Nonlinear control of an underwater vehicle ... - Semantic Scholar
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Nonlinear Control of an Underwater Vehicle/Manipulator with Composite Dynamics Carlos Canudas de Wit, Ernesto Olguín Díaz, Member, IEEE, and Michel Perrier, Associate Member, IEEE
Abstract—This paper is devoted to the problem of control design of an underwater vehicle/manipulator (UVM) system composed of a free navigating platform equipped with a robot manipulator. This composite system is driven by actuators and sensors having substantially different bandwidth characteristics due to their nature. Such difference allows for a mathematical setup which can be naturally treated by standard singular perturbation theory. On the basis of this analysis, two control laws are proposed. The first is a simplification of the computed torque control law which only requires partial compensation for the slow-subsystem (vehicle dynamics). Feedback compensation is only needed to overcome the coupling effects from the arm to the basis. The second aims at replacing this partial compensation by a robust nonlinear control that does not depend on the model parameters. The closed-loop performance of this controller is close to that of the model-based compensation. Both control laws are shown to be closed-loop stable in the sense of the perturbation theory. A comparative study between a linear partial derivative (PD) controller, a partial model-based compensation, and the nonlinear robust feedback is presented at the end of this paper. Index Terms—Control systems, nonlinear systems, singularly perturbed systems, underwater vehicle control.
I. INTRODUCTION
T
HE use of remotely operated vehicles (ROVs) in subsea tasks has been increasing recently especially in off-shore applications. Most of the systems used up to now are teleoperated from the surface and the vehicle generally has quite a few automatic functions. Autonomy is hampered by the difficulty of properly controlling the system position and orientation (attitude). The main control difficulties stem from the highly coupling model, the system nonlinearities, and the external unmodeled disturbances acting on the vehicle dynamics. To cope with these problems some control designs have been proposed in recent years [1]–[8]. For some special tasks, for example sampling recollection on the ocean bed or pipeline welding, a robot manipulator is needed. The vehicle/manipulator system has become a new and more complex problem to deal with. Although some of the proposed control designs account for many types of uncertainties
Manuscript received June 15, 1998; revised April 30, 1999. Recommended by Associate Editor, F. Ghorbel. C. C. de Wit is with the Laboratoire d'Automatique de Grenoble, Institut National Polytechnique de Grenoble, France (e-mail: [email protected]; [email protected]). E. Olguín Díaz is with the Electrical Engineering Department, ITESM-CCM, France (e-mail: [email protected]). M. Perrier is with the Subsea Robotics and Artificial Intelligence Laboratory, IFREMER, France (e-mail: [email protected]). Publisher Item Identifier S 1063-6536(00)03185-7.
TABLE I VORTEX/PA10's SENSOR AND ACTUATOR TIME RESPONSES
(i.e., parameter mismatches, external disturbances, etc.), they were designed for a singular underwater body. In recent works, feedback control designs for a vehicle/manipulator system have been proposed [9]–[11], but they require full model knowledge to compensate for coupling and nonlinear effects. Little attention has been paid to bandwidth constraints due to the specific type of actuators and sensors used in such applications. As an alternative to feedback linearizization and decoupled controllers, the composite dynamics of the system can be exploited via a singular perturbation model formulation. This composite dynamic characteristic arises from the different bandwidth limitations on the vehicle and the manipulator. For instance, the VORTEX/PA10 system is composed of a PA10 seven degree of freedom (7-DOF) robot manipulator mounted on a free navigating VORTEX platform. The VORTEX is driven by thrusters having a time response of about 0.16 s., between the commanded torque and the effective applied force. The VORTEX position and orientation sensors are of different nature and have quite slow dynamics. For example, the pitch and roll relative angles are sensed by inclinometers with a time response of about 0.1 s, whereas the depthmeter and the magnetic compass (yaw angle) provide a measure every second. In opposition, the PA10 has electrical dc motors driving the torque joints and the joint angles are measured by standard encoders whose dynamics are roughly 100 times faster than the VORTEX sensor's dynamic (see Table I). Actual technology of some of these sensors, like the magnetic compass, is such that it is impossible to obtain similar sensors with improved bandwidth. Hence, in this particular application, the sensors impose a difficult constraint to be implicitly considered during control design. The characteristics of each actuator/sensor subsystem clearly impose different control bandwidth limitations on each of the
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system states. The time response and the ability of rejecting disturbances may be quite different for each of these subsystems. It should be noted that the VORTEX/PA10 does not exhibit any time-scale separation in open loop, but it may do in closed loop if some of the subsystems have a substantially large closed-loop bandwidth. From the differences on the sensor and the actuator dynamics observed in Table I, it will be clearly possible to reach (throughout a linear controller) a larger closed-loop bandwidth for the PA10 manipulator, than for the VORTEX. It thus appears natural to consider the closed-loop VORTEX states as slow variables, and the closed-loop PA10 joint angles as fast variables. This natural time scale separation suggests the use of the singular perturbation theory [12], [13], which is well adapted for systems with different time scales. Taking advantage of this formulation, a control law can be designed for the reduced model which compensates only part of the nonlinear coupling effects. Although the complexity of the resulting control is reduced, it still depends on the partial knowledge of the model. To cope with this dependence, a nonmodel-based robust nonlinear control is proposed. This robust control does not change the bandwidth ratio of the coupled system (it only has high-order components), thus it is only needed to compensate the coupling effects of the arm to the vehicle. The closed-loop stability and performance are preserved. II. MODEL DESCRIPTION The VORTEX/PA10 system is an UVM which is composed of a PA10 7-DOF robot manipulator mounted on the bottom of the free navigating VORTEX platform (see Fig. 1). The VORTEX is driven by six thrusters for 5-DOF (the roll angle is not controlled), while the PA10 manipulator has electrical DC motors driving the torque joints. The model of such a system can be expressed, with respect to the inertial frame, by the solution of the Euler-Lagrange equation as (see [14])
Fig. 1. The VORTEX/PA10 system is composed of a PA10 7-DOF robot manipulator mounted on the bottom of the free navigating VORTEX platform.
