A street-skateboard practitioner changes the board’s orientation by pushing his feet down on either side of the board with the effect of modifying the wheels’ angle with respect to the board’s main axis. When keeping both feet on the board, propulsion is obtained by synchronizing body and arms rotation about the vertical axis with feet induced orientation changes. This is an interesting example of transfer from rotational motion to longitunal motion. This type of actuation can be simply modelled by two motors, one used to modify the board’s wheels orientation and the other to rotate a momentum bar mounted on a vertical axis placed at the center of the board and mimicking a rider’s torso. It then remains to control the two motors in order to create the desired overall motion for the board. This challenging problem, which involves a vehicle which is both nonholonomic and underactuated, has so far been little addressed by control specialists.
The transverse function control approach provides a solution to the automatic trajectory tracking problem for such a motorized skateboard. In the illustrating simulation videos, the objective is to have the red frame attached to the board track the yellow frame whose motion is specified by the user, without taking into account the board’s actuation particularities, nor its limitations, like the impossibility to accelerate or decelerate along a straight line, or even to follow this line with constant longitudinal velocity because of friction. As a matter of fact, none of the yellow frame trajectories shown in the videos are feasible for the skateboard. Therefore, exact tracking is not possible and only practical stabilization can be achieved in this case.
The first video illustrates the actuation principle, with the yellow frame accelerating and decelerating along a straight line, then rotating about the vertical axis.
The snakeboard is a variation of the skateboard in which the front and back wheels are steered independently, instead of being turned by opposite angles. This additional degree of freedom allows the board to crab, that is to move obliquely with respect to its main axis, while propulsion still relies on the rotation of a momentum bar mimicking the rider’s torso. The snakeboard is also nonholonomic and underactuated, and the transverse function control approach applies to it as well, the skateboard controller being extended so as to exploit the extra steering degree of freedom.
In the second video, the skateboard and the snakeboard are controlled in parallel, each of them tracking the same yellow frame, along a more complex trajectory.
The third video compares the two machines cruising along a straight path, their speed rising from zero to 5 m/s. No gait is prescribed, the transverse-function feedback by itself produces the undulating motion of skaters pumping on their boards.