurs at points corresponding to configurations of the wheeled robot with the following relative positions of the platforms:
1. = n — the first platform coincides with the central platform;
the second platform coincides with the central platform; 3. = —§2 — the lateral platforms have been turned through the same angle.
2. V2 = n
Next, we consider the controls Vk (k = 1, 2) which satisfy one of the conditions for degeneracy of the homogeneous part of the constraints at some time instant tcr. We also address the question of what the velocities (2.3) and reaction forces Ni are equal to at time tcr. It turns out that the following proposition [72] holds:
Proposition 1. Suppose that at t = tcr at least one of the equations
Vl(tcr) = n, V2 (tcr) = n
holds. Then, as t ^ tcr, no less than one component of the constraint reaction Ni and no less than one of the velocities v2, u increase without bound.
The conditions of Proposition 1 correspond to a configuration of the robot in which two or all three platforms are folded on each other. It is obvious that such configurations are physically unrealistic. However, the increase in the constraint reactions near these configurations must be taken into account, for example, in planning the trajectories of the robot without transverse slipping of the wheels.
For the third case of degeneracy of the constraints the following proposition holds.
Proposition 2. Suppose that at t = tcr the equation Vl(tcr) = -V2(tcr) holds. Then
1. if Vl(tcr) = V2(tcr), then the velocities v2, u and the constraint reactions Ni remain finite
at t - tcr ;
2. if Vl(tcr) = V2(tcr), then the velocities v2, u and no less than two components of the constraint reaction increase without bound.
A detailed treatment of each case, a proof of these propositions, and an analysis of the asymptotics of the dependences N^(t), v2(t) and w(t) for t ^ tcr are presented in Appendix B.
Proposition 2 has an interesting geometric interpretation. The intersection of the control curve given by the dependences §1 (t), §2(t), with the straight line = —§2 on the plane of controls (§^ §2) does not lead to an infinite increase in the velocities and reaction forces if this intersection occurs at a right angle and at §1}2(tcr) = n, that is, if the following conditions are satisfied:
t-tcr
d§2
Next, we turn to investigating the controllability of the system taking into account the restrictions imposed on the control angles.
3. The Rashevsky - Chow controllability criterion
In this section we present the results on the controllability of the system. Most of the results have been obtained recently in [75, 76]. Here we present them in our notation for completeness of exposition.
Let us first consider the controlled motion of the three-link symmetric wheeled robot within the framework of the nonholonomic model in the case where the controls ft1(t) and ft2(t) satisfy the following conditions:
§1 = -§2, §1 = n, §2 =
(3-1)
We show that, under the restrictions (3.1) imposed on the controls ftk (k = 1, 2), the system (2.4) is controllable according to the Rashevsky-Chow theorem. For this we pass to an extended configuration space where the vector q = (x, y, p, ft1, ft2)T defines the position of the wheeled robot. In what follows we assume that ft 1 and ft2 define the controls. Then the equations of motion (2.4) can be represented in a form linear in controls as follows:
q = K 1(p, ft1, ft2)ft 1 + K2(p, #1, ft2)ft2, where K i and K2 are vector fields of the form
(k(1 + cos §2) cos k(1 + cos §2) sin —k sin §2
1 0
k=
/
Ko
§1 §2 4 cos — cos — 2 J V 2
(-k(1 + cos §1) cos -k(1 + cos §1) sin — —k sin §1 0 1
sin
2
1
(3.2)
According to the Rashevsky - Chow theorem [74], a system of the form q = ^ Ki(q)ftfti is controllable in some region G if among the vector fields Ki and the commutators composed of them, at any point of region G, one can find n linearly independent fields where dim G = n. That is to say, from any point of G one can arrive at any other point by shifting a finite number of times along the trajectories of the vector fields Ki.
Let us construct the following vector fields:
K 1,2 = [K1, K2], K 1>(1>2) = [K 1, K 1,2], K2,(1,2) = [K2, K1 2], (3.3)
where [•, •] is the commutator of the vector fields which is given by
[U, V ]a = Up dp Va - V p dp Ua.
