Dynamic Pneumatic Actuator Model for a Model-Based Torque ...
Dynamic Pneumatic Actuator Model for a Model-Based Torque Controller Joachim Schr¨ oder∗ Center for Intelligent Systems Vanderbilt University Nashville, TN, USA http://eecs.vanderbilt.edu/CIS/ [email protected] Kazuhiko Kawamura, Ph.D. Center for Intelligent Systems Vanderbilt University Nashville, TN, USA http://eecs.vanderbilt.edu/CIS/
R¨ udiger Dillmann, Dr.-Ing. Industrial Applications of Informatics and Microsystems University of Karlsruhe (TH) Karlsruhe, Germany http://wwwiaim.ira.uka.de/
Abstract The modeling of pneumatic actuators, also known as artificial muscles, has been the main focus in several research papers in the past [1, 2, 3, 4, 5]. At the Center for Intelligent Systems (CIS) at Vanderbilt University, we have been using pneumatic actuators for our Intelligent Soft Arm Control (ISAC, Figure 1). For the position control of a joint, driven by two artificial muscles, a physical actuator model was designed and used as the basis for a subsidiary torque control. It is known that such actuators contain a high nonlinearity including a hysteresis. Experiments show that the static hysteresis is less important than the dynamic one. In similar models, damping properties of the muscles have not been considered. This research paper focuses on the modification of a physical static model and the extension with a dynamic part. The quality of the model was verified by implementing it as a torque controller and running experiments on a testbed.
1 1.1
Duygun Erol Center for Intelligent Systems Vanderbilt University Nashville, TN, USA http://eecs.vanderbilt.edu/CIS/
coders at each joint on both arms. The artificial muscles resemble human muscle movement.
Figure 1: Humanoid robot system ISAC
Introduction ISAC Humanoid Robot System
The Intelligent Soft Arm Control (ISAC) has been developed at the CIS. ISAC consists of a human-like torso with two six-degree-of freedom robot manipulators, called Soft Arms [6]. Each Soft Arm is a pneumatically-driven robotic manipulator that is actuated by artificial muscles. These muscles are pneumatic actuators whose lengths shorten as their internal air pressure increases and expand as their internal pressure decreases. The muscles are fed with compressed air through servo control valves. The joint angles are calculated with optical position en-
1.2
Artificial Muscles
Artificial muscles were first developed in the 1950’s by the physician Joseph L. McKibben. Thirty years later, Bridgestone Corporation commercialized the muscles called Rubbertuators for industry (Figure 3a). Rubbertuators feature hysteresis in their operation due to the original hysteresis inherent in the rubber, friction between the rubber and cords and friction between the cords themselves (Figure 2).
∗ Acknowledgement: This research is conducted with the financial support of the Center for Intelligent Systems (CIS) and the Institute for Industrial Applications of Informatics and Microsystems (IAIM). The research is part of Mr. Schr¨ oder’s Diploma Thesis at the University of Karlsruhe (TH). Currently, Mr. Schr¨ oder is a visiting scholar at the Vanderbilt University.
Figure 2: Structure of artificial muscles [8]
Rubbertuators are well suited to a number of robotic applications because they are lightweight, naturally compliant, and have a high force-to-weight ratio [7]. The biggest advantage of artificial muscles is the natural movement and the safe interaction with humans because of lower power and stiffness than industrial robots. Shadow Robot Company [8] in England began production of artificial muscles for robots in the 1990’s (Figure 3b). We replaced most of the Soft Arm muscles with Shadow Air Muscles, that is why this paper deals mainly with this type of muscles. The Air Muscle work principle is the same as Bridgestone Rubbertuators. Figure 3 shows both types of artificial muscles.
(a)
Describing variables for the static state of a joint are: • Torque T = (F2 − F1 )r • Joint angle α =
• Pressure difference ∆p to initial pressure (1) The muscles are activated so one of them is deflated to p1 = p0 − ∆p and the other one is inflated to p2 = p0 + ∆p. Thus, ∆p is not the difference between the two muscles, but the distance to the initial pressure p0 . This kind of symmetric muscle activation has proved to be qualified in the past and will be used in the experiments. The advantage of this procedure is that while one muscle shortens, the second one is lengthen approximately the same amount. This guarantees a high stiffness for the entire workspace of the joint.
2.2 (b) Figure 3: Bridgestone’s Rubbertuator (a) and Shadow Robot’s Air Muscle (b)
While Shadow muscles have many desirable properties, they exhibit the same disadvantages as Bridgestone Rubbertuators, e.g., the nonlinearity and the changing pressure-position relationship over time. Due to the actuators performing as a complex behavior, accurate modeling and a robust control technique are required [9]. Another manufacturer of artificial muscles is the Festo Company. First developed for industrial applications, the Festo muscles were later also used in humanoid robotics at the University of Karlsruhe (TH) [10]. Their most famous project based on artificial muscles is the six-legged walking machine Airbug. Several projects running on other Universities show that the interest of artificial muscles for robot applications is still high and will be subject of research in the next years.
