Model-based linear control of nonlinear pneumatic soft bending actuators

Advanced model-based control techniques hold great promise for the precise control of pneumatic soft bending actuators (PSBAs) with strong nonlinearities. However, most previous controllers were designed in a cumbersome nonlinear form. Considering the simplicity of linear system theory, this paper presents a novel perspective on using model-based linear control to handle nonlinear PSBAs, and for the first time, summarizes two methodologies, global linearization and pseudo-linear construction. Derived from them, Koopman-based and hysteresis-based linear control approaches are proposed, respectively. For the former, a novel fusion prediction equation is developed to build a high-fidelity Koopman model, realizing global linearization, and then the linear model predictive control (MPC) is deployed. For the latter, the inverse of the generalized Prandtl–Ishlinskii (GPI) model cascades with the PSBA to construct a pseudo-linear system, eliminating the asymmetric hysteresis, which activates the linear proportional-integral-derivative (PID) control. It is worth noting that the above two are based on data-driven models adapted to various PSBAs with material and structural customization. Finally, the two model-based linear control approaches are verified and compared through a series of experiments. The results show that the proposed linear controls, with more concise designs, achieve comparable or even superior performance than nonlinear controls.


Introduction
Soft robots, composed of highly deformable and elastic materials, exhibit adaptability and safety, promising for applications in unpredictable environments and human-robot interactions.Various actuation forms, such as shape memory alloys are challenging to apply to real-time control owing to computational complexity.The lumped parameter model, which approximates the dynamics of the PSBA as a standard secondorder equation, provides a simple method for system modeling [11].The low computational cost is suitable for model-based control design [12].However, the parameters of the lumped model have severe uncertainties that vary with the operating frequency and pressure amplitude.Advanced model-based control methods, such as sliding mode control and adaptive control [13][14][15][16], have to be designed in a complex nonlinear form to deal with the uncertainties, which is rather cumbersome and difficult to tune.
Considering the simplicity of the linear system theory, we present a novel perspective: using model-based linear control to handle nonlinear PSBAs.The two practical methodologies are summarized below.The first is to achieve global linearization of nonlinear PSBAs and then implement advanced linear control techniques, such as linear quadratic regulator (LQR) and model predictive control (MPC).The second is to transform the nonlinear PSBA into a pseudo-linear system by cascading an inverse model of the PSBA, and the classical proportional-integral-derivative (PID) controller can work effectively since the nonlinearities are nearly eliminated.
The critical challenge of the first methodology is to linearize the nonlinear PSBA on a global scale.Fortunately, Koopman operator theory proves that high-fidelity global linearization can be realized by mapping the nonlinear dynamics to a high-dimensional Koopman space via proper lifting functions [17,18].Recent work has attempted to use the Koopman operator to model and control robotic fish [19], soft arms [20][21][22], and wheeled mobile robots [23].Unlike the analysis modeling, the Koopman modeling is data-driven and generic for PSBAs with customized materials and structures.
