Generalized Kinematic Modeling of Wheeled Mobile Robots: A Unified Framework for Heterogeneous Architectures

Abstract

The increasing heterogeneity of wheeled mobile robot (WMR) architectures, including differential-drive, Ackermann, omnidirectional, and reconfigurable platforms, poses a major challenge for defining a unified, scalable kinematic representation. Most existing formulations are tailored to specific mechanical layouts, limiting analytical coherence, cross-platform interoperability, and the systematic reuse of modeling, odometry, and motion-related algorithms. This work introduces a generalized kinematic modeling framework that provides a mathematically consistent formulation applicable to arbitrary WMR configurations. Wheel–ground velocity relationships and non-holonomic constraints are expressed through a concise vector formulation that maps wheel motions to chassis velocities, ensuring consistency with established models while remaining independent of the underlying mechanical structure. A parameterized wheel descriptor encodes all relevant geometric and kinematic properties, enabling the modular assembly of complete robot models by aggregating wheel-level relations. The framework is evaluated through numerical simulations on four structurally distinct platforms: differential-drive, Ackermann, three-wheel omnidirectional (3, 0), and 4WD. Results show that the proposed formulation accurately reproduces the expected kinematic behavior across these fundamentally different architectures and provides a coherent and consistent representation of their motion. The unified representation further provides a common kinematic backbone that is directly compatible with odometry, motion-control, and simulation pipelines, facilitating the systematic retargeting of algorithms across heterogeneous robot platforms without architecture-specific reformulation. Additional simulation studies under realistic physics-based conditions show that the proposed formulation preserves coherent kinematic behavior during complex trajectory execution and supports the explicit incorporation of geometric imperfections, such as wheel mounting misalignments, when such parameters are available. By consolidating traditionally separate derivations into a single coherent formulation, this work establishes a rigorous, scalable, and architecture-agnostic foundation for unified kinematic modeling of wheeled mobile robots, with particular relevance for modular, reconfigurable, and cross-architecture robotic systems.

1. Introduction

Wheeled Mobile Robots (WMRs) represent one of the most mature and widely deployed locomotion technologies in robotics, offering high energy efficiency, mechanical simplicity, and control precision. They are fundamental in industrial automation, logistics, service robotics, and autonomous transportation. Central to their operation is the accurate representation of kinematics, i.e., the mapping between wheel actuation and global motion, which underpins localization, path planning, and control algorithms. Over decades of research, the kinematic modeling of WMRs has evolved through a variety of formulations tailored to specific drive mechanisms. Classic approaches model differential, Ackermann, or omnidirectional configurations independently, resulting in specialized equations that lack interoperability. This fragmentation limits model reuse and scalability when designing hybrid or reconfigurable robotic systems.

2. State of the Art and Related Work

The study of kinematic modeling for WMRs has evolved significantly over the last three decades, progressing from simple differential-drive formulations to highly reconfigurable and omnidirectional mechanisms. However, despite the remarkable mechanical and control innovations achieved, modeling methodologies remain largely fragmented, often tailored to specific locomotion schemes. This section reviews the main research trends in kinematic modeling, highlighting their strengths, limitations, and the lack of a unified representation across different architectures.

2.1. Historical Context and Foundational Works

The theoretical foundations of WMR kinematics were firmly established in the 1980s and 1990s, during which geometric and constraint-based descriptions of mobility were formalized. Early geometric treatments such as those by Dubins [1] and Reeds-Shepp [2] provided the first characterizations of curvature-constrained motion, anticipating later developments in non-holonomic robotics.

A major turning point occurred with the work of Samson [3], who consolidated controllability and stabilization results for non-holonomic systems, and with the influential survey by Campion et al. [4], which introduced a structural classification of WMRs based on mobility, steerability, and degree of non-holonomy. The geometric perspective formalized by Campion and colleagues remains one of the most authoritative references for understanding rolling constraints and the structure of admissible motions.

In parallel, the robotics community developed systematized treatments of nonholonomic mechanics and control. The works of Bloch et al. [5,6] formalized nonholonomic constraints within geometric mechanics, providing a unifying mathematical structure widely adopted in subsequent WMR analyses.

The transition from theory to practice was significantly shaped by authoritative textbooks such as those of Murray, Li, and Sastry [7], and later Siegwart and Nourbakhsh [8]. These resources consolidated kinematic modeling, perception, and control strategies into cohesive frameworks that influenced both research and industrial development of mobile robotics for nearly two decades.

With the rapid introduction of omnidirectional platforms in the 2000s, the need for unified kinematic descriptions intensified. Works such as those of Muir and Neuman [9] and Mandow et al. [10] provided early systematic analyses of omnidirectional wheel mechanisms and their velocity coupling. More recently, Li et al. [11] introduced a generalized mecanum-wheel kinematic model that formalizes roller geometry and coupling matrices, representing an important step toward architectural unification, though still limited to holonomic systems.

Contemporary surveys, such as those by Tagliavini et al. [12] and Alexa et al. [13], highlight the persistent fragmentation of kinematic formulations across architectures, including differential, Ackermann, omnidirectional, articulated, and hybrid WMRs, despite decades of development. This reinforces the need for a unified, parameterized, wheel-level modeling approach, such as the one advanced in the present work, capable of representing heterogeneous locomotion mechanisms within a common mathematical framework.

2.2. Classical and Non-Holonomic Architectures

Differential and Ackermann steering configurations represent the most traditional types of non-holonomic locomotion. Their modeling is based on geometric constraints that limit lateral motion while preserving rolling contact. Although these formulations are computationally efficient and well understood, they exhibit limited flexibility when extended to systems with independent steering or multiple drive mechanisms. Tao et al. [14] developed a precise kinematic model for a mobile robot designed for the machining of large-scale workpieces, demonstrating how differential drive principles can be adapted to industrial applications. Similarly, Benamar et al. [15] proposed a generalized differential model for articulated mobile robots, capable of representing chained mechanical structures with passive joints. Nevertheless, both studies remain constrained by their architecture-specific assumptions, making their direct application to holonomic or hybrid systems non-trivial.

2.3. Omnidirectional and Holonomic Mobility

Omnidirectional robots have been extensively studied because of their ability to achieve full planar mobility. Siradjuddin et al. [16] introduced a general inverse kinematic formulation for omnidirectional robots, providing a unified structure for several wheel arrangements. However, the scope of their generalization is limited to holonomic systems, as the approach relies on the full-rank property of the kinematic Jacobian, which does not hold for non-holonomic configurations.

Li et al. [17] analyzed the kinematics and dynamics of a mecanum wheel platform using RecurDyn multibody simulation software, validating the consistency between analytical and numerical results. Their study reinforced the importance of accurately modeling wheel–ground interaction in omnidirectional mobility. Similarly, Rijalusalam et al. [18] implemented real-time odometry for a four-omni-wheel robot on an embedded microcontroller, demonstrating the feasibility of executing complex motion control on resource-limited systems. Despite these advances, none of these works addresses the problem of integrating holonomic and non-holonomic motion under a unified mathematical framework.

2.4. Reconfigurable and Hybrid Platforms

In recent years, the growing demand for flexible, adaptable mobility has driven the development of reconfigurable wheel mechanisms that combine differential, steering, and omnidirectional capabilities. Zhao et al. [19] proposed the Omni-Differential Drive (ODD), a hybrid mechanism that enables a smooth transition between differential and omnidirectional behaviors. This design exemplifies the emerging class of morphologically adaptive robots capable of real-time reconfiguration. In parallel, Martínez-García et al. [20] presented a multi-configuration kinematic model for active-drive/steer four-wheel robots, providing a mathematical foundation for describing multiple locomotion modes within a common formulation. However, their method still requires discrete reparameterization for each configuration, preventing a continuous representation across architectures.

A related trend involves large or heavy-duty mobile manipulators with independent drive-steer systems. Tao et al. [14] and Benamar et al. [15] demonstrated that extending classical non-holonomic formulations to these complex systems often requires ad-hoc modifications, leading to reduced generality and limited compatibility with standard control frameworks.

2.5. Odometry Calibration and Model Consistency

Accurate odometry is essential for reliable motion estimation and control in WMRs. Sousa et al. [21] introduced OptiOdom, a generic approach to odometry calibration applicable to multiple robot architectures. Their method decouples geometric calibration from the control system, allowing for improved motion accuracy across different locomotion types. Nevertheless, as noted in [12,21], such calibration techniques remain largely disconnected from the underlying kinematic model, leading to redundant parameter tuning and suboptimal integration with control strategies.

Similarly, Jilek et al. [22] investigated the odometry of six-wheeled mobile robots and highlighted that traditional Jacobian-based derivations fail to capture the interdependencies between wheels in multi-contact systems. Their findings underscore the need for a more comprehensive kinematic framework that explicitly encodes wheel-to-body interactions and geometric couplings.

2.6. Limitations of Existing Models

Despite notable progress, current kinematic frameworks for WMRs exhibit four major structural limitations that hinder their applicability across heterogeneous platforms.

• Dependence on architecture-specific derivations: Most kinematic models are independently formulated for differential, Ackermann, omnidirectional, or hybrid systems [12,14,15]. Consequently, these models lack interoperability and require full re-derivation when the wheel arrangement changes.
• Separation between holonomic and non-holonomic modeling: Existing formulations treat non-holonomic constraints (e.g., no lateral slip) separately from holonomic wheel geometries (e.g., omni/mecanum rollers) [16,22]. This leads to two incompatible mathematical representations, which prevent the creation of a unified kinematic structure.
• Lack of modular wheel-level parameterization: Many classical derivations model the robot as a monolithic structure rather than as an assembly of wheel modules [19,20]. The absence of modularity complicates the modeling of reconfigurable or adaptive systems and hinders the automatic generation of new architectures from shared kinematic primitives.
• Weak integration with calibration and control: Calibration frameworks such as OptiOdom [21] operate independently of the kinematic modeling layer. This separation reduces the overall consistency between geometric parameter estimation, velocity reconstruction, and closed-loop control, especially in architectures with multiple constraints or wheel–ground interactions.

Collectively, these limitations underscore the need for a unified, modular, and computationally coherent kinematic representation that can model both holonomic and non-holonomic architectures within a single mathematical framework. The absence of such a general framework constitutes a fundamental bottleneck in achieving model portability, scalable control synthesis, and cross-platform compatibility.

2.7. Motivation and Contributions

This paper introduces a generalized kinematic modeling framework that unifies diverse WMR architectures through a common parametric structure. The core idea is to define a modular wheel descriptor that encapsulates geometry, orientation, and actuation properties for any wheel type. By systematically assembling wheel-level constraints, the global forward and inverse kinematics emerge naturally, eliminating the need for architecture-specific derivations.

