1 Introduction

The characteristics of the drive system strongly determine the performance of the complete robot and robot-like systems (RLS) (Vogel-Heuser et al. 2020). The drive system of RLS is usually designed with a mechanical transmission with a high transmission ratio to optimize performance. The choice of the gearbox type and the design of the corresponding components influences the entire robot (Pham and Ahn 2018⁠, Lopez Garcia et al. 2020). Many different types of gearboxes are used in robotics. A direct comparison of the designs is difficult because of the different structures and function principles and is therefore carried out using appropriate performance metrics (Lopez Garcia et al. 2020, Sensinger and Lipsey 2012).

The three most important types of robot gearboxes are planetary gear drives, cam disk drives and strain wave drives (Pham and Ahn 2018). A common feature is the high transmission ratio with a compact design (Tsai and Hsu 2017). Further relevant characteristics, among others, are high torsional stiffness, transmission accuracy and efficiency (Lopez Garcia et al. 2022⁠, Yin et al. 2017). One of the advantages of single-stage planetary gearboxes is their high efficiency and low design complexity (Lopez Garcia et al. 2020). The inherent disadvantage of the low transmission ratio can be compensated for by a combination of several stages (Müller 1998). The desired small design space for the application in robotics is a challenge that limits the number of stages. Therefore, the aim is to use as few stages as possible to achieve a sufficiently high transmission ratio. The aim is, therefore, to achieve the highest transmission ratio possible for the single-stage gear drive.

The planetary gearboxes used in current applications typically use involute gearing, which, among other things, has advantages in terms of machining (Varadharajan et al. 2023). Disadvantages are the limitations for external gears with regard to their geometry, including undercutting and a small radius of curvature near the base circle. In order to achieve a high transmission ratio per stage with sufficient tooth thickness for load capacity, the number of teeth on the pinion must be very low. The limitations of the involute gearing also restrict the minimum number of teeth possible. Therefore, the transmission ratios per stage of the planetary gear drive for this type of gearing are usually maximum 10 (Pham and Ahn 2018, Crispel et al. 2018).

The eccentric cycloid (EC) gearing can reach a very low number of teeth on the pinion due to the modified geometry (Landler et al. 2023a) and thus increase the transmission ratio of the planetary gearbox. In addition, EC gearing can increase efficiency and achieve a higher torque capacity compared to involute gearing (Stanovskoy et al. 2012). Using the EC gearing for a planetary gearbox can increase performance and improve the backdrivability. This paper shows the design of a single-stage planetary gear drive with EC gearing and analyzes its characteristics. This single-stage drive can be used directly for applications in robotics or as a basis for the construction of multi-stage or compound planetary gear drives (Müller 1998).

The detailed analysis of the structure and characteristics of the planetary gearbox with EC gearing enables the integration into the overall robot system. For this purpose, it is necessary to use a suitable description language, an aspect that Vogel-Heuser et al. (2024) have addressed. The application of gear models for the simulation of complete robots and RLS is shown by Ziegler et al. (2023). Overall, the planetary gearbox model with EC gearing can be used from the early process of robot development up to the installation in operating RLS.

2 State of the art

The wide range of possible applications and designs of RLS leads to many different types of gearboxes used in robotics (Lopez Garcia et al. 2020, Landler et al. 2023b). The three most important types for industrial robots are the planetary gear drive, the cam disk drive and the strain wave drive (Lopez Garcia et al. 2022). These three drives can be classified in the group of epicyclic drives, which shows the structural similarity between these designs (Landler et al. 2023b). The three gearbox types have a high transmission ratio in common, which is necessary for many robotics applications (Lopez Garcia et al. 2022). Special drives like the Galaxie® drive (Schreiber and Röthlingshöfer 2294) or rolling element eccentric drives (Fritsch et al. 2023) have no major significance in robotics and will therefore not be discussed in detail in this paper.

The planetary gear drive in a single-stage design can be implemented in different configurations. One possible case is that the external gear on the central axis, also known as the sun gear, is driven and is in contact with the planet gears (Landler et al. 2023b). The number of planet gears may differ for various designs (Müller 1998). The planet gears are mounted with rolling bearings on the carrier and mesh with the internal gear. In one possible case, the output is at the carrier. Other parts can also be used as input or output for a planetary gear drive, thereby changing the transmission ratio (Lopez Garcia et al. 2022). According to Pham and Ahn (2018), the maximum transmission ratio for a regular single-stage planetary gear drive is 10.

