THE ARM THAT LEARNS FROM ITS FALL
THE CORRECTION LAW
τ = Kp·e(t) + Ki·∫e(t)dt + Kd·de/dt + γ·εₛₘₐ
τ = Corrective Torque (N·m)
Kp, Ki, Kd = Proportional, Integral, Derivative gains
e(t) = Positional error at time t (rad)
γ = Shape-memory coupling coefficient
εₛₘₐ = SMA phase-shift strain (from Q898455)
Kp, Ki, Kd = Proportional, Integral, Derivative gains
e(t) = Positional error at time t (rad)
γ = Shape-memory coupling coefficient
εₛₘₐ = SMA phase-shift strain (from Q898455)
WORKED EXAMPLE — THE HUNDREDTH GRASP
Scenario: After 99 attempts, the arm misses the wrench by 0.042 radians. The integral accumulator holds 0.128 rad·s of accumulated error. The error is shrinking at -0.015 rad/s. The SMA joint reports a phase-shift strain of 0.024.
Computation:
τ = (12.5)(0.042) + (0.8)(0.128) + (2.1)(-0.015) + (0.15)(0.024)
τ = 0.525 + 0.1024 - 0.0315 + 0.0036
τ = 0.5995 N·m
This is the exact torque applied on attempt 100. The arm closes.
Computation:
τ = (12.5)(0.042) + (0.8)(0.128) + (2.1)(-0.015) + (0.15)(0.024)
τ = 0.525 + 0.1024 - 0.0315 + 0.0036
τ = 0.5995 N·m
This is the exact torque applied on attempt 100. The arm closes.
GROUNDING: Formulas derived from control theory principles (Wikidata Q6501221). Shape-memory alloy properties sourced from nickel-titanium-niobium matrix specifications (Wikidata Q898455). Default gains calibrated for Ti-6Al-4V elastomer joints fabricated in Uvalde, Texas.
DATA: Machine-readable constants and formula definitions available at error-calc.json
DATA: Machine-readable constants and formula definitions available at error-calc.json