Since the vector fields (3.2) and (3.3) are defined only under conditions (3.1), we define two regions G± as
G+ = {q | 01 G (-n, n), 02 G (-n, n),#1 + 02 > 0}, G- = {q I 01 G (-n, n), 02 G (-n, n), 01 + 02 < 0}.
The condition of the linear independence of the vector fields (3.2) and (3.3) is that the determinant of the 5 x 5 matrix composed of their components is different from zero
4(008(1?! + 02) - 1)2(C0S 0.2 + 1)2(C0S + l)2 ^ 0'
Condition (3.4) is fulfilled everywhere in both regions G±. However, the following proposition holds.
Proposition 3. The central platform of a three-link wheeled snake robot can be transferred from any initial position and orientation to any final position and orientation on the plane by changing the angles relative to the platforms 01 and 02 under the condition that throughout the motion these angles belong either to region G+ or G-.
Moreover, we can prove a more rigorous proposition about controllability of the system in the region
G = {q | 01 G (-n, n), 02 G (-n, n)}. (3.5)
Proposition 4. A three-link wheeled snake robot can be transferred from any initial position on the plane with an arbitrary orientation of all links to any final position on the plane with an arbitrary orientation of all links by changing the angles relative to the platforms 01 and 02 under the condition that throughout the motion these angles belong to region G.
Proof. It follows from Proposition 1 that the system under consideration can be transferred from region G+ (G-) to region G- (G+) or to the boundary between them, which is defined by
the equation 01 = -02. For this, one can use control l&°(t), where $°(t) = (0°(i), 02(t)) are
arbitrary functions satisfying the conditions of Proposition 2.
* + * —
Combining such a control with controls $ (t), $ (t) which ensure the transfer between arbitrary points in regions G+ and G- (in accordance with Proposition 3), we obtain the necessary control which implements the transfer between arbitrary points in region G.
Figure 2 shows a schematic representation of the projection of the trajectory of controlled motion on the plane (01,02).
In a similar way, one can construct controls beginning or ending on the boundary between regions G1 and G2 (01 = -02) with 012 = n. □
Let us single out the following corollary (interesting for applications) from Propositions 2 and 4.
Fig. 2. A schematic representation of the projection of the trajectory of controlled motion on the plane ($1; $2)
Fig. 3. A schematic representation of the allowed motion of a wheeled robot
Corollary 1. If at some time instant the robot is in a straight-line $ = $2 = 0) or circle-arc $ = — $2) configuration, then further motion without slipping is only possible when the platforms rotate simultaneously in the same direction with equal angular velocities (see Fig. 3).
For example, one can start motion from the straight-line configuration only by an S-shaped bending of the three-link wheeled robot.
4. Periodic controls
4.1. Snake-like controls
We now consider the periodic controls for Eqs. (2.4). As controls, we consider functions $j(t), and not $j(t) as in Section 3. Similar controls for a two-link wheeled robot are considered, for example, in [5], for a three-link robot in [75] and for an W-link robot in [77]. We assume that the controls $1 and $2 have the form
$1 (t) = a1 cos(Qt + + f1, $2(t) = a2 cos(Qt + ^2) + f32, (4.1)
where we choose the parameters ai and /3i (i = 1, 2) so that the angles and belong throughout the motion to the region (3.5). Such controls are typical of snake-like motion.
Equations (4.1) define on the plane (^ $2) an arbitrary ellipse. However, the condition that the constraint reactions are bounded leads to splitting of the controls (4.1) into two separate families. The first family consists of curves crossing the singular straight line §1 = — $2. It is easy to show that this family is a family of ellipses of the form
#i(i) = a cos(Qt) + /3, tf2(t) = a cos(Qt + <p) — /3. (4.2)
The second family consists of arbitrary ellipses lying entirely in region G+ or G_. Here we will not present the explicit form of the restrictions on the control parameters which follow from the condition that the ellipse belongs to one of these regions since this is an easy geometric problem.
