S S symmetry
Article Topological Design Methods for Mecanum Wheel Configurations of an Omnidirectional Mobile Robot
Yunwang Li 1,2,*, Sumei Dai 2,3, Lala Zhao 1, Xucong Yan 1 and Yong Shi 2
1 School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China; [email protected] (L.Z.); [email protected] (X.Y.) 2 Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA; [email protected] (S.D.); [email protected] (Y.S.) 3 School of Mechanical and Electrical Engineering, Xuzhou University of Technology, Xuzhou 221018, China * Correspondence: [email protected] or [email protected]
Received: 20 August 2019; Accepted: 5 October 2019; Published: 10 October 2019
Abstract: A simple and efficient bottom-roller axle intersections approach for judging the omnidirectional mobility of the Mecanum wheel configuration is proposed and proved theoretically. Based on this approach, a sub-configuration judgment method is derived. Using these methods, on the basis of analyzing the possible configurations of three and four Mecanum wheels and existing Mecanum wheel configurations of robots in practical applications, the law determining wheel configuration is elucidated. Then, the topological design methods of the Mecanum wheel configurations are summarized and refined, including the basic configuration array method, multiple wheels replacement method, and combination method. The first two methods can be used to create suitable multiple-Mecanum-wheel configurations for a single mobile robot based on the basic Mecanum wheel configuration. Multiple single robots can be arranged by combination methods including end-to-end connection, side-by-side connection, symmetrical rectangular connection, and distributed combination, and then, the abundant combination configurations of robots can be obtained. Examples of Mecanum wheel configurations design based on a symmetrical four-Mecanum-wheel configuration and three centripetal configurations using these topological design methods are presented. This work can provide methods and a reference for Mecanum wheel configurations design.
Keywords: Mecanum wheeled robot; omnidirectional motion; wheel configuration; symmetrical configuration; topological design method
1. Introduction Each Mecanum wheel has three degrees of freedom of motion in a plane [1,2], so a mobile robot system consisting of three or more than three Mecanum wheels can achieve omnidirectional motion in a plane only through the coordination of direction and rotation speed of wheels without the assistance of an auxiliary steering mechanism. Because of the simple structure and good motion flexibility, omnidirectional mobile robots with Mecanum wheels are widely used in various fields. According to application needs in different fields, a variety of Mecanum wheel configurations can be designed to develop various omnidirectional mobile robots. Some service robots usually adopt three or four-Mecanum-wheel configurations [3,4]. In the industrial field, an AGV (automated guided vehicle) with four Mecanum wheels, a kind of omnidirectional mobile robot, is also widely used [5–8]. For transporting large-scale equipment or components, a robot platform with multiple Mecanum wheels [9–11] or multiple-Mecanum-wheeled robot platforms are used cooperatively [12,13]. In order to design an omnidirectional mobile robot, it is necessary to select a reasonable Mecanum wheel
Symmetry 2019, 11, 1268; doi:10.3390/sym11101268 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1268 2 of 27 configuration for the robot. However, not all combinations of Macanum wheels can implement omnidirectional motion, and the arrangement of Mecanum wheels also influences the mobility performance of the robot [14]. Therefore, designing a reasonable configuration of Mecanum wheels constitutes the most basic and important technology problem in the design of omnidirectional mobile robots. Firstly, these configurations must satisfy the conditions of realizing omnidirectional movement. Secondly, motion performance, controllability, and structural rationality of these configurations must be evaluated in order to select the optimal Macanum wheel configuration. Some researchers have paid attention to the study of Mecanum wheel configurations. The kinematics and dynamics of a Mecanum-wheeled mobile robot form the basic premise to judge the robot to achieve omnidirectional movement in theory. Muir et al. [15,16] introduced a methodology for the kinematic modeling of wheeled mobile robots, studied an omnidirectional wheeled mobile robot with four Mecanum wheels, and derived the kinematic model of roller angle dead reckoning robot position wheel slip. Angeles [17] deduced a general kinematics model of the Mecanum-wheeled omnidirectional mobile system by vector method, and gave kinematics and dynamics equations of three-wheel and four-wheel robots, respectively. Campion [18] used a matrix transformation method to study the mobility characteristics of the robot under constraints, gave a unified description of modeling of a wheeled mobile robot, and deduced the kinematics equation of the three-wheeled robot. Gracia and Tornero [19,20] described the singular and heterogeneous types of mobile robots based on Mecanum wheels and Castor wheels using a descriptive geometry method, established the kinematics model of omnidirectional mobile robots under sliding conditions, and further established the Lagrange dynamics model. Zhang and Wang [21,22] analyzed the steering motion of a Mecanum-wheeled omnidirectional mobile platform, and established a control system model and dynamic model in MATLAB and RecurDyn software, respectively. Using joint simulation, the anisotropic motion characteristics of a mobile platform with different slip rates were obtained. Wang and Chang [23,24] analyzed the condition of omnidirectional motion of a Mecanum-wheeled mobile system and found that the Jacobian matrix of inverse kinematics velocity is a column full rank, discussed the possible singularities and solutions in motion, and showed six types of Mecanum wheels layouts and determined the four best Mecanum wheel layouts. Mishra et al. [25] proposed 10 possible configurations of the omnidirectional mobile robot with four Mecanum wheels. Gao et al. [26] studied a type of three-Mecanum-wheel omnidirectional mobile robot with symmetrical and concentric layout structure, and the motion simulation of the three-Mecanum-wheeled platform is compared with that of the symmetrical four-Mecanum-wheeled mobile robot platform. Zhang et al. [27] studied the three- and four-Mecanum-wheeled concentric layouts and analyzed the influence of the angles between wheel axes for a centered layout. He et al. [14] studied the two most used arrangement modes of Mecanum wheels, Type-X and Type-O, and used the inverse velocity Jacobian matrix of the arrangement to judge whether a vehicle can fulfill omnidirectional movement. The main contributions of these studies on wheel configuration include: (1) the method of establishing a kinematics equation of an omnidirectional mobile robot is proposed; (2) the method of judging omnidirectional mobility by rank of the Jacobian matrix of inverse kinematics is obtained; (3) the possible configuration of three or four Mecanum wheels is summarized and analyzed and compared. However, when using an inverse kinematics Jacobian matrix to analyze a multiple-Mecanum-wheeled mobile robot system, the calculation process is complex. Previous studies have not systematically summarized multiple-Mecanum-wheel (more than four wheels) configurations, and have not explicitly proposed a method to obtain the wheel configurations for omnidirectional mobile robots with more than four Mecanum wheels. This study explores a simple and efficient method to judge whether the wheel configurations possess omnidirectional mobility. On this basis, the common wheel configurations are judged and analyzed, the law of wheel configurations is explored, and the topological design methods of wheel configurations for an omnidirectional mobile robot are summarized and refined. This paper is organized as follows: In Section2, on the basis of studying the kinematic constraints of a single Mecanum wheel in a mobile system, the kinematics model of an n-Mecanum-wheeled mobile Symmetry 2019, 11, 1268 3 of 27
Symmetry 2019, 11, x FOR PEER REVIEW 3 of 29 robot is further deduced. In Section3, the relationship between the intersections of bottom-rollers kinematics of the mobile robot is established, and a bottom-rollers axles intersections approach for axles of any three Mecanum wheels and the column rank of the Jacobian matrix of inverse kinematics judging the omnidirectional mobility of Mecanum wheel configurations is proposed and proved of the mobile robot is established, and a bottom-rollers axles intersections approach for judging the theoretically, which is a simple and efficient geometric method. In Section 4, the four-Mecanum- omnidirectional mobility of Mecanum wheel configurations is proposed and proved theoretically, which wheel configurations are judged by using a bottom-rollers axles intersections approach, and the is a simple and efficient geometric method. In Section4, the four-Mecanum-wheel configurations are optimal four-Mecanum-wheel configuration is selected through comprehensive analysis and judged by using a bottom-rollers axles intersections approach, and the optimal four-Mecanum-wheel theoretical verification. In Section 5: firstly, the above method is used to judge the omnidirectional configuration is selected through comprehensive analysis and theoretical verification. In Section5: motion of a combined configuration consisting of two four-Mecanum-wheel configurations, and then firstly, the above method is used to judge the omnidirectional motion of a combined configuration the sub-configuration judgment method, which can be extended to N sub-configuration combinations consisting of two four-Mecanum-wheel configurations, and then the sub-configuration judgment is obtained. Secondly, this judgment method is used to analyze the Mecanum wheel configurations method, which can be extended to N sub-configuration combinations is obtained. Secondly, this and combination configurations for common omnidirectional mobile robots, and clarify the law judgment method is used to analyze the Mecanum wheel configurations and combination configurations determining wheel configuration. Finally, the topological design methods of the Mecanum wheel for common omnidirectional mobile robots, and clarify the law determining wheel configuration. configurations are summarized and refined, including the basic configuration array method, multiple Finally, the topological design methods of the Mecanum wheel configurations are summarized and wheel replacement method and combination method. Furthermore, based on the symmetrical four- refined, including the basic configuration array method, multiple wheel replacement method and Mecanum-wheel configuration, the Mecanum wheel configurations are generated by using the combination method. Furthermore, based on the symmetrical four-Mecanum-wheel configuration, the topological design methods. Mecanum wheel configurations are generated by using the topological design methods.
2. Kinematics2. Kinematics Model Model of anof Omnidirectionalan Omnidirectional Mobile Mobile Robot Robot with withn Mecanum n Mecanum Wheels Wheels
2.1.2.1. Mecanum Mecanum Wheel Wheel Configurations Configurations of the of Singlethe Single Omnidirectional Omnidirectional Mobile Mobile Robot Robot ForFor an independentan independent Mecanum-wheeled Mecanum-wheeled mobile mobile robot, robot, the the wheel wheel configurations configurations can can be mainlybe mainly divideddivided into into two categories:two categories: centripetal centripetal configuration configuration and symmetrical and symmetrical rectangular rectangular configuration configuration [24], as shown[24], as in shown Figure 1in. InFigure Figure 1.1 In, the Figure Mecanum 1, the wheelMecanum is represented wheel is represented by a rectangle by witha rectangle an oblique with an lineoblique in the middle, line in inthe which middle, the in oblique which line the represents oblique line the represents bottom roller the thatbottom contacts roller with that the contacts ground. with In thethe former ground. configuration, In the former theconfiguration, axles of all wheelsthe axles intersect of all wheels at the intersect same intersection at the same point, intersection as shown point, in Figureas shown1a. In in Figure Figure1a, 1a. the In centerline Figure 1a,OO thei of centerline the mobile OO roboti of coordinate the mobile system robotXOY coordinateand wheel system local coordinate system XiOiYi is collinear with coordinate axis Xi. In order to balance the load of each XOY and wheel local coordinate system XOYiii is collinear with coordinate axis Xi . In order to wheel, the wheels are evenly distributed in a 360◦ circumference. This centripetal configuration of an omnidirectionalbalance the load mobile of roboteach wheel, usually the composes wheels ofare three even [1ly,26 distributed] or four [27 in] Mecanum a 360° circumference. wheels. In the This symmetricalcentripetal rectangular configuration configuration of an omnidirectional in Figure1b, themobile Mecanum robot usually wheels composes are symmetrically of three arranged[1,26] or four on both[27] sidesMecanum of the wheels. line going In throughthe symmetrical the center rectangular of the robot, configuration and the overall in structureFigure 1b, is rectangular.the Mecanum Basedwheels on the are study symmetrically of the kinematics arranged constraints on both sides of a singleof the Mecanumline going wheel,through the the kinematics center of modelthe robot, of anandn-Mecanum-wheel the overall structure mobile is rectangu robot canlar. be Based further on derived,the study and of the then kinematics the omnidirectional constraints motionof a single characteristicsMecanum wheel, of the mobilethe kinematics systems model can be of analyzed. an n-Mecanum-wheel mobile robot can be further derived, and then the omnidirectional motion characteristics of the mobile systems can be analyzed. Mecanum wheel Y Y i Bottom roller β i X i
Yi O ϕ i γ &i i γ i X i ϕ& i l Y i θ& li Oi α θ& i α i X X O O
(a) (b)
FigureFigure 1. Wheel 1. Wheel configurations configurations of the of single-Mecanum-wheeledthe single-Mecanum-wheeled robot: robot: (a) centripetal(a) centripetal configuration; configuration; (b) symmetrical(b) symmetrical rectangular rectangular configuration. configuration.
