Pneumatic Artificial Muscle Actuator Under Parametric and Hard Excitations


 In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle (PAM) subjected to time varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales (MMS). The time and frequency response, phase portraits and basin of attraction have been plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters have been evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters have been illustrated.


Introduction
Pneumatic Artificial Muscle (PAM) is an actuator that converts the pneumatic pressure attains from the air pressure to a linear pulling force. PAM offers a number of actuator features which are advantageous compare to the other conventional pneumatic actuators. The main advantages of these muscle actuators are high force to mass ratio, soft and flexible structure, lightweight and cheaper, large pulling forces with the least amount of compressed-air consumption [1,2]. Due to these advantages, PAMs have been widely used in the field of bio-robotic as well as in industrial applications for its easy installation with safer human interactions. There are different types of 4 excitation has been studied by Kalita and Dwivedy [20] where a constant pressure inside the muscle gives rise to super-and sub-harmonic resonance conditions. In the present work, an attempt has been made to study the hard excitation of a parametrically excited pneumatically actuated artificial muscle system where a time varying muscle force is considered which gives rise to the super-and sub-harmonic resonance conditions. It may be noted that as the superposition rule cannot be applied to the nonlinear systems, hence, it is not possible to extend the work considered in [17][18][19][20] to obtain the results reported in this work.
From these literature and to the best of the authors' knowledge, it has been observed that no study has been made to investigate the artificial muscle under parametric and hard excitation considering super-and sub-harmonic resonance conditions. Hence, in this work, an attempt has been made to study the response of a single degree of freedom system with a PAM having periodically time varying pressure leading to a forced and parametrically excited system with super-and subharmonic resonance conditions. As mentioned earlier, it may be noted that the resonance conditions considered in this work are completely different from the earlier works of the authors [17][18][19][20]. Here, a second order nonlinear governing equation of motion with hard excitation is derived, which is solved by using the method of multiple scales to obtain the response of the system. This work will find several mechanical and bio-mechanical applications where the system is actuated through PAM, which include, the movement of upper arm or lower arm for elderly persons or persons with semi-paralytic disabilities [42][43][44] and for robotic manipulation in industrial applications [45,46]. Also, for PAM actuated medical instrumentation for laparoscopic surgery [12] or keyhole surgery [47], the present analysis will be useful, where more precision is required to control the muscle actuator. The mathematical modeling along with the analytical solutions, have been discussed in the following sections. Figure 1(a) shows a mass m connected to a PAM with control valve circuit. The equivalent single degree of freedom system is shown in Fig. 1(b) where the PAM is modeled as a nonlinear spring and a damper, subjected to a muscle force . The expression for mus F is given by the following equation which is similar to the work of Li et al. [12] and Kalita and Dwivedy [17,18].

