Parallel Simulation of Contact and Actuation for Soft Growing Robots

Abstract

Soft growing robots, commonly referred to as vine robots, have demonstrated a remarkable ability to interact safely and robustly with unstructured and dynamic environments. It is therefore natural to exploit contact with the environment for planning and design optimization tasks. Previous research has focused on planning under contact for passively deforming robots with preformed bends. However, adding active steering to these soft growing robots is necessary for successful navigation in more complex environments. To this end, we develop a unified modeling framework that integrates vine robot growth, bending, actuation, and obstacle contact. We extend the beam moment model to include the effects of actuation on kinematics under growth and then use these models to develop a fast parallel simulation framework. We validate our model and simulator with real robot experiments. To showcase the capabilities of our framework, we apply our model in a design optimization task to find designs for vine robots navigating through cluttered environments, identifying designs that minimize the number of required actuators by exploiting environmental contacts. We show the robustness of the designs to environmental and manufacturing uncertainties. Finally, we fabricate an optimized design and successfully deploy it in an obstacle-rich environment.

Keywords

modeling control learning for soft robots soft growing robots GPU-parallel simulation

Introduction

Soft growing robots, also known as vine robots, are a class of continuum robots that extend from the tip by pressure-driven eversion.¹ Since growth isolates the robot from the environment, vine robots have shown significant beneficial behaviors in tasks with natural pathways to follow, such as medical procedures,^2–4 especially for colonoscopies and endoscopies,^5–7 pipe inspection,⁸ and archaeology.⁹ These behaviors arise from the passive buckling of thin inflated tubes,^10,11 a challenging feature to accurately model.

Multiple works have examined kinematic models to predict and use this passive deformation.^12,13 However, the heuristic models in these works have been limited to purely passive behavior, primarily due to the increased difficulty of modeling the effects of active steering. Other works have also addressed the general simulation of vine robot growth with at most some preformed deformations, including fast, kinematic-only models¹⁴ and dynamic simulations.^15,16 While slow and accurate Finite Element Method (FEM) simulators¹⁷ have been used to model actuated vine robots,¹⁸ these are inefficient for downstream tasks such as planning, control, and design that require high-throughput simulation. Recent work has addressed fast parallel simulation of these robots that accurately captures the bending and buckling behavior intrinsic to vine robots.¹⁹ However, no simulation framework has addressed efficient modeling of actuated vine robots under contact forces.

While previous studies have captured the obstacle interactions of vine robots under passive deformation, this work achieves the integration of all key capabilities, including growth, pneumatic actuation, beam mechanics, and environmental contact. In this work, we provide for the first time the tools to design, model, control, and plan actuation for vine robots in contact-rich, complex environments. To validate our framework, we deploy the optimized designs in real environments and compare the performance with simulated predictions. Our major contributions include:

Unified modeling of growth, steering actuation, bending, and contact: We present a new modeling framework that captures how vine robots buckle, bend, deform, and extend while growing into obstacles with active steering. We chose series pneumatic artificial muscles (sPAMs)²⁰ as bending actuators for this study. Our modeling framework independently considers the effects of the actuator and inflated body when calculating the moment balance discretized along the length of the vine robot. Importantly, this leverages a continuous form of the thin-walled inflated tube buckling model that includes partial buckling reactions under small angle deflection²¹ called the wrinkling-based bending moment model. Thus, we can predict both distributed actuation seen during steering and concentrated actuation seen under environmental contact. Sec. 3 combines the models of sPAM actuation and wrinkling-based restoring moment to calculate the net restoring moment that the vine robot segment experiences.

Fast parallel simulation framework: GPU-parallel simulation has been enabled by efficient parallel computation libraries such as PyTorch and JAX. We build upon a prior parallel simulation framework¹⁹ for fast and accurate prediction of actuated vine robots navigating cluttered environments. We use our novel combined actuation, bending, and contact model and switch to a faster gradient-based penalty method to run a batch of simulations in parallel with orders of magnitude speedups. Sec. 4 describes the details of position-based dynamic simulation. To perform fast computations of sPAM actuators, a neural surrogate model is developed as detailed in Sec. 4.3. Sec. 5 describes the design optimization framework and the planner algorithm for batched SST∗.

Application for long-horizon design optimization task: To demonstrate the capability of our model for long-horizon planning, we apply it in a design optimization task: finding vine robot designs that minimize the number of actuators needed to navigate a complex, cluttered environment. The resulting designs successfully reach the target location when deployed in the real environment, demonstrating real-world applications of our framework. Sec. 6 describes the experimental and simulation results while analyzing the funneling and robustness behaviors.

In summary, our work presents a unified computational framework to optimize the design of vine robots. These advances enable the implementation of simulation tools to deploy optimized designs in real, cluttered environments. We release our simulator and design optimization tool as open source: https://github.com/CoMMALab/ActVineSimPy.

Related Work

The flexible, yet inextensible material used in vine robots necessitates the use of novel approaches for modeling and simulation of vine robot behavior. While early modeling focused primarily on describing the growth,²² first-principles models for vine robot buckling have been built on inflated thin-shelled beam models.^10,23 Discrete buckling models, which predict restoring moment independent of bend angle,^11,24 led to kinematic models for contact-induced bending,¹³ tip localization,²⁵ and obstacle mapping.¹² Recent work has further strengthened this model by considering partial buckling.²¹

For distributed surface wrinkling, these constant moment models are not as accurate, so heuristic models have been developed instead, using experimental parameter fitting to create kinematic models,²⁶ or geometrically relating surface strain to resulting general curves.^27,28 The geometric mappings are generally limited to quasi-static behaviors, though data-driven approaches with Koopman operators have shown that dynamic behaviors can also be empirically modeled.²⁹ While combined system models have been primarily heuristic, some first-principles actuator models have been developed for series pneumatic artificial muscles (sPAMs)²⁶ and a range of other common and vine robot-specific actuators.³⁰

While actuator-beam models describe free-space actuated vine robots, incorporating contact with the environment is challenging. Finite element models built on frameworks like SOFA¹⁷ have been used to successfully model vine robots.^16,18,31,32 However, these approaches are computationally expensive and ill-suited for real-time, at-scale simulation. Other work has simplified the simulations to rigid body models, either using minimal coordinates through virtual joint angles³³ or maximal coordinates with poses of frames in a global reference with implicit constraints.¹⁵ Previous work has proposed more realistic models for computing strain that respect geometric design parameters^28,34 and has made these scalable using GPU-accelerated computation frameworks.¹⁹ Overall, our work fills a gap for efficient models that generalize to external and internal forces.

Our design optimization approach is built upon sampling-based kinodynamic planning (SBKP),^35–37 which addresses the general problem of finding a sequence of controls that, when applied to a robot, achieves a goal state. Unlike previous works that use planning to design a vine robot³⁸, we are given only a black-box forward dynamic simulation that captures the vine robot evolution given actuation and environment interactions. Some planners, such as Kinodynamic-RRT* for linear dynamics,³⁹ arbitrary kinodynamic planning in combination with the AO-X meta algorithm,⁴⁰ and Stable Sparse RRT (SST),³⁷ provide asymptotic optimality guarantees, but with no guarantee on when high-quality solutions will be found. Thus, in practice, accelerating the speed of dynamic simulation, the bottleneck of SBKPs, helps them find better solutions faster. Recent work such as Kino-PAX⁴¹ has focused on highly parallel implementations to address these computational challenges. Our approach also leverages the large-scale batch GPU computation of our proposed simulator to improve performance.

