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 6
n 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, :M = 6n - \sum_{i=1}^j (6 - f_i) = 6(N-1 - j) + \sum_{i=1}^j\ f_i, 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 : M = \sum_{i=1}^j\ f_i . 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 : M = \sum_{i=1}^j\ f_i - 6. 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 (revolute-spherical-spherical-revolute) 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.
Planar and spherical movement exemplify a four-bar, one
degree of freedom mechanical linkage. The adjustable base pivot makes this a two degree-of-freedom
five-bar linkage. It is common practice to design the linkage system so that the movement of all of the bodies are constrained to lie on parallel planes, to form what is known as a
planar linkage. It is also possible to construct the linkage system so that all of the bodies move on concentric spheres, forming a
spherical linkage. In both cases, the degrees of freedom of the link is now three rather than six, and the constraints imposed by joints are now
c = 3 −
f. In this case, the mobility formula is given by :M = 3(N- 1 - j)+ \sum_{i=1}^j\ f_i, and we have the special cases, • planar or spherical simple open chain, :: M = \sum_{i=1}^j\ f_i, • planar or spherical simple closed chain, :: M = \sum_{i=1}^j\ f_i - 3. An example of a planar simple closed chain is the planar four-bar linkage, which is a four-bar loop with four one degree-of-freedom joints and therefore has mobility
M = 1. == Joints ==