Blame for scripts/robotusd.py

"""Shared helpers to rebuild robot USDs whose joints and links match a URDF."""

import math
import xml.etree.ElementTree as ET

import numpy as np
from pxr import Gf, Sdf, Usd, UsdGeom, UsdPhysics, Vt

# ---------------------------------------------------------------- URDF


def _origin(el):
    xyz = [float(v) for v in (el.get("xyz", "0 0 0") if el is not None else "0 0 0").split()]
    rpy = [float(v) for v in (el.get("rpy", "0 0 0") if el is not None else "0 0 0").split()]
    cr, sr = math.cos(rpy[0]), math.sin(rpy[0])
    cp, sp = math.cos(rpy[1]), math.sin(rpy[1])
    cy, sy = math.cos(rpy[2]), math.sin(rpy[2])
    T = np.eye(4)
    T[:3, :3] = [
        [cy * cp, cy * sp * sr - sy * cr, cy * sp * cr + sy * sr],
        [sy * cp, sy * sp * sr + cy * cr, sy * sp * cr - cy * sr],
        [-sp, cp * sr, cp * cr],
    ]
    T[:3, 3] = xyz
    return T


def _rot(axis, angle):
    axis = np.asarray(axis, float)
    axis = axis / np.linalg.norm(axis)
    x, y, z = axis
    c, s, C = math.cos(angle), math.sin(angle), 1 - math.cos(angle)
    T = np.eye(4)
    T[:3, :3] = [
        [c + x * x * C, x * y * C - z * s, x * z * C + y * s],
        [y * x * C + z * s, c + y * y * C, y * z * C - x * s],
        [z * x * C - y * s, z * y * C + x * s, c + z * z * C],
    ]
    return T


def _trans(v):
    T = np.eye(4)
    T[:3, 3] = v
    return T


def urdf_joints(path):
    root = ET.parse(path).getroot()
    joints = []
    for j in root.findall("joint"):
        axis = j.find("axis")
        lim = j.find("limit")
        mimic = j.find("mimic")
        joints.append(
            {
                "name": j.get("name"),
                "type": j.get("type"),
                "parent": j.find("parent").get("link"),
                "child": j.find("child").get("link"),
                "origin": _origin(j.find("origin")),
                "axis": [float(v) for v in axis.get("xyz").split()] if axis is not None else [1, 0, 0],
                "limit": (float(lim.get("lower", 0)), float(lim.get("upper", 0))) if lim is not None else None,
                "mimic": (mimic.get("joint"), float(mimic.get("multiplier", 1)), float(mimic.get("offset", 0)))
                if mimic is not None
                else None,
            }
        )
    links = [l.get("name") for l in root.findall("link")]
    return links, joints


def urdf_fk(path, q):
    """World pose of every link for joint values `q` (radians / meters)."""
    links, joints = urdf_joints(path)
    by_child = {j["child"]: j for j in joints}
    out = {}

    def value(j):
        if j["mimic"]:
            name, mult, off = j["mimic"]
            return q.get(name, 0.0) * mult + off
        return q.get(j["name"], 0.0)

    def world(link):
        if link in out:
            return out[link]
        j = by_child.get(link)
        if j is None:
            out[link] = np.eye(4)
            return out[link]
        T = world(j["parent"]) @ j["origin"]
        if j["type"] in ("revolute", "continuous"):
            T = T @ _rot(j["axis"], value(j))
        elif j["type"] == "prismatic":
            T = T @ _trans(np.asarray(j["axis"], float) * value(j))
        out[link] = T
        return T

    for l in links:
        world(l)
    return out


# ---------------------------------------------------------------- USD kinematics


def np_matrix(m):
    """Gf (row vectors) to numpy (column vectors)."""
    return np.array(m, dtype=float).T


def _quat(q):
    im = q.GetImaginary()
    return Gf.Quatd(q.GetReal(), Gf.Vec3d(im[0], im[1], im[2]))


def _frame(pos, rot):
    T = np.eye(4)
    T[:3, :3] = np.array(Gf.Matrix3d(Gf.Rotation(_quat(rot))), dtype=float).T
    T[:3, 3] = [pos[0], pos[1], pos[2]]
    return T


