Mathematical models are essential tools in the functional analysis of complicated biological structures such as the human knee joint. This paper presents two new mathematical models of the human knee for passive motion simulation. Both models comprise three scalar equations in four unknowns that can be reduced to one univariate polynomial equation, when a value of the mechanism’s generalized coordinate is given. The polynomial equation admits at most forty real solutions. Finally, the relationships to transform the employed joint coordinates into the joint coordinates usually employed in clinical practice are also reported.