Estimation of Motion Capabilities of Mobile Platforms with Three Omni Wheels Based on Discrete Bidirectionality Compliance Analysis

Abstract

This paper presents a procedure for estimating the motion capabilities of an omnidirectional mobile platform with three omni wheels arbitrarily distributed and oriented. This procedure is based on the analysis of the bidirectionality compliance between the inverse and forward kinematics of a mobile platform for a wide set of discrete motion commands. This procedure has been applied to analyze eleven alternative mobile platform configurations with three omni wheels. The estimation of the omnidirectional motion capabilities of these platforms agrees with state-of-the-art methods while providing new differentiated information on the translational capabilities of each platform. The procedure can be applied in the design stage of new omnidirectional mobile platforms in order to verify the motion capabilities of new designs with omni wheels.

1. Introduction

An omnidirectional mobile platform is characterized by its capability to simultaneously translate and rotate in any direction without kinematic constraints [1,2,3]. Omnidirectional wheels have free rollers on their perimeters [4] to combine rotation around the wheel axis and translation in the rotation direction of the rollers. The angular orientation of the free rolling direction of the rollers relative to the wheel hub ($\gamma$) determines the type of omnidirectional wheel. The most commonly used are Mecanum ($\gamma =\pm 45°$) [5,6,7] and omni wheels ($\gamma =0°$) [8,9,10]. Omnidirectional wheels are implemented in a significant number of robot applications [11,12,13]. Compared with conventional wheels, their main disadvantages are the generation of vibrations and the susceptibility to slippage.

The combination of several omnidirectional wheels in a system does not ensure omnidirectional motion; only when wheel placement and orientation is adequate, the platform can achieve three degrees of freedom (DOF). Proposals of omnidirectional mobile robots with different wheel configurations are presented in the scientific literature [14,15,16]; their selection depends on the field of application. The most frequent omnidirectional platforms are based on symmetric designs [17,18] with three omni wheels [19,20] or four Mecanum wheels [21,22,23]. Less common implementations include asymmetric distributions [17,24] and systems with six [25,26,27] or eight wheels [28]. Specific approaches such as navigation within tunnels to inspect nuclear sites rely on compact designs [29] or systems with joints to change the alignment of the Mecanum wheels [30]. Other designs include omnidirectional wheels on a bending platform so that robots can climb and inspect ferromagnetic structures such as walls and circular constructions [31]. Hybrid mobile robots combining legs and Mecanum wheels are convenient in heterogeneous environments with irregular terrains [32], and proposals such as the collinear Mecanum drive are interesting omnidirectional alternatives to typical two-wheeled inverted-pendulum robots [33]. Similarly, hybrid quadruped spherical robots combine the structure of spherical robots and four legged robots [34]. Other omnidirectional proposals extend the operation sceneries to terrain and water by implementing oloid-like paddlewheels based on Schatz linkages [35].

The analysis of omnidirectional mobile platforms is often addressed with the aim of refining their design and operability [36]. For example, Almasri et al. [18] proposed a model for systems with symmetrical distributions of N omni wheels to optimize robot trajectories, Barreto et al. [37] presented a model predictive control to compensate for frictional effects on trajectories, and Manzl et al. [38] considered orthotropic friction to model roller-ground contact of Mecanum wheels and developed an efficient wheel model. With respect to their constructive characteristics, Zhang et al. [39] analyzed the effect of the angles of the rollers on motion performances, Tagliavini et al. [40] presented an overview of three-wheeled omnidirectional motion strategies, and Hijikata et al. [41] proposed an alternative wheel arrangement of four omni wheels in a mobile robot, suggesting that wheel arrangement changes the effect of imperfect control on the motion. Reconfigurable and asymmetric wheel arrangements are useful in case of motor failure [24,42]. In this context, Leng and Cao [43] studied the maximum velocity of omnidirectional robots with arbitrary wheel arrangements, and Mohanraj et al. [19] analyzed the front and back movement of a three-wheeled omnidirectional robot by varying the angles between its wheels.

Several authors have analyzed the omnidirectional mobility of a system on the principle that if the Jacobian matrix of the inverse kinematics model is full rank, the system has three DOF [44]. Li et al. [17] proposed an alternative approach to determine if a mobile platform based on omnidirectional wheels is omnidirectional or not. Their proposal analyzes the intersections of the axes of the rollers in contact with the ground. They validated their method by analyzing a set of platforms with Mecanum wheels under different configurations and comparing the results with the bottom-roller axles approach and the column rank of the Jacobian matrix. When the number of intersections determines that the system is omnidirectional, verifications on dynamics are required to formulate restrictions on the position and orientation of wheels and rollers and guarantee operability and omnidirectionality [45]. Configurations to be avoided include wheels with their axes parallel to the axes of the rollers and distributions with the axes of the rollers coinciding in the center of the platform.

The aforementioned variety of approaches to omnidirectionality analysis manifests that the design of mobile platforms requires a procedure focused on the analysis of the configuration of the wheels to ensure omnidirectional motion without affecting system performance [44].

New Contribution

This paper presents a procedure for estimating the motion capabilities of mobile platforms with three omni wheels arranged in arbitrary locations and orientations. This procedure is based on the analysis of the bidirectionality compliance between the inverse and forward kinematics of a mobile platforms for a wide set of discrete motion commands. The procedure provides a differentiated estimation of the omnidirectional capabilities and of the translational capabilities mobile platforms. This information enables the comparison between diverse implementation alternatives to satisfy the requirements and constraints of an intended application [46]. This approach has been applied to analyze the motion performances of a set of mobile platform configurations. The results of the estimation of the motion capabilities agree with state-of-the-art methods and have been validated using simulations.