Modeling of Combined Actuation and Growth

Accurately modeling actuated soft growing robots under environmental contact presents a challenge due to the nonlinear coupling between actuator force and displacement, beam stiffness or restoring moment, and contact forces. While an actuated robot deployed in free space can be modeled heuristically as one or more constant curvature segments with radius of curvature inversely proportional to actuator pressure,²⁶ and the path of an unactuated vine robot under environmental contact can be robustly predicted based on the visibility graph of the obstacles,¹⁴ these heuristic models do not obviously or easily combine. To address this complexity, we develop a unified modeling approach that builds on analytical models of each component separately and then discretizes the vine robot along its length to locally capture the effects of contact and actuation. With this model, the vine robot’s dynamics can be simulated in cluttered environments, where contact interactions significantly influence robot behavior and thus can be exploited for improved navigation performance. In this section, we present each component model separately and then discuss the method to combine the models to accurately predict both actuation and contact with the environment.

Serial pneumatic artificial muscles

We employ serial pneumatic artificial muscles (sPAMs) to apply distributed actuation along the robot body. A sPAM can be fabricated using a pliable thin-film tube, which has an inflated radius $R_{act}$ , and constricting that tube to radius $R_{c}$ (where $R_{c} ≪ R_{act}$ ) at regular intervals $l_{0}$ , forming a chain of bubbles. When the actuators are completely deflated, they start at full extension (Fig. 1), with constrictions at interval lengths of $l_{0}$ . Upon pressurization, the actuator expands radially and, as a result, contracts. The radial expansion continues until the maximum radius of the actuator unit reaches the limit, $R_{act}$ , saturating the response for that part of the actuator cell. Further pressurization increases this saturated region, further shortening the actuator and decreasing the active region.

FIG. 1.

(a) and (b) Variable definitions for the sPAM and inflated beam model. (c) Illustration of sPAM states from completely deflated with zero strain (top) to saturated with maximal strain (bottom). Darker color indicates higher input pressure and greater contraction. sPAM, series pneumatic artificial muscle.

To make an actuator component model, we build upon the refined sPAM model in study by Wang and Blumenschein,³⁴ which adapts the ideal PPAM (Pleated Pneumatic Artificial Muscle) model in study by Daerden.⁴² The ideal PPAM model does not account for the saturation based on the tube radius, only predicting the cross-section profile, which maintains the material length and maximizes volume. The improved sPAM model incorporates actuator saturation by moving the saturated length to shrink the effective length of the actuator when predicting the force–strain output. For a given set of actuator design parameters and pressure inputs:

\frac{E (ϕ_{R c}, m) - \frac{1}{2} F (ϕ_{R c}, m)}{\sqrt{m} \cos ϕ_{R c}} = \frac{l_{a}}{2 R_{c}} (1 - \frac{l_{0}}{l_{a}} ε),

(1a)

\frac{F (ϕ_{R c}, m)}{\sqrt{m} \cos ϕ_{R c}} = \frac{l_{a}}{R_{c}} (1 + \frac{a}{2 m \cos^{2} ϕ_{R c}}),

(1b)

a = \frac{P_{act} R_{c}}{2 t E}, ϕ_{R c} = \cos^{- 1} (\frac{R_{c}}{R_{act}})

(1c)

F_{t} = π P_{act} R_{c}^{2} \frac{1 - 2 m}{2 m \cos^{2} ϕ_{R c}} .

(1d)

Here, $ε$ denotes the strain, $F_{t}$ represents the contraction force. $m \in (0, 0.5]$ and $ϕ_{R c} \in (0, \frac{π}{2}]$ are dimensionless model parameters, with the former denoting the extent of contraction relative to the full contraction and the latter proportional to the maximum radius of the sPAM unit at a given state. $l_{a}$ and $l_{0}$ denote the active and total actuator lengths, respectively, and $F (a, b)$ and $E (a, b)$ are incomplete elliptic integrals of the first and second kind. The pressure-dependent correction factor $a$ relates the actuator’s material properties to force–strain computation. While in the original PPAM model, this correlated with material elasticity, $E$ , and pressure, $P_{act}$ , with the relatively inextensible tube and constraint materials, we instead treat this parameter as an empirically determined constant ( $a = 0.0001$ ). The model accounts for saturation effects by distinguishing between $l_{a}$ and $l_{0}$ and by setting $ϕ_{R c}$ based on the saturation and constraint radii [Eq. (1c)]. Setting $l_{a} = l_{0}$ and treating $ϕ_{R c}$ as unknown instead recovers the PPAM model applicable to unsaturated scenarios. To generate the force–strain output pairs needed for the neural surrogate modeling described in Sec. 4.3, we vary $m$ over $m \in (0, 0.5]$ . For each $m$ , we solve the governing equation for $ε$ , and in the unsaturated case, for $ϕ_{R c}$ . We then compute the contraction force $F_{t}$ using these solutions and the prescribed design parameters.

Wrinkling-based restoring moment model

The body of the vine robot is made of a pressurized thin-film tube, which we can treat as an inflated beam. An inflated beam model developed by Comer and Levy¹⁰ demonstrates that the restoring moment of a bent inflated beam increases with surface wrinkling development at bending locations. Previous implementations of this model have focused on the maximum restoring moment when the surface is fully wrinkled, especially in works that discuss the buckling of the vine robot under environmental forces.¹³ However, for small deflection angles (≲10 degrees), like those seen in steering (Fig. 2a), this significantly overestimates the restoring moment. A wrinkling-based model described in study by Chen¹⁹ captures this angle-to-moment relationship by introducing a critical surface strain $ε_{critical}$ at which wrinkling initiates. The critical strain is obtained by fitting experimentally measured deflection-restoring moment data. The restoring moment $M$ is computed as follows:

M = π P_{vine} R_{vine}^{3} \frac{\sin (2 γ_{0}) + 2 π - 2 γ_{0}}{4 (\sin γ_{0} + (π - γ_{0}) \cos γ_{0})},

(2a)

γ_{0} = \cos^{- 1} (2 \frac{ε_{critical}}{\sin (θ_{curv} / 2)} - 1) .

(2b)

FIG. 2.

(a) Vine segment bending under sPAM actuation. (b) Vine segment wrinkled and tensioned regions depicting $γ_{0} .$ (c) Comparison of measured bending angles with theoretical estimations using wrinkling-based model and constant moment model.

Here, $P_{vine}$ denotes the pressure of the vine robot and $R_{vine}$ denotes the radius of the vine robot as shown in Figure 1. $θ_{curv}$ denotes the turning angle at the point of wrinkling. $γ_{0}$ represents the angle corresponding to the wrinkled surface of the vine robot as shown in Figure 2b and defined in Eq. (2b). Importantly, this model calculates the restoring moment as proportional to the total axial material tension, summed around the tube circumference, that is, $π P_{vine} R_{vine}^{2}$ . If the vine robot is growing, $P_{vine}$ cannot exceed the conditions of quasi-static equilibrium for growth, which are based on the vine robot material and geometry, the internal material tension, and the growth speed.²² In practice, the effects of growth speed are negligible, so a value for $P_{vine}$ can be set by modifying the internal material tension.

