Tracking Control with Hysteresis Compensation for Manipulator ...
Tracking Control with Hysteresis Compensation for Manipulator Segments driven by Pneumatic Artificial Muscles Frank Schreiber, Yevgen Sklyarenko, Kathrin Schl¨uter, Jan Schmidt, Sven Rost and Walter Schumacher
I. I NTRODUCTION Pneumatic muscles have matured to become reliable, highly durable low cost actuators. Due to favorable characteristics like high power-to-weight ratio and their inherent compliance, they are well-suited to be applied to construct light-weight versatile robots [1], [2]. A. Muscle characteristics Most of the pneumatic muscles in use today are based on the McKibben artificial muscle, consisting of a air proof so-called rubber bladder which is surrounded by a sheath of inextensible fibers and closed at both ends by caps [3]. Under pressure the bladder increases in volume, resulting in an expansion in radius and an axial contraction due to the inextensible sheath. When attached to an appropriate bearing the pressurized muscle is able to exert a pulling force. When pressure is released, only the relaxation of the deformed muscle slowly relaxes the muscle, allowing it to passively return to its original shape. Therefore, just like their natural counterparts, pneumatic muscles can only feasibly exert pulling forces and have to be used in an antagonistic setup. One of the main challenges in the application of artificial pneumatic actuators is their difficult controllability, as the F. Schreiber, Y. Sklyarenko and W. Schumacher are with the Institute of Control Engineering, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany. K. Schl¨uter and J. Schmitt are with the Institute of Machine Tools and Production Technology, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany. Sven Rost is with the National Technical University Donetsk, Department of Control Systems and Mechatronics, 83000 Donetsk, Ukraine. Corresponding author: [email protected]
0.02 0
Contraction ratio
Abstract— In this paper the feed-forward control structure is introduced for a manipulator joint driven by a set of symmetrical antagonistic muscles. The setup allows the specification of the desired joint angle, as well as the single muscle forces and produces the pressure signal for the muscles’ supplying valves based upon the manipulator kinematic and the inverse static relation in a single muscle. Applying this setup allows to compensate the nonlinear characteristic mean relation in a single muscle actuator, while the muscles’ hysteretic behavior introduces a hysteresis in the relation between the desired and the actual joint angles. It is shown that if the muscles are subject to the same forces the resulting hysteretic behavior can be described by a Preisach model. The inverse of this model is then utilized as a feed-forward term to compensate the hysteretic behavior. Experiments confirm the improvement in tracking control in comparison to the system solely controlled by a feedback controller. Therefore, hysteresis compensation with a feed-back controller significantly improves the tracking control accuracy of the manipulator joint.
−0.1
−0.2
−500
0 bar 1 bar 2 bar 3 bar 4 bar 5 bar 6 bar 7 bar 8 bar −400
−300
−200
−100
0
Force (N)
Fig. 1. Hysteresis behavior of the isobarically loaded muscle. Contraction forces and displacements are denoted with a negative sign.
available muscles exhibit a wide range of nonlinear effects. Apart from omnipresent creep effects, their deflection, which can reach values of up 25% of the muscle length in the unloaded case, is dependent upon the applied air pressure, the force, as well as the velocity of change. Due to frictional effects in the air path and the muscle material, the deflection exhibits an unsymmetrically hysteretic behavior, see filled areas in Fig. 1, which prohibits the precise open loop control of a single muscle. Two approaches can be distinguished to deal with the problem of nonlinear contraction forces and the complex hysteresis: one using simplified models or advanced nonlinear control algorithms [4], [5]. The gray contours in Fig. 1 depict the load force and contraction loops measured at different air pressures at a frequency of 0.1 Hz, acquired for an pneumatic muscle type DMSP-10-150N by FESTO. To derive a description for a mean relation between contraction and load force for a given air pressure, the mean force for a given length is calculated from the flanks of loops and displayed as a colored line. The resulting mapping is given in Fig. 2. An abundance of approaches has been suggested to model the resulting mapping [6], [7]. Boblan et al. compare several possible approaches to describe the static relation between contraction, pressure and muscle force [5]. After evaluating power series, separation model and sine model approaches he concludes that the static relation on a single muscle can be described with best accuracy by a sine model, consisting of a superposition of linear and sinusoid components. Therefore
Contraction ratio
0.1 0 −0.1 −0.2
−0.3 −500 −400 −300 −200 −100 Force (N)
Fig. 2.
0
0
2
4
6
8
Air pressure (bar)
Static characteristics of used muscle type.
a sine model approach is chosen and parameterized using the measured data displayed in Fig. 2. Consequently, this model was inverted to provide the necessary air pressure for an admissible combination of muscle contraction and muscle force.
B. Manipulator setup Fig. 3 (left) depicts the used manipulator structure as introduced by Schmitt et al. in [1]. A single stage consists of a pair of antagonistic pneumatic muscles manufactured by FESTO of the type DMSP-10-150N, possessing a working length of 150 mm and a muscle diameter of 10 mm. From the module design depicted in Fig. 3 (right) the relation between the joint angle ϕ and the corresponding effective actuator
Fig. 4.