Specifically, the key to Koopman modeling is to design proper lifting functions consisting of several nonlinear functions related to the measurement coordinates (observable states).Many efforts have been made to design lifting functions using monomials [20,21], polynomials [24], higherorder derivatives [19,25], or machine learning methods [26,27], each with benefits according to the situation.However, almost all focus on the diversity of the nonlinear functions while ignoring the importance of choosing measurement coordinates.A poor choice requires much cost to construct large-scale lifting functions.In contrast, a linear combination of the correct measurement coordinates is sufficient to describe the nonlinear dynamics, leading to a concise form.For PSBAs, the measurement coordinates are typically single and empirical, making it hard to design proper lifting functions and thus impairing the Koopman model's accuracy.
The second methodology, transforming the nonlinear PSBA into a pseudo-linear system, is relatively easy to implement compared to the first.By cascading an inverse model of the PSBA, the original nonlinearities are almost eliminated, resulting in the output of the cascade system being close to the input (pseudo-linear), which activates the linear PID control.This methodology has been successfully applied to piezoceramic actuators [28], where the key is to model the system and determine its inverse.Since the nonlinearities of PSBAs mainly originate from hysteresis effects, it is theoretically feasible to construct the pseudo-linear system using the inverse hysteresis model.
In detail, a popular class of hysteresis models is phenomenological operator-based models, such as Presch [29], Prandtl-Ishlinskii [30], and Maxwell-slip [31] models.Among them, the classical Prandtl-Ishlinskii (CPI) model is the most widely used in hysteresis compensation control due to its simple analytical inversion [32], and it has also been applied to PSBAs [33].Nonetheless, the CPI model fails to describe the asymmetric hysteresis loops, resulting in sluggish control performance.Some modified Prandtl-Ishlinskii models have been developed and applied to conventional pneumatic muscles [34,35], but rarely used in PSBAs.
In short, to achieve precise control while avoiding cumbersome nonlinear designs, two methodologies of model-based linear control are summarized as shown in figure 1, along with their respective limitations.We proposed Koopman-based and hysteresis-based linear control approaches to overcome the above limitations.For the former, a novel fusion prediction equation is developed to derive correct measurement coordinates, realizing high-fidelity global linearization, and then a linear MPC controller is deployed.For the latter, the generalized Prandtl-Ishlinskii (GPI) model eliminates the asymmetric hysteresis, building a pseudo-linear system that activates a linear PID controller.The two proposed model-based linear control approaches are validated and compared through a series of experiments.The contributions of this work are as follows: • We present a novel perspective on using model-based linear control to handle nonlinear PSBAs, and for the first time, systematically summarize two methodologies that promise to inspire research on precise control of PSBAs.
where x[k] represents the system state x ∈ R n at the discrete time step k, and f is the flow map.In Koopman operator theory, proper lifting functions Ψ : R n → R N , N ≫ n can be found to lift x, resulting in an infinite-dimensional Koopman operator K to advance the lifted states in linear form where K d is the discrete operator.However, the theory is difficult to apply to real systems due to its infinite dimensionality.Fortunately, the EDMD algorithm [36] was recently proposed to approximate K d with a finite-dimensional representation Kd in a data-driven manner.It consists of three steps: collecting data, lifting data, and approximating operator.
Step 1: Collecting data.For each j ∈ {1, 2, • • • , p}, collect the snapshot pairs (x[j], x[j + 1]) and integrate them into two matrices Step 2: Lifting data.The lifting functions Ψ(x), composed of several nonlinear real-valued functions: {ψ i : R n → R} N i =1 , are defined as Also, Ψ(x) enables the projection of the Koopman space back to the initial state space due to the form of the first n terms.
Lift X 1 and X 2 via (5) to get X 1li f t and X 2li f t Step 3: Approximating operator.Kd is obtained by minimizing the cost function (8)

