A Correct Detection Can Already Be Out of Date

State Estimation Connects Measurements Across Time

BOTIE

A sensor reports where something was when the measurement was taken. The object can keep moving while an image is transferred and a detector processes it. Treating the answer as a measurement of the present position mixes two different times.

Kalman's 1960 work on filtering and prediction described a changing system through its state and a rule for how that state evolves. The estimator could carry its prediction forward and combine it with observations, accounting for uncertainty. That made the interval between observations part of the model rather than an unmentioned gap.

We will begin with the prediction part alone. A conveyor moves at a supplied constant velocity, so Botie can advance an old position by the time that has passed. This is a short-interval teaching approximation. It does not estimate unknown velocity or perform the full Kalman measurement correction introduced later.

Sources for this history

The Camera Saw the Part Before the Belt Moved It

Botie detects a part on a moving conveyor. By the time the prediction arrives, the belt has moved. The coordinate conversion is correct, but it describes where the part was.

A short-term motion model can estimate the current position. We need to state its assumptions and decide when an observation is too old to use.

Constant velocity and valid nonnegative ages are assumed in the circuit. Python adds a maximum-age rejection rule.

Step 1 of 1

The estimate refers to capture time

All values use the robot frame.

The predicted displacement is [0.2,0] metres.
Quantity	Value
Captured position	[1,2] metres
Velocity	[0.5,0] metres/second
Observation age	0.4 seconds

Captured position
Value
[1,2] metres
Velocity
Value
[0.5,0] metres/second
Observation age
Value
0.4 seconds

The predicted displacement is [0.2,0] metres.

The capture timestamp belongs to the measurement

The propagation makes the motion assumption explicit.

Move the estimate to the action time

A delayed detector result describes the part’s earlier position.

BOTIE

A delayed detector result describes the part’s earlier position.

Choose an example

Captured position

1

2

Velocity in the same frame

1

2

Elapsed seconds

Position propagated to the action time

01
12

Follow the operations

1. Captured position
1. 01
2. 12
2. Velocity in the same frame
1. 00.5
2. 10
3. Elapsed seconds
```
0
```
4. Multiply
1. 00
2. 10
5. Add
1. 01
2. 12
6. Result
1. 01
2. 12

Constant velocity and valid nonnegative ages are assumed in the circuit. Python adds a maximum-age rejection rule.

1. The capture timestamp belongs to the measurement

Latency includes sensor readout, inference, queues and transport. Use capture time rather than the time the response happened to arrive.

Convert position and velocity into a common frame. Adding a camera-frame velocity to a robot-frame position is another coordinate mismatch.

A clock error can look like a motion error. Timestamp quality is part of the measurement contract.

2. Predict displacement over the elapsed interval

Under constant velocity, displacement is velocity multiplied by elapsed time. Add it to the captured position.

age = action time - capture time
displacement = [0.5,0] × 0.4 = [0.2,0]
predicted position = [1,2] + [0.2,0] = [1.2,2]

Negative age is invalid under this shared-clock contract. Excessively old observations should be rejected, not extrapolated indefinitely.

Velocity supplies the displacement

The two entries remain separate coordinate components.

Python

import numpy as np
velocity = np.array([-.5,1.])
print((velocity*.4).tolist())

Output

[-0.2, 0.4]

This is a displacement in metres for the specified interval, not an absolute position.

Check yourself

How old is the observation?

Question 1 of 2

What is the propagated position?

Use the information below.

Position 1 m, velocity 0.5 m/s, age 0.4 s.

3. Motion uncertainty grows while we wait

Acceleration, slipping belts and changing velocity make the approximation less reliable over longer intervals. A tracker would propagate uncertainty as well as position.

Choose a maximum age from task requirements and validation. If a record is too old, the application can reacquire a measurement or stop the action path.

A maximum-age check bounds exposure to stale evidence; it does not prove the remaining prediction is accurate.

4. Test timing independently of geometry

Zero age should preserve the captured position. Reversing velocity should reverse displacement. Changing only delay should move the estimate proportionally under this model.

Then test real timing traces and motion changes outside the toy model. A constant-velocity unit test cannot establish safe interception of physical objects.

The Python function adds a freshness policy to the numerical propagation built in the workshop.