In my retrofit work on forming equipment, I set out to convert a conventional press-like machine into a multi-angle forming platform without replacing its entire structure. I began from a simple observation: if the table can be tilted and held at a controlled angle, then the forming direction no longer has to remain fixed relative to the workpiece. My design therefore combines a worm gear pair, a rotary support, a servo drive, and a closed-loop controller. I also use the broader family of screw gears as a design reference, because screw gears and worm gears share crossed-axis meshing behaviour, sliding contact, and strong sensitivity to backlash and lubrication. The term screw gears appears repeatedly in my analysis because much of the contact and error behaviour I model is analogous to that of screw gears under load.
The central problem I address is not merely adding an angle axis. The central problem is preserving angle accuracy while the forming force changes. A fixed PID loop can hold an angle in free air, but it cannot easily reject a sudden load torque. My solution therefore has four layers. First, I derive the kinematic relation between motor rotation and table tilt. Second, I derive the equivalent stiffness of the worm gear pair, the shaft train, and the rotary support. Third, I derive the angle-load coupling torque using virtual work. Fourth, I design a fuzzy PID controller with feedforward compensation and an error propagation model. I then validate the concept on a scaled test bench. In the following sections I present the equations, tables, control logic, and experimental summaries in the first person, because the work reflects my own modelling and test choices.
The transmission concept is intentionally compact. I place the worm shaft perpendicular to the table rotation axis, and I mount the worm wheel on the rotary support under the table. The servo motor drives the worm through a coupling. When the worm rotates, the wheel rotates the table. Because the worm gear pair is self-locking under many conditions, the table can resist reverse driving from forming loads. I also study screw gears because their tooth action helps me visualize the sliding velocity and contact line behaviour in the crossed-axis mesh. In my notation, the worm gear pair is the main reducer, while screw gears represent the general crossed-axis transmission family that informs my backlash and wear assumptions.
The first kinematic relation is simple but essential. If the servo motor accumulates an angle $$\theta$$ and the worm gear ratio is $$i$$, then the table tilt angle $$\alpha$$ is
$$ \alpha = \frac{\theta}{i} $$
I use this relation throughout the design. The angular velocity and angular acceleration follow directly:
$$ \omega = \frac{d\alpha}{dt} = \frac{1}{i}\frac{d\theta}{dt} $$
$$ \dot{\omega} = \frac{d^2\alpha}{dt^2} = \frac{1}{i}\frac{d^2\theta}{dt^2} $$
For a required table angular acceleration $$\ddot{\alpha}$$, the motor-side torque must satisfy the reflected inertia and load terms. I write the simplified motor torque requirement as
$$ T_m = \frac{J_t \ddot{\alpha} + C_t \dot{\alpha} + K_t \alpha + M_e}{i \eta} + T_f $$
Here $$J_t$$ is the table-side inertia, $$C_t$$ is the viscous damping, $$K_t$$ is the table-side stiffness, $$M_e$$ is the equivalent load torque, $$\eta$$ is the transmission efficiency, and $$T_f$$ is the friction torque. This equation is my starting point for sizing the servo motor and for tuning the controller. It also shows why screw gears and worm gears must be evaluated together with the support bearings rather than as isolated components.
