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joints -- Kinematic Joints definition

Usual joints

Ball(solids, center, local=None, default=None)

Bases: Joint

Joint for rotation all around a point.

ball joint

Classical definition: Ball (point) the initial state doen't require more data the class holds a point for each side

Source code in madcad/joints.py 
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def __init__(self, solids, center, local=None, default=None):
	self.solids = solids
	local = local or center

	if default is not None:
		self.default = default

	if isinstance(center, mat4):
		self.post = center
		self.pre = affineInverse(local)
		self.position = center[3].xyz, local[3].xyz
	elif isinstance(center, vec3):
		self.post = translate(center)
		self.pre = translate(-local)
		self.position = center, local
	else:
		raise TypeError("ball can only be placed on a vec3 or mat4")
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

post = center instance-attribute

pre = affineInverse(local) instance-attribute

position = (center[3].xyz, local[3].xyz) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

bounds = (quat(-2, -2, -2, -2), quat(2, 2, 2, 2)) class-attribute instance-attribute

increment = quat(0.5, 0.5, 0.5, 0.5) class-attribute instance-attribute

default = quat() class-attribute instance-attribute

__repr__()

Source code in madcad/joints.py 
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def __repr__(self):
	return '{}({}, ({:.3g},{:.3g},{:.3g}))'.format(self.__class__.__name__, self.solids, *self.post[3].xyz)

normalize(orient)

Source code in madcad/joints.py 
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def normalize(self, orient):
	return normalize(quat(orient))

direct(orient)

Source code in madcad/joints.py 
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def direct(self, orient):
	return self.post * mat4(normalize(quat(orient))) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	return quat(affineInverse(self.post) * matrix * affineInverse(self.pre))

grad(orient)

Source code in madcad/joints.py 
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def grad(self, orient):
	w,x,y,z = orient
	return (
		self.post * mat4(
			0,    2*z, -2*y, 0,
			-2*z,  0,    2*x, 0,
			2*y, -2*x,  0,   0,
			0,    0,    0,   0,
			) * self.pre,
		self.post * mat4( 
			0,   2*y,  2*z, 0,
			2*y, -4*x,  2*w, 0,
			2*z, -2*w, -4*x, 0,
			0,   0,    0,   0,
			) * self.pre,
		self.post * mat4(
			-4*y,  2*x, -2*w, 0,
			2*x,  0,    2*z, 0,
			2*w,  2*z, -4*y, 0,
			0,    0,    0,   0,
			) * self.pre,
		self.post * mat4(
			-4*z,  2*w,  2*x, 0,
			-2*w, -4*z,  2*y, 0,
			2*x,  2*y,  0,   0,
			0,    0,    0,   0,
			) * self.pre,
		)

transmit(force, parameters=None, velocity=None)

Source code in madcad/joints.py 
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def transmit(self, force, parameters=None, velocity=None):
	return Screw(vec3(0), force.momentum, self.center)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, world_solid, kinematic_color

	from .primitives import ArcCentered

	size = settings.display['joint_size']
	radius = size/3
	sch = Scheme()
	resolution = ('div', 16)

	center = self.post[3].xyz
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(center), maxsize/size))
	if attach_start:
		x,y,z = dirbase(normalize(center - attach_start))
	else:
		x,y,z = mat3()
	profile = ArcCentered(Axis(O,y), x*radius, -z*radius).mesh(resolution=resolution)
	emisphere = gt.revolution(profile, Axis(O,z), resolution=resolution)
	sch.add(emisphere, shader='ghost')
	sch.add(emisphere.outlines(), shader='line')
	if attach_start:
		sch.set(shader='line')
		sch.add(-z*radius)
		sch.add(attach_start, space=world_solid(self.solids[0]))

	center = affineInverse(self.pre)[3].xyz
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(center), maxsize/size))
	sch.add(gt.icosphere(O, radius*0.8, resolution=('div', 3)), shader='ghost')
	if attach_end:
		sch.set(shader='line')
		sch.add(radius*0.8*normalize(attach_end-center))
		sch.add(attach_end, space=world_solid(self.solids[1]))

	return sch

Revolute(solids, axis, local=None, default=None, bounds=None)

Bases: Joint

Joint for revolution around an axis

revolute joint

Source code in madcad/joints.py 
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def __init__(self, solids, axis: Axis, local=None, default=None, bounds=None):
	self.solids = solids
	local = local or axis

	if default is not None:
		self.default = default
	if bounds is not None:
		self.bounds = bounds

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
		self.position = axis[3].xyz, local[3].xyz
	elif isinstance(axis, Axis):
		self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
		self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
		self.position = axis[0], local[0]
	else:
		raise TypeError('Revolute can only be placed on an Axis or a mat4')
	self.scale = max(glm.abs(self.post[3]))

bounds = (-inf, inf) class-attribute instance-attribute

default = 0 class-attribute instance-attribute

increment = 1 class-attribute instance-attribute

solids = solids instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

position = (axis[3].xyz, local[3].xyz) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

