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RESEARCH PAPERS

# Power Efficiency of the Rotational-to-Linear Infinitely Variable Cobotic Transmission

[+] Author and Article Information
Eric L. Faulring

Chicago PT, LLC, 2510 Gross Point Road, Evanston, Illinois 60201eric.faulring@ieee.org

J. Edward Colgate

Department of Mechanical Engineering,  Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208colgate@northwestern.edu

Michael A. Peshkin

Department of Mechanical Engineering,  Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208peshkin@northwestern.edu

Haptic displays convey touch and kinesthetic feedback to a human operator by dynamically varying the relationship between motion and force, often in the form of virtual constraint surfaces. Haptic displays are used to interact with computer aided design models, flight simulators, telerobotic surgery, micro-∕nanomanipulation, undersea salvage, as well as maintenance of nuclear plants and other hazardous environments.

We take lateral to mean transverse to the rolling direction and longitudinal to mean tangent to the rolling direction.

Kinematic creep is not present in the rotational-to-linear IVT analyzed here, as the rolling elements of our geometry do not sustain significant tractive loads.

Karnopp et al.  provide an excellent review of bond graph notation (27).

While the incorporation of the wheel bearing losses here may appear arbitrary, we endeavor to fit the transmission plant model into the format of Fig. 1.

The sloping upper boundary is the motor’s maximum velocity given the operating voltage and applied torque. This sloping boundary would intersect the horizontal axis at the momentary stall torque achievable by the motor.

Maxon provides a useful reference for motor dynamics at http:∕∕www.maxonmotorusa.com∕media∕maxontechnology∕02̱selandcaḻe.pdf. (31).

The rms power is evaluated from instantaneous power, $Ẇ(t)=l̈(t)ml̇(t)=mα2ϖ3sin(ϖt)cos(ϖt)$.

J. Mech. Des 129(12), 1285-1293 (Dec 07, 2006) (9 pages) doi:10.1115/1.2779885 History: Received February 19, 2006; Revised December 07, 2006

## Abstract

Cobots are a class of robots that use infinitely variable transmissions to develop high fidelity programmable constraint surfaces. Cobots consume very little electrical power even when resisting high forces, and their transmissions are highly power efficient across a broad range of transmission ratios. We have recently introduced the Cobotic Hand Controller, a haptic display that illustrates the high dynamic range and low-power consumption achievable by cobots. In this paper, we present models of the rotational-to-linear rolling contact transmissions utilized in the Cobotic Hand Controller. We compare their efficiency to fixed-ratio gear trains. We also compare the overall power efficiency of the cobotic architecture to the power efficiency of a conventional electromechanical actuation scheme, for both constant and dynamic power flows. The cobotic architecture is shown to be more efficient at frequencies and power levels characteristic of voluntary human motions.

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## Figures

Figure 1

A series of snapshots of the rotational-to-linear transmission as utilized to drive a prismatic link (tube) via a rotational input (cylinder). The transmission consists of a steered wheel pressed against a rotating cylinder, and traveling with a prismatic link.

Figure 2

A bond graph representation of the lossless rotational-to-linear transmission

Figure 3

Velocity kinematics of the rotational-to-linear transmission, including flow loss due to lateral creep

Figure 4

Force balance of the rotational-to-linear transmission, including rolling losses due to inelastic materials, and wheel axle bearing friction. Note that fl is defined as the force applied by the transmission to the link.

Figure 5

Typical nondimensionalized lateral creep curve. This curve predicts lateral creep velocity u, relative to forward rolling velocity rθ̇, given the amount of lateral force applied to the wheel fu relative to the force that will cause gross slip μP. The true experimental creep curve is well approximated by a linear model up until 60–70% of μP.

Figure 6

Transmission plant portion of the cobotic transmission system. The modeled rolling losses, δWin∕dt and δWwa∕dt, and lateral creep loss, δWdef∕dt, have now been added to the lossless transmission plant introduced in Fig. 2.

Figure 7

Theoretical and experimental efficiencies of the rolling contact reduction element of the cobotic transmission, operating at various transmission ratios and levels of maximum effort. Experimental efficiencies are not reported for reduction ratios larger than 100:1, since accurate measurements become a difficulty, although the device is capable of rendering ∞:1 ratios, or a completely clutched state. We are also unable to experiment with the highest possible loading condition, fl∕(μP)=0.95, due to the unpredictable nature of friction near the breakaway force.

Figure 8

Efficiency of the rotational-to-linear rolling contact reduction element versus conventional gears sampled at random from the internet. Only very expensive, very high-torque planetary gear trains (not useful for low-power applications in the proximity of human operators) or single gear pairs can achieve efficiencies above 90%. The cobotic efficiencies are reported for various percentages of maximum effort throughput fl∕(μP) at a given flow ratio 1∕tan(ϕ). The cobotic efficiencies fall off steeply at large reduction ratios (e.g., 1000:1), as rolling friction losses become large relative to power throughput, and also at small reduction ratios (e.g., 1:10), as lateral creep losses become large relative to power throughput.

Figure 9

(A) Cobotic architecture. (B). Conventional architecture. The output for each drive train is the flow l̇ of the mass m. The conventional system requires electrical power InVn, while the cobotic system requires steering power IsVs and cylinder power IcVc. Both systems allow an external force f to apply effort to the load.

Figure 10

A general cobotic system is composed of four plants detailed in Figs.  111213. The plants allow two current sources and a human effort source to do work on a mass m. The steering plant adjusts the modulus of the transformer in the transmission plant, but no power flows between the steering plant and transmission.

Figure 11

The steering plant portion of the cobotic system converts electrical power into the angle ϕ, the modulus of the transformer in the cobotic transmission. Losses include Ohmic heating, dissipation at the contact patch (δWshear∕dt), and bell bearing (δWbell∕dt), both of which sustain preload P and power required to drive inertia of the steering motor (Jm,s) and housing of the wheel (Jb). We ignore any losses in the small gear reduction ns.

Figure 12

The cylinder plant portion of the cobotic system converts electrical power into mechanical power τω. Power is lost to Ohmic heating, friction in the cylinder bearings and gear train (δWnominal∕dt), and to the cylinder motor and cylinder inertias, Jm,c and Jc, respectively.

Figure 13

The link plant portion of the cobotic system. Power is lost to friction in the link, δWlg∕dt, and to accelerate the mass of the link ml before any external load can be driven.

Figure 14

A conventional fixed-ratio rotary electric to linear system as depicted in Fig. 9. Power is lost to Ohmic heating, friction in the gear train and cabling δWgpc∕dt, and to the inertia of the motor, gearing, pulley, and cable, Jgpc, before power can be transmitted to external loads.

Figure 15

The ranges of operation of some conventional drive-train designs and contours of power efficiency of the Motor 2 design. The darker shaded region indicates the desired operating regime, and the lighter shaded region is the continuous operating regime of Motor 2. l̇desired and fdesired are the desired maximum velocity and force specifications. Motor 2 is 2.7 times larger than Motor 1.

Figure 16

The range of operation of a cobotic drive train using the same Motor 1 as the conventional drive train design in Fig. 1. The feasible cobotic design regime (indicated by the lighter shaded region) is bounded where the required effort would cause the IVT wheel to slip, and where the required motor torque exceeds the motor’s thermal limit. l̇desired and fdesired are the desired maximum velocity and force specifications.

Figure 17

Comparison of the power efficiency contours of cobotic drive trains and conventional drive trains at driving a mass sinusoidally. Neither system can work in the upper right portion of the plot and only the cobotic system can work in the upper left portion.

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