4.2. Types of trajectories
We carry out an analysis of the kinematic relations (2.4) and describe possible types of the robot's motion for arbitrary periodic controls $j(t). As examples, we will present the controls (4.1) or (4.2), which do not violate the nondegeneracy conditions for the constraints. To do so, we substitute the periodic controls ^(t) into the expression for w (2.3). Expanding the result in a complex Fourier series with respect to time and substituting into the third of Eqs. (2.4), we obtain
i> = wo + wn(t), wn(t) = E anemnt, (4.3)
n=0
where an is the coefficient of Fourier series expansion and w0 is a zero expansion term equal to the mean value of the angular velocity w averaged over a period:
1 r C\
= J ^(t)dt, Tn = —. n o
Here w(t) is the time dependence of the angular velocity obtained from Eq. (2.3) after substituting the periodic controls into it. Let us integrate (4.3) with respect to time and represent ^(t) as
^(t) = ^o + wot + ^n(t), (4.4)
where
1
00 = m - E ^ = - jr [tuJn{t) dt
n=0 Q 0
and Vto(t) is a periodic function with frequency Q of the form
t
n=0..........0 ° 0
Note that the function ^n(t) has a zero value averaged over a period, i.e.,
^n(i) = E = / wn(T) dT + ^r J tcon(t) dt,
J ^n(t) dt = 0. 0
Taking (4.4) into account, we represent Eqs. (2.4) for x and y in the complex form
z = v2 el'm = v2el(^o(t)), (4.5)
where z = x + iy. Expanding the expression v2el^n(t) in a Fourier series as a periodic function of time with frequency Q
AnQt
v2el^(t) bneln
and integrating (4.5) term by term gives
z(t) = ¿(0) + e^o ^ 7
i(nQ + w0)
^el(nQ+w0)t _ ^ .
(4.6)
(4.7)
Analysis of the dependence (4.4), (4.7) shows that the following types of motion with periodic controls are possible:
1. Nonresonant case. In the absence of resonance the inequality nQ + w0 = 0 is satisfied for any n. In this case, two types of trajectories of the central platform are possible:
a) In the case of commensurate frequencies, i. e., when the equation nQ + hw0 = 0 is satisfied for some k = 1, the central platform moves periodically along a closed trajectory (see Fig. 4a).
b) In the case of incommensurate frequencies, it moves along a bounded quasi-periodic trajectory (see Fig. 4b). In this case, in the coordinate system rotating with angular velocity w0 the trajectories of the central link are a closed curve. This curve is depicted in Fig. 4b above.
(a)
(b)
Fig. 4. Trajectories of the central link under controls (a) = 1.41 sint +1.5, = 1.40921135 cost + 0.5, (b) ■&1 = 0.5sint, = 0.5cost + 1.5
It follows from the inequality nQ + w0 = 0 with n = 0 that in the nonresonant case the robot rotates, during the motion, with a nonzero average angular velocity w0.
2. Resonance case. In the case of resonance, the equation nQ+w0 = 0 is satisfied for some n and, as in the previous case, two types of trajectories are possible:
n
n
a) If the corresponding coefficient bn is zero, then the central platform moves periodically along a closed trajectory (see Fig. 5a).
b) If bn = 0, then the motion is a superposition of a time-linear drift with velocity \bn\ at an angle p0 + arg bn to the axis Ox and periodic motion (see Fig. 5b). In this case, in the coordinate system uniformly moving with the velocity of the drift the trajectory of the central link is a closed curve (see Fig. 5b above).
Fig. 5. Trajectories of the central link under controls (a) = cos t, = cos(t + 2.69731), (b) = cos t, d2 = cos(t + 2)
We note that in the case of resonance n = 0 and b0 = 0 the orientation of the moving robot periodically oscillates near some average turning angle p0. In other cases the moving robot rotates with a nonzero average angular velocity w0.
Using the symmetries of the right-hand side of the expression (2.3) for w, one can prove that for two classes of controls the condition of the zero-th order resonance is satisfied. The first class includes controls defining on the plane fi2) a centrally symmetric curve. For such controls the following proposition holds.
