1 Introduction The Gyroscopic Pendulum is shown schematically in Figure 1. It is a physical pendulum with a symmetric mass (disk) attached to the end

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1 Nonlinear Control of the Gyroscopic Pendulum Mark W. Spong Coordinated Science Laboratory University of Illinois at Urbana{Champaign 18 W. Main Street Urbana, Ill. 181 Peter Corke Division of Manufacturing Science & Technology CSIRO Kenmore, Australia, 49 Rogelio Lozano Heudiasyc UMR CNRS 599 Universite de Technologie de Compiegne BP 59 5 Compiegne cedex I watched it for some minutes, somewhat in fear, but more in wonder.... it seemed massy and heavy, tapering from the edge into a solid and broad structure above. It was appended to a weighty rod of brass, and the whole hissed as it swung through the air. - Edgar Allen Poe, \The Pit and the Pendulum" (184) [8]. Abstract In this paper we consider the nonlinear control of the Gyroscopic Pendulum, a novel mechanical system consisting of a physical pendulum with a rotating mass at the end. We study the problem of swingup and balance of the pendulum, where the control input is the coupling torque produced by the angular acceleration of the end-mass. We show that the system is locally feedback linearizable by a local dieomorphism in state space and nonlinear feedback. We compare the feedback linearization control with a linear pole-placement control for the problem of balancing the pendulum about the inverted position. For the swingup problem we compare two dierent energy based controllers; one based on passivity of the Lagrangian dynamics of the system and the other based on collocated partial feedback linearization, and passivity of the resulting zero dynamics. A hybrid/switching control strategy is used to switch between the swingup and the balance control. Both simulations and experimental results are presented. This research was partially supported by the National Science Foundation under grants CMS-971 and ECS and by a CNRS/UIUC Collaborative Research Agreement

2 1 Introduction The Gyroscopic Pendulum is shown schematically in Figure 1. It is a physical pendulum with a symmetric mass (disk) attached to the end which is free to spin about an axis parallel to the axis of rotation of the pendulum. It is thus a two-degree-of-freedom system. The disk is actuated by a DC-motor and the coupling torque generated by the angular acceleration of the disk can be used to actively control the system. The control problems for the Gyroscopic Pendulum are reminiscent of those for the Acrobot [, 9], but are distinct enough to warrant a separate investigation. Because of the symmetric mass distribution of the disk, precise analytical statements are more readily obtainable for the Gyroscopic Pendulum than for the Acrobot[14] or Pendubot[15]. From a pedagogical standpoint, the Gyroscopic Pendulum is one of the simplest nonlinear systems that can be used to illustrate advanced control designs based on recently developed geometric methods. y q 1 x l c1 l 1 m 1, I 1 m, I q Figure 1: Coordinate conventions for the gyroscopic pendulum. We consider the problem of swinging the pendulum up and balancing it about the inverted position. This is accomplished with a supervisory hybrid/switching control strategy which uses a passivity based nonlinear controller for swingup and a local controller for balance. The nonlinear swingup controllers are designed so that trajectories are guaranteed to eventually enter the basin of attraction of the balance controller, which is in turn designed to asymptotically stabilize the inverted equilibrium state. The supervisor determines when to switch between the swingup and balance controllers based on an estimate of the basin of attraction of the balance controller. The switch includes hysteresis to avoid chattering between the two controllers. For the design of the balance controller we consider two approaches; a linear pole placement design

