page_title: Linkage (mechanical)
section_title: Mobility
hierarchical_section_title: Linkage (mechanical) / Mobility
caption_attribution_description: English: 4 bar linkage
context_page_description: A mechanical linkage is an assembly of bodies connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a kinematic chain.
  Linkages may be constructed from open chains, closed chains, or a combination of open and closed chains. Each link in a chain is connected by a joint to one or more other links. Thus, a kinematic chain can be modeled as a graph in which the links are paths and the joints are vertices, which is called a linkage graph.
  The movement of an ideal joint is generally associated with a subgroup of the group of Euclidean displacements. The number of parameters in the subgroup is called the degrees of freedom of the joint.
  Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement.
context_section_description: The configuration of a system of rigid links connected by ideal joints is defined by a set of configuration parameters, such as the angles around a revolute joint and the slides along prismatic joints measured between adjacent links.  The geometric constraints of the linkage allow calculation of all of the configuration parameters in terms of a minimum set, which are the input parameters.  The number of input parameters is called the mobility, or degree of freedom, of the linkage system.
  A system of n rigid bodies moving in space has 6n degrees of freedom measured relative to a fixed frame.  Include this frame in the count of bodies, so that mobility is independent of the choice of the fixed frame, then we have M = 6(N − 1), where N = n + 1 is the number of moving bodies plus the fixed body.
  Joints that connect bodies in this system remove degrees of freedom and reduce mobility. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom.  It is convenient to define the number of constraints c that a joint imposes in terms of the joint's freedom f, where c = 6 − f.  In the case of a hinge or slider, which are one degree of freedom joints, we have f = 1 and therefore c = 6 − 1 = 5.
  Thus, the mobility of a linkage system formed from n moving links and j joints each with fᵢ, i = 1, ..., j,  degrees of freedom can be computed as,
  where N includes the fixed link.  This is known as Kutzbach–Grübler's equation
  There are two important special cases: (i) a simple open chain, and (ii) a simple closed chain. A simple open chain consists of n moving links connected end to end by j joints, with one end connected to a ground link.  Thus, in this case N = j + 1  and the mobility of the chain is
  For a simple closed chain, n moving links are connected end-to-end by n+1 joints such that the two ends are connected to the ground link forming a loop.  In this case, we have N=j and the mobility of the chain is
  An example of a simple open chain is a serial robot manipulator.  These robotic systems are constructed from a series of links connected by six one degree-of-freedom revolute or prismatic joints, so the system has six degrees of freedom.
  An example of a simple closed chain is the RSSR spatial four-bar linkage.  The sum of the freedom of these joints is eight, so the mobility of the linkage is two, where one of the degrees of freedom is the rotation of the coupler around the line joining the two S joints.
caption: Simple linkages are capable of producing complicated motion.