Real-time force/position control of soft growing robots: A data-driven model predictive approach

2.1 Structure description

The structural design of the vine-growing robot examined in this study is illustrated in Figure 1. The robot comprises several key components: a pressure chamber, a spool, a flexible tube, and a steering mechanism that enables control of the robot’s tip local roll and yaw angles. The flexible tube, typically made from thin-walled low-density polyethylene, is sealed at one end and inverted, passing through the center of the existing tube before being rolled onto the spool. The open end is securely attached to the nozzle of the pressure chamber. When pressurized air from a compressor is introduced into the pressure chamber, the internal pressure propels the material through the robot’s body, causing it to evert at the tip and extend in length. This process is referred to as growth.

Figure 1

Schematic diagram of vine growing robot depicting its structure.

The servomotor, attached to the spool, manages the feeding of material necessary for the robot’s growth and retracts the material after task completion. While some studies use a servomotor to control the robot’s elongation, in this research, the servomotor is employed solely for two purposes: to feed the material required for growth without causing tension in the “tail” – the material that travels within the robot’s body – and to retract the material after the task is completed.

The steering system typically employs distributed strain actuation, using flexible actuators evenly spaced along the robot’s backbone to achieve distributed strain. Alternatively, it may use concentrated strain actuation, where actuators are placed at specific intervals along the robot’s length. Regardless of the approach, the primary purpose of the steering mechanism is to control the robot’s growth direction by adjusting the local roll and yaw angles. This is accomplished through asymmetric growth, which creates a length differential on either side of the robot’s body, mimicking how a vine bends due to uneven cell growth on its sides.

2.2 Kinematic analysis

To determine the tip’s position and orientation of the soft growing robot, it is essential to establish its kinematics within the chosen configuration space. Various methods such as piecewise constant curvature [18,20,30] and the classical Denavit–Hartenberg (D-H) coordinate system [11] have been utilized to elucidate the kinematics of vine robots. The piecewise constant curvature approach is particularly favored for modeling a broad array of continuum robots due to its analytical elegance and the capability to decompose kinematics into two distinct mappings: from actuator space to configuration space and from configuration space to task space [16]. This method is especially accurate for soft robots with a single curvature and actuators distributed along their backbone, proving effective when gravitational effects and external loads are negligible [31]. However, it is less suitable for soft robots with minimal bending curvatures [32], and the constant curvature assumption becomes unreliable at longer lengths [33].

Conversely, the classical D-H coordinate approach, while providing a homogeneous transformation matrix, relies on parameters that are difficult to measure directly in real-time settings. Therefore, this study adopts a comprehensive and general approach to model the kinematics of the growing robot, integrating the strengths of both methods to accurately capture the robot’s kinematic behavior in various conditions.

Figure 2 illustrates the detailed kinematic modeling of a vine-growing robot in free space. The global frame G ( O X Y Z ) is anchored at the center of the robot’s base, with the Y -axis tangent to the backbone’s base, providing a global orientation reference. The body frame B ( O x y z ) , attached to the robot’s tip, ensures that the y -axis remains tangent to the end of the robot’s backbone, regardless of environmental contacts or disturbances. The position of the origin of the body frame B ( O x y z ) relative to the global frame G ( O X Y Z ) is represented by the position vector p = [ X Y Z ] T , and the everted length of the robot is denoted by l . The angles α k and β k represent the bending angles around the local x - and z -axes, respectively, resulting from the asymmetrical growth due to the steering mechanism or environmental interactions and disturbances.

Figure 2

Schematic diagram of the kinematic model of the vine robot.

The spool, rotating at an angular velocity ω , controls the extension and retraction of the robot’s body. At time step k , the local frame B ( O x y z ) undergoes a sequence of rotations: first, a rotation by angle α k around the current x -axis, described by the rotation matrix R x ( α k ) ∈ S O ( 3 ) , followed by a rotation by angle β k around the current z -axis, described by the rotation matrix R z ( β k ) ∈ S O ( 3 ) . Consequently, the rotation matrix from the tip to the global coordinate system is expressed as follows:

where

and T represents the total number of discrete time steps. Therefore, after substitution, the transformation matrix R B G is given as follows:

In this study, the final orientation of the body frame B ( O x y z ) relative to the global frame G ( O X Y Z ) after a series of finite rotations is represented using Euler angles φ , θ , and ψ . These angles correspond to the rotation around the global Z -axis, the rotation around the local x -axis, and the rotation around the local z -axis, respectively. The transformation matrix from B ( O x y z ) to G ( O X Y Z ) is given by:

Let r m n denote the element in row m and column n of the rotation matrix in (2). The Euler angles at time step k can then be determined as follows:

For sin θ k ≠ 0 , the angles φ k and ψ k are given by:

However, if sin θ k = 0 , the angles simplify to:

The rate at which the everted length l increases in free space must equal the velocity of the robot’s tip, as the everted length directly translates to the linear displacement of the tip. This relationship ensures that the physical extension of the robot’s body is accurately represented by its motion in space. Mathematically, this is expressed as follows:

To prevent tension in the tail, the rate at which the spool feeds the robot’s body during growth must be at least equal to the rate at which material is added at the robot’s tip. This ensures that the material supply keeps pace with the extension demands. This constraint is crucial to avoid buckling caused by the force applied at the tip due to tension in the tail and to simplify the dynamic model. Consequently, the velocity of the tip is controlled solely by internal air pressure. Maintaining this balance is essential for ensuring smooth and uninterrupted growth, as well as preventing mechanical failure. The mathematical relationship governing this balance is expressed as follows:

where r s p is the radius of the spool and ω is the angular velocity at which the spool rotates. Ensuring this relationship holds true allows for consistent material feeding, thereby enhancing the robot’s performance and reliability during operation.

3.1 The growth model

This section explores the modeling of the everting vine robot during its growth phase, focusing on the dynamics and interactions involved. It examines the forces exerted by both the environment and internal pressure. Figure 3 illustrates the free body diagram of the vine-growing robot, which has a concentrated fictitious mass m at its tip. The force acting on the tip, B F = [ 0 F 0 ] T in the body frame, is generated by the internal absolute pressure P and acts along the y -axis of the body frame. In addition, the force exerted by the vine robot on the environment due to environmental interactions is represented by F e G = [ F e x F e y F e z ] T in the global frame. To model the bending behavior of the system caused by disturbances and environmental interactions, we introduce a torque τ at the tip. Air with a volumetric flow rate Q is introduced into the pressure chamber by the compressor. The dynamics of the pressure-driven everting vine robot is expressed as follows:

where F G = A B G F B . By substituting this into the equation of motion, we obtain:

The terms involving the angular variables φ , θ , and ψ in equation (10) represent the orientation of the vine robot in space and describe how the internal forces are projected along the global axes. These angles are derived from the robot’s kinematic configuration and evolve with its motion during growth and steering.

Figure 3

Dynamic model of the vine-growing robot, illustrating forces, actuation inputs, and environmental interactions.

The force F corresponds to the internal force generated by the pressure-driven eversion mechanism, and it is distributed across the global X , Y , and Z directions according to the robot’s orientation. The terms sin ψ , cos φ , and cos θ allow the projection of this internal force onto the respective global coordinate axes.

The external forces F e x , F e y , and F e z represent interactions with the environment, such as contact forces from obstacles. These forces act in opposition to the robot’s internal forces and are subtracted accordingly. In this model, the external forces are subject to the following assumptions: First, the contact point where the external force is applied must always lie outside the vine robot; no external forces can act within its structure. Second, the normal forces exerted by obstacles must be nonnegative, meaning that obstacles can only push the robot away and cannot pull it toward them. In addition, it is assumed that the material properties of the vine robot remain unchanged, with no degradation due to fatigue caused by repeated interactions with the environment. The angular variables also indirectly reflect external perturbations, as any environmental interaction (e.g., obstacle contact) can alter the robot’s configuration, which in turn affects the values of φ , θ , and ψ .

The force F exerted on the tip due to internal pressure is given by:

where A v represents the cross-sectional area of the tip, P atm is the atmospheric pressure, and P Y is the yield pressure, which is the minimum pressure required to initiate the eversion process. The yield pressure P Y depends on several factors, including the material properties of the vine robot’s membrane, such as stiffness and thickness. While the yield pressure is dynamically complex, it is typically treated as a constant in our model for computational simplicity and to align with prior research on vine robots [23].

In practice, P Y could vary based on environmental conditions or material fatigue. However, in this study, we assume it remains constant throughout the simulation. The specific value used for P Y in our model is based on experimental data from the literature [10], which provides validated pressure values for materials with similar characteristics.