2 2.1
Arm Control Actuator
The entire arm consists of six joints. In each joint, two artificial muscles are combined to one actuator, as shown in Figure 4. This is the same principle human joints are built, where two muscles, called agonist and antagonist are working against each other to allow movement in both directions and to stabilize the joint.
Control Architecture
The Soft Arm is not a highly dynamic system. The arm parts are connected in a serial way, therefore the control needs to be fast and robust. Such tasks can normally be completed with a regular PID controller. The problem is that a one-loop PID controller has to compute the manipulated value ∆p according to the desired position value. The variable ∆p is not directly related to the joint angle, but causes a torque first. This torque is responsible for any movement and the resulting joint angle. To improve speed and quality of control, a cascaded controller was chosen to build a subsidiary torque control loop. The structure shown in Figure 5 should help minimize errors given to the slow outer position control loop. A dynamic model of the arm can be added to consider masses of inertia and gravity of all parts. A inner torque control loop with real torque feedback could result in the best control quality, but additional torque or force sensors and the IO interfaces to the PC would be required. In this case, no exact information about the actuator would be necessary, and no model to describe the behavior would have to be built. The hysteresis and nonlinearity of the pneumatic muscles could be controlled easily. Due to the extra effort in additional hardware and costs, wiring and placement of all the sensors, the torque control feedback is realized with an actuator model. This model must be exactly the opposite to the real muscle actuator, calculating the actual torque. The disadvantage of this procedure is that errors on the torque level cannot be measured and are given to the next control loop. Therefore, we need to find a convenient actuator model.
3
Figure 4: Artificial muscle actuator
l1 −l2 r
Actuator Testbed
To find the relationship between the described values of the muscles, to generate an actuator model, and to verify it, a testbed was built. (Figure 6) To measure the resulting torque, two force sensors (Futek L2353 [11]) are connected between the end of the muscles and the testbed frame. External pressure sensors (Futek P4010, [11]) are placed as close to the muscles as possible to measure the real muscle pressures and to not having time delays or pressure loss because of long tubes. An optical encoder (Sumtek, 8000
Figure 5: Position control architecture
and 290mm. In the testbed, the 210mm muscles (lmax ) are mounted with a pre-stressing of about 15%, the middle of the maximum contraction ratio of 30%. The diameter of the chain wheel is r = 0.019m. The muscles are driven with a initial pressure of 3kg/cm2 , ∆p varies between ±2kg/cm2 . The initial muscle length is l0 = 0.85lmax , because of a chosen pre-tension of 15%.
4
Actuator Modeling
4.1 Figure 6: Actuator Testbed
steps/turn) is used to determine the actual joint angle. Over an encoder buffer, the encoder signal is read by a PC interface board, which has been developed in the CIS lab [12]. Figure 7 illustrates an overview of the components. Pressure is given from a Bridgestone servo valve unit (SVO 102-A06) that receives the desired pressures via a 4-20mA input signal from the PC interface card. A National Instruments multi-IO card (Lab PC 1200) reads the signal of the force and pressure sensors in the PC.
Static Modeling
Most existing models are physical static models based on the approach of energy conservation [1, 3, 6]. All are very similar and can partly be transformed from one into another. One possibility to describe the relationship between force F , pressure p and contraction ratio of one muscle is the Bridgestone model: F = D0 2 p[a(1 − )2 − b]
(2)
Where F is the force generated by the muscle, p is the internal muscle pressure, and is the contraction ratio. D0 is the muscle diameter before displacement and a and b are parameters. Combining the two muscles and using the relations 1, as well as =
lmax − l lmax
(3)
and l1,2 = l0 ± rα
(4)
the new formula for the joint is:
T =
2D0 2 (p1 + p2 )ar2 2D0 2 ar3 ∆pα2 − α 2 l0 l0 +D0 2 r(2a − 2b)∆p
(5)
or the simplified form by transferring D02 , l0 and p1 + p2 = 2p0 into the parameters a and b: T = a0 ∆pr3 α2 + b0 r2 α + c0 ∆pr Figure 7: Testbed Components
The Shadow Robot Company offers three different types of muscles with initial lengths of 150mm, 210mm
(6)
Where a0 , b0 and c0 are the new joint parameters, r is the radius of the chain wheel, l1,2 are the muscle lengths and l0 is the initial muscle length. The radius r of the chain wheel is different, in the testbed and real joints,
10 8 6 4 Torque [Nm]
therefore it cannot be identified within the parameters a0 , b0 and c0 . In the first experiment, a constant pressure was applied to the muscles in the testbed. The arm of the joint was moved very slowly by hand to record the quasi-static relation between torque and joint angle. The result was a small hysteresis, shown in Figure 8, which resulted in an error of up to 0.4N m. The measurement error of 0.1N m was in the acceptable range, but was too much to gather data for an accurate hysteresis modeling. The modeling of the hysteresis curve was therefore abandoned and it was approximated with a single line.
