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0Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 2, pp. 311-336. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd240502
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70E18, 70E55, 70E60, 70Q05
No-Overturn Conditions for Omnivehicle Motion
G. N. Moiseev
We consider the dynamics of an omnidirectional vehicle moving on a perfectly rough horizontal plane. The vehicle has three omniwheels controlled by three direct current motors.
To find out the limits of the scope where the bilateral constraints model is applicable, we study the normal reactions of the vehicle. We present a step-by-step algorithm for finding out reaction components in the case of controlled motion. Based on these results, no-overturn conditions of the vehicle are proposed.
We apply this approach to study a specific model, that of a symmetrical omnivehicle. As a consequence, vehicle design recommendations are proposed.
Keywords: omnidirectional vehicle, omniwheel, Mecanum wheel, reaction force, nonholo-nomic model, unilateral constraints, constraint violation
1. Introduction
Omnidirectional vehicles are a special group of vehicles that can move in an arbitrary direction without turning about a vertical [1]. Such vehicles are useful in the case of operations in a limited space, so they serve in lots of areas as transporters or manipulator bases.
One approach to building an omnidirectional vehicle is to use a platform with omniwheels [1, 2]. In general, an omniwheel consists of a circular basis that carries a number of light rollers on the periphery. The omniwheel model is defined by the angle between the roller axes and the central wheel axis. There are Mecanum (or Swedish) wheels, where this angle is 45° [3]. There are also wheels where this angle is 90° and the roller axes lie in the basis plane, which are usually called just "omniwheels", in the narrow sense [4, 5].
In the papers covering omnivehicle motion and control, the model with the bilateral constraints between the wheels and the surface is widespread [2, 6-25]. The majority of these papers can be divided into two vast groups: those considering motion with friction between the wheels and the surface [13, 19, 23-25] and those dealing with motion on a perfectly rough plane [2, 612, 17, 18, 20-22]. In the first group of papers, in the majority of works reactions are examined
Received February 22, 2024 Accepted April 19, 2024
Georgy N. Moiseev moiseev.georgii@gmail.com
Lomonosov Moscow State University Leninsikie gory 1, Moscow, 119991 Russia
only to build a friction model. Mamaev et al. have also studied the no-slip conditions of the wheel rollers [23]. Saipulaev et al. have studied changes in normal reactions depending on the motion parameters of the four-wheeled Mecanum platform [25]. The paper also describes the limits for the maximum values of the control torques under which the contacting rollers of the Mecanum wheels do not lose contact with the supporting surface. In the second group of papers, while sometimes the constraint reactions are used to derive the dynamic equations [11], the majority of papers use various general purpose methods (Lagrange multipliers' method [2, 12], Chaplygin's equations [17], Ferrers equations [16], Appel-Gibbs method [14, 18, 21], Tatarinov laconic equations [6-8]), so there is no explicit examination of reactions. Borisov et al. have studied motion on a sphere, also with bilateral constraints [16]. The normal reactions in these papers are assumed to be positive for all motions under consideration.
Kosenko, Gerasimov et al. have studied omnivehicle motion on a surface with unilateral constraints [26-28]. The computer modeling system was used to study the motion. In the experiments, the constraints were examined to track the computational error and model consistency.
Following [6, 15, 21, 23], we study the motion on a perfectly rough plane with bilateral constraints. This approach can be applied only in the case of positive normal reactions at the point of contact. If this condition is violated, dynamic equations can no longer be applied to describe the motion, and the vehicle may slip or overturn. This paper studies the normal reactions of the system with bilateral constraints and boundaries in which these constraints can be applied. The results of this paper can be used as a criterion in path planning for the controlled motion, as well as for the vehicle design.
The structure of the paper is as follows. In Section 2, we consider the dynamics of the system in general and derive the reactions. In Section 3, we consider various no-overturn conditions at rest and in motion. The results are summarized in a step-by-step algorithm. In Section 4, we study a specific vehicle model — a symmetrical omnivehicle [2, 6, 15, 21] — by applying the results of Section 3. Some actual motions of this vehicle are studied to illustrate no-overturn conditions. The Appendix A recalls the procedure for deriving the equations for normal reactions.
2. Normal reactions
2.1. Vehicle design
The vehicle consists of a horizontal base platform and three omniwheels (see Fig. 1a) [2]. Each wheel can rotate independently about its horizontal axis. The wheels' planes are vertical.
Let S denote the system's center of mass. Let (x, y) be the coordinates of the center of mass on a supporting plane in the inertial frame. Let Q be some fixed point on the base platform and Pi be the ith wheel center, where i = 1, 2, 3. We will attach the frame Q(nz to the base platform: Qz is vertical, Qn is co-directional to qS, Q( completes the right-handed system. If Q coincides with S, Qn could be chosen arbitrarily. Let ni be co-directional with the ith wheel axis (see Fig. 1b). Let x. be the ith wheel rotation angle. Let the positive direction be counterclockwise from the platform's point of view, so the wheel's relative angular velocity is equal to — x.n. (see Fig. 1c). Let d be a base platform course angle — an angle between Qx and Q(. The platform's angular velocity is dez.
The vehicle geometry is defined by the following parameters: ai is the angle between Q( and QpP. (see Fig. 1a), (ii is the angle between Q( and ni (see Fig. 1b), A = iQ^, 5i = \C——Pi\ (see Fig. la), 7j defines the ith wheel rollers axis angle (see Fig. Id). For example, % = ±f define Mecanum wheels.
Let the mass of the whole system be m and the wheel radius be R. h is the height of the system's center of mass above the support horizontal plane.
Hence, the vehicle consists of 4 rigid bodies and the state of the system is described by 6 generalized coordinates: (x, y, 6, x1, X2, X3).
O x
(a) Vehicle geometry
—^Xi
ni
Mi
(c) Wheel geometry
(b) Wheel position
(d) Rollers geometry
Fig. 1. Omnivehicle
2.2. Dynamic equations
We consider the controlled motion on a perfectly rough horizontal plane. Let the fth wheel be controlled by the torque
(ciUi - c2Xi)(-ni),
where Ui is voltage input, and c1 > 0, c2 > 0 are constants defining the motor's characteristics.
Let the pseudovelocities be
v1 = x cos 6 + y sin 6, v2 = —x sin 6 + y cos 6,
V3 = A6,
(2.1) (2.2) (2.3)
where A2 is the system's moment of inertia about a vertical passing through the center of mass Sez. Essentially, v1 and v2 are projections of the center of mass velocity on the base platform frame axes Q(n and v3 is proportional to the value of the platform's angular velocity.
