Visual Motion Observer-based Pose Control with Panoramic Camera via Passivity Approach

Transcription

1 Visual Motion Observer-based Pose Control with Panorac Camera via Passivity Approach Hiroyuki Kawai, Toshiyuki Murao and Masayuki Fujita Abstract This paper considers the vision-based estimation and control with a panorac camera via passivity approach. First, a hyperbolic projection of a panorac camera is presented. Next, using standard body-attached coordinate frames the world frame, rror frame, camera frame and object frame, we represent the body velocity of the relative rigid body motion position and orientation. After that, we propose a visual motion observer to estimate the relative rigid body motion from the measured camera data. We show that the estimation error system with a panorac camera has the passivity which allows us to prove stability in the sense of Lyapunov. After that, stability and L 2-gain performance analysis for the closedloop system are discussed. Finally, simulation and experimental results are shown in order to confirm the proposed method. I. INTRODUCTION Vision based control uses the computer vision data to control the motion of a robot in an efficient manner. The combination of mechanical control with visual information, so-called visual feedback control or visual servoing, is important when we consider a mechanical system working under dynacal environments 1. Recently, Lippiello et al. 2 presented a position-based visual servoing for a hybrid eye-in-hand/eye-to-hand multicamera configuration by using the extended Kalman filter and a multiarm robotic cell. Gans and Hutchinson 3 proposed a hybrid switched-system control which utilizes image-based and position-based visual feedback control. Hu et al. 4 considered a homography-based robust visual servo control for the uncertainty of the camera calibration. In our previous works, we discussed the dynac visual feedback control for 3D target tracking based on passivity 56. Although these previous works give us the new vision-based robot control theory systematically, most of the works use a simple perspective projection by a pinhole camera. On the other hand, omnidirectional cameras are useful for recognizing unknown surroundings widely. Geyer and Daniilidis 7 presented a unifying theory for all central panorac systems, i.e., an equivalence of catadioptric and spherical projections. Mariottini et al. 8 reviewed the several epipolar geometry estimation algorithms by using a omnidirectional camera and give us Epipolar Geometry Toolbox which is a simulation environment with MATLAB. Fomena and Chaumette 9 considered improvements on modeling features for visual servoing using a spherical projection. Although these novel methods need a desired image a priori, these works focus on the theoretical contributions. From the more practical point of view, Vidal et al. 1 H. Kawai is with Department of Robotics, Kanazawa Institute of Technology, Ishikawa , Japan hiroyuki@neptune.kanazawa-it.ac.jp T. Murao is with Master Program of Innovation for Design and Engineering Advanced Institute of Industrial Technology, Tokyo 14-11, Japan M. Fujita is with Department of Mechanical and Control Engineering, Tokyo Institute of Technology, Tokyo , Japan Fig. 1. Omnidirectional vision-based formation of mobile robots. used motion segmentation techniques to estimate the position of each leader for the vision-based formation control of mobile robots with central panorac cameras. Although the vision-based pose synchronization has been proposed in our previous work 11, a perspective projection model of a pinhole camera is used in order to estimate relative poses. This paper deals with vision-based control by using a panorac camera for mobile robot systems as depicted in Fig. 1. We propose the visual motion observer with a panorac camera in order to estimate the relative rigid body motion position and orientation. The main contribution of this paper is to show that the estimation error system with a panorac camera has the passivity which allows us to prove stability in the sense of Lyapunov. After that, stability and L 2 -gain performance analysis for the closed-loop system are discussed. Finally, simulation and experimental results are shown in order to confirm the proposed method. II. PANORAMIC CAMERA PROJECTION A. Hyperbolic Projection of Panorac Camera In this paper, we consider a panorac camera which consists of a pinhole camera and a hyperbolic rror as shown in Fig. 2. So, the pinhole camera catches reflected images through the hyperbolic rror. We first review the panorac camera model 8 in order to represent a image feature in our framework. Visual feedback systems by using a panorac camera use four coordinate frames which consist of a world frame Σ w, a rror frame Σ m, a camera frame Σ c, and a object frame Σ o as in Fig. 2. Let p mo R 3 and eˆξθ mo SO3 be the position vector and the rotation matrix from the rror frame Σ m to the object frame Σ o. Then, the relative rigid body motion from Σ m to Σ o can be represented by g mo p mo,eˆξθ mo SE3 1. Silarly, g wm p wm,eˆξθ wm, g wc p wc,eˆξθ wc and g wo p wo,eˆξθ wo denote the rigid body motions from the world frame Σ w to the rror frame Σ m, from the world frame Σ w to the camera frame Σ c and from the world frame Σ w to the object frame Σ o, respectively. 1 The notation of the homogeneous transform is referred to 5.