Symmetry 2019, 11, 1268 4 of 27
2.2. Kinematics Constraint Model of a Single Mecanum Wheel and Kinematics Model of an n-Mecanum- Wheel Robot The kinematics research of the Mecanum wheel is similar to that of a traditional wheeled mobile system. The kinematics model of the Mecanum-wheel mobile system can also be built by a bottom-up process. Each of the relatively independent Mecanum wheels contributes to the motion of the system and is relatively constrained by the motion of the system. Because the Mecanum wheels are installed on the chassis of a mobile system, the kinematic constraints of each wheel can be combined to describe the kinematic constraints of the whole mobile system. In this section, the kinematic constraints of a single Mecanum wheel are studied first, and then, the linear mapping relationship between the velocity of the mobile system and the velocity of a single wheel is obtained. Then, the kinematic constraints of each wheel are combined to describe the kinematic constraints of the entire mobile system. In order to reduce the difficulty of system kinematics modeling, several assumptions are usually introduced to discuss the motion constraint relationship of wheels under ideal conditions. (1) Assuming that the whole mobile robot, especially the wheels, is rigid, it will not undergo mechanical deformation; (2) the entire range of motion is confined to a 2D plane, ignoring the impact of irregular ground; (3) ignoring the factor of rollers slipping, that is, the roller has enough friction with the ground; (4) assuming that the contact point between the roller and the ground is located directly below the wheel center. Based on the above assumptions, the kinematic constraints of a single Mecanum wheel will be derived by a vector method [17] and matrix transformation method [18]. (a) Vector Method In order not to lose generality, a mobile robot consisting of n Mecanum wheels is designed, in which the i-th wheel is mounted on the body at a certain angle, as shown in Figure2. R and r are the radius of the wheel and the radius of the roller, respectively; Oi is the center of the i-th wheel; Zi represents the direction passing through the wheel center Oi and perpendicular to the ground; Pi is the center of the roller contacting the ground, Qi is the contact point between the roller and the ground, according to the hypothesis, both of them are under Oi at the same time; zi represents the direction passing through the roller center Pi and perpendicular to the ground. Xi and hi represent the rotation axis direction of an active Mecanum wheel and passive roller, respectively. The two angular . . velocity vectors are ϕi and φi, and Xi and Yi constitute the right-handed Cartesian coordinate system O X Y Z , h and g constitute the right-handed Cartesian coordinate system P g h z . (l , α ) is used i − i i i i i i − i i i i i to describe the relative installation orientation of the origin O of the body coordinate system and the center Oi of the wheel; the angle between the Xi axis and the li is βi, which is defined as the installation attitude angle of the local coordinate system of the wheel; the velocity of the motion center is vo in the . current state, and the angle between the vo and the X axis is θo; θ is the rotation angular velocity of the system when moving in the plane. The angle between the projection of Xi and hi on the plane is the tilt angle γi(0◦ < γi < 90◦) of the roller. According to the above definition, the motion relationship between the active wheel and the passive roller can be expressed by the formula
voi = vpi + vi. (1)
In this formula, voi is the velocity vector of the center of the i-th wheel; vpi is the velocity vector of the roller in contact with the ground; vi is the relative velocity vector of point Pi and Oi. SymmetrySymmetry2019 2019,,11 11,, 1268 x FOR PEER REVIEW 55 of 2729
Y X i r β γ i hi i Zi ϕ &i R Yi Yi
O ϕ i &i gi vo O X i zi i li pi θ o α θ& i γ gi Bottom roller i
hi Qi O X (a) (b) Y X YI y i ′ Y X ′ θ& h li i γ β i i ψ ϕ x &i Yi Y O′()o D y& ϑ vgi Oi X θ vR= ϕ& yI l ii i O α θ& i α + β ii
O OI xI x& X X I
(c) (d)
FigureFigure 2.2. KinematicKinematic constraintsconstraints ofof aa MecanumMecanum wheelwheel andand thethe coordinatecoordinate systemssystems ofof aa mobilemobile system:system: ((aa)) structuralstructural principleprinciple ofof aa MecanumMecanum wheel;wheel; ((bb)) KinematicKinematic constraintsconstraints diagramdiagram ofof aa MecanumMecanum wheelwheel onon thethe robotrobot usingusing aa vectorvector method;method; ((cc)) KinematicKinematic constraintconstraint diagramdiagram ofof aa MecanumMecanum wheelwheel usingusing aa matrixmatrix transformationtransformation method; method; (d ()d Location) Location of of the the mobile mobile robot robot in thein the global global coordinate coordinate system system and and the relationshipthe relationship regarding regarding position position between between two localtwo local coordinate coordinate systems. systems.
ω and ω represent the rotational angular velocity vectors of the active wheel and the passive ωo and ωp represent the rotational angular velocity vectors of the active wheel and the passive roller, respectively,o p as roller, respectively, as . . . ωo = θZi + ϕiXi, ωp = ωo + φihi. (2) then ω =+θϕ& & ωω=+φ& oiii. ZX. , poiih . . (2) v = ωp Q P = r(ϕ Y φ g ), v = ωo P O = (R r)ϕ Y . (3) pi × i i − i i − i i i × i i − − i i thenFrom Formulas (1) and (3), we obtain vQPYg=×ω =−−r()ϕφ& & . vPOY=×. ω =−−()Rrϕ& pi p i iv i= iR iϕ i Y, +ioiirφ g . ii. (3) (4) oi − i i i i From Formulas (1) and (3), we obtain If the known moving system moves in the plane, the relation between the wheel center Oi and the origin O of the body coordinate system can be=− expressedϕφ + as& vYgoiRr& i i i i . (4) . v = v + l If the known moving system moves in oithe ploane,θξ thei. relation between the wheel center Oi and (5) the origin O of the body coordinate" system# can be expressed as 0 1 In this formula,ξ = − , which means that the vector l is rotated 1 0 =+θ&ξ i vvoi o l i . (5) 90 degrees counterclockwise.
Symmetry 2019, 11, 1268 6 of 27
The following formula can be obtained from Formulas (4) and (5).
. . . Rϕ Y + rφ g = vo + θξl . (6) − i i i i i According to the definition of vectors, we can obtain
h g = 0, h Y = sin γ . (7) i · i i · i i . Since the roller rotates passively, its angular velocity φi is an uncontrollable variable. According to the calculation result defined by the vector in Formula (7), multiplying the vector hi at the same time . on both sides of Formula (6), the subformula containing the term φi can be eliminated. . . T T R(sin γ )ϕ = h vo + h θξl . (8) − i i i i i Then, the inverse kinematics equation of the i-th Mecanum wheel is . . 1 h T T i θ ϕi = hi ξli hi . (9) −R(sin γi) vo
Given the kinematic constraint equation of any Mecanum wheel in the plane, the inverse kinematics equation of the omnidirectional motion system composed of n Mecanum wheels whose radii are R can be expressed as . h . . . iT = ϕ ϕ1 ϕ2 ϕn ··· S = diag 1 , 1 , , 1 sin γ1 sin γ2 sin γn ··· T T h1 ξl1 h1 T T h ξl h 2 2 2 M = . . . (10) . . . . T T hn ξln hn 1 J = SM − R h . iT t = θ vo . ϕ = Jt . In the formula, ϕ is the angular velocity matrix of the wheel; J is the Jacobian matrix of the inverse kinematics velocity of the mobile robot, including the matrix S of tilt angle of rollers and the matrix M of wheel installation orientation; t is the rotation matrix of the mobile system. In this section, three coordinate transformation matrices—including translation transformation, rotation transformation, and composite transformation—are introduced, which form an important theoretical basis for studying the kinematics constraints of mobile systems. The kinematic constraints of a single Mecanum wheel are derived by the vector method. On this basis, the general kinematic model of the mobile system composed of n Mecanum wheels is obtained. (b) Matrix Transformation Method Matrix transformation is another common method for kinematics analysis of a wheeled mobile system, which can be used for kinematics modeling of an omnidirectional wheel. The precondition of using this method to study a single Mecanum wheel still needs to satisfy the above assumptions and start with the study of rolling and sliding constraints of the wheel. The motion constraints of one Mecanum wheel are shown in Figure2c [23,24]. Based on the above assumptions, the motion between the roller and the ground satisfies the condition of pure rolling, the contact point between the roller and the ground does not slip, and the Symmetry 2019, 11, 1268 7 of 27 instantaneous velocity is 0. According to the constraints of rolling and sliding, the following formulas can be obtained . . . . x sin(αi + βi) y cos(αi + βi) liθ cos βi = Rϕi vgi cos γi . − . − . − . (11) x cos(α + β ) + y sin(α + β ) + l θ sin β = v sin γ i i i i i i − gi i
. . . T In the formula, x y θ is the motion state of the mobile system in its own local coordinate system; vgi is the central velocity of the roller contacting the ground on the i-th Mecanum wheel. Because the rollers rotate passively, the velocity of motion vgi is an uncontrollable variable, which is usually not taken into account. By eliminating vgi from Formula (11), we obtain
. . . . x cos(α + β + γ ) + y sin(α + β + γ ) + l θ sin(β + γ ) = Rϕ sin γ . (12) i i i i i i i i i − i i The inverse kinematics matrix equation of any Mecanum wheel is . x h i . . cos(αi + βi + γi) sin(αi + βi + γi) li sin(βi + γi) y + Rϕ sin γi = 0. (13) . i θ
The motion state in a local coordinate system can be mapped to a global coordinate system, as shown in Figure2d, which is expressed as
h i . . cos(αi + βi + γi) sin(αi + βi + γi) li sin(βi + γi) Rot(θ)ζI + Rϕi sin γi = 0. (14) where cos θ sin θ 0 Rot(θ) = sin θ cos θ 0 . − 0 0 1 . x . . 1 1 . ζI = Rot− (θ)ζ = Rot− (θ) y . . θ
3. Bottom-Roller Axle Intersections Approach for Judging Robot’s Omnidirectional Mobility
3.1. Conditions for Omnidirectional Motion of a Mecanum-Wheeled Mobile Robot System If the Mecanum wheel configuration of a robot cannot achieve omnidirectional movement, it will lose practical value. Therefore, it is necessary to study the relationship between the wheel configuration and the realization of the omnidirectional movement of the mobile system. The inverse kinematics velocity Jacobian matrix of a mobile system consisting of n(n 3) Mecanum wheels is Jn 3. According ≥ × to the kinematics principle of the robot, if the Jacobian matrix is a column full rank matrix, that is, rank(Jn 3) = 3, the mobile robot system will have three degrees of freedom in the plane. The Jacobian × matrix Jn 3 is written into the form of block matrix, which is expressed as × ! J3 3 Jn 3 = × . (15) × J(n 3) 3 − ×
Assuming that the third-order square matrix J3 3 is an invertible matrix, i.e., rank(J3 3) = 3. × × According to the elementary transformation theory of a matrix, the simplest matrix of the reversible matrix J3 3 is the unit matrix I3 of the third order. ×
J3 3 I3. (16) × → Symmetry 2019, 11, 1268 8 of 27
Extending this conclusion to the whole Jacobian matrix Jn 3, then × ! I3 Jn 3 . (17) × → (n 3) 3 − × Therefore, in the mobile system composed of n Mecanum wheels, the system can achieve omnidirectional movement, as long as the inverse kinematics velocity Jacobian matrix of any three wheels is a column full rank matrix. According to the basic theory of coordinate transformation, when the coordinate system changes, the description of the motion state of the mobile system will change accordingly, and the Jacobian matrix of its inverse kinematics velocity will change, which can be expressed by the following formula
J0n 3 = Jn 3K3 3. (18) × × × where, J0n 3 is the inverse kinematics velocity Jacobian matrix in the new coordinate system; Jn 3 is the × × inverse kinematics velocity Jacobian matrix in the original coordinate system; K3 3 is reversible square × matrix of the third order, then, rank(K3 3) = 3. × Let C = JK, given K is an invertible matrix, K , 0, the inverse matrix exists, then CK 1 = J. | | − According to the properties of matrices—the rank of the product of the matrices is not greater than the rank of each matrix—the following formulas can be derived ( rank(JK) rank(J) ≤ . (19) rank(J) = rank(CK 1) rank(C) = rank(JK) − ≤ According to the above formula, the rank of the product of J and K is equal to the rank of J, that is
rank(J0) = rank(JK) = rank(J). (20)
The above deduction shows that the change of the coordinate system will not change the rank of the Jacobian matrix in the mobile system. Under certain circumstances, the appropriate coordinate system can be selected to simplify the calculation of the Jacobian matrix rank.
