If you’re building any holonomic drive, you’ll come back to the same kinematics problem: you know the robot motion you want (move here, turn there), but your motors can only follow commands for individual wheel speeds. This Omni-Wheel Robot Velocity Resolver breaks your desired X velocity, Y velocity, angular velocity, and robot radius into what each wheel actually needs to do for both 3- and 4-wheel omni-drive setups. Small errors here show up fast in tight environments—warehouse robots, competition bots, anywhere you can’t afford accumulated mistake in position. Below you’ll find the exact formula, a step-by-step worked example, some background on where it comes from, and a FAQ targeted at real use.
What is omni-wheel velocity resolution?
Omni-wheel velocity resolution means turning the robot’s intended movement—forward, sideways, spin—into specific wheel speeds that motors can track. Each wheel doesn’t care about the whole robot, just about how fast it should turn to help with the overall move.
Simple Explanation
It’s like having a bunch of oars on a boat, most pointing in different directions—each oar only pushes its own way, but if you get the push amounts right, the boat will go where you want. Each omni-wheel rolls in its own direction. The trick is working out what speed to tell each one to achieve your total desired motion. This calculator handles that part for you.
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Omni-Wheel Robot Velocity Resolver Interactive Calculator
See for yourself how a target robot motion (X, Y, rotation) breaks down into the speed each wheel will need to achieve it. The tool covers both 3- and 4-wheel omni drive layouts, visually combining wheel vectors for clarity.
WHEEL 1 SPEED
-1.00 m/s
WHEEL 2 SPEED
0.50 m/s
WHEEL 3 SPEED
0.50 m/s
FIRGELLI Automations — Interactive Engineering Calculators
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
This calculator is intended for education, concept evaluation, and preliminary design. Results are based on the equations and assumptions described on this page, but cannot account for every real-world load case, tolerance, material property, environmental condition, installation detail, safety factor, code, or regulatory requirement. Verify all inputs, assumptions, units, and results independently before selecting components or using the result in a real application. Safety-critical, structural, medical, lifting, transportation, or regulated applications must be reviewed by a qualified engineer.
📹 Video Walkthrough — How to Use This Calculator
Mathematical Equations
This calculator uses trigonometry to split up the movement commands into something each wheel can contribute. For each wheel, you use:
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
Holonomic (omni-directional) robots are designed for movement in any direction at any time, not just forward and backward. The underlying math is about breaking down your motion command (how you want the robot to move) into speeds for several wheels, each with its own drive angle. Figuring out those individual wheel speeds correctly is fundamental to making an omni-robot actually go where you want, not just where the motors “think” they should.
The Physics of Omni-Wheel Motion
Standard wheels only roll one way: forwards or backwards. Omni-wheels have little rollers around the rim, letting them slide sideways as easily as forward. Once you mount several of these at set angles around your robot, you have a holonomic drive. Each wheel can only push in its rolling direction, so you have to command just the right combination of wheel speeds for a specific net robot movement.
Behind the scenes, each wheel’s speed is the total effect of X-velocity, Y-velocity, and spin, projected into its unique rolling axis. That’s why you see sines and cosines in the velocity resolution formula: each wheel only “cares” about how much your desired move projects onto its axis.
Mathematical Foundation
You start by specifying the X velocity, Y velocity, and angular (spin) velocity for the robot as a whole. For every wheel, you use these values, the angle the wheel is mounted at, and the robot radius to calculate each speed. The formula ties each wheel’s contribution to robot motion directly to its geometry, not to some arbitrary coordinate system.
Get these calculations right, and each motor runs the speed needed to make the robot do exactly what you asked—no wasted effort, no drift (within real-world limits).
Practical Implementation Considerations
In real robots, actual positioning can drift due to slip, uneven floors, wear, or just control lag. You’ll get closest to theory if you use feedback: encoders for wheel speed, IMUs for heading, and a control algorithm to close the loop. No open-loop system, even with perfect math, will be perfectly accurate outside a lab.
If you’re comparing layouts: three wheels is the bare minimum for full omni-directionality, and it keeps things mechanically simple. Four wheels give you better payload and redundancy. If stability or robustness are important, four wheels do better. But keep in mind, control software and precision in mounting/fabrication must be tighter for four wheels.
For robots that also need vertical or arm movement, adding FIRGELLI linear actuators is a way to get precise linear positioning without complicating your holonomic drive math.
Worked Example: 3-Wheel Robot Navigation
Picture a 3-wheel warehouse robot, radius 0.25 meters, moving diagonally while spinning to face a shelf. Suppose Vx = 0.5 m/s, Vy = 0.3 m/s, ω = 0.2 rad/s.
With wheels 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
With these wheel speeds, the controller will move the robot along the commanded path and orientation (within whatever error caused by real-world friction and lag).
Advanced Applications and Control Strategies
Some advanced systems go beyond open-loop wheel speed commands. Examples: robots driving along complex paths while spinning (trajectory following), or multiple robots moving together. These jobs need algorithms that recalculate wheel velocities on the fly, usually using feedback from encoders and vision or position sensors to stay on track.
For fine motion (like in factories), most robots use feedback loops and sensors—wheel encoders, IMUs, sometimes cameras—to keep position error from creeping up. The more feedback, the less you’ll depend on the “perfect frictionless world” assumption behind the equations.
Adding manipulators to an omni-base means you’ll often have two control loops: one for the drive kinematics (wheels, as above), and separately for the actuators on arms or lifters.
Design Optimization and Performance Factors
Improving an omni-wheel robot’s performance comes down to optimizing a handful of parameters. Larger wheels let you go faster, but can limit acceleration. Expanding the robot’s radius improves stability but makes tight turns wider. You can’t have it all—your job is to trade off these factors for what matters in your use case.
Pick motors that can smoothly and accurately follow the full range of calculated wheel speeds, including at low speed and under real load. Brushless DC motors with encoders are common for reliability and repeatability. But even the best motor won’t save you from loose bearings, chassis flex, or sloppily aligned wheels—these cause real error that no formula will fix.
If you take care with wheel mounting, structure rigidity, and bearing quality, the actual achieved path will stay closest to your theoretical prediction.
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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