UAV Rotatable Antenna under Rotation Delay

I. INTRODUCTION

• Mechanical rotation consumes finite-slot transmission time, creating a tradeoff between communication improvement and the time remaining for data transmission.Existing RA studies generally optimize after reconfiguration without explicitly modeling this finite-slot time loss.Source: §I · p. 1
• The rotation-delay-aware model accounts for the maximum rotation angle under parallel antenna rotation and jointly designs RA orientations, transmit beamforming, and rotation budget.The objective is to maximize the minimum effective transmission rate under finite communication slots.Source: §I · p. 2
• For one user, the optimum has no-, partial-, or full-rotation regimes under a sufficient strict-concavity condition, with a second-order closed-form approximation.For multiple users, the problem separates into rotation-budget selection and fixed-budget max-min SINR optimization, with AG-ZF using analytical gradients and spherical-cap feasible updates.Source: §I · p. 2
• Numerical results report near-optimal effective-rate performance for the closed-form design and favorable performance across multi-user settings for AG-ZF.The paper also reports a computational advantage for AG-ZF in its theoretical complexity analysis.Source: §I · p. 2

II. SYSTEM MODEL

• The modeled system has one UAV with a planar array of directional RAs serving K users with independent data streams over quasi-static communication slots.The UAV follows a predetermined trajectory; the study jointly designs antenna orientations and transmit beamforming within a given slot.Source: §II · p. 3

A. Channel and Signal Model

• RA orientation changes directional antenna gains, while the received signal contains both the desired stream and multi-user interference.The model assumes narrowband free-space line-of-sight propagation and, for subsequent theoretical analysis, a far-field approximation.Source: A. Channel and Signal Model · pp. 3–4

B. Rotation Delay Model and Problem Formulation

• Parallel rotation suspends data transmission until reconfiguration ends, with delay set by the largest antenna rotation angle divided by angular speed.The common budget δ bounds every antenna's rotation, and δ/ω must fit within the slot duration.Source: B. Rotation Delay Model and Problem… · p. 4
• The objective maximizes the minimum effective transmission rate across users by jointly choosing beamforming, RA orientations, and the common rotation budget.Increasing the budget expands feasible orientations but reduces data-transmission time.Source: B. Rotation Delay Model and Problem… · p. 4
• The joint problem is nonconvex because beamforming and orientations determine SINRs while the rotation budget affects feasible orientations and transmission time.At the optimum, the budget equals the maximum actual antenna rotation angle.Source: B. Rotation Delay Model and Problem… · p. 4

III. SINGLE-USER SYSTEM

• For one user, the section studies how RA rotation gain trades off against mechanical rotation delay in the joint design.Source: §III · p. 4

A. Problem Reformulation

• The optimal single-user beamformer for given orientations is maximum-ratio transmission, maximizing SNR for the fixed channel.Source: A. Problem Reformulation · p. 4
• For a fixed rotation budget, each antenna's orientation can be optimized independently because the SNR is a sum of individual gain terms and the orientation constraints are independent.The resulting pointing problem has the same structure as the single-user RA pointing problem in.Source: A. Problem Reformulation · p. 5
• After all antennas are fully aligned, further rotation cannot improve maximum SNR but reduces time available for transmission, so larger budgets need not be considered.Source: A. Problem Reformulation · p. 5
• Once the fixed-budget beamformer and orientations are determined, the original joint design reduces to a one-dimensional optimization over the rotation budget.Source: A. Problem Reformulation · p. 5