2 0 −2 −4 −6 −8
10 left ⋅ ⋅ ⋅ right
8 6
Torque [Nm]
4
∆p = −2 kg/cm2 ∆p = −1 kg/cm2 2 ∆p = 0 kg/cm 2 ∆p = 1 kg/cm ∆p = 2 kg/cm2
−10 −3/8
−1/4
−1/8
0 Angle [pi]
1/4
3/8
Figure 9: Wider hysteresis for higher velocity
2 0 −2
with a bundle of lines resulted from this data, only the points with constant velocities were extracted and plotted in the angle-torque diagram. Figure 10 shows the results for the two different velocities.
−4 −6 −8 −10 −1/2
−3/8
−1/4
−1/8
0 1/8 Angle [pi]
1/4
3/8
1/2
4 3
The first part in equation 6 would have only an appreciable effect on the torque for larger joint angles and higher ∆p’s. According to the experimental results in Figure 8, this part does not deserve any interest and does not affect the behavior of the actuator significantly. The formula 6 is therefore changed into:
2
T = b0 r2 α + c0 ∆pr
(7)
ω = 1.0 ± 0.1 pi/s ω = 2.0 ± 0.1 pi/s
1 0 −1 −2 −3
or, written with new parameters: T = Ar2 α + B∆pr
Torque [Nm]
Figure 8: Quasi-static behavior for different ∆p’s
(8)
Parameters A and B have been identified as 2 N A = −73670 m·pi and B = 174.2N cm . kg
4.2
1/8
−4 −1/4
−1/8
0 Angle [pi]
1/8
1/4
Figure 10: Constant velocities on straight lines
Dynamic Modeling
For the next experiment, the velocity of the movement was chosen up to 2 pi . One portion of an oscillating s movement is shown in Figure 9. The hysteresis was, depending on the velocity, much wider than in the static movement. For a distance between the two paths of 1N m for the highest velocity, the hysteresis cannot be approximated with a single line and has to be modeled itself. Because the arm is moved by a hand and restricted to the end positions, the velocity of the joint changes during the entire movement and cannot be kept constant. This allows no conclusion about the form of the hysteresis curve and the relation to the driven velocity. To gain further information and to plot only regions with constant velocities, the arm was moved for a longer period of time, approximately 5min, in all positions and with different speeds. Later, a completely filled hysteresis curve
The points with constant velocity lay on straight lines with the same gradient than the static main line once again. The velocity-dependence is finally only an offset to the static model. Equation 8 is therefore extended to: T = Ar2 α + Br∆p + f (α) ˙
(9)
To find out the relationship between this offset and the corresponding velocity, the bundle of hysteresis was cut at α = 0pi and the section plotted (Figure 11). The middle position of the joint was chosen because this point contained most of the appearing velocities. To make sure that the allocation did not change over the joint position, the results were verified with cuts at other positions. The following function is qualified for interpolating the data in Figure 11.
The first experiment was repeated with different ∆p to guarantee the functionality for extreme joint positions. As long as the arm was not moved until one of the artificial muscles lost the pre-tension completely, the actuator model prediction and the real measurement results resembled each other.
1.5
0.5
0
6
−0.5
4
−1
2
−1.5 −2
−1.5
−1
−0.5 0 0.5 Joint Velocity [pi/s]
1
1.5
Torque [Nm]
Torque Offset [Nm]
1
2
2.0/−2.0 Nm 4.0/−4.0 Nm
0
−2
Figure 11: Torque offset and joint velocity −4
−6 0
3
f (α) ˙ = C α˙ + Dα˙
Finally, the complete model representing the torque controller is: T = Ar2 α + Br∆p + C α˙ 3 + Dα˙
(11)
Experimental Results
To verify the torque controller in equation 11, two different experiments were performed. First, the arm of the joint was moved by hand so the torque controller had to predict the applied torque. At the same time, the force sensors measured the real torque. The pressure of kg the muscles was adjusted to ∆p = 0 cm 2 . The result is illustrated in Figure 12.
