Joints Framework (Docstrings)

Joint (Docstrings)

class sympy.physics.mechanics.joint.Joint(
name,
parent,
child,
coordinates=None,
speeds=None,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
parent_axis=None,
child_axis=None,
parent_joint_pos=None,
child_joint_pos=None,
)[source]

Abstract base class for all specific joints.

Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

coordinates : iterable of dynamicsymbols, optional

Generalized coordinates of the joint.

speeds : iterable of dynamicsymbols, optional

Generalized speeds of joint.

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_axis : Vector, optional

Deprecated since version 1.12: Axis fixed in the parent body which aligns with an axis fixed in the child body. The default is the x axis of parent’s reference frame. For more information on this deprecation, see New Joint intermediate frames.

child_axis : Vector, optional

Deprecated since version 1.12: Axis fixed in the child body which aligns with an axis fixed in the parent body. The default is the x axis of child’s reference frame. For more information on this deprecation, see New Joint intermediate frames.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

parent_joint_pos : Point or Vector, optional

Deprecated since version 1.12: This argument is replaced by parent_point and will be removed in a future version. See Change in joint attachment point argument for more information.

child_joint_pos : Point or Vector, optional

Deprecated since version 1.12: This argument is replaced by child_point and will be removed in a future version. See Change in joint attachment point argument for more information.

Explanation

A joint subtracts degrees of freedom from a body. This is the base class for all specific joints and holds all common methods acting as an interface for all joints. Custom joint can be created by inheriting Joint class and defining all abstract functions.

The abstract methods are:

  • _generate_coordinates

  • _generate_speeds

  • _orient_frames

  • _set_angular_velocity

  • _set_linear_velocity

Notes

When providing a vector as the intermediate frame, a new intermediate frame is created which aligns its X axis with the provided vector. This is done with a single fixed rotation about a rotation axis. This rotation axis is determined by taking the cross product of the body.x axis with the provided vector. In the case where the provided vector is in the -body.x direction, the rotation is done about the body.y axis.

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates.

speeds

(Matrix) Matrix of the joint’s generalized speeds.

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_axis

(Vector) The axis fixed in the parent frame that represents the joint.

child_axis

(Vector) The axis fixed in the child frame that represents the joint.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

kdes

(Matrix) Kinematical differential equations of the joint.
property child

Child body of Joint.

property child_axis

The axis of child frame.

property child_point

Attachment point where the joint is fixed to the child body.

property coordinates

Matrix of the joint’s generalized coordinates.

property kdes

Kinematical differential equations of the joint.

property name

Name of the joint.

property parent

Parent body of Joint.

property parent_axis

The axis of parent frame.

property parent_point

Attachment point where the joint is fixed to the parent body.

property speeds

Matrix of the joint’s generalized speeds.

class sympy.physics.mechanics.joint.PinJoint(
name,
parent,
child,
coordinates=None,
speeds=None,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
parent_axis=None,
child_axis=None,
joint_axis=None,
parent_joint_pos=None,
child_joint_pos=None,
)[source]

Pin (Revolute) Joint.

joint_axis q, u child.masscenter attachment points parent.frame parent_interframe parent.masscenter child.frame child_interframe
Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

coordinates : dynamicsymbol, optional

Generalized coordinates of the joint.

speeds : dynamicsymbol, optional

Generalized speeds of joint.

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_axis : Vector, optional

Deprecated since version 1.12: Axis fixed in the parent body which aligns with an axis fixed in the child body. The default is the x axis of parent’s reference frame. For more information on this deprecation, see New Joint intermediate frames.

child_axis : Vector, optional

Deprecated since version 1.12: Axis fixed in the child body which aligns with an axis fixed in the parent body. The default is the x axis of child’s reference frame. For more information on this deprecation, see New Joint intermediate frames.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

joint_axis : Vector

The axis about which the rotation occurs. Note that the components of this axis are the same in the parent_interframe and child_interframe.

parent_joint_pos : Point or Vector, optional

Deprecated since version 1.12: This argument is replaced by parent_point and will be removed in a future version. See Change in joint attachment point argument for more information.

child_joint_pos : Point or Vector, optional

Deprecated since version 1.12: This argument is replaced by child_point and will be removed in a future version. See Change in joint attachment point argument for more information.

Explanation

A pin joint is defined such that the joint rotation axis is fixed in both the child and parent and the location of the joint is relative to the mass center of each body. The child rotates an angle, θ, from the parent about the rotation axis and has a simple angular speed, ω, relative to the parent. The direction cosine matrix between the child interframe and parent interframe is formed using a simple rotation about the joint axis. The page on the joints framework gives a more detailed explanation of the intermediate frames.

