Experimental Control of Flexible Robot Manipulators

Transcription

1 Experimental Control of Flexible Robot Manipulators 8 A. Sanz an V. Etxebarria Universia el País Vasco Spain. Introuction Flexible-link robotic manipulators are mechanical evices whose control can be rather challenging, among other reasons because of their intrinsic uner-actuate nature. This chapter presents various experimental stuies of iverse robust control schemes for this kin of robotic arms. The propose esigns are base on several control strategies with well efine theoretical founations whose effectiveness are emonstrate in laboratory experiments. First it is experimente a simple control metho for trajectory tracking which exploits the two-time scale nature of the flexible part an the rigi part of the ynamic equations of flexible-link robotic manipulators: a slow subsystem associate with the rigi motion ynamics an a fast subsystem associate with the flexible link ynamics. Two experimental approaches are consiere. In a first test an LQR optimal esign strategy is use, while a secon esign is base on a sliing-moe scheme. Experimental results on a laboratory two-of flexible manipulator show that this composite approach achieves goo close-loop tracking properties for both esign philosophies, which compare favorably with conventional rigi robot control schemes. Next the chapter explores the application of an energy-base control esign methoology (the so-calle IDA-PBC, interconnection an amping assignment passivity-base control) to a single-link flexible robotic arm. It is shown that the metho is well suite to hanle this kin of uner-actuate evices not only from a theoretical viewpoint but also in practice. A Lyapunov analysis of the close-loop system stability is given an the esign performance is illustrate by means of a set of simulations an laboratory control experiments, comparing the results with those obtaine using conventional control schemes for mechanical manipulators. The outline of the chapter is as follows. Section covers a review on the moeling of flexible-link manipulators. In subsection. the ynamic moelling of a general flexible multilink manipulator is presente an in subsection. the methoology is applie to a laboratory flexible arm. Next, some control methoologies are outline in section 3. Two control strategies are applie to a two-of flexible robot manipulator. The first esign, in subsection 3., is base on an optimal LQR approach an the secon esign, in subsection 3., is base on a sliing-moe controller for the slow subsystem. Finally, section 4 covers the IDA-PBC metho. In subsection 4. an outline of the metho is given an in subsection

2 56 Robot Manipulators 4. the propose control strategy base on IDA-PBC is applie to a laboratory arm. The chapter ens with some concluing remarks.. Moeling of flexible-link manipulator. The ynamic moeling of a general flexible multilink manipulator In orer to obtain a ynamic moel for a multilink flexible robot manipulator, it becomes necessary the introuction of a convenient kinematic escription of the manipulator, incluing the eformation of the links. In orer to limit the complexity of the erivation, it is assume that rigi motion an link eformation happen in the same plane, without torsional effects. A sketch of a two-link manipulator of this kin is shown in Fig. with an appropriate coorinate frame assignment. > X X Y -θ > Y Y > Y 0 w (x ) θ X > X 0 Figure. Planar two-link flexible manipulator The rigi motion is escribe by the joint angles θ, θ, while w (x ) enotes the transversal eflection of link at x with 0 x l, being l the link length. This eflection is expresse as a superposition of a finite number of moes where the spatial an time variables are separate: n e w(,) xt = ϕ() xqt () () i= where φ i (x) is the moe shape an q i (t) is the moe amplitue. To obtain a finite-imensional ynamic moel, the assume moes link approximation can be use. By applying Lagrange's formulation the ynamics of any multilink flexible-link robot can be represente by i i Hrr () + Vrr (,) + Kr + Fr () + Gr () = Br () τ () where r(t) expresses the position vector as a function of the rigi variables θ an the eflection variables q. H(r) represents the inertia matrix, Vrr (,) are the components of the Coriolis vector an centrifugal forces, K is the iagonal an positive efinite link stiffness matrix, F() r is the friction matrix, an G(r) is the gravity matrix. B(r) is the input matrix, which epens on the particular bounary conitions corresponing to the assume moes. Finally, τ is the vector of the input joint torques. The explicit equations of motion can be erive computing the kinetic energy T an the potential energy U of the system an then forming the Lagrangian.

3 Experimental Control of Flexible Robot Manipulators 57 The kinetic energy of the entire system is n n T = T + T (3) hi li i= i= where T hi is the kinetic energy of the rigi boy locate at hub i of mass m hi an moment of inertia I hi ; r i inicates the absolute position in frame (,) XY ˆˆ of the origin of frame (X i,y i ) an α is the absolute angular velocity of frame (X i i,y i ). 0 0 T Thi = mrr hi i i + Iα hi i (4) An T li is the kinetic energy of the ith element where ρ i is the mass ensity per unit length of the element an p is the absolute linear velocity of an arm point. i li T Tli = ()() () ρ 0 ixpx i i ipxx i i i (5) Since the robot moves in the horizontal plane, the potential energy is given by the elastic energy of the system, because the gravitational effects are supporte by the structure itself, an therefore they isappear insie the formulation: The elastic energy store in link i is n U = U (6) i= w() ixi ei ()()( i i ) i xi The total elastic energy of the system can be written as: ei U = EI x x (7) T UU = e = qkq (8) where K is the iagonal stiffness matrix that only affects to the flexible moes. As a result, the ynamical equation () can be partitione in terms of the rigi, θ i (t), an flexible, q ij (t), generalize coorinates. Hθθ(,) θq Hθq(,) θ cθ(,,,) θq θq 0 τ T + + = Hθq(,) Hqq(,) θq q cq(,,,) θq θq Dq + Kq 0 Where H θθ, H θq y H qq are the blocks of the inertia matrix H, which is symmetric an positive efinite, c θ, c q are the components of the vector of Coriolis an centrifugal forces an D is the iagonal an positive semiefinite link amping matrix. (9). Application to a flexible laboratory arm The above analysis will now be applie to the flexible manipulator shown in Fig., whose geometric sketch is isplaye in Fig. 3. As seen in the figures, it is a laboratory robot which

