overconstrained

Overconstrained describes a condition in a system where there are more independent constraints than are necessary to determine its state. In practice this can mean that constraints rigidly fix variables or render some configurations impossible. The term is used across disciplines, including mathematics, engineering, and design.

In mathematics and computation, an overconstrained system has more equations than unknowns. For a linear system Ax = b with m > n, a solution exists only if the equations are consistent; otherwise the system is typically infeasible. If it is consistent, there is at most one solution. In numerical methods, inconsistent overdetermined systems are often addressed by least squares, which finds the x that minimizes the error |Ax − b|.

In engineering and mechanics, an overconstrained (or redundantly constrained) mechanism has more constraints than necessary for a prescribed motion. This can lead to internal stresses if constraints do not align perfectly due to manufacturing tolerances, making the mechanism stiff or prone to jamming. Some designs deliberately incorporate redundancy for stability or precision, but such systems require careful tolerance management and sometimes compliant elements to avoid binding. In kinematics, a system with redundant constraints may be statically indeterminate, providing more equations than are needed to determine reaction forces.

Practical handling of overconstrained systems often involves relaxing or reordering constraints, introducing compliance, or using optimization to find a state that best satisfies a possibly conflicting set of constraints. The opposite condition is underconstrained, where too few constraints leave multiple possible states.