Identification of a system with dry friction

Identification of a system with dry friction Claudiu Iurian, Fayçal Ikhouane, José Rodellar, Robert Griñó IOC-DT-P-2005-20 Setembre 2005 Contents 1 The State of the Art in Friction Modeling for Systems Identification 3 1.1 INTRODUCTION ............................................ 3 1.2 STATICFICTIONMODELS....................................... 3 1.2.1 CoulombFriction......................................... 4 1.2.2 ViscousFriction ......................................... 5 1.2.3 Stiction.............................................. 5 1.2.4 StribeckEffect .......................................... 5 1.2.5 Continuous zero-velocity crossing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.6 Amoregeneralstaticmodel................................... 6 1.2.7 KarnoppModel.......................................... 6 1.2.8 Thesevenparametermodel ................................... 7 1.3 FRICTIONPHENOMENA........................................ 7 1.3.1 PreslidingDisplacement ..................................... 7 1.3.2 FrictionalLag........................................... 8 1.3.3 VaryingBreak-AwayForce.................................... 9 1.3.4 Stick-SlipMotion......................................... 9 1.3.5 Time-dependent, position dependent, and direction-dependent friction . . . . . . . . . . . . . 9 1.4 DYNAMICFICTIONMODELS..................................... 10 1.4.1 DahlModel............................................ 10 1.4.2 TheBristleModel ........................................ 11 1.4.3 TheResetIntegratorModel ................................... 12 1.4.4 Models for lubricated contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.5 Bliman-SorineModel....................................... 13 1.4.6 LuGreModel........................................... 13 1.4.7 LeuvenModel .......................................... 15 1.5 CONCLUSIONS ............................................. 16 2 Summary of the Work Done 17 2.1 M220 - THE EQUATIONS OF MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Two-gearbeltmechanism .................................... 17 2.1.2 Three-gear belt mechanism - M220 Industrial Emulator . . . . . . . . . . . . . . . . . . . . . 21 2.1.3 TheInertiaBalance........................................ 26 2.1.4 Plant continuous and discrete transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.5 M220-DCBrushlessMotor................................... 32 2.1.6 Zero-order hold for DC motor and plant together . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 STEP RESPONSE IDENTIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 COULOMB AND VISCOUS FRICTION IDENTIFICATION . . . . . . . . . . . . . . . . . . . . . 44 2.4 REFERENCES.............................................. 46 Part 1 The State of the Art in Friction Modeling for Systems Identification 1.1 INTRODUCTION Friction is a complex nonlinear phenomenon. A great deal is known about friction in specific circumstances, but not in the general case, if there is such a thing. There is still much to be learned about its nature and how it changes under different circumstances, and how it can be predicted and controlled. Indeed, it is difficult to find a process, in nature or industry, that is entirely free of friction [1]. Examples of frictional effects in everyday life are, without a doubt, endless. What is the definition of friction? Before anything else, we must say that the term friction is used more as a descriptive convenience rather than as a rigorous definition of friction phenomenon. The term friction comes from the Latin verb fricare, that is, to rub. The word tribology (Greek word for the study of rubbing), however, includes not only friction but also lubrication and wear. So, tribology is the science of the mechanisms of friction, lubrication, and wear of interaction bodies that are in relative motion. Tribology addresses and answers questions such as the true contact area, the relationships between friction, material properties and lubricating processes, wear mechanisms, and so on. In [2] is presented a timeline of the scientific study of friction starting with Leonardo Da Vinci (1452) giving the 1994 state of the art in friction for control purposes. Until recently, in tribology frictional dynamics has not been a focus [2]. Nevertheless, for the control engineer the dynamics of friction is of greatest interest; and here, lately, the control literature has become a major developer. We will focus only on just a group of frictional phenomena: static and dynamic friction between solid bodies, friction modelling, and compensation methods for machines with friction. Usually friction is not wanted, so a great deal has been done to reduce it by design, or by control. Friction introduces significant limitations to achieving good performance in controlled mechanical systems. In this respect, a widely used principle of friction control is model-based friction compensation which is utilized to apply a force or torque command equal and opposite in sign to the instantaneous friction force. An accurate friction model is needed for this purpose. Thus, various mathematical relationships have been developed to determine the influence of friction on machine behavior. Due to the fact that friction is such a common phenomenon, when feedback control methods are applied to machines and bodies in motion, friction appears invariably among the forces of motion [2] and which, oftentimes, cannot be simply ignored. The number of friction models proposed in literature is huge, see [2] for a complete literature survey, and can be subdivided with respect to their detail in describing surface contact properties occurring on a microscopic and macroscopic level. In the past decade major effort and contributions [3], [4], [5] are made in the development of friction models, suitable for analysis and controller synthesis, which have relatively limited complexity but a relevant similarity to practically observed friction phenomena. 