Multi-Point Alignment System

The driver includes an alignment engine based on Singular Value Decomposition (SVD). This system fits a mathematical model of the mount to observed star positions.

How it Works

The alignment process solves the Orthogonal Procrustes Problem: finding a rotation matrix \(R\) that maps “sky” unit vectors \(\{s_i\}\) to “mount” (encoder) unit vectors \(\{m_i\}\).

The driver minimizes the squared error: $\(\text{minimize} \sum w_i \| R s_i - m_i \|^2\)$

Characteristics

  • Robustness: Handles \(N \ge 2\) points and is mathematically stable.

  • RMS Feedback: Calculates the Root Mean Square (RMS) Error in arcseconds.

  • Local Bias: Allows weighting points near the current position to compensate for local mechanical errors.

Operation

1. Initial Alignment

  1. Set Coord Set Mode to SYNC.

  2. Center a star in the eyepiece or camera.

  3. Issue a Sync command from the client software.

  4. Repeat for 2-3 stars distributed across the sky.

2. Refinement

You can add points throughout a session. Performing a Sync after a plate solve will update the model.

3. Grid Thinning

To maintain sky coverage, the driver uses a grid-based algorithm:

  • The sky is divided into 15° x 15° sectors.

  • Each sector holds a maximum of 2 points.

  • If a sector is full, the driver keeps the points with the lowest residuals.

4. Model Complexity

The complexity scales with the number of points:

  • 1-2 Stars: Rotation only (SVD).

  • 3-5 Stars: 4-parameter model (Rotation + Zero Point Offsets).

  • 6+ Stars: 6-parameter geometric model, compensating for:

    • Cone Error (\(CH\)): Non-perpendicularity between OTA and Dec axis.

    • NP (\(NP\)): Non-perpendicularity between axes.

    • Altitude Index (\(ID\)): Zero-point offset in the Altitude axis.