Examples

A single pin joint is created from two bodies and has the following basic attributes:

>>> from sympy.physics.mechanics import RigidBody, PinJoint
>>> parent = RigidBody('P')
>>> parent
P
>>> child = RigidBody('C')
>>> child
C
>>> joint = PinJoint('PC', parent, child)
>>> joint
PinJoint: PC  parent: P  child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
P_masscenter
>>> joint.child_point
C_masscenter
>>> joint.parent_axis
P_frame.x
>>> joint.child_axis
C_frame.x
>>> joint.coordinates
Matrix([[q_PC(t)]])
>>> joint.speeds
Matrix([[u_PC(t)]])
>>> child.frame.ang_vel_in(parent.frame)
u_PC(t)*P_frame.x
>>> child.frame.dcm(parent.frame)
Matrix([
[1,             0,            0],
[0,  cos(q_PC(t)), sin(q_PC(t))],
[0, -sin(q_PC(t)), cos(q_PC(t))]])
>>> joint.child_point.pos_from(joint.parent_point)
0

To further demonstrate the use of the pin joint, the kinematics of simple double pendulum that rotates about the Z axis of each connected body can be created as follows.

>>> from sympy import symbols, trigsimp
>>> from sympy.physics.mechanics import RigidBody, PinJoint
>>> l1, l2 = symbols('l1 l2')

First create bodies to represent the fixed ceiling and one to represent each pendulum bob.

>>> ceiling = RigidBody('C')
>>> upper_bob = RigidBody('U')
>>> lower_bob = RigidBody('L')

The first joint will connect the upper bob to the ceiling by a distance of l1 and the joint axis will be about the Z axis for each body.

>>> ceiling_joint = PinJoint('P1', ceiling, upper_bob,
... child_point=-l1*upper_bob.frame.x,
... joint_axis=ceiling.frame.z)

The second joint will connect the lower bob to the upper bob by a distance of l2 and the joint axis will also be about the Z axis for each body.

>>> pendulum_joint = PinJoint('P2', upper_bob, lower_bob,
... child_point=-l2*lower_bob.frame.x,
... joint_axis=upper_bob.frame.z)

Once the joints are established the kinematics of the connected bodies can be accessed. First the direction cosine matrices of pendulum link relative to the ceiling are found:

>>> upper_bob.frame.dcm(ceiling.frame)
Matrix([
[ cos(q_P1(t)), sin(q_P1(t)), 0],
[-sin(q_P1(t)), cos(q_P1(t)), 0],
[            0,            0, 1]])
>>> trigsimp(lower_bob.frame.dcm(ceiling.frame))
Matrix([
[ cos(q_P1(t) + q_P2(t)), sin(q_P1(t) + q_P2(t)), 0],
[-sin(q_P1(t) + q_P2(t)), cos(q_P1(t) + q_P2(t)), 0],
[                      0,                      0, 1]])

The position of the lower bob’s masscenter is found with:

>>> lower_bob.masscenter.pos_from(ceiling.masscenter)
l1*U_frame.x + l2*L_frame.x

The angular velocities of the two pendulum links can be computed with respect to the ceiling.

>>> upper_bob.frame.ang_vel_in(ceiling.frame)
u_P1(t)*C_frame.z
>>> lower_bob.frame.ang_vel_in(ceiling.frame)
u_P1(t)*C_frame.z + u_P2(t)*U_frame.z

And finally, the linear velocities of the two pendulum bobs can be computed with respect to the ceiling.

>>> upper_bob.masscenter.vel(ceiling.frame)
l1*u_P1(t)*U_frame.y
>>> lower_bob.masscenter.vel(ceiling.frame)
l1*u_P1(t)*U_frame.y + l2*(u_P1(t) + u_P2(t))*L_frame.y

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates. The default value is dynamicsymbols(f'q_{joint.name}').

speeds

(Matrix) Matrix of the joint’s generalized speeds. The default value is dynamicsymbols(f'u_{joint.name}').

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_axis

(Vector) The axis fixed in the parent frame that represents the joint.

child_axis

(Vector) The axis fixed in the child frame that represents the joint.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

joint_axis

(Vector) The axis about which the rotation occurs. Note that the components of this axis are the same in the parent_interframe and child_interframe.

kdes

(Matrix) Kinematical differential equations of the joint.
property joint_axis

Axis about which the child rotates with respect to the parent.

class sympy.physics.mechanics.joint.PrismaticJoint(
name,
parent,
child,
coordinates=None,
speeds=None,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
parent_axis=None,
child_axis=None,
joint_axis=None,
parent_joint_pos=None,
child_joint_pos=None,
)[source]

Prismatic (Sliding) Joint.

../../../../_images/PrismaticJoint.svg
Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

coordinates : dynamicsymbol, optional

Generalized coordinates of the joint. The default value is dynamicsymbols(f'q_{joint.name}').

speeds : dynamicsymbol, optional

Generalized speeds of joint. The default value is dynamicsymbols(f'u_{joint.name}').