1.2 STATIC FICTION MODELS Various mathematical models have been proposed in the literature that describe the important friction phenomena observed. Most of them are still used now, which model is preferred now depends on the purpose of it, but the model that accurately describes all the observed phenomena is in general to be preferred. Besides the question of effectiveness 3 The State of the Art in Friction Modeling for Systems Identification 4 also the model efficiency, e.g., the required computational resources in terms of time, can be of importance when for instance the model will be used in simulation studies. Due to the complexity of the physical mechanisms underlying friction, most models are of an empirical nature. Furthermore, a distinction was made between static and dynamic models depending on the inclusion of frictional memory. For static friction models, this frictional memory is not take into account, whereas for dynamic friction models this memory behavior is described with additional dynamics between velocity and the friction force. The observed friction phenomena during the early days of scientific study of friction, Amontons in [6], then Coulomb in [7], and then others, have led to models of Coulomb, viscous, static friction (stiction), and their possible combinations, which are often referred to as classical models of friction. The Stribeck model, which models the Stribeck effect [8], can also be classified nowadays as belonging to this set of models. Four possible combinations are shown below. (a) Coulomb (b) Coulomb + viscous (c) Coulomb + viscous + stiction (d) Coulomb + viscous + Stribeck effect The static friction models are those that give the friction forces a function of velocity. These models only describe the steady-state behavior between velocity and friction force. One drawback of the above models is discontinuity at zero velocity that allows the friction rate to take on an infinite number of possible values. The discontinuity does not reflect the real friction behavior in a good way and causes errors or even instability in the algorithms used to compensate friction. 1.2.1 Coulomb Friction Independent of the area of contact, the Coulomb friction always opposes relative motion and is proportional to the normal force of contact. The Coulomb friction is the friction phenomenon that is only dependent of the direction of the velocity, not of the magnitude of the velocity. It is modelled as a static map between velocity and friction force that depends on the sign of the velocity. F=FCsgn(v) where FC=µ|fn|,µbeing the coefficient for friction and fnthe normal force. Coulomb friction is called also kinetic friction because it defines friction for non-zero velocities. For zero velocities The State of the Art in Friction Modeling for Systems Identification 5 the above Coulomb friction depends upon the signum function definition. A common use of the switching function is sgn(v) =          +1∀v>0 0∀v=0 −1∀v<0 A question that immediately arises is with Coulomb friction and which is very important because can create instability in the algorithms that depend on the true zero velocity to compensate friction is: When precisely the velocity reaches and crosses the zero level? In fact, this is the fundamental problem with all classical models of friction they are not causal, i.e., the discontinuity at zero velocity permits friction to take an infinite number of values [9]. 1.2.2 Viscous Friction Viscous friction results from the viscous behavior of a fluid lubricant layer between two rubbing surfaces. As shown in above, viscous friction is represented as a linear function of velocity. F=Fvv Viscous friction is the friction component that is proportional to velocity and goes to zero at zero velocity. 1.2.3 Stiction Experimentally has been observed that friction force at rest is higher than the kinetic force or Coulomb friction. If the system is in sticking an externally applied force is need that is equal or greater than the stiction force to put the body in motion, i.e. in slipping. This has been studied intensely in the 1950s, see the study of Rabinowicz [10] about the nature of the static and kinetic coefficients of friction. Static friction (stiction) is the force required to initiate motion from rest. Typically, the magnitude of static friction is greater than the magnitude of Coulomb friction which can lead to intermittent motion known as stick-slip motion. Stiction is assumed to be independent of the velocity, however varies as a function of the dwell-time when sticking and the rate of increase of the applied force. Thus, friction at rest cannot be described as a function of only velocity. Instead, it has to be captured in the model an external force Feas in the following description. F=   Feif v=0 and |Fe|<Fs FSsgn(Fe)if v=0 and |Fe|≥Fs 1.2.4 Stribeck Effect Stribeck effect is the friction phenomenon that arises from the use of fluid lubrication and gives rise to decreasing friction with increasing velocity at low velocity. For low velocities the friction force decreases with increasing velocity. Stribeck effect is needed to correctly predict initial conditions leading to stick-slip motion. Stribeck curve [8] is a continuous drop in the friction force for small velocities, which originates from the transition of boundary lubrication to full fluid lubrication through partial fluid lubrication [2]. The Stribeck effect could not be ignored for most of the cases and had been introduced and used extensively into the new frictional models developed. 