Ventil 2 / 2020 • Letnik 26 Abstract: A design and development of a planocentric gearbox to be used in robot arm joints, namely for a col- laborative robot, is presented in the papers. Besides strict limitations regarding the near zero backlash, the output position and torque should be available as well. So, a spatial-awareness encoder and a torque sensor are incorporated in a gearbox. The first assures an accurate absolute output position and the latter ensures information on actual output torque. The presented solution is based on the S-shaped tooth flank geometry. The near zero backlash requirement requires a tolerance analysis, which was accomplished by simulations in KissSoft software. The results of the analysis enabled a successful modification of the gearbox. A sophisticated testing rig was developed to verify actual gearbox characteristics, and to test its short-term and long-term behaviour. Backlash, stiffness, kinematic error, and dynamic behaviour of produced gear trains are measured in this way. Keywords: planocentric gear train, backlash, tolerance analysis, S-gearing, torque sensor, spatial-awareness sensor, testing rig 114 1 Introduction Planocentric gearboxes are used in robotics, ma- chine tools, aeronautics, aircraft, marine and many other industries. The efficiency can be above 90 % and gear ratios can achieve up to 160 : 1 (160 rotati- ons of the input shaft for a single turn of the output shaft). Basic arrangements of this type are descri- bed in well-known references [1]. The available indu- strial solutions include Sumitomo cyclo gearboxes [2], Spinea drives [3], Nabtesco [4], Onvio [5] and many others. Gearings are usually cycloidal or lan- tern. Some solutions combine a classic planetary gear train and a cycloidal stage with three eccentri- cs having origins in the centers of the planets. Some other devices incorporate a threefold eccentric ro- tating three planets. Such gearboxes are more com- pact but also more complex. All producers claim near zero or zero backlash. There is also a strong patent activity all over the world, especially in Rus- sia and China, which indicates the importance of the field. The mentioned companies are holders of such patents as well. The expected characteristics are low hysteresis for accurate positioning, low lost motion, compact design, high torsional stiffness, low inertia, high efficiency, overload capacity, easy assembly, and lifetime lubrication. Some robot producers de- velop their own gearboxes in order to achieve the best possible accuracy and reliability. Potential advantages of planocentric gear boxes, namely a high-speed reduction ratio combined with t owards i ntelliGent p lanocentric Gear t rain for r obotic i ndustry – p art 1 Gorazd Hlebanja, Miha Erjavec, Luka Knez, Simon Kulovec, Jože Hlebanja MEHANSKI SKLOPI V MEHATRONIKI dr. Gorazd Hlebanja, univ. dipl. inž., University of Novo Mesto, FME, Slovenia and Podkrižnik, d. o. o., Nazarje, Slovenia; Miha Erjavec, univ. dipl. inž., dr. Luka Knez, univ. dipl. inž., dr. Simon Kulovec, univ. dipl. inž., vsi Podkrižnik, d. o. o., Nazarje, Slovenia; Prof. em. dr. Jože Hlebanja, univ. dipl. inž., Univer- sity of Ljubljana, FME, Slovenia Figure 1 : Gear ring and planet gear during meshing [6]. Ventil 2 / 2020 • Letnik 26 115 MEHANSKI SKLOPI V MEHATRONIKI high output torque in a relatively small volume, were a driving force for a new development alrea- dy twenty years ago. So, the modified lantern gears were developed for use in planocentric gear drives with eccentric [6] and patented [7]. The internal gear pair during meshing is illustrated in Fig. 1, cle- arly indicating the value of eccentricity. The teeth of the ring gear are designed as semi-cir- cular extremities, whereas the planetary gears are designed with corresponding semi-circular spaces, adapted in size for a tolerance. Design aims were focused in automatic production lines and CNC- -machinery. The planocentric lantern gear box de- sign was robust. The transmission of rotation from the planet gears through a pin composition to the output shaft was provided by a single sided cage with double bearing output shaft. Small series were produced with various gear ratios up to 100 and with various