Inspired by human motor control theory, stiffness control is highly effective in manipulation and human-interactive tasks. The implementation of stiffness control in robotic systems, however, has largely been limited to closed-loop control, and suffers from multiple issues such as limited frequency range, potential instability, and lack of contribution to energy efficiency. Variable-stiffness actuator represents a better solution, but the current designs are complex, heavy, and bulky. The approach in this paper seeks to address these issues by using pneumatic actuator as a variable series elastic actuator (VSEA), leveraging the compressibility of the working fluid. In this work, a pneumatic actuator is modeled as an elastic element with controllable stiffness and equilibrium point, both of which are functions of air masses in the two chambers. As such, for the implementation of stiffness control in a robotic system, the desired stiffness/equilibrium point can be converted to the desired chamber air masses, and a predictive pressure control approach is developed to control the timing of valve switching to obtain the desired air mass while minimizing control action. Experimental results showed that the new approach in this paper requires less expensive hardware (on–off valve instead of proportional valve), causes less control action in implementation, and provides good control performance by leveraging the inherent dynamics of the actuator.

Introduction

Stiffness control, proposed by Salisbury in 1980s, is a highly effective control approach for robotic manipulators in their interaction with humans or the environment [1]. Unlike position control or force/torque control, stiffness control regulates the behavior of a robotic manipulator to follow the desired behavior of an artificial spring. Similar to a real mechanical spring, an artificial spring generates an output force when there is a deflection from the equilibrium position. Later, this approach was generalized by Hogan to formulate the concept of impedance control, in which an artificial damper was added [2].

A major reason for stiffness/impedance control's popularity in modern robotics is its close match to the biological motor control theory. From a dynamics perspective, a biological muscle functions as a source of controllable force and impedance. The widely accepted equilibrium position hypothesis presented by Feldman and Lewis [3] supports this observation via physiological evidences. According to the theory, with an agonist–antagonist musculoskeletal structure, a pair of muscles provides independent position and stiffness control. The stiffness of the joint is determined via the sum of muscle activations, while the output torque and subsequent output position are determined by the difference in muscles activations. Prior research presented by Hogan [46] has shown that such a capability is critical in providing stability in the humans' interaction with the environment. Research has also shown that variable stiffness is a major contributing factor for the high energy efficiency in mammalian locomotion [7,8]. Inspired by such findings, stiffness/impedance control has been used in a wide variety of robotic applications. For example, impedance control has been used in conjunction with a finite-state machine to obtain a highly effective walking controller for lower-limb prosthesis [9].

For the implementation of stiffness control, current approaches have largely been limited to the traditional closed-loop control. Specifically, the desired force is calculated according to the deviation from the equilibrium point in combination with the desired stiffness, and closed-loop force control is used to regulate the actuator output to obtain the desired actuation force. Such strategy is conceptually simple, and the implementation does not rely on the specific actuator type. On the other hand, this strategy also suffers from the common problems associated with closed-loop control, especially the limited frequency range and potential stability issue due to the time delay in the control loop. Furthermore, the closed-loop strategy does not contribute to the energy efficiency of the robotic system, since closed-loop control is energetically nonconservative in general.

Motivated by these issues, researchers have also attempted to develop actuators with physically existing stiffness. Series elastic actuator (SEA) is a typical example [1012]. Connecting elastic elements in series with the electric motor-transmission assembly, an SEA incorporates the desired elasticity. However, the package is bulky and complex due to the added components. Furthermore, the elasticity of the SEA is fixed, without the capability of adjusting the stiffness in use. Under similar strategies, actuators with variable stiffness have also been developed, most of which based on nonlinear springs in combination with two independently controlled motors. Among them, some use the agonist–antagonist configuration, inspired by the human musculo–skeletal structure (e.g., Refs. [1317]); others use a configuration with each motor dedicated to a certain function (generating force/torque output or regulating output stiffness) (e.g., Refs. [18] and [19]). Obviously, these actuator designs also suffer from similar problems as the SEA, e.g., added complexity and excessive weight/size.

