Robots with Flexible Links: Modeling and Control

Transcription

1 Scuola di Dottorato CIRA Controllo di Sistemi Robotici per la Manipolazione e la Cooperazione Bertinoro (FC), Luglio 23 Robots with Flexible Links: Modeling and Control Alessandro De Luca Dipartimento di Informatica e Sistemistica (DIS) Università di Roma La Sapienza

2 Outline Motivation for considering distributed link flexibility Dynamic modeling of FL robots: single link and multiple link cases Formulation of control problems Controllers for regulation tasks Controllers for joint and end-effector trajectory tracking tasks Controllers for rest-to-rest motion tasks Conclusions References 2

3 Motivation distributed link deformation in robot manipulators arises when the design of very long and slender arms is needed for the application lightweight materials are used (without special care) link rigidity is always an ideal assumption and may fail when increasing payload-to-weight ratio motion speed control bandwidth flexible structures in motion are present in different domains: space manipulators, robots for underwater and underground waste sites, automated cranes,... as for joint elasticity, neglected link flexibility limits static (steady-state error) or dynamic (vibrations, poor tracking) task performance from the control point of view, there is an additional problem of non-colocation between input commands and typical outputs to be controlled 3

4 Robots with link flexibility SSRMS Space Shuttle Remote Manipulation System (Canadarm) telemanipulated by astronauts link bending due to fast motion (not gravity!) 4

5 Robots with link flexibility Sam II developed at Georgia Tech (W. Book) macro-micro concept for remote exploration and manipulation of nuclear waste sites 5

6 Robots with link flexibility prototypes in Roma DMA harmonic steel beam (.5 kg): DD-DC motor, encoder, 7 strain gauges DIS/DIA FLEXARM: planar two-link with flexible forearm (1.8 kg), DD-DC motors, encoders, on-board optical sensor measuring deformation at three points 6

7 Robots with link flexibility prototype in Waterloo WATFLEX planar 2R with both link flexible (each with 2 strain gauges), moving with air bearings on a glass table to support the weight of the second motor; encoders, tachometers, overviewing CCD camera 7

8 Robots with link flexibility prototypes in Japan spatial 3R flexible arm at Kyoto cooperating 6R flexible arms at Tohoku 8

9 Frequency identification of a single flexible link deg/s s [ left ] frequency sweep joint acceleration signal: plant vs. model [ right ] joint acceleration frequency response: plant vs. model (matching within 1% of resonances at f 1 = 14.4, f 2 = 34.2, and f 3 = 69.3 Hz) 9

10 Dynamic modeling of a single flexible link one-link flexible arm modeled as a Euler-Bernoulli beam in rotation Y m p J p x y CoM J θt θ θ c X length l, uniform density ρ, Young modulus cross-section inertia EI actuator inertia J, payload mass m p and inertia J p reference frames: (X, Y ) absolute; (x, y) moving with instantaneous CoM 1

11 Assumptions and definitions Euler-Bernoulli theory applies to slender arm design (length vs. section) beam undergoes small deformations of pure bending type in the plane of motion (no torsion or compression) bending deformation w(x, t), with x [,l], is directed along the y direction (no shear) rotational inertia of beam sections is neglected ( Timoshenko theory) as well as the isoperimetric constraint ( extension of beam neutral axis is negligible) definition of other relevant angular variables: position θ(t) of the CoM (not measurable, but convenient) position θ c (t) of the tangent to the link base (measured by motor encoder) position θ t (t) of a line pointing to the beam tip (measurable in several ways) 11

12 build the Lagrangian from kinetic and elastic potential energy of the beam using Hamilton principle and calculus of variations, the bending deformation w(x, t) and angle θ(t) satisfy the linear differential equations EIw (x, t)+ρ(ẅ(x, t)+x θ(t)) = τ(t) J θ(t) = i.e., a PDE and an ODE (rigid motion), where J = J +(ρl 3 )/3+J p +m p l 2 and τ = torque input geometric/dynamic boundary conditions associated to the PDE w(,t) = EIw (,t) = J ( θ(t)+ẅ (,t)) τ(t) (balance of moments at base) EIw (l, t) = J p ( θ(t)+ẅ (l, t)) (balance of moments at tip) EIw (l, t) = m p (l θ(t)+ẅ(l, t)) (balance of shear forces at tip) 12

