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Systems & Control LaboratoryKanazawa University 1 Passivity-based Control and Estimation of Visual Feedback Systems with a Fixed Camera University of Karlsruhe September 30 th, 2004 Masayuki Fujita Department of Electrical and Electronic Engineering Kanazawa University, Japan

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Systems & Control LaboratoryKanazawa University 2 Application of Visual Feedback System Fig. 2: Some Examples of Visual Feedback System Swing Automation for Dragline Autonomous Injection of Biological Cells etc. Introduction Automatic Laparoscope (Surgical Robot) etc. Eye-in-Hand Configuration Fixed-camera Configuration Fig. 1(a): Eye-in-Hand Camera Image Robot Target Object World Frame Camera Image Robot Target Object World Frame Fig. 1(a): Fixed-camera In the previous work, almost all the proposed methods depend on the camera configuration. Our proposed methods have the same strategy for both camera configurations. One of methods for the eye-in-hand configuration has been discussed in ACC, In this research, it will be extended to the fixed-camera configuration.

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Systems & Control LaboratoryKanazawa University 3 Position and Orientation Homogeneous Representation Fig. 3: Visual Feedback System Objective of Visual Feedback Control Direction of Rotation: Angle of Rotation: Rodrigues formula Objective of Visual Feedback Control One of the control objective is to track the target object in the 3D workspace. The relative rigid body motion must tend to the desired one Unknown Information ( Target motion is unknown.) Known Information Composition Rule

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Systems & Control LaboratoryKanazawa University 4 Fundamental Representation Fig. 3: Visual Feedback System Fundamental Representation of Relative Rigid Body Motion Body Velocity of Camera Body Velocity of Target Object : Translation : Orientation is the difference between and. Relative Rigid Body Motion in Fixed Camera Configuration : OMFC ( Camera is static. ) not measurable OMFC Fig. 4: Block Diagram of OMFC (Object Motion from Camera) (1) (2)

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Systems & Control LaboratoryKanazawa University 5 Target Object World Frame Camera Frame Image Plane : Focal Length Perspective Projection Relative Feature Points Fig. 5: Pinhole Camera Image Information (m points) Image information f includes the relative rigid body motion. Fig. 6: Block Diagram of OMFC with Camera Camera Model and Image Information ( depends on.) measurablenot measurable OMFCCamera (3) (4) (5)

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Systems & Control LaboratoryKanazawa University 6 Model of Estimated Relative Rigid Body Motion EsOMFC Estimated OMFC : Input for Estimation Error Estimated Image Information Fig. 7: Block Diagram of Estimated OMFC Nonlinear Observer in VFS Estimated Body Velocity Fig. 3: Visual Feedback System estimated EsOMFC Camera Model (6) (2) (7) (8)

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Systems & Control LaboratoryKanazawa University 7 Fig. 8: Block Diagram of OMFC and Estimated OMFC Nonlinear Observer in VFS continued Estimation Error (Vector Form) Relation between Estimation Error and Image Information measurablenot measurable OMFCCamera + EsOMFC Camera Model estimated (Error between Estimated State and Actual One) Fig. 3: Visual Feedback System estimated (9) Estimation Error System (10)

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Systems & Control LaboratoryKanazawa University 8 Desired RRBM Control Error Fig. 9: Block Diagram of HMFC and Reference Control Error System (Vector Form) (Error between Estimated State and Desired One) Fig. 3: Visual Feedback System Estimated Information by Composition Rule Relative Rigid Body Motion from to (HMFC: Hand Motion from Camera ) (11) Control Error System (This is dual to the estimation error system.) (12) HMFC +

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Systems & Control LaboratoryKanazawa University 9 Control Error System Estimation Error System Visual Feedback System with Fixed Camera Configuration Visual Feedback System Fig. 10: Block Diagram of Control and Estimation Error Systems OMFC Camera EsOMFC Camera Model + (13) HMFC +

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Systems & Control LaboratoryKanazawa University 10 Visual Feedback System with Fixed Camera Configuration Visual Feedback System Fig. 11: Block Diagram of Visual Feedback System Controller + OMFC Camera EsOMFC Camera Model + (13) HMFC +

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Systems & Control LaboratoryKanazawa University 11 Property of Visual Feedback System Fig. 11: Block Diagram of Visual Feedback System Controller + Camera Model + Lemma 1 If the target is static, then the visual feedback system (13) satisfies where is a positive scalar. OMFCCamera EsOMFC HMFC (14) +

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Systems & Control LaboratoryKanazawa University 12 (Proof) Differentiating the energy function (15) with respect to time along the trajectories of the visual feedback system yields is a skew-symmetric matrix skew-symmetric matrices Integrating both sides from 0 to T, we can obtain (14) Passivity Property of Manipulator Dynamics (Q.E.D.) Property of Visual Feedback System continued Energy Function (15) Error Function of Rotation Matrix

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Systems & Control LaboratoryKanazawa University 13 If, then the equilibrium point for the closed -loop system (14) and (16) is asymptotically stable. Passivity-based Visual Feedback Control Law Stability Analysis Theorem 1 Gain Fig. 11: Block Diagram of Visual Feedback System Controller + Camera Model + OMFCCamera EsOMFC HMFC (16) +

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Systems & Control LaboratoryKanazawa University 14 Tracking Problem L 2 -gain Performance Analysis Based on the dissipative systems theory, we consider L 2 -gain performance analysis in one of the typical problems in the visual feedback system. Disturbance Attenuation Problem Fig. 12: Generalized Plant of Visual Feedback System + + OMFCCamera EsOMFC HMFC + Camera Model

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Systems & Control LaboratoryKanazawa University 15 Tracking Problem L 2 -gain Performance Analysis Based on the dissipative systems theory, we consider L 2 -gain performance analysis in one of the typical problems in the visual feedback system. Disturbance Attenuation Problem Given a positive scalar and consider the control input (31) with the gains and such that the matrix P is positive semi-definite, then the closed-loop system (28) and (31) has L 2 -gain. Theorem 2 represents a disturbance attenuation level of the visual feedback system. Other problems can be considered by constructing the adequate generalized plant.

