## Abstract

This work presents a novel approach for the design and control of a two degrees-of-freedom (DOF) robotic manipulator driven by one pneumatic artificial muscle (PAM) and one passive spring for each of its DOFs. The required air pressure is supplied to the PAMs using fast-switching on/off type pneumatic flow control valves. The proposed control architecture uses a proportional-derivative (PD) controller with a feed-forward term in the outer control loop to correct the position errors using an approximate model of the system dynamics and approximate PAM force-contraction characteristics. An inner pressure regulator loop tracks the reference pressure signals supplied by the outer loop using a pulse width modulation (PWM) scheme to control the pneumatic valves based on the approximated inflation–deflation characteristics for the given pneumatic flow circuit. The proposed controller is unique for PAM actuated robots that simultaneously consider three levels of complications, namely, coupled dynamics of multi-degrees-of-freedom system, non-linearities in the force-contraction characteristics of PAMs, and non-linearities involved in the use of on/off type pneumatic flow control valves. Experiments carried out using a laboratory prototype validate the effectiveness of the proposed control scheme.

## 1 Introduction

Pneumatic artificial muscle (PAM) actuators are preferable over other conventional actuators such as electric motors or hydraulic drives for driving rehabilitation robots, exoskeletons, and applications involving human–robot interaction [1,2] since these actuators display compliant behavior [3], high power to weight ratio, lighter weights, and lower costs. These properties also make them suitable for a variety of industrial and aerospace applications as well as applications involving environmental interactions [4]. However, the compliant behavior is accompanied by load and position dependent non-linearities [5] resulting from air-compressibility, non-linear force-contraction characteristics, friction between the rubber tube and the braid, and friction in between the threads of the braid [6]. Additionally, the phenomenon of hysteresis causes varying force-contraction characteristics during the inflation and the deflation stages of actuator motion [7,8]. All such factors add up to make accurate control of pneumatic artificial muscles a challenging task. The application of such actuators to drive the joints of a robotic manipulator with two degrees-of-freedom (2DOF) further adds to the control complexity since robot dynamics equations of motion are highly coupled and dependent on joint velocities and positions.

Previous studies have made use of antagonistic arrangement of two PAMs to drive a single joint which expedites the cost of the system and the controller. Noritsugu and Tanaka [9] used antagonistic arrangement of muscles for a two degrees-of-freedom manipulator to serve as a therapy robot and used an impedance control scheme to implement various physical therapy modes. Tondu et al. [10] designed seven-degrees-of-freedom anthropomorphic robot-arm for teleoperation entirely actuated by antagonistic McKibben PAM pairs. Many previous studies such as Refs. [11–13] have used proportional control valves for PAM actuation control which are quite expensive. Alternatively, using simple on/off pneumatic valves using pulse width modulation (PWM) provides a cost effective and simpler solution. The reduced cost comes at the price of increased control complexity because of additional issues to be addressed such as valve dead time, and varying the duty cycle and frequency of PWM for controlling the effective valve orifice area.

The non-linear and time varying nature of PAMs necessitates a framework to model their dynamic behavior. As proposed in Ref. [14], models for PAM dynamics can be broadly classified into two categories, namely, theoretical models and phenomenological models. The former models are derived by relating muscle dynamic behavior with the geometrical shape and material properties of the muscles. The latter makes use of input–output relationships for modeling complex PAM dynamics. Experimentally identified models of force–pressure–displacement relationships representing PAM behavior in various operating regimes are widely used as well [15]. Ganguly et al. [16] conducted experiments to establish an empirical model for quasi-static and dynamic characteristics of PAMs and used this model with an extended PID position controller for position tracking of a single degree-of-freedom manipulator. The above model was an improvement over the previously proposed model [17] which uses virtual work principle to express the force exerted by the PAM as a function of muscle contraction and pressure. Active model-based control for pneumatic artificial muscle [14] uses a three-step control scheme for the control of PAM displacement involving a simplified three-element reference model for the PAM, followed by the use of a Kalman filter to actively estimate the modeling errors in the previously described reference model and finally using a nominal controller to reject those errors. It uses proportional valves. In Ref. [18], the authors have argued that hysteresis compensation can achieve a better position control of the PAM actuators and hence developed pressure/length hysteresis model of a single PAM using a Maxwell slip model and experimentally verified the effectiveness of such controller and its robustness. A variety of other approaches used for the control of PAMs can be found in the literature such as non-linear PID control [5], neural network-based PID control [19], adaptive feed-forward-PID control with neural network [20], fuzzy logic controllers [21], and sliding mode controllers with non-linear disturbance observers [13]. Particularly, experimental findings for controlling a single DOF PAM-based system in Ref. [22] show that for most of the applications, two independent 2/2 on/off solenoid valves perform slightly better or at least at par with other configurations such as using a single 3/2 high-speed on/off solenoid valve or using a proportional valve.