The main technical contributions of this work can be summarized as follows:

• A generalized wheel descriptor that encapsulates all geometric and kinematic parameters of an arbitrary wheel in a unified mathematical structure. This includes the position vector ${\mathit{r}}_i$, mounting angle ${\beta }_i$, roller orientation ${\alpha }_i$, admissible rolling direction ${\mathit{u}}_i$, and the mapping between the commanded wheel velocity $v_i$ and the effective ground-contact velocity $v_{c_i}$. This descriptor provides the fundamental abstraction enabling architecture-independent modeling.
• A universal closed-form wheel kinematic equation applicable to any actuated wheel type: standard, omni, mecanum, steerable, or caster-constrained. The formulation:

  $$v_i={\displaystyle \frac{1}{cos{\alpha }_i}}\left(u_{i,x}{\dot{x}}_m+u_{i,y}{\dot{y}}_m+c_i\omega \right)$$

  captures geometric effects, directional admissibility, and non-holonomic constraints within a single expression, eliminating the need for architecture-specific derivations at the wheel level.
• A systematic model-assembly procedure that constructs the full robot kinematic model by stacking the wheel-level equations into:

  $$\mathit{v}=J{\mathit{\xi }}_m,$$

  where the Jacobian J is generated directly from the descriptors rather than from platform-specific derivations. This enables automatic and consistent model generation for differential-drive, Ackermann, omnidirectional, 4WD/4WS, and hybrid architectures.
• A unified projection-based formulation of holonomic and non-holonomic constraints. The lateral no-slip condition is expressed using orthogonal projections onto ${\mathit{u}}_{n_i}$, allowing all constraint types to be handled consistently at the wheel level and propagated systematically to the chassis model, independent of steering or roller orientation.
• A cross-architecture modeling interface that provides a common kinematic backbone for heterogeneous WMR platforms. This abstraction provides a common kinematic backbone on which odometry, motion-control, and simulation algorithms can be formulated in a unified set of variables, constraints, and Jacobians, facilitating direct algorithmic reuse across robots with fundamentally different wheel arrangements and significantly improving modularity and scalability.

Taken together, these contributions constitute a comprehensive and fully unified kinematic framework for wheeled mobile robots that, to the best of our knowledge, has not been previously reported. Whereas traditional approaches derive kinematic models on an architecture-by-architecture basis, the proposed formulation establishes a single, scalable, and architecture-independent representation that applies uniformly to all wheel types and platform configurations. This unified treatment provides a robust foundation for reconfigurable mobility, the development and integration of model-based control and odometry calibration methods, and cross-platform simulation.

3. Generalized Kinematic Framework

The proposed framework provides a unified kinematic formulation applicable to any WMR architecture. In contrast to classical approaches, which are typically tailored to specific configurations such as differential, Ackermann, or omnidirectional drives, the present formulation is built upon a common set of wheel-level quantities: wheel position vectors, wheel–ground contact velocities, rolling directions, and actuation variables. By expressing all wheel types, including roller-based mechanisms, within a shared mathematical structure, the model achieves architectural neutrality, analytical consistency, and seamless scalability.

This general framework enables the systematic derivation of wheel kinematics, non-holonomic constraints (when applicable), and inverse/forward mappings between wheel velocities and chassis motion. It also supports reconfigurable and heterogeneous platforms, as model construction reduces to assembling the individual wheel equations defined by their geometric and kinematic descriptors.

The stages that constitute this framework are presented in detail in the following sections.

3.1. Local Reference Frame Definition

The robot is modeled within a local reference frame ($X_m,Y_m$) whose origin can, in principle, be placed at any fixed point in the polygon defining the chassis. In practice, however, it is typically located at the midpoint between the drive wheels for standard differential and Ackermann configurations, and at the geometric center of the wheel contact points for omnidirectional platforms. This flexibility allows the proposed framework to provide a unified representation applicable to differential, Ackermann, tricycle, omnidirectional, and hybrid robots.

Moreover, although not mandatory, the proposed framework suggests orienting the $X_m$-axis along the nominal forward direction of motion, and the $Y_m$-axis completes a right-handed coordinate system. Adopting this local frame provides a consistent description of translational and rotational motions via homogeneous kinematic relationships, independent of the robot’s global pose.

3.2. Kinematic Quantities and Representation

The principal kinematic variables are illustrated in Figure 1. The local reference frame representation includes the following elements:

• The fixed reference point on the robot’s body (the center of the motion kinematic equation) is selected as the origin $(0,0)$ of the local frame $\{X_m,Y_m\}$, with the $X_m$-axis defining the forward direction of motion.
• The tangential wheel velocities $v_i$, corresponding to the motion ordered or resulting at each wheel. These may represent actuated inputs, passive responses, or reference velocities depending on the wheel type.
• Linear velocities of the chassis ${\dot{x}}_m$, ${\dot{y}}_m$, and yaw rate $\omega$, which arise from the combined effect of all wheel–ground interactions.
• The effective contact velocities $v_{c_i}$ at the wheel–ground interaction are defined as follows. For standard traction wheels (for example, the solid black wheel in the figure), one has $v_{c_i}=v_i$. In contrast, for wheels equipped with rollers, such as mecanum wheels (striped wheel), the roller axis introduces an additional lateral velocity component that modifies $v_{c_i}$. It is important to clarify that in Figure 1 the mecanum wheel is depicted from a top view, so the displayed rollers are in such a view. Although the wheel body is shown in that perspective, the actual ground-contact point corresponds to the roller in the bottom view, highlighted in the figure by the dotted orange line. For this type of wheel, the effective traction direction is always aligned with the top roller axis for visualization purposes only.

This unified representation captures both conventional wheels and roller-based mechanisms within a single framework. It provides the foundation for systematically deriving wheel kinematics, non-holonomic constraints, and motion control laws.

3.3. Wheel Position Vector

The position vector of the i-th wheel with respect to the local reference frame is defined as (see Figure 2):

$${\mathit{r}}_i=\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right],$$

(1)

which specifies the wheel’s fixed spatial location within the chassis and is essential for calculating the linear velocities induced by the body’s rotational motion.

3.4. Ground-Contact Velocity of the Wheels

The translational velocity of the mobile center is given by (see Figure 1):

$${\mathit{V}}_t=\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\end{array}\right].$$

(2)

For non-holonomic configurations, such as differential, Ackermann, or tricycle drives, lateral motion is restricted, leading to:

$${\dot{y}}_m=0.$$

(3)

On the other hand, a rotational motion of the chassis induces at the i-th wheel a linear velocity determined by:

$${\mathit{V}}_{r_i}=\mathit{\omega }\times {\mathit{r}}_i$$

where the angular velocity is $\mathit{\omega }={[0,0,\omega ]}^T$ and the wheel position vector expressed in 3D-space is ${\mathit{r}}_i={[x_i,y_i,0]}^T$.

Since the motion occurs in 2D-space, the induced linear velocity of the rotational motion results:

$${\mathit{V}}_{r_i}=\left[\begin{array}{c}-y_i\omega \\ x_i\omega \end{array}\right].$$

Thus, the total ground-contact velocity of the i-th wheel is obtained as the sum of the translational and rotational components, i.e., the rigid-body velocity relation between two points (see Figure 2b):

$${\mathit{V}}_i={\mathit{V}}_t+{\mathit{V}}_{r_i},$$

(4)

resulting in:

$${\mathit{V}}_i=\left[\begin{array}{c}{\dot{x}}_m-y_i\omega \\ {\dot{y}}_m+x_i\omega \end{array}\right].$$

(5)

3.5. Rolling Direction of Each Wheel

The unit vector $u_i$ defining the rolling direction of the i-th wheel with respect to the local reference frame $\{X_m,Y_m\}$ is given by:

$${\mathit{u}}_i={[u_{i,x},u_{i,y}]}^T=R_z\left({\beta }_i\right){\mathit{t}}_i,$$

(6)

where $R_z\left({\beta }_i\right)$ is the 2D rotation matrix about the Z-axis with a wheel orientation of ${\beta }_i$ relative to the $X_m$-axis of the local frame (i.e., the mounting angle), and ${\mathit{t}}_i={[cos{\alpha }_i,sin{\alpha }_i]}^T$ represents the unit direction vector of $v_{c_i}$ whose orientation is defined by the angle ${\alpha }_i$ with respect to the tangential velocity $v_i$, as shown in Figure 2a.

For standard or non-mecanum wheels, ${\alpha }_i=0$, implying that the rolling direction coincides with the tangential axis of the wheel.

3.6. Lateral Constraint on the Wheels

To determine the non-holonomic constraint that prevents lateral slip, a unit vector orthogonal to the rolling direction is introduced relative to the local reference frame as (see Figure 2a):

$${\mathit{u}}_{n_i}=R_z\left(\pi /2\right){\mathit{u}}_i.$$

(7)

where $R_z(\pi /2)$ is the rotation matrix around the Z-axis with a fixed value of $\pi /2$.

The corresponding non-slip constraint condition is then expressed as

$${\mathit{V}}_i·{\mathit{u}}_{n_i}=0,$$

(8)

ensuring that the component of the projection of ${\mathit{V}}_i$ along ${\mathit{u}}_{n_i}$ is not possible, i.e., the standard wheel motion in the lateral direction does not occur. This formulation is particularly useful for incorporating steering variables in architectures such as the Ackermann drive, where the wheels change their orientation while consistently maintaining the lateral no-slip constraint in any steering configuration. For omnidirectional wheels, movement along ${\mathit{u}}_{n_i}$ is admissible, so deriving this vector may not be necessary. When lateral slip is considered in the lateral constraint condition, a lateral slip velocity $v_{n_i}^{\mathrm{slip}}$ is introduced into (8). Consequently, the ideal no-slip constraint is relaxed by replacing the zero lateral velocity assumption with the slip-induced lateral velocity, leading to ${\mathit{V}}_i·{\mathit{u}}_{n_i}=v_{n_i}^{slip}$. Although the slip-induced lateral velocity is explicitly introduced in the formulation, its modeling and formal analysis are beyond the scope of this work.

3.7. Rolling Velocity and Wheel Actuation

The component of the wheel–ground velocity in the rolling direction (component of the projection of ${\mathit{V}}_i$ along ${\mathit{u}}_i$) is the following:

$$v_{c_i}={\mathit{V}}_i·{\mathit{u}}_i.$$

(9)

The mechanical relationship between the velocity (9) and the wheel tangential velocity $v_i$ of the wheel, commonly used as the input in mobile kinematics, is given by (see Figure 2a):

$$v_{c_i}=v_{i}cos{\alpha }_i,$$

(10)

with $v_i=r{\omega }_i$, r as the radius of the wheel and ${\omega }_i$ as the wheel angular speed.

Then, the tangential velocity of each wheel can be obtained in terms of its mounting configuration, the chassis velocities, and its rolling direction as follows:

$$v_i={\displaystyle \frac{1}{cos{\alpha }_i}}({\mathit{V}}_i·{\mathit{u}}_i).$$

(11)

As noted in Figure 2, for standard or non-mecanum wheels ${\alpha }_i=0$, implying that the rolling direction coincides with the tangential axis of the wheel, the quantities coincide $v_{c_i}=v_i$.

