This is the basic laboratory mobile robot, whose motion on a planar surface relies on the control of two wheels independently actuated. This is also the simplest mechanical system on which the transverse function approach can be tested.
The video shows a simulation of a trajectory tracking task as performed by a single feedback controller designed with the approach. The reference trajectory is given by the motion, in both position and orientation, of the blue frame. This motion is specified by the user and it does not have to be feasible for the mobile robot. For instance, the blue frame is allowed to move laterally whereas this type of motion is not possible for the robot, due to the nonholonomic constraints associated with wheels rolling on the ground with no slippage.
The scenario is as follows. First, the blue frame is motionless and the mobile robot, initially far away from this frame, has to catch up with it and, eventually, converge to its position and orientation. Then, the blue frame starts moving (forward-straight, backward-straight, then forward and backward along a curved path). This phase corresponds to a feasible motion for the mobile robot so that perfect tracking can be achieved. Classical feedback control techniques can perform as well during this phase. The next phase involves a pure lateral motion that the mobile robot can only track approximately by making maneuvers. These maneuvers are automatically generated by the controller. Tracking precision can be tuned via the choice of the parameters of the transverse function used in the control law, keeping in mind that smaller tracking errors necessarily involve more frequent maneuvers. After a short stop during which the mobile robot initiates convergence to the blue frame, this frame moves along a trajectory which is not feasible for the mobile robot due to a non-zero lateral velocity component. However, contrary to the phase of pure lateral motion, the mobile robot achieves practical tracking without making maneuvers. The reason is that the approach allows for tracking errors up to a threshold that can be specified via the choice of the transverse function parameters. Finally, the blue frame reaches a stop and the mobile robot initiates convergence to the motionless blue frame.

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Using smaller parameters in the transverse function yields smaller tracking errors along non-feasible reference trajectories, and more frequent and energy consuming maneuvers, as illustrated by the following video, whereas using larger parameters goes with larger tracking errors and less frequent maneuvers, as illustrated by this video.