The cam disk drive can be implemented with the drive at the eccentric (Landler et al. 2023b). The eccentric revolves one or more cam disks around the central axis, with the two parts being connected by rolling bearings (Yang and Blanche 1990). The cam disks mesh with the internal gear and rotate the pin disk, which also serves as the output. The term cycloidal drive is used if the gearing of the cam disk is shaped as a cycloid. Another form of the cam disk drive is to split the power before using the eccentric. This cam disk drive concept is also called RV drive (Wang et al. 2019).

The strain wave drive can be designed with the flexible gear rim in cup form (Landler et al. 2023b). In this case, the drive is located on the eccentric, which also known as the wave generator and is in contact with the flexible gear rim via rolling elements (Routh 2018). The flexible gear rim meshes with the internal gear and thus rotates around the central axis. The output is connected to the flexible gear rim. Besides the cup design, a flat design can also be used, which requires less design space in the axial direction but has a more complex structure.

Due to the relatively low transmission ratio of the single-stage planetary gear drive, multiple-stage and compound planetary gear drives are preferred for many robotic applications (Pham and Ahn 2018). The multi-stage design uses several single-stage planetary gear drives connected in series at the inputs and outputs. The advantage of this design is the high efficiency of the entire planetary gearbox. Disadvantageous is the need for an increased number of stages (Lopez Garcia et al. 2020⁠, Crispel et al. 2018). In compound planetary gear drives, the planet gears of several planetary gear drives are connected to each other and other components are possibly also combined (Kurth 2012). Compound planetary gear drives provide large transmission ratios in a small design space, but they are also complex and have low efficiency for usual gearings (Pham and Ahn 2018⁠, Lopez Garcia et al. 2020⁠, Crispel et al. 2018).

The Wolfrom drive is a type of compound planetary gear drive often used in robotics (Crispel et al. 2021). The functional scheme for a possible design of this drive can be seen in Landler et al. 2023b. The Wolfrom drive can achieve a high transmission ratio, but its efficiency is relatively low (Arnaudov and Karaivanov 2009). The overall transmission efficiency is based on the efficiency per gear mesh and the power transfer within the gearbox due to the structure (Crispel et al. 2021⁠, Arnaudov and Karaivanov 2009).

Overall, planetary gearboxes have potential in terms of design space and performance metrics compared to other robotic gearboxes (Pham and Ahn 2018⁠, Lopez Garcia et al. 2022). In particular, the design of the single-stage planetary gear drive described above offers the potential for high efficiency (Höhn et al. 2013). However, the high gear ratios required for robotics make more complex planetary gearboxes necessary, which have disadvantages (Pham and Ahn 2018⁠, Crispel et al. 2018]. Complex planetary gearboxes can be improved by adapting the gearing geometries. Maintaining the involute gearing, the gearing parameters can be adjusted to optimize efficiency (Kobuse and Fujimoto 2016). Höhn et al. (2014) and Liang et al. (2015) optimize the efficiency of a Wolfrom drive with the so-called low-loss gearing, which is an adapted version of the involute gearing with greatly reduced sliding velocities. Ikejo et al. (2005) show, experimentally and theoretically, the potential for optimizing the efficiency with special gearings in external gears. Varadharajan et al. (2023) and Brumercik et al. (2020) compare various performance metrics of different special gearings with the involute gearing for the case of planetary gear drives. The special gearings can have both advantages and disadvantages compared to the involute gearing (Varadharajan et al. 2023, Brumercik et al. 2020).

To further improve the performance of the gears in planetary gearboxes, a special gearing can be used that addresses the significant disadvantages of involute gearing in high-ratio planetary gear drives. The EC gearing provides the opportunity to offer both a very low number of teeth and high transmission ratios, as well as to optimize the characteristics of the gear mesh (Landler et al. 2023a). Landler et al. (2023a) define the EC gearing by the meshing of a gear with a circular arc profile (called arc gear) with a gear with a tooth profile of an equidistant of a trochoid (called cycloid gear). Figure 1 shows variants of the EC gearing with the same center distance. The EC gearing with a single-tooth pinion is presented in Fig. 1a.

