Stabilized Controller for Jet Actuated Cantilevered Pipe Using Damping Effect of an Internal Flowing Fluid

Fluid jet actuation is a potential actuation technique for continuum robots. It can generate and rapidly control a relatively large force using a small and lightweight structure because a significant amount of energy can be transported through its internal channels. Recently, jet-actuated flying continuum robots have been developed using this advantageous characteristic. However, a challenging issue in controlling the robot is the fluid structure interaction between the flexible body and the internal flowing fluid. This interaction often causes instability in the pipe conveying fluid. In this study, as a first step to address this issue, we propose a stabilized controller (vertical position control) for a jet-actuated two-dimensional cantilevered pipe with a nozzle unit at the tip using the damping effect of the internal flowing fluid and verify the controller with a real robot. Specifically, a model is constructed with the net force of the jets as the control input. A simple controller that can constantly decrease the energy function is proposed by utilizing the damping effect of the flowing fluid. Numerical simulations verify the stability of the system regardless of the flow velocity. In particular, fluid damping mainly suppresses the higher-order mode oscillations. Moreover, the stability of the system can be improved by adjusting the controller gains. We also conduct experiments using an actual robot to verify the simulation results. The vibrations can be damped by the fluid effect, and the stability can be improved using the proposed controller.


I. INTRODUCTION
Fluid jet actuation is one of the potential actuators for continuum robots because it can generate and rapidly control a relatively large force using a small and lightweight structure. The jet-actuated continuum robot uses the reaction forces of the fluid jets to transform the shape. The fluid is delivered through a channel in the body. It can generate a relatively large force with a lightweight and thin body structure because an energy source such as a pump can be located at the root. In particular, a large force can be controlled with a high responsivity if rotating nozzles are used because the jet The associate editor coordinating the review of this manuscript and approving it for publication was Hui Xie . reaction force can be rotated without any torque when the jet direction is perpendicular to the rotating axis [1]. Therefore, because of these advantages, some studies have proposed fluid jet actuation for continuum robots. For instance, some studies developed a water injection thin continuum robot, with a tip driven by three jets, to investigate nuclear reactors [2], and for rapid investigation of organs [3]. A fluid jet actuator that can be attached to the middle of a hose has also been developed [4]. Furthermore, to explore narrow debris, we developed an active scope camera that can elevate this tip using air injection [5].
Moreover, a flying continuum robot can be realized by mounting multiple nozzle units. We proposed a flying firefighting hose (Dragon Fire Fighter) driven by water jets [6]- [8]. A similar concept was proposed in [9]. Our goal is to float the hose and directly reach the fire source by controlling the reaction forces of the distributed nozzle units. The nozzle unit consists of multiple rotating nozzles and generates a net force with high responsivity by adjusting the nozzle angles. To date, we have achieved stable levitation and direction switching of 4-m robots by attaching one nozzle unit each on the tip and the middle [8].
A key issue in controlling the jet-actuated flying continuum robot is the fluid structure interaction between the flexible body and the internal flowing fluid. This is because the interaction becomes non-negligible as the body length increases. Generally, when a fluid flows inside an elastic structure, the fluid structure interaction causes a vibration phenomenon. In particular, when a fluid flows in a cantilevered elastic tube with an open end, the system becomes a non-conservative system. A flutter phenomenon, in which the deflection increases while vibrating, occurs when the flow velocity exceeds a specific limit.
However, these interactions have not been considered for jet-actuated continuum robots. This is because the robot is flexible and can be deformed nonlinearly into various shapes in a three-dimensional space, which complicates the interaction with the fluid. In addition, if there are multiple nozzle units, the flow path branches to make the system more complex.
In this study, as a first step, we propose a stabilized controller (vertical position controller) for a jet-actuated twodimensional cantilevered pipe with a nozzle unit at the tip using the damping effect of the internal flowing fluid, and verify the controller with an actual robot. First, we construct a two-dimensional cantilevered water pipe model with a head nozzle unit considering fluid motion. We assume that the nozzle unit can control the magnitude and direction of the net force by changing the directions of multiple jets. Next, based on the characteristics of the equations of motion, we propose a vertical position controller using the net force as an input. The controller can always decrease the energy function of the system, regardless of the flow velocity, using the damping effect of the flowing fluid. Through numerical simulations, we validate the stability of the controller, which can be improved by using the damping effect of the fluid and tuning the control gains. Furthermore, the validity of the proposed controller is demonstrated through experiments using actual robots.
The contributions of this study are as follows: (1) We propose a stabilized controller for a cantilevered pipe with multiple fluid injections for the first time. The controller can utilize the damping effect of the flowing fluid for stabilization. Although many studies have assessed the stabilization of the pipe, as discussed in the next section, no study has yet proposed a stabilized controller using multiple jets.
(2) The validity of the proposed controller is verified through simulations and experiments. Although these results cannot be directly applied to jet-actuated continuum robots owing to the difference in stiffness, they provide significant suggestions and insights for future control strategies of jet-actuated continuum robots. For example, the proposed control strategy, using the damping property of an internal flowing fluid, would be very useful for continuum robots, which are also distributed parameter systems with a limited number of actuators.