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Systems & Control LaboratoryKanazawa University 16 Experimental Testbed on 2DOF Manipulator

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Systems & Control LaboratoryKanazawa University 17 Stability Analysis L 2 -gain Performance Analysis Lyapunov Function Storage Function Future Works Visual Feedback System Fundamental Representation of Relative Rigid Body Motion Conclusions ( is the difference between and.) Nonlinear Observer in Visual Feedback System Passivity of Visual Feedback System Fig. 3: Visual Feedback System Dynamic Visual Feedback Control (with manipulator dynamics) Uncertainty of the camera coordinate frame (one of calibration problems) Energy Function Our proposed methods have the same strategy for both camera configurations.

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Systems & Control LaboratoryKanazawa University 18 Appendix

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Systems & Control LaboratoryKanazawa University 19 Application of Visual Feedback System Fig. 1: Some Examples of Visual Feedback System Swing Automation for Dragline Autonomous Injection of Biological Cells Automatic Laparoscope (Surgical Robot) etc. Introduction Control Vision Robotics Research Field of Visual Feedback System Control will be more important for intelligent machines as future applications. Fig. 2: Research Field of VFS (Machines + Visual Information) x Control Visual Feedback Control with Fixed-camera Passivity-based Control In this research Visual Feedback Control

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Systems & Control LaboratoryKanazawa University Introduction 2. Objective of VFC and Fundamental Representation 3. Nonlinear Observer and Estimation Error System 4. Control Error System 5. Passivity-based Control of Visual Feedback System Outline 6. Conclusions

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Systems & Control LaboratoryKanazawa University 21 Position and Orientation Homogeneous Representation Fig. a1: Visual Feedback System Relative Rigid Motion (a1) Homogeneous Representation Camera Motion Target Object Motion Relative Rigid Motion Direction of Rotation: Angle of Rotation: Rodrigues formula

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Systems & Control LaboratoryKanazawa University 22 (Ref.: R. Murray et al., A Mathematical Introduction to Robotic Manipulation, 1994.) Body Velocity Body Velocity by Homo. Rep. Body Velocity of Camera MotionBody Velocity of Target Object Motion Body Velocity of Relative Rigid Body Motion by Homo. Rep. : Translation : Orientation (a2) (a3) (a4) (a5) Fundamental Representation of Relative Rigid Body Motion

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Systems & Control LaboratoryKanazawa University 23 (by Adjoint Transformation) (Adjoint Transformation) (Ref.: R. Murray et al., A Mathematical Introduction to Robotic Manipulation, 1994.) RRBM Body Velocity of Relative Rigid Body Motion (by Homo. Rep.) Fig. a2: Block Diagram of RRBM (a6) (1) (a7) Fundamental Representation of Relative Rigid Body Motion continued (Fundamental Representation of Relative Rigid Body Motion RRBM

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Systems & Control LaboratoryKanazawa University 24 Estimation Error System (10) (a8) Estimation Error System ( Detail of Derivation of Estimation Error System ) (This system is obtained from (a8) by the property of Adjoint Transformation (a7).)

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Systems & Control LaboratoryKanazawa University 25 Image Jacobian Relation between Estimation Error and Image Information (9) Taylor Expansion with First Order Approximation

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Systems & Control LaboratoryKanazawa University 26 Image Jacobian continued Representation by 4 dimension Representation by 3 dimension

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Systems & Control LaboratoryKanazawa University 27 Image Jacobian continued Relation between Estimation Error and Image Information (9)

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Systems & Control LaboratoryKanazawa University 28 Object Frame World Frame Camera Frame Hand Frame Fig. 13: Experimental Testbed for Dynamic Visual Feedback Control Experimental Testbed on a 2DOF Manipulator

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Systems & Control LaboratoryKanazawa University 29 Experiment for Stability Analysis Fig. 14: Initial Condition Field of View Target Object 2DOF Manipulator Gains Fig. 15: Trajectory of Manipulator

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Systems & Control LaboratoryKanazawa University 30 Experimental Results Experimental Results (Stability Analysis) Fig.16: Control Errors Fig. 17: Estimation Errors Fig. 18: Joint Velocity Errors Fig. 19: Euclid Norm of States

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Systems & Control LaboratoryKanazawa University 31 Target Motion in Experiment The target moves on the xy plane for 9.6 seconds. Fig. 20(a): Straight Motion Fig.20(b): Figure 8 Motion L 2 -gain Performance Analysis in Experiment

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Systems & Control LaboratoryKanazawa University 32 Fig. 21: Norm of z Experimental Results for L 2 -gain Performance Analysis Experimental Results Gain A Gain B represents a disturbance attenuation level. Gain A Gain B

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