1.2.5 Continuous zero-velocity crossing model The discontinuity of friction at zero velocity can lead to unwanted consequences such as (1) non-uniqueness of the solutions to the equations of motion for the system [2] which can occur in real friction and (2) numerical problems if such a model is used in simulation like numerical chatter. Precautions have to be taken when numerically integrating a system model that includes discontinuities as introduced by the classical friction models. One approach to overcome The State of the Art in Friction Modeling for Systems Identification 6 (e) Stribeck curve of steady-state friction force (f) Continuous zero-velocity crossing model the discontinuity is approximating or smoothing the map by a curve with finite slope [11] as depicted in the figure below Nonetheless, a very steep slope around zero velocity can result in very short integration time steps which slows down simulation. Moreover, the body connected to the frictional contact surface will accelerate even if the external forces on the body are less than the peak static friction force Fs and therefore these alternate friction models will not provide true stiction. 1.2.6 A more general static model A more general model than the classical models of friction is given in [12] and it covers Coulomb, viscous, stiction, and Stribeck friction. F=       F(v)if v6=0 and |Fe|<Fs Feif v=0 and |Fe|<Fs FSsgn(Fe)otherwise where F(v)is a nonlinear function, e.g., of the form F(v) = FC+(FS−FC)e− v vs δ +Fvv with vsas the Stribeck velocity and δas the form factor. Such kind of friction models have been used for quite a long time. 1.2.7 Karnopp Model For control purposes, the main disadvantage when using a model such as the general model above, is the problem of detecting when exactly the velocity is zero. Karnopp in [13] developed a model to overcome the problems with zero The State of the Art in Friction Modeling for Systems Identification 7 velocity detection and to avoid switching between different state equations for sticking and sliding. The model defines a zero velocity interval, |v|<DV. For velocities within this interval the internal state of the system the velocity may change and be non-zero but the output of the block is maintained at zero by a dead-zone. The drawback with the model is that it is so strongly coupled with the rest of the system. The external force is an input to the model and this force is not always explicitly given. The model therefore has to be tailored for each configuration. Variations of the Karnopp model are widely used since they allow efficient simulations. Nevertheless, the zero velocity interval does not agree with real friction phenomenon and has not been used to extensively. 1.2.8 The seven parameter model The seven parameter model is a friction model designed to include all relevant experimentally observed friction phenomena and was proposed by Armstrong in [2], [11]. It consists of two separate models: a stiction model and a sliding model. During stiction, friction is modelled as a stiff spring to account for presliding displacements. Ff(x) = σx In sliding, the model is modelled as Coulomb + viscous + Stribeck effect with frictional memory: Ff(˙x,t) =      Fc+Fv|˙x|+Fs(γ,t2)1 1+˙x(t−τL) ˙xs2     sgn(˙x) where Fs(γ,t2) = Fs,a+(Fs,∞−Fs,a)t2 t2+γ describes the varying friction level at break-away. The level of the stiction force Fsvaries with the time at zero velocity t2(dwell time). A long dwell time means a high break-away force. The force Fs,ais the magnitude of the Stribeck friction at the end of the previous sliding period; γis an empirical parameter. The friction force for the sliding mode is equivalent to a static friction model where the velocity has been replaced with a delayed version and which has a time-dependent coefficient Fs[14]. The time delay τLmodels the desired frictional memory. When used in simulations some mechanism has to be employed to switch between the two modes for sliding and stiction. Also, the model states xand ˙xhave to be initialized appropriately every time a switch occurs. Although useful for analysis of stick-slip behavior [11], for simulation purposes the model seems to be less appropriate, see [15] where was performed a simulation study employing, amongst several other friction models, the seven parameter model. 1.3 FRICTION PHENOMENA Experimentally has been shown that friction in control applications exhibits phenomena that cannot be modelled with a static friction model as above, such as presliding displacement, frictional lag, varying break-away force, and stick-slip motion. In studying friction there is a clear necessity of using dynamic friction models to take those phenomena into account and, thus, the following properties of a friction model are considered to be very important for the control of friction in mechanical systems. 1.3.1 Presliding Displacement If a force is applied to two surfaces in contact a small displacement will always occur due to limited stiffness of contact asperities [8]. Friction force behaves like a spring (and that causes a displacement linear dependent on the applied force) if the externally applied force is less than the break-away force. Small motions in the elastic region in sticking are referred to as presliding displacement or the Dahl effect. Presliding displacement is a consequence of elastic deformation of the surface asperities where contact and sliding occur and it significantly influences friction forces during velocity reversal [11]. It is the compliance of presliding displacement that softens the hard nonlinearity of friction at zero velocity. Presliding displacement is needed to correctly predict small displacements while sticking (including velocity reversals). The State of the Art in Friction Modeling for Systems Identification 8 Contrary to the predictions derived from the classical friction model, researchers including Courtney-Pratt and Eisner [16] and others have found experimentally that small relative displacements between two bodies in contact do occur when the applied relative tangential force is less than the static friction. Although the magnitude of this pre-sliding displacement is small, with sufficient gain, as in a robot with a fairly long link, small displacements at the rubbing surface can translate into significant displacements elsewhere in the mechanism. Further, the nature of pre-sliding displacements provides insight into a difficult part of the control problem, the transition between sticking and sliding. Courtney-Pratt and Eisner interpreted the pre-sliding phenomenon within the framework of the theory of asperity junction adhesion, asperity junctions being the load bearing interfaces between rubbing surfaces. Specifically, as the shear force at the contact surfaces increases, the asperity junctions deform elastically and then plastically. When the applied force finally reaches the stiction level, the asperity junctions break and sliding begins. Because of the plastic deformation, alternate increases and decreases in applied tangential force result in friction hysteresis loops. The pre-sliding displacement phenomenon is illustrated in figure below, which shows friction Fas a function of displacement xbased on experimental results. 