Peer-Reviewed Publications

2024

In-situ calibration of six-axis force/torque transducers on a six-legged robot

Shai Revzen, and Ziyou Wu

ASME J Dynamic Systems, Measurement and Control, 2024

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Modeling multi-legged robot locomotion with slipping and its experimental validation

Ziyou Wu, Dan Zhao, and Shai Revzen

IEEE International Journal of Robotics Research, 2024

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Multi-legged robots with six or more legs are not in common use, despite designs with superior stability, maneuverability, and a low number of actuators being available for over 20 years. This may be in part due to the difficulty in modeling multi-legged motion with slipping and producing reliable predictions of body velocity. Here, we present a detailed measurement of the foot contact forces in a hexapedal robot with multiple sliding contacts, and provide an algorithm for predicting these contact forces and the body velocity. The algorithm relies on the recently published observation that even while slipping, multi-legged robots are principally kinematic, and employ a friction law ansatz that allows us to compute the shape-change to body-velocity connection and the foot contact forces. This results in the ability to simulate motion plans for a large number of contacts, each potentially with slipping. Furthermore, in homogeneous environments, this kind of simulation can run in (parallel) logarithmic time of the planning horizon.
Self-Righting Shell for Robotic Hexapod

Katelyn King, and Shai Revzen

In 2024 IEEE International Conference on Robotics and Automation (ICRA), May 2024

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Decimeter scale robots in human environments are small relative to obstacles they encounter, making them prone to flipping over and needing to self-right. We present a multifaceted shell that by its geometry alone enables the hexapedal robot MediumANT to passively self-right without the need for additional sensory feedback. We designed the shell by specifying the cross-sectional geometry in the yz and xy planes such that the robot returns to an upright position by rolling around the longitudinal (x) axis, and then tweaked the design to reduce the number of faces. We then attached the shell to the robot by modifying some of its chassis structural plates to extend to and support the shell. We evaluated the effectiveness of the shell in two experimental scenarios: passive righting ? balancing the robot on each face of the shell before releasing the robot ? and an intentional fall ? walking the robot off a ledge at various approach angles. As intended by our design, the robot recovered the upright orientation from all starting faces in the passive righting test and righted itself and continued walking in all falling trials. This work presents an example of using biologically inspired simplicity to solve what would otherwise be a technically challenging problem.
Phase response curves and the role of coordinates

Simon Wilshin, Matthew D Kvalheim, and Shai Revzen

Biological Cybernetics, May 2024

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The “infinitesimal phase response curve”(PRC) is a common tool used to analyze phase resetting in the natural sciences in general and neuroscience in particular. We make the observation that the PRC with respect to a coordinate v actually depends on the choice of other coordinates. As a consequence, a complete delay embedding reconstruction of the dynamics using v which would allow phase to be computed still does not allow the v PRC to be computed. We give a coordinate-free definition of the PRC making this observation obvious. This leads to an experimental protocol: first collect an appropriate ensemble of measurements by intermittently controlling neuron voltage. Then, for any suitable current carrier dynamic postulated, we show how the ensemble can be used to compute the voltage PRC with that current carrier. The approach extends to many oscillators measured and controlled through a subset of their coordinates.

2023

Contract Drafting, Programming Languages, and What They Can Teach Each Other

Shai Revzen

Wayne St. UJ Bus. L., May 2023

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2022

Walking is like slithering: A unifying, data-driven view of locomotion

D Zhao, B A Bittner, G Clifton, and 2 more authors

Proceedings of the National Academy of Science, May 2022

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Legged movement is ubiquitous in nature and of increasing interest for robotics. Most legged animals routinely encounter foot slipping, yet detailed modeling of multiple contacts with slipping exceeds current simulation capacity. Here we present a principle that unifies multilegged walking (including that involving slipping) with slithering and Stokesian (low Reynolds number) swimming. We generated data-driven principally kinematic models of locomotion for walking in low-slip animals (Argentine ant, 4.7% slip ratio of slipping to total motion) and for high-slip robotic systems (BigANT hexapod, slip ratio 12 to 22%; Multipod robots ranging from 6 to 12 legs, slip ratio 40 to 100%). We found that principally kinematic models could explain much of the variability in body velocity and turning rate using body shape and could predict walking behaviors outside the training data. Most remarkably, walking was principally kinematic irrespective of leg number, foot slipping, and turning rate. We find that grounded walk- ing, with or without slipping, is governed by principally kinematic equations of motion, functionally similar to frictional swimming and slithering. Geometric mechanics thus leads to a unified model for swimming, slithering, and walking. Such commonality may shed light on the evolutionary origins of animal locomotion control and offer new approaches for robotic locomotion and motion planning.
Representing and computing the b-derivative of the piecewise-differentiable flow of a class of nonsmooth vector fields

George Council, Shai Revzen, and Samuel A Burden

Journal of Computational and Nonlinear Dynamics, May 2022

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This paper concerns first-order approximation of the piecewise-differentiable flow generated by a class of nonsmooth vector fields. Specifically, we represent and compute the Bouligand (or B-)derivative of the piecewise-differentiable flow generated by a vector field with event-selected discontinuities. Our results are remarkably efficient: although there are factorially many ?pieces? of the derivative, we provide an algorithm that evaluates its action on a tangent vector using polynomial time and space, and verify the algorithm’s correctness by deriving a representation for the B-derivative that requires only exponential time and space to construct. We apply our methods in two classes of illustrative examples: piecewise-constant vector fields and mechanical systems subject to unilateral constraints.

