Nilpotent Bases for a Class of Non-Integrable Distributions with Applications to Trajectory Generation for Nonholonomic Systems

This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using "chained form" and cast that work into a coordinate-free setting

Real-valued average consensus over noisy quantized channels

This paper concerns the average consensus problem with the constraint of quantized communication between nodes. A broad class of algorithms is analyzed, in which the transmission strategy, which decides what value to communicate to the neighbours, can include various kinds of rounding, probabilistic quantization, and bounded noise. The arbitrariness of the transmission strategy is compensated by a feedback mechanism which can be interpreted as a self-inhibitory action. The result is that the average of the nodes state is not conserved across iterations, and the nodes do not converge to a consensus; however, we show that both errors can be made as small as desired. Bounds on these quantities involve the spectral properties of the graph and can be proved by employing elementary techniques of LTI systems analysis

A group-theoretic approach to formalizing bootstrapping problems

The bootstrapping problem consists in designing agents that learn a model of themselves and the world, and utilize it to achieve useful tasks. It is different from other learning problems as the agent starts with uninterpreted observations and commands, and with minimal prior information about the world. In this paper, we give a mathematical formalization of this aspect of the problem. We argue that the vague constraint of having "no prior information" can be recast as a precise algebraic condition on the agent: that its behavior is invariant to particular classes of nuisances on the world, which we show can be well represented by actions of groups (diffeomorphisms, permutations, linear transformations) on observations and commands. We then introduce the class of bilinear gradient dynamics sensors (BGDS) as a candidate for learning generic robotic sensorimotor cascades. We show how framing the problem as rejection of group nuisances allows a compact and modular analysis of typical preprocessing stages, such as learning the topology of the sensors. We demonstrate learning and using such models on real-world range-finder and camera data from publicly available datasets

Quantitative Performance Bounds in Biomolecular Circuits due to Temperature Uncertainty

Performance of biomolecular circuits is affected by changes in temperature, due to its influence on underlying reaction rate parameters. While these performance variations have been estimated using Monte Carlo simulations, how to analytically bound them is generally unclear. To address this, we apply control-theoretic representations of uncertainty to examples of different biomolecular circuits, developing a framework to represent uncertainty due to temperature. We estimate bounds on the steady-state performance of these circuits due to temperature uncertainty. Through an analysis of the linearised dynamics, we represent this uncertainty as a feedback uncertainty and bound the variation in the magnitude of the input-output transfer function, providing a estimate of the variation in frequency-domain properties. Finally, we bound the variation in the time trajectories, providing an estimate of variation in time-domain properties. These results should enable a framework for analytical characterisation of uncertainty in biomolecular circuit performance due to temperature variation and may help in estimating relative performance of different controllers

Extremal Properties of Complex Networks

We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph qualities for any arbitrary number of nodes and edges and analytically derive the form and properties of such networks

Decomposing GR(1) Games with Singleton Liveness Guarantees for Efficient Synthesis

Temporal logic based synthesis approaches are often used to find trajectories that are correct-by-construction for tasks in systems with complex behavior. Some examples of such tasks include synchronization for multi-agent hybrid systems, reactive motion planning for robots. However, the scalability of such approaches is of concern and at times a bottleneck when transitioning from theory to practice. In this paper, we identify a class of problems in the GR(1) fragment of linear-time temporal logic (LTL) where the synthesis problem allows for a decomposition that enables easy parallelization. This decomposition also reduces the alternation depth, resulting in more efficient synthesis. A multi-agent robot gridworld example with coordination tasks is presented to demonstrate the application of the developed ideas and also to perform empirical analysis for benchmarking the decomposition-based synthesis approach

Limits on the Network Sensitivity Function for Multi-Agent Systems on a Graph

This report explores the tradeoffs and limits of performance in feedback control of interconnected multi-agent systems, focused on the network sensitivity functions. We consider the interaction topology described by a directed graph and we prove that the sensitivity transfer functions between every pair of agents, arbitrarily connected, can be derived using a version of the Mason's Direct Rule. Explicit forms for special types of graphs are presented. An analysis of the role of cycles points out that these structures influence and limit considerably the performance of the system. The more the cycles are equally distributed among the formation, the better performance the system can achieve, but they are always worse than the single agent case. We also prove the networked version of Bode's integral formula, showing that it still holds for multi-agent systems

Hiding variables when decomposing specifications into GR(1) contracts

We propose a method for eliminating variables from component specifications during the decomposition of GR(1) properties into contracts. The variables that can be eliminated are identified by parameterizing the communication architecture to investigate the dependence of realizability on the availability of information. We prove that the selected variables can be hidden from other components, while still expressing the resulting specification as a game with full information with respect to the remaining variables. The values of other variables need not be known all the time, so we hide them for part of the time, thus reducing the amount of information that needs to be communicated between components. We improve on our previous results on algorithmic decomposition of GR(1) properties, and prove existence of decompositions in the full information case. We use semantic methods of computation based on binary decision diagrams. To recover the constructed specifications so that humans can read them, we implement exact symbolic minimal covering over the lattice of integer orthotopes, thus deriving minimal formulae in disjunctive normal form over integer variable intervals