Unlike these existing approaches, the variable-stiffness actuator presented in this paper does not rely on dedicated elastic components. Instead, a double-acting pneumatic actuator, with two pressure-controlled chambers, can function as a variable-stiffness actuator with much less complexity compared with the aforementioned motor-spring designs. Note that there have been works on the simultaneous stiffness and position/force control for pneumatic muscle-type and cylinder-type actuators [2023]. However, these works do not explicitly address the regulation of equilibrium point, affecting their efficacy in use. In Sec. 2, the modeling of the pneumatic actuator as a VSEA is presented, followed by the control and experimental demonstration of the proposed approach.

Pneumatic Actuator as a VSEA

With the compressible working fluid, a pneumatic actuator features a physically existing elasticity, which enables its use as a SEA, similar in concept to the aforementioned SEA actuator. Furthermore, for a double-acting cylinder-type actuator, two actuator chambers can be independently controlled with respect to its air pressure (Fig. 1). As a result, this actuator can be treated as two-input-two-output dynamic system, which provides the basis for its use as a VSEA. In this section, elastic behavior of a pneumatic actuator is modeled to formulate the two-input-two-output mapping for the implementation of stiffness control.

To derive the equation for actuator stiffness, we can start from the equation of actuator force
$F=PaAa−PbAb−PatmAr$
(1)
where Pa is the chamber a (rodless chamber) pressure, Aa is the piston area facing chamber a, Pb is the chamber b pressure, Ab is the piston area facing chamber b, Patm is the atmospheric pressure, and Ar is the rod cross-sectional area. The actuator stiffness is defined as the partial derivative of the actuator force with respect to displacement
$K=−∂F∂x=−Aa∂Pa∂x+Ab∂Pb∂x$
(2)
where K is the actuator stiffness and x is the displacement with respect to the middle point of the stroke as the reference. Assuming air is an idea gas, chamber pressure can be expressed as
$Pa=maRTVa=maRTAa(L2+x)+Vda$
(3)
$Pb=mbRTVb=mbRTAb(L2−x)+Vdb$
(4)
where R is the universal gas constant, T is the air temperature, Va and Vb are the total volumes in chamber a and chamber b, respectively, ma and mb are the air masses in chamber a and chamber b, respectively, L is the cylinder stroke, and Vda and Vdb are the dead volumes in chamber a and chamber b, respectively. The dead volumes are the volumes unaffected by the motion of the piston, e.g., volume in the air pathway in the actuator, internal volume of the connection tube, etc. They are included to improve the accuracy of the dynamic model. To facilitate the subsequent derivation, a new variable, namely, dead length, is introduced as the ratio of the dead volume versus the corresponding piston area
$Lda=VdaAa$
(5)
$Ldb=VdbAb$
(6)
Substituting Eqs. (5) and (6) into Eqs. (3) and (4) yields
$Pa=maRTAa(L2+Lda+x)$
(7)
$Pb=mbRTAb(L2+Ldb−x)$
(8)
Differentiating Eqs. (7) and (8) with respect to x yields
$∂Pa∂x=−maRTAa(L2+Lda+x)2$
(9)
$∂Pb∂x=mbRTAb(L2+Ldb−x)2$
(10)
Substituting Eqs. (9) and (10) into Eq. (2)
$K=maRT(L2+Lda+x)2+mbRT(L2+Ldb−x)2$
(11)
As can be clearly seen from this equation, the actuator stiffness K is a function of the chamber air masses ma and mb as well as the piston displacement x. When the chambers are closed, the air masses remain as constants, and thus the stiffness is nearly a constant when the displacement is small. Note that the stiffness still varies with the displacement, which requires special attention when the displacement is significant. Similar to the derivation above, the equilibrium position xe can also be expressed as a function of ma and mb. At equilibrium, there is no force output
$Fe=PaeAa−PbeAb−PatmAr=0$
(12)
where Pae and Pbe are the chamber pressures when the piston is at the equilibrium point. They can be expressed as functions of chamber air masses by assuming the air to be an ideal gas
$Pae=maRTAa(L2+Lda+xe)$
(13)
$Pbe=mbRTAb(L2+Ldb−xe)$
(14)
Substituting Eqs. (13) and (14) into Eq. (12) yields
$maRTL2+Lda+xe−mbRTL2+Ldb−xe−PatmAr=0$
(15)
This equation can serve as an implicit definition of the equilibrium point xe as a function of ma and mb. As such, Eqs. (11) and (15) constitute a dynamic model from the chamber air masses as the input to the stiffness/equilibrium point as the output. Conversely, for the implementation of stiffness control, the desired chamber air masses can be calculated from the desired stiffness/equilibrium point. This can be conducted by solving the following matrix equation, which is derived by manipulating and combining Eqs. (11) and (15):
$[A]{mamb}={B}$
(16)
where
$[A]=[RT(L2+Lda+x)2RT(L2+Ldb−x)2RTL2+Lda+xe−RTL2+Ldb−xe]$
(17)
${B}={KPatmAr}$
(18)
The result can be expressed as
${mamb}=[A]−1{B}$
(19)