We model an omniwheel as a rigid disk, assuming that the velocity of its lowest point is perpendicular to the rollers' axis [2, 8]. The constraint equations are
aii =
cos (A +7i)
R sin Yi '
ai2 =
X = -
sin (Pi + 7j)
R sin Yi '
v,
°i3 =
Acos(& + %) + Sj sin(/j- + 7. - q.) AR sin Yi
(2.4)
(2.5)
where v = (v1, v2, v3) are pseudovelocities and x = (Xi, X2, X3) are the wheels' rotation angles (Fig. 1c).
The equations of motion are
( V2V3 \
(M + X
2
m
"=A
—v v
1v3
V
0
+ c1E*U — c2-
iv,
(2.6)
/
where M = diag(m, m, 1) is the 3 x 3 diagonal matrix, A2 is the moment of inertia of each wheel about its proper axis Pini, and U = (U1, U2, U3) are control voltages. The sign □* denotes the transposition of a real matrix. For a step-by-step guide on obtaining Eqs. (2.6), see [6-8, 21].
By solving (2.6) with respect to pseudoaccelerations, we get the following equations:
i> = (STi/) + Wv + £VU,
A
(2.7)
where , , are matrices defined by the following expressions:
( 0 1 o\
= hj II = (M + X2E*E)
-10 0
0 0
0
I j I
2 -1
= —c2 (M + A2S*S)
2 -1
= II cj II = c1 (M + X2 2*2)
(2.8)
(2.9) (2.10)
The superscript □ indicates that the matrices and their components are related to pseudove-locities and pseudoaccelerations.
*
+
+
2.3. Reaction forces
Let us consider a system in which the reactions can be reduced to three reaction forces Ri between the ith wheel and the ground at the points Mi with no reaction torques. Let the components of each reaction force Ri be
R1 = RlS+ R1nen + N1ez ,
R2 = R2£e£ + R2nen + N2ez ,
R3 = R3£e£ + R3V en
We want to study the values of normal reaction forces (N1, N2, N3).
Statement 1. Normal reactions Ni (i = 1, 2, 3) are the solutions of the linear system of equations
(2.11)
AN = (SkV) TÙL + r&Av + tAU + dAg,
where
/ 1 1 1 N1
A = sin a1 — A ô2 sin a2 — A ô3 sin a3 — A , N = 2N to
V " -¿1 cos a1 -ô2cos a2 —ô3 cos a3 ! N3
I 0 0 0 \ / 0 0
(2.12)
Aa =
ij
= - A2
B
A
cos cos (32 cos ysin f sin (32 sin f3J
i 0 0 o\
A
m
\
—mh
— sin ff1 — sin ff2 — sin ff3
cos f 1 cos f 2 cos f 3
fo 0 o\
(
jA.I ij
= A2
0 1 0
x—1 0 0 0 0
Av - h
1 0 0 0 1 0 y
0
\
( 0 0 0\
CA = ||cA-1| = —A2
Cv - mh
cos f1 cos f2 cos f3 — mh
sin f 1 sin f 2 sin f 3
/ 0 0 0 \
cos f 1 cos f 2 cos f 3 sin f 1 sin f 2 sin f 3
/A
dA = ||dA || = m 0 0
0 1 0 —1 0 0y
/ 0 0 0\
0 1 0 —1 0 0y
Bv
(2.13)
(2.14)
(2.15)
(2.16)
The superscript indicates that the matrices are related to the normal reactions system of equations.
Proof. Refer to the Appendix A.
□
Definition 1. A three-wheeled vehicle is singular if all its support points He on the same straight line (Fig. 2) [2, 12, 16].
Statement 2. System (2.11) always has a unique solution for a nonsingular vehicle. Proof. The support point location Sp (i = 1, 2, 3) is as follows:
Sp = qS + Qp = ôi cos aie^ + (ôi sin ai — A)en + (R — h)ez (i = 1, 2, 3).
Three points form a line if vectors — p, — p, ——PS3 are linearly dependent. This condition is equivalent to the following one:
( ôi
det
cos a
ô2 cos a2 ô3 cos a3 \
ô1 sin a1 — A ô2 sin a2 — A ô3 sin a3 — A
V
1
1
1
= 0.
/
On the other hand,
det
cos a
ô1
ô1 sin a1 — A ô2 sin a2 — A ô3 sin a3 — A
ô2 cos a2
ô3 cos a3
V
1
1
1
= - det A.
/
Thus, the condition det A = 0 is equivalent to the linear independence of SP1, SP2, SP3 and the system (2.11) always has a unique solution if the vehicle is nonsingular. □
Further, for any nonsingular vehicle we will assume det A > 0. This condition holds true if the wheels are indexed counterclockwise, so any nonsingular vehicle that does not satisfy this condition can be relabeled to satisfy it.
By solving the linear system (2.11), we obtain normal reaction components. They have the following structure:
N = (?lNv) ^ + <BNu + tNU + DNg (i = 1, 2, 3),
where the matrices are defined by the following expressions:
An = A-1Aa, = A-1BA, = A-1CA, = A-1 dA.
iA
¡N
A
N
1-W A
N
-1 A
(2.17)
(2.18)
The superscript □N indicates that the components are related to normal reactions N.
All An, BN, CN, dN components are constants which depend only on the vehicle's parameters ak, (3k, , $k (k = 1) 2, 3), A, h, R, m, A, A and the motor's characteristics c1, c2.
3. No-overturn conditions
In Section 2 we have defined a model with bilateral constraints. This approach can be applied to describe the omnivehicle's motion only in the case of positive normal reactions Ni > 0 (i = 1, 2, 3) in the points of contact. If one of these conditions is violated, dynamic equations can no longer be applied to describe the motion, and the vehicle may slip or overturn.
In this section we derive conditions, exploring whether the vehicle in its current phase state may overturn or not. Some of these conditions predict the boundaries exactly, while others tend to be simple and concise.
3.1. No-overturn at rest
Let us start by studying the state of rest:
Vi = v2 = V3 = 0, U = U2 = U3 = 0.
The conditions here are equivalent to a known statics theorem (for example, see [29, p. 151]).
Statement 3. The vehicle does not overturn at rest if the center of mass is inside the support point triangle, i. e.,
P2 P3
x P—S^j, ez) det A > 0, x P—S^j, ez) det A > 0, x P—S^j, ez) det A > 0.
P3 P1
(3.1)
(Sex the counterexample in Fig. 3.)
Fig. 3. A vehicle in nonequilibrium
Proof. At rest we have Ni = g, which can be reduced to > 0 (i = 1, 2, 3). The reverse matrix A-1 has the following form:
A"1 =
1
det A
^(¿2 sin a2 — A)(—¿3 cos a3) — (¿3 sin a3 — A)(—¿2 cos a2) □ □ (¿3 sin a3 — A)(—¿1 cos a1) — (¿1 sin a1 — A)(—¿3 cos a3) □ □ ^(¿1 sina1 — A)(—¿2cosa2) — (¿2sina2 — A)(—¿1 cosa1) □□J
f
Let us consider the first component df. By definition (2.18), we have
df = m((52 sin a2 — A)(—53 cos a3) — (53 sin a3 — A)(-52 cos a2))(det A)-1. On the other hand, by definition (Fig. 1a), we have Q$ = Aev — hez,
Qpi = 5i cos a^e^ + 5i sin a^ev (i = 1, 2, 3).
