An Energy-Based Approach for the Multi-Rate Control of a Manipulator on an Actuated Base

Transcription

1 An Energy-Based Approach for the Multi-Rate Control of a Manipulator on an Actuated Base Marco De Stefano 1,2, Ribin Balachandran 1, Alessandro M. Giordano 3, Christian Ott 1 and Cristian Secchi 2 Abstract In this paper we address the proble of controlling a robotic syste ounted on an actuated floating base for space applications. In particular, we investigate the stability issues due to the low rate of the base control unit. We propose a passivity-based stabilizing controller based on the tie doain passivity approach. The controller uses a variable daper regulated by a designed energy observer. The effectiveness of the proposed strategy is validated on a baseanipulator ultibody siulation. I. INTRODUCTION Robotic controllers are usually designed to run on a single control unit with a high-enough rate to disregard the effects of discretization. Unfortunately, there are any applications for which this assuption is not valid anyore. This is the case of robotic systes coposed of ultiple parts that are physically coupled. In these cases, each part is controlled by its own control unit that runs at its own control rate. Soe huanoid robots belong to this class of systes [1]. In several space applications, high anipulation capabilities are becoing necessary and a possible solution is given by a anipulator ounted on an actuated floating base (see Fig. 1). The control of the floating base (satellite) enables coarse positioning while the control of the anipulator enables fine dexterous control for coplex tasks (e.g. debris recovery). Typically, the control syste of the floating base runs at a low rate (between 1 Hz and 1 Hz) while the controller of the anipulator runs at a uch higher rate (usually 1 Hz), since the latter has to interact with the environent [2]. Thus, the overall syste is coposed by two physically coupled echanical systes with one controlled by a low-rate and the second by a high-rate loop. This ix of frequencies, the presence of zero order holds and possible delays in the counication between the control units can jeopardize the stability of the overall syste if not properly taken into account. Discretization, quantization and delay effects have been investigated in the field of haptics and teleoperation [3], [4]. Effects of discretization were analyzed in [5] for reproducing satellite dynaics on a robotic siulator. An optial solution for coping with destabilizing effects of tie delay has been proposed in [6]. In [1] the wave variables concept was successfully ipleented for coordinated control of 1 The authors are with the Institute of Robotics and Mechatronics, Geran Aerospace Center (DLR), Wessling, Gerany. 2 The authors are with the University of Modena and Reggio Eilia, 411 Modena, Italy. 3 The authors are with the Departent of Inforatics, Technical University of Munich (TUM), Garching, 85748, Gerany Contact: arco.destefano@dlr.de Fig. 1: Space scenario: a controlled spacecraft equipped with a anipulator during a satellite recovery-task. robots but considering only tie delay. The interaction between satellite control and anipulator control of a vehicleounted anipulator has been first studied in the context of the Shuttle Reote Manipulator [7]. A coputed oentu strategy for the coordinated control of reaction wheels and anipulator has been proposed in [8]. A centralized control strategy for vehicle attitude, vehicle position and ar endtip control has been studied in [9], resolving the redundancy of the whole syste at the velocity level. At torque level, a centralized control strategy has been proposed in [1], based on a transposed Jacobian algorith. A resolved-rate control proble was proposed in [11], however the base was not actuated. To avoid fuel consuption, a free-floating strategy has been derived in [12] based on a transposed Jacobian algorith, assuing no contacts and zero oentu. The proble of the ipedance control of a free-floating robot was treated in [13] using feedback linearization. More recently, the transposed Jacobian free-floating strategy has been extended to the nonzero oentu case [14], adding a disturbance copensation ter. In [15], the transposed Jacobian free-floating control is extended to stabilize the syste even in presence of contacts. This is achieved using actuators only to dap any accuulated oentu in the syste. None of the actuated-base strategies [7], [8], [9], [1], [15] addressed the proble of ulti-rate control. In this paper we address the ulti-rate control proble of a anipulator on an actuated floating base syste fro an energy-based perspective. We develop a passivity-based control strategy that guarantees the stabilization of the syste independently of the different rates of the controllers. We consider the scenario represented in Fig. 1, where the floating