2021

Challenges in Dynamic Mode Decomposition

Z Wu, S L Brunton, and Shai Revzen

J of The Royal Society Interface, May 2021

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Dynamic Mode Decomposition (DMD) is a powerful tool for extracting spatial and temporal patterns from multi-dimensional time series, and it has been used successfully in a wide range of fields, including fluid mechanics, robotics, and neuroscience. Two of the main challenges remaining in DMD research are noise sensitivity and issues related to Krylov space closure when modeling nonlinear systems. Here, we investigate the combination of noise and nonlinearity in a controlled setting, by studying a class of systems with linear latent dynamics which are observed via multinomial observables. Our numerical models include system and measurement noise. We explore the influences of dataset metrics, the spectrum of the latent dynamics, the normality of the system matrix, and the geometry of the dynamics. Our results show that even for these very mildly nonlinear conditions, DMD methods often fail to recover the spectrum and can have poor predictive ability. Our work is motivated by our experience modeling multilegged robot data, where we have encountered great difficulty in reconstructing time series for oscillatory systems with slow transients, which decay only slightly faster than a period.
Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits

Matthew D Kvalheim, and Shai Revzen

Physica D: Nonlinear Phenomena, May 2021

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Generic properties of Koopman eigenfunctions for stable fixed points and periodic orbits

Matthew D Kvalheim, David Hong, and Shai Revzen

IFAC-PapersOnLine, May 2021

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2020

Multi-legged steering and slipping with low DoF hexapod robots

D Zhao, and Shai Revzen

Bioinspiration and Biomimetics, May 2020

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Thanks to their sprawled posture and multi-legged support, stability is not as hard to achieve for hexapedal robots as it is for bipeds and quadrupeds. A key engineering challenge with hexapods has been to produce insect-like agility and maneuverability, of which steering is an essential part. However, the mechanisms of multi-legged steering are not always clear, especially for robots with underactuated legs. Here we propose a formal definition of steering, and show why steering is difficult for robots with 6 or more underactuated legs. We show that for many of these robots, steering is impossible without slipping, and present experimental results which demonstrate the importance of allowing for slipping to occur intentionally when optimizing steering ability. Our results suggest that a non-holonomic multi-legged slipping model might be more appropriate than dynamic models for representing such robots, and that conventional non-slip contact models might miss significant parts of the performance envelope.
Data-Driven Geometric System Identification for Shape-Underactuated Dissipative Systems

B Bittner, R L Hatton, and Shai Revzen

Bioinspiration and Biomimetics, May 2020

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The study of systems whose movement is both geometric and dissipative offers an opportunity to quickly both identify models and optimize motion. Here, the geometry indicates reduction of the dynamics by environmental homogeneity while the dissipative nature minimizes the role of second order (inertial) features in the dynamics. In this work, we extend the tools of geometric system identification to "Shape-Underactuated Dissipative Systems (SUDS)" – systems whose motions are kinematic, but whose actuation is restricted to a subset of the body shape coordinates. A large class of SUDS includes highly damped robots with series elastic actuators, and many soft robots. We validate the predictive quality of the models using simulations of a variety of viscous swimming systems. For a large class of SUDS, we show how the shape velocity actuation inputs can be directly converted into torque inputs suggesting that, e.g., systems with soft pneumatic actuators or dielectric elastomers, could be controlled in this way. Based on fundamental assumptions in the physics, we show how our model complexity scales linearly with the number of passive shape coordinates. This offers a large reduction on the number of trials needed to identify the system model from experimental data, and may reduce overfitting. The sample efficiency of our method suggests its use in modeling, control, and optimization in robotics, and as a tool for the study of organismal motion in friction dominated regimes.

2019

Gait modeling and optimization for the perturbed Stokes regime

M Kvalheim, B Bittner, and Shai Revzen

J Nonlinear Dynamics, May 2019

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Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that for swimming at the “Stokesian” (viscous; zero Reynolds number) limit, the motion is governed by a reduced-order “connection” model that describes how body shape change produces motion for the body frame with respect to the world. In the “perturbed Stokes regime” where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer linear in shape change rate. We derive this model using results from singular perturbation theory and the theory of noncompact normally hyperbolic invariant manifolds. Using the theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a “gait”) directly from observational data of shape and body motion. This extends our previous work which assumed kinematic “connection” models. To compare the old and new algorithms, we analyze simulated swimmers over a range of inertia-to-damping ratios. Our new class of models performs well on the Stokesian regime and over several orders of magnitude outside it into the perturbed Stokes regime, where it gives significantly improved prediction accuracy compared to previous work. In addition to algorithmic improvements, we thereby present a new class of models that is of independent interest. Their application to data-driven modeling improves our ability to study the optimality of animal gaits and our ability to use hardware-in-the-loop optimization to produce gaits for robots.
Coulomb Friction Crawling Model Yields Linear Force–Velocity Profile

Ziyou Wu, Dan Zhao, and Shai Revzen

J Applied Mechanics, May 2019

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Conventional wisdom would have it that moving mechanical systems that dissipate energy by Coulomb friction have no relationship between force and average speed. One could argue that the work done by friction is constant per unit of distance travelled, and if propulsion forces exceed friction, the net work is positive, and the system accumulates kinetic energy without bound. We present a minimalistic model for legged propulsion with slipping under Coulomb friction, scaled to parameters representative of single kilogram robots and animals. Our model, amenable to exact solutions, exhibits nearly linear (R2 > 0.96) relationships between actuator force and average speed over its entire range of parameters, and in both motion regimes, it supports. This suggests that the interactions inherent in multilegged locomotion may lead to governing equations more reminiscent of viscous friction than would be immediately obvious.

2018

Geometrically Optimal Gaits: a Data-Driven Approach

B A Bittner, R A Hatton, and Shai Revzen

Nonlinear Dynamics, May 2018

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The study of optimal motion of animals or robots often involves seeking optimality over a space of cyclic shape changes, or gaits, specified using a large number of parameters. We show a data-driven method for computing the gradient of a cost functional with respect to a large number of gait parameters by employing geometric properties of the dynamics to efficiently construct a local model of the system, and then using this model to rapidly compute the gradients. Our modeling step specifically applies to systems governed by connection-like models from geometric mechanics, which encompass a number of high-friction regimes. We demonstrate using our method for optimizing gaits under noisy, experiment-like conditions by simulating planar multi-segment serpent-like swimmers in a low Reynolds number (viscous friction) environment. Our optimization results recover known results for 3-segment swimmers with a 66 dimensional gait parameterization, and extend to optimizing the motion of a 9 segment swimmer with a 264 dimensional gait space, using only 30 simulation trials of 30 gait cycles each. The data-driven geometric gait optimization approach we present is designed to operate on noisy, stochastically perturbed dynamics as noisy and variable as experimental data?and efficiently optimize a large number of parameters. We believe this approach has the potential to significantly advance our ability to optimize robot gaits with hardware in the loop and to study the optimality of animal gaits with respect to hypothesized cost functions.
Global linearization and fiber bundle structure of invariant manifolds

J Eldering, M Kvalheim, and Shai Revzen

Nonlinearity, May 2018

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We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a C^k disk bundle structure if the local stable foliation is assumed C^k. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global C^k linearizing conjugacy. We also prove a C^k global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel normal form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.

