Energy-Constrained Trajectory Planning Framework

• The paper presents a family of formulations that integrate energy as an optimization variable within dynamic trajectory planning, yielding significant efficiency gains.
• Energy-constrained trajectory planning is a framework that synthesizes feasible trajectories by jointly considering geometric, dynamic, and energy limitations across various applications.
• Algorithmic realizations range from learning-based residual correction and sequential convex programming to receding-horizon MPC, addressing practical demands in UAVs, vehicles, and robotics.

Energy-constrained trajectory planning frameworks treat energy or power as an explicit planning variable, either as an objective to be minimized, a budget to be respected, or a trade-off term coupled to communication, safety, latency, or task completion. In the literature represented here, the framework is not a single algorithmic template but a family of formulations spanning unmanned aerial vehicles, connected and automated vehicles, manipulators, marine vehicles, coverage robots, swarms, and planetary rovers. What unifies these formulations is the replacement of purely geometric path generation by trajectory synthesis under dynamics, actuation, sensing, and mission constraints, with energy feasibility embedded directly in the optimization or learning architecture (Zhu et al., 2021, Licea et al., 2020, Hu et al., 18 Sep 2025).

1. Problem class and scope

The core planning problem is to generate a state- and control-feasible trajectory that satisfies terminal or task requirements while minimizing a cost related to energy use. Depending on the application, that trajectory may include only motion variables, or it may jointly encode communication decisions, cluster-head selection, route assignment, timing, task offloading, or agent-to-goal assignment. A recurrent pattern is that trajectory planning is inseparable from another combinatorial or cyber-physical decision layer: UAV visiting order and cluster-head choice in wireless sensor networks, route recovery and intersection scheduling in mobility systems, or task assignment in UAV-assisted fog computing (Zhu et al., 2021, Bang et al., 2023, Liu et al., 2024).

The literature uses energy in several distinct senses. In some works, energy is a scalar objective over the whole mission, such as the integral of squared control or acceleration. In others, it is an application-specific physical quantity, such as electrical energy in manipulators, fuel consumption in road vehicles, or hybrid RTG-solar power compliance in planetary rovers. A further class couples energy to auxiliary mission criteria, including cumulative data throughput, wireless channel quality, latency, or safety margin, so that the planner must explicitly resolve competing objectives rather than optimize a single proxy (Dona' et al., 16 Jan 2025, Tian et al., 2024, Pradhan et al., 24 May 2026).

This scope also clarifies a common ambiguity in the phrase “energy-constrained.” In the surveyed work, it may denote minimization of a weighted energy term, satisfaction of cumulative energy availability, enforcement of instantaneous power limits, or maintenance of sufficient reserve for retreat and recharge. The online coverage algorithm $\epsilon^*$+ exemplifies the last interpretation by monitoring remaining energy continuously and retreating to the charging station before battery exhaustion, then resuming from a nearby unexplored waypoint rather than the previous retreat point (Shen et al., 2020).

2. Energy objectives and physical models

The most direct formulations optimize a mission-level energy functional under dynamics. For UAV-aided wireless sensor networks, the objective is a weighted sum of ground-network and UAV energy over one complete data-collection round,

$$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$

with $E_{\text{ground}}$ aggregating sensor-to-CH and CH-to-UAV communications and $E_{\text{UAV}}$ aggregating flight and hovering/communication energy (Zhu et al., 2021). For communication-aware drone planning with limited channel knowledge, the global cost combines control effort and expected transmitted data through a weight $\lambda$,

$$\mathcal{J}(\bar{\mathbf{u}},\lambda,t_f)=\frac{\lambda}{E_0}\int_0^{t_f}\|\bar{\mathbf{u}}(t)\|_2^2dt+(1-\lambda)\,w\!\left(\mathcal{D}_{\lfloor t_f/T\rfloor}\right),$$

thereby coupling mechanical energy and communication performance under only first-order channel statistics (Licea et al., 2020).

Manipulator planning introduces more explicitly electromechanical models. A fixed-time, point-to-point optimal control formulation minimizes

$$\mathcal{E}=\int_0^{t_f}\mathcal{P}(x(t),u(t))\,dt, \qquad \mathcal{P}=u^\top Ru+u^\top Nx,$$

where the state is $(q,v)$ and the control is motor torque $\tau$ (Dona' et al., 16 Jan 2025). By contrast, several transportation papers adopt energy surrogates tied to acceleration or fuel-rate models. The CAV intersection framework minimizes $\frac{1}{2}\int u_n^2(t)\,dt$ for each vehicle, using squared acceleration as the energy-associated objective within a double-integrator model (Bang et al., 2023). For autonomous driving, the differentiable fuel model is fitted offline as

$$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$0

so that the online nonlinear program directly minimizes predicted fuel consumption rather than a model-agnostic smoothness proxy (Tian et al., 2024).

