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VIP: Variation-based Iterative-learning Planning for Robotic Navigation

Shuli Lv, Pengda Mao, Chen Min, Li Hong, Runxiao Liu, Shuai Wang, Quan Quan

arXiv:2608.24618v1cs.RO

TL;DR

Robotic navigation needs safe and efficient planning in complex, obstacle-dense environments despite limited onboard computation and growing costs for conventional, especially multi-robot, planners. VIP updates planning commands as continuous functions with model- or robot-in-the-loop learning, and simulations and real-world experiments demonstrate reduced traversal time, bounded task error, and applicability across robotic platforms and swarm configurations.

  • Problem

    Safe, dynamically feasible, and efficient motion planning remains computationally difficult in complex environments, while trajectory resolution, prediction horizon, decision variables, and swarm size increase planning costs.

  • Method

    VIP performs function-level variation-based iterative learning of planning commands, using a unified energy formulation and execution feedback for model-in-the-loop or robot-in-the-loop planning.

  • Results

    Simulations and real-world experiments demonstrate reduced traversal time, bounded task error, and applicability across different robotic platforms and swarm configurations.

  • Takeaways & Limitations

    VIP provides a unified planning-level learning layer for improving traversal efficiency in both single-robot and robotic-swarm navigation.

  • Takeaways & Limitations

    The band [ζmin, ζmax] requires a tunable compromise between approximation accuracy and robustness to modeling uncertainty, measurement noise, and iteration fluctuations.

Abstract

from arXiv · show

Over the past decade, autonomous robotic systems have been increasingly deployed in applications such as surveying, search and rescue, and last-mile delivery. These applications require robots to generate safe and efficient motion plans in large, complex, and obstacle-dense environments, often under limited onboard computing resources. However, conventional planning methods commonly rely on finite-dimensional trajectory parameterization or increasingly long prediction horizons, leading to rapidly growing computational costs, particularly in multi-robot scenarios. This paper presents a novel variation-based iterative-learning planning (VIP) framework for efficient motion planning of both single robots and robotic swarms. Instead of optimizing a large number of discrete trajectory variables, VIP directly updates the planning command as a continuous function in an infinite-dimensional function space. The same variation-based update can be implemented in a model-in-the-loop manner for offline planning or in a robot-in-the-loop manner between online physical executions. By avoiding the computational burden associated with horizon expansion and high-dimensional trajectory discretization, VIP maintains a per-iteration computational complexity of $\mathcal{O}(n)$, where $n$ denotes the number of spatial discretization points. Extensive simulations and real-world experiments demonstrate that the proposed framework can efficiently generate and iteratively improve motion plans for different planning objectives, robotic platforms, and swarm configurations, highlighting its effectiveness, computational efficiency, and scalability as a general planning methodology.

1 Introduction

Robotic navigation requires safe, dynamically feasible, and efficient planning under limited onboard computation, with costs becoming especially difficult for dense environments and swarms. VIP addresses this challenge by optimizing planning commands as functions and learning them through energy variations across offline and online settings.

  • Robots must generate safe, dynamically feasible, and efficient motion commands despite environmental constraints and limited onboard computing resources.
  • Prediction-based planners incur rising costs with longer horizons, finer trajectory resolution, and more decision variables, while reactive planners offer less lookahead.
  • Swarm planning additionally requires inter-robot coordination, collision avoidance, and collective objectives, while centralized computation grows rapidly with swarm size.
  • VIP directly optimizes planning commands in an infinite-dimensional function space and represents objectives through corresponding system-energy functionals.
  • VIP supports model-in-the-loop offline planning and robot-in-the-loop online planning by updating commands from energy variations recorded during previous rollouts.
  • The framework applies one variation-based planning principle to individual robots and swarms while targeting low computational complexity across planning settings.

2 Related work

Prior work spans prediction-based, reactive, formation-based, top-down, and iterative-learning approaches to robot and swarm navigation. These methods trade computational cost, lookahead, coordination, scalability, and planning-level optimization in different ways.

  • Single robot motion planning: Prediction-based planning optimizes finite-horizon commands or parameterized trajectories but becomes more computationally demanding as resolution, constraints, or search space increase.
  • Single robot motion planning: Reactive planning reduces computation and communication requirements but can be sensitive to local environmental structure and limited lookahead.
  • Robotic swarm motion planning: Swarm planners use centralized or distributed trajectory coordination, yet scalability can be constrained by joint-state dimension, communication, and coordination overhead.
  • Robotic swarm motion planning: Formation-control methods coordinate swarms through structured relationships, but fixed or weakly adaptive formations may restrict feasible motion through clutter.
  • Robotic swarm motion planning: Top-down swarm methods describe collective behavior with macroscopic variables or PDEs rather than explicitly planning each robot's motion.
  • Iterative learning: Iterative learning updates inputs across repeated executions, but conventional ILC is not directly suited to navigation or planning objectives such as traversal time and motion efficiency.

