# Analyze Uncertainty¶

## Deliverables¶

We have observed some uncertainties in the system. Now we like to understand what are the major sources of such errors.

Deliverables

```
1. List possible sources/reasons of discrepancy between the analytical model and hardware
2. Derive how can you mathematically characterize such sources
3. Consider how you alter ***hardware*** to reduce such gap
4. Consider how you alter ***Analytical Model*** to reduce such gap
5. Apply the modifications and compare results between before and after the changes
```

## List of possible uncertainty sources¶

- Wheel radius
- Servo motor angular velocities
- Your answer 1
- Your answer...

### Wheel radius (Example)¶

In the model, we assume that the wheel is perfectly circular, which is not a case with hand cut cardboard wheel.

The wheel is also not perfectly vertically to the ground.

### Servo motor angular velocities¶

Your brief explanations

### Your answer 1¶

Your brief explanations

### Your answer ...¶

Your brief explanations

## Mathematically Characterize the sources¶

Important!

You *DO NOT* need to derive an exact mathematical equation for this activity. (No need to derive a closed form solution)

We like to understand input/output relationship. Hence, you can do such as:

\(a = f(b)\) where a is ..., b is ...

You can refer to the example answer.

### Wheel radius (Example)¶

#### Mathematical representation¶

This effect can be captured mathematically by measuring *effective* radius of the wheel.

Left: ideal model. Middle: more realistic wheel model. Right: wheel tilting in the hardware

Wheel tilt (left) and comparisons between hand-cut cardboard wheel and laser cut wood wheel

where:

- \(\omega\) is the angular velocity of the wheel
- \(r\) is radius of the wheel
- \(V_{wheel}\) is the tangential velocity of the wheel
- \(r_{mean}\) is the mean radius of the hardware non-circular wheel
- \(\theta_{tilt}\) is the angle offset from the wheel vertical position

Hence,

\(r_{effective} = f(r_{mean}, \theta_{tilt})\)

\(V_{effective} = g(r_{effective}, \omega)\)

#### Assumption and limitation¶

This is assuming no slip and the wheel is still relatively circular. This is only considering a constant wheel tilt in one orientation.

If the wheel tilt is variable or 3 dimensional, then we need to derive effective wheel tilt, which will be a non-linear function by itself.

### Servo motor angular velocities¶

#### Mathematical representation¶

Your brief mathematical explanations

#### Assumption and limitation¶

Your brief explanations

### Your answer 1¶

#### Mathematical representation¶

Your brief mathematical explanations

#### Assumption and limitation¶

Your brief explanations

## Alternate Hardware Designs¶

### Make wheel circular and attach rigidly (Example)¶

### Your answer 1¶

Your brief explanations

## Alternate Analytical Model¶

### Use calculated or measured *effective* radius¶

### Your answer 1¶

Your brief explanations

## Apply the modifications¶

Apply one of the modifications *you* have suggested above.
Then redo one of the previous experiments to see if there is any improvement in fact.

Try your idea!

```
Let's try modifying hardware this for R4A workshop due to limited time.
```

### What have you changed?¶

Your brief explanations on what you changed

### Conduct experiments¶

Conduct at least one of the experiment again to see any improvement due to the change

### Results¶

#### Observation¶

Your brief explanations on what change you observed

#### Quantitative results¶

Your brief explanations on how much improvement it caused i.e. the linear velocity gap improved by xx %

### Did the alternation worth doing?¶

Your brief explanations on whether your modification worth doing compared to efforts necessary