2 Fig. 3. Coordinate frames for mobile robots. Fig. 2. Panorac camera model pinhole camera and hyperbolic rror. Here, p wo is projected at s through the origin of the camera frame Σ c, after being projected at p mh through the origin of the rror frame Σ m asshowninfig.2.leta and b be the hyperbolic rror parameters which satisfy z mh + r 2 x2 mh + y2 mh 1 1 a 2 b 2 with eccentricity r a 2 + b 2. Eq. 1 means a constraint with respect to p mh. The transformation to obtain the projection s in the camera frame see, Fig. 2 is given by s 1 eˆξθ Λ cm αe ˆξθ wm p wo p wm + p cm 2 z ch where α is a scalar and defined as s : f x f y 1 T R 3 where f x and f y are the coordinates of x-axis and y- axis onto the image plane, respectively 8. In this paper, Λ is assumed as the ideal internal calibration matrix of the pinhole camera and defined as Λ:diag{λ, λ, 1} where λ is a focal length. From Fig. 2, the relation between the camera frame Σ c and the rror frame Σ m can be represented as p cm 2r T, eˆξθcm I 3 3 and it is assumed that these parameters are known. Moreover, p mh can be represented as follows: p mh αp mo 4 Because αp mo has to satisfy the constraint 1, the following relation holds αz mo + r 2 a 2 α2 x 2 mo + α 2 ymo 2 b Solving 5 for α, we obtain αp mo b2 rz mo ± a p mo 6 b 2 zmo 2 a 2 x 2 mo a 2 ymo 2 where αp mo represents that α depends on p mo explicitly 2. 2 Although αp mo has two solutions, the suitable solution will be selected by p mo 8. From Eqs. 2 and 3 and z ch 2r +z mh, the hyperbolic projection of the panorac camera can be represented as 1 s Λ αp mo p mo + p cm 7 2r + αp mo z mo where we exploit z mh αp mo z mo and the composition rule, i.e., p mo e ˆξθ wm p wo p wm. B. Image Feature for Panorac Camera Let p oi R 3 and p R 3 be the position vectors of the target object s i-th feature points i 1,,n,n 4 relative to Σ o and Σ m, respectively see Fig. 2. Using a transformation of the coordinates, we have p g mo p oi 8 where p and p oi should be regarded, with a slight abuse of notation, as p T 1T and p T oi 1T via the well-known homogeneous coordinate representation in robotics, respectively see, e.g., 12. The hyperbolic projection of the i-th feature point onto the image plane gives us the image plane coordinate f i : f xi f yi T R 2 as λαp x f i 9 2r + αp z y where αp means that p mo in Eq. 6 is replaced with p x y z T. It is straightforward to extend this model to n image points by simply stacking the vectors of the image plane coordinate, i.e., fg mo :f T 1 f T n T R 2n. 1 and p m : p T m1 p T mn T R 3n. We assume that multiple feature points on a known object are given. Although the problem of extracting the feature points from the target object is interesting in its own right, we will not focus on this problem and merely assume that the image feature are obtained by well-known techniques 13. From Eq. 8, the image feature f only depends on the relative rigid body motion g mo. III. BODY VELOCITY FOR PANORAMIC CAMERA The relative rigid body motion g mo is discussed in this section, because the image feature f only depends on g mo. We recall that visual feedback systems by using a panorac camera use four coordinate frames and the relative rigid

3 body motion from Σ m to Σ o can be represented by g mo p mo,eˆξθ mo as shown in Fig. 3. The relative rigid body motion from Σ m to Σ o can be led by using the composition rule for rigid body transformations 12, Chap. 2, pp. 37, eq as follows: g mo gwmg wo. 11 The relative rigid body motion involves the velocity of each rigid body. To this aid, let us consider the velocity of a rigid body as described in 12. We define the body velocity of the rror relative to the world frame Σ w as Vwm b vt wm ωt wm T,wherev wm and ω wm represent the velocity of the origin and the angular velocity from Σ w to Σ m, respectively 12 Chap. 2, eq Differentiating 11 with respect to time, the body velocity