0 Introduction

Compared with traditional actuators, such as motor drivers and hydraulic drivers, the pneumatic muscle actuator (PMA) has the advantages of simple structure, good flexibility, convenient installation, low noise, high power / weight ratio and so on^[1,2,3,4]. It has great potential for applications in bionic robot, mobility assistance^[5] and can greatly improve the rehabilitative protocols for paraplegic patients^[6]. Since the 1930s, many experts have done a lot of research on its structure design, characteristics of joints, control system and application, especially on the static characteristics. Chou and Hannaford^[7] considered the elastic deformation and used the energy conservation law to achieve the static model of output force and pressure. Klute et al.^[8] wrote a model including a non⁃linear Mooney⁃Rivlin mathematical description of the actuator’s internal bladder. Kothera et al.^[9] added the non⁃cylindrical tip shape and bladder thickness to improve the model that Ferraresi et al.^[10] set up by the force balance method. The working process of PMA is nonlinear, that is, the input pressure is not proportional to the contraction force. It is difficult to establish an accurate theoretical model based on the energy conservation law or the force balance method, and there is a great gap between theoretical results and experimental results. Even considering factors such as the influence of PMA ends, rubber elastic force, and friction between fiber networks, the actual theoretical model is either very complex or still inconsistent with the actual model.

To get a comparatively simple and accurate static model of PMA, a quasi⁃static test platform is designed and the concept of hybrid elastic modulus is proposed correspondingly. By combining the theoretical derivation with the experimental data, the model is established according to the material and structure characteristics of PMA and is compiled with the experimental data to obtain a completely new model of PMA, which can reflect its own characteristics and meet the precision requirements. In applications of PMA, it is common that two antagonistic PMAs were used to control a rotation joint^[5,11,12]. Liu et al.^[13] used four PMAs to control a 3⁃DOF parallel mechanism. At the same time, the PMA was widely used to drive the bionic arm^[14,15]. In order to gain a bionic shoulder joint with better working characteristics, a new 3⁃DOF parallel mechanism controlled by five PMAs is designed. This mechanism has some advantages of the parallel mechanism and PMA together, such as the good flexibility and compact structure, and it also can be used to be a wrist joint or a neck joint^[16].

In this paper, a new static model of PMA is firstly established by using the energy conservation law and the hybrid elastic modulus E which is fit out on the basis of experimental data. Then a 3⁃DOF parallel mechanism (a bionic shoulder joint) driven by five PMAs is designed. Finally, control experiments are done based on the new static model.

1 Static Characteristics of Pneumatic Muscle Actuator

PMA is composed of a rubber tube, a woven mesh, and two fixing components at the end. After the air is pumped in, the rubber tube will expand and pull the braid to make the artificial muscles produce the axial force. The geometric structure of PMA is illustrated in Fig.1. As shown from Fig.1

Fig.1 Geometric schematic of pneumatic artificial muscle

where L is the length of PMA, D _i the inner diameter of the rubber tube, D the diameter of fiber layer, b the fiber length around the artificial muscle, n the number of turns, θ the angle between the fiber and the axis line of rubber tube and w the thickness of the rubber tube.

Assuming that the volume of the rubber tube is unchangeable in the process of contraction and relaxation, this can be described as

where D ₀, L ₀ and w ₀ represent the initial diameter, the initial length and the initial thickness of PMA, respectively. Eqs.(1, 2) are brought into Eq.(3) to achieve the expression of the instantaneous thickness of rubber tube, which can be expressed as

The hybrid elastic modulus E is set up to represent its overall elastic property, and it can be easily achieved by fitting the static test curves of PMA. Besides, the energy conservation law can be used to the contracting and expanding process of PMA. The energy change in the process of transition is expressed as

where W _E is the energy that the fiber rubber tube have done in the axial direction, W _S the energy that the pressure have done in the radial direction, W _F the energy that external load have done and W _C the energy that the external pressure have done at two ends. Although the output force will change because of the nonlinearity of PMA, it can be regarded as a constant in small displacement.

Meanwhile, the hybrid elastic modulus E also can be considered as a constant in the tiny contraction. Eqs.(1)—(4) are brought into Eq.(5) to achieve the expression of W _E, W _S, W _F and W _C, which can be expressed as

The quasi⁃static test platform uses FESTO’s DMSP⁃10⁃100N⁃RM⁃CM PMA. The inner diameter is 10 mm, the initial length is 100 mm, the maximum allowable tension is 630 N, the maximum contraction percentage of the initial length is 25%, the initial braiding angle θ₀ is 45º and the outer diameter D ₀ is 13.5 mm. In Fig.2, curves with the same color represent the two processes of the expansion and the contraction in the condition of the same pressure.