AXES = {"X": [1, 0, 0], "Y": [0, 1, 0], "Z": [0, 0, 1]}


def usd_model(stage):
    cache = UsdGeom.XformCache()
    bodies, joints = {}, []
    for p in stage.Traverse():
        if p.HasAPI(UsdPhysics.RigidBodyAPI):
            bodies[str(p.GetPath())] = np_matrix(cache.GetLocalToWorldTransform(p))
    for p in stage.Traverse():
        if not p.IsA(UsdPhysics.Joint):
            continue
        j = UsdPhysics.Joint(p)
        b0 = j.GetBody0Rel().GetTargets()
        b1 = j.GetBody1Rel().GetTargets()
        kind = "fixed"
        axis, limit = None, None
        if p.IsA(UsdPhysics.RevoluteJoint):
            kind, s = "revolute", UsdPhysics.RevoluteJoint(p)
        elif p.IsA(UsdPhysics.PrismaticJoint):
            kind, s = "prismatic", UsdPhysics.PrismaticJoint(p)
        if kind != "fixed":
            axis = s.GetAxisAttr().Get()
            limit = (s.GetLowerLimitAttr().Get(), s.GetUpperLimitAttr().Get())
        # A joint may target a prim under its rigid body (a mesh): the body is the nearest ancestor
        # with RigidBodyAPI, and the joint frame is relative to the targeted prim.
        def owner(targets, local_frame):
            if not targets:
                return None, local_frame
            prim = stage.GetPrimAtPath(targets[0])
            body = prim
            while body and body.IsValid() and str(body.GetPath()) not in bodies:
                body = body.GetParent()
            if not body or not body.IsValid() or body.IsPseudoRoot():
                return str(targets[0]), local_frame
            offset = np.linalg.inv(bodies[str(body.GetPath())]) @ np_matrix(cache.GetLocalToWorldTransform(prim))
            return str(body.GetPath()), offset @ local_frame

        body0, J0 = owner(b0, _frame(j.GetLocalPos0Attr().Get(), j.GetLocalRot0Attr().Get()))
        body1, J1 = owner(b1, _frame(j.GetLocalPos1Attr().Get(), j.GetLocalRot1Attr().Get()))
        joints.append(
            {
                "name": p.GetName(),
                "path": str(p.GetPath()),
                "type": kind,
                "body0": body0,
                "body1": body1,
                "J0": J0,
                "J1": J1,
                "axis": axis,
                "limit": limit,
            }
        )
    return bodies, joints


def usd_fk(model, q):
    """World pose of every body for joint values `q` (degrees / meters), keyed by body path."""
    bodies, joints = model
    # A joint to a prim that isn't a body (or to nothing) anchors its body at rest.
    by_body1 = {j["body1"]: j for j in joints if j["body0"] in bodies and j["body1"]}
    out = {}

    def world(b):
        if b in out:
            return out[b]
        j = by_body1.get(b)
        if j is None:
            out[b] = bodies[b]
            return out[b]
        M = np.eye(4)
        v = q.get(j["name"], 0.0)
        if j["type"] == "revolute":
            M = _rot(AXES[j["axis"]], math.radians(v))
        elif j["type"] == "prismatic":
            M = _trans(np.asarray(AXES[j["axis"]], float) * v)
        out[b] = world(j["body0"]) @ j["J0"] @ M @ np.linalg.inv(j["J1"])
        return out[b]

    for b in bodies:
        world(b)
    return out


def check_against_urdf(stage, urdf, samples=200, seed=0):
    """Pose USD and URDF with the same random joint values and compare every link.

