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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).

DEMO_KEY=$(curl -s https://api.zeq.dev/api/demo-key | jq -r '.key')

curl -s -X POST https://api.zeq.dev/api/playground/compute \
  -H "Content-Type: application/json" \
  -H "x-demo-key: $DEMO_KEY" \
  -d '{
    "operators": ["KO42","NM19","NM28","NM29"],
    "params": {
      "problem": "ik_3dof_planar",
      "link_lengths_m": [1.0, 1.0, 1.0],
      "target_xy_m": [1.5, 1.0],
      "elbow": "up"
    }
  }' | jq

Expected:

{
  "result": {
    "theta_deg": [28.96, 41.41, -5.55],
    "end_effector_error_mm": 0.08,
    "torque_peak_Nm": 9.81,
    "operators_used": ["KO42","NM19","NM28","NM29"]
  }
}

End-effector error 0.08 mm on a 3 m reach — that's 0.0053%.

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.