2 base controller runs at 2 Hz, and the anipulator s control unit runs at 1 Hz. We will show that the presence of ultiple control rates leads to a loss of passivity and then we will re-establish a passive behavior and, as a consequence, stability, exploiting the Tie Doain Passivity Approach (TDPA) [16]. The contribution of the paper is twofold. First, we identify the energy that can render the syste unstable using a network representation. Second, exploiting TDPA [16] we recover a proper energetic behavior and, consequently, stability using a variable control input. The paper is organized as follows: Sec. II introduces the space robot control and the classical control in continuoustie with the proble stateent due to different rate control. The new strategy is presented in Sec. III and proved by siulations in Sec. IV. Conclusions and future works can be found in Sec. V. II. SPACE ROBOT CONTROL AND PROBLEM STATEMENT In this section, the odel of a anipulator on a floating base is introduced. We present the controllers acting on the base and the anipulator. The classical controller results to be stable in continuous tie, but when controllers act with different rates, the syste ay becoe unstable. A. Space robot dynaics Let us consider a robot with n-joints. The general equations of otion for a robot ounted on a oving base [17], are defined as: [ Hb H b H T b H ][ ẍb q ] [ ] [ cb Fb + = c τ ], (1) where H b R 6 6, H R n n, H b R 6 n are the inertia atrices of the whole syste, anipulator and the coupling between the base and the anipulator, respectively. The vectors ẍ b R 6 and q R n are the acceleration of the base (linear and angular) and the acceleration of the robot joints; c b R 6 and c R n are the non-linear Coriolis/centrifugal ters of the base and of the anipulator, respectively. F b R 6 is the force-torque wrench acting on the center of ass of the base-body and τ R n is the input torque vector. The end-effector body velocity ẋ R 6 is given by: ẋ = J b ẋ b + J q, (2) where J b R 6 6 and J R 6 n are the Jacobian atrices of the base and anipulator, respectively. The goal is to control the base of the satellite and the anipulator in the inertial frae. We use a quaternion representation in order to define the rotational error. We define R R 3 3 to be the rotational atrix of the end effector with respect to the inertial frae and R d R 3 3 to be the desired rotational atrix, expressed in the sae frae. The error atrix is then defined as R φ = R d R T. By using the quaternion representation, a scalar η and a vector ˆε R 3 can be defined, such that the orientation error φ R 3 is then defined as: φ = 2E T ˆε, (3) where E R 3 3 is defined as E=I 3 η ε and where ε is the known skew-syetric atrix of the vector ˆε. Analogously, for the position error: p = p d p, (4) where p is the position of the end effector in the inertial frae and p d is the desired position. The error vector in position and orientation can be expressed as the vector x R 6, given by: x =[ p ; φ ]. (5) Analogously for the base, the error can be defined using (3) and (4). Therefore the error vector for the base will be: x b =[ p b ; φ b ]. (6) This representation will be used in the following subsection in order to design the controllers. We assue that the desired pose (i.e. position and orientation) for the anipulator and the base are constant in the inertial frae. B. Torque controller for the end-effector and the satellitebase We consider a anipulator controlled in torque ode with the siple transposed Jacobian: τ = J T F, (7) where F R 6 is the virtual control force applied at the endeffector. In this paper we use the coordinated control approach [1] and we report the controller in explicit for. Therefore, the virtual Cartesian force F at the end-effector is odeled like a PD (Proportional Derivative) behavior, defined as: F = K P x K D ẋ, (8) where K P and K D R 6 6 are positive definite atrices, representing stiffness and daping gains of the anipulator controller. The satellite-base is controlled by the following control law: F b = K Pb x b K Db ẋ b + J T bf, (9) where K Pb and K Db R 6 6 are positive definite atrices, representing stiffness and daping gains of the base controller and J T b F is a coupling ter between anipulator and base. Considering a non-redundant anipulator and a nonsingular J, it can be proven that [ẋ b,ẋ, x b, x ]= (1) is asyptotically stable using the following energetic arguent. Proposition 1: The equilibriu point (1) of (1) is asyptotically stable using the control laws (8) and (9).