II. RELATED STUDIES
Many studies have investigated the dynamics and instability of the cantilevered pipe conveying fluid [10], [11]. A straight cantilevered pipe, with an open end, loses stability via flutter when the flow velocity increases. Thus far, rich dynamic properties have been investigated by changing various physical parameters. For example, chaotic motions occur by adding an end mass [12], spring supports [13], [14], or motion constraints [15]. In addition, some studies have investigated movements with a large deformation [16], [17], including in three-dimensions [18], with an initially curved tube [19], and with pulsatile flow [20], [21]. In particular, Sugiyama [22] pointed out that the flowing fluid inside a cantilevered pipe contributes to damping the body oscillation in the case of a slow-flowing speed. Our study also utilizes this damping effect. However, our proposed jet controller can ensure a damping effect regardless of the flow speed range.
In terms of the stabilized controller, although many studies have discussed methods to stabilize the cantilever pipe conveying fluid, no study has yet used multiple jets to stabilize the pipe. For example, Doki et al. proposed a method to damp vibrations by controlling the tendon mechanism [23]. Pisarski et al. proposed a damping method using electromagnetic devices [24]. Several researchers have attached a nonlinear energy sink to the pipe, which dissipates the vibrational energy of the pipe [25]- [27]. Yu et al. controlled the tip bending moment using piezoelectric actuators to increase the critical flow velocity [28]. Although Borglund attempted to dampen fluttering by controlling the jet direction using a proportional derivative (PD) controller at the tip [29], no study has yet used multiple jets. We control the magnitude of the net force by controlling multiple jets to stabilize the cantilevered pipe conveying fluid.

III. MODEL OF CANTILEVERED PIPE WITH A NOZZLE UNIT A. ASSUMPTIONS AND MODEL
Our targeted jet-actuated cantilevered pipe consists of an elastic beam and a nozzle unit, as shown in Fig. 1. The nozzle unit VOLUME 10, 2022 We assume a plug flow, and the pipe friction is sufficiently small. (A3) The nozzle unit is attached to the tip as the mass point of weight m n and has multiple direction controllable outlets. It generates a controllable force R (control input of the system) by adjusting these directions. Assumptions (A1 and A2) are frequently used to investigate the dynamics of a cantilevered pipe conveying water, as in [30]. The physical variables are shown in Fig. 1. The direction of gravity g is the −y-axis. The weights of the beam and fluid per unit length are m and M , respectively. i and j represent the unit vectors along the x and y axes, respectively.
The equation of motion (EQM) can be derived using the following three steps: First, we derive the EQM of a cantilevered pipe conveying the fluid (Section III-B). Then, we derive the boundary conditions by considering the reaction forces generated by the nozzle unit (Section III-C). Finally, combining steps one and two, we derive the EQM of the system (Section III-D). Note that the partial derivatives of position x and time t are represented as [] and[], respectively.

B. EQM FOR A CANTILEVERED PIPE CONVEYING FLUID
Here, we derive the EQM for a cantilevered pipe conveying fluid by referring to [30]. For a small element at x (Fig. 2), we derive the EQMs for the fluid and elastic beam parts independently. Subsequently, we integrate them to derive the EQM for the cantilevered pipe with a fluid. The formulation procedure is identical to that in [30], except for the direction of gravity and boundary conditions discussed in Section III-C.

