Coordinated Mobile Manipulator Point-Stabilization Using Visual-Servoing Techniques
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1 Coordinated Mobile Manipulator Point-Stabilization Using Visual-Servoing Techniques Marco Gilioli, Claudio Melchiorri DEIS - Department of Electronics, Computer Science, and Systems University of Bologna, via Risorgimento 2, 436 Bologna, Italy {mgilioli, cmelchiorri}@deis.unibo.it Abstract In this paper we consider the problem of stabilizing in a desired configuration a mobile manipulator; only the arm s joint displacements information and the measures provided by a camera mounted on the end-effector are used to stabilize the system. In particular, no knowledge about the position and the orientation of the mobile base is supposed to be available. An hybrid control algorithm, based on the concatenation of a sensor-based feedback control and an open-loop strategy, is proposed. A 3-dof planar manipulator mounted on a mobile base, modelled as an unicycle, is considered as case study, and simulation results are reported in order to demonstrate the capabilities of the proposed control algorithm. Introduction In this paper, we consider the control problem for mobile manipulators, i.e. systems composed by a robotic arm installed on a mobile base. These systems are able to reach and operate over targets which are initially outside the working space of the robot arm. Mobile manipulators can be divided into many categories, depending on the tipology of the mobile base. In particular, nonholonomic wheeled vehicles, subject to the rolling-without-slipping constraint, are considered,. As far as the control of these devices is concerned, one can identify two main problems: the control of the mobile base and the base/arm coordination. With respect to the mobile base, it is well known that a continuous state feedback law, able to exponentially stabilize the wheeled base to a desired configuration, cannot exist, 2. This problem has been solved by several authors with different approaches. For example, in 3, 4 (among may others) a time-varying state feedback control has been proposed. The base/arm coordination problem has been addressed in different manners. For example, in and 5 the authors, assuming that the manipulator s end-effector follows a pre-computed trajectory, control the base in order to maintain the arm in a preferred configuration, that is computed on the basis of its manipulability index. Recently, in order to obtain more robust input-output stabilization laws, visual data provided by a camera mounted on the robot have been considered for the design of feedback control loops. In 6, the problem of tracking a curve with a mobile base using only the camera information has been considered. In 4 and 7, a visual-servoing technique able to solve the pointstabilization problem of the mobile manipulator has been proposed. The base is stabilized by an hybrid time-varying technique, and its state is reconstructed on the basis of the camera information, assuming as known the position of a target. The camera measures are also used to stabilize the manipulator. In this paper, the stabilization problem for a mobile manipulator using a vision-based approach is considered. The proposed control strategy takes into consideration two typical problems arising in mobile manipulator control: the difficulties in the base state reconstruction and the coordination problem. In particular, if the base control inputs are computed by considering only the instantaneous arm joint displacements and the manipulator is controlled using the camera information, the base reconstruction problems are avoided. Moreover, an extra task must be solved if the camera measures are considered to stabilize the manipulator: the targets must be maintained inside the camera field of view during the whole stabilization process. The structure of this paper is the following. In Section 2, the kinematic model of the mobile manipulator and of the camera measures is reported. In Section 3, the control law, able to stabilize the system in a desired configuration, is discussed, while in Section 4 simulation results are reported to validate the proposed ap-