2. Materials and Methods

The materials and methods used are the kinematics of a general omnidirectional mobile platform, and a set of omnidirectional mobile platforms configurations.

2.1. Kinematics of an Omnidirectional Mobile Platform

This section describes the general kinematic model of a platform with omni wheels, which is used to estimate its translation and rotation performances. Figure 1 shows the system definition of a mobile platform. The system is defined in a static frame $\left\{s\right\}$. The mobile platform has its own body-attached frame $\left\{b\right\}$, and each wheel also has its corresponding frame $\left\{w_k\right\}$, being $k$ the identifying number the wheel. The position of the platform is defined by its frame coordinates $(x_b,y_b)$ and orientation $\theta$ relative to the world frame $\left\{s\right\}$. Likewise, the position of each wheel is defined by its frame coordinates $(x_{w_k},y_{w_k})$ and orientation relative to the body frame $\left\{b\right\}$. In the scientific literature, the orientation of the wheels (${\beta }_k$) is usually referenced as $x_b$ (Figure 1a) [47]. In this paper we propose an alternative representation that emphasizes the relative orientation of the wheels (${\phi }_k$), referenced to the perpendicular of the centerline, joining the centroids of the platform and each wheel (Figure 1b).

The direction of the $x_b$ axis in the platform frame defines the positive direction of translation (forwards), and the positive rotation of the platform is counterclockwise. In each wheel, $x_{w_k}$ coincides with the wheel plane, and $y_{w_k}$ is collinear with the wheel hub.

Figure 2 shows a top view of the definition of the wheels in an omnidirectional platform. The axes of the rollers are represented in grey, with thicker lines to depict the rollers in contact with the ground.

Figure 2a shows how wheels are enumerated: the first one is the closest to $x_b$ in the first quadrant of the platform frame and the others are defined in the counterclockwise direction. Figure 2b shows the dimensional parameters: $r_k$ is the radius of the wheel $k$, measured from the center of the wheel to the ground; $R_k$ is the centerline joining the centroids of the platform and wheel $k$; $x_k$ and $y_k$ are the distances between the centroids of the platform and wheel $k$ in the $x_b$ and $y_b$ directions; ${\delta }_k$ is the angular distribution of wheel $k$, measured between the $x_b$ axis and the centerline $R_k$; ${\phi }_k$ is the angular orientation of wheel $k$, measured between the perpendicular of the centerline $R_k$ and the $x_{w_k}$ axis; ${\gamma }_k$ is the angular orientation of the rollers of wheel $k$, measured as the angle of the free rolling direction relative to the $y_{w_k}$ axis.

The target velocity of the omnidirectional motion is defined in the platform frame by the motion command ${\left(v,\alpha ,\omega \right)}_{\left\{b\right\}}$, where $v_{\left\{b\right\}}$ is the norm of the translational velocity of the motion, ${\alpha }_{\left\{b\right\}}$ is the angular orientation of the translational velocity, and ${\omega }_{\left\{b\right\}}$ is the angular velocity of the platform. The components of this command can be decomposed in the platform frame as ${[v_x,v_y,\omega ]\prime }_{\left\{b\right\}}$:

$${v_x}_{\left\{b\right\}}=v_{\left\{b\right\}}·\cos \left({\alpha }_{\left\{b\right\}}\right),$$

(1)

$${v_y}_{\left\{b\right\}}=v_{\left\{b\right\}}·\sin \left({\alpha }_{\left\{b\right\}}\right).$$

(2)

The velocity of the platform ${[v_x,v_y,\omega ]\prime }_{\left\{b\right\}}$ must be expressed in the wheel frame as ${[v_x,v_y]\prime }_{\left\{w_k\right\}}$ by first translating ($T$) and then rotating ($R$), the velocity vector in the platform frame:

$${\left[\begin{array}{c}{v}_x\\ v_y\end{array}\right]}_{\left\{w_k\right\}}=A_{b\to w_k\left(R\right)}·A_{b\to w_k\left(T\right)}·{\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]}_{\left\{b\right\}},$$

(3)

where the translation matrix from the platform centroid to the wheel centroid is (4), the rotation from the platform to the wheel frame is ${{ϵ}_k=\phi }_k+{\delta }_k+90°$, and the rotation matrix is (5):

$$A_{b\to w_k\left(T\right)}=\left[\begin{array}{ccc}1& 0& -y_k\\ 0& 1& x_k\end{array}\right],$$

(4)

$$A_{b\to w_k\left(R\right)}=\left[\begin{array}{cc}\cos \left({ϵ}_k\right)& \sin \left({ϵ}_k\right)\\ -\sin \left({ϵ}_k\right)& \cos \left({ϵ}_k\right)\end{array}\right].$$

(5)