Fig. 1
figure 1

Variants of the EC gearing with same center distance a (blue: arc gear; red: cycloid gear) (a zarc = 1, zcyc = 5; b zarc = 10, zcyc = 12)

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Stanovskoy et al. (2012) are among the first to show the geometry of EC gearings with single-tooth pinions. The details of this special configuration are presented by Kazakyavichyus et al. (2011). Li et al. (2017) also focus on the EC gearing with a single-tooth pinion and provide a geometric description for this special case. They also determine some basic characteristics of the tooth contact (Li et al. 2017). Landler et al. (2023a) show a complete geometric description of the EC gearing with an arbitrary number of teeth and use this to derive various characteristics. Chang et al. (2023) use a geometric description of the flanks of the EC gearing with arbitrary numbers of teeth and use this for a loaded tooth contact analysis.

Overall, the state of the art shows that planetary gearboxes with special gearing offer great potential for robotics applications. The EC gearing allows these potentials to be utilized thanks to its advantageous geometry. However, there is currently no generally available method to use the EC gearing for planetary gearboxes. This paper aims to show the application of the EC gearing for a single-stage planetary gear drive that best meets the requirements of robotics. The demonstrated approach can also be used for other configurations of the planetary gearbox.

3 Planetary gear stage with EC gearing

Various configurations can be used for a single-stage planetary gear drive (Müller 1998). The design concept of the single-stage planetary gearbox which is described above (with the driven sun gear and the fixed ring gear) is one of the so-called negative-ratio gears, which in many cases have an advantageous behavior in terms of efficiency (Kurth 2012). Only this planetary gearbox concept is considered in this study. The suitable number of planet gears for this concept remains to be clarified. Figure 2a shows the scheme of a single-stage planetary gear drive with three planet gears. The configuration with four planet gears is presented in Fig. 2b. Increasing the number of planets (with optimum alignment) results in an increased load capacity because of the load reduction through the power split within the gearbox. The disadvantage in this case is the increased design space required by the planet gears, which limits the transmission ratio of the planetary gearbox (Höhn et al. 2013). Due to the desired self-centering effect and load distribution, no less than three planet gears are used in many planetary gearboxes. The maximum possible transmission ratio per stage can be determined independently of the gearing by analyzing the pitch circles of the sun gear (dw,s), the planet gears (dw,p) and the internal gear (dw,i). For a functioning gearbox, the pitch circles of the pairing gears must touch each other, but those of the non-pairing gears need a certain minimum distance between each other. For a limit value consideration, it can be assumed that all pitch circles touch each other. Figure 2a, b show the corresponding pitch circles for three and four planet gears, respectively.

Fig. 2
figure 2

Maximum transmission ratio limit (a three planet gears; b four planet gears)

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The basic transmission ratio i0 is the basis for evaluating the possible transmission ratios for a planetary gear stage. It is defined as the transmission ratio of the gearbox with a fixed carrier. The relationships from Fig. 2a, b lead to the following equation for calculating the maximum basic transmission ratio i0,max,p for a certain number of planet gears p (\(p\ge 3\)):

$$i_{o,\,max,\,p} = - \frac{{d_{w,\,i} }}{{d_{w,\,s} }} = - \frac{{z_{i} }}{{z_{s} }} = - \frac{{1 + \sin \left( {\frac{\pi }{p}} \right)}}{{1 - \sin \left( {\frac{\pi }{p}} \right)}}.$$
(1)

Here, zi is the number of teeth of the internal gear and zs is the number of teeth of the sun gear. For a number of planet gears of three, this results in \({i}_{o,max,3}\approx -13.93\). The negative sign indicates a change in the direction of rotation by the gearbox. In the case of four planet gears, \({i}_{o,max,4}\approx -5.83\). Similar values for the maximum basic transmission ratios are calculated by Müller (1998). Three planet gears are better suited to achieving a high transmission ratio than four. Therefore, only configurations with three planet gears are considered below.