4 58 Robot Manipulators moves in the horizontal plane with a workspace similar to that of common Scara inustrial manipulators. Links marke an 4 are flexible an the remaining ones are rigi. The flexible links' eflections are measure by strain gauges, an the two motor's positions are measure by stanar optical encoers. Figure. Photograph of the experimental flexible manipulator Link Link 3 w ( x ) Y 0 Link Link 5 θ w 4 ( x 4 ) θ4 Link 4 X 0 Figure 3. Planar two-of flexible manipulator The moel is obtaine by a stanar Lagrangian proceure, where the inertia matrices have the structure: hh hh 3 4 hh Hθθ(,) θq = Hθq(,) = Hqq(,) θq = hh h3 h4 h43 h44 with: h = Ih + mj( l + ϕq) + mj( l + ϕq + l ϕ 4q4) + Ih5 + Ih6 + ml + ςq + m(l + ϕq) 3 + m5( l + l ϕ 4q4) 3 3 h = mj(lcos( θ θ4) + l ϕ qsin( θ θ4) lϕqsin( θ θ4) + lcos( θ θ4) ϕϕ q lsin( θ θ4) ϕ 4q4 + lsin( θ θ) ϕq + lcos( θ θ) ϕ ϕ q) + ml ( cos( θ θ) + lsin( θ θ) ϕ q lsin( θ θ) ϕq lcos( θ θ4) ϕϕ q) + ml ( cos( θ θ) lsin( θ θ) ϕ q + lsin( θ θ) ϕq cos( + l θ θ4 ) ϕ4ϕ 4q4 )

5 Experimental Control of Flexible Robot Manipulators 59 h = h 4 4 h = I + mll ( + ϕ q) + I + I + ml ( + ϕq) + ml ( + ϕq) + m( l + l ϕ q) + ml + ς q ml ( + ϕ q) h j h3 h4 j 4 4 j h = mlϕ + m( lϕ + lcos( θ θ) ϕ lsin( θ θ) ϕϕ q) + υ + ml ( ϕ + lcos( θ θ) ϕ lsin( θ θ) ϕϕ q) 3 j j h = I ϕ + m ( l ϕ + lcos( θ θ) ϕ lsin( θ θ) ϕ ϕ q) + I ϕ + m( l ϕ lsin( θ θ) ϕ ϕ q 3 + l cos( θ θ4 ) ϕ4 ) Ih ϕ + mj( l ϕ + lcos( θ θ4) ϕ + lsin( θ θ4) ϕ ϕ q) h3 = 4 + Ih3 ϕ + ( m l ϕ + lsin( θ θ4) ϕ ϕ q 3 + lcos( θ θ) ϕ) 4 h5 4 j h h4 = ml j ϕ4 + mj( lϕ4 + lcos( θ θ4) ϕ 4 + lsin( θ θ4) ϕ4ϕ 4q4) + υ4 + ml 5( ϕ4 + lcos( θ θ4) ϕ 4 + lsin( θ θ4) ϕ4ϕ 4q4) 4 h33 = mjϕ + Ihϕ + mj( ϕ + l ϕ + cos( l θ θ4) ϕϕ ) + Ih3ϕ + ς+ m( ϕ + l ϕ + cos( l θ θ4) ϕϕ ) 3 h = h = h44 = mjϕ4 + Ih5ϕ 4 + mj( ϕ4 + l ϕ 4 + cos( l θ θ4) ϕϕ 4 4) + Ih6ϕ 4 + ς4 + m5( ϕ4 + l ϕ 4 + lcos( θ θ4) ϕϕ 4 4) 3 where i is the amplitue of the first moe evaluate in the en of the link i, ϕ i = ( x ϕ i/ i) x i = l i ; ij is the eformation moment of orer one of moe j of link i an ijk is the cross moment of moes j an k of link i. Also l is the links length, m j is the elbow joint mass, m j is the tip joint mass, m i (i=,4) is the flexible link mass, m k (k=,3,5) is the rigi link mass an q i is the eflection variable. 3. Control esign The control of a manipulator forme by flexible elements bears the stuy of the robot's structural flexibilities. The control objective is to move the manipulator within a specific trajectory but attenuating the vibrations ue to the elasticity of some of its components. Since time scales are usually present in the motion of a flexible manipulator, it can be consiere that the ynamics of the system is ivie in two parts: one associate to the manipulator's movement, consiere slow, an another one associate to the eformation of the flexible links, much quicker. Taking avantage of this separation, iverse control strategies can be esigne. 3. LQR control esign In a first attempt to esign a control law, a stanar LQR optimal technique is employe. The error feeback matrix gains are obtaine such that the control law τ () t = K Δ x minimizes the cost function: c ( * τ* τ) 0 J = ΔxQx Δ + R t (0)

