Spiral bevel gears transmit torque between intersecting shafts through curved tooth flanks. Their gradual engagement and high contact ratio suit helicopter reducers, where compactness, load capacity and reliability are decisive. The same curved geometry produces long, three-dimensional tooth edges that are difficult to deburr and chamfer after hardening. A controlled chamfer removes sharp edge discontinuities, lowers edge stress concentration and reduces the risk of chipping during handling and meshing.
Chamfering Requirements for Hardened Spiral Bevel Gears
In a helicopter reducer, the spiral bevel gear carries high torque at high speed and must remain reliable in a compact housing. The tooth edges are not straight lines but curved space curves, and the chamfer must follow the complex curved tooth profile without damaging the hardened flank. The target for the gear considered here was a C0.5–0.75 mm chamfer on an alloy-steel gear with a chamfer-zone hardness of HRA 82. At this hardness, manual filing is difficult to control and depends strongly on operator skill. Dedicated chamfering machines can be productive, but they require part-specific tooling and offer limited flexibility when the gear geometry changes.
A robot with a compliant spindle addresses the geometric problem differently. The arm provides the nominal path and reach, while the spindle provides local force and position compliance. This division is important for a hard, curved edge: the robot does not need to force a rigid tool into the profile, and the spindle can absorb small mismatches between the programmed path and the actual tooth surface.
System Configuration and Compliance
The system combines a six-axis industrial robot, a pneumatic floating spindle, an automatic rotary table and the workpiece. The robot is an ABB IRB4600 with a 2.05 m working range and a 60 kg maximum payload. The spindle is an Amtru FB-CUT110 with radial floating capability, a maximum floating force of 25 N, a maximum speed of 19,000 rpm and a compensation/retraction range of ±5 mm. A diamond grinding disc is held in a special holder at the robot end, and the gear is mounted on the rotary table. The robot and table are coordinated: the table indexes each tooth while the robot repeats the chamfer path.
| Element | Specification or function |
|---|---|
| Robot | ABB IRB4600, six axes, 2.05 m reach, 60 kg payload |
| Floating spindle | Amtru FB-CUT110, radial compliance, max. 25 N, max. 19,000 rpm, ±5 mm compensation |
| Tool | Diamond grinding disc on a special holder; edge contact |
| Worktable | Automatic rotary/indexing table for tooth-by-tooth positioning |
| Workpiece | Alloy-steel spiral bevel gear, HRA 82 at the chamfer zone, 31 teeth |
Calibration and Offline Path Generation
The process chain is calibration, offline programming, automatic machining and dimensional inspection. Tool-center-point (TCP) calibration uses the four-point method. The robot is taught four poses in which the same tool point reaches a reference; the points define a sphere whose center is the TCP. In frame form, \(B=AX\), so \(X=A^{-1}B\), where \(A\) is the robot pose, \(X\) the unknown tool pose and \(B\) the measured pose. Because the diamond disc cannot be used directly for point teaching, a pointer of the same length is substituted. The resulting TCP is the center of the disc end face.
The workpiece frame is established by the three-point method. The gear’s three-dimensional model is imported into ABB Robot Studio and translated to coincide with the actual workpiece frame. The chamfer path is generated in this frame and then optimized for reach, singularity avoidance and interference between the spindle and the gear. The tool rotates clockwise; in the path comparison, travel from the tooth root toward the tip was preferred over the reverse direction.
After a tooth is finished, the table rotates by one tooth pitch and the cycle repeats. This division of labor keeps the robot within a compact workspace and makes the tool orientation repeatable from tooth to tooth. The offline program also allows the path to be checked in simulation before execution, which reduces the risk of collision.
Floating Compensation and Force Control
The floating spindle is the key element for applying a robot to a hardened, curved edge. Its radial compliance absorbs small mismatches between the programmed path and the actual tooth profile, follows part deflection and prevents the rigid robot from forcing the disc into the gear. The nominal chamfer position is still defined by the robot path; the compliance acts over ±5 mm and is not a substitute for calibration. Contact force is set by air pressure. The spindle’s elastic element deforms until the force balance is reached, so the disc remains in contact with the edge while the tool can retract locally when the profile deviates.
