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9d86bb203e750f239c07dcf307ae79c5043da738
solver/tests/test_joints.py
forbes-0023 8e521b4519 fix(solver): use all 3 cross-product components to avoid XY-plane singularity 
The parallel-normal constraints (ParallelConstraint, PlanarConstraint,
ConcentricConstraint, RevoluteConstraint, CylindricalConstraint,
SliderConstraint, ScrewConstraint) and point-on-line constraints
previously used only the x and y components of the cross product,
dropping the z component.

This created a singularity when both normal vectors lay in the XY
plane: a yaw rotation produced a cross product entirely along Z,
which was discarded, making the constraint blind to the rotation.

Fix: return all 3 cross-product components. The Jacobian has a
rank deficiency at the solution (3 residuals, rank 2), but the
Newton solver handles this correctly via its pseudoinverse.

Similarly, point_line_perp_components now returns all 3 components
of the displacement cross product to avoid singularity when the
line direction aligns with a coordinate axis.
2026-02-22 15:51:59 -06:00

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"""Integration tests for kinematic joint constraints.
These tests exercise the full solve pipeline (constraint → residuals →
pre-pass → Newton / BFGS) for multi-body systems with various joint types.
"""
import math
import pytest
from kindred_solver.constraints import (
BallConstraint,
CoincidentConstraint,
CylindricalConstraint,
GearConstraint,
ParallelConstraint,
PerpendicularConstraint,
PlanarConstraint,
PointInPlaneConstraint,
PointOnLineConstraint,
RackPinionConstraint,
RevoluteConstraint,
ScrewConstraint,
SliderConstraint,
UniversalConstraint,
)
from kindred_solver.dof import count_dof
from kindred_solver.entities import RigidBody
from kindred_solver.newton import newton_solve
from kindred_solver.params import ParamTable
from kindred_solver.prepass import single_equation_pass, substitution_pass
ID_QUAT = (1, 0, 0, 0)
# 90° about Z: (cos(45°), 0, 0, sin(45°))
c45 = math.cos(math.pi / 4)
s45 = math.sin(math.pi / 4)
ROT_90Z = (c45, 0, 0, s45)
# 90° about Y
ROT_90Y = (c45, 0, s45, 0)
# 90° about X
ROT_90X = (c45, s45, 0, 0)
def _solve(bodies, constraint_objs):
"""Run the full solve pipeline. Returns (converged, params, bodies)."""
pt = bodies[0].tx # all bodies share the same ParamTable via Var._name
# Actually, we need the ParamTable object. Get it from the first body.
# The Var objects store names, but we need the table. We'll reconstruct.
# Better approach: caller passes pt.
raise NotImplementedError("Use _solve_with_pt instead")
def _solve_with_pt(pt, bodies, constraint_objs):
"""Run the full solve pipeline with explicit ParamTable."""
all_residuals = []
for c in constraint_objs:
all_residuals.extend(c.residuals())
quat_groups = []
for body in bodies:
if not body.grounded:
all_residuals.append(body.quat_norm_residual())
quat_groups.append(body.quat_param_names())
all_residuals = substitution_pass(all_residuals, pt)
all_residuals = single_equation_pass(all_residuals, pt)
converged = newton_solve(
all_residuals, pt, quat_groups=quat_groups, max_iter=100, tol=1e-10
)
return converged, all_residuals
def _dof(pt, bodies, constraint_objs):
"""Count DOF for a system."""
all_residuals = []
for c in constraint_objs:
all_residuals.extend(c.residuals())
for body in bodies:
if not body.grounded:
all_residuals.append(body.quat_norm_residual())
all_residuals = substitution_pass(all_residuals, pt)
return count_dof(all_residuals, pt)
# ============================================================================
# Single-joint DOF counting tests
# ============================================================================
class TestJointDOF:
"""Verify each joint type removes the expected number of DOF.
Setup: ground body + 1 free body (6 DOF) with a single joint.
