Burmester theory helps choose linkage pivots for prescribed rigid-body positions. The interactive example below shows a real, constrained four-bar layout and measures its sensitivity to a misplaced fixed pivot. It is a kinematic demonstration, not a general synthesis solver.
Burmester Theory Linkage Interactive Calculator
Change nominal frame length and the right ground-pivot vertical offset. The calculator closes the linkage at four reference input angles and measures the displacement of point C relative to the nominal mechanism.
Equation Used
- Rigid links, ideal revolute joints, the same continuous upper assembly branch and the same input angle for nominal and offset mechanisms.
- All link lengths scale with nominal frame length; the vertical error is applied only to D. Lengths do not change during motion.
- Point displacement is reported separately from body orientation. It is not a combined pose metric, manufacturing tolerance guarantee or full-path bound.
- This preselected example demonstrates analysis and sensitivity. It does not find Burmester curves or accept prescribed target poses.
The former fixed error multipliers have been replaced with circle-intersection kinematics. Four is the number of poses checked here, not the maximum allowed by Burmester theory.
Body guidance and Burmester theory
Burmester synthesis seeks fixed and moving pivots that allow a rigid body to visit prescribed positions and orientations. Four-position synthesis generally produces families of candidate dyads. Five-position synthesis is also possible, although usable real solutions and practical motion are not guaranteed. Candidate linkages must be checked for assembly branch, motion order and interference.
The animation demonstrates the resulting type of mechanism: two pivoted links guide a rigid coupler plate. It uses a selected example rather than solving a new synthesis problem. Pale plates show reference poses, and the marked point C gives a consistent point for error measurement.
Where finite-position guidance helps
Linkage synthesis is useful when a moving body must occupy several specified configurations. A complete design also needs clearance, load, joint, drive and motion-order checks. This small demonstrator covers only geometry and the effect of one fixed-pivot offset.
How this calculator measures error
The nominal frame pivots are O = (0,0) and D = (L,0). The offset case moves D to (L,e). Input link OA is 0.35L, coupler AB is 0.85L and output link DB is 0.65L. The input angle fixes A. B is found by intersecting circles of radii AB and DB about A and D, selecting the same upper branch.
Point C sits at the coupler midpoint with a perpendicular offset of 0.18L. For each of four input angles, the result is the Euclidean distance between the offset and nominal C positions. The lowest and highest of those distances are displayed. No universal error-amplification factor is assumed.
A point distance alone is not a complete rigid-body pose error; orientation can change too. Four samples also cannot certify the maximum error over all travel.
Try a controlled pivot-offset comparison
Start with L = 200 mm and e = 0.5 mm. The links are 70, 170 and 130 mm long, and the nominal fixed-pivot spacing is 200 mm. The moving reference point is 36 mm perpendicular to the coupler midpoint.
Set e to zero: both mechanisms coincide and every reported sample error becomes zero. Restore the offset to compare the four input positions. Changing L scales the mechanism while leaving the entered millimetre offset independent.
Analysis is different from synthesis
This calculator starts with fixed link proportions and computes motion. A synthesis program starts with required body poses and seeks suitable linkage geometry. Passing a set of pose constraints does not automatically give a practical continuous mechanism.
The present tool can illustrate sensitivity but does not replace a synthesis solver, tolerance study, collision check or structural calculation.
Questions about the linkage model
Are four poses the theoretical maximum?
No. Four is simply the number of sample positions checked here; Burmester theory also addresses five prescribed positions.
Does this calculator generate Burmester curves?
No. It evaluates a preselected connected linkage and the effect of one pivot offset.
What does maximum sample error mean?
It is the largest point-C displacement at the four stated input angles, not a bound over the complete motion or an orientation-error measure.
Why is the offset hard to see on the main drawing?
The main mechanism uses the same geometric scale for links and offset. The separate bars make the numerical differences readable without distorting the mechanism.
Reference and scope
Zhao et al., TGA-based solutions map method for four-position synthesis of planar 4R linkage, Mechanical Sciences (2022): synthesis context, rigid coupler poses, and branch/order checks. The calculator uses an independent illustrative geometry and does not implement that paper’s synthesis algorithm.
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