6 60 Robot Manipulators where Q an R are the stanar LQR weighting matrices; an Δ x is the applicable state vector comprising either flexible, rigi or both kin of variables. This implies to minimize the tracking error in the state variables, but with an acceptable cost in the control effort. The resulting Riccati equations can conveniently be solve with state-of-the-art tools such as the Matlab Control System Toolbox. The effectiveness of the propose control schemes has been teste by means of real time experiments on a laboratory two-of flexible robot. This manipulator, fabricate by Quanser Consulting Inc. (Ontario, Canaa), has two flexible links joine to rigi links using high quality low friction joints. The laboratory system is shown in Fig.. From the general equation (9) the ynamics corresponing to the rigi part can be extracte: h(,) θq h(,) θq θ c(,,,) θq θq τ + = h(,) θq h(,) θq θ4 c(,,,) θq θq τ4 () The first step is to linearize the system () aroun the esire final configuration or reference configuration. Defining the incremental variable Δ θ = θ θ an with a state-space representation efine by Δ x = ( Δθ Δ θ) T the approximate moel corresponing to the consiere flexible manipulator's rigi part can be escribe as: Δ x = Δx τ () 0000 H θθ 0000 where it has been taken into account the fact that the Coriolis terms are zero aroun the esire reference configuration. Table isplays the physical parameters of the laboratory robot. Using these values, the following control laws have been esigne. In all the experiments the control goal is to follow a prescribe cartesian trajectory while amping the excite vibrations as much as possible. Property Value Motor inertia, I h Kgm Link length, l Flexible link mass, m,m 4 Rigi link mass, m,m 3,m 5 Elbow joint mass, m j Tip joint mass, m j 0.3 m 0.09 Kg 0.08 Kg 0.03 Kg 0.04 Kg Table. Flexible link parameters

7 Experimental Control of Flexible Robot Manipulators 6 The control goal is to track a cartesian trajectory which comprises 0 straight segments in about secons. Each stretch is covere in approximately 0.6 secons, after which the arm remains still for the same time at the points marke to 0 in Fig. 4. As a first test, Fig. 5 shows the system's response with a pure rigi LQR controller that is obtaine solving the Riccati equations for R=iag ( ) an Q=iag( ). The values of R an Q have been chosen experimentally, keeping in min that if the values of Q are relatively large with respect to the values of R a quick evolution is specifie towar the esire equilibrium Δ x = 0 which causes a bigger control cost (see (0)). As seen in the figures, this kin of control, which is very commonly use for rigi manipulators, is able to follow the prescribe trajectory but with a certain error, ue to the links' flexibilities. It has been shown in the above test that, as expecte, a flexible manipulator can not be very accurately controlle if a pure rigi control strategy is applie. From now on two composite control schemes (where both the rigi an the flexible ynamics are taken into account) will be escribe. The non-linear ynamic system (9) is being linearize aroun the esire final configuration. The obtaine moel is: Hθθ(,) θq Hθq(,) Δ θ 00 Δθ Δτ T + Hθq(,) Hqq(,) θq = Δq 0 K Δq 0 (3) where again the Coriolis terms are zero aroun the given point. Reucing the moel to the maximum (taking a single elastic moe in each one of the flexible links), a state vector is efine as: Δ x = ( Δθ Δq Δ θ Δq ) T (4) an the approximate moel of the consiere flexible manipulator can be escribe as: where 0 0 I I 0 Δ x = Δ x+ Δτ (5) 0 LHK qq 0 0 LH θq T 0 LH θq K 0 0 LH θθ L = ( H H H H) T θθ θq θq qq T = ( θq θθ qq θq) L H H H H Following the same LQR control strategy as in the previous lines, a scheme is evelope making use of the values of Table, with R=iag( ) an Q=iag( ). From measurements on the laboratory robot the following values for the elements' inertias I h =I h3 =I h5 =I h6 =.0-5 Kg.m have been estimate. Also the elastic constant K=3, an the space form in the consiere vibration moes = 4 =0.00m an = φ 4 =0. have been measure.

8 6 Robot Manipulators In Fig. 6 the obtaine results applying a combine LQR control are shown. The tip position exhibits better tracking of the esire trajectory, as it can be seen comparing Fig. 5(a) an Fig. 6(a). Even more important, it shoul be note that the oscillations of elastic moes are now attenuate quickly (compare Fig. 5(b,c) an Fig. 6(b,c)) x(m) Y Time(s) (a) θ θ y (m) 0.35 (b) X Time (s) Figure 4. Cartesian trajectory of the manipulator. (a) Time evolution of the x an y coorinates. (b) Sketch of the manipulator an trajectory 3. Sliing-moe control esign The presence of unmoelle ynamics an isturbances can lea to poor or unstable control performance. For this reason the rigi part of the control can be esigne using a robust sliing moe philosophy, instea of the previous LQR methos. This control esign has been prove to hol soli stability an robustness theoretical properties in rigi manipulators (Barambones & Etxebarria, 000), an is teste now in real experiments. The rigi control law is thus change to: τ ri pi sgn( Si) if Si > βi = Si pi otherwise β i T where β = (,,), β βn βi > 0is the with of the ban for each sliing surface associate to each motor, p i is a constant parameter an S is a surface vector efine by S = E +λe, being E = θ θ the tracking error ( θ is the esire trajectory vector) an λ = iag(,,), λ λ λ > 0. n i