Thus,
df = m ( (QP2 — OS) x (qH — OS), ez) (det A)-1 = m x Sl^j, ez) (det A)-1.
Since —2 x Sp = (— S^s) x Sp = (P—H x SP^, after multiplying the condition df > 0 by m(det A)2 (we already assume that det A = 0) we get the first condition of (3.1). The other two conditions can be obtained using the same approach. □
3.2. No-overturn conditions in motion
In general, the no-overturn conditions are as follows.
Theorem 1. Let the vehicle have pseudovelocities vi, v2, v3 at a given instant. If the applied voltages satisfy the conditions
CfiUi + cfU2 + cf3Us + fi(Vi, V2, V3) > 0,
cfiUi + cf U2 + cfsU33 + f2(Vi, V2, V33) > 0, (3.2)
, cfiUi + cfU2 + cfsU33 + fs(vi, V2, V33) > 0, where
fi("i, "2, "3) = = ' ' (?; = 1'2'3)' M
\j=i J j=i
then all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. Conditions (3.2) express the inequalities Ni > 0 (i = 1, 2, 3) after the substitution from (2.17). □
Conditions (3.2) are exact — if at least one of them is violated, some reactions are either zero or negative. (More strictly, if at least one left-side expression is negative, the corresponding reaction is negative.) But in many cases it is rather hard to analyze or visualize these expressions. We also need to specify each pseudovelocity and voltage distinct value to use them. In this subsection we will state some corollary theorems with more concise no-overturn conditions as
Theorem 2. Let the vehicle have pseudovelocities vi, v2, v3 at a given instant. If the power of the applied voltages satisfies the condition
u[+u[+ui< min . ^ (3.4)
<=1,2,V(c«)2+(c«)2+ra2
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. For a specific vehicle, in the coordinates U1U2U3 conditions (3.2) describe an intersection of three semispaces. The specific values of pseudovelocities define the parallel shifts of these planes for each instant. In some cases it is possible to inscribe a ball with its center at the origin point in the no-overturn domain.
It is well known from analytic geometry that the distance from the origin point to a plane i is —;—Ifii^i'^'^)!-^^ j^. jg convenient, to introduce a signed distance — it is a positive distance
to the plane i when the point of origin satisfies the no-overturn condition for a specific i and a negative distance when it is not. Since it is essentially the sign of fi(v1, v2, v3), the signed
distance is -fi^i-^'^)-
'(N )+(cN) 2+(cN )2
Thus, condition (3.4) of the theorem defines a ball inscribed in the no-overturn domain. In case there is no such ball, at least the one of fi(v1, v2, v3) is negative, so the condition describes an empty set. In case there is such a ball, it touches the closest plane (or planes). □
The condition is sufficient, but not necessary, since the ball is just a part of the no-overturn domain. On the other hand, it always touches the domain boundary, so if the voltage power violates the condition, there are always voltages that will result in an overturn.
Theorem 3. Let the speed of the vehicle's center of mass be vs ^ 0 and let uez be the platform's angular velocity at a given instant. If the power of the applied voltages satisfies the condition
~ ' to)
Juf + m + Щ < min
V 1 z d ¿=1,2,3 // лгч27
where
w) = jminf^z/!, z/2, z/3) sjv\ + v% = vs, v3 = Aw j (-¿ = 1,2,3),
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Theorem 4. Let the vehicle have kinetic energy T = H at a given instant. If the power of the applied voltages satisfies the condition
фр.+Щ + Щ < mi
mm ■
i=1, 2, 3
where
Hi(H) = { min fi(vi, V2, V3) | T(vi, V2, V3) = H } (i = 1, 2, 3),
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. Similar to Theorem 2, the conditions of the theorem define a set of parallel planes. In some cases, it is possible to inscribe a ball in the no-overturn domain. To satisfy all possible pseudovelocity values, we must choose the worst case, defined by the minimal signed distance for each condition (for each i). □
Again, these conditions are sufficient, but not necessary. If the voltage power violates the condition, there always exist voltages and pseudovelocities that will result in an overturn.
3.3. Application of no-overturn theorems
The method of Lagrange multipliers can be used to find functions Vi(vS, w) from Theorem 3 and Hi (H) from Theorem 4.
Proposition 1. The following expression defines the function Vi(vs, w) for Theorem 3:
ZJiivs, u) = -vsyj(ma^u + bf )2 + (■niagu + bg)2 + (magw + Aw + (3.5) for each i = 1, 2, 3.
Proof. The definition of pseudovelocity (2.3) allows us to substitute all v3 entries with a known value Aw.
Stationary points (v1, v2) can be obtained from the following system of equations:
mafi w + bfi + 2^v1 = 0,
mafl w + bf2 + 2/J.V2 = 0, (3.6)
2 2 2 vl + v2 = vS,
where i is the Lagrange multiplier for each i = 1, 2, 3. There are two solutions of the system (3.6):
, N ■ mafiw + b* mag w + bg
K. U2)= , . ,VS>
^{magu + bg)2 + {magu + bg)2 ' ' /(ma> + bf)2 + (magu + b%f
,„ „ , I _ma^w + bg_ _magu + bg_
N\2 i „N, , i (,N\2 „N,, i (,N\2 i f™ „N, , i (,N
il)
if the denominator is nonzero. The extremum values will be
w) = /Kw + ^' + W^^)2 + M " + bg) Aw + of g. Since, by definition vs ^ 0, the minus sign defines the minimum value
ZJiivs, w) = -vs^ma^u + bg )2 + (mat> + bg)2 + (magw + bg) Aw + if g
for each i = 1, 2, 3, which defines the desired expression (3.5).
Finally, if the denominator is zero, aN = bN = af = bf = 0 and V(vS, w) does not depend on vS, then the expression (3.5) also takes place. □
Proposition 2. The solution of the following system defines the stationary points of the function Hi(H) from Theorem 4:
f Adf.V + = -bN,
1 t (3.7)
I vTAkmv = 2H,
where
m
= x
/ 0 0 \
0 0 aN
aN aN 2aN
\ail ai2 2ai3)
AMn = M + A2S*S, bN
fbfi\ bN
bN
(3.8)
H is the Lagrange multiplier, for each i = 1, 2, 3.
Finding out the general solution of Eq. (3.7) is rather laborious, but it is possible to find the solution in some specific cases. An example will be shown in Section 4.
3.4. Algorithm for finding no-overturn conditions
Let us organize the results of Sections 2.2-3.3 to form a step-by-step algorithm for finding out no-overturn conditions based on Theorem 1.
