Object-Pickup Minimum Energy Path Problem

• OMEPP is a path-planning problem where a robot selects an optimal pickup location to minimize energy consumption despite payload-induced changes on uneven terrain.
• The approach formalizes the problem on a Digital Elevation Model, incorporating payload-sensitive slope constraints and a specialized energy model based on friction and gravitational forces.
• Key contributions include an optimal Z*-based baseline and a faster, near-optimal PCPD-guided concurrent search that dramatically reduces online computation time.

Searching arXiv for the primary OMEPP paper and closely related work on energy-aware path planning and pickup/delivery planning. The Object-Pickup Minimum Energy Path Problem (OMEPP) is a path-planning problem for a battery-powered ground robot moving on uneven terrain, in which the robot must pick up an object with known weight at one of several candidate locations and then deliver it to a destination while minimizing total energy consumption. In its formalization, OMEPP extends minimum-energy routing by making payload a decision-dependent variable: the robot travels from the source to a selected pickup node under the initial payload, then from that pickup node to the target under the increased payload after collection. The formulation introduced in "Energy-Efficient Path Planning with Multi-Location Object Pickup for Mobile Robots on Uneven Terrain" defines the terrain as a directed graph derived from a Digital Elevation Model (DEM), incorporates payload-dependent slope feasibility and edge energy costs, and studies both an optimal baseline and a faster near-optimal search framework (Babakano et al., 7 Sep 2025).

1. Formal definition and graph-theoretic model

OMEPP is defined on a terrain represented by a Digital Elevation Model as a directed graph

$G=(V,E,D,\Theta),$

where each vertex corresponds to a DEM node with coordinates $(n.x,n.y,n.z)$, each node has up to 8 neighbors in an 8-connected grid, and each directed edge carries a Euclidean distance and a slope quantity. The robot is characterized by mass $m$, constant velocity $cv$ (denoted $v$ in experiments), maximum power $P_{\max}$, coefficient of dynamic friction $\mu$, coefficient of static friction $\mu_s$, and gravitational acceleration $g$. The payload $\rho$ is the sum of the initial payload $(n.x,n.y,n.z)$0 and, after pickup, the object payload $(n.x,n.y,n.z)$1 (Babakano et al., 7 Sep 2025).

The key structural feature of OMEPP is that payload affects both energy expenditure and traversability. A route is therefore not merely a geometric path but a concatenation of two minimum-energy path segments under different payload regimes. If the pickup candidates are $(n.x,n.y,n.z)$2, source is $(n.x,n.y,n.z)$3, and target is $(n.x,n.y,n.z)$4, the objective is to choose a pickup location $(n.x,n.y,n.z)$5 and minimize

$(n.x,n.y,n.z)$6

Equivalently, the optimal path is

$(n.x,n.y,n.z)$7

This formulation makes OMEPP a multi-location pickup problem coupled to terrain-aware energy-optimal routing. The paper does not provide a formal complexity proof such as NP-hardness; instead, it treats the problem operationally as a multi-source or multi-target energy-optimal search problem and emphasizes empirical computational behavior, especially the growth in cost as the number of pickup locations increases (Babakano et al., 7 Sep 2025).

A plausible implication is that OMEPP occupies an intermediate position between classical shortest-path search and integrated task-and-path planning. It retains a single-robot, single-object structure, but the choice of pickup node introduces a combinatorial layer analogous to pickup-and-delivery planning more broadly (Aryan et al., 2024).

2. Energy model, slope constraints, and payload effects

The OMEPP energy model is inherited from the Z* framework for uneven-terrain routing and is defined at edge level. For an edge from $(n.x,n.y,n.z)$8 to $(n.x,n.y,n.z)$9 with payload $m$0, Euclidean distance $m$1, and slope angle $m$2, the motion energy is

$m$3

However, the model is constrained by payload-sensitive slope feasibility (Babakano et al., 7 Sep 2025).