9 Experimental Control of Flexible Robot Manipulators 63 The philosophy of this control is to attract the system to S i =0 by an energetic control p sgn( S i ) an once insie a ban aroun S i =0, to regulate the position using a smoother control, roughly coincient with the one esigne previously. The sliing control can be tune by means of the sliing gain p an by ajusting the with of ban β which limits the area of entrance of the energetic part of the control. It shoul be kept in min that the external action to the ban shoul not be excessive, to avoi exciting the flexibilities of the fast subsystem as much as possible (which woul invaliate the separation hypothesis between the slow ynamics an the fast one). In this thir experiment a secon composite control scheme (sliing-moe for the slow part an LQR for the fast one) is teste. Fig. 7 exhibits the experimental results obtaine with the sliing control strategy escribe for the values β=( ) an λ=( ). β is the with of the ban for each sliing surface associate to each motor, which limits the area of entrance of the energetic part of the control. If the values of β are too big, the action of the energetic control is almost negligible; on the contrary if this values are too small, the action of the external control is much bigger an the flexible moes can get excite. The with of the ban is chosen by means of a trial-an-error experimental methoology. The values use of λ are such that the gain control values insie the ban match the LQR gains esigne in the previous section. In the same way as with the combine LQR control, the rigi variable follows the esire trajectory, an again, the elastic moes are attenuate with respect to the rigi control, (compare Fig. 5(b,c) an Fig. 7(b,c)). Also, it can be seen that this attenuation is even better than in the previous combine LQR control case (compare Fig. 6(b,c) an Fig. 7(b,c)). To gain some insight on the ifferences of the two propose combine schemes, the contribution to the control torque is compare in the sliing-moe case (rigi) with the LQR combine control case (rigi), both to (Fig. 8(a) an 8(b)). It is observe how in the intervals from to s., to 3s. (corresponing to the stretch between the points marke an 3 in Fig.4) an 7 to 8s. (corresponing to the iagonal segment between the points marke 7 an 8 in Fig.4), the sliing control acts in a more energetic fashion, favoring this way the pursuit of the wante trajectory. However, in the intervals from 5 to 6s. (point marke 5 in Fig.4), 6 to 7s. (marke 6), 9.5 to 0.5s. (marke 9) an to s. (marke 0), in the case of the sliing control, a smoother control acts, an its control effort is smaller than in the case of the LQR combine control, causing this way smaller excitation of the oscillations in these intervals (as it can be seen comparing Fig. 7(b,c) with Fig. 6(b,c)). Note that the sliing control shoul be carefully tune to achieve its objective (attract the system to S i =0) but without introucing too much control activity which coul excite the flexible moes to an unwante extent.

10 64 Robot Manipulators Figure 5. Experimental results for pure rigi LQR control. a) Tip trajectory an reference. b) Time evolution of the flexible eflections (link eflections). c) Time evolution of the flexible eflections (link 4 eflections)

11 Experimental Control of Flexible Robot Manipulators 65 Figure 6. Experimental results for composite (slow-fast) LQR control. a) Tip trajectory an reference. b) Time evolution of the flexible eflections (link eflections). c) Time evolution of the flexible eflections (link 4 eflections)

12 66 Robot Manipulators Figure 7. Experimental results for composite (slow-fast) sliing-lqr control. a) Tip trajectory an reference. b) Time evolution of the flexible eflections (link eflections). c) Time evolution of the flexible eflections (link 4 eflections)

13 Experimental Control of Flexible Robot Manipulators 67 Figure 8. a) Contribution to the torque control (rigi) with the LQR combine control squeme. b) Contribution to the torque control (rigi) in the sliing-moe case 4. The IDA-PBC metho 4. Outline of the metho The IDA-PBC (interconnection an amping assignment passivity-base control) metho is an energy-base approach to control esign (see (Ortega & Spong, 000) an (Ortega et al., 00) for complete etails). The metho is specially well suite for mechatronic applications, among others. In the case of a flexible manipulator the goal is to control the position of an uner-actuate mechanical system with total energy: T Hqp (,) = pm() qpuq + () (6) n n where q, p, are the generalize positions an momenta respectively, M(q)=M T (q)>0 is the inertia matrix an U(q) is the potential energy.

14 68 Robot Manipulators If it is assume that the system has no natural amping, then the equations of motion of such system can be written as: q 0 In qh 0 = + u p In 0 ph Gq () (7) The IDA-PBC metho follows two basic steps:. Energy shaping, where the total energy function of the system is moifie so that a preefine energy value is assigne to the esire equilibrium.. Damping injection, to achieve asymptotic stability. The following form for the esire (close-loop) energy function is propose: T Hqp (,) = pmqpuq () + () (8) T where M = M > 0 is the close-loop inertia matrix an U the potential energy function. It will be require that U have an isolate minimum at the equilibrium point q *, that is: In PBC, a composite control law is efine: q* = argmin U( q) (9) uuqpuqp = (,) + (,) (0) es where the first term is esigne to achieve the energy shaping an the secon one injects the amping. The esire (close-loop) ynamics can be expresse in the following form: i q p H H q = ( Jqp (,) Rqp (,)) p () where T 0 MM J = J = MM J(,) qp () 0 0 T RR = = 0 T 0 GKG v represent the esire interconnection an amping structures. J is a skew-symmetric matrix, an can be use as free parameter in orer to achieve the kinetic energy shaping (see Ortega & Spong, 000)). The secon term in (0), the amping injection, can be expresse as: T where KK = > 0. v v T i v p (3) u = KG H (4)

