Locomotion describes how mobile systems change their position by changing their shapes. For instance, the position of the three-link system shown above is the location of its middle link and the orientation of that link with respect to the inertial frame, and its shape is completely described by the two joint angles. By making coordinated changes to these joint angles, this system can move around the plane, even if it does not have actuators directly controlling the position variables.
If we directly control the shape of the system, then we can think of the system's actions as being trajectories through the shape space, called shape changes. Shape changes that start and end at the same shape correspond to gaits, or cyclic patterns of motion. At any point on a shape change or gait, the shape velocity is the tangent vector to the trajectory.
By changing its shape, a system can "pull" against constraints and so change its position. In body coordinates, the relationship between shape and position velocities is captured by the reconstruction equation. The body velocity is related to the shape velocity by the local connection. The generalized momentum captures how much the system is "coasting".
A kinematic system is one that cannot coast (has no generalized momentum); its body velocity only depends on the shape velocity and local connection. To visualize this relationship, we extract vector fields from the rows of the local connection, and plot them together with shape changes. The motions resulting from executing the shape changes are characterized by the aligment of the associated shape velocity with the connection vector fields, with positive, zero, or negative alignment with each field producing positive, zero, or negative motion in that body direction.
The "floating snake" system has three links and is confined to movement in the plane. It is completely isolated from its environment and only experiences internal forces. From conservation of linear momentum, the center of mass velocity can never change, so the first two rows of the local connection are zero. From conservation of angular momentum, the center link rotates in response to joint motion, to keep the net angular momentum constant; the third row of the local connection is consequently non-zero, and provides a connection vector field. The shape velocity has positive alignment with the connection vector field over the first stage of the gait, so the floating snake rotates positively. Over the second stage of the gait, the alignment is negative, and the floating snake rotates negatively.
The familiar differential drive car has two non-zero connection vector fields, for forward and rotational motion. In the shape change shown here, the first segment has both wheels turning forward equally; this motion is completely aligned with the forward vector field and perpendicular to the rotational vector field, so the car moves forward. The second stage has the wheels turning oppositely; the shape change is aligned with the rotational field and perpendicular to the forward field, so the car turns positively (counterclockwise). Finally, the third stage is negatively aligned with the forward field and positively aligned with the rotational field, so the car move backwards while turning counterclockwise.
A three-link kinematic snake is like the floating snake, but each link is constrained from moving laterally. The constraint on the middle link prevents the system from moving laterally, so, like the differential drive car, the kinematic snake has only forward and rotational connection vector fields. These vector fields have singularities, which result from interactions between the constraints. The two gaits shown for the kinematic snake trace out the same curve in the shape space, but with different starting points. Because changing the starting point changes the order of forward and rotation motions, the two gaits produce significantly different net displacements.
A physical implementation of the three-link kinematic snake. We track the motion of the robot with a pair of optical mice, which act as ground-scanning encoders.