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_axis : Vector, optional

Deprecated since version 1.12: Axis fixed in the parent body which aligns with an axis fixed in the child body. The default is the x axis of parent’s reference frame. For more information on this deprecation, see New Joint intermediate frames.

child_axis : Vector, optional

Deprecated since version 1.12: Axis fixed in the child body which aligns with an axis fixed in the parent body. The default is the x axis of child’s reference frame. For more information on this deprecation, see New Joint intermediate frames.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

joint_axis : Vector

The axis along which the translation occurs. Note that the components of this axis are the same in the parent_interframe and child_interframe.

parent_joint_pos : Point or Vector, optional

Deprecated since version 1.12: This argument is replaced by parent_point and will be removed in a future version. See Change in joint attachment point argument for more information.

child_joint_pos : Point or Vector, optional

Deprecated since version 1.12: This argument is replaced by child_point and will be removed in a future version. See Change in joint attachment point argument for more information.

Explanation

It is defined such that the child body translates with respect to the parent body along the body-fixed joint axis. The location of the joint is defined by two points, one in each body, which coincide when the generalized coordinate is zero. The direction cosine matrix between the parent_interframe and child_interframe is the identity matrix. Therefore, the direction cosine matrix between the parent and child frames is fully defined by the definition of the intermediate frames. The page on the joints framework gives a more detailed explanation of the intermediate frames.

Examples

A single prismatic joint is created from two bodies and has the following basic attributes:

>>> from sympy.physics.mechanics import RigidBody, PrismaticJoint
>>> parent = RigidBody('P')
>>> parent
P
>>> child = RigidBody('C')
>>> child
C
>>> joint = PrismaticJoint('PC', parent, child)
>>> joint
PrismaticJoint: PC  parent: P  child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
P_masscenter
>>> joint.child_point
C_masscenter
>>> joint.parent_axis
P_frame.x
>>> joint.child_axis
C_frame.x
>>> joint.coordinates
Matrix([[q_PC(t)]])
>>> joint.speeds
Matrix([[u_PC(t)]])
>>> child.frame.ang_vel_in(parent.frame)
0
>>> child.frame.dcm(parent.frame)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> joint.child_point.pos_from(joint.parent_point)
q_PC(t)*P_frame.x

To further demonstrate the use of the prismatic joint, the kinematics of two masses sliding, one moving relative to a fixed body and the other relative to the moving body. about the X axis of each connected body can be created as follows.

>>> from sympy.physics.mechanics import PrismaticJoint, RigidBody

First create bodies to represent the fixed ceiling and one to represent a particle.

>>> wall = RigidBody('W')
>>> Part1 = RigidBody('P1')
>>> Part2 = RigidBody('P2')

The first joint will connect the particle to the ceiling and the joint axis will be about the X axis for each body.

>>> J1 = PrismaticJoint('J1', wall, Part1)

The second joint will connect the second particle to the first particle and the joint axis will also be about the X axis for each body.

>>> J2 = PrismaticJoint('J2', Part1, Part2)

Once the joint is established the kinematics of the connected bodies can be accessed. First the direction cosine matrices of Part relative to the ceiling are found:

>>> Part1.frame.dcm(wall.frame)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])

>>> Part2.frame.dcm(wall.frame)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])

The position of the particles’ masscenter is found with:

>>> Part1.masscenter.pos_from(wall.masscenter)
q_J1(t)*W_frame.x

>>> Part2.masscenter.pos_from(wall.masscenter)
q_J1(t)*W_frame.x + q_J2(t)*P1_frame.x

The angular velocities of the two particle links can be computed with respect to the ceiling.

>>> Part1.frame.ang_vel_in(wall.frame)
0

>>> Part2.frame.ang_vel_in(wall.frame)
0

And finally, the linear velocities of the two particles can be computed with respect to the ceiling.

>>> Part1.masscenter.vel(wall.frame)
u_J1(t)*W_frame.x

>>> Part2.masscenter.vel(wall.frame)
u_J1(t)*W_frame.x + Derivative(q_J2(t), t)*P1_frame.x

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates.

speeds

(Matrix) Matrix of the joint’s generalized speeds.

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_axis

(Vector) The axis fixed in the parent frame that represents the joint.

child_axis

(Vector) The axis fixed in the child frame that represents the joint.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

kdes

(Matrix) Kinematical differential equations of the joint.
property joint_axis

Axis along which the child translates with respect to the parent.

class sympy.physics.mechanics.joint.CylindricalJoint(
name,
parent,
child,
rotation_coordinate=None,
translation_coordinate=None,
rotation_speed=None,
translation_speed=None,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
joint_axis=None,
)[source]

Cylindrical Joint.

../../../../_images/CylindricalJoint.svg
Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

rotation_coordinate : dynamicsymbol, optional

Generalized coordinate corresponding to the rotation angle. The default value is dynamicsymbols(f'q0_{joint.name}').

translation_coordinate : dynamicsymbol, optional

Generalized coordinate corresponding to the translation distance. The default value is dynamicsymbols(f'q1_{joint.name}').

rotation_speed : dynamicsymbol, optional

Generalized speed corresponding to the angular velocity. The default value is dynamicsymbols(f'u0_{joint.name}').

translation_speed : dynamicsymbol, optional

Generalized speed corresponding to the translation velocity. The default value is dynamicsymbols(f'u1_{joint.name}').