Mathematical model of the system
    (1) where 1 2 3 1 2 , , , , and c c c d d K are the constants which can be obtained from the experiments. P is the operating pressure in the muscle to actuate u is the displacement with respect to the unstressed position of the system and max l is the maximum possible length that can be attained by the muscle. In this case, 0 sin m P P P t   is considered, where m P and 0 P are static and dynamic pressure inside the muscle, respectively along with the frequency of the dynamic pressure is .
The nondimentional displacement can be written as u r x  by using a scaling factor r and the following non-dimensional parameters have been used for formulation.
Now, Eq. (3) can be simplified to the temporal equation of motion as follows.
In Eq. (6), the book keeping parameter  is less than 1 and  is the non-dimensional damping parameter. It may be noted that the nondimensional parameters 1 2 2 , and p p f are the function of 0 ; P 11 and pf are the function of . m P From the fourth term on the left-hand side of Eq. (6), it may clearly be observed that the coefficients of the response x contain time-varying terms with frequencies,  and 2 . Hence, this is a parametrically excited system with multi frequency excitation. Finally, the last term of the left hand side contains a cubic nonlinear term with the coefficient .
 In addition to this, the system is also a forced vibration system having a sinusoidally varying force of amplitude 2 f and frequency  . So, in this work, the governing equation Eq. (6) with cubic nonlinearity is subjected to hard excitation, where the magnitude of the forcing term is larger than that of the coefficient of the linear term. It may be noted that in the authors previous work [17,18], the forcing is considered to be weak and the forcing parameter 12 and ff are the function of the static pressure m P and dynamic pressure 0 , P respectively. Also, in some other works 7 by Kalita and Dwivedy [19,20], the authors investigated the nonlinear characteristics of PAM with weak [19] and hard [20] excitation by considering constant pressure P inside the muscle.
Due to the presence of various nonlinear terms in Eq. (6), it is very difficult to find the exact solution. So, there is a need to use the perturbation technique to find the approximate solution.
Hence, the method of multiple scales [40,41] Equating the coefficient of 0  and 1  from Eq. (7), the following equations are found respectively.
The general solution of Eq. (8) can be expressed as follows. where, A is a complex number and A is the complex conjugate of .
A Substituting the value of 0 x from Eq. (10) in Eq. (9) leads to the following equation.       The secular and nearly secular terms in Eq. (12), which should be eliminated to have bounded solution are given below. 00 33 Again, 00 33 3 2 2 2 11 Here, A is function of 1 T and considering the polar form i.e., 1 (17), the stability of the steady state response can be found by determining the eigenvalues of the Jacobian matrix ( J ). The Jacobian matrix ( J ) is as follows.
The system will be stable for the superharmonic resonance condition, if the real parts of all the eigenvalues of the Jacobian matrix ( J ) in Eq. (21) are negative.
For steady state response   00 , , a  following similar procedure as that in superharmonic resonance condition, one can obtain the following frequency response equation.
From Eq. (28), it may be noted that for subharmonic resonance condition, one may obtain both trivial state (i.e., 0 a  ) and nontrivial state (i.e., 0 a  ) responses. Hence, to obtain the nontrivial state response for the system consider The system will be stable for the subharmonic resonance condition, if the real parts of all the eigenvalues of the Jacobian matrix ( J ) in Eq. (29) are negative.

Stability of the trivial steady state response
The polar form of modulation Eq. (27) contains term like a  but for finding the stability of the trivial state the linearized equation will not contain the perturbation 1 Similarly, the stability of the steady state response   00 , pq for this case, can be achieved by examining the eigenvalues of the Jacobian matrix ( J ) found by perturbing Eq. (30) and Eq. (31). The Jacobian matrix can be written as follows.
Here, the system will be stable for trivial state responses when all the real parts the eigenvalues of the Jacobian matrix ( J ) in Eq. (32) are negative.

Results and discussions
In this section, the time and frequency responses for the system as shown in Fig.1 have been plotted by taking different set of system parameters values for super-and sub-harmonic resonance conditions. The value of the different system parameters are given in Table 1 which is similar to the work of Li et al. [12] and Kalita and Dwivedy [17,18]. Here, the steady state responses can be obtained from Eq. (21) for superharmonic resonance condition and Eq. (32) for subharmonic resonance condition from the method of multiple scales. In the following subsections results for super-and sub-harmonic resonance conditions have been discussed with the variation of system parameters. In all the frequency response plots red color depicts the unstable solution and blue color shows the stable solutions. 13

Superharmonic Resonance Condition
In this section, the responses are plotted for the superharmonic resonance condition, where the external excitation frequency is one third of the natural frequency of the system. A typical frequency response plot is shown Fig. 2. The stability of the steady state response of the system is examined with the help of the Jacobian matrix Eq. (21).  one can obtain the same results within a fraction of seconds and less memory space. Also, the unstable equilibrium points cannot be obtained by the conventional method.
In Fig. 3  In Fig. 4, the validation has been done for the frequency response plot in Fig. 2 Fig. 4(a). For the point N ( 2   ) marked in Fig. 2, the basin of attraction is shown in Fig. 4(b)   ) the system has single stable state.

 
which is increased by double as compared to Fig. 2.
From Fig. 2 and Fig. 5(d), it can be observed that by increasing in the value of nonlinearity ,  the maximum response amplitude can be achieved at higher value of detuning parameter .  In Fig. 5( Like the previous cases for the muscle parameters 1 c and 2 c , the maximum response amplitude decreases with either increase or decrease in the value of another muscle parameter 3  In this case also like in Fig. 5(e) and are same as that in Fig. 2 Fig. 5(i), the maximum response amplitude increases with increase in the value of another muscle parameter 2 d and decreases with decrease in the value of 2 d same as that in the cases for 0 P in Fig. 5(a) and  in Fig. 5 passively. This will help the designer to choose the system parameters according to the particular application. From Table 2, it can be noticed that in Case 3 with low value of 0 and , m PP one may obtain the same displacement as compared to the other cases. Also, Case 17 provides the desired amplitude at a low frequency of the system. So, one may use the different set of system parameters to effectively move the PAM for the exact application.

