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An Efficient Algorithm for Minimum-Pressure Growth Planning of Vine Robots
Andres C. Torres, Tobia Marcucci, Elliot W. Hawkes
TL;DR
Vine robots need growth paths that avoid polytopic obstacles while keeping pressure below burst limits, a criterion existing planners do not directly optimize. The paper derives a general pressure model and converts planning into an efficiently solved shortest-path problem, achieving global optimality in 2D and discretization-dependent approximation in 3D. Simulations and hardware experiments support the method’s speed, scalability, and path ranking.
Problem
The paper addresses finding minimum-pressure paths for vine robots growing between points around polytopic obstacles, because excessive growth pressure can rupture the robot.
Method
The authors derive a growth-pressure equation, characterize optimal paths, discretize 3D obstacle edges, and solve the resulting weighted line-graph problem with modified Dijkstra search.
Results
The algorithm is globally optimal in 2D and approaches global optimality in 3D as discretization becomes finer; a 15,000-obstacle instance runs in 21 seconds.
Takeaways & Limitations
Minimum-pressure planning enables paths that can reach farther into cluttered environments while remaining applicable to teleoperated and autonomous vine robots.
Takeaways & Limitations
In 3D, the returned path is only optimal up to discretization error, although the suboptimality gap vanishes with finer discretization.
Abstract
from arXiv · showhide
Vine robots navigate cluttered environments by extending from their tip. Although their ability to operate in such environments has been extensively demonstrated, little work has addressed growth planning, i.e., finding optimal growth paths. Moreover, existing planners do not account for the growth pressure necessary to follow a given path, which can cause the robot to burst when it is too high. In this paper, we address the problem of finding minimum-pressure paths for vine robots growing around polytopic obstacles. We propose an efficient algorithm that is guaranteed to find globally optimal solutions in 2D and approximate solutions in 3D, with an error that vanishes as a discretization parameter approaches zero. First, we derive a growth pressure equation for vine robots of arbitrary shape, which we use to show that there always exists a minimum-pressure path that is piecewise-linear and can bend only at specific points on the obstacles. We then leverage this observation to reduce the growth-planning problem to a shortest-path problem with time-dependent weights, which we efficiently solve using a modified Dijkstra's algorithm. We demonstrate the speed and scalability of our approach through numerical simulations. We also validate our algorithm with hardware experiments and provide an open-source and high-performance implementation in the Python package, VinePlanner: https://github.com/Ahsoka/VinePlanner.
I. INTRODUCTION
Vine robots must plan paths that avoid obstacles while keeping growth pressure below burst limits, but prior planners generally optimize other objectives. This paper introduces pressure-minimizing planning with a globally optimal 2D method and a 3D extension.
- I. INTRODUCTION: Growth pressure rises with extension and changes in direction, so paths exceeding the robot’s burst pressure may be infeasible.Only a small subset of obstacle-avoiding paths may safely reach the goal.
- I. INTRODUCTION: The paper proposes an efficient planner that finds globally optimal minimum-pressure paths in 2D and discretization-dependent approximate paths in 3D.The method is presented as an efficient growth-planning algorithm applicable to both settings.
- I. INTRODUCTION: Prior vine-robot planners target uncertainty or manufacturing complexity rather than growth pressure, and some assume pressure remains constant.These objectives therefore do not directly address pressure required along a path.
- I. INTRODUCTION: The authors derive a growth-pressure equation for vine robots of arbitrary shape.This generalizes pressure modeling beyond simpler path geometries.
- I. INTRODUCTION: The method is released as the open-source, high-performance Python package VinePlanner.The implementation is identified as one of the paper’s main contributions.
A. Previous growth pressure equation
The paper develops a pressure model for vine robots composed of multiple straight segments and turns. It accounts for yield pressure, length-dependent friction, tail tension, and multiplicative bend friction under slow growth.
- A. Previous growth pressure equation: The model combines yield pressure, length-dependent straight-section friction, tail tension, and bend friction through the Capstan coefficient and turn angle.The turn angle θ is measured in radians, while λ captures friction along straight sections.
- A. Previous growth pressure equation: Sequential turns multiply tension according to Tout,2 = Tholdeµθ1eµθ2, rather than contributing independently.Each bend applies another exponential Capstan factor.
- A. Previous growth pressure equation: The derivation assumes slow growth and therefore omits the velocity term present in the original equations.This is an explicit modeling assumption.