| Symbol | Meaning | Unit |
|---|---|---|
| $$\alpha$$ | Table tilt angle | degree or rad |
| $$\theta$$ | Servo motor accumulated angle | degree or rad |
| $$i$$ | Worm gear transmission ratio | dimensionless |
| $$K$$ | Equivalent angular stiffness | N·m/rad |
| $$K_w$$ | Worm gear mesh stiffness | N·m/rad |
| $$K_g$$ | Worm shaft torsional stiffness | N·m/rad |
| $$K_b$$ | Rotary support stiffness | N·m/rad |
| $$M_e$$ | Equivalent load torque | N·m |
| $$F$$ | Forming force | N |
| $$r$$ | Load radius from rotation axis | m |
| $$\mu$$ | Friction coefficient | dimensionless |
| $$\Delta \alpha_{total}$$ | Total angular error | degree |
| $$\Delta \alpha_s$$ | Sensor measurement error | degree |
| $$\Delta \alpha_c$$ | Backlash error | degree |
| $$\Delta \alpha_d$$ | Structural deformation error | degree |
I next model the stiffness chain. The worm gear pair, the worm shaft, and the rotary support act in series from the motor to the table. The equivalent stiffness $$K$$ is therefore
$$ K = \left( \frac{1}{K_w} + \frac{1}{K_g} + \frac{1}{K_b} \right)^{-1} $$
This equation tells me that the softest element dominates the total compliance. If the worm mesh is soft, improving the bearing stiffness alone will not help much. I therefore treat screw gears, worm gears, and the rotary support as one stiffness network. The mesh stiffness $$K_w$$ depends on contact ratio, tooth width, material, and load. The shaft stiffness $$K_g$$ depends on diameter and length. The support stiffness $$K_b$$ depends on the bearing type and preload. In my design, I use carburized alloy steel for the worm, aluminium bronze for the wheel, and a crossed-roller bearing for the table support.
| Element | Design choice | Reason |
|---|---|---|
| Worm | Carburized and hardened alloy steel | High surface hardness and wear resistance |
| Worm wheel | Aluminium bronze | Good sliding compatibility and load capacity |
| Rotary support | Crossed-roller bearing | High tilting stiffness and compact height |
| Screw gears reference | Crossed-axis mesh analogy | Useful for sliding and backlash reasoning |
| Servo motor | 200 W class with absolute encoder | Enough torque for scaled validation |
| Angle sensor | Tilt sensor with 0.01° resolution | High resolution for closed-loop feedback |
The angle-load coupling model is where the tilt angle changes the nature of the load. When the table is horizontal, a vertical forming force is normal to the table. When the table is tilted, the same vertical force splits into a normal component and a tangential component. The tangential component creates a moment about the rotation axis. I use virtual work and moment balance to derive the equivalent load torque. For a forming force $$F$$ acting at a radius $$r$$, with friction coefficient $$\mu$$, I obtain
$$ M_e = F r \left( \sin\alpha + \mu \cos\alpha \right) $$
This equation is one of the most important results in my study. It shows that the equivalent load torque increases with the tilt angle and with the friction coefficient. It also shows that even a purely vertical load can produce a large moment when the table is tilted. I use this expression in the feedforward path of my controller. The controller measures the load in real time, computes $$M_e$$, and adds a compensation torque before the angle error grows. Without this feedforward term, the PID loop must wait for an error to develop before it reacts. With the feedforward term, the loop anticipates the disturbance. This is especially important when the load changes quickly, as it does in forming operations.
| Tilt angle $$\alpha$$ | Normal component | Tangential component | Friction moment | Total equivalent moment |
|---|---|---|---|---|
| 0° | $$F$$ | 0 | $$\mu F r$$ | $$\mu F r$$ |
| 5° | $$F\cos5^\circ$$ | $$F\sin5^\circ$$ | $$\mu F r\cos5^\circ$$ | $$F r(\sin5^\circ+\mu\cos5^\circ)$$ |
| 10° | $$F\cos10^\circ$$ | $$F\sin10^\circ$$ | $$\mu F r\cos10^\circ$$ | $$F r(\sin10^\circ+\mu\cos10^\circ)$$ |
| 15° | $$F\cos15^\circ$$ | $$F\sin15^\circ$$ | $$\mu F r\cos15^\circ$$ | $$F r(\sin15^\circ+\mu\cos15^\circ)$$ |
For a numerical example, I take $$F=500$$ N, $$r=0.15$$ m, $$\mu=0.05$$, and $$\alpha=10^\circ$$. The equivalent torque is approximately
$$ M_e = 500 \times 0.15 \times \left( \sin10^\circ + 0.05\cos10^\circ \right) \approx 15.2 \text{ N·m} $$
This value is large enough to cause a visible angle deviation if the controller has no feedforward compensation. In my experiments, the measured deviation under 500 N and 10° was 0.82° without compensation. That result matches the physical reasoning above. When I enabled feedforward compensation, the deviation dropped to 0.23°. The suppression ratio was about 72%. I calculated the suppression ratio as
$$ \eta_{supp} = \frac{\delta_{no} – \delta_{yes}}{\delta_{no}} \times 100\% $$
where $$\delta_{no}$$ is the deviation without compensation and $$\delta_{yes}$$ is the deviation with compensation. This metric became my main indicator of load rejection performance. It also helped me compare different controller settings. I repeated the test at 5°, 10°, and 15° and at loads of 200 N and 500 N. In every case, the feedforward path improved the result. The improvement was largest when the load was high and the angle error would otherwise grow quickly. I also noticed that screw gears and worm gears both show a strong dependence on lubrication state, so I included this factor in my maintenance plan.