Subharmonic Resonance Condition
In this section, the numerical results for subharmonic resonance condition, where the external excitation frequency is three times of the natural frequency of the system have been explored. The frequency response have been plotted in Fig. 6 by taking the system parameters values as given in Table 1   In the nontrivial branch in Fig. 6, the jump up phenomenon can be noticed at 12  From Fig. 6, it can be observed that the system exhibits single stable solution in the trivial branch upto the saddle node bifurcation point C ( 7.928

 
) and after that the nontrivial branch arises with a stable and an unstable solution, i.e., the system shows bistable state. So, to validate the frequency response curve in Fig. 6, the basin of attraction have been plotted in a  plane for two different values of  in Fig. 8. For the point S ( 7.5   ) in Fig. 6, the system has only one stable solution, i.e., 0 a  in the trivial branch of the frequency response curve, which can be verified from the Fig. 8(a). From Fig. 8(b), it can be observed that the system has two stable solution with 0 a  and 0.1795 a  in the trivial and nontrivial branch of the frequency response curve in Fig. 6 for 11.

 
The system has also an unstable solution with 0.1668 a  at 11

 
in the nontrivial branch of the frequency response, which always shows an affinity to jump to the nearest stable solution with 0.1795. a  The frequency responses have been plotted for different set of system parameters and the trivial branch is found to be stable in all the cases for a wide range of .
 Hence, the nature of the nontrivial branch of the frequency response plots has been discussed extensively. In Fig. 9(a) the effect of the dynamic pressure in the muscle 0 P on the stability of the nontrivial branch has been studied with two different values of 0 P . With decrease in 0 29 kPa P  in Fig. 9(a), which is half to that in Fig Hence, from Fig. 6, Fig. 9(a) and Fig. 9(b), it can be observed that an optimum value of 0 and m PP should be chosen to operate the system at lower value of .  this case also the response amplitude is increased at a higher value of  in comparison to Fig. 6, whether the value of 3 c is increased or decreased.

Fig. 9
Frequency responses under subharmonic resonance condition with variation in the system parameters (a) 0 Like the case in Fig. 5(h) for superharmonic resonance condition, from Fig. 9(h) and Fig. 6, it can be observed that the effect of the muscle parameter 1 d is not distinct in case of subharmonic resonance condition. The response amplitude in Fig. 9  kPa in discrete steps of 50 kPa. At each pressure step, the PAM contraction is measured and noted as given in Table 4. The above steps are repeated for different loads in the range 0-3 kg (0-30 N).     The dynamic characteristics of the PAM have been analyzing with time response over a range of external forced excitation. Figure 11(a) and Fig. 11(b) show the schematic diagram and actual experimental setup respectively. The load pan of weight 3.5 N is used, which will be considered as dead-weight. The pan has been attached to the free end of the muscle first to measure the dynamic characteristics. The system is then excited vertically as shown in Fig. 11   The response function is then calculated from the Fast-Fourier transformed force and acceleration data. This experiment is repeated by varying the loads in the load pan with different input pressure inside the muscle. In Fig. 12 the time responses (Acceleration data) are shown for the muscle for lifting a load of 2.5 kg with 200 kPa input pressure into the muscle for above mentioned three external excitation frequencies.  Table 5  found to be in good agreement with the experimental results and the reduced equations from the method of multiple scales can be used to study system behaviour, which will reduce the computational time and memory.

Conclusion
In the present work, the nonlinear dynamics of a pneumatic artificial muscle is studied under hard excitation for super-and sub-harmonic resonance conditions. Using the method of multiple scales,