This research focuses on the dynamics of an under-actuated quadruped trotting robot with directional leg compliance. The equations of motion for such a robot consist of a system of 22 second-order nonlinear ordinary differential equations, making an analytical solution infeasible. To investigate whether such a dog-sized robot is capable of passive stable trotting, computer simulations were conducted to perform a limited search of the enormous design space, utilizing the design of experiments techniques. Results indicate that fast-paced stable trotting is achievable, and stability can be improved through the intelligent selection of leg parameters.