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1Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 803-815. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221219
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70B10, 70B15, 70E18, 70E55, 70E60, 70F25
Comparative Analysis of the Dynamics of a Spherical Robot with a Balanced Internal Platform Taking into Account Different Models of Contact Friction
G. R. Saypulaev, B. I. Adamov, A. I. Kobrin
The subject of this paper is a spherical robot with an internal platform with four classic-type omniwheels. The motion of the spherical robot on a horizontal surface is considered and its kinematics is described. The aim of the research is to study the dynamics of the spherical robot with different levels of detailing of the contact friction model. Nonholonomic models of the dynamics of the robot with different levels of detailing of the contact friction model are constructed. The programmed control of the motion of the spherical robot using elementary maneuvers is proposed. A simulation of motion is carried out and the efficiency of the proposed control is confirmed. It is shown that, at low speeds of motion of the spherical robot, it is allowed to use a model obtained under the assumption of no slipping between the sphere and the floor. The influence of the contact friction model at high-speed motions of the spherical robot on its dynamics under programmed control is demonstrated. This influence leads to the need to develop more accurate models of the motion of a spherical robot and its contact interaction with the supporting surface in order to increase the accuracy of motion control based on these models.
Keywords: spherical robot, dynamics model, kinematics model, omniwheel, omniplatform, multicomponent friction
Received August 05, 2022 Accepted November 30, 2022
This work is supported by the Russian Science Foundation under grant 22-21-00831.
Gasan R. Saypulaev
saypulaevgr@mail.ru,bot05_00@mail.ru
Boris I. Adamov adamoff.b@yandex.ru
Alexander I. Kobrin kobrinai@yandex.ru
National Research University "MPEI"
ul. Krasnokazarmennaya 14, Moscow, 111250 Russia
1. Introduction
In recent years, research on mobile wheeled robots that are capable of omnidirectional motion has become popular. One of the most widespread ways to achieve the possibility of omnidirectional motion is to use spherical or omnidirectional wheels [1]. In this paper, we will consider a mobile wheeled robot in the form of a sphere.
In the development and study of the mechanics of spherical robots, an important role is played by issues related to the solution of the problems of motion control and navigation of such robots. An overview of the principles of control of spherical robots is presented in more detail in [2, 3]. One of the most common control principles is associated with changing the center of mass of a spherical robot (for example, using a pendulum located inside the sphere), the creation of gyrostatic moments (for example, the rotation of rotors fixed inside the sphere), and equipping with internal omnidirectional platforms (omniplatforms) with omniwheels.
Studies of the dynamics and control of spherical robots are closely related to the research on the ball's motion and the classical problems of billiards or the Chaplygin ball [4-8]. In these studies, the dynamics of the ball is described using the methods of nonholonomic mechanics (under the conditions of no-slip contact points with the floor or the absence of spinning). Motion on a plane, a cylindrical surface and over an ellipsoid is considered. The papers [7, 8] discuss the controllability of the Chaplygin ball using three gyrostats in two cases: under the condition that the ball rolls without slipping at the contact point, and in the presence of friction forces. In the latter case, the viscous friction model and the Coulomb dry friction model are considered.
In [9, 10], for a ball with an internal pendulum, a feedback motion control law is proposed. Limitations are shown when using such a control, due to technological difficulties involved in manufacturing a spherical robot with a pendulum inside.
In this regard, a promising direction in the development of spherical robots is the creation of spherical robots driven by internal omniplatforms [11-14]. In these articles, a spherical robot with an internal platform equipped with three Mecanum wheels is considered (see Fig. 1). The work [11] presents a kinematic model that can be used to calculate the angular velocities of the platform wheels for rectilinear trajectories at arbitrary speeds, and for curvilinear trajectories, only at low speeds.
Fig. 1. Spherical robot with an internal platform having three Mecanum wheels [11]
Another avenue for further research is to develop a dynamic model which takes into account the slippage of the omniwheel rollers, the characteristics of the floor, and the design of a control algorithm based on this model.