The velocity of the platform in the wheel frame is then decomposed in the drive direction (translation in the direction of the wheel plane) and slide direction (free sliding in the direction of the rollers), as in Figure 3. The direction of $v_{x_{\left\{w_k\right\}}}$ coincides with the drive direction, so it is labelled as $v_{x_{\left\{w_k\right\}}}=v_{{xd}_{\left\{w_k\right\}}}$. The components of $v_{y_{\left\{w_k\right\}}}$ are labelled as $v_{{yd}_{\left\{w_k\right\}}}$ in the drive direction and $v_{{ys}_{\left\{w_k\right\}}}$ in the slide direction. The resulting velocity in the drive direction is

$${v_{drive}}_k=v_{{xd}_{\left\{w_k\right\}}}+v_{{yd}_{\left\{w_k\right\}}}=v_{x_{\left\{w_k\right\}}}+v_{y_{\left\{w_k\right\}}}\tan \left({\gamma }_k\right).$$

(6)

The angular velocities of the wheels are obtained from the translational velocity in the drive direction, assuming there is no skidding (${v_{drive}}_k={\omega }_k·r_k$):

$${\omega }_k={\displaystyle \frac{v_{x_{\left\{w_k\right\}}}}{r_k}}+{\displaystyle \frac{v_{y_{\left\{w_k\right\}}}\tan \left({\gamma }_k\right)}{r_k}}.$$

(7)

Equation (7) is expressed in a matrix form for wheel $k$ and as a function of the target velocity defined in the platform frame as:

$${\omega }_k=\left[\begin{array}{cc}{\displaystyle \frac{1}{r_k}}& {\displaystyle \frac{\tan \left({\gamma }_k\right)}{r_k}}\end{array}\right]·A_{b\to w_k\left(R\right)}·A_{b\to w_k\left(T\right)}·{\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]}_{\left\{b\right\}},$$

(8)

which can be written in a compact form as:

$${\omega }_k=A_k{\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]}_{\left\{b\right\}}.$$

(9)

In the case with $n$ wheels ($k=1\dots n$), the inverse kinematics of the mobile platform can be described as

$$\left[\begin{array}{c}{\omega }_1\\ {\omega }_2\\ ⋮\\ {\omega }_n\end{array}\right]=\left[\begin{array}{c}{A}_1\\ A_2\\ ⋮\\ A_n\end{array}\right]{\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]}_{\left\{b\right\}},$$

(10)

where the matrix containing the vectors $A_{k=1\dots n}$ is the Jacobian matrix $J$ of the inverse kinematics of the mobile platform.

Finally, the forward kinematics expression to compute the velocity of the platform from the velocity of its wheels is

$${\left[\begin{array}{c}{v}_x\\ v_y\\ \omega \end{array}\right]}_{\left\{b\right\}}=J^{+}\left[\begin{array}{c}{\omega }_1\\ {\omega }_2\\ ⋮\\ {\omega }_n\end{array}\right],$$

(11)

where $J^{+}$ is the Moore–Penrose generalized inverse matrix which, in case the Jacobian matrix $J$ is full rank, corresponds with the inverse of the Jacobian matrix, $J^{-1}$.

2.2. Set of Mobile Platform Configurations Assessed

This section describes eleven mobile platform configurations whose omnidirectional capabilities are analyzed in the paper. All of them have three identical omni wheels with the same radius ($r_k=r$), roller orientation (${\gamma }_k=0°$), and distance from the centroid of the mobile platform ($R_k=R$), but they have different wheel distributions (${\delta }_k$) and relative orientations (${\phi }_k$).

Table 1. Mobile platforms analyzed.

Configuration	Representation	${\mathit{\delta }}_1(°)$	${\mathit{\delta }}_2(°)$	${\mathit{\delta }}_3(°)$	${\mathit{\phi }}_1(°)$	${\mathit{\phi }}_2(°)$	${\mathit{\phi }}_3(°)$
3A		60	180	300	0	0	0
1A-2B		60	180	300	−90	0	−90
1A-2C		60	180	300	30	0	−30
1A-1B-1C		60	180	300	0	−90	−30
1A-1B-1D		60	180	300	0	−90	60
3E		60	180	300	49.37	49.37	49.37
3B		60	180	300	−90	−90	−90
2A-1B		60	180	300	0	−90	0
1B-2C		60	180	300	30	−90	−30
1B-2D		60	180	300	−60	−90	60
1A-2D		60	180	300	−60	0	60

shows the configurations assessed, providing an identification name, a top-view schematic representation of the distribution and orientation of the three omni wheels, and the angular distribution of each wheel (${\delta }_k$) and its relative orientation (${\phi }_k$). The axes of the platform frame are represented with green ($x_b$) and black ($y_b$) arrows, and the axes of the wheel frames are represented with pink ($x_{w_k}$) and black ($y_{w_k}$) arrows.

3. Procedure for Omnidirectionality Analysis

This section describes the procedure proposed to analyze the omnidirectional capabilities of a mobile platform with three omni wheels. The procedure is based on the hypothesis that the analysis of the bidirectionality compliance between the inverse and forward kinematics can be used to evaluate the omnidirectionality of a motion system. That is, given a set of target motion commands, the result of the sequential computation of the inverse and forward kinematics for each command is coincidental with its corresponding target motion command as long as the system is omnidirectional.