The control problem posed by a two DOF robot driven by PAM actuators with antagonistic linear springs, in which the air flows to the PAMs are regulated by on/off type solenoid valves, manifests itself in the form of complications at three levels. First, the uncertainty in the coupled dynamic behavior of robot joints is caused by inaccuracies in inertia and coriolis/centrifugal matrices which cause uncertain joint torque requirements. Second, the required torques are generated by PAM forces whose nature is highly non-linear, caused by various factors mentioned previously. Lastly, maintaining the desired pressures in the PAMs corresponding to the desired forces using on/off type solenoid valves is a non-linear control problem since the only control variable is the duty cycle of solenoid valves. Simplistic problems involving trajectory tracking of a single DOF system using proportional valves have been addressed in Refs. [14,23–25] thus avoiding the first and the third complication levels. Majority of researchers such as in Refs. [21,26] have addressed only the first two levels of complications using proportional control valves instead of on/off solenoid valves for controlling a multi-DOF robot. A few studies such as Ref. [16] have addressed only the last two levels of complications showing effective control of a single degree-of-freedom only with on/off type solenoid operated pneumatic valves.

In the current study, we present a novel design of a two degrees-of-freedom planar manipulator which uses only two PAM actuators combined with passive linear springs unlike the conventional designs which typically use antagonistically arranged PAMs which would have required four PAMs. The spring behavior is mostly well predictable when compared to PAM behavior. Therefore, we have kept the lowest possible number of PAMs to improve the system performance. The novel aspects of this study are:

It covers all three levels of complications mentioned previously which arise because of interaction of non-linearities due to the presence of multiple DOFs, non-linear PAM behavior, and non-linearities caused by on/off type pneumatic flow control valves.

Offline computation of feed-forward terms for the proportional-derivative (PD) control scheme to save online computational effort.

Mathematical approximation of valve flow characteristics which does not require as detailed experiments as in Ref. [16].

The following contents are organized as follows. Section 2 describes the physical model of the system, PAM force-contraction model, and system dynamics. Section 3 describes the controller design for trajectory tracking task. This is followed by Sec. 4 which describes the experimental results for pressure tracking by inner control loop and joint position tracking by outer control loop after appropriate parameter tuning using the values tabulated in this section. Finally, Sec. 6 concludes the paper with a discussion on scope of future work.

## 2 Modeling of the Physical System

### 2.1 Mechanical Construction of the System.

The proposed two-link manipulator presented here is driven by two PAMs with a combination of gears as shown in Fig. 1. The restoring force at the joints is provided by the two springs S1 and S2. PAM1 drives gear 2 which is meshed with gear 1. Gear 1, gear 2, and link 1 altogether form an epicyclic gear train. Since gear 1 is fixed, contraction of PAM1 results in clockwise rotation of link 1. During this motion, spring 1 gets stretched and applies a restoring torque at joint 1. Thus, in order to control the position of joint 1, a precise control on the force of PAM 1 and its contraction is required. Similarly, joint 2 is driven by gear 4 which is meshed with gear 3. Gear 3 in turn is driven by the contraction of PAM2. The number of teeth in gear 4 is less than gear 3 to increase the joint speed and joint range of motion for the given maximum contraction of PAM2. The whole system is mounted on a thrust bearing located at joint 1. To get feedback about the joint positions, two potentiometers (POT1 and POT2) are located on link 1, as shown in Fig. 1.