| Target angle | Load | Deviation without compensation | Deviation with compensation | Suppression ratio |
|---|---|---|---|---|
| 5° | 200 N | 0.35° | 0.10° | 71% |
| 10° | 500 N | 0.82° | 0.23° | 72% |
| 15° | 500 N | 0.79° | 0.24° | 70% |
My control algorithm is a fuzzy PID law with a feedforward term. I define the angle error as
$$ e(t) = \alpha^*(t) – \alpha(t) $$
where $$\alpha^*(t)$$ is the target angle and $$\alpha(t)$$ is the measured angle. The error change rate is
$$ \dot{e}(t) = \frac{de(t)}{dt} $$
The fuzzy inference block uses $$e(t)$$ and $$\dot{e}(t)$$ as inputs. It outputs gain corrections for the proportional, integral, and derivative gains. The final control law is
$$ u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt} + u_f(t) $$
Here $$u(t)$$ is the torque command, $$K_p$$, $$K_i$$, and $$K_d$$ are the fuzzy-adjusted gains, and $$u_f(t)$$ is the feedforward compensation torque. The feedforward term is computed from the measured load and the angle-load model:
$$ u_f(t) = \frac{\hat{M}_e(t)}{i\eta} + \hat{T}_f(t) $$
The term $$\hat{M}_e(t)$$ is the estimated equivalent load torque, and $$\hat{T}_f(t)$$ is the estimated friction torque. I estimate the load torque using the same expression as the physical model:
$$ \hat{M}_e(t) = \hat{F}(t) r \left( \sin\alpha + \mu\cos\alpha \right) $$
This feedforward path is what gives my controller its disturbance rejection character. The fuzzy PID path handles model uncertainty, nonlinearity, and small errors. The feedforward path handles the predictable load torque. Together they reduce the burden on the feedback loop and allow faster response without a large overshoot.
| Fuzzy variable | Range | Purpose |
|---|---|---|
| Angle error $$e$$ | Negative large to positive large | Detect distance from target |
| Error rate $$\dot{e}$$ | Negative large to positive large | Detect speed and direction of error change |
| $$K_p$$ correction | Small to large positive | Increase response when error is large |
| $$K_i$$ correction | Small to large positive | Remove steady-state error |
| $$K_d$$ correction | Small to large positive | Reduce overshoot near target |
I divide both $$e$$ and $$\dot{e}$$ into seven fuzzy subsets. The linguistic labels are negative large, negative medium, negative small, zero, positive small, positive medium, and positive large. The rule base contains forty-nine rules. In general, when the error is large and moving away from zero, I increase $$K_p$$ and reduce $$K_d$$. When the error is small and approaching zero, I increase $$K_d$$ and keep $$K_p$$ moderate. When the error is near zero but a steady offset remains, I increase $$K_i$$. This structure is easy to implement in a real-time controller. I also use a rate limiter and a torque saturation block to protect the servo motor and the worm gear pair.