Combined actuation and contact model

To combine the component models, we can recognize that the moments exerted by the sPAMs and by the bending of the vine robot segment should balance in free space and combine to produce a net restoring moment under dynamic conditions. Without external contact, this equilibrium relationship enables the determination of both the strain produced in the vine segment and the radius of curvature in free space. Compared with previous approaches, the wrinkling-based model improves the accuracy of curvature determination, as shown in Figure 2c.

For developing the forward dynamics simulation that includes contact with the environment, the vine robot is discretized into rigid, serially connected segments. This allows the bending behavior of each segment to be considered separately. The segments have a length of $l_{seg} = 25 mm$ , excluding the last growing segment $l_{last}$ , which grows with a predefined growth rate. The value 25 mm was chosen for the segment length because it afforded a good balance between granularity and computational cost, allowing enough discretized segments to provide a reasonably accurate model without introducing too much extra computation.

If the relative bending angle between any two consecutive segments of the vine robot is $θ$ , then the actuator strain $ε$ corresponding to the bending angle $θ$ and net restoring moment $M_{tot} (θ)$ for that segment at each time step is calculated as follows:

ε = \frac{(2 R_{vine} + R_{act}) θ}{l_{seg}},

(3a)

M_{tot} (θ) = M (θ) - F_{t} (m, ϕ_{R c}, P_{act}) \cdot (2 R_{vine} + R_{act}) .

(3b)

Here, the restoring moment of the vine robot segment $M (θ)$ is calculated using Eq. (2). The actuator force $F_{t}$ and the geometric parameters $(m, ϕ_{R c})$ for the actuator are calculated using Eq. (1).

Simulation of Actuation and Growth

Although the individual bending, growth, and contact behaviors are well defined, producing a realistic simulation requires unifying these forces into a single forward function. We combine these forces in a cost-minimization problem, which we use to solve for the next state with a position-based dynamics (PBD)⁴³ simulation method, which has shown success for rigid and soft body simulations alike.⁴⁴ The pipeline leverages GPU-accelerated batch processing to achieve the throughput requirements necessary for effective design optimization in complex environments. Algorithm 1 details the single simulation step that updates the vine robot position forward in time. This step essentially combines the effects of vine robot growth, bending, actuation, and obstacle contact to generate the next state of the vine robot.

Position-Based dynamic simulation

As described in Sec. 3.3, we chose to discretize the vine into a chain of bodies to produce a finite parameterization. We aim to find the dynamics ${θ}_{t + 1}, l_{last} = f ({θ}_{t}, l_{last})$ . $l_{last}$ is the length of the distal segment; all others are held at a fixed length $l_{seg}$ . ${θ}$ is the set of relative bending angles between consecutive bodies in the vine; they describe the local curvature at each point. This rigid body representation was also used in study by Jitosho et al. and Chen,^15,19 albeit with maximal coordinates. Note that maximal coordinates require extra constraints to ensure that successive bodies remain connected; our minimal coordinate formulation can only produce connected vines. Here, we formulate the simulation as finding the next state with minimum cost:

sim ({θ}^{*}) = \arg \min_{{θ}} w_{a} M_{tot} ({θ}) + w_{b} growth (l_{last}) + w_{c} contact ({θ}, l_{last}, X_{obs}),

(4)

where “growth” tends to zero as

l_{last}

increases with a predefined growth rate, “contact” measures the amount of penetration of bodies into obstacles, and

M_{tot}

is the restoring moment of the current state as found in Eq. (3). Often, such forces will be in opposition—for example, when the vine is growing into an obstacle, then the growth term will attempt to lengthen

l_{last}

while “contact” will attempt to shorten it. We empirically scaled the weights

w_{a}

w_{b}

, and

w_{c}

to ensure that in all common cases, contacts dominate over moments which in turn dominate growth, thus forbidding segments from growing into the obstacles, with

w_{a} = 160

w_{b} = 15

, and

w_{c} = 1

. An exception to this occurs when the tip gets stuck in a corner or head-on collision, causing growth to dominate bending moments as the vine coils up on itself. Note that we omit velocity and inertial terms as they are insignificant due to the high air resistance and low mass of the vine.

Equation (4) is highly nonlinear, and the relative magnitudes of each force can change rapidly. We use a penalty-based gradient method, which iteratively searches for the minimization of Eq. (4). We compute gradients using JAX autodifferentiation⁴⁵ and found 50 steps to be sufficient for convergence in all environments. The details of the implementation are presented in Algorithm 1.

Algorithm 1 Single simulation step

1: Input: Initial state ${θ}_{t}$ , last length l_last, cost weights $w_{a}$ , $w_{b}$ , $w_{c}$

neural surrogate parameters $ϵ$ , $l_{0}$ , learning rate $α$ , obstacle set $X_{obs}$

2: Initial vine state: ${θ} \leftarrow {θ}_{t}$

3: $m, ϕ_{R c} \leftarrow neural surrogate (ϵ, l_{0})$

4: Actuator force $F_{t} \leftarrow E q . (1 d) (m, ϕ_{R c})$

6: for k = 1 to 50 do

7: Restoring moment $M \leftarrow$ Eq.(2) $({θ})$

8: Total moment $M \leftarrow$ Eq.(3) $(M, F_{t})$

9: Optimization Objective:

$J = w_{a} M_{tot} + w_{b} growth (l_{last}) + w_{c} contact ({θ}, l_{last}, X_{obs})$

10: Gradient: $\nabla_{θ} J \leftarrow$ autodifferentiate(J, θ)

11: Gradient descent step: ${θ} \leftarrow {θ} - α \nabla_{θ} J$

12: end for

13: return Optimized vine state ${θ}$

Actuator design synthesis

Although it is desirable to parameterize actuators along the vine in terms of their produced turning angle $θ_{curv}$ , actuators are actually parameterized by pressure and length $P_{act}$ and $l_{0}$ , for simulation and fabrication. Finding these two parameters from $θ_{curv}$ is nontrivial, since multiple solutions of $P_{act}$ and $l_{0}$ can satisfy any given curvature $θ_{curv}$ . We address this redundancy by isolating the specific portion of the sPAM model that leads to this non-uniqueness and introducing a cost term to favor an unambiguous single solution out of the space of all possible $P_{act}$ and $l_{0}$ . First, we solve for the actuator length $l_{0}$ and strain $ε$ that satisfy the target curvature $θ_{curv}$ as specified by Eq. (2) and Eq. (1d). Second, we compute the corresponding parameters $ϕ_{R c}$ and $m$ by solving the nonlinear system in Eq. (1) using a gradient-based root-finding method; the Levenberg–Marquardt (LM) method⁴⁶ proves effective, achieving convergence within 1 × 10 ⁻ ⁶ tolerance. To avoid local optima inherent in gradient methods, we initialize the LM solver with the lowest-cost candidate from 1000 randomly sampled starting points. Each root-finding iteration requires evaluation of Legendre elliptic integrals of the first and second kind, which lack analytical forms and are approximated to 1 × 10⁻¹⁰ precision using 10 expansions of the Taylor series. This two-stage approach eliminates the one-to-many relationships exhibited in the first step, ensuring that a single solution is found in both stages of the actuator synthesis process.