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

joint_axis : Vector, optional

The rotation as well as translation axis. Note that the components of this axis are the same in the parent_interframe and child_interframe.

Explanation

A cylindrical joint is defined such that the child body both rotates about and translates along the body-fixed joint axis with respect to the parent body. The joint axis is both the rotation axis and translation axis. The location of the joint is defined by two points, one in each body, which coincide when the generalized coordinate corresponding to the translation is zero. The direction cosine matrix between the child interframe and parent interframe is formed using a simple rotation about the joint axis. The page on the joints framework gives a more detailed explanation of the intermediate frames.

Examples

A single cylindrical joint is created between two bodies and has the following basic attributes:

>>> from sympy.physics.mechanics import RigidBody, CylindricalJoint
>>> parent = RigidBody('P')
>>> parent
P
>>> child = RigidBody('C')
>>> child
C
>>> joint = CylindricalJoint('PC', parent, child)
>>> joint
CylindricalJoint: PC  parent: P  child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
P_masscenter
>>> joint.child_point
C_masscenter
>>> joint.parent_axis
P_frame.x
>>> joint.child_axis
C_frame.x
>>> joint.coordinates
Matrix([
[q0_PC(t)],
[q1_PC(t)]])
>>> joint.speeds
Matrix([
[u0_PC(t)],
[u1_PC(t)]])
>>> child.frame.ang_vel_in(parent.frame)
u0_PC(t)*P_frame.x
>>> child.frame.dcm(parent.frame)
Matrix([
[1,              0,             0],
[0,  cos(q0_PC(t)), sin(q0_PC(t))],
[0, -sin(q0_PC(t)), cos(q0_PC(t))]])
>>> joint.child_point.pos_from(joint.parent_point)
q1_PC(t)*P_frame.x
>>> child.masscenter.vel(parent.frame)
u1_PC(t)*P_frame.x

To further demonstrate the use of the cylindrical joint, the kinematics of two cylindrical joints perpendicular to each other can be created as follows.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import RigidBody, CylindricalJoint
>>> r, l, w = symbols('r l w')

First create bodies to represent the fixed floor with a fixed pole on it. The second body represents a freely moving tube around that pole. The third body represents a solid flag freely translating along and rotating around the Y axis of the tube.

>>> floor = RigidBody('floor')
>>> tube = RigidBody('tube')
>>> flag = RigidBody('flag')

The first joint will connect the first tube to the floor with it translating along and rotating around the Z axis of both bodies.

>>> floor_joint = CylindricalJoint('C1', floor, tube, joint_axis=floor.z)

The second joint will connect the tube perpendicular to the flag along the Y axis of both the tube and the flag, with the joint located at a distance r from the tube’s center of mass and a combination of the distances l and w from the flag’s center of mass.

>>> flag_joint = CylindricalJoint('C2', tube, flag,
...                               parent_point=r * tube.y,
...                               child_point=-w * flag.y + l * flag.z,
...                               joint_axis=tube.y)

Once the joints are established the kinematics of the connected bodies can be accessed. First the direction cosine matrices of both the body and the flag relative to the floor are found:

>>> tube.frame.dcm(floor.frame)
Matrix([
[ cos(q0_C1(t)), sin(q0_C1(t)), 0],
[-sin(q0_C1(t)), cos(q0_C1(t)), 0],
[             0,             0, 1]])
>>> flag.frame.dcm(floor.frame)
Matrix([
[cos(q0_C1(t))*cos(q0_C2(t)), sin(q0_C1(t))*cos(q0_C2(t)), -sin(q0_C2(t))],
[             -sin(q0_C1(t)),               cos(q0_C1(t)),              0],
[sin(q0_C2(t))*cos(q0_C1(t)), sin(q0_C1(t))*sin(q0_C2(t)),  cos(q0_C2(t))]])

The position of the flag’s center of mass is found with:

>>> flag.masscenter.pos_from(floor.masscenter)
q1_C1(t)*floor_frame.z + (r + q1_C2(t))*tube_frame.y + w*flag_frame.y - l*flag_frame.z

The angular velocities of the two tubes can be computed with respect to the floor.

>>> tube.frame.ang_vel_in(floor.frame)
u0_C1(t)*floor_frame.z
>>> flag.frame.ang_vel_in(floor.frame)
u0_C1(t)*floor_frame.z + u0_C2(t)*tube_frame.y

Finally, the linear velocities of the two tube centers of mass can be computed with respect to the floor, while expressed in the tube’s frame.

