13_control.pdf - COMS W4733 Computational Aspects of Robotics Lecture 13 Motion Control Instructor Tony Dear Motion Control ▪ Suppose we have a

13_control.pdf - COMS W4733 Computational Aspects of...

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COMS W4733: Computational Aspects of Robotics Lecture 13: Motion Control Instructor: Tony Dear
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Motion Control Suppose we have a desired pose or trajectory that our robot should follow We can compute controlled input velocities using inverse kinematics Is that all we need to do? What if there is noise in what we see or do? What if we deviate from our path? Robot most likely has sensors we need to complete the feedback loop!
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Manipulator Control Scheme
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Mobile Control Problems Trajectory tracking : Asymptotically follow a given trajectory (? ? ? , ? ? (?)) from some initial configuration 𝒒 0 We may never reach it exactly, but we can at least minimize error over time Posture regulation : Move from some initial 𝒒 0 to a given 𝒒 ? Trajectory is not given, but we may be able to plan one
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Unicycle Kinematics We derived the following: Suppose robot is currently at 𝒒(?) = ?, ?, 𝜃 𝑇 and wants to get to 𝒒 ? (?) = ? ? , ? ? , 𝜃 ? 𝑇 Represent the error 𝒒 ? − 𝒒 in robot’s body frame: ሶ? = ? cos 𝜃 ሶ? = ? sin 𝜃 𝜃 = ? ? ? ? = ሶ? ? ? 2 + ሶ? ? ? 2 ? ? ? = ሶ? ? ? ሷ? ? ? ሷ? ? ? ሶ? ? (?) ሶ? ? ? 2 + ሶ? ? ? 2 ? = 𝑒 ? 𝑒 ? 𝑒 𝜃 = ? 𝜃 ? 𝜃 0 −? 𝜃 ? 𝜃 0 0 0 1 ? ? − ? ? ? − ? 𝜃 ? − 𝜃
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Error Dynamics How does the error change over time? This is a nonlinear dynamical system of the form ሶ? ? = ?( ? ? , ? ? ) We want to find controllers ? ? to drive ? (?) to 0 Unfortunately, each component has dependencies on other components ? = 𝑒 ? 𝑒 ? 𝑒 𝜃 ? 𝜃 ? 𝜃 0 −? 𝜃 ? 𝜃 0 0 0 1 ? ? − ? ? ? − ? 𝜃 ? − 𝜃 ሶ? = ሶ𝑒 ? ሶ𝑒 ? ሶ𝑒 𝜃 = ? ? 𝑒 ? + ? 1 𝑒 ? ? 2 ? ? 𝑒 ? + ? ? sin 𝑒 𝜃 + 𝑒 ? ? 2 ? 2 ? 1 = ? ? cos 𝑒 𝜃 ? ? 2 = ? ? ?
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Linearization If this were a linear system it’d be easier to analyze Idea: Linearize ሶ? ? = ?(? ? , ? ? ) about ? ? , ? ? ሶ? ? = ?(? ? , ? ? ) ሶ? ? = ?(?)?(?) + ?(?)?(?) ? ? = ?? ?? ?=0,?=? 𝑑 ? ? = ?? ?? ?=0,?=? 𝑑 ? ? = 0 ? ? − ? 2 0 −?
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