15 Experimental Control of Flexible Robot Manipulators 69 To obtain the energy shaping term u es of the controller, (0) an (4), the composite control law, are replace in the system ynamic equation (7) an this is equate to the esire close-loop ynamics, (): 0 In qh 0 0 MM qh + ues In 0 ph = G MM Jqp (,) p H Gu = H M M H + J M p (5) es q q In the uner-actuate case, G is not invertible, but only full column rank. Thus, multiplying (5) by the left annihilator of G, G G=0, it is obtaine: { q q } G H MM H + JM p = (6) 0 If a solution (M,U ) exists for this ifferential equation, then u es will be expresse as: u = ( GG) G( H MM H + JM p) (7) T T es q q The PDE (6) can be separate in terms corresponing to the kinetic an the potential energies: T T { q q } G ( pmp) MM ( pmp) + JMp = 0 (8) { q q } G U MM U = 0 (9) So the main ifficulty of the metho is in solving the nonlinear PDE corresponing to the kinetic energy (8). Once the close-loop inertia matrix, M, is known, then it is easier to obtain U of the linear PDE (9), corresponing to the potential energy. 4. Application to a laboratory arm The object of the stuy is a flexible arm with one egree of freeom that accomplishes the conitions of Euler-Bernoulli (Fig.9). In this case, the elastic eformation of the arm w(x,t) can be represente by means of an overlapping of the spatial an temporary parts, see equation (). Figure 9. Photograph of the experimental flexible manipulator

16 70 Robot Manipulators If a finite number of moes m is consiere, Lagrange equations lea to a ynamical system efine by m+ secon orer ifferential equations: I 00 t 0 qt 0() qt 0() + = u 0 I qt () 0 K i f qt i() ϕ (0) (30) where q 0 (a x vector) an q i (an mx vector) are the ynamic variables; q 0 (t) is the rigi generalize coorinate, q i (t) is the vector of flexible moes, I t is the total inertia, K f is the stiffness mxm matrix that epens on the elasticity of the arm, an it is efine as K f =iag( ω ), where ω i is the resonance frequency of each moe; are the first spatial i erivatives of i (x) evaluate at the base of the robot. Finally u inclues the applie control torques. Defining the incremental variables as qqq =, where q is the esire trajectory for the robot such that q 0 0, q 0 = 0 an q i =0, then the ynamical moel is given by: I 00 t 0 qt 0() qt 0() + q0 u 0 I qt () + = 0 K f qt i() ϕ (0) i (3) The total energy of the mechanical system is obtaine as the sum of the kinetic an potential energy: (,) (3) T T Hqp = pm p+ qkq where piqiq = ( 0 ) T t i an M an K are (m+)x(m+) matrices I 0 00 t M = K = 0 I 0 Kf The point of interest is q * = (0,0), which correspons to zero tracking error in the rigi variable an null eflections. The controller esign will be mae in two steps; first, we will obtain a feeback of the state that prouces energy shaping in close-loop to stabilize the position globally, then it will be injecte the necessary amping to achieve the asymptotic stability by means of the negative feeback of the passive output. The inertia matrix, M, that characterizes to the system, is inepenent of q, hence it follows that it can be chosen J =0, (see (Ortega & Spong, 000)). Then, from (8) it is euce that the matrix M shoul also be constant. The resulting representation capturing the first flexible moe is M aa = aa 3 where the conition of positive efiniteness of the inertia matrix, lea to the following inequations: a > 0 aa > a (33) 3

17 Experimental Control of Flexible Robot Manipulators 7 Now, consiering only the first flexible moe, equation (9) corresponing to the potential energy can be written as: a ϕ (0) a U U + ( a3 ϕ (0) a) = q ω It q 0 q (34) where qqq 0 = 0 0 is the rigi incremental variable an qq = 0 is the flexible incremental variable. The equation (34) is a trivial linear PDE whose general solution is I ω( a ϕ (0) a) I ω U = q qq + Fz () (35) t 3 t 0 0 ( a + ϕ (0) a) ( a + ϕ (0) a) z = γq + q (36) 0 Ia t( 3 ϕ (0) a) γ = ( a + ϕ (0) a) (37) where F is an arbitrary ifferentiable function that we shoul choose to satisfy conition (9) in the points q *. Some simple calculations show that the necessary restriction U (0) = 0 is satisfie if Fz ((0)) = 0, while conition U (0) > 0 is satisfie if: q q ω F > ( a ϕ (0) a) 3 (38) Uner these restrictions it can be propose the conition: Fz () = (/) Kz, with K >0, which prouces a > ϕ (0) a (39) 3 So now, the term corresponing to the energy shaping in the control input (7) is given as u GG G U MM U Kq K q (40) T ( ) T es = ( q q ) = p0+ p where Iaa ( a) K (( K a (0)) a ) t 3 p = 3 + ϕ + ω ( a + ϕ (0) a) K p a ω Kaa ( 3 a) = ( a + ϕ (0) a) The controller esign is complete with a secon term, corresponing to the amping T injection. This is achieve via negative feeback of the passive output G H, (4). As p

