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Despite the neuromechanical complexity underlying animal locomotion, the steady-state center-of-mass motions and ground reaction forces of animal running can be predicted by simple spring-mass models such as the canonical spring-loaded inverted pendulum (SLIP) model. Such SLIP models have been useful for the fields of biomechanics and robotics in part because ground reaction forces are commonly measured and readily available for comparing with model predictions. To better predict the stability of running, beyond the canonical conservative SLIP model, more recent extensions have been proposed and investigated with hip actuation and linear leg damping (e.g., hip-actuated SLIP). So far, these attempts have gained improved prediction of the stability of locomotion but have led to a loss of the ability to accurately predict ground reaction forces. Unfortunately, the linear damping utilized in current models leads to an unrealistic prediction of damping force and ground reaction force with a large nonzero magnitude at touchdown (TD). Here, we develop a leg damping model that is bilinear in leg length and velocity in order to yield improved damping force and ground reaction force prediction. We compare the running ground reaction forces, small and large perturbation stability, parameter sensitivity, and energetic cost resulting from both the linear and bilinear damping models. We found that bilinear damping helps to produce more realistic, smooth vertical ground reaction forces, thus fixing the current problem with the linear damping model. Despite large changes in the damping force and power loss profile during the stance phase, the overall dynamics and energetics on a stride-to-stride basis of the two models are largely the same, implying that the integrated effect of damping over a stride is what matters most to the stability and energetics of running. Overall, this new model, an actuated SLIP model with bilinear damping, can provide significantly improved prediction of ground reaction forces as well as stability and energetics of locomotion.
In this paper, we explore an approach to actively plan and excite contact modes in differentiable simulators as a means to tighten the sim-to-real gap. We propose an optimal experimental design approach derived from information-theoretic methods to identify and search for information-rich contact modes through the use of contact-implicit optimization. We demonstrate our approach on a robot parameter estimation problem with unknown inertial and kinematic parameters which actively seeks contacts with a nearby surface. We show that our approach improves the identification of unknown parameter estimates over experimental runs by an estimate error reduction of at least $\sim 84\%$ when compared to a random sampling baseline, with significantly higher information gains.
This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We first motivate the use of representing nonlinear systems as linear Koopman operator systems by illustrating the improved model-based control performance with an actuated Van der Pol system. Information-theoretic methods are then applied to the Koopman operator formulation of dynamical systems where we derive a controller for active learning of robot dynamics. The active learning controller is shown to increase the rate of information about the Koopman operator. In addition, our active learning controller can readily incorporate policies built on the Koopman dynamics, enabling the benefits of fast active learning and improved control. Results using a quadcopter illustrate single-execution active learning and stabilization capabilities during free-fall. The results for active learning are extended for automating Koopman observables and we implement our method on real robotic systems.
We present a method for controlling a swarm using its spectral decomposition—that is, by describing the set of trajectories of a swarm in terms of a spatial distribution throughout the operational domain—guaranteeing scale invariance with respect to the number of agents both for computation and for the operator tasked with controlling the swarm. We use ergodic control, decentralized across the network, for implementation. In the DARPA OFFSET program field setting, we test this interface design for the operator using the STOMP interface—the same interface used by Raytheon BBN throughout the duration of the OFFSET program. In these tests, we demonstrate that our approach is scale-invariant—the user specification does not depend on the number of agents; it is persistent—the specification remains active until the user specifies a new command; and it is real-time—the user can interact with and interrupt the swarm at any time. Moreover, we show that the spectral/ergodic specification of swarm behavior degrades gracefully as the number of agents goes down, enabling the operator to maintain the same approach as agents become disabled or are added to the network. We demonstrate the scale invariance and dynamic response of our system in a field-relevant simulator on a variety of tactical scenarios with up to 50 agents. We also demonstrate the dynamic response of our system in the field with a smaller team of agents. Lastly, we make the code for our system available.
Exploration requires that robots reason about numerous ways to cover a space in response to dynamically changing conditions. However, in continuous domains there are potentially infinitely many options for robots to explore which can prove computationally challenging. How then should a robot efficiently optimize and choose exploration strategies to adopt? In this work, we explore this question through the use of variational inference to efficiently solve for distributions of coverage trajectories. Our approach leverages ergodic search methods to optimize coverage trajectories in continuous time and space. In order to reason about distributions of trajectories, we formulate ergodic search as a probabilistic inference problem. We propose to leverage Stein variational methods to approximate a posterior distribution over ergodic trajectories through parallel computation. As a result, it becomes possible to efficiently optimize distributions of feasible coverage trajectories for which robots can adapt exploration. We demonstrate that the proposed Stein variational ergodic search approach facilitates efficient identification of multiple coverage strategies and show online adaptation in a model-predictive control formulation. Simulated and physical experiments demonstrate adaptability and diversity in exploration strategies online.
Although a number of solutions exist for the problems of coverage, search and target localization---commonly addressed separately---whether there exists a unified strategy that addresses these objectives in a coherent manner without being application-specific remains a largely open research question. In this paper, we develop a receding-horizon ergodic control approach, based on hybrid systems theory, that has the potential to fill this gap. The nonlinear model predictive control algorithm plans real-time motions that optimally improve ergodicity with respect to a distribution defined by the expected information density across the sensing domain. We establish a theoretical framework for global stability guarantees with respect to a distribution. Moreover, the approach is distributable across multiple agents, so that each agent can independently compute its own control while sharing statistics of its coverage across a communication network. We demonstrate the method in both simulation and in experiment in the context of target localization, illustrating that the algorithm is independent of the number of targets being tracked and can be run in real-time on computationally limited hardware platforms.
In this paper, we consider the problem of improving the long-term accuracy of data-driven approximations of Koopman operators, which are infinite-dimensional linear representations of general nonlinear systems, by bounding the eigenvalues of the linear operator. We derive a formula for the global error of general Koopman representations and motivate imposing stability constraints on the data-driven model to improve the approximation of nonlinear systems over a longer horizon. In addition, constraints on admissible basis functions for a stable Koopman operator are presented, as well as conditions for constructing a Lyapunov function for nonlinear systems. The modified linear representation is the nearest \textit{stable} (all eigenvalues are equal or less than 1) matrix solution to a least-squares minimization and bounds the prediction of the system response. We demonstrate the benefit of stable Koopman operators in prediction and control performance using the systems of a pendulum, a hopper, and a quadrotor.