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Samuel A. Burden

Publications and source records attributed to Samuel A. Burden.

At least 19 recordsLinked to original sources
arXiv2025

Spring-Brake! Handed Shearing Auxetics Improve Efficiency of Hopping and Standing

Energy efficiency is critical to the success of legged robotics. Efficiency is lost through wasted energy during locomotion and standing. Including elastic elements has been shown to reduce movement costs, while including breaks can reduce standing costs. However, adding separate elements for each increases the mass and complexity of a leg, reducing overall system performance. Here we present a novel compliant mechanism using a Handed Shearing Auxetic (HSA) that acts as a spring and break in a monopod hopping robot. The HSA acts as a parallel elastic actuator, reducing electrical power for dynamic hopping and matching the efficiency of state-of-the-art compliant hoppers. The HSA\u2019s auxetic behavior enables dual functionality. During static tasks, it locks under large forces with minimal input power by blocking deformation, creating high friction similar to a capstan mechanism. This allows the leg to support heavy loads without motor torque, addressing thermal inefficiency. The multi-functional design enhances both dynamic and static performance, offering a versatile solution for robotic applications.

cs.RO↗
arXiv2025

A Learning Algorithm That Attains the Human Optimum in a Repeated Human-Machine Interaction Game

When humans interact with learning-based control systems, a common goal is to minimize a cost function known only to the human. For instance, an exoskeleton may adapt its assistance in an effort to minimize the human's metabolic cost-of-transport. Conventional approaches to synthesizing the learning algorithm solve an inverse problem to infer the human's cost. However, these problems can be ill-posed, hard to solve, or sensitive to problem data. Here we show a game-theoretic learning algorithm that works solely by observing human actions to find the cost minimum, avoiding the need to solve an inverse problem. We evaluate the performance of our algorithm in an extensive set of human subjects experiments, demonstrating consistent convergence to the minimum of a prescribed human cost function in scalar and multidimensional instantiations of the game. We conclude by outlining future directions for theoretical and empirical extensions of our results.

cs.GT↗
arXiv2024

Effect of Adaptation Rate and Cost Display in a Human-AI Interaction Game

As interactions between humans and AI become more prevalent, it is critical to have better predictors of human behavior in these interactions. We investigated how changes in the AI's adaptive algorithm impact behavior predictions in two-player continuous games. In our experiments, the AI adapted its actions using a gradient descent algorithm under different adaptation rates while human participants were provided cost feedback. The cost feedback was provided by one of two types of visual displays: (a) cost at the current joint action vector, or (b) cost in a local neighborhood of the current joint action vector. Our results demonstrate that AI adaptation rate can significantly affect human behavior, having the ability to shift the outcome between two game theoretic equilibrium. We observed that slow adaptation rates shift the outcome towards the Nash equilibrium, while fast rates shift the outcome towards the human-led Stackelberg equilibrium. The addition of localized cost information had the effect of shifting outcomes towards Nash, compared to the outcomes from cost information at only the current joint action vector. Future work will investigate other effects that influence the convergence of gradient descent games.

cs.AI↗
arXiv2023

Consistent Conjectural Variations Equilibrium: Characterization & Stability for a Class of Continuous Games

Leveraging tools from the study of linear fractional transformations and algebraic Riccati equations, a local characterization of consistent conjectural variations equilibrium is given for two player games on continuous action spaces with costs approximated by quadratic functions. A discrete time dynamical system in the space of conjectures is derived, a solution method for computing fixed points of these dynamics (equilibria) is given, local stability properties of the dynamics around the equilibria are characterized, and conditions are given that guarantee a unique stable equilibrium.

cs.GT↗
arXiv2023

Human adaptation to adaptive machines converges to game-theoretic equilibria

Adaptive machines have the potential to assist or interfere with human behavior in a range of contexts, from cognitive decision-making to physical device assistance. Therefore it is critical to understand how machine learning algorithms can influence human actions, particularly in situations where machine goals are misaligned with those of people. Since humans continually adapt to their environment using a combination of explicit and implicit strategies, when the environment contains an adaptive machine, the human and machine play a game. Game theory is an established framework for modeling interactions between two or more decision-makers that has been applied extensively in economic markets and machine algorithms. However, existing approaches make assumptions about, rather than empirically test, how adaptation by individual humans is affected by interaction with an adaptive machine. Here we tested learning algorithms for machines playing general-sum games with human subjects. Our algorithms enable the machine to select the outcome of the co-adaptive interaction from a constellation of game-theoretic equilibria in action and policy spaces. Importantly, the machine learning algorithms work directly from observations of human actions without solving an inverse problem to estimate the human's utility function as in prior work. Surprisingly, one algorithm can steer the human-machine interaction to the machine's optimum, effectively controlling the human's actions even while the human responds optimally to their perceived cost landscape. Our results show that game theory can be used to predict and design outcomes of co-adaptive interactions between intelligent humans and machines.

