In this Demonstration, links 1 and 3 are connected to the ground plane and are separated by a distance that can be adjusted by the slider "separation (

)". The other three link lengths are set by the interval slider.

The four-bar linkage has four links, which are connected in a closed chain. The four-bar linkage is therefore a parallel mechanism. Grübler's formula states that the number of degrees of freedom of a planar mechanism is given by

.

In this Demonstration,

is the number of degrees of freedom,

is the number of links,

is the number of joints, and

is the number of degrees of freedom provided by joint

. The

of the four-bar linkage is therefore

.

The kinematic loop constraint equations for the four-bar linkage are given in [1] as:

,

,

, and

.

Here

,

and

are the lengths of the three links and

is the separation distance of links 1 and 3. These equations can be found using the law of cosines. The

sign in the expression for

gives two solutions. The more positive solution of

is drawn with an orange line, and the more negative solution with a blue line.

Depending on the link lengths chosen, the configuration space curve may have bifurcation points where branches of the curve meet. Snapshot 1 with link lengths

has bifurcations at

,

,

and

. Snapshot 2 has no bifurcations, and Snapshot 3 with link lengths

has a bifurcation at

.

Vertical lines in the configuration space plot are spurious.

[1] K. M. Lynch and F. C. Park,

*Modern Robotics: Mechanics, Planning, and Control*, New York: Cambridge University Press, 2017.