- A. Previous growth pressure equation: The prior multi-segment equation does not accurately capture coupling between length and bends or multiplicative friction accumulation.The paper derives a new equation to model multiple straight segments more accurately.
B. Proposed growth pressure equation
The paper derives a recursive pressure equation for piecewise-linear vine-robot paths, combining friction from segment lengths with exponential tension increases at turns.
- B. Proposed growth pressure equation: Tension increases by segment-length friction and is multiplied exponentially at each turn according to the turn angle.The construction extends from a two-turn example to a robot with n segments and recursively computes endpoint pressure.
- B. Proposed growth pressure equation: The generalized pressure equation applies to piecewise-linear paths, with segment lengths and inter-segment angles as the geometric inputs.The segment length l_i is positive, while each angle θ_i lies in [0, π].
- B. Proposed growth pressure equation: Equation (3c) is restricted to piecewise-linear vine shapes, although a path-independent velocity term can be reintroduced when velocity is not negligible.
C. Extension to piecewise-smooth paths
The paper extends the pressure model from piecewise-linear paths to arbitrary piecewise-smooth paths by accumulating curvature-related angles and breakpoint turns along arc length.
- C. Extension to piecewise-smooth paths: The extension represents a vine robot as a piecewise-smooth path and approximates smooth curves with arbitrarily many straight segments.For a piecewise twice-differentiable path, the limiting construction yields a continuous tension formulation.
- C. Extension to piecewise-smooth paths: The accumulated angle includes both smooth-segment contributions and the angles at corners or kinks, with reversals accumulating rather than canceling.
- C. Extension to piecewise-smooth paths: Substituting the accumulated angle into the tail-tension equation and adding yield pressure produces the general pressure equation.
III. PROBLEM STATEMENT
The problem is to find a collision-free path between two points in 2D or 3D that minimizes the vine robot’s growth pressure at its endpoint.
- III. PROBLEM STATEMENT: Paths must avoid the interiors of finitely many pairwise-disjoint polytopic obstacles.The obstacle-avoidance constraint applies at every point along the path.
- III. PROBLEM STATEMENT: The planner seeks a curve connecting x_init to x_final that minimizes endpoint growth pressure.Both the path γ and its length L are optimization variables, and pressure is given by the generalized pressure equation.
IV. GROWTH PLANNING ALGORITHM
The growth-planning algorithm reduces the search to piecewise-linear paths through obstacle ridges and solves the resulting pressure problem with a line-graph shortest-path method.
- IV. GROWTH PLANNING ALGORITHM: The algorithm is globally optimal in 2D and optimal up to discretization error in 3D.
- IV. GROWTH PLANNING ALGORITHM: A feasible problem always has a globally optimal piecewise-linear solution whose breakpoints lie exclusively on obstacle ridges.In 2D, ridges are obstacle vertices; in 3D, they are obstacle edges.
- IV. GROWTH PLANNING ALGORITHM: A modified Dijkstra algorithm solves the resulting shortest-path problem because edge costs depend non-additively on accumulated tension.The line-graph formulation assigns turn- and path-dependent costs to model the pressure objective.
- IV. GROWTH PLANNING ALGORITHM: The method constructs a visibility graph, then a directed line graph whose edges represent physical turns and retain the angle information needed for pressure costs.
B. Shortest-path problem
The planner reformulates minimum-pressure growth as a time-dependent shortest-path problem and solves it with a generalized Dijkstra algorithm. In 2D, the resulting path is globally optimal, while 3D solutions approach optimality as obstacle-edge discretization becomes finer.
- B. Shortest-path problem: The line graph assigns initial tension at the source, terminal length cost at the target, and recursively computed tension increases to intermediate edges.These weights encode the pressure consequences of traversing path segments and bends.
- B. Shortest-path problem: A shortest path in the weighted line graph corresponds to minimum-pressure growth, with pressure obtained from the optimal cost, area, and yield pressure.The optimal shortest-path value is divided by A and added to Y.
- B. Shortest-path problem: Because edge costs depend on accumulated tension rather than adding independently, the planner uses a time-dependent shortest-path formulation and generalized Dijkstra algorithm.Tension plays the role of time in this formulation.
- B. Shortest-path problem: The generalized Dijkstra method is valid because the edge costs satisfy consistency and nonnegativity conditions.Consistency follows from positive exponential factors, while nonnegativity follows from nonnegative friction, angles, tension, and lengths.