>>> tube.masscenter.vel(floor.frame).to_matrix(tube.frame)
Matrix([
[       0],
[       0],
[u1_C1(t)]])
>>> flag.masscenter.vel(floor.frame).to_matrix(tube.frame).simplify()
Matrix([
[-l*u0_C2(t)*cos(q0_C2(t)) - r*u0_C1(t) - w*u0_C1(t) - q1_C2(t)*u0_C1(t)],
[                    -l*u0_C1(t)*sin(q0_C2(t)) + Derivative(q1_C2(t), t)],
[                                    l*u0_C2(t)*sin(q0_C2(t)) + u1_C1(t)]])

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

rotation_coordinate

(dynamicsymbol) Generalized coordinate corresponding to the rotation angle.

translation_coordinate

(dynamicsymbol) Generalized coordinate corresponding to the translation distance.

rotation_speed

(dynamicsymbol) Generalized speed corresponding to the angular velocity.

translation_speed

(dynamicsymbol) Generalized speed corresponding to the translation velocity.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates.

speeds

(Matrix) Matrix of the joint’s generalized speeds.

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

kdes

(Matrix) Kinematical differential equations of the joint.

joint_axis

(Vector) The axis of rotation and translation.
property joint_axis

Axis about and along which the rotation and translation occurs.

property rotation_coordinate

Generalized coordinate corresponding to the rotation angle.

property rotation_speed

Generalized speed corresponding to the angular velocity.

property translation_coordinate

Generalized coordinate corresponding to the translation distance.

property translation_speed

Generalized speed corresponding to the translation velocity.

class sympy.physics.mechanics.joint.PlanarJoint(
name,
parent,
child,
rotation_coordinate=None,
planar_coordinates=None,
rotation_speed=None,
planar_speeds=None,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
)[source]

Planar Joint.

child_point child_interframe planar_vectors parent_point parent_interframe q1, u1 q0, u0child.masscenter child.frame parent. masscenter parent.frame q2, u2 rotation_axis
Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

rotation_coordinate : dynamicsymbol, optional

Generalized coordinate corresponding to the rotation angle. The default value is dynamicsymbols(f'q0_{joint.name}').

planar_coordinates : iterable of dynamicsymbols, optional

Two generalized coordinates used for the planar translation. The default value is dynamicsymbols(f'q1_{joint.name} q2_{joint.name}').

rotation_speed : dynamicsymbol, optional

Generalized speed corresponding to the angular velocity. The default value is dynamicsymbols(f'u0_{joint.name}').

planar_speeds : dynamicsymbols, optional

Two generalized speeds used for the planar translation velocity. The default value is dynamicsymbols(f'u1_{joint.name} u2_{joint.name}').

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

Explanation

A planar joint is defined such that the child body translates over a fixed plane of the parent body as well as rotate about the rotation axis, which is perpendicular to that plane. The origin of this plane is the parent_point and the plane is spanned by two nonparallel planar vectors. The location of the child_point is based on the planar vectors (\(\vec{v}_1\), \(\vec{v}_2\)) and generalized coordinates (\(q_1\), \(q_2\)), i.e. \(\vec{r} = q_1 \hat{v}_1 + q_2 \hat{v}_2\). The direction cosine matrix between the child_interframe and parent_interframe is formed using a simple rotation (\(q_0\)) about the rotation axis.

In order to simplify the definition of the PlanarJoint, the rotation_axis and planar_vectors are set to be the unit vectors of the parent_interframe according to the table below. This ensures that you can only define these vectors by creating a separate frame and supplying that as the interframe. If you however would only like to supply the normals of the plane with respect to the parent and child bodies, then you can also supply those to the parent_interframe and child_interframe arguments. An example of both of these cases is in the examples section below and the page on the joints framework provides a more detailed explanation of the intermediate frames.

rotation_axis

parent_interframe.x

planar_vectors[0]

parent_interframe.y

planar_vectors[1]

parent_interframe.z

Examples

A single planar joint is created between two bodies and has the following basic attributes:

>>> from sympy.physics.mechanics import RigidBody, PlanarJoint
>>> parent = RigidBody('P')
>>> parent
P
>>> child = RigidBody('C')
>>> child
C
>>> joint = PlanarJoint('PC', parent, child)
>>> joint
PlanarJoint: PC  parent: P  child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
P_masscenter
>>> joint.child_point
C_masscenter
>>> joint.rotation_axis
P_frame.x
>>> joint.planar_vectors
[P_frame.y, P_frame.z]
>>> joint.rotation_coordinate
q0_PC(t)
>>> joint.planar_coordinates
Matrix([
[q1_PC(t)],
[q2_PC(t)]])
>>> joint.coordinates
Matrix([
[q0_PC(t)],
[q1_PC(t)],
[q2_PC(t)]])
>>> joint.rotation_speed
u0_PC(t)
>>> joint.planar_speeds
Matrix([
[u1_PC(t)],
[u2_PC(t)]])
>>> joint.speeds
Matrix([
[u0_PC(t)],
[u1_PC(t)],
[u2_PC(t)]])
>>> child.frame.ang_vel_in(parent.frame)
u0_PC(t)*P_frame.x
>>> child.frame.dcm(parent.frame)
Matrix([
[1,              0,             0],
[0,  cos(q0_PC(t)), sin(q0_PC(t))],
[0, -sin(q0_PC(t)), cos(q0_PC(t))]])
>>> joint.child_point.pos_from(joint.parent_point)
q1_PC(t)*P_frame.y + q2_PC(t)*P_frame.z
>>> child.masscenter.vel(parent.frame)
u1_PC(t)*P_frame.y + u2_PC(t)*P_frame.z

To further demonstrate the use of the planar joint, the kinematics of a block sliding on a slope, can be created as follows.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import PlanarJoint, RigidBody, ReferenceFrame
>>> a, d, h = symbols('a d h')

First create bodies to represent the slope and the block.