8
Testbed Model
6
Torque [Nm]
4 2 0 −2 −4 −6 −8 0
2
4
6 Time [s]
8
0.2
0.3
0.4
0.5 0.6 Time [s]
0.7
0.8
0.9
1
(10)
The parameters of this function were identified as s s 3 C = 0.07N m( pi ) and D = 0.45N m pi .
5
0.1
10
12
Figure 12: Measured torque vs. actuator model
Figure 13: Various step inputs for torque controller
In the second type of experiment, the arm of the joint was fixed in one position. A torque was given to the actuator model, which one had to calculate the needed ∆p. The desired value was given as a step input and the answer was recorded by the force sensors. Figure 13 shows the results for a fixed middle position of the joint and different input values. This is the most important factor because this will be exactly the way the joint is actuated on ISAC. In a certain joint position, the actuator model has to adjust ∆p to force a movement by applying the torque. Due to time delays of the air system and the compressible medium, it took about 0.15s to apply a torque of 2Nm and 0.25s for a desired torque of 4Nm. These time delays are very low for a pneumatic controlled system and may increase on ISAC where longer tubes are needed. The desired torque was reached with an accuracy of about 0.1Nm. To judge the influence of the dynamic part of equation 11, the experiment in Figure 12 was repeated and the prediction of the actuator model was generated with and without the dynamic part. Initially, the results do not differ very much from the ones in Figure 12. However, after a more in depth view in the areas with high velocity, the model without dynamic part predicts (Figure 14) a higher torque if a positive torque is applied, (means that the arm was moved in positive direction), and a lower torque on the decreasing path of the hysteresis. The error between static and dynamic behavior is caused exactly by the width of the hysteresis, so the full actuator model compensates this error almost totally.
6
Summary
Experimental results showed the actuator model predicts the torque very well for different velocities, joint positions and muscle pressures. Because the feedback of the torque is missing, it is very important to carefully identify the model parameters and to adjust the right pre-tension when mounting the muscles in the arm.
[6] Bridgestone Company. Rubbertuators and Applications for Robotics. Technical Guide No.1, 1986 1
Testbed Static Model Full Model
0.8
[7] K. Kawamura, R. A. Peters II, S. Bagchi, M. Iskarous and M. Bishay Intelligent robotic systems in service of the disabled IEEE Transactions on Rehabilitation Engineering,3(1) , pp. 14–21, March 1995
0.6
Torque [Nm]
0.4 0.2
[8] Shadow Robot Company. www.shadow.org.uk/
0
[9] N. Sarkar S. Thongchai, M. Goldfarb and K. Kawamura. A frequency modeling method of Rubbertuators for control application in an IMA framework. Proceedings of American Control Conference, Arlington, Virginia,pp. 1710–1714, June 2001.
−0.2 −0.4 −0.6 −0.8 −1 5.5
6
6.5
7
Time [s]
Figure 14: Comparison of static model and full model including static and dynamic parts
Otherwise the parameters will change and cause incorrect facts for the torque controller. Another issue is that the pressure cannot be measured at the muscles in ISAC. The real pressure may differ from the desired pressure in ISAC because of the longer air tubes used between the pressure control unit and the muscles in the testbed. In this case, the pressure control unit and the wires would have to be modeled to consider their time delay. The velocity of the joint depends mainly on the pressure control unit. With the Bridgestone servo valves for example, it is hardly possible to reach a joint velocity of more than 1 pi . However, it is important to have s a accurate model that fits a larger range of operation. The torque control is now ready to be implemented into the cascaded control. The position control itself will be implemented soon, and the recent experiments made us looking forward to getting good control results using the subsidiary torque control as well.
References [1] Ching-Ping Chou and Blake Hannaford. Static and dynamic characteristics of McKibben pneumatic artificial muscles. International Conference on Robotics and Automation, 1,pp. 281–286, 1994. [2] Ching-Ping Chou and Blake Hannaford. Measurement and modeling of McKibben pneumatic artificial muscles. IEEE Transactions on Robotics and Automation, 12(1),pp. 90–102, February 1996. [3] H.F. Schulte Jr. The characteristics of the McKibben artificial muscle. The Application of External Power in Prosthetics and Orthotics, National Acamy of Sciences - National Research Council, Washington D.C., 1961. [4] F. Groen P. van der Smagt and K. Schulten. Analysis and control of a Rubbertuator arm. Biological Cybernetics, Springer-Verlag, 75,pp. 433–440, 1996. [5] N. Tsagarakis and Darwin G. Caldwell. Improved modelling and assessment of pneumatic muscle actuators. IEEE Conference on Robotics and Automation,pp. 3641–3646, April 2000.
[10] Research Center for Information Technologies. University of Karlsruhe (TH). wwwneu.fzi.de/ids/eng/ [11] Futek Advanced www.futek.com/
Sensor
Technology.