- B. Shortest-path problem: In 2D, reconstructing the path from the visibility graph yields a globally optimal solution, whereas 3D requires discretizing obstacle edges.The 3D search space is infinite without discretization; the resulting suboptimality gap vanishes as discretization becomes finer.
V. EXPERIMENTS
Experiments evaluate VinePlanner’s implementation and runtime on simulated and hardware obstacle courses. The planner handles a dense 15,000-obstacle instance in 21 seconds, while a shorter path can require dangerously higher pressure.
- V. EXPERIMENTS: The experiments use VinePlanner, an open-source Python library optimized with Numba-parallelized visibility-graph construction.Simulations ran on a Windows 10 machine with an 8-core AMD Ryzen 7 7800X3D processor and 64 GB of DDR5 RAM.
- V. EXPERIMENTS: For courses with hundreds of obstacles, runtime is a few tens of milliseconds, and empirical scaling is approximately O(n^2).The scaling study uses randomly generated 2D courses spanning roughly 100 to 200,000 obstacles.
- V. EXPERIMENTS: On the 15,000-obstacle problem, the minimum-distance path reaches 19,430 kPa, over 60 times the planner’s 318 kPa, for only a 4% length reduction.The minimum-distance path accumulates roughly twice the total path angle, producing exponentially higher pressure and exceeding vine-robot burst pressures.
- V. EXPERIMENTS: Hardware validation uses a 24 mm-diameter, 0.05 mm-thick LDPE vine robot and a modular voxel-and-acrylic obstacle course.The robot grows through a 90° outlet in an airtight PVC tee fitting.
- V. EXPERIMENTS: The planner solves a dense course with 15,000 obstacles in 21 seconds, despite a visibility graph of 54,712 vertices and a line graph with 801,008 vertices and 12,738,916 edges.The dense-course result is also represented by the gray-dashed data points in Fig. 3.
1) Two-dimensional obstacle course:
The experiments validate that the pressure model correctly ranks candidate paths in both 2D and 3D, while the planner’s minimum-pressure solutions support operation under pressure constraints. The study also identifies assumptions and limitations affecting deployment scope.
- 3D obstacle course: 100% of six 3D trials recovered the correct ordering of four paths whose predicted maximum pressures differed by at least 1 kPa.The experiment added two uniformly spaced points per obstacle edge and observed intertrial pressure variation.
- Implications: The planner can address much more complicated environments and design paths that reach much farther into them by finding minimum-pressure paths or close approximations in 3D.The conclusion attributes the broader planning scope to the algorithm’s speed and pressure optimization.
- 3D obstacle course: Only the minimum-pressure path remains feasible below a hypothetical burst pressure of 21.5 kPa in the 3D hardware demonstration.The four candidate paths were separated by at least 1 kPa in predicted pressure.
- Limitations: The method assumes a fully known environment, lacks an exact 3D suboptimality gap, omits gravity, and does not test active actuation.The authors identify online planning, convergence quantification, gravity modeling, and actuator-inclusive experiments as future work.
- Limitations: The pressure model ranks paths correctly but shows some inaccuracy in predicted absolute pressures and requires experiments with lower variability across more paths.The reported experiments manually placed the robot in prescribed paths rather than actively steering it.
APPENDIX
The appendix proves that an optimal growth path can be chosen piecewise-linearly, with bends restricted to obstacle ridges. The proof uses local straight-line shortcuts that preserve or reduce pressure while shortening paths.
- Piecewise-linear optimality: A globally optimal solution exists that is piecewise-linear.Replacing a curved free-space segment with a straight segment does not increase the final pressure, so the replacement remains optimal.
- Piecewise-linear optimality: Straightening a curved segment preserves optimality because the original path’s pressure is at least that of the shortcut, while the paths coincide after the shortcut endpoint.The proof establishes the pressure inequality through tension comparisons and endpoint-tension monotonicity.
- Geometric consequence: The appendix’s shortcutting argument supports restricting the search to piecewise-linear paths rather than curved paths.The stated conclusion is that curving, rather than bending, never reduces growth pressure.
- Obstacle-ridge breakpoints: A globally optimal piecewise-linear solution can be chosen with breakpoints exclusively on obstacle ridges.If a breakpoint lies off a ridge, a collision-free straight shortcut is strictly shorter without increasing pressure, contradicting minimum length among optimal solutions.