4.1 Plant Design Variables

Time-independent physical decisions¶

The vector $p\in\mathcal P$ collects physical quantities selected during design and held fixed during one operating trajectory. Dimensions, material properties, component ratings, passive parameters, and placement coordinates are typical examples. “Time independent” does not mean immutable over the product lifecycle; it means that $p$ is not commanded at every instant of the modeled event.

For the electromechanical positioner, use

where $\ell$ is link length, $m_\ell$ link mass, $P_m$ motor rating, and $N$ gear ratio. These variables enter several models simultaneously:

Thus a motor rating is not merely an actuator limit. It also changes mass, thermal capacity, cost, rotor inertia, and sometimes efficiency. A gear ratio changes reflected inertia, available output torque, speed range, and mechanical losses. Good CCD formulations preserve every coupling that could alter the preferred design.

Independent variables versus parameterizations¶

Do not optimize redundant physical quantities independently. If mass is determined by geometry and density, either calculate it or introduce an equality constraint that enforces consistency. A lower-dimensional parameterization generally improves conditioning and interpretability.

Bounds define the candidate design domain:

Bounds should reflect manufacturing, catalogs, packaging, and model validity—not merely values that help the solver.

Activity 4.1: audit the plant vector¶