[12] Steve Northrup. A PC-based controller for the Soft Arm robot. Master’s thesis, Vanderbilt University, Center for Intelligent Systems, Nashville, Tennessee, March 1997.
R¨ udiger Dillmann, Dr.-Ing. Industrial Applications of Informatics and Microsystems University of Karlsruhe (TH) Karlsruhe, Germany http://wwwiaim.ira.uka.de/
Abstract The modeling of pneumatic actuators, also known as artificial muscles, has been the main focus in several research papers in the past [1, 2, 3, 4, 5]. At the Center for Intelligent Systems (CIS) at Vanderbilt University, we have been using pneumatic actuators for our Intelligent Soft Arm Control (ISAC, Figure 1). For the position control of a joint, driven by two artificial muscles, a physical actuator model was designed and used as the basis for a subsidiary torque control. It is known that such actuators contain a high nonlinearity including a hysteresis. Experiments show that the static hysteresis is less important than the dynamic one. In similar models, damping properties of the muscles have not been considered. This research paper focuses on the modification of a physical static model and the extension with a dynamic part. The quality of the model was verified by implementing it as a torque controller and running experiments on a testbed.
1 1.1
Duygun Erol Center for Intelligent Systems Vanderbilt University Nashville, TN, USA http://eecs.vanderbilt.edu/CIS/
coders at each joint on both arms. The artificial muscles resemble human muscle movement.
Figure 1: Humanoid robot system ISAC
Introduction ISAC Humanoid Robot System
The Intelligent Soft Arm Control (ISAC) has been developed at the CIS. ISAC consists of a human-like torso with two six-degree-of freedom robot manipulators, called Soft Arms [6]. Each Soft Arm is a pneumatically-driven robotic manipulator that is actuated by artificial muscles. These muscles are pneumatic actuators whose lengths shorten as their internal air pressure increases and expand as their internal pressure decreases. The muscles are fed with compressed air through servo control valves. The joint angles are calculated with optical position en-
1.2
Artificial Muscles
Artificial muscles were first developed in the 1950’s by the physician Joseph L. McKibben. Thirty years later, Bridgestone Corporation commercialized the muscles called Rubbertuators for industry (Figure 3a). Rubbertuators feature hysteresis in their operation due to the original hysteresis inherent in the rubber, friction between the rubber and cords and friction between the cords themselves (Figure 2).
∗ Acknowledgement: This research is conducted with the financial support of the Center for Intelligent Systems (CIS) and the Institute for Industrial Applications of Informatics and Microsystems (IAIM). The research is part of Mr. Schr¨ oder’s Diploma Thesis at the University of Karlsruhe (TH). Currently, Mr. Schr¨ oder is a visiting scholar at the Vanderbilt University.
Figure 2: Structure of artificial muscles [8]
Rubbertuators are well suited to a number of robotic applications because they are lightweight, naturally compliant, and have a high force-to-weight ratio [7]. The biggest advantage of artificial muscles is the natural movement and the safe interaction with humans because of lower power and stiffness than industrial robots. Shadow Robot Company [8] in England began production of artificial muscles for robots in the 1990’s (Figure 3b). We replaced most of the Soft Arm muscles with Shadow Air Muscles, that is why this paper deals mainly with this type of muscles. The Air Muscle work principle is the same as Bridgestone Rubbertuators. Figure 3 shows both types of artificial muscles.
(a)
Describing variables for the static state of a joint are: • Torque T = (F2 − F1 )r • Joint angle α =
• Pressure difference ∆p to initial pressure (1) The muscles are activated so one of them is deflated to p1 = p0 − ∆p and the other one is inflated to p2 = p0 + ∆p. Thus, ∆p is not the difference between the two muscles, but the distance to the initial pressure p0 . This kind of symmetric muscle activation has proved to be qualified in the past and will be used in the experiments. The advantage of this procedure is that while one muscle shortens, the second one is lengthen approximately the same amount. This guarantees a high stiffness for the entire workspace of the joint.
2.2 (b) Figure 3: Bridgestone’s Rubbertuator (a) and Shadow Robot’s Air Muscle (b)
While Shadow muscles have many desirable properties, they exhibit the same disadvantages as Bridgestone Rubbertuators, e.g., the nonlinearity and the changing pressure-position relationship over time. Due to the actuators performing as a complex behavior, accurate modeling and a robust control technique are required [9]. Another manufacturer of artificial muscles is the Festo Company. First developed for industrial applications, the Festo muscles were later also used in humanoid robotics at the University of Karlsruhe (TH) [10]. Their most famous project based on artificial muscles is the six-legged walking machine Airbug. Several projects running on other Universities show that the interest of artificial muscles for robot applications is still high and will be subject of research in the next years.
