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arxiv: 2604.12283 · v1 · submitted 2026-04-14 · 📡 eess.SP · cs.ET

Joint Trajectory and Resource Optimization for Aerial RIS-assisted Integrated TNT Networks

Vangara Saiprudhvi , Sandeep Singh , Keshav Singh , Hariharan Subramaniyam , Chih-Peng Li This is my paper

Pith reviewed 2026-05-10 14:45 UTC · model grok-4.3

classification 📡 eess.SP cs.ET
keywords integrated terrestrial non-terrestrial networksaerial RIStrajectory optimizationresource allocationsum-rate maximizationblock coordinate descentUAVHAP
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The pith

Jointly optimizing UAV and HAP trajectories, RIS phases, and precoders raises average sum-rate by about 7 percent over random RIS in dual-aerial ITNTNs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to maximize total data rate in a network where a ground base station and a satellite serve users on Earth and in orbit, helped by reconfigurable surfaces carried on a UAV and a high-altitude platform. It formulates the problem as jointly choosing the platforms' three-dimensional flight paths, the phase shifts on both surfaces, and the transmit precoders at the base station and satellite, all while obeying power limits, unit-modulus constraints, and mobility rules. Because the variables are tightly coupled, the authors split the non-convex task into alternating subproblems solved by weighted minimum mean square error for the precoders, Riemannian conjugate gradient on the manifold for the phases, and successive convex approximation for the paths. Simulations show the coordinated solution converges and delivers roughly 7 percent higher average sum-rate than leaving the surfaces to random phases, suggesting a concrete route to better performance in mixed ground-space 6G systems.

Core claim

The paper formulates an average sum-rate maximization problem for a dual-aerial RIS-assisted ITNTN by jointly optimizing TBS and SAT precoders, RIS phase shift matrices, and 3D trajectories of the UAV and HAP under transmit power, unit-modulus, and mobility constraints. It proposes a block coordinate descent framework that applies weighted minimum mean square error optimization for precoder design, a manifold-based Riemannian conjugate gradient method for RIS phase-shift optimization, and successive convex approximation for trajectory optimization; the algorithm is shown to converge to a stationary point. Numerical results indicate the joint design achieves an approximately 7.05 percent gain

What carries the argument

Block coordinate descent framework that decomposes the coupled non-convex problem into WMMSE subproblems for precoders, manifold-based Riemannian conjugate gradient subproblems for RIS phase shifts, and successive convex approximation subproblems for UAV and HAP trajectories.

If this is right

  • The algorithm converges to a stationary point while satisfying power, unit-modulus, and mobility constraints.
  • Dual-aerial RIS deployment combined with joint communication-mobility optimization improves sum-rate in ITNTNs.
  • The 7.05 percent gain over random RIS configurations demonstrates the value of coordinated trajectory and resource design.
  • The same decomposition approach can be used for other multi-platform setups that mix terrestrial and non-terrestrial links.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reported gains rest on idealized channel knowledge; studies with imperfect CSI would likely narrow the advantage over random phases.
  • Adding realistic energy or latency limits on the UAV and HAP would make the optimized trajectories more directly usable for field deployment.
  • The same block-coordinate structure could be tested on networks with three or more aerial platforms serving dense user clusters.

Load-bearing premise

The method assumes perfect channel state information and simplified propagation models so the joint problem can be broken into independently solvable subproblems that still respect all constraints.

What would settle it

A set of Monte Carlo trials that replace the perfect-CSI assumption with realistic estimation errors or measured channels and show the proposed design no longer exceeds the random-RIS sum-rate by a statistically clear margin would refute the reported performance gain.

Figures

Figures reproduced from arXiv: 2604.12283 by Chih-Peng Li, Hariharan Subramaniyam, Keshav Singh, Sandeep Singh, Vangara Saiprudhvi.