    USD bodies are matched to URDF links by name. Each link may differ from the URDF by a constant
    offset at rest (a different frame convention); after that, motion must agree exactly.
    """
    model = usd_model(stage)
    bodies, joints = model
    links, ujoints = urdf_joints(urdf)
    name_of = {p.rsplit("/", 1)[1]: p for p in bodies}
    # Frames that aren't bodies (tips) ride on their nearest ancestor body.
    cache = UsdGeom.XformCache()
    riders = {}
    for p in stage.Traverse():
        if p.GetName() in links and p.GetName() not in name_of and p.IsA(UsdGeom.Xformable):
            anc = p.GetParent()
            while anc and str(anc.GetPath()) not in bodies:
                anc = anc.GetParent()
            if anc:
                b = str(anc.GetPath())
                riders[p.GetName()] = (b, np.linalg.inv(bodies[b]) @ np_matrix(cache.GetLocalToWorldTransform(p)))
    def fk_all(model, q):
        out = usd_fk(model, q)
        for name, (b, rel) in riders.items():
            out["rider:" + name] = out[b] @ rel
        return out

    name_of.update({n: "rider:" + n for n in riders})
    bodies = {**bodies, **{"rider:" + n: bodies[b] @ rel for n, (b, rel) in riders.items()}}
    shared = [l for l in links if l in name_of]
    movable = [j for j in ujoints if j["type"] in ("revolute", "prismatic", "continuous") and not j["mimic"]]
    mimics = [j for j in ujoints if j["mimic"]]
    rest_u = urdf_fk(urdf, {})
    rest_s = fk_all(model, {})
    ref = shared[0]
    A = rest_s[name_of[ref]] @ np.linalg.inv(rest_u[ref])
    D = {l: np.linalg.inv(A @ rest_u[l]) @ rest_s[name_of[l]] for l in shared}
    rest_err = max(np.abs(rest_s[p] - bodies[p]).max() for p in bodies)
    rng = np.random.default_rng(seed)
    worst_pos, worst_rot = 0.0, 0.0
    for _ in range(samples):
        qu = {}
        for j in movable:
            lo, hi = j["limit"] if j["limit"] else (-math.pi, math.pi)
            qu[j["name"]] = rng.uniform(lo, hi)
        for j in mimics:
            name, mult, off = j["mimic"]
            qu[j["name"]] = qu[name] * mult + off
        qs = {k: (math.degrees(v) if any(j["name"] == k and j["type"] != "prismatic" for j in ujoints) else v) for k, v in qu.items()}
        fu = urdf_fk(urdf, qu)
        fs = fk_all(model, qs)
        for l in shared:
            expected = A @ fu[l] @ D[l]
            got = fs[name_of[l]]
            worst_pos = max(worst_pos, np.linalg.norm(expected[:3, 3] - got[:3, 3]))
            R = expected[:3, :3].T @ got[:3, :3]
            worst_rot = max(worst_rot, math.degrees(math.acos(max(-1, min(1, (np.trace(R) - 1) / 2)))))
    frame_offsets = {
        l: (round(float(np.linalg.norm(D[l][:3, 3])) * 1000, 2), round(math.degrees(math.acos(max(-1, min(1, (np.trace(D[l][:3, :3]) - 1) / 2)))), 2))
        for l in shared
    }
    return {
        "links_missing": [l for l in links if l not in name_of],
        "joints_missing": [j["name"] for j in ujoints if j["type"] != "fixed" and j["name"] not in {k["name"] for k in joints}],
        "rest_consistency_m": float(rest_err),
        "worst_pos_mm": worst_pos * 1000,
        "worst_rot_deg": worst_rot,
        "frame_offsets_mm_deg": frame_offsets,
    }