To model the system’s bending response, we incorporate torsion springs at the points of interaction with the environment. This approach can be mathematically expressed as follows:

where K = diag ( k , k ) is the stiffness matrix and θ e denotes the angles of rotation around the x and z axes of the body frame due to interaction with the environment. It is important to note that in this study, we assume a constant stiffness matrix for simplicity and computational efficiency, as validated in previous works [25]. However, we acknowledge that in real-world scenarios, the stiffness may vary along the robot’s length.

For extended lengths of the vine-growing robot, the weight becomes significant, as it is proportional to the free length l in space (excluding the inverted tail and neglecting the weight of the air within the tube). This weight generates a restorative moment at the nearest support point. In addition, the robot may carry payloads at its tip, such as cameras or other scientific instruments, which further increase the lateral weight. The total weight acting on the robot, as illustrated in Figure 4, can be expressed as follows:

where l is the unconstrained length, μ is the weight per unit length, m t is the mass attached at the tip, and g is the acceleration due to gravity. This total weight W generates a moment at the nearest support point, which is counteracted by a restorative moment M , provided by the weight of the pressure chamber.

Figure 4

Dynamic force balance of the vine robot. The diagram illustrates the restorative moment M generated by the structure to counteract the gravitational forces and other external forces acting on the robot’s extended length l .

The deflection of the vine robot due to gravitational effects and additional weight is negligible, primarily because the stiffness induced by the internal pressure within the robot’s membrane is significantly greater than the forces caused by these weights. For this assumption to hold, the internal pressure must be sufficiently high to ensure that deflections remain minimal, which justifies the exclusion of these effects in the development of the dynamic equation in (10). Specifically, the principal stress at any point along the backbone of the vine robot must be nonnegative to prevent any significant deformation [34].

To check this posteriori, the axial stress Σ X X at the point ( X = 0 , Y = 0 , Z = − R ) must satisfy:

where R is the radius of the vine robot, P is the internal pressure, P atm is atmospheric pressure, t is the membrane thickness, and M is the moment generated by the weight W . If the internal pressure P drops below a certain threshold governed by the inequality equation in (14), wrinkles may begin to form at the point ( X = 0 , Y = 0 , Z = − R ) , leading to bending. Such deformation would invalidate the assumptions made in the dynamic equation (10), as it does not account for these effects. This is why, under normal operating conditions, the pressure is maintained high enough to keep the robot’s structure stiff and the deflections negligible.

The vine-growing robot can be modeled as a fluid capacitor with a capacitance C f . When the robot is not fully extended, the volume flow rate, considering the vine robot as an ideal fluid capacitor, is given by:

where Q represents the volumetric flow rate of air supplied by the compressor, C f is the fluid capacitance, P ˙ is the rate of pressure change, A v is the cross-sectional area at the robot’s tip, and l ˙ is the rate of change of the everted length.

The fluid capacitance for a given volume of fluid is typically defined as follows:

where Δ V represents the change in the volume of the entrapped fluid and Δ P is the corresponding change in pressure.

Assuming the vine robot is in thermal equilibrium with the environment and experiences slow changes in pressure and volume, allowing it to maintain a constant temperature, the fluid capacitance of air under isothermal conditions is given by:

where the volume of air V is defined as follows:

In this context, l denotes the everted length of the vine robot and A v is the cross-sectional area at the robot’s tip.

When the vine-growing robot is fully extended, the volumetric flow rate into the robot can still be modeled as an ideal capacitor, described by:

where C fl represents the fluid capacitance, which consists of two components: one arising from the compressibility of air and the other from changes in volume due to the axial stiffness of the vine robot’s membrane. The fluid capacitance C fl is expressed as follows:

where C s is defined as follows:

In this context, k s represents the spring stiffness of the vine robot’s body. By substituting equation (21) into the equation for C fl , the total fluid capacitance becomes:

Combining equations from (15) to (22), the rate at which the internal pressure P changes is given by:

where l T is the length of the vine robot when it is fully extended. Furthermore, this length imposes physical constraints on the robot. When the robot is fully extended, the velocity and acceleration of the tip cease, as the materials predominantly used to construct the vine-growing robot, such as thin-walled low-density polyethylene, are flexible but inextensible with relatively high axial stiffness. This constraint can be mathematically represented as follows:

These conditions indicate that the robot’s tip cannot extend further and that the tip’s velocity and acceleration are zero once the maximum length is reached.