3 Preliminaries

The preliminaries define path-based navigation in obstacle-free configuration space, then introduce virtual tubes and the time-to-space transformation used for single robots and swarms.

  • 3.1 Definitions in robotic navigation: Robots plan a smooth collision-free path V within traversable space C from start point bp to goal point gp.The path is parameterized by γ(s), where s is arc length and L is total path length.
  • 3.1 Definitions in robotic navigation: The closest-point mapping indexes a robot’s position by path coordinate l and supplies the local tangent direction for navigation.The mapping associates each robot position with a closest point on γ and its arc-length coordinate.
  • 3.2 Virtual tube for navigation: Restricting robots to the virtual tube reduces collision checking to interactions with other robots and the tube boundary.The tube has previously been used for planning and passing-through control.
  • 3.2 Virtual tube for navigation: A virtual tube is constructed from terminal cross-sections, a generator curve, and intermediate cross-sections to provide a safe traversal space.Its formal representation uses the tuple (C0, C1, f, h) and the set T = {h(q0, qm, l)}.
  • 3.2 Virtual tube for navigation: Tube planning in obstacle environments uses Tube-RRT*, which supplies the virtual tube for the reported simulations and experiments.The tube surface is its boundary ∂T.
  • 3.3 Transformation from Time Domain to Space Domain: The time-to-space transformation uses progression speed v = D_tl, with l serving as the closest-point or swarm progression coordinate.For v > 0, l(t) is strictly increasing and has a global inverse, allowing time-domain signals to be expressed in space.

4 Problem formulations

The paper formulates single-robot and swarm navigation through distinct regulation errors but a shared command structure and unified energy dynamics. This supports a common optimization problem balancing traversal efficiency and navigation safety.

  • 4 Problem formulations: The single-robot formulation regulates finite-dimensional path error, while the swarm formulation regulates distribution error within a bounded virtual-tube domain.The swarm state is represented by a normalized density ρ(t,p).
  • 4.1 Single-robot formulation: The executable velocity is produced by radial saturation with limit vm, combining nominal regulation and traversal components.For the single robot, the nominal component corrects path error and the traversal component advances along the path tangent.
  • 4.1 Single-robot formulation: Increasing traversal speed vt may reduce κv under active saturation, leaving less command authority for path-error correction.The single-robot analysis assumes λs > 0, bounded εs, and bounded low-level velocity-execution error.
  • 4.2 Robotic-swarm formulation: The swarm density-regulation field is exponentially dissipative for stationary desired density, with coefficient CP depending only on the tube domain T.Velocity saturation, field interaction, and desired-density variation appear as bounded coefficients in the derivative analysis.
  • 4.2 Robotic-swarm formulation: For swarms, a saturated composite field regulates density while advancing the swarm, with density evolution described by a continuity equation.The desired density is constructed from occupied tube cross-sections and shifted forward to limit aggregate antagonism between regulation and traversal.
  • 4.3 Common energy system: Both formulations use scalar Lyapunov energies that are positive definite with respect to their corresponding regulation errors, rather than mechanical energy.The unified energy combines the single-robot energy Vs or swarm energy Vm into V.
  • 4.3 Common energy system: The common energy satisfies negative drift away from zero in the nominal case and perturbed dynamics under bounded nonideal effects.The dissipation term dominates the lower-order coupling term for sufficiently large command magnitude, yielding negative drift outside a bounded neighborhood.
  • 4.5 Problem formulation: VIP formulates navigation as a unified spatial integral that weights traversal time and safety through kV, using the same effective command scaling for energy and spatial progression.The strictly positive traversal command preserves the invertibility required by the time-to-space transformation.

5 VIP: variation-based iterative-learning planning

VIP develops model-based and model-free variation laws that iteratively update traversal commands in function space. The model-free law preserves the qualitative descent direction while reducing implementation requirements and maintaining linear per-iteration complexity.

  • Framework: VIP uses a model-based variation law as a theoretical reference and a model-free law that approximates its direction from rollout energy profiles.The model-free law avoids analytical derivatives during implementation.
  • Model-based variation: If the functional gradient is nonzero, its negative is a descent direction; if it vanishes, the command satisfies first-order stationarity.This establishes the optimization interpretation of the variation update.
  • Model-free learning: The model-free law accelerates when recorded energy is small and decelerates when regulation requires more command authority.Its energy band preserves the qualitative sign transition without online energy derivatives or adjoint calculations.
  • Convergence: Under a sufficiently small learning gain and a stationary point inside the admissible energy band, the model-free cost decreases monotonically and the learned energy approaches that band.The model-free update direction converges to zero in the L2 norm.
  • Computational complexity: Each learning iteration has O(n) computational and memory complexity because VIP replaces coupled trajectory optimization with pointwise scalar evaluations and algebraic updates.Physical rollout or simulation execution time is excluded from this complexity.