To resolve redundancies, we impose a cost function that favors higher pressure $P_{act}$ over larger actuator length $l_{0}$ , except where such preferences would yield infeasible solutions. When multiple valid combinations of $P_{act}$ and $l_{0}$ exist, we select the design with the highest pressure from the feasible set. This selection criterion is motivated by experimental observations that higher pressures produce more consistent curvatures despite fabrication tolerances and modeling uncertainties. Although the parameter sets ${{l}_{0}}$ and ${{P}_{act}}$ form continuous one-dimensional manifolds in the design space, we restrict sampling to discrete values due to physical fabrication constraints. Specifically, our fabrication process is limited by a 1 mm accuracy for $l_{0}$ , and we are only capable of providing pressures from a discrete set, which limits the available parameter values. This process ultimately yields optimal pressure $P_{act}^{*}$ and corresponding actuator length $l_{0}^{*}$ . We also note that throughout this work, we maintain constant values for several parameters that were excluded from the notation: constriction radius $R_{c} = 5$ mm, actuator tube radius $R_{act} = 17.18$ mm, vine segment radius $R_{vine} = 33.35$ mm, and vine robot growth pressure $P_{vine} = 1.5$ psi.

Neural surrogate

As detailed in Sec. 4.2, actuator design parameter generation requires precise solutions to the sPAM model in Eq. (1). While numerical methods provide reasonable performance for individual solutions (requiring only milliseconds), we wish to use our actuator model in a design optimization task, which demands hundreds of thousands of simulation steps, corresponding to millions of individual sPAM model evaluations. In this high-throughput regime, numerical methods become computationally prohibitive. We instead approximate the sPAM model using a neural network surrogate. Analysis of the function reveals smooth and monotonic behavior of the output with respect to input parameters. The absence of high-frequency features and the single-mode nature of the function make accurate approximation by a neural surrogate possible, as such networks excel at fitting smooth, well-behaved data. Furthermore, neural networks can be evaluated efficiently in parallel.

The input parameters for the neural surrogate are sPAM actuator strain ϵ and actuator length $l_{0}$ , and the outputs are geometric parameters $m$ and $ϕ_{R c}$ . This is illustrated as step 3 in Algorithm 1. sPAM parameters $m$ and $ϕ_{R c}$ are required to calculate the actuator force $F_{t}$ in Eq. (1d). This is step 4 in Algorithm 1. The training data for the neural surrogate is generated by uniformly sampling sPAM actuator parameters $m \in (0, 0.5]$ and $l_{0} \in [0.02, 0.08]$ m and running them through our full analytical model from Eq. (1) to get the corresponding values of $ϕ_{R c} \in (0, \frac{π}{2}]$ and $ϵ \in [0, 1]$ . Through hyperparameter optimization, we determined that a 2 × 32 × 2 multilayer perceptron with Rectified Linear Unit (ReLU) nonlinearities⁴⁷ and He initialization⁴⁸ provided the best balance of speed and accuracy. We trained the surrogate with an 80/10/10 train/test/validation split on 40,000 ground truth samples. Performance evaluation results are presented in Figure 3. The surrogate achieves four orders of magnitude improvement in throughput (at experimental batch sizes) while maintaining average MSE loss of 0.003 (with outputs normalized). These computational speeds enable simulation faster than real time, making our design optimization tractable for complex environments.

FIG. 3.

Left: The neural surrogate demonstrates a major speedup over the baseline unbatched method, especially as the batch size increases. Batched GPU processing can greatly accelerate parallelizable simulation tasks. Right: Distribution of neural surrogate squared error as compared with ground truth numeric method. We uniformly sampled 40,000 values over the surrogate’s input space and normalized the outputs.

Long-Horizon Design Optimization

In contact-rich environment navigation problems, contacts, nonlinear bending mechanics, and growth dynamics jointly determine vine behavior. Analytical or inverse numerical solutions for vine design under these coupled effects are either intractable or computationally challenging, necessitating approaches that use forward simulation. We formulate the design optimization problem for soft growing robots as a sampling-based kinodynamic planning (SBKP) problem.³⁶ SBKPs provide a framework for solving problems that can only be explored with forward dynamic simulation—in our problem, we simulate the vine robot under varying actuator design parameters and propagate the vine robot forward in time by simulating constant growth with sampled actuator designs.

Specifically, we use a modification to the Stable Sparse RRT* (SST*)³⁷ algorithm, which provides asymptotic near-optimality guarantees for finding dynamic paths between points given a cost. That is, as the number of samples drawn goes to infinity, the cost of the solution converges to $1 + ϵ$ times the optimal solution. Specifically, we use this approach to enable the discovery of robot designs that minimize actuator requirements to achieve navigation goals, often by exploiting environmental contact.

We modify SST* to make use of the simulation framework from Sec. 4 to provide batch simulation capabilities for the planner, expanding many nodes simultaneously, rather than one at a time. At each iteration, the planner chooses a varying time of execution and actuation parameters to propagate, sampling from existing nodes in the search tree to grow from—the nodes in the search tree are parameterized by the pose of the tip of the vine in space. An example of how the planner progresses its search with these batch rollouts is shown in Figure 4.

FIG. 4.

Progression of planning in the long environment, illustrating rapid initial exploration followed by convergent optimization. The planner identifies an initial solution by iteration 5, then refines the design through exploration of alternative contact-exploitation strategies. The colorbar on the right represents the number of actuated segments.

Reverse tree heuristic

To reduce SST*’s computational cost, we incorporate a reverse tree search heuristic that uses geometric (not dynamic) rollouts of designs to guide exploration toward goal-reaching areas. Cost-to-go heuristics improve search efficiency in methods like AIT*⁴⁹ and GBRRT,⁵⁰ which employ bidirectional trees where the reverse tree provides guidance rather than a direct connection. Our reverse tree uses RRT*⁵¹ in OMPL⁵² with biarc interpolation, providing asymptotic optimality. Biarcs are two tangentially continuous circular arcs and align well with the piecewise constant curvature behavior of actuated vine segments.²⁸ We modify SST*’s cost function to include both cost-to-come and estimated cost-to-go from the reverse geometric tree, which establishes upper bounds on path costs used by the kinodynamic planner. The objective function minimizes the number of curved segments (actuator count) with path length as a tie-breaker. For state validity checking, we verify that the curvature of sampled biarcs $θ_{curv}$ lies within numerically determined feasible minimum and maximum values, rather than computing explicit actuator designs, maintaining planning soundness while reducing computation.

Planner algorithm

The details of the planner are shown in Algorithm 2. The planner is initialized with an empty tree data structure $T$ and a predetermined start state $x_{start}$ , goal region $X_{goal}$ , obstacle list $X_{obs}$ , and batch size $B$ . States in the planner include a full robot state ${θ}$ and the pose of the tip of the robot $p \in S E (2)$ . The start state is inserted into $T$ . Next, $B$ states are selected for expansion $x_{exp}$ , prioritizing states with the lowest cost. This cost is the sum of the cost of actions taken to reach the state and the cost-to-goal heuristic of the nearest node in the reverse geometric tree. In the first iteration, only the start state can be expanded. The planner then samples random bending controls and the number of timesteps $T$ to propagate. After repeating the simulation step, Algorithm 1, $T$ times, the resultant states $x_{res}$ are found. However, because there are $B$ states, with many overlapping, a pruning step is necessary to maintain optimality. States are “overlapping” if the robot’s tip is within a diminishing radius $r$ according to an $S E (2)$ metric. This radius is reduced by a decay factor after certain iterations determined by a scheduling function; we choose a value of 60 for the initial radius (in the $S E (2)$ metric space, which combines millimeters and angular distance) and reduce it by multiplying by $α = 0.8$ . Among the overlapping states, the planner keeps the state with the lowest cost and prunes the others to maintain the sparsity of the tree. The remaining states after pruning are inserted into T. Finally, the planner checks if any of the inserted states fall within the goal region. If the goal is not reached, the planner reselects states to expand for the next iteration.