2 frag replacements y y b θ b F F b x b F e Camera Figure : Definition of the reference frames. proach. Finally, in Section 5 considerations and ideas for future works are discussed. 2 Kinematic Model In this paper a mobile manipulator, composed by a nonholonomic base modelled as an unicycle and equipped with a n-dof planar manipulator, is considered. Furthermore, a camera is mounted on the end effector of the robot arm, see Fig. (). If F is the reference frame, F b the mobile base and F e the end-effector frames, then c b = p T b, θ b T represents the configuration of the base in F, being p b = x b, y b T the position and θ b the angle between the x-axes of F b and F. Moreover, c e = p T e, θ e T represents the configuration of F e with respect to F and b c e = b p T e, b θ e T the position and the orientation of the end-effector with respect to F b. Finally, the camera frame is assumed coincident with F e. The kinematic differential model of the unicycle is given by ṗ b θb = J b ( c b ) u = cos θ b sin θ b x v ω () where u can be considered as the control input of the standard unicycle and J b its Jacobian matrix. Let q = T q... q n be the arm joint positions vector. The relationship between the end-effector configuration and the base can be written as: b c e (q) = b p e (q) b θ e (q) = b f ep (q) b f eθ (q) (2) By differentiating (2), one obtains the relationship between the velocity of the end-effector and the joint velocities q: bṗ e b θe = b f ep (q) q b f eθ (q) q q = b J e (q) q (3) In the planar case, the tip configuration in F can be easily computed as: { p e = p b + R( θ b ) b p e θ e = θ b + b θ e (4) cos( ) sin( ) where R( ) = is the rotation matrix about the z sin( ) cos( ) axis. Let now suppose to measure by the camera 3 fixed targets in the environment p i = x i y i T, i =, 2, 3. If Z i is the camera measure of the i-th target, then Z = Z Z 2 Z 3 T represents the camera measures set. Note that Z i depends on e p i, the position of the i-th target in the end-effector frame, and is given by the projection function f( e p i ) : IR 2 IR of this target on the camera plane: Z i = f( e p i ) = f( e x i, e y i ) (5) where e p i can be easily computed as: From (5), one obtains: e p i = R ( θ e )( p i p e ) (6) Z i = f( e p i ) eṗ e i (7) p i Note that only terms referred to F e appear in eq. (7), i.e. terms referred to the camera frame. Then, Ż does not depend on the reference frame F, that therefore can be arbitrarily chosen. For the sake of simplicity, we consider F F b, and therefore c b = T. From (6) and (7), the relationship between the movements of the end-effector and the temporal variation of the camera measure is derived as: Z i = J i f ( e p i ) R ( b θ e ) ċ e (8) where Jf i ( e p i ) = f f e p i. If the 3 targets are not aligned, e p i can be computed from Z and the a priori knowledge of the geometric relations between the targets, therefore Jf i = J f i (Z). From (), (3), and (4), ċ e becomes: ċ e = b y e b x e u + R( b θ e ) b J e (q) q = J b (q)u + J m (q) q (9)
3 If J f (Z) = J f (Z) J 2 f (Z) J 3 f (Z) () is the 3 3 interaction matrix of the camera measure set, from (9) and (8) the differential kinematic model of the measure set becomes: Ż = J f (Z) Jb (q)u + J m (q) q = J B (Z, q)u + J M (Z, q) q () where J R b (q) = ( b θ e ) J b (q) and J m (q) = R ( b θ e ) J m (q). Eq. () relates the variations of the camera measures with the base and arm control inputs (i.e. joint velocities). 3 Coordinated Hybrid Control In this Section we address the point-stabilization problem for a planar mobile manipulator using the data provided by a camera mounted on the end-effector. Let us define as Z d the desired camera measures, i.e. the camera measures when the mobile manipulator is in the desired configuration c md = c T bd, qt d T, being c bd the desired base pose and q d the desired manipulator joint displacement. The point-stabilization problem can be formulated as: Problem Definition. Find a control law U( ) = u T ( ) q T ( ) T such as lim Z T (t) q T (t) T = t Z T d qd T T. It is well known that it is not possible to stabilize a wheeled mobile robot to a desired pose by using a continuous state feedback, 2. In 4, the authors propose a time-varying feedback law able to stabilize the base/arm system. This approach is based on the concatenation of open-loop sequences computed taking into account the base configuration at the beginning of each sequence. From these results, we have developed an hybrid control law based on the concatenation of a coordinated feedback control U f with an open-loop time-varying control sequence U o : U f (Z, Z d, q, q d ) nt t < (n + )T U(t) = U o(q, q d, t) (n + )T t < (n + 2)T (2) In fact, if the arm configuration and the camera measure coincide with the desired ones, also the base is in the desired configuration being T a proper period and n =, 2, 4,.... The feedback control law U f considers the arm configuration and the camera measures and it is computed in order to coordinate the base with the manipulator movements. The strategy can be summarized as follows: while the manipulator is controlled in order to reduce the camera errors and maintain the targets inside the camera field of view, the base moves to maintain the robot arm far from singularities. The purpose of the open-loop time-varying control sequence U o is to bring the manipulator to the desired configuration. During this phase the manipulator is controlled only to compensate the movements of the base, i.e. in order to maintain the end-effector in the same position. On the other hand, the base is controlled using an open-loop strategy, computed on the basis of the arm configuration at the beginning of the sequence. 