Details for the errata of the paper "Design and Evaluation of a Balan ce Assistance Control Moment Gyroscope" DOI: 10.1115/1.4037255, published in t he ASME Journal of Mechanisms and Robotics. Erratum Description •τdis wrongly used in equations (11), (12) and (13) as well as in F igure 8 under "Allocation Law". The correct symbol is τIP,ref •Subsection " 3.2 Flywheel Aerodynamics and Actuation " presented an incorrect value for air density of ρg= 41.6kg/m3, which correspond to the air density per molar mass in mol/m3compared to the correct value ρg= 1.181kg/m3. In addition, estimated bearing friction did not account for tribological phenomena. Given the considerably eﬀect these values have in the estimation of the maximum ﬂywheel speed, c orrections on the analysis of aerodynamic drag and bearing friction are given below as r evised sections. Moreover, it is important to note that reported experimental results wer e unaﬀected by the erroneous estimations. Revised Sections Revised sections below include section 3. Control Moment Gyr oscope Hardware Design (partic- ularly subsections 3.2 Flywheel Aerodynamics and Actuatio n and 3.3 Flywheel Stress Analysis), 6. Discussion and 7. Conclusions of the original paper. 1 Hybrid bearingsh triroRiRo d Figure 3: Flywheel cross section Regime Flow Type Conditions δ K aKbKc IRea<3×105,s < δ5.5/parenleftbiggν Ω/parenrightbigg0.52π−1−1 II s > δ 3.7 0.1−0.5 IIIRea>3×105,s < δR3 5o/parenleftbiggν Ω/parenrightbigg0.20.08−1/6−0.25 IV s > δ 0.0102 0 .1−0.2 Table 1: Axial frictional torque coeﬃcient regimes [4] Control Moment Gyroscope Hardware Design Flywheel Aerodynamics and Actuation An analysis of the diﬀerent friction sources was performed t o estimate the maximum ﬂywheel speed. The power required to reach a given Ωcan be calculated as P=Pa+Pb, wherePaand Pbare the power required to overcome aerodynamic drag and bear ing friction respectively. The power to overcome aerodynamic drag for a rotating disc can be expressed as [2, 5] Pa=ρgΩ3R5 oCm,a+πρgΩ2R4 oh 2Cm,r (6) where the non-dimensional drag torque coeﬃcients Cm,aandCm,rdepend on the drag pro- duced by the lateral planar surfaces of the disc and the drag p roduced by the cylindrical face of the rim respectively. Estimation of the axial torque coeﬃcient Cm,awas based on the relationships determined by Daily and Nece [4] as 2 Regime Conditions KuKv Laminar Rer≤64 10 −1 Transitional Flow 64<Rer≤500 2 −0.6 Turbulent500<Rer≤1×1041.03−0.5 Rer>1×1040.065−0.2 Table 2: Radial frictional torque coeﬃcient regimes [2, 1] Cm,a=KaGKb aReKc a (7) being a function of the axial rotational Reynolds number Reaand the axial gap ratio Ga=sa Ro, withsathe axial clearance between the ﬂywheel and the casing [2]. Ka,KbandKcare constants determined by any of the four ﬂow regimes that can be present [ 4, 2] as shown in Table 1. These regimes are categorized depending on the ﬂow type and the nat ure of the boundary layers [4, 2]. To this end the radial rotational Reynolds number is calculat ed as Rea=R2 oΩ ν=ρgR2 oΩ η(8) whereρg= 1.181kg/m3,Ro= 0.1mandη= 1.84×10−5Pas, are the air density, ﬂywheel’s rim outer radius and the air dynamic viscosity, respectivel y. Subsequently the behavior of the boundary layers (i.e. whether the boundary layers are merge d or separate) was determined. Here, the boundary layer thickness δwas compared with the axial clearance s. Table 1 summa- rizes the constants used to estimate the axial frictional mo mentCm,a. Estimation of the radial torque coeﬃcient Cm,rwas based on the equations reported by Bilgen and Boulos [1] as Cm,r=KuG0.3 rReKv r (9) being a function of the circumferential rotational Reynolds numberRerand the radial gap ratioGr=sr Ro, withsrthe radial clearance between the ﬂywheel and the casing [2]. KuandKv are constants determined by any of the four ﬂow regimes that c an be present [1, 2] as shown in Table 2. These regimes are categorized based on the circum ferential rotational Reynolds number Rer=RoΩsr ν=ρgRoΩsr η. (10) The power required to overcome bearing friction was estimat ed based on The SKF model for calculating frictional moment [7]. This model estimates the bearing frictional moment as Pb= (Mrr+Msl+Mseal+Mdrag)Ω (11) 3 P Pa Pb Ω(rpm )Power (W) 0 1000 2000 3000 4000 