Koopman linear model.
Based on the Koopman operator theory, we attempt to construct a control-oriented Koopman linear model for the PSBA with control inputs where z d [k] is the lifted states z d ∈ R N and u[k] is the control input u ∈ R m at the discrete time step k.X 1lift and X 2lift are extended to incorporate the input terms Notably, the same ] are extended in (10) and (11) because we do not concern the evolution of the control inputs.
Then, approximate K du (the Koopman operator with control inputs) with a finite-dimensional Kdu by minimizing In fact, Kdu contains the best A d and Meanwhile, C d is obtained derived from ( 5) where I denotes an identity matrix, O denotes a zero matrix.
In a word, it is clear to construct the Koopman linear model for the PSBA with control inputs.Moreover, the data-driven modeling makes it generic for PSBAs with customized materials and structures.See more information about the Koopman operator theory in [18].

Koopman optimization by fusion prediction equation
The accuracy of the Koopman model is majorly determined by lifting functions Ψ.Compared to designing complex ψ i , choosing correct measurement coordinates x is more critical.The bending angle θ is usually chosen as the measurement coordinate of PSBAs for Koopman modeling, since it is the controlled variable and can be easily measured.However, that is too simple, leading to complicated lifting functions.
The fusion prediction equation is developed to find the variables that are significantly correlated with θ.Choosing them as measurement coordinates promise to simplify the lifting functions and to optimize the Koopman modeling.
Intuitively, we derive the fusion prediction equation from the common incremental equation The key for predicting θ (k + 1) is to find the angle increment ∆θ (k + 1), which can be estimated based on three assumptions below.Assumption 1: the angle increment is constant.The first estimate ∆θ 1 (k + 1) is derived by linear extrapolation Assumption 2: The angular velocity θ (k) is constant during sample time T. The second estimate ∆θ 2 (k + 1) is available where θ [k] can be smoothly estimated by the tracking differentiator (TD) proposed by Han [37].Assumption 3: the angle increment is proportional to the input increment, which is based on the phenomenon of PSBAs.
Note that the input of PSBAs is the command pressure rather than the current, and u [k] will act on θ [k + 1].Thus, we assume ∆θ 3 (k + 1) is proportional to ∆u (k).ε is the proportionality coefficient.
The above three assumptions are not entirely accurate.Inspired by the idea of data fusion in the Kalman filter, the optimal estimation of ∆θ (k + 1) is expressed as where α and β are model parameters.Substitute ( 19) into ( 15) Delay coordinates are introduced and denoted as θ , which can be obtained by the zero-order holder (ZOH).Equation ( 20) is rewritten as The fusion prediction equation of PSBAs is shown in (21).In order to predict θ (k + 1), the model makes full use of and the current input.We do not need to measure the noise of the model ( 16)-( 18) to determine the parameters in (21).It is meaningful that the model provides variables that are closely related to the bending angle, which are chosen as the measured coordinates to optimize the Koopman modeling.

Linear IMPC control
The fusion prediction equation facilitates the simplification of lifting functions to obtain a high-fidelity Koopman model.Then, model-based linear control techniques can be implemented conveniently.An IMPC is designed in this study.
Introduce the augmented state z ∈ R (N+m) and convert (9) into an incremental Koopman linear model where T , and the system matrices are redefined as A linear IMPC controller is designed based on the incremental Koopman model and transformed into the dense form min for the current kth step, X rk ∈ R N h n represents a sequence of reference values x r in the prediction horizon , and m constrain the bounds.D 1 , D 2 and D 3 are precomputed offline, which enables the computational cost to be independent of the augmented state's dimension.See [38] for more information.

Hysteresis-based linear control
This section proposes a hysteresis-based linear control approach derived from the methodology that converts the nonlinear PSBA into a pseudo-linear system and deploys a linear controller.First, the generalized Prandtl-Ishlinskii (GPI) model is used to describe the asymmetric hysteresis of PSBAs.Then, a pseudo-linear system is constructed by cascading the inverse GPI model.Finally, an incremental proportionalintegral-derivative (IPID) controller is activated.The control block diagram is shown in figure 3.

Generalized Prandtl-Ishlinskii hysteresis model
The hysteresis of PSBAs is usually asymmetric, leading to the invalidity of the CPI model.The GPI model is a modified CPI model that can describe asymmetric hysteresis loops [32].We first introduce the CPI model and its inverse.
The output of Π C is defined as where p 0 is a positive constant and the density function p (r) disappears for large values of r.Since the finite operators are sufficient to describe hysteresis in reality, Π C is typically used in discrete form The inverse CPI model can be derived analytically.It is also a CPI model whose parameters are functions of the original CPI model parameters where Γ zi is a play operator with threshold z i , and q i is the related density function ) where G determines the asymmetric hysteresis, and its output s is expressed as where Υ R and Υ L are envelope functions with respect to v.
Their difference is the key to reflecting the asymmetry.The most common form is the hyperbolic tangent function where a 0 , a 1 , a 2 , a 3 , b 0 , b 1 , b 2 , and b 3 are constants to be identified.
The inverse GPI model is a cascade structure of the inverse nonlinear model and the inverse CPI model where Π −1 C can be solved referring to (31).Then, solve G −1 to get the output of the inverse GPI model (41)

Parameters identification and pseudo-linear system
The parameters of the GPI model can be identified experimentally.The input and output data are normalized to accelerate the identification, and the thresholds for m play operators are chosen to be uniformly positive in 0 ∼ 1 The Modified Particle Swarm Optimization (MPSO) algorithm [39], which is known for preventing premature convergence to local optima, was used to identify GPI model parameters.The minimum cost function is chosen as where M denotes the number of experimental data.y a i and y i are actual and model outputs, respectively.
The inverse GPI model are solved by ( 38)∼(41), and cascades with the original PSBA to construct a new system.Theoretically, it is a linear system with equal inputs and outputs, where the input is the reference angle θ r and the output is the current angle θ In reality, the identified GPI model only approximates the PSBA system, and the inverse GPI model cannot completely remove the hysteresis nonlinearities.Therefore, the cascade system behaves pseudo-linearly, with the output being approximately equal to the input, as shown in figure 4.