### 2.2 Pneumatic Artificial Muscle Model.

*P*is the PAM internal pressure and $\epsilon =(l0\u2212l)/l0$ is the contraction ratio with

*l*

_{0}and

*l*being the length of PAM in fully deflated and in current state, respectively. There are other dynamic components of PAM force primarily due to the thread-on-tube friction and thread-on-thread friction inside the braided shell. The dynamic force components are modeled in the form dry coulomb friction force and viscous friction force both of which are dependent on the rate of contraction of the PAM as well as PAM internal pressure [6]. In the present work, only static component of force is supplied as a muscle model for determining the feed-forward terms in the control system.

### 2.3 Manipulator Dynamics Equations

#### 2.3.1 Newton–Euler Manipulator Dynamics Formulation.

*M*(

**Θ**) is the

*n*×

*n*mass matrix of the manipulator, $C(\Theta ,\Theta \u02d9)$ is the

*n*× 1 vector of Coriolis and centrifugal forces, and

**is the**

*g**n*× 1 gravity term vector. For the system under consideration,

*n*= 2 and

**has two components**

*τ**τ*

_{1}and

*τ*

_{2}corresponding to the net joint torques required at joint 1 and joint 2.

#### 2.3.2 Dynamics of Joint 1.

*T*

_{1}and

*T*

_{2}are the number teeth in gear 1 and gear 2, respectively, with Δ

*θ*

_{1}and Δ

*θ*

_{g2}being the angular change of link 1 and gear 2 from the home configuration of the manipulator to the current state. Here

*x*corresponds to the number of revolutions given to gear 2 as shown in Table 1 for carrying out the motion analysis of epicyclic gear train at joint 1. Solving Eqs. (3) and (4), we obtain

*τ*

_{g2eq}(the torque used for driving joint 1 and countering spring S1 torque) at joint 1 given as

*τ*

_{1}at joint 1 for a given trajectory is the algebraic sum of torques from spring 1 and PAM1 given as

*τ*

_{l1S1}is the torque applied by spring S1. Differentiating Eq. (5) and putting in Eq. (6) yields a relation for an equivalent torque on link 1 in the form of torque on gear 2 as

*τ*

_{g2PM1}is torque on gear 2 due to PAM1,

*I*

_{g2}is the mass moment of inertia of gear 2 about its centroidal axis, and $\theta \xa8g2$ relates to joint 1 acceleration (Eq. (5)) as follows:

*F*

_{PM1}is the force generated by PAM1, the torque applied by PAM1 on gear 2 is (Fig. 3)

*l*

_{AD}represents the distance between points A and D. Putting Eqs. (11) and (12) into Eq. (10) and using Eq. (5)

*x*is related to the change in length of PAM1 given by

*x*is given by Eq. (3). The torque acting on joint 1 by spring 1 is

*l*

_{s1}is length of spring 1 in a given configuration,

*k*

_{s1}is spring 1 stiffness, and

*l*

_{PR}represents the distance between points P and R. The expressions for

*l*

_{s1}and

*l*

_{PR}are given in Appendix A. Putting Eq. (13) into Eq. (9), we obtain

*τ*

_{l1S1}. Equations (14) and (16) provide the expressions for PAM1 contraction and force, respectively, which are used by the proposed controller.