2017

Rapidly Prototyping Robots: Using Plates and Reinforced Flexures

I Fitzner, Y Sun, V Sachdeva, and 1 more author

IEEE Robotics Automation Magazine, Mar 2017

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In this article, we present plate and reinforced flexure (PARF) fabrication, an inexpensive, rapid fabrication technique for robot mechanisms inspired by the smart composite manufacturing (SCM) used in VelociRoACH and RoboBee designs. PARF extends SCM to larger sizes at low costs. We used PARF to develop a meter-scale hexapedal robot, BigANT, a design we made publicly available. Manufacture of BigANT requires minimal tooling-foam board, tape, a saw blade, and a knife; the chassis costs <; US$20 in materials, encouraging its use in research, education, and recreation. We present a study of PARF joints, showing several variations spanning a range of fabrication effort and mechanical properties. PARF promises the possibility of quickly and inexpensively building robot mechanisms for tasks as the requirements arise, rather than relying on prefabricated robot bodies.
Why we need more degrees of freedom

Shai Revzen, and D E Koditschek

Procedia IUTAM, Mar 2017

24th International Congress of Theoretical and Applied Mechanics

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Mechanical systems encountered in biology typically have many more degrees of freedom (DOF) than the 6 DOF required to manipulate a body in space. Even the relatively rigid arthropods and crustaceans have at least 5 DOF in each limb; tentacles and human hands have many more. Robotics engineers are routinely required to choose the number of DOF in a robot in the early design stages, potentially limiting the robot’s future uses. We theoretically motivate the definition of “mechanical versatility” as the ability of a mechanical system to express distinct static configurations and switch among them rapidly. Requiring versatility and assuming that the systems are power and force limited, and must furthermore resist finite energy environmental disturbances to their state, we show that such multiuse 1 mechanical systems have a lower bound on the number of DOF they require. For bio-mechanics, this suggests which organs and organisms will be driven to become more complex mechanically by indicating domains where higher DOF systems would intrinsically out-compete lower DOF systems.
Morphology and the gradient of a symmetric potential predict gait transitions of dogs

S Wilshin, G C Haynes, J Porteous, and 3 more authors

Biological Cybernetics, Mar 2017

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Gaits and gait transitions play a central role in the movement of animals. Symmetry is thought to govern the structure of the nervous system, and constrain the limb motions of quadrupeds. We quantify the symmetry of dog gaits with respect to combinations of bilateral, fore-aft, and spatio-temporal symmetry groups. We tested the ability of symmetries to model motion capture data of dogs walking, trotting and transitioning between those gaits. Fully symmetric models performed comparably to asymmetric with only a 22% increase in the residual sum of squares and only one-quarter of the parameters. This required adding a spatio-temporal shift representing a lag between fore and hind limbs. Without this shift, the symmetric model residual sum of squares was 1700% larger. This shift is related to (linear regression, n=5, p=0.0328) dog morphology. That this symmetry is respected throughout the gaits and transitions indicates that it generalizes outside a single gait. We propose that relative phasing of limb motions can be described by an interaction potential with a symmetric structure. This approach can be extended to the study of interaction of neurodynamic and kinematic variables, providing a system-level model that couples neuronal central pattern generator networks and mechanical models.
Longitudinal quasi-static stability predicts changes in dog gait on rough terrain

S Wilshin, M A Reeve, G C Haynes, and 3 more authors

Journal of Experimental Biology, Mar 2017

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Legged animals utilize gait selection to move effectively and must recover from environmental perturbations. We show that on rough terrain, domestic dogs, Canis lupus familiaris, spend more time in longitudinal quasi-statically stable patterns of movement. Here, longitudinal refers to the rostro-caudal axis. We used an existing model in the literature to quantify the longitudinal quasi-static stability of gaits neighbouring the walk, and found that trot-like gaits are more stable. We thus hypothesized that when perturbed, the rate of return to a stable gait would depend on the direction of perturbation, such that perturbations towards less quasi-statically stable patterns of movement would be more rapid than those towards more stable patterns of movement. The net result of this would be greater time spent in longitudinally quasi-statically stable patterns of movement. Limb movement patterns in which diagonal limbs were more synchronized (those more like a trot) have higher longitudinal quasi-static stability. We therefore predicted that as dogs explored possible limb configurations on rough terrain at walking speeds, the walk would shift towards trot. We gathered experimental data quantifying dog gait when perturbed by rough terrain and confirmed this prediction using GPS and inertial sensors (n=6, P<0.05). By formulating gaits as trajectories on the n-torus we are able to make tractable the analysis of gait similarity. These methods can be applied in a comparative study of gait control which will inform the ultimate role of the constraints and costs impacting locomotion, and have applications in diagnostic procedures for gait abnormalities, and in the development of agile legged robots.