It was shown in [12] that the displacement of the center of mass of the internal omniplatform leads to a significant deviation from a given trajectory in the process of motion. The paper [13] presents the equations of dynamics of a spherical robot with an internal omniplatform, taking into account the displacement of the center of mass of the platform, described within the framework of nonholonomic mechanics, and develops a trajectory tracking algorithm. However, when using the proposed algorithm, after the rolling along the given trajectory is completed and the control is turned off, the spherical robot continues free motion, which in the general case is chaotic. To eliminate this shortcoming, it is proposed to use elementary maneuvers (gates) that allow the robot to switch from one stationary motion to another.
A multicomponent friction model can be used for describing the contact interaction forces of a spherical robot with a supporting surface (floor), taking into account sliding, rolling and spinning of the sphere. For example, in [15-17] the motion of a ball on a horizontal plane is described using the multicomponent friction model.
In this paper, the subject of research is a spherical robot with an internal platform equipped with four classic-type omniwheels. The aim of the research is to study the dynamics of a spherical robot with different levels of detailing of the contact friction model. To achieve the aim of the study, dynamic equations with various detailing of the contact friction model are constructed and program control laws for a spherical robot with an internal platform equipped with four omniwheels are obtained.
In the case of using an internal platform with three Mecanum wheels, the accuracy of the spherical robot may be reduced due to the influence of the real design of the Mecanum wheels, including the displacement of the contact point of the rollers along the axis of the wheels. And the use of classic omniwheels for the inner platform can help to avoid these effects, because the point of contact of the omniwheels does not move along the wheel axis and the simplified non-holonomic model of the omniwheels is closer to the real design of the classic-type omniwheels. However, a more detailed description of the advantages and disadvantages of using Mecanum wheels or omniwheels for the internal platform requires additional research, which will be carried out in future works.
The motivation for this study stems from the use of spherical mobile robots in observation, environmental monitoring, patrolling, underwater and planetary research.
It is expected that the contact friction model, which takes into account the sliding, rolling and spinning of the sphere, will significantly affect the dynamics of the robot.
2. Description of the spherical robot
2.1. Robot design and assumptions
In this paper, we consider a spherical robot with an internal omnidirectional platform (Fig. 2). In contrast to the known designs for the spherical robot, the omniplatform of the robot is equipped with four symmetrically arranged omniwheels of the classical type.
In describing the motion of the spherical robot, the following assumptions are made:
• the centers of mass of the spherical shell and the internal omniplatform coincide and lie at the geometric center of the sphere C;
• depending on the level of detailing of the contact interaction, the contact of the sphere with the supporting surface is a point contact or is represented by a contact area with the center at point P;
X
<P
(a)
pi ///// (b)
Fig. 2. Kinematic scheme of the spherical robot with an internal platform having four omniwheels: (a) side view; (b) front view; (c) top view
• the omniplatform performs translational motion [11, 12, 14];
• a simplified model of omniwheels [18] is used and there is no slippage at the contact points of the omniwheel rollers along their axes with a spherical shell.
The use of a simplified contact model of the omniwheels with a spherical shell is associated with a significantly smaller size of the contact patch of the omniwheels with the shell compared to the size of the contact patch of the spherical shell with the floor. Therefore, a more detailed friction model is used to describe the contact of a spherical shell with a supporting surface. As a direction for further research, it will be interesting to consider a detailed interaction model for describing the contact of omniwheels with a spherical shell.
To describe the motion, a moving coordinate frame CXYZ of the spherical robot with its center at the geometric center C of the sphere is introduced, and a fixed coordinate frame xyz, which differs by rotation by an angle ^ relative to the vertical z-axis.