3.2. Relation Between the Roller Axle Intersection Points Number on Three Mecanum Wheels and the Column Rank of the Jacobian Matrix The two straight lines in the plane have three positional relations: parallel, intersection, and coincidence, and the corresponding number of intersections is 0, 1, and infinite. In a plane, the number of intersections of three roller axles on three Mecanum wheels is 0, 1, 2, 3, and infinite. Next, we will discuss the relationship between the number of intersections and the rank of the Jacobian matrix. That is, the relationship between Mecanum wheel configurations and omnidirectional motion is studied. In this paper, infinite intersection points are specialized into one intersection point, which is discussed in detail below.
3.2.1. No Intersection of the Three Bottom-Rollers Axles In Figure3, the axles of any two bottom-rollers are parallel to each other, and the number of intersections is 0. The mobile system coordinate XOY is established by choosing any point on one of the roller axles as the origin, and then the local wheel coordinate systems XiOiYi (i = 1, 2, 3) are established in counterclockwise order. (αi, βi, li) is used to describe the positional state of each wheel relative to the coordinate system XOY of the mobile robot system. The radius of the Mecanum wheel is R, and the tilt angle of rollers of each Mecanum wheel is γi. The relationship of the parameters is shown in Table1. Symmetry 2019, 11, x FOR PEER REVIEW 9 of 29
The two straight lines in the plane have three positional relations: parallel, intersection, and coincidence, and the corresponding number of intersections is 0, 1, and infinite. In a plane, the number of intersections of three roller axles on three Mecanum wheels is 0, 1, 2, 3, and infinite. Next, we will discuss the relationship between the number of intersections and the rank of the Jacobian matrix. That is, the relationship between Mecanum wheel configurations and omnidirectional motion is studied. In this paper, infinite intersection points are specialized into one intersection point, which is discussed in detail below.
3.2.1. No Intersection of the Three Bottom-Rollers Axles In Figure 3, the axles of any two bottom-rollers are parallel to each other, and the number of intersections is 0. The mobile system coordinate XOY is established by choosing any point on one of = the roller axles as the origin, and then the local wheel coordinate systems X iiiOY ( i 1, 2, 3 ) are α β established in counterclockwise order. (,,)iiil is used to describe the positional state of each wheel relative to the coordinate system XOY of the mobile robot system. The radius of the Mecanum γ wheelSymmetry is2019 R, ,and11, 1268 the tilt angle of rollers of each Mecanum wheel is i . The relationship of9 ofthe 27 parameters is shown in Table 1. Y
Y1
X 2 γ β 1 γ O 1 2 1 β α l 2 2 1 X1 α 1
Y2 O 2 l2 α O X 3 Y3
l3 γ 3
O3 β X 3 3 Figure 3. Wheel configuration of the mobile robot with three Mecanum wheels whose bottom-roller Figure 3. Wheel configuration of the mobile robot with three Mecanum wheels whose bottom-roller axles are parallel to each other. axles are parallel to each other. Table 1. Relationship between the parameters of the three Mecanum wheels in Figure3. Table 1. Relationship between the parameters of the three Mecanum wheels in Figure 3 Serial Number |α | [0 ,360 ) |β | [0 , 180 ] |γ | (0 , 90 ) i ∈ ◦ ◦ i ∈ ◦ ◦ i ∈ ◦ ◦ Serial number α ∈[0°,360°) β ∈[0°，180°] γ ∈(0°，90°) 1 i α1 i β1 i γ1 2 α β γ 1 α 2 2β 2 γ 3 1α β 1 γ 1 α 3 β3 3 γ 2 2 2 2 α β γ The origin O of3 the coordinate system3 XOY is located3 on the bottom-roller3 axle of wheel O1, and the axles of any two rollers are parallel to each other, so the following relationship is established as The origin O of the coordinate system XOY is located on the bottom-roller axle of wheel O1 , and the axles of any two rollersβ1 are+ γ parallel1 = 0, α toi + eachβi + other,γi = c soi ( ithe= 1,following 2, 3). relationship is established (21) as The tilt angles of the axles of the three rollers are the same, so let c = α . βγ+= α ++=βγ = i 1 From Formulas (10), (13),11 and (21),0 the， inverseiiii kinematicsci ( velocity 1,2,3) Jacobian. matrix of the system(21) is obtained as cos α1 sin α1 l1 sin(β1+γ1) c = α The tilt angles of the axles of the three rollers are the same, so let i 1 . sin γ1 sin γ1 sin γ1 1 cos α sin α l2 sin(β2+γ2) 1 From Formulas (10), (13),J = and (21), the1 inverse1 kinematics velocity = SMJacobian. matrix of the system (22) R sin γ2 sin γ2 sin γ2 R is obtained as − ( + ) − cos α1 sin α1 l3 sin β3 γ3 sin γ3 sin γ3 sin γ3
where ! cos α1 sin α1 l1 sin(β1 + γ1) 1 1 1 S = M = l ( + ) diag , , , cos α1 sin α1 2 sin β2 γ2 . sin γ1 sin γ2 sin γ3 cos α1 sin α1 l3 sin(β3 + γ3) The roller tilt angle matrix S is a reversible square matrix of the third order. The rank of inverse kinematics velocity Jacobian matrix J depends on the matrix M that describes the installation orientation information of the Mecanum wheel, that is, rank(J) = rank(SM) = rank(M). According to Formula (22), the following formula can be obtained.
det(M) = 0, rank(M) , 3. (23)
According to the multiplication theorem of the determinant, we obtain
det(J) = 0, rank(J) , 3. (24)
According to the above analysis, in a mobile system composed of three Mecanum wheels, if the axles of rollers are parallel to each other and the number of intersection points is 0, then the mobile Symmetry 2019, 11, x FOR PEER REVIEW 10 of 29
αα βγ+ cos11111 sinl sin( ) sinγγ sin sin γ 11 1 11cosαα sinl sin( βγ+ ) JSM=−11222 =− γγ γ . (22) RRsin22 sin sin 2 αα βγ+ cos11 sin l333sin( ) γγ γ sin33 sin sin 3 where αα βγ+ cos11111 sinl sin( ) = 111 =+αα βγ S diag , , , Mlcos11222 sin sin( ) . sinγγγ sin sin 123 αα βγ+ cos11333 sinl sin( )
The roller tilt angle matrix S is a reversible square matrix of the third order. The rank of inverse kinematics velocity Jacobian matrix J depends on the matrix M that describes the installation orientation information of the Mecanum wheel, that is, rank(JSMM )== rank( ) rank( ) . According to Formula (22), the following formula can be obtained.
det(M )= 0 , rank(M )≠ 3 . (23)
According to the multiplication theorem of the determinant, we obtain
det(J )= 0 , rank(J )≠ 3 . (24)
SymmetryAccording2019, 11, 1268 to the above analysis, in a mobile system composed of three Mecanum wheels,10 if of the 27 axles of rollers are parallel to each other and the number of intersection points is 0, then the mobile systemsystem hashas singularity.singularity. TheThe inverseinverse kinematicskinematics velocityvelocity JacobianJacobian matrixmatrix ofof thethe systemsystem isis notnot aa columncolumn fullfull rankrank matrix,matrix, soso thethe mobilemobile systemsystem cannotcannot achieveachieve omnidirectionalomnidirectional movement.movement.
3.2.2.3.2.2. The Axles of the Three Bottom-RollersBottom-Rollers IntersectIntersect atat OneOne PointPoint InIn Figure Figure 44,, the the axles axles of of the the three three bottom-rollers bottom-rollers intersect intersect at one at point, one point, and the and coordinate the coordinate system systemXOY ofXOY the mobileof the robot mobile system robot is systemestablished is established with the intersection with the intersection point as the pointorigin, as and the the origin, local = andwheel the coordinate local wheel systems coordinateXOYiii systems (1,2,3)i XiOiYi (i =are1, 2,established 3) are established in a counterclockwise in a counterclockwise order. order. The Therelationship relationship of the of paramete the parametersrs is shown is shown in Table in Table 2. 2.
Y
Y1
β γ O1 1 1 Y 2 β X 2 2 α 2 l1 γ X1 2 α 1 O2 l2 α O X 3
l3
O3
β Y3 3 γ 3 X 3 Figure 4. Wheel Configuration of the mobile robot with three Mecanum wheels whose bottom-roller Figure 4. Wheel Configuration of the mobile robot with three Mecanum wheels whose bottom-roller axles intersect at one point. axles intersect at one point. Table 2. Relationship between the parameters of the three Mecanum wheels in Figure4.
Serial Number |αi| [0◦,360◦) |βi| [0◦, 180◦] |γi| (0◦, 90◦) ∈ ∈ ∈ 1 α γ γ 1 − 1 1 2 α 180 γ γ 2 − ◦− 2 2 3 α γ γ 3 − 3 3
According to the parameters in Table2, we can obtain
β + γ = c (c = 0◦ or 180◦ , i = 1, 2, 3) (25) i i i i ± then, sin(βi + γi) = 0. (26) The matrix M of wheel installation orientation is cos(α1 + c1) sin(α1 + c1) l1 sin(c1) cos(α1 + c1) sin(α1 + c1) 0 M = ( + c ) ( + c ) l (c ) = ( + c ) ( + c ) cos α2 2 sin α2 2 2 sin 2 cos α2 2 sin α2 2 0 . cos(α3 + c3) sin(α3 + c3) l3 sin(c3) cos(α3 + c3) sin(α3 + c3) 0
The values of the third column of matrix M are all 0, then,
det(M) = 0, rank(M) , 3. (27) then, det(J) = 0, rank(J) , 3. (28) In the mobile system composed of three Mecanum wheels, if the axles of the three bottom-rollers intersect at one point, the mobile system has singularity, and the inverse kinematics velocity Jacobian Symmetry 2019, 11, 1268 11 of 27 matrix does not satisfy the condition of a column full rank matrix; therefore, the mobile system cannot achieve omnidirectional motion. If the axles of any two bottom-rollers coincide with each other or the axles of three bottom-rollers coincide with each other, it can be concluded that the axles of two bottom-rollers intersect at one point, which also satisfies the inference that the axles of three bottom-rollers intersect at one point. There are four configurations of three Mecanum wheels whose axles intersect at one point, three of which have collinear roller axles.