1.3.2 Frictional Lag Frictional lag is the delay in the change of the friction force as function of a change in the velocity which may have a significant impact on dynamics as noted by Hess and Soom [17]. There is a hysteresis in the relationship between friction force and velocity. Namely, the friction force is lower for decreasing velocities than it is for increasing velocities. The width of the hysteresis loop increases with frequency and with higher rates of velocity changes [3]. Frictional lag is a dynamic behavior that results in a larger friction force for increasing velocities than for decreasing velocities and becomes more apparent for large acceleration and/or deceleration [17]. We note the considerable empirical evidence that has become available indicating that friction does not respond instantaneously to a change in velocity. The primary work here is due to geophysicists who use stick-slip for earthquakerelated predictions. Hess and Soom also found strong evidence of frictional lag, in their experiments on a flat steel button rubbing against a rotating steel disk. Frictional lag makes stick-slip instabilities less likely. Because a decrease in friction occurs slowly when velocity is increased, stiff systems will not experience stick-slip [11]. The State of the Art in Friction Modeling for Systems Identification 9 1.3.3 Varying Break-Away Force The break-away force is the force required to overcome stiction and initiate motion and it has been intensely studied for the last several decades [2]. Varying break-away force or rising static friction (sometimes also called static friction level) is the dependence of the break-away force on the rate of increase of the applied force [16]. The dwell-time when sticking is always related to the rate of increase of the applied force and the effects cause by these two connected factors cannot be separated. Also, the break-away force can be interpreted in the control systems as the minimum open-loop force. 1.3.4 Stick-Slip Motion Stick-slip motion manifests itself as repeated sequences of sticking between two surfaces with static friction followed by sliding or slipping of the two surfaces, for instance, when moving slowly, machines are likely to exhibit stickslip motion [11]. Stick-slip is a phenomenon which happens when sliding one body over another under a steady pulling force and the sliding velocity fluctuates widely. These fluctuations consist of sticking where the motion stops and slipping where the bodies suddenly accelerate again. In most practical sliding systems, these fluctuations of the sliding velocity are considered a serious nuisance. Eliminating or reducing the amplitude of the fluctuations is usually necessary. Stick-slip motion is caused by the fact that friction is larger at rest than during motion. When the applied force reaches the break-away force the body starts to slide and friction decreases rapidly due to the Stribeck effect. For the servo-mechanism control problem, stick-slip can diminish control accuracy. The stick-slip limit cycling can be avoided if damping and stiffness are sufficiently high. 1.3.5 Time-dependent, position dependent, and direction-dependent friction Time-dependent friction - From practical experiments can be seen that friction changes with time [1]. These changes of friction with time are due to such things as loss of lubricant, deformation of the surface material, change in temperature due to generated heat and accumulation of wear debris, and the like. Position-dependent friction - Friction exhibits also a dependence on the positioning of a system. This has been experimentally observed and is well known by a lot of tribology researchers. This position-dependency is caused by spatial inhomogeneities [1] in the transmission of the system due to contact geometry and loading which varies as a function of position. As the load varies, the normal force between the sliding surfaces varies, causing a varying friction. Direction-dependent friction - Again, experimentally, has been found the friction to be dependent on the direction of the motion of a system. Different Coulomb and viscous friction levels in the left and right directions of a single, linear motion have been observed experimentally on many occasions. This may be due to anisotropies in material or material geometry. The State of the Art in Friction Modeling for Systems Identification 16 1.5 CONCLUSIONS We have reviewed the mathematical models of friction used in controls literature starting from the simplest to the most complex, each of them trying in one way or another to capture the complex friction phenomena. It must be stressed that LuGre model is today one of the most used in the literature. LuGre model exhibits a rich behavior in terms of observed friction phenomena and in particular is able to model: stiction, the Stribeck effect, frictional lag or hysteresis, and stick-slip transitions. However, some of the practically observed hysteresis related phenomena can not be predicted accurately by the LuGre model as noted by Olsson et al. [14] and Swevers et al. [24]. The latter proposes an extension of the LuGre model to approach these hysteresis problems. The