13 in free evolution (τ(t) θ(t) =), PDE solved by separation of variables w(x, t) =φ(x)δ(t) EI ρ φ (x) φ(x) = δ(t) δ(t) = ω 2 for a positive constant ω 2 (self-adjoint problem) to be determined time solution δ(t) = ω 2 δ(t) δ(t) =c 1 sin ωt + c 2 cos ωt with c 1,c 2 depending on the initial conditions δ() and δ() space solution (try it!) φ (x) =β 4 φ(x) β 4 = ρω2 EI φ(x) =A sin βx + B cos βx + C sinh βx + D cosh βx with A, B, C, D obtained from the geometric/dynamic b.c. s on w(x, t) 13

14 using w(x, t) =φ(x)δ(t) and δ = ω 2 δ, and holding the b.c. s for any δ(t), these are rewritten in terms of φ(x) only φ() = EIφ () + J ω 2 φ () = EIφ (l) J p ω 2 φ (l) = EIφ (l)+m p ω 2 φ(l) = using the general solution φ(x), a system of linear homogeneus equations follows A(EI,ρ,l,J,m p,j p, β) A B C D = to exclude the trivial solution, the determinant of matrix A should be zero (eigenvalue problem) ( ) 14

15 det A(β) =at infinite (but countable!) positive increasing roots β i of the trascendental characteristic equation (csh sch) 2m p ρ β i ssh m p ρ 2 β4 i (J + J p )(csh sch) 2J p ρ β3 i cch J ρ β 3 i (1 + cch)+j J p ρ 2 β6 i (csh+ sch) J J p m p ρ 3 βi 7 where s = sin β i l, c = cos β i l, sh = sinh β i l, ch = cosh β i l (1 cch)= this is an exact result, that includes common physical approximations clamped-free model: m p = J p =, J 1+cch= pinned-free model: m p = J p = J = csh sch = 15

16 associated to each root β i we have: an eigenfrequency ω i = EIβi 4 /ρ, characterizing a system vibration an eigenvector φ i (x) a spatial deformation mode, defined up to a constant a deformation variable δ i (t), oscillatory in time a finite-dimensional approximation of the distributed bending deformation is obtained by truncation w(x, t) = i=1 φ i (x)δ i (t) n e i=1 φ i (x)δ i (t) where n e is the (arbitrary) number of orthogonal modes that we wish to include normalization of the modes can be chosen in different ways (some integral of the φ i (x) equal to 1, to m,...) 16

17 resulting dynamic model is particularly simple (after fairly complex analysis...) J θ = τ δ i + ω 2 i δ i = φ i ()τ i =1,...,n e remarkable properties: rigid motion θ(t) and each vibratory motion δ i (t) are decoupled in free evolution (τ(t) ) all modes are excited by an input command τ(t), with a weighting that depends on φ i () the tangent at the link base of each deformation mode arm stiffness is summarized by the (squared) eigenfrequencies ω i each vibratory motion is persistent during free evolution, if initially excited by δ i () (absence of damping) 17

18 modal damping can be easily included in the dynamic model J θ = τ δ i +2ζ i ω i δ i + ωi 2 δ i = φ i ()τ i =1,...,n e with coefficients ζ i [, 1) its matrix version, with generalized coordinates q = ( θδ 1...δ ne ) T IR n e+1, shows the classical mass-spring-damper form with M = [ J I ] D = M q + D q + Kq = Bτ [ ] 2ZΩ K = [ ] Ω 2 B = [ ] 1 Φ Ω=diag {ω 1,...,ω ne }, Z = diag {ζ 1,...,ζ ne }, Φ = ( φ 1 ()...φ n e () ) T 18

19 a different (but equivalent) choice of generalized coordinates may let the input τ appear in just one equation leads to [ J JΦ (θ, δ) =(θ, δ 1,...,δ ne ) (θ c,δ)=(θ + δ T Φ,δ)=(θ + φ i ()δ i,δ 1,...,δ ne ) JΦ T I + J 2 Φ Φ T ][ θ c δ ] [ + 2ZΩ ][ θ c δ ] [ + Ω 2 ][ ] θc with same (diagonal) damping D and stiffness K matrices, but full inertia matrix δ = [ ] 1 τ 19