Once the goal has been reached, the solution path can be found by following the parents of the leaf up to the root of the tree, which is the start state. Because the edges of the tree represent actions taken to go from the parent to child state, we can always find the series of actions needed to traverse from the starting state to any other state in the tree.

Algorithm 2 Batched SST*

1: Input: $x_{start}, X_{goal}, X_{obs}, B, r_{init}$

2: Initialize: tree $T \leftarrow {{x}_{start}}$ , solution $X_{sol} \leftarrow {}$ , radius $r \leftarrow r_{init}$

3: $i \leftarrow$ 1, $j \leftarrow$ 1, sst_iter₀ $\leftarrow$ 7

4: next_update_iter ← sst_iter₀

5: while not isGoal( $T, X_{goal}$ ) do

6: $x_{exp}^{0} \leftarrow sample (B, T)$

7: $ϵ, l_{0}, T \leftarrow$ sample-action (B)

8: for $t = 1$ to $T$ do

9: $x_{exp}^{t} \leftarrow$ Algorithm 1 $(x_{exp}^{t - 1}, ϵ, l_{0}, X_{obs})$

10: end for

11: $x_{res} \leftarrow x_{exp}^{t - 1}$

12: $x_{valid} \leftarrow$ prune( $x_{res}, r$ )

13: $T \leftarrow T \cup x_{valid}$

14: if $i$ = next_update_iter then

15: $r \leftarrow α r, j \leftarrow j + 1$

16: next_update_iter $\leftarrow i$ + ${sst_iter}_{0} \cdot (1 + \log_{2} (j)) \cdot α^{- (d + l + 1) j}$

17: end if

18: $i \leftarrow i + 1$

19: end while

20: $X_{sol} \leftarrow$ solution( $T$ )

21: return $X_{sol}$

Experimental Results

With our simulation framework established, we now seek to evaluate the model and simulator against physical vine robot behavior and assess the effectiveness of the simulator in enabling the design optimization framework across a number of challenging environments. In addition, inspired by the previous analysis of vine robot planning using contact to navigate environments,¹³ we use the simulator to evaluate how contact with the environment affects the robustness of planned robot designs and how it reduces down the variation in vine robot states after obstacle contacts.

Physical validation of simulator and planner

For initial validation, we constructed a single-obstacle environment to evaluate the accuracy of our contact and actuation models under well-defined conditions. We prepared three distinct contact interaction configurations: (1) an acute angle of contact, designed to bend into the contact normal (increasing curvature under contact); (2) an obtuse angle of contact, designed to bend away from the contact normal (decreasing curvature under contact); and (3) a head-on contact, which leads to buckling behavior as the beam bends away from the contact normal (Fig. 5). We calculate the root-mean-square error (RMSE) at each video frame compared to the corresponding simulator step, using the vine robot radius ( $R_{vine} = 33.35$ mm) as a reference dimension when evaluating the deviations. The RMSE distributions for an acute angle of contact, an obtuse angle of contact, and a head-on contact are 20.6 ± 3.8 mm, 22.2 ± 5.1 mm, and 15.8 ± 5.2 mm, respectively.

FIG. 5.

Single-obstacle environment experiment comparing simulated predictions for different obstacle contact conditions. (a) Acute angle of contact, (b) obtuse angle of contact, and (c) head-on contact. The robot buckles at a point near its base.

Additionally, to evaluate the physical match of the designs produced by our design optimization approach, we constructed a multiobstacle environment with many potential solutions that required the exploitation of contacts for successful navigation. First, we applied our design optimization framework to find a vine robot actuator design that would navigate the robot from a specified start pose to a target goal region. Second, we fabricated the generated designs. Third, we deployed these physical robots and compared their behavior against the simulator’s predictions. The design optimization framework was able to find multiple feasible paths for our real-life demonstrations. The simulated predictions superimposed on the deployed robot in Figure 6 demonstrate that our simulator accurately models both contact mechanics and bending behavior and that our generated designs translate effectively to physical implementations. Quantitative analysis of the deviation between simulated predictions and deployed robots shows RMSE of 29.8 ± 4.1 mm for solution 1 and 23.8 ± 5.5 mm for solution 2.

FIG. 6.

Physical validation experiment comparing simulated predictions (green) with real vine robot behavior in a complex multiobstacle environment. The designed actuator configuration successfully navigates the cluttered environment in both simulation and reality, with the mean RMSE of 29.8 ± 4.1 mm for solution 1 and 23.8 ± 5.5 mm for solution 2. The scale bar in the figure is equal to 100 mm. RMSE, root-mean-square error.

Figure 7 shows the frame-by-frame deviation between simulated predictions and deployed robot behavior for single-obstacle and multiobstacle environments, and Table 1 summarizes the results.

FIG. 7.

(a) Frame-by-frame RMSE. (b) Distribution of RMSE between simulated predictions and deployed robot behavior for single-obstacle and physical validation environments.

Table 1.

Mean and Standard Deviation (Std) of RMSE Between Simulated Predictions and Deployed Robot Behavior for Single-Obstacle and Physical Validation Environments (Note: $R_{vine} = 33.35$ mm)

Environment	Mean RMSE (mm)	Std RMSE (mm)
Obtuse angle contact	22.2	5.1
Acute angle contact	20.6	3.8
Head-on contact	15.8	5.2
Physical validation solution 1	29.8	4.1
Physical validation solution 2	23.8	5.5

RMSE, root-mean-square error.

Long-horizon design optimization

We assess the effectiveness of our design optimization framework across six different environments, each designed to test specific aspects of the planning approach and environmental interaction capabilities. The evaluation environments, illustrated in Figure 8, span a range of planning challenges:

FIG. 8.

Multiple solutions found for the six evaluation environments by our design optimization approach. Our approach successfully identified solutions for: complex navigation through clutter (Plus, Maze, and Pickone), navigation through tight constraints (Needle), precise contact exploitation (Long), and trivial solutions when they exist (Tube). Free-floating images display the final actuated vine shape in free space.

Plus: A uniform grid of plus-shaped obstacles creates multiple viable paths, testing the planner’s ability to identify better solutions among alternatives.

Maze: A structured environment with few solutions and many dead ends, evaluating planning robustness in constrained solution spaces.

Pickone: Multiple identical passages with only one leading to the goal, evaluating the effectiveness of heuristic guidance.

Needle: A series of narrow passages requiring precision to navigate, evaluating the planner’s ability to handle tight geometric constraints.

Long: An environment with strategically positioned contacts that must be exploited to avoid dead-end passages, testing contact-based navigation.

Tube: A minimal environment to test whether the planner can identify trivial solutions.

Our design optimization framework successfully identified feasible solutions for all evaluation environments, as demonstrated in Figure 8. Even in the more difficult environments such as Maze, Long, and Needle, which are cluttered, full of dead ends, and require precise movement, our planner effectively discovered minimum-actuator designs that exploited contact with the environment to achieve the task objective.