Action | Revolutions of link 1 | Revolutions of gear 2 | Revolutions of gear 1 |
---|---|---|---|

Arm (link 1) is locked | 0 | x | $\u2212T2T1x$ |

Whole gear train is locked | Δθ_{1} | Δθ_{1} | Δθ_{1} |

Adding the above two conditions | Δθ_{1} | x + Δθ_{1} | $\Delta \theta 1\u2212T2T1x$ |

Action | Revolutions of link 1 | Revolutions of gear 2 | Revolutions of gear 1 |
---|---|---|---|

Arm (link 1) is locked | 0 | x | $\u2212T2T1x$ |

Whole gear train is locked | Δθ_{1} | Δθ_{1} | Δθ_{1} |

Adding the above two conditions | Δθ_{1} | x + Δθ_{1} | $\Delta \theta 1\u2212T2T1x$ |

#### 2.3.3 Dynamics of Joint 2.

*θ*

_{g4}= Δ

*θ*

_{2}represent change in joint 2 angle from home configuration,

*T*

_{3}and

*T*

_{4}are number of teeth on gear 3 and gear 4, respectively. The moment equation for gear 3 (Fig. 4) is

*τ*

_{g3PM2}is torque on gear 3 due to PAM2,

*I*

_{g3}is the mass moment of inertia of gear 3 about its centroidal axis, and $\theta \xa8g3$ relates to joint 2 acceleration from Eq. (17) as follows:

*F*

_{PM2}is the force generated in PAM2, the torque applied by PAM2 on gear 3 is (Fig. 4)

*τ*

_{2}at joint 2 for a given trajectory is the algebraic sum of torques from spring 2 and PAM2 given as

*τ*

_{l2S2}is the torque applied by spring S2. Differentiating Eq. (17) and putting in Eq. (21) yields a relation for an equivalent torque on link 2 in the form of torque on gear 3 as

*τ*

_{2}since the value of

*τ*

_{l2S2}cannot be positive because of the physical mounting of spring 2 in the system. Any given trajectory must satisfy this condition and hence there is a limitation on joint 2 velocities when PAM2 deflates and joint 2 rotates clockwise.

*k*

_{s2}is stiffness of spring 2,

*l*

_{s2}and $ls20$ are stretched and unstretched lengths of spring 2, respectively. The expressions of

*l*

_{s2}and

*l*

_{LM}are given in Appendix B.

*θ*

_{g3}is given by Eq. (17), such that Δ

*θ*

_{g4}= Δ

*θ*

_{2}and Δ

*θ*

_{g2}is obtained from the desired trajectory. From Eq. (24), we obtain

## 3 Design of Controller

### 3.1 Architecture of Outer Control Loop.

The overall control architecture of the proposed system is shown in Fig. 5. The difference between the desired and the actual joint angle trajectories are fed as errors into each joint’s PD control block which are tuned for each joint. PD blocks output pressure corrections to be done in corresponding PAMs. Additionally, both the joints have a feed-forward term (operating on system dynamics and PAM characteristics) described later which reduces the corrective actions required from PD blocks, thus reducing the PD gains as well as chances of instability. Furthermore, feed-forward term improves performance by preventing large errors, which would otherwise have to be corrected by PD blocks.

### 3.2 Inner Loop Pressure Regulation.

*A*and the upstream and downstream pressures (

*P*

_{up},

*P*

_{dn}) for the pneumatic flow control valve. The rate of pressure change in the PAM is governed by the equation [16]

*u*is the applied control signal to the valve,

*A*is the valve orifice area which is a non-linear function of the control signal

*u*, and $P\u02d9$ is the rate of pressure change in the PAM. Here $h\xaf$ denotes the influence observed in the flow characteristics because of the upstream and downstream pressure difference. Equation (28) is a forward mapping of duty cycle or control signal

*u*to the rate of pressure change in the PAM. For the control of PAMs, an inverse mapping of required rate of pressure change $P\u02d9req$ and to duty cycle or control signal

*u*is required which is represented by the equation

*h*

^{−1}in Eq. (29). Therefore, in this work we use the approach of obtaining

*u*in Eq. (29) in the form of a lookup table as described in the following paragraphs.