cs.AI↗
arXiv2022

On infinitesimal contraction analysis for hybrid systems

Infinitesimal contraction analysis, wherein global asymptotic convergence results are obtained from local dynamical properties, has proven to be a powerful tool for applications in biological, mechanical, and transportation systems. The technique has primarily been developed for systems governed by a single, possibly nonsmooth, differential or difference equation. We generalize infinitesimal contraction analysis to hybrid systems governed by interacting differential and difference equations. Importantly, we leverage an intrinsic distance function to derive the first contraction results for hybrid systems without restrictions on mode sequence or dwell time. Our theoretical results are illustrated in several examples and applications.

eess.SY↗
arXiv2021

A pasting lemma for Lipschitz functions

We give a necessary and sufficient condition ensuring that any function which is separately Lipschitz on two fixed compact sets is Lipschitz on their union.

math.CA↗
arXiv2021

Representing and computing the B-derivative of an $EC^r$ vector field's $PC^r$ flow

This paper concerns the 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-$C^r$ flow generated by an event-selected $C^r$ vector field. Our results are remarkably efficient: although there are factorially many "pieces" of the desired derivative, we provide an algorithm that evaluates its action on a given 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.

math.DS↗
arXiv2021

Stability of Gradient Learning Dynamics in Continuous Games: Vector Action Spaces

Towards characterizing the optimization landscape of games, this paper analyzes the stability of gradient-based dynamics near fixed points of two-player continuous games. We introduce the quadratic numerical range as a method to characterize the spectrum of game dynamics and prove the robustness of equilibria to variations in learning rates. By decomposing the game Jacobian into symmetric and skew-symmetric components, we assess the contribution of a vector field's potential and rotational components to the stability of differential Nash equilibria. Our results show that in zero-sum games, all Nash are stable and robust; in potential games, all stable points are Nash. For general-sum games, we provide a sufficient condition for instability. We conclude with a numerical example in which learning with timescale separation results in faster convergence.

cs.GT↗
arXiv2020

Stability of Gradient Learning Dynamics in Continuous Games: Scalar Action Spaces

Learning processes in games explain how players grapple with one another in seeking an equilibrium. We study a natural model of learning based on individual gradients in two-player continuous games. In such games, the arguably natural notion of a local equilibrium is a differential Nash equilibrium. However, the set of locally exponentially stable equilibria of the learning dynamics do not necessarily coincide with the set of differential Nash equilibria of the corresponding game. To characterize this gap, we provide formal guarantees for the stability or instability of such fixed points by leveraging the spectrum of the linearized game dynamics. We provide a comprehensive understanding of scalar games and find that equilibria that are both stable and Nash are robust to variations in learning rates.

cs.GT↗
arXiv2020

A Hybrid Systems Model for Simple Manipulation and Self-Manipulation Systems

Rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs are ubiquitous simplifications introduced to reduce the complexity of mechanics models despite the obvious physical inaccuracies that each incurs individually. In concert, it is well known that the interaction of such idealized approximations can lead to conflicting and even paradoxical results. As robotics modeling moves from the consideration of isolated behaviors to the analysis of tasks requiring their composition, a mathematically tractable framework for building models that combine these simple approximations yet achieve reliable results is overdue. In this paper we present a formal hybrid dynamical system model that introduces suitably restricted compositions of these familiar abstractions with the guarantee of consistency analogous to global existence and uniqueness in classical dynamical systems. The hybrid system developed here provides a discontinuous but self-consistent approximation to the continuous (though possibly very stiff and fast) dynamics of a physical robot undergoing intermittent impacts. The modeling choices sacrifice some quantitative numerical efficiencies while maintaining qualitatively correct and analytically tractable results with consistency guarantees promoting their use in formal reasoning about mechanism, feedback control, and behavior design in robots that make and break contact with their environment.