3.2.3. The Axles of the Three Bottom-Rollers Intersect at Two Points According to the hypothesis, when two roller axles intersect at one point, the other roller axe must be parallel to one of the roller axles. As shown in Figure5, if any two roller axles coincide, the result will inevitably be transformed into the case of axles intersecting at one point. The coordinate system XOY is established by arbitrarily choosing one of the intersections as the origin O, and the local wheel Symmetrycoordinate 2019 system, 11, x FORXi PEEROiYi REVIEW(i = 1, 2, 3) is also established in counterclockwise order. The relationship12 of 29 of the parameters is shown in Table3.
Y Y1
O1 β γ 1 1 X 2 Y2 β 2 α l X γ 2 1 1 2 α O2 1 l X 2 O α 3 Y3
l3 O 3 γ 3
β X 3 3 Figure 5. Wheel configuration of the mobile robot with three Mecanum wheels whose roller axles Figure 5. Wheel configuration of the mobile robot with three Mecanum wheels whose roller axles intersect at two points. intersect at two points. Table 3. Relationship between the parameters of the three Mecanum wheels in Figure5. Table 3. Relationship between the parameters of the three Mecanum wheels in Figure 5 Serial Number |α | [0 , 360 ) |β | [0 , 180 ] |γ | (0 , 90 ) i ∈ ◦ ◦ i ∈ ◦ ◦ i ∈ ◦ ◦ Serial number 1 α ∈[0°，360°)α β ∈[0°γ ，180°] γ γ ∈(0°，90°) i 1 i − 1 1 i 2 α2 180◦ γ2 γ2 1 α − −γ− γ 3 1 α3 β3 1 γ3 1 α γ γ 2 2 −180°− 2 2 The system coordinate systemα XOY is established at theβ intersection of the rollerγ axles of wheel 3 3 3 3 O1 and wheel O2, then we can obtain The roller axles of wheels O1 and O3 are parallel to each other, and the following relations are established β + γ = c (c = 0◦ or 180◦, i = 1, 2), sin(β + γ ) = 0. (29) i i i i ± i i α ++=βγ α = iii1 ( i 1, 3 ). (30) The roller axles of wheels O1 and O3 are parallel to each other, and the following relations are establishedThus, the matrix M can be obtained ad αi + βi + γi = α1 (i = 1, 3). (30) αα cos(1111 ) sin( )lc sin( ) Mcclc=+cos(αα ) sin( + ) sin( ) 22 22 2 2 ααβγ cos(11333 ) sin( )l sin( + ) . αα cos(11 ) sin( ) 0 =cos(αα +cc ) sin( + ) 0 22 22 ααβγ cos(11333 ) sin( )l sin( + )
The determinant of matrix M is =+βγ α α +− α α + M lcc3331sin( )[cos( )sin( 22 ) sin( 1 )cos( 22 )] . =+βγ α +− α (31) lc333221 sin( )sin( )
Combining with the discussion in Section 3.2.2, the roller axle of wheel O3 does not coincide βγ+≠ ± o βγ+≠ with the straight line l3 , so 330 or 180 , then, sin(33 ) 0 . αα+≠ ± o Because the roller axles of wheels O1 and O2 intersect at one point, 210 or 180 and αα+− ≠ sin(22c 1 ) 0 , then, the following is established
det(M )≠ 0 , rank(M )= 3 . (32) then,
Symmetry 2019, 11, 1268 12 of 27
Thus, the matrix M can be obtained ad cos(α1) sin(α1) l1 sin(c1) M = ( + c ) ( + c ) l (c ) cos α2 2 sin α2 2 2 sin 2 cos(α1) sin(α1) l3 sin(β3 + γ3) cos(α1) sin(α1) 0 = ( + c ) ( + c ) cos α2 2 sin α2 2 0 . cos(α1) sin(α1) l3 sin(β3 + γ3)
The determinant of matrix M is
M = l sin(β + γ )[ cos(α ) sin(α + c ) sin(α ) cos(α + c )] | | 3 3 3 1 2 2 − 1 2 2 (31) = l sin(β + γ ) sin(α + c α ) 3 3 3 2 2 − 1
Combining with the discussion in Section 3.2.2, the roller axle of wheel O3 does not coincide with the straight line l , so β + γ , 0 or 180 , then, sin(β + γ ) , 0. 3 3 3 ± ◦ 3 3 Because the roller axles of wheels O and O intersect at one point, α + α , 0 or 180 and 1 2 2 1 ± ◦ sin(α + c α ) , 0, then, the following is established 2 2 − 1 det(M) , 0, rank(M) = 3. (32) Symmetry 2019, 11, x FOR PEER REVIEW 13 of 29 then, det(detJ )(≠J) 0,, rank(0, rankJ )(=J) 3 =. 3.(33) (33)
In theIn themobile mobile system system consisting consisting of three of three Mecanum Mecanum wheels, wheels, if the if theaxles axles of the of thethree three bottom-rollers bottom-rollers intersectintersect at two at two points, points, there there is no is singularity no singularity in the in thesystem, system, and and the theJacobian Jacobian matrix matrix of the of theinverse inverse kinematicskinematics velocity velocity is a is column a column full full rank rank matrix. matrix. The The mobile mobile system system can canrealize realize omnidirectional omnidirectional movementmovement in the in theplane. plane.
3.2.4.3.2.4. The The Axles Axles of the of theThree Three Bottom- Bottom-RollersRollers Intersect Intersect at Three at Three Points Points WhenWhen the theaxles axles of ofthe the three three bottom-rollers bottom-rollers intersect intersect at three points,points, no no two two axles axles of of the the bottom-rollers bottom- can be parallel or coincide with each other, as shown in Figure6. The coordinate system XOY is rollers can be parallel or coincide with each other, as shown in Figure 6. The coordinate system XOY established by arbitrarily selecting one of the intersections as the origin. The relationship of the is established by arbitrarily selecting one of the intersections as the origin. The relationship of the parametersparameters is shown is shown in Table in Table 4. 4.
Y
Y1
O1 β γ X 2 1 1 Y2 β 2 α 2 l1 γ X1 2 α 1 O 2 l X 2 O α 3
l3
O3
X 3 β 3 Y γ 3 3
Figure 6. Configuration of mobile robot with three Mecanum wheels whose roller axles intersect at Figurethree 6. points.Configuration of mobile robot with three Mecanum wheels whose roller axles intersect at three points.
Table 4. Relationship between the parameters of the three Mecanum wheels in Figure 6 α β γ Serial number i ∈[0°, 360°) i ∈[0°, 180°] i ∈(0°, 90°) α γ γ 1 1 − 1 1 α γ γ `2 2 −180°− 2 2 α β γ 3 3 3 3
The coordinate system XOY is established at the intersection point of roller axles of wheel O1 and wheel O2 . The following formula can be obtained βγ+= = ± o = βγ+= iiic ( ci 0 or 180 , i 1, 2 ), sin(ii ) 0 . (34)
The matrix M can be obtained αα++ cos(11cc ) sin( 11 ) 0 Mc=+cos(αα ) sin( + c ) 0 22 22 . αβγ++ αβγ ++ βγ + cos(333 ) sin( 3333 )l sin( 33 )
The determinant of matrix M is =+βγ α + α +−++ α α Ml3331122sin( )[cos( c )sin( c ) sin( 1122 c )cos( c )] . =+βγ α +−− α (35) lcc3332211 sin( )sin( )
Symmetry 2019, 11, 1268 13 of 27
Table 4. Relationship between the parameters of the three Mecanum wheels in Figure6.
Serial Number |α | [0 , 360 ) |β | [0 , 180 ] |γ | (0 , 90 ) i ∈ ◦ ◦ i ∈ ◦ ◦ i ∈ ◦ ◦ 1 α γ γ 1 − 1 1 ‘2 α 180 γ γ 2 − ◦− 2 2 3 α3 β3 γ3
The coordinate system XOY is established at the intersection point of roller axles of wheel O1 and wheel O2. The following formula can be obtained
β + γ = c (c = 0 or 180◦, i = 1, 2), sin(β + γ ) = 0. (34) i i i i ± i i The matrix M can be obtained cos(α1 + c1) sin(α1 + c1) 0 M = ( + c ) ( + c ) cos α2 2 sin α2 2 0 . cos(α3 + β3 + γ3) sin(α3 + β3 + γ3) l3 sin(β3 + γ3)
The determinant of matrix M is
M = l sin(β + γ )[ cos(α + c ) sin(α + c ) sin(α + c ) cos(α + c )] | | 3 3 3 1 1 2 2 − 1 1 2 2 (35) = l sin(β + γ ) sin(α + c α c ) 3 3 3 2 2 − 1 − 1 According to the discussion results in Sections 3.2.2 and 3.2.3, we can obtain
sin(β3 + γ3) , 0.