20 Choice of the controlled output joint level angular output (clamped angle) θ c = θ + n e i=1 φ i () δ i!! is always minimum phase (zeros in left-hand side of complex plane) tip level angular output θ t = θ + n e i=1 φ i (l)!! is typically non-minimum phase (at least for no tip payload) output at a point x [,l] along the link θ x = θ + n e i=1 l φ i (x) x δ i δ i 2

21 torque τ clamped joint angle θ c = P c (s) = 1 ne Js J i=1 Transfer functions φ i ()2 s 2 +2ζ i ω i s + ω 2 i ne i=1 (s2 +2ζ i ω i s + ω 2 i )+s2 n e i=1 φ i ()2 ne j i (s2 +2ζ j ω j s + ω 2 j ) torque τ tip angle θ t = P t (s) = 1 ne Js J i=1 s 2 n e i=1 (s2 +2ζ i ω i s + ω 2 i ) φ i () φ i(l) l s 2 +2ζ i ω i s + ω 2 i ne i=1 (s2 +2ζ i ω i s + ωi 2)+s2 n e i=1 φ i () φ i(l) ne l j i (s2 +2ζ j ω j s + ωj 2) s 2 n e i=1 (s2 +2ζ i ω i s + ωi 2) 21

22 A numerical example physical data J =.2, l =1, ρ =.5, EI =1 (m p = J p =) by considering up to n e =5modes (and no damping), we obtain Ω 2 = diag { , , , , } Φ [ T = ] Φ T l = [ ]!! note the alternating signs of φ i (l) s... 22

23 First four mode shapes phi_ x [m] 2 phi_ x [m] phi_3 -.5 phi_ x [m] x [m] at f 1 =3.2678, f 2 =8.8936, f 3 = , and f 4 = [Hz] 23

24 Pole-zero patterns 6 poli-zeri FdT di giunto (2 modi) 6 poli-zeri FdT di tip (2 modi) 15 poli-zeri FdT di giunto (3 modi) 15 poli-zeri FdT di tip (3 modi) Imag Axis Imag Axis Imag Axis Imag Axis Real Axis Real Axis Real Axis Real Axis two modes clamped joint and tip output three modes 24

25 Frequency responses 1 1 Magnitude (db) 5-5 Magnitude (db) Frequency (rad/sec) Frequency (rad/sec) Phase (deg) -9 Phase (deg) Frequency (rad/sec) Frequency (rad/sec) clamped joint three modes tip 25

26 Useful control-oriented remarks in pole-zero patterns of P c (s), zeros precede and alternate with poles on the imaginary axis passivity property zero patterns of P t (s) are symmetric w.r.t. the imaginary axis non-minimum phase property no direct system inversion is feasible while moving the output along the link (P x (s)), zeros migrate along imaginary axis and several phenomema occurr: total pole-zero cancellation when pointing at CoM (vibrations unobservable from rigid motion variable θ) for a particular x (,l), all zeros vanishes together at infinity (P x (s) has maximum relative degree equal to 2(n e +1)) beyond x (e.g., for x = l), all pairs of zeros reappear in Re + /Re 26

27 modal damping destroys perfect symmetry in zeros and poles (system is anyway asymptotically stable), but not the non-minimum phase property of P t (s) from the Bode plots, it follows that classical controller synthesis in the frequency domain is harder for the tip output multiple crossing of db axis of P t (jω) especially for high control gain increasing phase lag when adding modes 27

28 Dynamic modeling of robots with multiple flexible links a convenient kinematic description should be used, both for rigid body motion and flexible deformation recursive procedures can be set up for open chains with flexible links, similarly to the rigid case differential kinematic relationships are needed for computing kinetic and potential energy, within a Lagrangian approach modeling results from the one-link case can be embedded (with caution) in the description of each flexible link of the robot to limit complexity, we sketch here only the planar case (with gravity) of robots with N flexible links undergoing small bending deformations 28

