International Applied Mechanics, Vol. 44, No. 10, 2008 SLIDING MODE CONTROL OF A “SOFT” 2-DOF PLANAR PNEUMATIC MANIPULATOR M. Van Damme*, B. Vanderborght, P. Beyl, R. Versluys, I. Vanderniepen, R. Van Ham, P. Cherelle, F. Daerden, and D. Lefeber This paper presents a sliding mode controller for a “Soft” 2-DOF Planar Pneumatic Manipulator actuated by pleated pneumatic artificial muscle actuators. Since actuator dynamics is not negligible, an approximate model for pressure dynamics was taken into account, which made it necessary to perform full input–output feedback linearization in order to design a sliding mode controller. The design of the controller is presented in detail, and experimental results obtained by implementing the controller are discussed Keywords: Soft 2-DOF, planar pneumatic manipulator, sliding mode controller, artificial muscle actuator 1. Introduction. Manual material handling tasks such as lifting and carrying heavy loads, or maintaining static postures while supporting loads are a common cause of lower back disorders and other health problems. In fact, manual material handling has been associated with the majority of lower back injuries, which account for 16–19% of all worker compensation claims, while being responsible for 33–41% of all work-related compensations [1]. The problem has an important impact on the quality of life of affected workers, and it presents an important economic cost. The traditional solution is using commercially available manipulator systems, most of which use a counterweight. In order to increase the safety and productivity of human workers, several other approaches to robot-assisted manipulation have been studied in the robotics community [2–4]. The devices developed in the course of these studies belong to a class of materials handling equipment called Intelligent Assist Devices (IADs). Most of these systems, however, are heavy, complex, and expensive. We are working towards a multifunctional assistive device that can be used in direct interaction with a human operator, so safety is of paramount importance. As a first prototype, we have developed a small 2-DOF manipulator (shown in Fig. 1) powered by Pleated Pneumatic Artificial Muscles (PPAMs, see [5]). The compliance of these actuators, combined with the very low total weight of the system (» 2.6 kg) make the hardware intrinsically much safer than a stiff system would be. In this paper, we present a sliding mode controller for the 2-DOF manipulator. Sliding mode control has been considered before for systems actuated by pneumatic artificial muscles (see, e.g., [6–9]). Almost invariably, however, pressure dynamics is not considered, and only simulation results are provided. Practical experience shows that modeling pressure dynamics is necessary to obtain a usable controller. Other related works can be found in [10–13]. The paper is organized as follows. Sections 2 and 3 introduce the pleated pneumatic artificial muscle actuator and the manipulator, respectively. Section 4 gives a detailed account of how the controller was designed, and also provides some experimental results obtained from tests on the manipulator. Conclusions are drawn in Sec. 5. 2. Pleated Pneumatic Artificial Muscles. 2.1. Concept. Pneumatic artificial muscles (PAMs) are contractile devices whose core element is an inflatable membrane. When inflated they bulge, shorten, and thereby generate a contraction force. Over the years, different types have been developed, the most well known being the McKibben muscle [14]. It consists of a rubber tube, which expands when inflated, surrounded by a netting that transfers tension. Robotics and Multibody Mechanics Research Group Department of Mechanical Engineering, Vrije Universiteit Pleinlaan 2, 1050 Brussel, Belgium, e-mail: michael.vandamme@vub.ac.be. Published in Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 135–144, October 2008. Original article submitted January 25, 2007. 