Roller axles of wheel O and wheel O intersect at one point, α α , 180 m (m = 0, 1, 2, 3); 1 2 1 − 2 ◦ · ± ± ± therefore, sin(α + c α c ) , 0. The value is substituted into Formula (35), we can obtain 2 2 − 1 − 1 det(M) , 0, rank(M) = 3. (36) then, det(J) , 0, rank(J) = 3. (37) In a mobile system consisting of three Mecanum wheels, if the axles of the three bottom-rollers intersect at three points, there is no singularity in the system, and the Jacobian matrix of the inverse kinematics velocity satisfies the column full-rank condition, and the mobile system can achieve omnidirectional motion in the plane. According to the deduction of Formula (17) and the analysis results of the number of intersection points mentioned above, the condition for the omnidirectional motion of the mobile system composed of multiple Mecanum wheels (three wheels or more) is that any three axles of bottom-rollers in contact with the ground in three Mecanum wheels intersect at two or three points, which can be called a ‘bottom-roller axle intersections approach. Table5 illustrates the three-Mecanum-wheel configurations of the omnidirectional mobile robot and shows the relationship between the number of intersection points and the column full rank. Based on these illustrations, we can quickly judge whether the mobile system can achieve omnidirectional movement using only wheel configurations. Symmetry 2019, 11, x FOR PEER REVIEW 14 of 29 SymmetrySymmetry 20192019,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 1414 ofof 2929
According to the discussion results in Sections 3.2.2 and 3.2.3, we can obtain AccordingAccording toto thethe discussiondiscussion resultsresults inin SectionsSectionsβγ +≠ 3.2.23.2.2 andand 3.2.3,3.2.3, wewe cancan obtainobtain sin(βγβγ33+≠+≠ ) 0 . sin(sin(βγ3333+≠ ) ) 0 0.. 33 o Roller axles of wheel O and wheel O intersect at one point, αα−≠180oo ⋅m （ Roller axles of wheel OO1 and wheel OO2 intersect at one point, αααα12−≠−≠180180 ⋅ ⋅mm （（ RollerRoller axlesaxles ofof wheelwheel O11 andand wheelwheel O22 intersectintersect atat oneone point,point, 1212180 m （ m =±±±0, 1, 2, 3 ）; therefore, sin(αα+−−cc ) ≠ 0 . The value is substituted into Formula (35), mm=±±±=±±±0,0, 1, 1, 2, 2, 3 3））; therefore, sin(sin(αααα22+−−+−−cccc 11 ) ) ≠ ≠ 0 0 . The value is substituted into Formula (35), m 0, 1, 2, 3 ）;; therefore,therefore, sin(2222cc 11 11 ) 0.. TheThe valuevalue isis substitutedsubstituted intointo FormulaFormula (35),(35), we can obtain wewe cancan obtainobtain ≠ = det(M )≠≠ 0 , rank(M )== 3 . (36) det(det(MM ) )≠ 0 0,, rank(rank(MM ) )= 3 3.. (36)(36) then, then,then, ≠ = det(J )≠≠ 0 , rank(J )== 3 . (37) det(det(JJ ) )≠ 0 0,, rank(rank(JJ ) )= 3 3.. (37)(37) In a mobile system consisting of three Mecanum wheels, if the axles of the three bottom-rollers InIn aa mobilemobile systemsystem consistingconsisting ofof threethree MecanumMecanum wheels,wheels, ifif thethe axlesaxles ofof thethe threethree bottom-rollersbottom-rollers intersect at three points, there is no singularity in the system, and the Jacobian matrix of the inverse intersectintersect atat threethree points,points, therethere isis nono singularitysingularity inin thethe system,system, andand thethe JacobianJacobian matrixmatrix ofof thethe inverseinverse kinematics velocity satisfies the column full-rank condition, and the mobile system can achieve kinematicskinematics velocityvelocity satisfiessatisfies thethe columncolumn full-rankfull-rank condition,condition, andand thethe mobilemobile systemsystem cancan achieveachieve omnidirectional motion in the plane. omnidirectionalomnidirectional motionmotion inin thethe plane.plane. According to the deduction of Formula (17) and the analysis results of the number of intersection AccordingAccording toto thethe deductiondeduction ofof FormulaFormula (17)(17) andand thethe analysisanalysis resultsresults ofof thethe numbernumber ofof intersectionintersection points mentioned above, the condition for the omnidirectional motion of the mobile system pointspoints mentionedmentioned above,above, thethe conditioncondition forfor thethe omnidirectionalomnidirectional motionmotion ofof thethe mobilemobile systemsystem composed of multiple Mecanum wheels (three wheels or more) is that any three axles of bottom- composedcomposed ofof multiplemultiple MecanumMecanum wheelswheels (three(three wheelswheels oror more)more) isis thatthat anyany threethree axlesaxles ofof bottom-bottom- rollers in contact with the ground in three Mecanum wheels intersect at two or three points, which rollersrollers inin contactcontact withwith thethe groundground inin threethree MecaMecanumnum wheelswheels intersectintersect atat twotwo oror threethree points,points, whichwhich can be called a ‘bottom-roller axle intersections approach. Table 5 illustrates the three-Mecanum- cancan bebe calledcalled aa ‘bottom-roller‘bottom-roller axleaxle intersectionintersectionss approach.approach. TableTable 55 illustratesillustrates thethe three-Mecanum-three-Mecanum- wheel configurations of the omnidirectional mobile robot and shows the relationship between the wheelwheel configurationsconfigurations ofof thethe omnidirectionalomnidirectional mobilemobile robotrobot andand showsshows thethe relationshiprelationship betweenbetween thethe number of intersection points and the column full rank. Based on these illustrations, we can quickly numbernumber ofof intersectionintersection pointspoints andand thethe columncolumn fullfull rank.rank. BasedBased onon thesethese illustrations,illustrations, wewe cancan quicklyquickly Symmetryjudge 2019whether, 11, 1268 the mobile system can achieve omnidirectional movement using only wheel 14 of 27 judgejudge whetherwhether thethe mobilemobile systemsystem cancan achieveachieve omnidirectionalomnidirectional movementmovement usingusing onlyonly wheelwheel configurations. configurations.configurations. TableTable 5. 5.Three-Mecanum-wheel Three-Mecanum-wheel configurations configurations of the of theomnidirectional omnidirectional mobile mobile robot robot. TableTable 5.5. Three-Mecanum-wheelThree-Mecanum-wheel configurationsconfigurations ofof thethe omnidirectionalomnidirectional mobilemobile robotrobot NumberNumber of of Column NumberNumber ofof TypicalTypical Configurations Configurations ColumnColumnColumn Full Rank IntersectionIntersection Points Points TypicalTypical ConfigurationsConfigurations Full Rank IntersectionIntersection PointsPoints FullFull RankRank
Y Y YY 0 0 NoNo 0 O X No 00 O XX NoNo OO X
Y Y Y Y Y YY Y Y Y Y Y Y Y YY 1 X X X No 1 1 O X O X O X NoNo 11 O X O X O X NoNo O X OO X O X X O O O X O XX OO
Y Y YY 2 Yes 2 O X Yes 22 2 O XX YesYesYes OO X
Y Y YY
3 O X Yes 33 3 O XX YesYesYes 3 OO X Yes
3.3. Three-Mecanum-Wheel Configurations of the Mobile Robot 3.3.3.3. Three-Mecanum-WheelThree-Mecanum-Wheel ConfigurationsConfigurations ofof thethe MobileMobile RobotRobot 3.3.3.3. Three-Mecanum-Wheel Three-Mecanum-Wheel Configurations Configurations of ofthe the Mobile Mobile Robot Robot In order to balance the load, the three-wheel configuration usually adopts a circular array InIn orderorder toto balancebalance thethe load,load, thethe three-wheelthree-wheel configurationconfiguration usuallyusually adoptsadopts aa circularcircular arrayarray arrangement.In order to The balance common the three-Mecanum-wheel load, the three-wheel config configurationurations include usually a centripetal adopts circular a circular array array arrangement.arrangement. TheThe commoncommon three-Mecanum-wheelthree-Mecanum-wheel configconfigurationsurations includeinclude aa centripetalcentripetal circularcircular arrayarray configuration [26], a non-centripetal circular array configuration [1], and a star-type circular array arrangement.configurationconfiguration The [26],[26], common aa non-centripetalnon-centripetal three-Mecanum-wheel circularcircular arrayarray configurationconfiguration configurations [1],[1], andand include aa star-typestar-type a centripetal circularcircular circular arrayarray array configuration, as shown in Figure 7a–7c, respectively. Mecanum wheels in three-wheel configurationconfiguration,configuration, [ 26 asas], ashownshown non-centripetal inin FigureFigure circular 7a–7c,7a–7c, arrayrespectively.respectively. configuration MecanumMecanum [1], wheels andwheels a star-typeinin three-wheelthree-wheel circular array configuration, as shown in Figure7a–c, respectively. Mecanum wheels in three-wheel configurations usually have the same structure, whose rollers have the same inclination. Judging by the bottom-roller axle intersections approach, these configurations all have omnidirectional mobility performance. The wheels in the three-wheel configuration are, typically, special Mecanum wheels whose rollers’ axles are orthogonal to the hub axle. These are known as omniwheels, transwheels, or multidirectional wheels, as shown in Figure7f. In this article, we use an orthogonal Mecanum wheel to name this kind of wheel. The two typical orthogonal Mecanum wheel configurations are the centripetal circular array configuration [18,28,29] and T-configuration, as shown in Figure7d,e, respectively. The configuration in Figure7d is a rotational symmetry configuration, and this configuration is often used for indoor mobile service robots and light-duty handling robots. In this article, the orthogonal Mecanum wheel configurations are not studied in depth. Symmetry 2019, 11, x FOR PEER REVIEW 15 of 29
configurations usually have the same structure, whose rollers have the same inclination. Judging by the bottom-roller axle intersections approach, these configurations all have omnidirectional mobility performance. The wheels in the three-wheel configuration are, typically, special Mecanum wheels whose rollers’ axles are orthogonal to the hub axle. These are known as omniwheels, transwheels, or multidirectional wheels, as shown in Figure 7f. In this article, we use an orthogonal Mecanum wheel to name this kind of wheel. The two typical orthogonal Mecanum wheel configurations are the centripetal circular array configuration [18,28,29] and T-configuration, as shown in Figure 7d and 7e, respectively. The configuration in Figure 7d is a rotational symmetry configuration, and this Symmetry 2019, 11, 1268 15 of 27 configuration is often used for indoor mobile service robots and light-duty handling robots. In this article, the orthogonal Mecanum wheel configurations are not studied in depth.
(a) (b) (c)
(d) (e) (f)
Figure 7. Three-Mecanum-wheel configurations: (a) centripetal circular array configuration; (b) Non-centripetalFigure 7. Three-Mecanum circular array-wheel configuration; configurations: (c )(a star-type) centripetal circular circular array array configuration; configuration; ((db)) centripetalNon- circularcentripetal array configurationcircular array configuration; of orthogonal (c Mecanum) star-type wheels;circular array (e) T-configuration configuration; (d of) thecentripetal orthogonal circular array configuration of orthogonal Mecanum wheels; (e) T-configuration of the orthogonal Mecanum wheels; (f) an orthogonal Mecanum wheel. Mecanum wheels; (f) an orthogonal Mecanum wheel. 4. Symmetrical Wheel Configurations of the Four-Mecanum-Wheel Mobile Robot 4. Symmetrical Wheel Configurations of the Four-Mecanum-Wheel Mobile Robot 4.1. Judging the Four-Mecanum-Wheel Configurations by a Bottom-Roller Axle Intersections Approach 4.1. Judging the Four-Mecanum-Wheel Configurations by a Bottom-Roller Axle Intersections Approach At present, the mobile robot with four Mecanum wheels is the most popular configuration in At present, the mobile robot with four Mecanum wheels is the most popular configuration in both scientific research and industrial application. There are many possible wheel configurations both scientific research and industrial application. There are many possible wheel configurations for for thethe four-Mecanum-wheelfour-Mecanum-wheel robot robot [23,25,3 [23,250],,30 some], some of them of them are illustrated are illustrated in Figure in 8. Figure In Figure8. In 8, Figure the 8, the inclinedinclined lineline onon the wheel wheel in in the the figure figure represents represents the the inclined inclined direction direction of roller of roller in contact in contact with the with the ground.ground. Figure Figure8a–j 8a–8j show show 10 rectangular 10 rectangular configurations configurations of fourof four wheels wheels that that are are arranged arranged at at the the corner and whosecorner and axles whose areparallel axles are to parallel the centerline to the centerli of robot.ne of Figure robot.8 Figurek–n show 8k–8n four show possible four possible centripetal configurationscentripetal configurations of four wheels. of four Figure wheels.8o showsFigure 8o a centripetalshows a centripetal circular circular array array configuration configuration of four omniwheels.of four omniwheels. In the configurations In the configurations (a), (f), (a), and (f), (k), and any (k), threeany three roller roller axles axles are are parallel parallel to to each each other or coincideother or withcoincide each with other. each The other. number The number of intersection of intersection points points of theof the three three roller roller axlesaxles is 0 or or 1 1 (the (the overlapping axles are considered to intersect at one point). The column ranks of the Jacobian overlapping axles are considered to intersect at one point). The column ranks of the Jacobian matrix of matrix of these configurations are 2. These configurations obviously cannot achieve omnidirectional thesemovement. configurations In the are wheel 2. These configurations configurations (b)–(e), obviously(g)–(j), and cannot(l)–(n), the achieve axles omnidirectionalof the bottom-rollers movement. of In thethree wheel wheels configurations intersect at (b)–(e),two points, (g)–(j), so the and Jacobi (l)–(n),an thematrices axles of of these the bottom-rollers wheel configurations of three are wheels intersectcolumn at twofull-rank points, matrices. so the In Jacobian theory, these matrices configurations of these can wheel achieve configurations omnidirectional are movement column full-rank in matrices.the plane. In theory, these configurations can achieve omnidirectional movement in the plane. In practical applications, besides satisfying the conditions of the column full rank of the Jacobian matrix, the configuration also needs good operability and driving performance. Wheel configurations (b), (c), (d), (g), (h), (i), (l), and (m) can satisfy the conditions of omnidirectional motion, but the symmetry and the driving performance of the mobile system is poor. Considering the influence of dynamic factors, such as friction and moment of inertia, in actual operation, there will be a large deviation in the motion. Therefore, these configurations are generally not used. In addition, if the centers of four wheels in the configuration (j) form a square, the axles of the three bottom-rollers intersect at the one point, the column rank of Jacobian matrix in these configurations will change from 3 to 2, and it is no longer an omnidirectional mobile system. Configuration (n) has omnidirectional mobility, but the motion friction component of the configuration cannot offset itself in the course of movement, and there is a tendency to rotate in situ. Configuration (o) is the configuration (n) using orthogonal Mecanum wheels, mostly for small robotic mobile platforms. The symmetry of wheels configurations Symmetry 2019, 11, 1268 16 of 27
(e) and (j) is the best among these wheel configurations that can achieve omnidirectional motion. However, when rotating on the spot, the mobile robot system with the configuration (j) has a small driving torque and a weak driving effect. Therefore, the configuration (e) is the optimal configuration of a four-Mecanum-wheel system. The characteristics of the Mecanum wheel configurations in Figure8 are summarizedSymmetry 2019, 11 in, x Table FOR PEER6. REVIEW 16 of 29
(a) (b) (c) (d) (e)
(f) (g) (h) (i) (j)
(m) (n) (o) (k) (l) Figure 8. a – j FigureFour-Mecanum-wheel 8. Four-Mecanum-wheel configurations: configurations: (a))–((j)) rectangular rectangular configurations configurations of four of four wheels wheels that that are arrangedare arranged at the at the corner corner and and whose whose axles axles are are parallelparallel to to the the centerline centerline of ofthe the robot; robot; (k)– (kn) –centripetal(n) centripetal configurationsconfigurations of four of Mecanumfour Mecanum wheels; wheels; (o) centripetal (o) centripetal circular circular array configurationsarray configurations of four of orthogonal four Mecanumorthogonal wheels. Mecanum wheels.