29 Kinematics X 2 X 1 w θ 2 (x 2 ) 2 Y 2 Y 2 =Y 3 X 2 =X 3 Y 1 Y 1 Y w 1 (x 1 ) X 1 θ 1 X link i: rigid motion by clamped angle θ i (t); lateral bending w i (x i,t), x i [,l i ] position vectors and (rigid/flexible) rotation matrices (w ie = w i i p i (x i )= [ ] x i w i (x i ) i r i+1 = i p i (l i ) A i = [ cos θi sin θ i sin θ i cos θ i ] x i xi =l i ) E i = [ 1 w ie w ie 1 29 ]

30 recursive equations for absolute quantities in ( ˆX, Ŷ ) p i = r i + W i i p i r i+1 = r i + W i i r i+1 W i = W i 1 E i 1 A i differential kinematics absolute angular velocity of frame (X i,y i ) α i = i j=1 θ j + i 1 ẇ ke k=1 absolute linear velocity of a point on link i ṗ i =ṙ i + Ẇ i i p i + W iiṗ i where i ṗ i = [ ẇ i (x i )] T (link extension is neglected) 3

31 Kinetic energy hub i T = N i=1 T hi + N i=1 T li + T p T hi = 1 2 m hiṙ T i ṙi J hi α 2 i link i payload T li = 1 2 li ρ i(x i )ṗ i (x i ) T ṗ i (x i )dx i T p = 1 2 m pṙ T N+1ṙN J p( α N +ẇ Ne )2 31

32 U = N i=1 elastic energy of link i U ei = 1 2 U ei + Potential energy li N i=1 U ghi + (EI) i(x i ) gravitational energy of hub i and of link i N i=1 U gli + U gp ( d 2 ) 2 w i (x i ) dx i dx 2 i U ghi = m hi g T r i U gli = g T gravitational energy of payload li ρ i(x i )p i (x i )dx i U gp = m p g T r N+1 where g is the gravity acceleration vector 32

33 Euler-Lagrange equations introduce any finite-dimensional discretization for w i (x i,t) w i (x i,t)= n ei j=1 ϕ ij (x i )δ ij (t) i =1,...,N Lagrangian is in terms of N + M generalized coordinates, M = N i=1 n ei, L = T U = L ( {θ i (t),δ ij (t), θ i (t), δ ij (t)} ) and satifies to ( ) d L L = τ i i =1,...,N dt ( θ i ) θ i d L L = j =1,...,n ei i =1,...,N dt δ ij δ ij being τ i the torque delivered by the actuator at joint i 33

34 the general dynamic model of flexible robots (including modal damping) is then [ ] [ ] [ ] [ ] [ Mθθ (θ, δ) M θδ (θ, δ) ][ θ cθ (θ, δ, θ, δ) gθ (θ, δ) τ Mθδ T (θ, δ) M = δδ(θ, δ) δ c δ (θ, δ, θ, δ) g δ (θ, δ) D δ + Kδ with blocks of suitable dimensions (e.g., M θδ in the inertia matrix is (N M)), or in the more compact form M(q) q + c(q, q)+g(q)+ with q := (θ, δ) IR N+M [ D δ + Kδ vector of centrifugal/coriolis terms can be factorized using Christoffel symbols [ ][ ] Cθθ (q, q) C c(q, q) =C(q, q) q = θδ (q, q) θ C δθ (q, q) C δδ (q, q) δ ] = [ τ ] 34 ]

35 Model properties as usual, matrix Ṁ 2C is skew-symmetric also blockwise, e.g. Ṁ δδ 2C δδ spatial dependence in the kinetic and potential energy of the links can be resolved introducing a set of dynamic coefficients so that (De Luca, Siciliano, 1991) [ ] τ Y (θ, δ, θ, δ, θ, δ)a = where constant vector a summarizes the mechanical (rigid + flexible) properties of the links and can be computed off-line (or identified experimentally) choice of assumed modes basis functions ϕ ij (x i ) for bending deformation admissible functions satisfy only geometric b.c. s comparison functions (FE method, Ritz-Kantorovich) satisfy also natural b.c. s orthonormal eigenfunctions (links modeled as Euler-Bernoulli beams) leads to simplifications in inertia submatrix M δδ (block diagonal, constant) 35