(1) Writing the inertia matrix1 and its inverse as is the generalized coordinates vector in where the inertial frame composed of the position and attitude of the and the joint positions of the manipulator vehicle . Vehicle coordinates Manipulator coordinates. Inertia frame was selected because it allows for general task formulation including contact phases [16]. In the above model, is the force control vector, is the inertia matrix, and is the vector including: the Coriolis and centrifugal forces, the hydrodynamic damping, the gravity vector and the disturbance due to the sea currents (where is the fluid velocity).
The system (1) has the following properties [1], [11]: ; • due to its dissipative nature; • with . •
then the system (1) can be rewritten as (2) (3) The Coriolis matrix is represented as
where • • 1See
[15] for properties of the inertia matrix.
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Bandwidth limitations: The time response of the electrical motors of the PA10 manipulator is about 100 times faster than the thruster dynamics of the VORTEX platform. On the other hand, the manipulator joint angle encoders also have a dynamics of about 100 times faster than those of VORTEX sensors (inclinometers, depthmeter, and magnetic compass). This limitation leads to a composite dynamic system. The nature of this composite sensor/actuator system not only imposes control bandwidth limitations axis per axis [17], but also on the reachable closed-loop bandwidth ratio between the dynamics of the VORTEX, and the dynamics of the PA10. In this case it is natural to consider the VORTEX states as the slow variables, and the PA10 joint angles as the fast variables, and to resort to singular perturbation analysis [12], [13], which is well adapted for systems with different time scales. With this in mind, assume that the control has the form
and study new controllers with reduced complexity while preserving certain closed-loop properties. III. SINGULAR PERTURBED MODEL The purpose of this section is to set a singular perturbed model expressing the difference between the two time operation scale between the vehicle and the manipulator [12] (9) is a continuous and differentiable To this end, assume that , and introduce the folfunction with respect to the variables lowing error coordinates: (10)
(4) (11) (5) is the nonwhere is the linear part of the controller and linear part to be defined at a later stage. , due to the Assume that the linear part of the control law constraints mentioned above, has the following form:
In these coordinates, the error dynamic equations can be written as a singular perturbed model, satisfying the conditions of the , standard formulation (i.e., continuity, isolated roots for and time scale separation), as shown in (12) and (13) at the bottom of the page, where
(6)
(14)
(7)
(15)
and are positive definite matrices (which may be where constant and diagonal). The proportional and derivative gains and . Where and are the are: damping ratio and the induced natural frequency of the linwould be the induced natural freearized vehicle model. quency of the linearized manipulator. Finally, the dimensionless induced natural frequency ratio represents the bandwidth constraint for the composite dynamic system. (8) In our application, may be small due to the physical limitations of the actuator/sensors of the VORTEX/PA10 system. The bandwidth constraint appears only in the closed-loop equation. Ideally, feedback linearizable and decoupled controllers can be designed in our setup [18]. However, substantial efforts in modeling and computation burden may be necessary. As an alternative, we wish to take advantage of the above formulation
(16)
(17) The reduction of this system expressed in the singular perturas shown bation standard form is found at the limit when in (18) and (19) at the bottom of the next page, where should be found as one of the isolated roots (if any) of (19). Note that from definitions (10)–(11), the approximated solutions of the generalized coordinates are (20)
(21)
(12)
(13)
De WIT et al.: NONLINEAR CONTROL OF AN UNDERWATER VEHICLE/MANIPULATOR WITH COMPOSITE DYNAMICS
In this case, the inertia matrix and the nonlinear vector , when evaluated along the approximated solutions, are only dependent on the slow dynamic of the coupled system
The following particular solution for ensures the existence and uniqueness of a solution for (19). Lemma 1: Assume that the auxiliary nonlinear control satisfies the following conditions: 1)
2)
then there exists one and only one root for (19) given by (22), shown at the bottom of the page. The substitution of (19) in (18) yields to the approximation of the slow-dynamics error equation (23), shown at the bottom of the page. For this reduction to be valid, it is necessary to fulfill the conditions of Tikhonov's theorem [13]. Let and . These functions are described by (12)–(13)–(22), respectively. The conditions of the Tikhonov theorem are expressed as follows. and their first partial derivatives with 1) The functions must be continuous. The function respect to and the Jacobian must have first partial
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derivatives with respect to their arguments. The initial conditions and must be smooth functions of . 2) The reduced system (23) must have a unique solution , defined on , and for all . 3) The origin of the boundary-layer system (dynamic equa) must be tion of the boundary-layer correction: . exponentially stable, uniformly in In view of the analytical properties of the singular perturbed system (12)–(13) and the conditions of lemma 1, the first condition is fulfilled. The second condition is also fulfilled since the reduced system (23) is Lipschitz. This result comes from the fact that is bounded. the norm of the Jacobian Then the system (23) fulfills the Lipschitz condition
Only the third condition has still to be checked. For this purpose, the expression of the boundary-layer system is needed. It is defined in the -scale of the fast dynamics by the boundary. From this and from (13) when layer correction, , we obtain (24), shown at the bottom of the page, which, (see Lemma above) and after substitution of (15) with (22), yields to the following boundary layer system dynamics:
(25) The inertia matrix cause the variables
in this equation is constant beand are frozen in the -scale.
(18)
(19)
(22)
(23)
(24)
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Fig. 2. Time evolution of the error variables (x (t); z (t)) for the vehicle pitch angle and the first PA10 joint angle with (a) = 0:05 (continuous line) and (b) = 0:2 (dashed line). These responses are compared to its approximated values ( x (t); z (t)) obtained by simulating the approximated system (dotted line).