1) EQM FOR FLUID IN THE SMALL ELEMENT
We derive the EQM for the fluid that flows through the small element B with length δx, as shown in Fig. 2A. p represents the relative pressure. The normal and tangential forces applied by the corresponding pipe on the fluid are represented as Fδx and sδx, respectively. We set the flow speed and density as V and ρ, respectively.
From the momentum conservation law of the fluid, we derive the following relations, while considering the small deflection approximation (first-order approximation): where S B and n represent the surface of B and its normal vector, respectively. The first and second terms on the left-hand side represent the variation ratio of the momentum in B and loss ratio of the momentum through the channel sections, respectively. The right-hand term represents the force applied to the fluid. The left side of (1) can be transformed using the Gauss theorem and the equation of continuity (∇ · V = 0) as Because the internal flow speed relative to the pipe is constant U (assumption (A2)) and the pipe element moves at a speed ofẏj; therefore, the flow speed on the inertial frame V can be derived as follows: Equation (2) can be further transformed as follows by substituting (3) into (2) and using the expressionsU = 0, ρA = M under the small deflection approximation.
By substituting (2) and (4) into (1), we derive the following EQM of the fluid in the pipe: Equation (5) represents the equilibrium of forces in the x direction (forces applied by the pressure, pipe, and gravity). Equation (6) represents the equation of motion in the y direction. The three terms on the left-hand side represent the acceleration, Coriolis term, and y element of the axis force MU 2 applied by the fluid inflow and outflow. The right-hand terms represent the y elements of the forces applied by the pressure, pipe, and gravity. Note that the formulations are identical to those in [30], except for the direction of gravity.

2) EQM FOR PIPE OF SMALL ELEMENT
The EQM for the beam (pipe) is derived by considering a small element with length δx, as shown in Fig. 2B. T is the longitudinal tension, Q is the shear force, and µ is the bending moment. The normal and tangential forces applied by the corresponding fluid to the pipe are represented as Fδx and sδx, respectively. The EQM in the i and j directions can be written as follows by ignoring the deformation along the longitudinal axis: Iy , E is the Young's modulus, I is the moment of inertia, and E * represents the viscoelastic coefficient (Kelvin-Voigt type). Note that the formulations are identical to those in [30], except for the direction of gravity.

3) EQM FOR CANTILEVERED PIPE CONVEYING FLUID
We can derive the EQM for the cantilevered pipe conveying fluid by integrating (6) and (8): Moreover, the following condition is derived using (5) and (7): This equation determines the term T − pA of (9) by using the boundary condition p(L), T (L) as follows: The nozzle unit can control the magnitude and direction of the net force by adjusting the directions of multiple jets, as shown in Fig. 3. The unit vectors parallel and normal to the nozzle inlet are defined as e in and e shear , respectively. Multiple rotating nozzles are attached to the unit, and each nozzle outlet generates a reaction force r i = −ρA i U 2 i e i , where e i , U i , and A i represent the unit vector of the jet direction, outlet flow velocity, and outlet section, respectively. The net force applied to the nozzle unit considering the pressure and momentum gain at the inlet of the nozzle unit can be written as follows: where R = r i . Note that we ignore the rotational moment of the nozzle unit because the outlets and inlet are in close distance.
Because the nozzle unit and pipe are connected, the tension applied to the tip of pipe T (L) is determined by the e in element of the generated force on the nozzle unit: where R i = R · e in . By substituting this equation into (11), we obtain: The boundary condition of the e shear element of the force can be written as: where R s = R · e shear . Equations (14) and (15) are different from those in previous studies, such as [30].
In addition, the boundary conditions for the moment at the tip, and for the position and posture at the fixed root, are

D. EQM FOR CANTILEVERED PIPE WITH NOZZLE UNIT
We represent the mass of the pipe per unit length as m + m n δ(L − x) because the weight of the nozzle unit is m n .
. The boundary conditions are derived using (15)(16) as follows: In (17), the first four terms represent the motion of the Euler-Bernoulli beam under gravity. The fifth and sixth terms represent the Coriolis term of the fluid and the effect of the e in element of R, respectively. In (18), the first three terms are the ordinal constraints for the cantilevered beam. The fourth term represents the e shear element of R. The pressure p and longitudinal forces between the pipe and fluid s do not affect the motion in this model as [11]. Furthermore, the tip nozzle eliminates the momentum effect that flows along the pipe MU 2 and is different from the standard models [11]. We set the control inputs in this study as the net forces R i and R s . In addition, we derive another representation of the EQM, with boundary conditions identical to the Euler-Bernoulli beam for simulation, by using the Dirac delta function.
where the boundary conditions are