Linear IPID control
Controlling the pseudo-linear system is equal to controlling the PSBA, and the linear PID is available for the pseudo-linear system.An IPID controller is deployed in this study where θ e is the tracking error, k p , k i , and k d are the parameters of the IPID controller.Overall, the proposed hysteresis-based linear control approach is distinct from the previous studies [33,34].The hysteresis model is used to construct a pseudo-linear system and activate a linear controller, rather than acting as a feedforward compensator in parallel with a feedback controller.

Experimental verification
This section verified the two proposed model-based linear control approaches on a PSBA platform, as shown in figure 5.The PSBA is an extensile pneumatic artificial muscle with a strain-limiting layer on one side.A flex sensor (Bendlabs One Axis, USA; Dimensions: 3.94 in × 0.30 in × 0.05 in; Repeatability: 0.18 deg; Sensitivity: 0.274 pF deg −1 ) is fixed to the layer to measure the bending angle in real-time.The air compressor (discharge rate: 65 l min −1 ; maximum pressure: 8 bar) provides the pressure source, after which an electric proportional valve (Festo VPPE-3-1-1/8-E1, Germany; maximum rated pressure: 11 bar) is connected to regulate the internal pressure of the PSBA.During pressurization, the PSBA produce the bending motion due to the different length between the two sides.The data acquisition card (NI PCI-6289, USA) sends command pressure to the valve and collects feedback signals.The rapid control prototype is created using the Matlab/Simulink Desktop Real-Time Toolbox and operates at 20 Hz.
Three experiments were conducted on the PSBA platform.First, we verified the effectiveness of the developed fusion prediction equation for the Koopman-based approach.Second, we validated the feasibility of constructing a pseudo-linear system by cascading the inverse GPI model for the hysteresisbased approach.Finally, the control performance of the two proposed linear approaches was compared with the classical nonlinear control.

Koopman modeling and global linearization
Koopman modeling is data-driven, relying on massive stochastic data.For the PSBA system, the input u is the command pressure, and the output is the bending angle θ.Referring to previous studies [21,22], we employed random step and sinusoidal signals as inputs to adequately stimulate the fast and periodic dynamics of the system across its entire operational range.Given the open-loop bandwidth of the PSBA typically less than 1 Hz, we gathered 63 min of input and output data sampled at 20 Hz.This sampling strategy is sufficient to capture the dynamics and ensures the closed-loop control performance.
To verify the effectiveness of the developed fusion prediction equation, the measurement coordinates derived from the equation were compared with the other eight sets of coordinates.They were chosen as the lifting functions without additional nonlinear forms to build the Koopman model, which intended to clarify the role of each coordinate.Subsequently, the 10-fold cross-validation was used to evaluate the prediction accuracy of nine Koopman models, including short-(1 s) and long-term (100 s) predictions.The distinction is made because short-term accuracy is related to optimal control, while long-term is associated with simulation analysis.
Table 1 shows the root mean square error (RMSE) of the predictions.The Koopman model with the measurement coordinates [ θ θ D θD u D ] T derived from the fusion prediction equation exhibited the smallest error.In detail, compared the original coordinate θ, adding θ D and θ slightly improved the short-term prediction accuracy but weakened the longterm; and adding u D considerably improved the both.Thus, u D is the most critical measurement coordinate of the PSBA system for Koopman modeling.Besides, the combination of coordinates displayed a trade-off effect.If several measurement coordinates had the same influence, the combination was cumulative.Otherwise, a compromise was reached.In addition, when we introduced θD that were not included in the fusion prediction equation, the error was larger even though the measurement coordinates were more abundant.It indicates that the redundancy is counterproductive and further confirms the superiority of the developed fusion prediction equation.Since the fusion prediction equation enables the correct choice of measurement coordinates, it is practical to simplify the lifting functions using monomials.In order to find the optimal power, the RMSE of the model predictions with different highest powers are shown in table 2. With the highest power of 3 and a dimension of 34 for the lifting functions, the Koopman model is superior.Thus, mindlessly designing the high-dimensional lifting functions is not recommended, and the rank of the Koopman operator may be deficient.
As a result, the measurement coordinates were chosen as [ θ θ D θD u D ] T , and the lifting functions were designed as monomials with the highest power of 3 to build the Koopman model of the PSBA.The short-and long-term predictions of the Koopman model are shown in figure 6.The short-term prediction exhibited relatively high accuracy, which facilitated the implementation of optimal control techniques such as MPC.The RMSE of the long-term prediction was 7.29% of the full range.In short, the Koopman linear model can describe the nonlinear PSBA with tolerable error, especially in the short term, achieving high-fidelity global linearization.

