Radar antenna servos are commonly built as nested elevation-over-azimuth pedestals. The elevation axis, its torque motor, reducer, encoder, and cable harness are mounted on the rotating azimuth structure. This arrangement is mechanically direct and can be made stiff, but it imposes a large swept envelope and a high moving mass. It also forces the cable bundle to flex through the azimuth joint, which adds torque disturbance and reduces reliability. For a mobile export radar with strict size and weight limits, these penalties can dominate the design.
An alternative is to keep both drives and both position sensors on the stationary base and combine their motions through a differential. The two motor shafts are collinear. A spiral bevel gear pair converts the relative rotation of the two shafts into elevation of the antenna trunnion, while common rotation of both shafts produces azimuth. The arrangement is mechanically compact, but it makes the two axes kinematically coupled: the elevation motor must continuously follow the azimuth motion, and the output elevation is the difference between two motor angles.
Coaxial differential architecture
The platform uses two direct-drive AC torque motors, two absolute encoders, and two bearing sets mounted coaxially in a common base housing. The azimuth motor drives an azimuth adapter. The elevation motor drives an elevation cantilever that is coaxial with the adapter. The cantilever carries the driving spiral bevel pinion. The mating spiral bevel gear is fixed to the elevation output trunnion, which supports the antenna. The elevation trunnion axis is horizontal and intersects the vertical coaxial input axis at 90°.
The gears are Gleason-type spiral bevel gears. Their curved tooth trace produces gradual engagement and a localized contact pattern, which gives smoother motion and lower noise than a straight bevel pair. For a servo axis that must scan slowly and reversibly, this smoothness matters. The same geometry makes backlash and contact pattern sensitive to mounting distance and shimming, a point that returns in the error discussion.
Because the pinion is carried by the azimuth adapter, the gear pair has two inputs: rotation of the carrier and rotation of the pinion relative to the carrier. The output elevation is their difference. When both motors rotate at the same speed, the pinion and gear rotate together as a rigid body; the antenna changes azimuth without changing elevation. When the elevation motor turns relative to the azimuth motor, the pinion turns relative to the carrier and drives the gear, tilting the trunnion. The differential action is mechanical; no separate differential gearbox is added. Because the elevation drive no longer rides on the azimuth structure, the cable harness to the antenna follows a shorter and simpler flexure path.
Kinematics and command decoupling
Let α and β be the absolute angular positions of the azimuth and elevation motor shafts, z1 the number of teeth on the spiral bevel pinion, and z2 the number of teeth on the driven gear. The gear ratio is iG = z2/z1. Consider the two limiting cases. With the azimuth motor fixed, a pinion rotation θ1 produces an output rotation θ1/iG. With the elevation motor fixed, an azimuth rotation θ3 carries the pinion around the gear and produces an output rotation θ3/iG in the opposite sense. Cancelling that rotation requires the elevation motor to turn θ3 in the same direction as the azimuth motor. Superposition gives the output elevation:
$$ \theta_E = \frac{\beta-\alpha}{i_G} $$
Since the azimuth output is the carrier rotation, α = θA, and the command law is
$$ \beta = \theta_A + i_G \theta_E $$
The azimuth motor receives the azimuth command directly. The elevation motor receives the azimuth command plus the scaled elevation command. Velocity and acceleration commands follow by differentiation.
In the prototype, iG = z2/z1 = tan 35° = 0.7. The pair is speed-increasing: one output revolution requires 0.7 motor revolution. This reduces the motor speed demand but amplifies the effect of backlash and tooth-to-tooth errors on the output by 1/iG ≈ 1.43. It also means that during azimuth-only motion the elevation motor must turn one-to-one with the azimuth motor. The kinematic decoupling is exact, but the mechanical transmission and the two servo loops determine the residual error.
Acceleration and torque sizing
The velocity loop was designed for a small-signal bandwidth of 40 Hz at an amplitude of 0.5 °/s. For sinusoidal motion, the corresponding acceleration amplitude is
$$ \varepsilon_{bw}=A_m 2\pi f_{bw}=0.5\times 2\pi\times 40=125.66^\circ/\mathrm{s}^2 $$
The design acceleration was set to 150 °/s² to cover both bandwidth and response-time requirements.