Stiffness Control of Pneumatic Actuator

The model derived in Sec. 2 forms the foundation for the stiffness control using the pneumatic actuator as a VSEA. Desired chamber air masses, as calculated through Eq. (19), serve as the set point for the lower-level controller. Air masses, however, cannot be directly measured, and thus need to be converted to the chamber pressures according to Eqs. (7) and (8), based on the current position. Subsequently, the desired chamber pressures can be obtained through simple pressure control. The whole process is depicted in Fig. 2(a), and the entire dynamic system is depicted in Fig. 2(b).

A major advantage of the approach presented in this paper is its potential in minimizing control actions in the stiffness control process. When a set of stiffness/equilibrium point is implemented, the corresponding chamber masses remain constant, assuming small deviation from the equilibrium point (i.e., $x≈xe$). As such, a simple three-position, closed-center valve can be used for each chamber, and only one cycle of valve action (open to supply or exhaust, and then return to the closed position at the center) is needed to increase or decrease chamber air mass to the desired value, which is unaffected by the subsequent piston motion. This approach offers multiple advantages over the traditional closed-loop implementation of stiffness control, including significantly reduced control action, noise level, and energy consumption.

To realize the aforementioned one-cycle valve action, however, poses a challenge to the controller design. On–off valves usually have significant time delay between signal input and the corresponding spool movement. Such time delay, if not properly addressed, will cause severe overshoot and large control error. To address this problem, the authors developed a pressure prediction approach to control the timing of valve closing action. When a new set of stiffness/equilibrium point commands arrives, the desired air mass in a chamber is compared to the current air mass to determine the initial control action (connecting to the supply to increase the air mass, or connecting to the exhaust to decrease). In the meantime, a pressure prediction algorithm is used to predict the future air pressure in a switching cycle (i.e., the time for the valve to switching to the closed position after the valve control signal is received). If the predicted pressure reaches the desired pressure, the valve will be switched to the center closed position to complete the cycle. The flowchart for this process is shown in Fig. 3, and the details of the pressure prediction algorithm are described below.