Feed-forward joint angle control.
lengths lA and lB can be derived: ru cosϕ −sinϕ −ro lA = + · , lu sinϕ cosϕ lo −ru cosϕ −sinϕ l lB = + · o . lu sinϕ cosϕ ro
(1)
From the kinematics of a single segment in (1), as shown in Fig. 3, and the static mapping relating the air pressure and the applied force to the resulting mean contraction in Fig. 2 derived from the measurements described in Sec. I-A the inverse mapping can be calculated providing the necessary air pressures in each muscle for a combination of desired joint angle and individual muscle forces. The resulting feedforward control structure is displayed in Fig. 4. The described feed-forward structure compensates the static nonlinear relations between the mean values of the contraction force and displacement to the pressure. Depending on the combination of the muscle forces, drive torques as well as antagonistic torques will be produced. The antagonistic torques do not result in a motion as they merely cause a prestress in both actuators, that can be used to vary the joint’s stiffness [8].In the following the external force input is only used to generate a prestress in the system. In the antagonistic setup the same effects causing hysteretic behavior in a single muscle will result in a hysteretic behavior between the desired and the measured joint angle, due to the symmetric setup of the joint actuators the resulting hysteretic behavior will consequently be symmetric. II. P REISACH HYSTERESIS MODEL OF THE ANTAGONISTICALLY ACTUATED SEGMENT.
Fig. 3. Manipulator structure (left) and kinematic setup of a single manipulator segment (right).
A variety of hysteresis models have been applied in an attempt to model the inherent hysteretic behavior of pneumatic muscle actuators. Minh et al. successfully demonstrated the application of a Maxwell-slip-model for the description of the contraction length to force behavior for a single muscle [9], as well as for the torque hysteresis for the movement in a joint driven by an antagonistic muscle pair [10]. While the described modeling results provide a good reproduction of the behavior of single muscles, no control approach is presented that can be extended to incorporate the control of a joint driven by an antagonistic muscle pair. Attempts have
been made to apply classic hysteresis modeling approaches like Preisach models to the modeling and control of pneumatic muscles and pairs of pneumatic muscles [11], [12]. The results obtained in these works provided only limited, superficial modeling success, as an important necessary precondition as given by Mayergoyz [13] to ensure the applicability of a Preisach-approach, the congruency condition, is not fulfilled by the asymmetric hysteretic behavior of a single pneumatic muscle. Therefore the models derived in the mentioned works were only able to reproduce major loops similar to ones used in the identification process of the model. The main potential of a Preisach model, the ability to model the minor loop behavior and the model inversion for an approximate compensation of the hysteresis could therefore not be exploited. It will be shown that the use of the presented symmetrical antagonistic setup featuring a pneumatic muscle pair with a compensation of the static mean nonlinear force-contraction-pressure characteristics results in a symmetrical hysteresis behavior, which fulfills all necessary conditions to allow modeling by a Preisach model. A. Preisach model of hysteresis. The Preisach hysteresis model was first developed by Preisach in 1935 in an attempt to model the physical mechanisms of magnetization [14]. Although it was first regarded to be a physical model of hysteresis, the Preisach model turned out to be a phenomenological model that has mathematical generality and is applicable to phenomena from many disciplines. A rigid mathematical generalization has been presented by Mayergoyz, who also determined the necessary conditions for the applicability of such a model [15]. The simplest type of hysteresis operator γˆαβ can be represented as rectangular loops in the in-/output-plane, as shown in Fig. 5 (left). Its output switches between +1 and −1 depending on the initial output and the history of past inputs, representing a local memory. In addition to the set of operators γˆαβ , with α and β corresponding to the “up” and “down” switching values of the input u(t), a weighting function µ(α, β) has to be considered, which is called the Preisach function and can be identified for a given system. The resulting Preisach model with the system output f (t) is then given by Z Z f (t) = µ(α, β)ˆ γαβ u(t)dαdβ. (2) α≥β
The switching values are subject to the relation umax ≥ α ≥ β ≥ umin with umax and umin being determined by the system’s physical properties. The feasible combinations of α and β for the triangle T as displayed in Fig. 5 (right). Therefore the output of the Preisach model is determined by integrating the product of the weighting function µ(α, β) and the operator γˆαβ over the triangle T . The model output is dependent on the extremal values in the history of the input sequence u(t) as depicted in the example in Fig. 6. The set of dominant maxima and minima determines the output of the system, due to the wipe-out property of the Preisach model, input values larger than past dominant maxima or
1
g^α,βu
b
au umin
-1 Fig. 5. (right).
umax
a
umax u h1
umin
ul1
b umax
An elementary hysteresis operator γ ˆ (left) and Preisach plane
u(t) h1
h3
h2
t
l2 umin
l3
l1
umax u h1 u h3 u(t) umin
a
umin ul3 ul3
b umax
Fig. 6. Input sequence with dominant, as well as wiped out extremal values (left) and Preisach plane with correspondingly activated elemental operators (right).
smaller the past minima wipe out the effects of the older extrema. The dominant extrema are marked in the input sequence depicted in Fig. 6 (left). At any instant the Preisach plane can be divided into two regions, the one in which the relay operator outputs are +1, marked dark gray in Fig. 6 (right), and the one in which the relays’ outputs are −1. Both area are separated by a descending staircase, whose corners are determined by the past reversal points in the input sequence. After applying the identification algorithm presented by Mayergoyz to determine the Preisach function from first order descending curves [13], the Everett map E is defined, which contains the change of the output value f (t) as a function of α and β [16]. E(uα , uβ ) = fα − fαβ .