modules, down to m = 0.5 mm. Pure mechanical drives can conform to high tech industry requirements with regard to backlash, lost motion, stiffness, hysteresis, etc. However, supple- mentary features based on sensorics can add ad- ditional functionalities to such a gearbox. So, an accurate output shaft positioning and an output torque sensorics are installed in the device as an option. However new functionalities enable the in- corporation of such devices in collaborative robot’s arm joints and adaptive control. An upgrade to a self-aware condition monitoring system could inc- rease the overall reliability of the drive and the ef- fective predictive maintenance whereas a corre- sponding condition monitoring could enable safe human interactions which is of special importance e.g. in the field of robotics. Thus, the paper describes how the planet gear mo- ves based on the eccentric rotation and induces the output rotation. Since the gearing geometry is based on an S-gear flank profile, some ideas about this gear type are explained. Next, the gearbox pro- totype is revealed. The described testing rig ena- bles all necessary tests, e.g. stiffness, hysteresis, kinematic error and durability measurements, whi- ch are necessary in confirming the prototype de- sign or indicating possible improvements. A longer chapter discusses influences of tolerances. Findin- gs of tolerance analysis helped in improving the ge- arbox design furthermore. In the final chapter, the torque flange is described. The torque sensor is in its final stage of development, so the flange design, the strain-gauges, and electronics and signal tran- smission are already defined. 2 Kinematic Circumstances of a Planocentric Gear Train The planocentric gearbox has coaxial input and output shafts, and large transmission ratios can be achieved based on a gear ring with internal gea- ring in combination with usually two planet gears with external gearing, where the difference in the numbers of teeth between the gear ring zv and the planet gears z p rules the output gear ratio, Eq. (1). The difference in ring and planet numbers of teeth should be one. 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a (1) The planet gears are mounted on an eccentric shaft, where bearings separate the planet gears Figure 2 : Planetary gear movement in accordance with hypocycloid generated on the kinematic circle of the ring gear [8]. Ventil 2 / 2020 • Letnik 26 from the eccentric. The planet gears wobble aro- und the gear ring, that is, they reverse for one tooth in each revolution of the eccentric. The wobbling movement is in accordance with a hypocycloidal movement where the generating circle with the radius of the eccentric is rolling on the kinematic circle of the ring gear. At the same time, the planet gear kinematic circle rolls in the inner side of the ring gear kinematic circle, which is simultaneous with the rotation of the eccentric. In this way the planetary gears develop rotation superimposed on the wobble. So, the input rotation of the eccentric is transformed into the reduced output rotation of the cage with the pins according to the gear ratio in the reverse direction of the input shaft in the same axis. And the gear ring is fixed to the housing. The eccentric driven planocentric gear train can be regarded as a simple mechanism with two links. The first link size is the radius of the eccentric and its joint indicates its position. The second one con- nects the eccentric with a point on the planet gear (a rigid body), e.g. the contact point. The eccentric link rotates and induces movement of the chosen point on the planet gear, which is restricted by the following rule: 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a (2) r v and r p are the radii of the kinematic circles of the ring gear and the planet gear, respectively. If the ring gear rotates for ϕ v the planet rotates for ϕ p . Fig. 2 illustrates movement of the planet based on the ro- tation of the eccentric and rolling of the planet kine- matic circle on the fixed ring circle. A simple algorithm can be used to define move- ment of the planet based on the rotation of the eccentric and limited by Eq. (2).  