If is chosen positive definite, exponential stability of the boundary layer can be easily verified by defining as
and observing that
from which we conclude that , exponentially fast, based on the fact that the solutions of a linear time invariant system can only be exponentially stable if they converge to zero. Theorem 1 (Tikhonov Theorem ([13], Theorem 9.1)): If the above three conditions are fulfilled, then there exist positive such that and constants and , the singular perturbation system (12)–(13) has a on , and unique solution
Fig. 2 shows the time responses of two error coordinates (one fast and one slow), and compares the exact errors with the approximated ones . Note that the approximated and the real solutions are close to each other after a finite is, in this scale, time interval. Since the initial value of ), the evolution of this variable only appears too large ( after 0.3 s. In the original system coordinates we obtain, using the trans: formation (10)–(11), the following relations, for some (29) (30)
(26)
is just a linear control Using the fact that and the control is composed as mentioned above, of a linear , the approximated system feedback and a nonlinear term dynamics in the original coordinates is written as
(27)
(31)
, where is the solution of holds uniformly for the boundary-layer system. Moreover, after the boundary-layer transient (28) . holds uniformly for This equation set indicates that the difference between the solutions of the real and approximated error systems is of the . order of in the time interval
(32) . which is valid From the properties obtained above, it is possible to quantify the difference between the joint angle motion and the desired trajectory of the manipulator
De WIT et al.: NONLINEAR CONTROL OF AN UNDERWATER VEHICLE/MANIPULATOR WITH COMPOSITE DYNAMICS
where this expression is obtained using (30) and the following expression:
which reduces to (32) when . The error , will be . quantified after the design of the nonlinear control Remark 1: In the sense of this approximation, the linear control suffices to track the desired joint manipulator reference. However, it clearly appears from this formulation that the vehicle's coordinates are strongly perturbed by the motion of the manipulator; the manipulator-to-vehicle coupling effects dominate the vehicle-to-manipulator ones. This confirms the expected effect that the manipulator high bandwidth overcomes most of the nonlinear and coupling effects. The vehicle requires a more sophisticated controller that can cope with the dynamic coupling effects. The most important outcome of the above remark is that full decoupling and linearization may not be needed. The above approximated model can be used as a basis for designed simplified linearizable and partially decoupling feedback, as will be shown in the next two sections. IV. CONTROL DESIGN
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Two important cases will be considered here. 1) The control is based on a position dependent matrix , i.e., . . 2) The desired vehicle trajectory is constant, i.e., In the first case, the right-hand side (RHS) of (36) is zero and the inertia matrix appears as a left factor. The right factor is a second-order linear differential equation whose origin is exponentially stable. In the second case, the RHS of (36) is also zero but the matrix cannot be factorized. This makes it difficult to prove exponential stability. To obtain this property, the control law (33) is simplified to (37) whose approximated form is (38) . Note that Remark 4: This control law is simpler than (33) because it does not compensate the Coriolis terms associated with the vehicle. Remarks 2 and 3 still hold. Using either control law (37) or (38), the error equation becomes
The approximated model (31)–(32) suggests designing (39)
as (33)
which is, indeed, exponentially stable at the origin. This can be verified with the Lyapunov function
which has the approximated form (34) Remark 2: The equivalent control (34), can be obtained using various combinations of the arguments in (33). For example, if is used instead of the desired manipulator velocity vector the measured velocity (to avoid noise measurements), then the same expression for the equivalent control (34) is obtained. The approximation. The control difference is absorbed by the law (33) is refered to throughout this paper as partial singular perturbed model-based compensation. Remark 3: The structure of the control (33) is simpler and requires less information on the model equation than the control performing the exact compensation and decoupling, which would be designed without this approximation. For instance, the to be destandard computed torque control law will require fined as (35) Note that, in the context of our approximation, the term is replaced by . The control (35) is refered to throughout this article as partial model-based compensation. Either control (33) or (34) used in the approximated model -coordi(31)–(32) gives the following error equation in the nates: (36) describes the error between the approxiwhere mated solutions and the desired motion.
for a small positive (See [15]). As a consequence of the exponential stability of (36) and (39), we have the following. , has the following property 1) The difference :
Hence after some time , the vehicle error is of order of magnitude . [13, Th. 2) The above property can be extended to 9.4]. The following theorem summarizes our first result. be a smooth time function and consider Theorem 2: Let the following two cases. consider the con1) For the tracking case trol law
with and as in (6) and (7), with 2) For the regulation case control law:
. consider
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Fig. 3. Closed-loop errors of the VORTEX/PA10 system under: (a) Linear PD control (continuous line). (b) Partial singular perturbed model-based compensation (dotted line). (c) Partial model-based compensation (dashed line).
with and as in (6) and (7), where definite constant matrix. Then for a sufficiently small , a finite time that the solution of the closed-loop system following properties:
is a positive exists such , satisfies the
with bounded before the finite time is reached. Remark 5: From the continuity and existence of the closed-loop solutions (our system fulfills the Lipschitz local is bounded in the compact conditions), the evolution of . time interval A. Simulation Results Simulations have been performed with the 6-DOF planar VORTEX/PA10 simulator developed by LAG and IFREMER [14]. More detail on this simulations can also be found in [15]. The vehicle's 3-DOF are: the horizontal position, vertical posi. The maniption, and the vehicle orientation . ulator has three coplanar rotational angles The manipulator when fully extended is about 1 m long, from base to end effector. Its dry-weight is 40 kg. The vehicle is 1 m long and its dry-weight is 150 kg. The closed-loop bandwidth rad/s, while frequency for the VORTEX was set to rad/s. This gives a that of the PA10 was set to . Although smaller ratios can be bandwidth ratio of reached, we have selected the worst case here.