3. Coordinated Feedback Control The coordinated feedback control U f is applied in order to reduce the camera measure errors and, at the same time, to maintain the manipulator far from singularities. If e = Z Z d are the measure errors, then ė = Ż. The manipulator control input q can be computed in order to guarantee an exponential convergence to zero of the camera errors, i.e. ė = λe. From eq. () the manipulator control inputs q are computed as: q(t) = J + M (Z, q)( λe J B (Z, q)u) (3) where J + M is the pseudo-inverse of matrix J M. Note that each term in (3) depends only on the camera measures Z and on the arm joint displacements q. From the definition of J M and J B, if J f and J m are not singular, eq. (3) can be simplified as: q(t) = λj M (Z, q)e J m (q) J b (q)u (4) J m where J f does not appear in the second term ( J b u), fact that increases the overall control robustness. In the following, a 3-dof manipulator will be considered as case study. Eq. (4) can be used also to compute the base control input u. Given the desired joints displacement q d, then e 2 = q q d is the displacement error, and ė 2 = q. Then, from (4) one obtains ė 2 = λj M (q)e J m J b (q)u = λ b e 2 (5) and the base input u can be determined as: u = ( J J m b ) + (λ b e 2 λj M e ) (6) Since it is desirable to avoid singular configurations for the arm, it is necessary to introduce a motion coordination of the 2 systems. For this purpose, the error
4 convergency ratio λ in (6) is modulated as: { λ(q) = g( q qd ) if q q d < K max λ(q) = otherwise (7) where g( ) : IR + IR + is a strictly decreasing real function, for example g( ) = λ e K q qd, and K max is a threshold used to avoid singular configurations. In fact, if the manipulator is sufficiently far from the desired configuration (where also singular configurations are considered), then λ and ė : in this manner (see (3)), the manipulator is forced to compensate only the motion of the base, that is controlled in order to reduce q q d (see (6)). This control law is based on a continuous state feedback, so it is not able to stabilize the whole systems to a desired configuration. It guarantees a convergence of the measures errors and avoids singular configurations. To solve the point-stabilization problem, an open-loop control sequence is added to achieve the convergence of the whole system to the desired pose. 3.2 Open-Loop Control The purpose of the open-loop control sequence U o is to steer the mobile base in order to have the manipulator in the desired configuration q d, while the arm is controlled to maintain constant the camera measures. By defining S(q) = { q : det( b J e (q)) } as the set of all the non singular arm configurations, and given the arm joint displacements at the beginning of the control sequence q = q(t ), the control problem can be formulated as: Problem Definition. Find a control law U o (q, t) such that q(t ) = q d, q( ) S(q) and c e (q( )) = c e (q(t )), t t, t, being t = (n + )T and t = (n + 2)T. By posing λ = in (4), the arm control inputs are computed to maintain the end-effector in the same configuration with respect to the target (despite the base movements): q(t) = J m (q) J b (q)u(t) (8) Given a base control u(τ), the manipulator configuration q(t) can be computed as Assuming the reference frame coincident with the base frame, i.e. c b =, and recalling eq. (4) and (2), the final base configuration c bf = p bf, θ bf T (see Fig. 2), used for the open-loop control computation, becomes: p bf + R( θ bf ) b f ep (q d ) = b f ep (q ) (2) θ bf + b f eθ (q d ) = b f eθ (q ) c bf is computed at the beginning of the open-loop control sequence and depends only on q and q d. If the base reaches c bf, while the end-effector is maintained in the same configuration, q is forced to reach q d. Now, c bf can be used to define an open-loop time-varying sequence able to steer the mobile base to this configuration. In particular, u =, arctan( y bf ) T < t T T x bf arctan( ybf 2 T 2 T + x2 bf ), T T < t T 2, arcsin(α) T T 2 < t T T T 2 where < T < T 2 < T, t = t (n + )T and α = cos( θbf ) sin( θ bf ) 4 Simulations cos(arctan( y bf x bf )) sin(arctan( y bf x bf )) (2). (22) As a case study, a mobile manipulator composed by a unicycle-like mobile base equipped with a 3-dof planar manipulator has been considered. The manipulator Ja- p bf θ bf Final Base Configuration Manipulator Preferred Configuration t q(t) = q J m (q(τ)) J b (q(τ))u(τ)dτ (9) t PSfrag replacements then the joint trajectory is completely defined by q and u(t). If the arm is controlled with eq. (8), the control problem can be reformulated as: Problem Definition. Compute an open-loop base control ū(q, t) such as q(t ) = q d and q( ) S(q), t t t. F Initial Base Configuration Figure 2: Computation of the final configuration for the mobile base in the open-loop control sequence.
