H preview control of an active stabilizer for heart-beating surgery E. Laroche (1), W. Bachta (2), P. Renaud (1), J. Gangloff (1) (1) LSIIT (UMR CNRS-ULP 7005), U. Strasbourg & CNRS, France (2) ISIR, Univ. Paris 6, France French-Israeli Workshop on Delays and Robustness Haifa, Israel, April 3-5, 2011 H preview control of an active stabilizer 1
Outline 1 Context of beating-heart surgery 2 Control Issue 3 Control design 4 Robustness analysis 5 Conclusion H preview control of an active stabilizer 2
Outline 1 Context of beating-heart surgery 2 Control Issue 3 Control design 4 Robustness analysis 5 Conclusion H preview control of an active stabilizer 3
Beating-heart surgery Context of beating-heart surgery Context Coronary Artery Bypass Grafting (CABG): a frequent operation in the case of heart blood irrigation insufficiency Current most used procedure: stop the heart and implementation of an extra corporeal circulation CABG with heart-beating operation: reduce complications Use of mechanical stabilizers in order to immobilize the area of operation (ex: Octopus by Medtronic) Figure: Invasive stabilizer: Octopus 4.3 (Medtronic) Limitation of current stabilizers Residual displacement about 1 mm [Cattin04] Required accuracy : 0.1 mm Insufficient of endoscopic surgery [Loisance05] Investigated solution Figure: Endoscopic stabilizer: Octopus TE (Medtronic) H preview control of an active stabilizer 4
Cardiolock: an active cardiac stabilizer Cardiolock 1 Description Beam Diameter compatible with minimally invasive surgery (10 mm diameter) Sterilizable in autoclave Fixed on the heart by suction Actuation system H preview control of an active stabilizer 5
Cardiolock: an active cardiac stabilizer Cardiolock 1 Description Beam Actuation system Parallel mechanism Rotating compliant joints: no backlash Linear piezo actuator: high dynamics and accuracy Enclosed in a sterile bag H preview control of an active stabilizer 5
Cardiolock: an active cardiac stabilizer Cardiolock 2 Full system Detail on one DOF 2 DOF Each DOF is actuated by a parallel mechanism in quasi-singulaity H preview control of an active stabilizer 6
Outline 1 Context of beating-heart surgery 2 Control Issue 3 Control design 4 Robustness analysis 5 Conclusion H preview control of an active stabilizer 7
Cardialock under operation Cardialock under operation Perturbation p Video camera Drive ECG Control u Computer Position y Perturbation rejection (heart beating and respiration) Availability of the frequencies of the heart (ECG) and respiration (artificial ventilation) for constructing a model of the perturbation H preview control of an active stabilizer 8
Dynamic model Dynamic model l2 F q1 Modelling Under the PRBS assumption l1 q2 Figure: Simplified scheme with equivalent rigid deformation Control : u = q 1 Measurement by camera : y = position of the tip of the beam H preview control of an active stabilizer 9
Dynamic model G(s) u(k) ZOH y(t) y(k) camera z 1 v(k) F c u(t) = q 1 (t) Figure: Bloc-diagram of the system M 21 q 1 + M 22 q 2 = l 2 F c K 2 q 2 f 2 q 2 (1) q 1 = u (2) y = (l 1 + l 2 )q 1 + l 2 q 2 (3) Flexible non-minimum phase system H preview control of an active stabilizer 1
Control issue u(k) H(z) - + p(k) v(k) Figure: Simplified scheme for control design Camera + ZOH equivalent to a UOH [IFAC 2008] ( ) 1 z H(z) = z 1 1 2 ( ) ZL 1 G2 (s) T s 2 Control issue Rejection of an output perturbation Estimation of the perturbation with ˆp = H(z) u v H preview control of an active stabilizer 1
Control issue Frequency response Figure: Frequency response of identified H(z) H preview control of an active stabilizer 1
Perturbation model Prediction of the perturbation Two components: heart (dφ c/dt = 2πf c where f c is evaluated after each ECG period) ventilation (dφ r/dt = 2πf r where f r is given by the ventilation system) Perturbation signal p(t) = M r(t)+m c(t) Ventilation component only from ventilation phase: M r(t) = n r l=1 a l sin ( lφ r(t) ) + b l cos ( lφ r(t) ) Heart composante based on both heart and ventilation phases : M c(t) = C c(t)(1+c r(t)) where C c(t) = n c l=1 e l sin ( lφ ) c(t) + f l cos ( lφ ) c(t) C r(t) = n r l=1 g l sin ( lφ ) r(t) + h l cos ( lφ ) r(t) Change of variable in order to obtain a linear-parameter model (p(t) = n θ l=1 θ l φ(t)) and parameter estimation with recursive mean square Prediction ˆp(t +δ) = n θ l=1 ˆθl φ(t +δ) H preview control of an active stabilizer 1