T p0 and T v0 coincide with C. P 0 is a point on the planet also coinciding with C.  T p1 and T v1 are calculated according to Eq. (2). It is true: 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a  Eccentric turns for ϕ v to the new point O p1 . kkp rolls on kkv in such a way that T p1 coincides with T v1 . So, tangents and normals of kkv and kkp co- incide in T v1 .  The normal of the planet in this point runs thro- ugh O v and O p1 .  Since the planet is a rigid body, the right leg of the angle ϕ p rotates around O p1 in CW direction for the difference ∆ϕ=ϕ v -ϕ p .  The procedure is continuous, but it can be nu- merically calculated by any adequate number of steps. The above procedure can be formalized. Thus, su- ccessive points on the ring gear kinematic circle T vi are defined as follows: 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a (3) Similarly, successive position points O pi of the eccentric are 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a (4) The coordinates of the moving point Pi on the pla- net gear are (5) 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a 116 MEHANSKI SKLOPI V MEHATRONIKI Figure 3 : Planetary gear movement in accordance with hypocycloid generated on the kinematic circle of the ring gear [8]. Ventil 2 / 2020 • Letnik 26 The eccentricity e is defined by Eq. (6): 1 𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜 = 𝑧𝑧 𝑝𝑝 − 𝑧𝑧 𝑣𝑣 𝑧𝑧 𝑝𝑝 (1) 𝑟𝑟 𝑣𝑣 = 𝑟𝑟 𝑝𝑝 𝜑𝜑 𝑝𝑝 𝜑𝜑 𝑣𝑣 (2) p =  m = 𝐶𝐶𝐶𝐶 v 1 = 𝐶𝐶𝐶𝐶 p 1 . 𝑥𝑥 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 a a nd 𝑦𝑦 𝑣𝑣 𝑣𝑣𝑇𝑇 𝑇𝑇 𝑣𝑣𝑇𝑇 = 𝑟𝑟 𝑣𝑣 𝑟𝑟𝑖𝑖𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (3) 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 𝑣𝑣𝑇𝑇 a nd 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 = 𝑒𝑒 𝑟𝑟𝑖𝑖 𝑠𝑠 𝑟𝑟 𝑣𝑣𝑇𝑇 . (4) 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑥𝑥 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑟𝑟𝑟𝑟 ∆ 𝑟𝑟 𝑇𝑇 a nd 𝑥𝑥 𝑃𝑃𝑇𝑇 = 𝑦𝑦 𝑂𝑂𝑝𝑝 𝑇𝑇 + 𝑟𝑟 𝑝𝑝 𝑟𝑟𝑖𝑖𝑠𝑠 ∆ 𝑟𝑟 𝑇𝑇 . (5) 𝑒𝑒 = 𝑧𝑧 𝑣𝑣 − 𝑧𝑧 𝑝𝑝 2 ∙ 𝑚𝑚 (6) a (6) The planet gear tooth movement into a new ring gear tooth space is illustrated in Fig. 3 by 20 iterati- ons. So, each point and planet gear position in Fig. (2) is based on successive rotations of the eccentric for 18°. 3 Gear Tooth Flank Geometry Beside semicircular, many other gear flank geome- tries have been proposed for planocentric gear tra- ins, all in search for an optimal geometry. The involute gear shape can be used to compose a planocentric gear box. A research [9] with the goal to produce a robotic gear box in order to replace an existing cycloidal planocentric gearbox was con- ducted. The key point was also to be economic in manufacture due to little influence of manufacturing and assembly errors. However, due to possible gear interference, such gears exhibit rather high pressure angle α w around 30°, and ∆ z cannot amount to less than 5, or 4 based on the condition that the numbers of teeth of pinion and ring gear are rather high, whi- ch is z p =167 and z v =171 in this case. This also means essentially lower gear ratio according to Eq. 1, whi- ch amounts to i -1 = -41.75 for the above data. Some other attempts included a rectilinear tooth profile, where a tooth is defined by two lines enclosing an angle, by an inside or outside circle, by the root cir- cle and fillet arcs [10]. The power is transmitted by the arc at the pinion tooth tip which slides over the linear tooth part of the gear ring. The problem is that such a gear composition does not follow the law of gearing. If a high-speed rotation is required, then such a gear arrangement can develop a high noise and torque fluctuation. Yet another tooth design is trapezoidal [11], where the contact of teeth is sur- face-like. However, the efficiency of such gear box may be poor, due to lack of rolling, amount of sliding and (non)conformity to the law of gearing. A proposed solution is based on S-shaped tooth flank geometry for the meshing ring gear and pla- netary gears of the planocentric gear box. General ideas about S-gears have been described in several papers [12-14]. The S-gear configuration has several advantages, the most important being the following:  convex-concave contact in the vicinity of the meshing start and meshing end point;  a low amount of sliding