1. Describe the vehicle's parameters ai, /i, Yi, 5i (i = 1, 2, 3), A, h, R, m, A, A and the motor's characteristics ci, c2.
2. Ensure that the wheel numbering satisfies the condition det A > 0. Refer to (2.12) for the definition.
3. Compute the vehicle's constraint matrix 2. Refer to (2.5) for the definition.
4. Compute the components defining pseudoaccelerations: aV, bj, cj (i, j = 1, 2, 3). Refer to (2.8)-(2.10) for the definitions.
5. Compute the components defining normal reaction equations: aj, b j, cj, dV (i, j = 1, 2, 3). Refer to (2.13)-(2.16) for the definitions.
6. Compute the components defining normal reactions: af, bf, cf, df (i, j = 1, 2, 3). Refer to (2.18) for the definitions.
7. Substitute the values of the resulting components into the no-overturn conditions. Refer to (3.2) and (3.3) for expressions.
Theorems 2-4 provide a single concise condition instead of the system of conditions. To find a condition based on Theorem 2, the additional step is as follows.
it. Find the minimal value of fii^i'^'^) over j = i 2, 3 for za, v0, u-,.
/(MW
To find a condition based on Theorem 3, additional steps are as follows. 1tt. Compute the values of Vi(vS, w) (i = 1, 2, 3). Refer to (3.5) for the expression.
2tt. Find the minimal value of , 0 i 0 over i = 1, 2, 3 for vQ, uo.
V(^)
To find a condition based on Theorem 4, additional steps are as follows.
1ttt.
Find stationary points by solving Eq. (3.7) (i = 1, 2, 3). Refer to (3.8) for definitions.
2ttt. Find the Hi(H) functions by finding the minimal value of fi(vi, v2, v3) over stationary points obtained from (3.7).
+++ H (H)
3TTT. Find the minimal value of , 0 ' „ = over i = 1, 2, 3 for H.
(cN )2+(NY+(NY
4. Example: symmetrical omnivehicle
4.1. No-overturn conditions
Let us use the algorithm from Section 3.3 to find no-overturn conditions for one of the most common configurations of the omnivehicle [2, 6, 11, 15, 16, 21-24, 26, 28]. Consider a symmetrical vehicle with wheels which have rollers perpendicular to the wheel plane (Fig. 4):
A = 0, ^ = ô2 = ¿3 = p> 0, R = R> 0, h = h> 0,
n n 7n
®i=Pi = --, a2=f32 = -, a3 = /3 3 = —,
6' 2 ' 2 2
n
7i = 72 = 7s = g > m = m> 0, A = A > 0, A = A> 0, c1 = c1 > 0, c2 = c2 > 0.
6
O
Fig. 4. A symmetrical vehicle The vehicle's matrix A (2.12) is as follows:
( 111 \
A=
P
Vsp n Vsp \--2 U —)
Its determinant 0, so the vehicle does not overturn at rest and the wheels are numbered
counterclockwise.
The vehicle's constraint matrix 2 (2.5) is as follows:
2 R 2 R A R
1 0
R A R
1 V3 P
\ 2 R 2 R A R
The equations of motion (2.7) for the vehicle are
2mR2 v2v3 3c2 ci R . . . „ .
"l = ^TT^TTO ^ - OV^I + , ov^l - 2Ci2 + ^3), (4-1)
. _ 2mR2 ^г/3 Зс2 л/Зс^Д
= ~~2mR2 + ЗА2 Л " 2mR2 + ЗА2^2 + 2mR2 + ЗА2 " ^ з)' ( j
--A^Tsav"3 + л^да^+^+*«>• (4-3)
The vehicle's normal reactions (2.17) are as follows:
_ 2mhR + ЗА2 / A2 ¡^W^ c2 y^ - t/2 U2~U3\ , ™9 ,„ „ч
Nl p (2mR2 + за2) ^лё 2 + д 2 + j + ~3~' (44)
2mhR + 3A2 / A2 c2 - ^ \ me .
^ p(2m.g + ЗА2) ("Ля"'"3 + iH + j + 3 ■ (45)
2mhR + ЗА2 / A2 — \fbv2 c2 — у/Зг/1 — v2 U1 — U2 \ mg
3 = p (2mR2 + ЗА2) l ЛЯ 2 + Д 2 + Cl ^3 J + 1"'
Corollary 1. Lei the symmetrical omnivehicle have pseudovelocities v1, v2, v3 at a given instant. If the applied voltages satisfy the conditions
' TT TT + \pîv2 \f%vx - u2 U2 - U3 > --V3 - "1-1-- - "0,
U3 - Ui > U2V1V3 - U1V2 - Uo, (4.7)
ZA — yfZVo — \/3^i — V-
^ Ul - U2 > —u2'1 2V ~'v3 - Ui^-1^-* - Uo,
where
Уза2 л/Зс2 y/Smpg 2mR2 + за2
4i = —îT> uo
V 2mhR + ЗА2 ' (48)
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. The conditions are obtained by directly applying the steps described in Section 3.3 for Theorem 1. □
Corollary 2. Let the symmetrical omnivehicle have pseudovelocities иг, v2, v3 at a given instant. If the applied voltages satisfy the conditions
ф;2 + Щ + U2<^ min{fll, fl2, fl3}, (4.9)
where
+ л/Зг^о л/3т^1 — z/9
ill = 2- vi + U1-2- + U°'
fl2 = -U2V1V?, + U^ + Uo,
г/, — л/З^о —\[Ъул — v.
-
S3 = ^2 ~ ^-+ - + U0'
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. The conditions are obtained by directly applying the additional steps described in Section 3.3 for Theorem 2. □
Corollary 3. Let the speed of the center of mass of the symmetrical omnivehicle be vs ^ 0 and let wez be the angular velocity at a given instant. If the power of the applied voltages satisfies the condition
sjuf + U'i + U$ <
'^sju2 A2UJ2 + U2,
22
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. The conditions are obtained by directly applying the additional steps described in Section 3.3 for Theorem 3. In this case, V1(vs, w) = V2(vs, w) = V3(vs, w), so there are no additional steps required to find the minimal Vi(vs, w). □
Corollary 4. Let the symmetrical omnivehicle have kinetic energy T = H at a given instant. If the power of the applied voltages satisfies the condition
yjul + U2 + U'i < fj{H),
(4.11)
where
H(H ) =
H
2mR2 + 3A2:
73 lA2R2 + 3A2p2
c1 RV 2mR2 + 3A2 V^2R2 + 3A2p2
A2 R2
H +
2A2
for H ^ for H ^
c2 (A2E2 + 3A2p2)
2A4E2 c2 (A2E2 + 3A2p2) 2\4
all normal reactions are strictly positive and, consequently, the vehicle does not overturn.