In practicalTable applications, 6. Characteristics besides ofsatisfying the Mecanum the cond wheelitions configurations of the column in full Figure rank8 of. the Jacobian matrix, the configuration also needs good operability and driving performance. Wheel configurations Configurations in Figure8 abcdefghi Jklmn Intersections(b), (c), (d), (g), (h), (i), (l), and 0(m) 2can satisfy 2 2 the 2conditions 0 2 of omnidirectional 2 2 2 motion, 0 2 but 2 the 2 Columnsymmetry rank and the driving performance 2 3 of 3 the 3mobile 3 system 2 is 3 poor. 3 Considering 3 3 the 2 influence 3 3 of 3 Columndynamic full Rankfactors, such as friction N and Y moment Y Yof inertia, Y N in actual Y Yoperation, Y Y /thereNNYYY will be a large Omnidirectionaldeviation in the motion motion. capacity Therefore, n these B Bconfigurat B Gions nare generally B B not B us Ged./n In addition, n B if B the G Note:centers Y =ofthe four Jacobian wheels matrix in the is a configuration column full-rank (j) matrix,form a N square,= not; n the= the axles mobile of robotthe three system bottom-rollers does not have omnidirectionalintersect at the mobility one point, capacity; the Bcolumn= the omnidirectional rank of Jacobian motion matrix capacity in isthese not good; configurations G = good motion will capacity.change from 3 to 2, and it is no longer an omnidirectional mobile system. Configuration (n) has 4.2. Theoreticalomnidirectional Verification mobility, for but the the Symmetrical motion friction Rectangular component Configurations of the configuration with Four cannot Mecanum offset Wheels itself Thein the two course symmetrical of movement, rectangular and there configurations is a tendency of to the rotate four-Mecanum-wheel in situ. Configuration mobile (o) is robot the are configuration (n) using orthogonal Mecanum wheels, mostly for small robotic mobile platforms. The shown in Figure9. Choosing the geometric symmetry center as the origin O, the rectangular coordinate symmetry of wheels configurations (e) and (j) is the best among these wheel configurations that can systemachieveXOY omnidirectionalfixed with the motion. mobile However, robot is established. when rotating The on structuralthe spot, the parameters mobile robot of system each Mecanum with wheel are the same; therefore, γ1 = γ2 = γ3 = γ4 = γ (γ is positive) and γi = 45◦ (i = 1, 2, 3, 4). the configuration (j) has a small driving− torque and− a weak− driving effect. Therefore, the configuration (e) is the optimal configuration of a four-Mecanum-wheel system. The characteristics of the Mecanum wheel configurations in Figure 8 are summarized in Table 6.
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Table 6. Characteristics of the Mecanum wheel configurations in Figure 8
Configurations in Figure 8 a b c d e f g h i J k l m n Intersections 0 2 2 2 2 0 2 2 2 2 0 2 2 2 Column rank 2 3 3 3 3 2 3 3 3 3 2 3 3 3 Column full Rank N Y Y Y Y N Y Y Y Y/N N Y Y Y Omnidirectional motion capacity n B B B G n B B B G/n n B B G Note: Y = the Jacobian matrix is a column full-rank matrix, N = not; n = the mobile robot system does not have omnidirectional mobility capacity; B = the omnidirectional motion capacity is not good; G = good motion capacity.
4.2. Theoretical Verification for the Symmetrical Rectangular Configurations with Four Mecanum Wheels The two symmetrical rectangular configurations of the four-Mecanum-wheel mobile robot are shown in Figure 9. Choosing the geometric symmetry center as the origin O , the rectangular coordinate system XOY fixed with the mobile robot is established. The structural parameters of each Symmetry 2019, 11, 1268 γγγγγ=− = =− =− γ γ 17= of 27o Mecanum wheel are the same; therefore, 1234 ( is positive) and i 45 ( i = 1, 2, 3, 4 ). A
Y1 Y Y2 Y2 1 ϕ& 1 X γ O2 X 2 R 1 1 X 2 R ϕ& O1 γ 2 1 ϕ O &2 O ϕ& Y 2 Y 1 1 X1 H H θ& θ& X X O O D B Y Y4 Y 3 Y3 4 X X 3 4 X O X 4 3 4 O ϕ& O ϕ 3 3 4 &4 O3 ϕ& ϕ 3 &4 W W
C (a) (b)
Figure 9. Coordinate system assignments for the four-Mecanum-wheel robot with a symmetrical structure:Figure 9. (Coordinatea) wheel configuration system assignments in Figure7 e;for ( bthe) wheel four-Mecanum-wheel configuration in Figure robot7 j.with a symmetrical structure: (a) wheel configuration in Figure 7e; (b) wheel configuration in Figure 7j. In Figure9a, the bottom-rollers’ axles of the four Mecanum wheels intersect at points A, B, C, In Figure 9a, the bottom-rollers’ axles of the four Mecanum wheels intersect at points A, B, C, and D. According to the kinematics analysis of a single Mecanum wheel in Section 2.2, the relationship and D. According to the kinematics analysis of a single Mecanum wheel in Section 2.2, the between angle α1 and angle βα1 correspondingβ to wheel O1 can be determined as relationship between angle 1 and angle 1 corresponding to wheel O1 can be determined as
α1 + β1 = 0. (38) αβ+= 110 . (38) By substituting the conditions of Formula (38) into Formula (13), we can obtain By substituting the conditions of Formula (38) into Formula. (13), we can obtain x h i . . cos γ sin γ l sin(β + γ ) y + Rϕ sin γ = 0. (39) 1 1 1 1 . 1 1 x& θ []γγ βγ++& ϕγ& = cos11 sinlyR sin( 11 ) 11 sin 0 . (39) Formula (39) can be written as & θ . x Formula (39) can be written. as 1 h 1 H i . ϕ = tan 1 W tan y . (40) 1 −R γ1 − γ1 . θ
Similarly, the inverse kinematics equations of the other three wheels are expressed as . x . 1 h 1 H i . ϕ = tan 1 W tan y . (41) 2 −R γ2 − − γ2 . θ
. x . 1 h 1 H i . ϕ = tan 1 W + tan y . (42) 3 −R γ3 − γ3 . θ . x . 1 h 1 H i . ϕ = tan 1 W + tan y . (43) 4 −R γ4 γ4 . θ Symmetry 2019, 11, 1268 18 of 27
Because γ = γ = γ = γ = γ = 45 , the inverse kinematics equation of the 1 − 2 3 − 4 − − ◦ four-Mecanum-wheel mobile robot is [31]
. . . ϕ1 cot γ 1 W + H cot γ 1 1 W + H . − x − x ϕ 1 cot γ 1 W H cot γ . 1 1 1 (W + H) . . 2 = − − y = − y . (44) . ( ) . ϕ3 −R cot γ 1 W H cot γ −R 1 1 W + H . − − − θ − − θ ϕ4 cot γ 1 W + H cot γ 1 1 W + H
The inverse kinematics velocity Jacobian matrix is expressed as
1 1 W + H − 1 1 1 (W + H) J = − . −R 1 1 (W + H) − − 1 1 W + H
According to Formula (44), the symmetrical rectangular configuration of the four-Mecanum-wheel robot shown in Figure9a satisfies the column full rank of the inverse kinematics velocity Jacobian matrix; therefore, it can achieve omnidirectional motion. In the wheel configuration shown in Figure9b, the bottom-rollers’ axles of the four Mecanum wheels intersect at point O, therefore, W H cot γ = 0. According to Formula (45) and the structural − parameters of the configuration shown in Figure8b, the inverse kinematics equation of the mobile robot is . ϕ cot γ 1 W H cot γ . 1 1 0 . . 1 − x x ( ) . . ϕ2 1 cot γ 1 W H cot γ 1 1 1 0 . = − − − y = − y . (45) ϕ −R cot γ 1 (W H cot γ) . −R 1 1 0 . . 3 − − θ θ ϕ cot γ 1 W H cot γ 1 1 0 4 − − − In Formula (45), the third column of the Jacobian matrix is all 0, which limits the central rotation of the mobile robot. The Jacobian matrix of the inverse kinematics velocity is not a column full rank, and the mobile system cannot achieve omnidirectional motion. The above theoretical derivation verifies the correctness of the roller axle intersection approach. It can be judged whether the wheel configuration has omnidirectional movement performance according to the number of axle intersection points of the bottom-rollers in contact with the ground. The position of the intersection points can also be used to judge whether the wheel configuration has good or bad omnidirectional mobility performance. If the intersection position is symmetrical, the wheel configuration has good omnidirectional mobility. As shown in Figure9a, the axles of the bottom-rollers of the four wheels of the symmetrical rectangular configuration intersect at four points—A, B, C, and D—and the intersection points are located far from the geometric center and are symmetrical. Therefore, this configuration has good omnidirectional mobility characteristics and is the most widely used four-Mecanum-wheel configuration.