during meshing which is due to the curved path of contact;  an evenly distributed flank load, which is due to similar sizes of addendums and dedendums of both meshing gears. The other features include better lubrication due to high relative velocities and amount of rolling. In the case of internal-external gear pair some featu- res may become less pronounced. Additionally, the path of contact is less curved, which is on behalf of smaller dedendum and addendum heights. S-gear shape is ruled by two parameters, the height para- meter α p and the curve exponent n. By optimizing these two parameters one can shape this type of gearing in such a way to allow the gear and pla- net teeth number difference to be only one. So, it is possible to design gear boxes with small diameters and high reduction ratios. The S-gearing for plano- centric gear trains is illustrated in Fig. 4. Some time ago it was difficult for any basically better gear to- oth shape to compete with the involute gear shape, which was perfected during two centuries of deve- lopment. Now it is essentially easier to produce S- -tooth geometry, e.g. by machining with gear hobs based on the S-rack S-gear tooth profile. However, standard gear quality reports are based on E-geo- metry. So, CMM programs should be adapted. 117 MEHANSKI SKLOPI V MEHATRONIKI Figure 4 : S-gear flank geometry of the planocentric gear box [15]. Ventil 2 / 2020 • Letnik 26 4 Planocentric Gear Box Prototypes Several prototypes were produced and assembled during the development period. These prototypes were used for testing important characteristics and to acquire knowledge in design of a succe- eding gearbox. The gearbox is similar in functi- on to those already mentioned. It contains an in- put shaft with eccentrics. As a motor rotates the eccentric shaft rotates two planetary gears which wobble on the ring gear. The planetary gears are positioned in such a way that they enclose 180° for the sake of symmetry. Arrangements with three or more planetary gears are possible, which would impose high manufacturing skills regarding eccen- trics. A cage consists of a supporting ring and ou- tput ring (serving also as the output shaft) that are connected by pins in an interference fit. The cage is rotated by planetary gears, having appro- priate holes in which connecting pins with bearin- gs comply. The cage is fixed to the input shaft by bearings at the extremities and in a similar manner to the housing with the ring gear. In this way a compact low volume gearbox is achieved. Initial prototypes are devices with a reduction ratio of 80 (z v =81 and z p =80), an outer diameter of around 118 MEHANSKI SKLOPI V MEHATRONIKI Figure 5 : 3D schematic of the planocentric gearbox (left); photo of the prototype (right). Figure 6 : Ring gear (left) and a planet (right) of a specimen 01 after disassembly. Ventil 2 / 2020 • Letnik 26 Φ100 and having a module of 1 mm. The required maximal working torque is 120 Nm. The device is presented in Fig. 5, 3D schematic (left) and photo (right). The gearbox from Fig. 5 already includes an absolute output position enco- der, which is also an innovative Slovenian product, namely the AksIM absolute rotary encoder made by RLS [16]. Individual components (the eccentric, the ring gear and the planets) are measured on a CMM (Com- puterized measurement machine) before assembly. Assembled devices were tested. The tests include backlash, hysteresis and stiffness, kinematic error, vibrations and noise, as well as durability tests. The devices were disassembled afterwards, and critical components were inspected on the CMM and op- tically. Fig. 6 shows the ring gear and the planet of the spe- cimen 01 after the conducted durability tests. The hole in the planet (which is adapted for the cage of pins with bearing bushings) is slightly worn in the circumferential direction according to the acting force on the output bearing bushings of the pins. The specimen was submitted to high torques and speeds. The planet gears were made of 42CrMo4 and the ring of 25CrMo4, all gears plasma nitri- ded to HV700. The gears were carefully examined by an optical microscope. The gear teeth did not show any wear or damages. Initial wear appeared in some