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Exponential tracking kernels go flat exactly when the error is largest - pair them with an L2 term for the far field

A lesson the Coach cites as exp-kernel-needs-l2-far-field.
Mechanism understoodReward design

Never let an exp/Gaussian kernel be the only tracking pressure on a quantity that can drift far from target: pair it with an unbounded (L2) term sized as the "don't diverge" floor, and check which frame the kernel reads.

Symptom

With only an exp-type yaw tracking term (exp(-err/std^2), std 0.25), a robot whose heading had drifted badly received almost no corrective gradient: at error 0.6 rad/s the term evaluates to exp(-0.36/0.0625) = 0.003 - near zero AND flat.

Context

The exp kernel is excellent for fine tracking near zero error but its gradient vanishes at large error - precisely when correction matters most. Fix: add track_ang_vel_z_err_l2 (-0.5), a plain quadratic on the same quantity: "exp 管精细跟踪、L2 管'别发散', 互补". Both terms deliberately read WORLD-frame wz (matching the exp term's source), because this torso sways enough that body-frame wz means are systematically off (measured -0.039 while actually turning +0.152). The same far-field-gradient argument reappears in the v8 risk list: frozen joints could not climb back because their huge error put them on the exp plateau ("远端梯度消失是冻结自锁的帮凶").

Change

Added the L2 companion term at -0.5 alongside the existing exp term (a term that had been in an earlier draft and was lost in a rewrite - itself worth noticing).

Outcome

Corrective pressure restored across the whole error range; the exp+L2 pairing became the house pattern for tracking terms.

Mechanism

d/de[exp(-e^2/s^2)] -> 0 as e grows: the kernel saturates and cannot distinguish bad from terrible. A quadratic's gradient grows with error, covering the far field; summing the two yields monotone corrective pressure with fine shaping near the target.

Applies when

  • tracking rewards use exp/Gaussian kernels alone
  • a drifted or frozen state fails to recover during training
  • designing tracking terms for quantities with large transient errors
“exp 在误差大时梯度趋零, 恰好在最需要纠正的时候失灵。… 误差 0.6 → exp(-0.36/0.0625) = 0.003, 接近零且平坦。… exp 管精细跟踪、L2 管"别发散", 互补。”
train/WALK_V7_SPEC.md § ② track_ang_vel_z_err_l2 −0.5 —— 补 exp 的梯度洞