Legged robots are useful in tasks such as search and rescue because they can effectively navigate on rugged terrain. However, it is difficult to design controllers for them that would be stable and robust. Learning the control behavior is difficult because optimal behavior is not known, and the search space is too large for reinforcement learning and for straightforward evolution. As a solution, this paper proposes a modular approach for evolving neural network controllers for such robots. The search space is effectively reduced by exploiting symmetry in the robot morphology, and encoding it into network modules. Experiments involving physically realistic simulations of a quadruped robot produce the same symmetric gaits, such as pronk, pace, bound and trot, that are seen in quadruped animals. Moreover, the robot can transition dynamically to more effective gaits when faced with obstacles. The modular approach also scales well when the number of legs or their degrees of freedom are increased. Evolved non-modular controllers, in contrast, produce gaits resembling crippled animals that are much less effective and do not scale up as a result. Hand-designed controllers are also less effective, especially on an obstacle terrain. These results suggest that the modular approach is effective for designing robust locomotion controllers for multilegged robots.

SUMMARYDespite extensive research, optimal performance has not easily been available previously for matrix multiplication (especially for large matrices) on most architectures because of the lack of a structured approach and the limitations imposed by matrix storage formats. A simple but effective framework is presented here that lays the foundation for building high-performance matrix-multiplication codes in a structured, portable and efficient manner. The resulting codes are validated on three different representative RISC and CISC architectures on which they significantly outperform highly optimized libraries such as ATLAS and other competing methodologies reported in the literature. The main component of the proposed approach is a hierarchical storage format that efficiently generalizes the applicability of the memory hierarchy friendly Morton ordering to arbitrary-sized matrices. The storage format supports polyalgorithms, which are shown here to be essential for obtaining the best possible performance for a range of problem sizes. Several algorithmic advances are made in this paper, including an oscillating iterative algorithm for matrix multiplication and a variable recursion cutoff criterion for Strassen's algorithm. The authors expose the need to standardize linear algebra kernel interfaces, distinct from the BLAS, for writing portable high-performance code. These kernel routines operate on small blocks that fit in the L1 cache. The performance advantages of the proposed framework can be effectively delivered to new and existing applications through the use of object-oriented or compiler-based approaches.

Abstract-Controllers for multilegged robots are characterized by modularity and symmetry. However, the controller symmetries necessary for generating appropriate gaits are often difficult to determine analytically. This paper utilizes a nature-inspired approach called Evolution of Network Symmetry and mOdularity (ENSO) to evolve such controllers automatically. It uses group theory to mutate symmetry systematically, making it more effective than mutating symmetry randomly. This approach was evaluated by evolving modular neural network controllers for a quadruped robot in physically realistic simulations. On flat ground, the resulting controllers are as effective as those having hand-designed symmetries. However, they are significantly faster when evolved on inclined ground, where the appropriate symmetries are difficult to determine manually. The group-theoretic symmetry mutations of ENSO are also significantly more effective at evolving such controllers than random symmetry mutations. Thus, ENSO is a promising approach for evolving modular and symmetric controllers for multilegged robots, as well as solutions to distributed control problems in general.

Abstract-Symmetry is useful as a constraint in designing complex systems such as distributed controllers for multilegged robots. However, it is often difficult to determine which symmetries are appropriate. It is therefore desirable to design such systems automatically, e.g. by utilizing evolutionary algorithms that produce symmetry through developmental mechanisms. The success of these algorithms depends on how well they explore the space of valid symmetries. This paper presents an approach called Evolution of Network Symmetry and mOdularity (ENSO) that utilizes group theory to search the space of symmetries effectively. This approach was evaluated by evolving neural network controllers for a quadruped robot in physically realistic simulations. On flat ground, the resulting controllers are as fast as those having hand-designed symmetry, and significantly faster than those without symmetry. On inclined ground, where the appropriate symmetries are difficult to determine manually, ENSO produced significantly faster gaits that also generalize better than those of other approaches. On robots with a more complicated structure including knee joints, ENSO resulted in more regular gaits than the other approaches. These results suggest that ENSO is a promising approach for evolving complex systems with modularity and symmetry.

Evolving controllers for multilegged robots in simulation is convenient and flexible, making it possible to prototype ideas rapidly. However, transferring the resulting controllers to physical robots is challenging because it is difficult to simulate realworld complexities with sufficient accuracy. This paper bridges this gap by utilizing the Evolution of Network Symmetry and mOdularity (ENSO) approach to evolve modular neural network controllers that are robust to discrepancies between simulation and reality. This approach was evaluated by building a physical quadruped robot and by evolving controllers for it in simulation. An approximate model of the robot and its environment was built in a physical simulation and uncertainties in the real world were modeled as noise. The resulting controllers produced well-synchronized trot gaits when they were transferred to the physical robot, even on different walking surfaces. In contrast to a hand-designed PID controller, the evolved controllers also generalized well to changes in experimental conditions such as loss of voltage and were more robust against faults such as loss of a leg, making them strong candidates for real-world applications.