For each actuator chamber, the pressure dynamics can be expressed by
$P˙=RTVm˙−PVV˙$
(20)
where V is the chamber volume and $m˙$ is the mass flow rate. Pressure change usually occurs much faster than piston motion. As such, the velocity of piston movement can be assumed to be a constant, and the average chamber volume over the valve switch period and the rate of change of the chamber volume can be calculated accordingly. To calculate the mass flow rate, the flow through the valve can be modeled as the flow of an ideal gas through a converging nozzle, which yields the following equation:
$m˙(Pu,Pd)=±AvΨ(Pu,Pd)$
(21)
and
$Ψ(Pu,Pd)={γRT(2γ+1)(γ+1)/(γ−1)CfPu if PdPu≤Cr (choked)2γRT(γ−1)1−(PdPu)(γ−1)γ(PdPu)(1γ)CfPuotherwise (unchoked)$
(22)
where Av is the valve opening area, Pu and Pd are the upstream and downstream pressures, respectively, γ is the ratio of specific heats, Cf is the discharge coefficient of the valve (which accounts for irreversible flow conditions), and Cr is the pressure ratio that divides the flow regimes into unchoked (subsonic) and choked (sonic) flow through the orifice. To simplify the analysis, the valve opening area can be treated as a constant over the switching period. According to Eqs. (21) and (22), the mass flow rate can be expressed as a function of the chamber pressure. The specific form of the equation depends on the flow direction (charging/exhausting) and the flow regime (choked/unchoked), with four possible scenarios:
• (1)
Charging in the choked regime ($P/Ps≤Cr$, Ps is the supply pressure), which generates a constant flow rate
$m˙1=AvγRT(2γ+1)(γ+1)/(γ−1)CfPs=C1Ps$
(23)
where the constant C1 is introduced to simplify the expression
• (2)
charging in the unchoked regime ($P/Ps>Cr$), which yields the following function:
$m˙2=Av2γRT(γ−1)1−(PPs)(γ−1)γ(PPs)(1γ)CfPs$
(24)
• (3)
exhausting in the choked regime ($Patm/P≤Cr$), which yields a proportional function
$m˙3=−AvγRT(2γ+1)(γ+1)/(γ−1)CfP=−C1P$
(25)
• (4)
exhausting in the unchoked regime ($Patm/P>Cr$), which also yields the following function:
$m˙4=−Av2γRT(γ−1)1−(PatmP)(γ−1)γ(PatmP)(1γ)CfP$
(26)

Subsequently, the mass flow rate equations (23)(26) can be substituted into the pressure dynamics equation (20). The resulting equations and the corresponding solutions are summarized below:

• (1)
charging in the choked regime
$P˙+(V˙V)P=RTVC1Ps$
(27)
with the solution as
$P(t)=P0 exp(−V˙Vt)+RTV˙C1Ps[1−exp(−V˙Vt)]$
(28)
where P0 is the initial chamber pressure. From the equation above, the pressure at the end of the switching period can be calculated as P(TS) (TS is the length of the switching period).
• (2)
charging in the unchoked regime
$P˙=RTVm˙2−V˙VP$
(29)
with $m˙2$ defined by Eq. (24). Due to the complex form, it is difficult to obtain an analytical solution of this differential equation. As such, a numerical integration can be performed over the switching period to obtain P(TS). To reduce the computation load in the real-time implementation, a look-up table can be established for the $m˙2$ function to accelerate the operation.
• (3)
exhausting in the choked regime
$P˙=−(RTC1+V˙V)P$
(30)
with the solution as
$P(t)=P0 exp[−(RTC1+V˙V)t]$
(31)
• (4)
exhausting in the unchoked regime
$P˙=RTVm˙4−V˙VP$
(32)
with $m˙4$ defined by Eq. (26). Similar to scenario (2), it is difficult to obtain an analytical solution. As such, a numerical integration, performed with a look-up table on the $m˙4$ function, can be used to calculate P(TS).

After the predicted pressure is calculated, a simple comparison can be performed to determine if the control valve should be switched off at the current time step. Specifically, the desired pressure Pd is calculated according to the desired chamber mass md by using Eq. (7) (for chamber a) or Eq. (8) (for chamber b). Subsequently, if the predicted pressure P(TS) exceeds Pd (for pressurizing) or falls below Pd (for exhausting), the valve should be switched to the central closed position. Otherwise, the valve should stay at the current position for the current time step, and the calculation described previously should be repeated until the switching condition is met. Once the valve is switched to the central closed position, no further calculation is needed until the stiffness/equilibrium point changes to the next set of values.