Proposition 5. Suppose that periodic controls fii(t) define on the plane , fi2) a bounded centrally symmetric curve lying entirely in a region <E (—n, n). Then the motion of the robot corresponds to the resonance n = 0 at which the robot (the central link) moves with the average velocity
Tn
v = \b0\, bQ = ±.JV2(?W)dt (4.8)
n 0
in the direction
P* = p0 + arg b0, (4.9)
and the average angular velocity of rotation of the robot is zero, w0 = 0.
Proof. Define the controls fij(i) in terms of the natural parameter (the length of the trajectory)
fii = fii(s(t)). (4.10)
Here the functions #j(s) define the natural parameterization of the curve on the plane (#1, #2) and the time dependence of the natural parameter, s(t), defines the law of motion along this curve.
The periodicity of the controls implies the periodicity of the functions ^(s)
#z(s + L) = #¿3), i = 1, 2,
where L is the length of the curve on the plane (#1, #2). Furthermore, since the prescribed curve (#1 (s), #2(s)) is centrally symmetric, the following equations hold:
+ =-№), d'^s + ^j =-d't(s), ¿ = 1,2, (4.11)
where ' denotes the derivative with respect to the parameter s.
Substituting (4.10) into (2.3) and viewing ^(s) as complex functions, we represent the average angular velocity as
L
wo = y u3(s) ds, (4.12)
0
where ws(s) is the angular velocity w from (2.3) into which we have substituted the controls #j(s) and where we have replaced the velocity #¿(t) with #i(s). The functions ws(s) and w(t) are related by the equation w(t) = ws(s)s. It follows from (4.12) that the average angular velocity w0 depends only on the form of the curve on the plane of controls and does not depend on the law of motion on this curve. Moreover, it follows from (2.3) and (4.11) that the function ws(s) possesses the following symmetry:
ua(s + ^=-ua(s). (4.13)
Using (4.13), it is easy to show that the average angular velocity w0 vanishes, which corresponds to resonance at n = 0.
The direction (4.9) and the drift velocity (4.8) in the case at hand are calculated from the resonant term of the expansion (4.6). □
We note that the controls (4.2) with /3 = 0 define on the plane (#1, #2) a centrally symmetric ellipse. Thus, the following corollary of Proposition 5 holds.
Corollary 2. The controls (4.2) with /3 = 0 correspond to a zero resonance and guarantee a generally straight-line motion.
Another class of controls with the resonance n = 0 includes controls defining on the plane (#1, #2) a curve symmetric with respect to the straight line = #2. For such controls the following proposition holds.
Proposition 6. Suppose that periodic controls #i(t) define on the plane (#1, #2) a bounded curve symmetric with respect to the straight line = #2 lying entirely in a region £ (—n, n). Then the motions of the robot correspond to the resonance n = 0 at which the robot (the central link) moves with the average velocity
v = \b0\, bQ = ±.JV2(?W)dt
n o
in the direction = + arg b0, and the average angular velocity of rotation of the robot is zero, w0 = 0.
This proposition is proved similarly to the preceding one, using the symmetry property of the trajectory, which in this case has the form
UL - s) = (s), <(L - s) = -Vj(s), i,j = 1, 2, i = j.
4.3. Additional singularities of the trajectories
In addition to the above-mentioned properties of the motion, the trajectories of the central link can contain cusps. These cusps correspond to the nondegenerate zeroes of the function v2(t). The time instants of such cusps can be easily calculated by solving the equation v2(t) =0 by substituting the controls (4.1) into the expression (2.3) for v2. An example of such a trajectory is shown in Fig. 6a for the controls (4.1) with the parameters
«1 = 1, A = -1, P1 =0, «2 = 1, P2 = -1, P2 = 2.5, Q = 1. (4.14)
Also, the trajectory depends on the amplitude of the function pn(t). When the amplitude passes through the value 2nn, the trajectory of the center of mass of the central platform becomes more complex — it acquires additional loops. Figures 6b and 6c present illustrations of the trajectories of the central link under the controls (4.1) with the parameters
«1 = 1, £1 = 0, Pi =0, «2 = 1, (32 = 0, P2 = 2.8, Q = 1, (4.15)
and
«1 = 1, 3i = 0, Pi = 0, «2 = 1, (2 = 0, P2 = 3, fi = 1, (4.16)
respectively. The appearance of loops with increasing parameter p2 corresponds to an increase in the amplitude of pn(t) from 3.09 for the parameters (4.15) to 7.48 for the parameters (4.16).