planet teeth tips and at certain locations in teeth tips. The reason is in the meshing errors, which were discovered by measuring teeth of the planets and the ring gear with a CMM. 5 Conclusions The paper presents a gradual development of a planocentric gearbox from starting concepts and based on S-gearing principles. The gearbox design enables high gear ratios, with the developed proto- type having a reduction ratio of 80. Through gear shape optimization, design improvements, usage of CA tolerance analysis a substantial improvement of the planocentric gearbox mechanical performance was attained. The near zero backlash was accompli- shed, which enables usage of this product in robotic industry, i.e. robot arm joints. Severe durability tests showed no notable wear. The design is modular, so the gearbox can be purely mechanical, it can conta- in the absolute output encoder inside the gearbox body. It can also contain the torque flange, with elec- tronics at the output side. It should be noted that both sensors should be provided if such a gearbox is intended for collaborative robots or adaptive con- trol robots. The robotic companies do not sell such gearboxes, they are for internal use only. So, such a gearbox becomes even more interesting. Incorpora- tion of a servomotor is also being considered. Beside the gearbox with the reduction ratio 80, a smaller gearbox with a ratio of 48 and a bigger one with a ratio of 120 were designed. So, a gearbox family in a range of output torques from 40 Nm up to 400 Nm becomes available. Technological procedures for serial manufacturing are already being prepared and optimized. The project with the acronym SGU – S-Gearbox Ultra was therefo- re successfully brought to the end. Nevertheless, many tasks are still in progress, e.g. a setup of the serial mechanical and electronics production and serial assembly, automating calibration procedures, development of self-aware monitoring and many other. References [1] Radzevich, S.P., (2012), Dudley’s Handbook of Practical Gear Design and Manufacture, Second Edition, CRC Press, Taylor & Francis Group, Boca Raton, ISBN 978-1-4398-6602-3 (eBook – PDF) [2] Sumitomo Drive Technologies (2018), Fine Cyclo® – Zero Backlash Precision Gear-boxes, Catalog #991333. www.sumitomodrive.com, accessed 1/9/2019. [3] Spinea TwinSpin (2017) High Precision Re- duction Gears, Ed. I/2017 h t t p s : / / www.spinea.com/en/products/twinspin/in- dex, accessed 1/9/2019. [4] Nabtesco Precision Reduction Gear RVTM (2018) – E Series/C Series/Original Series CAT.180420, https://www.nabtesco.de/en/ downloads/product-catalogue/, accessed 1/9/2019. [5] Onvio Zero Backlash Speed Reducers (2005), www.onviollc.com, accessed 1/9/2019. [6] Hlebanja, J., Hlebanja, G., (1994) “Efficien- cy and Maximal Transmitted Load for Inter- nal Lantern Planetary Gears”. FAWCET, J.N. (Ed.). International gearing conference, pp. 117–120. [7] Hlebanja, J., Hlebanja, G., (1994): Patent No. 9300152, “Planetary Gear Train”. Slovenian Intellectual Property Office (SIPO) [8] Hlebanja, G., Erjavec, M., Kulovec, S, Hleban- ja, J. (2020) Optimization of Planocentric Gear Train Characteristics with CA-Tools. In Goldfarb, V., Trubachev, E., Barmina, N. (eds.) Mechanisms and Machine Science, Vol. 81: New Approaches to Gear Design and Pro- duction, ISBN 978-3-030-34945-5 (eBook), Springer Nature, p. 323–347. [9] Park, M.-W., et al. (2007) “Development of Speed Reducer with Planocentric Involute Gearing Mechanism”, Journal of Mechanical Science and Technology Vol. 21 (2007) pp. 1172–1177 [10] Kim, J.H. (2006) “Analysis of Planocentric Gear”, Agri.&Biosys.Eng. Vol. 7(1) pp.13–17 [11] Nam, W.K., Oh, S.