*p*

_{r}plots at a number of uniformly spaced constant duty cycles of inlet and outlet valves. Here

*p*represents absolute PAM pressure and

*p*

_{r}=

*p*

_{dn}/

*p*

_{up}such that

*p*

_{up}and

*p*

_{dn}are measured upstream and downstream pressures, respectively. That is,

*p*

_{r}=

*p*/

*p*

_{supply}for inflation and

*p*

_{r}=

*p*

_{atm}/

*p*for deflation, where

*p*

_{supply}is supply pressure and

*p*

_{atm}is atmospheric pressure. Observing the nature of these plots, the valve characteristic inflation–deflation curves for a given duty cycle have been approximated using

*α*and

*β*are parameters that are selected empirically for certain equally spaced control signal

*u*values. Carefully choosing an

*α*and a

*β*value for a given duty cycle gives a plot of left-hand side term of Eq. (30) which approximates the flow characteristics for that duty cycle. A systematic approach for selection of an

*α*and a

*β*value for a given duty cycle is discussed in Sec. 5.2. For our experiments, we selected values of these parameters as: for inflation

*α*= [2.8 2.5 2.2 2.0 1.5 1.3 1.1 0.7 0.4 0.2]

^{T}corresponding to duty cycles

*u*= [1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1]

^{T}. For deflation

*α*= [−3.6 −3.2 −2.9 −2.6 −1.9 −1.7 −1.4 −0.9 −0.5 −0.3]

^{T}corresponding to duty cycles

*u*= [−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1]

^{T}. Here a positive duty cycle indicates inlet valve is activated and a negative duty cycle denotes activation of outlet valve. The

*β*values were fixed at 1.92 for inflation and 1.8 for deflation. The left-hand side of Eq. (30) were plotted for each of these

*α*values at various

*p*

_{r}values chosen as

*p*

_{r}= [0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]

^{T}. Based on these values, the approximated inflation–deflation characteristics for the given valves and pneumatic circuit are plotted in Fig. 7. This plot is used to generate the lookup table for the duty cycle control signal

*u*similar to Eq. (29). Fine adjustments are done in the inflation–deflation characteristics during the tuning stage described in Sec. 3.4.

*u*to compensate for valve dead time which depends upon model of the valve. The saturated valve control signal is thus given as

*t*

_{d}is the valve dead time and

*t*

_{PWM}is the valve PWM duty cycle time. To decrease the fluctuations, a tolerance is provided between the actual PAM pressure and the reference pressure before providing a control signal to the valve as follows:

*u*

_{PWM}, the inlet valve or the exhaust valve is actuated as per the following specified condition:

*v*

_{i}and

*v*

_{o}denote the control signal (duty cycle) for inlet and the outlet valve, respectively.

The control architecture consists of a inner pressure regulator loop shown in Fig. 6 and an outer control loop shown in Fig. 5. The inner control loop takes the reference pressures to be tracked by the PAMs as an input from the outer control loop. It then uses a roughly estimated lookup table for generating valve control signal *u* in accordance with $u=h\u22121(Pup,Pdn,P\u02d9req)$.

### 3.3 Design of Feed-Forward Terms for the Controller.

The feed-forward terms in the outer control loop shown in Fig. 5 as dashed-rectangles derive their values from the motion dynamics of the given manipulator as well as force-contraction-pressure characteristics of PAMs from the manufacturer’s catalogue. The reference joint trajectories to be tracked are provided as input for offline computation of the feed-forward terms. Reference joint trajectories act as input to the dynamic motion equations of the manipulator described in Sec. 2.3 to obtain the required joint torques given by Eq. (2).

The required joint torques are then mapped into corresponding required PAM forces considering the moment arms of the PAMs and the joint gear ratios as given by Eqs. (16) and (27). The reference joint trajectories to be tracked are used for arriving at the corresponding PAM contractions using Eqs. (14) and (26). The PAM forces and the PAM contractions are then used to arrive at the required pressure to be maintained in a PAM using the force-contraction-pressure characteristic surface for the given PAM as shown in Fig. 8.