3.2 The steering model

In contrast to passive environment steering, where the soft growing robot naturally bends around obstacles and follows existing pathways in cluttered and narrow spaces due to its inherent compliance, an active steering mechanism is necessary to control the growth direction for specific tasks. This is especially important in applications that require the creation of deployable and reconfigurable structures. Various steering mechanisms have been developed to control the growth direction of soft growing robots, all based on the principle of altering the relative length of the material on opposite sides of the flexible thin-walled tube. This method induce asymmetric growth by shortening one side to change the robot’s shape and achieve turns, enabling it to curve in space. Recent innovative methods for steering the everting vine robot, which mimic how vine plants explore their surroundings in search of water and nutrients without moving already grown sections, are presented in [11,35].

This study considers a mutually independent, on-demand active steering mechanism as presented in [11]. This steering method uses four pairs of miniature electromagnets and circular plates, which are uniformly distributed at 9 0 ∘ intervals around the backbone’s cross section to store parts of the robot. These electromagnet units are attached along the robot’s body with double-sided tape. When energized, the electromagnets release the stored body sections between the electromagnets and circular plates, causing asymmetric growth. This process imitates how vines avoid obstacles and move toward their targets. The steering model that controls the motion trajectory of the vine-growing robot, as illustrated in Figure 5, relates the bending angle to the length of the stored section as follows:

where L m is the length of the body stored between the electromagnet and circular plate, θ m is the bending angle, R m is the radius of curvature of each pair, L 0 is the thickness of the electromagnet unit after absorption (i.e., when the electromagnet is in contact with the circular plate), and D is the diameter of the vine-growing robot. The turning mode utilized in this study ensures that at any given moment, only one electromagnet unit is activated at each joint to steer the robot, or none are activated, allowing the robot to continue moving in the same direction as mentioned earlier. When no electromagnet unit is activated ( L m = 0 ), the bending angle θ m is zero, and the radius of curvature R m is infinite, forming a straight line at each unit. The rotation angles α k and β k around the body frame’s x and y axes, respectively, in terms of the bending angles θ m are defined as follows:

Figure 5

Schematic diagram of the steering kinematic model [11].

The rotation angles α k and β k , necessary for the steering mechanism to control the growth direction, are computed as follows:

The bending angles θ = [ θ 1 θ 2 θ 3 θ 4 ] T and the lengths of the body sections L = [ L 1 L 2 L 3 L 4 ] T , stored between the electromagnetic units of the steering mechanism, are derived from Eqs (28) and (27), respectively.

This formulation allows for the comprehensive analysis and control of the vine robot’s behavior by incorporating both the growth and steering dynamics within a unified state-space framework.

3.3 Model description

The development of a precise model is crucial for the implementation of model-based control. In this study, we formulate the model using continuous ordinary differential equations and represent it in state-space form. The state vector is defined as x = [ p T p ˙ T l P ] T , encapsulating the system’s state. The input vector to the system is given by u = [ Q L T ] T . Consequently, the nonlinear model describing the dynamics of the everting vine robot in state-space form is expressed as follows:

This model effectively captures the complex interplay between the growth and steering mechanisms, enabling precise control of the vine robot’s movements and interactions with its environment.

6.1 Cost function

To achieve precise tracking of the desired reference trajectory while ensuring smooth control behavior, a cost function must be defined to penalize deviations in both the manipulated variable and the trajectory. This cost function is minimized by the optimizer and accounts for deviations from the reference trajectory r as well as changes in the manipulated variable over the prediction horizon N 2 . The cost function to be minimized at each time step k is defined as follows:

s.t.

where y ( k + i ∣ k ) represents the predicted control variable at time k + i given the system state at time step k , and Δ u k = u k − u k − 1 denotes the change in the manipulated variable. The slack variable ξ quantifies the permissible violation of output constraints, allowing for the softening of constraints within the cost function. The weight matrix W ξ controls the extent of this violation. In addition, the weight matrices W r ≥ 0 and W u > 0 are positive semi-definite and positive definite, respectively, controlling the trade-off between accurate trajectory tracking and smooth control behavior.

This formulation ensures that the system not only tracks the reference trajectory accurately but also does so with minimal abrupt changes in the control inputs, leading to more stable and efficient operation.

