A planar mechanism combines rigid links and joints that restrict their relative motion. Use the mobility count to compare available coordinates with independent constraints, and explore a separate four-bar example.
Generic Planar Mobility Count
Enter total links including ground, lower pairs and higher pairs. The worksheet reports each term of the generic planar count; actual mobility still requires checking constraint independence and geometry.
Equation Used
- All bodies are rigid and motion is planar.
- n includes one fixed ground link.
- All counted constraints are independent and there are no special geometric dependencies.
- The result is a generic count, not proof of assembly, range of motion or local mobility.
The worksheet counts coordinates and constraints for the entered numbers of links and joints. The animation shows one fixed four-bar example; changing the counts does not create a corresponding linkage drawing.
Coordinates minus independent constraints
A free rigid body in a plane has three coordinates: two translations and one rotation. With one link fixed as ground, n links begin with 3(n-1) moving-link coordinates. A revolute or prismatic lower pair leaves one relative freedom and therefore contributes two planar constraints; a higher-pair contact contributes one in this generic count.
The subtraction is valid only when the constraints are independent.
An initial mobility check
Use the worksheet to calculate a planar constraint count before checking the actual linkage geometry. Do not use it to conclude that a linkage assembles, has a desired motion, accepts one actuator, avoids singularities or carries a load.
Planar Kutzbach-Gruebler count
MIT sources present this count and also document its limitation: redundant constraints or special geometric dependencies can make actual mobility differ from the generic result.
Four-bar count example
A planar four-bar has n=4 links including ground, j1=4 revolute joints and j2=0 higher pairs. The worksheet gives 3(4-1)-2(4)=1. The separate animation shows this specific constant-length parallelogram example; the worksheet inputs do not change its link count.
What the count cannot establish
Actual mobility depends on the kinematic graph, joint freedoms, independent constraints and geometry; performance requires separate load and dynamic analysis.
Mobility-count questions
Does M_count=1 prove a one-DOF mechanism?
No. It is a generic count under the stated independence assumptions.
Can actual mobility differ?
Yes. Redundant constraints and special geometry are known exceptions.
Does a negative result prove the assembly cannot move?
No. It signals that the generic assumptions or count need closer examination.
Authoritative mobility references
- MIT 6.S080, Gruebler's equation: planar link and pivot constraint count with a four-bar example.
- MIT thesis, Reconfigurable mechanisms: extended 3(N-1)-2L-H form.
- MIT-hosted kinematic research: limitations for redundant constraints and special geometry.
- Rothenhofer et al., Precision Engineering: mobility count followed by geometry-specific constraint equations.
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