IV. PROPOSED CONTROLLER
We propose a controller based on the energy dissipation feature of the internal flow [22]. First, we explain the time evolution of the energy and subsequently discuss the proposed controller.

A. TIME EVOLUTION OF KINEMATIC ENERGY
The kinematic energy E p of the beam when U = 0 can be written as The derivative of the energy can be calculated as follows by using (19) and (20) under the assumption that R i is not a function of x: where y L = y(L). The first and second terms represent the energy dissipation caused by the viscoelastic term of the beam (Kelvin-Voigt damping effect) and fluid flow (Coriolis term 2MUẏ ), respectively. The third term represents the ratio of the mechanical work performed by the net force on the nozzle along the e shear direction. The fourth and fifth terms represent the ratio of the mechanical work along the e in direction (explained subsequently). The first and second terms always dissipate energy because they are non-positive values. The second term, which corresponds to the Coriolis effect, dissipates more energy for a high flow speed. Thus, if we design the control inputs, R i and R s for no energy gain, the controller can ensure energy dissipation regardless of the flow speed. Moreover, a high flow speed results in a high damping. This feature differs from that of previous studies. For an ordinal cantilevered pipe without a nozzle, such as in [22], [30], the net force is given as R i = −MU 2 and R s = 0. In this case, the fourth and fifth terms increase as the flow speed increases, which results in an undesired flutter phenomenon.
The fourth and fifth terms represent the ratio of the mechanical work executed by the nozzle along the e in direction. The fourth term itself appears to represent the e in element of the mechanical work ratio caused by the net force of the nozzle unit because it multiplies the e in element of the net force: R i and e in element of the tip velocity, y Lẏ L . However, it contains ostensible mechanical work related to the deformation of the beam length. This deformation is attributed to the approximation error of the Euler-Bernoulli beam; thus, this ostensible work should be excluded as the fifth term. This work can be derived by multiplying the length deformation ratio with the net force as follows:

B. PROPOSED CONTROLLER
We propose the controller as follows: where K and D are non-negative gain parameters, and y d is the desired height of the nozzle unit. This is a PD controller for the height of the nozzle unit. The gains K and D can adjust the responsiveness of the system. If we define the energy function V as the time derivative of V can be calculated as follows: The proposed simple PD controller can ensure energy dissipation (passivity). Moreover, the effect of the fluid structure interaction (the second term) can be used to improve the speed of energy dissipation. This energy dissipation feature is very useful because the energy function can be a Lyapunov function that ensures system stability. However, we do not discuss the theoretical stability analysis because the proof itself requires further mathematical analysis, which is beyond the scope of this study because of the distributed parameter system [31]. Thus, we numerically evaluate the system stability by linearizing the EQM using the Galerkin method in addition to conducting experiments.

V. DISCRETIZED EQUATION OF MOTION A. NON-DIMENSIONAL EQM AND CONTROLLER
To reduce the parameters for the simulation, the following non-dimensional quantities and equations are introduced using L and L 2 ((M + m)/EI ) 1/2 as the reference length and time, respectively.
where the other non-dimensional parameters are Therefore, the non-dimensional controller can be expressed as where

B. LINEARIZATION BY GALERKIN DISCRETIZATION
The standard Galerkin technique [11] is applied to transform the coupled partial differential equations into linearized ordinary differential equations. The comparison functions are set as the eigenfunctions of the normal cantilever beam η +η = 0 as where λ i is the solution of 1 + cosh λ i cos λ i = 0. The solution of EQM (28) is approximated as the summation of N eigenfunctions as follows: Here, q i fulfils the following condition: where [w N ] represents the error of EQM(28) when η = w N is substituted into (28). Consequently, the following N th linearlized equations are derived: where q = [q 1 q 2 · · · q N ] T , and the elements of M, C, K, g are as follows:

C. LINEARIZED DYNAMICS
If the inverse exists for the matrix M, the linearized EQM can be rewritten as follows (the inverse matrix always exists in the following simulations, although it could not be proved): where p = [q TqT ] T , and where E ∈ R N ×N is an identity matrix. The equilibrium point q * can be calculated as q * = −K −1 g, and the linear stability can be evaluated using the eigenvalues of A. We note that the desired height η d and gravity term γ only appear in g, which affects the equilibrium point q * and not linear stability. Therefore, the next section numerically investigates the stability by setting η d and γ to zero.