1063-7095/08/4410-1191 ©2008 Springer Science+Business Media, Inc. 1191 Fig. 1. The manipulator. The series arrangement of Pleated Pneumatic Artificial Muscles is clearly visible ft l0/R = 2 F(N) 4 1500 2.0 1.5 6 8 1.0 p = 300 kPa 250 200 150 1000 14 100 500 0.5 0 Fig. 2. Pleated muscle concept 10 20 30 40 e, % 0 10 20 30 40 e, % a b Fig. 3. f t 0 —dimensionless force function (a); force exerted by a PPAM (b) with l0 / R = 6 and l0 = 6 cm for different gauge pressures Although easy to make, the McKibben muscle has some important drawbacks, such as substantial hysteresis and a high threshold pressure under which no contraction occurs. Its total displacement is limited to just 20% to 30% of the initial length. To remedy these problems, a new type of PAM was developed, the PPAM [5, 15]. The PPAM (see Fig. 2) has a folded membrane that unfolds as it expands. Because of the unfolding, there is no threshold pressure, hysteresis is strongly reduced, and contractions of up to 50% are possible (depending on the slenderness (see further)). If one also takes into account the PPAMs low weight (under 100 g is possible), high power to weight (over 1 kW/kg), and the fact that it can be attached to the structure without reduction, it will be clear that this actuator can be useful for some robotic applications. 2.2. Characteristics. An accurate mathematical model that describes shape, volume, diameter, exerted force, and maximum contraction of PPAMs can be found in [5, 16]. Under the assumption of negligible membrane elasticity, the force exerted by the muscle is given by F = pl02 f t 0 ( e, l0 / R ). (1) In this expression, p is the applied gauge pressure, l0 is the muscle’s uncontracted length (or maximum length), R is its radius in uncontracted state (or minimum radius), and e is the muscle contraction (if l is the muscle length, we have e = 1- l / l0 ); f t 0 is a nonlinear, dimensionless function that depends on contraction and on the design-time parameter l0 / R (called the slenderness). The function f t 0 is shown in Fig. 2 for different values of l0 / R. As Fig. 2 and Eq. (1) show, there is a varying force-displacement relation at constant gauge pressure. This results in muscle-like behavior, with very high forces being generated at low contractions and very low forces at high contractions, as shown in Fig. 2 for a muscle with slenderness l0 / R = 6 and l0 = 6 cm. To avoid excessive material loading, contraction should be kept above 5%. 2.3. Creating a Revolute Joint. Since PAMs are contractile devices, they can only exert force in one direction (they can only pull, not push). In order to have a bidirectionally actuated revolute joint, two PAMs have to be used in what is generally called an antagonistic setup. This is illustrated in Fig. 4. The torque characteristics of such a joint are determined by the parameters of both muscles (slenderness and maximum length) and by the location of the four attachment points. 1192 q2 1 3 upper arm 2 4 lower arm 1 2 q1 Fig. 4. Antagonistic setup Fig. 5. The inverse elbow configuration 3. Manipulator. 3.1. Introduction. Since the developed manipulator is a first prototype, the length of both links was kept small: 30 cm. Figure 5 shows the conventions used in the rest of this document regarding to how both joint angles are defined and how the different pneumatic muscles are numbered. 3.2. Torque Characteristics. From the knowledge of the system’s kinematic parameters (i.e., muscle attachment point locations) and the PPAM actuator parameters, we can determine the torque characteristics of both joints. Using the nonlinear force–pressure–contraction relation of the PPAM muscle (see [5, 16]), the torque generated by a muscle can be written as t = pm( g ) (2) with g = q1 for muscles 1 and 2 and g = q 2 for muscles 3 and 4. The total actuator torque (in both joints) thus becomes é p m ( q ) + p 2 m2 ( q1 ) ù t=ê 1 1 1 ú, ë p 3 m3 ( q 2 ) + p 4 m4 ( q 2 )û (3) with p i (i = 1,…, 4) the gauge pressure in muscle i and mi the torque function associated with that muscle. Equation (2) provides a clear separation between the two factors that determine torque: gauge pressure and a torque function m that depends on the design parameters and the position. The torque functions mi are shown in Fig. 6. More details can be found in [17]. Note, however, that these torque functions provide a relatively rough approximation of reality. 