# ---------------------------------------------------------------- composition and namespace edits


def flatten(path, variants=None):
    """One layer holding the composed stage, every instance turned into its own prims."""
    stage = Usd.Stage.Open(path)
    stage.SetEditTarget(stage.GetSessionLayer())
    for name, selection in (variants or {}).items():
        stage.GetDefaultPrim().GetVariantSet(name).SetVariantSelection(selection)
    while True:
        instances = [p for p in stage.Traverse() if p.IsInstance()]
        if not instances:
            break
        for p in instances:
            p.SetInstanceable(False)
    return stage.Flatten(addSourceFileComment=False)


def _remap_list(listop, fn):
    for field in ("explicitItems", "prependedItems", "appendedItems", "addedItems", "deletedItems", "orderedItems"):
        items = list(getattr(listop, field))
        mapped = [fn(i) for i in items]
        mapped = [m for m in mapped if m is not None]
        if mapped != items:
            setattr(listop, field, mapped)


def remap_targets(layer, fn):
    """Applies `fn(path) -> path | None` to every relationship target and attribute connection."""

    def visit(path):
        if not path.IsPropertyPath():
            return
        spec = layer.GetObjectAtPath(path)
        if isinstance(spec, Sdf.RelationshipSpec):
            _remap_list(spec.targetPathList, fn)
        elif isinstance(spec, Sdf.AttributeSpec):
            _remap_list(spec.connectionPathList, fn)

    layer.Traverse(Sdf.Path.absoluteRootPath, visit)


def rename(layer, path, new_name):
    old = Sdf.Path(path)
    new = old.ReplaceName(new_name)
    edit = Sdf.BatchNamespaceEdit()
    edit.Add(Sdf.NamespaceEdit.Rename(old, new_name))
    if not layer.Apply(edit):
        raise RuntimeError(f"rename {old} -> {new_name}")
    remap_targets(layer, lambda p: p.ReplacePrefix(old, new) if p.HasPrefix(old) else p)
    return str(new)


def remove(layer, path):
    path = Sdf.Path(path)
    edit = Sdf.BatchNamespaceEdit()
    edit.Add(Sdf.NamespaceEdit.Remove(path))
    if not layer.Apply(edit):
        raise RuntimeError(f"remove {path}")
    remap_targets(layer, lambda p: None if p.HasPrefix(path) else p)


def add_frame(stage, parent_path, name, world):
    """An Xform under `parent_path` placed at `world` (numpy, column vectors)."""
    parent = stage.GetPrimAtPath(parent_path)
    parent_world = np_matrix(UsdGeom.XformCache().GetLocalToWorldTransform(parent))
    local = np.linalg.inv(parent_world) @ world
    x = UsdGeom.Xform.Define(stage, f"{parent_path}/{name}")
    x.AddTransformOp().Set(Gf.Matrix4d(local.T.tolist()))
    return x


# ---------------------------------------------------------------- geometry


def crease_normals(points, faces, crease_deg=30.0):
    """Splits vertices along edges sharper than `crease_deg` and returns per-vertex normals."""
    a, b, c = (points[faces[:, k]] for k in range(3))
    fn = np.cross(b - a, c - a)
    area = np.linalg.norm(fn, axis=1)
    keep = area > 1e-20
    faces, fn, area = faces[keep], fn[keep], area[keep]
    unit = fn / area[:, None]
    m = len(faces)
    corner_vertex = faces.reshape(-1)
    corner_face = np.repeat(np.arange(m), 3)
    order = np.argsort(corner_vertex, kind="stable")
    sorted_v = corner_vertex[order]
    starts = np.flatnonzero(np.r_[True, sorted_v[1:] != sorted_v[:-1]])
    ends = np.r_[starts[1:], len(sorted_v)]
    cos_t = math.cos(math.radians(crease_deg))
    corner_normal = np.zeros((3 * m, 3))
    for s, e in zip(starts, ends):
        corners = order[s:e]
        fs = corner_face[corners]
        u = unit[fs]
        similar = (u @ u.T) > cos_t
        corner_normal[corners] = similar.astype(float) @ fn[fs]
    corner_normal /= np.maximum(np.linalg.norm(corner_normal, axis=1, keepdims=True), 1e-30)
    q = np.round(corner_normal * 1024).astype(np.int32)
    key = np.concatenate([corner_vertex[:, None], q], axis=1)
    unique, first, inverse = np.unique(key, axis=0, return_index=True, return_inverse=True)
    new_points = points[unique[:, 0]]
    new_normals = corner_normal[first]
    new_faces = inverse.reshape(-1).reshape(m, 3)
    return new_points, new_faces, new_normals