cs.RO↗
arXiv2019

Nonsmooth optimal value and policy functions in mechanical systems subject to unilateral constraints

State-of-the-art approaches to optimal control use smooth approximations of value and policy functions and gradient-based algorithms for improving approximator parameters. Unfortunately, we show that value and policy functions that arise in optimal control of mechanical systems subject to unilateral constraints -- i.e. the contact-rich dynamics of robot locomotion and manipulation -- are generally nonsmooth due to the underlying dynamics exhibiting discontinuous or piecewise-differentiable trajectory outcomes. Simple mechanical systems are used to illustrate this result and the implications for optimal control of contact-rich robot dynamics.

cs.RO↗
arXiv2019

Convergence Analysis of Gradient-Based Learning with Non-Uniform Learning Rates in Non-Cooperative Multi-Agent Settings

Considering a class of gradient-based multi-agent learning algorithms in non-cooperative settings, we provide local convergence guarantees to a neighborhood of a stable local Nash equilibrium. In particular, we consider continuous games where agents learn in (i) deterministic settings with oracle access to their gradient and (ii) stochastic settings with an unbiased estimator of their gradient. Utilizing the minimum and maximum singular values of the game Jacobian, we provide finite-time convergence guarantees in the deterministic case. On the other hand, in the stochastic case, we provide concentration bounds guaranteeing that with high probability agents will converge to a neighborhood of a stable local Nash equilibrium in finite time. Different than other works in this vein, we also study the effects of non-uniform learning rates on the learning dynamics and convergence rates. We find that much like preconditioning in optimization, non-uniform learning rates cause a distortion in the vector field which can, in turn, change the rate of convergence and the shape of the region of attraction. The analysis is supported by numerical examples that illustrate different aspects of the theory. We conclude with discussion of the results and open questions.

math.OC↗
arXiv2019

Offline state estimation for hybrid systems via nonsmooth variable projection

A hybrid dynamical system switches between dynamic regimes at time- or state-triggered events. We propose an offline algorithm that simultaneously estimates discrete and continuous components of a hybrid system's state. We formulate state estimation as a continuous optimization problem by relaxing the discrete component and use a robust loss function to accommodate large changes in the continuous component during switching events. Subsequently, we develop a novel nonsmooth variable projection algorithm with Gauss-Newton updates to solve the state estimation problem and prove the algorithm's global convergence to stationary points. We demonstrate the effectiveness of our approach on simple piecewise-linear and -nonlinear mechanical systems undergoing intermittent impact.

math.OC↗
arXiv2016

Piecewise-differentiable trajectory outcomes in mechanical systems subject to unilateral constraints

We provide conditions under which trajectory outcomes in mechanical systems subject to unilateral constraints depend piecewise-differentiably on initial conditions, even as the sequence of constraint activations and deactivations varies. This builds on prior work that provided conditions ensuring existence, uniqueness, and continuity of trajectory outcomes, and extends previous differentiability results that applied only to fixed constraint (de)activation sequences. We discuss extensions of our result and implications for assessing stability and controllability.

math.OC↗
arXiv2016

A Hybrid Dynamical Extension of Averaging

We extend a smooth dynamical systems averaging technique to a class of hybrid systems with a limit cycle that is particularly relevant to the synthesis of stable legged gaits. After introducing a definition of hybrid averageability sufficient to recover the classical result, we provide a simple illustration of its applicability to legged locomotion and conclude with some rather more speculative remarks concerning the prospects for further generalization of these ideas.

cs.RO↗
arXiv2015

Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows

We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few domains of application. 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) that is piecewise-linear and continuous at every point. We exploit this first-order approximation to infer existence of piecewise-differentiable impact maps (including Poincaré maps for periodic orbits), show the flow is locally conjugate (via a piecewise-differentiable homeomorphism) to a flowbox, and assess the effect of perturbations (both infinitesimal and non-infinitesimal) on the flow. We use these results to give a sufficient condition for the exponential stability of a periodic orbit passing through a point of multiply intersecting events, and apply the theory in illustrative examples to demonstrate synchronization in abstract first- and second-order phase oscillator models.

math.DS↗