36 a common approximation evaluates total kinetic energy in the undeformed arm configuration δ = M = M(θ) and thus c = c(θ, θ, δ) c δ loses quadratic dependence on δ moreover, if M δδ is constant c θ loses quadratic dependence on δ c δ is a quadratic function of θ only if also M θδ is constant c δ c θ is a quadratic function of θ only finally, small deformation of each link implies g δ = g δ (θ) 36

37 Control problems regulation to a constant equilibrium configuration (θ, δ, θ, δ) =(θ d,δ d,, ) only the desired joint position θ d is assigned, while δ d has to be determined θ d may come from the kineto-static inversion of a desired cartesian pose x d, but no closed-form solution exists (see De Luca, Panzieri, 1994) direct kinematics of FL robots is in fact a complete function of rigid and flexible variables: x = kin(θ, δ) tracking of a joint trajectory θ d (t) the easy case tracking of a end-effector trajectory x d (t) the difficult case rest-to-rest motion in given time T (a trajectory planning problem in first place) in tracking problems, controllers try to stiffen the flexible arm at a point in a way or the other 37

38 Sensing requirements full state feedback requires sensing of joint/motor variables (θ, θ), deflections δ, and deflection rates δ (no direct sensor available) at least encoder + tachometer on the motor axis (sometimes is enough...) a range of sensors for measuring δ (or deformation related quantities), each with pros and cons: strain gauges, accelerometers, optical sensors, video camera (on-board or fixed in workspace), piezoelectric actuating/sensing devices,... 38

39 problems with camera: frame rate, field of view/accuracy 39

40 Regulation with joint PD + feedforward for regulation tasks, consider the control law τ = K P (θ d θ) K D θ + g θ (θ d,δ d ) with symmetric (diagonal) K P >, K D >, and the associated link deflection δ d := K 1 g δ (θ d ) Theorem (De Luca, Siciliano, 1993) If λ min ([ KP K ]) >α then the closed-loop equilibrium state (θ d,δ d,, ) is asymptotically stable 4

41 Remarks Lyapunov-based proof similar to the joint elastic case (not repeated here) determination of α in view of small deformation U e = 1 2 δt Kδ U e,max δ bound on the gradient of the gravitational term g q α + α 1 δ α + α 1 2Ue,max 2Ue,max λ max (K) λ max (K) =: α in absence of modal damping D =, special care in LaSalle analysis for tip regulation, compute θ d by solving via iterative techniques kin ( θ, K 1 g δ (θ) ) = x d 41

42 Numerical results a two-link flexible arm with two bending modes for each link with f 11 =1.4, f 12 =5.1, f 21 =5.2, f 22 =32.4 [Hz] point-to-point motion: θ() = ( 9, ) θ d =( 45, ) 5 joint angles 2 joint torques 1 deg Nm sec sec.1 1st link deflections.5 2nd link deflections m m sec sec 42

43 Joint trajectory tracking given a desired θ d (t) C 2, assuming that the state is measurable and the dynamic model of FL robot is available, we proceed by system inversion from the joint position output θ a nonlinear static state feedback is obtained that decouples and linearizes the input-output behavior, leaving an unobservable internal (nonlinear) dynamics exponential stabilization of the output tracking error is performed on the linear side of the problem some stability/boundedness of the internal system dynamics should be enforced original results in (De Luca, Siciliano, 1993b) 43

44 System inversion from second set of M equations in the dynamic model, solve (globally) for δ δ = M 1 δδ (c δ + g δ + Kδ + D δ + M T θδ θ) and plug in first set of N equations effects of flexible dynamics onto rigid dynamics ( Mθθ M θδ Mδδ 1 M θδ T ) θ + c θ + g θ M θδ Mδδ 1 ( cδ + g δ + Kδ + D δ ) = τ Matrix M θθ M θδ M 1 [ Mθθ M θδ M T θδ M δδ ][ δδ M T θδ I Mδδ 1 M θδ T I has always full rank, since ] = [ Mθθ M θδ Mδδ 1 M θδ T M θδ M δδ ] 44

45 system output θ has uniform vector relative degree {2, 2,..., 2} ( θ depends on τ in a nonsingular way) define the nonlinear control law τ = ( M θθ M θδ Mδδ 1 M θδ T ) a + cθ + g θ M θδ Mδδ 1 (c δ + g δ + Kδ + D δ) in which only the inversion inertia block M 1 the closed-loop system is δδ is required θ = a δ = Mδδ 1 ( M T θδ a + c δ + g δ + D δ + Kδ ) for stabilizing the output tracking error e = θ d θ, choose with (diagonal) K P >, K D > a = θ d + K D ( θ d θ)+k P (θ d θ) 45

