Control of a Spherical Robot with a Nonholonomic Omniwheel Hinge Inside Текст научной статьи по специальности «Медицинские технологии»

Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, pp. 179-193. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd240303

NONLINEAR ENGINEERING AND ROBOTICS

MSC 2010: 37N35, 70E60

Control of a Spherical Robot with a Nonholonomic

Omniwheel Hinge Inside

E. A. Mikishanina

This study investigates the rolling along the horizontal plane of two coupled rigid bodies: a spherical shell and a dynamically asymmetric rigid body which rotates around the geometric center of the shell. The inner body is in contact with the shell by means of omniwheels. A complete system of equations of motion for an arbitrary number of omniwheels is constructed. The possibility of controlling the motion of this mechanical system along a given trajectory by controlling the angular velocities of omniwheels is investigated. The cases of two omniwheels and three omniwheels are studied in detail. It is shown that two omniwheels are not enough to control along any given curve. It is necessary to have three or more omniwheels. The quaternion approach is used to study the dynamics of the system.

Keywords: dynamics, control, spherical robot, omniwheel, nonholonomic hinge, quaternion, trajectory

1. Introduction

This work is devoted to the control of a spherical robot consisting of a spherical shell with a balanced rigid body inside, which rotates around the geometric center of the shell. The inner body contacts the spherical shell by means of omniwheels, the rotation of which sets this mechanical system in motion. We call this design an omniwheel implementation of a nonholonomic hinge. If this mechanical system is controllable, we will call it a spherical robot.

The free dynamics of a similar mechanical design, but with the inner body with two "sharp" wheels contacting the shell at only one point, was studied in [1, 2]. The presence of sharp wheels creates a nonholonomic constraint between the inner body and the spherical shell, similar to the Suslov constraint [3], which prohibits relative rotations of the inner body in the direction

Received February 02, 2024 Accepted March 20, 2024

The work was supported by the grant of the Russian Science Foundation (No. 23-21-10019) and the Chuvash Republic (https://rscf.ru/en/project/23-21-10019/).

Evgeniya A. Mikishanina

[email protected]

I.N. Ulianov Chuvash State University

Moskovskii pr. 15, Cheboksary, 428015 Russia

fixed in it. Such an implementation of this constraint was first proposed by Wagner [4]. In the modern scientific literature one can also find a number of works on the dynamics of free systems consisting of a spherical shell with various moving bodies inside: for example, a Lagrange top [5] or a ball [6].

The problem of controlling the spherical robot (spherical shell) using various mechanisms that set it in motion is one of the urgent problems of nonholonomic mechanics and robotics. These are rotary mechanisms [7-12], pendulum mechanisms [13-17], mechanisms with omni-wheels [18-20] and some others [21, 22]. In these works, the equations of motion of the studied mechanical systems are given and control actions for the realization of motion are obtained.

Speaking about the control of the spherical robot, first of all we assume trajectory control, which is carried out by setting two (in the case of two omniwheels) or three (in the case of three omniwheels) servo-constraints [16, 23]. In this paper, trajectory control will be understood as motion control along a given smooth curve with a given velocity, that is, with a given law of motion. Motion control along a given smooth curve can also be the basis of an algorithm for the spherical robot pursuing some moving object [16]. Therefore, the problem of trajectory control of the spherical robot with certain design features is relevant and important.

In the case of controlling the spherical pendulum-type robot, due to the appropriate torque control of an unbalanced axisymmetric pendulum, trajectory control is quite possible. This is confirmed by the results of [13, 16].

In the case of control of a variable gyrostatic momentum due to internal rotors, two rotors are not enough for trajectory controllability for moving the spherical robot along any given trajectory [11]. At least three internal rotors are required.