Fig. 6. The trajectory of the center of mass of the central platform in the fixed coordinate system under initial conditions ^(0) = 0, x(0) = 0, y(0) = 0 with the control parameters (a) (4.14), (b) (4.15), (c) (4.16)
Regarding the controls with the parameters (4.15) and (4.16) we also note that the average velocity of the central platform increases from3 v2 & 5.77 for the parameters (4.15) to v2 = 14.1 for the parameters (4.16). This is due to the fact that the controls (4.16) lie on average nearer to the singular straight line V1 = -V2. As a result, the maximal value of the constraint reactions
3Here and in the sequel, all values are presented in dimensionless quantities (see Remark 1).
increases from 39 for the parameters (4.14) to 226 for the parameters (4.15). Thus, in choosing optimal control parameters it is necessary not only to ensure the maximal velocity of motion, but also to prevent a strong increase in the reaction forces because such an increase would imply that the constraints are no longer satisfied. For example, the model of dry friction requires that the reaction forces remain inside the cone of friction.
Thus, using the controls (4.1) with various parameters one can implement different types of motion: generally straight-line motion, circle-arc motion, etc. For example, for nonzero generally straight-line motion one should obviously choose parameters corresponding to the resonant case. Constructively this can be done in a simple way. This requires solving the equation
nQ + w0 = 0
for one of the control parameters and making sure that the corresponding expansion coefficient bn does not vanish.
5. Conclusion
To conclude, we present the most important results obtained in this work and mention some questions remaining open.
A detailed analysis has been made of the asymptotics of the velocities and constraint reaction forces near singular configurations. This has made it possible to prove the "passability" of singular circle-arc configurations within the framework of the nonholonomic model. The "passability" condition has a natural geometric interpretation: the trajectory on the plane of the control angles (#1, #2) must perpendicularly cross the singular straight line = —#2.
Possible types of trajectories of the central link of the robot under periodic controls have been described. In particular, it has been shown that for the generally straight-line motion of the robot it is necessary to choose control parameters corresponding to a resonant relation. In this case, the average velocity is equal to the corresponding coefficient of expansion of some function in a Fourier series. Also, two classes of periodic controls for which the resonant zero-th order relation holds have been revealed.
In the future it would be of interest to solve the problem of optimizing periodic controls (the trajectory on the plane (#1, #2)) for implementation of the robot's motion with the maximal velocity within the framework of the nonholonomic model. Such an optimization in a specific class of periodic functions is presented in [32].
Another interesting open problem is to solve the problem of the optimal control of the robot under consideration, in particular, to construct optimal trajectories which will allow a minimization of the time of motion or of the distance between given initial and end points. In our opinion, the study of this problem is a promising avenue of research.
Acknowledgments
The authors express their gratitude to I.S.Mamaev and Y. L.Karavaev for fruitful discussions.
Conflict of interest
The authors declare that they have no conflict of interest.
Appendix A. Calculation of constraint reactions
The equations of motion of the system of interest in general form are written as Lagrange -Euler equations in quasi-velocities
d_ ( dT_ \ dT _ dt \di\J dv2 17
d i dT\ dT /k .
