Pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
Domestic robotA domestic robot is a type of service robot, an autonomous robot that is primarily used for household chores, but may also be used for education, entertainment or therapy. While most domestic robots are simplistic, some are connected to Wi-Fi home networks or smart environments and are autonomous to a high degree. There were an estimated 16.3 million service robots in 2018. People began to design robots for processing materials and construct products, especially during the Industrial Revolution in the period about 1760 to around 1840.
Robot combatRobot combat is a mode of robot competition in which custom-built machines fight using various methods to incapacitate each other. The machines have generally been remote-controlled vehicles rather than autonomous robots. Robot combat competitions have been made into television series, including Robot Wars in the UK and BattleBots in the US. These shows were originally broadcast in the late 1990s to early 2000s and experienced revivals in the mid-2010s.
FeedbackFeedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to feed back into itself. The notion of cause-and-effect has to be handled carefully when applied to feedback systems: Simple causal reasoning about a feedback system is difficult because the first system influences the second and second system influences the first, leading to a circular argument.
Human–robot interactionHuman–robot interaction (HRI) is the study of interactions between humans and robots. Human–robot interaction is a multidisciplinary field with contributions from human–computer interaction, artificial intelligence, robotics, natural language processing, design, and psychology. A subfield known as physical human–robot interaction (pHRI) has tended to focus on device design to enable people to safely interact with robotic systems. Human–robot interaction has been a topic of both science fiction and academic speculation even before any robots existed.
Curvature of Riemannian manifoldsIn mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
Autonomous robotAn autonomous robot is a robot that acts without recourse to human control. The first autonomous robots environment were known as Elmer and Elsie, which were constructed in the late 1940s by W. Grey Walter. They were the first robots in history that were programmed to "think" the way biological brains do and meant to have free will. Elmer and Elsie were often labeled as tortoises because of how they were shaped and the manner in which they moved. They were capable of phototaxis which is the movement that occurs in response to light stimulus.
Model predictive controlModel predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification.
Riemannian geometryRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Linear–quadratic regulatorThe theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below. LQR controllers possess inherent robustness with guaranteed gain and phase margin, and they also are part of the solution to the LQG (linear–quadratic–Gaussian) problem.