3.1. Bidirectionality Error

The bidirectionality error is computed from the execution of a specific motion command. It is defined as the difference between the motion command applied to the mobile platform and the motion estimated from its kinematics. The bidirectionality error is computed in two steps. Firstly, the target motion command applied to the mobile platform ${\left(v,\alpha ,\omega \right)}_{\left\{b\right\}}$, expressed as ${\left(v_x,v_y,\omega \right)}_{\left\{b\right\}}$, is used to compute the angular velocities of the wheels using inverse kinematics (10). Secondly, the angular velocities of the wheels obtained in the first step are used to compute the motion of the mobile platform ${\left(v_x\prime ,v_y\prime ,\omega \prime \right)}_{\left\{b\right\}}$ using direct kinematics (11). Thirdly, the computed motion of the mobile platform, expressed as ${\left(v\prime ,\alpha \prime ,\omega \prime \right)}_{\left\{b\right\}}$, is compared with the target motion command to obtain the bidirectionality errors $\left(ev,e\alpha ,e\omega \right)$ using

$$ev=\left|v-v^{\prime }\right|,$$

(12)

$$e\alpha =\left|\alpha -{\alpha }^{\prime }\right|,$$

(13)

$$e\omega =\left|\omega -{\omega }^{\prime }\right|.$$

(14)

These errors are zero or numerical noise if the computed motion is equal to the original target motion command applied to the mobile platform, which means that the bidirectionality compliance is satisfied for this motion command.

Figure 4 shows the flowchart of the procedure used to estimate the bidirectionality error associated with one specific target motion command for a mobile platform with three omni wheels. The inputs of the procedure are the platform model (see Figure 2, Table 1), and a target motion command expressed in the platform base ${\left(v,\alpha ,\omega \right)}_{\left\{b\right\}}$. The outputs of the process are the bidirectionality error for each variable of the motion command, $ev,e\alpha ,e\omega$. For the sake of comparison, this procedure is executed in 0.522 ms on a testbench computer.

3.2. Omnidirectionality Analysis

The complete procedure proposed to evaluate the omnidirectional capabilities of a mobile platform is based on the assessment of the kinematics bidirectionality compliance for a whole set of motion commands. This requires the computation of the bidirectional errors for a range of motion commands explored to analyze under which motion commands the bidirectionality compliance is satisfied.

The selection of the set of motion commands to be evaluated consists of the definition of a feasible constant norm for the translational velocity $v$, and the definition of a feasible set of values for the angular orientation of the translational velocity ${\alpha }_{i=1\dots N}$ and for the angular velocity ${\omega }_{j=1\dots M}$, with $N\in \mathbb{N}-\left\{0\right\}$ and $M\in \mathbb{N}-\left\{0\right\}$. The values are ${\alpha }_i\in \left[\mathrm{0,360}\right](°)$ and ${\omega }_j\in \left[{-\omega }_{MAX},{\omega }_{MAX}\right](rad/s)$, with ${\omega }_{MAX}$ being the maximum angular velocity analyzed. In this procedure, the translational velocity $v$ is not explored as a set of values because there is a linear relationship between the angular velocity of the wheels (${\omega }_{k=1\dots n}$) and the translational velocity $v$ that makes it unnecessary to assess its influence in the computation. This linear relationship is described in Equation (8).

Figure 5 shows the diagram of the process used to assess the omnidirectionality of a mobile platform. The inputs of the process are the platform model and the range of motion commands expressed in the platform base. Specifically, the set of motion commands explore, for each value of the angular velocity (${\omega }_j$), the whole set of angular orientations (${\alpha }_{i=1\dots N}$) of the linear velocity of the platform.

Firstly, for each combination of $v$, ${\alpha }_i$, and ${\omega }_j$, the bidirectionality errors $e{v}_{ij}$, $e{\alpha }_{ij}$, $e{\omega }_{ij}$ are computed following the procedure described in Figure 4. The evaluation of the whole set of bidirectionality errors is stored in the matrixes $Ev$, $E\alpha$, and $E\omega$, defined as

$$Ev=e{v}_{i=1\dots N,j=1\dots M},$$

(15)

$$E\alpha =e{\alpha }_{i=1\dots N,j=1\dots M},$$

(16)

$$E\omega =e{\omega }_{i=1\dots N,j=1\dots M}.$$

(17)

The errors computed in this first step are represented in Figure 5 as a color-scaled image. The X-axis of the image denotes the range of ${\omega }_j$ values explored, and the Y-axis denotes the range of ${\alpha }_i$ values explored. In this case, the bidirectionality errors $Ev$, $E\alpha ,$ and $E\omega$ displayed in Figure 5 correspond to the case 2A-1B.

Secondly, three acceptability threshold values are applied to the bidirectionality errors:${ev}_{max}$ is the maximum acceptable error allowed in the linear velocity $v\prime$, ${e\alpha }_{max}$ is the maximum acceptable error allowed in the angular orientation $\alpha \prime$, and ${e\omega }_{max}$ is the maximum acceptable error allowed in the angular rotation of the mobile platform $\omega \prime$. The segmented errors are stored in the matrixes $Tv$, $T\alpha ,$ and $T\omega$, whose elements are defined as follows:

$$if\left(e{v}_{ij}<{ev}_{max}\right),then T{v}_{ij}=1,otherwise,T{v}_{ij}=0,$$

(18)

$$if\left(e{\alpha }_{ij}<{e\alpha }_{max}\right),then T{\alpha }_{ij}=1,otherwise,T{\alpha }_{ij}=0,$$

(19)

$$if\left(e{\omega }_{ij}<{e\omega }_{max}\right),then T{\omega }_{ij}=1,otherwise,T{\omega }_{ij}=0.$$

(20)

Again, the segmented errors computed in this second step are represented in Figure 5 as a color-scaled image, where white color represents errors above the threshold, and green depicts errors below it. The segmented errors $Tv$, $T\alpha ,$ and $T\omega$ displayed in Figure 5 also correspond to the case 2A-1B.

