Designing a holonomic drive system means solving the same kinematic problem every time — you know where you want the robot to go, but your motors only understand wheel speeds. Use this Omni-Wheel Robot Velocity Resolver to calculate individual wheel speeds for 3-wheel and 4-wheel omni-drive configurations using your desired X velocity, Y velocity, angular velocity, and robot radius. Getting this right matters in warehouse automation, competition robotics, and mobile manipulation platforms where any motion error compounds fast. This page covers the full velocity decomposition formula, a worked example with real numbers, the underlying theory, and a practical FAQ.
What is omni-wheel velocity resolution?
Omni-wheel velocity resolution is the process of converting a robot's desired overall motion — forward, sideways, and rotational — into the exact speed each individual wheel needs to spin. It's the core calculation that makes a robot move precisely in any direction without turning first.
Simple Explanation
Think of it like rowing a boat with multiple oars pointing in different directions — each oar can only push the water one way, but together they can move the boat anywhere. Each omni-wheel can only roll in its own direction, but when you calculate the right speed for each one, the combined effect moves the robot wherever you want. This calculator does that math for you instantly.
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Omni-Wheel Robot Velocity Resolver Interactive Calculator
Visualize how desired robot motion (X, Y, rotation) converts into individual wheel speeds for 3-wheel and 4-wheel omni-drive configurations. Watch wheel vectors combine to produce the exact robot movement you specify.
WHEEL 1 SPEED
-1.00 m/s
WHEEL 2 SPEED
0.50 m/s
WHEEL 3 SPEED
0.50 m/s
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How to Use This Calculator
- Enter your desired X velocity (forward/backward) and Y velocity (left/right) in m/s.
- Enter the angular velocity in rad/s and the robot radius (center to wheel) in meters.
- Select your wheel configuration — 3-wheel (120° spacing) or 4-wheel (90° spacing).
- Click Calculate to see your result.
Omni-Wheel Robot Configuration
Omni-Wheel Velocity Calculator
📹 Video Walkthrough — How to Use This Calculator
Mathematical Equations
The omni wheel velocity calculator uses trigonometric velocity decomposition to resolve robot motion into individual wheel speeds. For each wheel i, the velocity is calculated using:
Use the formula below to calculate individual wheel velocity from desired robot motion.
Where:
- Vwheel,i = Linear velocity of wheel i (m/s)
- Vx = Desired robot velocity in X direction (m/s)
- Vy = Desired robot velocity in Y direction (m/s)
- θi = Mounting angle of wheel i (radians)
- ω = Desired angular velocity (rad/s)
- R = Robot radius from center to wheel (m)
For common configurations:
3-Wheel Configuration: θ = [90°, 210°, 330°]
4-Wheel Configuration: θ = [45°, 135°, 225°, 315°]
Simple Example
A 3-wheel robot with a 0.2 m radius. Desired motion: Vx = 1.0 m/s, Vy = 0 m/s, ω = 0 rad/s.
- Wheel 1 (90°): −1.0 × sin(90°) + 0 × cos(90°) + 0 = −1.000 m/s
- Wheel 2 (210°): −1.0 × sin(210°) + 0 × cos(210°) + 0 = 0.500 m/s
- Wheel 3 (330°): −1.0 × sin(330°) + 0 × cos(330°) + 0 = 0.500 m/s
Understanding Omni-Wheel Robot Velocity Resolution
Omni-directional robots represent one of the most sophisticated achievements in mobile robotics, offering unprecedented maneuverability through their ability to move instantly in any direction without rotation. At the heart of these systems lies the critical process of velocity resolution – transforming desired robot motion into precise wheel speeds that work together to achieve seamless omnidirectional movement.
The Physics of Omni-Wheel Motion
Unlike conventional wheels that can only roll forward and backward, omni-wheels feature small rollers perpendicular to the main wheel circumference. This unique design allows them to slip laterally without resistance while maintaining traction in their primary rolling direction. When multiple omni-wheels are arranged at specific angles around a robot chassis, they create a holonomic drive system capable of independent translation and rotation.
The fundamental principle behind omni wheel velocity calculation involves decomposing the desired robot motion into components that each wheel can contribute to. Each wheel can only apply force in its rolling direction, but the vector sum of all wheel forces produces the desired robot motion. This is where trigonometric velocity decomposition becomes essential.