Load torque includes the inertia torque and bearing friction:
$$ M_{acc}=J\varepsilon,\qquad M_f=\mu P\frac{d}{2} $$
The reflected inertia J was obtained from the 3D model, combining motor and load inertia. Bearing friction was evaluated at a 2g overload with a rolling friction coefficient μ = 0.0015 and the nominal bearing bore d. The inertia term dominates. The resulting load torques and the motor torque requirements, using an engineering factor of 2.0, are given in the table.
| Axis | Load torque (N·m) | Engineering factor | Required motor torque (N·m) |
|---|---|---|---|
| Azimuth | 0.185 | 2.0 | 0.370 |
| Pitch | 0.108 | 2.0 | 0.216 |
These are low torque values because the antenna and trunnion are lightweight and the direct-drive motors operate at low speed. The pitch axis requires less torque than the azimuth axis because its reflected inertia is lower. The low gear ratio favors motor speed but makes output accuracy sensitive to gear errors.
Three-loop servo and field-oriented current control
The servo uses a cascade of position, speed, and current loops. The position and speed loops use PID control; the current loop uses PI control. Speed and acceleration feedforward reduce dynamic lag and improve the system error-rate constant. The current loop is field-oriented. In the synchronous d-q frame, the d-axis current regulator sets flux and the q-axis regulator sets torque.
The current regulators include active damping, back-EMF and cross-coupling feedforward, and vector anti-windup. The feedforward terms cancel the speed-dependent coupling between the d and q axes, so the two current regulators behave as independent first-order loops. Active damping improves stability. The anti-windup uses the inverter output model and limits the voltage vector to a circle of radius Vdc/3 in the stator synchronous frame. This preserves phase accuracy during saturation instead of clipping the d and q axes independently. Seven-segment SVPWM is used. Because the motors operate at low speed and no field weakening is required, the modulator remains in its linear range.
With the motor stalled, ignoring the current-sampling filter delay, and choosing Kp = Laωc and Ki = Raωc, the closed-loop current response is
$$ \frac{i_a}{i_a^*}=\frac{\omega_c}{s+\omega_c}=\frac{1}{T_c s+1},\qquad T_c=\frac{1}{\omega_c} $$
The bandwidth equals ωc, with no overshoot and no steady-state error. The current-loop bandwidth was set to 500 Hz (ωc = 2π × 500 rad/s). Typical servo drives use 500 Hz to 2 kHz; small drives with low electrical time constants and high-frequency PWM can reach 3–5 kHz. The 500 Hz choice was matched to the motor electrical time constant.
Integrated dual-axis drive electronics
The two axes are controlled by one DSP+FPGA board rather than two independent drives. The DSP runs host communication, kinematic decoupling, and the three-loop algorithms. The FPGA handles signal routing and acquisition. Cross-axis data stay inside the DSP, which removes the communication delay that would otherwise appear between the azimuth and elevation command paths. This matters because the elevation command contains the azimuth command: a delay between axes would produce a transient elevation error during azimuth maneuvers.
| Parameter | Value |
|---|---|
| Supply voltage | 18–36 V |
| Rated current | 6 A |
| Peak current | 12 A |
| Motor axes | 2 |
| Communication interfaces | RS422/485, CAN, PWM |
| Feedback inputs | Resolver, SSI, BiSS-C, Hall |
| Mass | 0.25 kg |
| Dimensions | 120 × 80 × 16 mm |
These ratings cover the two low-power torque motors, and the 0.25 kg package fits inside the base envelope. The controller also monitors communication, encoder, position, speed, current, overcurrent, open-circuit, overvoltage, undervoltage, and overtemperature faults.