2 2.1
Arm Control Actuator
The entire arm consists of six joints. In each joint, two artificial muscles are combined to one actuator, as shown in Figure 4. This is the same principle human joints are built, where two muscles, called agonist and antagonist are working against each other to allow movement in both directions and to stabilize the joint.
Control Architecture
The Soft Arm is not a highly dynamic system. The arm parts are connected in a serial way, therefore the control needs to be fast and robust. Such tasks can normally be completed with a regular PID controller. The problem is that a one-loop PID controller has to compute the manipulated value ∆p according to the desired position value. The variable ∆p is not directly related to the joint angle, but causes a torque first. This torque is responsible for any movement and the resulting joint angle. To improve speed and quality of control, a cascaded controller was chosen to build a subsidiary torque control loop. The structure shown in Figure 5 should help minimize errors given to the slow outer position control loop. A dynamic model of the arm can be added to consider masses of inertia and gravity of all parts. A inner torque control loop with real torque feedback could result in the best control quality, but additional torque or force sensors and the IO interfaces to the PC would be required. In this case, no exact information about the actuator would be necessary, and no model to describe the behavior would have to be built. The hysteresis and nonlinearity of the pneumatic muscles could be controlled easily. Due to the extra effort in additional hardware and costs, wiring and placement of all the sensors, the torque control feedback is realized with an actuator model. This model must be exactly the opposite to the real muscle actuator, calculating the actual torque. The disadvantage of this procedure is that errors on the torque level cannot be measured and are given to the next control loop. Therefore, we need to find a convenient actuator model.
3
Figure 4: Artificial muscle actuator
l1 −l2 r
Actuator Testbed
To find the relationship between the described values of the muscles, to generate an actuator model, and to verify it, a testbed was built. (Figure 6) To measure the resulting torque, two force sensors (Futek L2353 [11]) are connected between the end of the muscles and the testbed frame. External pressure sensors (Futek P4010, [11]) are placed as close to the muscles as possible to measure the real muscle pressures and to not having time delays or pressure loss because of long tubes. An optical encoder (Sumtek, 8000
Figure 5: Position control architecture
and 290mm. In the testbed, the 210mm muscles (lmax ) are mounted with a pre-stressing of about 15%, the middle of the maximum contraction ratio of 30%. The diameter of the chain wheel is r = 0.019m. The muscles are driven with a initial pressure of 3kg/cm2 , ∆p varies between ±2kg/cm2 . The initial muscle length is l0 = 0.85lmax , because of a chosen pre-tension of 15%.
4
Actuator Modeling
4.1 Figure 6: Actuator Testbed
steps/turn) is used to determine the actual joint angle. Over an encoder buffer, the encoder signal is read by a PC interface board, which has been developed in the CIS lab [12]. Figure 7 illustrates an overview of the components. Pressure is given from a Bridgestone servo valve unit (SVO 102-A06) that receives the desired pressures via a 4-20mA input signal from the PC interface card. A National Instruments multi-IO card (Lab PC 1200) reads the signal of the force and pressure sensors in the PC.
Static Modeling
Most existing models are physical static models based on the approach of energy conservation [1, 3, 6]. All are very similar and can partly be transformed from one into another. One possibility to describe the relationship between force F , pressure p and contraction ratio of one muscle is the Bridgestone model: F = D0 2 p[a(1 − )2 − b]
(2)
Where F is the force generated by the muscle, p is the internal muscle pressure, and is the contraction ratio. D0 is the muscle diameter before displacement and a and b are parameters. Combining the two muscles and using the relations 1, as well as =
lmax − l lmax
(3)
and l1,2 = l0 ± rα
(4)
the new formula for the joint is:
T =
2D0 2 (p1 + p2 )ar2 2D0 2 ar3 ∆pα2 − α 2 l0 l0 +D0 2 r(2a − 2b)∆p
(5)
or the simplified form by transferring D02 , l0 and p1 + p2 = 2p0 into the parameters a and b: T = a0 ∆pr3 α2 + b0 r2 α + c0 ∆pr Figure 7: Testbed Components
The Shadow Robot Company offers three different types of muscles with initial lengths of 150mm, 210mm
(6)
Where a0 , b0 and c0 are the new joint parameters, r is the radius of the chain wheel, l1,2 are the muscle lengths and l0 is the initial muscle length. The radius r of the chain wheel is different, in the testbed and real joints,
10 8 6 4 Torque [Nm]
therefore it cannot be identified within the parameters a0 , b0 and c0 . In the first experiment, a constant pressure was applied to the muscles in the testbed. The arm of the joint was moved very slowly by hand to record the quasi-static relation between torque and joint angle. The result was a small hysteresis, shown in Figure 8, which resulted in an error of up to 0.4N m. The measurement error of 0.1N m was in the acceptable range, but was too much to gather data for an accurate hysteresis modeling. The modeling of the hysteresis curve was therefore abandoned and it was approximated with a single line.
