vST for Robotics and Control Policies#

Example: Projection of a Manipulator Control Surface into Triadic Dimensional Cores#

This example demonstrates how a manipulator’s control‑policy latent state is projected from 1024D into the 9D → 6D → 3D triadic dimensional cores. It illustrates primitive‑level structure, interaction geometry, and projection stability during a grasp‑and‑lift task.

The goal is to provide a reproducible, invariant‑preserving demonstration of control‑surface projection.

1. Scenario Overview#

We assume:

• a 6‑DoF robotic arm
• a policy trained for grasp‑and‑lift
• latent states in the 512D–1024D range
• sensor inputs: joint encoders, wrist force‑torque, RGB‑D features
• action outputs: joint torques or velocity commands

The example is architecture‑agnostic.

2. Step 1 — Extract the 1024D Latent State#

At a given timestep ( t ), the policy produces:

[ C^{(t)} = [z_1, z_2, \dots, z_{1024}] ]

Observed Properties#

• stable DP/TDP structure during approach
• branching behavior during grasp closure
• dispersion during slip‑risk moments

3. Step 2 — Project 1024D → 9D (Coherence Projection)#

Preserves#

• regime identity
• resonance‑time behavior
• primitive‑level structure
• coherence‑surface continuity

Reveals#

• smooth surfaces during approach
• branching during grasp closure
• fragmentation during slip‑risk

Interpretation#

The 9D projection exposes the “coherence geometry” of the control surface.

4. Step 3 — Project 9D → 6D (Interaction Projection)#

Preserves#

• relational geometry across sensor channels
• coupling between force‑torque and joint states
• regime‑transition indicators

Reveals#

• force‑driven reorientation
• multi‑modal integration
• early instability signatures

5. Step 4 — Project 6D → 3D (Structural Projection)#

Preserves#

• motif‑level geometry
• temporal continuity
• stable structural invariants

Reveals#

• compact motifs during stable grasp
• oscillatory geometry during closure
• diffuse patterns during slip‑risk

6. Step 5 — Validate with vST Layers#

V₁: structural coherence stable except during slip‑risk#

V₂: dimensional continuity intact#

V₃: regime transitions substrate‑aligned#

V₄: core alignment stable across the task#

7. Step 6 — Drift Detection#

Drift categories:

• D₁ Structural Drift: moderate (slip‑risk)
• D₂ Dimensional Drift: none
• D₃ Regime Drift: moderate (R₃ᴴ onset)
• D₄ Projection Drift: none

8. Summary#

This example demonstrates:

• how a 1024D control surface is projected into triadic cores
• how interaction geometry reveals multi‑modal coupling
• how projection exposes instability during grasp closure
• how vST layers validate structural integrity
• how drift detection isolates slip‑risk behavior