The diagonal constant matrices used in the linear part of the control have been selected as the diagonal values of the inertia matrix when the vehicle orientation is zero and the arm kg kg kg m and is fully extended: kg m kg m kg m . Simulations were performed for the second case described above, when the desired vehicle position is constant. The references are set so that the vehicle must stay at an assigned fixed position while the manipulator performs a pendulum-like movement around the first joint. The first joint reference is a 90 amplitude with a 5-s period. The other joints are kept at rest. This movement is important enough to induce substantial hydrodynamic coupling (swimming). The following cases have been simulated: ; 1) Linear PD control: 2) Partial singular perturbed model-based compensation: (34) 3) Partial model-based compensation: (35). The results shown in Figs. 3 and 4 demonstrated that simple PD control does not suffice for good performance, while both partial singular perturbed model compensation and partial modelbased compensation provide better results. Note that our simplified partial singular perturbed model compensation obtained from the singular perturbation setup comes close to the partial model-based compensation after a few seconds. The difference in performance of these two controllers vanishes as the bandwidth ratio becomes smaller. Note also that the small deviation during the transient is in part due to the control saturation.
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Fig. 4. Control inputs corresponding to the simulation shown in Fig. 3. (a) Linear PD control (continuous line). (b) Partial singular perturbed model-based compensation (dotted line). (c) Partial model-based compensation (dashed line). Dot-dash line describes the actuator saturation levels.
Errors in the manipulator coordinates do not vary excessively in magnitude as it was expected.
trol to be designed and defined as follows:
is a state dependent disturbance
V. ROBUST NONLINEAR CONTROL In the last two sections, it has been shown that the vehicle/manipulator system can be decoupled partially with a linear feedback. In this case, the closed-loop system can be divided into two subsystems via approximated equations. One of these is fully independent of the other (manipulator) while the other is perturbed by the first subsystem (vehicle). It has been proved that the solutions of these approximated equations are close to the real ones in a time interval that can be extended to infinity under some conditions. In the last section, a control law was designed to compensate the perturbations on the vehicle due to the manipulator motion. Even if the closed-loop behavior of such a control law is acceptable, it continues to depend on the model knowledge of the system. In this section, a robust nonlinear control law will be proposed as an alternative. For this purpose, the approximated error equation resulting from the singular perturbation model will be used. The closed-loop error equation of system (23) can be written in the following state-space form: (40) where is the approximated error state vector [in the sense of the singular perturbation analysis, see (10)]. is a constant stable matrix whose eigenvalues are defined by the linear feedis the input matrix. is the auxiliary conback gains.
(41) The variables and are defined by (20)–(21). must be designed so that it can From this formulation, cope with the state dependent disturbance . Although the state disturbance may not be explicitly known, an upper bound can be obtained as follows: (42) and are some positive bounded scalar where is the bound of the quadratic terms: Coriolis functions. is the bound due to and the hydrodynamic damping. is the bound of the gravity the residual linear feedback. vector, the fluid velocity disturbance, and the desired velocity and acceleration of both the vehicle and the manipulator. The proposed robust control has the following form: (43) where is the real error variable defined in (10), , and is the unique solution of
,
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Fig. 6. Circle tracking with robust nonlinear control law where = 10 .
Fig. 5. Circle tracking with = 0. i.e. u = simple PD controller.
. In the context of the singular perturbation theory, the approximated control law would be (44) It is clear that if the control law (43) is used in the real system, the singular perturbation analysis would reduce it to (44). Therefore, the use of the reduced controller in the following analysis is consistent with the previous results. Theorem 3 [19]: Consider the error system (40) where the is bounded according to (42). state dependent disturbance is defined by Assume that the auxiliary control law (44), where and are defined as above. The error system (40) is globally exponentially stable if the is of the form scalar function
Fig. 7. Circle tracking with partial model-based compensation.
and the scalars
and
satisfy for all time (45)
Proof: This can be proven with the Lyapunov function . Remark 6: If the scalar function , the control (43) also minimizes the cost function [19]
where
is as follows:
As a consequence, the control (43) would fulfill the return difference condition [19]. This means that the closed-loop energy
dissipation of the system (40) is higher with the controller (43) than without it. The control law (43) is said to be robust because it reduces the effects of the state disturbance . Even if gravity and buoyancy effects are compensated, will never be zero for the tracking , the robust control (43) leads to case. Therefore, if uniformly bounded stability of the error system. The bounds of and the smallest invariant set are functions of the size of the gain . In this case, exponential stability is lost but stability in the Lyapunov sense remains. The robust control (43) mainly copes with the quadratic term of the state dependent disturbance . The linear term is handled by the linear feedback. So the linear gains must be chosen in accordance with the desired transient perwhich is not state dependent, formance. Finally, the term cannot be completely handled by this controller. This term will induce errors in the tracking problem. To reduce them the gain needs to be large.
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Fig. 8. Closed-loop errors of the VORTEX-PA10 system, corresponding to the first 10 s of simulations shown in Figs. 5 –7. (a) = 0 (dashed line). (b) = 10 (continuous line). (c) Partial model-based compensation (35) (dotted line).
Fig. 9. Closed-loop control inputs corresponding to the first 10 s of the simulations shown in Figs. 5–7. (a) = 0 (dashed line). (b) = 10 (c) Partial model-based compensation (35) (dotted line).
(continuous line).