Two slope thresholds govern motion. The first is the maximum permissible uphill slope angle

$m$4

where $m$5 is derived from traction and motor power and $m$6 captures static-friction-limited traction. The second is the critical downhill slope

$m$7

below which descending is treated as effectively free in the sense of zero net engine energy, ignoring braking. These thresholds yield the piecewise edge-cost model

$m$8

This construction captures the central asymmetry of OMEPP. Before pickup, the robot may traverse steeper slopes and incur lower energy expenditure because the payload is smaller. After pickup, the increased payload can both raise the energetic cost of the same terrain and render previously feasible slopes non-traversable. The practical consequence is that the energetically optimal pickup location need not be the geometrically nearest one. The paper explicitly notes that a closer pickup may be worse than a farther but topologically favorable pickup (Babakano et al., 7 Sep 2025).

This payload-sensitive notion of traversability distinguishes OMEPP from many minimum-energy path formulations that assume a fixed load. It also differs from energetic path problems for electric vehicles with battery-capacity state, where arc costs may be positive, zero, or negative and feasibility depends on residual charge rather than terrain traction limits (Dorfman et al., 2023). By contrast, OMEPP’s state transition is driven by a discrete payload jump at the pickup event.

3. Optimal baseline via Z* and its limitations

The baseline solution enumerates all pickup candidates and solves two point-to-point minimum-energy searches for each one using Z*, a variant of A* specialized to energy-efficient pathfinding on uneven terrain. Z* uses

$m$9

with accumulated energy cost $cv$0 and an admissible, consistent heuristic $cv$1 tailored to slope and payload constraints. The heuristic distinguishes three cases: infeasible steep direct ascent, effectively free shallow downhill motion, and direct feasible motion under the same energy model as the edge cost (Babakano et al., 7 Sep 2025).

For each pickup node $cv$2, the baseline computes the exact minimum-energy path from $cv$3 to $cv$4 under payload $cv$5, and the exact minimum-energy path from $cv$6 to $cv$7 under payload $cv$8. Since Z* is optimal for each fixed payload instance and the algorithm checks all pickup nodes, the resulting baseline is globally optimal for OMEPP. Its structure is simple: $cv$9 Z* searches for $v$0 pickup candidates.

The limitation is computational redundancy. Many searches repeatedly expand similar regions near the source and destination, but the baseline performs no reuse across pickup candidates. In the reported experiments on a real DEM with approximately 250,000 nodes, 1,993,008 directed edges, and 50 pickups, baseline runtimes are reported as typically 1448–2332 ms and one case 7099 ms per query (Babakano et al., 7 Sep 2025).

This establishes an important distinction between correctness and scalability. The baseline provides an exact reference solution, but its cost grows with the number of pickup nodes because it effectively treats each candidate independently. That behavior parallels broader pickup-and-delivery planning systems in which task choice and path computation are decoupled, often yielding optimality at substantial computational expense (Aryan et al., 2024).

4. Payload-Constrained Path Database and concurrent search

To mitigate repeated search, the paper introduces the Payload-Constrained Path Database (PCPD), an extension of the Compressed Path Database (CPD). Standard CPD precomputes shortest paths between all node pairs and stores only the first move of the optimal path, compressing the first-move table via run-length encoding after depth-first ordering of targets. PCPD generalizes this idea by constructing multiple CPDs, one for each representative payload bucket, where each CPD stores first moves of minimum-energy paths subject to payload-specific slope constraints (Babakano et al., 7 Sep 2025).

The payload domain $v$1 is partitioned into equal-sized buckets

$v$2

In the experiments, $v$3 kg and eight buckets are used:

$v$4

For each representative payload $v$5, the method computes the corresponding maximum traversable slope and runs a modified Dijkstra algorithm from every source vertex using energy cost rather than geometric distance. The result is an energy-aware CPD for that payload (Babakano et al., 7 Sep 2025).

At query time, PCPD is not used to certify optimality directly but to guide successor generation. For a current payload $v$6, the algorithm selects the nearest lower and upper precomputed payload buckets, queries both CPDs for the first move toward the current subgoal, validates feasibility where necessary, and keeps one or two candidate moves. This reduces branching sharply. In the concurrent search, naïve joint expansion could generate up to $v$7 combined successors, whereas PCPD guidance yields at most 4 joint successors (Babakano et al., 7 Sep 2025).

