Dynamics of a four‐wheeled mobile robot with Mecanum wheels

The paper deals with the dynamics of a mobile robot with four Mecanum wheels. For such a system the kinematical rolling conditions lead to non‐holonomic constraints. From the framework of non‐holonomic mechanics Chaplygin's equation is used to obtain the exact equation of motion for the robot. Solving the constraint equations for a part of generalized velocities by using a pseudoinverse matrix the mechanical system is transformed to another system that is not equivalent to the original system. Limiting the consideration to certain special types of motions, e.g., translational motion of the robot or its rotation relative to the center of mass, and impose appropriate constraints on the torques applied to the wheels, the solution obtained by means of the pseudoinverse matrix will coincide with the exact solution. In these cases, the constraints imposed on the system become holonomic constraints, which justifies using Lagrange's equations of the second kind. Holonomic character of the constraints is a sufficient condition for applicability of Lagrange's equations of the second kind but it is not a necessary condition. Using the methods of non‐holonomic mechanics a greather class of trajectories can be achieved.


INTRODUCTION
This paper relates to mechanics of wheeled locomotion. Beside biological inspired forms of locomotion like crawling, flying or swimming, this form of locomotion with a special propelling device is still in the focus of research. The demand for mobile platforms working in complex environments or personal robots with a high maneuverability for handicapped people lead to new kinds of wheels especially in the past fifty years. Starting with the first patent of J. Grabowiecki in 1919 in the U.S.A. [1], engineers developed wheels which cannot move only in the direction of the wheels plane, but also perpendicular to this plane, e.g. omnidirectional wheels. A key issue of an efficient application of these wheels and an optimal control of the whole mobile system is the understanding of the physical interaction between the wheels and the environment. For this reason, mechanics of wheeled locomotion draws attention of both mechanical engineers, see, e.g., [2][3][4][5][6] and control engineers, e.g., in [7][8][9][10][11]. In a number of studies e.g. [12][13][14][15][16][17] the motion of systems with so-called Mecanum wheels, as a special class of omnidirectional wheels, was investigated. The authors of the mentioned papers mainly use methods of analytical mechanics to obtain the equations of motion. In many cases the Lagrange equation of second kind is the selected tool, which is correct applicable for holonomic systems.
In this article we consider the classical kinematic constraint, involving point contact and rolling without slipping. There are more complex models of rolling bodies on the surface, taking into account the contact spot, the distribution of normal pressure F I G U R E 1 Four wheeled mobile robot with Mecanum wheels F I G U R E 2 Model of a Mecanum wheel forces on the contact spot and rolling resistance. Such studies are contained in [18][19][20], and others. Taking these phenomena into account is useful in solving a number of practical problems related to the dynamics of Mecanum wheels. The purpose of the present work was to analyze the method widely used in robotics related to the use of a pseudoinverse matrix and the subsequent compilation of Lagrange's equations of the second kind. We called this method approximate. These equations are compared with equations obtained using non-holonomic mechanics methods. Non-holonomic mechanics methods we called exact.
A Mecanum wheel is a wheel with rollers attached to its circumference. Each roller rotates about an axis that forms an angle of 45 degrees with the plane of the disk. Such a design provides additional kinematic advantages for the Mecanum wheels in comparison with the conventional wheels and leads to non-holonomic constraints. Thus, a full non-holonomic approach is used describing the dynamics of a mobile robot with four Mecanum wheels.

FORMULATION OF THE MECHANICAL PROBLEM AND ITS MATHEMATICAL MODEL
The dynamics of a four-wheeled robot with Mecanum wheels arranged on two parallel axles ( Figure 1) are studied. The robot moves so that all its wheels have permanent contact with a plane. The body of the robot has a mass of 0 , its center of mass lies on the longitudinal axis of symmetry of the body. The distance from the center of mass of the robot to each of its wheel axles is , the distance between the centers of the wheels is 2 . The coordinates of the center of mass in a fixed coordinate system are , , the angle formed by the longitudinal axis of symmetry of the body with axis is , each wheel has a mass of 1 . The angles of rotation of the wheels relative to the axes that are perpendicular to the planes of the respective wheels and pass through their centers are , and the torques applied to the wheels are ( = 1, … , 4).

