The name of these wheeled mechanisms comes from that they move via shape deformation, thus mimicking, to some extent, snake motion.

The first mechanism here considered is composed of three sequential bars equipped with non-actuated wheels and attached to each other via two actuated rotary joints. The kinematics of this mechanism can be modelled as a 5-dimensional control system with 2 control inputs.
The control objective is to have the red frame associated with the central bar track the blue frame whose motion is specified by the user. Alignment of two consecutive bars corresponds to a mechanical singularity that has to be avoided.
Click here to watch the simulation video.

The second mechanism differs from the first one in that the wheels at the extremities of the external bars can rotate about an axis perpendicular to the plane of motion. These added degrees of freedom allow for a smoother motion. The kinematics of this mechanism can be modelled as a 7-dimensional control system with 4 control inputs.
Click here to watch the simulation video.

The third mechanism is essentially the same as the second one, except for the central wheels whose direction is perpendicular to the central bar, instead of being aligned with it. The orientation of the red frame associated with the central bar is also different. These modifications drastically change the way the mechanism deforms to track the blue frame, eventhough the feedback control law is essentially the same.
Click here to watch the video.

Remark: A technical particularity of the controllers used for second and third mechanisms it that the involved transverse function is defined on SO(4).