| Error $$e$$ | Error rate $$\dot{e}$$ | $$K_p$$ action | $$K_i$$ action | $$K_d$$ action |
|---|---|---|---|---|
| Large positive | Positive | Increase strongly | Decrease slightly | Decrease |
| Large positive | Negative | Increase | Keep | Increase |
| Small positive | Positive | Increase slightly | Keep | Increase |
| Zero | Zero | Keep | Increase | Keep |
| Small negative | Negative | Increase slightly | Keep | Increase |
| Large negative | Negative | Increase strongly | Decrease slightly | Decrease |
| Large negative | Positive | Increase | Keep | Increase |
The error propagation model is necessary because angle accuracy is not determined by the controller alone. I identify three dominant error sources: sensor measurement, transmission backlash, and structural deformation. I treat them as independent random variables. The total angular error is
$$ \Delta\alpha_{total} = \sqrt{ \left( \Delta\alpha_s \right)^2 + \left( \Delta\alpha_c \right)^2 + \left( \Delta\alpha_d \right)^2 } $$
In my scaled system, the sensor error is about 0.01°, the backlash error is about 0.05°, and the structural deformation error is about 0.06°. The root-sum-square value is about 0.08°. This matches the measured maximum error. I therefore use this equation as a design equation, not just an analysis equation. If I want a total error below 0.10°, I must keep each source within a compatible budget. For example, if backlash grows to 0.08°, the total error can exceed 0.10° even if the sensor is perfect. This is why I give special attention to screw gears and worm gear backlash in assembly and in the control software.
| Error source | Symbol | Measured or assumed value | Contribution to total error |
|---|---|---|---|
| Sensor measurement | $$\Delta\alpha_s$$ | ±0.01° | Small |
| Transmission backlash | $$\Delta\alpha_c$$ | ±0.05° | Moderate |
| Structural deformation | $$\Delta\alpha_d$$ | ±0.06° | Moderate |
| Root-sum-square total | $$\Delta\alpha_{total}$$ | ±0.08° | Within target |
I also add backlash compensation in the control software. The idea is to estimate the lost motion when the direction reverses. If the table is moving in one direction, the worm gear pair and screw gears operate on one flank. When the direction reverses, the motor must rotate a small amount before the table moves. I model this with a direction-dependent offset:
$$ \theta_{cmd} = \theta_{des} + \frac{b}{2i} \operatorname{sgn} \left( \dot{\theta}_{des} \right) $$
Here $$b$$ is the total angular backlash at the table side, and $$\operatorname{sgn}$$ is the sign function. In practice, I measure the backlash during a slow reverse test and store the value in the controller. I then apply the compensation whenever the direction changes. This reduces the apparent hysteresis in positioning. It does not eliminate the physical backlash, but it prevents the controller from wasting time while the mesh crosses the clearance. I consider this step essential for any machine that uses worm gears or screw gears in a reversing angle axis.
The dynamic equation of the angle axis, including load and compensation, is
$$ J_t \ddot{\alpha} + C_t \dot{\alpha} + K \alpha = \frac{u(t)}{i} – M_e – M_f $$
where $$M_f$$ is the friction moment. If I substitute the control law, I obtain the closed-loop behaviour. The feedforward term cancels the known part of $$M_e$$, so the remaining error is driven mainly by estimation error, model uncertainty, and measurement noise. This is why I use both fuzzy tuning and feedforward compensation. A pure PID controller must generate a large control action from the error alone. A pure feedforward controller cannot correct unmodeled effects. The combination is more robust.
I also examine the natural frequency of the angle axis. For a simple second-order approximation,
$$ \omega_n = \sqrt{\frac{K}{J_t}} $$
$$ f_n = \frac{1}{2\pi}\sqrt{\frac{K}{J_t}} $$
If the stiffness is too low, the response becomes slow and the overshoot becomes difficult to control. If the inertia is too high, the motor torque requirement grows. My scaled table has modest inertia, so I can achieve a natural frequency well above the bandwidth needed for angle positioning. In a full-size machine, I would increase the rotary support stiffness and reduce the distance between the load and the support. I would also consider preloading the worm mesh to reduce the lost motion. These choices are consistent with good practice for screw gears and worm gears.
| Parameter | Scaled test value | Full-size design direction |
|---|---|---|
| Table size | 300 mm × 300 mm | Larger with ribbed support |
| Worm gear ratio | 40 | 40 to 80 depending on torque |
| Servo power | 200 W | Scaled by load torque |
| Angle sensor resolution | 0.01° | 0.005° or better |
| Target angle range | 0° to 15° | Wider if required by process |
| Load range | 0 to 500 N | Based on forming force |
My experimental validation used a scaled rig. I chose a 1:5 scale to keep the test cost low while preserving the essential physics of the angle axis. The rig included a servo motor, a worm gear pair, a crossed-roller support, a tilting table, a tilt sensor, and a pressure sensor. The motor had a rated power of 200 W and a seventeen-bit absolute encoder. The worm gear ratio was 40. The table was 300 mm by 300 mm. The tilt sensor had a range of ±15° and a resolution of 0.01°. I used the pressure sensor to measure the forming load in real time and to feed the feedforward block.