6 Single-robot realization and model-free implementation

The single-robot realization applies VIP to a desired velocity profile using measured path energy. Its update increases speed below the admissible energy band and decreases speed above it, without requiring the full robot dynamics online.

  • Single-robot realization: For single-robot navigation, VIP updates the traversal profile using the single-robot path energy as the generic learning-law input.The implementation acts on the desired velocity profile rather than requiring full robot dynamics.
  • Model-free implementation: Below the admissible path-energy band, the update increases traversal speed; above the band, it decreases traversal speed.Online implementation uses measured Vs,k(l) without requiring τ, λs, or εs.
  • Model-free implementation: The derivative-enhanced update uses ζd,k = keVs,k + kd ˙Vs,k with kd > 0.This implementation augments measured path energy with its rate of change.
  • Boundedness: Under Assumption 1 and the stated compact active-saturation conditions, the derivative-enhanced traversal profile remains uniformly ultimately bounded as k approaches infinity.The result assumes the controller and update satisfy the theorem’s gain condition.

7 Robotic-swarm model-free implementation and microscopic realization

The swarm realization applies VIP to density-distribution error while a distributed virtual-tube controller handles microscopic safety and distribution. VIP therefore learns a common traversal speed without optimizing each robot trajectory individually.

  • Swarm model-free implementation: The swarm update uses density-error energy computed from KDE density and does not require online identification of kρ, βρ, or ∂ρd/∂t.The common traversal command is scaled to an admissible band according to swarm-density tolerance.
  • Macroscopic model: Increasing swarm density generally reduces achievable traversal speed, creating a trade-off between common forward motion and distribution regulation.The swarm energy measures mismatch between current and desired density distributions.
  • Macro–micro realization: The macroscopic forward-speed field corresponds to the microscopic forward controller, while the density-regulation field corresponds to inter-robot dispersion.The tube-boundary component separately enforces local containment.
  • Macro–micro realization: VIP modifies only the common traversal magnitude, while the established microscopic controller maintains local safety and distribution.This separates swarm-level efficiency learning from local collision avoidance and tube keeping.

8 Simulations and comparisons

Simulations evaluate VIP for single robots and swarms in obstacle-rich environments. The results report lightweight computation, adaptive traversal-speed behavior, improved swarm progress, and successful planar and three-dimensional tube navigation.

  • Single-robot simulations: Across two random forest maps, single robots navigate obstacle-dense environments while VIP increases speed in open regions and becomes conservative in narrow or turning regions.The learned command balances traversal time against path convergence and safety regulation.
  • Single-robot simulations: 0.28ms for 50 waypoints and 0.52ms for 100 waypoints indicate approximately linear planning-time growth, while average speed reaches 2.98m/s under a 5m/s maximum.These measurements support the reported lightweight spatial-profile update.
  • Comparisons: 1.31-1.57% traversal-time improvements over MPCC and Polynomial methods are reported for the 60-obstacle scenario.The reported advantage varies across scenarios, including values from -1.31% to 1.37% compared with MPCC.
  • Swarm simulations: With VIP, the swarm moves faster along the virtual tube while maintaining a bounded spatial distribution and the tube constraint.Snapshots compare traversal with and without VIP at multiple time instants.
  • Swarm simulations: Swarm traversal time decreases over learning iterations and then stabilizes as the model-free update converges.VIP adjusts the common traversal component while density regulation continues to control swarm configuration.
  • Three-dimensional navigation: In a three-dimensional obstacle environment, the swarm passes through the planned virtual tube while maintaining feasible collective distribution.The learned profile improves traversal speed while the tube controller confines agents inside the safe corridor.

9 Experiments

Experiments evaluate VIP on single aerial robots, a lifting-wing quadcopter, and a three-robot swarm. Across these settings, repeated execution improves traversal commands while preserving feasible motion and reducing computational burden.

  • Single-drone experiments: 0.93ms computation time for 100 path points enabled VIP to update traversal speed alongside RRT*-based path replanning.RRT* determines where to fly, while VIP determines how fast to traverse the resulting path.
  • Lifting-wing quadcopter: The lifting-wing quadcopter reduced traversal time from 77.4s to 42.7s in hardware-in-the-loop simulation, then from 42.1s to 41.3s in two real-world iterations.Simulation iterations initialized the command before physical deployment, while model-free learning used recorded execution data.
  • Swarm experiments: After five iterations, the swarm’s maximum common traversal speed increased from 0.5m/s to 2m/s while relative configuration remained bounded.The virtual-tube controller maintained local distribution and boundary constraints during this increase.
  • Swarm experiments: Swarm traversal time decreased from 241s to 82s after six iterations without optimizing independent trajectories for all quadcopters.VIP updated the shared macroscopic traversal component, while the virtual-tube controller preserved microscopic distribution regulation and collision-safe motion.
  • Unified experimental findings: The experiments support one traversal-command learning structure across different energy definitions for single-robot path following and swarm distribution regulation.The broader results also support separating macroscopic traversal learning from microscopic interaction control.