__repr__()

Source code in madcad/joints.py 
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def __repr__(self):
	return '{}({}, Axis(({:.3g},{:.3g},{:.3g}), ({:.3g},{:.3g},{:.3g})))'.format(
		self.__class__.__name__, self.solids, *self.post[3].xyz, *self.post[2].xyz)

direct(angle)

Source code in madcad/joints.py 
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def direct(self, angle) -> mat4:
	return self.post * rotate(angle, Z) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	if close is None:
		close = 0
	m = affineInverse(self.post) * matrix * affineInverse(self.pre)
	angle = - atan2(m[1][0] - m[0][1], m[0][0] + m[1][1])
	angle += (2*pi) * ((angle + pi - close) // (2*pi))
	return angle

grad(angle, delta=1e-06)

Source code in madcad/joints.py 
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def grad(self, angle, delta=1e-6):
	return self.post * drotate(angle, Z) * self.pre,

transmit(force, parameters=None, velocity=None)

Source code in madcad/joints.py 
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def transmit(self, force, parameters=None, velocity=None) -> Screw:
	l = force.locate(self.axis[0])
	return Screw(l.resulting, project(l.momentum, self.axis[1]))

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, world_solid, kinematic_color

	size = settings.display['joint_size']
	radius = size/4
	sch = Scheme()
	resolution = ('div', 16)

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size))
	cylinder = gt.cylinder(
					-z*size*0.4, 
					+z*size*0.4, 
					radius=size/4, 
					resolution=resolution,
					fill=False,
					)
	sch.add(cylinder, shader='ghost')
	sch.add(cylinder.outlines(), shader='line')
	if attach_start:
		v = normalize(noproject(attach_start - o, z))
		if not isfinite(v):
			v = x
		p = v*size/4
		sch.add(gt.cone(p+size*cornersize*v, p, size*cornersize, fill=False, resolution=('div', 4)), shader='fill')
		sch.set(shader='line')
		sch.add(p + v*size*cornersize)
		sch.add(attach_start, space=world_solid(self.solids[0]))

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(o), maxsize/size))
	c1 = primitives.Circle(
					(-z*size*0.5, -z), 
					radius, 
					resolution=resolution,
					).mesh()
	c2 = primitives.Circle(
					(+z*size*0.5, z), 
					radius, 
					resolution=resolution,
					).mesh()
	# sch.add([z*size*0.5, o+z*size*0.5], shader='line')
	sch.add(c1, shader='line')
	sch.add(c2, shader='line')
	sch.add(gt.fill(c1), shader='ghost')
	sch.add(gt.fill(c2), shader='ghost')

	if attach_end:
		sch.set(shader='line')
		side = z * size * 0.5
		if dot(attach_end-o, side) < 0:
			side = -side
		if dot(attach_end, side)**2 < length2(attach_end)*length2(side) * 0.25:
			radial = normalize(noproject(attach_end-o, z))
			sch.add(side + radial*radius)
			sch.add(side + radial*2*radius)
		else:
			sch.add(side)
		sch.add(attach_end, space=world_solid(self.solids[1]))

	return sch

Cylindrical(solids, axis, local=None, default=None, bounds=None)

Bases: Joint

Joint with rotation and translation around an axis

cylindrical joint

Source code in madcad/joints.py 
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def __init__(self, solids, axis: Axis, local=None, default=None, bounds=None):
	self.solids = solids
	local = local or axis

	if default is not None:
		self.default = default
	if bounds is not None:
		self.bounds = bounds

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
		self.position = axis[3].xyz, local[3].xyz
	elif isinstance(axis, Axis):
		self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
		self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
		self.position = axis[0], local[0]
	else:
		raise TypeError('Pivot can only be placed on an Axis or a mat4')
	self.scale = max(glm.abs(self.post[3]))

bounds = (vec2(-inf), vec2(inf)) class-attribute instance-attribute

increment = vec2(1, inf) class-attribute instance-attribute

default = vec2(0) class-attribute instance-attribute

solids = solids instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

position = (axis[3].xyz, local[3].xyz) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

__repr__()

Source code in madcad/joints.py 
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def __repr__(self):
	return '{}({}, Axis(({:.3g},{:.3g},{:.3g}), ({:.3g},{:.3g},{:.3g})))'.format(
		self.__class__.__name__, self.solids, *self.post[3].xyz, *self.post[2].xyz)

direct(parameters)

Source code in madcad/joints.py 
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def direct(self, parameters) -> mat4:
	return self.post * rotate(parameters[0], Z) * translate(parameters[1]*Z) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = affineInverse(self.post) * mat4(matrix) * affineInverse(self.pre)
	angle = atan2(m[1][0] - m[0][1], m[0][0] + m[1][1])
	angle += (2*pi) * ((angle + pi - close) // (2*pi))
	return angle, m[2][2]

grad(parameters, delta=1e-06)