B. Optimal Rotation Design Under the Far-Field Approximation

• For a given rotation budget, the far-field model simplifies each RA’s residual off-boresight angle to α − δ.The feasible upper bound is δ̄ = min{δmax, α}.Source: B. Optimal Rotation Design Under the… · p. 5
• Increasing the rotation budget improves instantaneous SNR by reducing residual off-boresight angle but leaves less time for data transmission.For an interior optimum, the two effects exactly offset.Source: B. Optimal Rotation Design Under the… · p. 5
• A sufficient condition in (30) makes the effective-rate objective strictly concave over the feasible interval, yielding a unique global optimum.For an interior solution, δ0 is the unique solution to (29).Source: B. Optimal Rotation Design Under the… · p. 6
• The unique optimum is no rotation, partial rotation, or full rotation, depending on whether the rate gain from reorientation offsets the added rotation delay.At an interior optimum, the gain and delay-induced loss exactly balance; the interior solution can be found by one-dimensional bisection.Source: B. Optimal Rotation Design Under the… · pp. 5–6

C. Closed-Form Solution

• The approximation uses x = α − δ, with feasible range α − δ̄ ≤ x ≤ α.The closed-form budget is obtained from this residual-angle solution.Source: C. Closed-Form Solution · p. 6
• A second-order Taylor expansion of the logarithmic term around x = 0 yields a closed-form rotation-budget approximation without numerically solving (29).Here x is the residual off-boresight angle after rotation.Source: C. Closed-Form Solution · p. 6
• When a = ωTs − α > 0, the approximate objective is strictly concave for x ≥ 0 and has a unique nonnegative stationary point.The variable x denotes the residual off-boresight angle after RA rotation.Source: C. Closed-Form Solution · p. 6

IV. MULTI-USER SYSTEM

• Because each RA orientation affects multiple users, orientations cannot be optimized independently for a fixed budget.The orientations are coupled with transmit beamforming through the multi-user channels and interference.Source: §IV · p. 6
• For a fixed rotation budget, the design maximizes the minimum SINR, then optimizes over the budget; the fixed-budget feasible sets expand with δ, so the maximum minimum SINR is nondecreasing.The orientation and beamforming variables remain strongly coupled in the unrestricted fixed-budget problem.Source: §IV · p. 6

A. ZF-Based RA Pointing Optimization

• Under feasible ZF with M ≥ K and rank(H) = K, beamforming eliminates multi-user interference, reducing the fixed-budget problem to RA pointing optimization.The common ZF SINR is maximized by minimizing Tr[DG−1(F)].Source: §IV · p. 6, A. ZF-Based RA Pointing Optimization · p. 7
• The ZF pointing objective penalizes small eigenvalues of the channel Gram matrix, so orientation should account for both channel strength and user-channel separability.Proposition 1 provides the Euclidean gradient with respect to each RA orientation vector when H(F) has full column rank.Source: A. ZF-Based RA Pointing Optimization · p. 7

B. Proposed AG-ZF Algorithm

• For each candidate budget, AG-ZF optimizes orientations by analytical-gradient updates with backtracking, then evaluates the rate using the actual maximum rotation angle.The outer search refines budgets around the actual angle of the current best solution.Source: B. Proposed AG-ZF Algorithm · p. 7
• For a fixed budget, backtracking ensures the objective does not increase after accepted updates, so its positive objective sequence converges to a finite value.Iterations stop when relative objective change falls below ϵAG or the maximum iteration count is reached.Source: B. Proposed AG-ZF Algorithm · p. 7
• The delay-aware alternating-optimization comparison counts all fixed-budget problems evaluated during the outer rotation search.Its per-budget approach alternately solves SOCP-based beamforming and RA-pointing subproblems.Source: B. Proposed AG-ZF Algorithm · pp. 7–8
• AG-ZF replaces repeated SOCP solves in DA-AO with analytical-gradient updates and structured matrix operations, reducing computational complexity.One channel evaluation and update has dominant complexity O(MK^2 + K^3).Source: B. Proposed AG-ZF Algorithm · p. 8

• Unless otherwise specified, the reported average results are obtained over 100 independent user-location realizations.Source: §V · p. 8
• RAs rotate independently and in parallel at a common angular speed of ω = 2π rad/s, comparable to the reported maximum yaw-tracking speed of a Trace-G1 gimbal.Source: §V · p. 8
• The selected rotation speed provides a relatively conservative evaluation of rotation delay because higher rotation speed reduces that delay.Source: §V · p. 8