A. STABILITY DUE TO THE DAMPING EFFECT OF FLOWING FLUID
We conduct a simulation to verify the stabilization owing to the damping effect of the fluid. The damping and gravity terms of the body are assumed to be zero (α = 0, γ = 0), and the three weight ratios of water to the body (β) are set according to the literature [11] (β = 0.2, 0.295, 0.5). The discretization order is set to N = 10 based on [11]. Focusing on the effect of the flow speed, the control parameter was set to k = d = η d = 0. We investigate the change in the linear stability at q * = 0 against flow speed u. Note that if the tip water channel is opened (no nozzle), the pipe motion becomes unstable when the flow speed exceeds u cf = 5.60, 8.20, and 9.33 for β = 0.2, 0.295, and 0.5, respectively, [11].
The response of the eigenvalue against flow speed u is shown in Figs.4A, B, and C, which correspond to the values of β = 0.2, 0.295, and 0.5, respectively. The horizontal axis represents the imaginary part of each eigenvalue, and the vertical axis represents the negative real part of the eigenvalue; crossing the y-axis downward indicates destabilization of the corresponding mode. The value of u is indicated by the color of each point. Similarly, Figures 4A2, B2, and C2 exhibit a change in the maximum real part of the eigenvalue with respect to u. Instability occurs when the eigenvalue becomes larger than zero.
The results indicate that the flowing fluid does not cause instability but acts as a damping term for the system and improves the stability of the higher modes. For Figs. 4A2, B2, and C2, the maximum real part of the eigenvalues decreases monotonically as the flow speed increases from zero. In contrast, when the flow speed exceeds a certain value, the maximum real part of the eigenvalues approaches zero (not greater than zero). In this case, the eigenvalue of the first-order mode tends to approach the zero eigenvalue along the real axis, as shown in Figs. 4A1, B1, and C1. This means that the first-order mode is overdamped. In contrast, the real part of the eigenvalues of the second-order and higher modes reduces uniformly with an increase in the flow speed, indicating that the fluid damping effect improves the stability of these modes. This improvement in the stability of higher-order modes is more prominent when the fluid fraction is larger (i.e., β is larger).

B. STABILITY DUE TO THE CONTROL PARAMETERS
We examine the effect of the control parameters k and d on the stability under the same configuration as in the previous VOLUME 10, 2022   For all β, we confirm that an increase in the P gain k improves the stability at a high flow speed. This suggests that increasing the P gain prevents over-damping at high flow speeds. In contrast, increasing the D gain d deteriorates the stability for a high flow speed. The increase in the D gain causes an over-damping situation. From these results, we found that the stability of the system can be improved by appropriately setting the control parameters.