5. Design Method of Mecanum Wheel Configurations for the Omnidirectional Mobile Robot
5.1. Sub-Configuration Judgment Method for Judging the Omnidirectional Motion Capacity of the Wheel Combination Configurations In some applications, especially for moving large-scale components, it is necessary to combine multiple robots to transport objects cooperatively. Figure 10 is a tandem configuration composed of two four-Mecanum-wheel sub-configurations in Figure9b. By using a roller axle intersection approach, the omnidirectional movement performance of this configuration can be judged. If the bottom-rollers axles intersect at one point of any four-Mecanum-wheel configuration, the Mecanum configuration cannot achieve omni-directional motion. However, the four-Mecanum-wheel sub-configuration of 3, 4, 5, and 6 wheels does achieve omnidirectional movement, which is the same configuration as that Symmetry 2019, 11, x FOR PEER REVIEW 19 of 29
ϕ& cotγγ 1WH− cot 1 1 0 1 x&& x ϕ −−−γγ − &2 11cot 1（WH cot） 1 1 0 =−yy&& =− . (45) ϕγ& RRcot 1−−（）WH cot γ 1 1 0 3 θθ&& ϕ −−γγ − &4 cot 1WH cot 1 1 0
In Formula (45), the third column of the Jacobian matrix is all 0, which limits the central rotation of the mobile robot. The Jacobian matrix of the inverse kinematics velocity is not a column full rank, and the mobile system cannot achieve omnidirectional motion. The above theoretical derivation verifies the correctness of the roller axle intersection approach. It can be judged whether the wheel configuration has omnidirectional movement performance according to the number of axle intersection points of the bottom-rollers in contact with the ground. The position of the intersection points can also be used to judge whether the wheel configuration has good or bad omnidirectional mobility performance. If the intersection position is symmetrical, the wheel configuration has good omnidirectional mobility. As shown in Figure 9a, the axles of the bottom-rollers of the four wheels of the symmetrical rectangular configuration intersect at four points—A, B, C, and D—and the intersection points are located far from the geometric center and are symmetrical. Therefore, this configuration has good omnidirectional mobility characteristics and is the most widely used four-Mecanum-wheel configuration.
5. Design Method of Mecanum Wheel Configurations for the Omnidirectional Mobile Robot
5.1. Sub-Configuration Judgment Method for Judging the Omnidirectional Motion Capacity of the Wheel Combination Configurations In some applications, especially for moving large-scale components, it is necessary to combine multiple robots to transport objects cooperatively. Figure 10 is a tandem configuration composed of two four-Mecanum-wheel sub-configurations in Figure 9b. By using a roller axle intersection approach, the omnidirectional movement performance of this configuration can be judged. If the bottom-rollersSymmetry 2019, 11 axles, 1268 intersect at one point of any four-Mecanum-wheel configuration, the Mecanum19 of 27 configuration cannot achieve omni-directional motion. However, the four-Mecanum-wheel sub- configuration of 3, 4, 5, and 6 wheels does achieve omnidirectional movement, which is the same configurationshown in Figure as that9a, and shown the configurationin Figure 9a, ofand 1, 2,the 7, andconfiguration 8 also achieves of 1, omnidirectional2, 7, and 8 also movement achieves omnidirectionalability. The tandem movement configuration ability. shownThe tandem in Figure configuration 10 can move shown omnidirectionally. in Figure 10 Therefore, can move if a omnidirectionally.multiple-wheel configuration Therefore, if hasa multiple-wheel any sub-configuration configuration which has has any omnidirectional sub-configuration motion which capacity, has omnidirectionalit can also achieve motion omnidirectional capacity, it can motion. also achi Thiseve judgment omnidirectional method motion. is evolved This from judgment the approach method of isbottom-roller evolved from axle the intersection, approach whichof bottom-roller can be a called axle sub-configuration intersection, which judgment can method.be a called As shownsub- configurationbelow, the conclusion judgment is method. validated As by shown the Jacobian below, matrixthe conclusion of the inverse is validated kinematics by the velocity Jacobian of the matrix robot ofmobile the inverse system. kinematics velocity of the robot mobile system.
1 4 5 8 X1 X 2 Wx X Y Y1 2 O O1 2 θ& Y o θ& H y 1 2
2 3 6 7
Hd
Figure 10. A tandem configuration composed of two symmetrical Figurefour-Mecanum-wheel 10. A tandem sub-configurations.configuration composed of two symmetrical four-Mecanum-wheel sub- configurations. In Figure 10, the coordinate system XOY located in the symmetric center of the tandem configuration is established. Its offset relative to the two local coordinate systems X1O1Y1 and . . . T X2O2Y2 is Hd. Let x y θ be the generalized velocity of the system and the angular velocity . of the i-th Mecanum wheel be ϕ (i = 1, 2, , 8). The inverse kinematics equations of the two i ··· four-Mecanum-wheel configurations are derived when the original coordinate system is moved to the designated coordinate system XOY.
. . . ϕ1 cot γ 1 Hd cot γ . x − x ϕ . 1 cot γ 1 H cot γ . 2 1 y d y . = J1K1− (Hd, 90◦, 0) = − . (46) ϕ − . −R cot γ 1 Hd cot γ . . 3 θ − θ ϕ cot γ 1 H cot γ 4 − d . ϕ . cot γ 1 Hd cot γ . . 5 x x . . ϕ6 1 1 cot γ 1 Hd cot γ . = J2K2− (Hd, 90◦, 0) y = − − y . (47) ϕ . −R cot γ 1 Hd cot γ . . 7 θ θ ϕ cot γ 1 H cot γ 8 − − d where, cot γ 1 0 1 cot γ 1 0 J1 = J2 = − −R cot γ 1 0 cot γ 1 0 − 1 0 Hd 1 − K− (Hd, 90◦, 0) = 0 1 0 1 − 0 0 1 1 0 Hd 1 K− (Hd, 90◦, 0) = 0 1 0 1 0 0 1 In the above two formulas, the column rank of the Jacobian matrix of the inverse kinematics velocity of a mobile robot is still 2, which verifies that the change of coordinate system will not change the rank, Symmetry 2019, 11, 1268 20 of 27 and the two mobile robots still cannot achieve omnidirectional motion alone. If the motion of two mobile robots is considered as a whole, the inverse kinematics equation of the eight-Mecanum-wheel configuration is . ϕ cot γ 1 H cot γ 1 d . − ϕ2 cot γ 1 Hd cot γ . − ϕ . cot γ 1 H cot γ . 3 x d x . " 1 # − ϕ J1K− (Hd, 90◦, 0) . 1 cot γ 1 Hd cot γ . . 4 = 1 − y = − y 1( ) . R . ϕ5 J2K2− Hd, 90◦, 0 − cot γ 1 Hd cot γ . θ θ ϕ cot γ 1 Hd cot γ . 6 − − ϕ7 cot γ 1 Hd cot γ . ϕ cot γ 1 Hd cot γ 8 − − (48) 1 1 H d − 1 1 Hd − 1 1 H . d x − 1 1 1 Hd . = − y − R 1 1 H . d θ 1 1 Hd − − 1 1 H d 1 1 H − − d According to Formula (48), the inverse kinematics velocity Jacobian matrix of the eight-wheel configuration is a column full rank matrix. The above deduction verifies the correctness of the bottom-roller axle intersection approach and the sub-configuration judgment method. The coordinated motion of the tandem configuration can be realized by controlling the rotational velocities of each wheel of the configuration. This roller axle intersection approach can also be extended to the combination configurations of N four-Mecanum-wheel mobile robots.
5.2. Analysis of Mecanum Wheel Configurations and Combination Configurations for Common Omnidirectional Mobile Robots Figure 11 shows the Mecanum wheel configurations and their application examples in practical production. Based on the analysis of these configurations, the deduction method of multi-wheel configurations can be synthesized. In Section4, the optimal configuration of the four-Mecanum-wheel robot has been obtained, which is a symmetrical rectangular configuration, and is denoted as configuration W4. The front-right and rear-left wheels are right-handed wheels, and the four Mecanum wheels are mirror-symmetrical in both the front and rear, and are mirror-symmetrical for the left and right. The KUKA omniMove set with four Mecanum wheels, as shown in Figure 11a, adopts this configuration W4. In order to transport large objects, a single mobile robot system needs to use more wheels to improve its carrying and transportation capacity. Figure 11b is a 12-Mecanum-wheel configuration adopted by KUKA’s omniMove platform. This 12-Mecanum-wheel configuration consists of three four-Mecanum-wheel configurations W4, as shown in Figuration 11a, which are connected end and end to form a robot platform. In order to transport objects with larger volume and weight, the 12-Mecanum-wheel configuration robot shown in Figure 11b can be combined to form a 36-Mecanum-wheel combination mobile robot platform. The combined AGVs of the KUKA Omnimove AGV using configuration W36 can carry 63 tons of weight [10]. From Figure 11a–c, we can consider that the configuration W4 shown in Figure 11a is a sub-configuration of the configuration W12 shown in Figure 11b, while the configuration W12 is a sub-configuration of the configuration W36 shown in Figure 11c. Using the sub-configuration judgment method proposed above, both configuration W12 and configuration W36 achieve omnidirectional mobility. Symmetry 2019, 11, 1268 21 of 27 Symmetry 2019, 11, x FOR PEER REVIEW 22 of 29
Schematic Diagram of Wheel Configurations Application Examples
1 4
(a)
2 3 Configuration W4
(b)
3 W4 Configuration W12
(c) 3 W12 Configuration W36 Same wheels combination
(d) 6 W4 Configuration Same wheels combination W24
3-2 1-1 3-1 3-2 (e) 3-2 2-1 4-1 3-2
2 W4 Configuration W8
(f)
2 W8 Configuration W16
(g)
2 W4 (g1) (g2)
4 W6
(h)
2 W12
Configuration W24 Figure 11. Common symmetrical rectangular wheel configurations and application examples: (a) the Figure 11. Common symmetrical rectangular wheel configurations and application examples: (a) the b c typicaltypical four-Mecanum-wheel four-Mecanum-wheel configuration configuration W4; W4; ((b)) aa 12-Mecanum-wheel12-Mecanum-wheel configuration configuration W12; W12; (c () )a a combinationcombination configuration configuration W36 W36 with with three three W12; W12; ( d(d)) a a 24-Mecanum-wheel 24-Mecanum-wheel configuration configuration W24; W24; (e (e) )an an eight-Mecanum-wheeleight-Mecanum-wheel configuration configuration W8, W8, the example:the example: MC-Drive MC-Drive TP 200 ofTP CLAAS; 200 of (fCLAAS;) the combination (f) the configurationcombination W16 configuration with two W16 W8, with the two example: W8, the ex theample: combination the combination of two of MC-Drive two MC-Drive TP 200 TP 200 [12 ]; (g)[12]; an eight-Mecanum-wheel(g) an eight-Mecanum-wheel configuration configuration W8, W8, which which can can combine combine into into aa 16-Mecanum-wheel configurationconfiguration [32 ];[32] (h) a; rectangular(h) a rectangular combination combination configuration configuration consisting consisting of four of W6 four [13]. W6 The [13] examples. The in (examplesa–d,h) are in KUKA (a)–(d) omniMove and (h) are AGVs.KUKA omniMove AGVs.