Similarly, if slip in the tangential velocity is considered, the corresponding slip-induced velocity $v_{t_i}^{\mathrm{slip}}$ can be introduced along the rolling direction ${\mathit{u}}_i$, which accounts for the loss of effective rolling velocity [23]. In this case, the ideal rolling condition (9) is relaxed, and the effective contact velocity is expressed as ${\mathit{V}}_i·{\mathit{u}}_i=v_{i}cos{\alpha }_i+v_{t_i}^{\mathrm{slip}}$. However, its modeling and formal analysis are outside the scope of this work.

3.8. Wheel-Level Kinematic Equation

Substituting (5) into (11) yields the general wheel kinematic equation:

$$v_i={\displaystyle \frac{1}{cos{\alpha }_i}}\left(u_{i,x}{\dot{x}}_m+u_{i,y}{\dot{y}}_m+c_i\omega \right),$$

(12)

where:

$$c_i=-y_i{u}_{i,x}+x_i{u}_{i,y}.$$

(13)

Equation (12) is valid for all types of actuated wheels, including mecanum wheels, making it the core building block of the unified model.

3.9. Inverse Kinematics in the Mobile Frame

Stacking the kinematic Equation (12) of the n actuated wheels yields the inverse kinematics of the mobile:

$$\mathit{v}=J{\mathit{\xi }}_m,$$

(14)

with:

$$\mathit{v}={[v_1,\dots ,v_n]}^{\top },{\mathit{\xi }}_m={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^{\top },$$

(15)

and the Jacobian $J\in {\mathbb{R}}^{n\times 3}$:

$$J=BA,$$

(16)

with:

$$B=diag\left({\displaystyle \frac{1}{cos{\alpha }_1}},\dots ,{\displaystyle \frac{1}{cos{\alpha }_n}}\right),A=\left[\begin{array}{ccc}{u}_{1,x}& u_{1,y}& c_1\\ ⋮& ⋮& ⋮\\ u_{n,x}& u_{n,y}& c_n\end{array}\right]$$

(17)

that maps the chassis motion to the individual wheel tangential velocities.

3.10. Forward Kinematics in the Mobile Frame

Based on the inverse kinematics in (14), the chassis velocities ${\mathit{\xi }}_m$ can be obtained using the wheel velocities $\mathit{v}$ through the inverse or Moore-Penrose pseudo-inverse of the Jacobian J referred to as $J^{+}$:

$${\mathit{\xi }}_m=J^{+}\mathit{v}.$$

(18)

In case of fully-actuated mobiles ($\mathit{v}$ has the same dimension as ${\mathit{\xi }}_m$, e.g., the omnidirectional mobile robot with three wheels), the inverse Jacobian can be calculated directly, so:

$$J^{+}=J^{-1}.$$

(19)

For non-holonomic underactuated robots (the dimension of $\mathit{v}$ is larger than that of ${\mathit{\xi }}_m$, e.g., the standard differential and Ackermann platforms), the pseudo-inverse is obtained as:

$$J^{+}=J^T{\left(J{J}^T\right)}^{-1}.$$

(20)

While for holonomic redundant robots (the dimension of $\mathit{v}$ is smaller than that of ${\mathit{\xi }}_m$, e.g., the mecanum 4WD robot), it can be calculated with:

$$J^{+}={\left(J^{T}J\right)}^{-1}{J}^T.$$

(21)

These operations yield the forward kinematics for the robot, valid for both holonomic and non-holonomic systems, depending on the structure of J.

• Structural Properties of the Jacobian Matrix

Proposition 1

(Structural Invariance Under Wheel Reconfiguration). Let a wheeled mobile robot be modeled using the proposed generalized kinematic framework, with wheel positions ${\mathit{r}}_i\in {\mathbb{R}}^2$ and rolling direction unit vectors ${\mathit{u}}_i\in {\mathbb{R}}^2$. Any reconfiguration of the robot that modifies the wheel placement or orientation (i.e., changes in $r_i$ and/or $u_i$) preserves the functional structure of the kinematic mapping

$$\mathit{v}=J\left(\mathcal{P}\right)\mathit{\xi },$$

where $\mathcal{P}={\{r_i,u_i\}}_{i=1}^n$ is the set of geometric parameters.

Proof. 

Within the proposed framework, the velocity of the contact point of wheel i is obtained as

$${\mathit{V}}_i={\mathit{V}}_t+{\mathit{V}}_{r_i},$$

where ${\mathit{V}}_t$ is the translational velocity of the chassis center and ${\mathit{V}}_{r_i}$ is the rotational contribution induced by the angular velocity $\omega$.

The effective rolling velocity is defined through the geometric projection

$$v_{c_i}={\mathit{V}}_i·{\mathit{u}}_i,$$

which can be expanded as

$$v_{c_i}=u_{i,x}{\dot{x}}_m+u_{i,y}{\dot{y}}_m+\left(-y_i{u}_{i,x}+x_i{u}_{i,y}\right)\omega .$$

This expression depends exclusively on the geometric quantities $({\mathit{r}}_i,{\mathit{u}}_i)$ and the chassis twist $\mathit{\xi }={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^{\mathsf{T}}$. Collecting these relations for all wheels yields a Jacobian matrix whose i-th row is

$$J_i=\left[\begin{array}{ccc}{u}_{i,x}& u_{i,y}& -y_i{u}_{i,x}+x_i{u}_{i,y}\end{array}\right].$$

Therefore, any modification in wheel placement or orientation alters only the numerical values of ${\mathit{r}}_i$ and ${\mathit{u}}_i$, while the algebraic structure of each row of the Jacobian remains unchanged. Hence, the functional form of the kinematic mapping is invariant under wheel reconfiguration. □

• Discussion

Proposition 1 demonstrates that the proposed framework is not architecture-specific, but structurally invariant under wheel reconfiguration. Holonomic and nonholonomic behaviors naturally arise from the rank properties of J, rather than being imposed a priori. Importantly, this invariance is not a consequence of a fixed template Jacobian, but follows from the explicit construction of each row from geometric primitives, which distinguishes the proposed framework from prior unified formulations. Importantly, this invariance does not rely on a fixed Jacobian template, but results directly from constructing each row from the underlying wheel geometry.

This establishes a unified and general kinematic foundation applicable to heterogeneous wheeled mobile robots.

Proposition 2

(Holonomy as a Rank Property of the Generalized Jacobian). Let $J\in {\mathbb{R}}^{n\times 3}$ be the kinematic Jacobian matrix constructed according to the proposed framework. The wheeled mobile robot is holonomic if and only if

$$\mathrm{rank}\left(J\right)=3.$$

If $\mathrm{rank}\left(J\right)<3$, the system is nonholonomic, and the missing degrees of freedom cannot be generated by any combination of wheel velocities.

Proof. 

By construction, the generalized kinematic relation is

$$\mathit{v}=J\mathit{\xi },$$

where $\mathit{\xi }={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^{\mathsf{T}}$. If $\mathrm{rank}\left(J\right)=3$, the mapping is surjective, and any planar twist $\mathit{\xi }$ admits at least one solution in wheel velocities, implying holonomy. Conversely, if $\mathrm{rank}\left(J\right)<3$, the mapping is rank-deficient and at least one component of $\mathit{\xi }$ cannot be generated, resulting in nonholonomic behavior. □

• Discussion

Proposition 2 shows that holonomy and nonholonomy emerge naturally from the algebraic structure of the generalized Jacobian, rather than being imposed by architectural classification. This provides a unified criterion applicable to heterogeneous wheel configurations.

3.11. Transformation to the Global Frame

The forward and inverse kinematics of the mobile robot can be expressed in the global coordinate frame $\{X,Y\}$ (see Figure 3) by incorporating the rotation matrix $R_z\left(\theta \right)$ about the Z-axis in:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right]=R_z\left(\theta \right)\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\end{array}\right],\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\end{array}\right]=R_z^T\left(\theta \right)\left[\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right].$$

(22)

where $\theta$ denotes the robot’s orientation measured from the global X-axis and its time derivative satisfies $\omega =\dot{\theta }$.

This leads to:

$$\mathit{\xi }=H\left(\theta \right){\mathit{\xi }}_m,{\mathit{\xi }}_m=H^T\left(\theta \right)\mathit{\xi }$$

(23)

with the homogeneous transformation matrix:

$$H\left(\theta \right)=\left[\begin{array}{cc}{R}_z\left(\theta \right)& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{ccc}cos\theta & -sin\theta & 0\\ sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right],$$

(24)

its transpose:

$$H^T\left(\theta \right)=\left[\begin{array}{cc}{R}_z^T\left(\theta \right)& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{ccc}cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right],$$

(25)

and the vector of global mobile velocities:

$$\mathit{\xi }={[\dot{x},\dot{y},\omega ]}^{\top }.$$

(26)

Therefore, the inverse kinematics of the robot in the global frame is obtained from (14) as:

$$\begin{array}{|c|}\hline \mathit{v}=J{H}^T\left(\theta \right)\mathit{\xi },\\ \hline\end{array}$$

(27)

and the forward kinematics from (18) is:

$$H^T\left(\theta \right)\mathit{\xi }=J^{+}\mathit{v},$$

(28)

or:

$$\begin{array}{|c|}\hline \mathit{\xi }=H\left(\theta \right)J^{+}\mathit{v}.\\ \hline\end{array}$$

(29)

The above mapping is required for mobile simulation and control, trajectory planning, odometry, and navigation.

4. Modeling Representative Wheeled Robots

In this section, the proposed unified framework is utilized to systematically derive the kinematic models of four common mobile robot architectures, both holonomic and non-holonomic: differential drive, Ackermann steering, three-wheeled omnidirectional, and four-wheel-drive mecanum robots.

4.1. Differential Drive with Two Standard Wheels

4.1.1. Local Reference Frame Definition

The differential-drive mobile robot represents the canonical example of a non-holonomic wheeled platform. Its geometry consists of two independently actuated wheels mounted symmetrically along the chassis $X_m$-axis, with the body-frame origin located at the midpoint between them (Figure 4).

4.1.2. Kinematic Quantities and Representation

The robot layout depicted in Figure 4 satisfies all assumptions of the unified modeling framework, making it an ideal first application.

4.1.3. Wheel Position Vector

Let the left ($i=1$) and right ($i=2$) wheels be positioned at:

$${\mathit{r}}_1=\left[\begin{array}{c}0\\ \frac{L}{2}\end{array}\right],{\mathit{r}}_2=\left[\begin{array}{c}0\\ -\frac{L}{2}\end{array}\right],$$

(30)

where L is the lateral wheel separation (track width).