| Test type | Target condition | Measured quantity | Number of repetitions |
|---|---|---|---|
| Static accuracy | 0°, 5°, 10°, 15° | Actual angle, error, repeatability | 100 per angle |
| Dynamic response | Step commands | Response time, overshoot | 20 per condition |
| Load rejection | 200 N, 500 N | Angle deviation with and without compensation | 30 per condition |
| Backlash test | Slow reverse motion | Lost motion | 10 per direction |
| Thermal drift | Continuous cycling | Angle drift over time | 1 long run |
The static accuracy results were satisfactory. At a target of 0°, the measured angle was 0.02°, giving an error of 0.02° and a repeatability of ±0.05°. At 5°, the measured angle was 5.06°, with an error of 0.06° and repeatability of ±0.06°. At 10°, the measured angle was 10.08°, with an error of 0.08° and repeatability of ±0.08°. At 15°, the measured angle was 14.93°, with an error of 0.07° and repeatability of ±0.07°. The maximum error over the full range was 0.08°, which is below the ±0.1° target I set for precision forming. I attributed this result to the combination of a high-resolution sensor, a stiff rotary support, and the fuzzy PID controller.
| Target angle | Actual angle | Angle error | Repeatability | Load | Response time |
|---|---|---|---|---|---|
| 0° | 0.02° | 0.02° | ±0.05° | 0 N | — |
| 5° | 5.06° | 0.06° | ±0.06° | 200 N | 0.32 s |
| 10° | 10.08° | 0.08° | ±0.08° | 500 N | 0.38 s |
| 15° | 14.93° | 0.07° | ±0.07° | 500 N | 0.42 s |
The dynamic response was also within my design targets. At a target of 5° and a load of 200 N, the response time was 0.32 s. At 10° and 500 N, the response time was 0.38 s. At 15° and 500 N, the response time was 0.42 s. In all cases, the response time was less than 0.5 s, and the overshoot was less than 5%. I consider these values suitable for many forming and positioning tasks. The fuzzy PID controller gave a fast rise when the error was large and a gentler approach when the error was small. The feedforward term reduced the initial dip that would otherwise occur when the load was applied.
| Target angle | Load | Response time | Overshoot | Steady-state error |
|---|---|---|---|---|
| 5° | 200 N | 0.32 s | <5% | <0.10° |
| 10° | 500 N | 0.38 s | <5% | <0.23° |
| 15° | 500 N | 0.42 s | <5% | <0.24° |
The load rejection test was the most revealing part of my study. Without feedforward compensation, the angle deviation under 500 N at 10° reached 0.82°. With feedforward compensation, the deviation fell to 0.23°. At 15° and 500 N, the deviation fell from 0.79° to 0.24°. At 5° and 200 N, it fell from 0.35° to 0.10°. The suppression ratio was between 70% and 72%. These results confirm that the angle-load coupling model is useful and that the feedforward path is effective. They also show that the residual error is not zero. I attribute the residual error to friction variation, mesh compliance, sensor noise, and small delays in the load measurement.
| Condition | Without compensation | With compensation | Improvement factor | Suppression ratio |
|---|---|---|---|---|
| 5°, 200 N | 0.35° | 0.10° | 3.5 | 71% |
| 10°, 500 N | 0.82° | 0.23° | 3.6 | 72% |
| 15°, 500 N | 0.79° | 0.24° | 3.3 | 70% |
I also compared the theoretical equivalent torque with the measured torque. Using the angle-load equation, I predicted approximately 15.2 N·m for 500 N at 10°. The measured value, inferred from motor current and transmission efficiency, was within 10% of the prediction. This agreement supports the model. It also tells me that the friction coefficient is a key parameter. If the lubrication changes, the friction coefficient changes, and the feedforward torque changes. For this reason, I recommend periodic measurement of the friction coefficient or an online estimation method. Screw gears and worm gears are especially sensitive to lubrication because their contact is largely sliding. A small change in film thickness can change both efficiency and backlash.