In this paper, the possibility of controlling the spherical robot by controlling the omniwheels of the inner rigid body when the robot moves along a given trajectory with a given velocity is investigated. At first glance, this mechanical design is similar to that of a spherical shell with a balanced pendulum fixed in the center of mass. However, the presence of controlled omniwheels is a significant difference that affects the dynamics. We are considering the design with two omniwheels located diametrically opposite on the same axis. A similar design with two orthogonal oriented omniwheels was considered in [19], in which mathematical calculations concerning the outer spherical shell were performed in a fixed coordinate system, and calculations concerning the inner body (the spherical shell of a smaller radius) were performed in a moving coordinate system attached to the inner body. This approach somewhat complicates the mathematical calculations. The equations of motion given in [19] do not give an idea of how the control of omniwheels affects the control of the robot itself, and therefore do not give an idea of what structural changes are required to improve the maneuverability of the robot.

In this paper, we consider the design with two equally oriented omniwheels to compare controllability and coordinate the results with the results obtained in [19]. The equations of motion of the entire mechanical structure are obtained only in the moving coordinate system attached to the inner rigid body, which simplifies the solution of the problem. Nonholonomic constraints are imposed on the omniwheels, which do not allow the wheels to slip at the points of contact. Rolling of the spherical shell along the horizontal plane is also carried out without slipping. We believe that, at each moment of time, each omniwheel touches the spherical shell at only one point. It is shown that the arrangement of two omniwheels on the same axis will not allow motion along any given trajectory. The acceleration of the spherical robot in a straight line from rest to a given velocity in a finite period of time is studied in more detail. In this paper, we also consider the case of three omniwheels, when it is possible to realize the motion of the robot along any given trajectory. The analytical results obtained are confirmed by numerical

experiments. To analyze the dynamics of the inner body, we use the quaternion approach [24] and the method of constructing maps over a period in quaternions.

The results obtained may be of practical importance in the development and selection of optimal designs of the spherical robots that meet the tasks facing it. This can be protection of the territory, tracking of some mobile object, participation in land or underwater research, monitoring of the territory, among others.

2. Mathematical model of the spherical robot

2.1. Basic assumptions and notation

Consider the problem of rolling along an absolutely rough plane of a spherical robot, which is a homogeneous spherical shell of radius R, inside which the rigid body moves. The rigid body is in contact with the outer shell by means of N (N ^ 2) identical omniwheels of radius Rk. We assume that the center of mass of the spherical shell and the inner body (together with the omniwheels) coincide and are located in the geometric center of the shell C (Fig. 1).

We introduce two coordinate systems: • a fixed coordinate system OXYZ with an axis OZ perpendicular to the horizontal reference

• a moving coordinate system Cxyz attached to the center of mass C of the inner body, the axes of which coincide with the main axes of inertia of the inner body.

The orientation of the rigid body in space is given by a matrix

whose columns define the projections of the coordinate vectors of the fixed coordinate system a, 3, y in the moving one.

We introduce the following notation:

- (X, Y) are coordinates of the contact point of the spherical robot on the OXY plane;

- V is the velocity vector of the geometric center of the shell in the moving axes;

Fig. 1. Mechanical design

plane;

( ai pi Yi^ Q = «2 P2 Y2 e SO(3),

\a3 p3 Y3)

2

- O is the angular velocity vector of the shell in the moving axes;

- u is the vector of angular velocity of the inner body in the moving axes;

- IqE is the central tensor of inertia of the shell, E is the unit matrix of the third order;

- In = diag(I1, I2, I3) is the central tensor of inertia of the rigid body together with omni-wheels;

- m is the total weight of the spherical robot;

- xi is the angular velocity of the fth omniwheel in rotation relative to the inner body;

- j is the axial moment of inertia of each omniwheel.

To build a mathematical model of this mechanical system and to make the subsequent mathematical calculations clearer, we describe a nonholonomic model of the omniwheel in the spherical shell, the main provisions of which are set out in [25].

2.2. Nonholonomic omniwheel model

The design of the omniwheel is shown in Fig. 2, [25]. We assume that the planes of all omniwheels are parallel to the plane Cxy, the normals to the plane of the omniwheels are parallel to the axis Cz, the radius vector of the center of the fth omniwheel lies in the plane Cxy and is at an angle

2vr • (i - 1) . —-<t>i =-^-L, г = 1, N,

to the positive direction of the axis Cx.

We introduce the following notation: the vector ai is the unit vector directed along the axis of the roller in contact with the shell, the vector ni is the unit vector normal to the plane of the omniwheel, the £ is the angle between the axis of the roller and the normal to the plane of the omniwheel, ri is the radius vector directed from the center of the sphere to the center of the fth omniwheel, si = ri x ai.

