On the λ-equations for matching control laws

We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely characterized by a linear system of first order partial differential equations for one set of variables (λ) followed by a linear system of first order partial differential equations for the second set of variables (ĝ, V̂). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the λ-equations. We give four examples illustrating different features of matching control laws, The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case.