A. Single-User Performance

• The closed-form budget δcf yields an effective rate very close to the optimum despite differing from the exact optimal budget δ⋆.The comparison is plotted against rotation budget for a representative single-user realization.Source: A. Single-User Performance · p. 9, Figure 2 · p. 9
• The closed-form design remains near-optimal across considered configurations, although its relative effective-rate loss increases with both RA count M and SNR.The CDFs compare relative loss as M varies and as SNR varies with M = 4.Source: A. Single-User Performance · p. 9, Figure 3 · p. 9
• The exact and closed-form designs achieve the highest average effective rates with a consistently small gap; delay-unaware rotation incurs a penalty, and fixed antennas can outperform it.Fixed Antenna overtakes the delay-unaware design at larger M and sufficiently high SNR.Source: A. Single-User Performance · p. 9
• As normalized full-range rotation-time ratio ρ increases, the optimum transitions from full rotation to partial rotation and then no rotation.When rotation is no longer beneficial, the effective rate levels off because the optimal design incurs no mechanical rotation delay.Source: A. Single-User Performance · p. 9

B. Multi-user Performance

• With two refinement stages, local and full-interval refinement incur average relative effective-rate losses of 0.300% and 0.272%, respectively.At five stages, the respective losses are 0.249% and 0.012%; local refinement with two stages is used in remaining simulations.Source: Figure 7 · p. 10, B. Multi-user Performance · p. 10
• As the number of transmit RAs M increases, AG-ZF exceeds Delay-Aware Alternating Optimization across the considered range, while Fixed Antenna has the lowest rate.All schemes improve with M, attributed to higher array gain and enhanced spatial separation.Source: B. Multi-user Performance · p. 10, Figure 8 · p. 11
• As the number of users K increases, average minimum effective transmission rates decrease for all three schemes, while AG-ZF remains highest.The passage attributes the decrease to the greater challenge of separating user channels.Source: B. Multi-user Performance · pp. 10–11, Figure 9 · p. 11
• At ρ = 0.8, AG-ZF achieves approximately twice the effective rate of delay-unaware optimization, which eventually falls below Fixed Antenna.At small ρ, AG-ZF and delay-unaware optimization perform similarly; Fixed Antenna maintains a constant rate because it incurs no rotation delay.Source: Figure 10 · p. 11, B. Multi-user Performance · p. 11

VI. CONCLUSION

• The conclusion reports near-optimal effective-rate performance for the single-user closed-form approximation and performance advantages for AG-ZF over the considered benchmarks.It describes the single-user no-, partial-, and full-rotation regimes under a sufficient strict-concavity condition, and the multi-user Gram-matrix-based formulation with max-min fair power allocation.Source: §VI · p. 11
• Future work may examine inter-slot RA reconfiguration under UAV mobility and account for rotation acceleration, settling time, mechanical energy, and imperfect channel state information.Source: §VI · p. 11

APPENDIX A PROOF OF THEOREM 1

• Under condition (30), the single-user effective-rate function is strictly concave over the feasible rotation interval, so the optimization problem has a unique global optimum.The proof also establishes the derivative signs that locate the optimum at an endpoint or at the unique interior stationary point.Source: Appendix A · p. 12

APPENDIX B PROOF OF THEOREM 2

• The second-order Taylor expansion of ψ(x) is ln(1 + C) − Cp/(1 + C)x^2 + O(x^4).Retaining terms through second order yields the approximate rate function used in the closed-form design.Source: Appendix B · p. 12
• The approximate rate function is strictly concave over the feasible interval, making its unique stationary point the unconstrained maximizer.The closed-form rotation budget uses that maximizer when it lies within the feasible interval and otherwise uses the corresponding boundary.Source: Appendix B · p. 12