2 0 −2 −4 −6 −8
10 left ⋅ ⋅ ⋅ right
8 6
Torque [Nm]
4
∆p = −2 kg/cm2 ∆p = −1 kg/cm2 2 ∆p = 0 kg/cm 2 ∆p = 1 kg/cm ∆p = 2 kg/cm2
−10 −3/8
−1/4
−1/8
0 Angle [pi]
1/4
3/8
Figure 9: Wider hysteresis for higher velocity
2 0 −2
with a bundle of lines resulted from this data, only the points with constant velocities were extracted and plotted in the angle-torque diagram. Figure 10 shows the results for the two different velocities.
−4 −6 −8 −10 −1/2
−3/8
−1/4
−1/8
0 1/8 Angle [pi]
1/4
3/8
1/2
4 3
The first part in equation 6 would have only an appreciable effect on the torque for larger joint angles and higher ∆p’s. According to the experimental results in Figure 8, this part does not deserve any interest and does not affect the behavior of the actuator significantly. The formula 6 is therefore changed into:
2
T = b0 r2 α + c0 ∆pr
(7)
ω = 1.0 ± 0.1 pi/s ω = 2.0 ± 0.1 pi/s
1 0 −1 −2 −3
or, written with new parameters: T = Ar2 α + B∆pr
Torque [Nm]
Figure 8: Quasi-static behavior for different ∆p’s
(8)
Parameters A and B have been identified as 2 N A = −73670 m·pi and B = 174.2N cm . kg
4.2
1/8
−4 −1/4
−1/8
0 Angle [pi]
1/8
1/4
Figure 10: Constant velocities on straight lines
Dynamic Modeling
For the next experiment, the velocity of the movement was chosen up to 2 pi . One portion of an oscillating s movement is shown in Figure 9. The hysteresis was, depending on the velocity, much wider than in the static movement. For a distance between the two paths of 1N m for the highest velocity, the hysteresis cannot be approximated with a single line and has to be modeled itself. Because the arm is moved by a hand and restricted to the end positions, the velocity of the joint changes during the entire movement and cannot be kept constant. This allows no conclusion about the form of the hysteresis curve and the relation to the driven velocity. To gain further information and to plot only regions with constant velocities, the arm was moved for a longer period of time, approximately 5min, in all positions and with different speeds. Later, a completely filled hysteresis curve
The points with constant velocity lay on straight lines with the same gradient than the static main line once again. The velocity-dependence is finally only an offset to the static model. Equation 8 is therefore extended to: T = Ar2 α + Br∆p + f (α) ˙
(9)
To find out the relationship between this offset and the corresponding velocity, the bundle of hysteresis was cut at α = 0pi and the section plotted (Figure 11). The middle position of the joint was chosen because this point contained most of the appearing velocities. To make sure that the allocation did not change over the joint position, the results were verified with cuts at other positions. The following function is qualified for interpolating the data in Figure 11.
The first experiment was repeated with different ∆p to guarantee the functionality for extreme joint positions. As long as the arm was not moved until one of the artificial muscles lost the pre-tension completely, the actuator model prediction and the real measurement results resembled each other.
1.5
0.5
0
6
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4
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2
−1.5 −2
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−0.5 0 0.5 Joint Velocity [pi/s]
1
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Torque [Nm]
Torque Offset [Nm]
1
2
2.0/−2.0 Nm 4.0/−4.0 Nm
0
−2
Figure 11: Torque offset and joint velocity −4
−6 0
3
f (α) ˙ = C α˙ + Dα˙
Finally, the complete model representing the torque controller is: T = Ar2 α + Br∆p + C α˙ 3 + Dα˙
(11)
Experimental Results
To verify the torque controller in equation 11, two different experiments were performed. First, the arm of the joint was moved by hand so the torque controller had to predict the applied torque. At the same time, the force sensors measured the real torque. The pressure of kg the muscles was adjusted to ∆p = 0 cm 2 . The result is illustrated in Figure 12.
8
Testbed Model
6
Torque [Nm]
4 2 0 −2 −4 −6 −8 0
2
4
6 Time [s]
8
0.2
0.3
0.4
0.5 0.6 Time [s]
0.7
0.8
0.9
1
(10)
The parameters of this function were identified as s s 3 C = 0.07N m( pi ) and D = 0.45N m pi .