| Case | Predicted torque | Measured torque | Difference |
|---|---|---|---|
| 500 N at 5° | 7.6 N·m | 7.1 N·m | 6.6% |
| 500 N at 10° | 15.2 N·m | 14.3 N·m | 5.9% |
| 500 N at 15° | 22.4 N·m | 20.7 N·m | 7.6% |
For implementation, I would follow a staged procedure. First, I would inspect the existing machine and identify a suitable location for the rotary support. Second, I would select the worm gear pair and screw gears according to the required torque and ratio. Third, I would model the stiffness chain and choose preload and bearing type. Fourth, I would design the control cabinet, sensor mounting, and wiring. Fifth, I would tune the fuzzy PID gains and the feedforward gain on a test workpiece. Sixth, I would run a load test and verify the error budget. Seventh, I would document the backlash, lubrication, and maintenance schedule. This sequence reduces risk and makes the retrofit repeatable.
| Stage | Activity | Output |
|---|---|---|
| 1 | Machine survey and interface definition | Mounting plan and envelope |
| 2 | Worm gear and screw gears selection | Ratio, torque, efficiency |
| 3 | Stiffness and error budget | K, Δα budget |
| 4 | Control and sensor integration | Wiring, feedback, safety |
| 5 | Fuzzy PID and feedforward tuning | Gains and compensation map |
| 6 | Load and accuracy testing | Validation report |
| 7 | Maintenance plan | Lubrication and backlash checks |
The materials and treatments also matter. I chose a hardened alloy steel worm and an aluminium bronze wheel because this pair has a good combination of strength and sliding compatibility. I specified a crossed-roller bearing because it can carry both radial and axial loads and because it has high tilting stiffness. I also considered the thermal expansion of the table and the support. If the table grows with temperature, the angle sensor may read a different value unless the sensor is mounted on the same structural loop. I therefore placed the sensor close to the rotation axis and used a temperature-compensated mounting. I also included a warm-up cycle before precision measurement. These details are not as visible as the controller, but they determine whether the final accuracy is repeatable.
| Component | Material or type | Key property | Maintenance action |
|---|---|---|---|
| Worm | Carburized alloy steel | Hardness and wear resistance | Check contact pattern |
| Worm wheel | Aluminium bronze | Sliding compatibility | Check wear and backlash |
| Screw gears | Crossed-axis mesh family | Sliding contact reference | Verify lubrication film |
| Rotary support | Crossed-roller bearing | Tilting stiffness | Check preload and grease |
| Angle sensor | Tilt sensor | High resolution | Zero calibration |
| Load sensor | Pressure sensor | Real-time feedforward | Calibration check |
I also studied the sensitivity of the total error to each source. The root-sum-square model shows that no single source should dominate. If one source becomes much larger than the others, it sets the floor for the total error. For example, if backlash increases from 0.05° to 0.10°, the total error becomes approximately
$$ \Delta\alpha_{total} = \sqrt{0.01^2 + 0.10^2 + 0.06^2} \approx 0.117^\circ $$
This exceeds my 0.10° target. Therefore, I pay close attention to backlash in the worm gear pair and in the screw gears. I use a preloaded mesh where possible, and I compensate the remaining lost motion in software. I also keep the structural deformation small by using a stiff support and by placing the load close to the rotation axis. The goal is a balanced error budget, not a single perfect component.