>>> ground = RigidBody('G')
>>> block = RigidBody('B')

To define the slope you can either define the plane by specifying the planar_vectors or/and the rotation_axis. However it is advisable to create a rotated intermediate frame, so that the parent_vectors and rotation_axis will be the unit vectors of this intermediate frame.

>>> slope = ReferenceFrame('A')
>>> slope.orient_axis(ground.frame, ground.y, a)

The planar joint can be created using these bodies and intermediate frame. We can specify the origin of the slope to be d above the slope’s center of mass and the block’s center of mass to be a distance h above the slope’s surface. Note that we can specify the normal of the plane using the rotation axis argument.

>>> joint = PlanarJoint('PC', ground, block, parent_point=d * ground.x,
...                     child_point=-h * block.x, parent_interframe=slope)

Once the joint is established the kinematics of the bodies can be accessed. First the rotation_axis, which is normal to the plane and the plane_vectors, can be found.

>>> joint.rotation_axis
A.x
>>> joint.planar_vectors
[A.y, A.z]

The direction cosine matrix of the block with respect to the ground can be found with:

>>> block.frame.dcm(ground.frame)
Matrix([
[              cos(a),              0,              -sin(a)],
[sin(a)*sin(q0_PC(t)),  cos(q0_PC(t)), sin(q0_PC(t))*cos(a)],
[sin(a)*cos(q0_PC(t)), -sin(q0_PC(t)), cos(a)*cos(q0_PC(t))]])

The angular velocity of the block can be computed with respect to the ground.

>>> block.frame.ang_vel_in(ground.frame)
u0_PC(t)*A.x

The position of the block’s center of mass can be found with:

>>> block.masscenter.pos_from(ground.masscenter)
d*G_frame.x + h*B_frame.x + q1_PC(t)*A.y + q2_PC(t)*A.z

Finally, the linear velocity of the block’s center of mass can be computed with respect to the ground.

>>> block.masscenter.vel(ground.frame)
u1_PC(t)*A.y + u2_PC(t)*A.z

In some cases it could be your preference to only define the normals of the plane with respect to both bodies. This can most easily be done by supplying vectors to the interframe arguments. What will happen in this case is that an interframe will be created with its x axis aligned with the provided vector. For a further explanation of how this is done see the notes of the Joint class. In the code below, the above example (with the block on the slope) is recreated by supplying vectors to the interframe arguments. Note that the previously described option is however more computationally efficient, because the algorithm now has to compute the rotation angle between the provided vector and the ‘x’ axis.

>>> from sympy import symbols, cos, sin
>>> from sympy.physics.mechanics import PlanarJoint, RigidBody
>>> a, d, h = symbols('a d h')
>>> ground = RigidBody('G')
>>> block = RigidBody('B')
>>> joint = PlanarJoint(
...     'PC', ground, block, parent_point=d * ground.x,
...     child_point=-h * block.x, child_interframe=block.x,
...     parent_interframe=cos(a) * ground.x + sin(a) * ground.z)
>>> block.frame.dcm(ground.frame).simplify()
Matrix([
[               cos(a),              0,               sin(a)],
[-sin(a)*sin(q0_PC(t)),  cos(q0_PC(t)), sin(q0_PC(t))*cos(a)],
[-sin(a)*cos(q0_PC(t)), -sin(q0_PC(t)), cos(a)*cos(q0_PC(t))]])

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

rotation_coordinate

(dynamicsymbol) Generalized coordinate corresponding to the rotation angle.

planar_coordinates

(Matrix) Two generalized coordinates used for the planar translation.

rotation_speed

(dynamicsymbol) Generalized speed corresponding to the angular velocity.

planar_speeds

(Matrix) Two generalized speeds used for the planar translation velocity.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates.

speeds

(Matrix) Matrix of the joint’s generalized speeds.

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

kdes

(Matrix) Kinematical differential equations of the joint.

rotation_axis

(Vector) The axis about which the rotation occurs.

planar_vectors

(list) The vectors that describe the planar translation directions.
property planar_coordinates

Two generalized coordinates used for the planar translation.

property planar_speeds

Two generalized speeds used for the planar translation velocity.

property planar_vectors

The vectors that describe the planar translation directions.

property rotation_axis

The axis about which the rotation occurs.

property rotation_coordinate

Generalized coordinate corresponding to the rotation angle.

property rotation_speed

Generalized speed corresponding to the angular velocity.

class sympy.physics.mechanics.joint.SphericalJoint(
name,
parent,
child,
coordinates=None,
speeds=None,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
rot_type='BODY',
amounts=None,
rot_order=123,
)[source]

Spherical (Ball-and-Socket) Joint.