of the relative rigid body motion g mo can be written as follows See 5: Vmo b Ad g mo V wm b + Vwo. b 12 where Vwo b is the body velocity of the target object relative to Σ w. Eq. 12 is a standard formula for the relation between the body velocities of three coordinate frames 12 Chap. 2, pp. 59, Proposition Because the camera velocity Vwc b is adequate as a input rather than the rror velocity Vwo b in this framework, we lead the body velocity of the relative rigid body motion with the camera velocity. The body velocity of the rror frame relative to Σ w will be denoted as ˆV wm b gwmġ wm gwmġ wc g cm gwmg wc gwcġ wc g cm gcm ˆV wcg b cm. 13 From the property concerning the adjoint transformation, Vwm b can be transformed into Vwm b Ad gcm V b wc. 14 Thus, Eq. 12 can be transformed into Vmo b Ad gmo Ad gcm V b wc + V wo b. 15 This is the body velocity of the relative rigid body motion for the panorac camera. While g cm is known information from Eq. 3, g mo and g wo,i.e.vmo b and Vwo, b are unknown information in the visual feedback system. Then, the control objective is described as follows. Control Objective: The controlled mobile robot follows the target robot, i.e., the relative rigid body motion g co is coincided with the desired one g d. Because g cm is known a priori, g co can be obtained from g mo by the composition rule g co g cm g mo. Thus, we consider the estimate of g mo for the above control objective. IV. VISION-BASED ESTIMATION A. Image Jacobian for Panorac Camera Since the measurable information is only the image feature f from the panorac camera, we consider a visual motion observer in order to estimate the relative rigid body motion g mo from the image feature f. Using the body velocity of the relative rigid body motion 15, we choose estimates ḡ mo p mo,eˆ ξ θ mo and V mo b of the relative rigid body motion and velocity, respectively as V mo b Ad ḡ mo Ad gcm V wc b + u e. 16 The new input u e vue T ωue T T is to be deterned in order to drive the estimated values ḡ mo and V mo b to their actual values. Silarly to 8 and 9, the estimated image feature f i i 1,,n is defined as p ḡ mo p oi 17 λα p x f i 18 2r + α p z ȳ where p : x ȳ z T. fḡmo : f 1 T T f n T R 2n means the n image points case. In order to establish the estimation error system, we define the estimation error between the estimated value ḡ mo and the actual relative rigid body motion g mo as g ee : ḡmog mo 19 in other words, p ee e ˆ ξ θ mo p mo p mo and eˆξθ ee e ˆ ξ θ mo eˆξθ mo. We next define the error vector of the rotation matrix eˆξθ ab as e R eˆξθ ab : skeˆξθ ab where skeˆξθ ab denotes 1 2 eˆξθ ab e ˆξθ ab. Using this notation, the vector of the estimation error is given by e e : p T ee e T R eˆξθ ee T. From the above, we derive a relation between the actual and the estimated image feature. Suppose the attitude estimation error θ ee is small enough that we can let eˆξθ ee I + skeˆξθ ee. Then we have the following relation between the actual feature point p and the estimated one p p p eˆ ξ θ mo I ˆpoi p ee e R eˆξθ ee. 2 Using a first-order Taylor expansion approximation, the relation between the actual image feature and the estimated one can be expressed as f i f fi i x p p z p p y p p p p 21 where the partial differentiations / x, / y and / z are represented as Eqs at the bottom of the next page, respectively. Let us define the image feature error as f e : fg mo fḡ mo. Hence, the relation between the actual image feature and the estimated one can be given by f e Jḡ mo e e, 25 where Jḡ mo :SE3 R 2n 6 is defined as Jḡ mo :J1 T ḡ mo J2 T ḡ mo Jmḡ T mo T 26 fi J i ḡ mo : x p p y p p z p p eˆ ξ θ mo I ˆpoi. i 1,,n 27 We assume that the matrix Jḡ mo is full column rank for all ḡ mo SE3. Then, the relative rigid body motion can be uniquely defined by the image feature vector. Because this may not hold in some cases when n 3, it is known that