4.1.4. Ground-Contact Velocity of the Wheels

The chassis twist in the local coordinates is:

$${\mathit{\xi }}_m={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^T,$$

(31)

with ${\dot{y}}_m=0$.

Substituting ${\mathit{r}}_1$, ${\mathit{r}}_2$, and ${\dot{y}}_m$ in (5), the ground-contact velocities of the wheels are:

$${\mathit{V}}_1=\left[\begin{array}{c}{\dot{x}}_m-\frac{L}{2}\omega \\ 0\end{array}\right],{\mathit{V}}_2=\left[\begin{array}{c}{\dot{x}}_m+\frac{L}{2}\omega \\ 0\end{array}\right].$$

(32)

4.1.5. Rolling Direction of Each Wheel

Both wheels roll purely along the $X_m$ direction; thus, using the unified notation ${\mathit{u}}_i=R_z\left({\beta }_i\right){\mathit{t}}_i$ with ${\beta }_i=0$ and ${\alpha }_i=0$, the unit of roll vectors are reduced to:

$${\mathit{u}}_1={\mathit{u}}_2={[1,0]}^T$$

(33)

4.1.6. Lateral Non-Slip Constraint on the Wheels

Since there are no steering wheels in this mobile architecture, the calculation of vectors orthogonal to the rolling ones is omitted.

4.1.7. Rolling Velocity and Wheel Actuation

Using the unified wheel equation in (11) and considered that ${\alpha }_i=0$ for both wheels, we obtain the actuator-consistent expressions:

$$v_1={\dot{x}}_m-{\displaystyle \frac{L}{2}}\omega ,v_2={\dot{x}}_m+{\displaystyle \frac{L}{2}}\omega .$$

(34)

4.1.8. Wheel-Level Kinematic Equation

The resulting coefficients $c_i$ from wheel-level kinematic Equation (12) for the two actuated wheels are:

$$c_1=-{\displaystyle \frac{L}{2}},c_2={\displaystyle \frac{L}{2}}.$$

(35)

4.1.9. Inverse Kinematics in the Mobile Frame

Stacking the values from above actuated wheel equations, as in (16), leads to the following Jacobian:

$$J=\left[\begin{array}{ccc}1& 0& -\frac{L}{2}\\ 1& 0& \frac{L}{2}\end{array}\right],$$

(36)

which maps the chassis twist ${\mathit{\xi }}_m$ to the tangential velocities of the actuated wheels:

$$\mathit{v}={[v_1,v_2]}^T$$

(37)

Then, from (14), the inverse kinematics of the differential robot in the local frame is:

$$\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{ccc}1& 0& -\frac{L}{2}\\ 1& 0& \frac{L}{2}\end{array}\right]\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right].$$

(38)

4.1.10. Forward Kinematics in the Mobile Frame

Since J has full row rank and the platform is underactuated, its Moore-Penrose pseudo-inverse is:

$$J^{+}=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ 0& 0\\ -{\displaystyle \frac{1}{L}}& {\displaystyle \frac{1}{L}}\end{array}\right].$$

(39)

Therefore, from (18), the forward kinematics of the mobile is:

$$\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ 0& 0\\ -{\displaystyle \frac{1}{L}}& {\displaystyle \frac{1}{L}}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right].$$

(40)

4.1.11. Transformation to the Global Frame

As the final output of the framework, we obtain the inverse kinematics of the robot in the global frame according to (27):

$$\begin{array}{|c|}\hline \left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{ccc}cos\theta & sin\theta & -\frac{L}{2}\\ cos\theta & sin\theta & \frac{L}{2}\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right],\\ \hline\end{array}$$

(41)

and the corresponding forward kinematics from (29):

$$\begin{array}{|c|}\hline \left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}cos\theta & {\displaystyle \frac{1}{2}}cos\theta \\ {\displaystyle \frac{1}{2}}sin\theta & {\displaystyle \frac{1}{2}}sin\theta \\ -{\displaystyle \frac{1}{L}}& {\displaystyle \frac{1}{L}}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right].\\ \hline\end{array}$$

(42)

Summary and Remarks

The differential-drive kinematics follow directly from the proposed unified framework. These kinematic relations follow the classical non-holonomic unicycle model widely adopted in robotics [4,8,24]. The above ensures that the obtained model is fully consistent with the analytical models from the specialized literature. The geometry of the wheel encoded in ${\mathit{r}}_i$ and ${\mathit{u}}_i$ generates both the non-holonomic constraint and the velocity mapping without requiring case-specific derivations. The resulting model is compact, actuator-centric (expressed solely in terms of $v_1$ and $v_2$), and readily integrable into control, simulation, and trajectory-planning pipelines.

4.2. Ackermann Steering or Car-like Vehicle with Four Standard Wheels

4.2.1. Local Reference Frame Definition

The Ackermann, or car-like, steering configuration is also a canonical example of a non-holonomic wheeled platform. It consists of two rear driving wheels and a pair of front wheels that share a common steering axle. The center is located at the midpoint of the rear axle, and the front of this mobile is pointing to the $X_m$-axis (Figure 5).

4.2.2. Kinematic Quantities and Representation

For kinematic analysis, the two front wheels are replaced by a single equivalent steering wheel located at the midpoint of the front axle and oriented by the steering angle $\delta$. This abstraction preserves the essential geometric constraints and the vehicle’s non-holonomic structure [4,8,25,26]. Figure 5 shows the reduced model that satisfies the assumptions of the framework.

4.2.3. Wheel Position Vector

Let W denote the wheelbase (rear-front axle distance) and L the width of the track. Like the differential robot, the Ackermann mobile has a left ($i=1$) and right ($i=2$) actuated wheel in the rear axle. In addition, it includes the non-actuated equivalent single front wheel ($i=3$) at the midpoint of the steering axle. These three wheels are located at:

$${\mathit{r}}_1=\left[\begin{array}{c}0\\ {\displaystyle \frac{L}{2}}\end{array}\right],{\mathit{r}}_2=\left[\begin{array}{c}0\\ -{\displaystyle \frac{L}{2}}\end{array}\right],{\mathit{r}}_3=\left[\begin{array}{c}W\\ 0\end{array}\right].$$

(43)

4.2.4. Ground-Contact Velocity of the Wheels

As in the differential robot, the chassis twist in the local coordinates is:

$${\mathit{\xi }}_m={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^T,$$

(44)

and ${\dot{y}}_m=0$.

Using ${\mathit{r}}_1$, ${\mathit{r}}_2$, ${\mathit{r}}_3$, and ${\dot{y}}_m$ in (5), the following ground-contact velocities can be obtained:

$${\mathit{V}}_1=\left[\begin{array}{c}{\dot{x}}_m\\ \frac{L}{2}\omega \end{array}\right],{\mathit{V}}_2=\left[\begin{array}{c}{\dot{x}}_m\\ -\frac{L}{2}\omega \end{array}\right],{\mathit{V}}_3=\left[\begin{array}{c}{\dot{x}}_m\\ W\omega \end{array}\right].$$

(45)

4.2.5. Rolling Direction of Each Wheel

The rear wheels roll along the $X_m$ of the vehicle, hence ${\alpha }_1={\alpha }_2=0$, and:

$${\mathit{u}}_1={\mathit{u}}_2={[1,0]}^T$$

(46)

The equivalent front wheel is steered by ${\beta }_3=\delta$. So, its rolling direction is:

$${\mathit{u}}_3=R_z\left(\delta \right)\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}cos\delta \\ sin\delta \end{array}\right].$$

(47)

4.2.6. Lateral Non-Slip Constraint on the Wheels

The equivalent front wheel is used to change the vehicle’s direction. Therefore, the constraint of lateral movement of this wheel can be used to involve the steering angle $\delta$ in the kinematics.

In this sense, the unit vector orthogonal to the rolling direction ${\mathit{u}}_3$ is calculated from (7) as:

$${\mathit{u}}_{n_3}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right] {\mathit{u}}_3=\left[\begin{array}{c}-sin\delta \\ cos\delta \end{array}\right].$$

(48)

Based on the above vector, the no-slip condition requires that:

$${\mathit{u}}_{n_3}^{\mathsf{T}}{\mathit{V}}_3=0,$$

(49)

which yields:

$$-sin\delta {\dot{x}}_m+cos\delta \left(W\omega \right)=0.$$

(50)

Then, the angular velocity of the chassis $\omega$ can be written (if required by the model or application) in terms of $\delta$ as:

$$\omega ={\displaystyle \frac{{\dot{x}}_{m}tan\delta }{W}}.$$

(51)

4.2.7. Rolling Velocity and Wheel Actuation

From the unified wheel equation in (11) with ${\alpha }_1={\alpha }_2=0$ for the two actuated wheels, the tangential velocities are obtained:

$$v_1={\dot{x}}_m-{\displaystyle \frac{L}{2}}\omega ,v_2={\dot{x}}_m+{\displaystyle \frac{L}{2}}\omega .$$

(52)

4.2.8. Wheel-Level Kinematic Equation

The coefficients $c_i$ from wheel-level kinematic Equation (12) for the two actuated wheels are:

$$c_1=-{\displaystyle \frac{L}{2}},c_2={\displaystyle \frac{L}{2}}.$$

(53)

4.2.9. Inverse Kinematics in the Mobile Frame

Using the values of the wheel-level kinematic equations, the Jacobian of the inverse kinematics (16) is:

$$J=\left[\begin{array}{ccc}1& 0& -\frac{L}{2}\\ 1& 0& \frac{L}{2}\end{array}\right].$$

(54)

This Jacobian transforms the chassis twist ${\mathit{\xi }}_m$ to the tangential velocities of the actuated wheels:

$$\mathit{v}={[v_1,v_2]}^T$$

(55)

Then, the inverse kinematics for this robot in local coordinates is (14):

$$\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{ccc}1& 0& -\frac{L}{2}\\ 1& 0& \frac{L}{2}\end{array}\right]\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right].$$

(56)

4.2.10. Forward Kinematics in the Mobile Frame

Despite the inclusion of the steering angle $\delta$, this robot is also underactuated and J has full row rank. To compute forward kinematics, the Moore-Penrose pseudo-inverse of J is calculated:

$$J^{+}=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ 0& 0\\ -{\displaystyle \frac{1}{L}}& {\displaystyle \frac{1}{L}}\end{array}\right].$$

(57)

Using $J^{+}$, one can obtain the forward kinematics of the Ackermann mobile with (18):

$$\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ 0& 0\\ -{\displaystyle \frac{1}{L}}& {\displaystyle \frac{1}{L}}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right].$$

(58)

4.2.11. Transformation to the Global Frame

The final inverse kinematics of the Ackermann robot in the global frame is obtained with (27):

$$\begin{array}{|c|}\hline \left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{ccc}cos\theta & sin\theta & -\frac{L}{2}\\ cos\theta & sin\theta & \frac{L}{2}\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right],\\ \hline\end{array}$$

(59)

On the other hand, the global forward kinematics from (29) is:

$$\begin{array}{|c|}\hline \left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}cos\theta & {\displaystyle \frac{1}{2}}cos\theta \\ {\displaystyle \frac{1}{2}}sin\theta & {\displaystyle \frac{1}{2}}sin\theta \\ -{\displaystyle \frac{1}{L}}& {\displaystyle \frac{1}{L}}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right].\\ \hline\end{array}$$

(60)

Summary and Remarks

At this point, it is important to note that the kinematics of the Ackermann mobile are equivalent to those of a differential robot at the longitudinal motion level. The bicycle model represents the classical approximation for Ackermann steering and is fully consistent with the formulations commonly used in works on vehicle dynamics, autonomous driving, and mobile robotics [4,8,27,28,29]. It is also important to mention that if the $\delta$ steering angle needs to be incorporated into the model, it can be done in the global forward kinematics (and only in this mapping) by using $\omega$ from (51) and considering that ${\dot{x}}_m={\displaystyle \frac{1}{2}}(v_1+v_2)$ from (58), as observed next:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]=\left[\begin{array}{c}cos\theta \\ sin\theta \\ {\displaystyle \frac{1}{W}}tan\delta \end{array}\right]\left({\displaystyle \frac{v_1+v_2}{2}}\right).$$

(61)

This expression is intentionally written in a factorized form to emphasize that Ackermann steering is governed by an equivalent forward velocity, rather than by independently actuated wheel velocities, as in the differential-drive case.