2016

Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows

S A Burden, S S Sastry, D E Koditschek, and 1 more author

SIAM Journal of Applied Dynamical Systems, Mar 2016

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We study a class of discontinuous vector fields that arise in biomechanics and neuroscience. Under the conditions that (i) the vector field’s discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, we show that the vector field yields a well-defined flow that is Lipschitz continuous and piecewise-differentiable. This implies that although the flow is not classically differentiable, nevertheless it admits a first–order approximation (known as a Bouligand derivative). We exploit this first-order approximation to infer existence of piecewise-differentiable impact maps and assess structural stability of the flow
Bioinspired Legged Locomotion

M Kvalheim, and Shai Revzen

Mar 2016

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Bioinspired Legged Locomotion

Shai Revzen, and M Kvalheim

Mar 2016

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Bioinspired Legged Locomotion

P M Wensing, and Shai Revzen

Mar 2016

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2015

Model reduction near periodic orbits of hybrid dynamical systems

S A Burden, Shai Revzen, and S S Sastry

IEEE Transactions on Automatic Control, Mar 2015

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We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract super exponentially to a constant-dimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincaré map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior.
Thermal adaptation and phosphorus shape thermal performance in an assemblage of rainforest ants

M Kaspari, N A Clay, J A Lucas, and 3 more authors

Ecology, Mar 2015

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We studied the Thermal Performance Curves (TPCs) of 87 species of rainforest ants and found support for both the Thermal Adaptation and Phosphorus-Tolerance hypotheses. TPCs relate a fitness proxy (here, worker speed) to environmental temperature. Thermal Adaptation posits that thermal generalists (ants with flatter, broader TPCs) are favored in the hotter, more variable tropical canopy compared to the cooler, less variable litter below. As predicted, species nesting in the forest canopy 1) had running speeds less sensitive to temperature; 2) ran over a greater range of temperatures; and 3) ran at lower maximum speeds. Tradeoffs between tolerance and maximum performance are often invoked for constraining the evolution of thermal generalists. There was no evidence that ant species traded off thermal tolerance for maximum speed, however. Phosphorus-Tolerance is a second mechanism for generating ectotherms able to tolerate thermal extremes. It posits that ants active at high temperatures invest in P-rich machinery to buffer their metabolism against thermal extremes. Phosphorus content in ant tissue varied three-fold, and as predicted, temperature sensitivity was lower and thermal range was higher in P-rich species. Combined, we show how the vertical distribution of hot and variable vs. cooler and stable microclimates in a single forest contribute to a diversity of TPCs and suggest that a widely varying P stoichiometry among these ants may drive some of these differences.
Constructing predictive models of human running

H-M Maus, Shai Revzen, J M Guckenheimer, and 3 more authors

Journal of The Royal Society Interface, Mar 2015

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Running is an essential mode of human locomotion, during which ballistic aerial phases alternate with phases when a single foot contacts the ground. The spring-loaded inverted pendulum (SLIP) provides a starting point for modelling running, and generates ground reaction forces that resemble those of the centre of mass (CoM) of a human runner. Here, we show that while SLIP reproduces within-step kinematics of the CoM in three dimensions, it fails to reproduce stability and predict future motions. We construct SLIP control models using data-driven Floquet analysis, and show how these models may be used to obtain predictive models of human running with six additional states comprising the position and velocity of the swing-leg ankle. Our methods are general, and may be applied to any rhythmic physical system. We provide an approach for identifying an event-driven linear control- ler that approximates an observed stabilization strategy, and for producing a reduced-state model which closely recovers the observed dynamics.
Focused Modularity: Rapid Iteration of Design and Fabrication of a Meter-Scale Hexapedal Robot

D Miller, I Fitzner, SB Fuller, and 1 more author

In Assistive Robotics: Proceedings of the 18th International Conference on CLAWAR 2015, Mar 2015

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Data driven models of legged locomotion

Shai Revzen, and M Kvalheim

In Proc SPIE, Mar 2015

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Legged locomotion is a challenging regime both for experimental analysis and for robot design. From biology, we know that legged animals can perform spectacular feats which our machines can only surpass on some specially controlled surfaces such as roads. We present a concise review of the theoretical underpinnings of Data Driven Floquet Analysis (DDFA), an approach for empirical modeling of rhythmic dynamical systems. We provide a review of recent and classical results which justify its use in the analysis of legged systems.

2014

Deadbeat control with (almost) no sensing in a hybrid model of legged locomotion

G Council, S Yang, and Shai Revzen

In Advanced Mechatronic Systems (ICAMechS), 2014 International Conference on, Aug 2014

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Hybrid systems often appear as models of mechatronic devices, and are used to express the discontinuous transition in dynamics that occurs when mechanical contacts are made or broken. Contact state changes can coincide with dramatic shifts in control authority, and become sources of unwanted disturbance when they are exogenously driven. In the natural world, such systems appear in rapid legged locomotion. We present an insight derived from the analysis of running animals - namely that dynamics can be partially or fully controlled through geometric encoding of the desired control law in the feedforward trajectories of the appendages. The resulting systems use no sensing, or nearly so, yet can exhibit strong (deadbeat) stability. We provide a theoretical proof of a general result, then a simulation study of a simplified model of vertical hopping which rejects ground-height changes without sensing the ground. We thereby show that mechatronic controllers for non-trivial tasks such as manipulation and legged locomotion could be implemented mechanically, with little or no sensing, by encoding the control laws in the open-loop motions chosen. Our results highlight that identification of a control mechanism in an existing animal or machine must take into account that such geometric stabilization may exist without neural or computational feedback.

2013

SEAL Pack: Versatile, Portable, and Rapidly Deployable SEa, Air, and Land Vehicle

M Piccoli, Shai Revzen, and M Yim

In IEEE International Symposium on Safety Security and Rescue Robotics, Oct 2013

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There are thirteen categories in the Robotic USAR ontology covering land air and sea vehicles. We present a robot system that is capable of four of those categories including aerial, terrestrial and marine locomotion in a single package that is man-portable and low cost. Each mode of locomotion has useful capabilities that the others do not. The land vehicle can travel over 5km with a 500g load. The boat can travel 0.5 km over water or loiter for 140 minutes. The ﬂyer can traverse any terrain for short periods of time. The system packs into a small 33x20x14 cm package weighing 1 kg. We present design issues and experimental veriﬁcation.
Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches

Shai Revzen, S A Burden, T Y Moore, and 2 more authors

Biological Cybernetics, Oct 2013

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Instantaneous kinematic phase calculation allows the development of reduced-order oscillator models useful in generating hypotheses of neuromechanical control. When perturbed, changes in instantaneous kinematic phase and frequency of rhythmic movements can provide details of movement and evidence for neural feedback to a system-level neural oscillator with a time resolution not possible with traditional approaches. We elicited an escape response in cockroaches (Blaberus discoidalis) that ran onto a movable cart accelerated laterally with respect to the animals’ motion causing a perturbation. The specific impulse imposed on animals (0.50+/-0.04 m/s; mean, SD) was nearly twice their forward speed (0.25+/- 0.06 m/s) . Instantaneous residual phase computed from kinematic phase remained constant for 110 ms after the onset of perturbation, but then decreased representing a decrease in stride frequency. Results from direct muscle action potential recordings supported kinematic phase results in showing that recovery begins with self-stabilizing mechanical feedback followed by neural feedback to an abstracted neural oscillator or central pattern generator. Trials fell into two classes of forward velocity changes, while exhibiting statistically indistinguishable frequency changes. Animals pulled away from the side with front and hind legs of the tripod in stance recovered heading within 300 ms, whereas animals that only had a middle leg of the tripod resisting the pull did not recover within this period. Animals with eight or more legs might be more robust to lateral perturbations than hexapods.