### 3.4 Two-Stage Controller Tuning.

Two-stage tuning process involves tuning the inner loop pressure regulator gain separately to follow the desired pressure trajectories. The tuned inner control loop is then inserted as a block in the outer control loop. The outer control loop is then finally tuned. Inner loop tuning depends on valve characteristics during charging and discharging processes, valve switch on and switch off time, the actual length of the air flow paths, path diameters, presence of orifices like flow regulators, supply pressure, atmospheric pressure, the period used for PWM, dead time of valves, and volume of PAMs.

There are two separately tuned linear gains, i.e., *k*_{feed1} and *k*_{feed2}, used with the feed-forward terms of both the joints in the outer control loop. The reason for these gains is attributed to non-accounting of joint friction, dynamic components of PAM forces mentioned in Sec. 2.2, and other non-linearities such as PAM hysteresis while deriving the feed-forward terms.

## 4 Experimental Results and Analysis

### 4.1 Experimental Setup.

An experimental test bench consisting of a two-link manipulator was developed as shown in Fig. 9. The system uses two Festo DMSP-5-100N PAMs for controlling the two joints of the manipulator. Air flow to these PAMs is regulated by four (two for each PAM) fast-switching pneumatic solenoid valves with switching times of approximately 1.7 ms (opening time) and 2 ms (closing time). The control algorithms are implemented using matlab Simulink running on a Lenovo Ideapad 320 Laptop based on Intel Core i5 7th Gen processor. Simulink model running on this development computer (host) uses a time-step of 5 ms. The code generated by Simulink on the development computer is then deployed on an Arduino Due microcontroller board (target) using Simulink External Mode. The signals to be monitored are acquired from the Arduino’s buffer using XCP on Serial Communication Interface between the host and the target. Based on the algorithm deployed, the Arduino board makes time-dependent switching of four TIP31C transistors acting as a switch for the four solenoid valves. The pressures inside the two PAMs is sensed by two SMC digital pressure sensors ISE30A-01-E-LA1. The joint angles of the manipulator are measured by two analog potentiometers calibrated for mapping their voltage outputs to the joint positions.

### 4.2 System and Controller Parameters.

The system parameters for the used experimental setup are tabulated in Table 2. The inner loop controller (Fig. 6) gains are set as *k*_{inner} = 6.0 for both the joints. These inner loop gains are tuned like a regular proportional controller only after tuning the *α* and *β* values described in Sec. 5.2. The two-stage tuning process described in Sec. 3.4 is followed while deciding the inner loop gains. The PD gains in the outer control loop (Fig. 5) for joint 1 are taken as $kp1=0.85$ and $kd1=0.04$. For joint 2, the PD gains are selected as $kp2=0.4$ and $kd2=0.03$. Feed-forward gains for joint 1 and joint 2 are set as *k*_{feed1} = 3.5 and *k*_{feed2} = 5, respectively. The outer loop PD gains and the feed-forward gains are tuned simultaneously over a number of trials using sinusoidal joint angle reference signals of varying frequencies. The feed-forward gains are tuned in a way that the majority of the input to the inner loop controller comes in the form of feed-forward instead of coming from the outer loop PD controller as the proposed work involves detailed system modeling. This tuning is achieved by observing the contribution from the feed-forward block and the PD block for each joint over several trials with varying joint angle reference signals. Initially, the PD gains are kept as low as possible, and the feed-forward gains are increased gradually. The feed-forward gain for a joint is increased in small steps and it is verified that the errors in the joint angle keep on reducing. The PD gains are tuned once the error starts increasing with any further increase in the feed-forward gain. The typical force-contraction characteristics of PAMs supplied by the manufacturer (Festo) are shown in Fig. 2. A number of data points from these curves are fitted to a surface shown in Fig. 8 using the matlab function *fit()* to obtain the coefficients of PAM contraction force expressed in Eq. (1). The coefficients thus obtained with 95% confidence bounds are presented in Table 3.

