Velocity iterations

19 · Impacts and penetration

Boxes spawned 2 cm into the ground. Steel corrects penetration through a velocity bias β·φ/h, and that bias becomes real velocity. With β = 0.8 the box is launched. velocity_iterations re-solve the velocities without the bias after the positions are integrated, so the error is still removed but the box is not thrown.

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0.00 s

Cost

Time per step, CUDA graph replay, median of 7 interleaved rounds on one NVIDIA RTX PRO 6000 Blackwell Server Edition; ratios are against the first row. Solver allocation is the GPU memory the solver allocates at construction, before warm-up and graph capture. Scene: the 8-box stack in each world, 8 steps of 1/240 s per frame.

Option	1 world	64 worlds	1024 worlds	solver allocation, 1024 worlds	result in this tutorial
velocity_iterations=0	1.03 ms	1.21 ms	1.64 ms	215 MB	β 0.8: box hops 74 cm
velocity_iterations=4	1.27 ms 1.24×	1.47 ms 1.22×	2.00 ms 1.22×	217 MB	β 0.8: no hop

What to look at

baumgarte 0.2: a small hop of about 3 cm.
baumgarte 0.8: the box flies about 70 cm.
baumgarte 0.8 with 4 velocity iterations: no hop at all.

Key parameter

velocity_iterations

Velocity-only PGS iterations after the position solve. Default 0. baumgarte is the fraction of penetration removed per step, default 0.2.

Try changing

Use velocity_iterations=1.
Spawn the boxes 5 cm deep.

Concept

Baumgarte stabilization and velocity iterations

What it is

Penetration φ is corrected with a velocity bias of β·φ/h, removing a fraction β of the error per step. That bias becomes real velocity. velocity_iterations re-solve the velocities without the bias after positions are integrated, so the error is removed without leaving velocity behind.

target = −β · φ / h

In this scene

A box 2 cm deep hops 29 mm at β = 0.2 and 74 cm at β = 0.8; four velocity iterations remove the hop.

Considerations

Cost: 4 velocity iterations add about 25%: 1.0 → 1.3 ms per step in one world, 1.6 → 2.0 ms in 1024.
Interactions: Not allowed with sleeping, contact compliance or contact torsion; disables pre-elimination.
Use it when: Scenes with overlap at reset, or a raised β.

The script

cd tutorials && python 19_velocity_iterations.py prints each variant's final metric; --record out.json also writes the replay. Every tutorial imports replay.py from the same folder.

19_velocity_iterations.py
replay.py

# SPDX-FileCopyrightText: Copyright (c) 2026 Skild AI and the Steel contributors
# SPDX-License-Identifier: Apache-2.0

"""Penetration recovery: Baumgarte bias adds velocity; velocity_iterations remove it."""

import newton
import warp as wp
from newton.solvers import SolverSteel

from replay import Recorder

DT = 1.0 / 240.0
SUBSTEPS = 8


def build():
    builder = newton.ModelBuilder()
    builder.add_ground_plane()
    for i in range(3):
        # each box starts 2 cm into the ground
        box = builder.add_body(xform=wp.transform((0.25 * i - 0.25, 0.0, 0.03), wp.quat_identity()))
        builder.add_shape_box(
            box, hx=0.08, hy=0.08, hz=0.05, cfg=newton.ModelBuilder.ShapeConfig(mu=0.5)
        )
    return builder.finalize()


def simulate(rec, label, **options):
    model = build()
    solver = SolverSteel(model, **options)
    pipeline = newton.CollisionPipeline(model)
    contacts = pipeline.contacts()
    state_0, state_1, control = model.state(), model.state(), model.control()

    run = rec.variant(label, model)
    for _ in range(rec.frames(1.5)):
        run.frame(state_0, z_mm=1000.0 * (state_0.body_q.numpy()[1, 2] - 0.05))
        for _ in range(SUBSTEPS):
            pipeline.collide(state_0, contacts)
            solver.step(state_0, state_1, control, contacts, DT)
            state_0, state_1 = state_1, state_0
    run.check(solver)


if __name__ == "__main__":
    rec = Recorder()
    simulate(rec, "baumgarte=0.2 (default)")
    simulate(rec, "baumgarte=0.8", baumgarte=0.8)
    simulate(rec, "baumgarte=0.8, velocity_iterations=4", baumgarte=0.8, velocity_iterations=4)
    rec.save()