Finally, the bidirectionality compliance associated with a motion command is obtained from the logical conjunction (“AND”) for its corresponding segmented errors $Tv$, $T\alpha ,$ and $T\omega$:

$${Tv\alpha \omega }_{ij}={Tv}_{ij} AND {T\alpha }_{ij} AND {T\omega }_{ij}.$$

(21)

All the logical conjunctions corresponding to each combination of ${\alpha }_i$ and ${\omega }_j$ are stored in the matrix $Tv\alpha \omega$, which is analyzed to determine the omnidirectional capability of the mobile platform. The values in the matrix are represented as a two-dimensional image: the green color represents the errors below the thresholds, and white represents the cases in which at least one of the $T{v}_{ij}$, $T{\alpha }_{ij}$, and $T{\omega }_{ij}$ errors are above the thresholds. If this matrix $Tv\alpha \omega$ reveals no errors above the thresholds (the entire matrix has a green color), bidirectionality compliance is satisfied for the whole range of motion commands explored and thus the system is omnidirectional:

$$if({Tv\alpha \omega }_{ij}=1\forall {\alpha }_i\wedge \forall {\omega }_j),then OMD=1,otherwise,OMD=0,$$

(22)

where $OMD$ is the binary value that summarizes the omnidirectional capabilities of the mobile platform.

In the specific case with the errors below the threshold only in the whole set of ${\alpha }_i$ for ${\omega }_j=0 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, translation is possible and controllable in any direction but without rotation.

$$if\left({Tv\alpha \omega }_{ij}=1\forall {\alpha }_i\wedge {\omega }_j=0\right),then T=1,otherwise,T=0,$$

(23)

where $T$ is the binary value that summarizes the translational capabilities of the mobile platform.

Figure 5 includes an example representation of the $Tv\alpha \omega$ matrix from case 2A-1B. The outputs of the procedure are the platform translation ($T$) and omnidirectional ($OMD$) capabilities.

4. Results

4.1. Prediction of the Omnidirectional and Translation Capabilities

This section presents the results obtained from the application of the procedure proposed in this paper.

Table 2. Mobile platform configurations with the same negligible bidirectional errors in $Ev$, $E\alpha$, and $E\omega$. The maximum value in the color scale is 0.3 m/s for $Ev$, 20° for $E\alpha ,$ and 0.3 rad/s for $E\omega$. The schematic representation of the configurations is shown in Table 1.

Configuration	$\mathit{E}\mathit{v}$	$\mathit{E}\mathit{\alpha }$	$\mathit{E}\mathit{\omega }$	Scale
3A 1A-2B 1A-2C 1A-1B-1C 1A-1B-1D 3E

and

Table 3. Mobile platform configurations with different non-negligible bidirectional errors in $Ev$, $E\alpha$, and $E\omega$. The maximum value in the color scale is 0.3 m/s for $Ev$, 20° for $E\alpha ,$ and 0.3 rad/s for $E\omega$.

Configuration	$\mathit{E}\mathit{v}$	$\mathit{E}\mathit{\alpha }$	$\mathit{E}\mathit{\omega }$	Scale
3B
2A-1B
1B-2C
1B-2D
1A-2D

show the evaluation of the bidirectionality errors $Ev$, $E\alpha$, and $E\omega$ of the mobile platforms assessed in this paper (described in Table 1).

The results form Table 2 and Table 3 were obtained with the following parameters: $v=0.3 \mathrm{m}/\mathrm{s}$, ${\alpha }_i$ ranging within $\left[\mathrm{0,360}\right]°$, and ${\omega }_j$ ranging within $\left[-\mathrm{2,2}\right]\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$. These values are inspired by the motion commands used by a humanoid-sized assistance robot with three omni wheels using the 3A configuration [48].

Table 2 shows the graphical representation of the bidirectionality errors $Ev$, $E\alpha ,$ and $E\omega$ of mobile platform configurations 3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, and 3E. Given that the order of the bidirectionality errors in Table 2 is below 10⁻¹⁴, the errors can be considered numerical noise and are negligible, so they have the same matrixes $Ev$, $E\alpha ,$ and $E\omega$ and their representation is the same for all the platform configurations assessed. The color scale used to represent the bidirectionality errors is saturated with maximum values of 0.3 m/s for $Ev$, 20° for $E\alpha ,$ and 0.3 rad/s for $E\omega$. The term $max$ in the color scale in Table 2 is a general representation of the saturation values for $Ev$, $E\alpha ,$ and $E\omega$. Similarly, Table 3 shows the graphical representation of the bidirectionality errors of $Ev$, $E\alpha ,$ and $E\omega$ of configurations 3B, 2A-2B, 1B-2C, 1B-2D, and 1A-2D, which have non-negligible bidirectionality errors. In these cases, the errors obtained depend on the configuration of the mobile platform assessed and thus the configurations have different matrixes $Ev$, $E\alpha ,$ and $E\omega$. The color scale used in Table 3 is the same as in Table 2.

The bidirectionality errors $Ev$, $E\alpha$, $E\omega$ (Table 2 and Table 3) are segmented using the following thresholds: $e{v}_{max}=0.003 \mathrm{m}/\mathrm{s}$, ${e\alpha }_{max}=0.2°$, and $e{\omega }_{max}=0.003 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$. The images of the segmented errors $Tv$, $T\alpha$, $T\omega$ are displayed in

Table 4. Segmented errors $Tv$, $T\alpha$, $T\omega ,$ and $Tv\alpha \omega$ of the mobile platform configurations assessed. The green color depicts that all bidirectionality errors are below the acceptability thresholds.