6.2 Constraints and bounds

The system imposes physical constraints and limits on both the actuators and the output y to ensure the robot operates within safe and feasible regions, thereby preventing potential damage to the actuators. To achieve this, these constraints and bounds must be incorporated into the optimization loop, ensuring that the optimal trajectory of the manipulated variable u and the system states remain within acceptable ranges. The specific constraints and bounds considered in this study are as follows:

where l ˙ represents the rate of change of the everted length, P max denotes the maximum allowable pressure, Q max specifies the maximum volumetric flow rate that can be supplied by the compressor, and L max denotes the maximum stored length between the electromagnet units of the steering mechanism. These constraints are crucial for ensuring that the system operates both safely and efficiently, staying well within its physical and operational limits.

6.3 Implementation of MPC

The MPC approach proposed in this study for controlling the vine-growing robot is implemented using the do-mpc framework [37], a Python-based platform for nonlinear and robust MPC. This framework leverages CasADi [39] for symbolic computation and Ipopt [40] for solving nonlinear optimization problems, making it well suited for evaluating the performance and robustness of the proposed controller.

To integrate the DNN-based discrete-time dynamic model of the vine-growing robot, as detailed in Section 5, the model is first converted into a CasADi symbolic expression using the do-mpc ONNXConversion class, which facilitates the conversion from ONNX format. This conversion enables the model to operate as a discrete-time dynamic model within the MPC framework. The nonlinear optimization problem is addressed using Ipopt, which leverages the MA27 linear solver, while the dynamic model’s simulation is carried out using the cvodes tool. This approach ensures precise simulation and optimization, thereby supporting the robust real-time control of the vine-growing robot.

7.1 Performance of the DNN dynamic model of the vine robot

In this section, we evaluate the performance of the DNN-based discrete-time dynamic model for the pressure-driven everting vine robot, which is intended for integration into the MPC as a surrogate model. The DNN model is trained using Keras with TensorFlow as the backend engine, running on an Intel Core i5-4300M CPU @ 2.60 GHz with 7.4 GB of RAM. This setup allows us to assess the model’s accuracy and computational efficiency, ensuring it is suitable for real-time control applications within the MPC framework.

To evaluate the performance of the proposed DNN-based dynamic model for the vine robot, the training and validation loss were monitored over 200 epochs on normalized data. The Adam optimizer was employed with a learning rate of 0.001. Figure 9 illustrates the training and validation loss curves, both of which show the mean squared error (MSE) steadily converging. The rapid decrease in MSE during the initial epochs indicates effective learning, and the subsequent stabilization suggests that the model is not overfitting, maintaining consistent performance across both training and validation datasets.

Figure 9

Training performance of the DNN model.

Figure 10 illustrates the open-loop predictions of the system states using the DNN-based model, compared to the actual states derived from the first-principles model. The comparison shows that the differences between the DNN-predicted states and the first-principles model’s states are minimal, indicating a high level of accuracy in the DNN model’s predictions. This close alignment validates the use of the proposed DNN model as an effective surrogate for the more complex first-principles model, demonstrating its potential for integration into the MPC framework.

Figure 10

Comparison of the time evolution of system states between the nonlinear first-principles model and the DNN-predicted model.

To assess the generalization performance of the proposed DNN for the pressure-driven everting vine robot, a K-fold cross-validation approach was employed, as recommended by Marcot and Hanea [42]. This method is vital for identifying a robust architecture that exhibits minimal sensitivity to variations in training data partitions. In this study, the dataset was divided into five folds, and five distinct feed-forward DNN models were evaluated for performance. These models, varying in complexity, each comprise fully connected hidden layers as detailed in Table 2. The layers are denoted by “F” with subscripts indicating the number of neurons in each layer. All hidden layers utilize hyperbolic tangent activation functions, while the output layer employs a linear activation function. This systematic evaluation ensures the selection of a DNN architecture that balances complexity and performance, optimizing the model’s ability to generalize to unseen data.

The models were trained for 100 epochs using the Adam optimizer, with MSE as the primary loss metric. The convergence of the learning curves during five-fold cross-validation for all network configurations is depicted in Figure 11. The learning curves demonstrate that models with higher complexity, indicated by darker shades of red, tend to converge faster and stabilize more effectively, reflecting the models’ capacity to capture the underlying dynamics of the system more accurately.

Figure 11

Training losses for five DNN models during five-fold cross-validation.

