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

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

cs.ROAug 25, 2026

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

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

Organizations: School of Automation Science and Electrical Engineering, Beihang University, Beijing, P.R. China · Tianmushan Laboratory, Beihang University, Hangzhou, P.R. China

Abstract

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.

Figures & tables

Figure 1: Overview of the proposed variation-based iterative-learning planning (VIP) framework and its real-world experimental validation. (a) Euclidean optimization, where a trajectory is parameterized by a finite set of waypoints. (b) Functional optimization adopted by VIP, where the trajectory is directly optimized in an infinite-dimensional function space. (c) Offline flight in an unknown environment. (d) Online learning of a robot swarm. (e) Online learning of a lifting-wing quadcopter.

Paradigm	Subclass	Single robot	Robotic swarm
Prediction-based planning	Optimization-based	Explicit optimization ; constraint handling; model dependence; high computation	Coordinated trajectories; collision avoidance; increasing computation and communication
	Parameterized trajectory	Reduced variables; smooth trajectories; complex time optimization	Individual trajectory parameterization; demanding joint optimization
	Sampling-based	Nonlinear objectives; flexible search; sampling dependence	Flexible motion generation; conflict resolution; limited scalability
Reactive-based planning	Vector/potential field	Fast response ; low computation; convergence-oriented	Local interactions; distributed coordination; limited global optimization
	CBF/flocking	Safety constraints; reactive collision avoidance; online optimization	Collision avoidance; connectivity; cooperation; dense-swarm complexity
Iterative learning	Model-based ILC	Function-space update; model information; tracking-level learning	Repetitive collective tracking; predefined references; model dependence

Table 1: Comparison of planning paradigms for single robots and robotic swarms.

Paradigm	Principle	Main trade-off
Bottom-up	Generates collective motion through individual planning, control, or local interactions	Explicit individual coordination, but increasing computation and communication
Top-down	Describes the swarm using shared regions, macroscopic variables, or density evolution	High scalability, but limited direct control of individual motion performance

Table 2: Bottom-up and top-down swarm navigation paradigms.

Figure 2: Modified by ( [ Mao et al.(2024)Mao, Fu and Quan ] ). Definition of the virtual tube for spatial navigation. The tube is constructed from the terminal cross-sections $\mathcal{C}_{0}$ and $\mathcal{C}_{1}$ , the generator curve $\gamma$ , intermediate cross-sections $\mathcal{C}_{l}$ , providing a safe traversal space for subsequent single-robot and swarm navigation.

Figure 3: Geometric decomposition of the single-robot command. The nominal component regulates path error, the traversal component advances the robot along the tangent direction, and saturation produces the executable command.

Figure 4: Workflow of model-based optimization. The profile is initialized, propagated forward, corrected with the backward adjoint and gradient, and iteratively updated until convergence.

Figure 5: Admissible energy band and approximation offset in the model-free learning law.

Figure 6: Structure of VIP and its two rollout modes. The energy may describe path tracking error or swarm-distribution error. In the model-in-the-loop mode, rollouts are generated by a nominal simulator before deployment. In the robot-in-the-loop mode, the energy profile is recorded during a physical traversal and the command is updated before the next repeated traversal. Both modes use the same spatial model-free update, whose role is to determine how fast the robot or swarm traverses each path location.

Figure 7: Qualitative inverse relationship between swarm density and traversal speed revealed by the macroscopic fundamental diagram ( [ Safadi et al.(2023)Safadi, Fu, Quan and Haddad ] ).

Figure 8: VIP-based single-drone navigation in two random forest environments.