4.3. Three-Wheel Omnidirectional Platform (3, 0)

4.3.1. Local Reference Frame Definition

The $(3, 0)$ platform achieves full holonomic mobility via three equally spaced omni wheels. Each wheel provides a single rolling direction, while the orthogonal component is absorbed by passive rollers. The symmetric geometry produces a constant, invertible Jacobian, enabling direct mapping between wheel speeds and the chassis twist.

The wheels of this robot are positioned around a circle centered on the origin of the mobile reference system. Although not mandatory, it is suggested that the first wheel be located on the $X_m$-axis (Figure 6).

4.3.2. Kinematic Quantities and Representation

As observed in Figure 6, the three wheels are mounted at the radius L from the center of the chassis and separated by $2\pi /3$. The quantities shown in this figure can be used in the proposed framework.

4.3.3. Wheel Position Vector

The angular location of the i-th wheel relative to the body X-axis is:

$${\theta }_i={\displaystyle \frac{2\pi (i-1)}{3}},i=1,2,3,$$

(62)

so that the local coordinates are:

$${\mathit{r}}_i=\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right]=L\left[\begin{array}{c}cos{\theta }_i\\ sin{\theta }_i\end{array}\right].$$

(63)

Explicitly, the position vectors of all wheels of this robot are:

$${\mathit{r}}_1=L\left[\begin{array}{c}1\\ 0\end{array}\right],{\mathit{r}}_2=L\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right],{\mathit{r}}_3=L\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}\\ -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right].$$

(64)

Note: For any symmetric configuration with n omnidirectional wheels, the angular locations can be generalized as:

$${\theta }_i={\displaystyle \frac{2\pi (i-1)}{n}}.$$

(65)

4.3.4. Ground-Contact Velocity of the Wheels

The chassis twist in the local coordinates of this omnidirectional robot is:

$${\mathit{\xi }}_m={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^T,$$

(66)

Since the mobile is holonomic, the movement can be performed in any direction without restriction, and there are no further simplifications for this vector.

Substituting ${\mathit{r}}_1$, ${\mathit{r}}_2$, and ${\mathit{r}}_3$ in (5), the ground-contact velocities for the wheels are:

$${\mathit{V}}_1=\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m+L\omega \end{array}\right],{\mathit{V}}_2=\left[\begin{array}{c}{\dot{x}}_m-{\displaystyle \frac{L\sqrt{3}}{2}}\omega \\ {\dot{y}}_m-\frac{L}{2}\omega \end{array}\right],{\mathit{V}}_3=\left[\begin{array}{c}{\dot{x}}_m+{\displaystyle \frac{L\sqrt{3}}{2}}\omega \\ {\dot{y}}_m-\frac{L}{2}\omega \end{array}\right].$$

(67)

4.3.5. Rolling Direction of Each Wheel

Since omnidirectional wheels have ${\alpha }_i=0$, their rolling direction is tangent to the circle of radius L:

$${\beta }_i={\displaystyle \frac{\pi }{2}}+{\theta }_i,{\mathit{u}}_i=\left[\begin{array}{c}-sin{\theta }_i\\ cos{\theta }_i\end{array}\right].$$

(68)

Thus, ${\mathit{u}}_i$ is orthogonal to the radius vector ${\mathit{r}}_i$ and points in the direction of allowed motion. The obtained rolling directions of all the wheels are:

$${\mathit{u}}_1=\left[\begin{array}{c}0\\ 1\end{array}\right],{\mathit{u}}_2=\left[\begin{array}{c}-{\displaystyle \frac{\sqrt{3}}{2}}\\ -\frac{1}{2}\end{array}\right],{\mathit{u}}_3=\left[\begin{array}{c}{\displaystyle \frac{\sqrt{3}}{2}}\\ -\frac{1}{2}\end{array}\right].$$

(69)

4.3.6. Lateral Non-Slip Constraint on the Wheels

The omnidirectional robot does not have a lateral movement constraint. Then, the orthogonal vectors are not calculated.

4.3.7. Rolling Velocity and Wheel Actuation

From (11) with ${\alpha }_i=0$, the tangential velocities of the three wheels are:

$$v_1={\dot{y}}_m+L\omega ,v_2=-{\displaystyle \frac{\sqrt{3}}{2}}{\dot{x}}_m-\frac{1}{2}{\dot{y}}_m+L\omega ,v_3={\displaystyle \frac{\sqrt{3}}{2}}{\dot{x}}_m-\frac{1}{2}{\dot{y}}_m+L\omega .$$

(70)

4.3.8. Wheel-Level Kinematic Equation

From (12), the coefficients $c_i$ for this robot are:

$$c_1=c_2=c_3=L.$$

(71)

4.3.9. Inverse Kinematics in the Mobile Frame

Based on the above coefficients, the Jacobian is calculated with (16):

$$J=\left[\begin{array}{ccc}0& 1& L\\ -\frac{\sqrt{3}}{2}& -\frac{1}{2}& L\\ \phantom{-}\frac{\sqrt{3}}{2}& -\frac{1}{2}& L\end{array}\right].$$

(72)

The obtained Jacobian maps the chassis twist ${\mathit{\xi }}_m$ to the tangential velocities of the three actuated wheels:

$$\mathit{v}={[v_1,v_2,v_3]}^T$$

(73)

So, the inverse kinematics for the omnidirectional robot $(3, 0)$ in the local frame is achieved with (14):

$$\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]=\left[\begin{array}{ccc}0& 1& L\\ -\frac{\sqrt{3}}{2}& -\frac{1}{2}& L\\ \phantom{-}\frac{\sqrt{3}}{2}& -\frac{1}{2}& L\end{array}\right]\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right].$$

(74)

4.3.10. Forward Kinematics in the Mobile Frame

The Jacobian J from the local inverse kinematics is a square and non-singular matrix, so its inverse is calculated directly:

$$J^{+}={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}0& -\sqrt{3}& \sqrt{3}\\ 2& -1& -1\\ \frac{1}{L}& \frac{1}{L}& \frac{1}{L}\end{array}\right].$$

(75)

With the inverted Jacobian $J^{+}$, the local forward kinematics are derived based on (18):

$$\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right]=\left[\begin{array}{ccc}0& -\sqrt{3}& \sqrt{3}\\ 2& -1& -1\\ \frac{1}{L}& \frac{1}{L}& \frac{1}{L}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right].$$

(76)

4.3.11. Transformation to the Global Frame

As the ultimate result of the framework, we obtain the inverse kinematics of the robot in the global frame using (27):

$$\begin{array}{|c|}\hline \left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]={\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}-2sin\theta & 2cos\theta & 2L+2\\ sin\theta -\sqrt{3}cos\theta & -cos\theta -\sqrt{3}sin\theta & 2L-\sqrt{3}-1\\ sin\theta +\sqrt{3}cos\theta & \sqrt{3}sin\theta -cos\theta & 2L+\sqrt{3}-1\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]\\ \hline\end{array}.$$

(77)

By its part, the global forward kinematics obtained from (29) is:

$$\begin{array}{|c|}\hline \left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}-2sin\theta & sin\theta -\sqrt{3}cos\theta & sin\theta +\sqrt{3}cos\theta \\ 2cos\theta & -cos\theta -\sqrt{3}sin\theta & \sqrt{3}sin\theta -cos\theta \\ {\displaystyle \frac{1}{L}}+2& {\displaystyle \frac{1}{L}}-\sqrt{3}-1& {\displaystyle \frac{1}{L}}+\sqrt{3}-1\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]\\ \hline\end{array}$$

(78)

Summary and Remarks

The symmetric $(3, 0)$ omni platform yields a constant Jacobian and fully decoupled inverse kinematics. Its holonomy arises directly from the three non-collinear tangential rolling directions. The adopted model reproduces the characteristic behavior of Kiwi-drive platforms: when all wheel velocities are equal ($v_1=v_2=v_3$), the translational components cancel out, and the robot performs a pure rotation around its center, consistent with prior analyses of tricycle and omnidirectional-wheel platforms made in [4,8,30,31]. The formulation is fully compatible with the unified wheel model used throughout this paper.

4.4. Four-Wheel-Drive (4WD) Mecanum Platform

4.4.1. Local Reference Frame Definition

The mecanum 4WD configuration provides full holonomic mobility using four wheels equipped with passive rollers at $\pi /4$. Each wheel generates motion only along its roller-aligned rolling direction, while the perpendicular component is unconstrained. This mechanism enables independent control of ${\dot{x}}_m$, ${\dot{y}}_m$, and $\omega$, making it the canonical holonomic ground platform. The center of the robot is defined as the geometric centroid of the four contact points of the wheel, and its front is pointing to the $X_m$-axis (Figure 7).

4.4.2. Kinematic Quantities and Representation

Figure 7 illustrates the local geometry of this mobile robot. All required kinematic quantities are included for the framework’s use.

4.4.3. Wheel Position Vector

According to Figure 7, the wheel positions in the local frame are:

$${\mathit{r}}_1=\left[\begin{array}{c}W\\ L\end{array}\right],{\mathit{r}}_2=\left[\begin{array}{c}W\\ -L\end{array}\right],{\mathit{r}}_3=\left[\begin{array}{c}-W\\ L\end{array}\right],{\mathit{r}}_4=\left[\begin{array}{c}-W\\ -L\end{array}\right].$$

(79)

4.4.4. Ground-Contact Velocity of the Wheels

For the mecanum robot, the chassis twist in the local coordinates is:

$${\mathit{\xi }}_m={[{\dot{x}}_m,{\dot{y}}_m,\omega ]}^T.$$

(80)

The above vector cannot be simplified, since the holonomic robot’s motion is allowed in all directions.