Encoder interface and state estimation
Each axis uses an absolute encoder with an SSI interface. The SSI link runs at 2 Mbit/s with a 10 kHz update rate. The 24-bit frame is organized as 22-bit position data (D23–D2), one parity bit (D1), and one alarm bit (D0). If the encoder resolution is below 22 bits, the unused most significant bits are zero; when the alarm bit is set, the position field is undefined.
SSI was chosen for its high data rate, low latency, and noise immunity over the cable run. The encoder resolution is moderate, so the controller uses a PLL full-order observer to filter position and estimate speed and acceleration. The observer suppresses quantization noise that would otherwise be amplified by differentiation and provides smooth feedback for the 40 Hz speed loop. The same observer supplies the acceleration signal used for feedforward.
Static error budget
The static angle-measurement uncertainty is the root-sum-square of three terms: encoder accuracy ≤40″, quantization 1.236″ (half an LSB), and machining and assembly misalignment 5″. This gives
$$ \Delta_{m-abs}=\sqrt{40^2+1.236^2+5^2}=40.33\ \mathrm{arcsec} $$
The estimated position repeatability is 4.944″ from four resolution units. Because this is much smaller than the encoder term, the absolute pointing accuracy is dominated by the encoder and is approximately 40.33″ (0.0112°).
These are static estimates. The measured errors below include dynamic effects, backlash, structural deflection, and test-fixture loading.
Measured pointing accuracy
Pointing was measured with a laser precision angle platform. One axis was locked while the other was commanded to a sequence of setpoints. A photoelectric target ball on the moving load was tracked by the laser system, and the commanded and measured angles were compared. Azimuth was tested from −50° to +50° in 5° steps. Although the azimuth axis can travel ±90°, the target-ball mounting limited the valid measurement to ±50°. Pitch was tested from −90° to +90° in 5° steps.
| Axis | Test range | Max positive error | Max negative error | Max absolute error |
|---|---|---|---|---|
| Azimuth | −50° to +50° | +0.07° | −0.09° | 0.09° |
| Pitch | −90° to +90° | +0.15° | −0.16° | 0.16° |
Azimuth stayed within 0.09°, below the 0.1° pointing requirement. Pitch reached 0.16°, exceeding 0.1° but remaining within the prototype’s functional acceptance. The pitch error varied with angle and showed local extrema near −75°, −80°, and +90°, which is consistent with angle-dependent gear and backlash effects rather than a simple scale-factor error.
Interpretation and design limits
Three mechanisms explain the larger pitch error. First, the pitch train contains the spiral bevel pair and is speed-increasing. Backlash and tooth-to-tooth errors appear at the output amplified by 1/iG ≈ 1.43, and the contact pattern changes with load direction. Second, the output pitch angle is the difference of two motor angles. Encoder errors, servo following errors, and structural compliance on either motor axis add at the output; the azimuth axis is no longer independent of pitch. Third, the lightweight structure deflects under the eccentric moment of the test target mounted at the pitch-axis rim. This fixture-induced moment is not present in the same way during normal antenna operation, but it shows how sensitive the open, weight-optimized structure is to off-axis loads.
Azimuth error is smaller because the azimuth output is the common rotation of the whole assembly, not a differential. Gear errors in the bevel pair do not directly enter the azimuth channel; they affect elevation only. The azimuth encoder and motor errors still pass to the output, but there is no 1/iG amplification.
The coaxial differential is attractive when antenna envelope and mass dominate the design. It removes the elevation drive from the rotating structure, shortens the cable path, and allows both motors and sensors to sit on the base. The penalty is a coupled kinematic law, a speed-increasing gear stage, and error summation between axes. For medium-accuracy scanning and point-to-point pointing, the prototype’s 0.09° azimuth and 0.16° pitch errors place it in the 0.1°–0.2° class. For high-accuracy tracking, the design would need anti-backlash preload, higher gear accuracy, a stiffer trunnion, and model-based compensation of backlash and compliance. The differential command law itself is exact; the residual error comes from the mechanical transmission and the two-axis servo response, not from the kinematic decoupling. The spiral bevel gear pair is the enabling element: it turns two collinear inputs into one elevation output without a heavy orthogonal gearbox.