Constructing Parsimonious Analytic Models for Dynamic Systems via Symbolic Regression

Developing mathematical models of dynamic systems is central to many disciplines of engineering and science. Models facilitate simulations, analysis of the system's behavior, decision making and design of automatic control algorithms. Even inherently model-free control techniques such as reinforcement learning (RL) have been shown to benefit from the use of models, typically learned online. Any model construction method must address the tradeoff between the accuracy of the model and its complexity, which is difficult to strike. In this paper, we propose to employ symbolic regression (SR) to construct parsimonious process models described by analytic equations. We have equipped our method with two different state-of-the-art SR algorithms which automatically search for equations that fit the measured data: Single Node Genetic Programming (SNGP) and Multi-Gene Genetic Programming (MGGP). In addition to the standard problem formulation in the state-space domain, we show how the method can also be applied to input-output models of the NARX (nonlinear autoregressive with exogenous input) type. We present the approach on three simulated examples with up to 14-dimensional state space: an inverted pendulum, a mobile robot, and a bipedal walking robot. A comparison with deep neural networks and local linear regression shows that SR in most cases outperforms these commonly used alternative methods. We demonstrate on a real pendulum system that the analytic model found enables a RL controller to successfully perform the swing-up task, based on a model constructed from only 100 data samples

Evolving a Behavioral Repertoire for a Walking Robot

Numerous algorithms have been proposed to allow legged robots to learn to walk. However, the vast majority of these algorithms is devised to learn to walk in a straight line, which is not sufficient to accomplish any real-world mission. Here we introduce the Transferability-based Behavioral Repertoire Evolution algorithm (TBR-Evolution), a novel evolutionary algorithm that simultaneously discovers several hundreds of simple walking controllers, one for each possible direction. By taking advantage of solutions that are usually discarded by evolutionary processes, TBR-Evolution is substantially faster than independently evolving each controller. Our technique relies on two methods: (1) novelty search with local competition, which searches for both high-performing and diverse solutions, and (2) the transferability approach, which com-bines simulations and real tests to evolve controllers for a physical robot. We evaluate this new technique on a hexapod robot. Results show that with only a few dozen short experiments performed on the robot, the algorithm learns a repertoire of con-trollers that allows the robot to reach every point in its reachable space. Overall, TBR-Evolution opens a new kind of learning algorithm that simultaneously optimizes all the achievable behaviors of a robot.Comment: 33 pages; Evolutionary Computation Journal 201

Evolvability signatures of generative encodings: beyond standard performance benchmarks

Evolutionary robotics is a promising approach to autonomously synthesize machines with abilities that resemble those of animals, but the field suffers from a lack of strong foundations. In particular, evolutionary systems are currently assessed solely by the fitness score their evolved artifacts can achieve for a specific task, whereas such fitness-based comparisons provide limited insights about how the same system would evaluate on different tasks, and its adaptive capabilities to respond to changes in fitness (e.g., from damages to the machine, or in new situations). To counter these limitations, we introduce the concept of "evolvability signatures", which picture the post-mutation statistical distribution of both behavior diversity (how different are the robot behaviors after a mutation?) and fitness values (how different is the fitness after a mutation?). We tested the relevance of this concept by evolving controllers for hexapod robot locomotion using five different genotype-to-phenotype mappings (direct encoding, generative encoding of open-loop and closed-loop central pattern generators, generative encoding of neural networks, and single-unit pattern generators (SUPG)). We observed a predictive relationship between the evolvability signature of each encoding and the number of generations required by hexapods to adapt from incurred damages. Our study also reveals that, across the five investigated encodings, the SUPG scheme achieved the best evolvability signature, and was always foremost in recovering an effective gait following robot damages. Overall, our evolvability signatures neatly complement existing task-performance benchmarks, and pave the way for stronger foundations for research in evolutionary robotics.Comment: 24 pages with 12 figures in the main text, and 4 supplementary figures. Accepted at Information Sciences journal (in press). Supplemental videos are available online at, see http://goo.gl/uyY1R