(3)
Since α ≥ β during the measurement, only one half of the map can be constructed by measurements. The missing values are given by fαβ = −fβ,α . Thus the value of fαβ can be calculated for any single combination of α and β. With the sets of past dominant maxima H and dominant minima L, the Preisach model output can then be expressed for any given sequence of inputs u(t) by n(t)−1
f (t)
=
fmin +
X fHk ,Lk − fHk ,Lk−1 . . .
(4)
k=1
f − fHn ,Ln−1 + Hn ,u(t) fu(t),u(t) − fu(t),Ln−1
if u(t) ˙ ≤ 0, if u(t) ˙ ≥ 0.
with fmin being the output corresponding to u = umin (all relays set to −1). For a given Preisach model a computationally compact inverse can be derived from 5, if the first-order curve
5
20
20
0
10 0 −10 −20 −30 −30
−10 0 10 Desired joint angle (°) Fig. 7.
20
−5
−10
−15
−20 −20 −10 0 Desired joint angle (°)
Fig. 8. Congruency test. Magnification: The green plot is shifted to allow comparison for congruency.
(5)
With this mapping the unknown input u(f (t)) to achieve the desired output f (t) can be computed using k−1 X
−10
−20 0 20 Desired joint angle (°)
data surfaces fαβ are strictly monotonically increasing with respect to the parameters α and β. This will result in the inverse of the Everett map G, given by
u(f (t)) = umin +
0
−30 30
Wiping-out test.
G(fα , fαβ ) = uα − uβ .
10
−20
Cycle 1 Cycle 2 −20
Measured joint angle (°)
30
Measured joint angle (°)
Measured joint angle (°)
30
G Fdi+1 , Fdi + G (f (t), Fdk ) , (6)
i=1
with Fd being the set of past output extreme values. B. Applicability of Preisach model. Mayergoyz recognized that a hysteresis nonlinearity can only be represented by a Preisach model if it fulfills the wiping-out property and minor-loop congruence property conditions [15]. The fulfillment of the properties will be verified in the following. Wiping-out property If a system complies with this condition the output is only affected by the current input and the alternating series of previous dominant extremal values of the input, while the effect of all other inputs are wiped out. The antagonistic setup is tested for the wiping-out property by applying a sequence of two decaying sine input for the desired angle. The input is set to the largest possible input value in order to wipe out the effects of previous inputs, after one cycle the amplitude of the sine signal is linearly decreased over 10 oscillations. Consequently, the input sequence is repeated. As can be seen in Fig. 7 the output sequence as a result of the first and the second decaying sine input are equivalent. This is only possible if the large amplitude input in the second sequence wipes out the effects of the previous inputs. Therefore the wiping-out is fulfilled for the given system. Minor-loop congruency property Minor-loop congruency requires that equivalent minor hysteresis loops, resulting from an input varying between the
same two extrema, produce congruent minor loops that only differ by an offset due to the separate input histories. This property implies that the shape of a minor loop depends only on the two extremum values of the input path used to generate the loop. Using a test input, a minor loop was recorded following upon two different input histories. As depicted in Fig. 8, the resulting minor loops show the required congruency. C. Identification and feed-forward compensation. The hysteretic behavior of the system was identified by producing a series of first order descending curves. An input signal was chosen to lead the output along an ascending branch of the major loop, at distinct values, chosen as suggested in [17], the input reverses producing a descending curve in the input-output-diagram that terminates at negative saturation. Due to the symmetric hysteresis loop it is apparent, that the relation µ(α, β) = µ(−β, −α)
(7)
is valid. Therefore it is obvious, that for the identification of µ(α, β) also the first-order increasing curves could have been used which are attached to the limiting descending branch. After the derivation of the inverse Preisach model, the desired joint angle is fed into the inverse model whose output serves as the input to the feed-forward structure in Fig. 4. The resulting match between desired and measured angle is shown in Fig. 9. It can been seen, that the inverse model allows a significant reduction of the width of the resulting hysteresis between actual and desired angle and will, in combination with an appropriate feedback controller, allow a very precise control of the manipulator segment. III. C ONTROLLER DESIGN To ensure precise tracking in the presence of disturbances, creep, and model uncertainties a closed-loop controller it
20
30
15 10
10 Angle (°)
Measured joint angle (°)
20
0 −10
5 0 −5 −10
−20 −30 −30
Fig. 9.
open−loop (uncompensated) open−loop (compensated) −20
−10 0 10 Desired joint angle (°)
20
Desired angle Closed−loop control CL + hysteresis compensation
−15 −20
30
0
2
4
6
8
10
Time (s)
Open-loop hysteresis compensation using inverse Preisach model.
Fig. 11. Step response with/without inverse Preisach model feed-forward controller.
-1
IV. C ONCLUSION AND FUTURE WORKS . A. Conclusion.
Feed-forward joint angle plant jdes
-
PID
u
G(s)
jmeas
Fig. 10. Control structure comprising inverse Preisach model and PID feedback controller.
is imperative to complement the feed-forward compensator with a feed-back controller. The idea of the control scheme is to combine the hysteresis compensator and a linear controller to implement an effective tracking controller. Fig. 10 shows the resulting setup comprising the combination of the PID feed-back and the feed-forward hysteresis compensator. The output of the resulting discrete time control algorithm can thus be derived by the following equation: X u(k) = uIP M (k) + KP e(k) + KI e(k) + . . . KD (e(k) − e(k − q)), k = 1, 2, . . .