"""
def _setup(self, pos_b=(0, 0, 0), quat_b=ID_QUAT):
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, pos_b, quat_b)
return pt, a, b
def test_ball_3dof(self):
"""Ball joint: 6 - 3 = 3 DOF (3 rotation)."""
pt, a, b = self._setup()
constraints = [BallConstraint(a, (0, 0, 0), b, (0, 0, 0))]
assert _dof(pt, [a, b], constraints) == 3
def test_revolute_1dof(self):
"""Revolute: 6 - 5 = 1 DOF (rotation about Z)."""
pt, a, b = self._setup()
constraints = [RevoluteConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)]
assert _dof(pt, [a, b], constraints) == 1
def test_cylindrical_2dof(self):
"""Cylindrical: 6 - 4 = 2 DOF (rotation + translation along Z)."""
pt, a, b = self._setup()
constraints = [
CylindricalConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
assert _dof(pt, [a, b], constraints) == 2
def test_slider_1dof(self):
"""Slider: 6 - 5 = 1 DOF (translation along Z)."""
pt, a, b = self._setup()
constraints = [SliderConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)]
assert _dof(pt, [a, b], constraints) == 1
def test_universal_2dof(self):
"""Universal: 6 - 4 = 2 DOF (rotation about each body's Z)."""
pt, a, b = self._setup(quat_b=ROT_90X)
constraints = [
UniversalConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
assert _dof(pt, [a, b], constraints) == 2
def test_screw_1dof(self):
"""Screw: 6 - 5 = 1 DOF (helical motion)."""
pt, a, b = self._setup()
constraints = [
ScrewConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT, pitch=10.0)
]
assert _dof(pt, [a, b], constraints) == 1
def test_parallel_4dof(self):
"""Parallel: 6 - 2 = 4 DOF."""
pt, a, b = self._setup()
constraints = [ParallelConstraint(a, ID_QUAT, b, ID_QUAT)]
assert _dof(pt, [a, b], constraints) == 4
def test_perpendicular_5dof(self):
"""Perpendicular: 6 - 1 = 5 DOF."""
pt, a, b = self._setup(quat_b=ROT_90X)
constraints = [PerpendicularConstraint(a, ID_QUAT, b, ID_QUAT)]
assert _dof(pt, [a, b], constraints) == 5
def test_point_on_line_4dof(self):
"""PointOnLine: 6 - 2 = 4 DOF."""
pt, a, b = self._setup()
constraints = [
PointOnLineConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
assert _dof(pt, [a, b], constraints) == 4
def test_point_in_plane_5dof(self):
"""PointInPlane: 6 - 1 = 5 DOF."""
pt, a, b = self._setup()
constraints = [
PointInPlaneConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
assert _dof(pt, [a, b], constraints) == 5
def test_planar_3dof(self):
"""Planar: 6 - 3 = 3 DOF (2 translation in plane + 1 rotation about normal)."""
pt, a, b = self._setup()
constraints = [PlanarConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)]
assert _dof(pt, [a, b], constraints) == 3
# ============================================================================
# Solve convergence tests — single joints from displaced initial conditions
# ============================================================================
class TestJointSolve:
"""Newton converges to a valid configuration from displaced starting points."""
def test_revolute_displaced(self):
"""Revolute joint: body B starts displaced, should converge to hinge position."""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (3, 4, 5), ID_QUAT) # displaced
constraints = [RevoluteConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)]
converged, _ = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
pos = b.extract_position(env)
# Coincident origins → position should be at origin
assert abs(pos[0]) < 1e-8
assert abs(pos[1]) < 1e-8
assert abs(pos[2]) < 1e-8
def test_cylindrical_displaced(self):
"""Cylindrical joint: body B can slide along Z but must be on axis."""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (3, 4, 7), ID_QUAT) # off-axis
constraints = [
CylindricalConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
converged, _ = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
pos = b.extract_position(env)
# X and Y should be zero (on axis), Z can be anything
assert abs(pos[0]) < 1e-8
assert abs(pos[1]) < 1e-8
def test_slider_displaced(self):
"""Slider: body B can translate along Z only."""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (2, 3, 5), ID_QUAT) # displaced
constraints = [SliderConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)]
converged, _ = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
pos = b.extract_position(env)
# X and Y should be zero (on axis), Z free
assert abs(pos[0]) < 1e-8
assert abs(pos[1]) < 1e-8
def test_ball_displaced(self):
"""Ball joint: body B moves so marker origins coincide.
Ball has 3 rotation DOF free, so we can only verify the
world-frame marker points match, not the body position directly.