2012

Finding the dimension of slow dynamics in a rhythmic system

Shai Revzen, and J M Guckenheimer

Journal of The Royal Society Interface, May 2012

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Dynamical systems with asymptotically stable periodic orbits are generic models for rhythmic processes in dissipative physical systems. This paper presents a method for reconstructing the dynamics near a periodic orbit from multivariate time-series data. It is used to test theories about the control of legged locomotion, a context in which time series are short when compared with previous work in nonlinear time-series analysis. The method presented here identifies appropriate dimensions of reduced order models for the deterministic portion of the dynamics. The paper also addresses challenges inherent in identifying dynamical models with data from different individuals.
Dynamical trajectory replanning for uncertain environments

Shai Revzen, B D Ilhan, and D E Koditschek

May 2012

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We propose a dynamical reference generator equipped with an augmented transient "replanning" subsystem that modulates a feedback controller’s efforts to force a me- chanical plant to track the reference signal. The replanner alters the reference generator’s output in the face of unanticipated disturbances that drive up the tracking error. We demonstrate that the new reference generator cannot destabilize the tracker, that tracking errors converge in the absence of disturbance, and that the overall coupled reference-tracker system cannot be destabilized by disturbances of bounded energy. We report the results of simulation studies exploring the performance of this new design applied to a two dimensional point mass particle interacting with fixed but unknown terrain obstacles.
Linear iterative strategies to identify and overcome malicious links in wireless networks

S Sundaram, Shai Revzen, and G Pappas

Automatica, Nov 2012

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We consider a network where every node has a value that we wish to disseminate to all other nodes. A certain number of communication links between nodes are allowed to be under the control of an attacker who maliciously chooses the messages carried by these links. We study linear iterative strategies that disseminate information despite such attacks. In such strategies, at each time-step each node in the network broadcasts a value to its neighbors that is a linear combination of its previous value and the values received from its neighbors. As long the number of incoming links to any node and the total number of other nodes with incoming malicious links is no greater than f , we show that linear strategies are almost always resilient to malicious behavior provided that vertex-connectivity of the network is at least 2f + 1. Furthermore, we show that each node can identify the exact set of malicious links that directly enter that node, and can communicate this information to the other nodes via the linear strategy.

2011

Dimension reduction near periodic orbits of hybrid systems

S Burden, Shai Revzen, and S S Sastry

IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Nov 2011

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When the Poincare map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, there exists a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in finite time. This result shows that the long-term behavior of a hybrid model with a large number of degrees-of-freedom may be governed by a low-dimensional smooth dynamical system. The appearance of such simplified models enables the translation of analytical tools from smooth systems to the hybrid setting and provides a bridge between the efforts of biologists and engineers studying legged locomotion.
Structure synthesis on-the-fly in a modular robot

Shai Revzen, M Bhoite, J A Macasieb, and 1 more author

In IEEE International Conference on Intelligent Robots and Systems (IROS), Nov 2011

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We describe a modular robot system that can generate foam to make structural elements. The modular mobile system uses CKBot modules and carries extra modules along with a foam generation device. In this paper, we demonstrate the system synthesizing new robot morphologies - such as snake-like or legged robots or use the conforming foam-based structures to encapsulate objects or modify the environment and discuss the issues in building and using this technique.
A general stiffness model for programmable matter and modular robotic structures

P J White, Shai Revzen, C E Thorne, and 1 more author

Robotica, Nov 2011

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The fields of modular reconfigurable robotics and programmable matter study how to compose functionally useful systems from configurations of modules. In addition to the external shape of a module configuration, the internal arrangement of modules and bonds between them can greatly impact functionally relevant mechanical properties such as load bearing ability. A fast method to evaluate the mechanical property aids the search for an arrangement of modules achieving a desired mechanical property as the space of possible configurations grows combinatorially. We present a fast approximate method where the bonds between modules are represented with stiffness matrices that are general enough to represent a wide variety of systems and follows the natural modular decomposition of the system. The method includes nonlinear modeling such as anisotropic bonds and properties that vary as components flex. We show that the arrangement of two types of bonds within a programmable matter systems enables programming the apparent elasticity of the structure. We also present a method to experimentally determine the stiffness matrix for chain style reconfigurable robots. The efficacy of applying the method is demonstrated on the CKBot modular robot and two programmable matter systems: the Rubik’s snake folding chain toy and a right angle tetrahedron chain called RATChET7mm. By allowing the design space to be rapidly explored we open the door to optimizing modular structures for desired mechanical properties such as enhanced load bearing and robustness.