Obstacle Number	Traversal Time (s)				Avg. Replanning Time (ms)
Obstacle Number	MPCC	Polynomial	VIP	vs. Others	MPCC	Polynomial	VIP	vs. Others
60	34.30	34.39	33.85	-1.31% vs. MPCC -1.57% vs. Poly	5.32	17.66	2.37	-55.5% vs. MPCC -86.6% vs. Poly
80	25.55	27.79	25.90	+1.37% vs. MPCC -6.80% vs. Poly	4.98	21.32	2.62	-47.4% vs. MPCC -87.7% vs. Poly
100	25.05	26.77	25.25	+0.80% vs. MPCC -5.68% vs. Poly	5.44	19.81	2.45	-55.0% vs. MPCC -87.6% vs. Poly

Table 3: Performance comparison of different algorithms (VIP results in bold with relative change)

Figure 9: Numerical comparisons of single robot autonomous navigation in unknown environments.

Figure 10: Numerical comparison of robot swarm navigation in a virtual tube.

Figure 11: Passing-time descent process of swarm navigation under VIP. The curve shows that the learned common traversal command progressively reduces traversal time and then settles as the model-free IL update converges.

Figure 12: Three-dimensional swarm navigation in an obstacle environment. Tube-RRT* constructs a virtual tube through random obstacles, and VIP improves swarm traversal efficiency while the virtual-tube controller preserves safety and distribution regulation.

Figure 13: Experimental platform for LiDAR-based drone navigation.

Figure 14: Aerial view of single-drone fast navigation in an unknown outdoor environment. The recorded flight trajectory covers approximately

150\mathrm{m}

and verifies real-flight execution of the VIP traversal-speed profile under onboard perception and localization.

Figure 15: Repeated single-drone navigation trials with reconstructed point clouds and recorded trajectories. The bird’s-eye and 3-D views show repeatable fast traversal in an unknown outdoor environment using onboard LiDAR perception and VIP-based traversal-command learning.

Figure 16: Experimental platform of the lifting-wing quadcopter.

Figure 17: Outdoor flight trajectory of the lifting-wing quadcopter. The recorded trajectory demonstrates continuous execution of the VIP traversal command through straight, turning, and crossing regions on a physical platform.

Figure 18: Visualization of the lifting-wing quadcopter flight experiment. The image sequence illustrates the real-flight execution process and verifies the compatibility between the learned traversal command and the onboard flight-control architecture.

Figure 19: Iteration-wise traversal time reduction for the lifting-wing quadcopter experiment. The traversal time is first reduced through hardware-in-the-loop simulation iterations and is then further refined in real-world flight iterations, demonstrating the transferability and executability of the learned VIP traversal command on the physical platform.

Figure 20: Image sequence of three-quadcopter swarm flight inside an outdoor virtual tube. The snapshots show coordinated forward traversal while maintaining bounded relative configuration and virtual-tube safety constraints.

Figure 21: Iteration-wise traversal time reduction for the three-quadcopter swarm experiment. The common traversal command learned by VIP reduces the swarm traversal time across iterations, while the virtual-tube controller maintains inter-robot spacing, distribution regulation, and boundary constraints during real flight.

Figure 22: Recorded trajectories of the three-quadcopter swarm in the reconstructed point cloud and bird’s-eye view. The results demonstrate virtual-tube-constrained swarm motion and the execution of the learned common traversal command in real flight.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 23: Kernel-density reconstruction and diffusion-gradient field for swarm-distribution regulation.

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Autonomous mobile robots require timeefficient planning and safety-critical dynamic obstacle avoidance under constrained onboard computation. While Iterative Learning Planning (ILP) offers lightweight and efficient traversal planning, it lacks explicit mechanisms for dynamic obstacle perception and avoidance. This article extends ILP to safety-critical navigation in dynamic environments by integrating an anticipatory risk-blended control barrier function (ARB-CBF). The extended ILP learns traversal-speed and steering-bias profiles via a fractionalpower update based on local obstacle risk, generating nominal control commands that ARB-CBF modifies at runtime for real-time safety guarantees. Algorithmic analysis demonstrates that the ILP replanning stage scales at O(kN) for k iterations and N waypoints, while ARB-CBF executes with linear complexity. Comprehensive simulations and real-world experiments validate the framework, demonstrating superior temporal efficiency and safety with lower computational overhead compared to optimizationbased baselines, making it highly suitable for resourceconstrained platforms.

Abstract

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