With ${\mathit{r}}_1$, ${\mathit{r}}_2$, ${\mathit{r}}_3$, and ${\mathit{r}}_4$, the ground-contact velocities for the mecanum wheels can be obtained using (5):

$${\mathit{V}}_1=\left[\begin{array}{c}{\dot{x}}_m-L\omega \\ {\dot{y}}_m+W\omega \end{array}\right],{\mathit{V}}_2=\left[\begin{array}{c}{\dot{x}}_m+L\omega \\ {\dot{y}}_m+W\omega \end{array}\right],{\mathit{V}}_3=\left[\begin{array}{c}{\dot{x}}_m-L\omega \\ {\dot{y}}_m-W\omega \end{array}\right],{\mathit{V}}_4=\left[\begin{array}{c}{\dot{x}}_m+L\omega \\ {\dot{y}}_m-W\omega \end{array}\right].$$

(81)

4.4.5. Rolling Direction of Each Wheel

The mecanum wheels are mounted with angles ${\beta }_i$ and equipped with passive rollers whose orientations are defined as

$${\alpha }_1={\alpha }_3=-\frac{\pi }{4},{\alpha }_2={\alpha }_4=\frac{\pi }{4}.$$

(82)

Substituting these roller orientations into the generic expression (6) yields the corresponding unit rolling-direction vectors for each mecanum wheel as follows:

$${\mathit{u}}_1=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\end{array}\right],{\mathit{u}}_2=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right],{\mathit{u}}_3=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right],{\mathit{u}}_4=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\end{array}\right].$$

(83)

4.4.6. Lateral Non-Slip Constraint on the Wheels

The orthogonal vectors are not obtained, since this holonomic mobile robot does not have movement constraints.

4.4.7. Rolling Velocity and Wheel Actuation

Using the values of ${\alpha }_i$ in (82) in (11), the tangential velocities of the four mecanum wheels are:

$$\begin{array}{c}\hfill v_1={\dot{x}}_m-{\dot{y}}_m-(L+W)\omega ,v_2={\dot{x}}_m+{\dot{y}}_m+(L+W)\omega ,\\ \hfill v_3={\dot{x}}_m+{\dot{y}}_m-(L+W)\omega ,v_4={\dot{x}}_m-{\dot{y}}_m+(L+W)\omega .\end{array}$$

(84)

4.4.8. Wheel-Level Kinematic Equation

Using (12), the following coefficients $c_i$ are calculated:

$$c_1=c_3=-\frac{1}{\sqrt{2}}(L+W),c_2=c_4=\frac{1}{\sqrt{2}}(L+W).$$

(85)

4.4.9. Inverse Kinematics in the Mobile Frame

With the above coefficients, the Jacobian for the inverse kinematics is (16):

$$J=\sqrt{2}\left[\begin{array}{ccc}{\displaystyle \frac{1}{\sqrt{2}}}& -{\displaystyle \frac{1}{\sqrt{2}}}& -{\displaystyle \frac{1}{\sqrt{2}}}(L+W)\\ {\displaystyle \frac{1}{\sqrt{2}}}& {\displaystyle \frac{1}{\sqrt{2}}}& {\displaystyle \frac{1}{\sqrt{2}}}(L+W)\\ {\displaystyle \frac{1}{\sqrt{2}}}& {\displaystyle \frac{1}{\sqrt{2}}}& -{\displaystyle \frac{1}{\sqrt{2}}}(L+W)\\ {\displaystyle \frac{1}{\sqrt{2}}}& -{\displaystyle \frac{1}{\sqrt{2}}}& {\displaystyle \frac{1}{\sqrt{2}}}(L+W)\end{array}\right],$$

(86)

or else:

$$J=\left[\begin{array}{ccc}1& -1& -(L+W)\\ 1& 1& (L+W)\\ 1& 1& -(L+W)\\ 1& -1& (L+W)\end{array}\right].$$

(87)

This Jacobian provides the mapping required to compute the tangential velocities of the four actuated wheels when the chassis twist ${\mathit{\xi }}_m$ is prescribed:

$$\mathit{v}={[v_1,v_2,v_3,v_4]}^T$$

(88)

Then, the inverse kinematics for the 4WD mecanum robot in the local coordinates is (14):

$$\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right]=\left[\begin{array}{ccc}1& -1& -(L+W)\\ 1& 1& (L+W)\\ 1& 1& -(L+W)\\ 1& -1& (L+W)\end{array}\right]\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right].$$

(89)

4.4.10. Forward Kinematics in the Mobile Frame

The obtained Jacobian J has full row rank, so its Moore-Penrose pseudo-inverse is obtained to derive the forward kinematics:

$$J^{+}=\left[\begin{array}{cccc}1& 1& 1& 1\\ -1& 1& 1& -1\\ -{\displaystyle \frac{1}{L+W}}& {\displaystyle \frac{1}{L+W}}& -{\displaystyle \frac{1}{L+W}}& {\displaystyle \frac{1}{L+W}}\end{array}\right].$$

(90)

With $J^{+}$ in (18), the forward kinematics of the mecanum mobile is presented next:

$$\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right]=\left[\begin{array}{cccc}1& 1& 1& 1\\ -1& 1& 1& -1\\ -{\displaystyle \frac{1}{L+W}}& {\displaystyle \frac{1}{L+W}}& -{\displaystyle \frac{1}{L+W}}& {\displaystyle \frac{1}{L+W}}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right].$$

(91)

4.4.11. Transformation to the Global Frame

The final result of the proposed framework is the kinematics of the 4WD mecanum robot in global coordinates. Then, the global inverse kinematics is (27):

$$\begin{array}{|c|}\hline \left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right]=\left[\begin{array}{ccc}cos\theta +sin\theta & sin\theta -cos\theta & -(L+W)\\ cos\theta -sin\theta & cos\theta +sin\theta & 2+L+W\\ cos\theta -sin\theta & cos\theta +sin\theta & 2-(L+W)\\ cos\theta +sin\theta & sin\theta -cos\theta & L+W\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]\\ \hline\end{array}.$$

(92)

Finally, the global forward kinematics is (29):

$$\begin{array}{|c|}\hline {\displaystyle \left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \omega \end{array}\right]=\left[\begin{array}{cccc}cos\theta +sin\theta & cos\theta -sin\theta & cos\theta -sin\theta & cos\theta +sin\theta \\ sin\theta -cos\theta & cos\theta +sin\theta & cos\theta +sin\theta & sin\theta -cos\theta \\ -\frac{1}{L+W}& 2+\frac{1}{L+W}& 2-\frac{1}{L+W}& \frac{1}{L+W}\end{array}\right]\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\\ v_4\end{array}\right]}\\ \hline\end{array}$$

(93)

Summary and Remarks

The following remarks are obtained after applying the proposed framework to the 4WD mecanum mobile:

• The obtained kinematic model is consistent with the formulations of similar mecanum platforms in [4,8,32,33].
• The formulation is fully consistent with the unified wheel model used for differential and Ackermann architectures.
• The holonomic nature arises directly from the structure of the rolling directions $u_i$.
• The Jacobian retains the same projection-based interpretation used for all wheel types within the unified framework.

5. Simulation Results and Comparative Analysis

This section presents a comprehensive simulation-based evaluation of the proposed unified kinematic framework across multiple wheeled mobile robot architectures. The analysis progresses from qualitative validation of canonical motion primitives to the execution of complex trajectories under realistic physics-based conditions, and finally to the assessment of geometric imperfections such as wheel mounting misalignments. Together, these studies aim to verify the consistency, flexibility, and practical modeling capabilities of the formulation under increasingly realistic assumptions.

To facilitate reproducibility and further experimentation, an open-source implementation of the proposed unified kinematic framework, together with the simulation scripts used in this study, is available at source code https://github.com/ldmo-cidetec/unified-wheeled-robot-kinematics (accessed on 21 January 2026).

5.1. Qualitative Kinematic Consistency Analysis

In this section, the kinematic models of the four mobile architectures, previously obtained from the generalized framework, are validated through simulation. Forward kinematics in each robot’s global reference frame is used for this purpose.

In every case, the robot state is chosen as:

$$\mathit{z}={[x,y,\theta ]}^T,$$

(94)

i.e., the minimum set of variables required to describe the mobile behavior that includes the position and orientation of the system in the global frame.

Then, the kinematic state-space model common to all mobile robots is:

$$\begin{array}{c}\dot{\mathit{z}}\left(t\right)=f(\mathit{z}\left(t\right),\mathit{u}\left(t\right))\\ y\left(t\right)=z\left(t\right)\end{array},$$

(95)

where $\mathit{u}\left(t\right)=\mathit{v}\left(t\right)$ is the input vector given by the tangential velocities of the actuated wheels corresponding to each architecture, $\mathit{y}\left(t\right)=\mathit{z}\left(t\right)$ is the output indicating that all mobile states are measurable or observable, and the function $f(\mathit{z}\left(t\right),\mathit{u}\left(t\right))=H\left(\theta \right)J^{+}\mathit{v}$ defines robot kinematics. No feedback control law is considered; instead, the inputs $\mathit{u}\left(t\right)$ are constant open-loop commands, representing a perfect velocity controller, in order to analyze the kinematic behavior of each robot architecture.

An initial condition $z_0$ is imposed on (95) for each mobile robot, and the Initial Value Problem (IVP) obtained is solved by the Euler numerical integration method (ode1) with an integration step of $\Delta t=0.1\left(\mathrm{s}\right)$. Each simulation runs for $t_f=10\left(\mathrm{s}\right)$, and different fixed inputs $\mathit{v}\left(t\right)$ are applied to each platform to cover and observe a wide range of kinematic behaviors.

5.1.1. Simulation of the Differential Drive with Two Standard Wheels

The simulation employs the global forward kinematic model of the differential-drive previously derived in (42), which maps the left and right velocities $\mathit{v}={[v_1,v_2]}^T$ to the linear and angular velocities in the global frame $\mathit{\xi }={[\dot{x},\dot{y},\omega ]}^T$.

Consequently, the global velocities $\dot{\mathit{z}}=\mathit{\xi }$ at each integration step $\Delta t$ are obtained by evaluating the closed-form expressions of (42). By solving the associated IVP, one can find the states of the mobile $\mathit{z}={[x,y,\theta ]}^T$ at every instant.

To validate the kinematic relations derived for the differential-drive architecture, a set of canonical motion primitives is simulated. Each primitive corresponds to a specific combination of the left and right wheel velocities, capturing the characteristic motions of this non-holonomic platform. In this sense, four representative pairs of wheel velocities were selected. The sets of velocity commands are listed in

Table 1. Wheel velocity patterns used for the simulation of differential-drive motion primitives.

