Studies of Systems with Nonholonomic Constraints: the Segway and the Chaplygin Sleigh

In this thesis, two systems with nonholonomic systems are investigated: the Segway and the Chaplygin sleigh. Using Lagrangian mechanics, the constrained nonlinear equations of motion for both systems are derived. By use of the nullspace of the constraint matrices, the unconstrained equations of motion can be obtained. For the Segway, these equations are linearized about a zero equilibrium state, placed into state space form and decoupled. A feedback controller is designed about the velocity and heading angle rate reference commands. To compare to the real data from the built Segway, measurement noise was also included in the model. Experimental data is taken for the case of both zero and constant reference commands. The data is then compared to the simulated results. The model is shown to be satisfactory, but better parameter measurements of the Segway is needed for a more conclusive comparison. The unconstrained equations of motion for the Chaplygin sleigh can not be linearized. Thus Lyapunov stability theory was used for analysis. The Chaplygin sleigh with constant input was shown to spiral outward and settle into a circle. If a PD feedback controller was designed about the heading angle, then the Chaplygin sleigh would be driven to the angle, but would eventually coast to a stop. From simulations, the addition of a sinusoidal component appears to move in the desired direction without slowing down. A sinusoidal component was also added to a constant input to result in roulette like paths in the simulation. Future investigation would require a more definite analysis of the sinusoidal term in the input