# Optimal Experiment Design

Both [calibration](https://thanhndv212.github.io/figaroh-plus/tutorials/calibration_walkthrough/index.md) and [identification](https://thanhndv212.github.io/figaroh-plus/tutorials/identification_walkthrough/index.md) need measurement data — but *which* poses to measure and *which* trajectory to execute has a huge effect on how much data you need and how good the resulting parameters are. Measurement count alone does not establish observability. FIGAROH provides optimization-based configuration and trajectory design tools; their benefit must be assessed for the selected parameter scope, constraints and sensor model. Use [Plan, Fit and Validate](https://thanhndv212.github.io/figaroh-plus/example_workflow/index.md) to plan acquisition and independent validation together.

## Optimal configuration generation (for calibration)

**Goal:** pick the smallest set of robot poses that maximizes observability of the kinematic parameters.

Uses **D-optimal design**: select configurations that maximize the determinant of the Fisher information matrix built from the kinematic regressor `R`:

```
maximize:   det(Σ w_i · Rᵢᵀ Rᵢ)^(1/n)
subject to: Σ w_i ≤ 1,  w_i ≥ 0,  kinematic feasibility
```

Solved as a Second-Order Cone Program (SOCP) over candidate weights `w_i` for a pool of feasible candidate poses. Inspect the selected set's rank and information spectrum under the chosen model. Numerical design quality does not guarantee real measurement accuracy or recovery of parameters outside the observable subspace.

```
cd examples/tiago
python optimal_config.py --end-effector hey5
# → results/tiago_optimal_configurations_hey5.yaml
```

## Optimal trajectory generation (for identification)

**Goal:** generate a single smooth motion that "excites" all the dynamic coupling effects (inertial, Coriolis, gravitational, friction) needed to observe the dynamic parameters, while respecting every physical limit.

Formulated as a constrained optimization over cubic-spline waypoints:

```
minimize:   condition_number(W_base(trajectory))
subject to: joint position/velocity/acceleration limits
            joint torque/power limits
            self-collision avoidance
            C² trajectory smoothness
```

Solved with IPOPT (interior-point). Record convergence and verify which constraints the selected implementation enforces. Compare the candidate with a feasible baseline under the same scaling and assess the acquired data's held-out prediction; improved conditioning alone is insufficient evidence.

```
cd examples/tiago
python optimal_trajectory.py
# → candidate trajectory; check actual robot/acquisition constraints before execution
```

## Why this matters

Both problems reduce to the same idea: the [regressor](https://thanhndv212.github.io/figaroh-plus/tutorials/identification_walkthrough/index.md) or parameter Jacobian determines which directions the sampled model observes, under the declared measurement assumptions. A poorly chosen measurement set can remain rank-deficient despite many samples. Inspect conditioning alongside rank, scaling and the chosen design objective; D-optimal information and condition number are different criteria. The condition-number check in [`verify()`](https://thanhndv212.github.io/figaroh-plus/reporting_and_verification/index.md) is one diagnostic, not a complete experiment-quality or accuracy certificate.

## Next steps

- Run the [Calibration](https://thanhndv212.github.io/figaroh-plus/tutorials/calibration_walkthrough/index.md) or [Identification](https://thanhndv212.github.io/figaroh-plus/tutorials/identification_walkthrough/index.md) walkthrough using the configurations/trajectory generated here.
- [Examples Gallery](https://thanhndv212.github.io/figaroh-plus/examples/index.md) — `optimal_config.py` / `optimal_trajectory.py` are shipped for UR10 and TIAGo.