Case	${[{\mathit{v}}_1,{\mathit{v}}_2]}^{\mathit{T}}$	Description
(a)	${[0.10,0.10]}^T$	Forward translation
(b)	${[-0.10,-0.10]}^T$	Backward translation
(c)	${[-0.01,0.08]}^T$	Pure rotation (counter-clockwise)
(d)	${[0.08,-0.01]}^T$	Pure rotation (clockwise)

. It is important to note that similar motion-primitive validations are commonly observed in odometry and calibration studies [12,21].

The resulting trajectories are shown in Figure 8, following the notation (a) to (d). Comparable simulations and experimental analyses of differential drive kinematics can be found in [4,8,13,24,34,35,36,37,38,38]. These trajectories reproduce the expected behaviors of forward/backward translation and in-place rotation, consistent with the kinematic characteristics of non-holonomic differential-drive robots [4,35,37].

5.1.2. Simulation of the Ackermann Steering or Car-like Vehicle with Four Standard Wheels

The simulation of the Ackermann-steered platform is based on the bicycle-model global forward kinematics previously derived in (60) involving the steering angle $\delta$. These relationships establish the mapping between the left and right velocities $\mathit{v}={[v_1,v_2]}^T$ of the wheels at the rear axle and the angle $\delta$ to the linear and angular velocities $\mathit{\xi }={[\dot{x},\dot{y},\omega ]}^T$.

As in the case of the differential platform, the Ackermann model is subjected to different canonical driving conditions. For this, six representative motion primitives are simulated. These primitives consist of combinations of longitudinal velocity (given by equal wheel speeds at the rear axle) and steering angle that excite the platform’s characteristic steering behavior. The selected motion primitives used for simulation are summarized in

Table 2. Input patterns used for the simulation of Ackermann motion primitives.

Case	${[{\mathit{v}}_1,{\mathit{v}}_2]}^{\mathit{T}}$	$\mathit{\delta }$	Description
(a)	${[0.05,0.05]}^T$	$0^{\circ }$	Straight forward motion
(b)	${[-0.05,-0.05]}^T$	$0^{\circ }$	Straight backward motion
(c)	${[0.05,0.05]}^T$	$22.5^{\circ }$	Forward left turn
(d)	${[0.05,0.05]}^T$	$-22.5^{\circ }$	Forward right turn
(e)	${[-0.05,-0.05]}^T$	$22.5^{\circ }$	Backward left turn
(f)	${[-0.05,-0.05]}^T$	$-22.5^{\circ }$	Backward right turn

. These are consistent with previous validation practices in steering and odometry studies [12,34,39].

The resulting trajectories can be observed in Figure 9 following the notation (a) to (f). The resulting paths include straight segments and circular arcs of different curvature radii that match the analytical predictions.

5.1.3. Simulation of the Three-Wheel Omnidirectional Platform (3, 0)

This simulation employs the global forward kinematic mapping derived in (78), which relates the wheel angular velocities $\mathit{v}={[v_1,v_2,v_3]}^{\mathsf{T}}$ to $\mathit{\xi }={[\dot{x},\dot{y},\omega ]}^T$.

To validate the holonomic behavior predicted by the global forward kinematics formulation, a set of six canonical motion primitives is simulated. These primitives reproduce the characteristic motions achievable with an omnidirectional three-wheel platform with wheels arranged at $2\pi /3$. Each primitive is a representative wheel-velocity triplet, elected to span the motions of its configuration. The corresponding inputs are summarized in

Table 3. Wheel-velocity patterns for the 3R-omnidirectional platform, consistent with the kinematic model.

Case	${[{\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3]}^{\mathit{T}}$	Description
(a)	${[0.10,0.10,0.10]}^T$	Pure clockwise rotation around the mobile center
(b)	${[0.00,0.10,-0.10]}^T$	Pure translation along $-X$-axis
(c)	${[0.00,-0.10,0.10]}^T$	Pure translation along $+X$-axis
(d)	${[0.10,-0.10,-0.10]}^T$	Pure translation along $+Y$-axis
(e)	${[-0.10,0.10,0.10]}^T$	Pure translation along $-Y$-axis
(f)	${[0.10,-0.13,0.03]}^T$	Diagonal translation along $+X$ and $+Y$-axes

.

On the other hand, the resulting trajectories are in Figure 10, labeled as (a) to (f). As shown, the simulated trajectories are fully consistent with the kinematic characteristics of a holonomic $(3, 0)$ platform. The results include pure rotation (case a), pure lateral translations along X-axis (cases b and c), pure longitudinal translations along Y-axis (cases d and e), and diagonal motion without drift (case f). All motion primitives were obtained by analytically enforcing the corresponding velocity constraints on the inverse kinematic model, ensuring straight-line behavior without curvature drift.

5.1.4. Simulation of the Four-Wheel-Drive (4WD) Mecanum Platform

Finally, the simulation of the mecanum platform utilizes the global forward kinematics obtained in (93), which links the wheel angular velocities $\mathit{v}={[v_1,v_2,v_3,v_4]}^{\mathsf{T}}$ with $\mathit{\xi }$.

The forward kinematics of this mobile are also validated through a set of 10 canonical motion primitives in simulation. Each primitive corresponds to a specific combination of wheel angular velocities, together capturing the characteristic motions achievable with a 4WD mecanum platform. These primitives enable the mobile to perform representative movements with the robot, including longitudinal, lateral, diagonal, and rotational motions. The complete set of command inputs used in simulation is provided in

Table 4. Wheel velocity patterns used for the simulation of mecanum motion primitives.

Case	${[{\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,{\mathit{v}}_4]}^{\mathit{T}}$	Description
(a)	${[0.10,0.10,0.10,0.10]}^T$	Forward translation
(b)	${[-0.10,-0.10,-0.10,-0.10]}^T$	Backward translation
(c)	${[-0.10,0.10,0.10,-0.10]}^T$	Lateral motion (along $+Y$-axis)
(d)	${[0.10,-0.10,-0.10,0.10]}^T$	Lateral motion (along $-Y$-axis)
(e)	${[0,0.10,0.10,0]}^T$	Diagonal (forward-right)
(f)	${[0.10,0,0,0.10]}^T$	Diagonal (forward-left)
(g)	${[-0.10,0,0,-0.10]}^T$	Diagonal (backward-left)
(h)	${[0,-0.10,-0.10,0]}^T$	Diagonal (backward-right)
(i)	${[0.10,-0.05,0.10,-0.05]}^T$	Rotation (clockwise)
(j)	${[-0.05,0.10,-0.05,0.10]}^T$	Rotation (counter-clockwise)

.

The resulting trajectories are depicted in Figure 11 and faithfully reproduce the expected motion patterns, including longitudinal translations (cases a and b), pure lateral motions (cases c and d), diagonal displacements (cases e and h), and pure rotations (cases i and j). This agreement confirms the internal consistency of the kinematic model and its ability to predict the platform’s omnidirectional behavior.

5.2. Open-Loop Execution of Complex Trajectories in Realistic Simulation

Based on the qualitative consistency analysis of the reference motion primitives presented in Section 5.1, this subsection extends the evaluation to more complex trajectories involving continuous curvature variations and sustained coordination among multiple actuated wheels into the obtained kinematic model and the model provided by the Gazebo physics engine where physical effects such as wheel–ground interaction, friction, contact dynamics, and slip naturally arise. These scenarios provide a more demanding test for the proposed unified kinematic framework beyond elementary motions.

All experiments are conducted under an open-loop execution strategy, where the angular velocities of the actuated wheels are computed exclusively from the inverse kinematic model derived in Section 4. No feedback control, trajectory tracking algorithm, or error compensation mechanism is employed. This intentional choice allows the analysis to focus on the intrinsic behavior induced by the kinematic formulation itself, without introducing additional layers related to control design or state estimation.

For each trajectory, three different motion representations are considered: (i) the ideal reference path defined in task space, (ii) a purely kinematic simulation obtained by numerically integrating the forward kinematic model, and (iii) a realistic simulation implemented in ROS 2 Jazzy Jalisco and Gazebo, where the wheel joint velocities are then integrated into the Gazebo physics engine to generate the resulting robot motion governed by the multibody dynamics and the wheel–ground interaction. In all cases, the reported robot pose is assumed to be ideal. For the kinematic simulations, the pose is obtained by direct integration of the kinematic model, whereas in the ROS + Gazebo simulations, it corresponds to the ground-truth pose provided by the physics engine. Sensor noise, bias, and estimation errors are not considered. In ROS + Gazebo simulations, wheel–ground interaction, including friction and contact dynamics, is considered. The robots used for the realistic simulation are shown in Figure 12, and the utilized parameters are summarized in Appendix A.

Two representative smooth planar trajectories are also considered: a circular path and a Gerono lemniscate. Both trajectories are continuously differentiable, bounded, and periodic, which facilitates open-loop wheel command generation via inverse kinematics while introducing nontrivial curvature profiles. Each trajectory is executed by the four mobile robot architectures using identical open-loop wheel velocity commands generated using the same kinematic formulation, with only the wheel configuration parameters adapted to each platform.

The circular trajectory, with radius R and center at $\mathbf{c}={[0,R]}^T$, is defined as:

$$\begin{array}{cc}\hfill x\left(t\right)& =Rsin\left(\overline{\omega }t\right),\hfill \\ \hfill y\left(t\right)& =R\left(1-cos\left(\overline{\omega }t\right)\right),\hfill \end{array}$$

(96)

where $\overline{\omega }=\frac{2\pi }{T}$, $R=1$ m, and $T=10$ s. This parametrization starts at the Cartesian origin and completes exactly one full revolution over the interval $t\in [0,T]$.

The Gerono lemniscate introduces a more demanding and time-varying curvature profile, which is defined as:

$$\begin{array}{cc}\hfill x\left(t\right)& =-R\left(1-cos\left(2\overline{\omega }{t}^{\prime }\right)\right),\hfill \\ \hfill y\left(t\right)& =Asin\left(\overline{\omega }{t}^{\prime }\right),\hfill \end{array}$$

(97)

with $t^{\prime }=t-\frac{T}{4}$, $\overline{\omega }=\frac{2\pi }{T}$, $R=1$ m, $A=1$ m, and $T=10$ s. The phase shift $t^{\prime }$ is introduced to ensure a well-defined initial motion direction and to avoid an initial stationary configuration when generating open-loop wheel commands.

While the circular path provides a constant-curvature reference, commonly used as a baseline for wheeled mobile robots, the lemniscate requires continuous changes in wheel velocities and coordinated motion throughout execution. Together, these trajectories enable a qualitative assessment of whether the same open-loop inverse-kinematics formulation can generate coherent motions across heterogeneous wheel configurations under both purely kinematic and realistic physics-based simulations.

