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  "title": "Feedback closes the action loop",
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  "revisionDate": "2026-10-07",
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  "topic": "Observations become physical actions",
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  "version": "1.3.0",
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      "title": "Feedback closes the action loop",
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      "title": "Feedback closes the action loop · explicit relationship",
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      "title": "For the ideal sampled model, what happens when KΔt = 2.4?",
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    {
      "title": "Lynch and Park · Modern Robotics 11.3",
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  "content": "Observations become physical actions · Atlas v1.3.0\n\n# Feedback closes the action loop\n\nSampled controller, actuator limits, delayed sensing and disturbance.\n\n[Explore in Atlas](https://semiaifoundry.com/ai-atlas/#robot-feedback-control)Logical state, task evidence and declared units; no physical assembly scale\n\nLearning objectiveDerive and evaluate a sampled position-error recurrence under an explicit velocity-input model.\n\n## The mechanism\n\nFeedback 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.\n\n1.  A target and measured position define error at the start of each sample.\n2.  The proportional controller maps that error into a commanded velocity with a gain measured per second.\n3.  The ideal plant integrates that velocity for the sample duration before the next observation.\n4.  The resulting recurrence is checked for convergence within the declared no-delay, unsaturated model.\n\n## Science and assumptions\n\n### Feedback closes the action loop · explicit relationship\n\n$$\ne_{n+1}=(1-K\\Delta t)e_n,\\qquad |1-K\\Delta t|<1\\ \\Longleftrightarrow\\ 0<K\\Delta t<2\n$$\n\nIn 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.\n\nSymbols and units\n\neₙ\n\nTarget minus current position at sample n\n\nm\n\nK\n\nProportional velocity gain\n\n1/s\n\nΔt\n\nPositive fixed sample duration\n\ns\n\nLet 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.\n\nAssumptions Fixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence.\n\n## Engineering example\n\nLet 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.\n\n## Distinctions and tradeoffs\n\nA positive proportional gain alone does not establish stability for every sampled or delayed physical system.\n\nLarger gain can reduce error faster within a valid model but can amplify noise, hit limits or destabilize sampled/delayed behavior.\n\nRobotic action depends on control rate and physical feedback, beyond the semantic choice of a target.\n\n## Specifications and interfaces\n\nMechanism boundary\n\nHow does measured error change the next command?\n\nFixed target, exact sampled integrator, no sensing delay, disturbance, saturation or additional plant dynamics. The stated stability interval applies only to this recurrence.\n\nEvidence type\n\nAuthored worked example\n\nWorked explanation with a bounded numeric or state example. This lesson does not claim an executing implementation of an external model or platform.\n\n### Interfaces\n\nDeclared input and state Input\n\nA target and measured position define error at the start of each sample.\n\nChecked outcome Output\n\nDerive and evaluate a sampled position-error recurrence under an explicit velocity-input model.\n\n## Check your understanding\n\n### For the ideal sampled model, what happens when KΔt = 2.4?\n\n1.  Error shrinks monotonically\n2.  Error alternates and grows because the multiplier is −1.4\n3.  Error is always zero\n\nAnswer and reasoning\n\nError alternates and grows because the multiplier is −1.4\n\nThe multiplier has magnitude greater than one, so the ideal recurrence grows while changing sign.\n\n## Evidence and scope\n\nThis 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.\n\nThe worked example explains this mechanism. General numerical labs are a reference; they do not execute this lesson’s full mechanism.\n\n## Sources\n\n-   [Lynch and Park · Modern Robotics 11.3](https://modernrobotics.northwestern.edu/nu-gm-book-resource/11-3-motion-control-with-velocity-inputs-part-1-of-3/)\n\n[Published lesson data](https://semiaifoundry.com/ai-atlas/lesson-data/robot-feedback-control-78bca5e6f894bd9d.json) · [Atlas releases and licensing](https://semiaifoundry.com/ai-atlas/licenses.html)\n\n## Continue exploring\n\n### Foundations\n\n-   [Robot observations have coordinates and age](https://semiaifoundry.com/learn/atlas/robot-observation-frame/)\n\n### Related lessons\n\n-   [Tools, memory & agent loop](https://semiaifoundry.com/learn/atlas/tool-loop/)\n-   [Evaluation & the AGI frontier](https://semiaifoundry.com/learn/atlas/evaluation/)"
}