3 based on the linearized approximation of the nonlinear dynamics about the inverted equilibrium, and a full state nonlinear feedback linearizing controller. The fact that the dynamics of the gyroscopic pendulum are feedback linearizable, the proof of which is a new contribution of the present paper, is interesting in its own right. Other underactuated nonlinear systems of this type, such as the Acrobot, Pendubot, and cart-pole system do not satisfy the conditions for feedback linearization without some simplifying approximations [9, 11]. For the design of the swingup controller we compare two alternatives. The rst is based on the notion of collocated partial feedback linearization of underactuated systems from [1]. Passivity from acceleration of the disk to velocity of the pendulum of the resulting zero dynamics is used to design a Lyapunov function that is positive denite in the pendulum energy and the disk kinetic energy. The control then drives the pendulum energy and disk velocity to zero. The second swingup controller exploits passivity of the complete system dynamics from input torque to the velocity of the disk and does not use partial feedback linearization. A Lyapunov function, positive denite in the total energy of the system, is used to determine a control that also drives the pendulum energy and velocity of the disk to zero. In both cases, the proof of convergence relies on LaSalle's Invariance Principle. We compare the performance of the swingup and balance controllers in both simulation and experiment. Dynamics An easy way to derive the dynamic equations of the Gyroscopic Pendulum is to notice that the system may be modelled as a two-degree-of-freedom robot, where the pendulum forms the rst link and the rotating disk forms the second link. We assume that the center of mass of the disk is coincident with its axis of rotation and we measure the angle of the pendulum clockwise from the vertical. Under these assumptions the equations of motion can be taken from any standard text, for example [1], as q 1 + d 1 q + (q 1 ) = (1) d 1 q 1 + d q = () where q 1 is the pendulum angle, q is the disk angle, is the motor torque input and = m 1` c1 + m ` d 1 = d 1 = d = I 1 + I 1 + I () (q 1 ) =?(m 1`c1 + m `1)g sin(q 1 ) = mg sin(q 1 ) (5) with the various parameters as shown in Figure 1 and m := m 1`c1 + m `1. The system can be expressed compactly as (4) Dq + g(q) = u ()

4 where D = " d 1 d 1 d # ; q = " q 1 q # ; g(q) = " (q 1 ) # ; u = " # (7) The total energy of the system () is given by E = 1 _qt D _q + mg(cos q 1? 1): (8) From (8) and the system equation () it follows that _E = _q T Dq? mg sin(q 1 ) _q 1 = _q (9) which is just the familiar passivity property of Lagrangian mechanical systems from torque to velocity [1]. The gyroscopic pendulum with zero control input ( = ) has an unstable equilibrium at (q 1 ; _q 1 ; _q ) = (; ; ) with energy E(; ; ) =, and a stable equilibrium at (q 1 ; _q 1 ; _q ) = (; ; ) with energy E(; ; ) =? mg. Note that the disk position q can be arbitrary since the energy does not depend on q, i.e. q is a cyclic variable. Hence these points are not isolated equilibrium points in the four dimensional state space of the system. When _q =, Equation (8) reduces to E = 1 _q 1 + mg(cos q 1? 1) (1) which denes the energy of the pendulum alone. A similar calculation as above shows that _E = _q 1 ( q 1? mg sin(q 1 )) =? _q 1 d 1 q (11) from (), which shows that the gyroscopic pendulum also denes a passive mapping from (?d 1 times) the disk acceleration to pendulum velocity. These two passivity relationships will be used later in the design of the swingup controllers. The set dened by E = is the homoclinic orbit of the pendulum in the (q 1 ; _q 1 ) phase plane, which is a two-dimensional subspace of the four dimensional state space of the complete system. We will refer to the set dened by E = as the homoclinic orbit even when referring to the full order system. Although this is an abuse of terminology we assume that no confusion will arise in this case..1 Reduced Order Model Since the disk angular position, q, is arbitrary in the inverted equilibrium, it makes sense to ignore it in the sequel and dene a reduced order model with states x 1 = q 1, x = _q 1, and x = _q. Our 4