"""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (5, 5, 5), ID_QUAT)
constraints = [BallConstraint(a, (1, 0, 0), b, (-1, 0, 0))]
converged, _ = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
# Verify marker world points match
wp_a = a.world_point(1, 0, 0)
wp_b = b.world_point(-1, 0, 0)
for ea, eb in zip(wp_a, wp_b):
assert abs(ea.eval(env) - eb.eval(env)) < 1e-8
def test_universal_displaced(self):
"""Universal joint: coincident origins + perpendicular Z-axes."""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
# Start B with Z-axis along X (90° about Y) — perpendicular to A's Z
b = RigidBody("b", pt, (3, 4, 5), ROT_90Y)
constraints = [
UniversalConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
converged, _ = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
pos = b.extract_position(env)
assert abs(pos[0]) < 1e-8
assert abs(pos[1]) < 1e-8
assert abs(pos[2]) < 1e-8
def test_point_on_line_solve(self):
"""Point on line: body B's marker origin constrained to line along Z.
Under-constrained system (4 DOF remain), so we verify the constraint
residuals are satisfied rather than expecting specific positions.
"""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (5, 3, 7), ID_QUAT)
constraints = [
PointOnLineConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
converged, residuals = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
for r in residuals:
assert abs(r.eval(env)) < 1e-8
def test_point_in_plane_solve(self):
"""Point in plane: body B's marker origin at z=0 plane.
Under-constrained (5 DOF remain), so verify residuals.
"""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (3, 4, 8), ID_QUAT)
constraints = [
PointInPlaneConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)
]
converged, residuals = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
for r in residuals:
assert abs(r.eval(env)) < 1e-8
def test_planar_solve(self):
"""Planar: coplanar faces — parallel normals + point in plane."""
pt = ParamTable()
a = RigidBody("a", pt, (0, 0, 0), ID_QUAT, grounded=True)
# Start B tilted and displaced
b = RigidBody("b", pt, (3, 4, 8), ID_QUAT)
constraints = [PlanarConstraint(a, (0, 0, 0), ID_QUAT, b, (0, 0, 0), ID_QUAT)]
converged, _ = _solve_with_pt(pt, [a, b], constraints)
assert converged
env = pt.get_env()
pos = b.extract_position(env)
# Z must be zero (in plane), X and Y free
assert abs(pos[2]) < 1e-8
# ============================================================================
# Multi-body integration tests
# ============================================================================
class TestFourBarLinkage:
"""Four-bar linkage: 4 bodies, 4 revolute joints.
In 3D with Z-axis revolutes, this yields 2 DOF: the expected planar
motion plus an out-of-plane fold. A truly planar mechanism would
add Planar constraints on each link to eliminate the fold DOF.
"""
def test_four_bar_dof(self):
"""Four-bar linkage in 3D has 2 DOF (planar + fold)."""
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
link1 = RigidBody("l1", pt, (2, 0, 0), ID_QUAT)
link2 = RigidBody("l2", pt, (5, 3, 0), ID_QUAT)
link3 = RigidBody("l3", pt, (8, 0, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, link1, (0, 0, 0), ID_QUAT),
RevoluteConstraint(link1, (4, 0, 0), ID_QUAT, link2, (0, 0, 0), ID_QUAT),
RevoluteConstraint(link2, (6, 0, 0), ID_QUAT, link3, (0, 0, 0), ID_QUAT),
RevoluteConstraint(link3, (4, 0, 0), ID_QUAT, ground, (10, 0, 0), ID_QUAT),
]
bodies = [ground, link1, link2, link3]
dof = _dof(pt, bodies, constraints)
assert dof == 2
def test_four_bar_solves(self):
"""Four-bar linkage converges from displaced initial conditions."""