5 cobian b J e is: L i sin( q j ) i= j= b Je = L i cos( q j ) i= j= L i sin( q j ) i=2 j= L i cos( q j ) i=2 j= L 3 sin( q j ) j= L 3 cos( q j ) j= (23) with det( b J e ) = L L 2 sin q 2, and L =.8 m, L 2 =.5 m, L 3 =.3 m. The desired configuration for the arm π T is q d =, in order to maximize its manipulability index det( b J e ). Furthermore, the robot 2 is equipped with a planar pin-hole camera, whose projection function f(x, y) is given by: f(x, y) = F y x (24) where F is the focal length of the camera. In the simulations the 3 targets have been assumed in.7, 7 T, 2.4, 9 T,, 7 T, the initial configuration of the mobile base was the origin of the reference frame and the manipulator initial configuration has been chosen close to the desired final configuration. The desired final base configuration has been defined as q bd = In the coordinated feedback sequence, the following parameters have been chosen: λ =.5, K =, λ b =.5. For the hybrid control sequence we have assigned T = 5, T 2 = s, and the period length of each sequence is T = 5s. In Fig. 3 the manipulability index is shown: note how it decreases during the hybrid open-loop sequence (it is still under development a control law able to maintain this index close to the maximum value). In Fig. 4 the angular displacements of the targets, computed with respect to the camera frame, are shown. The trajectories of the angular displacements are monotonic, then the overall trajectories will be maintained inside the camera field of view, i.e. the targets will be maintained visible by the camera, if either the initial and the final angular displacements are inside the camera field of view. In Fig. 5(a) the camera measure errors are reported: when the feedback control law is applied the errors are strictly decreasing, while, during the open-loop phase, the errors are maintained constant. In Fig. 5(b) and in Fig. 5(c) the cartesian position errors of the base and its cartesian trajectory are shown. In Fig. 5(d) the mobile base control input is reported. To avoid limit cycles due to the open loop choice, we apply this sequence only when the base configuration errors are larger than a threshold. 5 Conclusion The proposed control algorithm has been validated by simulations. The control law is able to solve the pointstabilization problem for a mobile manipulator without needing any explicitly localization of the mobile robot. Moreover, the mobile manipulator extra-dof are used to accomplish an added problem: maintain the targets inside the camera field of view during the task execution. These capabilities are useful for many applications, like navigation in unknown environments using graph-based navigation approach, as Topological Map, or like each navigation strategy which does not need a fine mobile robot localization. If nonholonomic constraints act on the base, the feedback control, proposed in (5), is only able to steer the base in order to maintain the arm far from the singularity configurations but does not guarantee the convergency to the desired base configuration. In other words, the base does not reach the desired pose even if the measure are going to zero. From these considerations, we have added an hybrid open-loop phase able to bring the base in the correct position and orienta- Manipulability Index Angular Displacement ( ) Manipulator Manipulability Index Figure 3: Manipulator Manipulability Index Target Angular displacements of the targets with respect the camera frame Time (s) Figure 4: Angular displacement of the targets with respect the camera frame.
6 .7 Camera Measure Errors 2 Mobile Base Errors Trend.6 e xbe ybe.5 e 2 e 3.4 Errors.3.2. Errors (m) (a) Mobile Base Trajectory (b) Mobile Base Controls v(t) ω(t) 4.5 yb (m) 3 2 Velocities xb (m) (c) (d) Figure 5: (a) Camera Measure Errors, (b) Mobile Base Pose Errors, (c) Mobile Base Cartesian Trajectory, (d) Mobile Base Control Inputs tion, while the manipulator is controlled to maintain the measure errors constant. Future work will deal with the development of a control strategy defined in the joint space-state, in order to be able to maintain the manipulability index close to the maximum value during the open-loop phase, and also to the extension of this technique to robot arms with different kinematic configurations. Moreover, experimental activity is planned to test the proposed control strategies on a mobile platform available in our laboratory. References Y. Yamamoto and X. Yun. Coordinating locomotion and manipulation of a mobile manipulator. IEEE Trans. on Automatic Control, 39(6): , June R.W. Brockett. Asymptotic Stability and Feedback Stabilization. Differential Geometric Control Theory. R.S. Millman and H.J. Sussmann, Birkhauser, Boston, J.B. Pomet and C. Samson. Time-varying exponential stabilization of nonholonomic systems in power form. Internal Report 226, INRIA, Sophia-Antipolis, D.P. Tsakiris, P. Rives, and C. Samson. Extending visual servoing techniques to nonholonomic mobile robots. The Confluence of Vision and Control, Lecture Notes in Control and Information Systems, 998. Eds. G. Hager, D. Kriegerman, S. Morse. 5 Y. Yamamoto and X. Yun. Effects of the dynamic interaction on coordinated control of mobile manipulators. IEEE Trans. on Robotics and Control, 2(5):86 824, Oct Yi Ma, J. Kosecká, and S. Sastry. Vision guided navigation for a nonhlonomic mobile robot. IEEE Trans. on Robotics and Automation, 5(3):52 536, June D.P. Tsakiris, P. Rives, and C. Samson. Applying visual servoing techniques to control nonholonomic mobile robots. Internation Conference on Intelligent Robots and Systems, pages 2 32, September Grenoble, France.
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