Bounded function
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A bounded function is a mathematical function whose output values remain confined within a finite range — that is, there exist real numbers M and m such that m ≤ f(x) ≤ M for all inputs x. In robotics and AI, bounded functions appear throughout control theory, motion planning, and neural network design. Controllers for robot manipulators, mobile robots, and autonomous vehicles frequently rely on bounded activation functions, barrier Lyapunov functions, or bounded potential fields to ensure that outputs — such as control torques, tracking errors, or neural network approximations — never grow unbounded, which would destabilize the system. Sliding mode controllers, adaptive neural network controllers, and trajectory planners all exploit bounded function properties to provide formal guarantees on tracking performance, safety constraints, and stability margins. Bounded artificial potential functions, for example, enable exact robot navigation while avoiding local minima and obstacle collisions. The concept matters because real physical systems have finite actuator limits, safety envelopes, and stability requirements; mathematical guarantees that key signals remain bounded are essential for certifying that a robot will behave predictably and safely under uncertain, dynamic real-world conditions.
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