Exact solutions for the spatial four-bar with dynamic friction: Position, force, Painlevé paradox, and singularity asymptotics

Despite extensive research on spatial mechanisms, works analytically predicting their behavior with friction remain elusive. While tests and numerical methods are employed, they cannot fully explain the observed behavior. The RSSR spatial four-bar, perhaps the most widely used spatial mechanism, alone or integrated in longer kinematic chains, is an iconic example of this situation. Friction can negatively impact mechanical systems, causing oscillations, instabilities, jamming, or structural failures. Incorporating friction in rigid-body systems typically results in complex, non-deterministic equations. In 1895, Painlevé showed that rigid-body systems with Coulomb friction can be well-posed (have a unique solution), indeterminate (multiple solutions), or inconsistent (no solution). This work provides closed-form solutions for key problems in the spatial RSSR mechanism with friction by (a) deriving a compact solution to the position problem, (b) solving the quasi-static load problem, (c) establishing conditions for well-posedness, inconsistency, and indeterminacy, (d) determining mobility (e.g., free, self-locked), and (e) describing the asymptotic behavior of singularities. Comparisons with multibody simulations are presented. While solutions to the position problem are well-known, the friction-dependent results (b)–(e) are, to the best of the author’s knowledge, presented here for the first time.