2.2. Kinematic model
Let us find expressions that relate the velocities of the platform to the rotation velocities of the wheels. A vector equation that expresses the condition of no-slip of the contacting rollers of the ith omniwheel (i = 1,4) relative to the spherical shell is used:
(up x rco. + vi x ra.K. — vs x Tck) • ei = 0, (2.1)
where the following notation is introduced:
• vp = 0, vS = (&sx, wsy, wsy)T, vi are the angular velocity vectors of the omniplatform, the spherical shell and the fth omniwheel, respectively;
• rCc = rCK — rC k is the radius vector connecting the geometric center of the sphere and
i i i i
the center of mass of the fth omniwheel Ci;
• rc k is the radius vector connecting the point Ci and the point of contact Ki of the fth
i i
omniwheel with the sphere;
• rCK = (jf3-^) rc.K. is the radius vector connecting the geometric center of the sphere and the point Ki;
• ei is the unit vector of the axis of the contact roller of the fth omniwheel;
• RW is the radius of the omniwheel;
• RS is the radius of the spherical shell;
The projections of these vectors on the moving axes CXYZ have the form
^S = (uSX, uSY, uSY)T, VC = (VX, VY, 0)T,
= (p1 cos a, 0, Pp1 sin a)T, e1 = (0, —1, 0)T, rCiK = RW(sin a, 0, — cos a)T,
u2 = (0, p2 cos a, p2 sin a)T, e2 = (—1, 0, 0)T, rC K = RW(0, sin a, — cos a)T, (2.2)
u3 = (—pi3 cos a, 0, p3 sin a)T, e3 = (0, 1, 0)T, rCiiK3 = RW(— sin a, 0, — cos a)T,
= (0, —p4 cos a, p4 sin a)T, e4 = (1, 0, 0)T, rC = RW(0, — sin a, — cos a)T
where a =
Z fCZ, rCC.
is the angle between the planes of the omniwheel and the vertical
axis CZ; <pi is the angle of rotation of the ith omniwheel (% = 1, 4) relative to the platform.
Taking into account expressions (2.2), Eqs. (2.1) referred to the moving axes CXYZ can be written as
<¿1 = TT-(usxcosa + ujsz sin a), <p2 = -^-(ujSYcosa +ujsz sin a),
Rw rw (2.3)
RS RS
<p3 = ——(-U)SX cos a + UJSZ sin a), <p4 = —— (-u)SY cos a + wsz sin a).
RW RW
Considering the nonholonomic model of the motion of the spherical shell, the following no-slip conditions for the contact point of the sphere with the supporting surface are used:
vx = rs usy , vy = —rs^sx. (2.4)
The resulting kinematic equations (2.3) and (2.4) are used to describe the model of the spherical robot dynamics.
3. Nonholonomic dynamic models of the spherical robot
Consider two cases of describing the dynamics of the spherical robot: the case of the sphere moving without slipping at the point of contact with the supporting surface, and the case where sliding, spinning and rolling during the motion of the spherical robot are taken into account. To obtain the equations of dynamics of the spherical robot, the Appel formalism [19] is used.
3.1. Model of dynamics of the spherical robot taking into account sliding, spinning and rolling friction in the contact spot
The Appel equation of the robot motion has the form
I-"-
where n is the vector of the generalized forces, n = (VX, VY, USX, USY, USY)T is the quasi-acceleration vector, and the expression for the acceleration energy of the system can be calculated
as the sum of the acceleration energies of the spherical shell, the internal platform and four omniwheels, according to the formula
s = -js [(w^y - (jjsz(jjsy)2 + (w^y + ujsz(jjsx)2 + u)gz] +
+ \jw (<pi + <tl + $ + + \rnr [(Fy - uszvy)2 + (vy + ujszvx)2], (3.2)
where mR = mS + mP + 4mW is the total mass of the sphere, the platform and four omniwheels, Js is the moment of inertia of the spherical shell, Jw is the moment of inertia of the omniwheel around its own axis.
Taking into account the equations of the nonholonomic constraints (2.3), we can write the acceleration energy of the spherical robot (3.2) in the form
s = -js - uszusy)2 + (¿sy + uszusx)2 + usz] +
1
H—mR 2 R
(vx - ^szvy)2 + (vy + uszvx)2
+
1 ( R \ 2
+ Jw (j^-j [2 (üsx + 4 Y) cos2 a + 4üj2sz sin2 a]. (3.3)
The following equation for the power of the active forces is used to find the generalized forces:
Na = E (Mi - nw+ FXr(Vx - Rs^sy) + Ffr(VY + Rs^sx) +
i=1
+ MfruSX + +MfruSY + M fruSZ, (3.4)
where Mi are the control torques produced by the omniwheel drives, ¡iW is the coefficient of the linear friction in the platform-wheel joints, , Ffr are the projections of the sliding friction force on the axes CX and CY, Mx , Mfr are the projections of the moment of rolling friction on the axes CX and CY, Mfr is the moment of spinning friction.
