Underactuated Vehicle Control Backstepping Simulations Using Godhavn's Model Report Number 7

Overview This brief report presents two gures and shows we have been able to replicate the results established by Godhavn 1]. We followed the backstepping process as developed by God-havn and used it to track a trajectory similar to the example in his paper. Our vehicle had the same parameter values and initial displacement from the desired trajectory. The only signiicant diierence between the two approaches is that we used Mathematica's built-in interpolation function to calculate a seventh order polynomial to approximate the desired trajectory instead of Godhavn's polynomial approximation technique. The diierences between the polynomials should be small, since they are based on the same information. We initially had trouble getting the Mathematica software to converge on a solution. The problem came from using using normalized parameter values for the masses and moment of inertia and not making the corresponding adjustments to the forces and moments. To solve the problem, we normalized the desired trajectory and velocity for the vehicle. We then were able to solve the diierential equations for the normalized system. To display the output with the correct scale, we just multiplied the resulting trajectory by the appropriate scale factor. Figure 1 shows the resulting trajectory for the vehicle and the desired path. The vehicle exponentially converges to the desired path provided the forward velocity remains positive. Figure 2 shows the actual and desired orientation of the vehicle versus normalized time. The vehicle maintains an orientation close to the desired direction, but does not converge to it exactly. If the trajectory was straight instead of curving, the actual and desired orientations would eventually align. We note that the equations of motion for this simulation are similar to the ones we use, but they do not incorporate the non-minimum phase property we have included in our model.