notion of stiction is re-addressed by Dupont et al. [19], who discusses the difference between stiction and presliding displacement. In their analysis both the dynamic Dahl and LuGre friction models are considered to possess presliding displacement but no stiction. A new elasto-plastic state variable friction model is proposed which models both stiction and presliding displacement. However, the proposed Leuven model by Swevers et al. [24] is more complex than the standard parametrization of the LuGre model due to the use of a hybrid hysteresis model and therefore more difficult to be used for control design and analysis. On the other hand, the elasto-plastic state variable friction model proposed by Dupont et al. [19] is mainly based on simulation studies and the presented ideas are not yet confirmed experimentally. Hence, the standard parametrization of the LuGre model is still most wildly used nowadays. All these experimentally observed phenomena and their modelation are necessary to fully understand the problems present in controlled mechanical systems with friction, since a motion system faces one or more of the above described frictional stages for each task performed. Moreover, the simplest friction model combining all these properties has to contain extra dynamics for the modelling of varying static friction and hysteresis due to frictional lag and should at the same time be nonlinear to capture the Stribeck curve. However, for the analysis of controlled mechanical systems, simplification of the friction model is often necessary due to the limited applicability of the used analysis tools. On the other hand, some friction induced phenomena can successfully be described with less complex friction models, which is desirable from a conceptual point of view and is also an important research issue. The estimation of the model parameters is important for obtaining quantitatively accurate friction models, which can be used as a mathematical representation of the friction. In general, it is not possible to measure the friction force directly and therefore the identification of friction in a mechanical system is far from trivial. Since the friction force can not be observed directly, experiments for the identification procedure are performed by sensing quantities that are influenced indirectly by the friction force, such as displacements, velocities or acceleration of the mass connected to the frictional contact surface. To estimate the model parameters, often extended with parameters to describe other dynamics in the system, such as mass and stiffness, different dedicated and time-consuming experiments must to be conducted. Each experiment must be designed to visualize one of the friction phenomena as described above by excluding other dynamics in the system. However, the time-varying nature of friction due to wear and exogenous variables such as changing load or operating temperatures might limit the applicability of an estimated friction model considerably. At a macroscopic level friction forces vary in time due to microscopic effects such as deformation of contact surfaces, accumulation of wear particles or changes in the properties of the lubricant. Since these influences are hard to measure it is also difficult to model these time varying phenomena. Hence, the estimated friction model is expected to capture at best an averaged behavior of the actual friction over time. Validation of the friction model with its estimated parameters can either focus on the ability of the identified model to predict the friction characteristics of interest or on the closed-loop performance when the identified friction model is incorporated in the controller design. Typical errors caused by friction in control loops are steady-state errors in position regulation, tracking lags, and limit cycles. The models can be validated in an open-loop or closed-loop setting and results might be improved, if desirable, in an iterative procedure, by choosing a more complex friction model. Part 2 Summary of the Work Done 2.1 M220 - THE EQUATIONS OF MOTION For the purpose of studying friction phenomena in all detail our investigation group CoDAlab from UPC Barcelona has purchased an industrial emulator made by ECP Systems, see [28], which will be referred to from now on as M220. This machine is a solid tool and testbed used primarily for experimentation in systems identification and control. M220 consists of a drive inertia and a load inertia, namely, it has a motor connected to a first smaller disk or "drive inertia" which in turn is connected to an idler pulley by an inelastic belt whose shaft is connected to the second larger disk or "load inertia". The main advantages offered by M220 are: 1) it can be used for studying the nonlinear dry friction phenomena that occur in mechanical systems, and 2) it provides accurate and reliable data by means of high resolution encoders controlled by a DSP controller and a specialized C-like software which interface with a PC. 2.1.1 Two-gear belt mechanism Let us consider the two-gear belt mechanism as shown below. We have: θ1is the angular position of the driving gear (motored). θ2is the angular position of the driven gear (load). r1is the radius of the driving gear. r2is the radius of the driven gear. Gear ratio is the number of turns the driving gear (motored - or first gear) does per one single turn of the driven gear (load - or second gear). ng=r2 r1is the gear ratio, and because ng>1 is called speed reducer. J1is the driving gear inertia. J2is the driven gear inertia. 