Kinematic pair

In the field of classical mechanics, a kinematic pair is a vital connection between two physical objects. These objects are known as rigid bodies. This connection is important because it imposes constraints on how the objects move relative to one another. The study of these movements is called kinematics. Some scientists refer to kinematics as the "geometry of motion." This branch of mechanics describes the motion of points, bodies, and systems of bodies. Notably, kinematics focuses on the motion itself without considering the forces that cause it.

To understand how machines work, we must look at the history of this concept. German engineer Franz Reuleaux introduced the kinematic pair as a new way to study machines. His approach was a major advancement over older ideas that focused on simple machines. Reuleaux identified two distinct kinds of connections between rigid bodies. He categorized them as higher pairs and lower pairs. This distinction helped engineers understand how different parts of a mechanism interact.

Lower pairs are connections where the two parts have area contact. This means they touch over a surface rather than just a single point. Because of this surface contact, the relative motions of the parts are considered similar. If you were to swap the elements between the links, the relative motion would not change. Common examples include pin connections, crossheads, and ball-and-socket joints. A nut and screw is another example of a lower pair.

Engineers use specific types of lower pairs to control movement through degrees of freedom. A revolute joint, or R joint, acts like a hinge. It has only one degree of freedom because it imposes five constraints on the links. A prismatic joint, or P joint, allows a part to slide along a line. This also has one degree of freedom. A screw joint, or helical H joint, uses cut threads to combine turning and sliding motions. This joint also possesses one degree of freedom. A cylindrical joint, or C joint, combines revolute and prismatic motions and has two degrees of freedom.

Other lower pairs allow for even more complex movement. A spherical joint, or S joint, is also called a ball-and-socket joint. This joint has three degrees of freedom, allowing rotation around three different axes. A planar joint allows a part to slide in two dimensions and rotate on an axis. This joint also provides three degrees of freedom. There is also a universal joint, or U joint, which consists of two intersecting revolute joints. Finally, a parallelogram joint, or Pa joint, uses four links and four revolute joints to form a parallelogram shape.

In contrast, higher pairs work through point or line contact. In these connections, the two elements touch at a very small area. Because the contact is so small, the relative motions of the points are dissimilar. A common example is a ball bearing. Another example is a disk cam and a follower. You can also see higher pairs in the way gear teeth mesh together. Even a wheel rolling on a flat surface is considered a higher pair. Some connections, like belts or chains, are called wrapping pairs. These are similar to higher pairs but involve multiple points of contact.

In robotics and mechanism design, these links and joints form a specific arrangement called topology. Engineers use joint notation to describe this topology clearly. They use abbreviations like P, R, U, C, S, and Pa. For a serial manipulator, like a SCARA robot, the notation shows the sequence from the base to the end effector. For example, the notation RRP describes a robot with two active revolute joints and one active prismatic joint. Parallel manipulators, such as the Gough-Stewart mechanism, use multiple serial chains called limbs. Their notation, such as 6-UPS, describes the specific joints in each limb.