5 goal will thus be to control only the pendulum position, pendulum velocity and disk velocity and leave the disk position unspecied. In terms of the state vector x = (x 1 ; x ; x ) T we can write the system () as _x 1 = x (1) _x =? d detd (x 1)? d 1 _x = detd (1) d 1 detd (x 1) + detd (14) where detd = d? d 1 d 1 >. This can be written as _x = f(x) + g(x) (15) in terms of the vector elds f(x) = 4? d x detd (x 1) d 1 detd (x 1) 7 5 ; g(x) = 4? d 1 detd detd 7 5 (1) We will assume that the pendulum angle is computed modulo so that the system state trajectory evolves on the manifold S 1 <, where S 1 is the unit circle. Therefore, boundedness of x 1 = q 1 is not an issue and the state will be bounded whenever the velocities x and x are bounded. We will use this reduced order model (15) in the remainder of the paper. Feedback Linearization (FL) In this section we show that the Gyroscopic Pendulum is locally feedback linearizable in a neighborhood of the inverted equilibrium, x =, using a nonlinear change of coordinates and nonlinear feedback. To show feedback linearizability we will follow the approach of [5] and seek an output function with respect to which the system relative degree equals the dimension of the state space, in this case, three. With the system dened as in (15), we dene an output equation y = h(x) = x + d 1 x : (17) The function h(x) is the rst component of the generalized momentum of the system (). The derivative of the output function y satises _x = g(x) = L f h + L g h (18) 5

6 where L f h and L g h denote the Lie derivatives of h with respect to f and g, respectively. In local coordinates, L f h and L g h are given as and h L f h = h L g h = i ; ; d 1 4 i ; ; d 1 4? d x detd (x 1) d 1 detd (x 1)? d 1 detd detd 7 5 =?(x 1 ) = mg sin(x 1 ) (19) 7 5 = : () Continuing in this fashion it is straightforward to compute the higher derivates of y as where y = L f h + L gl f = L f h ; L gl f h = y () = L f h + L gl f h ; L gl f h = (1) L f h = mg cos(x 1)x () L f h =? mg sin(x 1)x + ( mg) d detd cos(x 1) sin(x 1 ) () L g L f h =? d 1 detd mg cos(x 1) (4) Thus the system has a well dened relative degree of three with respect to the output y = x + d 1 x since L g L f h is nonzero in the region?= < q 1 < =. We can therefore dene new state variables 1 ; : : : as follows 1 = h(x) = x + d 1 x = L f h(x) = mg sin(x 1 ) (5) = L f h(x) = mg cos(x 1)x It is easy to see that this state space transformation denes a local dieomorphism T : S 1 <! S 1 <. Indeed, the Jacobian of this transformation at the origin is = 4 d 1 mg mg 7 5 () which clearly has rank. The inverse of the dieomorphism T can be explicitly computed as x 1 = sin?1 ( mg ) (7)

7 x = mg cos(sin?1 ( mg )) (8) x = 1 d 1 1? d 1 mg cos(sin?1 ( mg )) (9) In terms of these new state variables, the system becomes _ 1 = _ = _ = L f h + LgL f h () (1) () We can now dene the feedback transformation = 1? L g L f h(x)[u L f h(x)] () so that the system becomes the linear chain of integrators _ 1 = _ = _ = u (4) (5) () The new control variable u can then be taken as u =?k 1 1? k? k (7) to place the closed loop poles (in the -coordinates) arbitrarily. The state transformation and hence the feedback linearization control strategy is valid as long as jx 1 j < =. We have shown that the gyroscopic pendulum is feedback linearizable in the region jx 1 j < =. Thus, as long as the pendulum angle, q 1, is above the horizontal, the feedback linearizing control strategy can balance it. In practice, of course, limitations on the available input torque will reduce the region in which the pendulum can be stabilized using this control. We leave it to the reader to verify that the region jx 1 j < = is the largest possible region in which the system can be feedback linearizable. This can be shown by computing the involutivity and rank conditions that are necessary and sucient for feedback linearizability [5]. It turns out that involutivity of the set fg; ad f gg for the third order system holds globally but the rank condition, i.e. rank[g; ad f g; ad f g] = fails at jx 1 j = =. We also remark that the full fourth order system, including the disk angular position, is also locally feedback linearizable in the above fashion using the output equation y = q 1 + d 1 q 7