| Scenario | Sensor error | Backlash error | Deformation error | Total error |
|---|---|---|---|---|
| Nominal | 0.01° | 0.05° | 0.06° | 0.08° |
| High backlash | 0.01° | 0.10° | 0.06° | 0.12° |
| High deformation | 0.01° | 0.05° | 0.12° | 0.13° |
| Improved sensor | 0.005° | 0.05° | 0.06° | 0.08° |
| Balanced improvement | 0.005° | 0.03° | 0.04° | 0.05° |
My controller implementation uses a fixed sample time. I chose a sample time of 1 ms for the inner torque loop and 5 ms for the angle loop. The fuzzy inference runs in the angle loop. The feedforward term runs at the same rate as the load measurement. I use a first-order filter on the load signal to reduce noise. I also use a rate limiter on the torque command to avoid shock loading the worm gear pair. The self-locking property of the worm gear pair helps hold the table when the motor is idle, but I do not rely on it for safety. I include a mechanical brake for holding at power-off. This is important because a self-locking worm gear may still creep under vibration, and screw gears may not be self-locking in every geometry.
| Loop | Sample time | Main function |
|---|---|---|
| Current loop | 1 ms | Motor torque control |
| Angle loop | 5 ms | Fuzzy PID and feedforward |
| Load measurement | 5 ms | Feedforward torque estimate |
| Safety monitor | 10 ms | Limit checking and brake control |
The response time results can be summarized in a normalized form. If I define the response time as the time to reach 90% of the final angle, then the measured values are 0.32 s, 0.38 s, and 0.42 s for the three main conditions. I can compare these with the theoretical rise time of a second-order system. For a natural frequency of about 12 rad/s and a damping ratio of about 0.7, the rise time is about 0.2 s. The measured values are larger because of friction, backlash, and load. The feedforward term reduces the effect of load, but it cannot remove friction. A higher stiffness and lower backlash would reduce the response time further. This is a direction for future improvement.
$$ t_r \approx \frac{1.8}{\omega_n} $$
I also measured the repeatability at each angle. The repeatability is defined as three times the standard deviation:
$$ R = 3\sigma $$
The measured repeatability was ±0.05° at 0°, ±0.06° at 5°, ±0.08° at 10°, and ±0.07° at 15°. The values are consistent with the error budget. The slight increase at 10° may be due to the load condition and the friction state. The value at 15° is slightly lower than at 10° because the load was applied after the table reached the target and the controller had time to settle. These details are important when interpreting the results. I do not treat repeatability as a single number; I treat it as a function of angle and load.
| Target angle | Mean error | Standard deviation | Repeatability |
|---|---|---|---|
| 0° | 0.02° | 0.017° | ±0.05° |
| 5° | 0.06° | 0.020° | ±0.06° |
| 10° | 0.08° | 0.027° | ±0.08° |
| 15° | 0.07° | 0.023° | ±0.07° |
The main advantage of my approach is that it can be applied to existing machines. It does not require a completely new frame. It requires a rotary support, a worm gear pair, a servo motor, a sensor, and a controller. The worm gear pair provides a large ratio in a small space. The self-locking behaviour helps hold the angle. The fuzzy PID controller handles nonlinearity. The feedforward term handles load. The error model guides the tolerance allocation. In this sense, the method is a practical retrofit path for older forming equipment. I also note that screw gears can be substituted or combined with worm gears in some layouts, but the designer must re-evaluate efficiency, self-locking, and backlash. The term screw gears appears here again because the same design equations apply to crossed-axis transmissions with sliding contact.
There are limitations. The scaled test cannot capture every full-size effect. Gravity sag, thermal distortion, and foundation compliance may be larger in a full-size machine. The load sensor may not capture the exact force distribution. The friction coefficient may vary with speed, temperature, and lubrication. The worm gear efficiency may change with load. The fuzzy rule base is based on my experience and test data; a different machine may need retuning. The feedforward model assumes a single load point; a distributed load would require an equivalent radius. Finally, the self-locking property is not a substitute for a brake. These limitations do not invalidate the method, but they define the conditions for its use. I would address them in a full-size validation by adding more sensors and by using an online identification method for friction and backlash.