4 n 4 is desirable for the full column rank of the image Jacobian. The above discussion shows that we can derive the vector of the estimation error e e from image feature f and the estimated value of the relative rigid body motion ḡ mo, e e J ḡ mo f e 28 where denotes the pseudo-inverse. Therefore the estimation error e e can be exploited in the 3D visual feedback control law using image feature f obtained from the panorac camera. Remark 1: If we select one and the imaginary unit as a and b numerically, i.e., a 1, b i, then the image Jacobian for the panorac camera 27 equals to the pinhole s one f 5 as follows: i x λ z 1 T f, i y λ z 1 T, z λ x z 2 y T. Thus our previous work 5 can be regarded as a special case of this study, although the pinhole camera has different applications from the panorac one in the practical view. B. Passivity of Estimation Error System In the same way as Eq. 12, the estimation error system can be represented by Vee b Ad g ee u e + Vwo. b 29 Then, we have the following lemma relating the input u e to the vector form of the estimation error e e. Lemma 1: If Vwo b, then the following inequality holds for the estimation error system 29. u T e e edt β e 3 where β e is a positive scalar. Proof: Consider the positive definite function V e 1 2 p ee 2 + φeˆξθ ee 31 where φeˆξθ ab : 1 2 tri eˆξθ ab is the error function of the rotation matrix 14. Evaluating the time derivative of V e along the trajectories of 29 gives us V e p T ee eeeˆξθ e ˆξθ eeṗ ee + e T ee Reˆξθ eˆξθ ee ω ee e T e Ad e ˆξθee V ee b u T e e e. 32 Integrating 32 from to T yields u T e e edτ Vdτ V β e 33 where β e is the positive scalar which only depends on the initial state of g ee. Remark 2: Let us consider the vector form of the estimation error e e as its output. Then, Lemma 1 says that the estimation error system 29 is passive from the input u e to the output e e. In fact, the body velocity of the relative rigid body motion 15 has passivity, the estimation error system preserves its passivity. C. Visual Motion Observer Based on the above passivity property of the estimation error system, we consider the following control law. u e K e e e 34 where K e : diag{k e1,, k e6 } is the positive gain matrix of x, y and z axes of the translation and the rotation for the estimation error. Theorem 1: If V b wo, then the equilibrium point e e for the closed-loop system 29 and 34 is asymptotic stable. Proof: Theorem 1 can be easily proved by Lemma 1, hence the proof is otted. Fig. 4 shows a block diagram of the visual motion observer with a panorac camera. By the proposed visual motion observer, the unmeasurable motion g co will be exploited as the part of control law. Our proposed visual motion observer is composed just as Luenberger observer for linear systems. Remark 3: The estimation error vector is configured by available information i.e.,the measurement and the estimate though it is defined by unavailable one. This is one of the main contributions of this paper. where 2rλα x p x 2r + αp z 2 2rλα y p y 2r + αp z 2 2rλα z p z 2r + αp z 2 x λαp 1 + y 2r + αp z λαp + y 2r + αp z 1 λα 2 p x y 2r + αp z 2 y x x α x p αp 2a2 b 2 rx z p ±ab 2 x r 2 z 2 + a2 p 2 x b2 z 2 a2 x 2 a2 y 2 2 p α y p αp 2a2 b 2 ry z p ±ab 2 y r 2 z 2 + a2 p 2 y b2 z 2 a2 x 2 a2 y 2 2 p α z p αp b2 rb 2 z 2 + a2 x 2 + a2 y 2 p ab 2 z r 2 x 2 + y2 +b2 p 2 z b2 z 2 a2 x 2 a2 y 2 2 p