We evaluated the planner with and without the reverse tree heuristic over 30 independent runs on each environment; statistics are reported in Table 2. The planner had consistent performance; all environments achieved initial solutions within 60 s on an AMD Ryzen 9 5900 CPU and NVIDIA RTX 3050 GPU, enabling interactive design iteration and deployment with a desktop computer.

Table 2.

Comparison of Planning Performance with (Default) and Without (No Geo) Geometric Heuristic Guidance

	Default			No Geo
Env	Solve time (s)	Best cost	Iter time (s)	Solve time (s)	Best cost	Iter time (s)
Plus	20.63	4.24	5.65	19.50	4.04	5.71
Maze	19.95	3.82	6.86	23.64	3.91	6.49
Pickone	15.94	3.08	6.06	14.66	3.16	5.88
Needle	46.08	5.09	4.97	50.50	5.38	5.00
Long	27.95	5.07	6.07	27.88	5.21	5.44
Tube	6.25	1.24	5.88	5.73	1.24	5.82

Solve time: Average time to first solution. Best cost: Average final cost after optimization. Iter time: Average time per iteration of SST*. Surprisingly, the geometric heuristic provides minimal performance improvement across evaluated environments. We hypothesize that contact forces the vine robot into “funnels,” guiding it along corridors in the environment.

Robustness of designs

Figure 9 presents the success rate of reaching the goal region as a function of uncertainty in both the environment and actuation parameters, in the same environment as Figure 6. We model environmental uncertainty through Gaussian perturbations of obstacle positions and sizes with gradually increasing variance levels. Similarly, actuation uncertainty is modeled through perturbations of actuator pressure and length parameters. For each level of uncertainty, we evaluate success rates across 1000 trials.

FIG. 9.

Left: Robustness test of a nominal design over Gaussian noise applied to obstacles and produced design. Scale for obstacle shift is the percent size difference and position change. The scale for bending control shift is the percent change to actuator pressure and actuator length. Each division was run with a sample size of 1000, and the fraction of successes is reported. Right: Examples of 10 sampled environments at the highest obstacle deviation and 60 rollouts with varying bending control deviation from the same nominal design. The “funneling” effects of contacts are clear: even with variations in obstacles and designs, the vine robot is pushed down similar trajectories.

Results show graceful degradation of task satisfaction with increasing uncertainty levels and that designs generated by our framework have inherent tolerance to uncertainties, particularly when contact interactions provide environmental guidance that compensates for actuation variations. This supports the deployment of found designs in real-world environments where the perfect transfer of simulation to reality is unattainable.

Funneling behavior

A separate study was run to investigate the vine robots’ inherent tendency to display funnel-like behaviors after contact with obstacles, which causes trajectories with minor deviations to converge.¹³ Figure 10 illustrates 50 vine rollouts with uniformly distributed starting angles in 4 environments as obstacles gradually become denser. Some variance was introduced to the bending controls to bias growth towards the top right corner. Dispersion statistics were run on the endpoints of the trajectories and are displayed in Figure 11.

FIG. 10.

Vine rollouts are run with slight deviations to start angle. The environment with no obstacles shows the even spread of vine trajectories. The other environments show the funneling behavior, forming highways where many trajectories converge.

FIG. 11.

K-nearest neighbors concentration density ratio across multiple values of k and the four environments shown in Figure 10.

Dispersion analysis was done using the k-nearest neighbor density concentration ratio of the top 10 percent of clusters. Given an endpoint of a vine trajectory $i$ , the average distance $r_{k} (i)$ between point $i$ and its $k$ nearest neighbors is calculated. The local density estimate around point $i$ would be:

p_{i} = \frac{k}{π r_{k} {(i)}^{2}}

The larger $p_{i}$ is, the denser the cluster around point $i$ . The density concentration ratio is the ratio between the average of the 90th percentile of $p_{i}$ and its mean. A higher value indicates a higher degree of funneling of vine trajectories, as seen by comparing the free space results to all three environments with obstacles.

Discussion

This work introduces a unified modeling framework for soft growing robots, effectively capturing the interactions between pneumatic actuation, beam mechanics, and environmental contact. The innovative neural surrogate approach allows for efficient simulation of vine robots, achieving a computational speedup of four orders of magnitude compared to traditional methods. This advancement enables faster-than-real-time simulation, which is crucial for developing a contact-aware design optimization framework. In this section, we discuss the performance and limitations of our framework and highlight major advances.

Capturing actuator–contact interactions

The single-obstacle environment experiments in Sec. 6.1 demonstrate that our simulator captures the fundamental contact-actuation coupling accurately across the range of contact conditions. The simulator is able to capture both the constant curvature bends produced by free space actuation or contact in a direction of bending and the discrete buckling when actuation and contact direction are opposed. Due to the nonlinearity of the sPAM bending model and contact mechanics, these configurations exhibit asymmetrical behaviors and yield substantially different final robot configurations. This match is maintained for multiobject interactions, with only a slight increase in RMSE, likely due to the longer length between contact events.

It should be noted that small deviations in the simulator before obstacle contact, such as start position and heading angle, can cause divergence on contact, as contact direction and force often dominate and can “funnel” the robot into different homotopy classes of motion through the environment (e.g., hitting on one side of an obstacle vs. another). This sensitivity is underscored by our funneling behavior evaluation in Sec. 6.4. While many neighboring trajectories seen in Figure 10 clumped up in final tip position, starting angles hitting on either side of an obstacle corner ended up further apart after contact.

Application to long-horizon design optimization task

To assess the performance of the design optimization framework, we selected six different simulated environments. These environments were explicitly designed to capture important aspects of the planner’s ability, such as (1) picking a better solution from multiple alternatives, (2) handling tight geometric constraints, and (3) identifying trivial solutions.

The Long environment validates the core hypothesis of our approach: the planner successfully identified designs that exploit deliberate contact interactions to navigate passages that would be inaccessible without environmental support. This demonstrates the practical advantage of contact-aware design optimization over traditional collision-avoidance approaches. This is also evidenced in the Tube environment, where the planner consistently produced near-optimal solutions with only one or two segments, indicating successful identification of minimal-actuation designs despite the availability of more complex alternatives. For environments with multiple viable solution paths (e.g., Plus and Maze), the planner demonstrated breadth of exploration, identifying diverse design alternatives that utilize different environmental corridors. This capability provides designers with multiple implementation options with different potential robustness against environmental variations.

Surprisingly, the reverse tree heuristic did not provide substantial performance improvements across the evaluated environments, even for more complex scenarios like Pickone and Maze that feature dead-ends and branching passages where heuristic guidance would be expected to provide benefit. We hypothesize that the natural funnel-like behavior exhibited by vine robot dynamics—particularly the tendency for contact interactions to guide the robot toward feasible solutions—reduces the need for explicit guidance in environments of the scale we evaluated. This observation suggests that the inherent physics of vine robots implicitly guides the system, obviating the need for other heuristics.

The real-world multiobstacle environment in Figure 6 shows a concrete example of the practical applications of our fast simulation framework in a contact-aware design optimization task. The planner finds designs of vine robots with minimal numbers of actuators within the constraints of feasible actuator fabrication and control. This example specifically highlights the ability to navigate through inaccessible passages (due to limits in actuator strain) by leveraging environmental contact. The validation of the model and simulator through real robot experiments highlights their robustness, with results indicating that the mean RMSE is of the same order as the vine robot radius. Deviations of this magnitude are within acceptable tolerance for the navigation tasks considered, particularly given the inherent uncertainties in fabrication, deployment, and measurement.