8 We omit the details..1 Approximate Linearization (AL) For later comparison in simulation and experiment we also present here the linear approximation about the origin of the nonlinear system (15). Since the Lagrangian equations of motion are linear apart from the gravity term, linearizing about x 1 = gives the controllable linear system _x = Ax + B (8) where A = 4 1 mgd detd? mgd 1 detd 7 5 ; B = 4?d 1 detd detd 7 5 (9) In the later section on simulation and experimental results we will compare the performance of the linear approximation and the exact feedback linearization for balance control using parameter values measured for our laboratory apparatus. 4 Partial Feedback Linearization Based Control (PFLBC) Since the feedback linearization control derived in the previous section is only valid locally around the origin, it cannot be used to swing the pendulum up from the vertically downward position, x 1 =. Indeed, the initial angle of the pendulum is restricted to the interval (?=; +=), since the coordinate transformation (5) loses rank at x 1 = =. In this section we derive a swingup control based on the notion of Collocated Partial Feedback Linearization from [1] and passivity of the resulting zero dynamics. Interestingly enough, this swingup controller cannot be used to balance the pendulum, as we shall see, but it can swing the pendulum up to a neighborhood of the equilibrium starting from (almost) arbitrary initial conditions. This is the reason that our complete control strategy must switch between the swingup and the balance control. The partial feedback linearization strategy proceeds, as in the previous section, by choosing an output function and its derivatives as new state variables. In this case we take the output equation as y = h(x) = x (4) which is the velocity of the disk, and compute _y = _x = L f h + Lgh (41) 8

9 where, from (14), L f h = d 1 detd (x 1) (4) L g h = detd Since L g h is nonzero globally, the system has relative degree one with respect to the output y, resulting in two dimensional zero dynamics which we compute below. Taking as state variables (4) = h(x) = x 1 = x 1 = x (44) (45) (4) and dening the control input = 1 L g h (u? L f h) (47) we have, after some algebra, _ = u _ 1 = (49) _ =? 1 ( 1 )? d 1 u (5) We note that, with u =, the system _ 1 = (51) _ =? 1 ( 1 ) (5) denes the dynamics of the undamped pendulum. We shall dene the additional control input u both to stabilize the disk angular velocity,, and to render the homoclinic orbit of the pendulum attractive. To accomplish this we rst set E = 1 Z 1 + (48) ()d; (5) the energy of the free pendulum corresponding to u = and compute E_ =?d 1 u (54) which is the passivity property of the pendulum with input?d 1 u and output that we computed previously. We then choose as a Lyapunov function V V ( 1 ; ; ) = 1 k ee + 1 k v (55) 9

10 It is easy to show that V is positive denite and that _V =?(d 1 k e E? k v )u (5) If we therefore choose the control input u according to u = k u (d 1 k e E? k v ) (57) we have that _V =?k u u (58) We use LaSalle's Invariance Principle to compute the invariant set to which all trajectories converge. For completeness, we recall LaSalle's Theorem from []. Theorem [LaSalle] Let D be any region of the state space S 1 < containing the origin and let D be a compact subset of D that is positively invariant with respect to the system considered. Let V : D! < be a continuously dierentiable function such that _ V (x) in. Let E be the set of all points in where _ V (x) =. Let M be the largest invariant set in E. Then every solution starting in approaches M as t! 1. We now claim that with the control law (57) all solutions of the gyroscopic pendulum converge to the set M := C 1 [ C where and C 1 = f( 1 ; ; ) j E ( 1 ; ) = and = g: (59) C = f( 1 ; ; ) j 1 = or and = g: () Proof: Let x denote ( 1 ; ; ) T for convenience and take D = S < 1. Since V (x) dened by (55) is a Lyapunov function, we may take = V?1 (x()). is positively invariant since _ V. Setting _V identically zero in (58) yields u which implies, from Equation (48), that is constant and, from Equation (54), that E is constant. Now, from Equation (57), it follows that E Constant: (1) Dierentiating (1), with E constant, yields E : () 1