Configuration	$\mathit{T}\mathit{v}$	$\mathit{T}\mathit{\alpha }$	$\mathit{T}\mathit{\omega }$	$\mathit{T}\mathit{v}\mathit{\alpha }\mathit{\omega }$	Scale
3A 1A-2B 1A-2C 1A-1B-1C 1A-1B-1D 3E

and

Table 5. Segmented errors $Tv$, $T\alpha$, $T\omega ,$ and $Tv\alpha \omega$ of the mobile platform configurations assessed. The white color depicts bidirectionality errors above the acceptance thresholds.

Configuration	$\mathit{T}\mathit{v}$	$\mathit{T}\mathit{\alpha }$	$\mathit{T}\mathit{\omega }$	$\mathit{T}\mathit{v}\mathit{\alpha }\mathit{\omega }$	Scale
3B
2A-1B
1B-2C
1B-2D
1A-2D

. These tables also display the matrixes $Tv\alpha \omega$ storing the logical conjunction “AND” of the segmented errors of each motion command, which are used to evaluate the omnidirectional capabilities of the mobile platform configurations. The green color represents the combinations in which the errors are smaller than their corresponding threshold, and white color represents the combinations in which the errors are greater than their corresponding threshold. The term $e_{max}$ in the color scale used in Table 4 and Table 5 represents the thresholds $e{v}_{max}$ for $Tv$, ${e\alpha }_{max}$ for $T\alpha ,$ and $e{\omega }_{max}$ for $T\omega$.

Table 4 shows the segmented errors of configurations 3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, and 3E, which have the bidirectionality errors below the thresholds, thus having the same matrixes $Tv$, $T\alpha ,$ and $T\omega$ and the same combination $Tv\alpha \omega$. Table 5 shows the segmented errors of configurations 3B, 2A-2B, 1B-2C, 1B-2D, and 1A-2D, which have bidirectionality errors above the thresholds, thus having different matrixes $Tv$, $T\alpha ,$ and $T\omega$ and a different combination $Tv\alpha \omega$.

The bidirectionality errors of the mobile platform configurations shown in Table 2 (3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, 3E) describe negligible numerical noise, which is represented with a uniform color. The application of the error thresholds results in the segmented errors shown in Table 4, which also includes the matrixes $Tv\alpha \omega$ with the logical conjunctions that combine the segmented results for each motion command. The results in Table 4 show that the segmented errors are below the thresholds for the whole set of motion commands explored ($Tv\alpha \omega$ is all green), so the bidirectionality compliance is satisfied and the configurations have omnidirectional capabilities. That is to say, all the set of motion commands assessed (all combinations of $v$, ${\alpha }_i$, and ${\omega }_j$) can be executed by the mobile platforms, so they can translate in any direction ${\alpha }_i$ with or without rotation ${\omega }_j$. In short, configurations 3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, 3E (Table 1) have been classified as omnidirectional according to the procedure proposed in this paper.

The bidirectionality errors in the mobile platform configurations shown in Table 3 (3B, 2A-2B, 1B-2C, 1B-2D, 1A-2D) are case-dependent and their representations show a non-uniform color distribution that will be analyzed case by case.

In Table 3, configuration 3B, the bidirectionality errors $Ev$ obtained from comparing the target and calculated linear velocities of the mobile platform ($v,v\prime$) are negligible. Similarly, the bidirectionality errors $E\alpha$ obtained from comparing the target and calculated angular orientations of the motions ($\alpha ,\alpha \prime$) from its kinematics are also negligible. However, the bidirectionality errors $E\omega$ obtained from comparing the target and calculated angular speeds of the motions ($\omega ,\omega \prime$) are negligible only in the set of ${\alpha }_{i=1\dots N}$ with ${\omega }_j=0 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, which indicates that the mechanism is uncontrollable with respect to rotation.

Table 5, configuration 3B, shows these errors segmented according to the defined thresholds and the logical conjunction “AND” in $Tv\alpha \omega$. In this case, some points in $Tv\alpha \omega$ have bidirectionality errors above the thresholds (represented in white color), which means that bidirectionality compliance is not satisfied and an omnidirectional motion is not possible. Nevertheless, in the cases in the set of ${\alpha }_{i=1\dots N}$ with ${\omega }_j=0 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, $Tv\alpha \omega$ show bidirectionality errors below the thresholds (represented as a green vertical line), meaning that the mobile platform is able to translate with any angular orientation without rotation. This new information about the translational capabilities of the platform is only provided by the procedure proposed in this paper.

In Table 3, configurations 2A-1B, 1B-2C, 1B-2D, and 1A-2D, the bidirectionality errors $Ev$, $E\alpha ,$ and $E\omega$ obtained from comparing the target and calculated motions of each mobile platform (comparing $v$ and $v\prime$, $\alpha$ and $\alpha \prime$, and $\omega$ and $\omega \prime$), have irregular non-zero distributions. After applying the acceptability thresholds, the representations of the combined segmented errors $Tv\alpha \omega$ in Table 5 depict the following information. In cases 2A-1B and 1B-2D, the mobile platforms can only translate in the directions of the angular orientations ${\alpha }_i=0°$ and ${\alpha }_i=180°$ (forwards and backwards) with ${\omega }_j=0 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, so the applicability of these configurations is very limited. In cases 1B-2C and 1A-2D, the mobile platforms can only move in the direction of two specific angular orientations: ${\alpha }_i=0°$ or $180°$ for the case 1B-2C, and ${\alpha }_i=90°$ or $270°$ for the case 1A-2D. In both cases, they can rotate with any angular speed ${\omega }_j$ while performing these displacements, but their applicability is also limited because only two angular orientations are possible. Finally,

Table 6. Comparative analysis of the omnidirectional capabilities of the platforms obtained with two state-of-the-art methods and the omnidirectional and translation capabilities obtained with the procedure proposed in this paper.