Fig. 2. Omniwheel design The specified vectors are set in the coordinate system Cxyz as follows:

((R - Rk) • cos (R - Rk) •sin 0i V 0 )

ai

(— sin 0i • sin cos 0i • sin £ \ cos £

r

ni

M 0

w

^ (R — Rk) • sin $i ■ cos £ ^ —(R — Rk) ■ cos ■ cos £ y (R — Rk) ■ sin £ )

Then the nonholonomic constraints which ensure that the omniwheels do not slip at the contact points have the form [25]

R

(«¿> ni) -Xi = ~7>~(si> n - w). Rk

(2.1)

Remark 1. In the cases of £ = 0 and £ = n, the problem is reduced to the problem of rolling the spherical shell with a classical nonholonomic hinge that is the rigid body in contact with the shell with "sharp" wheels. The dynamics of such a system for two wheels is studied in [1, 26]. Next, we assume that £ = 0 and £ = n.

2.3. Equations of motion

The no-slip constraints at the point of contact of the shell with the reference plane are

V = RO x y. (2.2)

The Lagrangian of the system, which coincides with its kinetic energy, takes the form

N N

2...... 2"u~" ' 2 v~' ' ' T

L = -mV2 + -I0n2 + -(w, I„u;) + -./^ U • ./

i=1

i=1

The equations of motion of the spherical shell with the rigid body inside, which contacts the shell by means of omniwheels, have the form (see [18])

d_ fdL_

dt V du>

dL_ d u

dL dV

()L_ dY

x ^ + t^tt xV + — x 7 + 77- ^ Hi

R_ ~R

N

E

k i=i

d ( dL\

N

dL R _

+ Rk x 7'

dt\d O dO Rk

N / k i=1

d_ ( dL_\ _ dL

dt \dV

(2.3)

dV

x U + K,

d i dL

dL

Y = Y x u,

where L is the Lagrangian of the system, ¡i1, ¡i2, ..., Hn, k1 ,

, K3 are undefined multipliers

that determine the reaction of nonholonomic constraints, Ki is the torque applied to the axis of

the ith omniwheel, s,- = 7—

' i (si > ni)

, K = (k1 , K2, K3)

T

The controlled rotation of the omniwheels drives the inner rigid body relative to the spherical shell, which creates a torque that is transmitted to the sphere

M

R

N

R

(2.4)

k i=1

s

Remark 2. If we express the terms ji from the fourth equation of the system (2.3) taking into account (2.1)

AH = H - w) - j(u>, nj - A'i;

Rk

we get the relationship between the vector M and the control torques on the omniwheels Ki, which is given by the formulas (2.4). Equations (2.4) can be solved with respect to Ki only in the case of two and three omniwheels, and Ki are determined in a unique way. In the case of more than three omniwheels, the control torques on the wheels will not be determined uniquely.

From the third vector equation of the system (2.3), taking into account (2.2), we find

k = mRO x y + mRY x (O x u).

Excluding k from Eqs. (2.3) and making the transformation (2.4), we obtain the closed subsystem in the variables O, u, y :

J(O + u x O) = -M, (^w + (j ^ x^J n j = ^w + (j X^J n j X u + M, (2.5)

Y = Y x u,

where J = (Iq + mR2) E — mR2Y ® Y, n = ni Vi, and the angular velocities ~xi are given by the formulas (2.1).

For 2 ^ N ^ 4

V^ • N-Rfn ^ ¿^Xi = -5—(^3

i=i rk and the system (2.5) is converted to the form

J(O + u x O) = —M, (Inu + aNQ3n)' = (Inu + aNQ3n) x u + M, (2.6)

Y = Y x u,

where ln = diag^, /2, I3 - aN) = diag{Iv /2, 73), aN = Obviously, I3 > 0.

k

To determine the orientation of the spherical robot and its location in absolute space, it is also necessary to add the equations describing the evolution of other coordinate vectors of the fixed coordinate system

a = a x u, ¡3 = /3 x u, (2.7)

and the equations for the trajectory of the contact point

X = R(O, ¡), Y = —R(O, a). (2.8)

Equations (2.8), which represent the standard law of motion of the system along the trajectory, can be considered as servo-constraints that set the robot's motion program.

