Tension Field Dynamics & Control

Spatioz controls systems by propagating mathematical tension through a network. Rather than commanding absolute speeds or positions, it alters the "tension potential" of the connections, letting actuators flow naturally to the lowest energy state.

1. Tension Propagation

Let $T_{ij}$ denote the tension between node $i$ and node $j$. The tension is a function of the normalized distance $d(x_i, xj)$ and the configured weight $w{ij}$:

$$T{ij} = w{ij} \cdot \Phi(d(x_i, x_j))$$

Where $\Phi$ is the activation function. The net force $F_i$ acting on any single node $i$ is the vector sum of its connection tensions:

$$Fi = \sum{j \in \text{Neighbors}(i)} T{ij} \mathbf{u}{ij}$$

where $\mathbf{u}_{ij}$ is the unit vector pointing from $i$ to $j$ in the non-Euclidean manifold.

graph TD
    NodeA((Node A)) -- Tension T_AB --> Actuator((Actuator Node))
    NodeB((Node B)) -- Tension T_BC --> Actuator
    Obstacle((Obstacle)) -- Repulsion F_obs --> Actuator
    Actuator -->|Resulting Net Force| Homeostasis[Localized Motor Adjustment]

2. Shadow Variables (Interference Management)

To smooth transitions between opposing actions (e.g., accelerating vs. braking), Spatioz uses Shadow Variables. These variables are mathematically simulated inside the relational graph but correspond to no physical hardware. They act as filters or buffers:

• Primary Intents: Active physical outputs (e.g., Throttle, Steering).
• Shadow Intents: Inactive or buffer states (e.g., Coast, Stabilize).

When the primary intent increases, it must overcome the baseline tension of the shadow variable via the Relative Tension Distribution (RTD). This mimics inertia and physical damping, preventing jittery oscillations.