Model of a Mecanum wheel
A Mecanum wheel is a wheel with rollers fixed on its outer rim. The axis of each of the rollers forms the same angle (0 • < ≤ 90 • ) with the plane of the wheel. As a rule, the angle is equal to 45 • . Each roller may rotate freely about its axis, while the wheel may roll on the roller. We will model a Mecanum wheel by a thin disk of radius ; the velocity of the point of contact of the disk with the supporting plane is orthogonal to the axis of the roller, see Figure 2. The rollers are densely attached to the circumference of the wheel (disk). The dimensions of the rollers are much less than the diameter of the disk and are comparable with the disk's thickness. Within the framework of these assumptions we can use a model of the Mecanum wheel in which the rollers have infinitesimal dimensions.
Let be the unit vector of the roller's axis. The wheels move without slip, which implies the constraint If is the velocity of the wheel's center , then where is the angular velocity of the wheel and = ⃖⃖⃖⃖⃖⃖ ⃗ . Let be the angle of rotation of the wheel about the axis that is perpendicular to the wheel's plane and passes through its center. Then expression (1) can be represented as follows: where is the unit vector tangent to the wheel's rim at the point of contact. From expression (3) we find

Kinematic constraint equations
In what follows, we assume that = ∕4. Let denote the velocity of the center of mass of the mobile robot and let ⃖⃖⃖⃖⃖⃖ ⃗ = . Then where is the velocity of the center of the respective wheel and is the angular velocity of the body of the robot. Then the kinematic constraint that follows from (4) and represents the condition for rolling without slip along the roller's axis can be written as Here, is the unit vector that points along the axis of the roller of the respective wheel that has contact with the plane at the current time instant. Since constraint equation (6) can be rewritten as follows: Introduce a robot-attached coordinate system with origin at the canter of mass of the cart. We point axis along the longitudinal symmetry axis of the cart, axis along the lateral symmetry axis, and axis vertically upward. Denote by and the projections of the velocity of the center of mass onto the movable axes and , respectively, and represent expression (8) as follows: Since the constraint equations become − − ( + )̇=̇1, The four Equations (11) can also be represented equivalently in form of the following four equations: The character of constraints is essential for deriving the dynamic equations. If the constraints imposed on a mechanical system restrict its position, then these constraints are called holonomic (geometrical) constraints. For the systems subject to holonomic constraints, Lagrange's equations of the second kind can be used. The constraints that restrict the velocities but do not restrict the coordinates are called non-holonomic constraints. Such constraints cannot be integrated and reduced to geometrical constraints. Non-holonomic constraints require different methods for deriving the dynamic equations for the mechanical system. The answer to the question of whether the system of equations that defines constraints is integrable (holonomic) is given by Frobenius theorem. [21] The quantities and are related to the componentṡanḋof the velocity vector of the center of mass in the fixed reference frame by =̇cos +̇sin , Theṅ= cos − sin , Taking into account the first two equations of (12), we can represent (14) as follows: The other two equations (12) can be integrated to obtain = 2( + ) ( 2 − 3 ) + 1 , where 1 , and 2 are constants. Therefore, the system under consideration has two holonomic constraints that allow two generalized coordinates and 4 to be eliminated. As to the constraints of (15), they, with reference to (16), can be represented bẏ Here The fact that constraints (17) are non-holonomic (non-integrable) implies that not all of the six skew-symmetric quantities are equal to zero. For the case under consideration we have which implies that Equations (15) correspond to non-holonomic constraints.