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A. Simulation Results for the Robust Nonlinear Control Simulations were performed for different values of the gain and the vehicle tracking case. In this paper only the most significant are reported. The vehicle/manipulator references are as follows: the desired vehicle trajectory is a 4-m diameter circle with an 80-s period. The vehicle orientation reference is set to zero degrees. The manipulator reference is still the pendulum-like movement described in Section IV-A. was calculated as the solution of the Lyapunov Matrix equality
Hence, the robust nonlinear control law (43) becomes (46) The reported simulations concern the following three cases: ; 1) Linear PD control: (46); 2) Robust nonlinear control where 3) Partial model-based compensation (35). The simulations were performed without actuator saturation. Figs. 5–7 show the system evolution in the plane for the three controllers. Figs. 8 and 9 show the tracking error and the control torques, respectively. The most significant result, is that the VORTEX orientation error droops from almost 20 amplitude to almost 0.1 when the robust control is applied (when changing the gain from zero to 10 ), see Fig. 8. Note that the apparently “large” magnitude of results from the fact that this gain weights the third-order to be efficient, it is polynomial controller. For the control necessary for the product of this gain with the high-order function shown in (46), to have an order of magnitude similar to the linear feedback (6)–(7). Again, it is important to stress that the proposed nonlinear law does not belong to the category of high-gain controllers which are, in general, locally not lipschitz (i.e., infinite local slope). Indeed, due to the high-order polynomial nature of this law, the first and second partial derivative evaluated at the origin are zero. This ensures low noise sensicomtivity near to the origin. The main contribution of the ponent is thus when the tracking error is large. Also note that the tracking performance and the control are torques obtained with the robust control and similar to those obtained by means of the partial model-based compensation, as shown in Figs. 8, and 9 (continuous and dotted lines are almost superimposed). The similarity in the control law evolution of these two approaches clearly shows that the proposed robust control strategy does not belong to the high-gain control class as discussed in the introduction. When the gain increases, the robust control approaches the partial model-based controller. Also note in the lower left plot of Fig. 9 that the VORTEX orientation torque for both controllers (the robust control and the partial model-based compensation) exhibits the saturation limits. Saturation would normally affect the performance of both controllers.
Fig. 10. (a) Circle tracking with robust nonlinear control law and actuator saturation. = 10 . (b) Circle tracking with partial model-based compensation.
When saturation is included, the overall performance of the VORTEX was slightly reduced for the robust control as well as for the partial model-based compensation as can be seen in Figs. 10(a) and (b). The VORTEX orientation error for both cases (with and without saturation) is shown in Fig. 11. This figure shows that both controllers achieve a similar performance even in the presence of saturation. VI. CONCLUSION We have presented, on the one hand, an alternative control design for the UVM composite system with substantially different bandwidth characteristics. For this purpose we have resorted to a singular perturbed model that allows reduction of control complexity. We have presented a formal study of the asymptotic properties of the closed-loop system solution. As a result of this study, we have shown that the error variables are of the order of magnitude of the bandwidth ratio, and that this bound holds for a time horizon that can be extended to
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Fig. 11.
VORTEX orientation error and control errors. (a) robust control where = 10
infinity. The more meaningful result concerns the case where the vehicle reference is constant and the manipulator reference is time varying. Extension to the complete tracking case may be possible if the desired vehicle acceleration can be scaled down to slow motions of magnitude . Simulations have shown that a good precision is obtained when compared to partial model-based compensation. Moreover, we have presented a robust nonlinear feedback control for the UVM system with composite dynamics. Singular perturbed model was used as a basis for this study. The approximation results of this analysis are only valid in an in. The extension of terval of time can be obtained if , i.e., if and the regulation case is studied. However, these results are only sufficient, and it seems possible that this extension can be achived via a different analysis, using the same control law. Simulations have , the performance obtained with shown that even if this nonmodel-based control is similar to the one resulting from model-based feedback. It was also shown by simulation that this is still true in the case of saturation. In this case, the robust nonlinear control proposed in this paper is a good compromise between control complexity and closed-loop performance.
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(continuous line). (b) Partial model-based compensation (dotted line).
REFERENCES [1] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 1994. [2] C. C. de Wit, D. Williamson, and R. Bachmayer, “Performance-oriented robust control for a class of mechanical systems: A study case,” presented at the Proc. Int. Conf. Syst., Man, Cybern., Le Touquet, France, Oct. 17–20, 1993. [3] A. Healey and D. Lienard, “Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles,” IEEE J. Ocean. Eng., vol. 18, 1993. [4] M. Perrier and C. C. de Wit, “Experimental comparison of PID versus PID plus nonlinear controller for subsea robots,” J. Autonomous Robots (Special Issue on Autonomous Underwater Robots), 1996. [5] M. Perrier, V. Rigaud, C. C. de Wit, and R. Bachmayer, “Performanceoriented robust nonlinear control for subsea robots,” presented at the Proc. IEEE Int. Conf. Robot. Automat., San Diego, CA, 1994. [6] C. C. de Wit, E. O. Díaz, and M. Perrier, “Control of underwater vehicle/manipulator system with composite dynamics,” presented at the Proc. Amer. Contr. Conf., Philadelphia, PA, 1998. [7] , “Robust nonlinear control of an underwater vehicle/manipulator system with composite dynamics,” presented at the Proc. Int. Conf. Robot. Automat., Leuven, Belgium, 1998. [8] H. Mahesh, J. Yuh, and R. Lakshmi, “A coordinated control of underwater vehicle and robot manipulator,” J. Robot. Syst., vol. 8, no. 3, pp. 339–370, June 1991. [9] T. W. McLain, S. M. Rock, and M. J. Lee, “Coordinated control of an underwater robotic system,” presented at the Proc. IEEE Int. Conf. Robot. Automat. (Video Submission), Minneapolis, MN, 1996.