Mathematical Foundation
The velocity resolution process begins with the robot's desired motion state, defined by three parameters: forward/backward velocity (Vx), left/right velocity (Vy), and rotational velocity (ω). These must be transformed into individual wheel speeds using the kinematic relationship that accounts for each wheel's mounting angle and distance from the robot center.
For each wheel, the required velocity combines three components: the projection of linear X-velocity onto the wheel's rolling direction, the projection of linear Y-velocity, and the tangential velocity due to rotation. The mathematical relationship ensures that when all wheels rotate at their calculated speeds, the vector sum produces exactly the desired robot motion.
Practical Implementation Considerations
Real-world omni-wheel robot systems require careful attention to several engineering factors beyond basic velocity calculation. Wheel slip, surface friction variations, and mechanical tolerances all affect the actual robot motion. Many successful implementations incorporate feedback control systems that continuously adjust wheel speeds based on actual robot position and orientation.
The choice of wheel configuration significantly impacts robot performance. Three-wheel systems offer mechanical simplicity and cost advantages but may have reduced load capacity and stability. Four-wheel configurations provide better load distribution and redundancy but require more complex control algorithms and higher-precision manufacturing.
Modern omni-wheel robots often integrate FIRGELLI linear actuators for auxiliary functions such as lifting mechanisms, arm positioning, or adjustable wheel assemblies. These actuators complement the omni-wheel drive system by providing precise linear motion for manipulator arms or height adjustment mechanisms.
Worked Example: 3-Wheel Robot Navigation
Consider a warehouse automation robot with a 3-wheel omni-drive system. The robot has a radius of 0.25 meters and needs to move diagonally while rotating to align with a shelf. The desired motion is: Vx = 0.5 m/s, Vy = 0.3 m/s, ω = 0.2 rad/s.
For the three wheels positioned at 90°, 210°, and 330°:
Wheel 1 (90°):
V₁ = -0.5 × sin(90°) + 0.3 × cos(90°) + 0.2 × 0.25 = -0.5 + 0 + 0.05 = -0.45 m/s
Wheel 2 (210°):
V₂ = -0.5 × sin(210°) + 0.3 × cos(210°) + 0.2 × 0.25 = 0.25 - 0.26 + 0.05 = 0.04 m/s
Wheel 3 (330°):
V₃ = -0.5 × sin(330°) + 0.3 × cos(330°) + 0.2 × 0.25 = 0.25 + 0.26 + 0.05 = 0.56 m/s
These calculated wheel speeds, when implemented by the motor control system, will produce the exact desired robot motion.
Advanced Applications and Control Strategies
Modern omni-wheel systems extend far beyond basic velocity control. Advanced applications include trajectory following, where the robot follows complex paths while maintaining specific orientations, and formation control for multi-robot systems. These applications require sophisticated control algorithms that continuously calculate wheel speeds based on position feedback and path planning algorithms.
Industrial automation applications often demand precise positioning accuracy, requiring closed-loop control systems that monitor actual robot position and continuously adjust wheel speeds to compensate for disturbances. Many systems incorporate encoders on each wheel motor, IMU sensors for orientation feedback, and vision systems for absolute positioning.
The integration of omni-wheel drive systems with robotic manipulators creates mobile manipulation platforms capable of performing complex tasks while maintaining precise positioning. These systems often use multiple control loops: one for base mobility using omni wheel velocity calculation, and another for manipulator control using actuator positioning systems.
Design Optimization and Performance Factors
Optimizing omni-wheel robot performance requires balancing multiple design parameters. Wheel radius affects maximum speed and acceleration capabilities, while robot radius influences maneuverability and stability. The trade-off between these parameters depends heavily on the specific application requirements.
Motor selection plays a crucial role in system performance. Each wheel requires a motor capable of precise speed control across the full range of calculated velocities. Brushless DC motors with integrated encoders are common choices, offering high precision and reliability. The motor control system must respond rapidly to changing velocity commands to maintain smooth robot motion.
Mechanical design considerations include wheel mounting precision, chassis rigidity, and bearing quality. Small angular errors in wheel mounting can significantly affect the accuracy of velocity resolution calculations. High-quality bearings and rigid mounting systems ensure that actual wheel motion matches the calculated values.
Frequently Asked Questions
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About the Author
Robbie Dickson
Chief Engineer & Founder, FIRGELLI Automations
Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.
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