DYNAMIC EQUATIONS OF MOTION OF THE MECHANICAL SYSTEM
Let us assume that the configuration of a mechanical system is defined by generalized coordinates, , = 1, … , of which − coordinates can be expressed in terms of the remaining coordinates by using the non-holonomic constraint equations. Then the non-holonomic constraint equations can be represented as follows: Let the coefficients , + in these equations be functions of only the independent coordinates 1 , … , . Since the constraints are independent of time (i.e. scleronomic), the kinetic energy of the system is expressed by a quadratic form of the generalized velocities. Let the coefficients of this quadratic form depend only on the coordinates 1 , … , . Such a mechanical system is called Chaplygin system, since they can be described by Chaplygin's equations for non-holonomic systems. The equations of motion for such a mechanical system can be represented as follows [22][23][24][25] : where * is the kinetic energy of the system in which the dependent generalized velocitieṡ+ 1 , … ,̇have been expressed in terms of the independent velocitieṡ1, … ,̇by using expressions (21), and are the generalized forces. The additional terms that are accounted by non-holonomic constraints and distinguish these equations from Lagrange's equations of the second kind are given by where is the expression for the kinetic energy of the system in which the dependent generalized velocities have not been expressed in terms of the independent velocities according to (21). Expression (23) implies that the additional terms vanish if the constraints are holonomic. In this case, However, the terms may vanish even if the constraints are non-holonomic. In this case, not all of expressions (24) are equal to zero, but the sum of these expressions multiplied by ∕̇+ vanishes. The respective example is given below.
Chaplygin system allow the dynamic equations of motion to be separated from non-integrable constraint equations, [26] and therefore, the dynamic equations form a closed system of equation. In addition, these equations have a form of Lagrange's equations of the second kind with additional terms due to non-holonomic constraints. These equations are convenient; in particular, they allow identifying special cases where equations of motion for non-holonomic systems coincide in form with Lagrange's equations of the second kind.
The configuration of the system under consideration is defined by seven generalized coordinates: , , , 1 , 2 , 3 , 4 . Since we have four constraint equations, the system has three degrees of freedom. Notice that the coefficients , , = 1, … , in equations of non-holonomic constraints (17) depend only on independent generalized coordinates , = 1, … , (only on 2 , 3 , in our case) and are independent of and . For the mechanical system under consideration, Chaplygin's equations are given by Here, * is the function of the kinetic energy of the system from which the velocities , have been eliminated using Equations (17) for non-holonomic constrains, is the kinetic energy of the unconstrained system, are the generalized forces, and is the number of independent generalized coordinates. The additional (as compared with Lagrange's equations of the second kind) terms vanish, if all quantities and are zero, i.e, if the constraints are holonomic. However, as follows from (25), the terms may vanish also for nonzero and , if the respective sums are equal to zero, i.e., for some cases of non-holonomic constrains. Then Chaplygin's equations coincide in form with Lagrange's equations of the second kind.
To derive the dynamic equations we, first of all, calculate the kinetic energy for the system under consideration. The total kinetic energy is the sum of the kinetic energy 0 of the translational motion, and the kinetic energy 1 of the rotational motion, where 0 is the moment of inertia of the body about the center of mass , 1 is the moment of inertia of the wheel about the axis that is perpendicular to the plane of the wheel and passes through its center of mass, and 2 is the moment of inertia of the wheel about the vertical axis passing through the center of mass of the wheel. Since the kinetic energy of the mechanical system can be represented as where = 0 + 4 1 is the total mass of the mechanical system and = 0 + 4( 2 + 1 ( 2 + 2 )) is the moment of inertia of the system about the vertical axis passing through the center of mass.
We will call the equations of motion for deriving which the non-holonomic character of constraints have been taken into account the exact equations, while the equations derived by another technique that is frequently used in robotics will be called the approximate equations.

An approximate technique for deriving the equations of motion
In this section, we will analyze in detail the technique for compiling dynamic equations of a robot with the Mecanum wheels, which is widely used in robotics.