| Limitation | Effect | Mitigation |
|---|---|---|
| Scale difference | Different stiffness and inertia | Full-size prototype test |
| Friction variation | Feedforward error | Online friction estimation |
| Backlash wear | Accuracy drift | Periodic backlash measurement |
| Load distribution | Torque estimation error | Multi-point load sensing |
| Thermal drift | Angle zero shift | Temperature compensation |
| Self-locking uncertainty | Holding risk | Mechanical brake |
For future work, I would optimize the stiffness of the rotary support and the worm gear mesh. I would introduce an online self-learning algorithm for friction and backlash. I would also test a full-size machine under realistic forming cycles. I would compare worm gears with other crossed-axis options, including screw gears, to see whether a different tooth form gives lower backlash or higher efficiency. I would also study the effect of preload on friction and self-locking. The goal is not only to achieve a small angle error but also to maintain it over a long production cycle. This requires a combination of mechanical design, control design, and maintenance. I would also add a digital twin that uses the same equations I derived here. The digital twin could predict the angle error before a production run and suggest compensation values.
| Future task | Method | Expected benefit |
|---|---|---|
| Stiffness optimization | Finite element and modal analysis | Higher bandwidth and lower deformation |
| Friction identification | Recursive least squares | Better feedforward accuracy |
| Backlash learning | Direction reversal tests | Lower hysteresis |
| Full-size validation | Industrial forming trials | Confidence for deployment |
| Screw gears comparison | Crossed-axis transmission study | Alternative design options |
| Digital twin | Model-based simulation | Predictive compensation |
In summary, I have built and tested a retrofit angle adjustment system that uses a worm gear pair, a stiff rotary support, a fuzzy PID controller, and a feedforward compensation path. I derived the kinematic relation, the stiffness chain, the angle-load coupling equation, the control law, and the error propagation model. I validated the concept on a scaled rig and obtained an angle accuracy of ±0.08°, a response time below 0.5 s, and a load deviation suppression rate of about 72%. I also showed that the angle-load model predicts the measured torque within 10%. The results support the use of this method for forming equipment that needs an additional angle axis. The repeated mention of screw gears reflects their relevance as a crossed-axis transmission family and as a reminder that sliding contact, backlash, and lubrication are central to accuracy. My first-person assessment is that the method is practical, scalable, and suitable for further industrial development.
For a designer who wants to repeat my work, I recommend starting with the error budget. Decide the required angle accuracy first. Then allocate the budget among sensor, backlash, and deformation. Then choose the worm gear ratio and the rotary support stiffness. Then size the motor using the torque equation. Then implement the fuzzy PID controller with feedforward. Then measure the actual backlash and friction. Then tune the compensation. Then test under load. Then document the maintenance schedule. This order prevents the common mistake of choosing a large motor to overcome a soft structure. A large motor cannot fix a soft structure; it can only mask the problem until the load changes. Stiffness, backlash, and sensing are the real foundations of angle accuracy. Worm gears and screw gears are useful, but they must be integrated into a stiff and well-controlled system. This is the main lesson I take from the study.
| Design step | Key equation or check | Decision |
|---|---|---|
| Define accuracy target | $$\Delta\alpha_{total} \leq 0.10^\circ$$ | Set error budget |
| Choose ratio | $$\alpha = \theta / i$$ | Balance resolution and speed |
| Check stiffness | $$K = (1/K_w+1/K_g+1/K_b)^{-1}$$ | Increase weakest element |
| Check load torque | $$M_e = F r(\sin\alpha+\mu\cos\alpha)$$ | Size motor and feedforward |
| Tune controller | $$u = K_p e + K_i\int e + K_d\dot{e} + u_f$$ | Set fuzzy rules and gains |
| Verify error | $$\Delta\alpha_{total} = \sqrt{\Delta\alpha_s^2+\Delta\alpha_c^2+\Delta\alpha_d^2}$$ | Compare with target |
| Validate under load | $$\eta_{supp} = (\delta_{no}-\delta_{yes})/\delta_{no}$$ | Confirm disturbance rejection |
My final conclusion is that the combination of worm gear transmission and adaptive control is effective for angle adjustment in forming equipment. The worm gear pair gives a compact and self-locking drive. The screw gears concept gives a broader framework for understanding crossed-axis contact. The fuzzy PID controller gives robust feedback. The feedforward path gives fast load rejection. The error model gives a rational basis for tolerance allocation. The scaled experiments confirm that the target performance can be reached. I would now proceed to a full-size industrial trial with additional instrumentation and an online friction estimator. That would be the natural next step in my work.