Now we present the generalized forces taking into account the expressions for the rotation velocities of the omniwheels (2.3):
nyx = FXr, nVy = Ffr
il,v = m? + f{:rs + - M3) cos « - 2^e|cos2%y,
RW RW
T-r „ rfr RS * 2aW RS cos2 a (3.5) IL,V = M{: - f{ R s + tt-(M2 - M4) cos a - -i-^s-^
RW RW
fr Rq , , 4uwR2 sin2 a n = mfzr + -^(m, + M2 + m, + M4) sin a - J-vls-Usz.
SZ RW RW
Here, the reduction of friction coefficients to pseudovelocities is performed similarly to how it was done in [20].
4
Differentiating the energy of accelerations (3.3) taking into account expressions (3.5), we obtain equations of dynamics in the form
mR(Vx - wSZVY) = Fxi mR(VY + wSZVX) = FY >
+ 2jw^f cos2 a^j (usx - ujszujsy) = ,\/(r + FpRg-
2aWRS cos2 a RS ,
uSx + ~fr-(Mi ~ Ms)cos <*>
- 2 Sx
RW RW
+ ZJw-fif COs2 a) ^SY + Usz^sx) = MY - FXRS- (3 6)
2uW RS cos2 a „ „■ x ""iJLl§-+ TT^ - M4) cos a,
RW RW
, t RS 2 \ ,,fr 4uW RS sin2 a
Js + sm2 a usz = M|r - -usz+
RW RW
R
+jt^(M1 + M2 + M3 + M4) sin a.
RW
In the case where the sliding, rolling and spinning of the spherical shell relative to the floor (contact patch) are taken into account, the motion of the spherical robot can be described using a system of five dynamics equations (3.6) and four kinematic equations (2.3). In this case, to close the system of equations (3.6), it is necessary to redefine the friction model.
3.2. Model of the dynamics of the spherical robot obtained under the
condition that there is no slipping of the sphere at the contact point
In the absence of slipping at the contact point of the sphere with the floor (2.4), the expression for the acceleration energy (3.3) can be written as
S = t ( mR + ^f R
(Vx - ^szVy)2 + (Vy + wszVx)2
, JS -2
+ yUsz+
+
2 ft'2 2RW
2 ( VX + VY ) cos2 a + 4wSZ sin2 a
(3.7)
Now we present the generalized forces taking into account the expressions for the rotation velocities of the omniwheels (2.3):
n
v.
* R
1 2nW cos2 a^T (M2 - M4) cos a - -\ x-
W
2
RW
n
V
^ R
1 2nW cos2 a^T (M3 - Mj) cos a - -Vr>
W
2
RW
n.
°SZ R
„,-x 4uW RS sin2 a
s (M1 + M2 + M3 + M4) sin a - hw S-U)SZ.
W
2
RW
(3.8)
Differentiating the energy of accelerations (3.3) and taking into account expressions (3.5), we obtain the equations of dynamics in the form
Js\ /t> t^ 2Jw cos2 a • 2^W cos2 a
cos a
'"/,- + $ )(vx - UszVy) + ; + hWT ~VX = ^(M2 - M4),
Rs' RW RW RW
Js \ ,T> 2JW cos2 a • 2jj,W cos2 a
cos a
'"/,- + 777 (VY + WszVx) + " ; Vy + hWT ~Vy = ^(M3 - A/,). (3.9)
RC / RW RW RT,
s/ RW RW W
/ R% 2 \ 4uwR% sin2 a RQ , Js + 4 Jw-J- sin2 a)usz + jljl1i-ujsz = + M2 + M3 + M4) sin a.