Effects of contact on robustness through funneling

The robustness tests (Sec. 6.3) quantified the effect of uncertainties in the environment, such as obstacle positioning, and vine robot actuation parameters. The results from the robustness tests indicate that the vine robot’s behavior in an obstacle-rich environment is highly robust to these uncertainties. Only with greater than 10% variation in obstacle or 12% variation in bending control does the success rate drop below 90%. Eventually, greater deviance in either environment or design will cause more significant failures, as the deviance in rollout becomes more likely to change the initial wall contacted, leading to a different homotopy of the final path. The funneling behavior tests (Sec. 6.4) showed that the vine robots indeed display funnel-like behavior that creates concentrated trajectories after contact with obstacles. These results together show that minor deviations in vine bending control have little effect on overall vine trajectory due to effects from contact forces, though this behavior shifts dramatically after a certain point.

These evaluations also reinforce our earlier observation that contact-guided navigation provides natural error correction that maintains overall success despite small deviations in behavior. An interesting direction for future work would be to explicitly target designs that exploit behaviors that are inherently more robust to uncertainty due to this effect.

Generalizing to different actuator and control models

While we used sPAMs as active steering actuators, other actuation methods,^18,30 such as the serial pouch motor, the cylindrical pneumatic artificial muscle (cPAM), the fabric PAM, and tip steering, can be used in our simulator due to the modular feature of our framework. The neural surrogate approach used for sPAM model integration can be easily replaced with similar neural surrogate or first principle models for cPAMs and pouch motors.

The current framework is also implemented as an open-loop control task, where the vine robot design (vine robot length, actuator placement, and actuation pressures) is selected during fabrication and remains fixed throughout the deployment. However, this can be easily extended to enable time-dependent or position-dependent steering change. Through initial simulation examples in the Supplementary Data, we showed how tip steering could be added to the modeling framework by temporarily penalizing growth and changing steering command.

Limitations

Despite these promising results, the current simulator and planner have limitations. It is confined to planar environments and relies on several assumptions about pressure dynamics, which become less realistic with longer robot growth. The current framework does not take into account the mass and inertia of the vine robot, thus limiting its use for vine robots without a tip mass or tip-based retraction mechanism.⁵³ For the planner, due to its planning-as-design approach, it is currently unable to generate solution that generalize to multiple environments. While this is not an inherent limit, more work would likely be needed to expand the actuator design space to perform multiobjective design optimization. Additionally, the design framework was tested in relatively simple scenarios due to constraints on the total length of the robot in the physical experiment in order to maintain a consistent tail tension and therefore growth pressure, limiting its ability to demonstrate long-horizon guidance through geometric heuristics. Longer lengths of vine robots tend to have internal friction, not currently addressed by the constant growth pressure assumption.²² In practice, pressurized base stations offer better control over vine robot pressure over long lengths, so future work will look to integrate the planner designs with the closed-loop control of growth rate enabled by a pressurized base station. A future model would need to consider the effect of tail tension to simulate controlled growth and retraction.

Conclusion

The unified modeling framework and fast simulation approach introduced in this work significantly advance the field of soft growing robots, allowing for fast, efficient, and accurate simulations. The contact-aware design optimization framework provides a novel method for creating robust vine robot designs to navigate a known environment. More research is needed to address the current limitations, such as the model’s restriction to planar environments and the simplicity of the tested scenarios. Additionally, exploring cost functions that utilize the characterized funneling effects of contact forces could enhance robustness and reduce uncertainty. Another interesting area for research could be the integration of sensing modules in the simulation framework to enable real-time feedback-based closed-loop control of vine robots.

Our work opens new avenues for exploration in addressing the challenges of soft robot path planning and control. A natural extension of this framework is to tackle multiobjective optimization tasks involving high-level reasoning and long-horizon navigation, such as payload deployment, manipulation of objects, and navigation in unknown environments. Ultimately, the advances in simulation frameworks will be the basis on which intelligent, autonomous vine robots are built.

Authors’ Contributions

Y.G. and L.C. developed and evaluated the simulation and planning framework and assisted in physical validation. P.B. and S.W. developed the combined actuation and contact moment model and physically validated the approach. Z.K. and L.H.B. supervised the project. All authors discussed the results and contributed to the final article.

Footnotes

Acknowledgment

The authors thank Gilbert Chang for feedback on this article.

Author Disclosure Statement

The authors have no conflicts of interest to declare.

Funding Information

This work was supported in part by NSF FRR 2308653. This material is based upon work supported by the United States Air Force (AFMC AFRL/RXNW) under Air Force Contract No. FA2394-24-C-B060.

Supplemental Material

References

Hawkes

, Blumenschein

, Greer

, et al. A soft robot that navigates its environment through growth. Sci Robot, 2017; 2(8):eaan3028.

Berthet-Rayne

, Hadi Sadati

, Petrou

, et al. Mammobot: A miniature steerable soft growing robot for early breast cancer detection. IEEE Robot Autom Lett, 2021; 6(3):5056–5063.

Borvorntanajanya

, Treratanakulchai

, Rodriguez y Baena

, et al. Model-based tracking control of a soft growing robot for colonoscopy. IEEE Trans Med Robot Bionics, 2024; 6(4):1354–1362.

Girerd

, Alvarez

, Hawkes

, et al. Material scrunching enables working channels in miniaturized vine-inspired robots. IEEE Trans Robot, 2024; 40:2166–2180.

Giri

, Girerd

, Cervera-Torralba

, et al. Inchi-grab: An inchworm-inspired guided retraction and bending device for vine robots during colonoscopy. IEEE/ASME Trans Mechatron, 2025; 30(6):6870–6881.

Kim

, Park

, Seo

, et al. A soft growing robotic endoscope for painless and strain-free insertion. Soft Robot, 2025.

Shi

, Borvorntanajanya

, Chen

, et al. Design, control, and evaluation of a novel soft everting robot for colonoscopy. IEEE Trans Robot, 2025; 41:4843–4859.

Qin

, Okamura

, Salathe

. Design, control, and modeling for soft growing robot deployment for nuclear material inspection. In: Institute of Nuclear Materials Management Annual Meeting, INMM; 2024.

Coad

, Blumenschein

, Cutler

, et al. Vine robots. IEEE Robot Automat Mag, 2020; 27(3):120–132.

10.

Comer

, Levy

. Deflections of an inflated circular-cylindrical cantilever beam. AIAA Journal, 1963; 1(7):1652–1655.

11.

Haggerty

, Naclerio

, Hawkes

. Characterizing environmental interactions for soft growing robots. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE; 2019; pp. 3335–3342.

12.

Fuentes

, Blumenschein

. Mapping unknown environments through passive de-formation of soft, growing robots. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE; 2023; pp. 2522–2527.

13.

Greer

, Blumenschein

, Alterovitz

, et al. Robust navigation of a soft growing robot by exploiting contact with the environment. Int J Rob Res, 2020; 39(14):1724–1738.

14.

Fuentes

, Diagne

, Kingston

, et al. Exteroception through proprioception: Sensing through improved contact modeling for soft growing robots. International Journal of Robotics Research, 2025.