Energy models become still more explicit when supply constraints are part of the mission. In planetary rover planning under hybrid RTG-solar power,

$$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$1

and the planner accounts for translational, rotational, and resistive motion power together with baseline subsystem loads (Hu et al., 18 Sep 2025). A closely related UAV-assisted fog-computing framework decomposes system energy into UAV movement, transmission energy, and computation energy, and couples that energy to a latency term in a joint system objective (Liu et al., 2024).

These models show that “energy” is not a single universal quantity across the field. It may represent thrust and torque effort, electrical losses, fuel rate, communication-induced expenditure, or mission power feasibility. A plausible implication is that cross-domain transfer of algorithms is easier than transfer of cost models: many solvers recur, but the meaning of the optimized energy is strongly application-dependent.

3. Constraints, feasibility, and trajectory structure

Energy-constrained planners are defined as much by their constraints as by their objectives. In UAV-WSN data collection, the problem is a constrained combinatorial optimization in which one cluster head must be selected in each cluster and the UAV must visit each selected CH exactly once, without sub-tours, in a round trip from and back to base (Zhu et al., 2021). In communication-constrained urban UAV planning, the free-final-time optimal control problem includes full 12-state nonlinear rigid-body dynamics, actuator bounds, obstacle avoidance, and a cumulative throughput requirement

$$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$2

which binds mission duration and geometric path to data-relay performance (Pradhan et al., 24 May 2026).

Safety and actuation constraints are equally central in ground and manipulation systems. The E-APF manipulator framework imposes joint velocity limits $$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$3 rad/s and acceleration limits $$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$4 rad/s$$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$5, while its trajectory optimizer jointly minimizes integrated joint jerk and execution time under those limits (Uppal et al., 10 Aug 2025). In the hierarchical CAV framework, longitudinal dynamics are coupled with rear-end safety constraints $$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$6 and lateral temporal-clearance constraints $$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$7 at conflict points, producing constrained and unconstrained trajectory regimes at intersections (Bang et al., 2023). Marine vehicle planning adds hard obstacle-clearance constraints to an EMPC problem that already contains actuator and state limits, while using deviation from a maximum-safety-margin reference as the operational safety measure (Liang et al., 2021).

Several papers analyze the geometry of feasible optimal trajectories rather than only their numerical solution. In polygonal environments with double-integrator dynamics, energy-optimal motion is characterized by unconstrained arcs joined at instantaneous contacts with obstacle vertices, allowing path planning to be reduced to an integer program over sequences of active constraints (Beaver et al., 2022). In swarm motion planning with integrator dynamics, continuity of the energy-optimal control input at a transition between unconstrained and constrained arcs is shown to hold if and only if the right-hand derivative of the tangency vector vanishes at the junction. The same work formulates barycentric motion and collision avoidance as explicit path constraints and derives analytical conditions for gluing constrained and unconstrained arcs (Beaver et al., 2020).

These constraint structures matter because they determine whether the energy problem is a smooth optimal control problem, a combinatorial sequencing problem, a hybrid system, or a path-constrained variational problem. This suggests that “trajectory planning” in the energy-constrained literature is often shorthand for a broader feasibility architecture in which geometry, timing, and resource admissibility are co-designed.

4. Algorithmic realizations

The algorithmic landscape is correspondingly heterogeneous. One prominent line combines structured combinatorial search with learning. The Ptr-A* framework models UAV visiting order as a sequence-to-sequence prediction task with a pointer network, while A* selects the cluster head in each visited cluster for minimum cumulative energy under the proposed order. The parameters are trained by actor-critic reinforcement learning in an unsupervised manner, and inference can be performed by greedy decoding, sampling, or active search (Zhu et al., 2021). A different learning-based strategy, constraint-informed residual learning, learns only the residual that maps a standard boundary-condition-satisfying trajectory to a near minimum-energy trajectory. Hard endpoint constraints are embedded through a scaling function such as $$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$8, yielding fast online generation from offline OCP solutions (Dona' et al., 16 Jan 2025).