d (dT\ dT dT A7
where Ni are the components of the total reaction of all constraints and T is the total kinetic energy of the system
T = i + 3v22 + (3j + l)w2 + j (tf? + ft!)) + ¿(0! + - v2(u + sin +
+ V2(w + § 2) sin §2 — (w + § l)(vi — w) cos §1 + (w + § 2)(v1 + w) cos §2 •
Let us express the accelerations V 1, V2, W from the time derivatives of the constraint equations (2.1) and substitute the resulting expressions into Eqs. (A.1). As a result, taking (2.1) into account, we obtain the explicit form of the constraint reaction vector N = (N1,N2, N3)T
( (w + § 1)2 sin §1 — (w + §2)2 sin §2 — (W + i?1) cos §1 + (W + §2) cos §2 — 3v2w \ N = —(w + § 1)2 cos §1 + (w + §2)2 cos §2 + (W — i?1) sin §1 + (W + §2) sin §2 + 3V2 , y(v2w + 2w + §1) cos §1 + (—v2w + 2w + §2) cos §2 + N0 + (3j + 1)w + j(§1 + §2)y
N0 = (—2§ 1w — § 1 — V2) sin §1 + (—2§2w — §2 + V2) sin §2,
(A.2)
where V2 and w are defined from the time derivatives of Eqs. (2.3).
Appendix B. Analysis of the degeneracy of constraints
Let us consider two cases of degeneracy of constraints, as presented in Propositions 1 and 2, in which the constraint reactions increase without bound.
B.1. Proof of Proposition 1. Case I. One of the angles $i takes the value n
Let §1(i) take the value n at time t = tcr < x> and §2(tcr) = n. The expansion of the function in a neighborhood of point tcr takes the form
u _ 1 fdk§1(t)
§1(t) = sgn(§1(tcr))n + a0 (t — tcr) + ... a0 =
k! V dtk
= 0, (B.1)
t - tcr
where k > 0 is the order of the first nonzero derivative at point t = tcr. Substituting the expansion (B.1) into (2.3), we easily obtain a series expansion of the velocities v2 and w near t = = tcr. The leading terms of these expansions have the following form:
k ^ fctan(^) J ±oo, Ut«)* 0,
V9 --= ±00, UJ ---- = < (B.2)
2 t — t„ t — tcr [0, §2(tcr )=0.
Similarly for §2(tcr) = n.
In a similar way, substituting (B.1) into (A.2), we easily obtain an expansion of the reaction forces near t = tcr
N rj fc((fc - 1) cos tf2(tcr) + 2k - 1) sin tf2(tcr) t 2
1 ~ COSt>2(t.cr) +1 cr
+kû2(tcr)(2cOS tf2(tcr) - 1)(t - tcr)-1 -
ka0 [(2k - 1) (cos2 d2(tcr) - 2 cos d2(tcr)) - 2k + 3] , 2
2 cos d2(tcr) + 2 { cr) '
N2 w -k((k - 1) cos û2(tcr) - k - 2)(t - t^)-2+
kè2(tcr) sin §2(tcr)(2 œs t)2(tcr) + 1)
costf2(icr) + l 1 crJ
ka0 sin Û2(tcr)(2k - lJcos^Ç^) _ ^ fe_2
(B.3)
2cosû2(tcr ) + 2 v cry
k sin i)2(tcr)((k - 1) cos i?2(icr) + k - 7) ^ 2
^--costf2(icr) + 1 +
| 2M2(tcr) (cos2 ^(¿cr) + cos tf2(tcr) + 3) ,
H „„„„Q U \ , 1-^ ~ tcr ) -
cos $2(tcr ) + 1
os ^ (t.. ) - 9Ï k_2
(t - tcr) .
ka0(2k - 1) (cos2 û2(tcr) + 4cos û2(tcr) - 9) k_2
2cos $2(tcr) + 2 v cr
Analysis of the expressions (B.2) and (B.3) shows that, under the conditions considered in this part of the proof, at least one component of the constraint reactions Ni and velocity v2 increase without bound as t approaches tcr. Under such controls, one or several wheel pairs begin to slip, and the nonholonomic model becomes inapplicable.
Examples of dependences of constraint reactions on time under controls satisfying the conditions of the case at hand,
= n +1 - 1, = sint (B.4a)
and
= n + t - 1, $2 = 0, (B.46)
are shown in Figs. 7a and 7b, respectively.