Configuration	Representation	Omnidirectional		Proposed Procedure
		$\mathbf{Rank} (\mathit{J}$), Full	Y. Li et al. [17]	Omnidirectional	Translation
3A		YES	YES	YES	YES
1A-2B		YES	YES	YES	YES
1A-2C		YES	YES	YES	YES
1A-1B-1C		YES	YES	YES	YES
1A-1B-1D		YES	YES	YES	YES
3E		YES	YES	YES	YES
3B		NO	NO	NO	YES
2A-1B		NO	NO	NO	NO
1B-2C		NO	NO	NO	NO
1B-2D		NO	NO	NO	NO
1A-2D		NO	NO	NO	NO

summarizes the analysis of the motion capabilities of all the set of mobile platform configurations described in Table 1. Table 6 shows that the omnidirectional capability results obtained with the procedure proposed in this paper agree with the state-of-the-art methods while a new differentiated estimation of the translational capabilities is also provided.

Specifically, Table 6 presents the configuration name, a schematic representation of each platform configuration, the analysis of its omnidirectional capabilities based on two state-of-the-art methods [49], and the estimation of the omnidirectional and translation capabilities obtained with the procedure proposed in this paper. The first state-of-the-art method is based on computing the rank of the Jacobian matrix of the kinematic model of the mobile platform: if the matrix is full rank, the system is omnidirectional. The second state-of-the-art method is based on the bottom-roller axle intersections approach proposed by Y. Li et al. [17] applied to omni wheels with ${\gamma }_k=0°$. This method analyzes the intersections between the axes of the passive rollers in contact with the ground to determine geometrically if a system is omnidirectional: when the axes intersect at two or three points, the platform is omnidirectional, whereas when the axes do not intersect or intersect only at one point, it is non-omnidirectional. As stated before, the results of Table 6 shows that the estimation of the omnidirectional capabilities provided by these state-of-the-art methods agree with the procedure proposed in this paper.

4.2. Validation of the Omnidirectionality and Translation Capabilities

This section presents the experiments conducted to validate the application of the procedure proposed for estimating the omnidirectional and translational capabilities of the mobile platforms described in Table 1. This validation is based on the simulation of the motion of the different omnidirectional platforms assessed. The simulator used in the paper is described in [50]. The simulator uses the information of the configuration of the wheels of the platform and the target velocity assigned to each wheel to compute the trajectory and displacement of the mobile platform. The simulator is limited to indoor conditions and no wheel slippage.

The validation experiments consist of the definition of a set of trajectories and on the implementation of these trajectories by the mobile platforms assessed. The execution of the motion defined in these trajectories requires a combination of translation and rotation capabilities in the mobile platform.

Table 7. Definition of the motion commands used in the validation experiments.

Experiment	Motion Command $(\mathit{v},\mathit{\alpha },\mathit{\omega },\mathit{t})$$(\mathbf{m}/\mathbf{s},°,\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{s},\mathbf{s})$	Motion Required		Final Position $({\mathit{x}}_{\mathit{f}},{\mathit{y}}_{\mathit{f}},{\mathit{\theta }}_{\mathit{f}})$$(\mathit{m},\mathit{m},°)$
		Translation	Rotation
E1	$(0.1,0,0,10)$	X	-	$(1,\mathrm{0,0})$
E2	$(0.1,90,0,10)$	X	-	$(\mathrm{0,1},0)$
E3	$(0.1,45,0,14.14)$	X	-	$(1,\mathrm{1,0})$
E4	$(0.1,0,0.15,14)$	X	X	$(0.58,\mathrm{1.00,120.32})$
E5	$(0.1,90,-0.15,14)$	X	X	$(1.00,0.58,-120.32)$
E6	$(0,0,0.15,10.5)$	-	X	$(0,\mathrm{0,90.24})$

details the experiments, the target motion command applied to each mobile platform, the translation or rotation capabilities required to execute the motion, and the expected final position of the platform after the motion $(x_f,y_f,{\theta }_f)$. In all these experiments, the initial position of the mobile platform in the simulation $(x_i,y_i,{\theta }_i)$ is the same: (0 m, 0 m, 0°). The last parameter $t$ of the motion command $(v,\alpha ,\omega ,t)$ is the time defined to execute the motion in order to reach the final position of the motion $(x_f,y_f,{\theta }_f)$. Additionally, Figure 6 shows the expected trajectory that must follow the mobile platform in case of executing the motion commands defined in Table 7. Trajectories E1 and E2 only require translational capabilities in the $x_s$ and $y_s$ axes of the world frame. Trajectory E3 requires a combination of translational capabilities in both axes in the world frame. Trajectories E4 and E5 require translation and rotation capabilities. Finally, trajectory E6 only requires rotation capabilities as it depicts a single rotation of approximately 90° to the left side of the mobile platform.