5000 6000 7000020406080100120140 Figure 4: Flywheel mechanical power as function of ﬂywheel s peed. Here the contributions of both aerodynamic drag and bearing friction are depicted (da shed and dot-dashed lines respec- tively). whereMrrandMslare rolling and sliding frictional moments, and MsealandMdragare frictional moments caused by the bearing seals and drag due t o external lubrication respectively. Given that the formulas used in this model are rather large an d complex, we refer the reader to the complete model in [7] for a more detailed description. Given size constraints in the ﬂywheel’s hub, a 120W Maxon EC 4-pole 22 brushless motor and high-precision hybrid angular contact ball bearings (G MN HY KH 61910 TA P4 L 252) were selected as ﬂywheel actuator and suspension respectiv ely. A mechanical power of 108W was estimated taking into account the losses due to the motor eﬃciency ηFM=90% . Bearing friction in Eq. (11) was estimated using the grease viscosit y range reported by the manufacturer (26mm2/s@40◦Cand6mm2/s@100◦Cfor Turmogrease HS L252). Due to the complexity of the expressions to calculate both aerodynamic and bearin g friction from Eqs. (6) and (11), a script was created to estimate the maximum ﬂywheel speed Ωbased on total power P=Pa+Pb. Figure 4 shows the power Prequired to overcome both aerodynamic drag and bearing fric tion against ﬂywheel speed Ω, where a maximum ﬂywheel speed of 5250rpm was estimated for the selected ﬂywheel motor and bearings. Flywheel Stress Analysis To evaluate the mechanical strength of the ﬂywheel under nom inal operation, the maximum ﬂywheel angular rate Ω =5250rpm in addition to the maximum designed gyroscopic moment of90Nm were used. Due to the complexity of the ﬂywheel geometry Fini te Element Method (FEM) analysis was performed using ANSYS Workbench and its s tatic structural module. As the CMG is not used in reaction wheel mode (i.e the ﬂywheel ang ular rate remains constant in nominal operation), the imposed condition over the model is constant rotational speed. To 4 Figure 5: FEM equivalent (von Mises) stress for the designed ﬂywheel geometry. validate the FEM results, a simpliﬁed model keeping the same overall dimensions but using only centrifugal loads was simulated and compared with anal ytical results. FEM analysis of the simpliﬁed model showed good agreement with the analytic rad ial and tangential stresses with Root Mean Square Error (RMSE) values of σr,RMSE=0.07MPa andσθ,RMSE=0.13MPa . Using the same simulation parameters and boundary conditio ns and including the load due to the gyroscopic moment, the ﬂywheel geometry as shown in Fi g. 3 was used to perform the FEM analysis. Figure 5 shows the equivalent (von Mises) s tress, where the maximum of 50MPa is much lower than the yield strength of the ﬂywheel material (Aluminum 7075 - T6, Sy= 503MPa ), giving a safety factor of 10 against the combined centrifu gal and gyroscopic loads. In addition to the stress analysis, it is necessary to assess the critical speed, since the ﬂywheel can experience several types of vibrations being shaft whir l, lateral and torsional vibrations the most important [6]. This critical speed must be avoided, sin ce the resulting deﬂections might cause stresses beyond the strength of the material. The ANSY S Workbench modal analysis module was used to perform the critical speed analysis given the complexity of the ﬂywheel geometry. Shaft whirl and lateral vibration eﬀects were neg lected as the ﬂywheel was thoroughly balanced after manufacturing and the deﬂection of the shaft is very small at the ﬂywheel location during bending due to the gyroscopic moment. Thus, Torsiona l vibrations were analyzed using the same parameters and boundary conditions as the static st ructural analysis, where the modal analysis results showed a minimum critical speed of 200778rpm , giving a margin of 38 when compared with the intended rotational speed ( Ω = 5250rpm ). 