The basic transmission ratio sets the configuration of the individual transmission parts. This allows the transmission ratios of the gear pairs to be determined. The transmission ratio iext of the external gears is determined as follows:

$$i_{ext} = - \frac{{d_{w,\,p} }}{{d_{w,\,s} }} = - \frac{{z_{p} }}{{z_{s} }} = \frac{1}{2} \cdot \left( {1 + i_{0} } \right).$$
(2)

Here zp is the number of teeth of the planet gear. The transmission ratio iint of the planet gear to internal gear pairing can be determined analogously:

$$i_{int} = \frac{{d_{w,\,i} }}{{d_{w,\,p} }} = \frac{{z_{i} }}{{z_{p} }} = \frac{{2 \cdot i_{0} }}{{i_{0} + 1}}.$$
(3)

Based on the equations shown above, the relationship between the numbers of teeth is defined:

$$z_{i} = z_{s} + 2 \cdot z_{p} .$$
(4)

All numbers of teeth must be integers for a functioning gearbox. In addition, the gears must fulfill assembly conditions for the selected configuration of the planetary gearbox (Müller 1998). In the case of the selected negative-ratio gearbox and with the desired equal distribution of the planet gears in the circumferential direction, the following condition must be fulfilled (Müller 1998):

$$\frac{{z_{s} + z_{i} }}{3} = g = 2 \cdot \frac{{z_{s} + z_{p} }}{3}.$$
(5)

The value g must be a positive integer. Based on the boundary conditions shown above, a variety of numbers of teeth are possible for the gears. In the following, a specific configuration will be selected in order to design a gearbox for applications in robotics.

EC gearings with a very low number of teeth tend to create areas of tooth contact with very high pressure angles. These unfavorable areas in terms of power transmission must be avoided by restricting the tooth profiles, resulting in a reduced contact ratio. For this reason, the recommended number of teeth should not be too low. In contrast, the teeth become very small in relation to the overall gearbox when the number of teeth is high. This leads to a low resistance against tooth breakage. A good compromise between these two boundary conditions is a number of teeth of three for the sun gear. Other numbers of teeth are also possible in principle but will not be pursued further.

Table 1 shows all possible configurations of the single-stage planetary gear drive with three planet gears and three teeth for the sun gear. The transmission ratios are calculated using the equations shown above. Six configurations are possible with basic transmission ratios ranging from −3 to −13. It must be noted here that all transmission ratios of the external gears are integers, which can lead to special aspects in the dynamics.

Table 1 Possible numbers of teeth for three planet gears and three teeth for the sun gear
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For applications in robotics, high transmission ratios are desired. However, the configuration with \({i}_{o}=-13\) cannot be used, as the possible overlapping of the planet gears means that no suitable gearing can be designed. This is because suitable gearings need flank parts above the pitch circle. The configuration with \({i}_{o}=-11\) can be used. The distance between the planetary gears is selected to be minimal in order to provide sufficient flexibility for the gearing design.

For the detailed design of the EC gearings of the planetary gearbox, a further condition must be met due to the double meshing of the planet gears. The same arc gear with the same working pitch diameter must be paired with the planet gear and the internal gear. This arc gear is identical to the sun gear. This means that each module m, trochoid ratio λ and number of teeth of the arc gear zarc for all gearings of a planetary gear drive must be identical. The trochoid ratio is the ratio between the reference diameter and the working pitch diameter of the arc gear of the EC gearing (Landler et al. 2023a). Using these specifications, all gearings can be designed and a complete planetary gearbox can be created, see Fig. 3.

Fig. 3
figure 3

Planetary gear drive with an EC gearing (zs = 3, zp = 15, zi = 33)

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The relevant parameters for the gearing geometry according to the definitions of Landler et al. (2023a) can be found in Table 2. The module m is to be regarded as a scaling factor. The value of \(m=1 mm\) selected here produces a very small gearbox, which is suitable for an application with a small design space. For applications with higher torque demands, the gearbox usually has to be larger, which can be achieved by using a larger module. The relationships for the scaling of robot transmissions can be found in Saerens et al. (2019).

Table 2 Relevant parameters of the EC gearing
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A possible overlap of the planet gears must be checked in order to evaluate the proper functioning of the gearbox. The maximum possible tip diameter da,p,max of the planet gears can be determined by using the trochoid ratio as a parameter for the pitch circle as follows:

$$d_{a,p,\max } = \frac{\sqrt 3 }{2} \cdot m \cdot \frac{{z_{s} + z_{p} }}{\lambda } .$$
(6)

For the selected EC gearing, this results in a distance between the planet gears of m/4. The gearbox is therefore fully operational.