5 Fig Block diagram of visual motion observer. V. VISUAL MOTION OBSERVER-BASED POSE CONTROL A. Pose Control Error System Let us consider the dual of the estimation error system, which we call the pose control error system, in order to achieve the control objective. First, we define the pose control error as follows: g ec g d g co, 35 which represents the error between the relative rigid body motion g co and the reference one g d. It should be remarked that g co can be calculated by using the estimated relative rigid body motion ḡ mo, the estimation error vector e e p T ee e T R eˆξθ ee and the known rror parameter g cm equivalently, although g co can t be measured directly see 6 for more details. Using the notation e eˆξθ R, the vector of the pose control error is defined as e c : p T ec e T R eˆξθ ec T. The reference of the relative rigid body motion g d is assumed to be constant in this paper, i.e., ġ d and hence Vec b Vco. b Thus, the pose control error system can be represented as V b ec Ad g ec Ad g V wc b d + Vwo. b 36 This is dual to the estimation error system. Silar to the estimation error system, the pose control error system also preserves the passivity property. B. Passivity of Visual Motion Error System Combining the estimation error system 29 and the pose control one 36, we construct the visual motion observerbased pose control error system we call the visual motion error system as follows: V b ec V b ee Adg ec Ad g ee I u + V b I wo 37 where u : Ad g V wc b T u T T e. Let us define the error d vector of the visual motion error system as x : e T c et e T which consists of the pose control error vector e c and the estimation error vector e e. It should be noted that if the vectors of the pose control error and the estimation one are equal to zero, then the actual relative rigid body motion g co tends to the reference one g d when x. Next, we show an important relation between the input and the output of the visual motion error system. Lemma 2: If Vwo b, then the visual motion error system 37 satisfies u T xdt β, T > 38 where β is a positive scalar. Proof: Lemma 2 can be proved by using the positive definite function V 1 2 p ec 2 + φeˆξθ ec p ee 2 + φeˆξθ ee, hence the proof is otted. Remark 4: Let us take u as the input and x as its output. Thus, Lemma 2 implies that the visual motion error system 37 is passive from the input u to the output x. C. Visual Motion Pose Control and Stability Analysis Based on the above passivity property of the visual motion error system, we consider the following control law. Kc u Kx, K : 39 K e where K c : diag{k c1,,k c6 } is the positive gain matrix of x, y and z axes of the translation and the rotation for the pose control error. Theorem 2: If Vwo b, then the equilibrium point x for the closed-loop system 37 and 39 is asymptotic stable. Proof: Theorem 2 can be easily proved by Lemma 2, hence the proof is otted. Theorem 2 shows Lyapunov stability for the closed-loop system. If the camera velocity is decided directly, the control objective is achieved by using the proposed control law 39. Although the dead angle problem 9 is not considered explicitly in this framework, our proposed method will overcome this problem by dealing with a collision avoidance as in 11. D. L 2 -Gain Performance Analysis In this subsection, we utilize L 2 -gain performance analysis to evaluate the tracking performance of the control scheme in the presence of a moving target robot. The motion of the target robot is regarded as an external disturbance. In order to derive a simple and practical gain condition, we redefine K e k e I where k e is a positive scalar. Theorem 3: Given a positive scalar γ, assume k c,n > γ2 2k e 1 + 2k e 1 2 {γ 2 4 2k e 1 1} k e > γ2 +1 2γ 2 41 where k c,n means the nimum value in K c. Then the closed-loop system 37 and 39 has L 2 -gain γ. Proof: The proof is otted due to its silarity to the proof of Theorem 3 in 5. The estimation gain does not have to be restricted to a scalar, although it cause slightly complicated gain conditions compared with Eqs. 4 and 41. for the reason that Schur complement can not be used. In this framework, γ can be considered as an indicator of the tracking performance. VI. SIMULATION AND EXPERIMENTAL RESULTS Both the proposed estimation and control methods are confirmed by the simulation, while the experiment is carried out by using the proposed visual motion observer only.

