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Research Papers

# Parametric Design of Parallel Force/Velocity Actuators: Force Distribution Analysis

[+] Author and Article Information
Dinesh Rabindran2

Mechanical Engineering Department, The University of Texas at Austin, 1 University Station C2200, Austin, TX 78712–0292dineshr@mail.utexas.edu

Delbert Tesar

Mechanical Engineering Department, The University of Texas at Austin, 1 University Station C2200, Austin, TX 78712–0292tesar@mail.utexas.edu

The linear weighted sum of two normally distributed variables is a normal distribution (see Ref. 20, p. 11).

Note that the correlation between $ϕ̂v$ and $ϕ̂f$ is zero because the encoder uncertainties in the FA and VA inputs are independent of each other.

Our sign convention is such that an input shaft has positive power associated with it. In other words, the torque and angular velocity of an input shaft have the same sign. This is similar to the convention in Ref. 8.

For multi-input-multi-output systems this is not a scalar.

For a given load, the selected motor’s size and inertia depends on the gear ratio and gear train type.

This criterion has sometimes been called torque-to-inertia ratio by motor manufacturers (see Ref. 16, p. 2).

It is important to make a distinction between the acceleration responsiveness at the input versus that at the output. In the former (latter) case, we are looking at the rate at which the motor (output) shaft can be accelerated or decelerated by the motor, given the machine and actuator parameters.

Closed-form inverse for a $2×2$ matrix was used.

Acceleration responsiveness can be calculated using either the continuous or the peak torque rating of the motor. The former (latter) calculation signifies how much maximum acceleration can be achieved for continuous (peak) operation.

Testing done by Kevin Crouchley at the Naval Sea Systems Command, Philadelphia, PA.

2

Corresponding author.

J. Mechanisms Robotics 2(1), 011013 (Jan 11, 2010) (14 pages) doi:10.1115/1.4000526 History: Received February 23, 2009; Revised June 26, 2009; Published January 11, 2010

## Abstract

In this paper we present the force distribution analysis for a dual input actuator called parallel force/velocity actuator (PFVA). We present five physical quantities that are relevant to the design and operation of PFVA-based systems. For each of them we (i) follow a first principles approach to develop a model, (ii) define dimensionless parameters and criteria that indicate the relative distribution of the quantity between the two inputs of the PFVA, (iii) express the basic model in terms of these dimensionless parameters, (iv) provide numerical examples using five candidate designs with commercial off-the-shelf components, (v) investigate the limiting case as the two inputs become more and more kinematically distinct, and (vi) suggest design guidelines based on our analysis. We studied four aspects of PFVA design: (i) mixing of position uncertainties of the two inputs, i.e., force actuator (FA) and velocity actuator (VA), (ii) distribution of static and inertia torques between the inputs for a given output loading condition, (iii) acceleration responsiveness, and lastly, (iv) effective stiffness of the PFVA system with respect to some basic design parameters of the PFVA. As an example result, we observed that the PFVA's effective stiffness will be at least 40% greater than that of the VA if the FA is 85% as efficient as the VA, the FA is 17% less stiff than the VA, and the kinematic scaling between the two inputs (FA and VA) is approximately 11.5. The results we obtained are organized into five design guidelines for the PFVA. To demonstrate the utility of this analysis and the guidelines, we present a design case study that describes a PFVA prototype. The results of this paper assist in better designing PFVA-based systems with a focus on the coupling between the two inputs.

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## Figures

Figure 1

Schematic of a rotational (rr⪢rs) and linear PFVA (r1⪢r2)(8)

Figure 2

Schematic showing RSFs ρ̃ for various positive ratio drives

Figure 3

Sectioned-view of the SR-20 differential (courtesy of Andantex, Inc., Wanamassa, NJ (14)) used in the UTRRG PFVA prototype

Figure 4

Surface plot of relative accuracy factor α̃fj as a function of RSF ρ̃ and α̃fv. Note that limρ→∞ α̃fj=1. The relative accuracy factor corresponding to the five PFVA designs are shown.

Figure 5

Schematic of a PFVA (based on a negative-ratio differential) with an output work function

Figure 6

Schematic of the procedure for force distribution analysis of the PFVA

Figure 7

Mechanical efficiency analysis of a DISO PFVA as the superposition of a SISO FA and VA (adapted from Ref. 8)

Figure 8

Surface representing the variation in STDR with respect to the relative efficiency ratio and the RSF. The circular markers represent the five PFVA designs. To demonstrate how a point on this surface should be interpreted, the square markers show two example PFVA designs with the same relative efficiency ratio and different RSFs.

Figure 9

A single-link manipulator with lumped output inertia and reflected input inertia. (Courtesy Newtonium for providing the RoboWorks software which was used to generate this 3D model)

Figure 10

Surface representing the variation in ITDR with respect to the acceleration mixing ratio and the RSF. The circular markers represent the five PFVA designs. The different surfaces correspond to various load inertia settings, namely, 2×, 1×, 0.5×, 0.25×, and 0×; the output link inertia, i.e., 0.225 kg m2.

Figure 11

Surface representing the variation in relative acceleration responsiveness with respect to the rated motor torque ratio and the RSF. The different surfaces correspond to various load inertia settings, namely, 2×, 1×, and 0.5×; the output link inertia, i.e., 0.225 kg m2.

Figure 12

(a) Schematic representation of transmission stiffness including soft and hard stiffness zones (29). (b) Experimental stiffness data from a 14 scale weapons elevator actuator using a hypocyclic gear train (courtesy of Kevin Crouchley, NAVSEA, Philadelphia, PA).

Figure 13

Representation of the PFVA as an in-series coupled-spring system. The broken arrow on IL∗ indicates that this parameter is a variable depending on the configuration of the mechanism. (a) Complete lumped spring-mass model identifying all principal mechanical stiffness elements on the input side of the PFVA and (b) simplified in-series spring system model with the kinematic transformation for each input.

Figure 14

Variation in the relative joint stiffness (K̃j) with respect to the relative stiffness (K̃) and the RSF (ρ̃) for ηj→v varying as a function of RSF. 3D surfaces representing this data for various settings of η̃b, namely, 0.85, 1, and 1.15 (corresponding to the three surfaces in the figure) are also shown.

Figure 15

Variation in SISO efficiencies with respect to the RSF ρ̃ for the commercially available SA series differentials from Andantex Inc. (see Ref. 8 for a detailed discussion on SISO efficiencies). The subscript o refers to the output or the machine joint. Therefore, ηv→o=ηv→j is the basic efficiency of the drive train.

Figure 16

Contour plot representing the data in Fig. 1 for η̃b=0.85

Figure 17

A 2-DOF motor drive model for a direct-drive motor. The parameters shown are all lumped and referenced to the rotor shaft (adapted from Ref. 31).

Figure 18

Laboratory prototype of the PFVA built in the Robotics Research Laboratory at UT Austin in Austin, TX

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