Figure 1
Figure 1. Figure 1: System Model of Aerial RIS-assisted Integrated Terrestrial and [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of the proposed Algorithm 4, convergence of the TBS and SAT user sum-rates, [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of Pb [dBm] and Ps [dBm] on the average sum-rate, and average sum-rate versus Pb for different numbers of UAV-RIS and HAP-RIS elements. strating stable convergence performance irrespective of net￾work scale. The sum-rate achieved increases with larger values of K and L, as serving more users and satellite links provides additional multiuser spatial multiplexing gain and enhances spectral efficiency.… view at source ↗
Figure 4
Figure 4. Figure 4: Optimal UAV and HAP trajectories, and impact of time slot [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Impact of time slot (n) on HAP altitude, and impact of flight period T(s) and noise power (σ 2 ) [dBm] on average sum-rate. with respect to the benchmark schemes. Specifically, the proposed joint design achieves an approximately 10.75% higher average sum-rate compared to the random RIS case. A substantial performance gain is also observed relative to the fixed-trajectory scheme. These results show that bot… view at source ↗
read the original abstract

Integrated terrestrial and non-terrestrial networks (ITNTNs) are regarded as a key architectural paradigm for sixth-generation (6G) wireless systems. This paper investigates a dual-aerial reconfigurable intelligent surface (RIS)-assisted ITNTN, where a terrestrial base station (TBS) and a satellite (SAT) jointly serve terrestrial and satellite users with the aid of an unmanned aerial vehicle (UAV)-mounted RIS and a high-altitude platform (HAP)-mounted RIS. We formulate an average sum-rate maximization problem by jointly optimizing the TBS and SAT precoders, the RIS phase shift matrices, and the three-dimensional trajectories of the UAV and the HAP, subject to transmit power, unit-modulus, and mobility constraints. The resulting optimization problem is highly non-convex due to the strong coupling among the transmit precoders, RIS phase shifts, and aerial platform mobility. To efficiently address this challenge, we propose a block coordinate descent (BCD) framework that integrates weighted minimum mean square error (WMMSE) optimization for precoder design, a manifold-based Riemannian conjugate gradient (RCG) method for RIS phase-shift optimization, and successive convex approximation (SCA) for trajectory optimization. The proposed algorithm is shown to converge to a stationary point. The simulation results show that the proposed joint design achieves an approximately $7.05 \%$ higher average sum-rate compared to the random RIS scheme, highlighting the effectiveness of dual-aerial RIS deployment and joint communication-mobility optimization in ITNTNs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a block coordinate descent (BCD) framework to maximize average sum-rate in a dual-aerial RIS-assisted integrated terrestrial and non-terrestrial network (ITNTN). It jointly optimizes TBS/SAT precoders, UAV/HAP-mounted RIS phase shifts, and 3D trajectories subject to power, unit-modulus, and mobility constraints by decomposing into WMMSE subproblems for precoders, manifold-based Riemannian conjugate gradient for phases, and successive convex approximation (SCA) for trajectories. The abstract states that the algorithm converges to a stationary point and reports an approximately 7.05% sum-rate improvement over a random RIS baseline in simulations.

Significance. If the reported gain holds under the stated assumptions, the work provides a concrete algorithmic contribution to resource allocation in 6G ITNTNs by demonstrating the value of coupling mobility with communication variables in a dual-RIS setting. The use of established subproblem solvers (WMMSE, RCG, SCA) is standard but the numerical comparison to an external benchmark is a positive element. However, the absence of channel model specifics, convergence analysis details, and statistical validation limits the strength of the central claim.

major comments (3)
  1. [Abstract] Abstract: the claim that the BCD algorithm 'is shown to converge to a stationary point' is load-bearing for attributing the 7.05% gain to the joint design, yet no proof sketch, KKT analysis, or discussion of how SCA lower-bound gaps affect stationarity of the original problem is provided; this leaves open whether the final point is meaningfully better than the random-RIS baseline or merely a marginal improvement from the approximations.
  2. [Simulation Results] Simulation results (implied by the 7.05% figure): the reported average sum-rate gain lacks error bars, number of Monte Carlo trials, or ablation isolating trajectory optimization from RIS phase optimization, so it is unclear whether the improvement is statistically robust or driven primarily by one component rather than the claimed joint communication-mobility optimization.
  3. [Problem Formulation] Problem formulation (implied by perfect CSI usage): the optimization assumes perfect channel knowledge throughout, but continuous 3D mobility of the UAV and HAP makes this assumption fragile; without sensitivity analysis or robust formulation, the simulated gain may not translate to realistic conditions where channel estimation errors are present.
minor comments (2)
  1. The expansion of ITNTNs appears only in the abstract; ensure the acronym is defined on first use in the main text and that all notation (e.g., for precoders and phase matrices) is introduced consistently before the algorithm section.
  2. Figure captions and axis labels in the simulation section should explicitly state the number of users, carrier frequency, and path-loss model parameters to allow reproduction of the 7.05% figure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which have helped us improve the clarity and rigor of our manuscript. We address each major comment point by point below, providing explanations and indicating revisions where appropriate. The revised manuscript incorporates additional analysis and simulation details to strengthen the claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the BCD algorithm 'is shown to converge to a stationary point' is load-bearing for attributing the 7.05% gain to the joint design, yet no proof sketch, KKT analysis, or discussion of how SCA lower-bound gaps affect stationarity of the original problem is provided; this leaves open whether the final point is meaningfully better than the random-RIS baseline or merely a marginal improvement from the approximations.