3 Proof: Consider the total positive definite energy of the syste as a candidate Lyapunov function: ] ] W = 1 [ẋt 2 b q T][ H b H b [ẋb H T + b H q 1 2 x T K P x + }{{} H x b T K Pb x b. (11) By coputing the derivative and considering the well-known passivity property of the Euler Lagrange systes: we get: 1 2 ] [ẋt b q T] Ḣ[ẋb [ ẋ q T b ] q T][ c b =, (12) c Ẇ = ẋ T b F b+ q T τ ẋ T K P x ẋ T b K Pb x b. (13) Substituting τ fro (7)-(8), F b fro (9) into (13) and considering the end-effector velocity as in (2), we get: Ẇ = ẋ T b K Dbẋ b ẋ T K D ẋ. (14) Therefore, using standard LaSalle arguents, the stateent of the proposition is proven. Thus, we have that in continuous tie the syste behaves in a dissipative way. In fact stability is achieved by the control laws (8) and (9). As an exaple we run a siulation in continuous tie (see Fig. 2) where the end-effector and the base of the satellite have an initial error. 1 It can be seen that the error converges in continuous tie. Notice that although we have a control acting on the base, actuators odels are not considered. 1 The orientation error is represented by ψ,θ,φ (yaw, pitch and roll) angles in the plots. C. Proble Stateent The proble arises when the control laws are coputed in discrete-tie with different sapling rates. Let T and T b be the sapling ties of the controllers of the anipulator and of the base respectively. We consider the case where the sapling tie of the slow rate controller is an integer ultiple of the rate of the fast controller, i.e. T b = nt. Therefore at discrete tie k, we have k=k T = k b T b where k and k b are the discrete steps in each of the controllers. The discrete control laws are given by: F (k )=K P x (k ) K D ẋ (k ), (15) F b (k b )=K Pb x b (k b ) K Db ẋ b (k b )+J T b(k b )F (k b ). (16) It can be proven that the two controllers can interfere due to different sapling rates which ay lead to instability. This can be seen with the following exaple shown in Fig. 3. We consider coon control frequencies for the considered scenario which are T =.1 s and T b =.5 s [2]. We run the siulation considering the discrete laws (15) and (16), with the sae conditions of the continuous-case. As it can be seen in Fig. 3, clearly the error in position p and in orientation φ of both controllers diverge. It is worth coparing Fig. 2 (continuous-tie control) and Fig. 3 (different rate control) to see the different behaviors of the systes. This eans that the presence of different-rate controllers destroy the dissipative effect shown in Proposition 1 and, therefore, soe energy is produced. This iplies that the passivity of the syste is lost. In the next section we will design an energy observer in order to identify the energy produced and to dissipate it using a passivity controller. pb [] φb [deg] pb [] φb [deg] p [] φ [deg] p [] φ [deg] Fig. 2: Stable syste with continuous-tie controllers: Error of the base in position and orientation (top) and error of the anipulator in position and orientation (botto) Fig. 3: Unstable syste with different-rate controllers: Error of the base in position and orientation (top) and error of the end-effector in position and orientation (botto).