2010

CKBot Platform for the ICRA 2010 Planetary Challenge

Shai Revzen, J Sastra, N Eckenstein, and 1 more author

In Proceedings of IEEE International Conference on Robotics and Automation, Workshop "Modular Robots: The State of the Art", Nov 2010

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The ICRA Planetary Contingency Challenge 2010 will include 3 teams that will be using the new CKBot module and software package. This paper will present some of the new aspects of this hardware, underlying software architecture for quickly programming the modules and will include a tutorial for these teams.
Insects running on elastic surfaces

A J Spence, Shai Revzen, J Seipel, and 2 more authors

Journal of Experimental Biology, Nov 2010

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In nature, cockroaches run rapidly over complex terrain such as leaf litter. These substrates are rarely rigid, and are frequently very compliant. Whether and how compliant surfaces change the dynamics of rapid insect locomotion has not been investigated to date; largely due to experimental limitations. Here we test the hypothesis that a running insect can maintain average forward speed over an extremely soft elastic surface (10 N/m) equal to 2/3 of its virtual leg stiffness (15 N/m). Cockroaches (Blaberus discoidalis) running from a rigid surface onto an elastic surface were able to maintain forward speed (39.6 +/- 0.7 cm/s, rigid substrate, versus 43.3 +/- 1.1 cm/s, elastic substrate; p = 0.0064). Step frequency was unchanged (25.5 +/- 0.3 steps/sec, rigid surface, versus 26.1 +/- 0.5 steps/sec, elastic surface; p = 0.28). To uncover the mechanism we measured the animal’s COM dynamics using a novel miniature backpack, consisting chiefly of a 3-axis MEMs accelerometer attached very near the COM. Vertical acceleration of the COM on the elastic surface had smaller peak-to-peak amplitudes (13.95 +/- 0.41 m/s2, rigid, vs. 10.07 +/- 0.50 m/s2 on the elastic surface; p < 0.0001). The observed change in COM acceleration over an elastic surface requires no change in effective stiffness when duty factor and ground stiffness are taken into account. Due to the lowering of the COM towards the elastic surface the swing leg lands earlier and increases the period of double support. A simple feedforward control model explains the experimental result and provides one plausible model mechanism. We find no evidence for active neural control of effective leg stiffness.

2008

Active tails enhance arboreal acrobatics in geckos

A Jusufi, D I Goldman, Shai Revzen, and 1 more author

PNAS, Nov 2008

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Geckos are nature’s elite climbers. Their remarkable climbing feats have been attributed to specialized feet with hairy toes that uncurl and peel in milliseconds. Here, we report that the secret to the gecko’s arboreal acrobatics includes an active tail. We examine the tail’s role during rapid climbing, aerial descent, and gliding. We show that a gecko’s tail functions as an emergency fifth leg to prevent falling during rapid climbing. A response initiated by slipping causes the tail tip to push against the vertical surface, thereby preventing pitch-back of the head and upper body. When pitch-back cannot be prevented, geckos avoid falling by placing their tail in a posture similar to a bicycle’s kickstand. Should a gecko fall with its back to the ground, a swing of its tail induces the most rapid, zero-angular momentum air-righting response yet measured. Once righted to a sprawled gliding posture, circular tail movements control yaw and pitch as the gecko descends. Our results suggest that large, active tails can function as effective control appendages. These results have provided biological inspiration for the design of an active tail on a climbing robot, and we anticipate their use in small, unmanned gliding vehicles and multisegment spacecraft.
Estimating the phase of synchronized oscillators

Shai Revzen, and J M Guckenheimer

Physical Review E, Nov 2008

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The state of a collection of phase-locked oscillators is determined by a single phase variable or cyclic coordinate. This paper presents a computational method, Phaser, for estimating the phase of phase-locked oscillators from limited amounts of multivariate data in the presence of noise and measurement errors. Measurements are assumed to be a collection of multidimensional time series. Each series consists of several cycles of the same or similar systems. The oscillators within each system are not assumed to be identical. Using measurements of the noise covariance for the multivariate input, data from the individual oscillators in the system are combined to reduce the variance of phase estimates for the whole system. The efficacy of the algorithm is demonstrated on experimental and model data from biomechanics of cockroach running and on simulated oscillators with varying levels of noise.
Progress in motor control - a multidisciplinary perspective

Shai Revzen, D E Koditschek, and R J Full

Nov 2008

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Our objective is to provide experimentalists with neuromechanical control hypotheses that can be tested with kinematic data sets. To illustrate the approach, we select legged animals responding to perturbations during running. In the following sections, we briefly outline our dynamical systems approach, state our over-arching hypotheses, define four neuromechanical control architectures (NCAs) and conclude by proposing a series of perturbation experiments that can begin to identify the simplest architecture that best represents an animal?s controller.

Patents

2021

Wireless power transfer in modular systems

Al-Thaddeus Avestruz, Akshay Sarin, Shai Revzen, and 1 more author

Nov 2021

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2006

Analysis of Electrocardiogram Signals

B Shani, Shai Revzen, and A Frimerman

Nov 2006

WO/2006/123334

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Apparatus for graphical representation of a train of ECG complexes, said ECG complexes comprising an R wave and a T-P interval and having variable isoelectric levels, the apparatus comprising: an isoelectric alignment unit for aligning the complexes in terms of isoelectric level by aligning respective T-P intervals, thereby to provide a graphical representation of said train of ECG complexes; and a temporal alignment unit for aligning said complexes temporally using a predetermined point of respective R waves. The aligned units are superimposed to provide a distribution of a normalized ECG signal over a series of pulses or heartbeats.

2003

Lossless data compression

E Frachtenberg, and Shai Revzen

Feb 2003

20030030575

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Dictionary based data compression apparatus comprising: a library of static dictionaries each optimized for a different data type, a data type determiner operable to scan incoming data and determine a data type thereof, a selector for selecting a static dictionary corresponding to said determined data type and a compressor for compressing said incoming data using said selected dictionary. The apparatus is useful in providing efficient compression of relatively short data packets having undefined contents as may be expected in a network switch.

