Samuel A. Burden

Programmable parts: a demonstration of the grammatical approach to self-organization

In this paper, we introduce a robotic implementation of the theory of graph grammars (Klavins et al., 2005), which we use to model and direct self-organization in a formal, predictable and provably-correct fashion. The robots, which we call programmable parts, float passively on an air table and bind to each other upon random collisions. Once attached, they execute local rules that determine how their internal states change and whether they should remain bound. We demonstrate through experiments how they can self-organize into a global structure by executing a common graph grammar in a completely distributed fashion. The system also presents a challenge to the grammatical method (and to distributed systems approaches in general) due to the stochastic nature of its dynamics. We conclude by discussing these challenges and our initial approach to addressing them.

Bio-inspired design and dynamic maneuverability of a minimally actuated six-legged robot

Rapidly running arthropods like cockroaches make use of passive dynamics to achieve remarkable locomotion performance with regard to stability, speed, and maneuverability. In this work, we take inspiration from these organisms to design, fabricate, and control a 10cm, 24 gram underactuated hexapedal robot capable of running at 14 body lengths per second and performing dynamic turning maneuvers. Our design relies on parallel kinematic mechanisms fabricated using the scaled smart composite microstructures (SCM) process and viscoelastic polymer legs with tunable stiffness. In addition to the novel robot design, we present experimental validation of the lateral leg spring (LLS) locomotion models prediction that dynamic turning can be achieved by modulating leg stiffness. Finally, we present and validate a leg design for active stiffness control using shape memory alloy and demonstrate the ability of the robot to execute near-gymnastic 90° turns in the span of five strides.

Optimal Rules for Programmed Stochastic Self-Assembly

We consider the control of programmable selfassembling systems whose dynamics are governed by stochastic reaction-diffusion dynamics. In our system, particles may decide the outcomes of reactions initiated by the environment, thereby steering the global system to produce a desired assembly type. We describe a method that automatically generates a program maximizing yield based on tuning the rates of experimentally determined reaction pathways. We demonstrate the method using theoretical examples and with a robotic testbed. Finally, we present, in the form of a graph grammar, a communication protocol that implements the generated programs in a distributed manner.

Setpoint regulation for stochastically interacting robots

We present an integral feedback controller that regulates the average copy number of an assembly in a system of stochastically interacting robots. The mathematical model for these robots is a tunable reaction network, which makes this approach applicable to a large class of other systems, including ones that exhibit stochastic self-assembly at various length scales. We prove that this controller works for a range of setpoints and how to compute this range both analytically and experimentally. Finally, we demonstrate these ideas on a physical testbed.

Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems

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.

Metrization and Simulation of Controlled Hybrid Systems

The study of controlled hybrid systems requires practical tools for approximation and comparison of system behaviors. Existing approaches to these problems impose undue restrictions on the systems continuous and discrete dynamics. Metrization and simulation of controlled hybrid systems is considered here in a unified framework by constructing a state space metric. The metric is applied to develop a numerical simulation algorithm that converges uniformly, with a known rate of convergence, to orbitally stable executions of controlled hybrid systems, up to and including Zeno events. Benchmark hybrid phenomena illustrate the utility of the proposed tools.

Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches

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

Dimension reduction near periodic orbits of hybrid systems

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