18 7 Robot Manipulators H (/) T pmp U( q) = +, an U only epens on the q variable, u i only epens on the appropriate election of M : u = Kq + Kq (4) i v0 v with Ia ( + ϕ (0) a) K = K t 3 v v ( aa 3 a) ( a ϕ (0) a) K = K v v ( aa 3 a) A simple analysis on the constants K p an K v with the conitions previously impose, implies that both shoul be negative to assure the stability of the system. To analyze the stability of the close-loop system we consier the energy-base Lyapunov function caniate (Kelly & Campa, 005), (Sanz & Etxebarria, 007) ( (0) ) Vqq (,) = pm pu + = ( (0) ) T Iqa t 03 Iqqa t 0 + qa Itωa3 ϕ aq 0 aa 3 a a+ ϕ a Iqq tω 0 + Kqq ( γ + ) a + ϕ (0) a 0 (4) which is globally positive efinite, i.e: V(0,0)=(0,0) an Vqq (,) > 0 for every (,) qq (0,0). The time erivative of (4) along the trajectories of the close-loop system can be written as ( Iq t ( a + ϕ (0))( a + a ϕ (0))) aq Vqq (,) = K 0 (43) v 0 3 ( aa 3 a) where K v >0, so V is negative semiefinite, an (,) qq = (0,0) is stable (not necessarily asymptotically stable). By using LaSalle's invariant set theory, it can be efine the set R as the set of points for which V = 0 q 0 q = :(,)0 = =, & + = 0 q 0 q { 0 γ 0 } 4 R Vqq qq q q (44) Following LaSalle's principle, given R an efine N as the largest invariant set of R, then all the solutions of the close loop system asymptotically converge to N when t. Any trajectory in R shoul verify: γ qq = (45)

19 Experimental Control of Flexible Robot Manipulators 73 an therefore it also follows that: γ qq = (46) γ qqk + = (47) 0 Consiering the close-loop system an the conitions escribe by (46) an (47), the following expression is obtaine: γ γω ( + ϕ(0) Kk f) ( (0)) q0 0 I + + ϕ ( a ϕ(0) a) = (48) + t As a result of the previous equation, it can be conclue that qt 0 () shoul be constant, in consequence qt () = 0 0, an replacing it in (45), it also follows that qt () = 0. Therefore qq 0 = 0 = an replacing these in the close-loop system this leas to the following solution: qt 0() 0 = qt () 0 In other wors, the largest invariant set N is just the origin (,) qq = (0,0), so we can conclue that any trajectory converge to the origin when t, so the equilibrium is in fact asymptotically stable. To illustrate the performance of the propose IDA-PBC controller, in this section we present some simulations. We use the moel of a flexible robotic arm presente in (Canuas et al., 996) an the values of Table which correspon to the real arm isplaye in Fig. 9. The results are shown in Figs. 0 to. In these examples the values a =, a =0.0 an a 3 =50 have been use to complete the conitions (33) an (39). In Fig. 0 the parameters are K =0 an K v =000. In Figs. an the effect of moifying the amping constant K v is emonstrate. With a smaller value of K v, K v =0, the rigi variable follows the esire trajectory reasonably well. For K v =000, the tip position exhibits better tracking of the esire trajectory, as it can be seen comparing Fig. 0(a) an Fig. (a). Even more important, it shoul be note that the oscillations of elastic moes are now attenuate quickly (compare Fig. 0(c) an Fig. (b)). But If we continue increasing the value of K v, K v =00000, the oscillations are attenuate even more quickly (compare Fig. 0(c) an Fig. (b)), but the tip position exhibits worse tracking of the esire trajectory. Property Value Motor inertia, I h 0.00 kgm Link length, L Link height, h Link thickness, Link mass, M b Linear ensity, Table. Flexible link parameters 0.45 m 0.0 m m 0.06 kg kg/m Flexural rigiity, EI 0.6 Nm

20 74 Robot Manipulators Figure 0. Simulation results for IDA-PBC control with K v =000: (a) Time evolution of the rigi variable q 0 an reference q ; (b) Rigi variable tracking error; (c) Time evolution of the flexible eflections; () Composite control signal The effectiveness of the propose control schemes has been teste by means of real time experiments on a laboratory single flexible link. This manipulator arm, fabricate by Quanser Consulting Inc. (Ontario, Canaa), is a spring steel bar that moves in the horizontal plane ue to the action of a DC motor. A potentiometer measures the angular position of the system, an the arm eflections are measure by means of a strain gauge mounte near its base (see Fig. 9). These sensors provie respectively the values of q 0 an q (an thus q 0 an q are also known, since q 0 an q are preetermine). The experimental results are shown on Figs. 3, 4 an 5. In Fig. 3 the control results using a conventional PD rigi control esign are isplaye: ukqq = ( ) + Kqq ( ) P 0 0 D 0 0 where K P =-4 an K D = These gains have been carefully chosen, tuning the controller by the usual trial-an-error metho. The rigi variable tracks the reference (with a certain error), but the naturally excite flexible eflections are not well ampe (Fig. 3(b)).

