Artificial Intelligence in Robotics

Introduction

ROBOTICS is a very interdisciplinary field, among others, involving areas from engineering, computer science, artificial intelligence, cognition, logics and, of course, mathematics. Robotics is a rich source of demanding mathematical problems arising in Kinematics, Singularities, Dynamics, Control, Path Planning, Task Planning and others. Methods from analysis, differential equations, algebra, geometry, topology, differentiable manifolds, Lie groups are of relevance in a natural way (Wyatt & Spiegel 3-7).

Discussion

Real-world applications of intelligent machines involve scientific, mathematical, and engineering problems with enormous practical, theoretical, and economic interest. The ultimate goal of current robotics research is the creation of physical machines with near or even beyond human levels of perception, physical capability, practical intelligence, and behavior, as well as the creation of intelligent and efficient human interfaces to these complex systems (Harris & Owens 30-34). Indeed, in some areas, such as computer aided surgery, human capabilities can be surpassed. The wider deployment of robots awaits further advances in basic unsolved problems of motion planning and contact, particularly in the area of robust solutions. 

There is a tradition of mathematicians working with roboticists. Many problems in robotics, or in the disciplines that are core to what we call robotics, have attracted mathematicians to this field. As far back as the 19th century, algebraic geometers like Kemp and Tschebyshev were drawn to the beautiful mechanics of linkages (the predecessors of today's complex articulated manipulators) and geometers like Poincare were attracted by the dynamics of machines. Clearly, the development of the foundations of many classical and modern mathematical tools was spurred by technological advances in machines and mechanisms. This tradition has continued and there are several examples in the last two decades where some of the most fundamental and enduring results in robotics have come from mathematicians and their interactions with engineers.

The classical general motion planning results in robotics were developed by Schwartz and Sharir, which in turn were improved on by Canny using methods from algebraic geometry. Similarly, Milgram and Trinkle have used results from modern algebraic topology to obtain an improved understanding of the configuration spaces of closed chain mechanisms, leading to improved algorithms for motion planning for such systems. Marsden, Brockett, and Sastry have used differential geometry and Lie theory to formalize the kinematics, dynamics and control of spatial linkages. This work has also sensitized the engineers to such important mathematical ideas as frame invariance and invariance with respect to parameterization.

Other areas where there are strong developing ties between mathematics and robotics include, for example, Bayesian statistics to develop algorithms for perception and learning, ...