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[10] S. McMillan, D. E. Orin, and R. McGhee, “Efficient dynamic simulation of an underwater vehicle with a robotic manipulator,” IEEE Trans. Syst., Man., Cybern., vol. 25, pp. 1194–1206, Aug. 1995. [11] I. Schjølberg and T. I. Fossen, “Modeling and control of underwater vehicle-manipulator systems,” presented at the Proc. 3rd Conf. Marine Craft Maneuvering Contr. (MCMC'94), Southampton, U.K., 1994. [12] P. Kokotovic, H. K. Khalil, and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design. New York: Academic, 1986. [13] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [14] E. O. Díaz, “Underwater Mobile Manipulator Modeling,”, Int. Rep. 1-9LJ2/1-95-3-321285 LAG, Oct. 1996. [15] E. O. Díaz, C. C. de Wit, and M. Perrier, “Nonlinear Control of an Underwater Vehicle/Manipulator with Composite Dynamics: Appendices,”, Int. Rep. 99-100 LAG, Feb. 1999. [16] J. B. Marion and S. P. Thornton, Classical Dynamics of Particles and Systems, 4th ed. Orlando, FL: Harcourt Brace, 1995. [17] K. J. Åström, “Limitations on control systems performance,” presented at the IEEE Proc. Eur. Contr. Conf., Bruxels, Belgium, June 1997. [18] C. C. de Wit, B. Siciliano, and G. Bastin, Eds., Theory of Robot Control. New York: Springer-Verlag, 1996. [19] D. Williamson and C. C. de Wit, “Performance-oriented robust control for a class of nonlinear systems,” presented at the Proc. Eur. Contr. Conf., Rome, Italy, 1995.
Carlos Canudas de Wit was born in Villahermosa, Tabasco, Mexico, in 1958. He received the B.Sc. degree in electronics and communications from the Technological Institute of Monterrey, Mexico, in 1980. He received the M.Sc. degree from the Polytechnic of Grenoble, Department of Automatic Control, Grenoble, France, in 1984. He received the Ph.D. degree in automatic control from the Polytechnic of Grenoble in 1987. He was Visiting Researcher in 1985 at Lund Institute of Technology, Lund, Sweden. Since then he has been working at the same department as “Directeur de recherche at the CNRS,” where he teaches and conducts research in the area of adaptive and robot control. He teaches undergraduate and graduated courses in robot control and stability of nonlinear systems. He is also responsible for a team on Ccontrol of electromechanical systems and robotics. His research interests include adaptive control, identification, robot control, nonlinear observers, control of systems with friction, AC and CD drives, Automotive control, nonholonomic systems. He has written a book on Adaptive Control of Partially Known Systems: Theory and Applications (Amsterdam, The Netherlands: Elsevier). He also edited a book on Advanced Robot Control (New York: Springer-Verlag, 1991), and a joint 12–author book Theory of Robot Control (New York: Springer-Verlag, 1997). He has been Associate Editor of Automatica since March 1999. Dr. Canudas de Wit was Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, from January 1992 to December 1997.
Ernesto Olguín-Díaz (M’00) was born in Mexico City, Mexico, in 1968. He received the B.S. degree in mechanical and electrical engineering from the Technological Institute of Monterrey (ITESM), Mexico, in December 1991. He received the the M.S. degree in automation and the Ph.D. degree from the National Polytechnic Institute of Grenoble (INPG), Grenoble, France, in June 1995 and January 1999, respectively. His research includes mobile manipulation, underwater vehicles, nonlinear systems, nonlinear control theory, and robust control.
Michel Perrier (A’93) received the Ph.D. degree in robotics and automation in 1991 from the University of Montpellier II, France. In 1992, he spent one year as a Postdoctoral Researcher in the Underwater Vehicles Laboratory, Massachusetts Institute of Technology (MIT) Sea Grant College Program, Boston, MA, where he worked on the design and implementation of the low and high-level contol of the Autonomous Underwater Vehicle “Odyssey I.” Since December 1992, he had been a full-time Research Engineer in the Subsea Robotics and Artificial Intelligence Laboratory at the French Research Institute for the Exploitation of the Sea (IFREMER), La Seyne-sur-Mer, France. His interests include trajectory planning and path following for autonomous land mobile robots, using artificial intelligence technics for fusing sensor data and world model information, underwater vehicle systems including underwater vehicle control, coupled underwater vehicle-manipulator system control, telemanipulation systems, mission programming and planning, and control architecture.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER 2000
Nonlinear Control of an Underwater Vehicle/Manipulator with Composite Dynamics Carlos Canudas de Wit, Ernesto Olguín Díaz, Member, IEEE, and Michel Perrier, Associate Member, IEEE
Abstract—This paper is devoted to the problem of control design of an underwater vehicle/manipulator (UVM) system composed of a free navigating platform equipped with a robot manipulator. This composite system is driven by actuators and sensors having substantially different bandwidth characteristics due to their nature. Such difference allows for a mathematical setup which can be naturally treated by standard singular perturbation theory. On the basis of this analysis, two control laws are proposed. The first is a simplification of the computed torque control law which only requires partial compensation for the slow-subsystem (vehicle dynamics). Feedback compensation is only needed to overcome the coupling effects from the arm to the basis. The second aims at replacing this partial compensation by a robust nonlinear control that does not depend on the model parameters. The closed-loop performance of this controller is close to that of the model-based compensation. Both control laws are shown to be closed-loop stable in the sense of the perturbation theory. A comparative study between a linear partial derivative (PD) controller, a partial model-based compensation, and the nonlinear robust feedback is presented at the end of this paper. Index Terms—Control systems, nonlinear systems, singularly perturbed systems, underwater vehicle control.