46 Analysis of internal dynamics zero dynamics, when output θ(t) (or constant): δ = Mδδ 1 ( cδ + g δ + D δ + Kδ ) asymptotically stable (via Lyapunov argument) whole closed-loop system too clamped dynamics, when output θ(t) θ d (t): δ = A 2 (t) δ A 1 (t)δ + f δ (t) where (in the case M independent of δ) f δ (t) = Mδδ 1 d)(mθδ T (θ d) θ d + c δ (θ d, θ d )+g δ (θ d )) A 1 (t) = Mδδ 1 d)k A 2 (t) = Mδδ 1 d)d all time-varying functions are bounded closed-loop stability is ensured 46

47 Remarks on joint trajectory tracking the input-output linearization result is the nonlinear/mimo counterpart of the transfer function τ θ c with minimum phase zeros (stable zero dynamics) the more rigid is the tracking of a desired joint trajectory, the less vibration energy is taken out from (or the more is injected into) the rest of flexible arm!! a nominal feedforward is computed by forward integration of flexible dynamics δ = M 1 δδ (θ d, δ)(c δ (θ d, δ, θ d, δ)+g δ (θ d )+D δ + Kδ + M T θδ (θ d, δ) θ d ) from δ() = δ, δ() = δ nominal (and bounded) evolution δ d (t), δ d (t) substitution of (θ d (t),δ d (t), θ d (t), δ d (t)) in the expression of the control law (w/out feedback) yields τ d (t) and the simple local tracking controller τ = τ d (t)+k P (θ d θ)+k D ( θ d θ) 47

48 End-effector trajectory tracking accurate end-effector trajectory tracking is the toughest control problem for flexible robots direct extension of inversion strategies to end-effector output closed-loop instabilities linear (single-link) case: non-minimum phase tip transfer function nonlinear (multilink) case: unstable zero dynamics for end-effector motion main ideas proposed in the literature: resort to suitable feedforward strategy (non-causal solutions) use feedback, but avoid cancellation (causal solutions) selection of suitable end-effector trajectories that induce smaller arm deflections is of interest in any case 48

49 Inversion in frequency domain idea: desired motion trajectory as being part of a periodic profile use Fourier transforms (Bayo, 1987) single-link flexible arm (with generic variables) [ mθθ m T ] [ ][ ] [ ][ ] [ ] δθ ][ θ θ θ τ + + = δ D δ K δ m δθ M δδ tip position output rewrite in terms of (y, δ) [ mθθ m T δθ m θθc T e m δθ M δδ m δθ c T e y(t) =[1 c T e ] ][ÿ ] δ + [ D [ ] θ δ ][ẏ ] δ + [ K ][ ] y δ = [ τ 49 ]

50 take bilateral Fourier transforms Ÿ (ω) = and obtain m θθ exp(jωt)ÿ(t)dt (ω) = T (ω) = exp(jωt)τ(t)dt m T δθ m θθc T e m δθ M δδ m δθ c T e + 1 jω D 1 ω 2 K solve for accelerations [ ] Ÿ (ω) (ω) = exp(jωt) δ(t)dt [ Ÿ (ω) (ω) [ g11 (ω) g12 T (ω) ][ T (ω) g 21 (ω) G 22 (ω) ] = ] [ T (ω) ] 5

51 torque is obtained through inversion (in the frequency domain) T (ω) = 1 g 11 (ω)ÿ (ω) =r(ω)ÿ (ω) for a given zero-mean ÿ d (t), with ÿ d (t) =for t T/2 and t T/2, this can be embedded into a periodic signal from (, + ) ÿ d (t) Ÿ d (ω) T d (ω) τ d (t) from finite inverse Fourier transform τ d (t) = r(t τ)ÿ d(τ)dτ = T/2 T/2 r(t τ)ÿ d(τ)dτ expanding beyond [ T/2,T/2] (non-causal inverse system) 51

