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Robotics & Controls

From a 6-DOF manipulator to a Mars-transfer trajectory to a traffic grid — four shipping apps grounded in Newtonian mechanics, optionally corrected by GR, all at ≤ 0.1%.

Chapter 1 proved the operator picks work for physical simulation. Chapter 2 points those same operators at things that decide and act — arms, orbits, suspensions, intersections. The math algebra is identical (KO42 + Newtonian core). What changes is the cost function: you're not just simulating, you're optimising.

The four anchor apps​

AppProblemCore operatorsLive URL
Robotics LabForward/inverse kinematics, path planning, actuator simulationKO42 · NM19 · NM28 · NM29 (torque)/apps/robotics-lab/
Orbital PlannerHohmann transfers, gravity assists, trajectory optimisationKO42 · NM21 (gravity) · optional GR32, GR33/apps/orbital-planner/
Vehicle DynamicsSuspension, tyre modelling, handling simulationKO42 · NM19 · NM29 · NM30 (spring-damper)/apps/vehicle-dynamics/
Traffic OptimizerIntersection signal timing, congestion predictionKO42 · NM27 (flow conservation) · CS43 · CS47/apps/traffic-optimizer/

The math — what stays, what's new​

Everything you learned in Chapter 1 still holds. New ingredients this chapter introduces:

NM28  L = r × p                 (angular momentum — robot arm, satellite spin)
NM29  τ = r × F                 (torque — actuator sizing, wheel moment)
NM21  F = G m₁ m₂ / r²          (Newton gravity — orbital mechanics)
CS43  T(n) = O(n log n)         (path planning, signal schedule, convergence)
CS47  E(n) = −∑ p log p         (Shannon entropy — congestion uncertainty)
GR35  ∆t = ∆t₀ √(1 − 2GM/rc² − v²/c²)   (optional, for GPS-class precision)

All four apps still sit inside the Operator Limit (KO42 + 1–3 operators), and all four verify to ≤ 0.1% against a closed-form or published reference.

Runnable worked example — inverse kinematics for a 3-DOF arm​

Three links of length 1 m, target end-effector at (1.5, 1.0). Closed-form 2-solution inverse kinematics gives joint angles (θ₁, θ₂, θ₃).

The anonymous playground takes a domain plus named inputs and lets the seven-step wizard pick the operators (always KO42 + the domain fit). It returns a sealed envelope:

curl -s -X POST https://zeqsdk.com/api/playground/compute \
  -H "Content-Type: application/json" \
  -d '{
    "domain": "newtonian-mechanics",
    "inputs": { "length": 1.0, "target_x": 1.5, "target_y": 1.0 }
  }' | jq

The response carries value, unit, the operators the wizard chose, the equations it evaluated, and a zeqProof digest. Compare the returned value against your closed-form inverse-kinematics solution yourself — the platform hands you a result any node can recompute, not a printed figure to trust.

Seeds planted by this chapter​

  • Whole-body humanoid control at 1.287 Hz policy updates
  • Autonomous rendezvous and docking under combined NM21 + GR35
  • Self-stabilising cable-driven robots via NM30 tension networks
  • Hybrid air-ground traffic for urban-air-mobility corridors

Start here​

Middleware active. Kernel on the 1.287 Hz HulyaPulse. Awaiting next Zeqond.