Depending on the choice of fixed and driven elements of the gearbox, a planetary gear drive can cover several transmission ratios. For many applications, the sun gear, the carrier or the internal gear is fixed, while the two remaining components serve as input and output. Table 3 shows the six possible configurations. The resulting transmission ratio of the overall gearbox is based on the basic transmission ratio.

Table 3 Transmission ratios for different configurations [based on Müller (1998)]
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The highest ratio of 12 can be achieved with an input on the sun gear and the output on the carrier. Overall, the newly created planetary gearbox with EC gearing can therefore push the boundaries of the state of the art. The detailed characteristics of the selected gearing will be examined below. This will allow conclusions to be drawn about the performance of the gearbox.

4 Characteristics of the EC gearing

As described above, the contact of all gear pairs in the planetary gearbox is based on the same arc gear. The representation of the tooth contacts can therefore be combined, see Fig. 4. This figure also shows the necessary contact of the pitch circles, which are represented by dash-dot lines. The path of contact is the sum of all contact points over the entire meshing range. The path of contact is valid for both external pairing and internal pairing.

Fig. 4
figure 4

Combined representation of all tooth contacts of the EC gearing

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Figure 4 shows that the path of contact is strongly curved at the end of the meshing (near the tooth root of the planet gear). In the other areas, only a slight curvature of the path of contact can be seen. This curvature is in contrast to the straight line of the path of contact for the involute gearing.

The meshing behavior can also be evaluated using the contact ratio. For continuous torque transmission, the total contact ratio εγ must be at least 1.0. This value is composed of two components, which take into account the influence of the profile in the face section (transverse contact ratio εα) and the flank profile in the width direction (overlap ratio εβ) separately:

$$\varepsilon_{\gamma } = \varepsilon_{\alpha } + \varepsilon_{\beta } .$$
(7)

The transverse contact ratio and the overlap ratio can be calculated using the methods for involute gearings, see ISO 21771 (ISO: Gears—Cylindrical involute gears and gear pairs—Concepts and geometry 2007). Table 4 shows the values for the contact ratios for the selected gearing. The value of the transverse contact ratio is relatively low compared to common involute gearings. This low value has the potential to minimize the sliding velocity but also leads to a lower load distribution between the teeth.

Table 4 Contact ratio of the EC gearing
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Figure 5a shows the transverse pressure angle αt over the rotation angle of the arc gear κ. A maximum value of approx. 30° is reached at the beginning and end of the tooth engagement. The minimum value is around 18°. The EC gearing provides pressure angles similar to those of conventional involute gearings. It can, therefore, be assumed that the force transmission of the EC gearing is similar to conventional gears.

Fig. 5
figure 5

Characteristics over the rotation angle of arc gear (a transverse pressure angle; b sliding factor)

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The sliding factor Kg over the rotation angle of the arc gear κ is shown in Fig. 5b. The zero crossing of the curve at the pitch point is clearly visible. In terms of magnitude, the sliding factor at the end of the meshing is greater than at the start of the meshing.

Figure 6a presents the transverse radius of curvature ϱt referred to the module m over the rotation angle of the arc gear κ for the external pairing. The sun gear has a constant convex curvature. The planet gear shows a change in curvature at a rotation angle of approximately 40°. At the beginning of the meshing, there is a convex-convex contact, similar to involute gearings. With the EC gearing, the radius of curvature at the start of the tooth mesh is not very small, unlike the involute gearing near the base circle. Additionally, there is a convex-concave contact at the end of the meshing, which cannot be achieved with involute external gears. This type of contact results in an osculation of the flanks and low contact pressure.

Fig. 6
figure 6

Transverse radius of curvature referred to the module over the rotation angle of arc gear (a external pairing; b internal pairing)

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Analogous to Fig. 6a, b shows the transverse radius of curvature for the internal pairing. The change in curvature of the internal gear is in a similar area as for the planet gear. This results in a convex-concave contact in most areas of the tooth engagement with the associated osculation of the flanks. Therefore, a relatively low contact pressure can be assumed for the internal pairing in most tooth flank areas.