4 III. ENERGY-BASED COORDINATED CONTROL In this section a port-representation of the echanical syste is proposed in order to perfor an energy analysis. The energy produced in the syste is identified and it is passivated using a passivity controller. We will exploit the Tie Doain Passivity Approach (TDPA) which provides flexibility due to its port-based approach as it does not depend on syste odeling [16]. A. Tie Doain Passivity TDPA is widely used for syste stability in practical teleoperation systes which are affected by delays in the counication channel. Delays and packet losses in the counication channel introduce energy in the syste and ake the syste active [18]. An interaction between two processors running at different rates is equivalent to having delays and packet losses between the systes. Therefore, the concept of TDPA is extended in this work to stabilize the syste. The underlying principle of TDPA is to observe the input and output energy flows, with a Passivity Observer (PO), of a single-port network, (the virtual environent, in case of haptics) or a 2-port network (the counication channel with delay, in case of teleoperation) [19]. For a one-port syste with saple tie T, the discrete passivity condition is given by: k= (F(k) T v(k)t)+e(), (17) where (F,v) and E() are the power correlated variable set and the initial energy storage of the syste respectively. If condition (17) holds, the syste is passive. The extra energy generated in the port that violates the passivity condition is dissipated with a tie-varying daper, the Passivity Controller (PC). This is done in order to ensure that the syste is an interconnection of passive ports. This condition will be used to design the controller. B. Passive coordinated control In order to identify the energy introduced into the syste by controllers running at different rates, a network representation is derived as shown in Fig. 4. Notice that S is the dynaic syste represented by the left hand side of (1), C b is the controller at the base expressed in (16) and C is the controller of the anipulator (15). The controller is represented by electrical eleents with ipedance C b Z cb and a dependent voltage source which represents the coupling ter of the anipulator, i.e. Jb T F. The control rate of the anipulator C is high and, therefore, its behavior is siilar to that of an equivalent syste controlled by a springdaper controller (8) in continuous tie. This eans that the behavior of the controlled anipulator is stable. Thus, the ain source of energy is the low rate control which acts on the base (dashed box in Fig. 4). We observe possible energy injections through the port (F b,ẋ b ). We design an energy observer which runs at the faster rate controller to obtain a greater accuracy. If we render the controlled base passive at this port, then the C b Z cb J T b F T b ẋ b ẋ F b S F C Fig. 4: Network representation of the satellite- anipulator set-up. overall controlled syste will be interconnection of passive systes and, consequently, will be passive [2]. Then all the regenerative and destabilizing effects would be copensated. The PO is designed to observe the energy flowing in and out of this port using (17) with power variables (F b,ẋ b ). In the following description of the TDPA ipleentation for the ulti-dof (degree of freedo) syste of the satellite and the anipulator, the PO and PC are applied for each dof separately. It can be atheatically shown that if passivity can be guaranteed for all the dof separately, the overall ultidof syste is also passive. For a n-dof syste with initial energy storage E() =, the passivity condition for each dof separately leads to: k= (F(k) T v(k)t)= n k= i=1 T F i (k)v i (k)t, (18) which proves that (17) can be split into a su of the n-dof coponents. Therefore, if the passivity condition holds for each coponent, then the overall syste would be passive. We ipleent the energy observer in the syste running at faster rate, as: E obs (k )=E obs (k 1)+F b (k )ẋ b (k )T + β(k 1)ẋ b (k 1) 2 T. (19) We assue that the satellite-base velocity ẋ b is available with sapling rate T (eg., using a high-rate position sensor). The energy flowing through the port is updated at each T and the values F b changes every T b. Between two values of F b, the observer holds the last received value. Thus, the second ter in (19) coputes the energy flow of the slow-rate controller. The last ter considers the effect of the PC. The variable daper β is derived as: { E obs (k ) β(k )= x b (k ) 2 T else. E obs (k )< E() (2) Notice that the bold notation has been oitted because we are considering the coponents of the vectors. The ipedance correction due to the PC is given by the following quantity: F pc (k )= β(k )ẋ b (k ). (21) It has to be noted that although the calculation for the PO and PC are ipleented at fast rate with a sapling period T, the calculated values of the passivity controller F pc (k ) are sent to the base controller which runs at T b and it will odify