-H., (2011) “A design of 119 MEHANSKI SKLOPI V MEHATRONIKI Ventil 2 / 2020 • Letnik 26 120 MEHANSKI SKLOPI V MEHATRONIKI Razvoj inteligentnega planocentričnega prenosnika za robotsko industrijo Razširjeni povzetek: Za planocentrične zobniške prenosnike sta značilna visoka redukcija vhodne rotacijske hitrosti in veliko povečanje izhodnega navora v najmanjšem mogočem volumnu, zato so zanimivi za visokotehnološko in- dustrijo. Predstavljena rešitev je zasnovana na S-obliki bokov zob in posebej fokusirana na robotsko indu- strijo. Zahtevane mehanske karakteristike prenosnika pomenijo striktne omejitve za sestavljeni proizvod, npr. največjo zračnost pod 1 kotno minuto oz. blizu nične zračnosti. Dodatna kvaliteta predstavljenega prenosnika pa je senzorika, ki je vgrajena opcijsko, na modularen način, zgolj z dodatnimi elementi. Tako je lahko prenosnik zgolj mehanski, lahko pa vsebuje absolutni rotacijski enkoder AksIM 2 firme RLS, ki po- sreduje točno izhodno pozicijo. Dodatno pa se lahko prigradi senzor izhodnega navora, ki je zasnovan kot posebna prirobnica z dovolj veliko torzijsko deformacijo, ki omogoča korekten signal. V ta namen je upo- rabljen ustrezen sistem merilnih lističev. Poznavanje lege in navora pa je podlaga za uporabo prenosnikov v sklepih rok v t. i. sodelujočih robotih ali pri adaptivnem krmiljenju. Prenosnik iz tega članka je bil v osnovi zasnovan pred skoraj 30 leti z drugačnim ozobjem, t. i. paličnim ozobjem s cilindričnimi vdolbinami in ustreznimi izdolbinami. V tem prispevku je predstavljeno kinematsko delovanje bistveno izboljšanega prenosnika z drugačnim S-ozobjem in strukturo. Za namene testiranja zračnosti, histereze, kinematske napake, vibracij in obremenitvenih testov različnih prenosnikov – od tistih v razvoju do raznih na tržišču, npr. Spinea, Harmonic drive itd. – je bilo zgrajeno sofisticirano preizkuševa- lišče. Rezultati testiranj so pripomogli k hitri konvergenci v razvoju. Vse bistvene komponente prototipov so bile izmerjene na CMM pred uporabo in po končanih trajnostnih preizkusih. Komponente so bile pregle- dane tudi na mikroskopu, kjer so se ugotavljale morebitne poškodbe. Nujna pa je simulacija obnašanja prenosnika na osnovi toleranc. Sistem KissSoft je bil uporabljen za anali- zo vplivov toleranc, za kontaktno analizo in za vpliv toleranc ležajev in nosilcev. Ta analiza je pokazala na potrebo po parjenju zobnikov, tj. venca in planetnikov iz diametralnih tolerančnih mej – zgornja/spodnja ali spodnja/zgornja, po debelejših zobeh in povečani medosni razdalji. Kontaktna analiza razkriva potrebo po toplotno obdelanih legiranih jeklih. Vpliv kontaktnega tlaka na možno interferenco pa je zanemarljiv. Vpliv toleranc ležajev in ležišč ekscentra na skrajne lege planetnikov ob predvideni nominalni obremenitvi prenosnika pokaže na možne interference, kar vodi do konstrukcije zob z ustreznimi zaokrožitvami. V tem primeru se je za KissSoftovo analizo uporabljala modificirana geometrija bokov zob, ki je omogoča- la računske postopke na osnovi korekcije evolventnih bokov v S-geometrijo. Zaradi omejene višine vrhov in vznožij je bila ta analiza dovolj natančna. Ključne besede: planocentrični prenosnik, zračnost, analiza toleranc, S-ozobje, senzor navora, senzor zaznavanja lege, pre- izkuševališče speed reducer with trapezoidal tooth profile for robot manipulator”, Journal of Mechan- ical Science and Technology Vol. 25 (1) pp. 171–176 [12] Hlebanja, G., (2011) “Specially shaped spur gears: a step towards use in miniature me- chatronic applications”. 7th Int. Sci. Conf. Re- search and Development of Mechanical Ele- ments and Systems – IRMES 2011, April, 2011, Zlatibor, Serbia, Miltenović, V. (Ed.), Proceed- ings, Niš, pp. 475–480. [13] Hlebanja, G., Hlebanja, J., (2013) “Contribu- tion to the development of cylindrical gears”. Dobre, G., Vladu, M.R. (Eds.), Power transmis- sions: Proc. 4th Int. Conf., Sinaia, Mechanisms and machine science, ISSN 2211-0984, Vol. 13. Dordrecht [etc.]: Springer, pp. 309–320 [14] Hlebanja, G., Kulovec, S., Hlebanja, J., Du- hovnik, J. (2014). “S-gears made of poly- mers”. Ventil, ISSN 1318-7279. 10. 2014, Vol. 20, No. 5, p. 358–367. [15] Hlebanja, G., Kulovec, S. (2015) Development of a planocentric gear box based on S-gear geometry. In Lüth, T. (Edt.). 11. Kolloquium Getriebetechnik, Garching, 28.–30.9.2015. München: Technische Universität, p. 205–216. [16] AksIM 2 absolute rotary encoder, datasheet, https:/ /www.rls.si/en/aksim-2-off-axis-rota- ry-absolute-encoder, accessed 24/2/2020. [17] KissSoft (2019). KISSsoft Release 2019 User Manual. KISSsoft AG – A Gleason Company, Bubikon. Acknowledgment The investment is co-financed by the Republic of Slovenia and the European Union under the Euro- pean Regional Development Fund, no. SME 2/17-3/2017 and C3330-18-952014