Tracking Control: A Differential Geometric Approach

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Tracking Control: A Differential Geometric Approach Torsten Scholt y, Britta Riege z y University of Duisburg, Dep of Measurement and Control D 4748 Duisburg, Germany Phone: ++49(3) 379 33 Fax: ++49

1 Tracking Control: A Differential Geometric Approach Torsten Scholt y, Britta Riege z y University of Duisburg, Dep of Measurement and Control D 4748 Duisburg, Germany Phone: ++49(3) Fax: ++49 (3) scholt@uni-duisburgde z DaimlerChrysler AG D 7546 Stuttgart, Germany Phone: ++49(711) Fax: ++49 (711) 17-5 BrittaRiege@DaimlerChryslercom Keywords: model-based control, differential geometry, multi-variable systems, tracking the output, nonlinear control Abstract Tracking a trajectory is a demanding task when flexible robots are being considered To tackle this problem this paper introduces a model-based differential geometric approach In the design process a control structure will be proposed that allows the choice of an arbitrary dynamic response of the controlled system within certain limits The result is a complex feedback structure with inherent stability 1 Introduction Many industrial applications involve the control of multibody systems, eg the control of manipulators or robots Usually these systems are assumed to be rigid to avoid too complex control structures In many cases this assumption does not hold A lot of applications have to deal with very slender and/or long structures that deform under the influence of gravity or dynamic processes On the other hand, the restriction to lightweight structures permits the utilization of smaller actuators which consume less energy To outline the process of designing such a control scheme this paper will introduce a model-based approach to this problem which shows inherent stability First, the creation of the needed analytic model and its numerical approximation will be described To reduce the complexity of the problem a system with two kinematic degrees of freedom will be considered The design of the control scheme itself can be found in the third section In the last section some simulated results are presented Analytical Model The surveyed problem is the control of the joint angles of a robot with flexible links as described in [1, 7, ] The robot is designed in a way such that the revolute joint s elasticities can be omitted Due to their slender design, the links can be modelled as Euler-Bernoulli-beams wich can only be deformed in one dimension Nonlinear deformations and internal friction are neglected To minimize friction between the table s surface and the flexible arm air suspension was employed Two electric drives in the joints axes actuate the system A picture of the testbed system is given in figure 1 Clearly the airbearings and the electric drives can seen Figure 1: Flexible Manipulator (testbed) Assuming that the robot has the structure as shown in figure, one can choose the generalized coordinates to be

2 Y 1 Y X 1 w 1(x 1) Gage 1 Y u Y 1 Gage ϑ Y Y 3 w (x ) X X 1 X X 3 Due to the complexity of the highly nonlinear analytical model it its hardly possible to make use of such a model in a real time application With the help of a series expansion [1] all nonlinearities of the model are represented by polynomials of the maximum degree of All higher order terms are omitted The computational tool used was introduced in [6] ϑ 1 X This procedure yields a model of the generic form: 1 u 1 Figure : Flexible Manipulator (schematic) _x = A 1 x + A x Ω x + B u + B 1 x Ω u + B x () Ω u y = Cx; x R n ; u R m ; y R p : (5) q =[# 1 # ffi1;1 ffi;1 ] T : (1) # 1 and # describe the angles of the links with respect to the joints Variables ffi 1;1 and ffi ;1 refer to the respective elastic degrees of freedom of the system These describe the first vibrational modes of the flexible links by assuming the modes of a clamped-free beam [3] The joint angles are measured with the help of pulse generators and the deformations by using strain gages attached to the links (see Fig ) The local dynamics are set up in the coordinate systems (X i ;Y i );i = 1; These are transformed into a general description ( X ; Y ) by utilizing corresponding matrices For the surveyed testbed the detailed procedure of creating the analytic model can be found in [7] The result is a system of four second order differential equations: [H(q) +J] q + h c (q; _q) +K e q + D _q = u = K m U: () The entries of the inertia matrices H(q) and J and the vector of Coriolis and centrifugal forces h c (q; _q) can be determinded from the robot s physical characteristics, ie the equations of motion and the corresponding values for the mass and inertia of the robot s parts Experiments were performed to fill the entries of the damping matrix D, the stiffness matrix K e and the gain matrix K m which computes the torque applied to the system from the voltage applied to the electric drives The model data was validated in [7] After rearranging the equations () one obtains the vector of accelerations: q(t) =(H + J) 1 [h c K e q D _q + K m U] : (3) Defining a state vector x = [q; _q] T, the state-space model can be determined as: _x = _q (H + J) 1 (h c + K e q + D _q) (H + J) 1 U K m y = c(x) ; x R n ; u R m ; y R p : (4) + 3 Controller Design The controller design is based upon the exact linearization in conjunction with a stabilizing state and output feedback A model consisting of the original system (4) or its nonlinear approximation and a linear reference model is exactly linearized In addition the error between the two is fed back By doing this, the error can be reduced to zero and the entire system is stabilized[9] The problem is referred to in [5] as tracking the output First, a reference model is added to the original nonlinear model of the plant Definition 1 [8] Given are a system (4) and a dynamic reference model: ff x m (t) = Ax m (t) +Bu m (t) μy(t) = c T x m (t); x m R m : (6) With x m = x m () The tracking problem for u m 6=;t t, where the error signal e(t) between the system s output y(t) and the reference model s output μy(t) satisfies the following equations: lim e(t) = lim [μy(t) y(t)] = ; (7) t!1 t!1 and is called The Problem of Tracking a Model s Output for MIMO systems The model now under consideration is depicted in figure 3 The term QLC refers to the nonlinear quadratic approximation which possesses a linear control term Yet LS refers to a linear state-space model Consideration has to be made as to whether a solution to the problem stated in definition 1 actually exists A generic decoupled and exactly linearized 1 The tensor product or Kronecker product of two vectors is defined as follows: For two vectors a R m and b R n the tensor product yields a Ω b = a 1 b T ;a b T ;:::;amb T Λ T R nm For one vector a R m the multiple tensor product yields (i) a = a Ω a Ω Ωa z } i times