Experimental Results

Experiments were conducted to demonstrate the performance of the proposed approach. The experimental setup includes a double-acting pneumatic cylinder (092.59-DPV, Bimba Manufacturing Company, University Park, IL) mounted on a vibration-isolated table. The piston in the pneumatic cylinder is connected to a moving block mounted on a linear slide, which defines the piston motion and allows a human operator to manually move the piston. Each of the two chambers of the cylinder is connected to a three-position, closed-centered solenoid valve (VQ1300K-5B1, SMC Corporation, Tokyo, Japan) for independent pressure control. Sensors in the experimental setup include a pair of pressure transducers (SDET-22 T-D25-G14-U-M12, FESTO, Esslingen, Germany) for the measurement of chamber air pressures and a linear potentiometer (LP-100F, Midori Precisions Co., Tokyo, Japan) for the measurement of the piston position. A photo of the experimental setup is shown in Fig. 4, and the model and control parameters are listed in Table 1.

The primary purpose of the experiments is to demonstrate the advantage of using the pneumatic actuator as a VSEA. With the approach in this paper, the pneumatic VSEA enables the implementation of stiffness control without the high-frequency control action required for closed-loop control. In the experiments, the pneumatic VSEA, starting from an at-rest state (both chambers at atmospheric pressure), implemented the stiffness control as described in Sec. 3 to obtain a desired set of stiffness/equilibrium point. The piston of the actuator was subject to a reciprocating motion imposed by a human operator. The output force of the actuator recorded in this process is then compared to the desired reaction force calculated from the artificial spring behavior
$Fd=−K(x−xe)$
(33)

Such comparison serves the purpose of demonstrating the accuracy of stiffness control of the pneumatic VSEA. A set of typical results are shown in Figs. 58, including the plots of piston motion (Fig. 5), comparison of desired versus measured actuator forces (Fig. 6, in which the forces are inverted to better match the motion plot), position–force relationship (Fig. 7), and valve commands (Fig. 8, in which 1 represents pressurizing, 0 represents being closed, and −1 represents exhausting). In this experiment, an artificial spring with the stiffness of 15 N/mm and equilibrium point at 5 mm was implemented starting at t = 2 s. It can be clearly observed that the measured output force closely matches the desired output force in Fig. 6, indicating that the pneumatic VSEA is able to provide the desired elastic behavior of the artificial spring. Such elastic behavior is more clearly shown in the position–force plot (Fig. 7). In this figure, the desired spring behavior is represented by a straight line, indicating a linear spring with a fixed equilibrium point. The measured actuator behavior, as represented by the force trajectory, closely matches the desired spring behavior. Small hysteresis loops are present, but the deviation is very small, demonstrating the validity of the elasticity model derived in Sec. 2. In the valve command plots (Fig. 8), only one short cycle of valve switching is displayed for each valve, indicating minimum control action and minimal involvement of closed-loop control.

The authors also conducted a set of experiments that involves the transition between two different sets of stiffness/equilibrium point, with the purpose of demonstrating the controller's capability in modulating the dynamic characteristics of the pneumatic VSEA in real-time. Typical results of such experiments are shown in Figs. 9 and 10. In this experiment, an artificial spring with the stiffness of 5 N/mm and equilibrium point at 5 mm was implemented starting at t = 2 s. Eight seconds later (t = 10 s), the springs parameters switched to 7 N/mm and 18 mm for the stiffness and equilibrium point, respectively. The piston motion, as compared with the equilibrium point, is shown in Fig. 9, with the corresponding forces shown in Fig. 10. As can be seen in Fig. 10, except for the short transitional period following t = 10 s, the measured output force closely matches the desired output force, demonstrating that the pneumatic VSEA is able to modulate the artificial spring behavior in real-time.