The grinding mode is approximately vertical surface grinding: the edge of the disc contacts the tooth edge, and material removal is controlled by the contact force and the kinematic parameters. Let \(F_t\), \(F_n\) and \(F_a\) be the tangential, normal and axial components. For a single active grain, \(F’_t=\sigma a_m\) and \(F’_n=\lambda F’_t\), where \(\sigma\) is the specific grinding force, \(a_m\) the mean chip area and \(\lambda\) the ratio of normal to tangential force. Summing over the active grains gives
$$
F_t=\sigma b a_p \frac{v_w}{v_s},\qquad
F_n=\lambda \sigma b a_p \frac{v_w}{v_s},\qquad
F_a=\sigma b a_p \frac{v_w v_a}{v_s^2}.
$$
Here \(b\) is the contact width, \(a_p\) the axial infeed of the disc, \(v_w\) the workpiece speed, \(v_s\) the wheel speed at the contact point and \(v_a\) the axial component of \(v_s\). The relations show the control direction. Increasing the floating pressure raises \(a_p\), the removed volume per unit time and the chip thickness, so the grinding force rises. Increasing \(v_s\) reduces the chip thickness per grain and lowers the force. The axial force is smaller than the tangential force by the ratio \(v_a/v_s\). In practice, the pressure loop sets the force level, while the robot path and spindle speed set the geometric and kinematic conditions. The equations are a kinematic estimate of force trends; the pressure-force characteristic of a given spindle and tool should be calibrated before production.
Trial on a Helicopter Reducer Gear
The trial used an alloy-steel spiral bevel gear for a helicopter reducer. The chamfer zone had a hardness of HRA 82, and the gear had 31 teeth. The target chamfer was C0.5–0.75 mm. The process followed the same chain: tool and workpiece calibration, offline path generation in Robot Studio, simulation for singularity and interference avoidance, automatic chamfering with table indexing, and inspection. A contacting profilometer (model 334006) was used to measure the chamfer. The inspection covered chamfer size and surface condition; the dimensional results are summarized below. Eight of the 31 teeth were selected at random; on each tooth, measurements were taken at top, middle and bottom positions along the tooth flank.
Measured Dimensions and Consistency
| Tooth no. | Top | Middle | Bottom | Mean |
|---|---|---|---|---|
| 1 | 0.722 | 0.723 | 0.723 | 0.7227 |
| 5 | 0.722 | 0.725 | 0.722 | 0.7230 |
| 9 | 0.724 | 0.722 | 0.722 | 0.7227 |
| 13 | 0.723 | 0.726 | 0.726 | 0.7250 |
| 17 | 0.722 | 0.725 | 0.721 | 0.7227 |
| 21 | 0.722 | 0.722 | 0.723 | 0.7223 |
| 25 | 0.722 | 0.721 | 0.726 | 0.7230 |
| 29 | 0.723 | 0.720 | 0.722 | 0.7217 |
Across the 24 readings, the mean was 0.7229 mm and the full range was 0.720–0.726 mm, a spread of 0.006 mm, or 2.4% of the 0.25 mm target band. The tooth means ranged from about 0.7217 to 0.7250 mm. All measured values fell inside the target band, and the tooth-to-tooth and position-to-position variation was small relative to the band width. The mean lies in the upper half of the C0.5–0.75 mm band, so the available margin is larger toward the lower limit than toward the upper limit.
Process Capability and Application Scope
For the sampled teeth, the robot grinding process produced chamfers within the target band with a spread of a few thousandths of a millimeter. The combination of offline programming, automatic indexing and floating force control replaces manual filing with a repeatable path and a controlled contact condition. The controlled contact force supports surface quality by limiting edge chipping, which is important for a hardened spiral bevel gear. The automatic cycle is the basis for efficiency: the table indexes and the robot repeats the path without re-teaching each tooth. The measured spread supports consistency at the sampled locations.
The measurement covers eight of 31 teeth at three positions each; the result describes the sampled set. The position of the mean within the band also has a practical consequence: because the values sit nearer the upper limit than the lower limit, production control should monitor the upper side of the tolerance. Extending the process to other spiral bevel gear sizes requires new fixtures, recalibration and part-specific path generation, while the compliant spindle and force-control principle remain reusable.