Fig.2 Static test curves of pneumatic artificial muscle

According to Eqs.(5)—(9), if the force F and other variables are known, the elastic modulus E will be calculated in the process of expansion as well as the process of contraction by the same way. As shown in Fig.3, six pictures severally describe the relationship between the elastic modulus E and the contraction percentage ε when the input pressure p is steady. The graphic relation is certainly clear and Eq.(10) is used to fit out the elastic modulus E. The coefficients a, b, c and d are expressions related to the input pressure

Fig.3 Relationship between elastic modulus ε and contraction percentage ε

Since Fig.3shows the relationship between the hybrid elastic modulus and the contraction percentage, values of the coefficients (a, b, c and d) can be achieved by the module of fitting linear curves in the origin software. Then, the relationship between the pressure and the coefficients is fitted to get the expressions of the coefficients a(p), b(p), c(p) and d(p), whose results are shown in Table1.

Since a(p), b(p), c(p) and d(p) have been expressed in Table1, the expression of the elastic modulus E can be achieved by Eq.(10). Then, Eqs.(6)—(9) are brought into Eq.(5) to get an expression, which is the static model of PMA. For the convenience of the control, Eq.(5) and Eqs.(10)—(13) need to be integrated. Assuming that A=W _E/E, B=W _S/E, C=W _F, D=W _C/p, A, B, C and D are brought into Eq.(5) to get Eq.(11)

Assuming that a(p)=a ₁+a ₂ p, b(p)=b ₁+b ₂ p, c(p)=c ₁+c ₂ p, d(p)=d ₁+d ₂ p, a(p), b(p), c(p), d(p) and Eq.(10) are brought into Eq.(11) to get Eq.(12)

where K ₁=(b ₂ ε+c ₂ ε ² +d ₂ ε ³)(B-D), K ₂=Aa ₂+(1+b ₁ ε+c ₁ ε ²+d ₁ ε ³)(B-D)-(b ₂ ε+c ₂ ε ²+d ₂ ε ³)C, K ₃=Aa ₁-(1+b ₁ ε+c ₁ ε ²+d ₁ ε ³)C. According to Eq.(12), if the force F and contraction percentage ε are known, the input pressure p will be figured out.

2 Design and Analysis of Parallel Mechanism

The structure design of the parallel experimental platform is shown in Fig.4. The large motion platform is connected to the fixed platform by three pneumatic muscle actuators (PMA1, PMA2 and PMA3), which make the large motion platform have two rotational degrees of freedom. Besides, the small motion platform is linked to the fixed platform by a spherical joint and is connected to the big motion platform by a revolute joint and two pneumatic muscle actuators (PMA4 and PMA5), which make the small motion platform have one rotational degree of freedom to the big motion platform.

Fig.4 Photograph and structure diagram of the 3⁃DOF parallel mechanism

2.1 Kinematic analysis of parallel mechanism

The X⁃Y⁃Z Euler angle (α, β, γ) is taken to represent the posture of the small motion platform. The angle of coordinate system rotating around the O ₁ X ₁ axis is γ; the angle of coordinate system rotating around the O ₁ Y ₁ axis is β; and the angle of coordinate system rotating around the O ₁ Z ₁ axis is α. So, 21 T , the transformation matrix from the moving coordinate system O ₂ X ₂ Y ₂ Z ₂ to the fixed coordinate system O ₁ X ₁ Y ₁ Z ₁, can be easily solved. Then the length L _PMA1, L _PMA2, L _PMA3 of pneumatic muscle actuators (PMA1, PMA2, PMA3) can be expressed as

According to the moving coordinate system O ₂ X ₂ Y ₂ Z ₂, the coordinate point B ₄, B ₅, C ₁ and C ₂ can be figured out. Then, the length of L _PMA4 and L _PMA5 are expressed as

2.2 Dynamic analysis of parallel mechanism

Lagrange method is used to establish the dynamic equation of large motion platform^[17]. The known Lagrange equation format is as follows.

where K _E is the kinetic energy, P _E the potential energy, and q the generalized coordinate vector of system. The following expression can be obtained.

where τ denotes the generalized force (torque) vector acting on generalized coordinate. The roll, pitch, yaw (RPY) angle mentioned above is viewed as the generalized coordinate of the system, namely q =[γ β α]^T. The following expression can be obtained by Eq.(16)

where M ( q ) denotes the inertia matrix of the system, V(q,q˙) the Coriolis force and centrifugal force, and G ( q ) the gravity. According to the principle of virtual work, the following equation can be obtained.

where F is the force vector composed of the output forces of PMA1, PMA2, PMA3, and J ₁ is the Jacobian matrix of large motion platform.