11 Thus, either E or ( 1 ) or both. It follows from (57) that, if E then. If E =, then, since ( 1 ), we must have and so from u. In either case, we have that. The above discussion shows that that the disk angular velocity, converges to zero globally. The pendulum trajectory converges to the set where the energy E =, which is the homoclinic orbit or to ( 1 ; ) = (; ), in which case the energy satises E (; ; ) =? mg. This completes the proof of the claim. Remark: The open loop equilibrium point, (; 1 ; ) = (; ; ), is also an equilibrium of the closed loop system so, if the system starts out in this state, it cannot swing up. Also, LaSalle's Theorem cannot rule out the possibility that the pendulum comes to rest at the downward position for initial conditions away from this equilibrium point, since it is in the invariant set. We can analyze this situation further to discover the behavior of the system near this equilibrium point. Combining the equations (48,49,5,5,57) yields, after some algebra _ =?k v +?d 1 k e mg + d 1 k e mg cos( 1 ) + 1 d 1 k e () _ 1 = _ = d 1 k v + d 1 k e mg + mg sin( 1 )? 1 d 1 k e? d 1 (4) k e mg cos( 1 ) (5) Computing the Jacobian A of this nonlinear system at (; 1 ; ) = (; ; ) yields, A = 4?k v?d 1 k e mg 1 d 1 k v? mg d 1 k e mg Using the parameters from Table 1 gives A = 4?k v?1:59 1?5 k e 1 :7k v?78:58 1:8 1?7 k e 7 5 () 7 5 (7) We can compute the root locus of eigenvalues of A in terms of k v for given k e. For a range of k e up to about 4: the root locus is qualitatively as shown in Figure, while for larger values of k e, the root locus has the shape shown in Figure. We see that, for positive values of k v, the closed loop system always has one stable and two unstable eigenvalues. Thus the linearization of the closed loop system about (; 1 ; ) = (; ; ) will have a one dimensional stable subspace and a two dimensional unstable subspace. Hence, the nonlinear system will have a one dimensional stable invariant manifold and a two dimensional unstable invariant manifold tangent to these stable and unstable subspaces, respectively [4]. For initial conditions on the stable invariant manifold, the trajectory will converge to the lower equilibrium. However, since this set of initial conditions has measure zero, so that for \almost all" initial conditions the 11

12 1 8 kv =1; ke= 5e5 ku=1 4 Imag Axis Real Axis Figure : Locus of eigenvalues of A in terms of k v for k e 4: kv=1; ke=5e ku=1 4 Imag Axis Real Axis Figure : Locus of eigenvalues of A in terms of k v for k e > 4: system trajectory will converge to the homoclinic orbit of the pendulum. It also means that convergence to the lower equilibrium conguration will not be seen in practice due to noise, parameter uncertainty, computational round-o, etc. 5 Passivity Based Control (PBC) In this section we present an alternative swingup control strategy that is not based on partial feedback linearization. Instead, the control design in this section exploits the passivity property of the complete Lagrangian dynamics. Similar control strategies have been used to control other underactuated mechanical systems in [7] for the cart and pole system and in [] for the Pendubot. Although the control design in this section is distinct from that in the previous section, the strategy is essentially the same, i.e. to guarantee that the disk angular velocity and the total energy both 1

13 converge to zero. Therefore, consider again the system (15) and let E represent the total energy as dened in (8). Dene the Lyapunov function candidate V = 1 k EE + 1 k vx (8) where k E and k v are positive constants. Dierentiating V and using (9) we obtain _V = k E Ex + k v x _x (9) Substituting _x from the state equations (14) we obtain _V = x [(k E E + k v det D ) + k vd 1 det D mg sin(x 1)] (7) We therefore propose the control input = k E E + k v det D?1? k vd 1 det D mg sin(x 1)? k q x (71) so that _V =?k q x (7) In view of (8), E? mg: Therefore, in order to avoid division by zero in (71), k E and k v should be chosen so that the following inequality holds for some " > k E E + k v det D?k E mg + k v det D " > (7) From (8) and (7) we conclude that V V (). This implies that the energy E remains bounded and so will x and x. Thus the closed-loop state x is bounded and we can apply LaSalle Invariance Principle. Setting V _ it follows from (7) that x x = we have from (14) and hence that E is constant from (8). When d 1 =? mg sin(x 1 ) The control input in (71), when x = ; can also be rewritten as (E k E det D k v d 1 + d 1 ) =? mg sin(x 1 ) (74) Subtracting the two equations above it follows that E = : From (71) it follows that if = then sin(x 1 ) = which implies that x 1 = or. If on the other hand E = then the system moves 1