The objective of this analysis is not to achieve precise trajectory tracking, but rather to observe how different mobile platforms respond to complex motion commands under realistic physical conditions. In particular, the focus is on assessing whether the resulting motions remain qualitatively consistent with the intended trajectory shapes and expected kinematic behavior of each robot, despite the presence of unmodeled dynamics and contact-related effects.

The results obtained for the circular trajectory are shown in Figure 13, where the ideal reference path is compared against both the purely kinematic simulation and the realistic ROS 2 simulation for all four mobile platforms. In the kinematic case, the trajectories closely follow the ideal reference, as expected from direct numerical integration of the forward kinematic model. When realistic physical effects are introduced, deviations from the ideal path are noticeable due to slip, friction, and contact dynamics. Nevertheless, the overall circular shape and curvature are preserved across all platforms, indicating that the open-loop inverse kinematic formulation generates coherent motion commands even under non-ideal conditions.

Figure 14 presents the results for the Gerono lemniscate trajectory, which imposes a more demanding motion profile due to its time-varying curvature and self-intersection. Compared to the circular case, larger deviations between the ideal, kinematic, and realistic trajectories are observed, particularly in the realistic simulations. These deviations are consistent with the increased sensitivity of open-loop execution to unmodeled dynamics and wheel slip when fast curvature changes are required. In the Ackermann platform, the more pronounced trajectory deformation can be attributed to geometric steering constraints and the reliance on front-wheel steering angles, which amplify the effects of small deviations in wheel orientation and lateral slip under open-loop operation. Despite this, all platforms reproduce the characteristic shape of the lemniscate, demonstrating qualitatively consistent behavior across heterogeneous wheel configurations.

A quantitative summary of the planar position errors is reported in

Table 5. RMSE of planar position for different mobile robot architectures tracking circular and lemniscate trajectories, comparing kinematic and realistic simulations against the ideal reference.

Robot	Trajectory	Kinematics vs. Ref.	Realistic vs. Ref.	Realistic vs. Kinematics
Differential	Circle	4.4407 × 10−3	4.9559 × 10−2	5.3536 × 10−2
	Lemniscate	1.0955 × 10−2	1.0041 × 10−1	9.6891 × 10−2
Ackermann	Circle	4.4407 × 10−3	3.7452 × 10−1	3.7015 × 10−1
	Lemniscate	3.5662 × 10−3	4.3000 × 10−1	4.2811 × 10−1
Omni3	Circle	4.4407 × 10−3	6.6455 × 10−2	6.2896 × 10−2
	Lemniscate	8.0055 × 10−3	1.7775 × 10−1	1.7148 × 10−1
Mecanum	Circle	4.4407 × 10−3	1.7946 × 10−1	1.7918 × 10−1
	Lemniscate	8.0055 × 10−3	1.8028 × 10−1	1.7797 × 10−1

, where the root-mean-square error (RMSE) is computed with respect to the ideal reference trajectory. These results reveal a clear separation between numerical and physical sources of deviation. The errors between the ideal and kinematic trajectories remain small and are primarily attributed to numerical integration effects, whereas significantly larger deviations are observed in the ROS + Gazebo simulations due to dynamic effects such as wheel–ground interaction and slip. For circular motion, the kinematic simulations exhibit very small errors across all platforms, reflecting the consistency of the forward and inverse kinematic formulations. In contrast, the realistic simulations show increased RMSE values, with magnitude variations across platforms that can be attributed to differences in wheel configuration and sensitivity to slip and frictional effects. For the lemniscate trajectory, the RMSE values increase substantially in both kinematic and realistic simulations, primarily due to sharper curvature changes and self-intersection of the path. Notably, the difference between the realistic and kinematic RMSE remains bounded in all cases, indicating that physical effects introduce deviations without compromising the structural consistency of the resulting motion. These results reinforce the observation that, while open-loop execution under realistic conditions does not yield exact trajectory tracking, the unified kinematic formulation produces consistent and physically meaningful motion patterns across all evaluated mobile robot architectures.

Overall, the results presented in this subsection indicate that the proposed unified kinematic formulation extends naturally from elementary motion primitives to more complex trajectories under both purely kinematic and realistic simulation conditions. While open-loop execution in a physics-based environment inevitably leads to deviations from the ideal reference, the observed motion patterns remain qualitatively consistent across all evaluated platforms. This behavior highlights the robustness of the unified formulation at the kinematic level and motivates its use as a common foundation for more advanced control or estimation strategies.

5.3. Effect of Wheel Misalignment Modeling in Open-Loop Trajectory Execution

Following the analysis of complex trajectories under ideal wheel mounting conditions presented in Section 5.2, this subsection investigates the effect of wheel mounting misalignments on open-loop trajectory execution. The objective is to illustrate how the proposed unified kinematic framework naturally accommodates such geometric deviations when they are known or can be measured, and to assess their impact on both kinematic and realistic simulations.

The analysis is limited to the differential-drive mobile robot and the circular trajectory, using the same simulation settings and open-loop execution strategy described in Section 5.2. Artificial misalignments are introduced in the wheel mounting angles, with a deviation of ${\beta }_1=25$ $\mathrm{deg}$ applied to the left wheel and ${\beta }_2=-15$ $\mathrm{deg}$ applied to the right wheel. These values are intentionally exaggerated to clearly expose the effect of misalignment on the resulting motion.

In this sense, two different modeling cases are considered. In the first case, the wheel misalignments are explicitly incorporated into the inverse kinematic model when computing the wheel velocity commands. In the second case, the same physical misalignments are present in the robot model, but they are neglected in the inverse kinematic formulation used to generate the control inputs. In both cases, the resulting commands are applied to the purely kinematic simulation and to the realistic ROS 2 and Gazebo simulation.

Using the unified framework, the inverse kinematics of the differential-drive robot in the local frame when ${\beta }_1$ and ${\beta }_2$ are considered is:

$$\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{ccc}cos{\beta }_1& 0& -\frac{L}{2}cos{\beta }_1\\ cos{\beta }_2& 0& \frac{L}{2}cos{\beta }_2\end{array}\right]\left[\begin{array}{c}{\dot{x}}_m\\ {\dot{y}}_m\\ \omega \end{array}\right].$$

(98)

The resulting trajectories for both cases are shown in Figure 15. When wheel misalignments are accounted for in the inverse kinematic model, the kinematic simulation closely follows the ideal circular reference, and the realistic simulation preserves the intended trajectory shape despite physical effects. In contrast, when misalignments are ignored during the inverse kinematic computation, significant deviations from the reference trajectory are observed at the purely kinematic level, which are further amplified in the realistic simulation.

This behavior is quantitatively observed in the RMSE values reported in

Table 6. RMSE of planar position for a differential-drive robot following a circular trajectory, comparing kinematic and realistic simulations with and without explicit modeling of wheel misalignments.

Case	Kinematics vs. Ref.	Realistic vs. Ref.	Realistic vs. Kinematics
Considering misalignments	4.4407 × 10−3	1.2411 × 100	1.2437 × 100
Without considering misalignments	5.1398 × 10−1	1.3870 × 100	9.2707 × 10−1

. When misalignments are properly modeled, the kinematic simulation exhibits a very small error relative to the ideal reference due to integration time, which is comparable to the ideal mounting case. Conversely, neglecting misalignments in the inverse kinematic formulation leads to a substantial increase in error, even before considering realistic physical effects. Although the realistic simulations present larger absolute errors in both cases due to open-loop control execution and contact dynamics, the results clearly indicate that explicitly modeling wheel misalignments significantly improves consistency at the kinematic level.

These results highlight one of the practical advantages of the proposed unified framework: geometric imperfections such as wheel mounting misalignments can be incorporated directly into the kinematic model without altering the overall formulation. When such parameters are available, their inclusion leads to more coherent motion generation and mitigates systematic trajectory distortions in open-loop control execution.

5.4. Brief Summary

The results presented in Section 5.1, Section 5.2 and Section 5.3 validate the proposed unified kinematic formulation across a range of motion scenarios and modeling assumptions. The simulated behaviors confirm that the global forward kinematics accurately captures the mobility characteristics and structural constraints inherent to each wheeled mobile robot architecture.

The differential-drive and Ackermann platforms exhibit non-holonomic behavior due to constraints on lateral motion, both under ideal kinematic assumptions and in physics-based simulations. In contrast, the omnidirectional $(3, 0)$ platform allows full planar mobility by enabling independent control of translational velocities. The four-wheel mecanum system, while kinematically holonomic, may exhibit effective motion constraints in realistic simulations due to wheel–ground interaction effects in the absence of explicit slip compensation mechanisms.

Beyond reproducing canonical motion primitives, the framework demonstrates coherent behavior under complex trajectories, realistic physical interactions, and geometric imperfections such as wheel mounting misalignments. The ability to explicitly incorporate such geometric parameters into the kinematic model, when available, highlights a key practical advantage of the proposed formulation.

Overall, these results show that the unified kinematic framework provides a consistent and flexible mathematical structure for modeling, simulation, and analysis of heterogeneous wheeled mobile robots, supporting both analytical reasoning and numerical experimentation within a single formulation.

6. Conclusions and Future Work

This paper presented a unified and parameter-driven kinematic modeling framework for wheeled mobile robots. By abstracting geometry and motion constraints at the wheel level into a consistent vectorized structure, the approach unifies diverse locomotion types, both holonomic and non-holonomic, within a single formulation. The proposed framework offers substantial versatility and provides a common kinematic foundation compatible with odometry formulations, kinematic mappings used in control design, and simulation environments.

A key theoretical contribution of this work is the formal characterization of the structural properties of the generalized kinematic mapping. In particular, the proposed Jacobian construction was shown to be invariant under wheel reconfiguration, preserving its functional structure regardless of the number, placement, or orientation of the wheels. Furthermore, holonomy and nonholonomy were rigorously identified as intrinsic properties emerging from the rank of the generalized Jacobian, rather than as architecture-dependent assumptions. These results were formalized through explicit propositions and proofs, establishing the framework as a genuine generalization of classical kinematic models rather than a mere reformulation.

This generalized framework represents a significant step toward the long-term goal of universal, architecture-independent kinematic mobility models for robotic platforms. Beyond reproducing canonical motion primitives, the proposed framework was shown to preserve coherent kinematic behavior under complex trajectories, realistic physics-based simulation conditions, and geometric imperfections such as wheel mounting misalignments. These results indicate that the formulation captures not only the structural mobility properties of different platforms but also provides a practical kinematic modeling tool capable of incorporating non-ideal geometric conditions commonly encountered in robotic systems, when such parameters are known or can be estimated. The extension of the proposed framework toward dynamic formulations of motion, as well as the explicit modeling and analysis of wheel–ground interaction effects, including comprehensive slip representations, constitute the primary directions for future work.