Perturbation model Evaluation on experimental data 180 160 pixels 140 120 100 80 0 2 4 6 8 10 12 temps (s) Figure: Residual displacement measured with a passive stabiliser (plain) and 3-samples ahead prediction (dashed, T e = 3 ms) H preview control of an active stabilizer 1
Outline 1 Context of beating-heart surgery 2 Control Issue 3 Control design 4 Robustness analysis 5 Conclusion H preview control of an active stabilizer 1
Simple feedback controller Several approaches for rejection of quasi-periodic perturbation Dynamic output feedback Estimate and compensate Least-square recursive estimation Kalman filter Repetitive control (in discrete-time) Adaptive compensation (direct adaptation of the parameters of a perturbation model [Bodson 2001]) H preview control of an active stabilizer 1
Simple feedback controller Simple feedback controller (synthesis in continuous time) W 1 (s) z 1 W 2 (s) z 2 p K(s) u H(s) + - v Figure: 2-blocs synthesis scheme (for tuning modulus margin, accuracy, bandwidth and roll-off) Transfert de p vers y The Transfer from p to u 10 10 gain (db) 0 10 20 30 40 gain (db) 0 10 20 30 40 50 50 10 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) 60 10 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) Figure: Frequency response (features: dot; system behavior: plain) H preview control of an active stabilizer 1
Simple feedback controller Resonant feedback controller W 1 (s) is modified with a resonant filter adapted to the cardiac frequency 20 The Transfer from p to e 20 The Transfer from p to u 10 10 0 0 gain (db) 10 20 30 40 50 gain (db) 10 20 30 40 60 10 0 10 1 10 2 10 3 10 4 frequency (rad/sec) 50 10 0 10 1 10 2 10 3 10 4 frequency (rad/sec) H preview control of an active stabilizer 1
Preview controller 2-DOF controller (feedback + feedforward) p(t) K(z) u H(z) v Figure: Control scheme with measured perturbation K(z) = [K 1 (z) K 2 (z)], H(z) = [H 1 (z) H 2 (z)] T vp(z) = (I H 2 (z) K 2 (z)) 1 (H 1 (z)+h 2 (z)k 2 (z)) Feedback K 2 (z) for rejection in low frequency (robust to the model errors) Feedforward K 1 (z) to enhance the rejection at higher frequency (K 1 (z) = H 1 2 (z) H 1 (z)) (sensitive to the model errors) Restriction if H 1 (z) have non proper or non sable inverse (both in our case) Solution: synthesis of K(z) in one shot (idem as an additional measurement) Limitation: the information comes too late for an efficient control action H preview control of an active stabilizer 1
Preview controller Preview controller p(t) p(t +τ) avance u K(z) H(z) - + v Figure: Principle of preview control Anticipation made possible by the prediction model Controller synthesis in one shot Similitude with predictive control: requires to know in advance the future samples of the exogenous signal (i.e. reference or perturbation) Equivalent for reference tracking H preview control of an active stabilizer 2
Preview controller Full control scheme ˆp(t +τ) advance ˆp(t) Ĥ(z) p(t) + - K(z) u H(z) + - v Figure: Control scheme with estimation of the perturbation H preview control of an active stabilizer 2
Preview controller Synthesis scheme v 2 W 3 (s) p delay w 1 + - e K(s) u H(s) - + v W 2(s) z 2 W 1 (s) z 1 Figure: Synthesis scheme for the 2-DOF controller allowing to tune separately the feedback and feedforward effects ( p(t) = e τs p(t)) Synthesis in continuous-time (continuous-to-discrete conversion with the bilinear transform) Pade approximation of the delay Advance from the prediction model H preview control of an active stabilizer 2
Preview controller Transfer de r vers e Transfert de p vers e 10 20 gain (db) 0 10 20 30 40 gain (db) 0 20 40 60 50 10 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) 10 0 10 Transfert de r vers u 80 10 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) 20 10 0 Transfert de p vers u gain (db) 20 30 gain (db) 10 20 40 30 50 40 60 10 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) 50 10 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) Figure: Frequency response with the 2-DOF preview controller (features: dots; realized system: plain) H preview control of an active stabilizer 2