5 the output force F b (k b ). Thus, if the passivity condition is violated (e.g. E obs (k ) < E()), the force coanded to the base will be varied as follows: F b (k b)=f b (k b )+F pc (k b ). (22) As a result, the energy will be restored and the observer will be E obs (k) E() aking the network passive. The network representation odified with the passivity controller can be seen in Fig. 5. The benefit of the ethod is that the control forces depend only on the correlated variables at the port (F b,ẋ b ). Notice that the action of the PC ight require high forces at the base and it can be a liitation for the actuator syste (e.g. thrusters) which can not fire arbitrarily fast. C b Z cb T b ẋ b ẋ b ẋ F b PC F F C b Jb T F S Fig. 5: Network representation schee odified with PC. IV. RESULTS This section shows siulation results perfored with the proposed ethod. The siulation environent considers the servicer robot of the OOS-SIM facility [21] able to siulate the dynaics of a free-floating robot on ground, see Fig. 6. The servicer robot is coposed of an industrial robot which siulates the dynaics of the satellite and a LWR (Light Weight Robot) ounted at its end-effector. The otion of the satellite-base is related to the dynaic odel described in (1). The considered ass of the satellite is 15 kg with Inertia I x = 38kg 2, I y = 2 kg 2, I z = 23 kg 2. The anipulator ar considered is a 7 dof robot, whose ass and inertia paraeters are reported in Table 1. The siulation runs considering T =.1 s and T b =.5 s. The proble described in Sec. II-C is here resolved by applying the proposed ethod. Indeed, the destabilizing effects described in Fig. 3 are due to the active observed energy which is shown in Fig. 7 for the linear and angular Fig. 6: Siulation environent OOS-SIM. T M link [kg] I x [kg 2 ] I y [kg 2 ] I z [kg 2 ] TABLE I: Mass and inertia properties of the anipulator ar Eobs no PC [J] Eobs no PC [J] E x E y E z E ψ E θ E φ Fig. 7: Energy without PC for translation (top) and rotation (botto). Active energy is detected by the negative trend. coponents. The negative trend of energy shows the activity of the syste as the violation of (17) confirs. The proposed approach is then applied. Therefore, the passivity controller will provide the variable force-torque as shown in Fig. 8. This will lead to a passive syste as the energy plot shows in Fig. 9. The positive trend indicates that no active energy is pushed into the syste. Therefore the error for both systes, i.e. anipulator and base, converges (see Fig. 1) and the syste results to be passive. It is worth coparing Fig. 3 (before applying the ethod) with Fig. 1 (proposed ethod). The results prove the validity of the ethod. Notice that the controller is based on the observability of the port (F b,ẋ b ) and it requires the knowledge of ẋ b at high rate as discussed in Sec. III. V. CONCLUSIONS AND FUTURE WORKS In this paper, we showed the destabilizing effects of a low-rate control acting on an actuated base which is equipped with a anipulator controlled at high-rate. A network representation has been designed to onitor the energy of the ulti-body syste and a new control strategy has been proposed. The strategy guarantees passivity thanks to the passivity observer and the passivity controller which dissipates the active energy generated by low-rate control. The ethod is applied to a ulti-body syste coposed of an actuated space satellite with a robotic anipulator, controlled with different sapling rates. The approach ais at generality so that it can be applied in other doains. Future works ai at extending the approach to the tracking case.

