Odometry Position Update — Y Component
Solve the Y-component of a position update Δy = v · Δt · sin(θ) for any of its four quantities — Y displacement, speed, time interval, or heading angle — with a live 2D vector diagram, a sine response chart, and companion navigation metrics.
The Y-component is a projection
The displacement along y is the projection of the distance vector d = v · Δt onto the y-axis. sin θ measures how much of the motion points north. At θ = 0° (due east) the projection is zero; at θ = 90° (due north) the entire motion is in y. This is the geometric essence of resolving a 2D vector into components.
Sin θ is symmetric and periodic
sin(180° − θ) = sin θ, so a heading of 30° and 150° give the same Y-displacement. sin θ is also periodic with 360°, so a heading of 45° and 405° are identical. This matters for solving for θ: the equation Δy = v · Δt · sin(θ) generally has two solutions in [0°, 360°) — θ and 180° − θ. The calculator reports the principal value and flags the second.
Sign carries direction
Positive Δy means the vehicle moved north, negative means south. The sign comes from sin θ: headings in the upper half-plane (0° < θ < 180°) give positive Δy, and headings in the lower half-plane give negative Δy. The magnitude is the same for ±θ — a symmetric heading gives equal-sized displacements in opposite directions along y.
Companion X-component completes the picture
The Y-component is only half of the 2D update. The complementary relation is Δx = v · Δt · cos θ, and Pythagoras guarantees Δx² + Δy² = (v · Δt)² for any heading. This is why odometry updates both x and y at each step: together they fully describe the displacement in the plane. The companion metrics panel below reports Δx alongside Δy.
How to Use This Calculator
The Y-component update links four quantities: Δy, v, Δt, and θ. Pick which one you want the tool to solve, then fill in the other three.
1. Choose what to solve for
Pick Δy, v, Δt, or θ from the toggle. The calculator rearranges Δy = v · Δt · sin(θ) automatically for whichever you choose. Solving for θ uses the inverse sine.
2. Use realistic presets
Presets cover typical values: Y displacements from −5 to +10 m, speeds from 0.5 to 5 m/s, time intervals from 0.1 to 5 s, and headings from −90° to +90° — spanning ground vehicles, robots, and drones.
3. Watch the angle convention
θ is measured counter-clockwise from east (the +x axis): 0° = east, 90° = north, 180° = west, −90° = south. This is the standard math convention and matches the formula Δy = v · Δt · sin(θ). If your system uses north as zero, subtract 90° from your heading.
4. Read the result and verdict
The result card shows the solved value with equivalents in km, ft, and the companion Δx. A colour-coded verdict classifies the Y-component fraction |sin θ| from pure eastward through gentle, balanced, steep, to pure north/south.
5. Check the diagram and chart
The 2D diagram shows the velocity vector, its X and Y components, and the heading angle, with the Y-component highlighted. The chart plots Δy against θ for three distances, revealing the characteristic sine curve.
6. Switch to Advanced for depth
Advanced mode adds a chart of Δy vs. Δt, a sensitivity check on a +1% input change, and companion metrics including Δx, total distance, displacement magnitude, and bearing.
What This Tool Offers
Built around the classic Y-component relation Δy = v · Δt · sin(θ), the fundamental link between speed, time, heading, and northward displacement in 2D odometry.
Four-way solving
Solve for Y displacement, speed, time interval, or heading angle. The calculator rearranges the formula for whichever variable you choose, including the inverse-sine case.
Live 2D vector diagram
A vector diagram showing the velocity vector, its X and Y projections, the heading angle, and a readout panel — all updating with your inputs.
Component-class verdict
Each result is classified by |sin θ| from pure eastward through gentle, balanced, steep, to pure north/south motion, with a coloured badge.
Multi-unit hints
Each field shows the value converted to related units — km and ft for Δy; km/h, kn and mph for v; ms and min for Δt; rad and sin θ for the heading.
Sine response chart
Δy vs. θ for three distances, revealing the characteristic sine curve, plus an advanced Δy vs. Δt chart for three speeds.
Reference tables
A stage-by-stage Δy lookup grid for various headings and distances, plus a full variable reference table and unit conversion card.
Reference Tables & Conversions
A distance and angle converter, a Y-component lookup grid, and typical values for odometry updates.
