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# NVIDIA CORPORATION and its licensors retain all intellectual property
# and proprietary rights in and to this software, related documentation
# and any modifications thereto. Any use, reproduction, disclosure or
# distribution of this software and related documentation without an express
# license agreement from NVIDIA CORPORATION is strictly prohibited.
"""Helper functions for computing rigid body inertia properties.
"""
import warp as wp
import numpy as np
import math
from typing import List, Union
@wp.func
def triangle_inertia(
p: wp.vec3,
q: wp.vec3,
r: wp.vec3,
density: float,
com: wp.vec3,
# outputs
mass: wp.array(dtype=float, ndim=1),
inertia: wp.array(dtype=wp.mat33, ndim=1),
):
pcom = p - com
qcom = q - com
rcom = r - com
Dm = wp.mat33(pcom[0], qcom[0], rcom[0], pcom[1], qcom[1], rcom[1], pcom[2], qcom[2], rcom[2])
volume = wp.determinant(Dm) / 6.0
# accumulate mass
wp.atomic_add(mass, 0, 4.0 * density * volume)
alpha = wp.sqrt(5.0) / 5.0
mid = (com + p + q + r) / 4.0
off_mid = mid - com
# displacement of quadrature point from COM
d0 = alpha * (p - mid) + off_mid
d1 = alpha * (q - mid) + off_mid
d2 = alpha * (r - mid) + off_mid
d3 = alpha * (com - mid) + off_mid
# accumulate inertia
identity = wp.mat33(1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0)
I = wp.dot(d0, d0) * identity - wp.outer(d0, d0)
I += wp.dot(d1, d1) * identity - wp.outer(d1, d1)
I += wp.dot(d2, d2) * identity - wp.outer(d2, d2)
I += wp.dot(d3, d3) * identity - wp.outer(d3, d3)
wp.atomic_add(inertia, 0, (density * volume) * I)
return volume
@wp.kernel
def compute_solid_mesh_inertia(
# inputs
com: wp.vec3,
weight: float,
indices: wp.array(dtype=int, ndim=1),
vertices: wp.array(dtype=wp.vec3, ndim=1),
# outputs
mass: wp.array(dtype=float, ndim=1),
inertia: wp.array(dtype=wp.mat33, ndim=1),
volume: wp.array(dtype=float, ndim=1),
):
i = wp.tid()
p = vertices[indices[i * 3 + 0]]
q = vertices[indices[i * 3 + 1]]
r = vertices[indices[i * 3 + 2]]
vol = triangle_inertia(p, q, r, weight, com, mass, inertia)
wp.atomic_add(volume, 0, vol)
@wp.kernel
def compute_hollow_mesh_inertia(
# inputs
com: wp.vec3,
density: float,
indices: wp.array(dtype=int, ndim=1),
vertices: wp.array(dtype=wp.vec3, ndim=1),
thickness: wp.array(dtype=float, ndim=1),
# outputs
mass: wp.array(dtype=float, ndim=1),
inertia: wp.array(dtype=wp.mat33, ndim=1),
volume: wp.array(dtype=float, ndim=1),
):
tid = wp.tid()
i = indices[tid * 3 + 0]
j = indices[tid * 3 + 1]
k = indices[tid * 3 + 2]
vi = vertices[i]
vj = vertices[j]
vk = vertices[k]
normal = -wp.normalize(wp.cross(vj - vi, vk - vi))
ti = normal * thickness[i]
tj = normal * thickness[j]
tk = normal * thickness[k]
# wedge vertices
vi0 = vi - ti
vi1 = vi + ti
vj0 = vj - tj
vj1 = vj + tj
vk0 = vk - tk
vk1 = vk + tk
triangle_inertia(vi0, vj0, vk0, density, com, mass, inertia)
triangle_inertia(vj0, vk1, vk0, density, com, mass, inertia)
triangle_inertia(vj0, vj1, vk1, density, com, mass, inertia)
triangle_inertia(vj0, vi1, vj1, density, com, mass, inertia)
triangle_inertia(vj0, vi0, vi1, density, com, mass, inertia)
triangle_inertia(vj1, vi1, vk1, density, com, mass, inertia)
triangle_inertia(vi1, vi0, vk0, density, com, mass, inertia)
triangle_inertia(vi1, vk0, vk1, density, com, mass, inertia)
# compute volume
a = wp.length(wp.cross(vj - vi, vk - vi)) * 0.5
vol = a * (thickness[i] + thickness[j] + thickness[k]) / 3.0
wp.atomic_add(volume, 0, vol)
def compute_sphere_inertia(density: float, r: float) -> tuple:
"""Helper to compute mass and inertia of a solid sphere
Args:
density: The sphere density
r: The sphere radius
Returns:
A tuple of (mass, inertia) with inertia specified around the origin
"""
v = 4.0 / 3.0 * math.pi * r * r * r
m = density * v
Ia = 2.0 / 5.0 * m * r * r
I = wp.mat33([[Ia, 0.0, 0.0], [0.0, Ia, 0.0], [0.0, 0.0, Ia]])
return (m, wp.vec3(), I)
def compute_capsule_inertia(density: float, r: float, h: float) -> tuple:
"""Helper to compute mass and inertia of a solid capsule extending along the y-axis
Args:
density: The capsule density
r: The capsule radius
h: The capsule height (full height of the interior cylinder)
Returns:
A tuple of (mass, inertia) with inertia specified around the origin
"""
ms = density * (4.0 / 3.0) * math.pi * r * r * r
mc = density * math.pi * r * r * h
# total mass
m = ms + mc
# adapted from ODE
Ia = mc * (0.25 * r * r + (1.0 / 12.0) * h * h) + ms * (0.4 * r * r + 0.375 * r * h + 0.25 * h * h)
Ib = (mc * 0.5 + ms * 0.4) * r * r
I = wp.mat33([[Ia, 0.0, 0.0], [0.0, Ib, 0.0], [0.0, 0.0, Ia]])
return (m, wp.vec3(), I)
def compute_cylinder_inertia(density: float, r: float, h: float) -> tuple:
"""Helper to compute mass and inertia of a solid cylinder extending along the y-axis
Args:
density: The cylinder density
r: The cylinder radius
h: The cylinder height (extent along the y-axis)
Returns:
A tuple of (mass, inertia) with inertia specified around the origin
"""
m = density * math.pi * r * r * h
Ia = 1 / 12 * m * (3 * r * r + h * h)
Ib = 1 / 2 * m * r * r
I = wp.mat33([[Ia, 0.0, 0.0], [0.0, Ib, 0.0], [0.0, 0.0, Ia]])
return (m, wp.vec3(), I)
def compute_cone_inertia(density: float, r: float, h: float) -> tuple:
"""Helper to compute mass and inertia of a solid cone extending along the y-axis
Args:
density: The cone density
r: The cone radius
h: The cone height (extent along the y-axis)
Returns:
A tuple of (mass, inertia) with inertia specified around the origin
"""
m = density * math.pi * r * r * h / 3.0
Ia = 1 / 20 * m * (3 * r * r + 2 * h * h)
Ib = 3 / 10 * m * r * r
I = wp.mat33([[Ia, 0.0, 0.0], [0.0, Ib, 0.0], [0.0, 0.0, Ia]])
return (m, wp.vec3(), I)
def compute_box_inertia(density: float, w: float, h: float, d: float) -> tuple:
"""Helper to compute mass and inertia of a solid box
Args:
density: The box density
w: The box width along the x-axis
h: The box height along the y-axis
d: The box depth along the z-axis
Returns:
A tuple of (mass, inertia) with inertia specified around the origin
"""
v = w * h * d
m = density * v
Ia = 1.0 / 12.0 * m * (h * h + d * d)
Ib = 1.0 / 12.0 * m * (w * w + d * d)
Ic = 1.0 / 12.0 * m * (w * w + h * h)
I = wp.mat33([[Ia, 0.0, 0.0], [0.0, Ib, 0.0], [0.0, 0.0, Ic]])
return (m, wp.vec3(), I)
def compute_mesh_inertia(
density: float, vertices: list, indices: list, is_solid: bool = True, thickness: Union[List[float], float] = 0.001
) -> tuple:
"""Computes mass, center of mass, 3x3 inertia matrix, and volume for a mesh."""
com = wp.vec3(np.mean(vertices, 0))
num_tris = len(indices) // 3
# compute signed inertia for each tetrahedron
# formed with the interior point, using an order-2
# quadrature: https://www.sciencedirect.com/science/article/pii/S0377042712001604#br000040
# Allocating for mass and inertia
I_warp = wp.zeros(1, dtype=wp.mat33)
mass_warp = wp.zeros(1, dtype=float)
vol_warp = wp.zeros(1, dtype=float)
if is_solid:
weight = 0.25
alpha = math.sqrt(5.0) / 5.0
wp.launch(
kernel=compute_solid_mesh_inertia,
dim=num_tris,
inputs=[
com,
weight,
wp.array(indices, dtype=int),
wp.array(vertices, dtype=wp.vec3),
],
outputs=[mass_warp, I_warp, vol_warp],
)
else:
weight = 0.25 * density
if isinstance(thickness, float):
thickness = [thickness] * len(vertices)
wp.launch(
kernel=compute_hollow_mesh_inertia,
dim=num_tris,
inputs=[
com,
weight,
wp.array(indices, dtype=int),
wp.array(vertices, dtype=wp.vec3),
wp.array(thickness, dtype=float),
],
outputs=[mass_warp, I_warp, vol_warp],
)
# Extract mass and inertia and save to class attributes.
mass = float(mass_warp.numpy()[0] * density)
I = wp.mat33(*(I_warp.numpy()[0] * density))
volume = vol_warp.numpy()[0]
return mass, com, I, volume
def transform_inertia(m, I, p, q):
R = wp.quat_to_matrix(q)
# Steiner's theorem
return R @ I @ wp.transpose(R) + m * (wp.dot(p, p) * wp.mat33(np.eye(3)) - wp.outer(p, p))
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