Hysteresis modeling and pseudo-linear system
The GPI model was used to describe the asymmetric hysteresis of the PSBA.Since the input was the command pressure, which was always non-negative, the one-sided play operator and the simplified envelope functions were chosen  To speed up the identification, the input and output data were normalized to 0-1.Table 3 lists the identified model parameters using the MPSO algorithm, which is configured with 50 particles, a weighting size of 6, a mutation probability of 0.75, and iterated 100 generations to ensure an optimal balance between exploration and exploitation in the parameter search space.The different a i and b i indicates the GPI model exhibits asymmetric.The comparison of the normalized model output with the experimental data is shown in figure 7.Over the full range of motion, the maximum error is less than 6%, and the RMSE is 1.94%.The GPI model well characterizes the asymmetric hysteresis of the PSBA.
Then, the identified GPI model was inverted and cascaded with the PSBA.In figure 8, the input and output data of the cascade system were collected and compared with the original PSBA.The output of the cascaded system was nearly a replica of the input.It indicates that the hysteresis nonlinearity of the original PSBA is almost eliminated.A simple linear controller can handle this pseudo-linearity.Therefore, it is feasible  to convert the nonlinear PSBA into a pseudo-linear system by cascading the inverse GPI model.

Comparison of control performance
The first experiment demonstrated that the Koopman modeling method achieved high-fidelity global linearization of the PSBA.The second experiment proved the feasibility of constructing a pseudo-linear system.These results serve as the basis for implementing the two proposed model-based linear control approaches.We also designed the adaptive sliding mode control (ASMC) as a classical nonlinear controller for comparison with the proposed linear controllers.

Nonlinear control design.
Referring to previous studies [13,14], the ASMC is designed using the lumped parameters model, which is identified as a standard To simplify the design process, we define new system states, x 1 (t) := θ (t), x 2 (t) := θ (t), and convert (49) to where . These parameters shown in table 4 are obtained using the Matlab System Identification Toolbox.It is observed that these parameters vary with the operating frequency.Nonetheless, the adaptive sliding mode controller is capable of handling such uncertainties to ensure the stability of the closed-loop system.
Define the sliding mode function s as The ASMC control law is designed as where γ1 , γ2 and γ3 are the estimates of the model parameters in (50).Note that sign in (52) can be replaced with a saturation function to reduce oscillation.The estimates γ1 , γ2 and γ3 are updated according to the adaptive law where β 1 , β 2 and β 3 determine the update rate.To avoid overestimation, the mapping algorithm is used to modify the adaptive law.Details of the mapping algorithm and the stability analysis of the closed-loop system can be seen in appendix.