I. INTRODUCTION
T
HE use of remotely operated vehicles (ROVs) in subsea tasks has been increasing recently especially in off-shore applications. Most of the systems used up to now are teleoperated from the surface and the vehicle generally has quite a few automatic functions. Autonomy is hampered by the difficulty of properly controlling the system position and orientation (attitude). The main control difficulties stem from the highly coupling model, the system nonlinearities, and the external unmodeled disturbances acting on the vehicle dynamics. To cope with these problems some control designs have been proposed in recent years [1]–[8]. For some special tasks, for example sampling recollection on the ocean bed or pipeline welding, a robot manipulator is needed. The vehicle/manipulator system has become a new and more complex problem to deal with. Although some of the proposed control designs account for many types of uncertainties
Manuscript received June 15, 1998; revised April 30, 1999. Recommended by Associate Editor, F. Ghorbel. C. C. de Wit is with the Laboratoire d'Automatique de Grenoble, Institut National Polytechnique de Grenoble, France (e-mail: [email protected]; [email protected]). E. Olguín Díaz is with the Electrical Engineering Department, ITESM-CCM, France (e-mail: [email protected]). M. Perrier is with the Subsea Robotics and Artificial Intelligence Laboratory, IFREMER, France (e-mail: [email protected]). Publisher Item Identifier S 1063-6536(00)03185-7.
TABLE I VORTEX/PA10's SENSOR AND ACTUATOR TIME RESPONSES
(i.e., parameter mismatches, external disturbances, etc.), they were designed for a singular underwater body. In recent works, feedback control designs for a vehicle/manipulator system have been proposed [9]–[11], but they require full model knowledge to compensate for coupling and nonlinear effects. Little attention has been paid to bandwidth constraints due to the specific type of actuators and sensors used in such applications. As an alternative to feedback linearizization and decoupled controllers, the composite dynamics of the system can be exploited via a singular perturbation model formulation. This composite dynamic characteristic arises from the different bandwidth limitations on the vehicle and the manipulator. For instance, the VORTEX/PA10 system is composed of a PA10 seven degree of freedom (7-DOF) robot manipulator mounted on a free navigating VORTEX platform. The VORTEX is driven by thrusters having a time response of about 0.16 s., between the commanded torque and the effective applied force. The VORTEX position and orientation sensors are of different nature and have quite slow dynamics. For example, the pitch and roll relative angles are sensed by inclinometers with a time response of about 0.1 s, whereas the depthmeter and the magnetic compass (yaw angle) provide a measure every second. In opposition, the PA10 has electrical dc motors driving the torque joints and the joint angles are measured by standard encoders whose dynamics are roughly 100 times faster than the VORTEX sensor's dynamic (see Table I). Actual technology of some of these sensors, like the magnetic compass, is such that it is impossible to obtain similar sensors with improved bandwidth. Hence, in this particular application, the sensors impose a difficult constraint to be implicitly considered during control design. The characteristics of each actuator/sensor subsystem clearly impose different control bandwidth limitations on each of the
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system states. The time response and the ability of rejecting disturbances may be quite different for each of these subsystems. It should be noted that the VORTEX/PA10 does not exhibit any time-scale separation in open loop, but it may do in closed loop if some of the subsystems have a substantially large closed-loop bandwidth. From the differences on the sensor and the actuator dynamics observed in Table I, it will be clearly possible to reach (throughout a linear controller) a larger closed-loop bandwidth for the PA10 manipulator, than for the VORTEX. It thus appears natural to consider the closed-loop VORTEX states as slow variables, and the closed-loop PA10 joint angles as fast variables. This natural time scale separation suggests the use of the singular perturbation theory [12], [13], which is well adapted for systems with different time scales. Taking advantage of this formulation, a control law can be designed for the reduced model which compensates only part of the nonlinear coupling effects. Although the complexity of the resulting control is reduced, it still depends on the partial knowledge of the model. To cope with this dependence, a nonmodel-based robust nonlinear control is proposed. This robust control does not change the bandwidth ratio of the coupled system (it only has high-order components), thus it is only needed to compensate the coupling effects of the arm to the vehicle. The closed-loop stability and performance are preserved. II. MODEL DESCRIPTION The VORTEX/PA10 system is an UVM which is composed of a PA10 7-DOF robot manipulator mounted on the bottom of the free navigating VORTEX platform (see Fig. 1). The VORTEX is driven by six thrusters for 5-DOF (the roll angle is not controlled), while the PA10 manipulator has electrical DC motors driving the torque joints. The model of such a system can be expressed, with respect to the inertial frame, by the solution of the Euler-Lagrange equation as (see [14])
Fig. 1. The VORTEX/PA10 system is composed of a PA10 7-DOF robot manipulator mounted on the bottom of the free navigating VORTEX platform.
(1) Writing the inertia matrix1 and its inverse as is the generalized coordinates vector in where the inertial frame composed of the position and attitude of the and the joint positions of the manipulator vehicle . Vehicle coordinates Manipulator coordinates. Inertia frame was selected because it allows for general task formulation including contact phases [16]. In the above model, is the force control vector, is the inertia matrix, and is the vector including: the Coriolis and centrifugal forces, the hydrodynamic damping, the gravity vector and the disturbance due to the sea currents (where is the fluid velocity).
The system (1) has the following properties [1], [11]: ; • due to its dissipative nature; • with . •
then the system (1) can be rewritten as (2) (3) The Coriolis matrix is represented as
where • • 1See
[15] for properties of the inertia matrix.
