For a connected planar linkage containing only one-degree-of-freedom lower pairs and no redundant constraints, mobility is M = 3(n − 1) − 2j. A classical six-link, seven-revolute-joint topology therefore has one degree of freedom.
Six-bar Linkage and Planar Mobility
Explore a moving Watt-type six-bar linkage and calculate a separate planar mobility count.
Equation Used
- The mechanism is connected and planar.
- All counted joints are one-DOF lower pairs represented as binary-joint equivalents.
- Constraints are independent and nonredundant.
- Special geometric dependencies, higher pairs and singularities are excluded.
The moving example has six links and seven pin joints. The count controls change only the separate mobility arithmetic.
Mobility count and six-bar topology example
The Watt-type animation has two adjacent rigid triangular links, a rotating input crank, two upper connecting links and a fixed frame. Seven pin joints connect the six links. The triangular links each carry three joints and remain rigid as the crank turns. Selected count controls affect the mobility arithmetic, not the illustrative link dimensions.
What the calculator determines
Select total links n, including ground, and revolute-joint equivalents j. The calculator reports moving links, two scalar constraints per joint, planar mobility and the nonnegative count of independent drivers implied by that mobility.
Planar Grübler count
M = 3(n − 1) − 2j for the declared all-revolute lower-pair assumptions.
Six-bar example
For n = 6 and j = 7, M = 3(5) − 14 = 1.
Topology count only
The count can fail for disconnected assemblies, redundant or dependent constraints, special geometry, higher pairs, coincident joints interpreted incorrectly, or mechanisms at singular configurations. It does not synthesize link lengths or motion.
Questions
Does every n=6, j=7 linkage move correctly?
No. The formula is a generic mobility count under stated assumptions.
Does a negative result mean negative physical motion?
No; it flags an overconstrained selected count.
References & Further Reading
- Bob Williams, Ohio University, An Atlas of Structures, Mechanisms, and Robots, page 11, Watt I topology (illustrative proportions here). Wikipedia contributors. Six-bar linkage. Wikipedia
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