Comparative experiment and results
. Three control methods, Koopman-based linear control (Linear Control 1), hysteresis-based linear control (Linear Control 2), and ASMC (Nonlinear Control) were compared by tracking four trajectories, including step, triangle, low-frequency sine (0.05 Hz), and high-frequency sine (0.1 Hz).For the Linear Control 1, an IMPC controller was employed and the prediction horizon was set to 20.The matrices D 1 , D 2 , and D 3 were determined offline from the system model as well as the weight coefficients in receding horizon optimization (Q = 0.1, F = 1, R = 0.5).For the Linear Control 2, an IPID controller was used to handle the cascaded pseudo-linear system, and the parameters were tuned by Ziegler-Nichols rule: k p = 0.8, k i = 0.1 and k d = 0.03.More challengingly, up to six parameters of the Nonlinear Control were turned by trial and error to obtain the relatively good performance: k a = 1.5, η = 0.6, c = 3, β 1 = 0.1, β 2 = 0.01 and β 3 = 0.01.
Figure 9 presents the experimental results of tracking control for the PSBA.Note that the initial bending angle was set at 30 • for two primary reasons: (1) The soft material tends to naturally bend at an angle of roughly 30 • due to changes in its properties after multiple bending motions.(2) The requirement of the PSBA for pre-filling with a starting pressure of approximately 0.5 bar to maintain stability, which experimentally corresponds to a bending angle close to 30 • .The RMSE of tracking all four cases is shown in 5.Among these, tracking the step proves most challenging.The results show that Nonlinear Control has a minor tracking error 6.8905 with an impressively quick response time.Similarly, Linear Control 2 also exhibits a faster response speed than Linear Control 1.The underlying reason is that both Nonlinear Control and Linear Control 2 use the model for some degree of feedforward compensation.However, there exhibit significant overshoots and oscillations in both methods, particularly in Nonlinear Control, due to inaccurate feedforward models and slow parameter adaptation.These phenomena can hardly be avoided, no matter how the controllers are tuned.On the other hand, although the RMSE of Linear Control 1 appears worse than the other methods, it shows almost no overshoot or static error, with excellent dynamic and steady-state performance.
In the remaining three cases, both proposed linear controllers were deemed comparable or superior to the nonlinear controller.Specifically, Linear Control 1 outperformed Linear Control 2 and Nonlinear Control.This is due to the fact that the IMPC controller utilizes a short-term Koopman model with high precision to solve for the optimal control input, while the IPID and ASMC controllers rely on feedback that inevitably comes with a delay.Additionally, it is worth noting that the Nonlinear Control could not completely eliminate oscillation, resulting in a certain degree of high-frequency oscillation in the control input and actual angle.In contrast, the two proposed linear control methods, subject to constraints on the control increments, yield smoother inputs and outputs.

Discussion
Discovering a unified yet concise control framework for PSBAs with material and structural customization is meaningful.The nonlinear control strategies based on lumped parameter models, although feasible, tend to be complex and cumbersome.This paper is dedicated to discuss a novel perspective that employs model-based linear control to deal with nonlinear  Derived from the two methodologies, the Koopman-based and the hysteresis-based approaches are proposed and validated in comparison to the nonlinear controller based on lumped parameters model.The performance of the two model-based linear controls is comparable or superior to the classical nonlinear control.Moreover, the linear control design is standard and concise, involving merely three parameters, as opposed to the nonlinear controller that demands intricate design based on Lyapunov stability theory and management of up to six parameters.This simplicity in deploy and parameter tuning underscores the practical advantages of the proposed linear approaches.Additionally, both model-based linear controls utilize data-driven models, which implies high universality and can be adapted to various soft actuators with customized structures and materials.
Another contribution is developing a fusion prediction equation to derive the correct measurement coordinates, thus improving the accuracy of the Koopman linear model.Previous studies proved that state delay and derivative coordinates were effective [19,21].The fusion prediction equation supports these arguments and highlights the crucial role of the delay input coordinate.On this basis, the Koopman model describes the nonlinear PSBA in the long term with tolerable errors.Also, as shown in figure 6, the short-term prediction is excellent, which facilitates the design of the MPC as only the next few steps are needed.In addition, the fusion prediction equation also holds the potential to be applied to alternative pneumatic soft actuators, not limited to PSBAs.
Revisiting these two model-based linear control approaches has led us to the discovery that the types and usages of the models differ.The first approach implements a highdimensional linear Koopman model, which enables advanced linear control techniques such as LQR and MPC.The second approach employs a GPI hysteresis model that is inverted to create a pseudo-linear system.We compared the prediction accuracy of the Koopman and GPI models, along with the lumped parameter model, as shown in figure 10.The RMSE and the determination coefficient (R 2 ) are shown in table 6.The results indicate that the Koopman and GPI models surpass the lumped parameter model in terms of precision, suggesting that utilizing the lumped model for nonlinear control design may be more expensive.The high accuracy of the GPI model's predictions is particularly noteworthy.While it may present challenges when applied to advanced control designs due to its static nature, the GPI model is more suitable for simulation analysis and offers a promising alternative to the conventional finite element method.
Although this work introduces a novel perspective, methodologies, and specific approaches to control nonlinear PSBA with the concise linear manner, certain limitations must be recognized and addressed.For instance, the robustness of the control of soft actuators under varying loads, a pivotal challenge for the practical deployment of soft actuators and soft robots, needs to be thoroughly examined.We envisage that the incorporation of advanced control technologies could be instrumental, such as implementing disturbance observers to estimate external loads or adopting robust control strategies may offer viable solutions.