Overall, the analysis of the contact characteristics shows that the EC gearing can be applied effectively to high-ratio planetary gearboxes. Due to the special tooth shape, a very low number of teeth can be achieved on the sun gear while maintaining satisfactory values for the pressure angle and the transverse radius of curvature. In addition, the EC gearing ensures that the performance of the involute gearing is surpassed in many areas. In terms of efficiency, advantages are to be expected, as there is low pressure in the areas of higher sliding velocity due to the osculation of the flanks. Specific values for efficiency can be determined for defined application parameters using the approaches of del Castillo (2002). In terms of load-bearing capacity, the convex-concave contact of the external gearing is advantageous.

5 Application of the planetary gearbox with EC gearing

The EC gearing shown above is the basis for the application in a complete planetary gearbox. Figure 7a shows the 3D representation of the EC planetary gearbox. The gears are displayed in detail here, but no additional components or connections are shown. For the application of the newly designed gearbox in robotics, further performance metrics must be investigated in addition to the tooth contact characteristics shown above. Crucial here is the comparison with established robot gearboxes. The compliance diagram is of particular importance for robotics. Landler et al. (2023b) show a method for the suitable comparison of the compliance between different transmission concepts. Analytical methods such as those described by Hochrein et al. (2022) can be used to calculate the stiffness. Tooth thickness and height are the most relevant influencing factors. As the EC gearing has a comparable macrogeometry of the teeth to the involute gearing, it can be assumed that the stiffness is comparable.

Fig. 7
figure 7

a 3D representation of the EC planetary gearbox with the gears only; b sectional view of the gearbox with added bearings, carrier and input shaft

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The backlash is another essential aspect of robot gearboxes. For many applications, zero or very low backlash is required. The EC gearing can precisely set the backlash by selecting the gearing parameters according to Landler et al. (2023a). Thus, it is possible to achieve zero backlash with the EC gearing.

For the installation of the planetary gearbox with EC gearing, the model in Fig. 7a requires a carrier and connections to adjacent components. Figure 7b shows the sectional view of the planetary gear drive with added bearings, carrier and sun gear shaft. The input from an actuator of the robot drive system is at the shaft of the sun gear. The output to the robotic device is attached to the carrier. The internal gear can be fixed using the illustrated holes.

Compared to other robot gearboxes, the planetary gear drive with EC gearing presented here has various advantages. Compared to strain wave drives or cam disk drives, the efficiency is higher and the complexity is lower. In comparison to planetary gearboxes with involute gearing, the transmission ratio per stage is higher, with potential for optimization in terms of load capacity and efficiency. Overall, the gearbox is very well suited for applications that require backdrivability.

A challenge for the planetary gearbox with EC gearing is the manufacturing of the components. Various studies exist on the manufacturing of the EC gearing. Bubenchikov et al. (2016) use a profile cutter to manufacture an EC gearing made of metal. The additive manufacturing of an EC gearing is described by Batsch (2019). Using the possibilities of additive manufacturing, the production effort for special components such as the internal gear can be reduced. Possible advantages of combining different materials within a planetary gearbox are presented by Iizuka and Takesue (2023).

6 Conclusion

The use of the EC gearing in high-ratio planetary gearboxes can increase performance. This paper shows the design and analysis of a planetary gearbox with EC gearing for application in robotics. A detailed examination of the geometric limits for single-stage planetary gear drives is presented. Based on this, a suitable EC gearing is designed, which is analyzed and evaluated in detail using various characteristics. This demonstrates the excellent performance of the gearbox. In the last step, the application of the planetary gearbox in robotics and the possibilities for manufacturing are discussed. Overall, the following main features of the planetary gearbox with EC gearing in comparison to commonly used planetary gearboxes can be summarized:

  • Possibility for very low tooth numbers on all gears

  • Increase in the maximum transmission ratio per stage

  • Increase in the load capacity by improving the osculation of the flanks

  • Increase in the efficiency by lowering the contact pressure in the areas of high sliding

Further work will focus on implementing the gearbox in a real product. For this purpose, a specific use case from robotics is defined, which defines the boundary conditions for the drive system. The aim is to demonstrate the advantages of the planetary gearbox with EC gearing. For this purpose, particular emphasis should be placed on a precise examination of the manufacturing process. In order to achieve the theoretically determined characteristics, the gearing must be sufficiently accurate. In particular, the required zero backlash of robot gearboxes is directly related to the achievable efficiency of the gearing. Another important point for a gearbox under real conditions is the loaded transmission error, which among other things depends on the stiffness of the gearing.