Consider kinematic constraints (11) as a system of four linear equations for three unknowns, , , anḋ. This system is overdetermined and does not have a solution for arbitrary values oḟ1,̇2,̇3, anḋ4. This system may have a solution only if the equations are linearly dependent. The compatibility condition is given bẏ In a number of studies on robotics, e.g., [27][28][29] the authors, all following and referencing, [14] proceed as follows. Represent the equations of non-holonomic kinematic constraints (11) in matrix forṁ Here, vectoṙhas a dimension of 4 × 1, matrix a dimension of 4 × 3, and vector a dimension of 3 × 1 Premultiply relation (31) by the transpose of the matrix to obtaiṅ The 3 × 3 matrix has the inverse ( ) −1 . Then from (33) we find The 3 × 4 matrix + is called the pseudoinverse of the matrix The values found in such a way do not satisfy system (31) for arbitrarẏ1,̇2,̇3,̇4. However, of all possible triples of quantities , ,̇, the triple of (36) provides a minimum for the sum of the squared discrepancies, i.e., the sum of the squared differences of the left-hand and right-hand sides of the equations of system (31). [30] Of course, if the compatibility relations (30) hold, then the exact solution of the system of linear equations (11) coincides with the solution constructed by using the pseudoinverse matrix: Consider then our mechanical system as a system subject to three constrains (36). In this case, the configuration of the system will be characterized, as previously, by seven generalized coordinates , , , 1 , 2 , 3 , and 4 , but now we have three constraint equations and, hence, the mechanical system has four degrees of freedom. Then the authors of the cited papers use Lagrange's equations of the second kind. Setting aside for a while the issue of applicability of Lagrange's equations of the second kind to systems with constraints in the form of (36) let us write down these equations. The kinetic energy * obtained by substituting expressions (36) into (29), taking into account (28), is given by The respective Lagrange's equations have the form Substitute expression (38) for the kinetic energy and into (40) to obtain Equations (41) are a system of linear equations for the angular accelerations of rotation of the wheels̈, = 1, … , 4. By solving these equations we obtain̈1 and the angular velocity of the body= Then, having determined the angle , we can readily find the coordinates , and of the center of mass from expressions (14). Let us revisit the constraint equations (36). Let us have a mechanical system subjected to such constraints. The third equation can be integrated to obtain the holonomic constraint where 3 is a constant. Therefore, our system has one holonomic constraint. Using (14), we rewrite the remaining two constraint Equations (36) as follows:̇= where ) , For this case, non-holonomicity (non-integrability) of constraints (47) implies that some of the twelve skew-symmetric quantities  ) .
(50) Therefore, relations (36) are equations of non-holonomic constraints. The coefficients and , = 1, … , 4 depend only on 1 , 2 , 3 , 4 and, hence the system under consideration is a Chaplygin system. The additional terms due to nonholonomic constraints have the form where the kinetic energy is defined by expression (29).
Using relations (50) and (14), we find Then, using the constraint equations (36) we obtain Similarly, Finally, the system of dynamic equations, with non-holonomic constraints being taken into account, becomes Unlike system (41), due to the additional terms, the system of Equations (55) is a set of nonlinear differential equations for the angular velocities of rotation of the wheels. By solving these equations with respect to the highest derivative we obtain where 1 =
The angular velocity of rotation of the bodẏiṡ From system (56) it follows that the expression for the angular velocity coincides with expression (45), obtained without taking into account the non-holonomic nature of the constraints. Thus, it is shown that if we use the approximate equations of kinematic constraints (36) obtained using the pseudo-inverse matrix, they still remain non-holonomic and the Lagrange's equations of the second kind are generally unapplicable to such systems.

Exact equations of motion for a non-holonomic system
Consider again the exact system subject to constraints (15) and (16). For this mechanical system, Chaplygin's equations (25) are given by Calculate the kinetic energy * of the system using expression (29) and constraint equations (11). Sincė we obtain * = 1 2 ) .