Proof. The conditions are obtained by following the additional steps of the algorithms described in Section 3.3 for Theorem 4. Let us recall the system (3.7):
Afv + Mkinv = -b vT Akin v = 2H.
N
Consider the first group of equations for reaction N1 (corresponds to i = 1). For a symmetrical vehicle, the components (3.8) are as follows:
_ A2 2mhR + 3A2 ~ ApR 2m R2 + 3A2
A) 0 i\
0 0^
I VI o \ 2 2 u
Akin =
m +
3A 2ÏÏ1
m +
3A 2ÏÏ1
/
0 0
0 i +
N _ c2 2mh.R + 3A2
pR 2mR2 + 3A2
2
1
2 0
(4.12)
(4.13)
(4.14)
t
c
0
2
2
c
1
c
0
2
Case I. Consider the case where Adf + iAkin is nonsingular. In this case, the Lagrange
df
multiplier i can be obtained by solving the following equation:
Adf1 + Akin (yAdf1 + ^Akin) b
2H,
(4.15)
where the sign □* denotes the transposition of a real matrix.
After substituting (4.12)-(4.14) into (4.15), we get the following equation:
2c2 (2mhR + ЗЛ2)2 ¿t2p2 (2rnR2 + 3A2)Î
2H.
(4.16)
After obtaining two Lagrange multipliers from the solution of Eq. (4.16), we can find two stationary points by solving the system (3.7):
W
= ±
HR2
2mR2 + 3Л2
1
V 0 J
(4.17)
At these stationary points, the values of the function f1(v1, v2, v3) are as follows:
mg __ о c2 2mhR + 3Л2
HR2
3 pR 2mR2 + 3Л2 V 2mR2 + 3Л2 '
(4.18)
The minus sign value of (4.18) f1 is the first candidate for the minimum value and the desired H1 (H) function:
„ , . mg c2 2mhR + 3Л2
= — -2 2
3 pR 2mR2 + 3Л2 V 2mR2 + 3Л2
HR2
(4.19)
Case II. Now let us consider the case where Af + ¡Akin is singular. In this case, the
Lagrange multiplier ¡i can be obtained by solving the following equation:
det (Adfi + ¡Akin) =0. After substituting (4.12)-(4.14) into (4.20), we get the following equation:
(4.20)
3A2W p2A2\ 2 ( A2 2mhR + 3A2\2!/ 3A2 \ ^ ,
^ m+2R2) (1 + 3£w)'' -(mpRA2m# + 3A>) }(m+2#)" = 0- (421)
There are no solutions of (3.7) for i = 0. The other two solutions give the following family of solutions for the first equation of (3.7):
= ±-
A2R2 + ЗА V ( А2г/Зя + УЗс2Л'
H
2Л2А\ 2mR2 + 3Л2
x.2/ ~ , , ^ v oX2u3 — C2AJ
The second equation of (3.7) gives us the last piece of solution:
v3 = v3
(4.22)
vH = ±
2HЛ4R2
y/2A _
2A2 V A2R2 + ЗА2fi2
c
(4.23)
v
2
v
1
Note that the signs ± in (4.22) and (4.23) are independent. This solution exists only if
2H\AR2 K2R2 + 3A V
- c2 ^ 0.
If a^+3aV — = 0, the solution is the same as (4.17). If ^igi^^ipi — c2 > 0, we have four more stationary points.
At these stationary points, the values of the function f1(v1, v2, v3) are as follows:
_ mg V2 2>mhR + 3\2 /A2R2 + 3A2p2 ( A2R2 c2 \
fi("i, "2, "3) — ± ~pR 2mR2 3A2 V 2mR* + 3A2 ^Â^TW + 2X2) " (424)
The minus sign value of (4.18) f" is the second candidate for the minimum value and the desired H1 (H) function:
„......, mg y/2 2mhR + 3A2 /A2R2 + 3A2p2 ( A2R2
fi K, - — " ^2mE2 + 3A2V 2mR2 + 3A2 U2^2 + SAV^ + 2A^J" (425)
Finally, since the following inequality takes place:
_„ 2
,„ v 2riihR + 3A2 IA2R2 + 3A2p2 / / A2E2 I
fl fl pE 2mR2 + 3A2 V 2mE2 + 3A2 (j A2E2 + 3A2p2 ^ V 2A2 J ^
fi1 defines the minimum value Hi(H), if it exists, otherwise fi defines the minimum value Hi(H).
Due to the symmetric nature, Hi(H) = H2(H) = H3(H), and (4.11) defines the desired condition. □
As a byproduct, the proofs of Corollary 3 and Corollary 4 define the most "dangerous" motions among the corresponding classes.
Corollary 5. Among all motions with the speed of the center of mass, vs ^ 0, and the angular velocity wez, the following takes place:
• (1/1, v9) = ( — A "I^J'g, —| derives the minimum normal reaction N-,,
V 2> \ 2V/A4W2+C| 2V/A4W2+C| bJ 1
• (za, V0) = -r&L—Vc--, ,cl derives the minimum normal reaction N,
1 2 y a/A4w2+C| 6 ^JX^^+cl bJ
2
• (1/1, v9) = ( — X l-^t'o, ) derives the minimum normal reaction No. V 2> \ 2v/A4w2+c2 2v/A4w2+c2 ^
Corollary 6. Among all motions without rotation (v3 = 0), moving backwards in the direction of the ith wheel derives the minimum normal reaction Ni (i = 1, 2, 3).
c2 (A2 R2 +3A2 p2)
Corollary 7. Among all motions with kinetic energy H ^ ——2\iR'}- ' ^ie following
takes place:
• u2, 1/3) = ^—^2wdP+3A2' \Jdenves the minimum normal reaction Nx,
' (u1, u2, v3) = ^0, —2\Jz^'J^sx-> derives the minimum, normal reaction N2,
• (u1, u2, v3) = sJ^njyP^-ix1> \J2> _R2^-3A2' ^ derives the minim,urn, normal reaction N3.
c2 (^2 R2 +3^2 p2 )
Corollary 8. Among all motions with kinetic energy H ^ ——2XiR2- > the following
takes place :
(vl, u2, us) = (^——^^ß-2—, — —vf^j derives the minimum norm al reaction Nlt
(vl, u2, us) = , —vf^j derives the minimum normal reaction N2,
(vl, u2, us) = 2>r ^' derives the minimum normal reaction Ns,
where x - , /MZ _ I VIA / 2iJA^2
umeiex- 2 y 2mi?2+3A2 ' - ^ 2A2 y A2.R2+3AV 2-
We can use these results to make some recommendations on the vehicle design.
Conclusion 1. The normal reactions of a symmetrical omnivehicle do not depend on the system's moment of inertia A2.
Proof. Reactions (4.4)-(4.6) have the only component depending on A, but the component actually depends on the Av3 product, which is essentially d. □
Conclusion 2. If the center of mass is at the same height as the wheels' centers (h = R) and there is no rotational motion (v3 = 0), the normal reactions of a symmetrical omnivehicle do not depend on the wheel's moment of inertia X2.