It is required to investigate the possibility of controlling the spherical robot along a given trajectory with a given velocity. We believe that the further motion of the spherical robot after passing the trajectory is carried out by inertia. For such a control to be possible, it is necessary that the components of the vector M be defined at each moment in time. We take a closer look at the control of the system in the case of two and three omniwheels.

3. Trajectory control with two omniwheels

We assume that the radius vectors of the centers of the omniwheels lie on the axis Cx, that is, 01 = 0, = n. Then

Si = (0, — ctg Z, 1)T, = (0, ctg Z, 1)T.

3.1. The case of ^ f, fe € Z

The vector of torque M takes the form

M = (0, M2, M3)T. The first equation of the system (2.6) is rewritten as

=____M - mE2(M' 7 (31)

We find projections of the vector Eq. (3.1) on the X-axis and Y-axis of the fixed coordinate system

(17, a) + («, a x w) = - (M'

(O, ¡) + (O, / x u) = —

Iq + mR2' (M, ¡)

Iq + mR2'

Taking into account (2.8), the following expressions are true:

(M, a) = vY, (M, ¡) = — vX, v

Iq + mR2

R

from which we define the components of the vector M:

The proposed design of the spherical robot with two omniwheels does not allow for motion along any given trajectory (smooth curve), since it cannot be guaranteed that M2 and M3 are determined at each moment in time due to the fact that 71 can go to zero. A similar result occurs with rotary control of the sphere by the spherical robot with two rotors [11].

In [19], it was stated that, if the axis connecting the geometric centers of the wheels never occupies a horizontal position, then it is possible to ensure motion along a given smooth curve. According to the expressions (3.2), this statement is equivalent to the condition

Yi = 0.

However, it is impossible to guarantee the fulfillment of this condition when the spherical robot moves along an arbitrary predetermined trajectory, which is confirmed by a large number of numerical experiments.

If the centers of mass are not located on the axis Cx, that is, 01 = = 0, = + n, the vector M has the form

M = (M1, M2, M3)T,

where the components M1 = -Rs'n^'o ct.g^(a*i — /¿2), M2 = _-Rc°s^o ct.g^(/x1—/x2) are proportional.

fc fc This property again prevents the robot from moving along any given trajectory.

However, we can talk about control in the sense of Rashevsky-Zhou, when the robot needs

to be moved from one given point in absolute space to another given point. Such a motion

can always be performed in a straight line if it is allowed by the initial orientation of the robot

and there are no obstacles along the straight line in which the robot moves. In particular, it is

possible to accelerate the motion of the robot in the straight line to a given velocity for a certain

period of time from the state of rest.

3.2. Acceleration in a straight line from rest to a given velocity

Let the robot be at rest at the initial time, that is,

w(0) = 0, 0(0) = 0.

We assume

X = -kfi2, Y = ka2, k = const = 0.

Then

M1 = 0, M2 = kv, M3 = 0.

For convenience, we choose the coordinate axes so that y2 (0) = 0, which ensures that the condition a2(0)2 + (32(0)2 = 0 is fulfilled. Then the angular velocity vector of the inner body is

f kv xT <*=(0,-t,0

From Eqs. (2.7), we have

a2 (t) = a2 (0) = const, ^(t) = ^2 (0) = const, 72 (t) = 0. Therefore, the angular velocity vector of the spherical shell is

and the coordinates of the contact point are

t2 t2

X = -k/32( 0)-, Y = ka2( 0)-.

Then the angular velocities of the omniwheels are determined as follows:

k f I0 + mR2s

By varying the parameter k, it is possible to accelerate the robot to a given velocity in a limited period of time. Obviously, in order to increase the absolute velocity of the robot, it is necessary to increase the absolute rotation velocity of the omniwheels in direct proportion. Thus, the high-speed capabilities of the robot are limited by the capabilities of the motors rotating the omniwheels.