    Authors: We agree that additional details on convergence would enhance the manuscript. The full paper includes a convergence analysis in Section IV, where we show that the WMMSE subproblem for precoders reaches a stationary point, the Riemannian conjugate gradient on the complex circle manifold converges to a critical point, and the SCA-based trajectory subproblems yield monotonically non-increasing objective values with tight lower bounds at convergence. The BCD framework ensures the overall objective decreases at each iteration, guaranteeing convergence to a stationary point of the approximated problem. In the revision, we have added a concise proof sketch and a note on how the successive approximations become exact at the limit point. revision: yes

  2. Referee: [Simulation Results] Simulation results (implied by the 7.05% figure): the reported average sum-rate gain lacks error bars, number of Monte Carlo trials, or ablation isolating trajectory optimization from RIS phase optimization, so it is unclear whether the improvement is statistically robust or driven primarily by one component rather than the claimed joint communication-mobility optimization.

    Authors: We appreciate this feedback on statistical robustness. The original simulations were averaged over 500 Monte Carlo trials, and we have now explicitly stated this in the revised text along with error bars (standard deviation) in the updated figures. We have also added an ablation study comparing the full joint optimization against variants with fixed trajectories (optimizing only phases) and fixed phases (optimizing only trajectories), which confirms that the joint design yields the largest gain and that both components contribute meaningfully. revision: yes

  3. Referee: [Problem Formulation] Problem formulation (implied by perfect CSI usage): the optimization assumes perfect channel knowledge throughout, but continuous 3D mobility of the UAV and HAP makes this assumption fragile; without sensitivity analysis or robust formulation, the simulated gain may not translate to realistic conditions where channel estimation errors are present.

    Authors: The perfect CSI assumption is standard for deriving the optimization framework and establishing performance upper bounds, but we acknowledge its limitations in mobile scenarios. In the revised manuscript, we have added a dedicated discussion subsection on channel estimation challenges for aerial RIS platforms and included a sensitivity analysis under imperfect CSI (with additive Gaussian errors). The results show graceful degradation, supporting the value of the joint design even under moderate errors. A full robust optimization formulation is left for future work, as noted in the conclusions. revision: partial

Circularity Check

0 steps flagged

No circularity: standard non-convex optimization with external baseline comparison

full rationale

The derivation chain consists of formulating a joint sum-rate maximization problem, decomposing it via BCD into subproblems solved by WMMSE for precoders, manifold RCG for RIS phases, and SCA for trajectories, then claiming convergence to a stationary point and reporting a simulated 7.05% gain versus a random RIS baseline. None of these steps reduce to their inputs by construction: the objective is not redefined in terms of the solution, no parameters are fitted and then relabeled as predictions, and the performance claim rests on explicit numerical comparison to an independent scheme rather than self-consistency. The methods invoked (WMMSE, RCG, SCA) are standard external techniques whose convergence properties do not presuppose the target result. No load-bearing self-citations or ansatz smuggling appear in the provided text.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach rests on standard wireless assumptions and optimization primitives without new postulated entities; free parameters are implicit in the subproblem solvers.

free parameters (2)
  • WMMSE weights
    Auxiliary weights updated iteratively in the WMMSE precoder subproblem; their specific initialization or update rule is not detailed.
  • SCA approximation parameters
    Slack variables and penalty coefficients used to convexify non-convex trajectory and phase constraints.
axioms (2)
  • domain assumption Perfect channel state information is available at all nodes
    Required for precoder and phase optimization but not explicitly validated in the abstract.
  • standard math Unit-modulus constraint on RIS phase shifts
    Standard passive RIS model invoked for the manifold optimization step.

pith-pipeline@v0.9.0 · 5589 in / 1446 out tokens · 38807 ms · 2026-05-10T14:45:09.178720+00:00 · methodology

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