Symmetry 2019, 11, 1268 22 of 27
The 24-Mecanum-wheel configuration W24 in Figure 11d is adopted by KUKA, and the eight-Mecanum-wheel configuration W8 shown in Figure 11e is adopted by AGV of CLAAS. Configurations W24 and W8 have the same structural characteristics. The middle four-Mecanum-wheel combination in W8 and W24 can be considered as configuration W4, as shown in Figure 11a. The four wheels on both sides of the middle-four wheels constitute the W4 configuration. The configuration is extended outward from the intermediate configuration W4. Configuration W16 in Figure 11f is a combination of the two configurations of W8 in Figure 11e. The middle configuration W4 and the outer layer configuration W4 are also considered as sub-configurations of the entire Mecanum wheels configuration. The configuration W8 in Figure 11e is a sub-configuration of configuration W16 in Figure 11f. The configurations in Figure 11d and 11e can be considered to be obtained by replacing the Mecanum wheels of the basic configuration W4 with the wheel combination including a row of wheels of the same specification. Mecanum wheels 1, 2, 3, and 4 of configuration W4 in Figure 11a are replaced by wheels 1-1 and 1-2, 2-1 and 2-2, 3-1 and 3-2, 4-1 and 4-2, respectively. Thus, the wheel configuration shown in Figure 11e is evolved. In this way, the configuration W4 is evolved into the wheel configuration W8, which is shown in Figure 11e. In the same way, the configuration W24 in Figure 11d can also be analyzed with the method. The configuration shown in Figure 11g1 can also be obtained by replacing the wheel with the coaxial tandem wheel combination. The combination configuration shown in Figure 11g2 is an end-to-end connection combination of the two configurations in Figure 11g1. Figure 11h shows a 24-Mecanum-wheel omnidirectional mobile system consisting of four six-Mecanum-wheel configurations W6. The four KUKA omniMove platforms using this configuration W24 are applied to move a railcar body at the Siemens plant. These four mobile robot platforms with configuration W6 are not connected to each other, but each robot platform is connected to the railcar body, thus achieving a fixed connection of the four robot platforms. Two six-Mecanum-wheel configurations W6 are vertically connected to form the 12-Mecanum-wheel subsystem. Two 12-Mecanum-wheel subsystems are connected horizontally to form the 24-Mecanum-wheel configuration W24. The six-Mecanum-wheel configuration W6 consists of a W4 sub-configuration with omnidirectional mobility and a wheelset. Although the configuration W6 is not symmetrical in the longitudinal direction, the configuration W6 is used for combination. The 12-Mecanum-wheel configuration W12 and configuration W24 have symmetrical structures and omnidirectional motion capacity.
5.3. Topological Design Methods of Multi-Mecanum Wheel Configuration for Omnidirectional Mobile Robot The bottom-roller intersection approach can be used to judge whether the wheel configuration of a single mobile robot system has omnidirectional mobility. The omnidirectional mobility of a multiple-wheel configuration can be judged by the sub-configuration judgment method. Through comprehensive analysis and evaluation, the symmetrical wheel configurations are more conducive to design, manufacture, and motion control. The symmetrical rectangular four-Mecanum-wheel configuration is the optimal wheel configuration in the four-wheel configurations. Based on this basic configuration, the multiple-Mecanum-wheel configuration of an omnidirectional mobile robot can be deduced. In this paper, the topological design methods of the Mecanum-wheel configurations are summarized and refined, including the basic configuration array method, multiple wheel replacement method, and combination method. The first two methods can be used to create suitable multiple-Mecanum-wheel configurations for a single mobile robot based on the basic Mecanum wheel configuration. Next, the topology methods are introduced in conjunction with Figure 12. The basic configuration array method is obtained based on the sub-configuration judgment method, and it includes a linear array method and a circular array method. The former can be used for new wheel configuration designs base on the symmetrical rectangular wheel configuration, which includes end-to-end connection method and side-by-side connection method. The latter can be used for the design of new wheel configurations based on the centripetal wheel configuration. Using the basic configuration array Symmetry 2019, 11, 1268 23 of 27 method, the four-Mecanum-wheel basic configuration is linearly arrayed in two directions; then, the configurations with end-to-end connection and side-by-side connection are obtained, as shown in Figure 12a,b. Using the multiple wheels replacement method, based on the basic four-Mecanum-wheel configuration, each wheel of the four-Mecanum-wheel configuration is replaced with multiple identical wheels. The wheel can be replaced by two types of wheel combinations: a series of longitudinally arranged wheels and a series of coaxial wheels, and the new configurations can be obtained, as shown in Figure 12c–e. If the number of longitudinally arranged wheels in the front- and rear-array wheels is the same, a symmetrical Mecanum-wheel configuration can be obtained, which is symmetrical in both left and right and in front and back, as shown in Figure 12c. If not, the Mecanum wheel configuration is only symmetrical in left and right, and not in front and back, as shown in Figure 12d. Among these configurations, configurations in Figure 12a,c are more commonly used. Using the above two methods, the suitable wheel configurations for the single mobile robot can be obtained. Multiple single robots can be arranged by combination methods, including end-to-end connection, side-by-side connection, symmetrical rectangular connection, and distributed combination methods; and then, the abundant combination configurations of robots can be obtained, including end-to-end combination configuration (Figure 12f), side-by-side combination configuration (Figure 12h), rectangular combination configuration (Figure 12g), and distributed combination configuration (Figure 12i). The distributed combination configuration means that there is no physical connection among the independent mobile robots. This belongs to the research field of multi-robot cooperative motion. In this article, a detailed analysis of the distributed combination configuration is not carried out. Using the topological design method for Mecanum wheel configurations proposed in this article, abundant wheel configurations can also be deduced based on other basic Mecanum wheel configurations. Using the circular array method, the centripetal three-Mecanum-wheel configuration shown in Figure7a can be arrayed as a centripetal six-Mecanum-wheel configuration, as shown in Figure 13a1. Using multiple coaxial wheels replacement method, the wheel configuration in Figure 13a2 can be obtained based on configuration in Figure7a. Using the multiple wheels replacement method, the configurations in Figure 13b1,b2 are obtained from the basic configurations in Figure8n and the configurations in Figure 13c1 is obtained from the basic configuration in Figure8o. Using end-to-end combination method, a combination configuration shown in Figure 13c2 can be obtained based on the configuration in Figure 13c1. Symmetry 2019, 11, 1268 24 of 27 Symmetry 2019, 11, x FOR PEER REVIEW 25 of 29
Mecanum wheel configurations of a single robot
(a)
End-to-end connection method
(b)
Side-by-side connection method Basic configuration array method
(c)
Multiple longitudinally arranged (d) wheels replacement method
(e)
Multiple coaxial wheels replacement method Multiple wheels replacement method A single mobile robot
End-to-end connection method Side-by-side Distributed connection method combination method (f)
End-to-end combination configuration Symmetrical rectangular connection method (h) (i)
(g)
Distributed Side-by-side combination combination Rectangular combination configuration configuration configuration
Combination configurations of multiple robots
Figure 12. Topological design methods for wheel configurations based on a symmetrical Figure 12. Topological design methods for wheel configurations based on a symmetrical four- four-Mecanum-wheel configuration: (a) end-to-end connection configuration; (b) side-by-side Mecanum-wheel configuration: (a) end-to-end connection configuration; (b) side-by-side connection connection configuration; (c) a front-back symmetric configuration; (d) a front-back asymmetric configuration; (c) a front-back symmetric configuration; (d) a front-back asymmetric configuration; (e) configuration;a symmetric (configuratione) a symmetric with configuration four coaxial with wheels four series; coaxial (f) wheelsend-to-end series; combination (f) end-to-end configuration; combination configuration;(g) rectangular (g )combination rectangular configuration; combination configuration;(h) side-by-side ( combinationh) side-by-side configuration; combination (i) configuration;distributed (i)combination distributed combinationconfiguration. configuration.
Symmetry 2019, 11, x FOR PEER REVIEW 26 of 29
Using the topological design method for Mecanum wheel configurations proposed in this article, abundant wheel configurations can also be deduced based on other basic Mecanum wheel configurations. Using the circular array method, the centripetal three-Mecanum-wheel configuration shown in Figure 7a can be arrayed as a centripetal six-Mecanum-wheel configuration, as shown in Figure 13a1. Using multiple coaxial wheels replacement method, the wheel configuration in Figure 13a2 can be obtained based on configuration in Figure 7a. Using the multiple wheels replacement method, the configurations in Figure 13b1–13b2 are obtained from the basic configurations in Figure 8n and the configurations in Figure 13c1 is obtained from the basic configuration in Figure 8o. Using Symmetry 2019, 11, 1268 25 of 27 end-to-end combination method, a combination configuration shown in Figure 13c2 can be obtained based on the configuration in Figure 13c1.
(a1) (a2)
Basic configuration Basic configuration circular Multiple wheels replacement in Figure 7a array method method (a)
(b1) (b2)
Basic configuration in Figure 8n Multiple wheels replacement method (b)
(c1) (c2)
Basic configuration Multiple wheels End-to-end combination method in Figure 8o replacement method (c)
Figure 13. The examples of deducing new wheel configurations based on three basic Mecanum wheel configurationsFigure 13. The examples by using theof deducing topological new method: wheel configur (a) newations configurations based on deducedthree basic from Mecanum a centripetal wheel circularconfigurations array configuration by using the of topological three Mecanum method: wheels (a) in new Figure configurations7a; ( b) new configurations deduced from deduced a centripetal from centripetalcircular array circular configuration array configuration of three Mecanum of four Mecanum wheels in wheelsFigure in7a; Figure (b) new8n; configurations ( c) new configurations deduced deducedfrom centripetal from centripetal circular circular array arrayconfiguration configurations of four of four Mecanum orthogonal wheels Mecanum in Figure wheels 8n; in Figure (c) new8o. configurations deduced from centripetal circular array configurations of four orthogonal Mecanum 6. Conclusionswheels in Figure 8o. The condition that the Mecanum wheeled robot can achieve omnidirectional movement is that 6. Conclusions the inverse kinematics Jacobian matrix of any three Mecanum wheels on the robot is a column full-rank matrix. In this paper, the relationship between the intersections of bottom-rollers axles of any three Mecanum wheels on the robot and the column rank of the Jacobian matrix is established. A bottom-rollers axles intersections approach for judging the omnidirectional mobility of Mecanum wheel configurations is proposed and proved theoretically, which is a simple and efficient geometric method. If the number of axles intersections is 2 or 3, the column rank is full and the robot can achieve omnidirectional motion in a plane; if the number of axles intersections is 0 or 1, this is not the case. A sub-configuration judgment method for judging whether a Mecanum wheel configuration has omnidirectional mobility is evolved based on the bottom-roller axle intersections approach. According Symmetry 2019, 11, 1268 26 of 27 to this method, if the multiple-Mecanum-wheel configuration has any individual sub-configuration with omnidirectional motion capacity, it can also achieve omnidirectional motion. The topological design methods of the Mecanum wheel configurations are summarized and refined, including basic configuration array method, multiple wheel replacement method, and combination method. The first two methods can be used to create suitable multiple-Mecanum-wheel configurations for a single mobile robot based on the basic Mecanum wheel configuration. Multiple single robots can be arranged by combination methods including end-to-end connection, side-by-side connection, symmetrical rectangular connection and distributed combination, and then, the abundant combination configurations of robots can be obtained.
Author Contributions: Methodology, Y.L. and S.D.; Validation, L.Z. and Y.S.; Formal analysis, Y.L. and S.D.; Investigation, Y.L. and X.Y.; Writing—original draft preparation, Y.L., S.D., and X.Y.; Writing—review and editing, Y.S. and L.Z.; Project administration, Y.L. Funding: This work was financially supported by the National Natural Science Foundation of China (no. 51675518), Six Talent Peaks Project in Jiangsu Province (no. JXQC-008), China Scholarship Council (no. 201706425041), Jiangsu Government Scholarship for Overseas Studies (no. JS-2018-152), Science Foundation of Xuzhou University of Technology (no. XKY2018129), and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Acknowledgments: We would like to thank Robert Bauer of Stevens Institute of Technology for his advice. Conflicts of Interest: The authors declare no conflict of interest.
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