The additional terms due to nonholonomic constraints are defined by Define now the generalized forces , = 1, 2, 3. Using the first expression of (16), we find Finally, the dynamic equations become By solving these equations with respect to the highest derivative we obtain For given torques ( = 1, … , 4) and initial conditions, we determine the angular velocitieṡ1,̇2,̇3 by integrating the system of differential equations (65). Then we use the constraint equations (12) to finḋ 4 =̇1 +̇2 −̇3, From the system of the Equations (65) follows that expression foṙin case of the exact equations coincides with expression (45) too. Then we use expressions (15) to calculate the velocitieṡ,̇, and the coordinates of the system's center of mass .

COMPARISON OF THE EXACT AND APPROXIMATE TECHNIQUES
We will find out how the exact and approximate solutions relate to each other. Notice first of all that if the system of dynamic equations derived by the approximate method is subject to additional compatibility conditions (30) for constraint equations, then, as mentioned previously, the constraint equations obtained by means of the pseudoinverse matrix coincide with the exact constraint equations. Since these constraints remain non-holonomic, the exact system of the Equations (64) or (65) should be compared with the system of the Equations (55) or (56). Relation (30) imposes a constraint on the torques applied to the wheels (these torques appear on the right-hand side of system (55)): With this constraint, four equations of the systems (55), (56) subject to respective initial conditions are equivalent to three equations of the systems (64), (65) combined with the relation (30). In fact, by adding the first two equations of the system (55) and subtracting from the resulting sum the sum of the remaining two equations we obtain where − 2 − = 1 . This implies for the systems (64), (65) that relations (30) hold if and only if (68) holds. Subject to the condition of (68), the system of dynamic equations (55), (56) with the constraint equations (36) is equivalent to the system of dynamic equations (64), (65) with the constraint equations (12).
Compare now the systems of Equations (41) and (64). For these systems to be equivalent, it is required, apart from relation (68), that the additional terms due to nonholonomic constraints vanish. As follows from Equations (64) this is the case iḟ For the case (70), as follows from the third relation of (12), the body moves translationallẏ Such a motion subjects the torques to the additional constraints: Then the system of equations together with the equations of holonomic constraints becomes where + = 2 ∕4 + 1 .
For the case (71), wherė3 = −̇2 =̇1, the respective motion is a rotation of the body about the center of mass that remains fixed. In fact, using relations (37) we find that = = 0, and then from (14) conclude thaṫ=̇= 0 . The additional constraint for this case is as follows: (75) The system of dynamic equations together with the equations of holonomic constraints becomes where + 2 − = ∕(4( + ) 2 ) + 1 . As a result, we can draw the following conclusions: 1. If the torques applied to the wheels satisfy relation (68), i.e., 1 + 2 = 3 + 4 , then for appropriate initial conditions, the exact system of Equations (64) or (65) is equivalent to the approximate system (55) or (56), where the kinematic relations are obtained by using the pseudoinverse matrix but the non-holonomic constraints (36) are taken into account.
2. If the torques are subjected to relation (73), i.e., 4 = 1 , 3 = 2 , (in this case, relation (68) is satisfied automatically), then the terms accounted for by the non-holonomic constraints disappear, the constraints become holonomic and Lagrange's equations of the second kind become applicable. Then the systems of Equations (42) and (56) coincide and are equivalent to the exact system (65)). In this case, the robot's body moves translationally ( = 0 ).
3. If the conditions imposed on the torques have the form of (75), then relation (68) again is satisfied automatically, since in this case, 4 = − 3 = 2 = − 1 . Then the terms accounted for by non-holonomic constraints are absent, and the motion of the robot is its rotation about the center of mass (̇=̇= 0).
Let us find out, whether the translational motion of the robot's body or its rotation about the center of mass are possible for the exact and approximate models if relation (68) for the torques does not hold and, hence, the exact and approximate systems of equations are not equivalent to each other.