Published Datasets and Supplemental Code

2021

Challenges in Dynamic Mode Decomposition – figures

Ziyou Wu, S L Brunton, and Shai Revzen

Aug 2021

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The code generates and processes a randomized data-set. To make the result reproducible, we fix the random number seed. These codes were produced as part of the Army Research Office Multi-University Research Initiative ARO MURI W911NF-17-1-0306 "From Data-Driven Operator Theoretic Schemes to Prediction, Inference, and Control of Systems" The code can be run using the runAll.sh shell script (in Linux and OS-X); code should work similarly under windows.
BigAnt v6 robot motion tracking data - RAW dataset

BIRDS-Lab

Aug 2021

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We used a commercial motion tracking system (Qualisys QTM with 10 cameras) to track 3 different BigAnt v6 robots with motion tracking markers attached. The dataset contains picture of one of the robots with the markers highlighted. We ran each robot at multiple "turning parameter" values and gait frequencies, and some robots were run on both High friction (rubber mats) and Low friction (flat linoleum) ground. This is the "raw" dataset. It contain all markers tracked on each of the robot, and is prior to various data conditioning a clean-up (e.g. removing z changes due to slightly non-flat floors) These data were produced in an attempt to characterize the turning and steering behaviors of 1-DoF multi-legged (hexpedal in this case) robots. Such turning behaviors require sliding contact points. The .tar file contains multiple trials in .csv.gz format, with names following an informative naming convention documented in the README. Additional metadata for the trials is given in the metadata.py file in both machine and human readable form.
BigAnt v6 robot motion tracking data - processed dataset

BIRDS-Lab

Aug 2021

Abs DOI

This dataset is derived from the raw dataset found at doi:10.7302/024q-kk06 using the global phase estimation based on the Phaser algorithm doi: 10.1103/PhysRevE.78.051907 and additional processing. These data were produced in an attempt to characterize the turning and steering behaviors of 1-DoF multi-legged (hexpedal in this case) robots. Such turning behaviors require sliding contact points. All the data is provided in a single, large .csv.gz file (416256 rows); additional details and example code in the README
BIRDS Lab Multipod robot motion tracking data - RAW dataset

BIRDS-Lab

Oct 2021

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We used a commercial motion tracking system (Qualisys QTM with 10 cameras) to track Multipod robots with 4,6,8,10 and 12 legs. These robots had motion tracking markers attached. The dataset contains picture of one of the robots with the markers highlighted, and an example video of unmarked robots moving. We ran each robot at multiple inter-segment phase differences and gait frequencies, and some robots were run with two different roll angle amplitudes for the leg motions. This is the "raw" dataset. It contain all markers tracked on each of the robot, and is prior to various data conditioning a clean-up. These data were produced for ARO W911NF-14-1-0573 "Morphologically Modulated Dynamics" to explore the trade-offs between various oscillator coupling models in modeling multilegged locomotion. The data were also used extensively in examining multi-contact slipping, in the studying the influence of number of legs on otherwise identical locomotion patterns, and in the use of geometric mechanics models for multilegged locomotion. Folder and file names encode the meta-data, with names following an informative naming convention documented in the README.
BIRDS Lab Multipod robot motion tracking data - processed data and code

BIRDS-Lab

Oct 2021

Abs DOI

This dataset is derived from the raw dataset found at doi:10.7302/m05a-0d90 using the global phase estimation based on the Phaser algorithm doi: 10.1103/PhysRevE.78.051907 and additional processing. These data were produced for ARO W911NF-14-1-0573 "Morphologically Modulated Dynamics" to explore the trade-offs between various oscillator coupling models in modeling multilegged locomotion of Multipod robots with 6,8,10 and 12 legs. The data is stored in .csv.gz files, one file for each robot morphology. Details of how to run the processing code on the raw dataset to generate the processed files found here, as well as example code for loading the data found here, are in the README. This dataset is self contained and can be used on its own without running any of the provided code.
Walking Like a Worm : dataset and figures

BIRDS-Lab

Aug 2021

Abs DOI

The data here is from multiple sources. It is consolidated in this one location to facilitate easy reproduction of the figures in the authors’ manuscript titled "Walking like a worm : Stokesian kinematics govern legged locomotion". Each constituent data-set is separately available. Ant data was prepared for doi:10.1098/rsos.192068 BigANT robot data was prepared for doi:10.1088/1748-3190/ab84c0 and is available at doi:10.7302/024q-kk06 and doi:10.7302/jh82-fh69
BIRDS Lab Multipod robot motion tracking data - videos

BIRDS-Lab

Oct 2021

Abs DOI

We filmed the motion tracking trials of https://doi.org/10.7302/m05a-0d90 (the "raw" dataset), which track Multipod robots with 4,6,8,10 and 12 legs. We ran each robot at multiple inter-segment phase differences and gait frequencies, and some robots were run with two different roll angle amplitudes for the leg motions. This deposit is the "videos" dataset, which contains video directories for each of the raw data directories.

Pre-prints and Technical Reports

2024

Rapid Integrator for a Class of Multi-Contact Systems

Marion Anderson, and Shai Revzen

2024

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2023

Modeling multi-legged robot locomotion with slipping and its experimental validation

Ziyou Wu, Dan Zhao, and Shai Revzen

2023

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2022

Estimating Phase from Observed Trajectories Using the Temporal 1-Form

Simon Wilshin, Matthew D Kvalheim, Clayton Scott, and 1 more author

2022

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Recovery of Behaviors Encoded via Bilateral Constraints

George Council, and Shai Revzen

2022

This is v3 of the document; first version published 2020

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If robots are ever to achieve autonomous motion comparable to that exhibited by animals, they must acquire the ability to quickly recover motor behaviors when damage, malfunction, or environmental conditions compromise their ability to move effectively. We present an approach which allowed our robots and simulated robots to recover high-degree of freedom motor behaviors within a few dozen attempts. Our approach employs a behavior specification expressing the desired behaviors in terms as rank ordered differential constraints. We show how factoring these constraints through an encoding template produces a recipe for generalizing a previously optimized behavior to new circumstances in a form amenable to rapid learning. We further illustrate that adequate constraints are generically easy to determine in data-driven contexts. As illustration, we demonstrate our recovery approach on a physical 7 DOF hexapod robot, as well as a simulation of a 6 DOF 2D kinematic mechanism. In both cases we recovered a behavior functionally indistinguishable from the previously optimized motion.