\ / RW RW
According to the model obtained, it is clear that the motion of the spherical robot can be described by a system of three dynamic equations (3.9) and six kinematic equations (2.3), (2.4) in the case of the absence of slipping.
4. Control of the motion of the spherical robot
To perform elementary maneuvers, the control torques can be calculated in the form of program control. For this, it is convenient to use the dynamics model (3.9) after rewriting them
as
R R R
Mo - MA = Mo - M, = ——Fy, M, + Mo + Mo + M4 =-^—My. (4.1)
cos a cos a Rs sin a
Here FX, FY, MZ denote the left-hand sides of the equations of dynamics (3.9), which are functions of velocities VX, VY, wsz, which are given as functions of time for the programmed motion.
Using the pseudoinverse matrix method, the system (4.1) is solved with respect to the control torques:
RW ( 2 p , 1 ^ \ _RW ( 2 - 1
Mi = —!z---Fr +-M7 , M, — -Fx +-My
4 V cos a Rs sin a ! 4 Vcos a Rs sin a ,
) S \ / S \ (4.2)
M, = ^ (—Fy +---My] , M4 = ^ (--—Fx +---My
4 \cos a Rs sin a J 4 \ cos a Rs sin a
To evaluate the performance of the control obtained, we numerically integrate the equations of dynamics (3.6) and (3.9) for one of the motions of the spherical robot with control actions given by formulas (4.2). In this case, for the model described by equations (3.6), as a model of contact interaction, we consider the model of multicomponent friction [21], which takes into
account the sliding and spinning of the spherical shell (yMx = Mfr = :
Ffx = ~fm,Rg-
sJv2x + V2Y + s+{i-)x\uSz\
F? = -fmsg . ^-,
sJv2x + V2Y + e+(^)X\^sz I (4-3)
Mf = -fm,RgX XUJsz-,
^V2x + V2y + S+(^)x\Usz\
VPX = VX - RswsY> VPY = VY + Rswsx. RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(5), 803 815_
Here e is a small parameter introduced to regularize expressions (4.3) near zero, % is the radius of the contact spot, and f is the coefficient of friction. The adequacy of this model of sliding and spinning friction was verified experimentally in [22, 23].
5. Simulation results
Let us simulate the motion of the spherical robot with the found program control. As an example of motion, consider a programmed motion along a straight line along the CX axis at the constant speed V. For comparison, we consider the motion with low (V = 0.6 m/s) and high (V = 3.0 m/s) values of speed.
During the simulation, we numerically integrate the equations of motion for the model (3.9), which does not take into account slippage at the point contact of the sphere with the support surface, and the model (3.6), which takes into account the sliding and spinning of the sphere, and control torques (4.2). The following numerical values of the other parameters are used:
mR = 7.8 kg, JS = 0.144 kg • m2, RS = 0.2 m, f = 0.1, RW = 0.04 m, a = 45°, JW = 0.25 • 10"3 kg • m2, % = 0.03 m, /j.W = 0.05 N • m • s,e = 10"8 m2/s2.
The simulation results are shown in Fig. 3 (for the case of low speed) and in Fig. 4 (for the case of high speed).
Vx, m/s
0.6 0.5 0.4 0.3 0.2 0.1
2 4 6 8 10 12 14
(a) Dependencies VX (t)
UJ
SY
, rad/s
t, s
2 4 6 8 10 12 14
(b) Dependencies uSY(t)
t, s
Fig. 3. Results of modeling Eqs. (3.9) (gray solid line) and Eqs. (3.6) (black dashed line) with control torques calculated at low program speed
2 4 6 8 10 12 14
(a) Dependencies VX ( t)
LJ
SY
, rad/s
t, s
2 4 6 8 10 12 14
(b) Dependencies uSY(t)
t, s
Fig. 4. Results of modeling Eqs. (3.9) (gray solid line) and Eqs. (3.6) (black dashed line) with control torques calculated at high program speed
It can be seen from the graphs that the found program control provides programmed motion at the end of transients. Based on the simulation results, it can be concluded that at low speeds of the spherical robot (see Fig. 3) it is allowed to use model (3.9), obtained under the condition of non-slippage of the sphere and under the assumption of point contact; and at high speeds of motion of the spherical robot (see Fig. 4), the influence of the model of contact friction of the spherical robot with the supporting surface is manifested. This conclusion is consistent with the results obtained in [10] when considering the spherical robot with an internal platform equipped with three mecanum wheels.
