# Traceability and Spectral Analysis of Transmission Error in Rotary Vector Planetary Gear Reducers
## 2 Transmission Characteristics
### Motion Transmission Path  
The primary motion of the RV reducer is depicted in Figure 2, which illustrates a configuration with two crankshafts; a three-crankshaft configuration follows a similar structure. In Figure 2, component 1 represents the pin housing, with pin 2 semi-embedded within it; component 3 is the crankshaft, equipped with planet gear 4 at its end to form a planetary gear transmission with sun gear 5; and component 6 is the assembled flange, featuring installation holes in its front and rear sections for mounting crankshaft 3. Several cycloidal gears 7 are installed on the eccentric part of crankshaft 3, creating a small-tooth-difference cycloidal-pin gear transmission with pin 2.  
The motion is characterized by two stages of reduction: the first stage involves planetary gear reduction, and the second stage involves small-tooth-difference cycloidal-pin gear reduction. There are two distinct transmission paths: one path transmits motion from the gear shaft to the pin gear output, while the other transmits motion from the gear shaft to the flange output. The specific transmission path depends on the two possible fixation configurations.  
Figure 1: Structure diagram of the RV reducer: (**a**) front view; (**b**) rear view; (**c**) side view; and (**d**) exploded view.  
Condition 1: As depicted in Figure 2, when pin housing 1 is fixed, pin gear 2 is also stationary. Sun gear 5 rotates at an angular velocity of \(\omega_{\text{sun}}\), which drives planet gear 4 to rotate at \(\omega_{\text{plan-s}}\). Consequently, crankshaft 3, rigidly connected to planet gear 4, rotates at the same speed, \(\omega_{\text{cra-s}}=\omega_{\text{plan-s}}\). Crankshaft 3 and cycloidal gear 7 together form a planar four-bar linkage mechanism, as illustrated in Figure 3. In this configuration, crankshaft 3 drives cycloidal gear 7 in an orbital motion around the system's overall rotation center at a speed of \(\omega_{\text{cyc-g}}\). During this orbital motion, cycloidal gear 7 meshes with fixed pin gear 2, causing cycloidal gear 7 to self-rotate at \(\omega_{\text{cyc-s}}\) due to the tooth count difference. This self-rotation of cycloidal gear 7 further drives crankshaft 3 to revolve at \(\omega_{\text{cra-g}}\). The revolving motion of crankshaft 3, in turn, causes flange 6 to rotate at \(\omega_{\text{fin}}\). Ultimately, the motion is output through the self-rotation of flange 6 at \(\omega_{\text{fin}}\).  
Condition 2: When the flange is fixed, the scenario simplifies. With flange 6 held stationary (\(\omega_{\text{fin}}=0\)), crankshaft 3 loses its degree of freedom to revolve (\(\omega_{\text{cra-g}}=0\)), and cycloidal gear 7 also loses its self-rotation freedom (\(\omega_{\text{cyc-s}}=0\)). Motion is input through gear shaft 5 at a speed of \(\omega_{\text{sun}}\), forming a stationary-axis planet gear system in the first stage. Planet gear 4 rotates in place at a speed of \(\omega_{\text{plan-s}}\). Planet gear 4 then drives crankshaft 3 into rotation at the same speed, \(\omega_{\text{cra-s}}=\omega_{\text{plan-s}}\). This rotation of crankshaft 3, in turn, drives cycloidal gear 7 into an orbital motion at a speed of \(\omega_{\text{cyc-g}}\). Cycloidal gear 7 engages with pin 2, causing pin housing 1 to rotate at a speed of \(\omega_{\text{pin}}\). Ultimately, the motion is output as the rotation of the pin tooth housing at speed \(\omega_{\text{pin}}\).  
Figure 3: Primary kinematic relationships of the RV reducer.  
Figure 2: Motion diagram of the RV reducer: (**1**) pin housing; (**2**) pin; (**3**) crankshaft; (**4**) planet gear; (**5**) gear shaft; (**6**) flange; and (**7**) cycloidal gear.