Distance & speed converter
Y-Component Lookup — Δy = v · Δt · sin(θ)
Computed for v · Δt = 1 m (distance d = 1 m). Values in metres. Multiply by your own distance d.
| θ (deg) | sin θ | d = 0.5 m | d = 1 m | d = 2 m | d = 5 m | d = 10 m | d = 20 m |
|---|
Typical Odometry Update Values by Platform
Representative values at a mid-range sampling rate. Actual values depend on vehicle speed, control loop rate, and mission.
| Platform | Typical v | Typical Δt | d = v · Δt | Typical θ range | Notes |
|---|---|---|---|---|---|
| Differential-drive robot | 0.2–1.0 m/s | 20–50 ms | 4–50 mm | 0°–360° | Per control cycle, indoor |
| Ackermann ground robot | 0.5–3 m/s | 50–100 ms | 25–300 mm | −90°–+90° | Car-like, forward only |
| Autonomous car (city) | 5–15 m/s | 50–100 ms | 0.25–1.5 m | −45°–+45° | Lane following |
| Autonomous car (highway) | 25–33 m/s | 50–100 ms | 1.25–3.3 m | −5°–+5° | Gentle lane changes |
| Drone (hover, GPS-denied) | 0.5–5 m/s | 10–20 ms | 5–100 mm | 0°–360° | High-rate IMU |
| Marine vessel | 2–10 m/s | 1–10 s | 2–100 m | 0°–360° | Low-rate updates |
| Agricultural robot | 0.5–2 m/s | 100–500 ms | 50–1000 mm | −90°–+90° | Slow, high accuracy |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| Δy | Y (north) displacement | −10⁴ – 10⁴ | m |
| v | Forward speed | 0 – 50 | m/s |
| Δt | Time interval / sampling period | 0.001 – 60 | s |
| θ | Heading angle from +x (east) | −180 – +180 | deg (or rad) |
| d | Distance travelled = v · Δt | 0 – 10⁴ | m |
| Δx | X (east) displacement = d · cos θ | −10⁴ – 10⁴ | m |
What Does Δy = v · Δt · sin(θ) Measure?
The Y-component update relation answers a simple but essential question in odometry: given that a vehicle is moving at speed v along heading θ for a time Δt, how far has it moved north (or along the +y axis)? The answer is the product of three quantities: the distance travelled d = v · Δt and the sine of the heading angle. The first two together give the magnitude of the displacement vector; multiplying by sin θ extracts the portion that lies along y.
This is the 2D vector decomposition applied to motion. The displacement vector d has two orthogonal components, Δx = d · cos θ and Δy = d · sin θ, and together they satisfy the Pythagorean identity Δx² + Δy² = d². In any odometry implementation that tracks position in a plane — wheel odometry, visual odometry, or IMU-aided dead reckoning — both components are updated at every step. This calculator focuses on the Y component, which is the northward (or forward-along-axis) part of the motion.
Y-Component Formula
The defining relationship is:
Rearranged to solve for the other three quantities:
Δt = Δy / (v · sin θ)
θ = arcsin( Δy / (v · Δt) )
The complementary X-component is computed in the same way, using cosine instead of sine:
Together they give the full planar displacement, and Pythagoras guarantees that Δx² + Δy² = (v · Δt)² regardless of heading.
Y-Component Variables
| Symbol | Meaning | Notes |
|---|---|---|
| Δy | Y displacement | Positive = north, negative = south; in metres |
| v | Forward speed | Magnitude of the velocity vector, ≥ 0 |
| Δt | Time interval | Sampling period or turn duration |
| θ | Heading angle | Measured CCW from +x (east); 0° = east, 90° = north |
| d | Distance travelled | d = v · Δt; magnitude of the displacement vector |
| Δx | X displacement | Complementary component, d · cos θ |
How Each Variable Affects Δy
The relation is linear in v and Δt but sinusoidal in θ. Doubling the speed doubles Δy. Doubling the time interval doubles Δy. But changing the heading does not scale Δy proportionally — it traces out a sine curve. At θ = 0° (due east) Δy = 0, at θ = 90° (due north) Δy = d, and at θ = 180° (due west) Δy returns to 0 again. The maximum northward displacement is achieved when heading due north, and the maximum southward displacement when heading due south.
The sine curve also means that near θ = 0° or θ = 180°, small changes in heading produce very small changes in Δy — the vehicle is mostly moving along x. Near θ = 90°, small changes in heading produce very small changes in Δy too — but for the opposite reason: the vehicle is already almost entirely moving along y, and the derivative of sin θ is cos θ, which is small near the peak. The greatest sensitivity of Δy to heading occurs at θ = 0° and θ = 180°, where the derivative cos θ is ±1.
