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Trajectory Tracking Control of the Bionic Joint Actuated by Pneumatic Artificial Muscle Based on Robust Modeling

To simply and effectively realize the trajectory tracking control of a bionic joint actuated by a single pneumatic artificial muscle (PAM), a cascaded control strategy is proposed based on the robust modeling method. Firstly, the relationship between the input voltage of the proportional directional control valve and the inner driving pressure of PAM is expressed as a nonlinear model analytically. Secondly, the nonlinear relationship between the driving pressure input of PAM and the angular position output of the bionic joint is described as a second-order linear time-invariant model (LTI) accompanied by parametric perturbations, equivalently, and then the parameters of the model are identified by the robust modeling method. Then, a hybrid model is established based on the two models (the nonlinear model and the LTI model) and corresponding to it, a cascaded controller is developed, the outer loop of which is an H-infinite controller for the angular position tracking designed by loop-shaping design procedure (LSDP) and the inner loop is a nonlinear controller based on the feedback linearization theory for the PAM driving pressure control. Finally, the experiment is accomplished within the joint rotation range of 90 degrees and with the working frequency upper bound of 1.25 rad/s. And the joint with the developed cascaded controller tracks given reference trajectories with steady-state errors smaller than 2%. Results show that the trajectory tracking control of a highly nonlinear system is highly efficient using the proposed strategy in the case of relatively low work frequency.


Introduction
 For the friendly interaction with human living environment, humanoid robot is always attractive for researchers of robotics. To achieve good performance and adaptability in various fields, the improvement on efficient of bionic joint is imperative. For the inherent advantages, such as high power-to-weight ratio, compliance and nonpollution, PAM has been used as a novel actuator for bionic system. However, the nonlinearity of the model between the force output, contraction displacement, and inner pressure of PAM is a great challenge for controller design to realize a perfect trajectory tracking [1].
To deal with the nonlinearity of PAM system, nonlinear control strategies are applied such as feedback linearization [2], backstepping control [3], adaptive control based on neural networks [4], and sliding mode control [5][6].
Besides these nonlinear control strategies, many linear and mixed strategies are also investigated such as predictive robust control [7] and ℋ ∞ control [8].
Although the tracking accuracy is relatively low, linear control strategies are easier to implement and can adopt more useful skills in frequency domain. Therefore, in this article, a practical and effective method is proposed to reduce the effect of nonlinearity of a bionic system with PAM actuator and realize a trajectory tracking based on robust control theory.
In this paper, the system model between the actual voltage input and joint motion output is divided into two cascaded subsystems, an inner pneumatic subsystem and an outer mechanical subsystem. The former model is analytic and the later is modelled by robust identification method. Accordingly, a cascaded controller is developed, where the inner controller is based on feedback linearization for tracking the driving pressure of PAM and the outer ℋ∞ controller synthesized by LSDP method for suppressing the effect of nonlinearity of mechanical motion [9].
The layout of this paper is shown as follows: Section 2 illustrates and models the pneumatic circuit and dynamics systems. Section 3 designs an ℋ∞ controller cascaded with a nonlinear controller of the PAM actuator system for trajectory tracking control. Section 4 implements and analyses the tracking experiment of the controlled platform. Thereby, the trajectory tracking of a PAM system is realized with good performance.