A. DEVELOPED ROBOT
The robot developed for this experiment is shown in Fig. 6. The robot consists of a nozzle unit at the tip and a 2.4 m long body. The body part consists of a PVC pipe (VU40) with a water hose (inner diameter: 25 mm) fixed inside.  Multiple inertial measurement units (IMUs) are mounted on the outside of the pipe. The root of the pipe is fixed. The root of the water hose is connected to a flowmeter (Keyence, FD-R50) and a pressure gauge (Nagano Keiki, GC31). The detailed physical parameters of the robot are listed in Table 1, and the non-dimensional parameters are listed in Table 2. To install the proposed controller on the robot, the position and velocity of the nozzle unit should be measured, and the nozzle unit should realize a normal net force.
To estimate the position and velocity of the nozzle unit, the pipe was approximated as a connection of seven rigid links with six joints (0.2 m long links at the end and tip, 0.4 m long links for others) whose attitudes and angular velocities are measured by the IMUs. Six IMUs (TDK, ICM-20948) were mounted on the nozzle unit and pipe at 0.4 m intervals. Each IMU measured the angular velocity and acceleration at 1 kHz, which were converted to attitude using the Madgwick filter [32] at 100 Hz. We note that the approximation error of this discrete position estimation is affordable for this experiment. The circular curve with a curvature of 0.3 /m (which is estimated from the attitudes of IMUs) can be approximated with a head position error less than 0.003 m, which is almost 30 times smaller than the given disturbance.
The structure and coordinate system of the nozzle unit are shown in Fig. 6C. The nozzle unit consists of two nozzles that can rotate around one axis (x n axis), and each nozzle has a diameter of 7 mm. The rotation angles of the nozzles are represented by φ 1 and φ 2 . The angle is zero when each nozzle faces directly down (in the −y n direction). The position-controlled servomotors (ROBOTIS, Dynamixel MX28) control the angles φ 1 and φ 2 . Owing to geometric constraints, we set −π < φ 1 ≤ 0 and 0 ≤ φ 2 ≤ π. These constraints were always fulfilled during the experiment.
The nozzle angles can be determined to realize the normal force R s when the roll posture of the nozzle unit is given as θ rn , where φ a = arccos R s 2F n , and F n is the reaction force of each nozzle. The reaction force F n is determined by F n = 28.3p, where p is the relative pressure (derived from a preliminary experiment). The net force R s (control input) is realized at 100 Hz by measuring p and commanding φ 1,2 at 100 Hz.

B. VALIDATION OF THE FLUID DAMPING EFFECT
We investigated the disturbance response for various flow speeds with control gains K = D = 0 to verify the fluid damping effect. For each flow speed, the root pressure (flow speed) was fixed, six initial disturbances were applied to the tip using a rod, and the time responses of the tip position were measured. The root pressure was changed for all seven patterns. Examples of the measured time responses to the disturbances are shown in Fig. 7A (see supplementary video). We found that the convergence time decreased as the flow speed increased, which exhibited a damping effect.
To investigate this tendency quantitatively, we define the following four indices for the time responses of the head height against disturbance.
(a) Settling time (the time when the deviation is less than 15% of the initial error) (b) Rise time (the time when the deviation changes from 90% to 10% of the initial error) (c) Overshoot (d) Equilibrium position (mean value of y L for 10 s without a disturbance) In addition, we numerically calculate the time response of the head height and the above four indices to compare and validate the simulation. We numerically solve the linearized equation (35) with the parameters in Table 2. The initial state VOLUME 10, 2022 as a disturbance is given as the equilibrium shape when the head is applied with a 10 N downward external force.
The responses of these four indices against the flow speed U are shown in Figs. 8A(a)-(d). The simulation results are shown in the same figures. The results indicated that the settling time decreased and the stability improved as the flow speed increased. This trend is also consistent with the simulation results. The rise time was almost constant, similar to the simulation results. The overshoot tends to decrease with the flow velocity except at the point where U = 0, which is also consistent with the simulation results. These results show that the stability can be improved according to the flow speed, and that the experimental results are in good agreement with those of the simulation.
In contrast, the equilibrium height was significantly different from the simulation results. This is because of the initial bending tendency in the pipe. However, the equilibrium height is almost constant with respect to the flow speed, which is consistent with the simulation results. Moreover, the overshoot value at U = 0 was significantly different from that of the simulation. This may be attributed to the influence of the internal fluid movement according to the pipe oscillation in the experiment. We note that the settling time in the simulation decreased stepwise because the time response of y(L) oscillates, as shown in Fig. 7A, meaning that the settling timing may jump by one wavelength when U changes.