# SPDX-FileCopyrightText: Copyright (c) 2026 Skild AI and the Steel contributors
# SPDX-License-Identifier: Apache-2.0

"""Record Steel tutorial runs as replay JSON for the tutorials page, or print a summary.

Every tutorial imports this module. ``python NN_name.py`` runs the variants and prints
the final metric of each; ``--record out.json`` also writes body poses, shapes and metric
series that the page replays in 3D.
"""

import argparse
import base64
import json
import os
import subprocess
import sys
import time

import newton
import numpy as np
import warp as wp

from source_hash import source_sha256

_RENDERED_TYPES = {
    newton.GeoType.PLANE,
    newton.GeoType.SPHERE,
    newton.GeoType.CAPSULE,
    newton.GeoType.ELLIPSOID,
    newton.GeoType.CYLINDER,
    newton.GeoType.BOX,
    newton.GeoType.MESH,
    newton.GeoType.CONE,
    newton.GeoType.CONVEX_MESH,
}


class Recorder:
    """Collect the variants of one tutorial; ``fps`` is the replay frame rate."""

    def __init__(self, fps: int = 30):
        parser = argparse.ArgumentParser()
        parser.add_argument(
            "--record", metavar="OUT.json", help="write a replay file for the tutorials page"
        )
        parser.add_argument("--device", default=None, help="Warp device, for example cuda:0")
        self.args = parser.parse_args()
        if self.args.device:
            wp.set_device(self.args.device)
        self.fps = fps
        self.script = os.path.abspath(sys.argv[0])
        self.variants: list[Variant] = []

    def frames(self, seconds: float) -> int:
        """Number of replay frames in ``seconds``, including the first."""
        return int(round(seconds * self.fps)) + 1

    def variant(
        self,
        label: str,
        model: newton.Model,
        *,
        ghost: bool = False,
        slot: int | None = None,
        scene: str | None = None,
    ) -> "Variant":
        """Start one run; ``ghost`` overlays run ``slot``, ``scene`` groups runs for charts."""
        variant = Variant(label, model, self.fps, ghost, slot, scene)
        self.variants.append(variant)
        return variant

    def save(self) -> None:
        """Print a summary and, with ``--record``, write the replay file."""
        for variant in self.variants:
            last = variant.metrics[-1] if variant.metrics else {}
            flag = "  (rows dropped)" if variant.overflow else ""
            print(f"{variant.label:<48} {json.dumps(last)}  {variant.wall:.2f} s{flag}")
        if not self.args.record:
            return
        source_sha = source_sha256(self.script)
        doc = {
            "script": os.path.basename(self.script),
            "script_sha256": source_sha,
            "newton_commit": _newton_commit(),
            "newton_version": newton.__version__,
            "warp_version": wp.config.version,
            "gpu": wp.get_device().name,
            "fps": self.fps,
            "variants": [v.to_json() for v in self.variants],
        }
        with open(self.args.record, "w") as f:
            json.dump(doc, f, separators=(",", ":"))
        print(f"wrote {self.args.record}")


class Variant:
    """One recorded run: a pose per frame for every body, plus named metric values."""