Genetically evolved dynamic control for quadruped walking

The aim of this dissertation is to show that dynamic control of quadruped locomotion is achievable through the use of genetically evolved central pattern generators. This strategy is tested both in simulation and on a walking robot. The design of the walker has been chosen to be statically unstable, so that during motion less than three supporting feet may be in contact with the ground. The control strategy adopted is capable of propelling the artificial walker at a forward locomotion speed of ~1.5 Km/h on rugged terrain and provides for stability of motion. The learning of walking, based on simulated genetic evolution, is carried out in simulation to speed up the process and reduce the amount of damage to the hardware of the walking robot. For this reason a general-purpose fast dynamic simulator has been developed, able to efficiently compute the forward dynamics of tree-like robotic mechanisms. An optimization process to select stable walking patterns is implemented through a purposely designed genetic algorithm, which implements stochastic mutation and cross-over operators. The algorithm has been tailored to address the high cost of evaluation of the optimization function, as well as the characteristics of the parameter space chosen to represent controllers. Experiments carried out on different conditions give clear indications on the potential of the approach adopted. A proof of concept is achieved, that stable dynamic walking can be obtained through a search process which identifies attractors in the dynamics of the motor-control system of an artificial walker

Knowledge-based vision and simple visual machines

The vast majority of work in machine vision emphasizes the representation of perceived objects and events: it is these internal representations that incorporate the 'knowledge' in knowledge-based vision or form the 'models' in model-based vision. In this paper, we discuss simple machine vision systems developed by artificial evolution rather than traditional engineering design techniques, and note that the task of identifying internal representations within such systems is made difficult by the lack of an operational definition of representation at the causal mechanistic level. Consequently, we question the nature and indeed the existence of representations posited to be used within natural vision systems (i.e. animals). We conclude that representations argued for on a priori grounds by external observers of a particular vision system may well be illusory, and are at best place-holders for yet-to-be-identified causal mechanistic interactions. That is, applying the knowledge-based vision approach in the understanding of evolved systems (machines or animals) may well lead to theories and models that are internally consistent, computationally plausible, and entirely wrong

Generating walking behaviours in legged robots

Many legged robots have boon built with a variety of different abilities, from running to liopping to climbing stairs. Despite this however, there has been no consistency of approach to the problem of getting them to walk. Approaches have included breaking down the walking step into discrete parts and then controlling them separately, using springs and linkages to achieve a passive walking cycle, and even working out the necessary movements in simulation and then imposing them on the real robot. All of these have limitations, although most were successful at the task for which they were designed. However, all of them fall into one of two categories: either they alter the dynamics of the robots physically so that the robot, whilst very good at walking, is not as general purpose as it once was (as with the passive robots), or they control the physical mechanism of the robot directly to achieve their goals, and this is a difficult task.In this thesis a design methodology is described for building controllers for 3D dynam¬ ically stable walking, inspired by the best walkers and runners around — ourselves — so the controllers produced are based 011 the vertebrate Central Nervous System. This means that there is a low-level controller which adapts itself to the robot so that, when switched on, it can be considered to simulate the springs and linkages of the passive robots to produce a walking robot, and this now active mechanism is then controlled by a relatively simple higher level controller. This is the best of both worlds — we have a robot which is inherently capable of walking, and thus is easy to control like the passive walkers, but also retains the general purpose abilities which makes it so potentially useful.This design methodology uses an evolutionary algorithm to generate low-level control¬ lers for a selection of simulated legged robots. The thesis also looks in detail at previous walking robots and their controllers and shows that some approaches, including staged evolution and hand-coding designs, may be unnecessary, and indeed inappropriate, at least for a general purpose controller. The specific algorithm used is evolutionary, using a simple genetic algorithm to allow adaptation to different robot configurations, and the controllers evolved are continuous time neural networks. These are chosen because of their ability to entrain to the movement of the robot, allowing the whole robot and network to be considered as a single dynamical system, which can then be controlled by a higher level system.An extensive program of experiments investigates the types of neural models and net¬ work structures which are best suited to this task, and it is shown that stateless and simple dynamic neural models are significantly outperformed as controllers by more complex, biologically plausible ones but that other ideas taken from biological systems, including network connectivities, are not generally as useful and reasons for this are examined.The thesis then shows that this system, although only developed 011 a single robot, is capable of automatically generating controllers for a wide selection of different test designs. Finally it shows that high level controllers, at least to control steering and speed, can be easily built 011 top of this now active walking mechanism