To provide a quantitative comparison of the proposed approach versus the traditional closed-loop implementation of stiffness control, the authors conducted an additional set of experiments, in which the pneumatic actuator was used as a traditional force-controlled actuator for the closed-loop implementation of stiffness control. The same artificial spring was implemented in this set of experiments. To improve the force control performance, the solenoid valves were replaced with a high-bandwidth proportional valve (MPYE-5-M5-010-B, FESTO, Esslingen, Germany). The force control was conducted by using a simple proportional integral derivative controller, with the desired force calculated from the desired artificial spring behavior according to Eq. (33). The results are shown in Figs. 1114, including the plots of piston motion (Fig. 11), comparison of desired versus measured actuator forces (Fig. 12, in which the forces are inverted to better match the motion plot), position–force relationship (Fig. 13), and valve command (Fig. 14, in which the valve command is normalized to the range of −1 to 1). Since the motion was generated by the human operator, there is a slight difference in piston motion among different sets of experimental results. Through closed-loop control, the force-controlled pneumatic actuator is able to provide the desired elastic behavior (Fig. 12), although the performance is slightly inferior to that provided by the pneumatic VSEA (Fig. 6). The position–force plot (Fig. 13) leads to a similar conclusion, with a slightly bigger deviation from the desired spring behavior when compared with the position–force plot for the pneumatic VSEA (Fig. 7). Although the closed-loop implemented stiffness control provides a similar performance as the VSEA, such performance is obtained through the nonstopping action of the proportional valve (Fig. 14), which is typical for closed-loop control. Through a comparison between the valve actions (Fig. 8 versus Fig. 14), it clearly shows that the proposed pneumatic VSEA provides a superior way of implementing stiffness control and avoiding the multiple issues with closed-loop control such as continuous high-frequency control action.

Conclusions and Future Works

This paper presents a new approach to implement stiffness control for robotic systems. With its compressible working fluid, a pneumatic actuator features a physically existing elasticity that can be modulated with the controlled chamber air masses. Leveraging this unique feature, the authors developed a model that characterizes the stiffness and equilibrium point as functions of the chamber air masses in the actuator. Subsequently, a stiffness control approach was developed, in which a predictive pressure control algorithm is used to improve pressure control performance while minimizing the valve action. This enables the pneumatic actuator to be used as a VSEA. Experimental results showed that the proposed approach is able to provide the desired elastic characteristics of an artificial spring in stiffness control. Compared with the traditional closed-loop control-based implementation, the pneumatic VSEA is an open-loop system, and thus is free of the multiple issues that affect the closed-loop systems (time delay, limited bandwidth, etc.). On the other hand, due to the nonlinearity of the air pressure dynamics, the stiffness of the pneumatic VSEA varies with the piston displacement, which requires special attention while the displacement is significant.

The pneumatic VSEA in this paper has a potential of replacing the existing force-controlled robotic actuators in interactive tasks such as manipulation and human assistance. A typical example is the actuation of lower-limb prostheses, which involve constant interaction with the human users. For the control of such prosthetic devices, the finite-state impedance control approach is especially effective in enabling human–robot interaction and restoring normal walking gait [9]. This approach requires the implementation of artificial impedance (primarily an artificial spring, complemented with a small artificial damper), with the parameters switching between different sets of values during phase transitions. Using the pneumatic VSEA to replace the existing force-controlled actuator, the implementation of the finite-state impedance approach is expected to be greatly simplified while generating better control performance. Leveraging the research in this area [2426], the authors' group will investigate such possibility and further compare the performances of the pneumatic VSEA versus the traditional force-controlled actuator in actual robotic applications.

Acknowledgment

This work was supported by the National Institutes of Health under Grant No. R01HD075493 and the National Science Foundation under Grant No. CBET-1351520.

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