The paper’s concurrent PCPD search is a two-level best-first procedure. For each pickup point $v$8, it maintains a child queue over paired frontier states $v$9 corresponding to the pre-pickup and post-pickup branches. A global queue then selects which pickup-specific child queue to advance next. The low-level evaluations on each side remain Z*-style, using the exact energy model and heuristic; the approximation arises only from restricting successor generation to PCPD-suggested moves (Babakano et al., 7 Sep 2025).

Because of that restriction, the concurrent PCPD search is not guaranteed to return the globally optimal OMEPP solution. It is complete and optimal only in the reduced graph defined by the PCPD-guided successor relation, and the paper states that no formal suboptimality bound is proven. This suggests that PCPD should be interpreted as a search-space reduction device rather than a heuristic estimator in the strict A* sense.

5. Empirical performance and scaling behavior

The experimental environment is a 250 m × 250 m Fairfax County DEM patch with approximately 250,000 nodes and 1,993,008 directed edges under 8-connectivity. The robot parameters are Husky A300-like: $P_{\max}$0 kg, velocity $P_{\max}$1 m/s, $P_{\max}$2 W, $P_{\max}$3, and $P_{\max}$4. The maximum payload in experiments is 70 kg. Query generation uses 1000 random source-target queries with 50 randomly scattered pickup nodes, random $P_{\max}$5, and random $P_{\max}$6; each query is run 10 times, the best and worst are removed, and the average of the remaining 8 is reported (Babakano et al., 7 Sep 2025).

The PCPD preprocessing cost is substantial but offline. Eight CPDs are built, one per payload bucket. Build time per CPD is approximately 99–107 minutes, memory per CPD is approximately 0.81–1.3 GB, and total memory for all eight CPDs is approximately 8–10 GB. The paper notes that memory decreases for higher payloads because more slopes become non-traversable and run-length encoding becomes more effective (Babakano et al., 7 Sep 2025).

The online speedups are the central empirical result.

Method	Runtime	Solution quality
Baseline Z* enumeration	typically 1448–2332 ms, one case 7099 ms	globally optimal
Concurrent PCPD search	typically 5.2–7.2 ms	average suboptimality 0.00024–0.01065

Average suboptimality ranges from 0.00024 to 0.01065, i.e., 0.024%–1.065%, and is reported as below 1.1% across tested payload configurations. The corresponding speedups are described as one to two orders of magnitude in the abstract and approximately 200× to 1300× in the detailed exposition (Babakano et al., 7 Sep 2025).

Scaling with the number of pickup points further differentiates the methods. As the number of pickup points increases from 25 to 100, baseline runtime grows approximately linearly because it performs two Z* searches per pickup. By contrast, concurrent PCPD search remains almost flat, around 5–7 ms across pickup counts, because the global queue quickly suppresses unpromising pickup candidates and PCPD keeps low-level branching small (Babakano et al., 7 Sep 2025).

These results place OMEPP within a broader family of planning problems where offline structure is traded for online responsiveness. Similar architectural trade-offs appear in integrated task-and-path planning for pickup and delivery, where heavy preprocessing or decomposition can shift computational effort away from online decision time (Aryan et al., 2024).

6. Relation to adjacent research areas

OMEPP is closely related to, but distinct from, several existing research strands. First, it extends energy-aware terrain routing by introducing a payload discontinuity at an unknown pickup choice. The immediate predecessor is minimum-energy pathfinding on uneven terrain under fixed payload, for which Z* provides the heuristic search substrate used by the OMEPP baseline and concurrent method (Babakano et al., 7 Sep 2025).

Second, OMEPP differs from manipulator-centric minimum-energy path formulations. In manipulator path placement optimization, the path shape may be fixed and the optimization concerns where that path is embedded in the workspace so as to minimize actuator energy under kinematic and dynamic constraints. For the Orthoglide 3-axis case, the objective is total electric energy used by the actuators, subject to geometric, kinematic, and dynamic limits (0910.4000). That line of work is analogous in its attention to realistic energy models, but OMEPP concerns a mobile ground robot on terrain rather than a fixed-base manipulator.

Third, OMEPP is structurally connected to general energetic path problems on graphs. For electric cars, energetic path planning has been formalized with a battery-capacity state and possibly negative edge costs due to regeneration, leading to Bellman–Ford and Dijkstra/A*-style algorithms adapted to nonstandard energy accumulation (Dorfman et al., 2023). A plausible implication is that OMEPP and such battery-constrained energetic routing share the broader principle that path feasibility and optimality cannot be separated from the robot’s evolving resource state. In OMEPP, the relevant state is payload-dependent slope feasibility rather than residual battery charge.

Fourth, OMEPP touches integrated pickup-and-delivery planning, especially where choice of pickup node is entangled with route cost. In multi-robot warehouse settings, integrated task planning and path planning have been combined using SMT and Conflict-Based Search with Precedence Constraints to obtain optimal collision-free pickup-and-drop plans under makespan or total-cost objectives (Aryan et al., 2024). OMEPP is narrower in scope—single robot, single object, uneven terrain, minimum energy—but it shares the same general feature that combinatorial pickup decisions and motion planning cannot be optimized independently without loss.

Finally, OMEPP should not be conflated with uncertainty-aware picking in clutter. Work on safe and effective picking under discrete pose uncertainty frames planning as a stochastic variant of Minimum Constraint Removal and optimizes a success probability that combines survivability and target reachability (Wang et al., 2020). That problem addresses uncertain collision geometry and grasp success in manipulator motion, whereas OMEPP assumes a static DEM and known object payload but uncertain choice among pickup locations.

7. Significance, limitations, and open directions

OMEPP’s principal significance lies in making payload change an explicit component of minimum-energy route planning on uneven terrain. This addresses practical autonomous mobile robot missions in which object collection occurs en route rather than at the start or destination. The paper lists delivery robots selecting among storage locations, search-and-rescue robots picking up equipment in rough terrain, and agricultural or environmental robots collecting samples from multiple candidate sites as representative applications (Babakano et al., 7 Sep 2025).

The formulation also clarifies a common misconception in energy-aware navigation: minimizing geometric distance or solving a fixed-payload energy-optimal path is not sufficient when carrying state changes during the mission. In OMEPP, the pickup decision changes both the edge cost function and the feasible subset of the terrain graph after the pickup. This is why enumerating pickup points with a fixed-payload solver is conceptually valid only if the payload switch is modeled explicitly in the two path segments (Babakano et al., 7 Sep 2025).

The current formulation has clear limitations. The environment is static; dynamic obstacles and time-varying terrain are not considered. The setting is restricted to a single robot and a single collected object. The energy model assumes constant velocity and uses frictional and gravitational forces, without explicit drivetrain efficiency maps, detailed motor models, or regenerative braking. PCPD relies on payload discretization, and the concurrent PCPD search has no formal suboptimality bound, even though empirical suboptimality is below 1.1% on the reported instances (Babakano et al., 7 Sep 2025).

The authors explicitly identify several future directions: multi-object scenarios, multiple trips, more general logistics and autonomy problems, multi-robot coordination, time windows for pickup and delivery, dynamic environments, richer energy and powertrain models, regenerative braking, and uncertainty in terrain, payload, and friction parameters (Babakano et al., 7 Sep 2025). This suggests that OMEPP may serve as a foundational single-robot, single-object problem class from which more general energy-aware pickup-and-delivery formulations can be developed.

In that sense, OMEPP can be viewed as a specialized but conceptually important bridge problem. It is richer than standard source-to-target energy-efficient routing because the pickup choice is endogenous, yet more structured than full integrated task-and-motion planning because the mission consists of exactly one pickup and one delivery. That intermediate position is precisely what makes the problem both analytically tractable enough for specialized algorithms and practically relevant for outdoor AMR deployment (Babakano et al., 7 Sep 2025).