{"id":"robot-feedback-control","name":"Feedback closes the action loop","shortName":"Feedback closes the action loop","role":"Sampled controller, actuator limits, delayed sensing and disturbance.","description":"Feedback compares observed state with a target and changes the next control command according to that error. A velocity-input model assumes a lower-level system can achieve the commanded velocity; it does not directly model forces, torques or actuator dynamics. Proportional control commands velocity in proportion to position error. The gain and sample interval jointly affect a discrete implementation. For an ideal sampled integrator with no delay or saturation, the error recurrence can be derived exactly from its update rule. Its convergence condition must not be generalized to a delayed or mechanically flexible robot. The example exposes that dependence with numbers: a gain that reduces error at one sampling rate can alternate and grow at another effective gain.","mechanism":["A target and measured position define error at the start of each sample.","The proportional controller maps that error into a commanded velocity with a gain measured per second.","The ideal plant integrates that velocity for the sample duration before the next observation.","The resulting recurrence is checked for convergence within the declared no-delay, unsaturated model."],"learningObjective":"Derive and evaluate a sampled position-error recurrence under an explicit velocity-input model.","engineeringExample":"Let the target be 1 m, initial position 0 m, sample duration 0.1 s and gain 2/s. The error multiplier is 1−2×0.1=0.8, so the first errors are 1, 0.8 and 0.64 m. With gain 24/s at the same interval, the multiplier becomes −1.4 and the errors are 1, −1.4 and 1.96 m: they alternate and grow. This ideal discrete example has no velocity cap, noise or delay. A real actuator may saturate before those commands are achieved, changing the dynamics and requiring a different analysis.","misconception":"A positive proportional gain alone does not establish stability for every sampled or delayed physical system.","check":{"question":"For the ideal sampled model, what happens when KΔt = 2.4?","options":["Error shrinks monotonically","Error alternates and grows because the multiplier is −1.4","Error is always zero"],"answer":1,"explanation":"The multiplier has magnitude greater than one, so the ideal recurrence grows while changing sign."},"checks":[],"prerequisites":["robot-observation-frame"],"related":["tool-loop","evaluation"],"specs":[{"label":"Mechanism boundary","value":"How does measured error change the next command?","context":"Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence."},{"label":"Evidence type","value":"Authored worked example","context":"Worked explanation with a bounded numeric or state example. This lesson does not claim an executing implementation of an external model or platform."}],"interfaces":[{"name":"Declared input and state","direction":"in","signal":"A target and measured position define error at the start of each sample."},{"name":"Checked outcome","direction":"out","signal":"Derive and evaluate a sampled position-error recurrence under an explicit velocity-input model."}],"tradeoff":"Larger gain can reduce error faster within a valid model but can amplify noise, hit limits or destabilize sampled/delayed behavior.","whyAI":"Robotic action depends on control rate and physical feedback, beyond the semantic choice of a target.","sourceIds":["automation-robot-control"],"sources":[{"title":"Lynch and Park · Modern Robotics 11.3","url":"https://modernrobotics.northwestern.edu/nu-gm-book-resource/11-3-motion-control-with-velocity-inputs-part-1-of-3/"}],"science":[{"title":"Feedback closes the action loop · explicit relationship","equation":"In this ideal sampled velocity-input model, error is multiplied by one minus gain times sample duration; convergence requires that multiplier’s magnitude be below one.","explanation":"Let the target be 1 m, initial position 0 m, sample duration 0.1 s and gain 2/s. The error multiplier is 1−2×0.1=0.8, so the first errors are 1, 0.8 and 0.64 m. With gain 24/s at the same interval, the multiplier becomes −1.4 and the errors are 1, −1.4 and 1.96 m: they alternate and grow. This ideal discrete example has no velocity cap, noise or delay. A real actuator may saturate before those commands are achieved, changing the dynamics and requiring a different analysis.","assumptions":"Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence."}],"evidenceNotes":"This is a worked explanation, not an executing external platform, model or robot. The quantities and fixtures are authored. Source scope is recorded in the automation learning review manifest; no source benchmark or private system behavior is claimed. Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence.","level":"model","branch":"Observations become physical actions","subbranch":"Feedback closes the action loop","scale":"Logical state, task evidence and declared units; no physical assembly scale","kind":"system","workstream":"W11","visualFamily":"software-flow","labId":"library","labScope":"The worked example explains this mechanism. General numerical labs are a reference; they do not execute this lesson’s full mechanism.","history":[],"workedTrace":{"initialErrorM":1,"sampleS":0.1,"stableGainPerS":2,"stableErrorsM":[1,0.8,0.64],"unstableGainPerS":24,"unstableErrorsM":[1,-1.4,1.96]}}