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The dynamic realizer

The plan to replace the held-pose realizer Λqs with one that plans the motion with its contact dynamics and certifies what is executed.

NoteStatus

A plan. The architecture and its staging are adopted; none of it is built yet. The last section describes a direction under evaluation, not a decision. The full design is Chapter 16 of the technical report.

Why

Λqs (the local solver) plans a sequence of held poses and tracks them with a PD on the disk and a static feed-forward. The motion between the knots is left to the simulator. Four measured problems come back to that.

  • No executed motion is certified. The certificates (Certifying a whole box) cover plans and poses. Λqs’s measured tracking is 8.1° in tilt and 17 mm in position (95/95), against the certificate’s resolution of 0.25° × 0.5 mm. Nothing connects the certified plans to the executed motion, and the certified regions are narrow: the pivot holds the box only between 46° and 74° of tilt.
  • The chatter. While the box pivots on the disk, the step corner and the floor, both environment contacts slide. In MuJoCo a loaded sliding contact floats: its penetration falls with the slip speed and reaches zero at v*(N), about 8.8·N mm/s at the floor and 4.5·N mm/s at the step corner (N in newtons). The contact then drops out for a step while still loaded, the box falls about 1 mm and bounces, every 0.2–0.3 s. Keeping the slip below v*(N) needs the Coulomb forces of the motion, and Λqs’s hand-built disk path has no quasi-static Coulomb solution on the steep pivots. The imposed path, not the physics, is what has no solution.
  • The disk’s own force. The gate’s hold test leaves out the disk’s force. In one diagnosed call the approach drove the disk’s reference 16 mm into the box; the hold margin with the disk’s measured force turned negative while the gate’s stayed positive, and the box rolled over the step corner.
  • Knife-edge outcomes across processors. Replays on one machine are bit-exact. Across two processors none of 80 stratified calls is, and 2 of 80 flip between success and failure. The sensitive calls are near-critical: a tip-off moment shifted by a few steps, a disk sliding at its sticking limit, a landing at 90°. A realizer whose outcome hangs on such events cannot give a stable verdict.

The architecture

The search still calls the realizer once per edit. The new realizer has a certified part that carries the claim and an online part that does the tracking.

  • The B-tube: a certified tube per edit. Offline, the edit’s motion is planned in the formation’s reduced coordinates (on the pivot: the box’s tilt and the disk’s offset along the face), together with a feedback law and a tube around the plan inside which the planned contact modes provably hold. The ingredients:
    • contact modes fixed by the formation and the plan: the formation says which contacts are closed, the plan’s kinematics say whether each one sticks or slides; nothing is chosen online;
    • Coulomb forces: a sliding contact’s force lies on the edge of its cone that opposes the slip; the disk’s path is chosen together with its forces, not imposed;
    • slip below v*(N): every loaded sliding contact’s slip stays below a declared fraction of v*(N), with v* calibrated per contact pair from the simulator’s own contact model;
    • robust over μdisk ∈ [0.3, 0.5], with the interval machinery of the box certificate;
    • MJX’s deviations as bounded disturbances: the measured penetration, the sliding float, the per-step residual against the rigid model and the disk’s placement error enter the tube as bounded disturbances.
    The proposed tube method is interval reachability, with contraction as the fallback. The target is a theorem that extends Theorem S from certified plans to executed motion; it is a draft.
  • The A-tracker, inside the tube. A fixed-mode MPC: at every step it linearizes the contact dynamics about the measured state and the reference. Because the modes are fixed, each solve is a convex quadratic program. Its horizon runs across edit boundaries, so the motion does not stop between edits, and its reference forces are the Coulomb solution, so the disk’s force is planned with the box’s balance.
  • A shield between them. The executed command is the tube’s certified law plus the A-tracker’s deviation from it, clipped to a budget. Whatever the tracker proposes is a bounded input deviation, which the tube already covers.
  • A sampling-MPC baseline: sampling MPC in MJX over the formation chain, compared with the A-tracker, beside Λqs (the incumbent) and an end-to-end RL baseline.
  • Learning only as a clipped residual inside the tube. A learned residual on the A-tracker is clipped so that it is one more bounded disturbance; the tube then keeps the planned modes for every policy that respects the clip, a mode-preserving shield. Learning never carries the certificate. RL rollouts may also warm-start the offline plan, which is certified independently.

The staging

  1. The design note: Chapter 16 of the technical report.
  2. A, the A-tracker, beside Λqs and without the clip. Its tracking width at 95/95 is measured on the replayed calls against the certified regions, then on the 34 held-out searches. That width tells the B-tube how much it must absorb.
  3. B, the B-tube: the offline plan with robust statics, the interval tube, the certified law, composition across edits and the shield; first on the certified indirect lift.
  4. The A3 check: executed MJX motions against their tubes, with the disturbance bounds measured at 95/95.

The largest risk is the tube’s width: it must fit inside the certified margins for every friction in the band, and the indirect lift’s thinnest certified margin is 0.0029.

Under consideration: certificates on the graph

WarningA direction being evaluated, not a decision
  • A certified funnel graph. Each edge of the contact-formation graph would be a certified dynamic transition: a tube and a feedback law that take every state of the edge’s entry set into the next edge’s entry set, for every μdisk in the band. The search would compose such funnels, so a returned plan would be certified as executed, and edges would no longer need to hold the box at every instant: motions that use momentum would become plannable.
  • Learned certificates, exactly verified. A graph network over the contact graph would propose the ingredients of a certificate (the tube’s shape, feedback gains, a statics margin); an exact interval verifier accepts or refuses them, and its counterexamples return to training. Soundness would come from the verifier alone.

The literature survey finds both parts new as a combination, not as ideas: certified funnels through contact-mode switches exist, with fixed friction and without object–environment formation changes, and learner–verifier loops exist, without contact-mode or friction certificates.

Back to the overview