Symbol | Physical meaning | Value |
---|---|---|

m_{1} | Mass of link 1 | 0.2419 kg |

m_{2} | Mass of link 2 | 0.1998 kg |

l_{1} | Length of link 1 | 0.448 m |

l_{2} | Length of link 2 | 0.370 m |

I_{zz1} | Mass moment of inertia of link 1 about the mass center | 0.0041 kg m^{2} |

I_{zz2} | Mass moment of inertia of link 2 about the mass center | 0.0562 kg m^{2} |

Symbol | Physical meaning | Value |
---|---|---|

m_{1} | Mass of link 1 | 0.2419 kg |

m_{2} | Mass of link 2 | 0.1998 kg |

l_{1} | Length of link 1 | 0.448 m |

l_{2} | Length of link 2 | 0.370 m |

I_{zz1} | Mass moment of inertia of link 1 about the mass center | 0.0041 kg m^{2} |

I_{zz2} | Mass moment of inertia of link 2 about the mass center | 0.0562 kg m^{2} |

Parameter | Value | Parameter | Value |
---|---|---|---|

a_{1} | 1.2290 | a_{5} | 3.0612 × 10^{4} |

a_{2} | 27.0229 | a_{6} | −115.5416 |

a_{3} | −2.2295 × 10^{3} | a_{7} | −1.7789 × 10^{5} |

a_{4} | −75.8333 | a_{8} | 3.8063 × 10^{5} |

Parameter | Value | Parameter | Value |
---|---|---|---|

a_{1} | 1.2290 | a_{5} | 3.0612 × 10^{4} |

a_{2} | 27.0229 | a_{6} | −115.5416 |

a_{3} | −2.2295 × 10^{3} | a_{7} | −1.7789 × 10^{5} |

a_{4} | −75.8333 | a_{8} | 3.8063 × 10^{5} |

### 4.3 Pressure Tracking Results by the Inner Loop Pressure Regulator.

The results for pressure tracking by the inner loop controller for different reference signals with varying frequencies are shown in Figs. 10 and 11. Two types of reference signals are used: impulse waves (peak to peak value of 2 bar with frequencies 0.5 Hz and 1 Hz) and sine waves (peak to peak value of 4 bar with frequencies *π*/2 rad/s and *π* rad/s). As observed, the inner loop pressure regulator is able to track the reference signals with sufficient accuracy. Although there is a continuous presence of high-frequency low amplitude noise in the actual PAM pressure, they do not affect the length contraction of PAMs since the change in PAM lengths is a relatively slow phenomenon when compared to rate of change of pressure inside the PAM. High-frequency noise in pressure signals centered around the reference pressure signal do not produce a significant deviation from the required PAM contraction. This is evident from the noise free position tracking results presented in Sec. 4.4.

### 4.4 Results for Joint Position Control.

In Ref. [26], the authors have shown satisfactory tracking of joint angles varying in a sinusoidal manner with a frequency of 0.05 Hz which is extremely low compared to the frequency of *π*/2 Hz used for tracking experiments in this work. The same can be argued for the tracking performance reported in Ref. [21] for joint reference trajectories varying in triangular and sinusoidal manner with a frequency of 0.1 Hz. In Ref. [29], tracking performance is reported for sinusoidal and triangular reference signals of 0.066 Hz. All of these studies make use of proportional valves for maintaining the reference pressure in the PAMs contrary to the on/off valves used for this work. Similarly, very low frequencies are reported in various other studies. Since the present work involves an extra level of complexity over the existing results available in the literature, a direct quantitative comparison between the results obtained for this study and the previous studies is not reasonable.