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
# Initial positions slightly displaced from valid config
link1 = RigidBody("l1", pt, (2, 1, 0), ID_QUAT)
link2 = RigidBody("l2", pt, (5, 4, 0), ID_QUAT)
link3 = RigidBody("l3", pt, (8, 1, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, link1, (0, 0, 0), ID_QUAT),
RevoluteConstraint(link1, (4, 0, 0), ID_QUAT, link2, (0, 0, 0), ID_QUAT),
RevoluteConstraint(link2, (6, 0, 0), ID_QUAT, link3, (0, 0, 0), ID_QUAT),
RevoluteConstraint(link3, (4, 0, 0), ID_QUAT, ground, (10, 0, 0), ID_QUAT),
]
bodies = [ground, link1, link2, link3]
converged, residuals = _solve_with_pt(pt, bodies, constraints)
assert converged
# Verify all revolute constraints are satisfied
env = pt.get_env()
for r in residuals:
assert abs(r.eval(env)) < 1e-8
class TestSliderCrank:
"""Slider-crank mechanism: crank + connecting rod + piston.
ground --[Revolute]-- crank --[Revolute]-- rod --[Revolute]-- piston --[Slider]-- ground
Using Slider (not Cylindrical) for the piston to also lock rotation,
making it a true prismatic joint. In 3D, out-of-plane folding adds
extra DOF beyond the planar 1-DOF.
3 free bodies × 6 = 18 DOF
Revolute(5) + Revolute(5) + Revolute(5) + Slider(5) = 20
But many constraints share bodies, so effective rank < 20.
In 3D: 3 DOF (planar crank + 2 fold modes).
"""
def test_slider_crank_dof(self):
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
crank = RigidBody("crank", pt, (1, 0, 0), ID_QUAT)
rod = RigidBody("rod", pt, (3, 0, 0), ID_QUAT)
piston = RigidBody("piston", pt, (5, 0, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, crank, (0, 0, 0), ID_QUAT),
RevoluteConstraint(crank, (2, 0, 0), ID_QUAT, rod, (0, 0, 0), ID_QUAT),
RevoluteConstraint(rod, (4, 0, 0), ID_QUAT, piston, (0, 0, 0), ID_QUAT),
SliderConstraint(piston, (0, 0, 0), ROT_90Y, ground, (0, 0, 0), ROT_90Y),
]
bodies = [ground, crank, rod, piston]
dof = _dof(pt, bodies, constraints)
# With full 3-component cross products, the redundant constraint rows
# eliminate the out-of-plane fold modes, giving the correct 1 DOF
# (crank rotation only).
assert dof == 1
def test_slider_crank_solves(self):
"""Slider-crank converges from displaced state."""
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
crank = RigidBody("crank", pt, (1, 0.5, 0), ID_QUAT)
rod = RigidBody("rod", pt, (3, 1, 0), ID_QUAT)
piston = RigidBody("piston", pt, (5, 0.5, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, crank, (0, 0, 0), ID_QUAT),
RevoluteConstraint(crank, (2, 0, 0), ID_QUAT, rod, (0, 0, 0), ID_QUAT),
RevoluteConstraint(rod, (4, 0, 0), ID_QUAT, piston, (0, 0, 0), ID_QUAT),
SliderConstraint(piston, (0, 0, 0), ROT_90Y, ground, (0, 0, 0), ROT_90Y),
]
bodies = [ground, crank, rod, piston]
converged, residuals = _solve_with_pt(pt, bodies, constraints)
assert converged
env = pt.get_env()
for r in residuals:
assert abs(r.eval(env)) < 1e-8
class TestRevoluteChain:
"""Chain of revolute joints: ground → body1 → body2.
Each revolute removes 5 DOF. Two free bodies = 12 DOF.
2 revolutes = 10 constraints + 2 quat norms = 12.
Expected: 2 DOF (one rotation per hinge).
"""
def test_chain_dof(self):
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
b1 = RigidBody("b1", pt, (3, 0, 0), ID_QUAT)
b2 = RigidBody("b2", pt, (6, 0, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, b1, (0, 0, 0), ID_QUAT),
RevoluteConstraint(b1, (3, 0, 0), ID_QUAT, b2, (0, 0, 0), ID_QUAT),
]
assert _dof(pt, [ground, b1, b2], constraints) == 2
def test_chain_solves(self):
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
b1 = RigidBody("b1", pt, (3, 2, 0), ID_QUAT)
b2 = RigidBody("b2", pt, (6, 3, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, b1, (0, 0, 0), ID_QUAT),
RevoluteConstraint(b1, (3, 0, 0), ID_QUAT, b2, (0, 0, 0), ID_QUAT),
]
converged, residuals = _solve_with_pt(pt, [ground, b1, b2], constraints)
assert converged
env = pt.get_env()
# b1 origin at ground hinge point (0,0,0)
pos1 = b1.extract_position(env)
assert abs(pos1[0]) < 1e-8
assert abs(pos1[1]) < 1e-8
assert abs(pos1[2]) < 1e-8
class TestSliderOnRail:
"""Slider constraint: body translates along ground Z-axis only.