Proof. See (4.4)-(4.6). □
Conclusion 3. The following statements take place for a symmetrical omnivehicle:
• the heavier the vehicle (m increases),
• the wider the platform (p increases),
• the lower the center of mass (h decreases),
• the less the viscous friction in the wheels' axes and self-EDS (c2 decreases), the less strict is the voltage power restriction.
Proof. Refer to conditions (4.10)-(4.11). □
Conclusion 4. The following statement takes place for a symmetrical omnivehicle with "turned off" motors (U1 = U2 = U3 = 0): the bigger the vs, the less is allowed w2, and vice versa.
Proof. In case U1 = U2 = U3 = 0, the no-overturn conditions (4.10) are as follows:
un
vs <
VuiA2W2 + u2 ' D
4.2. Example of motion
Consider some controlled motions of the symmetrical vehicle. We will explore the vehicle with a high center of mass, since it is easier to illustrate the overturn using such a model. (In real cases, it can be a vehicle with stretched-out cargo on it.) The following parameter values were used for numerical computations:
Parameter Value Unit
m 3 kg
P 0.15 m
h 0.20 m
R 0.05 m
Parameter Value Unit
A2 5•10"2 kg • m2
A2 5•10"4 kg • m2
Cl 10"2 kg • m2 • s"2 • V-1
c2 2.5 • 10"4 kg • m2•s_1
Example I (Fig. 5). Let the vehicle move from the state of rest with constant voltages
U1(t) = -20 V, U2(t) = 30 V, U3(t) = -12 V
(see Fig. 5d).
The result will be as follows. After t = 0.4 s, the power restriction condition of Corollary 4 for H is violated. After t = 2 s, the power restriction condition of Corollary 3 for vS, w is violated as well (Fig. 5e), and later at t = 2.7 s (depicted by a cross in the circle) the normal reaction N3 becomes negative and the model breaks up (Fig. 5c).
Example II (Fig. 6). Consider more sophisticated controlled motion. Let us program the vehicle to move in circles with acceleration (Fig. 6a). The desired motion can be achieved by moving with the following pseudovelocities:
vi(t) = 0,
"2® =
v3(t) = 2nAetrajt,
where Rtraj is the radius of the circle and etraj is the angular acceleration (see Fig. 6b). To control the vehicle, we use the voltages obtained from the direct substitution of the desired pseudovelocities into the equations of motion (4.1)-(4.3) (see Fig. 6d).
For motion with the parameters Rtraj =1 m and etraj = 0.04 s"2 we have the following results. After t = 6 s, the power restriction condition of Corollary 4 for H is violated (Fig. 6f). After t = 7.5 s, the power restriction condition of Corollary 3 for vS, w is violated (Fig. 6e), and later at t = 8.2 s (depicted by a cross in the circle), during the second "lap", the normal reaction N2 becomes negative and the model breaks up (Fig. 6c).
These examples show that the usefulness of the laconic conditions varies from motion to motion. The power restriction of Corollary 3 for vs, w is more precise than the power restriction of Corollary 4 for H, which is expected since the latter has less information. The same holds for the power restriction of Corollary 3 for vS, w and the exact conditions for the same reason.
5 4 3
S â 2
1
0
50 40 ^ 30
Sh g
I 20
a
10 0 -10
3 -4 x, m
(a) Trajectory
1.5
t, s
(c) Reactions
advice real
30 20 10 0 -10 -20
0.5
1.5
t, s
50 40
K 30
h
g
t 20 a
10 0 -10
(m/s) -(D-l'l (m/s) -©(kg V^m/s) ® f3
1.5
t, s
(b) Pseudovelocities
t, S
(d) Control
—a 9— -<D- £/i u2 ® u3
a - D -
0 0.5 1.5 2 2.5 3
2 2.5 3
(e) Power restrictions (vS, u)
Fig. 5. Example I: explicit motion control
\ : \ _ \
\ \ \
\ \ \ \
\ \ \
Y \ \ _L idvicc _ •eal
0 0.5 1.5 2 2.5 3
t, s
(f) Power restrictions (H)
5. Conclusion
We consider the motion of an omnivehicle controlled by direct current motors. For an arbitrary three-wheeled vehicle moving on a perfectly rough horizontal plane, the explicit form
-1.5 -1 x, m
(a) Trajectory
4
t, s
(b) Pseudovelocities
-©- u2 /
® U3
4
t, s
(c) Reactions
4
t, s
(d) Control
0 2 4 6
t, s
(e) Power restrictions (vS, u)
Fig. 6. Example II: moving in circles
4
t, s
(f) Power restrictions (H)
of normal reaction components was obtained. We also present a step-by-step algorithm for finding out components for a specific vehicle, which can be used to make a computer program.
We also state several theorems which provide restrictions on motor voltages in case we know the current state of a vehicle: pseudovelocity values, or the speed of the center of mass and the
platform's angular velocity, or the system's kinetic energy. Explicit algorithms for finding out these conditions are also presented, where it is possible to solve the general problem.
Based on these theorems, we study the motion of a symmetrical omnivehicle. We present various conclusions on the control decision and vehicle design approaches to prevent an overturn, based on these conditions. We study some examples of controlled motion together with their no-overturn conditions.
Steps used to study the system's reactions can also be reused to explore the system with Coulomb friction.
Appendix A. Normal reactions equations
The theorem on the motion of the center of mass gives
m
d(v1 eç + v2ev )
3
dt
i=i
mg + Ri, (A.1)
d(v1 eç + v2en) / v2v3 x / Ulv3
n—-I'^Th + ^ + TK <A-2>
By substituting the derivatives (A.2) into equation (A.1), we have the following scalar equations:
^^V V
+ + = (A.3)
Riv + R2V + RiV = riii>2 + ^f1, (A.4)
N1 + N2 + N3 = mg. (A.5) Finally, let us substitute the pseudovelocitites (2.7) into equations (A.3)-(A.4):
+ R2? + = m IE aiivi I ^r "+m E biivi + m E cvui>
33
i] 3 I a A ' ij j ij 31
J=1 ) 3=1 3=1
3 \ 3 3
mv3 , mv1v3 , v„ , „v
Riv + R2V + = m a23U3 ~T + ~~tA + m E bVvi + m E cVUi>
or, in vector form,
where
23 3 \ a A Z—✓ 23 3 3y
3=1 3=1 3=1
Rit + A* + = ^ + + tRU, (A.6)
+ R2n + R3n/ A
R /1 0 0\ 0 -1 0\
Ar = m Av + , (A.7)
\o 1 0y \i 0 0y
R (1 0 0\
BR = mi , (A.8)
^0 1 0y
R (1 0 0\
CR = ml . (A.9) 10 10/
The superscript indicates that the matrices are related to reaction components. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2024, 20(2), 311 336
3
The theorem on the momentum of the center of mass is
dK
—7— = M, (A.10)
dt y '
where K is the system's angular momentum and M is the moment of external forces, both with respect to the center of mass S.
