Joint springs and damping

36 · Joints and drives

Four hinged arms without gravity, started at 60° with −1 rad/s. The first is free. The others have a passive spring toward rest_q = 0 (stiffness 6 N·m/rad) with no, light or heavy viscous damping. Springs and damping are applied as explicit joint forces, like SolverFeatherstone, independently of drives and limits.

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What to look at

The free arm keeps turning.
The undamped spring oscillates forever.
Light damping rings down, and heavy damping returns to rest without overshoot.

Key parameter

stiffness / rest_q / damping

Joint arguments, applied explicitly: springs stay stable for roughly dt·√(k/m) < 2, and damping while dt·d·λmax(H⁻¹) < 2, where λmax(H⁻¹) is 1/I for one link but much larger for chains of light links. For stiff springs or heavy damping on light links use target_ke and target_kd, which Steel integrates implicitly.

Try changing

Set stiffness to 600 and see the explicit limit.
Use target_ke instead for the same spring, applied implicitly.

Concept

Explicit springs and damping

What it is

Joint stiffness and damping are applied as forces from the step's starting state, like SolverFeatherstone. Explicit forces stay stable only while they are weak relative to the inertia they act on.

τ = k (q_rest − q) − d q̇

In this scene

The undamped spring oscillates at constant amplitude; damping 0.3 rings down; damping 2 returns without overshoot.

Considerations

Stability: Keep h·√(k/m) < 2 and h·d·λmax(H⁻¹) < 2. A six-link chain of 0.1 kg capsules has λmax(H⁻¹) ≈ 4×10⁴, so at 240 Hz damping above about 0.012 diverges; while recording, damping 0.05 on it went to NaN within 0.06 s. SolverFeatherstone has the same limit; MuJoCo treats joint damping implicitly.
Use it when: Soft passive springs. For stiff springs or strong damping on light links use target_ke and target_kd, which Steel integrates implicitly.

The script

cd tutorials && python 36_springs_damping.py prints each variant's final metric; --record out.json also writes the replay. Every tutorial imports replay.py from the same folder.

36_springs_damping.py
replay.py

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

"""Passive joint springs (stiffness, rest_q) and viscous joint damping."""

import math

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

from replay import Recorder

DT = 1.0 / 240.0
SUBSTEPS = 8
JOINTS = [(0.0, 0.0), (6.0, 0.0), (6.0, 0.3), (6.0, 2.0)]  # (stiffness N m/rad, damping N m s/rad)


def tf(x, y, z):
    return wp.transform((x, y, z), wp.quat_identity())


def build():
    builder = newton.ModelBuilder(gravity=(0.0, 0.0, 0.0))
    for i, (stiffness, damping) in enumerate(JOINTS):
        y = 0.3 * i - 0.45  # each arm turns in its own xz plane
        link = builder.add_link(xform=tf(0.2, y, 0.4))
        builder.add_shape_box(
            link, hx=0.2, hy=0.02, hz=0.02, cfg=newton.ModelBuilder.ShapeConfig(density=400.0)
        )
        builder.add_shape_sphere(
            link,
            xform=tf(0.2, 0.0, 0.0),
            radius=0.04,
            cfg=newton.ModelBuilder.ShapeConfig(density=2000.0),
        )
        j = builder.add_joint_revolute(
            -1,
            link,
            axis=newton.Axis.Y,
            parent_xform=tf(0.0, y, 0.4),
            child_xform=tf(-0.2, 0.0, 0.0),
            stiffness=stiffness,
            damping=damping,
            rest_q=0.0,
        )
        builder.add_articulation([j])
    model = builder.finalize()
    model.joint_q.fill_(math.radians(60.0))
    model.joint_qd.fill_(-1.0)
    return model


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

    run = rec.variant(label, model)
    for _ in range(rec.frames(5.0)):
        q = np.degrees(state_0.joint_q.numpy())
        run.frame(state_0, **{f"k{k:g}_d{d:g}": q[i] for i, (k, d) in enumerate(JOINTS)})
        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, "free | spring | spring + light damping | spring + heavy damping")
    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.