14 in the homoclinic orbit. It follows that the invariant set M is identical to that given by (59)-(). The remaining details are omitted. Remarks: We leave it to the reader to verify as we did for the controller of the previous section that almost all trajectories, i.e. except those starting on a one-dimensional invariant manifold of the downward equilibrium conguration, converge to the homoclinic orbit of the pendulum with zero disk velocity. We also note that another way to prevent the pendulum from converging to the downward equilibrium is to further restrict the initial conditions as follows. Recall that the pendulum's energy is E(; ; ) =? mg at the stable equilibrium point. Thus, if the initial state is constrained such that V () < 1 k E( mg) (75) then, in view of (8) and given that V V (), the energy E will never reach the value? mg which characterizes the stable equilibrium point for x =. The inequality (75) imposes upperbounds on jx j and jx j. Hybrid/Switching Controller The nal controller is a hybrid control to switch between the swingup controller (either PBC or PFLBC) and the balancing controller (either the FL or AL). The balance controller must only be activated when the system state is within its basin of attraction. We will dene the switch in terms of a weighted norm of the state = s x1 x 1 max + x x max + x x max (7) In order to avoid chattering between the two controllers we introduce hysterisis if < 1 if > switch to linear controller switch to swingup controller The determination of of x i;max and i will be discussed in the next section using the parameter values of the actual laboratory hardware. Since the stable equilibrium conguration of the open loop system is an equilibrium point for both of the non-linear swingup controllers, it is required that the arm be displaced in order to start the swingup motion. This displacement could be accomplished by a third `push o' controller which is active at this point in state space. In fact we were able to use the balancing controller for this purpose, since it is unstable at the downward equilibrium point. Taking the angle x 1 modulo, instead of modulo, maps both equilibrium points to x 1 = where the linear controller is active. When the error is suciently large the swingup controller is activated. 14

15 7 Simulation and Experimental Results In this section we present the results of Matlab and Simulink simulations and compare them with experimenal results obtained on a hardware setup in the College of Engineering Control Systems Laboratory at the University of Illinois at Urbana-Champaign. The model parameters used in the simulations were the parameters from the actual system. These parameters are shown in Table 1. Parameter Value Units l 1.15 m l c1. m m 1. kg m. kg I ? kg:m I 1? kg:m 4:8 1? kg:m d 1 1? kg:m d 1 1? kg:m d 1? kg:m det D 155 1?9 kg:m m 8:7 1? kg Km 5:5 1? Nm=V Table 1: Estimated parameters for the laboratory apparatus. K m is a lumped gain representing motor torque constant and amplifer transconductance. The two controllers were simulated using Matlab and Simulink. The initial conditions for the pendulum were x 1 = :8 rad and x = x =. The AL and FL balancing controllers were both designed to place the closed-loop poles at?4 j and?8 rad=s. For the AL controller the statefeedback gain matrix is?[:788; :8; 5: 1? ] and for the FL controller is [1; 84; 1]. We estimate the basin of attraction for the linear balance controller (AL) case by assuming that states on its periphery result in saturating motor torque. First consider the case where the velocity errors are zero, x = x =, but there exists a nite position error. The maximum value of x 1 can then be estimated as x 1 k 1 = max where k 1 is the appropriate state feedback gain coecient and the maximum available torque is max = 1Km = 55 1? Nm which leads to an estimate of x 1 max = :7 rad. Similarly consider the case where there is zero position error, but nite velocity error, that is, x 1 = x =. We can write x k = max 15

