Locomotion is the means by which systems use internal shape changes to move through the world, and is one of the most fundamental actions performed by robots and living organisms.
Geometric mechanics is the application of differential geometry to problems in classical mechanics. By studying locomotion with geometric tools, we can make rigorous statements about systems’ motion capabilities.
The three-link kinematic snake
shown above propels itself by using shape changes (changes to the
internal joint angles) to push against the passive wheels on the links.
These wheels act as nonholonomic constraints
that prevent lateral translation while allowing free longitudinal and rotational
motion of each link. Using the framework of geometric mechanics, this
locomotory effect can be quantified and qualified, leading to the
identification of patterns of motion that produce a desired net
This research focuses on two aspects of geometric mechanics in locomotion: improving
approachability of the material to non-specialists and
optimally choosing the coordinates used in these analyses.
studies have produced powerful tools for system analysis. Unfortunately,
the inherent basis of these studies in differential geometry often acts
as a barrier to entry for non-specialists who otherwise might benefit
from applying the resulting tools to a broader set of systems. In many
cases, however, the largest barrier is one of language rather than
concept, in that the differential geometric terms in which the results
are presented have dual concepts in vector calculus.
In my thesis work I
emphasize these more familiar vector-calculus terms wherever possible,
noting the more formal nomenclature but keeping it out of the main
development if a simpler description will suffice. Some of these
simplified developments also lead to novel graphical representations of
system kinematics. For instance, a one-form
is a structure commonly encountered in differential geometry, defined
as a “linear map from vectors to scalars.” This structure is
functionally equivalent to the more familiar concept of a vector field
with a dot product, and thinking in such vector terms provides an avenue
for a newcomer to the field to draw on prior knowledge and intuition.
Making this vector analogy led us to develop the connection
vector field as an illustration of the local connection,
a set of one-forms used in geometric mechanics to describe the
relationship between shape velocities and the position velocities they
The connection vector
fields for the three-link system at the top of this page are shown
above. Each of the longitudinal, lateral, and rotational components of
the position velocity has a corresponding vector field defined over the
shape space (the space of joint angles), and the velocity produced by a
given input shape velocity is found by taking its dot product with each
field. Here, the vector a is aligned
with the longitudinal field and orthogonal to the rotational field, so
it produces a pure positive longitudinal translation. In contrast, the
vector b is orthogonal to the longitudinal
field and anti-aligned with the rotational field, so it produces a pure
negative rotation with no translation. The field for lateral motion has
zero magnitude because the lateral direction of the body frame is
always aligned with the constraint on the middle link.
As in many other areas of
math and engineering, differential geometry makes heavy use of
coordinate invariance - the idea that structures exist independently of
the coordinates used to describe them and are unaltered by a change of
coordinates. What is less well recognized, however, is that
coordinate invariance is in many cases an inherently local
property, and that the choice of coordinates can play an important role
in analysis of systems' macroscopic behavior. A primary result in my
thesis is finding optimal coordinates for locomotion study.
Locomotion is often accomplished by gaits, cyclic shape changes with
characteristic net displacements. Lie bracket averaging is
a powerful technique for finding gaits that produce desired net
displacements without simulating each one individually. In vector
calculus terms, the Lie bracket is a set of functions over the shape
space that are related to the curls of the connection vector fields. The
average position velocity of the system produced by a differential
oscillation (small-amplitude gaits) in the shape corresponds to the
magnitudes of these functions at the center of the oscillation - each
function corresponding to one direction in the body frame.
For larger, non-differential gaits, which are closed curves in the shape
space, the average velocity (or net displacement over each cycle) can be
approximated by integrating the Lie bracket over the area of the shape
space enclosed by the gait. This principle is closely related to
Stokes’s theorem, which equates the line integral of a vector field
along a closed curve to the area integral of the curl of the vector
field over the area bounded by the curve. The quality of this
approximation is inversely proportional to the amount the system rotates
during the gait, which until recently was regarded as a property of the
system and the gait.
research, I have found that the choice of body frame also affects the
amount of intermediate rotation and that this choice can be
systematically optimized to minimize the rotation. For instance,
consider two choices of body frame, one taken from the middle link of
the system and one based on its medial line (and center of mass):
In each figure, the three-link system is shown with Θ = 0 for different shapes. At a basic level, the medial-line coordinates on the right provide a more intuitive representation of the system's position, with the head-to-tail relationship much more consistent across the shapes than it is with the middle-link coordinates on the left.
medial line also rotates much less than the middle link in response to
shape changes. Qualitatively, the middle link generally counter-rotates
with respect to the outer links when the joints are moved; the medial
line counter-rotates to the middle link, and so ends up rotating very
little as the system changes shape. Consequently, the Lie bracket
approximation techniques work significantly better when used with the
medial line coordinates rather than the link-based coordinates.
lines for a three-link system are weighted averages of the orientations
of the three links. To find the best set of weights, or to choose
coordinates for other systems, I have developed a quantitative means of
optimizing the coordinates. This approach is based on the
Hodge-Helmholtz decomposition of a vector field (or one-form) into a divergence-free
field that is the vector field with smallest mean squared magnitude that has the same
curl as the original field and a curl-free field that is the
conservative (gradient) field that best approximates the original field.
this decomposition is applied to the connection vector field for
rotation in arbitrarily chosen coordinates, the divergence-free
component is the smallest-possible rotation connection vector field for
the system (producing the least-possible rotations), and the curl-free
component is the gradient of the coordinate change with respect to the
1) R. L. Hatton and H. Choset. Optimizing Coordinate Choice for Locomoting Systems. In the Proceedings of the IEEE International Conference on Robotics and Automation, May 2010. PDF
2)R. L. Hatton and H. Choset. Approximating Displacement with the Body Velocity Integral. In the Proceedings of Robotics: Science and Systems, Seattle, USA, June 2009. PDF
3) R. L. Hatton and H. Choset. Connection Vector Fields for Underactuated Systems. In the Proceedings of the IEEE BioRobotics Conference, October 2008. PDF