4. Control. 4.1. Introduction. In general, controlling the manipulator is not straightforward. Difficulties encountered when designing a controller include the following: Non-linear force–contraction relation of the PPAM actuator (see [5, 16]). Hysteresis in the force–contraction relation of the PPAM. Although the hysteresis effect is less pronounced in the PPAM than in other types of pneumatic artificial muscles, it still makes it difficult to estimate the actual force exerted by the actuator. Imprecise knowledge of PPAM parameters. Non-linearity in the pressure regulating valves. Actuator gauge pressures can take a relatively long time to settle (around 100 ms for large pressure steps). The coupling between actuator gauge pressures and link angles and angular velocities. This means that the system cannot be modeled as a cascade of a pneumatic system followed by a mechanical system. These factors make it difficult to efficiently control the system. In this paper, a sliding mode approach [18–20] is used for controller design. 4.2. Dp-approach. To reduce the number of actuator outputs that have to be calculated, the Dp-approach was used (see [16, 21]). This involves choosing an average pressure p m for both muscles of an antagonistic pair, and having the controller calculate a pressure difference Dp that is added in one muscle ( p + Dp) and subtracted in the other ( p - Dp). The choice of p m influences compliance while Dp determines joint position. 1193 m, Nm/100 kPa m, Nm/100 kPa m1 m3 10 6 4 5 –m2 0 0.6 –m4 2 0.9 1.2 q1, rad 0 –2.1 –1.8 –1.5 –1.2 –0.9 q2, rad a b Fig. 6. Torque functions The control of the actuator pressures themselves is handled by off-the-shelf proportional pressure regulating valves with internal PID controllers. 4.3. Sliding Mode Controller. 4.3.1. Mechanical Model. The dynamical model of a 2-DOF planar arm is well known, and can be written as H ( q )&&q + C ( q, q& )&q + G ( q ) = t, (4) where q = [ q1 q 2 ] T is the vector of joint angles, H is the inertia matrix, C is the centrifugal matrix (centrifugal and Coriolis forces), G is the gravitational force vector, t is a vector representing the actuator torques, and is given by Eq. (3). 4.3.2. Pressure Dynamics. Since actuator gauge pressures can take a relatively long time to settle, we must include its dynamics into the system model. Ignoring the pressure dynamics would lead to an unusable controller. There are two main factors that influence the actual gauge pressure in the muscles: the pressure regulating valves and the coupling between actuator gauge pressures and link angles and angular velocities. We will not attempt to develop a full model of the electro-pneumatic pressure regulating valves, as that would make the full system model far too complex. Instead, we will simply approximate the valve response by a first-order system. If p is the pressure inside a muscle and p d is the desired pressure then the approximated first order valve dynamics can be written as p& = - p / T + p d / T . (5) A change in joint angle q also changes the contractions of the muscles driving that joint, which means their volume changes as well. The pressure difference caused by this effect can be approximated as (see [22]) dp 1 dV ( q ) , = -n( Patm + p ) dq V ( q ) dq (6) with V ( q ) the muscle volume, Patm the atmospheric pressure, and n the polytropic coefficient (in the isentropic limit we have n =1.4 for dry air),V ( q )can be rewritten asV ( e ) (with e the muscle’s contraction), which can be found in [16]. Equation (6) leads to p& = dp dq 1 dV ( q ) = -n( Patm + p ) q&, dq dt V ( q ) dq (7) which describes the coupling between p, q and q&. By combining (5) and (7), we obtain the gauge pressure dynamics p& = - p pd 1 dV ( q ) + - n( Patm + p ) q&. T T V ( q ) dq Since in practice the desired pressures are determined using the Dp-approach (see Sec. 2), we can write this for all muscles as 1194 p& 1 = - p1 p m1 + Dp1 1 dV1 q& 1 , + - n(Patm + p1 ) T1 T1 V1 dq1 p& 2 = - p 2 p m1 - Dp1 1 dV2 q& 1 , + - n(Patm + p 2 ) T2 T2 V2 dq1 p& 3 = - p 3 p m 2 + Dp 2 1 dV3 q& 2 , + - n(Patm + p 3 ) T3 T3 V3 dq 2 p& 4 = - p 4 p m 2 - Dp 2 1 dV4 q& 2 , + - n(Patm + p 4 ) T4 T4 V4 dq 2 (8) Dp1 and Dp 2 (the desired values of Dp) are the control inputs for the upper and lower arm joints, respectively, and p m1 and p m 2 are the average pressures used in both joints. 