Physiological structure of PAM platform
In this paper, a bionic elbow joint is built to simulate the back-and-forth movement of forearm in the pitching plan as shown in Fig 1. The structure of PAM commonly used is depicted in Fig 2[10]. The main part of it is the cylindrical membrane, fixed between two flanges, which is of isotropic flexibility and made by textile-fiber. While the muscle inflated by compressed air, its contracts in length direction, expands in radial and output tension force. The mechanical platform is based on the 'slider-crank' structure which can realize the transmission between rectilinear motion/force and rotatory motion/torque as human elbow musculoskeletal system with compact size, as shown in Fig 3 [11]. The pendulum of it represents the forearm as the load of the joint. The parameters of mechanical platform are listed in Table 1.
For the compliance of PAM and unsymmetrical structure, the pair of antagonistic forces for driving the joint is the pulling force of PAM and the gravity on the pendulum. Thus the theoretical workspace is from horizontal position to vertical position.  Considering the gas is ideal and isothermal and mass constant inside the PAM, the model between the pressure and volume of the isentropic and polytropic process is described as [12]: where and denotes pressure and volume at the th state, and the polytropic index. Meanwhile, the relation between mass, pressure, temperature, and volume is given by: where is the mass of gas inside the muscle, the gas temperature and the gas constant. Combined with (1), the dynamic model of the pressure can be expressed by the total differential form of (2) as: where s is the contraction displacement of PAM and ̇ is the mass flow rate which is determined by the structure of valve and can be identified by experiment. Set ̇= ( , ) which can be calculated as: where is the input voltage of valve, the discharge coefficient, / the ratio between the downstream pressure and the upstream pressure .
In this paper, the gas flow is controlled with an electronic proportional directional control valve in 5/3-way produced by the pneumatic manufacturer FESTO [13][14]. The setpoint value input as analogue voltage signal is from 0v to 10v and the standard nominal rated flow 350L/min. The signals is collected with the software of NI LABVIEW and hardware of NI cDAQ-9178 compact data acquisition platform with NI 9269 and NI 9205 modules.
In (1) and (2), coefficients , , and are defined by experience, and the discharge coefficient determined by fitting the relationship between theoretical and experimental port area, as shown in Fig 4. All coefficients are listed in Table 2.

Modelling the dynamics system
To represent the nonlinear relationship between the motion output and inner driving pressure input with linear where where denotes the driving pressure, the angular position of the pendulum, and the real nonlinear and approximated linear parameter-varying model, (•) denotes the nonlinear submodel related to different order, , , and the parameter-varying linear submodel, and the high-order term.
Then, the transfer function of the linear model is expressed as And the nonlinearity of the system is model into the effect of parameters deviation related to angular position, as 1, accompanied with a high order part, as 2.
To describe the 1, the model set modeling(MEM) method is applied, by which the real model can be described with a set of possible submodels [17]. Based on it, a set of second order submodels are identified in corresponding subworkspaces. In the modelling experiment, the range of input pressure is divide into 10 successive parts, and these signals of input and output are plotted in Fig 5. The parameters of characteristic polynomial of each model are listed in Table  3 and the mean value ̅ ( ̅ = ∑ 1 ⁄ , z=j, c, and k) and the maximum relative error ̂ ( ̂= max with −1 ≤ , , ≤ 1 where , , and denotes the mean value error of each parameters, and , and the maximum relative of them respectively.

Controller design
Cascaded control strategy is effective when the relationship between the actual input and output of the system is complicated but can be simplified by described

Pressure control
Feedback linearization control strategy is broadly used for nonlinear system control, which can rearrange the system by introducing an external reference related to the final output indirectly. Based on it and combined with (3), (4) and (5), the derivative of pressure is chosen as the external reference signal of the inner loop for pressure tracking: =̇ (11) where denotes the external reference input. To diminishing the steady error exponentially, the external input is written as: where is the control parameter and the desired pressure. And the final input voltage of valve is given as:

Trajectory tracking control
For suppress the effect of parametric perturbations, an ℋ∞ controller is synthesize based on the loop-shaping design procedure (LSDP), and the principle diagram of it shown in  W1 and W2 is the pre-and post-compensator respectively, which are used to reshape the frequency response of open-loop transfer function so that the closed-loop system can achieve the required performance in different frequencies.
∞ is the shaped plant and formed as ∞ = , ∞ the ℋ∞ controller to stabilize the reshaped plant robustly, and the final feedback controller formed as: Consider the robust stabilization criterion [20]: Let =̃− 1̃ be the nominal model and the perturbed system can expressed as where △ is the perturbed system. △ and △ are the perturbations of model, unknown but stable. ̃,̃, △ , △ ∈ ℛℋ ∞ , and ‖[△ △ ]‖ ∞ < . And the perturbed system is robustly stable if and only if where is the feedback controller mentioned before and ( > 0) the stability margin.
Applying linear factional transformation (LFT), the three perturbed coefficients of (10) can be expressed as (17) and the block diagram of it is shown in  Ignoring the influence from high order part, as 2, the post-compensator W2 is set to 1 in the article. For the nominal model is only second order, the pre-compensator W1 is set to first order, expressed as: To eliminate the steady-state error, increasing the gain of shaped plant over low frequency range is an effective way. Meanwhile, to improve the transient performance, a higher cross-over frequency is accepted, and yet exaggerates the influence of parameters varying with high frequencies. Thus a tradeoff between them must be faced. However, for the bandwidth of the controlled inner pneumatic circuit is found about 3Hz by sine sweep frequency experiment. For the bandwidth is lower than the natural frequency of the mechanical platform, the 3Hz is set as the cross-over frequency of the controlled system. To study the effect of the gain over high frequency-range on the transient performance, three different W1s are selected and listed in Table 3. The bode diagrams of the three shaped plants(SP) are shown in Fig 10. For illustrating the transient performance of these controllers synthesized with different Comparing the three shaped systems, the three cases have the same steady-state error and rise time. However, with the gain lower over high-frequency range, the output is smoother, which can be confirmed by comparing the angular acceleration as shown in Fig 12. During the rise phase, the rate of velocity change of case 3 is smaller than the rest. Thus, the controller K3 synthesized by 1 3 is chosen. And the 1/γ is 0.7154 satisfying the requirement of inverse stability margin . The singular value of each components in the left of (21) is also calculated and shown in Fig 13, where the sensitivity of closed-loop over highfrequency range and the sensitivity complementary over low-frequency are all smaller than 1, and the theoretical bandwidth of the closed-loop system is about 1.2 rad/s.

Experiment and Discussion
To demonstrate the performance of this closed-loop system, a signal composed with 10 different desired steadystates ranging from 0.1rad to 1.4rad is designed, as shown in Fig 14. For observing the error for velocity tracking, cosine curve is used to connect every two neighboring steady-states, which makes the desired trajectory is differentiable during the whole tracking process. The performance of angular position tracking and velocity tracking are shown in Fig 14, 15, and 16. The Fig 15 only shows the relative error of steady-states for trajectory tracking. Except for □ 4 , the maximum of relative error during steady-state is smaller than 2% as shown in Fig 15, which shows the good robustness to the effect of parameter variation over low-frequency range. And, the small gain over high-frequency range results good smoothness and non-overshoot. However, for the transient performance sacrificed, a time delay exists between the desired and actual trajectory, which is the main source of tracking error for position and velocity. It leads to the maximums relative tracking errors for angular position appear when the sign of velocity tracking error changes as shown in Fig 14 and 16. The maximum of tracking error for trajectory is -0.46 during the interval from 0.1rad to 1.27rad for climb and 0.25 during the interval from 0.75rad to 0.23rad for descent Meanwhile, some details for transient performance can be found. although there is a positive relation between the maximum of tracking error and the stroke between every two neighboring steady-states as shown by □ 2 and □ 3 in Fig  14, the relative errors decreases and velocity error stable when the position of these strokes rising, as shown by □ 1 , which indicates the transient performance is robustly stable when the real model varying. The maximum of tracking error for velocity during acceleration and deceleration is 1.458rad/s and 1.261rad/s respectively. After that these errors converge to zero exponentially, as shown in Fig 15  and 16.
For the leakage of the electronic proportional is unavoidable, the performance is deteriorated when the inner driving pressure low. And, the effect of it can be counteracted by applying symmetrical structure and using two PAMs in an agonist-antagonist way. 1) For simulating the motion mechanism of elbow musculoskeletal system, an unsymmetrical bionic joint actuated with a single PAM is built, and a back-and-forth movement of forearm in pitching plane is realized.
2) Based on the robust modeling theory, the nonlinear relationship between the inner driving pressure of PAM and the motion output is model as a second order linear timeinvariant model accompanied with parametric perturbations, by which a modeling for the nonlinear system is realized 3) A cascaded controller, constructed with an outer ∞ controller synthesized by LSDP to reduce the effect of parametric perturbations and an inner controller based on the feedback linearization theory for the driving pressure control, is designed. And a trajectory tracking of the bionic joint is realized with it.