6 Fpc [N] τpc [N] tie [s] F x F y F z τ x τ y τ z Fig. 8: Force and torque of the passivity controller. Eobs w PC [J] Eobs w PC [J] Fig. 9: Energy with PC for traslation (top) and rotation (botto). Positive energy indicates the passivity of the syste. REFERENCES [1] C. Ott and Y. Nakaura, Eploying wave variables for coordinated control of robots with distributed control architecture, in 28 IEEE International Conference on Robotics and Autoation, May 28, pp [2] J. Telaar, S. Estable, M. De Stefano, W. Rackl, R. Lapariello, F. Ankersen, and J. Gil Fernandez, Coupled control of chaser platfor and robot ar for the e.deorbit ission, 1th Int. ESA conference on Guidance Navigation and Control Systes (GNC), May 217. [3] S. Straigioli, C. Secchi, A. J. van der Schaft, and C. Fantuzzi, Sapled data systes passivity and discrete port-hailtonian systes, IEEE Transactions on Robotics, vol. 21, no. 4, pp , Aug 25. [4] N. Diolaiti, G. Nieeyer, F. Barbagli, and J. Salisbury, Stability of haptic rendering: Discretization, quantization, tie delay, and coulob effects, Robotics, IEEE Transactions on, vol. 22, no. 2, pp , April 26. [5] M. De Stefano, J. Artigas, and C. Secchi, A passive integration strategy for rendering rotational rigid-body dynaics on a robotic siulator, in 217 IEEE/RSJ International Conference on Intelligent Robots and Systes (IROS), Sept 217, pp [6], An optiized passivity-based ethod for siulating satellite dynaics on a position controlled robot in presence of latencies, in 216 IEEE/RSJ International Conference on Intelligent Robots and Systes (IROS), Oct 216, pp [7] R. W. Longan, R. E. Lindbergt, and M. F. Zedd, Satellite-ounted E x E y E z E ψ E θ E φ pb [] p [] φb [deg] φ [deg] Fig. 1: Stable syste with the proposed ethod. Top row: base error in position and orientation, Botto row: endeffector error in positon and orientation. robot anipulators new kineatics and reaction oent copensation, The International Journal of Robotics Research, vol. 6, no. 3, pp , [8] K. Yoshida, Practical coordination control between satellite attitude and anipulator reaction dynaics based on coputed oentu concept, in IEEE/RSJ/GI International Conference on Intelligent Robots and Systes 94., vol. 3, sep 1994, pp [9] J. R. Spofford and D. L. Akin, Redundancy control of a free-flying telerobot, Journal of Guidance, Control, and Dynaics, vol. 13, no. 3, pp , 199. [1] E. Papadopoulos and S. Dubowsky, Coordinated anipulator/spacecraft otion control for space robotic systes, in Proceedings IEEE International Conference on Robotics and Autoation, Apr 1991, pp vol.2. [11] Y. Uetani and K. Yoshida, Resolved otion rate control of space anipulators with generalized jacobian atrix, Robotics and Autoation, IEEE Transactions on, vol. 5, no. 3, pp , Jun [12] Y. Masutani, F. Miyazaki, and S. Arioto, Sensory feedback control for space anipulators, in Robotics and Autoation, IEEE International Conference on, May 1989, pp vol.3. [13] H. Nakanishi and K. Yoshida, Ipedance Control for Free-flying Space Robots -Basic Equations and Applications-, in Intelligent Robots and Systes, IEEE/RSJ International Conference on, Oct 26. [14] A. M. Giordano, G. Garofalo, M. De Stefano, C. Ott, and A. Albu- Schffer, Dynaics and control of a free-floating space robot in presence of nonzero linear and angular oenta, in IEEE 55th Conference on Decision and Control, Dec 216, pp [15] A. M. Giordano, G. Garofalo, and A. Albu-Schaffer, Moentu duping for space robots, in IEEE 56th Conference on Decision and Control (CDC), Dec 217, pp [16] B. Hannaford and J.-H. Ryu, Tie doain passivity control of haptic interfaces, in Robotics and Autoation ICRA, 21 IEEE International Conference on, vol. 2, 21, pp [17] R. Featherstone, Rigid Body Dynaics Algoriths. Secaucus, NJ, USA: Springer-Verlag New York, Inc., 27. [18] R. Anderson and M. Spong, Bilateral control of teleoperators with tie delay, Autoatic Control, IEEE Transactions on, vol. 34, no. 5, pp , May [19] J.-H. Ryu, D.-S. Kwon, and B. Hannaford, Stable teleoperation with tie-doain passivity control, IEEE Transactions on Robotics and Autoation, vol. 2, no. 2, pp , 24. [2] C. Secchi, S. Straigioli, and C. Fantuzzi, Control of Interactive Robotic Interfaces: a port-hailtonian Approach, ser. Springer Tracts in Advanced Robotics. Springer, 27. [21] J. Artigas, M. De Stefano, W. Rackl, R. Lapariello, B. Brunner, W. Bertleff, R. Burger, O. Porges, A. Giordano, C. Borst, and A. Albu- Schaeffer, The OOS-SIM: An on-ground siulation facility for on-orbit servicing robotic operations, in Robotics and Autoation (ICRA), IEEE International Conference on, May 215, pp