Optimized input shaping for a single flexible robot link Page: 4 of 6
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Second Ordwr Model System Identjilcilon
1 - ---
t t- Experlmnial Response
High FidesKy Model System tdentdficaton
0 1 2 3 4 5
Figure 2: Model Identification Calibration Plots
Optimized Input Shaped Trajectory
We formulated a constrained optimization problem for the single flexible
link using both the second-order model and the high fidelity model. Solving
the trajectory optimization problem required the use of a Recursive Quadratic
Programming (RQP) algorithm, VF02AD (Hopper78). The problem used 20
discretized temporal control inputs as parameters. The second-order model
used the following performance index: 4(C) = 5(0(tf))2, with the following
constraints; Viz() = 8(t1) - z and 2(C) = 00(tf) - 2. The definition of a
third state sets the controlled variable to the commanded input rate; u = Or.
To determine a .starting point, the high fidelity model formulation used
the following performance index: (() =fj [O(t)2 -4 q(t)2 + 4(t)2jdt, with the
following constraints: io( ) = 6(tf) - a and jiz(G) = UG(tf) - a, which ap-
proximately converged. This initial run helped put the parameters into the
neighborhood of an optimal trajectory. The results of the previous run led to
new parameter values, and the performance index reconfigured to the follow-
ing: c(s) = 0.5 fc' u2di + 20(8(tf))2, with the following constraints: t'(I)
9(i1) - ., b2(.) q(t) - 0.0, 03( ) =(t1) - 0.0, and 4() 8c(tj) -
The control variable remained the same, e.g., u = 6.
Experimental / Predicted Results
The UNM flexible testbed consists of a flexible aluminum link, the dimen-
sions of which are 45.72 cm x 7.62 cm x 0.8128 mm; motor/hub/link mounting
hardware; an electric DC servo motor; an incremental encoder; and a VME
real-time control computer. The trajectories used as command inputs to the
servo system, at 50 Hz sampling, were the result of the optimization proce-
dure. Included in this report are two different runs. The first run used the
optimized trajectory found with the second order model, while the second run
used that found with the high fidelity model. Both runs arc for a final time
of t = 2.0 seconds. Figure 3 shows each run and contains the plots of the
shaped input; experimental and simulated responses. The improved results of
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Wilson, D.G.; Stokes, D.; Starr, G. & Robinett, R.D. Optimized input shaping for a single flexible robot link, article, March 1, 1996; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc665715/m1/4/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.