For translational motions (̇= 0), the terms accounted for by non-holonomic constraints disappear and the approximate systems of Equations (42) and (56) coincide. For this case, the third kinematic constraint of (36) implieṡ Then, from the system of Equations (56) we obtain Since 1 − 2 1 + 1 = 1∕( + 2 − ) = 1∕( 1 + 2 ∕( + ) 2 ) ≠ 0, we find In this case, the angular accelerations of the wheels are expressed bÿ In accordance with relations (36), the value of the velocity of the center of mass is defined by Solve Equations (80) subject to zero initial conditions to find We will find a condition, subject to which the system governed by the exact system of Equations (65) moves translationally. From the third and fourth kinematic relations (12) we finḋ Then, the system (65) implies the relation Since 2 − 2 = 1∕(2( + 2 − )) = 1∕(2 1 + 2 ∕(2( + ) 2 )) ≠ 0, the condition for the translational motion of the robot's body coincides with relation (79). In this case, since 2 + 2 = 1 − 1 = 1∕( + ), we obtain and the velocity of the center of mass in accordance with relations (12) is expressed by Solve Equations (85) subject to zero initial conditions to find the same expression (82). Thus, the conditions of translational motion for the exact and approximate models coincide. The velocities of the center of mass and the trajectory of the motion of the center of mass are the same. However, the angular accelerations and the angular velocities of the wheels are different.
Consider now the rotation of the robot about the center of mass. In this case,̇=̇= 0 or = = 0 , and the terms accounted for by nonholonomic constraints disappear from the systems of Equations (42) and (56).
For the approximate model, the kinematic relations (36) implẏ 2 +̇3 = 0,̇1 +̇4 = 0, Then from the equations of motion (42) we conclude that and, hence,̈1 The angular velocity of the rotation about the center of mass subject to zero initial conditions iṡ For the exact model, from the kinematic relations (12) we finḋ and, taking into account system (65), we again arrive at relations (88). For this case,̈1 and the angular velocity of the rotation of the body about the center of mass coincide with (90).
As it was the case for the translational motion, the conditions for the torques, subject to which the body of the robot rotates about the center of mass, coincide for the exact and approximate models. However, the angular accelerations of the wheels and the angular velocity of the rotation of the body about the center of mass are different for these models. Of course, the results of calculations according to the exact and approximate models coincide if the condition of (30) is imposed.
Therefore, the exact equations coincide with the approximate equations if the torques applied to the wheels satisfy appropriate relations and the character of the motion is determined in advance.
In the studies on robotics listed previously, only the translational motion of the body, witḣ= 0 and its rotation about the center of mass witḣ=̇= 0, when the angular velocities satisfy the condition, (30) are considered. For these cases, the constraint equations obtained by means of the pseudoinverse matrix coincide with the exact constraint equations and, in addition, the constraints become holonomic, which enables Lagrange's equations of the second kind to be applied.
For these cases, all constraints on the torques presented above are valid. For this reason, the results obtained by using the pseudoinverse matrix and Lagrange's equations of the second kind appear to be correct.

Calculations according to the exact and approximate models
Consider a particular case where all torques ( = 1, … , 4) applied to the wheels are constant. Let us find the trajectory of the system for this case on the basis of the approximate model. Solve Equations (44) subject to zero initial conditions to find where = 4( + ) ( 1 + 2 + 3 + 4 ), Since Equations (93) subject to zero initial conditions imply The trajectory of the center of mass in this case is a circumference Therefore, for any constant torques ( = 1, … , 4) and zero initial conditions, the trajectory according to the approximate model is a circumference of radius This means that the center of mass of the system moves along a circumference with a velocity that is increasing in magnitude: | ⃗ | =  Figure 5 shows the solution of the linear system that corresponds to the approximate model, the nonholonomic constraints are not being taken into account. For this case, the trajectory of the center of mass for arbitrary constant torques applied to the wheels (102) is a circumference (Figure 5a). The dependencies of coordinates and and angular velocities of wheels ( = 1, … , 4) on time are presented on Figures 5b, and 5c, respectively. As it was already noted, the dependence robot's body angular rotation on time coincides with Figure 4c. The experiments with the prototype, qualitatively confirms the calculations on the basis of the exact nonholonomic model.