    def __init__(self, label, model, fps, ghost, slot, scene):
        self.label, self.model, self.fps, self.ghost, self.slot = label, model, fps, ghost, slot
        self.scene = scene
        self.poses: list[np.ndarray] = []
        self.metrics: list[dict] = []
        self.awake: list[list[int]] = []
        self.particles: list[np.ndarray] = []
        self.overflow = False
        self.start = time.perf_counter()
        self.wall = 0.0

    def frame(self, state: newton.State, awake=None, particles=None, **metrics) -> None:
        """Record ``state.body_q``, optional awake flags, particle positions and metric values."""
        self.poses.append(state.body_q.numpy().astype(np.float32).copy())
        if particles is not None:
            self.particles.append(particles.numpy().astype(np.float32).copy())
        self.metrics.append({k: _round(v) for k, v in metrics.items()})
        if awake is not None:
            self.awake.append([int(x) for x in awake.numpy()])
        self.wall = time.perf_counter() - self.start

    def check(self, solver) -> None:
        """Note whether the solver dropped rows; call once after the run."""
        try:
            solver.check_constraint_capacity()
        except RuntimeError:
            self.overflow = True

    def to_json(self) -> dict:
        poses = (
            np.stack(self.poses)
            if self.poses
            else np.zeros((0, self.model.body_count, 7), np.float32)
        )
        return {
            "label": self.label,
            "fps": self.fps,
            "ghost": self.ghost,
            "slot": self.slot,
            "scene": self.scene,
            "bodies": int(self.model.body_count),
            "frames": int(poses.shape[0]),
            "poses": base64.b64encode(poses.astype(np.float32).tobytes()).decode(),
            "shapes": _shape_table(self.model),
            "metric": self.metrics,
            "awake": self.awake or None,
            "particles": self._particles_json(),
            "wall_s": round(self.wall, 3),
            "overflow": self.overflow,
        }

    def _particles_json(self):
        if not self.particles:
            return None
        frames = np.stack(self.particles)
        tris = self.model.tri_indices
        radius = self.model.particle_radius.numpy()
        return {
            "count": int(frames.shape[1]),
            "frames": base64.b64encode(frames.astype(np.float32).tobytes()).decode(),
            "radius": round(float(radius.mean()), 5) if radius.size else 0.0,
            "tris": [int(v) for v in tris.numpy().reshape(-1)]
            if tris is not None and tris.shape[0]
            else None,
        }


def _shape_table(model):
    types = model.shape_type.numpy()
    scales = model.shape_scale.numpy()
    xforms = model.shape_transform.numpy()
    bodies = model.shape_body.numpy()
    flags = model.shape_flags.numpy()
    shapes = []
    for i, t in enumerate(types):
        if int(t) not in _RENDERED_TYPES or not flags[i] & int(newton.ShapeFlags.VISIBLE):
            continue
        shape = {
            "b": int(bodies[i]),
            "t": int(t),
            "s": [round(float(v), 5) for v in scales[i]],
            "x": [round(float(v), 6) for v in xforms[i]],
        }
        if int(t) in (newton.GeoType.MESH, newton.GeoType.CONVEX_MESH):
            mesh = model.shape_source[i]
            shape["v"] = [round(float(v), 5) for v in np.asarray(mesh.vertices).reshape(-1)]
            shape["i"] = [int(v) for v in np.asarray(mesh.indices).reshape(-1)]
        shapes.append(shape)
    return shapes


def _round(value):
    value = float(value)
    return round(value, 5) if np.isfinite(value) else None


def _newton_commit():
    if os.environ.get("NEWTON_COMMIT"):
        return os.environ["NEWTON_COMMIT"]
    try:
        root = os.path.dirname(os.path.dirname(newton.__file__))
        return subprocess.check_output(["git", "-C", root, "rev-parse", "HEAD"], text=True).strip()
    except (OSError, subprocess.CalledProcessError):
        return None

Recorded with SolverSteel on Newton 1.9.0.dev0, Warp 1.19.0, NVIDIA RTX PRO 6000 Blackwell Server Edition. Replays read the state back every frame, so they are not a speed benchmark. Script · page text

Steel docs and tutorials are licensed under Apache-2.0. Third-party notices. "Skild", "Skild AI" and the Skild logo are trademarks of Skild AI and are not licensed under Apache-2.0.