Eq.(18) is the dynamic model of large motion platform. The derivation of the kinematic Eq.(13) can be obtained as

It is worth to point out that V(q,q˙) has little or even no effect on the dynamics. Thus, Eq.(18) can be simplified as

Because the small motion platform has only one rotational DOF around Z⁃axis, according to theorem of the rotating rigid body round fixed axis, the following equation can be obtained.

where τS is torque needed to drive the small platform, IS the moment of inertia of the small platform around Z⁃axis, and c _S the damping of the small platform.

On the other hand

Thus, according to Eqs.(21,22), F _S can be calculated.

3 Control of Parallel Mechanism

As shown from the structure diagram of the test system (see Fig.5), the 3⁃DOF parallel mechanism is a system of multiple inputs and three outputs. Pneumatic pump provides servo valves with air pressure. According to the input analog signal(0—10 V), the control card (ART Technology, Xian, Shanxi, model number: PCI8201) controls the output pressure(0—0.8 MPa) of five servo valves by the channel 0—4. Five PMAs are severally connected to the corresponding servo valves by pipeline. Three rotational angles (α,β,γ) are measured by the gyroscope and fed back to the computer.

Fig.5 Block diagram of control system of 3⁃DOF parallel mechanism

3.1 Rotation control of the large motion platform

The process of the closed⁃loop control is shown in Fig.6, and it is a kind of feed⁃forward control based on deviation. Two experiments of closed⁃loop control have been done (experiment 1: β=0º, γ=0º—-10º; experiment 2: γ=0º, β=0º—-10º), and the experimental results are shown as Figs.7 and 8. Besides, the maximum angle errors and average angle errors are calculated and shown in Table2.

Fig.6 Block diagram of closed loop control of motion platform

In Figs.7(a), 8(a), their average angle errors are 0.62º and 0.67º shown in Table2, which indicates that the new mathematical model can be applied to this new parallel mechanism. But the change of pressure in closed⁃loop control has bigger oscillation due to the existence of compensations of the input pressure. According to Figs.7(b), 8(b), the bias pressure has some palpable jumps because of the large inertia when the mechanism starts to move.

Fig.7 Error of the angle γ and change of the pressure in the process of γ from 0º to -10º (β=0º)

Fig.8 Error of the angle β and change of the pressure in the process of β from 0º to -10º (γ=0º)

3.2 Rotation control of the manipulator

In the feed⁃forward, the contraction percentage ε _PMA4 and ε _PMA5 can be figured out by the target rotation angle θ _d in Fig.9. According to the contraction percentage (ε _PMA4 and ε _PMA5) and the given contraction force (F _PMA5, F _PMA6), the instantaneous pressure (p _PMA4, p _PMA5) can be obtained and then applied to two pneumatic muscle actuators (PMA4 and PMA5).

Fig.9 Block diagram of closed⁃loop control of the manipulator

For the convenience of control, set F _PMA4=F _PMA5=58 N and ΔF _PMA4=ΔF _PMA5=ΔF=30 N during the test. The curves of closed loop⁃control are shown in Fig.10. Besides, the maximum angle error and average angle error are calculated and shown in Table2.

Fig.10 Error of the angle α and change of the pressure in the process of α from 39º to 89º

In Fig.10(a), the maximum angle error of closed⁃loop control calculated is 1.95º, and the average angle error is 1.02º. Both the maximum angle error and the average angle error are bigger than that of the large motion platform because the rotation angle of the manipulate changes from 39º to 89º, which is bigger than the rotation angle of the large motion platform. And, the tracking accuracy of the closed⁃loop control has great accuracy.

4 Conclusions

This article puts forward a new idea of establishing the static model of PMA, assuming that the energy is conserved in the instantaneous expanding or contracting process of PMA, and derives the quantitative value of the elastic modulus in the process of the shrinking or stretching of PMA. Then hybrid elastic modulus is fitted out to predict the relationship between contraction force, contraction percentage and the input pressure. Besides, the hybrid elastic modulus E is set up to avoid considering multiple nonlinear factors separately and simplifies the process of building this new model.

To validate the control accuracy of the new static model, a special parallel mechanism driven by five PMAs is designed. This parallel mechanism with three rotational degrees of freedom can be used as bionic shoulder joint and has advantages of good flexibility, no pollution, low cost, high power / weight ratio and so on. Afterwards, the inverse kinematics of this parallel mechanism is analyzed, and the parallel mechanism by using this new static mathematical model is effectively controlled. According to many experiments, the average angle error is almost 0.6º in the closed⁃loop control system, which verifies the effectiveness of this static model.

References