Energy-Based Trajectory Objectives

• Energy-Based Trajectory Objectives are criteria that incorporate physical energy metrics, such as torque, power, and battery depletion, into trajectory optimization to enforce efficiency and feasibility.
• They integrate energy terms as costs, constraints, or shaping mechanisms using methods like L2 surrogates, physics-based power models, and hybrid constraints to balance smoothness with performance.
• Applications span robotics, autonomous vehicles, and multi-agent systems, utilizing optimization and learning techniques such as direct collocation, reinforcement learning, and inverse optimal control for energy-aware performance.

An energy-based trajectory objective is any criterion for trajectory generation, prediction, tracking, or control that directly incorporates physical energy or a related work-functional as a cost, constraint, or shaping mechanism. Such objectives arise throughout robotics, autonomous vehicles, multi-agent systems, optimal control, reinforcement learning, and imitation learning. They encode performance, resource efficiency, or physical feasibility by penalizing cumulative expended energy (torque, power, battery depletion) or by explicitly shaping the energetic behavior of a system according to hardware, environmental, or task-specific requirements.

1. Mathematical Formulations of Energy-Based Trajectory Objectives

Energy-based objectives are instantiated as integrals or sums of physically-motivated quantities over trajectories. Canonical forms include:

• Mechanical/Electrical Energy Cost: For actuated systems, minimize

$$E_{\text{mech}} = \int_0^T \sum_{i} \tau_i(t)\,\dot{q}_i(t)\,dt \quad \text{or} \quad \sum_{n=1}^N P_{\mathrm{prop}}[n] \delta t$$

where $\tau_i$ are joint torques, $\dot{q}_i$ joint velocities, or $P_{\mathrm{prop}}[n]$ instantaneous propulsion power at discrete time $n$ (Ji et al., 2023, Sun et al., 2019, Hu et al., 18 Sep 2025).

• $L_2$ Surrogates on Control Inputs: For smooth, convex optimization, often use

$$J = \sum_k \left( \|\dot{q}_k\|_{Q_1}^2 + \|\tau_k\|_{R_1}^2 \right)$$

or, in continuous state-space,

$J = \int_0^T \|\mathbf{u}(t)\|^2\,dt$

where $Q_1, R_1$ are (semi)definite (Ji et al., 2023, Beaver, 2024).

• Physics-Based Power-Flow Models: Direct inclusion of empirical or analytic expressions for power demand $P(v,a)$ as a function of velocity $\tau_i$0 and acceleration $\tau_i$1, often including friction, aerodynamic drag, or battery dynamics (Zöllner et al., 7 Nov 2025, Li et al., 23 Mar 2026).
• Hybrid Energy Constraints: Objective functions may combine smoothness (e.g., jerk, acceleration), trajectory tracking/accuracy, and explicit energy terms:

$\tau_i$2

with $\tau_i$3 the energetic term (Hussain et al., 13 Mar 2025).

• Energy-Efficiency Metrics: Ratio of useful work (e.g., transmitted bits, delivered energy) to consumed energy:

$\tau_i$4

instantiated by, for example, bits/Joule in UAV communications (Zeng et al., 2016), or min-max form for fairness in wireless energy transfer (Xu et al., 2017).

• Energy-Based Statistical Models: In learning, trajectories are viewed as samples from

$\tau_i$5

where $\tau_i$6 is a learned energy, either based on expert demonstrations (Xu et al., 2019) or parameterized in latent spaces (Pang et al., 2021).

Energy-based trajectory objectives can appear as the primary cost, as regularizers, as feasibility constraints ("do not exceed battery capacity"), or as shaping terms in feedback control and learning.

2. Optimization Methods for Energy-Based Trajectories

A diversity of methods are employed to solve for energy-optimal or energy-aware trajectories across domains:

• Direct Collocation and Nonlinear Programming (NLP): Discretization of the horizon, state, and control, subject to dynamic and energetic constraints, yields large-scale NLPs. Used in multi-DOF robots, brachiation, and manipulator arms (Ji et al., 2023, Hussain et al., 13 Mar 2025).
• Sequential Convex Programming & Successive Convexification: For non-convex, nonlinear energy objectives, approximate locally as convex and iteratively refine (Sun et al., 2019, Zeng et al., 2016).
• Semi-Analytical Methods: Euler–Lagrange (or Pontryagin) necessary conditions reduce the infinite-dimensional optimization to low-dimensional nonlinear programs over polynomial/spline coefficients or switching times, especially for restricted dynamical classes or with bang-bang/constant arcs (Zöllner et al., 7 Nov 2025, Beaver, 2024).
• Reinforcement Learning (RL) with Energy Shaping: RL frameworks embed per-step energy costs or penalties (often in multi-objective settings with communication, delay, or handoff trade-offs), shaping policy search toward energy-frugal behaviors (Cherif et al., 2023, Song et al., 2022).
• Global-Optimal Control (Game-Theoretic): Exploiting structural decompositions, e.g., in multi-agent navigation, where constraint-activation schedules parameterize solutions, enabling efficient computation of energy-optimal Nash equilibria (Beaver, 2024).
• Convex Relaxations via Sum-of-Squares (SoS) and SDP: When the per-step or per-slot cost is a high-degree polynomial (e.g., in PCRB minimization with energy constraints), semidefinite relaxations yield tractable, globally optimal subproblems under strict convexity/concavity conditions (Jiang et al., 2024).
• Approximation and Surrogate Bidding (Resource-Aware Task Allocation): For auction-based multi-robot assignment, physics-based surrogates approximate OCP energy costs to enable rapid online bidding, with explicit quantification of ranking error and field-dependence (Li et al., 23 Mar 2026).

Method selection is governed by system dynamics, nonlinearity, real-time/online versus batch/offline operation, and required fidelity of the energy model.

3. Energy-Based Objectives in Learning and Inverse Optimal Control

Energy-based methods underpin recent advances in both imitation learning and trajectory prediction:

• Energy-Based Inverse Optimal Control (IOC): The expert cost $\tau_i$7 defining the probability of observing trajectory $\tau_i$8 is learned via maximum likelihood, with "analysis by synthesis" alternately generating negative samples and fitting parameters for feature-matching. When sampling is intractable, trajectory optimization via iLQR or gradient descent is used (Xu et al., 2019).
• Latent Energy-Based Models for Prediction/Multi-Modality: Human trajectory prediction uses highly expressive EBMs over latent variables $\tau_i$9, parameterizing the cost as $\dot{q}_i$0 conditioned on pooled social/motion history, optimized by Langevin or variational methods (Pang et al., 2021). These setups yield multi-modal prediction outputs and encode social compliance at the level of belief optimization.
• Energy-Based Prioritization for Replay Buffers: In RL for robotic manipulation, summing per-step positive energy increments (mechanical work on an object) provides a metric for prioritizing episodes in experience replay, shown to correlate strongly with TD-error and task-completeness, enabling significant gains in sample efficiency and final task success (Zhao et al., 2018).
• Cooperative Learning and Amortized Generators: Coupling generative models (e.g., trajectory-predicting MLPs) with energy-based scoring accelerates sampling and model iteration in learning pipelines, notably for high-dimensional, real-robot domains (Xu et al., 2019).

Within learning, the energy-based cost often serves not only as a learning target but also regularizes or guides the search through trajectory or action spaces.

4. Physical Energy Models and Domain-Specific Constraints

Effective energy-based trajectory objectives demand explicit, accurate modeling of the physical energy flows in the system and environment:

• Rotary- and Fixed-Wing UAVs: Aerodynamic power models include induced, profile, and parasite drag components, with energy as a function of speed and, for fixed-wing, heading and acceleration. Profiled in both rotary UAV delivery (Cherif et al., 2023) and aerial base stations (Sun et al., 2019), constraints may also include battery energy budgets, safety reserves, and instantaneous speed/acceleration bounds.
• Ground Mobile Robots and AMR Fleets: Battery depletion is captured by OCV-based models that couple mechanical power demands (friction, acceleration) and SOC dynamics, allowing for physics-based energy-aware bidding/logistics (Li et al., 23 Mar 2026).
• High-DOF Manipulators: Actuation and velocity-based costs reflect combined joint-torque squares and velocity penalties, sometimes extended to include motor efficiency or battery conversion losses (Hussain et al., 13 Mar 2025).
• Stacker Cranes and Electrical Machines: Power-flow maps $\dot{q}_i$1 describe irreversible losses, recuperation, and higher-order kinematics (velocity, acceleration, jerk). Objectives can be net-consumption or recuperation-maximizing, with practical distinctions for up- and down-movements (Zöllner et al., 7 Nov 2025).
• Planetary Rovers (Hybrid RTG-Solar): Cumulative and instantaneous power constraints, sourced from variable solar input plus RTG, enforce power-compliance and terrain-aware operation, with explicit inclusion of subsystem loads and dynamic feasibility (Hu et al., 18 Sep 2025).

These models are incorporated both in direct constraints (e.g., not exceeding battery or power capacity) and as primary optimization targets.