def weld(points, faces):
    unique, inverse = np.unique(points, axis=0, return_inverse=True)
    return unique, inverse.reshape(-1)[faces]


def author_mesh(stage, path, points, faces, normals, material=None):
    mesh = UsdGeom.Mesh.Define(stage, path)
    mesh.CreatePointsAttr(Vt.Vec3fArray.FromNumpy(points.astype(np.float32)))
    mesh.CreateFaceVertexCountsAttr(Vt.IntArray.FromNumpy(np.full(len(faces), 3, dtype=np.int32)))
    mesh.CreateFaceVertexIndicesAttr(Vt.IntArray.FromNumpy(faces.reshape(-1).astype(np.int32)))
    mesh.CreateNormalsAttr(Vt.Vec3fArray.FromNumpy(normals.astype(np.float32)))
    mesh.SetNormalsInterpolation(UsdGeom.Tokens.vertex)
    mesh.CreateSubdivisionSchemeAttr(UsdGeom.Tokens.none)
    mesh.CreateExtentAttr(Vt.Vec3fArray([Gf.Vec3f(*points.min(0)), Gf.Vec3f(*points.max(0))]))
    if material is not None:
        from pxr import UsdShade

        UsdShade.MaterialBindingAPI.Apply(mesh.GetPrim()).Bind(UsdShade.Material(stage.GetPrimAtPath(material)))
    return mesh


def compare_visual_bounds(source, out, renames=None):
    """Worst corner offset (mm) between each body's drawn geometry in `source` and in `out`."""
    renames = renames or {}

    def bounds(stage):
        cache = UsdGeom.BBoxCache(Usd.TimeCode.Default(), ["default"], useExtentsHint=False)
        found = {}
        for p in stage.Traverse(Usd.TraverseInstanceProxies()):
            if p.HasAPI(UsdPhysics.RigidBodyAPI):
                # Only this body's own meshes, not those of bodies nested under it.
                lo, hi = np.full(3, np.inf), np.full(3, -np.inf)
                it = iter(Usd.PrimRange(p, Usd.TraverseInstanceProxies()))
                for q in it:
                    if q != p and q.HasAPI(UsdPhysics.RigidBodyAPI):
                        it.PruneChildren()
                        continue
                    if q.IsA(UsdGeom.Mesh) and UsdGeom.Imageable(q).ComputePurpose() == "default":
                        r = cache.ComputeWorldBound(q).ComputeAlignedRange()
                        lo, hi = np.minimum(lo, r.GetMin()), np.maximum(hi, r.GetMax())
                if np.isfinite(lo).all():
                    found[renames.get(p.GetName(), p.GetName())] = (lo, hi)
        return found

    a, b = bounds(source), bounds(out)
    return {k: round(float(max(np.abs(a[k][0] - b[k][0]).max(), np.abs(a[k][1] - b[k][1]).max())) * 1000, 2) if k in b else "missing" for k in a}


# ---------------------------------------------------------------- URDF to USD


def read_stl(path):
    """Points and triangles of a binary STL."""
    data = open(path, "rb").read()
    n = int(np.frombuffer(data, np.uint32, 1, 80)[0])
    tris = np.frombuffer(data, np.dtype([("n", "<f4", 3), ("v", "<f4", (3, 3)), ("a", "<u2")]), n, 84)
    return tris["v"].reshape(-1, 3).astype(float), np.arange(3 * n).reshape(n, 3)


def srgb_to_linear(c):
    c = np.asarray(c, float)
    return np.where(c <= 0.04045, c / 12.92, ((c + 0.055) / 1.055) ** 2.4)