A REMARK ABOUT LAGRANGE'S EQUATIONS OF THE SECOND KIND AND NON-HOLONOMIC CONSTRAINTS
The holonomic nature of the constraints imposed on a mechanical system is a sufficient condition subject to which Lagrange's equation of the second kind can be applied. However, it is not a necessary condition. As has already been mentioned, the additional terms in Chaplygin's equations may vanish even if not all coefficients , , are equal to zero. In this case, despite the constraints are nonholonomic, equations of motion coincide with Lagrange's equation of the second kind. Such cases are known but occur seldom. [31] As an example, consider the rolling of a wheel pair along a plane. Both wheels are conventional, have the same mass 1 and the same radius . The wheels are set on the common axle that has a mass of 0 , and a length of 2 and can freely rotate about this axle. Let , be the coordinates of the axle midpoint and let 1 , 2 denote the angles of rotation of the wheels. The conditions for this system to roll without slip can be represented as follows [32] : cos +̇sin −̇=̇1, cos +̇sin +̇=̇2, −̇sin +̇cos = 0. The configuration of the system is characterized by five generalized coordinates , , , 1 , 2 ; the system is subject to three constraints (105) and, hence, has two degrees of freedom. The constraint equations (105) can be represented as follows: The third equation of (106) characterizes the holonomic constraint where = 0 + 2 1 is the total mass of the mechanical system, = 0 + 2( 2 + 1 2 ), and 0 is the moment of inertia of the axle about its midpoint, 1 and 2 are the moments of inertia of the wheels. are not equal to zero and, hence, the first two constraints of (106) are non-holonomic. However, the additional terms in Chaplygin's equations vanish. In fact, 1 =̇2 1̇2 +̇2 1̇2 = (̇2 1 +̇2 1 )̇2 = 3 8 (̇1 +̇2)(cos sin − sin cos )̇2 = 0.
Similarly, 2 = 0. Therefore, for the case under consideration, the equations of motion coincide with Lagrange's equations of the second kind, although the constraints imposed on the system are non-holonomic.

CONCLUSION AND THE FUTURE WORK
The condition subject to which a robot with four Mecanum wheels moves without slip leads to non-holonomic constraints. To describe the dynamics of such a system one should use equations of motion that are appropriate for mechanical systems with nonholonomic constraints, for example, Chaplygin's equations, Voronets's equations, Appel's equations, Lagrange's equations with multipliers (Lagrange's equations of the first kind), etc. Lagrange's equations of the second kind do not apply to non-holonomic systems in the general case. Apparently, for Chaplygin systems, Chaplygin's equations should be preferred, since in this case, the dynamic equations form a closed system with respect to the generalized velocities treated as independent variables. The holonomic character of the constraints is a sufficient condition for applicability of Lagrange's equations of the second kind but it is not a necessary condition. Therefore, the additional terms that distinguish Chaplygin's equations from Lagrange's equations of the second kind may vanish for some systems with non-holonomic constraints. However, such occurrences are rather rare. In particular, this is not the case for a robot with four Mecanum wheels. In the general case, solving the constraint equations for a part of the generalized velocities by using the pseudoinverse matrix reduces the mechanical system under consideration to a system that is not equivalent to the original system, because the number of degrees of freedom of the reduced system is larger than the number of degrees of freedom of the original system. However, if we confine our consideration to certain special types of motions, e.g., translational motion of the robot or its rotation relative to the center of mass, and impose appropriate constraints on the torques applied to the wheels, the solution obtained by means of the pseudoinverse matrix will coincide with the exact solution. In these cases, the constraints imposed on the system become holonomic constraints, which justifies using Lagrange's equations of the second kind. It is just the motions and constraints that are considered in the papers on robotics cited above, however, it is not stated explicitly. In the general case, the mathematical methods of non-holonomic mechanics should be used.
Subsequent studies are expected to evaluate the effect of the finite linear dimensions of the rollers and the associated body vibrations.