In addition, Figure 12 illustrates the average mean squared error and their standard deviations across different model architectures. The results show a clear trend where increased model complexity leads to lower MSE values, indicating improved predictive performance. However, this comes with the trade-off of increased computational cost and potential overfitting, as evidenced by the greater variability in some of the more complex models. These findings highlight the critical need to balance model complexity and generalization capability when selecting the optimal architecture for the vine robot’s dynamic model, ensuring both accuracy and robustness in real-world applications.

Figure 12

Average MSE and standard deviation.

7.2 Closed-loop simulation

To evaluate the performance and capabilities of integrating the DNN-based discrete-time dynamic model with the MPC framework, we conducted a comparative analysis between the DNN dynamic model and the first-principles model across three tracking control scenarios: set-point stabilization, obstacle avoidance, and trajectory tracking. The MPC was designed using both the first-principles model and the DNN model, allowing us to assess the performance improvements offered by a data-driven approach in controlling a nonlinear dynamic system in real-time applications. This comparison highlights the potential advantages of utilizing DNN-based models for enhanced accuracy and efficiency in complex control tasks.

7.2.1 Set-point stabilization

The performance of the proposed MPC, utilizing both DNN-based and first-principles dynamic models, is evaluated based on its ability to precisely track a desired setpoint in space. This capability is particularly vital for applications such as medical procedures, where the vine robot may be required to deliver drugs accurately to a specific organ from its base. In this study, the control objective is to guide the robot to a specific position in Cartesian space, ensuring that it comes to rest with the desired force. The target state is set as r = [ 0.3 − 0.8 − 0.1 0 0 0 2 × 1 0 5 ] T , which includes the desired position, velocity, and pressure.

The weighting matrices used in the MPC formulation are W r = I 7 and W u = 0.01 I 5 , where I n denotes an n × n identity matrix, reflecting a balanced emphasis on state tracking and control effort. These matrices were carefully tuned to achieve optimal tracking performance, ensuring a balance between accurate trajectory tracking and smooth control input adjustments. The simulation is conducted with a time step of T s = 0.08  s, and the prediction horizon is set to N 2 = 20 steps. To ensure safe operation, the maximum allowable pressure P max and air volume flow rate Q max are constrained to 241,325 Pa and 0.01 m³/s, respectively.

In addition to the system constraints outlined in Section 6, the robot’s position is further constrained within the following bounds:

− 4 − 4 − 4 ≤ p ≤ 4 4 4 .

These constraints ensure that the robot operates within a safe and feasible region while achieving the control objectives. This scenario highlights the effectiveness of the proposed MPC in maintaining precise control of the vine robot under the specified conditions.

The dynamic model is initialized with initial state of

x 0 = [ 0 0.05 0 0 0 0 0.05 101,325 ] T + 1 × 1 0 − 6 .

Figure 13 compares the performance of the DNN-based dynamic model with the first-principles model during closed-loop control, illustrating the evolution of system states, stresses, and actuation inputs over time.

Figure 13

Comparison of the closed-loop control performance of the DNN-based dynamic model and the first-principles model for set-point stabilization simulation. (a) Position tracking p . (b) Length l of the robot. (c) Pressure P of the system. (d) Stress response in longitudinal ( σ L ) and circumferential ( σ θ ) directions. (e) Volumetric flow rate ( Q ). (f) Angular velocity ( ω ). (g) Stored lengths ( L 1 , L 2 , L 3 , and L 4 ) during the actuation process.

In this comparison, both the DNN-based model and the first-principles model achieve effective set-point stabilization, with the DNN model showing slightly faster convergence in the position states ( X , Y , Z ). The system length also stabilizes more rapidly in the DNN model. In addition, the figure highlights the efficiency of the actuation inputs required for control. In the DNN model, the modulation of the volumetric flow rate ( Q ) and angular velocity ( ω ) is smoother, resulting in more efficient transitions compared to the first-principles model. The stored lengths ( L 1 , L 2 , L 3 , and L 4 ) exhibit a more controlled response under the DNN model, particularly in minimizing the overall material usage required to complete the task.

The results indicate that the average time taken by the MPC solver to find the optimal control with the DNN dynamic model is 2.2 times faster than using the first-principles model. This improvement in computational efficiency is particularly useful in real-time applications, where the prediction horizon can be five times shorter compared to that of the analytical model.

Overall, these figures highlight the advantages of the DNN-based model in achieving quicker and more stable control responses while requiring less aggressive actuation. This suggests that the DNN model can offer superior performance in real-time control scenarios, providing a viable alternative to more computationally intensive first-principles models.

