Configuration Space for Four-Bar Linkage

separation (

)

0.75

link lengths (

)

{1.35,2}

A planar four-bar linkage has four joints, so its configuration is described by four numbers. However, because the linkage is a closed chain, the resulting constraints make most configurations unattainable. The linkage's configuration space is a one-dimensional curve embedded in a four-dimensional space. The plot at left projects the configuration space onto the joint angles

(θ,ϕ)

of the first and third links. You can drag the locator to explore the set of reachable configurations. Because

and

are angles, this configuration space wraps around from

-π

. If any link is greater than the sum of the other three links, no valid configurations exist.

Details

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

DOF=3(N-1-J)+

∑

i=1

In this Demonstration,

DOF

is the number of degrees of freedom,

N=4

is the number of links,

J=4

is the number of joints, and

is the number of degrees of freedom provided by joint

. The

DOF

of the four-bar linkage is therefore

3(4-1-4)+4=1

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

ϕ=arctan(α,β)±arcos

where

α=2

-2

cosθ

β=-2

sinθ

, and

γ=

cosθ

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

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

[1,1,0.5,0.5]

has bifurcations at

(0,π)

(0,-π)

(π,π)

and

(-π,-π)

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

[0.9,1.2,0.5,0.8]

has a bifurcation at

(0,0)

Vertical lines in the configuration space plot are spurious.