21 Experimental Control of Flexible Robot Manipulators 75 Figure. Simulation results for IDA-PBC control with K v =0: (a) Time evolution of the rigi variable q 0 an reference q ; (b) Time evolution of the flexible eflections Figure. Simulation results for IDA-PBC control with K v =00000: (a) Time evolution of the rigi variable q 0 an reference q ; (b) Time evolution of the flexible eflections In Fig. 4 the results using the IDA-PBC esign philosophy are isplaye. The values a =, a =0.0, a 3 =50, K =0, K v =000 an ϕ (0) =. have been use. As seen in the graphics, the rigi variable follows the esire trajectory, an moreover the flexible moes are now conveniently ampe, (compare Fig. 3(b) an Fig. 4(c)). It is shown that vibrations are effectively attenuate in the intervals when q reaches its upper an lower values which go from to secons, 3 to 4 s., 5 to 6 s., etc. The PD controller might be augmente with a feeback term for link curvature: ukqq = ( ) + Kqq ( ) + K w P 0 0 D 0 0 C where w represents the link curvature at the base of the link. This is a much simpler version of the IDA-PBC an oesn't require time erivatives of the strain gauge signals.

22 76 Robot Manipulators Figure 3. Experimental results for a conventional PD rigi controller:: (a) Time evolution of the rigi variable q 0 an reference q ; (b) Time evolution of the flexible eflections Figure 4. Experimental results for IDA-PBC control: (a) Time evolution of the rigi variable q 0 an reference q ; (b) Rigi variable tracking error; (c) Time evolution of the flexible eflections; () Composite control signal

23 Experimental Control of Flexible Robot Manipulators 77 Figure 5. Performance comparison between a conventional rigi PD controller, augmente PD an IDA-PBC controller. (a) Time evolution of the rigi variable q 0 an reference q (PD); (b) Time evolution of the flexible eflections (PD); (c) Time evolution of the rigi variable q 0 an reference q (IDA-PBC); () Time evolution of the flexible eflections (IDA- PBC); (e) Time evolution of the rigi variable q 0 an reference q (augmente PD); (f) Time evolution of the flexible eflections (augmente PD); (g) Comparison between the strain gauge signals with the IDA-PBC controller an the augmente PD controller

24 78 Robot Manipulators Fig. 5 shows the comparison between the conventional rigi PD controller, augmente PD an the IDA-PBC controller for a step input. The K P an K D gains are the same both for the PD an the augmente PD case. Gain K C has been carefully tune to effectively amp the flexible vibrations in the augmente PD control. Comparing Fig. 5(a,b) to Fig. 5(c,) it is clearly seen how with the IDA-PBC controller the tip oscillation improves notably. Moreover, from Fig. 5(e,f) it can be observe that the augmente PD effectively attenuates the flexible vibrations, but at the price of worsening the rigi tracking. Also in Fig. 5(g), where the flexible responses of both the augmente PD an the IDA-PBC controllers have been amplifie an plotte together for comparison, it can be observe that the effective attenuation time is longer in the augmente PD case ( secons) than in the IDA-PBC case ( secon). Notice that the final structure of all the consiere controls is similar but with very ifferent gains an all of them imply basically the same implementation effort, although the IDA-PBC gains calculation may imply solving some involve equations. In this sense, the IDA-PBC metho gives for this application an energy interpretation of the control gains values an it can be use, in this particular case, as a systematic metho for tuning these gains (outperforming the trial-an-error metho). Finally, it shoul be remarke that for a more complicate robot the structure of the IDA-PBC an the PD controls can be very ifferent. 5. Conclusion An experimental stuy of several control strategies for flexible manipulators has been presente in this chapter. As a first step, the ynamical moel of the system has been erive from a general multi-link flexible formulation. If shoul be stresse the fact that some simplifications have been mae in the moelling process, to keep the moel reasonably simple, but, at the same time, complex enough to contain the main rigi an flexible ynamical effects. Three control strategies have been esigne an teste on a laboratory two-of flexible manipulator. The first scheme, base on an LQR optimal philosophy, can be interprete as a conventional rigi controller. It has been shown that the rigi part of the control performs reasonably well, but the flexible eflections are not well ampe. The strategy of a combine optimal rigi-flexible LQR control acting both on the rigi subsystem an on the flexible one has been teste next. The avantage that this type of combine control offers is that the oscillations of the flexible moes attenuate consierably, which emonstrates that a strategy of overlapping a rigi control with a flexible control is effective from an experimental point of view. The thir experimente approach introuces a sliing-moe controller. This control inclues two completely ifferent parts: one sliing for the rigi subsystem an one LQR for the fast one. In this case the action of the energetic control turns out to be effective for attracting the rigi ynamics to the sliing ban, but at the same time the elastic moes are attenuate, even better than in the LQR case. This metho has been shown to give a reasonably robust performance if it is conveniently tune. The IDA-PBC metho is a promising control esign tool base on energy concepts. In this chapter we have also presente a theoretical an experimental stuy of this metho for controlling a laboratory single flexible link, valuing in this way the potential of this technique in its application to uner-actuate mechanical systems, in particular to flexible manipulators. The stuy is complete with the Lyapunov stability analysis of the closeloop system that is obtaine with the propose control law. Then, as an illustration, a set of simulations an laboratory control experiments have been presente. The experimente