Source code in madcad/joints.py 
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def grad(self, parameters, delta=1e-6):
	return (
		self.post * drotate(parameters[0], Z) * self.pre,
		self.post * dtranslate(Z) * self.pre,
		)

transmit(force, parameters=None, velocity=None)

Source code in madcad/joints.py 
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def transmit(self, force, parameters=None, velocity=None) -> Screw:
	l = force.locate(self.axis[0])
	return Screw(l.resulting, project(l.momentum, self.axis[1]))

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, world_solid, kinematic_color

	size = settings.display['joint_size']
	radius = size/4
	sch = Scheme()
	resolution = ('div', 16)

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size))
	cylinder = gt.cylinder(
					o-z*size*0.4, 
					o+z*size*0.4, 
					radius=size/4, 
					resolution=resolution,
					fill=False,
					)
	sch.add(cylinder, shader='ghost')
	sch.add(cylinder.outlines(), shader='line')
	if attach_start:
		v = normalize(noproject(attach_start - o, z))
		if not isfinite(v):
			v = x
		p = v*size/4
		sch.add(gt.cone(p+size*cornersize*v, p, size*cornersize, fill=False, resolution=('div', 4)), shader='fill')
		sch.set(shader='line')
		sch.add(p + v*size*cornersize)
		sch.add(attach_start, space=world_solid(self.solids[0]))

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(o), maxsize/size))
	sch.add([o-z*size*0.5, o+z*size*0.5], shader='line')

	sch.set(shader='fill')
	if attach_end:
		sch.set(shader='line')
		side = z * size * 0.5
		if dot(attach_end-o, side) < 0:
			side = -side
		if dot(attach_end, side)**2 < length2(attach_end)*length2(side) * 0.25:
			radial = normalize(noproject(attach_end-o, z))
			sch.add(side + radial*radius)
			sch.add(side + radial*2*radius)
		else:
			sch.add(side)
		sch.add(attach_end, space=world_solid(self.solids[1]))

	return sch

Prismatic(solids, axis, local=None, default=None, bounds=None)

Bases: Joint

Joint for translation only in a direction

prismatic joint

Classical definition: Prismatic (direction vector) the initial state requires more parameters: the relative placements of the solids this class holds a base for each of the solids, bases are (0,X,Y) where X,Y are constrained to keep the same direction across bases, and the bases origins lays on their common Z axis

Source code in madcad/joints.py 
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def __init__(self, solids, axis: Axis, local=None, default=None, bounds=None):
	self.solids = solids
	local = local or axis

	if default is not None:
		self.default = default
	if bounds is not None:
		self.bounds = bounds

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
		self.position = axis[3].xyz, local[3].xyz
	elif isinstance(axis, vec3):
		self.post = mat4(quat(Z, axis))
		self.pre = mat4(quat(local, Z))
		self.position = axis, local
	elif isinstance(axis, Axis):
		self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
		self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
		self.position = axis[0], local[0]
	else:
		raise TypeError('Track can only be placed on a vec3, Axis or mat4')
	self.scale = max(glm.abs(self.post[3]))

bounds = (-inf, inf) class-attribute instance-attribute

increment = inf class-attribute instance-attribute

default = 0 class-attribute instance-attribute

solids = solids instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

position = (axis[3].xyz, local[3].xyz) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

__repr__()

Source code in madcad/joints.py 
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def __repr__(self):
	return '{}({}, vec3({:.3g},{:.3g},{:.3g}))'.format(
		self.__class__.__name__, self.solids, *self.post[3].xyz)

direct(translation)

Source code in madcad/joints.py 
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def direct(self, translation):
	return self.post * translate(translation*Z) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	return (affineInverse(self.post) * matrix * affineInverse(pre))[2][2]

grad(translation, delta=1e-06)

Source code in madcad/joints.py 
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def grad(self, translation, delta=1e-6):
	return self.post * dtranslate(Z) * self.pre

transmit(force, parameters=None, velocity=None)

Source code in madcad/joints.py 
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def transmit(self, force, parameters=None, velocity=None):
	z0, z1 = cross(b0[1],b0[2]), cross(b1[1],b1[2])
	normal = normalize(z0 + z1)
	return Screw(noproject(action.resulting, normal), action.momentum, action.position)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, world_solid, kinematic_color

	size = settings.display['joint_size']
	sch = Scheme()

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size),
		)
	profile = gt.parallelogram(0.4*size*x, 0.4*size*y, origin=o, alignment=0.5, fill=False)
	profile.tracks = typedlist(range(len(profile.edges)))
	exterior = gt.extrusion(profile, size*z, alignment=0.5)
	exterior.splitgroups()
	sch.add(exterior, shader='ghost')
	sch.add(exterior.outlines(), shader='line')
	if attach_start:
		v = attach_start - o
		if abs(dot(v,x)) > abs(dot(v,y)):
			v = x if dot(v,x)>0 else -x
		else:
			v = y if dot(v,y)>0 else -y
		p = o + v*0.2*size
		sch.add(gt.cone(p+size*cornersize*v, p, size*cornersize, fill=False, resolution=('div', 4)), shader='fill')
		sch.set(shader='line')
		sch.add(p+size*cornersize*v)
		sch.add(attach_start, space=world_solid(self.solids[0]))