3.3. The case of £ G {f,

By similar reasoning, the vector M takes the form

M = (0, 0, M3)T.

Multiplying the value of (3.1) by a and f3 and setting Y = ka3, X = -kfi3, a2 + ¡33 = 0, we

M3 = kv.

With initial conditions

Y3 (0) = 0, w(0) = 0, 0(0) = 0, the angular velocity vectors take the form

n = (o,o

V R>3 )

The trajectory of the spherical robot is set as follows:

t2 t2

X = -k/33(0)-, Y = ka3( 0)-.

The angular velocities of the omniwheels are

k ( vR + a2\ ,

= + — J"

4. Trajectory control of the spherical robot by three omniwheels

The ability of the spherical robot to move along a given trajectory is determined by the presence of three independent components of the control torque that are not identically equal to zero. That is, the design of the spherical robot with two omniwheels located diametrically opposite on the same axis significantly limits the maneuverability of the robot. Obviously, greater maneuverability of the spherical robot can be achieved if the inner body is equipped with three omniwheels.

We consider a spherical robot with a nonholonomic three-omniwheel hinge.

Remark 3. When the rollers are positioned at the angle £ g {f, ^f}, the equations of motion (2.6) for the spherical robot with three omniwheels are equivalent to the equations of motion of the spherical robot with two omniwheels with a similar roller arrangement.

Therefore, we consider ( / y, fc £ Z. Then all the components of the torque M are disproportionate in the aggregate and none of the components is identically zero. Two Eqs. (2.8) are not enough to determine them for trajectory control [27]. Another condition needs to be introduced. We set it in the form

(o, y) = 0. (4.1)

This condition is a servo-constraint and specifies the absence of rotation of the spherical shell. Thus, the "rubber" rolling is reproduced not due to the nonholonomic constraint, but due to the introduced servo-constraint (4.1). From (2.8) and (4.1) we have

(M, a) = vY, (M, /3) = -vX, (M, 7) = 0.

The vectors O and M are decomposed according to the varieties of the fixed coordinate system

O = R_1(-Ya + X3), M = vYa - vX/.

Thus, the study of the dynamics of the mechanical system with servo-constraints (2.8) and (4.1) reduces to investigating the system

W = W x u + vYa - vX3,

• (4.2)

a = a x u, /3 = 3 x u Y = 7 x u,

where W = Inu + a3Q3n is the modified angular momentum. Equations (4.2) have six geometric integrals and three integrals that are generalizations of area integrals:

(a, a) = 1, (3, 3) = 1, (7, 7) = 1,

(a, y) = 0, (3, Y) = 0, (a, 3) = 0, (4.3)

(W, a) - vF = Ca, (W, 3) + vX = Cp, (W, 7)= ^.

Then the vector W is also decomposed according to the coordinate vectors of the fixed coordinate system

W = (vr + Ca)a + (-vX + Cp )3 + Cy 7. (4.4)

To reduce the system (4.2) to the fixed level sets of the integrals (4.3), we apply the quaternion approach [24]. The quaternion parameterization of elements from SO(3) has the form

Q

+ - ^2 - A3 2(A0A3 + AiA2) 2(AiA3 - A0A2) ^ 2(AiA2 - A0A3) A0 - A° + A2 - A2 2(A0A1 + A2A3) ^ 2(A0A2 + A1A3) 2(A2A3 - A0A1) A0 - A1 - A2 + A3)

The evolution of quaternions is given by the equations

Ao = -^(w, A), A = + ^A x a>, where A = (A1, A2, A3)T, which have the first integral

(4.5)

A2 + A2 = 1. (4.6)

We investigate the dynamics of the system on the zero level sets of the integrals Ca = Cp = = CY = 0. To do this, we express the vector u from (4.4):

u = vA(Ya - X3),

where A = diag (if1, /2"\ I3l ■ (1 +1)) = diag^, a2, a3), e = jfc.

4.1. Symmetric case

We put a1 = a2 = a3 = a. Then the system (4.5) takes the form

va.

A0 = Y(-Y\1+X\2),

va,

A1 = —(rAo-XA3),

va

A2 = t(-YX3-XXO),

(4.7)

va

A3 = yO^+XAi).