1 free body, 1 slider = 6 - 5 = 1 DOF.
"""
def test_slider_on_rail(self):
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
block = RigidBody("block", pt, (3, 4, 5), ID_QUAT)
constraints = [
SliderConstraint(ground, (0, 0, 0), ID_QUAT, block, (0, 0, 0), ID_QUAT)
]
converged, _ = _solve_with_pt(pt, [ground, block], constraints)
assert converged
env = pt.get_env()
pos = block.extract_position(env)
# X, Y must be zero; Z is free
assert abs(pos[0]) < 1e-8
assert abs(pos[1]) < 1e-8
# Z should remain near initial value (minimum-norm solution)
# Check orientation unchanged (no twist)
quat = block.extract_quaternion(env)
assert abs(quat[0] - 1.0) < 1e-6
assert abs(quat[1]) < 1e-6
assert abs(quat[2]) < 1e-6
assert abs(quat[3]) < 1e-6
class TestPlanarOnTable:
"""Planar constraint: body slides on XY plane.
1 free body, 1 planar = 6 - 3 = 3 DOF.
"""
def test_planar_on_table(self):
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
block = RigidBody("block", pt, (3, 4, 5), ID_QUAT)
constraints = [
PlanarConstraint(ground, (0, 0, 0), ID_QUAT, block, (0, 0, 0), ID_QUAT)
]
converged, _ = _solve_with_pt(pt, [ground, block], constraints)
assert converged
env = pt.get_env()
pos = block.extract_position(env)
# Z must be zero, X and Y are free
assert abs(pos[2]) < 1e-8
class TestPlanarWithOffset:
"""Planar with offset: body floats at z=3 above ground."""
def test_planar_offset(self):
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
block = RigidBody("block", pt, (1, 2, 5), ID_QUAT)
# PlanarConstraint residual: (p_i - p_j) . z_j - offset = 0
# body_i=block, body_j=ground: (block_z - 0) * 1 - offset = 0
# For block at z=3: offset = 3
constraints = [
PlanarConstraint(
block, (0, 0, 0), ID_QUAT, ground, (0, 0, 0), ID_QUAT, offset=3.0
)
]
converged, _ = _solve_with_pt(pt, [ground, block], constraints)
assert converged
env = pt.get_env()
pos = block.extract_position(env)
assert abs(pos[2] - 3.0) < 1e-8
class TestMixedConstraints:
"""System with mixed constraint types."""
def test_revolute_plus_parallel(self):
"""Two free bodies: revolute between ground and b1, parallel between b1 and b2.
b1: 6 DOF - 5 (revolute) = 1 DOF
b2: 6 DOF - 2 (parallel) = 4 DOF
Total: 5 DOF
"""
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
b1 = RigidBody("b1", pt, (0, 0, 0), ID_QUAT)
b2 = RigidBody("b2", pt, (5, 0, 0), ID_QUAT)
constraints = [
RevoluteConstraint(ground, (0, 0, 0), ID_QUAT, b1, (0, 0, 0), ID_QUAT),
ParallelConstraint(b1, ID_QUAT, b2, ID_QUAT),
]
assert _dof(pt, [ground, b1, b2], constraints) == 5
def test_coincident_plus_perpendicular(self):
"""Coincident + perpendicular = ball + 1 angle constraint.
6 - 3 (coincident) - 1 (perpendicular) = 2 DOF.
"""
pt = ParamTable()
ground = RigidBody("g", pt, (0, 0, 0), ID_QUAT, grounded=True)
b = RigidBody("b", pt, (0, 0, 0), ROT_90X)
constraints = [
CoincidentConstraint(ground, (0, 0, 0), b, (0, 0, 0)),
PerpendicularConstraint(ground, ID_QUAT, b, ID_QUAT),
]
assert _dof(pt, [ground, b], constraints) == 2
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