5
0.1
10
12
Figure 12: Measured torque vs. actuator model
Figure 13: Various step inputs for torque controller
In the second type of experiment, the arm of the joint was fixed in one position. A torque was given to the actuator model, which one had to calculate the needed ∆p. The desired value was given as a step input and the answer was recorded by the force sensors. Figure 13 shows the results for a fixed middle position of the joint and different input values. This is the most important factor because this will be exactly the way the joint is actuated on ISAC. In a certain joint position, the actuator model has to adjust ∆p to force a movement by applying the torque. Due to time delays of the air system and the compressible medium, it took about 0.15s to apply a torque of 2Nm and 0.25s for a desired torque of 4Nm. These time delays are very low for a pneumatic controlled system and may increase on ISAC where longer tubes are needed. The desired torque was reached with an accuracy of about 0.1Nm. To judge the influence of the dynamic part of equation 11, the experiment in Figure 12 was repeated and the prediction of the actuator model was generated with and without the dynamic part. Initially, the results do not differ very much from the ones in Figure 12. However, after a more in depth view in the areas with high velocity, the model without dynamic part predicts (Figure 14) a higher torque if a positive torque is applied, (means that the arm was moved in positive direction), and a lower torque on the decreasing path of the hysteresis. The error between static and dynamic behavior is caused exactly by the width of the hysteresis, so the full actuator model compensates this error almost totally.
6
Summary
Experimental results showed the actuator model predicts the torque very well for different velocities, joint positions and muscle pressures. Because the feedback of the torque is missing, it is very important to carefully identify the model parameters and to adjust the right pre-tension when mounting the muscles in the arm.
[6] Bridgestone Company. Rubbertuators and Applications for Robotics. Technical Guide No.1, 1986 1
Testbed Static Model Full Model
0.8
[7] K. Kawamura, R. A. Peters II, S. Bagchi, M. Iskarous and M. Bishay Intelligent robotic systems in service of the disabled IEEE Transactions on Rehabilitation Engineering,3(1) , pp. 14–21, March 1995
0.6
Torque [Nm]
0.4 0.2
[8] Shadow Robot Company. www.shadow.org.uk/
0
[9] N. Sarkar S. Thongchai, M. Goldfarb and K. Kawamura. A frequency modeling method of Rubbertuators for control application in an IMA framework. Proceedings of American Control Conference, Arlington, Virginia,pp. 1710–1714, June 2001.
−0.2 −0.4 −0.6 −0.8 −1 5.5
6
6.5
7
Time [s]
Figure 14: Comparison of static model and full model including static and dynamic parts
Otherwise the parameters will change and cause incorrect facts for the torque controller. Another issue is that the pressure cannot be measured at the muscles in ISAC. The real pressure may differ from the desired pressure in ISAC because of the longer air tubes used between the pressure control unit and the muscles in the testbed. In this case, the pressure control unit and the wires would have to be modeled to consider their time delay. The velocity of the joint depends mainly on the pressure control unit. With the Bridgestone servo valves for example, it is hardly possible to reach a joint velocity of more than 1 pi . However, it is important to have s a accurate model that fits a larger range of operation. The torque control is now ready to be implemented into the cascaded control. The position control itself will be implemented soon, and the recent experiments made us looking forward to getting good control results using the subsidiary torque control as well.
References [1] Ching-Ping Chou and Blake Hannaford. Static and dynamic characteristics of McKibben pneumatic artificial muscles. International Conference on Robotics and Automation, 1,pp. 281–286, 1994. [2] Ching-Ping Chou and Blake Hannaford. Measurement and modeling of McKibben pneumatic artificial muscles. IEEE Transactions on Robotics and Automation, 12(1),pp. 90–102, February 1996. [3] H.F. Schulte Jr. The characteristics of the McKibben artificial muscle. The Application of External Power in Prosthetics and Orthotics, National Acamy of Sciences - National Research Council, Washington D.C., 1961. [4] F. Groen P. van der Smagt and K. Schulten. Analysis and control of a Rubbertuator arm. Biological Cybernetics, Springer-Verlag, 75,pp. 433–440, 1996. [5] N. Tsagarakis and Darwin G. Caldwell. Improved modelling and assessment of pneumatic muscle actuators. IEEE Conference on Robotics and Automation,pp. 3641–3646, April 2000.
[10] Research Center for Information Technologies. University of Karlsruhe (TH). wwwneu.fzi.de/ids/eng/ [11] Futek Advanced www.futek.com/
Sensor
Technology.
[12] Steve Northrup. A PC-based controller for the Soft Arm robot. Master’s thesis, Vanderbilt University, Center for Intelligent Systems, Nashville, Tennessee, March 1997.
