state-space model

A state-space model is a tidy, standardized way of writing down everything a control engineer needs to know about how a robot moves and responds. The central idea is the state: a short list of numbers that captures the complete, current condition of the system — for a single joint, that might be just its angle and how fast that angle is changing. The promise of the state is that if you know it right now, plus whatever commands you send from here on, you can predict the entire future without needing any history of how you got here. The state is the system's complete memory squeezed into a handful of variables.

The model then packages the dynamics into two compact equations. The first, the state equation, says how the state is changing right now — its rate of change — given the current state and the inputs you apply (the motor commands). The second, the output equation, says which parts of the state your sensors actually measure and report. By convention these are organized so the rates of change depend only on the present state and present input — first-order form — which is exactly the shape that the whole toolbox of modern control design, like placing poles or building observers, is built to work with.

Writing a robot this way is powerful because it turns a tangle of physics into clean bookkeeping that scales. A two-number state describes one joint; stack up positions and velocities and you can describe a whole multi-joint arm, a wheeled base, or a drone in one unified frame. Once a system is in state-space form, an enormous library of design methods becomes available off the shelf, which is why this representation is the common language underneath most advanced robot control.