## 5 Discussion

### 5.1 Novel Aspects of the Design.

The use of PAMs for driving any manipulator is accompanied with certain limitations on the speed of manipulation, available workspace, and limitations on joint torques depending upon the manipulator configuration. The primary reason for these limitations is the limited contraction and unidirectional force generation capabilities of PAMs illustrated in Fig. 2. The proposed design provides some flexibility to work around with these limitations. For instance, decreasing the stiffness of springs S1 or S2 increases the range of motion available for the corresponding joint, thus increasing the workspace. However, this would be accompanied by a decrease in the maximum available speed of manipulation from a particular configuration towards the home configuration as well as an increase in the maximum available speed of manipulation away from the home configuration. The provision of gears (Fig. 1) allows for better adjustment of available workspace, available joint torques, and speed of manipulation suited for the application under consideration. This is implemented by changing the gear ratio used at the joints. Most of the past research deals with two PAMs actuating one joint. However, in the proposed design, a single PAM is used for actuating a joint. Additionally, the use of two inexpensive on/off type valves (Festo MHE2-M1H-3/2G-M7-K, ∼50$ each) instead of one proportional control valve (Festo MPYE-5-M5-010-B, ∼750$) for each joint reduces the cost of the system. Additional costs incurred in the proposed design include the use of digital pressure sensors (SMC ISE30A-01-E-LA1, 50$). Still the overall cost of control for the proposed system is much less in comparison to that for a system designed with proportional control valves. Hence, the cost savings are not only for the simple on/off valve but also for the use of one PAM for one joint instead of two PAMs per joint. This however increases complexity in terms of control as the PAMs are unidirectional and the valves can attain discrete states only.

### 5.2 Approximation of Characteristic Inflation–Deflation Curves for Inlet and Exhaust Valves.

Referring to Eq. (30) and Fig. 7 while considering the inflation phase of a given PAM, an increase in the parameter *α* shifts the $p\u02d9/pup$ plot for a given duty cycle *u* further towards the positive side. The *β* value decides the rate of decay of this plot with increasing *p*_{r} or increasing pressure in the PAM. The basic idea is to tune *α* and *β* values for a given duty cycle such that this plot approximately matches the actual valve flow characteristics for a given duty cycle. Experimentally, this process can be followed during the tuning stage of the inner loop pressure regulator. While plotting the response of the inner loop pressure regulator (Figs. 10 and 11 for instance), if an abruptly high error repeatedly occurs at a particular value of the reference pressure signal, it indicates an incorrect selection of valve duty cycle *u* by the controller for that *p*_{r} value. By tuning the *α* and *β* using the logic described above, the inner loop controller is made to choose a more accurate value of *u* for that particular *p*_{r} value. This process is repeated over a number of trials using a variety of reference signals until an almost uniform error is obtained throughout the range of reference pressure signal to be tracked. The remaining error is corrected by tuning the *k*_{inner} gain of the pressure regulator (Fig. 6). This tuning follows the same principles as used for tuning any basic proportional controller over several trials. The metric used for evaluating the tuned inner loop pressure regulator is the root mean square of the error between actual and reference pressure signals with a maximum threshold of 0.25 bar. The results presented in Figs. 10 and 11 are tested against this metric to decide that the inner loop tuning yields satisfactory results.

## 6 Conclusion

This paper proposed a new PD and feed-forward-based position control scheme for a two DOF robotic manipulator actuated by one pneumatic artificial muscle and one passive spring per joint. The proposed control scheme features benefits such as: (1) offline feed-forward computations based on the desired task which needs smaller time-steps for controller calculations in comparison to other online schemes, (2) calculation of feed-forward terms provide a limitation on joint rates due to the use of passive springs as stated in Eq. (22), and (3) intuitive design of control loops which can be easily extended to systems with higher DOFs due to independent gain tuning provision for each joint.

The offline feed-forward computation scheme takes care of cross-coupling effects which appear in a multi-DOF system, whereas use of PAM force-contraction characteristics compensate for the non-linearity arising in a PAM-based system. Independent inner loop (for pressure tracking) and outer loop (for joint position tracking) parameter tuning allows for a quick and intuitive gain tuning.

The results show that the system is able to track the specified trajectories satisfactorily, thus verifying the proposed algorithm. In future we plan to replace the fixed gain PD control in the outer control loop with an adaptive PD control scheme to further improve the system performance in those pressure regimes where significant error is observed. The cyclical nature of the errors justifies the possibility of such an improvement. Furthermore, a study on disturbance rejection and load carrying abilities of this control scheme is planned.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.