Asymmetric Behavior in the Geometric Model of a Transformable Four-Legged Wheel Mechanism

Transformable wheels offer a geometric means of extending the reach of wheeled locomotion systems, but their design requires explicit relationships between transformation parameters and singularity constraints. This study presents a geometric and kinematic model of a transformable four-legged wheel driven by a trajectory-disc mechanism. Closed-form expressions describe follower displacement, disc rotation, pivot position, leg rotation, and the maximum theoretical leg-tip position. Three geometric cases, comprising two asymmetric cases and one symmetric case, are examined using a scalar Jacobian and the design constraint ΔXr<Leq to characterize admissible configurations. The analysis reveals different constraints in the asymmetric cases: Case 1 requires larger design angles to avoid singularities, reducing usable follower displacement, whereas Case 2 provides a wider design margin. The analytical Jacobian agrees with finite-difference estimates, with discrepancies on the order of 10−6 mm/deg. SolidWorks simulations evaluate five configurations covering the three geometric cases. Across the five configurations, discrepancies reach 3.80% for follower displacement and 0.62% for disc rotation. Simulated leg-tip positions remain below the theoretical upper bounds, with differences of 0.42–5.41% across the five configurations. A simple harmonic motion approximation yields position fits of R2=0.96–0.99, while velocity fits vary from approximately 0.53 to 0.96, indicating geometry-dependent accuracy. The model supports geometric design and singularity assessment; physical prototyping and dynamic analysis remain necessary to establish practical performance.