(8)
with KP , KI , and KD the proportional, integral and derivative gains, respectively and uIP M the output of the inverse Preisach model. The controller parameters were derived after evaluation the step response and slightly adjusted heuristically to allow sufficient damping in spite of varying dynamic parameters of the plant. Fig. 11 shows a comparison of the step response behaviors of the controlled joint angle. It can be seen, that the system’s tracking dynamics clearly improves as a result of the addition of the feed-forward hysteresis compensator. Apart from the reduces rise time the system output is able settle significantly faster around the desired value if the hysteresis compensator is in place.
This paper presents Preisach modeling of hysteresis and precision tracking control of a manipulator segment driven by a set of antagonistic pneumatic muscles system using inverse Preisach model and a feedback controller. Due to their unsymmetric hysteretic behavior, single pneumatic muscles do not lend themselves to a feed-forward control via an inverse Preisach model. A control structure was presented that allows the approximate compensation of the underlying force-displacement-pressure relationships for two muscles in a symmetric antagonistic setup. The resulting setup still exhibits a dominantly hysteretic behavior, but is symmetric under the mentioned conditions. A series of tests was conducted to study the hysteresis properties of the controlled actuator system. A classical Preisach model is then applied to simulate the static hysteresis behavior of this system. After identifying this model, its inverse is applied to allow feedforward compensation of the hysteretic behavior of the joint actuation system. In combination with an appropriate feed-back controller the feed-forward compensator is shown to significantly improve the performance of the joint angle tracking controller. B. Future Works Future works will include the setup of underlying force control loops and will be directed to study the possibility of a stiffness variation utilizing the antagonistic muscle forces. R EFERENCES [1] J. Schmitt, F. Grabert, and A. Raatz, “Design of a hyper-flexible assembly robot using artificial muscles,” in Robotics and Biomimetics (ROBIO), 2010 IEEE International Conference on, dec. 2010, pp. 897 –902. [2] I. Boblan, R. Bannasch, A. Schulz, and H. Schwenk, “A human-like robot torso ZAR5 with fluidic muscles: Toward a common platform for embodied AI,” vol. 4850, pp. 347–357, 2007.
[3] F. Daerden and D. Lefeber, “Pneumatic artificial muscles: actuators for robotics and automation,” European journal of Mechanical and Environmental Engineering, vol. 47, pp. 10–21, 2000. [4] T. Minh, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, “Cascade position control of a single pneumatic artificial musclemass system with hysteresis compensation,” Mechatronics, vol. 20, no. 3, pp. 402–414, Apr. 2010. [Online]. Available: http://dx.doi.org/10.1016/j.mechatronics.2010.03.001 [5] I. Boblan, “Modellbildung und Regelung eines fluidischen Muskelpaares,” Ph.D. dissertation, Fakult¨at III - Prozesswissenschaften - der Technischen Universit¨at Berlin, 30. November 2009. [6] D. Schindele and H. Aschemann, “Disturbance compensation strategies for a high-speed linear axis driven by pneumatic muscles,” pp. 436–441, August 2009. [7] C.-P. Chou and B. Hannaford, “Measurement and modeling of mckibben pneumatic artificial muscles,” Robotics and Automation, IEEE Transactions on, vol. 12, no. 1, pp. 90–102, Feb. 1996. [8] I. Sardellitti, G. Palli, N. Tsagarakis, and D. Caldwell, “Antagonistically actuated compliant joint: Torque and stiffness control,” in Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, oct. 2010, pp. 1909 –1914. [9] T. Minh, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, “A new approach to modeling hysteresis in a pneumatic artificial muscle using the maxwell-slip model,” Mechatronics, IEEE/ASME Transactions on, vol. 16, no. 1, pp. 177 –186, feb. 2011. [10] T. Minh, B. Kamers, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, “Modeling torque-angle hysteresis in a pneumatic muscle manipulator,” in Advanced Intelligent Mechatronics (AIM), 2010 IEEE/ASME International Conference on, july 2010, pp. 1122 –1127. [11] M. Van Damme, P. Beyl, B. Vanderborght, R. Van Ham, I. Vanderniepen, R. Versluys, F. Daerden, and D. Lefeber, “Modeling hysteresis in pleated pneumatic artificial muscles,” in Robotics, Automation and Mechatronics, 2008 IEEE Conference on, sept. 2008, pp. 471 –476. [12] T. Kosaki and M. Sano, “Control of a parallel manipulator driven by pneumatic muscle actuators based on a hysteresis model,” Journal of Environment and Engineering, vol. 6, no. 2, pp. 316–327, 2011. [13] I. Mayergoyz, “Mathematical models of hysteresis,” Magnetics, IEEE Transactions on, vol. 22, no. 5, pp. 603 – 608, sep 1986. ¨ [14] F. Preisach, “Uber die magnetische Nachwirkung,” Zeitschrift fur Physik, vol. 94, pp. 277–302, May 1935. [15] I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, 2nd ed. Academic Press, 2003. [16] D. H. Everett, “A general approach to hysteresis. part 4. an alternative formulation of the domain model,” Trans. Faraday Soc., vol. 51, pp. 1551–1557, 1955. [17] F. Li and J. Zhao, “Discrete methods based on first order reversal curves to identify Preisach model of smart materials,” Chinese Journal of Aeronautics, vol. 20, no. 2, pp. 157 – 161, 2007. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1000936107600259