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(o), maxsize/size),
		)
	interior = gt.extrusion(Web([
		o-0.2*size*(x+y),
		o+0.2*size*(x+y),
		o-0.2*size*(x-y),
		o+0.2*size*(x-y),
		],
		[(0,1), (2,3)],
		), size*z, alignment=0.5)
	sch.add(interior.outlines(), shader='line')
	if attach_end:
		v = attach_end - o
		v = z if dot(v, z) > 0 else -z
		p = o + v*size/2
		sch.add(p)
		sch.add(attach_end, space=world_solid(self.solids[1]))

	return sch

Planar(solids, axis, local=None, default=None, bounds=None)

Bases: Joint

Joint for translation in 2 directions and rotation around the third direction

planar joint

Classical definition: Planar (direction vector) the initial state requires an additional distance between the solids this class holds an axis for each side, the axis origins are constrained to share the same projections on the normal

Source code in madcad/joints.py 
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def __init__(self, solids, axis, local=None, default=None, bounds=None):
	self.solids = solids
	local = local or axis

	if default is not None:
		self.default = default
	if bounds is not None:
		self.bounds = bounds

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
		self.position = axis[3].xyz, local[3].xyz
	elif isinstance(axis, Axis):
		self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
		self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
		self.position = axis[0], local[0]
	else:
		raise TypeError('Planar can only be placed on an Axis or a mat4')
	self.scale = max(glm.abs(self.post[3]))

bounds = (vec3(-inf), vec3(inf)) class-attribute instance-attribute

increment = vec3(inf) class-attribute instance-attribute

default = vec3(0, 0, 0) class-attribute instance-attribute

solids = solids instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

position = (axis[3].xyz, local[3].xyz) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

__repr__()

Source code in madcad/joints.py 
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def __repr__(self):
	return '{}({}, Axis(({:.3g},{:.3g},{:.3g}), ({:.3g},{:.3g},{:.3g})))'.format(
		self.__class__.__name__, self.solids, *self.post[3].xyz, *self.post[2].xyz)

direct(position)

Source code in madcad/joints.py 
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def direct(self, position: tuple):
	return mat4(self.post) * translate(vec3(position.xy, 0)) * rotate(position.z, Z) * mat4(self.pre)

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = mat4(transpose(self.post)) * matrix * mat4(transpose(self.pre))
	angle = atan2(m[0][1], m[0][0])
	angle += (2*pi) * ((angle + pi - close) // (2*pi))
	return vec3(m[3].xy, angle)

grad(position, delta=1e-06)

Source code in madcad/joints.py 
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def grad(self, position: vec3, delta=1e-6):
	return (
		self.post * dtranslate(X) * rotate(position.z, Z) * self.pre,
		self.post * dtranslate(Y) * rotate(position.z, Z) * self.pre,
		self.post * translate(vec3(position.xy,0)) * drotate(position.z, Z) * self.pre,
		)

transmitable(force, parameters=None, velocity=None)

Source code in madcad/joints.py 
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def transmitable(self, force, parameters=None, velocity=None):
	a0,a1 = solidtransform_axis(self.solids[0], self.axis[0])
	normal = normalize(a0[1] + a1[1])
	return Screw(project(action.resulting, normal), noproject(action.momentum, normal), action.position)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, world_solid, kinematic_color

	size = settings.display['joint_size']
	sch = Scheme()
	gap = 0.1

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size),
		)
	square = gt.parallelogram(x*size, y*size, origin=o-gap*size*z, alignment=0.5)
	sch.add(square.outlines(), shader='line')
	sch.add(square, shader='ghost')

	if attach_start:
		sch.add(gt.cone(o-size*(gap+cornersize)*z, o-size*gap*z, size*cornersize, fill=False, resolution=('div', 4)), shader='fill')
		sch.set(shader='line')
		sch.add(o-size*(gap+cornersize)*z)
		sch.add(attach_start, space=world_solid(self.solids[0]))

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	sch.set(
		track = index[self.solids[1]],
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(o), maxsize/size),
		)
	square = gt.parallelogram(-x*size, y*size, origin=o+gap*size*z, alignment=0.5)
	sch.add(square.outlines(), shader='line')
	sch.add(square, shader='ghost')

	if attach_end:
		sch.add(gt.cone(o+size*(gap+cornersize)*z, o+size*gap*z, size*cornersize, fill=False, resolution=('div', 4)), shader='fill')
		sch.set(shader='line')
		sch.add(o+size*(gap+cornersize)*z)
		sch.add(attach_end, space=world_solid(self.solids[1]))

	return sch

PointSlider(solids, axis, local=None)

