A kinematic chain connects rigid links with joints so that motion of one link constrains the others. The animation uses a canonical closed four-bar: a rotating input crank, coupler and rocking output return to the fixed ground link with exact geometric closure.
Kinematic Chain Interactive Calculator
The entered totals feed a separate planar Grübler scalar count. They do not define the animated topology.
Equation Used
- One link is ground and all counted motion is planar.
- Every entered joint is a one-DOF lower pair imposing two independent constraints.
- No redundant constraints, higher pairs or special geometry are represented by the scalar inputs.
Constraint independence and special geometry are not checked. A negative scalar count does not by itself prove physical overconstraint.
Closed-chain motion
The animated four-bar has four links and four revolute joints. Once the input-crank angle is set, circle intersection closes the coupler and rocker at their fixed lengths. The red tracing point is fixed to the triangular coupler extension. Its dashed path is the locus of that point over one crank revolution. The demonstration lengths are 120, 345 and 255 with a fixed span of 430, in arbitrary units; they are not dimensions inferred from the entered link and joint counts. The stacked bars are shown in a schematic projection, without a joint-hardware or collision-clearance model.
Use mobility as an initial count
The four-bar is the simplest common closed lower-pair linkage. For an arbitrary chain, topology and constraint rank still must be checked separately.
Grübler count
M=3(n−1)−2j under the stated assumptions. Each moving planar rigid link contributes three freedoms and each ideal revolute or prismatic lower pair removes two.
Four-bar example
Four links, including ground, and four one-DOF joints give M=3(4−1)−2(4)=1. Once an assembly branch is chosen, one input angle determines this example’s pose. The count alone would not select between the two circle-intersection branches.
Limits of the mobility count
The four-bar terminology and classification follow Carnegie Mellon University's Introduction to Mechanisms. The scalar count omits redundancy, higher pairs, multiple joints, special geometry, singular configurations and Jacobian rank.
Questions
Do the link and joint totals redraw my chain?
No. Totals alone do not specify connectivity or dimensions; the animation remains a labeled four-bar example.
Does negative M prove overconstraint?
No. Constraint dependence and geometry must be analyzed.
References & Further Reading
- Wikipedia contributors. Kinematic chain. Wikipedia
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