7.2.2 Obstacle avoidance

In this section, the effectiveness of the proposed MPC is evaluated in the context of obstacle avoidance, a crucial capability for navigating cluttered environments such as archaeological excavation sites. The obstacle is positioned at x obs = [ 0.2 − 0.5 − 0.1 ] T along the robot’s path to its desired setpoint. The initial state x 0 , prediction horizon N 2 , desired state r , and sampling time T s are consistent with those utilized in Section 7.2.1.

To ensure the robot successfully avoids the obstacle, a nonlinear inequality constraint is introduced into the optimization problem. This constraint enforces that the Euclidean distance between the tip of the vine robot and the center of the obstacle remains above a specified safety threshold:

(36) ∥ p − x obs ∥ ≥ r o + r v ,

where r o = 0.2 m and r v = 0.0399 m represent the radii of the robot’s tip and the obstacle, respectively. This constraint is crucial for ensuring that the robot maintains a safe distance from the obstacle, thereby preventing collisions and ensuring smooth navigation toward the desired setpoint.

The simulation results using both the DNN-based discrete-time dynamic model and the first-principles model, depicted in Figure 14, demonstrate that the proposed MPC successfully navigated the vine robot around the obstacle and achieved the desired setpoint. The corresponding actuation inputs and system states during the obstacle avoidance maneuver are shown in Figure 15.

Figure 14

Simulation results of vine robot avoiding an obstacle in Cartesian space.

Figure 15

Closed-loop control performance comparison between the DNN-based dynamic model and the first-principles model during obstacle avoidance. (a) Position p . (b) Length l . (c) Internal pressure P . (d) Stress distribution. (e) Flow rate Q . (f) Angular velocity ω . (g) Stored lengths for each of the four electromagnet units ( L 1 , L 2 , L 3 , L 4 ).

The results confirm that the proposed MPC effectively guides the vine robot in avoiding the obstacle while maintaining the desired trajectory. Notably, the DNN-MPC architecture significantly outperformed the MPC framework based on the first-principles model in terms of computation time, with the DNN-MPC being 7.2 times faster than its counterpart. This substantial improvement in computational efficiency highlights the DNN-based approach’s potential for real-time applications, where quick decision-making is crucial for effective obstacle avoidance and trajectory control.

7.2.3 Trajectory tracking

In this final evaluation, a time-varying trajectory from the dataset is used to assess the effectiveness of the proposed MPC in tracking dynamic paths with the surrogate models. This scenario is particularly relevant for applications requiring a soft, reconfigurable, and deployable antenna, such as those that need to assume various shapes like a helix. The ability to accurately follow such trajectories is crucial for ensuring the antenna’s functionality in diverse operational environments.

All tuning parameters for the MPC, including the prediction horizon, weighting matrices, and constraints, are consistent with those presented in Section 7.2.1. This consistency allows for a direct comparison of the controller’s performance in both set-point stabilization and more complex trajectory tracking tasks, demonstrating the robustness and versatility of the proposed MPC framework.

Figure 16 illustrates the motion of the vine robot in space while tracking a time-varying trajectory. The performance of the proposed MPC, utilizing both the DNN-based dynamic model and the first-principles model, is presented in Figure 17, along with the corresponding actuation inputs to the system. The results show that the DNN-MPC architecture tracks the trajectory more effectively, achieving faster convergence and more precise alignment with the reference trajectory compared to the first-principles model. Furthermore, the DNN-MPC architecture demonstrates smoother and more efficient actuation, resulting in better overall control performance compared to the first-principles model.

Figure 16

3D motion of the vine robot while tracking a time-varying trajectory.

Figure 17

Closed-loop control performance comparison between the DNN-based dynamic model and the first-principles model during for the trajectory tracking task of the vine growing robot. (a) Position p . (b) Length l . (c) Internal pressure P . (d) Stress distribution. (e) Flow rate Q . (f) Angular velocity ω . (g) Stored lengths for each of the four electromagnet units ( L 1 , L 2 , L 3 , L 4 ).

Significantly, the DNN-MPC architecture outperformed the first-principles model-based MPC framework in terms of computation time, with the DNN-MPC being 11 times faster. This substantial improvement highlights the DNN model’s suitability for real-time control applications, where computational efficiency is critical for responsive and accurate trajectory tracking.