Let us look at the influence of the contact friction model on the accuracy of motion along the trajectory (using the example of motion along a rhombus), at different values of program speeds (for example, 0.111 m/s, 0.444 m/s and 1.777 m/s), but with the same time of one round of the trajectory (for example, 10 s). The simulation results are shown in Figs. 5 and 6.
Vx,
0.3 0.2 0.1
i/s
-0.1 -0.2 -0.3
10
15
20
25
30
t, s
(a) Dependencies VX (t) uiSY, rad/s
-5
10
20
(c) Dependencies uSY(t)
VY, m/s
0.3 0.2 0.1
-0.1 -0.2 -0.3
5 2jb 25 3
t, s
(b) Dependencies VY (t)
Jsx
, rad/s
-— t, s 30
-5
/ V '''
^ I 1 fv___Jj 5 2 h <y 5 3
t, s
(d) Dependencies uSX (t)
Fig. 5. Results of modeling Eqs. (3.9) (gray solid line) and Eqs. (3.6) (black dashed line) with control torques calculated at program speed 0.444 m/s for motion along a rhombus
Figure 5 shows that the influence of the contact friction model leads to deviations of the longitudinal and transverse velocities from the program values in the model (3.6). And in the model (3.9) the trajectory of the geometric center of the spherical robot corresponds to the programmed motion. It can be noted that, with an increase in the programm speed of the spherical robot for the model (3.6), which takes into account the multicomponent friction of the spherical shell with the support surface, the deviations of the robot's trajectory from the programmed motion increase.
A quantitative assessment of the deviations of the coordinates of the spherical robot from the program trajectory in the model (3.6) depending on the value of the program speed V is given in Table 1. The following designations are introduced in the table:
ex , ey are deviations
of the coordinates of the geometric center of the spherical shells from programmed motion,
yc> m
(a) Program speed modulus is (b) Program speed modulus is (c) Program speed modulus is equal 0.111 m/s equal 0.444 m/s equal 1.777 m/s
Fig. 6. Results of modeling Eqs. (3.9) (gray solid line) and Eqs. (3.6) (black dashed line) with control torques calculated at different program speeds for motion along a rhombus
Table 1. Deviations of the coordinates of the geometric center of the sphere from the program trajectory at different program speeds
V, m/s (e*c)' 111 (evc)'m (ller - (er)l|co), 111
0.111 0.005 0.004 0.012
0.444 0.106 0.039 0.124
1.777 1.643 0.773 1.617
er = yeXc, eycj is the vector of coordinate deviations, (x) is the mean value of x, and ||a||^ =
= max ¡a] is the ^-norm of an arbitrary vector a. It is shown that the average deviations of the coordinates of the geometric center increase nonlinearly with an increase in the program speed for the programmed motion considered.
6. Conclusion
This paper proposes a design of the spherical robot with an internal platform equipped with four classic-type omniwheels. The motion of the spherical robot on a horizontal surface is considered. Dynamic models of the spherical robot with different levels of detailing of the contact friction model are constructed.
A program control of the motion of the spherical robot using elementary maneuvers is proposed. A simulation of motion is carried out and the operability of the proposed control is confirmed. It is shown that at low speeds of the spherical robot it is allowed to use the model obtained under the condition that there is no slippage between the sphere and the surface. The influence of the model of contact friction at high speeds of the spherical robot on its dynamics under program control is demonstrated. This influence leads to the need to refine the models of the motion of the spherical robot and its contact interaction with the supporting surface for the subsequent synthesis of control, which provides a higher accuracy of motion.
Further work will take into account the displacement of the center of mass of the internal platform and the construction of the control that compensates for the effect of slippage at the points of contact.
Conflict of interest
The authors declare that they have no conflict of interest.
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