def _axis_frame(axis):
    """The USD axis token for a URDF axis along X, Y or Z, and the rotation turning that token into it."""
    a = np.asarray(axis, float)
    i = int(np.argmax(np.abs(a)))
    if not np.allclose(np.abs(a), np.eye(3)[i]):
        raise ValueError(f"axis {axis} is not along X, Y or Z")
    R = np.eye(4) if a[i] > 0 else _rot(np.eye(3)[(i + 1) % 3], math.pi)
    return "XYZ"[i], R


def _set_matrix(xformable, M):
    xformable.ClearXformOpOrder()
    xformable.AddTransformOp().Set(Gf.Matrix4d(M.T.tolist()))


def _pose(M):
    q = Gf.Matrix4d(M.T.tolist()).ExtractRotationQuat()
    im = q.GetImaginary()
    return Gf.Vec3f(*M[:3, 3]), Gf.Quatf(q.GetReal(), Gf.Vec3f(im[0], im[1], im[2]))


def from_urdf(urdf, name, colors=None):
    """A stage drawing the URDF's visual meshes, with a rigid body per link a moving joint drives.

    Links fixed to another are frames under its body, so every link keeps the URDF's name and frame;
    joints are UsdPhysics joints (limits in degrees), the root link at the origin. `colors` maps a
    link to an sRGB color replacing its URDF material.
    """
    from pxr import UsdShade

    root_el = ET.parse(urdf).getroot()
    mesh_dir = urdf.rsplit("/", 1)[0] if "/" in urdf else "."
    links, joints = urdf_joints(urdf)
    by_child = {j["child"]: j for j in joints}
    rest = urdf_fk(urdf, {})
    named_colors = {m.get("name"): [float(v) for v in m.find("color").get("rgba").split()[:3]] for m in root_el.findall("material") if m.find("color") is not None}

    stage = Usd.Stage.CreateInMemory()
    UsdGeom.SetStageUpAxis(stage, UsdGeom.Tokens.z)
    UsdGeom.SetStageMetersPerUnit(stage, 1.0)
    UsdPhysics.SetStageKilogramsPerUnit(stage, 1.0)
    robot = UsdGeom.Xform.Define(stage, f"/{name}")
    stage.SetDefaultPrim(robot.GetPrim())
    Usd.ModelAPI(robot.GetPrim()).SetKind("component")
    UsdGeom.Scope.Define(stage, f"/{name}/Materials")
    UsdGeom.Scope.Define(stage, f"/{name}/Geometry")
    UsdGeom.Scope.Define(stage, f"/{name}/Joints")

    materials = {}

    def material(rgb):
        key = tuple(round(c, 5) for c in rgb)
        if key not in materials:
            path = f"/{name}/Materials/material_{len(materials)}"
            mat = UsdShade.Material.Define(stage, path)
            shader = UsdShade.Shader.Define(stage, f"{path}/PreviewSurface")
            shader.CreateIdAttr("UsdPreviewSurface")
            shader.CreateInput("diffuseColor", Sdf.ValueTypeNames.Color3f).Set(Gf.Vec3f(*srgb_to_linear(rgb)))
            shader.CreateInput("metallic", Sdf.ValueTypeNames.Float).Set(0.0)
            shader.CreateInput("roughness", Sdf.ValueTypeNames.Float).Set(0.5)
            mat.CreateSurfaceOutput().ConnectToSource(shader.ConnectableAPI(), "surface")
            materials[key] = path
        return materials[key]