25 Experimental Control of Flexible Robot Manipulators 79 scheme, base on an IDA-PBC philosophy, has been shown to achieve goo tracking properties on the rigi variable an efinitely superior amping of the unwante vibration of the flexible variables compare to conventional rigi robot control schemes. In view of the obtaine experimental results, Fig. 3, 4 an 5, it has also been shown that the propose IDA-PBC controller leas to a remarkable improvement in the tip oscillation of the robot arm with respect to a conventional PD or an augmente PD approach. It is also worth mentioning that for our application the propose energy-base methoology, although inclues some involve calculations, finally results in a simple controller as easily implementable as a PD. In summary, the experimental results shown in the chapter have illustrate the suitability of the propose composite control schemes in practical flexible robot control tasks 6. References O. Barambones an V. Etxebarria, Robust aaptive control for robot manipulators with unmoele ynamics. Cybernetics an Systems, 3(), 67-86, 000. C. Canuas e Wit, B. Siciliano an G. Bastin, Theory of Robot Control. Europe: The Zoiac, Springer, 996. R. Kelly an R. Campa, Control basao en IDA-PBC el pénulo con ruea inercial: Análisis en formulación lagrangiana. Revista Iberoamericana e Automática e Informática Inustrial,, pp ,January 005. P. Kokotovic, H. K. Khalil an J.O'Reilly, Singular perturbation methos in control. SIAM Press, 999. Y. Li, B. Tang, Z. Zhi an Y. Lu, Experimental stuy for trajectory tracking of a two-link flexible manipulator. International Journal of Systems Science, 3():3-9, 000. M. Moallem, K. Khorasani an R.V. Patel, Inversion-base sliing control of a flexible-link manipulator. International Journal of Control, 7(3), , 998. M. Moallem, R.V. Patel an K. Khorasani, Nonlinear tip-position tracking control of a flexible-link manipulator: theory an experiments. Automatica, 37, pp , 00. R. Ortega an M. Spong, Stabilization of uneractuatemechanical systems via interconnection an amping assignment, in Proceeings of the st IFAC Workshop on Lagrangian an Hamiltonian methos in nonlinear systems, Princeton, NJ,USA, 000, pp R. Ortega an M. Spong an F. Gómez an G. Blankenstein, Stabilization of uneractuate mechanical systems via interconnection an amping assignment. IEEE Transactions on Automatic Control, AC-47, pp , 00. R. Ortega an A. van er Schaft an B. Maschke an G. Escobar, Interconnection an amping assignment passivity-base control of port-controlle Hamiltonian systems. Automatica, 38, pp , 00. A. Sanz an V. Etxebarria, Experimental Control of a Two-Dof Flexible Robot Manipulator by Optimal an Sliing Methos. Journal of Intelligent an Robotic Systems, 46: 95-0, 006. A. Sanz an V. Etxebarria, Experimental control of a single-link flexible robot arm using energy shaping. International Journal of Systems Science, 38(): 6-7, 007.

26 80 Robot Manipulators M. W. Vanegrift, F. L. Lewis an S. Q. Zhu, Flexible-link robot arm control by a feeback linearization singular perturbation approach. Journal of Robotic Systems, (7):59-603, 994. D. Wang an M. Viyasagar, Transfer functions for a flexible link. The International Journal of Robotic Research, 0(5), pp , 99. J.X. Xu an W.J. Cao, Direct tip regulation of a single-link flexible manipulator by aaptive variable structure control.international Journal of Systems Science, 3(): -35, 00. J.H. Yang, F.L. Lian, L.C. Fu, Nonlinear aaptive control for flexible-link manipulators. IEEE Transactions on Robotics an Automation, 3(): 40-48, 997. The Mathworks, Inc, Control System Toolbox.Natick, MA, 00. S. Yoshiki an K. Ogata an Y. Hayakawa, Control of manipulator with flexible-links via energy moification metho by using moal properties, in Proceeings of the 003 IEEE/ASME International Conference on Avance Intelligent Mechatronics (AIM 003), pp

27 Robot Manipulators Eite by Marco Ceccarelli ISBN Har cover, 546 pages Publisher InTech Publishe online 0, September, 008 Publishe in print eition September, 008 In this book we have groupe contributions in 8 chapters from several authors all aroun the worl on the several aspects an challenges of research an applications of robots with the aim to show the recent avances an problems that still nee to be consiere for future improvements of robot success in worlwie frames. Each chapter aresses a specific area of moeling, esign, an application of robots but with an eye to give an integrate view of what make a robot a unique moern system for many ifferent uses an future potential applications. Main attention has been focuse on esign issues as thought challenging for improving capabilities an further possibilities of robots for new an ol applications, as seen from toay technologies an research programs. Thus, great attention has been aresse to control aspects that are strongly evolving also as function of the improvements in robot moeling, sensors, servo-power systems, an informatics. But even other aspects are consiere as of funamental challenge both in esign an use of robots with improve performance an capabilities, like for example kinematic esign, ynamics, vision integration. How to reference In orer to correctly reference this scholarly work, feel free to copy an paste the following: A. Sanz an V. Etxebarria (008). Experimental Control of Flexible Robot Manipulators, Robot Manipulators, Marco Ceccarelli (E.), ISBN: , InTech, Available from: InTech Europe University Campus STeP Ri Slavka Krautzeka 83/A 5000 Rijeka, Croatia Phone: +385 (5) Fax: +385 (5) InTech China Unit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Roa (West), Shanghai, 00040, China Phone: Fax: