Kane's Method, Kane's Equations, Avoiding Lagrange Multipliers - Quasivelocities & Dynamic Equations

Introduction of topics.
Usual method of handling constraints using Lagrange multipliers in Lagrange's equations. If we have n generalized coordinates and S constraints, we end up with n+S equations and n+S unknowns..
Quasivelocities are introduced, and some examples mentioned. (1) Body-axis components of the angular velocity for Euler's rigid body dynamics; (2) Body-axis components of the inertial velocity in aircraft dynamics..
General approach: defining the last S quasivelocities as the constraints, and formulating the dynamics of the remaining unconstrained n-S quasivelocities. The main thing is we get to skip the use of Lagrange multipliers, and simulate the dynamics using a smaller number of dynamic ODEs (n-S instead of n+S, so a savings of twice the number of constraints!)..
Kane's method of getting the equations of motion for the n-S unconstrained quasivelocities, based on d'Alembert's principle. See also the Jourdain Principle..
Example using this method. The 2-particle baton with a wheel or skate under one mass. For the 2 unconstrained quasivelocities, we get fairly simple 1st order ODEs. A Matlab simulations shows that we get the same results as before..
Example: vehicle stability in a skid; Chaplygin sleigh. The resulting equations can be analyzed in a phase plane which shows lines of equilibria..
Example: model of semi-tractor-trailer truck or roller racer. Analysis of equilibria reveals the jackknife instability..