Classical optimization remains dominant where model fidelity is high. Communication-aware UAV planning with limited channel knowledge uses quantization of expected bit rate, top-down region decomposition, minimum-norm control between waypoints, and simulated annealing over boundary points and segment times (Licea et al., 2020). For full rigid-body urban quadrotor planning, the nonconvex free-final-time problem is solved by sequential convex programming with time scaling, linearization about reference trajectories, trust regions, and virtual controls (Pradhan et al., 24 May 2026). The planetary rover framework also uses sequential convex programming with an augmented Lagrangian and allows insertion of zero-velocity wait segments when energy must be accumulated before motion can continue (Hu et al., 18 Sep 2025).

Receding-horizon architectures appear when online responsiveness is critical. The marine EMPC planner first generates a feasible reference trajectory with the most safety margin and then solves a receding-horizon economic MPC problem whose stage cost contains both practical consumed energy and deviation from the safe reference (Liang et al., 2021). EHMPP for autonomous electric vehicles adopts a hybrid DP+QP structure: dynamic programming produces a coarse collision-free path and convex feasible region, and quadratic programming refines path and speed with phase-specific objectives for acceleration, cruising, and deceleration, explicitly targeting optimal power and passive recovery (Ding et al., 2024). In autonomous driving with traffic and slope prediction, a direct nonlinear program is solved online with IPOPT/CasADI under hard ACC constraints and a differentiable fuel model (Tian et al., 2024).

A separate strand reduces online optimization burden by offline precomputation. The optimization-free approximation framework for CAV eco-trajectory planning constructs an offline batch of fuel-optimal trajectories indexed by initial speed and travel time, then performs online selection, translation, feasibility checking, and truncation/smoothing without solving a new optimization problem in real time (Lei et al., 2023). At a more abstract level, Planning as Descent learns a goal-conditioned energy function over latent trajectories and performs planning by gradient-based refinement in that learned energy landscape, using the same computation during training and inference (García et al., 19 Dec 2025).

5. Representative application families

The breadth of application domains has produced several recurring coupling patterns. The following table summarizes representative families in the surveyed literature.

Application family	Energy-coupled planning variables	Representative papers
UAV communications and sensing	CH selection, visiting order, channel quality, throughput, mission time	(Zhu et al., 2021, Licea et al., 2020, Pradhan et al., 24 May 2026)
Road mobility systems	Route recovery, intersection timing, fuel rate, regenerative recovery, slope and traffic prediction	(Bang et al., 2023, Tian et al., 2024, Ding et al., 2024, Lei et al., 2023)
Manipulation and high-DoF robotics	Jerk, execution time, velocity/acceleration limits, residual correction to OCP solutions	(Uppal et al., 10 Aug 2025, Dona' et al., 16 Jan 2025, Hussain et al., 13 Mar 2025)
Swarms and multi-agent systems	Assignment, local priorities, barycentric constraints, pairwise safety	(Beaver et al., 2019, Beaver et al., 2020)
Marine, coverage, and planetary missions	Safety margin, recharge retreat, hybrid power availability, terrain risk	(Liang et al., 2021, Shen et al., 2020, Hu et al., 18 Sep 2025)
Networked cyber-physical systems	Attitude, trajectory, task execution location, delay-energy trade-off	(Liu et al., 2024)

Within UAV and communication-centric applications, energy is inseparable from information transfer. UAV-aided WSN collection optimizes cluster-head selection and UAV order jointly because choosing a CH that reduces ground communication energy can increase flight cost, and vice versa (Zhu et al., 2021). Limited-channel-knowledge planning likewise makes path and velocity profile depend on expected bit rate rather than deterministic SNR maps (Licea et al., 2020). Urban relay missions extend this idea to cumulative throughput constraints under full rigid-body dynamics and obstacle-rich environments (Pradhan et al., 24 May 2026).

In mobility systems, the planning stack often becomes hierarchical. The integrated CAV framework separates network-level flow optimization, heuristic route recovery, and local energy-optimal intersection coordination, thereby linking macroscopic travel-time objectives to microscopic feasibility (Bang et al., 2023). Road-vehicle energy planning also divides between explicit model-aware nonlinear optimization under slope and traffic forecasts (Tian et al., 2024), hybrid MPC structures designed to maximize regenerative effects (Ding et al., 2024), and offline-online schemes intended for limited onboard compute (Lei et al., 2023).