    # A link is a body unless a fixed joint holds it to its parent.
    def body_of(link):
        j = by_child.get(link)
        return link if j is None or j["type"] != "fixed" else body_of(j["parent"])

    paths = {}
    for link in links:
        body = body_of(link)
        if body == link:
            paths[link] = f"/{name}/Geometry/{link}"
            x = UsdGeom.Xform.Define(stage, paths[link])
            _set_matrix(x, rest[link])
            UsdPhysics.RigidBodyAPI.Apply(x.GetPrim())
            if by_child.get(link) is None:
                UsdPhysics.ArticulationRootAPI.Apply(x.GetPrim())
    for link in links:
        body = body_of(link)
        if body != link:
            paths[link] = f"{paths[body]}/{link}"
            _set_matrix(UsdGeom.Xform.Define(stage, paths[link]), np.linalg.inv(rest[body]) @ rest[link])

    for el in root_el.findall("link"):
        link = el.get("name")
        inertial = el.find("inertial")
        if inertial is not None and body_of(link) == link:
            mass = UsdPhysics.MassAPI.Apply(stage.GetPrimAtPath(paths[link]))
            mass.CreateMassAttr(float(inertial.find("mass").get("value")))
            mass.CreateCenterOfMassAttr(Gf.Vec3f(*_origin(inertial.find("origin"))[:3, 3]))
        for k, visual in enumerate(el.findall("visual")):
            mesh_el = visual.find("geometry/mesh")
            if mesh_el is None:
                raise ValueError(f"{link}: only mesh visuals are supported")
            points, faces = read_stl(f"{mesh_dir}/{mesh_el.get('filename')}")
            scale = [float(v) for v in mesh_el.get("scale", "1 1 1").split()]
            T = _origin(visual.find("origin"))
            points = (T[:3, :3] @ (points * scale).T).T + T[:3, 3]
            points, faces = weld(points, faces)
            points, faces, normals = crease_normals(points, faces)
            mat_el = visual.find("material")
            rgb = (colors or {}).get(link)
            if rgb is None and mat_el is not None:
                color_el = mat_el.find("color")
                rgb = [float(v) for v in color_el.get("rgba").split()[:3]] if color_el is not None else named_colors[mat_el.get("name")]
            author_mesh(stage, f"{paths[link]}/visual" + (f"_{k}" if k else ""), points, faces, normals, material(rgb) if rgb else None)

    for j in joints:
        if j["type"] == "fixed":
            continue
        kind = {"revolute": UsdPhysics.RevoluteJoint, "continuous": UsdPhysics.RevoluteJoint, "prismatic": UsdPhysics.PrismaticJoint}[j["type"]]
        joint = kind.Define(stage, f"/{name}/Joints/{j['name']}")
        token, R = _axis_frame(j["axis"])
        parent = body_of(j["parent"])
        frame = rest[j["child"]] @ R
        pos0, rot0 = _pose(np.linalg.inv(rest[parent]) @ frame)
        pos1, rot1 = _pose(R)
        joint.CreateBody0Rel().SetTargets([paths[parent]])
        joint.CreateBody1Rel().SetTargets([paths[j["child"]]])
        joint.CreateLocalPos0Attr(pos0)
        joint.CreateLocalRot0Attr(rot0)
        joint.CreateLocalPos1Attr(pos1)
        joint.CreateLocalRot1Attr(rot1)
        joint.CreateAxisAttr(token)
        if j["type"] != "continuous" and j["limit"]:
            lo, hi = j["limit"]
            if j["type"] == "revolute":
                lo, hi = math.degrees(lo), math.degrees(hi)
            joint.CreateLowerLimitAttr(lo)
            joint.CreateUpperLimitAttr(hi)
        if j["mimic"]:
            # PhysX's mimic constraint, position + gearing * reference + offset = 0, so a `<mimic>` joint
            # still follows its joint without the URDF.
            reference, multiplier, offset = j["mimic"]
            if offset:
                raise NotImplementedError(f"{j['name']}: a mimic offset needs its units settled first")
            dof = ("trans" if j["type"] == "prismatic" else "rot") + token
            prim = joint.GetPrim()
            prim.AddAppliedSchema(f"PhysxMimicJointAPI:{dof}")
            prim.CreateRelationship(f"physxMimicJoint:{dof}:referenceJoint").SetTargets([f"/{name}/Joints/{reference}"])
            prim.CreateAttribute(f"physxMimicJoint:{dof}:gearing", Sdf.ValueTypeNames.Float).Set(-multiplier)
    return stage