Bases: Joint

Joint for rotation all around a point belonging to a plane.

pointslider joint

Classical definition: Punctiform/Sphere-Plane (axis) The initial state does not require more data The class holds a normal axis for the plane side, and a point for the sphere side (that defaults to the axis' origin)

Source code in madcad/joints.py 
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def __init__(self, solids, axis, local=None):
	self.solids = solids
	self.axis = axis
	local = local or axis

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
	elif isinstance(axis, Axis):
		self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
		self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
	else:
		raise TypeError("PointSlider can only be placed on Axis or mat4")
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

axis = axis instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

default = [0] * 6 class-attribute instance-attribute

increment = [inf] * 2 + [0.5] * 4 class-attribute instance-attribute

bounds = ([-inf] * 2 + [-2] * 4, [inf] * 2 + [2] * 4) class-attribute instance-attribute

direct(parameters)

Source code in madcad/joints.py 
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def direct(self, parameters):
	translation = vec2(parameters[0:2])
	rotation = quat(parameters[2:6])
	return self.post * translate(vec3(translation, 0)) * mat4(rotation) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = affineInverse(self.post) * matrix * affineInverse(self.pre)
	return (
		*vec2(m[3]),
		*quat(m),
		)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, kinematic_color

	size = settings.display['joint_size']
	radius = size * 0.2
	resolution = ('div', 16)
	sch = Scheme()

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size),
		)
	square = gt.parallelogram(x*size, y*size, origin=o+radius*z, alignment=0.5)
	sch.add(square.outlines(), shader='line')
	sch.add(square, shader='ghost')

	if attach_start:
		attach = o+radius*z
		sch.add([attach, attach+z*0.1*size, attach+(x+y)*0.1*size], shader='fill')
		sch.add([attach+cornersize*size*z, attach_start], shader='line')

	center = affineInverse(self.pre)[3].xyz
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(center), maxsize/size),
		)
	sch.add(gt.icosphere(center, radius, resolution), shader='ghost')

	if attach_end:
		x,y,z = dirbase(attach_end)
		attach = radius*z
		sch.add([attach, attach+z*cornersize*size, attach+(x+y)*cornersize*size], shader='fill')
		sch.add([attach+cornersize*size*z, attach_start], shader='line')

	return sch

EdgeSlider(solids, axis, local=None)

Bases: Joint

Joint for rotation all around a, edge belonging to a plane.

Classical definition: Punctiform/Sphere-Plane (axis) The initial state does not require more data The class holds a normal axis for the plane side, and a point for the sphere side (that defaults to the axis' origin)

Source code in madcad/joints.py 
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def __init__(self, solids, axis, local=None):
	self.solids = solids
	self.axis = axis
	local = local or axis

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
	else:
		raise TypeError("EdgeSlider can only be placed on Axis or mat4")
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

axis = axis instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

default = [0] * 4 class-attribute instance-attribute

increment = [inf] * 2 + [1] * 2 class-attribute instance-attribute

bounds = ([-inf] * 4, [inf] * 4) class-attribute instance-attribute

direct(parameters)

Source code in madcad/joints.py 
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def direct(self, parameters):
	translation = vec2(parameters[0:2])
	rotation = quat(vec3(*parameters[2:4], 0))
	return self.post * translate(vec3(translation, 0)) * mat4(rotation) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = affineInverse(self.post) * matrix * affineInverse(self.pre)
	q = quat(m)
	return (
		*vec2(m[3]),
		pitch(q),
		roll(q),
		)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, kinematic_color

	size = settings.display['joint_size']
	radius = 0.2*size
	resolution = ('div', 16)
	sch = Scheme()

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size),
		)
	square = gt.parallelogram(x*size, y*size, origin=o+radius*z, alignment=0.5)
	sch.add(square.outlines(), shader='line')
	sch.add(square, shader='ghost')

	if attach_start:
		attach = o+radius*z
		sch.add([attach, attach+cornersize*size*z, attach+cornersize*size*(x+y)], shader='fill')
		sch.add([o+cornersize*size*z, attach_start], shader='line')

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(o), maxsize/size),
		)
	cylinder = gt.cylinder(o-x*0.5*size, o+x*0.5*size, radius, fill=False, resolution=resolution)
	sch.add(cylinder, shader='ghost')
	sch.add(cylinder.outlines(), shader='line')

	if attach_end:
		x,y,z = dirbase(noproject(attach_end, x))
		attach = radius*z
		sch.add([attach, attach+z*cornersize*size, attach+(x+y)*cornersize*size], shader='fill')

	return sch

Ring(solids, center, local=None)