Conclusion
In this paper, we proposed two model-based linear control to handle nonlinear PSBAs.For the Koopman-based approach, the fusion prediction equation was developed to derive correct measurement coordinates, and it demonstrated the critical role of delay input.A high-fidelity Koopman model of the PSBA was constructed based on the correct measurement coordinates, and a linear IMPC controller was implemented.For the hysteresis-based approach, the GPI model described the asymmetric hysteresis with high accuracy.Its inverse cascaded with the nonlinear PSBA to construct a pseudo-linear system, which was handled by a linear IPID controller.Both linear approaches offer more concise control designs with fewer parameters compared to nonlinear ones, simplifying deployment and tuning process.
Four tracking experiments validated that the performance of the two model-based linear controls is comparable or even superior to the classical nonlinear control.Both are based on data-driven models that are generic for various PSBAs with customized structures and materials.In addition, the GPI model outperforms the Koopman model in prediction accuracy and is more suitable for simulation analysis.Future work includes enhancing the robustness of the proposed method to control PSBAs under varying loads, integrating them into the soft rehabilitation robot, and investigating human-robot interaction control.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study.The data that support the findings of this study are available upon reasonable request from the authors.

Figure 1 .
Figure 1.Overview on precise control of PSBAs.Model-based nonlinear control designs are usually cumbersome, while linear control exhibits natural convenience.To implement model-based linear control techniques, two methodologies are summarized and derive two promising approaches along with their limitations.

3. 1 . 1 .
CPI model and its inverse.The CPI model Π C consists of play operators and density functions.For the monotone input

Figure 3 .
Figure 3. Hysteresis-based linear control.The pink block is the inverse GPI hysteresis model, which cascades with the PSBA to construct a pseudo-linear system.The blue block contains a linear IPID controller.

Figure 4 .
Figure 4.The inverse GPI model cascades with the PSBA to eliminate the original asymmetric hysteresis, constructing a pseudo-linear system where the output θ is approximately equal to the input θr.

Figure 5 .
Figure 5.The experimental platform of the PSBA system.

Figure 6 .
Figure 6.Experimental results of Koopman model prediction: comparison of prediction performance and error for the short-term and long-term.

Figure 7 .
Figure 7. Verification of GPI model: comparison of the normalized model output with the experimental data.

Figure 8 .
Figure 8. Verification of pseudo-linearity: comparison of the original PSBA with the cascade system.

Figure 9 .
Figure 9. Experimental results of bending angle control for the PSBA: comparison of tracking performance, error, and control input for the two proposed model-based linear controls and the nonlinear control.(a) Step.(b) Triangle.(c) Low-frequency sine.(d) High-frequency sine.

Figure 10 .
Figure 10.Experimental results of model prediction: comparison of prediction performance and error for the Koopman model, GPI model and lumped parameter model.

•
Derived from the two methodologies, Koopman-based and hysteresis-based linear control approaches are proposed and Koopman-based linear control.The pink dashed area corresponds to the fusion prediction equation utilized to derive the measurement coordinates x, which is helpful to simplify lifting functions.The Koopman model of the PSBA is built based on the measurement coordinates in the orange dashed area.In the blue dashed area, a linear IMPC controller is designed using the Koopman model.The solid orange and blue lines signify the offline and online transmission of data, respectively.
compared.Both linear controls, with more concise designs, achieve comparable or even superior performance than nonlinear controls.•A fusion prediction equation is developed to derive correct measurement coordinates, simplifying the lifting functions and improving the Koopman model's accuracy.Figure 2.

Table 1 .
RMSE of Koopman models with different measurement coordinates.

Table 2 .
RMSE of Koopman model with different highest powers.

Table 3 .
Identified parameters of GPI model.

Table 4 .
Identified parameters with different operating frequencies.

Table 5 .
RMSE of tracking four trajectories.
PSBAs.Based on this perspective, two methodologies, namely global linearization and pseudo-linear construction, are systematically summarized for the first time.The proposed perspective and methodologies hold the potential to inspire future research in the precise and efficient control of PSBAs.

Table 6 .
RMSE and R 2 of Koopman, GPI and Lumped models.