Manipulation and high-DoF robotics display a different emphasis. E-APF addresses collision avoidance and local-minima behavior by combining position- and velocity-dependent potentials and then refining the resulting path with a hybrid jerk-and-time optimizer (Uppal et al., 10 Aug 2025). The high-DoF sinusoidal-and-velocity-scaling framework emphasizes energy consumption, motion smoothness, and trajectory accuracy in simulation (Hussain et al., 13 Mar 2025). Constraint-informed residual learning occupies a middle ground between model-based optimal control and data-driven approximation, retaining hard boundary conditions while accelerating online inference (Dona' et al., 16 Jan 2025).

Multi-agent and long-duration autonomy add resource logistics to the trajectory problem. The decentralized assignment-and-trajectory framework for teams of autonomous agents uses a binary program for local assignment, a dynamic priority ordering, and constrained energy-optimal trajectories under safety constraints, with guarantees of convergence to a unique assignment (Beaver et al., 2019). The $$E=\omega \cdot E_{\text{ground}}+(1-\omega)\cdot E_{\text{UAV}}, \qquad 0\leq \omega \leq 1,$$9+ coverage algorithm operationalizes energy reserve as a retreat condition and recharge policy (Shen et al., 2020). Planetary rover planning goes further by enforcing instantaneous power compliance under hybrid RTG-solar supply rather than treating energy only as a terminal budget (Hu et al., 18 Sep 2025).

6. Empirical performance, recurring misconceptions, and research directions

The reported empirical results indicate that explicit energy modeling can materially change both trajectory shape and solution quality. In UAV-WSN collection, models trained on 20-cluster and 40-cluster instances generalize to different numbers of clusters without retraining, and the DRL-active-search variant reduces energy by 17–33% versus a genetic algorithm and 8–21% versus nearest neighbor at 50 clusters (Zhu et al., 2021). In manipulator planning, constraint-informed residual learning achieves 87.3% of OCP savings near the training dataset and 50.8% far from the dataset, while being two to three orders of magnitude faster than the OCP-based planner (Dona' et al., 16 Jan 2025). In model-aware autonomous driving, explicit energy modeling improves fuel efficiency by 3.71% for sedans and 7.15% for diesel trucks relative to model-agnostic quadratic programming (Tian et al., 2024).

A separate set of results concerns power-aware feasibility rather than pure optimality. EHMPP increases passive recovery energy by 11.74% while tracking motor speed and acceleration at optimal power in simulation (Ding et al., 2024). For lunar-like rover terrain, the proposed power-aware planner keeps peak planned power within 0.55 percent of the prescribed limit, whereas existing methods exceed limits by over 17 percent (Hu et al., 18 Sep 2025). In polygonal environments, graph-based energy-optimal planning outperforms both RRT* and PRM in energy cost and computational time by up to an order of magnitude, while yielding implementable Crazyflie trajectories (Beaver et al., 2022). In learned latent-space planning, PaD reaches 95% success on narrow expert demonstrations, whereas prior methods peak at 68%, and training on noisy, suboptimal data further improves success and plan efficiency (García et al., 19 Dec 2025).

The literature also corrects several recurrent simplifications. First, shortest or smoothest path planning is not equivalent to energy-optimal planning: the graph-based polygonal-environment study matches RRT* and PRM in path length while outperforming both in energy cost (Beaver et al., 2022). Second, model-agnostic surrogates such as acceleration or jerk penalties are not identical to explicit fuel or electrical models, as shown by the fuel-rate-based autonomous driving framework (Tian et al., 2024). Third, position-only obstacle fields can induce local minima and oscillatory motion; the E-APF framework responds by adding velocity-dependent terms and a hybrid optimizer (Uppal et al., 10 Aug 2025). Fourth, planning under average or cumulative energy alone may be insufficient in settings with hard power envelopes, a point made particularly sharply by the rover framework with instantaneous power constraints (Hu et al., 18 Sep 2025).

Current research directions, as indicated within these works, point toward tighter coupling of optimization layers and stronger runtime guarantees. Several papers explicitly move from path-only to joint path-and-velocity planning, from motion-only to motion-communication co-design, or from single-vehicle planning to routing-and-trajectory hierarchies (Licea et al., 2020, Bang et al., 2023). Others aim to reconcile real-time deployment with high-fidelity energy models through approximation, hybrid MPC, or learned residual corrections (Lei et al., 2023, Ding et al., 2024, Dona' et al., 16 Jan 2025). A plausible implication is that the next generation of energy-constrained trajectory planning frameworks will be judged less by nominal cost reduction alone than by how well they unify physical realism, computational tractability, and hard feasibility under communication, safety, and power constraints.