Bases: Joint

ring joint

Source code in madcad/joints.py 
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def __init__(self, solids, center, local=None):
	self.solids = solids
	local = local or center

	if isinstance(center, mat4):
		self.pre = center
		self.post = local
	elif isinstance(center, vec3):
		self.pre = translate(-center)
		self.post = translate(local)
	else:
		raise TypeError("ball can only be placed on a vec3 or mat4")
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

pre = center instance-attribute

post = local instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

bounds = ((-inf, -1, -1, -1, -1), (inf, 1, 1, 1, 1)) class-attribute instance-attribute

increment = (inf, 0.5, 0.5, 0.5, 0.5) class-attribute instance-attribute

default = (0, 1, 0, 0, 0) class-attribute instance-attribute

direct(orient)

Source code in madcad/joints.py 
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def direct(self, orient):
	return self.post * translate(params[0]) * mat4(quat(params[1:5])) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	return quat(affineInverse(self.post) * matrix * affineInverse(self.pre))

transmit(force, parameters=None, velocity=None)

Source code in madcad/joints.py 
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def transmit(self, force, parameters=None, velocity=None):
	return Screw(vec3(0), force.momentum, self.center)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, kinematic_color

	from .primitives import ArcCentered

	size = settings.display['joint_size']
	radius = size/3
	sch = Scheme()
	resolution = ('div', 16)

	center = self.post[3].xyz
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(center), maxsize/size))
	if attach_start:
		x,y,z = dirbase(attach_end - center)
	else:
		x,y,z = mat3()
	profile = ArcCentered(Axis(center,y), center+x*radius, center-z*radius).mesh(resolution=resolution)
	emisphere = gt.revolution(profile, Axis(center,z), resolution)
	sch.add(emisphere, shader='ghost')
	sch.add(emisphere.outlines(), shader='line')

	center = affineInverse(self.pre)[3].xyz
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(center), maxsize/size))
	sch.add(gt.icosphere(center, radius*0.8, resolution), shader='ghost')

	return sch

Universal(solids, axis, local=None, bounds=None)

Bases: Joint

universal joint

Source code in madcad/joints.py 
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def __init__(self, solids, axis, local=None, bounds=None):
	self.solids = solids
	local = local or axis

	if bounds is not None:
		self.bounds = bounds

	if isinstance(axis, mat4):
		self.post = axis
		self.pre = affineInverse(local)
	elif isinstance(axis, Axis):
		self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
		self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
	else:
		raise TypeError('Universal joint can only be placed on an Axis or a mat4')
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

post = axis instance-attribute

pre = affineInverse(local) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

bounds = (vec2(-inf), vec2(inf)) class-attribute instance-attribute

increment = vec2(1) class-attribute instance-attribute

default = vec2(0) class-attribute instance-attribute

direct(orient)

Source code in madcad/joints.py 
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def direct(self, orient):
	return self.post * mat4(quat(vec3(orient, 0))) * self.pre

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = quat(affineInverse(self.post) * matrix * affineInverse(self.pre))
	return vec2(pitch(m), yaw(m))

grad(orient)

Source code in madcad/joints.py 
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def grad(self, orient):
	pitch, yaw = orient
	return (
		self.post * drotate(pitch, X) * self.pre,
		self.post * drotate(pitch, Y) * self.pre,
		)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import scale_solid, world_solid, kinematic_color
	from .primitives import ArcCentered

	size = settings.display['joint_size']
	radius = size/3
	sch = Scheme()
	resolution = ('div', 16)

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = scale_solid(self.solids[0], fvec3(o), maxsize/size))
	if attach_start:
		o = self.post[3].xyz
		x,y,z = dirbase(attach_end - center, x)
		sch.add(attach_start, space=world_solid(self.solids[0]))
		sch.add(o - z*radius)
	sch.add(ArcCentered(Axis(o,x), y*radius, -y*radius).mesh(resolution), shader='line')
	sch.add([y, -y])

	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = scale_solid(self.solids[1], fvec3(o), maxsize/size))
	x,y,z,o = affineInverse(self.pre)
	if attach_end:
		o = affineInverse(self.pre)[3].xyz
		x,y,z = dirbase(attach_end - center, y)
		sch.add(attach_end, space=world_solid(self.solids[1]))
		sch.add(o - z*radius)
	sch.add(ArcCentered(Axis(o,y), x*radius, -x*radius).mesh(resolution), shader='line')
	sch.add([x, -x])

	return sch

ConstantVelocity(*args, default=0, **kwargs)

Bases: Joint

not implemented yet

Source code in madcad/kinematic/solver.py 
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def __init__(self, *args, default=0, **kwargs):
	if isinstance(args[0], Solid) and isinstance(args[1], Solid):
		self.solids = args[:2]
		args = args[2:]
	self.default = default
	self.init(*args, **kwargs)

Complex joints

Project(solids, projection, target)

Bases: Joint

not implemented yet

Source code in madcad/joints.py 
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def __init__(self, solids, projection:mat4, target:mat4):
	raise NotImplementedError("In development")

Rack(solids, radius, rack, axis)

Bases: Joint

Rack to gear interaction

radius is the ratio between the gear rotation and the rack translation

Attributes:

Name Type Description
radius

ratio between the gear rotation and the rack translation
rack

rack axis tangent to the gear
axis

gear axis
Source code in madcad/joints.py 
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def __init__(self, solids, radius:float, rack:Axis, axis):
	self.solids = solids
	self.radius = radius
	self.axis = axis
	self.rack = rack
	self.pre = mat4(quat(axis[1], Z)) * translate(-axis[0])
	self.post = translate(rack[0]) * mat4(quat(X, rack[1]))
	self.position = self.rack[0], self.axis[0]
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

radius = radius instance-attribute

axis = axis instance-attribute

rack = rack instance-attribute

pre = mat4(quat(axis[1], Z)) * translate(-axis[0]) instance-attribute

post = translate(rack[0]) * mat4(quat(X, rack[1])) instance-attribute

position = (self.rack[0], self.axis[0]) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

bounds = (-inf, inf) class-attribute instance-attribute

increment = 1 class-attribute instance-attribute

default = 0 class-attribute instance-attribute

direct(translation)

Source code in madcad/joints.py 
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def direct(self, translation):
	return self.post * translate(translation*X) * rotate(-translation/self.radius,Z) * self.pre

grad(translation)

Source code in madcad/joints.py 
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def grad(self, translation):
	return (
		+ self.post * dtranslate(X) * rotate(-translation/self.radius,Z) * self.pre
		+ self.post * translate(translation*X) * drotate(-translation/self.radius,Z)/-self.radius * self.pre
		)

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = affineInverse(self.post) * matrix * affineInverse(self.pre)
	return atan(*m[0].xy)

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import world_solid, kinematic_color
	from .generation import revolution
	from .primitives import Circle, Segment
	sch = Scheme()

	x,y,z,o = dmat4x3(self.post)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = world_solid(self.solids[0]))
	sch.add([o-self.radius*y-self.radius*x, o-self.radius*y+self.radius*x], shader='line')

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	h = 0.1*self.radius*z
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = world_solid(self.solids[1]))
	sch.add(Circle(Axis(o,z), self.radius).mesh(), shader='line')
	sch.add(revolution(Segment(o+self.radius*x+h, o+self.radius*x-h), Axis(o,z)), shader='ghost')

	return sch

Gear(solids, ratio, centerline, axis, local)

Bases: Joint

Gear interaction between two solids.

ratio is the factor from the rotation of s1 to the rotation of s2 The two pinions are considered circular, but no assumption is made on their radius or relative inclination. The interaction is symetric.

Attributes:

Name Type Description
ratio

the gear rotation transmission ratio
centerline

the relative positioning of the gears axis, could be 'float', vec3, mat3, quat, mat4
axis

first gear axis
local

second gear axis
Source code in madcad/joints.py 
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def __init__(self, solids, ratio:float, centerline:mat4, axis, local):
	self.solids = solids
	self.ratio = ratio
	if isinstance(centerline, (int,float)):
		centerline = centerline*X
	self.centerline = transform(centerline)
	self.axis = axis
	self.pre = mat4(quat(local[1], Z)) * translate(-local[0])
	self.post = translate(axis[0]) * mat4(quat(Z, axis[1]))
	self.position = self.axis[0], local[0]
	self.scale = max(glm.abs(self.post[3]))

solids = solids instance-attribute

ratio = ratio instance-attribute

centerline = transform(centerline) instance-attribute

axis = axis instance-attribute

pre = mat4(quat(local[1], Z)) * translate(-local[0]) instance-attribute

post = translate(axis[0]) * mat4(quat(Z, axis[1])) instance-attribute

position = (self.axis[0], local[0]) instance-attribute

scale = max(glm.abs(self.post[3])) instance-attribute

bounds = (-inf, inf) class-attribute instance-attribute

increment = 1 class-attribute instance-attribute

default = 0 class-attribute instance-attribute

direct(angle)

Source code in madcad/joints.py 
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def direct(self, angle):
	return self.post * rotate(angle,Z) * self.centerline * rotate(-self.ratio*angle,Z) * self.pre

grad(angle)

Source code in madcad/joints.py 
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def grad(self, angle):
	return (
		+ self.post * drotate(angle,Z) * self.centerline * rotate(-self.ratio*angle,Z) * self.pre
		+ self.post * rotate(angle,Z) * self.centerline * drotate(-self.ratio*angle,Z)*-self.ratio * self.pre
		)

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	m = affineInverse(self.post) * matrix * affineInverse(self.pre)
	angle = atan(*m[0].xy) / (1+self.ratio)
	return (angle - close + pi) % (2*pi) - pi

scheme(index, maxsize, attach_start, attach_end)

Source code in madcad/joints.py 
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def scheme(self, index, maxsize, attach_start, attach_end):
	from .scheme import Scheme
	from .kinematic.displays import world_solid, kinematic_color
	from .generation import revolution
	from .primitives import Circle, Segment
	sch = Scheme()