../../../../_images/SphericalJoint.svg
Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

coordinates: iterable of dynamicsymbols, optional

Generalized coordinates of the joint.

speeds : iterable of dynamicsymbols, optional

Generalized speeds of joint.

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

rot_type : str, optional

The method used to generate the direction cosine matrix. Supported methods are:

  • 'Body': three successive rotations about new intermediate axes, also called “Euler and Tait-Bryan angles”

  • 'Space': three successive rotations about the parent frames’ unit vectors

The default method is 'Body'.

amounts :

Expressions defining the rotation angles or direction cosine matrix. These must match the rot_type. See examples below for details. The input types are:

  • 'Body': 3-tuple of expressions, symbols, or functions

  • 'Space': 3-tuple of expressions, symbols, or functions

The default amounts are the given coordinates.

rot_order : str or int, optional

If applicable, the order of the successive of rotations. The string '123' and integer 123 are equivalent, for example. Required for 'Body' and 'Space'. The default value is 123.

Explanation

A spherical joint is defined such that the child body is free to rotate in any direction, without allowing a translation of the child_point. As can also be seen in the image, the parent_point and child_point are fixed on top of each other, i.e. the joint_point. This rotation is defined using the parent_interframe.orient(child_interframe, rot_type, amounts, rot_order) method. The default rotation consists of three relative rotations, i.e. body-fixed rotations. Based on the direction cosine matrix following from these rotations, the angular velocity is computed based on the generalized coordinates and generalized speeds.

Examples

A single spherical joint is created from two bodies and has the following basic attributes:

>>> from sympy.physics.mechanics import RigidBody, SphericalJoint
>>> parent = RigidBody('P')
>>> parent
P
>>> child = RigidBody('C')
>>> child
C
>>> joint = SphericalJoint('PC', parent, child)
>>> joint
SphericalJoint: PC  parent: P  child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
P_masscenter
>>> joint.child_point
C_masscenter
>>> joint.parent_interframe
P_frame
>>> joint.child_interframe
C_frame
>>> joint.coordinates
Matrix([
[q0_PC(t)],
[q1_PC(t)],
[q2_PC(t)]])
>>> joint.speeds
Matrix([
[u0_PC(t)],
[u1_PC(t)],
[u2_PC(t)]])
>>> child.frame.ang_vel_in(parent.frame).to_matrix(child.frame)
Matrix([
[ u0_PC(t)*cos(q1_PC(t))*cos(q2_PC(t)) + u1_PC(t)*sin(q2_PC(t))],
[-u0_PC(t)*sin(q2_PC(t))*cos(q1_PC(t)) + u1_PC(t)*cos(q2_PC(t))],
[                             u0_PC(t)*sin(q1_PC(t)) + u2_PC(t)]])
>>> child.frame.x.to_matrix(parent.frame)
Matrix([
[                                            cos(q1_PC(t))*cos(q2_PC(t))],
[sin(q0_PC(t))*sin(q1_PC(t))*cos(q2_PC(t)) + sin(q2_PC(t))*cos(q0_PC(t))],
[sin(q0_PC(t))*sin(q2_PC(t)) - sin(q1_PC(t))*cos(q0_PC(t))*cos(q2_PC(t))]])
>>> joint.child_point.pos_from(joint.parent_point)
0

To further demonstrate the use of the spherical joint, the kinematics of a spherical joint with a ZXZ rotation can be created as follows.

>>> from sympy import symbols
>>> from sympy.physics.mechanics import RigidBody, SphericalJoint
>>> l1 = symbols('l1')

First create bodies to represent the fixed floor and a pendulum bob.

>>> floor = RigidBody('F')
>>> bob = RigidBody('B')

The joint will connect the bob to the floor, with the joint located at a distance of l1 from the child’s center of mass and the rotation set to a body-fixed ZXZ rotation.

>>> joint = SphericalJoint('S', floor, bob, child_point=l1 * bob.y,
...                        rot_type='body', rot_order='ZXZ')

Now that the joint is established, the kinematics of the connected body can be accessed.

The position of the bob’s masscenter is found with:

>>> bob.masscenter.pos_from(floor.masscenter)
- l1*B_frame.y

The angular velocities of the pendulum link can be computed with respect to the floor.

>>> bob.frame.ang_vel_in(floor.frame).to_matrix(
...     floor.frame).simplify()
Matrix([
[u1_S(t)*cos(q0_S(t)) + u2_S(t)*sin(q0_S(t))*sin(q1_S(t))],
[u1_S(t)*sin(q0_S(t)) - u2_S(t)*sin(q1_S(t))*cos(q0_S(t))],
[                          u0_S(t) + u2_S(t)*cos(q1_S(t))]])

Finally, the linear velocity of the bob’s center of mass can be computed.