Table 8. Results of the validation experiments of the mobile platforms identified as omnidirectional. OK means that the trajectory has been successfully executed by the platform (it has arrived at the planned destination).

Platform	Experiment (Type of Motion Required)						Summary of the Simulations
	E1 (T)	E2 (T)	E3 (T)	E4 (T+R)	E5 (T+R)	E6 (R)	OMD	T	Trajectories
3A 1A-2B 1A-2C 1A-1B-1C 1A-1B-1D 3E	OK	OK	OK	OK	OK	OK	YES	YES

and

Table 9. Results of the validation experiments of the mobile platforms identified as non-omnidirectional. OK means that the trajectory has been successfully executed by the platform.

Platform	Experiment (Type of Motion Required)						Summary of the Simulations
	E1 (T)	E2 (T)	E3 (T)	E4 (T+R)	E5 (T+R)	E6 (R)	OMD	T	Trajectories
3B	OK	OK	OK	FAILED	FAILED	FAILED	NO	YES
2A-1B	OK	FAILED	FAILED	FAILED	FAILED	FAILED	NO	NO
1B-2C	OK	FAILED	FAILED	OK	FAILED	OK	NO	NO
1B-2D	OK	FAILED	FAILED	FAILED	FAILED	FAILED	NO	NO
1A-2D	FAILED	OK	FAILED	FAILED	OK	OK	NO	NO

summarize the results of the simulation experiments performed with all the mobile platforms assessed. Table 8 shows the results of the mobile platforms identified as omnidirectional in the previous analysis (provided in Table 6), while Table 9 shows the results of the mobile platforms identified as non-omnidirectional. Both tables show a CAD design of the mobile platform and the evaluation of the validation experiments defined in Table 7 and Figure 6. The result of each validation experiment is labeled as OK in the case where the final position reached by the mobile platform differs less than 0.01 m from the expected destination, and as FAILED if it differs more. Additionally, Table 8 and Table 9 provide a summary of the trajectories simulated. In these images, a yellow circle depicts that the mobile platform has reached the expected final position within the distance threshold of 0.01 m while an empty circle depicts that the final position has not been reached. The parameter OMD summarizes the omnidirectional capabilities of the mobile platform obtained by simulation: labeled as YES in the case that the mobile platform is able to successfully execute all trajectories requiring a rotation (E4, E5, and E6), and NO otherwise. Similarly, the parameter T summarizes the translational capabilities of each mobile platform: labeled as YES in the case that the mobile platform is able to successfully execute all trajectories requiring only translation (E1, E2, and E3), and NO otherwise.

The simulation results displayed in Table 8 show that the mobile platforms identified as omnidirectional by the procedure proposed in this paper are able to execute successfully all the validation trajectories. Alternatively, the simulation results displayed in Table 9 show that the mobile platforms identified as non-omnidirectional are not able to execute successfully all the validation trajectories. Results for the 3B platform indicate that it is not controllable when there is rotation of the platform ($\omega \ne 0 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$), so the system is stable only when translational trajectories without rotation are implemented. Both 2A-1B and 1B-2D platforms are only able to execute the E1 command (move forward without rotation), and do not successfully implement the rest of the commands, including the rotation-only motions. Both 1B-2C and 1A-2D platforms are able to implement specific translation-only trajectories and specific trajectories combining translation and rotation, but other motions requiring the same capabilities fail. Such results indicate that both platforms act like a differential-drive one, with its corresponding motion restrictions. Finally, these results obtained by simulation (Table 9) agree with the motion capabilities predicted in this paper (provided in Table 6).

5. Conclusions

This paper presents a procedure for estimating the motion capabilities of mobile platforms with three omni wheels, which is the minimum number of wheels required to achieve omnidirectionality. This procedure is able to differentiate between omnidirectional capabilities and translational capabilities by analyzing the bidirectionality compliance between the inverse and forward kinematics for a wide set of motion commands.

The procedure has been assessed with eleven mobile platforms, each one with different omni wheel configurations. The estimation of the motion capabilities of these platforms agrees with state-of-the-art methods [17] while providing a new specific estimation of the translation capabilities and combined visual information of the motion capabilities of the platforms. These estimations have been validated by simulating the trajectories followed by the mobile platforms while executing a reduced set of motion commands.

Results have shown that the procedure proposed in this paper for estimating the omnidirectionality capabilities is 869-fold faster than performing alternative trajectory simulations. This is because the computation of the bidirectionality error of a motion command requires an average time of 0.522 ms on a testbench computer, as it is based on two matrix operations (see Figure 4). Alternatively, the simulation of the execution of a trajectory by a mobile platform to verify if a motion command can be executed or not requires an average time of 0.454 s on the same testbench computer. This performance allows this procedure to be applied during the design stage of new mobile platforms with three omni wheels to validate their motion performances.

Relative to the related scientific literature [17,18,51], the advantage of this proposal is that it indicates which platform configurations become uncontrollable when the angular velocity of the platform is different from zero, which indirectly conveys specific information on the translational performance of the platform under evaluation. Additionally, the graphical information on the motion capabilities of a mobile platform provides direct visual information about which motions may be successfully executed by the platform.

This proposal has the limitation of not considering dynamic effects in the omnidirectional motion, which will be addressed in future works. Future work will apply the procedure proposed in this paper on the optimal design of translation-only mobile platforms tailored for heavy-duty industrial applications where rotation is not needed during transportation.