52 Remarks outside the given interval T of output motion, the computed input torque has a precharging action, bringing internal flexible state from rest to a suitable initial state at t = T/2 discharging action, bringing internal flexible state from the final state at t = T/2 to rest obtained initial condition is the unique state from which inversion control does lead to bounded internal evolution for the desired end-effector output trajectory! truncations (in time or frequency domain) inherent to actual computations (FFT) can be extended to the nonlinear (multilink flexible) setting, by repeated linear approximations along nominal trajectory (starting from rigid body motion) 52

53 Extension to nonlinear case: End-effector bang-bang trajectory gradi / sec ^ sec Nm sec y (m) x (m) acceleration profiles command torques FLEXARM motion 53

54 Extension to nonlinear case: End-effector sinusoidal trajectory gradi / sec ^2 Nm.5 y (m) sec sec x (m) acceleration profiles command torques FLEXARM motion 54

55 Regulation theory end-effector trajectory tracking in robots with flexible links is an instance of asymptotic output tracking with internal state stability (regulation problem) well-established technique in linear case and, by now, also in nonlinear case idea: compute the (bounded!) state trajectory associated to the desired output trajectory (generated by an autonomous antistable system, the exosystem) in linear case, write state-space equations (with x =(q, q)) for the flexible arm ẋ = Ax + Bτ e = y y d and for the generator of desired output ẇ = Sw y d = Qw 55

56 when (A, B) is stabilizable, the problem has a solution if and only if the following regulator equations are solvable in Π and Γ ΠS = AΠ+BΓ CΠ+Q = a state feedback + feedforward controller is then τ = F (x Πw)+Γw with feedback matrix F such that (A + BF) is Hurwitz the computed Πw is the desired state trajectory; x d () = Πw() is the unique initial state from which inversion control does lead to bounded internal evolution! control solutions with dynamic output feedback are also available 56

57 Numerical results for sinusoidal end-effector trajectory 1 autovalori in -1 (quattro) e -1 (quattro) 7 autovalori in -1 (quattro) e -1 (quattro) uscita tip (deg) 2-2 errore al tip (deg) sec sec tip output tip error 57

58 .3 autovalori in -1 (quattro) e -1 (quattro) 1.5 autovalori in -1 (quattro) e -1 (quattro).2 1 variabili deformazione delta (m) controllo (Nm) sec sec deformation variables torque input 58

59 1 autovalori in -1 (quattro) e -1 (quattro) uscita al giunto (deg) sec clamped joint angle (and desired tip output) 59

60 Rest-to-rest motion task: execute a slew motion with a FL robot arm between two undeformed configurations in given time problem: fast transfers induce residual oscillations extending the actual task completion time strategy: design a suitable output and plan output trajectories (and associated torque profiles) inducing complete absence of vibrations at the desired final time solution: output with maximum relative degree (no zeros); closed-form algorithm in the linear case; direct extension to MIMO nonlinear case (DFL or flat output, no zero dynamics) 6

61 Time-based algorithm for a single flexible link choose a parametric output y (unknown c i s) (De Luca, Di Giovanni, 21) y = θ + n e i=1 c i δ i = θ + c T δ impose τ-independence of (even) output derivatives ÿ = ( 1 n e J + i=1 y [4] =: d4 ne y dt 4 = c i φ n e i ())τ i=1 i=1 c i ωi 2 n e φ i () τ + c i ωi 2 δ i c i φ i () = 1/J i=1 c i ω 4 i δ i c i ω 2 i φ i () = and so on, until a set of n e equations are generated (torque τ appears in the 2(n e +1)-th output derivative) 61

62 solve for the coefficients c =(c 1,...,c ne ) V diag{φ 1 (),...,φ n e ()} c = [ 1/J...] T with Vandermonde matrix V generated by (ω 2 1,...,ω2 n e ) nominal torque τ d (t) computed by inversion on highest derivative imposing y [2(n e+1)] = y [2(n e+1)] d for a suitably planned output trajectory y d (t), t [, T ] (given trasfer time) for the output trajectory y d (t), solve a simple interpolation problem d i y y d () = θ i y d (T )=θ d f dt i () = di y d dt i (T )= i =1,...,2n e +1 e.g., a polynomial of degree 4n e +3will be sufficient 62