3 X QLC \ with i = 1; ;:::; m; z(t) = z 1 1 ;:::;z1 d1 ;z 1 ;:::;z d ;:::;zm dmλ T : If in the proximity of z = t(x ) the m m matrix D(x) (11) has full rank, ie X P LS rank D(z) =rank D(x)j x=t 1 (z) = m ; (14) Figure 3: Supplemental model multi-variable system has a new input w which shows linear input/output behaviour: _x(t) = a(x) +B(x)r(x) +B(x)V(x)w(t) y(t) = c(x) Where r(x) and V(x) are determined by r(x) = D(x)f (x) ff : (8) V(x) = D(x) 1 : (9) For the surveyed case the matrix D(x) and vector the f (x) are calculated by: and f (x) = f1 (x) f (x) = L a c 1(x) L a c (x) Lb1 L a c 1 (x) L b L a c 1 (x) D(x) = L b1 L a c (x) L b L a c (x) (1) ; (11) with respect to the output vector y = c(x) which refers to the difference between the nonlinear and the linear reference system System (8) has a relative degree of d, so that there exists a local coordinate transformation [8] at x : t i r (x) =Lr1c a i (x) ; i =1; ;:::;m r =1; ;:::;d i : (1) For the transformed system the equations defining the system s dynamics can be rewritten as: _z i 1 (t) = zi _z i di1 (t) = zi di _z i di (t) = f i(z) + P m j=1 d ij(z)u j (t) y i (t) = z i 1 (t) h The Lie derivatives are defined as Lf The multiple Lie derivative yields: L k k1 f fl(x) = fl(x) 9 >= >; (13) i T f (x) T f (x) the following equation can be solved for u: This yields w(t) =f (z) +D(z)u(t): (15) u(t) =D 1 (z)[f (z) +w(t)] : (16) For the setup specific to this application the nonlinear quadratic approximation is enlarged by two second order systems with an amplification of one to serve as reference systems To begin with, an expanded state vector has to be defined as: x m (t) =[x 9 ;x 1 ;x 11 ;x 1 ] T and ~x(t) = x T ; x T mλ T : This yields a new state-model: _~x(t) = (17) a(x) x 1 (T a+t b ) T x at 1 (t) b 1 T (x at 9 (t) b u m1 (t)) x 1 (Tc+T d) TcT x 1 (t) d 1 TcT (x 11 (t) d u m (t)) mx i=1 ~b i (x)u i (t) y(t) = ~c(~x(t)) = [μy(t) y(t)] (18) x1 (t) x 1 (t) = x 1 (t) x (t) ~x(t) R 1 ; u(t); y(t) R : Vector ~a(x; x m ; u m ) contains the plant eigendynamics and the reference system s ODE s Vectors b i (x) have to be enlarged to b ~ i (x) R 1 The desired trajectory for the angles # 1 and # is fed to the controller by u m (t) y(t) is referred to as the tracking error Furthermore the following equation has to hold: ÿ(t) =w(t) : (19) Substitute w in Equation (16) and solve for u(t) to gain the desired feedback terms For the original coordinates this yields: u(t) =D 1 (x; x m ; u m )[f (x; x m ; u m )+] : () The constants of the second order systems can be chosen within certain limits They should reflect the plant s capabilities In order to achieve acceptable performance, the 3 7 5

