# Driving Strategies for Omnidirectional Mobile Robots with Offset Differential Wheels

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## Abstract

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## 1. Introduction

## 2. Description of the Robot

## 3. Kinematic Analysis

- The angular velocity of the right wheel, ${\dot{\alpha}}_{R}$, denoting the angular position of the right wheel as ${\alpha}_{R}$.
- The angular velocity of the left wheel, ${\dot{\alpha}}_{L}$, denoting ${\alpha}_{L}$ as its angular position from a reference.
- The angular velocity of the platform about the vertical axis, $\dot{\phi}$, with respect to the chassis, with an angular position of the platform with respect to the chassis $\phi $.

- The position vector of the center of the robot with respect to a fixed frame, $\mathbf{p}=\{x,y\}$, and its derivative $\dot{\mathbf{p}}=\{\dot{x},\dot{y}\}$.
- The angular position of the platform with respect to the fixed frame, ${\theta}_{p}$, and its corresponding angular velocity, ${\dot{\theta}}_{p}$.
- The angular position of the chassis with respect to the fixed frame, ${\theta}_{c}$, and its derivative ${\dot{\theta}}_{c}$.
- The radius of the actuated wheels, R.
- The distance between the rotation axis of the actuated wheels and the vertical rotation axis, ${d}_{1}$.
- The distance between the actuated wheels, ${d}_{2}$.

- The distance from the center of the platform to the joint of the caster fork, ${d}_{4}$.
- The length of the caster fork from its joint to the axis of the caster wheel, ${d}_{3}$.
- The radius of the caster wheel, ${R}_{c}$.
- The angle of the caster fork with respect to the chassis, $\beta $.
- The rotational velocity of the caster wheel, ${\dot{\alpha}}_{c}$.

#### 3.1. Forward Kinematics

#### 3.2. Inverse Kinematics

#### 3.3. Passive Degrees of Freedom

## 4. Kinematic Behavior

#### 4.1. Driving Stability

#### 4.2. Kinematic Behavior

## 5. Driving Strategy

- The trajectory must be smooth, that is, it must be continuous as well as continuously differentiable. This extends to the initial position of the robot. The vector analogous to the pure pull direction must be tangent to the velocity at the starting point; otherwise, significant compensation will be necessary to remain in the pull region over the long term. If the first part of the path is curved, $\gamma $ and ${\gamma}_{goal}$ should be equal.
- Turns with a radius smaller than ${\mathbf{d}}_{\mathbf{1}}$ are possible but should be avoided. While momentarily turning tightly does not guarantee a change in velocity region, sustained rotation with such a small radius will eventually result in a change in the direction of the chassis.

## 6. Dynamic Behavior

#### 6.1. Lateral Stability

#### 6.2. Dynamic Model

- The mass, ${m}_{c}$, and the moment of inertia, ${I}_{c}$, of the chassis (excluding the actuated wheels).
- The mass, ${m}_{W}$, and the moments of inertia, ${I}_{W}$, of the actuated wheels, which were assumed to be the same for both.
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- ${I}_{W,ax}$ is the moment of inertia of the wheel spinning through its axis.
- –
- ${I}_{W,rad}$ is the moment of inertia as it relates to the contribution of the wheel when the whole chassis is rotating. It was calculated as the moment of inertia of the wheel spinning about its radius plus the additional inertia brought about by the distance from the center of rotation of the chassis, which is taken into account through the parallel axis theorem.

- The mass, ${m}_{p}$, and the moment of inertia, ${I}_{p}$, of the platform.
- The mass, ${m}_{cf}$, and the moment of inertia, ${I}_{cf}$, of the caster fork.
- The mass, ${m}_{cw}$, and the moments of inertia, ${I}_{cw,ax}$ and ${I}_{cw,rad}$, of the caster wheel.

#### 6.3. Forward Dynamics

#### 6.4. Inverse Dynamics

## 7. Tests

#### 7.1. Simulations

#### 7.2. Experiments

#### 7.3. Uncompensated Pull-Region Motion

#### 7.4. Compensated Pull-Region Motion

## 8. Results

#### 8.1. Simulations

#### 8.2. Experiments

## 9. Conclusions

#### Future Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The structure of MOBY robot. The actuated wheels have radius R and the distance between the wheels is ${d}_{2}$. The vertical axis of rotation is located at a distance ${d}_{1}$ from the axis of the wheels.

**Figure 2.**The instant velocity of the center of the robot (in blue) is a function of the velocities of the wheels (in red and green).

**Figure 6.**Simulation showing the reversal of the direction of the chassis. From left to right, the robot accelerates backwards, holds its speed, switches from pull to push, and comes to a stop.

**Figure 7.**The corrected motion of the chassis when reversing. The red arrow shows the point in the trajectory (the dashed black line) where the correction starts being applied. The dashed green line shows the originally intended trajectory.

**Figure 8.**Regions and main directions of the robot. The compensation region is defined by the user and can even include part of the push region. The dark red arrow shows an example of a velocity in the push region.

**Figure 9.**Dynamic simulation of the response of the robot to an impact on the fork of the caster wheel. The blue arrow indicates the position of the center of the robot when the force is applied and its direction. The green dashed line shows the expected trajectory had there been no impact.

**Figure 10.**The trajectory determined by concatenating arcs and a straight line. The dashed green line is the pre-defined trajectory. The path begins with a 60° arc with a radius of 5 m; then, it applies a 90° arc with a radius of 8 m and finally a 5-m-long straight line. The angular velocity changes uniformly in the stretches between arcs. The shapes of the chassis and the wheels of the robot show the position and orientation of the robot at evenly spaced instants.

**Figure 11.**Detail of the previous figure showing how tighter turns are prone to causing an increase in position error due to the discrete nature of the velocity commands.

**Figure 12.**The stability tetrahedron is formed by the contact points of the wheels and the center of inertia of the robot.

**Figure 14.**The torque required on the actuated wheels to perform the motion in Figure 6. (

**Left**), uncompensated; (

**right**), applying the compensation every instant.

**Figure 15.**The magnitude of the friction with the ground needed for the robot to perform the motion in Figure 6.

**Figure 17.**Dynamic effects of the caster wheel when it exited its unstable position during the second phase of compensated motion.

**Figure 18.**Velocities and chassis angle during the experiments: (

**left**), the first experiment; (

**right**), the second experiment (with compensation). Note the difference in scale between the plots.

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**MDPI and ACS Style**

Badia Torres, J.; Perez Gracia, A.; Domenech-Mestres, C.
Driving Strategies for Omnidirectional Mobile Robots with Offset Differential Wheels. *Robotics* **2024**, *13*, 19.
https://doi.org/10.3390/robotics13010019

**AMA Style**

Badia Torres J, Perez Gracia A, Domenech-Mestres C.
Driving Strategies for Omnidirectional Mobile Robots with Offset Differential Wheels. *Robotics*. 2024; 13(1):19.
https://doi.org/10.3390/robotics13010019

**Chicago/Turabian Style**

Badia Torres, Joan, Alba Perez Gracia, and Carles Domenech-Mestres.
2024. "Driving Strategies for Omnidirectional Mobile Robots with Offset Differential Wheels" *Robotics* 13, no. 1: 19.
https://doi.org/10.3390/robotics13010019