	# radial vector
	r0 = length(self.centerline[3].xy) / abs(self.centerline[2].z - 1/self.ratio)
	r1 = r0 / abs(self.ratio)

	x,y,z,o = dmat4x3(self.post)
	h = 0.1*min(r0, r1) * (z * r0 + self.centerline[2].xyz * r1) / (r0 + r1)
	sch.set(
		track = index[self.solids[0]], 
		color = kinematic_color(index[self.solids[0]]),
		space = world_solid(self.solids[0]))
	sch.add(Circle(Axis(o,z), r0).mesh(), shader='line')
	sch.add(revolution(Segment(o+r0*x+h, o+r0*x-h), Axis(o,z)), shader='ghost')

	x,y,z,o = dmat4x3(affineInverse(self.pre))
	h = 0.1*min(r0, r1) * (self.centerline[2].xyz * r0 + z * r1) / (r0 + r1)
	sch.set(
		track = index[self.solids[1]], 
		color = kinematic_color(index[self.solids[1]]),
		space = world_solid(self.solids[1]))
	sch.add(Circle(Axis(o,z), r1).mesh(), shader='line')
	sch.add(revolution(Segment(o+r1*x+h, o+r1*x-h), Axis(o,z)), shader='ghost')

	return sch

Helicoid(s0, s1, step, b0, b1=None, position=None)

Bases: Joint

Screw a solid into an other.

helicoid joint

step is the translation distance needed for that one solid turn by 2*pi around the other The interaction is symetric.

not implemented yet

Source code in madcad/joints.py 
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def __init__(self, s0, s1, step, b0, b1=None, position=None):
	self.step = step	# m/tr
	self.solids = s0, s1
	if isaxis(b0):			b0 = b0[0], *dirbase(b0[1])[:2]
	if b1 and isaxis(b1):	b1 = b1[0], *dirbase(b1[1])[:2]
	self.bases = b0, b1 or b0
	self.position = position or (self.bases[0][0], self.bases[1][0])

step = step instance-attribute

solids = (s0, s1) instance-attribute

bases = (b0, b1 or b0) instance-attribute

position = position or (self.bases[0][0], self.bases[1][0]) instance-attribute

direct(depth)

Source code in madcad/joints.py 
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def direct(self, depth):
	raise NotImplementedError("In development")

inverse(matrix, close=None)

Source code in madcad/joints.py 
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def inverse(self, matrix, close=None):
	raise NotImplementedError("In development")

scheme(solid, size, junc)

Source code in madcad/joints.py 
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def scheme(self, solid, size, junc):
	if solid is self.solids[0]:
		radius = size/4
		o,x,y = self.bases[0]
		z = cross(x,y)
		center = o + project(self.position[0]-o, z)
		cyl = generation.extrusion(
					web(primitives.Circle(
						(center-z*size*0.5, z), 
						radius, 
						resolution=('div', 16),
						)),
					z*size, 
					)
		l = len(cyl.points)
		v = junc - center
		v = normalize(noproject(v,z))
		if glm.any(isnan(v)):
			v,_,_ = dirbase(z)
		p = center + v*size/4
		cyl.points.append(p)
		cyl.points.append(p + z*size*cornersize)
		cyl.points.append(p + v*size*cornersize)
		cyl.points.append(junc)

		return Scheme(
				cyl.points, 
				cyl.faces, 
				[(l,l+1,l+2)], 
				(	[(l, l+2), (l+2, l+3)] 
				+	[(i-1,i) for i in range(1,l//2)] 
				+	[(i-1,i) for i in range(l//2+1,l)] 
				+	[(l//2-1,0), (l-1,l//2)]),
				)

	elif solid is self.solids[1]:
		radius = size/6
		o,x,y = self.bases[1]
		z = cross(x,y)
		center = o + project(self.position[1]-o, z)
		attach = side = z * size

		heli = []
		div = 32
		a = 4*pi
		for i in range(-div,div+1):
			t = i/div
			heli.append( radius*(cos(a*t)*x + sin(a*t)*y) + center + side*t )
		heli.extend([center+side, junc])
		heli = web(Wire(heli))
		return Scheme(heli.points, [], [], heli.edges)

Mesh-based joints

Cam(*args, default=0, **kwargs)

Bases: Joint

not implemented yet

Source code in madcad/kinematic/solver.py 
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def __init__(self, *args, default=0, **kwargs):
	if isinstance(args[0], Solid) and isinstance(args[1], Solid):
		self.solids = args[:2]
		args = args[2:]
	self.default = default
	self.init(*args, **kwargs)

Contact(*args, default=0, **kwargs)

Bases: Joint

not implemented yet

Source code in madcad/kinematic/solver.py 
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def __init__(self, *args, default=0, **kwargs):
	if isinstance(args[0], Solid) and isinstance(args[1], Solid):
		self.solids = args[:2]
		args = args[2:]
	self.default = default
	self.init(*args, **kwargs)