Y-Component Calculation Examples
Δy = 2 × 3 × sin(30°) = 6 × 0.5 = 3 m
v = 5 / (2 × sin 45°) = 5 / (2 × 0.7071) = 3.536 m/s
Δt = 4 / (2 × sin 60°) = 4 / (2 × 0.8660) = 2.309 s
sin θ = 3 / (2 × 3) = 0.5 → θ = arcsin(0.5) = 30° (or 150°)
Δy = 2 × 3 × sin 30° = 3 m
Δx = 2 × 3 × cos 30° = 5.196 m
d = √(3² + 5.196²) = 6 m ✓
Frequently Asked Questions
Because the Y-component is a projection of the total displacement vector onto the y-axis. If the vehicle moves distance d at heading θ, the northward part of that motion is d · sin θ. This is the same geometric operation as resolving any 2D vector into its components: the component along y is the magnitude times the sine of the angle from x. When θ = 90° (due north), sin θ = 1 and the entire displacement is along y; when θ = 0° (due east), sin θ = 0 and there is no y displacement.
d = v · Δt is the total distance travelled — the length of the displacement vector, always non-negative. Δy is the projection of that vector onto the y-axis, which can be positive (north) or negative (south). They are equal only when the vehicle is heading due north (θ = 90°), in which case sin θ = 1. In all other cases |Δy| < d, and the difference is accounted for by the X-component Δx.
Because the sine function is not one-to-one over a full revolution. For any value of sin θ in (0, 1), there are two angles in [0°, 360°) that produce it: θ itself (in the first quadrant) and 180° − θ (in the second quadrant). Both give the same Δy because sin(180° − θ) = sin θ. Geometrically this means a vehicle moving northeast and a vehicle moving northwest can have the same northward displacement. The calculator reports the principal value from arcsin (between −90° and +90°) and the note flags the second solution when relevant.
The equation has no solution. Since |sin θ| ≤ 1 for all real θ, the magnitude of Δy cannot exceed the distance travelled d = v · Δt. If you enter values that violate this — for example, Δy = 5 m with v = 1 m/s and Δt = 2 s so d = 2 m — the calculator flags an error. In a real odometry system this kind of inconsistency indicates a sensor fault, a time-stamp mismatch, or a data association error, and it should be treated as a signal to reject the update.
The two updates are complementary. Δx = v · Δt · cos θ gives the eastward displacement; Δy = v · Δt · sin θ gives the northward displacement. Together they satisfy Δx² + Δy² = (v · Δt)² for all θ — the Pythagorean identity. Every odometry implementation updates both at each step; this calculator focuses on the Y-component because it is often the more safety-critical one in automotive applications (lane-keeping, longitudinal control) while the X-component tends to dominate lateral error.
θ is measured counter-clockwise from the positive x-axis (east), following the standard mathematical convention. This means θ = 0° points east, θ = 90° points north, θ = 180° points west, and θ = −90° (or 270°) points south. This is the convention that makes Δx = d · cos θ and Δy = d · sin θ. Some navigation systems measure heading clockwise from north instead; in that case, convert by θmath = 90° − θnav. The calculator notes this in the heading field tooltip.
The formula Δy = v · Δt · sin θ assumes θ is constant over the interval, which is the standard discrete integration approximation. If the vehicle is turning, θ changes during Δt and the exact displacement is the integral of v(t) · sin θ(t) dt. For small Δt — a few tens of milliseconds — the error from this approximation is negligible and the update can be applied as-is. For larger intervals, split the interval into sub-steps and apply the update once per sub-step, updating θ between them. This is exactly what a real odometry loop does.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| delta_y | Change in y-position over the time step | m |
| v | Robot's linear velocity | m/s |
| delta_t | Time step duration | s |
| theta | Robot's current heading angle | degrees |
What it means
Similar to the x component, the incremental displacement in the y‑direction is v·Δt·sin(θ). Together with Δx and orientation update, it forms the odometry update equations. These are used to predict the robot’s pose between measurements. Understanding these helps in implementing navigation algorithms.
Worked example
Odometry Update (Y Component) – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| v (m/s) | 1 |
| Δt (s) | 0.1 |
| θ (deg) | 90 |
| Parameter | Value |
|---|---|
| v | 3 |
| Δt | 0.2 |
| θ | 30 |
Common mistakes
- Odometry position update (Y): Δy = v·Δt·sin(θ) – same considerations as X.
- Sign: θ measured from the x‑axis – positive counter‑clockwise.
- Units: Δy in metres.
- Assumes: No wheel slip and perfect encoders.
- Error accumulation: Odometry drifts over time due to integration of small errors.
Applications
Odometry position update (Y component), Δy = v·Δt·sin(θ), computes the change in the robot's y‑position over a time step. This complements the X component and is used in the same localisation framework. Engineers use it to maintain a continuous position estimate. Accurate odometry is crucial for autonomous navigation, especially when GPS is unavailable. This formula is also used in SLAM algorithms. Understanding both components is essential for implementing robot motion estimation.
- Dead‑reckoning localisation for ground robots
- Path tracking and feedback control
- Fusion with external position references
- Mapping and exploration in unknown environments
- Simulation and testing of motion models
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