16 leading to x max = : rad=s. Finally, the case where x 1 = x = yields the estimate x max = 88 rad=s. The switching parameters were selected as 1 = 1 and = in order to provide sucient hysteris to prevent chattering between the two control modes. Simulation results for the hybrid controller PFBLC+AL are shown in Figures 4 and 5. The phase portrait, shown in Figure 4, clearly shows the pendulum state approaching the homoclinic orbit and being captured by the balancing controller close to the origin d/dt q1 (rad/s) q1 (rad) Figure 4: Simulated phase plane trajectory for PFLBC+AL hybrid controller. K e = 5 1 5, K v = 1:7. The time evolution of the signals, Figure 5, shows that capture occurs at time t = s and that the rate of convergence is low after time t = 1 s. The gains K e = 5 1 5, K v = 1:7, and K u = 1 were determined empirically. The large value of K e is required due to the very small values of inertia in our apparatus. Figure compares the two energy measures; E is the total energy of the system, while E is the energy of the pendulum alone. Since the PFBLC is based on E it is this quantity which converges to zero in a non-decreasing fashion. Simulation results for the hybrid controller PBC+AL are shown in Figure 7. The PBC clearly has a much slower rate of convergence and it was much more dicult to nd an eective set of gains. We found it important to set K v as small as possible while also meeting the constraint (7). K q greater than one is also critical to swingup. 7.1 Experimental results The experimental apparatus is shown in Figure 8. A limitation of this design are the wires connected to the motor and encoder mounted on the end of the arm. These exert a signicant a spring-like disturbance torque on the joint making the results unrepeatable. A new version of the apparatus is currently under construction and will use slip rings to transfer power and signals to the rotating arm. The apparatus has a high-resolution encoder tted to each axis giving (4 count=rev on q 1 and 1

17 τ 1 q d/dt q Ebar x 15 1 V Figure 5: Simulated trajectories for PFLBC+AL hybrid controller showing capture by the balance controller at t = s. K e = 5 1 5, K v = 1:7. count=rev on q. A D/A converter provides a current demand voltage to an amplifer which drives the permananent magnet DC motor. The controller was implemented in Simulink using the WinCon real-time extension with a sample interval of 5 ms. This sample rate is suciently fast that the continuous time state-feedback gain matrices are used. Angular rates were estimated using a rst-order dierence with no ltering. Experimental results for the hybrid PFBLC+AL controller are shown in Figures 9 and 1. The control was able to reach the inverted position in only 5 swings. The control parameters used, K e = 1, K v = :1, K u = 1, had to be changed from those used in simulation in order to achieve swing up. We postulate that this is due to unmodelled frictional dissipation in the real plant. The switching function shown in Figure 1 indicates which controller is active the linear controller is active during both initial pusho and the nal balance phase. The linear controller, actually a regulator, should bring all states to zero but we can see in the Figure that _q has adopted a steady state value of nearly rad=s. Disturbance forces on the arm result in q 1 = and at steady-state = _q 1 = the state feedback control law k 1 q 1 + k _q = relates q 1 error to disk velocity _q. For this particular example q 1 :15 rad, and using the state-feedback gains from above we would estimate _q 18 rad=s as observed. We can also 17

18 E Ebar Figure : Comparison of simulated E and E for hybrid controller based on partial feedback linearization. K e = 5 1 5, K v = 1:7. observe in Figure 1 that neither the Lyapunov or energy functions exhibit the expected monotonic characteristic, and which we postulate is due to model error. Experimental results for the hybrid PFBLC+FL controller are shown in Figure 1 and are not signicantly dierent to the PFBLC+AL case. Qualitative investigations showed they both balancing controllers exhibited similar robustness to external disturbances, which is ultimately limited by the nite torque capability of the motor. The FL controller performs dierently in the pusho phase, taking longer to enter the operating region of the swingup controller, and doing so with a much higher disk velocity. The dierences between the two swingup controllers, PFBLC and PBC, is even more pronounced in experiment than simulation. In fact with PBC we were not able to swing the arm above horizontal. The very slow convergence of the passivity-based controller manifested in simulation, combined with energy dissipation due to friction in the real plant, meant that it entered a stable orbit on the phase plane. 8 Conclusion This paper has discussed hybrid nonlinear control of a Gyroscopic Pendulum to achieve both swingup and balancing. We investigate several dierent control strategies based on feedback linearization, partial feedback linearization, and energy/passivity methods. We have shown that the system is locally feedback linearizable by a local dieomorphism in state space and nonlinear feedback. In practice the performance of feedback linearization control is comparable to approximate linearization and pole-placement control the performance being limited by nite acuator torque capability. For the swingup problem we compared two dierent energy based controllers; one based on passivity of the Lagrangian dynamics of the system (PBC) and the other based on collocated partial 18