5. Multi-Objective and Application-Specific Extensions

Energy-based trajectory objectives are often embedded within multi-objective frameworks, balancing energy against throughput, timing, tracking error, safety, or communication objectives:

• Multi-objective RL for Communication, Task Collection, Delay: In mobile edge computing, trajectory policies are optimized over the Pareto front of energy, task delay, and number of tasks served, with evolutionary multi-policy RL yielding sets of nondominated policies for user selection (Song et al., 2022).
• Joint Trajectory and Cell-Association: For cargo UAVs, trajectory plans must simultaneously minimize energy, handoffs, and disconnectivity, with scalarized rewards reflecting weighted sum design and per-objective trade-offs (Cherif et al., 2023).
• Coverage versus Energy in UAV Sensing: Trajectories are selected to maximize coverage or min-max fairness for ground users or receivers, subject to energy budgets and flight time (Xu et al., 2017, Sun et al., 2019).
• Trajectory Planning under Power Constraints: Rovers or robots may face both cumulative energy budgets and instantaneous power caps, requiring smooth, dynamically feasible speed/heading profiles, enforced through softplus penalties or direct constraints (Hu et al., 18 Sep 2025).
• Task Allocation Auction Metrics: In multi-robot systems, bid selection for task assignment (energy- vs distance-based) is empirically shown to be domain-dependent; simulation reveals when energy-aware assignment yields net savings compared to Euclidean proxy (Li et al., 23 Mar 2026).

These multi-objective problems often admit solution methods that enable explicit traceability of the trade-off structure and direct policy selection for system designers.

6. Impact on Trajectory Characteristics and System Behavior

The integration of energy-based objectives fundamentally alters system behavior:

• Trade-Offs with Smoothness and Timing: Imposing energy constraints produces smoother, longer-duration motions with reduced peak torques, as opposed to minimum-time, bang-bang, or time-penalized profiles, which may provoke larger actuation, more abrupt transitions, and greater wear (Hussain et al., 13 Mar 2025, Ji et al., 2023).
• Gravity and Passive Phase Exploitation: In underactuated and brachiating robots, energy objectives exploit natural dynamics, reserving actuation for essential reorientation events while letting gravity supply most of the move (Ji et al., 2023).
• Fairness versus Total Energy: For resource delivery and wireless energy transfer, sum-energy maximization can create severe under-service for distant recipients. Incorporation of max-min (fairness) criteria, possibly via multi-location hovering and constrained motion, balances utility at the expense of aggregate performance (Xu et al., 2017).
• Trajectory Pacing and Power Compliance: For planetary rovers, energy-compliant optimization produces peak-shaved, terrain-aware profiles, leveraging rest or decelerated segments to avoid overloading hybrid power sources (Hu et al., 18 Sep 2025).
• Sample Efficiency and Learning Dynamics: In RL, energy-aware replay or trajectory prioritization improves curriculum learning and accelerates convergence, empirically verified through strong correlation with learning signal magnitude (Zhao et al., 2018).

System practitioners are advised to tailor energy-based objectives – and their surrogates or relaxation schemes – to balance mission constraints, resource limitations, and task-driven performance metrics.

7. Future Directions and Generalizations

Across all application areas, trends and open directions include:

• Enhanced Energy Models: Integrating detailed battery/motor efficiency, thermal modeling, or electrochemical wear for more realistic estimation (Hussain et al., 13 Mar 2025, Li et al., 23 Mar 2026).
• Real-Time Adaptive and MPC Incorporation: Hierarchical and model-predictive controllers that update energy-based objectives online, tractably handling hard constraints and surging task-loads (Hussain et al., 13 Mar 2025, Hu et al., 18 Sep 2025).
• Hybrid Energy-Physical-Latent Objectives: Combining physically grounded energy terms with learned, high-level or latent-derived cost functions, e.g., via hybrid imitation/RL paradigms (Xu et al., 2019, Pang et al., 2021).
• Scenario-Dependent Objective Selection: Empirically calibrated guidance on when and where energy-based auctioning or allocation yields net gains versus structure-induced proxies (distance) (Li et al., 23 Mar 2026).
• Safety, Wear, and Lifetime Extensions: Increasing attention to trajectory characteristics that trade-off energy with hardware longevity and fail-safe operation (peak power, sudden actuation, battery depth-of-discharge) (Hu et al., 18 Sep 2025).
• Unified Planning-Control-Learning Pipelines: Embedding energy-based objectives throughout integrated perception, prediction, planning, and control stacks for full-system autonomy.

The impact and suitability of energy-based trajectory objectives are determined jointly by the quality of energy models, optimization/learning algorithms, and operational regime of the target system. Their principled use is essential for the next generation of resource-aware, robust, and long-lived mobile and robotic platforms.