Kinematics applies geometry to the analysis of movement, or motion, of a mechanical system. The rotation and sliding movement central a mechanical system is modeled mathematically as Euclidean, or rigid, transformations. The set of rigid transformations in three dimensional space forms a Lie group, denoted as SE(3).
While all motion in a mechanical system occurs in three dimensional space, planar motion can be analyzed using plane geometry, if all point trajectories are parallel to a plane. In this case the system is called a planar mechanism (or robot). The kinematic analysis of planar mechanisms uses the subset of SE(3) consisting of planar rotations and translations, denoted SE(2).
The group SE(2) is three dimensional, which means that every position of a body in the plane is defined by three parameters. The parameters are often the x and y coordinates of the origin of a coordinate frame in M measured from the origin of a coordinate frame in F, and the angle measured from the x-axis in F to the x-axis in M. This is described saying a body in the plane has three degrees-of-freedom. SE(2) is the configuration space for a planar body, and a planar motion is a curve in this space.
It is possible to construct a mechanism system such that the point trajectories in all components lie in concentric spherical shells around a fixed point. An example is the gimbaled gyroscope. These devices are called spherical mechanisms. Spherical mechanisms are constructed by connecting links with hinged joints such that the axes of each hinge passes through the same point. This point becomes center of the concentric spherical shells. The movement of these mechanisms is characterized by the group SO(3) of rotations in three dimensional space. Other examples of spherical mechanisms are the automotive differential and the robotic wrist.
Select this link for an animation of a Spherical deployable mechanism.
The rotation group SO(3) is three dimensional. An example of the three parameters that specify a spatial rotation are the roll, pitch and yaw angles used to define the orientation of an aircraft. SO(3) is the configuration space for a rotating body, and a spherical motion is a curve in this space.
A mechanical system in which a body moves through a general spatial movement is called a spatial mechanism. An example is the RSSR linkage, which can be viewed as a four-bar linkage in which the hinged joints of the coupler link are replaced by rod ends, also called spherical joints or ball joints. The rod ends allow the input and output cranks of the RSSR linkage to be misaligned to the point that they lie in different planes, which causes the coupler link to move in a general spatial movement. Robot arms, Stewart platforms, and humanoid robotic systems are also examples of spatial mechanisms.
Select this link for an animation of Bennett's linkage, which is a spatial mechanism constructed from four hinged joints.
The group SE(3) is six dimensional, which means the position of a body in space is defined by six parameters. Three of the parameters define the origin of the moving reference frame relative to the fixed frame. Three other parameters define the orientation of the moving frame relative to the fixed frame. SE(3) is the configuration space for a body moving in space, and a spatial motion is a curve in this space.