17 Summary of the Work Done 18 T1is the torque of the driving gear (generated by the driving motor). T2is the torque of the driven gear (load torque). Definition 1 Inertia is the resistance an object has relative to changes in velocity. Definition 2 The moment of inertia J of a body is a measure of how hard it is to get it rotating about some axis. The moment Jis to rotation as mass mis to translation. The larger the J, the more work required to get the object spinning, just as the larger the mass m, the more work required to get it moving in a straight line. We make a torque balance, i.e. a force balance. A torque balance is usually done at the drive motor side, not at, for instance, the load side. If so, we have to express all the other relevant torques in the system to the drive motor shaft, i.e. "in terms of" the drive motor shaft torque. Definition 3 Areflected inertia is a given inertia (at any point of the system) as seen (through the respective pulleys and/or belts) by the drive motor shaft. Definition 4 Reflected load inertia is the load inertia as seen by the drive motor shaft through pulleys and/or belts. Definition 5 Reflected SR pulley assembly inertia is the calculated SR pulley assembly inertia as seen by the drive motor shaft through the respective pulleys and/or belts. Definition 6 Gears that both drive and are, in turn, driven are called idler gears. The driving gear angular displacement and the driven gear angular displacement and their derivatives (angular velocity and angular acceleration) are related by the gear ratio ngin the following manner: r1θ1=arc =r2θ2 θ1 θ2=r2 r1=ng ˙ θ1 ˙ θ2=r2 r1=ng ¨ θ1 ¨ θ2=r2 r1=ng Therefore: θ1=ngθ2(1) Summary of the Work Done 19 ˙ θ1=ng˙ θ2(2) ¨ θ1=ng¨ θ2(3) Aspeed reducer gear train is also a torque multiplier. Power =Torque ∗Angular Velocity Power in =Power out Torque in ∗Angular Velocity in =Torque out ∗Angular Velocity out Tin ˙ θin =Tout ˙ θout T1˙ θ1=T2˙ θ2 T2=˙ θ1 ˙ θ2T1=r2 r1T1 Therefore: T2=ngT1(4) The torque balance on the load: T2=J2¨ θ2(5) Substituting (3) and (4), we can rewrite (5) as : ngT1=J2 ng ¨ θ1(6) rearranging: T1=J2 n2 g ¨ θ1or T1=Jreflected ¨ θ1(7) Jreflected =J2 n2 g(8) Equation (7) is the torque equation in "motor coordinates". Jreflected is the driven gear inertia (load inertia) reflected by the driving gear (driving motor) or the load inertia "felt" by the driving motor. This means that the load inertia has been reduced by a factor of n2 gdue to the "gear train" from the motor’s point of view. r2 r1=N2 N1=θ1 θ2=˙ θ1 ˙ θ2=¨ θ1 ¨ θ2=ng Tis the torque on the shaft of the driving gear (motor torque). The equation of motion for the free body in diagram (a) is: J1¨ θ1=T+r1F1−r1F2−c1˙ θ1 J1¨ θ1+c1˙ θ1−r1F1+r1F2=T J1¨ θ1+c1˙ θ1+r1(F2−F1) = T Summary of the Work Done 20 Fbelt =F2−F1 J1¨ θ1+c1˙ θ1+r1Fbelt =T(9) Respectively, the equation of motion for the free body in diagram (b) is: J2¨ θ2+c2˙ θ2+r2F2−r2F1=0 J2¨ θ2+c2˙ θ2−r2(F1−F2) = 0 J2¨ θ2+c2˙ θ2−r2Fbelt =0 (10) r2 r1=θ1 θ2=˙ θ1 ˙ θ2=¨ θ1 ¨ θ2(11) Fbelt =J2 r2 ¨ θ2+c2 r2 ˙ θ2(12) The three relevant equations are:            J1¨ θ1+c1˙ θ1+r1Fbelt =T J2¨ θ2+c2˙ θ2−r2Fbelt =0 r2 r1=θ1 θ2=˙ θ1 ˙ θ2=¨ θ1 ¨ θ2=ng Substituting (14) in (12) J1r2 r1¨ θ2+c1r2 r1˙ θ2+Fbelt r1=T(13) Substituting (15) in (16), and then multiplying by r2 r1 J1r2 r1¨ θ2+c1r2 r1˙ θ2+J2r1 r2¨ θ2+c2r1 r2˙ θ2=T∗r2 r1(14) It results: hJ1r2 r12+J2i¨ θ2+hc1r2 r12+c2i˙ θ2=r2 r1T(15) or hJ1n2 g+J2i¨ θ2+hc1n2 g+c2i˙ θ2=ngT(16) Summary of the Work Done 21 2.1.2 Three-gear belt mechanism - M220 Industrial Emulator The M220 diagram shows a three-gear belt mechanism and is presented in the figure below: It can be observed that the system can be decomposed into three parts, see body-free diagram for more details. Tis the torque on the shaft of the driving gear (motor torque). Firstly, the equation of motion for the free body in diagram (a) is: J1¨ θ1=T+r1F1−r1F2−c1˙ θ1 J1¨ θ1+c1˙ θ1−r1F1+r1F2=T J1¨ θ1+c1˙ θ1+r1(F2−F1) = T Fbelt1=F2−F1 J1¨ θ1+c1˙ θ1+r1Fbelt1=T(17) Summary of the Work Done 22 Secondly, the equation of motion for the free body in diagram (b) is: J2¨ θ2=0−r2F1+r2F2+r3F3−r3F4−c2˙ θ2 J2¨ θ2+c2˙ θ2+r2F1−r2F2−r3F3+r3F4=0 J2¨ θ2+c2˙ θ2−r2(F2−F1)+r3(F4−F3) = 0 Fbelt1=F2−F1Fbelt2=F4−F3 J2¨ θ2+c2˙ θ2−r2Fbelt1+r3Fbelt2=0 r2 r1=θ1 θ2=˙ θ1 ˙ θ2=¨ θ1 ¨ θ2=npd n=n0 g Thirdly, the equation of motion for the free body in diagram (c) is: J3¨ θ3=0+r4F4−r4F3−c3˙ θ3−r5Fcsgn(˙ θ3) J3¨ θ3+c3˙ θ3−r4F4+r4F3+r5Fcsgn(˙ θ3) = 0 J3¨ θ3+c3˙ θ3−r4(F4−F3)+r5Fcsgn(˙ θ3) = 0 Summary of the Work Done 23 J3¨ θ3+c3˙ θ3−r4Fbelt2+r5Fcsgn(˙ θ3) = 0 (18) r4 r3=θ2 θ3=˙ θ2 ˙ θ3=¨ θ2 ¨ θ3=N npl =n00 g(19) The five relevant equations are:                            J1¨ θ1+c1˙ θ1+r1Fbelt1=T J2¨ θ2+c2˙ θ2−r2Fbelt1+r3Fbelt2=0 r2 r1=θ1 θ2=˙ θ1 ˙ θ2=¨ θ1 ¨ θ2=npd n=n0 g J3¨ θ3+c3˙ θ3−r4Fbelt2+r5Fcsgn(˙ θ3) = 0 r4 r3=θ2 θ3=˙ θ2 ˙ θ3=¨ θ2 ¨ θ3=N npl =n00 g And it results: Fbelt2=J3 r4 ¨ θ3+c3 r4 ˙ θ3+r5 r4Fcsgn(˙ θ3) Fbelt1=−1 r2h−r3Fbelt2−J2¨ θ2−c2˙ θ2i=1 r2hr3Fbelt2+J2¨ θ2+c2˙ θ2i J1¨ θ1+c1˙ θ1+r11 r2hr3Fbelt2+J2¨ θ2+c2˙ θ2i=T J1¨ θ1+c1˙ θ1+r11 r2hr3J3 r4 ¨ θ3+c3 r4 ˙ θ3+r5 r4Fcsgn(˙ θ3)+J2¨ θ2+c2˙ θ2i=T r2 r1=θ1 θ2 r4 r3=θ2 θ3 θ2=r4θ3 r3 r2 r1=θ1 r4θ3 r3 θ1=r2r4θ3 r1r3=n0 gn00 gθ3=ngθ3 θ2=r4θ3 r3 J1r2r4¨ θ3 r1r3+c1r2r4˙ θ3 r1r31+r11 r2hr3J3 r4 ¨ θ3+c3 r4 ˙ θ3+r5 r4Fcsgn(˙ θ3)+J2r4¨ θ3 r3+c2r4˙ θ3 r3i=T hJ1r2r4 r1r3+J3r1r3 r2r4+J2r1r4 r2r3i¨ θ3+hc1r2r4 r1r3+c3r1r3 r2r4+c2r1r4 r2r3i˙ θ3+r1r3 r2r4r5Fcsgn(˙ θ3) = T Then multiplying by r2 r1 r4 r3 hJ1r2r4 r1r32+J3+J2r4 r32i¨ θ3+hc1r2r4 r1r32+c3+c2r4 r32i˙ θ3+r5Fcsgn(˙ θ3) = r2 r1 r4 r3T hJ1n2 g+J2n002 g+J3i¨ θ3+hc1n2 g+c2n002 g+c3i˙ θ3+r5Fcsgn(˙ θ3) = ngT Summary of the Work Done 24 Jtotal ¨ θ+c˙ θ+fsgn(˙ θ) = ngT where: Jtotal =J1n2 g+J2n002 g+J3 c=c1n2 g+c2n002 g+c3 f=r5Fc θ=θ3 Tis the torque on the shaft of the driving gear (motor torque). nis the number of teeth on the drive disk pulley, fixed number;n=12. Nis the number of teeth on the load disk pulley, fixed number;N=72. npd is the number of teeth on the bottom pulley. npl is the number of teeth on the top pulley. ngis the total gear ratio. where ng=N npl npd n=72 npl npd 12 =6npd npl n0 gis the partial first gear ratio. where n0 g=npd n=npd 12 n00 gis the partial second gear ratio. where n00 g=N npl =72 npl If we write: Jtotal =J1n2 g+J2n002 g+J3 as Jtotal n2 g=J1+J2n002 g n2 g+J3 n2 g J=Jtotal n2 g=J1+J2 n02 g+J3 n2 g Now, if we look at the reflected inertia in the equation (10): Jd=Jdd +Jwd +Jp (n0 g)2+Jdl +Jwl (ng)2 We can identify the following: Jd=Jtotal n2 g=J J1=Jdd +Jwd J2=Jp J3=Jdl +Jwl Summary of the Work Done 25 From the M220 manual [28]: Jdd =0.00040 kg-m2 Jdrive disk =0.000381 kg-m2 Jmotor =3.8∗10−5kg-m2- (from manufacturer specifications) Diameter of the drive disk is: 13.21 cm. Thickness of the drive disk plate = 0.47 cm. ρaluminum =2.71g/cm3 Abrass weight has 500 g (including attachment bolts and nuts). The inertia of the cutout slots on the drive load and disks is negligible. The inertia of the encoder, and the belt and pulleys between the motor and the drive disk can be neglected too. The calculated mass of drive disk is 0.174 kg. Summary of the Work Done 32 2.1.5 M220 - DC Brushless Motor I. DC Brushless Motor Overview The DC brushless motor is known also as permanent magnet synchronous motor. The main advantage over the conventional DC brush motor is the elimination of brush friction associated wear. Other advantage is greater volume-to-power ratio. The permanent magnets are fixed on the rotor. The three phase windings are distributed in slots of the stator. In a DC brushless motor, a rotor positioning sensor is used and the commutation procedure is done electronically: it can be rectangular or sinusoidal. A brushless dc motor is essentially a dc motor turned inside out. However, the basic principle of operation of a brushless dc motor is similar to that of a normal dc motor. A brushless dc motor has a rotor with permanent magnets and a stator with windings that are connected to the control electronics. The control electronics act as a replacement for the commutator and perform its function of energizing the proper winding. The windings are energized in a pattern which rotates around the stator. Then the energized stator winding leads to the rotor magnet and switches just as the rotor aligns with the stator. In contrast to the conventional dc motor, the permanent magnets of the brushless dc motor are affixed directly onto the rotor itself. The phase windings (there are typically 3 phases) are distributed in the slots found on the stator. This arrangement provides for greater heat dissipation, which in turn leads to improved life and typically greater volume-to-power ratios for brushless motors than for brush motors. In any continuously rotating motor, to provide a continuous torque, the current must be successively altered or switched depending on the absolute position of the rotor. In a dc brushless motor, a rotor-positioning (opto) sensor is used and the commutation procedure is performed electronically. A brushless dc motor is actually cleaner, faster, more efficient, less noisy, and more reliable than a brush motor because there are no sparks involved in a brushless dc motor, and because a brushless dc motor is not restricted by the limitations of brush life, brush residue, maximum speed, and electric noise. One advantage of the brushless dc motor is its ability to use opto sensors that can provide reliable position sensing in harsh environments where high vibration, moisture and relatively high temperatures exist and because they are also known for their long life, fast response time, high repeatability, and zero speed-sensing capabilities. where ω(t)- is the motor shaft angular velocity. T(t)- is the torque of the motor. va(t)- is the applied voltage. vb(t)- is the back (induced) voltage. Kbis the motor constant. Ktis the motor torque constant (called also the motor armature constant). Kfis the motor viscous friction (damping), a function of motor’s angular velocity. Lis the armature winding inductance. Ris the armature resistance. Jmis the motor’s moment of inertia. Summary of the Work Done 33 2. Electrical Equations The torque T(t)seen at the shaft of the motor is proportional to the current i(t)induced by the applied voltage, va(t),T(t) = Kti(t) where Ktis the armature constant, is related to physical properties of the motor, such as magnetic field strength, the number of turns of wire around the conductor coil, and so on. The back (induced) electromotive force, vb(t), is a voltage proportional to the angular velocity ω(t)seen at the motor shaft, vb(t) = Kbω(t) where Kb, the motor constant, also depends on certain physical properties of the motor. va(t)−vb(t) = Ldi dt +Ri(t) va(t) = Ldi dt +Ri(t)+Kbω(t) di dt =−R Li(t)−Kb Lω(t)+ 1 Lva(t) II. Mechanical Equations The mechanical part of the motor equations is derived using Newton’s law, which states that the inertial load Jm times the derivative of angular rate (i.e. angular velocity) equals the sum of all the torques about the motor shaft. Jmdω dt =∑Ti=−Kfω(t)+ T(t) T(t) = Jmdω dt +Kfω(t) Jmdω dt =−Kfω(t)+Kmi(t) where −Kfω(t)is a linear approximation for viscous friction. dω dt =−Kf Jmω(t)+ Km Jmi(t) III. State-Space Equations for the DC Brushless Motor Given the two differential equations derived in the last section, we can now develop a state-space representation of the DC motor as a dynamic system. The current i(t)and the angular velocity ω(t)are the two states of the system. The applied voltage, va(t), is the input to the system, and the angular velocity ω(t)is the output.          