The system (4.7) is a system of linear differential equations with time-dependent coefficients. The solution of such a system is always regular. And if the coefficients are periodic functions of time, then the solution is always quasi-periodic with two frequencies.

In Fig. 3, on the fixed level set of the first integral (4.6) and for the trajectory

X = cos 0.5t(cos 0.5t - 1), Y = sin0.5t(cos 0.5t - 1),

(4.8)

the values of constants a = 2, v = 1 and the zero values of constants Ca, C^, CY, maps for the period of the dynamical system (4.7) in variables (A1, X2, A3) and phase curves in the space of variables (w1, u2, w3) are constructed.

Ua

(a)

(b)

(c)

Fig. 3. Control of the spherical robot with a = 2 and v =1 and given initial conditions Ax(0) = 1, A0(0) = = A2(0) = A3(0) = 0: (a) three-dimensional map for a 4^-period in the space (Ax, A2, A3); (b) phase curves in space (w1, u2, w3); (c) trajectory of the spherical robot

In the symmetric case, the system exhibits quasi-periodic regimes. We study maps for the period in the absence of the indicated symmetry.

4.2. General case

Figures 4-6 show the results of numerical experiments for the trajectory (4.8). The system again demonstrates quasi-periodic modes at sufficiently small values of the parameter v. With an increase in the value of the parameter v, quasi-periodic modes are replaced by chaotic ones. In a physical sense, this means that, if the ratio of the moments of inertia of the spherical shell to the moments of inertia of the inner body is small enough, then the system (4.5) demonstrates quasi-periodic modes, with an increase in this ratio, quasi-periodic modes are replaced by chaotic

(a)

0

(b)

(c)

Fig. 4. Control of the spherical robot with A = (2.5, 2, 1.5), v = 0.2 along the trajectory (4.8): (a) three-dimensional map for 4^-period in space A, A2, A3); (b) projection of the three-dimensional map onto the plane (Ax, A2); (c) graphs Ax, A2, A3 with initial conditions Ax(0) = —0.3, A2(0) = 0.3, A3(0) = 0

(a)

-1

rigl ft PilSSft. Pllli

+ M V Wt&SsB^M-'

0

(b)

(c)

Fig. 5. Control of the spherical robot with A = (2.5, 2,1.5), v = 0.5 along the trajectory (4.8): (a) three-dimensional map for 4^-period in space (AX,A2,A3); (b) projection of the three-dimensional map onto the plane (A1, A2); (c) graphs A1, A2, A3 with initial conditions Ax(0) = —0.3, A2(0) = 0.3, A3(0) = 0

ones. This is confirmed by the results of numerical experiments, as well as by an estimate of the value of the largest Lyapunov exponent, which in the quasi-periodic case is zero, and in the chaotic case takes positive values.

(a) (b) (c)

Fig. 6. Control of the spherical robot with A = (2.5, 2, 1.5), v = 0.6 along the trajectory (4.8): (a) three-dimensional map for 4^-period in space (A1, A2, A3); (b) projection of the three-dimensional map onto the plane (A1, A2); (c) graphs A1, A2, A3 with initial conditions A1(0) = —0.3, A2(0) = 0.3, A3(0) = 0

5. Conclusion

The equations of motion of the spherical robot with a nonholonomic omniwheel hinge inside, the introduced servo-constraints and additional analytical calculations obtained in the work allow us to draw the following conclusions:

- the design of the spherical robot with two omniwheels located on the same diametrical axis does not provide trajectory controllability for motion along any given trajectory;

- the design of the spherical robot with three omniwheels provides trajectory controllability for motion along any given trajectory.

To illustrate the conclusions made, the results of numerical experiments are presented. In the case of three omniwheels, the dynamics of the inner body is investigated within the framework of the quaternion approach. According to the results of numerical integration, the motion of the inner body is accompanied by the presence of quasi-periodic modes, which can be replaced by chaotic modes with a relative increase in the moment of inertia of the spherical shell compared with the moments of inertia of the internal structure.

Acknowledgment

The author thanks Prof. I. S. Mamaev for useful discussions.

Conflict of interest

The author declares that she has no conflict of interest.

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