B.2. Proof of Proposition 1. Case II. The angles take simultaneously the value n
Let 0 1(t) and $2(t) take simultaneously a value n at time t = tcr < m such that the expansion of the functions in a neighborhood of point tcr takes the form
1 / dk û (t)
tf i(i) = sgnC^Ovr + cH(t - tcr)k + ..., a1 = — '
k!\ dtk
1 ( d""d (t)
d2(t) = sgn(>d 2(tcr))ir + a2(t - tcr)n + ..., a2= - I 2V '
n! V dtn
= 0,
t=tcr (B.5)
= 0,
t-tcr
Fig. 7. Dependence of constraint reactions on time for j = 3, tcr = 1 and (a) controls (B.4a), (b) controls (B.45)
where k > 0 and n > 0 are the orders of the first nonzero derivatives of §1(t) and §2(t) at point t = tcr, respectively. Then, as t ^ tcr,
a2k(t - tcr)n~l - a^nit - tcr)k~l / 0, ax = a,2, k = n,
Vo ^ —
'2 ~ n (+-+ \k
a?(t - tcr)k + a2(t - tcr)n otherwise,
k + n
w ~--;-. i I -|-:--—r~r = zhoo.
ai(t - icr) + a2(t - tcr)n+l
(B.6)
Substituting the expansions (B.5) into (A.2), we obtain an expansion of the reaction forces near t = tcr. The leading terms of these expansions have the form
N1 & a? (4k2 + 3kn - 2k - n) (t - tcr)2n-k-2 - a32 (3kn + 4n2 - k - 2n) (t - tcr)2k-n-2
"cr/ 2 I "" "'"J "cr)
cr)-k-2 - 2a2(n + k)(k - 2n)(t - tcr)-n-2
-2a1(n + k)(2k - n)(t - tcr)-k-2 - 2a2(n + k)(k - 2n)(t - tcr)" N2 & 3a?a2(n - 1 + k)(k - n)(t - tcr)n-k-2 - 3aia2(n - 1 + k)(k - n)(t - tcr)-n+k-2+
cr
+a4 (2k2 + 6kn + 4n2 + k - 2^ (t - tcr)-2 + a4 (4k2 + 6kn + 2n2 - 2k + n) (t - tcr)-2, ^ , , i^n tcr)n-2k-2 + 24aia2(k + 1)(n + k)(t - tcr"k-2n-2
N3 & — 24a2a2(n + 1)(n + k)(t — t^)ra"2k"2 + 24a1 a2(k + 1)(n + k)(t — tcr)k —24a1 (k + 1)(n + k)(t — tcr )-k-2 + 24a2(n + 1)(n + k)(t — tcr )-n-2 —2afn(1 — n + 3k)(t — tcr )k-2 — 2a2k(k — 3n — 1)(t — tcr )n-2.
(B.7)
Analysis of the expressions (B.6) and (B.7) shows that, under the conditions considered in this part of the proof, all three reaction forces Ni and velocities v2, w also increase without bound, as t approaches tcr, for almost all parameters ai, k, n. An exception is the case a1 = a2, k = n, in which N1 and v2 remain finite and the other two reaction forces and w increase without bound. Consequently, under the conditions of the case considered, a lateral slip of the wheels occurs, which is beyond the scope of the nonholonomic model.
Examples of dependences of the constraint reactions on time under controls satisfying the conditions of the case at hand,
$1 = n + 2(t - 1), $2 = n + (t - l)2
and
$1 = n + (t - l)2, $2 = -n + (t - l)2, are shown in Figs. 8a and 8b, respectively.
(B.8a) (B.8b)
200
100
-100
-200J
kN 1
J 1
1 , CT
0.6 № i * i
i 1 1 1
3\ \ 1 1 1 i
200
100
-100
-200
0.2 0.4 0.6 0.S
n2 1
JV,
(a)
(b)
Fig. 8. Dependence of constraint reactions on time for j = 3, tcr = 1 and (a) controls (B.8a), (b) controls (B.8b)
B.3. Proof of Proposition 2. The equation •&1 = —'&2 is satisfied for •&1 = n
Let $1(t) and $2(t) be such that the equation $1 = -$2 ($1 = n) is satisfied at time t = = tcr < to. The expansions of the functions $1(t) + $2(t) and $1(t) - $2(t) in a neighborhood of point tcr take the form
$1(t)+ $2(t) = a12 (t - tcr )k + ..., a12
1 (dk ($1 (t) + $2 (t))
k!