63 Laplace-based algorithm (only in the linear case) impose that the transfer function has no zeros y(s) τ(s) = 1 ne Js 2 + c i φ i () i=1 s 2 + ωi 2 partial fractions expansion yields closed-form expressions = K s 2 n e i=1 (s2 + ω 2 i ) K = 1 J n e i=1 ω 2 i c i = 1 Jφ i () ne j=1 j i ω 2 j ω 2 j ω2 i (i =1,...,n e ) set y = y d and invert in the transformed domain (then back to time τ d (t)) τ d (s) = s 2 n e (s 2 + ωi 2 ) y d (s) J ne i=1 ω2 i i=1 63

64 Remarks method applies to any linear (controllable) model of a single-link flexible arm output structure for modal damping (De Luca, Caiano, Del Vescovo, 23) y = θ + n e i=1 c i δ i + γ θ + design output is (in the limit) a specific point x on the physical beam: for a given n e, c i = φ i (x n e )/x n e while lim ne x n e = x for improved torque/time performance, modified method generates smoothed bang-bang/bang-coast-bang torque profiles, with polynomial interpolating phases n e i=1 d i δ i trajectory planning (feedforward) combined with feedback control τ = τ d (t)+k P (θ c,d (t) θ c )+K D ( θ c,d (t) θ c ) with clamped joint reference θ c,d (t) computed from the algorithm 64

65 Numerical results n e =3modes with f 1 =4.5, f 2 =12.34, f 3 =22.87 [Hz] θ f θ i =9 in T =2s 19th degree polynomial (continuous torque derivative) deg 5 Nm s output trajectory s rest-to-rest torque 65

66 deg 5 m s s m clamped ( ), tip angle (- -) deformation variables arm motion 66

67 Experiment on DMA single flexible link 67

68 Extension to nonlinear case: Rest-to-rest motion time-based algorithm for two-link with flexible forearm (De Luca, Di Giovanni, 21b).25 delta 8 u1, u Nm m s s m first deformation mode command torques FLEXARM motion 68

69 Conclusions extra effort in dynamic modeling pays off model-based controllers for accurate trajectory tracking proof of stability for model-inpedendent regulation controllers conventional control strategies tend to suppress vibrations wherever they arise outcome of the analysis: controlled system should be brought to a vibratory behavior compatible with the given output task EJ robots are similar to FL robots in mechanical modeling, but intrinsically different from the control point of view 69

70 What has been left out... singular perturbation modeling and control (for joint or link stiffness K ), including corrective and invariant manifold controllers iterative learning control that yields same accuracy (for all types of tasks) without using a dynamic model but assuming repetitive tasks model uncertainties, disturbances,... 7

71 References (Bayo, 1987) A finite-element approach to control the end-point motion of a single-link flexible robot, JRS, 4, (De Luca, Caiano, Del Vescovo, 23) Experiments on rest-to-rest motion of a flexible arm, in Experimental Robotics VIII (Siciliano, Dario Eds), STAR 5, Springer, (De Luca, Di Giovanni, 21) Rest-to-rest motion of a one-link flexible forearm, IEEE/ASME AIM 1, (De Luca, Di Giovanni, 21b) Rest-to-rest motion of a two-link robot with a flexible forearm, IEEE/ASME AIM 1, (De Luca, Lanari, Ulivi, 1991) End-effector trajectory tracking in flexible arms: Comparison of approaches based on regulation theory, in Advanced Robot Control (Canudas de Wit Ed), LNCIS 162, Springer, (De Luca, Panzieri, 1994) An iterative scheme for learning gravity compensation in flexible robot arms, Automatica, 3(6),

72 (De Luca, Panzieri, Ulivi, 1998) Stable inversion control for flexible link manipulators, IEEE ICRA, (De Luca, Siciliano, 1991) Closed-form dynamic model of planar multi-link lightweight robots, IEEE SMC, 21(4), (De Luca, Siciliano, 1993) Regulation of flexible arms under gravity, IEEE TRA, 9(4), (De Luca, Siciliano, 1993b) Inversion-based nonlinear control of robot arms with flexible links, AIAA JGCD, 16(6), (De Luca, Siciliano, 1996) Flexible links, in Theory of Robot Control (Canudas de Wit, Siciliano, Bastin Eds), Springer,