4.3.3. Feedback Linearization. From (4), we can write q&& = -H -1Cq& - H -1G + H -1 t = - Aq& - B + H -1 t with A = H -1C and B = H -1G. Together with (3) and (8) we get the full model of the system to be controlled: é p m ( q ) + p 2 m2 ( q1 ) ù q&& = - Aq& - B + H -1 ê 1 1 1 ú, ë p 3 m3 ( q 2 ) + p 4 m4 ( q 2 )û p& 1 = - p1 p m1 + Dp1 1 dV1 q& 1 , + - n(Patm + p1 ) T1 T1 V1 dq1 p& 2 = - p 2 p m1 - Dp1 1 dV2 q& 1 , + - n(Patm + p 2 ) T2 T2 V2 dq1 p& 3 = - p 3 p m 2 + Dp 2 1 dV3 q& 2 , + - n(Patm + p 3 ) T3 T3 V3 dq 2 p& 4 = - p 4 p m 2 - Dp 2 1 dV4 q& 2 . + - n(Patm + p 4 ) T4 T4 V4 dq 2 (9) The system has two inputs (Dp1 and Dp 2 ) and two outputs (q1 and q 2 ). We can consider (9) to consist of two (coupled) input-affine SISO systems, writing them as x& i = f i + g i u i , (10) yi = hi ( xi ) (11) with i = 1, 2 (1 for upper arm, 2 for lower arm), x i = [ q i wi p 2 i -1 p 2 i ] T being the state vectors (with wi = q& i ), u i = Dp i the scalar inputs, y i = hi ( x i ) = q i the scalar system outputs, and wi é ù ê - A w - A w - B + H -1 t + H -1 t ú i ,1 1 i ,2 2 i i ,1 1 i ,2 2 ê ú ê p m i - p 2 i -1 dV2 i -1 ú n fi = ê q& i ú , (Patm + p 2 i -1 ) T dq V i i 2 1 2 1 i ê ú pmi - p2i dV2 i ê ú n q& i (Patm + p 2 i ) ê ú T2 i V2 i dq i ë û é 0 ù ê 0 ú ú. gi = ê ê1/ T2 i -1 ú ê -1/ T ú ë 2i û 1195 Because of the presence of the pressure dynamics these SISO systems are not in controllability canonical form*. This means we cannot immediately apply standard sliding mode control techniques. We have to feedback-linearize the systems first. Both 4th order SISO systems (10), (11) have relative degree 3, which means the output (q i ) has to be differentiated three times for the input (Dp i ) to appear. More formally, it implies that** Lg i L jf hi (x i ) = 0 j = 0, 1, (12) i Lg i L2f hi ( x i ) ¹ 0. (13) i Since both systems have relative degree 3, the coordinate transformation x i1 = hi ( x i ), (14) x i 2 = L f i hi ( x i ), (15) x i 3 = L2f hi ( x i ), (16) i h i ( x i ) with Lg i h i ( x i ) º 0 (17) transforms (10), (11) to the required form (see, e.g., [23]): &x = x , i1 i2 (18) &x = x , i2 i3 (19) &x = b ( x , h ) + a ( x , h )u , i3 i i i i i i i (20) h& i = ri ( x i , h i ) (21) with bi ( x i , h i ) = L3f hi ( x i ), a i ( x i , h i ) = Lg i L2f hi ( x i ), and ri ( x i , h i ) = L f i h i ( x i ). Since the practical calculations quickly i i become very complex, all expressions were calculated with symbolic mathematics software. The same software was used to show that (14)–(17) is in fact a diffeomorphism (by showing that its Jacobian is nonsingular), as is required for feedback linearization [23, 19]. 4.3.4. Controller. Systems (18)–(21) are now in a form suitable for sliding mode control techniques. To design a sliding mode controller that makes these systems track their respective desired output trajectories y im ( t ) = q im ( t ), we use e i 0 ( t ) = y im ( t ) - y i ( t ) to define si ( x i , t ) = &&e i 0 + a i1 e& i 0 + a i 0 e i 0 . (22) The state-space surfaces defined by si ( x i , t ) = 0are called the sliding surfaces. The coefficients a i 0 and a i1 are chosen so that the polynomials p 2 + a i1 p + a i 0 are Hurwitz***. This way, if the state trajectory is on the sliding surface (if si = 0), the error will tend to zero according to the error dynamics (22). In view of Eqs. (12), (13), we have * ** *** 1196 A SISO system is said to be in controllability canonical form (see for instance [19], chapter 6) if its dynamics can be written as x ( n ) = f (x ) + b(x )u with u the scalar control input, x the scalar output, x = [ x x&L x ( n -1) ]T the state vector and f (x ) and b(x ) nonlinear functions of the state. Note that no derivatives of the control input u are present. ¶h The scalar function L f h(x ) = f stands for the Lie derivative of the scalar function h with respect to the vector field f . ¶x L2f h(x ) = L f L f h(x ) = L f (L f h)(x ). L0f h(x ) = h(x ). See for instance [19, 23] for more information. A polynomial with real positive coefficients and roots which are either negative or pairwise conjugate with negative real parts. &y i = ¶hi x& i = L f i hi ( x i ) + Lg i hi ( x i )u i = L f i hi ( x i ), ¶x i &&y i = L2f hi ( x i ) + Lg L f hi ( x i )u i = L2f hi ( x i ) i i i i so (22) becomes si ( x i , t ) = &&y im - L2f hi ( x i ) + a i1 ( &y im - L f i hi ( x i )) + a i 0 ( y im - hi ). i When on the sliding surface the system dynamics are described by s& i = 0, which becomes (using a i = Lg i L2f hi ( x i )and i Lg i L f i hi ( x i ) = 0): ( ) s& i ( x i , t ) = &&&y im - L3f hi ( x i ) - Lg i L2f hi ( x i )u i + a i1 &&y im - L2f hi ( x i ) - Lg i L f i hi ( x i ) + a i 0 ( &y im - h& i ) (23) = &&&y im - bi - a i u i + a i1 (&&y im - &&y i ) + a i 0 ( &y im - h& i ) (24) = &&&y im - bi - a i u i + a i1&&e i 0 + a i 0 e& i 0 (25) = 0. (26) i i i The equivalent control (or continuous control law that would maintain s& i = 0if the dynamics was exactly known) is thus 1 given by u eq ,i = (&&&y im - bi + a i1&&e i 0 + a i 0 e& i 0 ). ai By adding a discontinuous switching term, we finally get the sliding mode control law: ui = 1 ( &&&yim - bi + a i1&&ei 0 + a i 0 e& i 0 + K i sgn(si (xi , t ))). ai (27) Substitution of this control law into (25) gives s& i ( x i , t ) = -K i sgn( si ( x i , t ) )so if K i is large enough to overcome system uncertainty and perturbations, we have s& i si < 0 which implies that the sliding surface will be attractive and will be reached in finite time. As is well known, once the sliding surface is reached, the term Ksgn( si ( x i , t ) ) in (27) will cause excessive control chattering. To reduce this problem, a boundary layer (see [19]) is introduced by replacing sgn ( si ) with sat ( si / Gi ), where | z | £ 1, ì z, sat( z ) = í î sgn( z ), otherwise, and Gi are constants determining the width of the boundary layers. Of course, these boundary layers diminish tracking precision. 4.3.5. Experimental Results and Discussion. When testing the above described sliding mode controller in practice, the chattering problem turned out to be quite severe. An obvious contributing cause is the modeling of the pressure control valves as simple first order systems, which is a rough approximation at best. Simulation showed that choosing the valve time constants Ti 20 to 50 percent lower than the (estimated) actual value significantly reduced chattering without affecting tracking performance too much. Although this approach also worked on the real system, significant boundary layers were nevertheless necessary (G1 = 4, G2 = 3). To illustrate performance, the position error e recorded during a representative experiment is shown in Fig. 7, where e = ( x - x d ) 2 + ( y - y d ) 2 , x and y indicate the Cartesian position of the tool center point, x d and y d give the desired position. During the experiment, the system had to track a circular trajectory (diameter 20 cm) in Cartesian space during a period of 5 seconds. Three periods are shown. It is clear that the necessary introduction of the boundary layers causes a significant tracking error. Although the controller’s performance is not entirely satisfactory, the result is nevertheless acceptable. More research is necessary to improve tracking performance. It is clear that the designed controller is not adaptive. This has a number of disadvantages, such as not being able to cope with unknown payloads. For control methods that do not suffer from this problem see [24–27]. 1197 Position error e, cm 1.5 1.0 0.5 0 5 10 time, sec Fig. 7. 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