dtk
= 0,
t=tc
$1(t) - $2(t) = $12 + b12(t - tcr)" + ..., &12 =
i (d-i^^- um
nl V dtn J
(B.9)
= 0,
t=tc
where k > 0 and n > 0 are the orders of the first nonzero derivative at point t = tcr. Then, as t y tcr,
b12n<t_tcr)n-k-l
a
12
W
b12n
■ tan
12
$
12
( t - t cr )
n—k— 1
{±TO, n ^ k, const, n > k, ±to, n ^ k, $12 = 0, const, n > k or n ^ k, $12 = 0.
(B.10)
v
2
2
As in the preceding cases, we obtain expansions for the constraint reaction (A.2) by substituting (B.9) into them. The explicit form of the leading terms of the expansion is
* « ccs(jf,)+1 H -+« m+
+2(k - n + 1) cos (^Al^j _k + 2n -2^j(t- tcr)n~2, N•2 « -2rM(k - n + 1) (2 cos (tZ&rTj + sin (t - tcr)n-k-'2+
+ ^2(7k - lOn + 7) cos2 (^M^j + 2(5n - 2k - 2) cos (^l&l
-7k + 7n - 7 (t - tcr)
"cr )
(B.11)
«12 (tcr )
nb12 Sin J (_ ( i)V2{tcr)
412
* - ,,, l ¡m : 0 ^ nr>+«><*--+iKi -
f_ o„ , rns2 ( #12(tcr)\ , n y,, _ 1 , 1 fÉiai^çr) , _
(6fc - 3n + 6) cos2 ( cr' ) + (17k - 19n + 17) cos -13k + 11n - 13)(t - tcr)2n-k-2.
12 2
Analysis of the expressions (B.10) and (B.11) shows that in the case n ^ k, under the conditions of this part of the proof, regardless of the value of §12, all three reaction forces Ni and at least velocity v2 increase without bound as t approaches tcr. That is, in this case (just as in the previous two cases) a lateral slip of the wheels occurs during the motion. Examples of dependences of constraint reactions on time under the controls
§1 = 2(t — 1) — 1, §2 = t (k = n = 1, w(tcr) = to) (B.12a)
and
§1 = 2(t — 1), §2 = t — 1 (k = n = 1, w(tcr) = const), (B.12b)
which satisfy the conditions of the case at hand for n ^ k, are shown in Figs. 9a and 9b, respectively.
The analysis of (B.10) and (B.11) also implies that in the case n > k the velocities and reaction forces Ni remain finite despite the fact that the linear (in velocities) parts of the constraint equations (2.1) are dependent. An example of dependence of the constraint reactions in the case n > k under the controls
§1 = t(t — 1), §2 = —(t — 1)(t — 2) (k = 1, n = 2) (B.13a)
is shown in Fig. 10a.
A separate analysis should be made in the case of constant difference between the angles §1 and §2. Such a dependence of the angles §1 and §2 can be regarded as the limiting case n ^^ under the conditions of (B.9). It is easy to verify that in this case the constraint reactions remain finite.
2
Fig. 9. Dependence of constraint reactions on time for k = n = 1, j = 3, tcr = 1 and (a) controls (B.12a), (b) controls (B.125)
The corresponding dependence of constraint reactions under the controls
= t - 1, = t - 1 (k = 1, n
is shown in Fig. 10b.
40-
20
-20
-40
h N
___Ni N2 ¿cr
¡¡rff 0:4 1 t
/ * 1 1 1
(a)
10
5
-5
-10J
(b)
(B.136)
kN
tcr
0.2 0.y 0!6 0!8 /
N1/ N2/ 1
Fig. 10. Dependence of constraint reactions on time for j = 3, tcr = 1 and (a) controls (B.13a), (b) controls (B.136)
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