15.

Jitosho

, Agharese

, Okamura

, et al. A dynamics simulator for soft growing robots. In: IEEE International Conference on Robotics and Automation. IEEE; 2021; pp. 11775–11781.

16.

, Zhang

, et al. A bioinspired soft robot combining the growth adaptability of vine plants with a coordinated control system. Research, 2021.

17.

François

, Christian

, Hervé

, et al. SOFA: A multi-model framework for interactive physical simulation. In: Soft Tissue Biomechanical Modeling for Computer Assisted Surgery. Springer: Berlin, Heidelberg; 2012; pp. 283–321.

18.

Du Pasquier

, Jeong

, Okamura

. Finite element modeling of pneumatic bending actuators for inflated-beam robots. IEEE Robot Autom Lett, 2023; 8(11):7416–7423.

19.

Chen

, Gao

, Wang

, et al. Physics-grounded differentiable simulation for soft growing robots. In: IEEE International Conference on Soft Robotics. IEEE; 2025; pp. 1–8.

20.

Greer

, Morimoto

, Okamura

, et al. Series pneumatic artificial muscles (spams) and application to a soft continuum robot. In: IEEE International Conference on Robotics and Automation. IEEE; 2017; pp. 5503–5510.

21.

Wang

, Frias-Miranda

, Valdivia

, et al. Anisotropic stiffness and programmable actuation for soft robots enabled by an inflated rotational joint. arXiv Preprint, 2024, arXiv:2410.13003.

22.

Blumenschein

, Okamura

, Hawkes

. Modeling of bioinspired apical extension in a soft robot. In: Conference on Biomimetic and Biohybrid Systems. Springer: Cham; 2017; pp. 522–531.

23.

Fichter

. A theory for inflated thin-wall cylindrical beams, Volume D-3466. National Aeronautics and Space Administration; 1966.

24.

Hwee

, Lewis

, Raines

, et al. Kinematic modeling of a soft everting robot from inflated beam theory. In: IEEE International Conference on Soft Robotics. IEEE; 2023; pp. 1–6.

25.

Frias-Miranda

, Srivastava

, Wang

, et al. Vine robot localization via collision. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE; 2023; pp. 2515–2521.

26.

Greer

, Morimoto

, Okamura

, et al. A soft, steerable continuum robot that grows via tip extension. Soft Robot, 2019; 6(1):95–108.

27.

Blumenschein

, Koehler

, Usevitch

, et al. Geometric solutions for general actuator routing on inflated-beam soft growing robots. IEEE Trans Robot, 2022; 38(3):1820–1840.

28.

Wang

, Blumenschein

. A geometric design approach for continuum robots by piecewise approximation of freeform shapes. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE; 2022; pp. 5416–5423.

29.

Haggerty

, Banks

, Kamenar

, et al. Control of soft robots with inertial dynamics. Sci Robot, 2023; 8(81):eadd6864.

30.

Kübler

, Du Pasquier

, Low

, et al. A comparison of pneumatic actuators for soft growing vine robots. Soft Robot, 2024; 11(5):857–868.

31.

Vartholomeos

, Wu

, Sadati

SMH

, et al. Lumped parameter dynamic model of an eversion growing robot: Analysis, simulation and experimental validation. In: IEEE International Conference on Robotics and Automation. IEEE; 2024; pp. 12734–12740.

32.

, De Iturrate Reyzabal

, Sadati

SMH

, et al. Towards a physics-based model for steerable eversion growing robots. IEEE Robot Autom Lett, 2023; 8(2):1005–1012.

33.

El-Hussieny

, Mehmood

, Mehdi

, et al. Development and evaluation of an intuitive flexible interface for teleoperating soft growing robots. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE; 2018; pp. 4995–5002.

34.

Wang

, Blumenschein

. Refined modeling for serial pneumatic artificial muscles enables model-based actuation design. In: IEEE International Conference on Soft Robotics. IEEE; 2024; pp. 800–807.

35.

Donald

, Xavier

, Canny

, et al. Kinodynamic motion planning. J ACM, 1993; 40(5):1048–1066.

36.

LaValle

. Planning Algorithms. Cambridge University Press; 2006.

37.

, Littlefield

, Bekris

. Asymptotically optimal sampling-based kinodynamic planning. Int J Rob Res, 2016; 35(5):528–564.

38.

Selvaggio

, Ramirez

, Naclerio

, et al. An obstacle-interaction planning method for navigation of actuated vine robots. In: IEEE International Conference on Robotics and Automation. IEEE; 2020; pp. 3227–3233.

39.

Webb

, van den Berg

. Kinodynamic RRT*: Asymptotically optimal motion planning for robots with linear dynamics. In: IEEE International Conference on Robotics and Automation. IEEE; 2013; pp. 5054–5061.

40.

Hauser

, Zhou

. Asymptotically optimal planning by feasible kinodynamic planning in a state–cost space. IEEE Trans Robot, 2016; 32(6):1431–1443.

41.

Perrault

, Ho

, Lahijanian

. Kino-PAX: Highly parallel kinodynamic sampling-based planner. IEEE Robot Autom Lett, 2025; 10(3):2430–2437.

42.

Daerden

. Conception and realization of pleated pneumatic artificial muscles and their use as compliant actuation elements. PhD thesis. Vrije Universiteit Brussel; 1999.

43.

Bender

, Müller

, Otaduy

, et al. A survey on position-based simulation methods in computer graphics. Computer Graphics Forum, 2014; 33(6):228–251.

44.

Macklin

, Müller

, Chentanez

. XPBD: Position-based simulation of com-pliant constrained dynamics. In: International Conference on Motion in Games. Association for Computing Machinery: New York, NY; 2016; pp. 49–54.

45.

Bradbury

, Frostig

, Hawkins

, et al. JAX: Composable transformations of Python+NumPy programs. 2018.

46.

Moré

. The Levenberg-Marquardt algorithm: Implementation and theory. In: Numerical Analysis. ( Watson

G. A

., eds) Springer Berlin Heidelberg: Berlin, Heidelberg; 1978; pp. 105–116.

47.

Nair

, Hinton

. Rectified linear units improve restricted boltzmann machines. In: International Conference on Machine Learning. Omnipress: Madison, WI; 2010; pp. 807–814.

48.

, Zhang

, Ren

, et al. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In: IEEE International Conference on Computer Vision. IEEE; 2015; pp.1026–1034.

49.

Strub

, Gammell

. Adaptively informed trees (AIT*): Fast asymptotically optimal path planning through adaptive heuristics. In: IEEE International Conference on Robotics and Automation. IEEE; 2020.

50.

Nayak

, Otte

. Bidirectional sampling based search without two point boundary value solution. ArXiv, 2020, arXiv:2010.14692.

51.

Solovey

, Janson

, Schmerling

, et al. and Revisiting the asymptotic optimality of RRT. In: IEEE International Conference on Robotics and Automation. IEEE; 2020; pp. 2189–2195.

52.

Şucan

, Moll

, Kavraki

. The open motion planning library. IEEE Robotics and Automation Magazine, 2012; 19(4):72–82. Available from: https://ompl.kavrakilab.org

53.

Coad

, Thomasson

, Blumenschein

, et al. Retraction of soft growing robots without buckling. IEEE Robot Autom Lett, 2020; 5(2):2115–2122.

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