The system's angular momentum with respect to the center of mass S is
K = Kp + Klw + K2w + K3w, (A.11)
where Kp is the platform's angular momentum and K1w, K2w, K3w are the wheels' angular momenta, all with respect to the center of mass of the whole system, S. The platform's angular momentum is
Kp = 6A2vez, (A.12)
where A^ is the platform's moment of inertia with respect to the vertical Sez. The wheels' angular momenta are
Kiw = 9A\wez - \2xini, (A.13)
K2w = ¿A2wez - A2x2n2, (A.14)
K3w = (>A3wez - \2x3n3, (A.15)
where A2w, A2w, A3w are the wheels' inertia momenta with respect to the vertical Sez. By substituting Eqs. (A.12)-(A.15) into Eq. (A.11), we have
33
K = (A^ + A?w + A2w + A2 J 6ez - ^ A2Xn = A20ez - ^ A2xX.
i=1 i=1
Let us compute the derivative of K:
3
dt ----3 ~
A26ez -Y A2fen + XXin). (A.16)
i=1
Since the vectors ni (see Fig. 1a) and their derivatives are
ni = cos /3ie^ + sin /3iev, rti = 6 cos /3iev - 6 sin /3ie^, the momentum derivative (A.16) is
dK \ 2 „\ 2 3
A2 Y(cos PiXi - sin Pi&xi)e^ - A2 ^(sin (3iXi + cos Pi9xi)ev + A26ez.
i=1 i=1 In vector form, the first two components of the derivative (A. 16) are
f^lr, et)\ ,, /cos/ii cos B9 cosBA —sin/i, — sin/39 — sin/3o\
) K ( = "A • ^ • R • 7 b " A M / / R * (A"17)
Vsin/^l sm@2 smPsJ V cosPi cosP-2 cosfh J
By substituting the definition of pseudovelocities (2.1)-(2.3), the constraints (2.4) and the form of pseudoaccelerations (2.7) into the components (A.17), we can obtain the following form:
if' 6?f) = (**") nJr + (»*") + ^' № s) / A
where
AK = —X2 cos 81 cos 82 cos 83
sin 81 sin 82 sin 83
BK = —X2 cos 81 cos 82 cos 83
sin 81 sin 82 sin 83
CK = —X2 cos 81 cos 82 cos 83
sin 81 sin 82 sin 83
X2 (— sin81 — sin82 — sin83\
2T - — 1 2 3 S, (A.18)
m \ cos 81 cos 82 cos 83 )
IBV, (A.19)
ICV. (A.20)
The superscript DA indicates that the matrices are related to the derivative of the system's momentum.
On the other hand, the momentum of external forces with respect to the center of mass S is
3
M = , R]. (A.21)
i=1
By definition (see Fig. 1a),
SM>i = Q]]i — qS + P—Ii = Si cos aie^ + (Si sin ai — A)en — hez. In this case, the first two components of the momentum of external forces (A.21) are
(M, es) = h(Rm + R2V + R3V) + sin — A)Nt,
i=1 3
(M, ev) = —h(R^ + R2t + R3t) — E ôi cos aiNi,
i=1
or, in vector form,
/<M, eA = h ( 0 A iRv + R2, + R-31 \ + ia2i a222 a2=A N (A.22)
V(M, enV \ — 10J R + R2n + R3VJ Va31 a32 a33j
where
a2i = 6i sin ai — A, a3i = —5i cos ai (i = 1, 2, 3) (A.23)
and N = (N1, N2, N3) are normal reactions.
By substituting the reactions (A.6) into the equation of the momentum of external forces (A.22), we can simplify the equation:
f(M' = (a^) + + u + °22 M N, \(M, ev)) A ^ a:i2 a33
where
AM = h
BM = h
CM = h
0 1\ „ / 0 1 0\ /1 0
AR = mhl + h
-1 0 \-1 0 0 0 1 0,
' 0 V
'0 1 -1 0,
BR = mh
CR = mh
/010'
l"10 '0 1 0\ -1 0 0/
Cv.
(A.24) (A.25) (A.26)
The superscript DM indicates that the matrices are related to the momentum of external forces. Finally, Theorem (A.10) gives us the following two equations for the normal reactions N:
a21 a22 a23 ,a31 a32 a33.
N = ((SlA - 5lM) u) ^ + (<BA - <BM) + (£a' - £M) t/
A
(A.27)
Together, Eqs. (A.5) and (A.27), with the definitions (A.7)-(A.9), (A.18)-(A.20), (A.23) and (A.24)-(A.26) taken into account, form the desired system:
AN = ^ + <BAv + <tAU + dAg,
A
where
/ 1 1 1 (N\
A = sin a1 - A ¿2 sin a2 - A ¿3 sin a3 - A , N = N to
V " -¿1cos a1 S2 cos a2 S3 cos a3 1 w
Aa = - A2
/ 0 0 0 N
cos /31 cos (32 cos f3
^sin /31 sin (32 sin ff3j
/ 0 0 0\ /0 0 0\
a"--
m
/
\
- sin f " sin (32 - sin (33 ^ cos /31 cos ff2 cos f3 J
mh
0 1 0 "1 0 0y
Av - h
0
BA = - A2
CA = - A2
1 0 0 0 1 0 J
( 0 0 0 \
cos /31 cos (32 cos (33 ^sin /31 sin (32 sin f33J
( 0 0 0 \
cos f 1 cos f 2 cos f 3 sin f 1 sin f 2 sin f 3
/ 0 0 0\
IBV - mh
ICV - mh
0 1 0 "1 0 0y
/ 0 0 0\
0 1 0 "1 0 0 J
d
A
m
1
0 0
The superscript indicates that the matrices are related to the system of equations for the normal reactions.
Conflict of interest
The author declares that he has no conflict of interest.
References
[1] Taheri, H. and Zhao, C.X., Omnidirectional Mobile Robots, Mechanisms and Navigation Approaches, Mech. Mach. Theory, 2020, vol. 153, no. 2, 103958, 28 pp.
[2] Campion, G., Bastin, G., and d'Andrea-Novel, B., Structural Properties and Classification of Kinematic and Dynamic Models of Wheeled Mobile Robots, IEEE Trans. Robot. Autom, 1996, vol. 12, no. 1, pp. 47-62.
[3] Ilon, B., Wheels for a Course Stable Selfpropelling Vehicle Movable in Any Desired Direction on the Ground or Some Other Base, Patent US No. 3 876 255 (13 Nov 1972).
[4] Grabowfecki, J., Vehicle Wheel, Patent US No. 1 305535 (11 May 1918).