>>> bob.masscenter.vel(floor.frame).to_matrix(bob.frame)
Matrix([
[                           l1*(u0_S(t)*cos(q1_S(t)) + u2_S(t))],
[                                                             0],
[-l1*(u0_S(t)*sin(q1_S(t))*sin(q2_S(t)) + u1_S(t)*cos(q2_S(t)))]])

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates.

speeds

(Matrix) Matrix of the joint’s generalized speeds.

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

kdes

(Matrix) Kinematical differential equations of the joint.
class sympy.physics.mechanics.joint.WeldJoint(
name,
parent,
child,
parent_point=None,
child_point=None,
parent_interframe=None,
child_interframe=None,
)[source]

Weld Joint.

attachment points parent.masscenter child.masscenter parent.frame child.frame attachment frames
Parameters:

name : string

A unique name for the joint.

parent : Particle or RigidBody or Body

The parent body of joint.

child : Particle or RigidBody or Body

The child body of joint.

parent_point : Point or Vector, optional

Attachment point where the joint is fixed to the parent body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the parent’s mass center.

child_point : Point or Vector, optional

Attachment point where the joint is fixed to the child body. If a vector is provided, then the attachment point is computed by adding the vector to the body’s mass center. The default value is the child’s mass center.

parent_interframe : ReferenceFrame, optional

Intermediate frame of the parent body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the parent’s own frame.

child_interframe : ReferenceFrame, optional

Intermediate frame of the child body with respect to which the joint transformation is formulated. If a Vector is provided then an interframe is created which aligns its X axis with the given vector. The default value is the child’s own frame.

Explanation

A weld joint is defined such that there is no relative motion between the child and parent bodies. The direction cosine matrix between the attachment frame (parent_interframe and child_interframe) is the identity matrix and the attachment points (parent_point and child_point) are coincident. The page on the joints framework gives a more detailed explanation of the intermediate frames.

Examples

A single weld joint is created from two bodies and has the following basic attributes:

>>> from sympy.physics.mechanics import RigidBody, WeldJoint
>>> parent = RigidBody('P')
>>> parent
P
>>> child = RigidBody('C')
>>> child
C
>>> joint = WeldJoint('PC', parent, child)
>>> joint
WeldJoint: PC  parent: P  child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
P_masscenter
>>> joint.child_point
C_masscenter
>>> joint.coordinates
Matrix(0, 0, [])
>>> joint.speeds
Matrix(0, 0, [])
>>> child.frame.ang_vel_in(parent.frame)
0
>>> child.frame.dcm(parent.frame)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> joint.child_point.pos_from(joint.parent_point)
0

To further demonstrate the use of the weld joint, two relatively-fixed bodies rotated by a quarter turn about the Y axis can be created as follows:

>>> from sympy import symbols, pi
>>> from sympy.physics.mechanics import ReferenceFrame, RigidBody, WeldJoint
>>> l1, l2 = symbols('l1 l2')

First create the bodies to represent the parent and rotated child body.

>>> parent = RigidBody('P')
>>> child = RigidBody('C')

Next the intermediate frame specifying the fixed rotation with respect to the parent can be created.

>>> rotated_frame = ReferenceFrame('Pr')
>>> rotated_frame.orient_axis(parent.frame, parent.y, pi / 2)

The weld between the parent body and child body is located at a distance l1 from the parent’s center of mass in the X direction and l2 from the child’s center of mass in the child’s negative X direction.

>>> weld = WeldJoint('weld', parent, child, parent_point=l1 * parent.x,
...                  child_point=-l2 * child.x,
...                  parent_interframe=rotated_frame)

Now that the joint has been established, the kinematics of the bodies can be accessed. The direction cosine matrix of the child body with respect to the parent can be found:

>>> child.frame.dcm(parent.frame)
Matrix([
[0, 0, -1],
[0, 1,  0],
[1, 0,  0]])

As can also been seen from the direction cosine matrix, the parent X axis is aligned with the child’s Z axis: >>> parent.x == child.z True

The position of the child’s center of mass with respect to the parent’s center of mass can be found with:

>>> child.masscenter.pos_from(parent.masscenter)
l1*P_frame.x + l2*C_frame.x

The angular velocity of the child with respect to the parent is 0 as one would expect.

>>> child.frame.ang_vel_in(parent.frame)
0

Attributes

name

(string) The joint’s name.

parent

(Particle or RigidBody or Body) The joint’s parent body.

child

(Particle or RigidBody or Body) The joint’s child body.

coordinates

(Matrix) Matrix of the joint’s generalized coordinates. The default value is dynamicsymbols(f'q_{joint.name}').

speeds

(Matrix) Matrix of the joint’s generalized speeds. The default value is dynamicsymbols(f'u_{joint.name}').

parent_point

(Point) Attachment point where the joint is fixed to the parent body.

child_point

(Point) Attachment point where the joint is fixed to the child body.

parent_interframe

(ReferenceFrame) Intermediate frame of the parent body with respect to which the joint transformation is formulated.

child_interframe

(ReferenceFrame) Intermediate frame of the child body with respect to which the joint transformation is formulated.

kdes

(Matrix) Kinematical differential equations of the joint.