vST for Robotics and Control Policies#

Example: Latent‑Space Regime Shift During a Quadruped Gait Transition#

This example demonstrates how a control policy undergoes a latent‑space regime shift during a quadruped robot’s transition from a walk to a trot. It illustrates how high‑dimensional latent states evolve, how coherence surfaces deform, and how the vST substrate classifies regime transitions using the 1024D latent substrate.

The goal is to provide a reproducible, invariant‑preserving demonstration of regime behavior in embodied control‑policy dynamics.

1. Scenario Overview#

We assume:

• a quadruped robot controlled by a recurrent or attention‑based RL policy
• latent states in the 256D–1024D range
• sensor inputs: IMU, joint encoders, foot contacts
• action outputs: joint torques or target positions
• a gait transition triggered by velocity increase

The example is architecture‑agnostic and applies to any locomotion policy.

2. Step 1 — Extract Latent States Across Time#

At each timestep ( t ), the policy produces a latent vector:

[ L^{(t)} = [h_1^{(t)}, h_2^{(t)}, \dots, h_{1024}^{(t)}] ]

Observed Properties#

• early timesteps: compact, low‑variance latent structure
• mid‑transition: branching and oscillatory latent behavior
• late timesteps: new stable coherence surface

Interpretation#

The latent trajectory reflects the robot’s internal reorganization during the gait shift.

3. Step 2 — Identify Regime Behavior#

Using variance distribution, coherence‑surface continuity, and primitive‑level stability, classify each timestep’s regime.

Example Regime Timeline#

Interpretation#

The policy moves through a structured triadic sequence as the gait changes.

4. Step 3 — Project Latent States into 9D#

Project each 1024D latent state into the 9D coherence core.

Reveals#

• smooth surfaces during walking (R₁ᴴ)
• branching surfaces during transition (R₂ᴴ)
• fragmented surfaces during instability (R₃ᴴ)

Interpretation#

The 9D projection exposes the “shape” of the policy’s internal reorganization.

5. Step 4 — Project 9D → 6D → 3D#

6D Interaction Projection#

Shows:

• sensor‑to‑action coupling changes
• reorientation of balance‑related features
• early instability signatures

3D Structural Projection#

Shows:

• compact motifs in stable gaits
• oscillatory geometry during transition
• diffuse patterns during instability

6. Step 5 — Validate with vST Layers#

V₁: structural coherence preserved except during R₃ᴴ#

V₂: dimensional continuity intact#

V₃: regime transitions smooth and substrate‑aligned#

V₄: core alignment stable across the transition#

7. Step 6 — Drift Detection#

Drift categories:

• D₁ Structural Drift: moderate (instability window)
• D₂ Dimensional Drift: none
• D₃ Regime Drift: moderate (R₃ᴴ onset)
• D₄ Projection Drift: none

Interpretation#

The instability is expected and resolves cleanly.

8. Summary#

This example demonstrates:

• how latent‑space trajectories encode gait transitions
• how regime behavior evolves during reorientation
• how projection reveals coherence and instability
• how vST layers validate structural integrity
• how drift detection isolates transient instability