19 τ q d/dt q E x 14 1 V Figure 7: Simulated trajectories for the PBC+AL hybrid controller. K e = 4 1 5, K v = 9:74 and K q =. feedback linearization, and passivity of the resulting zero dynamics (PFBLC). In both simulation and experiment we have found that the PFBLC is able to achieve swingup whereas PBC cannot in a practical period of time. Acknowledgements We would like to thank Dan Block, Manager of the College of Engineering Control Systems Laboratory at the University of Illinois for valuable assistance in constructing the gyroscopic pendulum and implementing the control algorithms derived in this paper. 19

20 Figure 8: Photograph of the experimental apparatus balancing at the unstable equilibrium point d/dt q1 (rad/s) q1 (rad) Figure 9: Experimental phase plane trajectory for the hybrid controller PFBLC+AL. The circular marker shows the location at which the switch to balancing control took place. K e = 5 1 5, K v = :1.

21 1 q d/dt q τ E x 17 V swingup Figure 1: Experimental trajectories for the hybrid controller PFBLC+AL. K e = 5 1 5, K v = :1. 1

22 1 q d/dt q τ E x 18 V swingup Figure 11: Experimental trajectories for the hybrid controller PFBLC+FL. K e = 5 1 5, K v = :1.

23 References [1] Arai, H. and Tachi, S. \Position Control of a Manipulator with Passive Joints Using Dynamic Coupling," IEEE Trans. Robotics and Automation, Vol. 7, No. 4, pp. 58{54, Aug [] Borto, S., and Spong, M.W., \Pseudolinearization of the Acrobot Using Spline Functions," IEEE Conf. on Decision and Control, Tucson, AZ, pp , Dec [] I. Fantoni, R. Lozano and M. W. Spong. Passivity based control of the Pendubot. To appear in the IEEE Transactions on Automatic Control. [4] Hartman, Ordinary Dierential Equations, Birkhauser, Boston, nd Edition, 198. [5] Isidori, A., Nonlinear Control Systems, Third Edition, Springer-Verlag, New York, [] Khalil, H.K., Nonlinear Systems, Second Edition, Prentice Hall, Upper Saddle River, NJ, 199. [7] Lozano R., I. Fantoni. Passivity based control of the inverted pendulum. Proceedings of the 4th IFAC NOLCOS'98, Enschede, The Netherlands, July [8] Mabbott, Thomas Ollive, ed. The Collected Works of Edgar Allan Poe, The Belknap Press of Harvard University Press Cambridge, MA, 199. [9] Murray, R.M., and Hauser, J., \A Case Study in Approximate Linearization: The Acrobot Example," Proc. American Control Conference, 199. [1] Saito, F., Fukuda, T., and Arai, F., \Swing and Locomotion Control for Two{Link Brachiation Robot," Proc. 199 IEEE Int. Conf. on Robotics and Automation, pp , Atlanta, GA, 199. [11] Spong, M.W., \Modeling and Control of Elastic Joint Manipulators," ASME J. of Dyn. Sys., Meas., and Control, December, [1] Spong, M.W., \The Control of Underactuated Mechanical Systems", Plenary lecture at the First International Conference on Mechatronics, Mexico City, January -9, [1] Spong, M.W., \Swing Up Control of the Acrobot using Partial Feedback Linearization", Proc. of the SY{ROCO`94, Capri, Italy, pp. 8-88, September, [14] Spong, M.W., \Swing Up Control of the Acrobot," IEEE Control Systems Magazine, Feb., [15] Spong, M.W., and Block, D., \The Pendubot: A Mechatronic System for Control Research and Education," Proc. IEEE CDC, New Orleans, Dec [1] Spong, M.W., and Vidyasagar, M., Robot Dynamics and Control, John Wiley & Sons, Inc., New York, [17] Wiklund, M., Kristenson, A., and Astrom, K.J., \A New Strategy for Swinging up an Inverted Pendulum," Proc. IFAC Symposium, Sydney, Australia, 199.