Benchmarks

Residual SAC under model mismatch (Pendulum-v1)

The plant's mass is 1.4; the MPC's Crocoddyl model keeps the nominal 1.0 — a 40% model error. All arms are evaluated on the same 15 fixed episodes; SAC arms use 3 seeds (band = min–max).

Arm	0 steps	3k steps	15k steps
MPC alone (nominal model)	−629	−629	−629
Residual SAC over MPC	−738	−406	−273
SAC from scratch	−1452	−832	−236

Two claims the figure supports: the residual agent never passes through the catastrophic exploration phase (from-scratch SAC sits near −1450 for its first ~2,500 steps — on hardware, that phase is crashes), and it beats the mismatched MPC from ~2k steps on.

Reproduce with benchmark/residual_pendulum/ (about 5 CPU-minutes per arm); full configuration and honest caveats — including why there is no oracle line and why residual authority matters — in that directory's README.

Expert distillation

BC student (2×64 MLP) cloned from the MPC expert; 15 shared eval seeds:

Arm	mean	worst	latency / action
MPC expert	−320	−1498	0.6 ms
BC student (10k pairs)	−423	−1519	24 µs (25×)
BC + 1 DAgger round	−558	−1576	24 µs

Naive MSE-DAgger hurts here: the bang-bang expert gives multimodal labels on student-visited states, and regression averages them into useless mid-torques. Details: benchmark/distill_pendulum/.

Policy warm start

The BC student seeds the MPC's cold solves (PolicyWarmStartMPC):

Arm	mean	worst	upright	iters/step
plain MPC	−320	−1498	14/15	2.04
policy seed (cold)	−373	−1492	13/15	1.95
policy seed (every solve)	−772	−1816	9/15	2.64
policy seed + best-of-two	−320	−1498	14/15	2.00

Seeding is basin selection: it rescues the start that traps plain MPC and loses a different one. Best-of-two reveals the deeper truth — from a near-hanging start the hanging plan genuinely has lower open-loop cost over 1.5 s, so no warm start can fix what is really objective myopia. Details: benchmark/warmstart_pendulum/.

Learned terminal cost

Cost-to-go V(x) as the terminal node — the fix for that myopia:

Arm	mean	worst	upright
H=30 plain	−320	−1498	14/15
H=8 plain	−1536	−1614	0/15
H=8 + V(MC-1500)	−414	−970	14/15
H=8 + V*(value iteration)	−434	−790	15/15
H=30 + V*(VI)	−649	−1390	13/15

An 8-step MPC with a learned terminal out-robusts the 30-step MPC — and V helps only as a replacement for the missing tail (at H=30 it adds bias, not information). Implementation traps (cost-unit consistency, float32 finite-difference Hessians, stationary targets, input normalization) are documented in benchmark/terminal_pendulum/.

Quadruped balance under payload mismatch (Go2)

The residual pattern at robot scale: whole-body Crocoddyl MPC (four feet in rigid contact, 37-dimensional state, 12 torques, ~3 ms per solve at 50 Hz) on the MuJoCo menagerie Go2 whose trunk mass is doubled — about the rated payload, unmodeled.

Arm	return
MPC, true-mass model (oracle)	−2.49
Residual SAC over nominal MPC	−3.50 (best seed −2.69)
MPC, nominal model	−5.67
SAC from scratch	−12.79 (starts at −92)

Two findings that differ from the pendulum, documented in benchmark/quadruped_balance/: residual authority is a stability hazard on a platform with unrecoverable states (0.3 authority collapsed evaluation to −149 before recovering; 0.1 works), and per-step rewards that are small against SAC's entropy bonus quietly pay the policy to stay noisy — training rewards are scaled ×100, evaluation is not. Also: Crocoddyl's ShootingProblem.quasiStatic returned uninitialized memory for this contact problem; blendmpc.envs.go2.quasi_static_torque computes static torques analytically with Pinocchio instead.

Quadruped trot under overload (Go2)

The locomotion milestone: CrocoddylCyclicMPC executes a trot-in-place gait (periodic contact schedule advanced with circularAppend) while the plant carries 3× its trunk mass unmodeled.

Arm	return
Residual SAC over nominal trot MPC	−5.81
Trot MPC, true-mass model (oracle)	−11.79
Trot MPC, nominal model	−24.12

The residual passes the true-model controller at ~10k steps: the remaining error is contact timing and soft-contact effects that a rigid-contact OCP cannot represent, but a learned correction absorbs. Also documented in benchmark/quadruped_trot/: the gait-degeneration trap (a soft swing cost produces a "trot" that never lifts its feet — assert on measured contacts, not plans) and a double oracle inversion (the informed model is worse at 2× payload, better at 3×).

Distilling the trot MPC (Go2)

A phase-conditioned MLP clone of the gait MPC (benchmark/quadruped_distill/):

Arm	return	survival	FL airborne	latency
Trot MPC (expert)	−6.23	5/5	22%	4.0 ms
BC student	−6.75	5/5	22%	28 µs (142×)

The student reproduces the gait itself — identical airborne fraction — not just the pose. Phase conditioning is essential: without it the regression target is multivalued and the student collapses to weight-shifting. A horizon ablation (in the trot benchmark README) explains why there is no learned-terminal-cost benchmark at robot scale: with an imposed contact schedule, an 8-node window trots as well as a full-cycle one, and the balance task is better at horizon 2 than 25 — terminal value functions fix discovered long-horizon structure, not scheduled structure.