C. RESULT FOR D GAIN CHANGE
To validate the adjustability of the responsiveness using the D gain of the proposed controller, we measured the disturbance response for various D gains, while the root pressure and controller P gain were fixed at 0.6 MPa (flow velocity of 3.66 m/s) and K = 0, respectively. The disturbance was created six times at each D gain, and the D gain was varied for seven patterns. Examples of the time response to the disturbances are shown in Fig. 7B (see supplementary video). As the D gain increased, the convergence time decreased, and the overshoot tended to disappear. At D = 15, we observe an over-damping case.
For quantitative analysis, we introduced the four indices discussed in the previous section. The responses of these four indices to D are shown in Figs. 8B(a)-(d). The simulation results for identical cases are illustrated in the same figure.
To evaluate the motor response delay in the experiment, we also conducted simulations where first-order filters with time constants of 0.03 and 0.07 s are applied to the output of the control equation (24).
The settling time tends to decrease and then increases slightly as the D gain increases, indicating that the stability can be improved by selecting an appropriate gain. This trend is in good agreement with the simulations. Moreover, the rise time tends to increase with the D gain, which is consistent with the simulation results. The overshoot tended to decrease with the D gain, which is also consistent with the simulation results. These results show that the stability and responsiveness of the controller can be adjusted according to the D gain.
In contrast, the equilibrium height is significantly different from the simulation, which is speculated to be due to the same reason as in the previous section. In addition, for the rise time, the experimental and simulation values differed significantly for a high D gain. We believe that this is because the time delay in the motor response becomes significant when the D gain increases. In fact, when a first-order filter was added to the simulation, the tendency was contiguous with the experimental results. However, there is still a considerable error in the rise time. This is probably because the nozzle angles do not correspond to the commanded values due to backlash.

D. RESULT FOR P GAIN CHANGE
To validate the adjustability of the responsiveness using the P gain, we measured the disturbance responses for different P gains K while the root pressure, controller D gain, and desired position were fixed at approximately 0.6 MPa (flow velocity of 3.72 m/s), D = 0, and y d = −0.66, respectively. The disturbance was introduced six times at each P gain, and the P gain was changed for eight patterns. Examples of the time response to the disturbances are shown in Fig. 7C (see supplementary video). As the P gain increased, the rise time decreased, and the overshoot tended to increase.
For quantitative analysis, we introduced the four indices described in the previous section. The responses of these four indices to K are shown in Figs. 8C(a)-(d). The simulation results for the same cases are shown in the same figure. To evaluate the delay of the motor response in the experiment, we also conducted simulations with first-order filters, as in the previous section. Note that we set y d = −0.54 in the simulation, so as not to change the simulation's equilibrium point.
The settling time increased with the P gain in a stepwise manner, which was also observed in the simulations. In addition, the rise time decreases, and the overshoot increases with the P gain, which is consistent with the simulation results. These results show that the responsiveness can be adjusted according to the P gain in this controller.
In contrast, the equilibrium height differs significantly from that of the simulation, but this is speculated to be due to the same reason as in the previous sections. For the convergence time, the location of the step was different for the experiment and simulation. This may be attributed to the nozzle backlash, which prevents the commanded force from realization and lowers the P gain in the experiment. However, the overall tendency for the convergence time to increase is consistent.

VIII. CONCLUSION
In this study, we proposed a stabilized controller (vertical position controller) for a jet-actuated two-dimensional cantilevered pipe with a nozzle unit at the tip using the damping effect of the internal flowing fluid, and verified the controller with a real robot. We derived a model based on several assumptions. The time evolution characteristics of the model energy function were analyzed, and a controller was proposed that can constantly decrease the energy function. The proposed controller was verified using simulations, and it was confirmed that the system was stable regardless of the flow velocity. In particular, the fluid effect contributed to suppressing higher-order mode oscillations. Moreover, we confirmed that the stability could be improved by tuning the gains of the proposed controller. Finally, the proposed controller and simulations were verified through experiments. The experiments proved that the vibrations could be damped by the fluid effect and that the stability and responsiveness of the system could be improved by tuning the gains of the controller.
The results provide significant suggestions and insights for future control strategies of jet-actuated continuum robots. For example, the proposed controller can stabilize the higher-order modes by utilizing fluid effects, even though the number of control inputs is small. This control strategy, using the damping property of the flowing fluid, would be very useful for continuum robots, because they are also distributed parameter systems with a limited number of actuators, such as the DFF controlled by the discrete nozzle units [6]- [8].
In the future, we aim to improve the model to handle significant deformations, for applications in jet-actuated continuum robots, such as DFF. Moreover, we aim to develop a controller for robots, based on the results of this study. Furthermore, we plan to create a three-dimensional model. Additionally, the theoretical proof of stability, which was not discussed in this paper, should be an area of focus.