5 Discussion The implemented controller successfully tracked the refer ence virtual stiﬀnesses, while keeping the inverted pendulum within its equilibrium position φ= 0. Even though no damping was emulated in the controller, dry friction was present due to t he inherent mechanical construction of the pendulum hinge joint. This resulted in a dead band beha vior in the vicinity of the equilibrium point where the pendulum angular velocity was v ery low. This eﬀect can be seen in Fig.9 where the colored regions showing the tracked stiﬀn ess seem to have a slight oﬀset compared to the reference stiﬀness represented by the solid line. Note that the oﬀset slightly changes as the stiﬀness tracking passes the equilibrium pos ition showing the eﬀect of the dead band caused by the dry friction in the hinge joint. Despite the rather heavy weight of our device, it showed a sub stantially better gyroscopic- torque to weight ratio against the only other comparable dev ice reported in literature ( 7Nm/kg vs.3.4Nm/kgby Chiu and Goswami [3]), although the target gyroscopic mom ent of90Nm was not reached in this speciﬁc experiment. To improve the ge nerated gyroscopic moment, we presume that implementation of a partial vacuum ﬂywheel c hamber could considerably diminish the eﬀect of the aerodynamic drag as lower air densi ty can be achieved, decreasing even more power consumption and weight, and increasing top speed , thus producing higher angular momentum. Further improvements in weight reduction can be m ade using diﬀerent material and geometry selection for the ﬂywheel, where higher moment of inertia can be achieved with a smaller wheel size using materials with higher density (e.g steel, iron, tungsten, etc). This could have a direct impact on the size and weight of the gimbal struc ture as well as on aerodynamic drag in the ﬂywheel given that it increases rapidly with the p eripheral speed. This reduction in aerodynamic drag implies at the same time a reduction in th e power requirement for the ﬂywheel motor. Experimental results showed that our estimations were rath er conservative. Measured max- imum ﬂywheel speed was higher ( Ω =5400rpm @118.56W of electrical power), although in the same oder of magnitude, than the estimated maximum speed (ˆΩ = 5250rpm ). This can be explained from (i) the eﬀect of the bearings running-in peri od and increase of running tempera- ture [7] and (ii) due to assumptions in ﬂywheel geometry and a ir properties in the aerodynamic drag estimation. Frictional moment in the bearings could be either similar or lower than the estimated values, once grease distributes itself within th e free space of the bearing and lubricant temperature rises over our assumed temperature lowering it s viscosity. Aerodynamic drag are aﬀected by changes in the air properties. Air density and vis cosity were calculated assuming constant temperatures and pressure which can vary in the clo sed ﬂywheel chamber. However, given the close agreement with our results we presume that ou r assumptions of turbulent ﬂow, merged boundary layers in addition to and estimated consume d power were correct. Although, special care must be taken in the aerodynamic anal ysis if a more complex ﬂywheel geometry is used, such as variable thickness ﬂywheels with s pokes or conical constructions, the estimations presented here could be used as reference for ﬂy wheel actuator selection during early design stages. It is recommended that these estimatio ns are validated using CFD or experimentally. 6 As our prototype was conceived as wearable device, safety re quirements were set to comply with directives for medical devices. Thus, pilot tests coul d be conducted to assess how humans react to transmitted moments in the upper body. Finally, the setup used to evaluate the capabilities of the C MG as balance assistance device is rather simplistic, as it is based on a overly simple human mod el. Future research should involve tests with more realistic experimental platforms, for exam ple emulating falls in all directions using a set of 2 or more CMGs in a 2 DoF inverted pendulum. Conclusions By including the bearing friction and aerodynamic behavior of the spinning wheel and the in- duced dynamics of the wearer, we demonstrated that, with the proposed design methodology, a CMG-based human balance assistance device can be built to c omply with the designed spec- iﬁcations given a proper selection of the actuators. We show ed that our device is capable of producing up to 70Nm with a total weight of approximately 10kg.