Joints Framework in Physics/Mechanics

sympy.physics.mechanics provides a joints framework. This system consists of two parts. The first are the joints themselves, which are used to create connections between bodies. The second part is the System, which is used to form the equations of motion. Both of these parts are doing what we can call “book-keeping”: keeping track of the relationships between bodies.

Joints in Physics/Mechanics

The general task of the joints is creating kinematic relationships between bodies. A joint is generally described as shown in the image below.

As can be seen in this image, each joint needs several objects in order to define the relationships. First off it needs two bodies: the parent body (shown in green) and the child body (shown in blue). The transformation made by the joint is defined between the joint attachments of both bodies. A joint attachment of a body consists of a point and a body-fixed frame. In the parent body the point is called parent_point and the frame parent_interframe. For the child body these are called child_point and child_interframe. For most joints it is the case that when the generalized coordinates are zero, that there is no rotation or translation between the parent and child joint attachments. So the child_point is at the same location as the parent_point and the child_interframe is in the same orientation as the parent_interframe.

For describing the joint transformation the joint generally needs dynamicsymbols() for the generalized coordinates and speeds. Some joints like the PinJoint, PrismaticJoint also require a joint_axis, which consists of the same components in the parent_interframe and child_interframe. This means that if for example the joint axis is defined in the parent_interframe as \(2\hat{p}_x + 4\hat{p}_y + 3\hat{p}_z\), then this will also be \(2\hat{c}_x + 4\hat{c}_y + 3\hat{c}_z\) in the child_interframe. Practically this means that in the case of the PinJoint, also shown below, the joint_axis is the axis of rotation, with the generalized coordinate \(q\) as the angle of rotation and the generalized speed \(u\) as the angular velocity.

With the information listed above, the joint defines the following relationships. It first defines the kinematic differential equations, which relate the generalized coordinates to the generalized speeds. Next, it orients the parent and child body with respect to each other. After which it also defines their velocity relationships.

The code below shows the creation of a PinJoint as shown above with arbitrary linked position vectors. In this code the attachment points are set using vectors, which define the attachment point with respect to the body’s mass center. The intermediate frames are not set, so those are the same as the body’s frame.

>>> from sympy.physics.mechanics import *
>>> mechanics_printing(pretty_print=False)
>>> q, u = dynamicsymbols('q, u')
>>> parent = RigidBody('parent')
>>> child = RigidBody('child')
>>> joint = PinJoint(
...     'hinge', parent, child, coordinates=q, speeds=u,
...     parent_point=3 * parent.frame.x,
...     child_point=-3 * child.frame.x,
...     joint_axis=parent.frame.z)
>>> joint.kdes
Matrix([[u - q']])
>>> joint.parent_point.pos_from(parent.masscenter)
3*parent_frame.x
>>> joint.parent_interframe
parent_frame
>>> joint.joint_axis.express(child.frame)
child_frame.z
>>> child.masscenter.pos_from(parent.masscenter)
3*parent_frame.x + 3*child_frame.x
>>> child.masscenter.vel(parent.frame)
3*u*child_frame.y

System in Physics/Mechanics

After defining the entire system you can use the System to parse the system and form the equations of motion. In this process the System only does the “book-keeping” of the joints. It uses another method, like the KanesMethod, as its backend for forming the equations of motion.

In the code below we form the equations of motion of the single PinJoint shown previously.

>>> system = System.from_newtonian(parent)
>>> system.add_joints(joint)
>>> system.form_eoms()
Matrix([[-(child_izz + 9*child_mass)*u']])
>>> type(system.eom_method)  # The method working in the backend
<class 'sympy.physics.mechanics.kane.KanesMethod'>