Understanding Linear and Yaw Modes in a Gantry

Problem

A gantry beam driven by two parallel motors with two encoders is a coupled two-input two-output system in drive coordinates: every drive command moves both encoders. Designing two independent SISO loops on $u_1 \to y_1$ and $u_2 \to y_2$ ignores this coupling and can go unstable at gains that each loop individually would tolerate. How should the coordinates be chosen?

Likely Causes

The coupling is structural, not accidental:

• the beam connects both drives, so force at one end accelerates the other;
• the payload rarely sits at the geometric center, so the mass matrix is not diagonal in drive coordinates;
• beam flexibility adds frequency-dependent coupling on top of the rigid-body terms.

Measurement Method

Transform to sum/difference (linear/yaw) coordinates:

$$x_{lin} = \tfrac{1}{2}(y_1 + y_2), \qquad \theta_{yaw} = \tfrac{1}{L}(y_1 - y_2)$$

with the matching force transformation (total force for linear, force couple for yaw). Then measure the 2×2 FRF matrix in both coordinate frames and compare diagonal dominance: the frame with the smaller off-diagonal terms over the servo band is the better design frame.

Engineering Interpretation

For a symmetric gantry, linear and yaw are close to the true eigenmodes: the mass, damping and stiffness matrices are near-diagonal in these coordinates, so two SISO loop designs become legitimate. The transformation silently assumes (1) rigid encoder-to-mode geometry, (2) a known effective separation $L$, and (3) matched drive gains. Payload offset skews the mode shapes so the clean sum/difference is only an approximation — the residual shows up as off-diagonal FRF content that bounds the achievable independent bandwidths.

What to Test Next

• Measure the off-diagonal FRFs at several payload positions to map how mode separation degrades with offset.
• Verify drive-gain matching (a pure gain mismatch masquerades as constant coupling across frequency).
• Evaluate whether a static decoupling matrix identified from low-frequency data improves diagonal dominance enough, before considering dynamic decoupling.