di dt =−R Li(t)−Kb Lω(t)+ 1 Lva(t) dω dt =−Kf Jmω(t)+ Km Jmi(t) Summary of the Work Done 34 IV. Block Diagrams Diagram 1 Applying Laplace transform to the following equation va(t)−vb(t) = Ldi dt +Ri(t) we have Va(s)−Vb(s) = sLI(s)+RI(s) I(s) Va(s)−Vb(s)=1 sL+R Diagram 2 From T(t) = Kti(t) we get T(s) = KtI(s) Diagram 3 In the same manner, applying Laplace to T(t) = Jmdω dt +Kfω(t) it results T(s) = JmsΩ(s)−KfΩ(s) Ω(s) T(s)=1 sJm+Kf Diagram 4 From vb(t) = Kbω(t) we get Vb(s) = KbΩ(s) Summary of the Work Done 35 DC Motor Block Diagram Connecting all the four block diagrams above we obtain the DC Motor block diagram. V. Transfer Functions A. The transfer function between the applied voltage and the angular velocity of the shaft of the motor From the above diagram block we can write the transfer function between the input of the motor, the applied voltage Va(s)and its output, the angular velocity of the shaft of the motor, Ω(s)as: H1(s) = Ω(s) Va(s)= Kt (sL+R) 1 (sJm+Kf) 1+KbKt (sL+R) 1 (sJm+Kf) =Kt (sL+R)(sJm+Kf)+KbKt B. The transfer function between the applied voltage and the torque of the motor The transfer function between the input of the motor, the applied voltage Va(s)and its output, the torque of the motor, T(s)can be written modifying the last diagram block as follows. H2(s) = T(s) Va(s)= Kt (sL+R) 1+Kb1 (sJm+Kf) Kt (sL+R) =Kt(sJm+Kf) (sL+R)(sJm+Kf)+KbKt=order 1 order 2 H2(s) = T(s) Va(s)=sKtJm+KtKf s2LJm+s(LKf+RJm)+ RKf+KbKt= sKt L+KtKf LJm s2+sLKf+RJm LJm+RKf+KbKt LJm Summary of the Work Done 36 2.1.6 Zero-order hold for DC motor and plant together H(s) = Hm(s)Hp(s) = sKt L+KtKf LJm s2+sLKf+RJm LJm+RKf+KbKt LJm ng Js2+cs = sngKt L+ngKtKf LJm s(Js+c) s2+sLKf+RJm LJm+RKf+KbKt LJm! Simplifying the notations m1=ngKt Lm2=ngKtKf LJm n1=LKf+RJm LJmn2=RKf+KbKt LJm we have H(s) = sm1+m2 s(Js+c)(s2+sn1+n2) m CASE 1 - The factor s2+sn1+n2has two complex conjugated roots H(z) = (1−z−1)Z"L−1(H(s) s)#= (1−z−1)Z"L−1(sm1+m2 s(Js+c)(s2+sn1+n2) s)#= (1−z−1)L−1(sm1+m2 s2(Js+c)(s2+sn1+n2)) sm1+m2 s2(Js+c)(s2+sn1+n2)=A s2+B s+C Js +c+Ds+E s2+sn1+n2 where A=m2 c n2 B=m1c n2−m2(J n2+c n1) c2n22 C=J3(m2J−m1c) c2J2n2−J c n1+c2 D=m1n2(Jn1−c)+ m2c n1−Jn12−n2 n22(J2n2−J c n1+c2) E=m1n2Jn12−n2−c n1−m2Jn1n12−2n2+cn2−n12 n22(J2n2−J c n1+c2) So we have H(z)=(1−z−1)Z"L−1(sm1+m2 s2(Js+c)(s2+sn1+n2))#= (1−z−1)Z"L−1(A s2+B s+C Js +c+Ds+E s2+sn1+n2)# From the table above we have Z"L−1(A s2)#=ATsz (z−1)2 Z"L−1(B s)#=Bz z−1 Z"L−1(C Js +c)#= C Jz z−e−c JTs Let us calculate Summary of the Work Done 37 Z"L−1(Ds+E s2+sn1+n2)# L−1(Ds+E s2+sn1+n2)=e−n1 2t"Dcos(bt) + E−Dn1 2 bsin(bt)# Z"e−n1 2tDcos(bt)#=Dz2−ze−n1 2Tscos(bTs) z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts where b=v u u tn2− n1 2!2 Z"e−n1 2tE−Dn1 2 bsin(bt)#= E−Dn1 2 b ze−n1 2Tssin(bTs) z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts where b=v u u tn2− n1 2!2 Z"L−1(Ds+E s2+sn1+n2)#=Dz2−ze−n1 2Tscos(bTs) z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts + E−Dn1 2 b ze−n1 2Tssin(bTs) z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts Summary of the Work Done 38 Z"L−1(Ds+E s2+sn1+n2)#= Dz2−ze−n1 2Tscos(bTs)+ E−Dn1 2 bze−n1 2Tssin(bTs) z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts Z"L−1(Ds+E s2+sn1+n2)#= Dz2−z e−n1 2Ts"Dcos(bTs)− E−Dn1 2 bsin(bTs)#! z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts where b=v u u tn2− n1 2!2 H(z)=(1−z−1)"ATsz (z−1)2+Bz z−1+ C Jz z−e−c JTs + Dz2−z e−n1 2Ts"Dcos(bTs)− E−Dn1 2 bsin(bTs)#! z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts# H(z) =  z−1  z"ATs z (z−1) 2+B z  z−1+(z−1) C J z z−e−c JTs +(z−1) Dz 2− z e−n1 2Ts"Dcos(bTs)− E−Dn1 2 bsin(bTs)#! z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts# H(z) = ATs z−1+B+(z−1) C J z−e−c JTs +(z−1) Dz− e−n1 2Ts"Dcos(bTs)− E−Dn1 2 bsin(bTs)#! z2−2ze−n1 2Tscos(bTs)+e−2n1 2Ts where b=v u u tn2− n1 2!2 Using the notations below, we can rewrite: H(z) = ATs z−1+B+(z−1) C J z+α+(z−1)Dz+β z2+γz+δ where α=−e−c JTs β=− e−n1 2Ts"Dcos(bTs)− E−Dn1 2 bsin(bTs)#! γ=−2e−n1 2Tscos(bTs) δ=e−2n1 2Ts H(z) = ATs(z+α)(z2+γz+δ) + B(z−1)(z+α)(z2+γz+δ)+ C J(z−1)2(z2+γz+δ)+(Dz+β)(z−1)2(z+α) (z−1)(z+α)(z2+γz+δ) H(z) = z4(B+C J+D)+z3(ATs+B(α+γ−1)+ C J(γ−2)+ D(α−2) + β)+ z4+z3(α+γ−1)+ z2(α(γ−1)−γ+δ)−z(α(γ−δ) + δ)−αδ +z2(ATs(α+γ)+B(α(γ−1)−γ+δ)−C J(2γ−δ−1)+ D(1−2α) + β(α−2))+ z4+z3(α+γ−1)+ z2(α(γ−1)−γ+δ)−z(α(γ−δ) + δ)−αδ Summary of the Work Done 39 +z(ATs(αγ+δ)−B(α(γ−δ)+δ)+ C J(γ−2δ)+ Dα−β(2α−1))+ z4+z3(α+γ−1)+ z2(α(γ−1)−γ+δ)−z(α(γ−δ) + δ)−αδ +ATsαδ−Bαδ+C Jδ+αβ z4+z3(α+γ−1)+ z2(α(γ−1)−γ+δ)−z(α(γ−δ) + δ)−αδ H(z) = b1z4+b2z3+b3z2+b4z+b5 z4+a1z3+a2z2+a3z+a4=order 4 order 4 where b1=B+C J+D b2=ATs+B(α+γ−1)+ C J(γ−2)+ D(α−2) + β b3=ATs(α+γ)+B(α(γ−1)−γ+δ)−C J(2γ−δ−1)+ D(1−2α) + β(α−2) b4=ATs(αγ+δ)−B(α(γ−δ)+δ)+ C J(γ−2δ)+ Dα−β(2α−1) b5=ATsαδ−Bαδ+C Jδ+αβ a1=α+γ−1 a2=α(γ−1)−γ+δ a3=−(α(γ−δ)+ δ) a4=−αδ m CASE 2 - The factor s2+sn1+n2has two real roots H(z) = (1−z−1)Z"L−1(H(s) s)#= (1−z−1)Z"L−1(sm1+m2 s(Js+c)(s−a)(s−b) s)#= (1−z−1)L−1(sm1+m2 s2(Js+c)(s−a)(s−b)) sm1+m2 s2(Js+c)(s−a)(s−b)=A s2+B s+C Js +c+D s−a+E s−b where a=−n1+pn12−4n2 2and b=−n1−pn12−4n2 2 A=m2 abc B=a(b(cm1−Jm2)+ cm2) + bcm2 a2b2c2 C=J3(Jm2−cm1) c2(aJ+c)(bJ +c) D=am1+m2 a2(a−b)(aJ+c) Summary of the Work Done 40 E=bm1+m2 b2(b−a)(bJ+c) H(z)=(1−z−1)Z"L−1(sm1+m2 s2(Js+c)(s−a)(s−b))#= (1−z−1)Z"L−1(A s2+B s+C Js +c+D s−a+E s−b)# From the Laplace table we have Z"L−1(A s2)#=ATsz (z−1)2 Z"L−1(B s)#=Bz z−1 Z"L−1(C Js +c)#= C Jz z−e−c JTs Z"L−1(D s−a)#=Dz z−eaTs Z"L−1(E s−b)#=E z z−ebTs H(z) = z−1 z"ATsz (z−1)2+Bz z−1+ C Jz z−e−c JTs +Dz z−eaTs+E z z−ebTs# H(z) = ATs z−1+B+ C J(z−1) z−e−c JTs +D(z−1) z−eaTs+E(z−1) z−ebTs H(z) = (ATs)(z+α)(z+β)(z+γ)+B(z−1)(z+α)(z+β)(z+γ)+ C J(z−1)2(z+β)(z+γ)+ (z−1)(z+α)(z+β)(z+γ) +D(z−1)2(z+α)(z+γ)+ E(z−1)2(z+α)(z+β) (z−1)(z+α)(z+β)(z+γ) where α=−e−c JTs β=−eaTs γ=−ebTs H(z) = z4(B+C J+D+E)+z3(ATs+B(α+β+γ−1)+ C J(β+γ−2))+ D(α+γ−2) + E(α+β−2)+ z4+z3(α+β+γ−1)+ z2(α(β+γ−1) + β(γ−1)−γ) + z(α(β(γ−1)−γ)−βγ)−αβγ +z2(ATs(α+β+γ)+B(α(β+γ−1)+β(γ−1)−γ)+ C J(β(γ−2)−2γ+1))+ D(α(γ−2)−2γ+1) + E(α(β−2)−2β+1)+ z4+z3(α+β+γ−1)+ z2(α(β+γ−1) + β(γ−1)−γ) + z(α(β(γ−1)−γ)−βγ)−αβγ +z(ATs(α(β+γ)+βγ)+B(α(β(γ−1)−γ)−βγ)+ C J(γ−β(2γ−1)))+ D(γ−α(2γ−1))+E(β−α(2β−1))+ z4+z3(α+β+γ−1)+ z2(α(β+γ−1) + β(γ−1)−γ) + z(α(β(γ−1)−γ)−βγ)−αβγ Summary of the Work Done 41 +ATsαβγ−Bαβγ+C Jβγ+Dαγ+Eαβ z4+z3(α+β+γ−1)+ z2(α(β+γ−1) + β(γ−1)−γ) + z(α(β(γ−1)−γ)−βγ)−αβγ H(z) = b1z4+b2z3+b3z2+b4z+b5 z4+a1z3+a2z2+a3z+a4=order 4 order 4 where b1=B+C J+D+E b2=ATs+B(α+β+γ−1)+ C J(β+γ−2)+ D(α+γ−2) + E(α+β−2) b3=ATs(α+β+γ)+B(α(β+γ−1)+β(γ−1)−γ)+ C J(β(γ−2)−2γ+1)+D(α(γ−2)−2γ+1)+ +E(α(β−2)−2β+1) b4= (ATs(α(β+γ)+βγ)+B(α(β(γ−1)−γ)−βγ)+ C J(γ−β(2γ−1))+ D(γ−α(2γ−1))+ +E(β−α(2β−1)) b5=ATsαβγ−Bαβγ+C Jβγ+Dαγ+Eαβ a1=α+β+γ−1 a2=α(β+γ−1)+ β(γ−1)−γ a3=α(β(γ−1)−γ)−βγ a4=−αβγ m CASE 3 - The motor is considered as being only a constant kM Hm(s) = sKt L+KtKf LJm s2+sLKf+RJm LJm+RKf+KbKt LJm =kM And therefore the plant becomes: H(s) = ngkM s(Js+c)