6 Fig. 5. Estimation error e e Gain A: Dashed, Gain B: Solid. Fig. 6. Pose control error e c Gain A: Dashed, Gain B: Solid. Fig. 9. Image features and estimated ones w.r.t. the x-axis. Fig. 1. Estimation error e e for the static target object with k e 5. we can confirm that the estimation error e e tends to zero by using the proposed visual motion observer. Fig. 7. Panorac camera. Fig. 8. Target mobile robot. A. Simulation Results In this subsection, we present simulation results for stability and L 2 -gain performance analysis in the case of a moving target robot. The target robot has four feature points and moves by t 4.8 s. The gains for the control law u 39 were empirically selected as follows: Gain A γ.123, K c 5I, k e 1 Gain B γ.82, K c 1I, k e 3. The simulation results are presented in Figs. 5 and 6 which illustrate the estimation error e e and the pose control one e c, respectively. In these figures, we focus on the errors of the translations of x and y and the rotation of z. In Figs. 5 and 6, the dashed line and the solid line are the errors in the case of γ.123 and γ.82, respectively. In the case of the static target robot, i.e., after t 4.8 s, all errors in Figs. 5 and 6 tend to zero. Therefore, asymptotic stability can be confirmed through the simulation. In the presence of the moving target robot as disturbances by t 4.8 s, the tracking performance is improved for the smaller values of γ from Figs. 5 and 6. Thus the simulation results show that L 2 -gain is adequate for the performance measure of the visual motion observer-based pose control. B. Experimental Results In this subsection, we describe experimental results with respect to the proposed visual motion observer with a panorac camera for a static target object as shown in Figs. 7 and 8. A panorac camera consists of a MTV- 731 camera and a hyperbolic rror. The video signals are acquired by a frame graver board PICOLO DILLI- GENT Euresys and an image processing software HAL- CON MVTech Software GmbH. The experiment was carried out with an appropriate initial estimation error. The experimental results are presented in Figs. 9 and 1. In Fig. 9, the dashed lines and the solid ones mean the image features and the estimated ones with respect to the x-axis, i.e., f xi and f xi i 1,, 4, respectively. The estimated image features coincide with the actual ones after t 1s. From Fig. 1 which illustrates the estimation error of the translations of x and y and the rotation of z, VII. CONCLUSIONS This paper considers the vision-based estimation and control with a panorac camera based on the passivity. The main contribution of this paper is to show that the estimation error system with a panorac camera has the passivity which allows us to prove stability in the sense of Lyapunov. The visual motion error system which consists of the estimation error system and the pose control one preserves the passivity. Our previous work 5 can be regarded as a special case of this study. REFERENCES 1 F. Chaumette and S. A. Hutchinson, Visual Servoing and Visual Tracking, In:B. Siciliano and O. Khatib Eds, Springer Handbook of Robotics, Springer Verlag, pp , V. Lippiello, B. Siciliano and L. Villani, Position-Based Visual Servoing in Industrial Multirobot Cells Using a Hybrid Camera Configuration, IEEE Trans. on Robotics, Vol. 23, No. 1, pp , N. R. Gans and S. A. Hutchinson, Stable Visual Servoing Through Hybrid Switched-System Control, IEEE Trans. on Robotics, Vol. 23, No. 3, pp , G. Hu, W. MacKunis, N. Gans, W. E. Dixon, J. Chen, A. Behal, and D. Dawson, Homography-Based Visual Servo Control With Imperfect Camera Calibration, IEEE Trans. on Automatic Control, Vol. 54, No. 6, pp , M. Fujita, H. Kawai and M. W. 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