[5] Blumrich, J., Omnidirectional Wheel, Patent US No. 3 789 947A (17 Apr 1972).
[6] Moiseev, G.N. and Zobova, A. A., Dynamics-Based Piecewise Constant Control of an Omnivehicle, Russian J. Nonlinear Dyn, 2022, vol. 18, no. 4, pp. 651-670.
[7] Zobova, A. A., Application of Laconic Forms of the Equations of Motion in the Dynamics of Non-holonomic Mobile Robots, Nelin. Dinam., 2011, vol. 7, no. 4, pp. 771-783 (Russian).
[8] Zobova, A. A. and Tatarinov, Ya.V., The Dynamics of an Omni-Mobile Vehicle, J. Appl. Math. Mech., 2009, vol. 73, no. 1, pp. 8-15; see also: Prikl. Mat. Mekh, 2009, vol. 73, no. 1, pp. 13-22.
[9] Yang, S., Lu, Y., and Li, S., An Overview on Vehicle Dynamics, Int. J. Dynam. Control, 2013, vol. 1, no. 4, pp. 385-395.
[10] Shen, J. and Hong, D., OmBURo: A Novel Unicycle Robot with Active Omnidirectional Wheel, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (ICRA '2020), pp. 8237-8243.
[11] Liu, Y., Jim Zhu, J., Williams, R. L., and Wu, J., Omni-Directional Mobile Robot Controller Based on Trajectory Linearization, Robot. Auton. Syst., 2008, vol. 56, no. 5, pp. 461-477.
[12] Kilin, A. A. and Bobykin, A.D., Control of a Vehicle with Omniwheels on a Plane, Nelin. Dinam., 2014, vol. 10, no. 4, pp. 473-481 (Russian).
[13] Rani, M., Kumar, N., and Singh, H. P., Force/Motion Control of Constrained Mobile Manipulators including Actuator Dynamics, Int. J. Dyn. Control, 2019, vol. 7, no. 3, pp. 940-954.
[14] Andreev, A. S. and Peregudova, O. A., On Global Trajectory Tracking Control for an Omnidirectional Mobile Robot with a Displaced Center of Mass, Russian J. Nonlinear Dyn., 2020, vol. 16, no. 1, pp.115-131.
[15] Kalmar-Nagy, T., D'Andrea, R., and Ganguly, P., Near-Optimal Dynamic Trajectory Generation and Control of an Omnidirectional Vehicle, Robot. Auton. Syst., 2004, vol. 46, no. 1, pp. 47-64.
[16] Borisov, A. V., Kilin, A. A., and Mamaev, I.S., An Omni-Wheel Vehicle on a Plane and a Sphere, Nonlinear Dynamics & Mobile Robotics, 2013, vol. 1, no. 1, pp. 33-50; see also: Nelin. Dinam., 2011, vol. 7, no. 4, pp. 785-801.
[17] Zeidis, I. and Zimmermann, K., Dynamics of a Four-Wheeled Mobile Robot with Mecanum Wheels, ZAMM Z. Angew. Math. Mech., 2019, vol. 99, no. 12, e201900173, 22 pp.
[18] Adamov, B. I., A Study of the Controlled Motion of a Four-Wheeled Mecanum Platform, Russian J. Nonlinear Dyn., 2018, vol. 14, no. 2, pp. 265-290.
[19] Adamov, B.I. and Saypulaev, G.R., Research on the Dynamics of an Omnidirectional Platform Taking into Account Real Design of Mecanum Wheels (As Exemplified by KUKA youBot), Russian J. Nonlinear Dyn., 2020, vol. 16, no. 2, pp. 291-307.
[20] Kilin, A., Bozek, P., Karavaev, Yu., Klekovkin, A., and Shestakov, V., Experimental Investigations of a Highly Maneuverable Mobile Omniwheel Robot, Int. J. Adv. Robot. Syst, 2017, vol. 14, no. 6, pp. 1-9.
[21] Martynenko, Yu. G. and Formal'skii, A. M., On the Motion of a Mobile Robot with Roller-Carrying Wheels, J. Comput. Sys. Sc. Int., 2007, vol. 46, no. 6, pp. 976-983; see also: Izv. Ross. Akad. Nauk. Teor. Sist. Upr, 2007, no. 6, pp. 142-149.
[22] Gerasimov, K. V. and Zobova, A. A., On the Motion of a Symmetrical Vehicle with Omniwheels with Massive Rollers, Mech. Solids, 2018, vol. 53, suppl. 2, pp. 32-42; see also: Prikl. Mat. Mekh, 2018, vol. 82, no. 4, pp. 427-440.
[23] Mamaev, I. S., Kilin, A. A., Karavaev, Yu.L., and Shestakov, V. A., Criteria of Motion without Slipping for an Omnidirectional Mobile Robot, Russian J. Nonlinear Dyn, 2021, vol. 17, no. 4, pp. 527-546.
[24] Williams, R. L. II, Carter, B.E., Gallina, P., and Rosati, G., Wheeled Omni-Directional Robot Dynamics including Slip, in Proc. of the Internat. Design Engineering Technical Conf. and Computers and Information in Engineering Conf. (ASME'2002, Montreal, QC, Canada, 2002): Vol. 5, pp. 201-207.
[25] Saipulaev, G.R., Adamov, B.I., and Kobrin, A. I., A Dynamicly Consistent Model of Normal Reactions at Points of a Mobile Platform Contact with a Surface Taking Account of the Design of Mecanum Wheels and Multicomponent Friction, Mech. Solids, 2023, vol. 58, no. 5, pp. 1447-1457; see also: Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2023, no. 5, pp. 15-26.
[26] Kosenko, I.I. and Gerasimov, K.V., Physically Oriented Simulation of the Omnivehicle Dynamics, Nelin. Dinam, 2016, vol. 12, no. 2, pp. 251-262 (Russian).
[27] Kosenko, I., Stepanov, S., Gerasimov, K., and Stavrovskiy, M., Virtual Testbench for the Omni Wheel Dynamics Simulation: New Contact Tracking Algorithm, in Proc. of the 7th European Congr. on Computational Methods in Applied Sciences and Engineering (ECC0MAS'2016), M. Pa-padrakakis, V. Papadopoulos, G. Stefanou, V. Plevris (Eds.), Crete Island, Greece: ECCOMAS, 2016, pp. 4553-4571.
[28] Kosenko, 1.1., Stepanov, S.Ya., and Gerasimov, K.V., Contact Tracking Algorithms in Case of the Omni-Directional Wheel Rolling on the Horizontal Surface, Multibody Syst. Dyn., 2019, vol. 45, no. 3, pp. 273-292.
[29] Appell, P., Traité de mécanique rationelle: Vol. 1. Statique. Dynamique du point, 6th ed., Paris: Gauthier-Villars, 1941.