2021

Dandelion-Picking Legged Robot

Sandilya Sai Garimella, and Shai Revzen

2021

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Phase Response Curves and the Role of Coordinates

Simon Wilshin, Matthew D Kvalheim, and Shai Revzen

2021

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The “Phase Response Curve” (PRC) is a common tool used to analyze phase resetting in the natural sciences. We make the observation that the PRC with respect to a coordinate y∈\R actually depends on the full choice of coordinates (x,y), x∈\R^d. We give a coordinate-free definition of the PRC making this observation obvious. We show how by controlling y, using delay coordinates of y, and postulating the dynamics of x as a function of x and y, we can sometimes reconstruct the PRC with respect to the (x,y) coordinates. This suggests a means for obtaining the PRC of, e.g., a neuron via a voltage clamp.
Optimizing Gait Libraries via a Coverage Metric

Brian Bittner, and Shai Revzen

2021

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2020

Generic Properties of Koopman Eigenfunctions for Stable Fixed Points and Periodic Orbits

M D Kvalheim, D Hong, and Shai Revzen

2020

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Our recent work established existence and uniqueness results for \mathcalC^k,α_\textloc globally defined linearizing semiconjugacies for \mathcalC^1 flows having a globally attracting hyperbolic fixed point or periodic orbit (Kvalheim and Revzen, 2019). Applications include (i) improvements, such as uniqueness statements, for the Sternberg linearization and Floquet normal form theorems; (ii) results concerning the existence, uniqueness, classification, and convergence of various quantities appearing in the "applied Koopmanism" literature, such as principal eigenfunctions, isostables, and Laplace averages. In this work we give an exposition of some of these results, with an emphasis on the Koopmanism applications, and consider their broadness of applicability. In particular we show that, for "almost all" \mathcalC^∞flows having a globally attracting hyperbolic fixed point or periodic orbit, the \mathcalC^∞Koopman eigenfunctions can be completely classified, generalizing a result known for analytic systems. For such systems, every \mathcalC^∞eigenfunction is uniquely determined by its eigenvalue modulo scalar multiplication.
Data-Driven Geometric System Identification for Shape-Underactuated Dissipative Systems

B Bittner, R L Hatton, and Shai Revzen

2020

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The study of systems whose movement is both geometric and dissipative offers an opportunity to quickly both identify models and optimize motion. Here, the geometry indicates reduction of the dynamics by environmental homogeneity while the dissipative nature minimizes the role of second order (inertial) features in the dynamics. In this work, we extend the tools of geometric system identification to "Shape-Underactuated Dissipative Systems (SUDS)" – systems whose motions are kinematic, but whose actuation is restricted to a subset of the body shape coordinates. A large class of SUDS includes highly damped robots with series elastic actuators, and many soft robots. We validate the predictive quality of the models using simulations of a variety of viscous swimming systems. For a large class of SUDS, we show how the shape velocity actuation inputs can be directly converted into torque inputs suggesting that, e.g., systems with soft pneumatic actuators or dielectric elastomers, could be controlled in this way. Based on fundamental assumptions in the physics, we show how our model complexity scales linearly with the number of passive shape coordinates. This offers a large reduction on the number of trials needed to identify the system model from experimental data, and may reduce overfitting. The sample efficiency of our method suggests its use in modeling, control, and optimization in robotics, and as a tool for the study of organismal motion in friction dominated regimes.

2019

Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits

Matthew D Kvalheim, and Shai Revzen

2019

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We consider C^1 dynamical systems having a globally attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for C^k,α_\textloc globally defined linearizing semiconjugacies, of which Koopman eigenfunctions are a special case. Our main results both generalize and sharpen Sternberg’s C^k linearization theorem for hyperbolic sinks, and in particular our corollaries include uniqueness statements for Sternberg linearizations and Floquet normal forms. Additional corollaries include existence and uniqueness results for C^k,α_\textloc Koopman eigenfunctions, including a complete classification of C^∞eigenfunctions assuming a C^∞dynamical system with semisimple and nonresonant linearization. We give an intrinsic definition of “principal Koopman eigenfunctions” which generalizes the definition of Mohr and Mezić for linear systems, and which includes the notions of “isostables” and “isostable coordinates” appearing in work by Ermentrout, Mauroy, Mezić, Moehlis, Wilson, and others. Our main results yield existence and uniqueness theorems for the principal eigenfunctions and isostable coordinates and also show, e.g., that the (a priori non-unique) “pullback algebra” defined in [MM16b] is unique under certain conditions. We also discuss the limit used to define the “faster” isostable coordinates in [WE18, MWMM19] in light of our main results.
Gait modeling and optimization for the perturbed Stokes regime

Matthew D Kvalheim, Brian Bittner, and Shai Revzen

2019

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Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that for swimming at the "Stokesian" (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection" model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime" where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer linear in shape change rate. We derive this model using results from singular perturbation theory, and the theory of noncompact normally hyperbolic invariant manifolds (NHIMs). Using the theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. This extends our previous work which assumed kinematic "connection" models. To compare the old and new algorithms, we analyze simulated swimmers over a range of inertia to damping ratios. Our new class of models performs well on the Stokesian regime, and over several orders of magnitude outside it into the perturbed Stokes regime, where it gives significantly improved prediction accuracy compared to previous work. In addition to algorithmic improvements, we thereby present a new class of models that is of independent interest. Their application to data-driven modeling improves our ability to study the optimality of animal gaits, and our ability to use hardware-in-the-loop optimization to produce gaits for robots.

2018

A Data-Driven Approach to Connection Modeling

Shai Revzen B Bittner

2018

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2016

Reverse-engineering invariant manifolds with asymptotic phase

Matthew Kvalheim, and Shai Revzen

2016

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We present a recipe for rendering a submanifold normally hyperbolic and invariant within a stability basin. The construction includes the ability to choose the asymptotic phase map. We are motivated by the notion of "templates and anchors" – the biomechanical observation that animal motions are often governed by low dimensional dynamics – and the growing applications in robotics which desire means for making such biologically derived templates govern the dynamics of robots. Our approach is fairly universal, in the sense that a broad range of model reduction constructions must be normally hyperbolic if they are robust, and a broad range of such normally hyperbolic systems can be produces from our construction.
Computing the Bouligand derivative of a class of piecewise-differentiable flows

Shai Revzen, and S A Burden

2016

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2014

Event-selected vector field discontinuities yield piecewise-differentiable flows

S A Burden, S S Sastry, D E Koditschek, and 1 more author

2014

2013

Model reduction near periodic orbits of hybrid dynamical systems

S A Burden, Shai Revzen, and S S Sastry

2013