This project enabled the creation of new mathematical solution techniques named "Random Monodromy Loops" and "Saddle Graph Optimization" which in turn, created new engineering devices that can enhance energetic output during legged locomotion.  These two techniques combined form a framework for "Homotopy-based Optimization" which was used for the discovery of closed chain mechanical structures useful for dynamic legged locomotion.  The research culminated with the discovery, prototyping, and testing of an energy enhanced robotic leg.  The closed chain mechanism of the leg can be adapted and integrated into other robot joints to enhance energetic output.

The leg mechanism enables a lightweight low-powered motor to output high-energy movements beyond its own power limits.  This is accomplished by combining a spring element with a carefully placed kinematic singularity, a special configuration of the mechanism.  While the leg is not performing work, the motor is able to slowly energize the spring over hundreds of milliseconds.  When the leg needs to perform work, the spring energy is expended in tens of milliseconds.  Cyclic loading and unloading of the spring unfold automatically due to a regulating singularity in a process we call "dynamic wind-up locomotion."  These results were experimentally verified in a dynamic hopping gait.  The device can be said to exhibit a high-powered mechanical reflex.  More broadly, this is an example of embodied intelligence, a longstanding ambition of roboticists.  Although demonstrated for legged locomotion, the general paradigm has broad applicability.

The "Random Monodomy Loops" technique is a new way to compute solutions to a set of equations.  This is important for solving the governing physics equations of robot legs. Compared to existing techniques, such as Newton's method, its enabling advantage is its ability to find nearly all solutions to a large system of equations.  In recent decades, the techniques of numerical homotopy continuation have enabled the systematic computation of all solutions, but waste time computing "roots at infinity."  These useless solutions are subsequently filtered out. For engineering problems, more than 95% of computing power is spent on roots at infinity.  The "Random Monodromy Loops" technique circumvents these wasted computations, with the trade-off that solution discovery is more stochastic, but still systematic.  Very roughly speaking, the new technique cuts compute time by about an order of magnitude, but provides less guarantee that all solutions have been found.  This trade-off is well worth it for practical engineering problems.  More broadly, "Random Monodromy Loops" is a worthwhile tributary of study as a general purpose solver in itself.  Engineers routinely benefit from blackbox solvers integrated into programs like Matlab, among many others.  Behind the scenes, those solvers are getting more powerful, and the continued development of "Random Monodromy Loops" would represent progress in this direction.

The "Saddle Graph Optimization" method is powered by the "Random Monodromy Loops" method for solving optimization problems.  In an optimization problem, some quantity should be minimized by picking some set of parameters.  For example, this approach informs an engineer how to pick the dimensions of a device to produce the best performance.  Performance is framed as a "cost function" to be minimized, and the device dimensions which minimize that cost are called "a minimum."  Such optimization problems are traditionally solved with "gradient descent" methods.  Like Newton's method mentioned above, gradient descent only finds one minimum of many.  There may be superior minima out there not found by gradient descent.  The "Random Monodromy Loops" technique is used to find all minima, of which the very best can be chosen.  However, "Random Monodromy Loops" finds more than the minima, it also finds all "saddles."  Saddles cannot be obtained through gradient descent and do not minimize the cost function.  They are important placemarkers on the optimization landscape as they serve as gateways between different minima.  By using "Random Monodromy Loops" to obtain all minima and saddles, we are able to map the entire optimization landscape, called a "Saddle Graph."  An engineer can then peruse this map to find a point of greatest utility.  Saddle Graphs possess greater utility than a single minimum since there is often a disconnect between optimization cost functions and true engineering utility.  The "Saddle Graph" technique recognizes this and presents to the engineer a range of solutions.  More broadly, "Saddle Graph Optimization" is worthwhile to develop as a software package to be integrated into engineering softwares such as Matlab, among others.  The general engineering workforce is well trained in gradient descent methods, but there are no blackbox tools for quickly getting a complete view of a high-dimensional optimization landscape.

Along the way, this project also formulated and solved several outstanding problems in mechanism optimization, leading to the development of a variety of new mechanical devices.  These efforts were executed by graduate and undergraduate students, enhancing their training as engineers and mathematicians.  Expertise in robotics and mathematics is particularly important to our national interests.

Last Modified: 04/06/2025
Modified by: Mark M Plecnik