4 time constants of the reference model should be chosen to reflect the plant s maximum capabilities, ie as fast as possible This can be estimated eg by using step responses of the plant with appropriate step heights The constants in Equation (17) are arranged in a way that they form the coefficients of the characteristic polynomials In [4] the asypmtotic stabilty was proven for bilinear systems There a Lypunov function is constructed with which it is shown that the error vector y(t) asymptotically tends to zero Remark: It should be mentioned that the added model can be chosen arbitrarily Even a nonlinear behaviour can be imposed on the system if so desired Due to the fact that the plant model has a relative degree of d = f; g the output of the reference model and its derivative have to be known This scheme does not guarantee a stable tracking of the desired trajectory since the error between the plant or its nonlinear model and the linear system used to create the controller might increase dramatically soon after the controller is engaged This drawback can be overcome by feeding back the error between the nonlinear model and the linear reference model on the position and the speed level of the system If the new output y(t) and its derivative _y(t) are fed back, the error between the nonlinear model (or the original plant respectively) can be forced to zero The error s dynamic behavior can be chosen within certain limits using pole-placement ÿ Λ (t) = y(t) fls1 _y 1 (t) +fl s y 1 (t) fl s3 _y (t) +fl s4 y (t) (1) Constants fl si ; i =1; ; 3; 4 are the coefficients of the characteristic polynomials of the error dynamics The faster the error dynamics are chosen, the higher is the control effort to force the error to zero This yields a setup as shown in fig 4 feedback X ALC \ plant in eq (4) using the quadratic approximation to generate the feedback law Some simulated system responses are included Experiments are in progress 41 System Dynamics This section presents some simulated responses to given trajectories Figure 5 show a simulated response for a step of the input of the linear reference system u m; The second input u m;1 was set to zero It is clear that the second beam s motion still has an impact on the behavior of the first beam, due to the mechanic coupling of the links However, due to the controller, the deviation from the desired trajectory is negligible The second order system response can be seen very well in figure 5 Figures 6 and 7 show responses of the #i/[rad]! t=[sec]! #1 # Figure 5: Response of # 1 for a unit step of # at t = system to sine-shaped trajectories The signal consists of two superimposed signals One is a pulse sequence and the other is a sine signal with! = rad/sec and a = :1 rad Figure 8 shows the control effort necessary to force the system to the desired trajectory The values for u i (t); i =1; range between ±1 V Table 1 lists the values chosen for the 4 X P [ LS [, X P P \ #1/[rad]! desired value actual value 4 Results Figure 4: Determination of error dynamics To be able to evaluate the controller scheme s capabilities some tests were performed on the nonlinear model of the t=[sec]! Figure 6: Response of # 1 to a sine oscillation simulated results presented in the section

5 4 Error Dynamics #/[rad]! t=[sec]! desired value actual value Figure 7: Response of # to a sine oscillation To demonstrate the dynamics exposed by the error feedback one of the plant s integrators was given an offset of ffi# 1 = : rad In Fig 9 it can be seen that this error tends to zero at the given rate y(t)/[rad]! t=[sec]! Figure 9: Error Dynamics 1 5 ui/[v]! 5 u1 u 43 Implementation The previous theoretical work has then been implemented on a fast computer involving a DSP TMS3C31 from Texas Instruments 5 Conclusion t=[sec]! Figure 8: Control effort for u i (t);i =1; Name Value Unit T a sec T b sec T c 47 sec T d 47 sec fl fl fl fl Table 1: Coefficients This paper proposes a design procedure for a tracking controller Starting point is a flexible structure with its complex highly nonlinear mathematical representation A nonlinear approximation is generated which is used to create the state and output feedback This feedback decouples and exactly linearizes a substitute model which consists of the approximated model and a reference model The difference between these two models is supposed to vanish Due to uncertainties this is a rather improbable assumption To work around this problem the error is fed back and a certain dynamic behavior is imposed upon the error This guarantees the stable tracking of a trajectory References [1] Mingli Bai Modeling, simulation and control of flexible robots [in german], volume 7 of VDI Fortschrittberichte Reihe 8 VDI Verlag, Düsseldorf, 1998 [] C de Wit, B Siciliano, and G Bastin Theory of Robot Control Springer-Verlag, London, 1996 [3] AR Fraser and RW Daniel Perturbation Techniques for Flexible Manipulators Kluwer Academic Publishers, Boston, 1991

6 [4] L Guo Bilinear System Control for Hydraulic Systems [in german], volume 45 of VDI Fortschritt-Berichte Reihe 8 VDI, 1991 [5] A Isidori Nonlinear Control Systems Springer Verlag, Berlin, 1995 [6] Markus Lemmen, Torsten Wey, and Mohieddine Jelali NSAS a computer-algebra-pack for analysis and synthesis of nonlinear systems Technical Report /95, Department of Measurement and Control, University of Duisburg, 1995 [7] Britta Riege and M A Arteaga Pérez Experimental modeling of a twolink flexible manipulator In IFAC Conference on System, Structure, and Control, pages , Nantes, France, 1998 [8] H Schwarz Nichtlineare Regelungsysteme - Systemtheoretische Grundlagen Oldenbourg, München, 1991 [9] Markus Senger Algebraic Formulation and Solution of essential systemtheoretic Issues [in german] VDI Fortschritt-Berichte, volume 77 of Reihe 8 VDI Verlag, Düsseldorf, 1999 [1] W Vetter Matrix calculus operations and Taylor expansions SIAM Review, 15:35369, 1973

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