Nominal evaluation Laboratory experimental setup light source cardiolock camera video spring heart simulator Heart movement emulated by a pan-tilt robot H preview control of an active stabilizer 2
Nominal evaluation Experimental results Correcteur sans connaissance a priori Correcteur avec prise en compte de la fréquence cardiaque Correcteur avec modèle de mouvement cardiaque 1.4 1.4 0.2 1.3 1.3 0.1 1.2 1.2 0 1.1 1.1 0.1 mm 1 mm 1 mm 0.2 0.9 0.9 0.8 0.8 0.3 0.7 0.7 0.4 0 5 10 15 20 25 30 temps (sec) 0 5 10 15 20 25 30 temps (sec) 0.5 0 5 10 15 20 25 30 temps (sec) Simple feedback Resonant feedback 2-DOF with preview In-vivo tests were also made H preview control of an active stabilizer 2
Nominal evaluation Evaluation in simulation with experimental data Control method RMS displacement (pixel) No control 22.3 Simple feedback 2,57 Resonant feedback 1,69 2-DOF with preview with perfect prediction 0,064 2-DOF with preview with estimated prediction 1,21 Table: Residual displacement obtained with the nominal model (prediction made with n c = 10 and n r = n r = 4) H preview control of an active stabilizer 2
Nominal evaluation Residual movement frequency analysis 10 0 Position (pixels) 10 1 10 2 10 3 10 4 0 2 4 6 8 10 Frequency (Hz) blue: simple feedback; red: resonant feedback; purple: 2-DOF with preview with perfect prediction; green: 2-DOF with preview with estimated prediction H preview control of an active stabilizer 2
Outline 1 Context of beating-heart surgery 2 Control Issue 3 Control design 4 Robustness analysis 5 Conclusion H preview control of an active stabilizer 2
Uncertain model Robustness issue Modification of the system behavior when in contact with the heart Interaction model F = F c k c y f c kẏ m c ÿ (4) F c: exogenous perturbation Nominal values: m c = 2 g, K c = 250 N/m and f c = 0.1 N.s/m Consider variations from 0 à 200 % H preview control of an active stabilizer 2
Uncertain model µ-analysis context Constants uncertains parameters LFR model Use of a performance criterion Robust if µ < 1 H preview control of an active stabilizer 3
Simple feedback control LFR model (stability + performance) z 1 w 1 c w 2 r z 2 W 1 (s) K(s) u H u (s) p + - v Figure: Structure du modèle LFR incertain c real diagonal; r full complex H preview control of an active stabilizer 3
Repetition index of the uncertain parameters Parameter Direct Reduction Robust toolbox m c 9 3 1 K c 3 2 1 f c 3 1 1 Table: Repetition index of the uncertain parameters of the LFR model obtained with different methods (Direct and Reduction: with LFR toolbox H preview control of an active stabilizer 3
µ plot 0.9 0.8 bornes sup. et inf. de µ 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 1 10 2 10 3 10 4 pulsation (rad/sec) Figure: Structured singular value µ < 1: robust to the considered uncertainties H preview control of an active stabilizer 3
2-DOF preview control Non causal model T yp(s) = lft(g(s), e τs ) that cannot be factorized (i.e. T yp(s) = e τ 1s G 1 (s)e τ 2s ) usual tools cannot be used Evaluation in simulation with p k = ρ p k0 where p k0 is the nominal value and ρ [0;2] valeur efficace du mouvement résiduel (pixel) 3 2.5 2 1.5 1 0 0.5 1 1.5 2 ρ Figure: Variation of the residual motion with respect to the parameter value ρ (plain: feedback control; dashed: preview control) H preview control of an active stabilizer 3
Outline 1 Context of beating-heart surgery 2 Control Issue 3 Control design 4 Robustness analysis 5 Conclusion H preview control of an active stabilizer 3
Conclusion Simple and efficient procedure for synthesis of preview H control Improvement thanks to the prediction of the pertuabation Obtained accuracy in accordance with the requirements for heart-beating surgery Future work Robustness analysis for the 2-DOF prevew controller with estimation Evaluation of Cardiolock 2 Comparison with GPC H preview control of an active stabilizer 3