10 Conclusion

The paper concludes that VIP unifies energy-variation planning, function-level optimization, and execution-feedback learning across single-robot and swarm navigation. Its analyses and experiments report convergence-related properties, reduced traversal time, bounded task error, and applicability across platforms, while larger onboard learning and uncertainty-aware extensions remain future work.

  • Conclusion: VIP represents task-dependent objectives through a common energy function for single-robot path following and robotic-swarm density regulation.The framework applies the same spatial learning principle to both task types.
  • Conclusion: O(n) computational complexity per model-free update is obtained through function-level planning optimization without online adjoint solution or horizon-based optimization.The update acts directly on the planning command rather than solving a high-dimensional trajectory optimization problem.
  • Conclusion: The same update law supports simulated rollouts in model-in-the-loop mode and recorded physical rollouts in robot-in-the-loop mode.This provides both offline and execution-feedback implementations of the planning framework.
  • Conclusion: Simulations and real-world experiments demonstrate reduced traversal time, bounded task error, and applicability across robotic platforms and swarm configurations.The theoretical analysis also establishes descent and neighborhood-approximation properties.
  • Conclusion: Future work will investigate within-rollout adaptation, dynamic environments, explicit uncertainty-aware safety constraints, and larger robot swarms with fully onboard learning.These items define the stated scope boundary for the current framework.

D Detailed conditions and proof of Proposition 2

The proof of Proposition 2 derives swarm convergence-related inequalities from the command constructions and the integral acute-angle design. It does not require pointwise acute-angle or pointwise non-opposition conditions.

  • Proof conditions: For the swarm, the partial derivative with respect to V is evaluated using a radial scalarization while the physical density profile remains time varying.The scalarization freezes only the normalized field shape during partial differentiation.
  • Proof conditions: The saturation calculation uses separate active and unsaturated regions with weighted self-products and the forward-shift design.The forward-shift design supplies the required nonnegative integral term.
  • Proof conclusion: The unsaturated set contributes directly to the progression derivative, and combining the regional inequalities proves the swarm result in Proposition 2.The proof integrates the relevant terms over the active region before combining them.
  • Proof conclusion: No pointwise acute-angle or pointwise non-opposition condition is required; the swarm result follows from the integral criterion.The required signs arise from integral acute-angle design rather than pointwise constraints.

E Proof of Theorem 1

Theorem 1 is supported by a descent inequality showing that the objective sequence is monotonically non-increasing and convergent. The update direction vanishes asymptotically, and accumulation points satisfy first-order stationarity, with convergence to a unique stationary solution in the neighborhood.

  • Proof of Theorem 1: A Lipschitz-continuous Fréchet derivative yields the descent inequality used in the convergence proof.The step-size condition 0 < b1 < 2/Fm provides the stated positive descent factor.
  • Proof of Theorem 1: The objective sequence is monotonically non-increasing and convergent because the objective is bounded from below.This establishes the sequence-level convergence claim in the proof.
  • Proof of Theorem 1: The update direction vanishes asymptotically, and every accumulation point satisfies the first-order stationarity condition.If the stationary solution is unique in the considered neighborhood, the generated sequence converges to it.

F Proof of Theorem 2

The proof establishes descent and convergence of the iterative objective, then interprets the limiting correction and derives boundedness of the velocity coordinate.

  • Descent consistency: The model-free correction g_k is constructed as an energy-based surrogate for the model-based descent direction, preserving descent consistency when nonzero.The update direction is analyzed through the recorded-energy-based surrogate and a sufficiently small learning gain.
  • Descent and convergence: The objective sequence {J(v_t,k)} is monotonically non-increasing and convergent because J is bounded from below.This follows after establishing positivity of the relevant coefficient under the learning-gain condition.
  • Descent and convergence: Summability of the correction norms implies that ||g_k||_L2(0,L) tends to zero as iterations increase.The proof obtains this by summing the descent inequality over k.
  • Limit interpretation: Because the zero set of g is the admissible energy band, vanishing correction means the learned energy approaches that band in the L2 sense.Since the band contains the zero point of the model-based descent direction, the result is interpreted as a neighborhood approximation of the stationary solution.
  • Boundedness: Under the stated contraction condition with bounded input, v_t,k is uniformly ultimately bounded.The scalar recursion has contraction factor 0 < 1 + b3kgkdλsγ1 < 1.
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