Abstract
Concept of closed loop control appears in many fields of engineering sciences, where the output quantity of some physical system must be forced to follow some prescribed function over time, e.g., when a robotic arm endpoint must track a desired trajectory or path given as timed series of spatial coordinates. The classic approach for solving this kind of problem involves a PID compensation block, and the necessary input signal for keeping the controlled process in the vicinity of the desired trajectory is calculated as the weighted sum of momentary deviation, deviation integral, and deviation derivative relative to the reference path. However, despite the obvious advantages, practical usability, and simplicity of the PID controllers, their performance is limited when they are utilized for controlling nonlinear systems. Even with linear systems, their proper operation requires an accurate system model and precise tuning process for finding the best weight values for the proportional, integral, and derivative effects, and the planned closed loop behavior might change significantly as the parameters of the controlled plant change over time. In this article, a computed torque-based controller is presented, which has only one adjustable parameter ensuring precise trajectory tracking even with significantly alternated model constants. The practical usability of the offered algorithm is evaluated and verified by simulations and experiments performed on a simple mechanical bi-rotor testbed playing the role of controlled plant.
Highlights
The traditional formulation of the Computed Torque Control (CTC) assumes the possession of a precise system model and the lack of unknown or unobserved external disturbances, known nominal trajectory to be tracked qN(t), the actual trajectory q(t) according to (1) in which Q(t) denotes the generalized forces exerted by the robot drives, H(q) being a positive definite inertia matrix of the robot arm, and h(q, q) containing Coriolis and gravitational forces: t e(t) := eN(t) − q(t), eint(t) = e(ξ)dξ, (1a)
The details of the realized CTC control scheme as it was first tested in Matlab Simulink is illustrated by Figure 7
We have developed and implemented an effective single parameter CTC control algorithm which was tested by various simulations and experiments that are ideal for applications where a specific nominal trajectory has to be followed precisely
Summary
The performance of the one-parameter CTC algorithm is studied by comparing the simulated and measured (when the algorithm was running on a microcontroller, controlling the real-world testbed) trajectory tracking results. These temporal peaks can be totally eliminated by choosing a less “aggressive” PID parameter set option, but, in this case, the settling time (the amount of time necessary to reach a constant value) expanded to 7–8 s, which generally resulted in even worse quality trajectory tracking.
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