I. I NTRODUCTION Pneumatic muscles have matured to become reliable, highly durable low cost actuators. Due to favorable characteristics like high power-to-weight ratio and their inherent compliance, they are well-suited to be applied to construct light-weight versatile robots [1], [2]. A. Muscle characteristics Most of the pneumatic muscles in use today are based on the McKibben artificial muscle, consisting of a air proof so-called rubber bladder which is surrounded by a sheath of inextensible fibers and closed at both ends by caps [3]. Under pressure the bladder increases in volume, resulting in an expansion in radius and an axial contraction due to the inextensible sheath. When attached to an appropriate bearing the pressurized muscle is able to exert a pulling force. When pressure is released, only the relaxation of the deformed muscle slowly relaxes the muscle, allowing it to passively return to its original shape. Therefore, just like their natural counterparts, pneumatic muscles can only feasibly exert pulling forces and have to be used in an antagonistic setup. One of the main challenges in the application of artificial pneumatic actuators is their difficult controllability, as the F. Schreiber, Y. Sklyarenko and W. Schumacher are with the Institute of Control Engineering, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany. K. Schl¨uter and J. Schmitt are with the Institute of Machine Tools and Production Technology, Technische Universit¨at Braunschweig, 38106 Braunschweig, Germany. Sven Rost is with the National Technical University Donetsk, Department of Control Systems and Mechatronics, 83000 Donetsk, Ukraine. Corresponding author: [email protected]
0.02 0
Contraction ratio
Abstract— In this paper the feed-forward control structure is introduced for a manipulator joint driven by a set of symmetrical antagonistic muscles. The setup allows the specification of the desired joint angle, as well as the single muscle forces and produces the pressure signal for the muscles’ supplying valves based upon the manipulator kinematic and the inverse static relation in a single muscle. Applying this setup allows to compensate the nonlinear characteristic mean relation in a single muscle actuator, while the muscles’ hysteretic behavior introduces a hysteresis in the relation between the desired and the actual joint angles. It is shown that if the muscles are subject to the same forces the resulting hysteretic behavior can be described by a Preisach model. The inverse of this model is then utilized as a feed-forward term to compensate the hysteretic behavior. Experiments confirm the improvement in tracking control in comparison to the system solely controlled by a feedback controller. Therefore, hysteresis compensation with a feed-back controller significantly improves the tracking control accuracy of the manipulator joint.
−0.1
−0.2
−500
0 bar 1 bar 2 bar 3 bar 4 bar 5 bar 6 bar 7 bar 8 bar −400
−300
−200
−100
0
Force (N)
Fig. 1. Hysteresis behavior of the isobarically loaded muscle. Contraction forces and displacements are denoted with a negative sign.
available muscles exhibit a wide range of nonlinear effects. Apart from omnipresent creep effects, their deflection, which can reach values of up 25% of the muscle length in the unloaded case, is dependent upon the applied air pressure, the force, as well as the velocity of change. Due to frictional effects in the air path and the muscle material, the deflection exhibits an unsymmetrically hysteretic behavior, see filled areas in Fig. 1, which prohibits the precise open loop control of a single muscle. Two approaches can be distinguished to deal with the problem of nonlinear contraction forces and the complex hysteresis: one using simplified models or advanced nonlinear control algorithms [4], [5]. The gray contours in Fig. 1 depict the load force and contraction loops measured at different air pressures at a frequency of 0.1 Hz, acquired for an pneumatic muscle type DMSP-10-150N by FESTO. To derive a description for a mean relation between contraction and load force for a given air pressure, the mean force for a given length is calculated from the flanks of loops and displayed as a colored line. The resulting mapping is given in Fig. 2. An abundance of approaches has been suggested to model the resulting mapping [6], [7]. Boblan et al. compare several possible approaches to describe the static relation between contraction, pressure and muscle force [5]. After evaluating power series, separation model and sine model approaches he concludes that the static relation on a single muscle can be described with best accuracy by a sine model, consisting of a superposition of linear and sinusoid components. Therefore
Contraction ratio
0.1 0 −0.1 −0.2
−0.3 −500 −400 −300 −200 −100 Force (N)
Fig. 2.
0
0
2
4
6
8
Air pressure (bar)
Static characteristics of used muscle type.
a sine model approach is chosen and parameterized using the measured data displayed in Fig. 2. Consequently, this model was inverted to provide the necessary air pressure for an admissible combination of muscle contraction and muscle force.
B. Manipulator setup Fig. 3 (left) depicts the used manipulator structure as introduced by Schmitt et al. in [1]. A single stage consists of a pair of antagonistic pneumatic muscles manufactured by FESTO of the type DMSP-10-150N, possessing a working length of 150 mm and a muscle diameter of 10 mm. From the module design depicted in Fig. 3 (right) the relation between the joint angle ϕ and the corresponding effective actuator
Fig. 4.
Feed-forward joint angle control.
lengths lA and lB can be derived: ru cosϕ −sinϕ −ro lA = + · , lu sinϕ cosϕ lo −ru cosϕ −sinϕ l lB = + · o . lu sinϕ cosϕ ro
(1)
From the kinematics of a single segment in (1), as shown in Fig. 3, and the static mapping relating the air pressure and the applied force to the resulting mean contraction in Fig. 2 derived from the measurements described in Sec. I-A the inverse mapping can be calculated providing the necessary air pressures in each muscle for a combination of desired joint angle and individual muscle forces. The resulting feedforward control structure is displayed in Fig. 4. The described feed-forward structure compensates the static nonlinear relations between the mean values of the contraction force and displacement to the pressure. Depending on the combination of the muscle forces, drive torques as well as antagonistic torques will be produced. The antagonistic torques do not result in a motion as they merely cause a prestress in both actuators, that can be used to vary the joint’s stiffness [8].In the following the external force input is only used to generate a prestress in the system. In the antagonistic setup the same effects causing hysteretic behavior in a single muscle will result in a hysteretic behavior between the desired and the measured joint angle, due to the symmetric setup of the joint actuators the resulting hysteretic behavior will consequently be symmetric. II. P REISACH HYSTERESIS MODEL OF THE ANTAGONISTICALLY ACTUATED SEGMENT.
Fig. 3. Manipulator structure (left) and kinematic setup of a single manipulator segment (right).
A variety of hysteresis models have been applied in an attempt to model the inherent hysteretic behavior of pneumatic muscle actuators. Minh et al. successfully demonstrated the application of a Maxwell-slip-model for the description of the contraction length to force behavior for a single muscle [9], as well as for the torque hysteresis for the movement in a joint driven by an antagonistic muscle pair [10]. While the described modeling results provide a good reproduction of the behavior of single muscles, no control approach is presented that can be extended to incorporate the control of a joint driven by an antagonistic muscle pair. Attempts have
been made to apply classic hysteresis modeling approaches like Preisach models to the modeling and control of pneumatic muscles and pairs of pneumatic muscles [11], [12]. The results obtained in these works provided only limited, superficial modeling success, as an important necessary precondition as given by Mayergoyz [13] to ensure the applicability of a Preisach-approach, the congruency condition, is not fulfilled by the asymmetric hysteretic behavior of a single pneumatic muscle. Therefore the models derived in the mentioned works were only able to reproduce major loops similar to ones used in the identification process of the model. The main potential of a Preisach model, the ability to model the minor loop behavior and the model inversion for an approximate compensation of the hysteresis could therefore not be exploited. It will be shown that the use of the presented symmetrical antagonistic setup featuring a pneumatic muscle pair with a compensation of the static mean nonlinear force-contraction-pressure characteristics results in a symmetrical hysteresis behavior, which fulfills all necessary conditions to allow modeling by a Preisach model. A. Preisach model of hysteresis. The Preisach hysteresis model was first developed by Preisach in 1935 in an attempt to model the physical mechanisms of magnetization [14]. Although it was first regarded to be a physical model of hysteresis, the Preisach model turned out to be a phenomenological model that has mathematical generality and is applicable to phenomena from many disciplines. A rigid mathematical generalization has been presented by Mayergoyz, who also determined the necessary conditions for the applicability of such a model [15]. The simplest type of hysteresis operator γˆαβ can be represented as rectangular loops in the in-/output-plane, as shown in Fig. 5 (left). Its output switches between +1 and −1 depending on the initial output and the history of past inputs, representing a local memory. In addition to the set of operators γˆαβ , with α and β corresponding to the “up” and “down” switching values of the input u(t), a weighting function µ(α, β) has to be considered, which is called the Preisach function and can be identified for a given system. The resulting Preisach model with the system output f (t) is then given by Z Z f (t) = µ(α, β)ˆ γαβ u(t)dαdβ. (2) α≥β
The switching values are subject to the relation umax ≥ α ≥ β ≥ umin with umax and umin being determined by the system’s physical properties. The feasible combinations of α and β for the triangle T as displayed in Fig. 5 (right). Therefore the output of the Preisach model is determined by integrating the product of the weighting function µ(α, β) and the operator γˆαβ over the triangle T . The model output is dependent on the extremal values in the history of the input sequence u(t) as depicted in the example in Fig. 6. The set of dominant maxima and minima determines the output of the system, due to the wipe-out property of the Preisach model, input values larger than past dominant maxima or
1
g^α,βu
b
au umin
-1 Fig. 5. (right).
umax
a
umax u h1
umin
ul1
b umax
An elementary hysteresis operator γ ˆ (left) and Preisach plane
u(t) h1
h3
h2
t
l2 umin
l3
l1
umax u h1 u h3 u(t) umin
a
umin ul3 ul3
b umax
Fig. 6. Input sequence with dominant, as well as wiped out extremal values (left) and Preisach plane with correspondingly activated elemental operators (right).
smaller the past minima wipe out the effects of the older extrema. The dominant extrema are marked in the input sequence depicted in Fig. 6 (left). At any instant the Preisach plane can be divided into two regions, the one in which the relay operator outputs are +1, marked dark gray in Fig. 6 (right), and the one in which the relays’ outputs are −1. Both area are separated by a descending staircase, whose corners are determined by the past reversal points in the input sequence. After applying the identification algorithm presented by Mayergoyz to determine the Preisach function from first order descending curves [13], the Everett map E is defined, which contains the change of the output value f (t) as a function of α and β [16]. E(uα , uβ ) = fα − fαβ .
(3)
Since α ≥ β during the measurement, only one half of the map can be constructed by measurements. The missing values are given by fαβ = −fβ,α . Thus the value of fαβ can be calculated for any single combination of α and β. With the sets of past dominant maxima H and dominant minima L, the Preisach model output can then be expressed for any given sequence of inputs u(t) by n(t)−1
f (t)
=
fmin +
X fHk ,Lk − fHk ,Lk−1 . . .
(4)
k=1
f − fHn ,Ln−1 + Hn ,u(t) fu(t),u(t) − fu(t),Ln−1
if u(t) ˙ ≤ 0, if u(t) ˙ ≥ 0.
with fmin being the output corresponding to u = umin (all relays set to −1). For a given Preisach model a computationally compact inverse can be derived from 5, if the first-order curve
5
20
20
0
10 0 −10 −20 −30 −30
−10 0 10 Desired joint angle (°) Fig. 7.
20
−5
−10
−15
−20 −20 −10 0 Desired joint angle (°)
Fig. 8. Congruency test. Magnification: The green plot is shifted to allow comparison for congruency.
(5)
With this mapping the unknown input u(f (t)) to achieve the desired output f (t) can be computed using k−1 X
−10
−20 0 20 Desired joint angle (°)
data surfaces fαβ are strictly monotonically increasing with respect to the parameters α and β. This will result in the inverse of the Everett map G, given by
u(f (t)) = umin +
0
−30 30
Wiping-out test.
G(fα , fαβ ) = uα − uβ .
10
−20
Cycle 1 Cycle 2 −20
Measured joint angle (°)
30
Measured joint angle (°)
Measured joint angle (°)
30
G Fdi+1 , Fdi + G (f (t), Fdk ) , (6)
i=1
with Fd being the set of past output extreme values. B. Applicability of Preisach model. Mayergoyz recognized that a hysteresis nonlinearity can only be represented by a Preisach model if it fulfills the wiping-out property and minor-loop congruence property conditions [15]. The fulfillment of the properties will be verified in the following. Wiping-out property If a system complies with this condition the output is only affected by the current input and the alternating series of previous dominant extremal values of the input, while the effect of all other inputs are wiped out. The antagonistic setup is tested for the wiping-out property by applying a sequence of two decaying sine input for the desired angle. The input is set to the largest possible input value in order to wipe out the effects of previous inputs, after one cycle the amplitude of the sine signal is linearly decreased over 10 oscillations. Consequently, the input sequence is repeated. As can be seen in Fig. 7 the output sequence as a result of the first and the second decaying sine input are equivalent. This is only possible if the large amplitude input in the second sequence wipes out the effects of the previous inputs. Therefore the wiping-out is fulfilled for the given system. Minor-loop congruency property Minor-loop congruency requires that equivalent minor hysteresis loops, resulting from an input varying between the
same two extrema, produce congruent minor loops that only differ by an offset due to the separate input histories. This property implies that the shape of a minor loop depends only on the two extremum values of the input path used to generate the loop. Using a test input, a minor loop was recorded following upon two different input histories. As depicted in Fig. 8, the resulting minor loops show the required congruency. C. Identification and feed-forward compensation. The hysteretic behavior of the system was identified by producing a series of first order descending curves. An input signal was chosen to lead the output along an ascending branch of the major loop, at distinct values, chosen as suggested in [17], the input reverses producing a descending curve in the input-output-diagram that terminates at negative saturation. Due to the symmetric hysteresis loop it is apparent, that the relation µ(α, β) = µ(−β, −α)
(7)
is valid. Therefore it is obvious, that for the identification of µ(α, β) also the first-order increasing curves could have been used which are attached to the limiting descending branch. After the derivation of the inverse Preisach model, the desired joint angle is fed into the inverse model whose output serves as the input to the feed-forward structure in Fig. 4. The resulting match between desired and measured angle is shown in Fig. 9. It can been seen, that the inverse model allows a significant reduction of the width of the resulting hysteresis between actual and desired angle and will, in combination with an appropriate feedback controller, allow a very precise control of the manipulator segment. III. C ONTROLLER DESIGN To ensure precise tracking in the presence of disturbances, creep, and model uncertainties a closed-loop controller it
20
30
15 10
10 Angle (°)
Measured joint angle (°)
20
0 −10
5 0 −5 −10
−20 −30 −30
Fig. 9.
open−loop (uncompensated) open−loop (compensated) −20
−10 0 10 Desired joint angle (°)
20
Desired angle Closed−loop control CL + hysteresis compensation
−15 −20
30
0
2
4
6
8
10
Time (s)
Open-loop hysteresis compensation using inverse Preisach model.
Fig. 11. Step response with/without inverse Preisach model feed-forward controller.
-1
IV. C ONCLUSION AND FUTURE WORKS . A. Conclusion.
Feed-forward joint angle plant jdes
-
PID
u
G(s)
jmeas
Fig. 10. Control structure comprising inverse Preisach model and PID feedback controller.
is imperative to complement the feed-forward compensator with a feed-back controller. The idea of the control scheme is to combine the hysteresis compensator and a linear controller to implement an effective tracking controller. Fig. 10 shows the resulting setup comprising the combination of the PID feed-back and the feed-forward hysteresis compensator. The output of the resulting discrete time control algorithm can thus be derived by the following equation: X u(k) = uIP M (k) + KP e(k) + KI e(k) + . . . KD (e(k) − e(k − q)), k = 1, 2, . . .
(8)
with KP , KI , and KD the proportional, integral and derivative gains, respectively and uIP M the output of the inverse Preisach model. The controller parameters were derived after evaluation the step response and slightly adjusted heuristically to allow sufficient damping in spite of varying dynamic parameters of the plant. Fig. 11 shows a comparison of the step response behaviors of the controlled joint angle. It can be seen, that the system’s tracking dynamics clearly improves as a result of the addition of the feed-forward hysteresis compensator. Apart from the reduces rise time the system output is able settle significantly faster around the desired value if the hysteresis compensator is in place.
This paper presents Preisach modeling of hysteresis and precision tracking control of a manipulator segment driven by a set of antagonistic pneumatic muscles system using inverse Preisach model and a feedback controller. Due to their unsymmetric hysteretic behavior, single pneumatic muscles do not lend themselves to a feed-forward control via an inverse Preisach model. A control structure was presented that allows the approximate compensation of the underlying force-displacement-pressure relationships for two muscles in a symmetric antagonistic setup. The resulting setup still exhibits a dominantly hysteretic behavior, but is symmetric under the mentioned conditions. A series of tests was conducted to study the hysteresis properties of the controlled actuator system. A classical Preisach model is then applied to simulate the static hysteresis behavior of this system. After identifying this model, its inverse is applied to allow feedforward compensation of the hysteretic behavior of the joint actuation system. In combination with an appropriate feed-back controller the feed-forward compensator is shown to significantly improve the performance of the joint angle tracking controller. B. Future Works Future works will include the setup of underlying force control loops and will be directed to study the possibility of a stiffness variation utilizing the antagonistic muscle forces. R EFERENCES [1] J. Schmitt, F. Grabert, and A. Raatz, “Design of a hyper-flexible assembly robot using artificial muscles,” in Robotics and Biomimetics (ROBIO), 2010 IEEE International Conference on, dec. 2010, pp. 897 –902. [2] I. Boblan, R. Bannasch, A. Schulz, and H. Schwenk, “A human-like robot torso ZAR5 with fluidic muscles: Toward a common platform for embodied AI,” vol. 4850, pp. 347–357, 2007.
[3] F. Daerden and D. Lefeber, “Pneumatic artificial muscles: actuators for robotics and automation,” European journal of Mechanical and Environmental Engineering, vol. 47, pp. 10–21, 2000. [4] T. Minh, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, “Cascade position control of a single pneumatic artificial musclemass system with hysteresis compensation,” Mechatronics, vol. 20, no. 3, pp. 402–414, Apr. 2010. [Online]. Available: http://dx.doi.org/10.1016/j.mechatronics.2010.03.001 [5] I. Boblan, “Modellbildung und Regelung eines fluidischen Muskelpaares,” Ph.D. dissertation, Fakult¨at III - Prozesswissenschaften - der Technischen Universit¨at Berlin, 30. November 2009. [6] D. Schindele and H. Aschemann, “Disturbance compensation strategies for a high-speed linear axis driven by pneumatic muscles,” pp. 436–441, August 2009. [7] C.-P. Chou and B. Hannaford, “Measurement and modeling of mckibben pneumatic artificial muscles,” Robotics and Automation, IEEE Transactions on, vol. 12, no. 1, pp. 90–102, Feb. 1996. [8] I. Sardellitti, G. Palli, N. Tsagarakis, and D. Caldwell, “Antagonistically actuated compliant joint: Torque and stiffness control,” in Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, oct. 2010, pp. 1909 –1914. [9] T. Minh, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, “A new approach to modeling hysteresis in a pneumatic artificial muscle using the maxwell-slip model,” Mechatronics, IEEE/ASME Transactions on, vol. 16, no. 1, pp. 177 –186, feb. 2011. [10] T. Minh, B. Kamers, T. Tjahjowidodo, H. Ramon, and H. Van Brussel, “Modeling torque-angle hysteresis in a pneumatic muscle manipulator,” in Advanced Intelligent Mechatronics (AIM), 2010 IEEE/ASME International Conference on, july 2010, pp. 1122 –1127. [11] M. Van Damme, P. Beyl, B. Vanderborght, R. Van Ham, I. Vanderniepen, R. Versluys, F. Daerden, and D. Lefeber, “Modeling hysteresis in pleated pneumatic artificial muscles,” in Robotics, Automation and Mechatronics, 2008 IEEE Conference on, sept. 2008, pp. 471 –476. [12] T. Kosaki and M. Sano, “Control of a parallel manipulator driven by pneumatic muscle actuators based on a hysteresis model,” Journal of Environment and Engineering, vol. 6, no. 2, pp. 316–327, 2011. [13] I. Mayergoyz, “Mathematical models of hysteresis,” Magnetics, IEEE Transactions on, vol. 22, no. 5, pp. 603 – 608, sep 1986. ¨ [14] F. Preisach, “Uber die magnetische Nachwirkung,” Zeitschrift fur Physik, vol. 94, pp. 277–302, May 1935. [15] I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, 2nd ed. Academic Press, 2003. [16] D. H. Everett, “A general approach to hysteresis. part 4. an alternative formulation of the domain model,” Trans. Faraday Soc., vol. 51, pp. 1551–1557, 1955. [17] F. Li and J. Zhao, “Discrete methods based on first order reversal curves to identify Preisach model of smart materials,” Chinese Journal of Aeronautics, vol. 20, no. 2, pp. 157 – 161, 2007. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1000936107600259
