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Research Papers

A Closed-Form Solution for Selecting Maximum Critically Damped Actuator Impedance Parameters

[+] Author and Article Information
Nicholas Paine

Department of Electrical and
Computer Engineering,
University of Texas,
Austin, TX 78705
e-mail: npaine@utexas.edu

Luis Sentis

Assistant Professor
Department of Mechanical Engineering,
University of Texas,
Austin, TX 78705
e-mail: lsentis@austin.utexas.edu

While the mechanical impedance is typically defined as Z = F/V (see Ref. [3]), in this work we use the form of mechanical impedance defined by the relationship Z = F/X, following the convention used in Ref. [27].

The authors spent a year working with the Valkyrie robot during the 2012–2013 portion of the DARPA Robotics Challenge.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 18, 2014; final manuscript received July 26, 2014; published online November 7, 2014. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 137(4), 041011 (Apr 01, 2015) (10 pages) Paper No: DS-14-1080; doi: 10.1115/1.4028787 History: Received February 18, 2014; Revised July 26, 2014

This paper introduces a simple and effective method for selecting the maximum feedback gains in PD-type controllers applied to actuators where feedback delay and derivative signal filtering are present. The method provides the maximum feedback parameters that satisfy a phase margin criteria, producing a closed-loop system with high stability and a dynamic response with near-minimum settling time. Our approach is unique in that it simultaneously possesses: (1) a model of real-world performance-limiting factors (i.e., filtering and delay), (2) the ability to meet performance and stability criteria, and (3) the simplicity of a single closed-form expression. A central focus of our approach is the characterization of system stability through exhaustive searches of the feedback parameter space. Using this search-based method, we locate a set of maximum feedback parameters based on a phase margin criteria. We then fit continuous equations to this data and obtain a closed-form expression which matches the sampled data to within 2–4% error for the majority of the parameter space. We apply our feedback parameter selection method to two real-world actuators with widely differing system properties and show that our method successfully produces the maximum achievable nonoscillating impedance response.

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Figures

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Fig. 3

Step response of ψ(s) for various values of fn. The phase margin (Pm) of each response is shown. Response deformation begins to occur at a phase margin of 39.6 deg and large oscillations are visible for a phase margin of 9.54 deg.

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Fig. 4

Phase margin of ψ(s) for various values of fn and fv. The system is destabilized either by heavily filtering the derivative term (lower fv values) or by increasing feedback gains (higher fn values). A phase margin threshold is shown at 50 deg. This threshold is determined by observing the minimum phase margin step response which does not exhibit oscillatory distortion in Fig. 3. Parameter combinations producing phase margins above this line are represented by an “o” in Fig. 5 while those below are represented by an “x.”

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Fig. 2

A spring–damper (K–B) impedance control model with delay (eTs) and velocity filtering (Qv). No force feedback is used, desired forces are simply translated into desired currents.

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Fig. 7

Impedance frequency response (Fext/X) of an actuator with various closed-loop gains (K, B, ζd = 1) determined by fn. With K and B set to zero, the impedance response is that of the passive actuator. The open-loop passive corner frequency (fp) is shown as well as the closed-loop natural frequency (fn) for fn = 10 Hz. The difference between maximum and passive impedance, Z-width, is also illustrated.

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Fig. 1

(a) An impedance interaction between an actuator and a human. (b) Model of actuator including forces from external sources.

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Fig. 5

A parametric search of phase margins of ψ(s) across values of fn and fv. Combinations producing phase margins (Pm) above the Pm threshold are represented by an “o” while those below are represented by an “x.” The line represents the maximum values of fn which pass the phase margin criteria and is analogous to the dashed line seen in Fig. 4.

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Fig. 6

A parametric search similar to the search in Fig. 5 except with higher vertical resolution and an added dimension showing sensitivity to time delay, T. Because fn represents specific values of K and B, this plot can be used to find the maximum values of K and B, given fv and T, which produces an impedance controller with a phase margin of 50 deg.

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Fig. 11

A graphical representation of the fitting accuracy of the fnmax equation compared to ground truth simulation values. The 240 sample points are marked on the contour plot. Error values are averaged along the fv dimension to simplify data representation. Error percentage remains below 5% for values of T < 0.005 s. Maximum error (21%) occurs in the corner case where T = 0.01 s and fp = 0.025 Hz.

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Fig. 8

Parametric search of maximum impedance for various values of fp and fv. A relation which is strongly linear can be seen between fnmax and fp. The markers represent simulation data while the dashed lines represent values calculated using Eq. (16).

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Fig. 9

(a) Search space of m and b for the experiment shown in Fig. 8. (b) Solution space of K and B for the experiment shown in Fig. 8. Each point represents a K–B pair producing a system with the target phase margin.

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Fig. 10

An example of the fitting process used to match the continuous fnmax equation to data points gathered from simulation. This fit represents the d term in Eqs. (16) and (17b).

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Fig. 12

A comparison between maximum impedance for gains selected by the proposed approach and gains selected by a passivity approach [2].

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Fig. 13

Ball screw pushrod actuator used in experimental tests. For the standalone motor experiment, the belt was removed so that the motor could spin freely. In the full actuator experiment, the belt was connected, coupling motor motion to ball screw and load arm motion. The depicted load cell is unused in these tests.

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Fig. 14

Experimental identification of inertial (m) model parameter.

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Fig. 15

Experimental identification of damping (b) model parameter using a step response. The deviation seen after 0.5 s may be attributed to the presence of unmodeled friction.

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Fig. 16

Standalone motor experiment. (a) Step responses for two different sets of parameters are shown. The first set (proposed method) was obtained using the fnmax equation (16). The second set (2 × B value) used double the B parameter from the first set, and selected K using the critically damped constraint. The higher gains produce a deformed step response which exhibits small oscillation and therefore exceeds the maximum achievable actuator impedance with a phase margin of 50 deg. The step displacement for this test was four motor rotations. Due to the high gains used, a higher displacement would cause current saturation to occur (30 amp limit). (b) Discrepancy between simulation and experimental results. Error peaks at 6% showing the simulation accurately represents real-world effects.

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Fig. 17

Full actuator experiment. The same experiment was performed as was described in Fig. 16 except with the full actuator. While the experimental data closely matches simulation data, larger discrepancies can be seen compared to Fig. 16. The cause of this increase is likely due to the drivetrain dynamics (particularly the belt). As was the case in the motor experiment, our method again correctly chooses the maximum critically damped control parameters with a phase margin of 50 deg. The step displacement for this test was 2 mm. Due to the high gains used, a higher displacement would cause current saturation to occur (30 amp limit).

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