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Bandwidth limitations: The time response of the electrical motors of the PA10 manipulator is about 100 times faster than the thruster dynamics of the VORTEX platform. On the other hand, the manipulator joint angle encoders also have a dynamics of about 100 times faster than those of VORTEX sensors (inclinometers, depthmeter, and magnetic compass). This limitation leads to a composite dynamic system. The nature of this composite sensor/actuator system not only imposes control bandwidth limitations axis per axis [17], but also on the reachable closed-loop bandwidth ratio between the dynamics of the VORTEX, and the dynamics of the PA10. In this case it is natural to consider the VORTEX states as the slow variables, and the PA10 joint angles as the fast variables, and to resort to singular perturbation analysis [12], [13], which is well adapted for systems with different time scales. With this in mind, assume that the control has the form
and study new controllers with reduced complexity while preserving certain closed-loop properties. III. SINGULAR PERTURBED MODEL The purpose of this section is to set a singular perturbed model expressing the difference between the two time operation scale between the vehicle and the manipulator [12] (9) is a continuous and differentiable To this end, assume that , and introduce the folfunction with respect to the variables lowing error coordinates: (10)
(4) (11) (5) is the nonwhere is the linear part of the controller and linear part to be defined at a later stage. , due to the Assume that the linear part of the control law constraints mentioned above, has the following form:
In these coordinates, the error dynamic equations can be written as a singular perturbed model, satisfying the conditions of the , standard formulation (i.e., continuity, isolated roots for and time scale separation), as shown in (12) and (13) at the bottom of the page, where
(6)
(14)
(7)
(15)
and are positive definite matrices (which may be where constant and diagonal). The proportional and derivative gains and . Where and are the are: damping ratio and the induced natural frequency of the linwould be the induced natural freearized vehicle model. quency of the linearized manipulator. Finally, the dimensionless induced natural frequency ratio represents the bandwidth constraint for the composite dynamic system. (8) In our application, may be small due to the physical limitations of the actuator/sensors of the VORTEX/PA10 system. The bandwidth constraint appears only in the closed-loop equation. Ideally, feedback linearizable and decoupled controllers can be designed in our setup [18]. However, substantial efforts in modeling and computation burden may be necessary. As an alternative, we wish to take advantage of the above formulation
(16)
(17) The reduction of this system expressed in the singular perturas shown bation standard form is found at the limit when in (18) and (19) at the bottom of the next page, where should be found as one of the isolated roots (if any) of (19). Note that from definitions (10)–(11), the approximated solutions of the generalized coordinates are (20)
(21)
(12)
(13)
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In this case, the inertia matrix and the nonlinear vector , when evaluated along the approximated solutions, are only dependent on the slow dynamic of the coupled system
The following particular solution for ensures the existence and uniqueness of a solution for (19). Lemma 1: Assume that the auxiliary nonlinear control satisfies the following conditions: 1)
2)
then there exists one and only one root for (19) given by (22), shown at the bottom of the page. The substitution of (19) in (18) yields to the approximation of the slow-dynamics error equation (23), shown at the bottom of the page. For this reduction to be valid, it is necessary to fulfill the conditions of Tikhonov's theorem [13]. Let and . These functions are described by (12)–(13)–(22), respectively. The conditions of the Tikhonov theorem are expressed as follows. and their first partial derivatives with 1) The functions must be continuous. The function respect to and the Jacobian must have first partial
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derivatives with respect to their arguments. The initial conditions and must be smooth functions of . 2) The reduced system (23) must have a unique solution , defined on , and for all . 3) The origin of the boundary-layer system (dynamic equa) must be tion of the boundary-layer correction: . exponentially stable, uniformly in In view of the analytical properties of the singular perturbed system (12)–(13) and the conditions of lemma 1, the first condition is fulfilled. The second condition is also fulfilled since the reduced system (23) is Lipschitz. This result comes from the fact that is bounded. the norm of the Jacobian Then the system (23) fulfills the Lipschitz condition
Only the third condition has still to be checked. For this purpose, the expression of the boundary-layer system is needed. It is defined in the -scale of the fast dynamics by the boundary. From this and from (13) when layer correction, , we obtain (24), shown at the bottom of the page, which, (see Lemma above) and after substitution of (15) with (22), yields to the following boundary layer system dynamics:
(25) The inertia matrix cause the variables
in this equation is constant beand are frozen in the -scale.
(18)
(19)
(22)
(23)
(24)
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Fig. 2. Time evolution of the error variables (x (t); z (t)) for the vehicle pitch angle and the first PA10 joint angle with (a) = 0:05 (continuous line) and (b) = 0:2 (dashed line). These responses are compared to its approximated values ( x (t); z (t)) obtained by simulating the approximated system (dotted line).
If is chosen positive definite, exponential stability of the boundary layer can be easily verified by defining as
and observing that
from which we conclude that , exponentially fast, based on the fact that the solutions of a linear time invariant system can only be exponentially stable if they converge to zero. Theorem 1 (Tikhonov Theorem ([13], Theorem 9.1)): If the above three conditions are fulfilled, then there exist positive such that and constants and , the singular perturbation system (12)–(13) has a on , and unique solution
Fig. 2 shows the time responses of two error coordinates (one fast and one slow), and compares the exact errors with the approximated ones . Note that the approximated and the real solutions are close to each other after a finite is, in this scale, time interval. Since the initial value of ), the evolution of this variable only appears too large ( after 0.3 s. In the original system coordinates we obtain, using the trans: formation (10)–(11), the following relations, for some (29) (30)
(26)
is just a linear control Using the fact that and the control is composed as mentioned above, of a linear , the approximated system feedback and a nonlinear term dynamics in the original coordinates is written as
(27)
(31)
, where is the solution of holds uniformly for the boundary-layer system. Moreover, after the boundary-layer transient (28) . holds uniformly for This equation set indicates that the difference between the solutions of the real and approximated error systems is of the . order of in the time interval
(32) . which is valid From the properties obtained above, it is possible to quantify the difference between the joint angle motion and the desired trajectory of the manipulator
De WIT et al.: NONLINEAR CONTROL OF AN UNDERWATER VEHICLE/MANIPULATOR WITH COMPOSITE DYNAMICS
where this expression is obtained using (30) and the following expression:
which reduces to (32) when . The error , will be . quantified after the design of the nonlinear control Remark 1: In the sense of this approximation, the linear control suffices to track the desired joint manipulator reference. However, it clearly appears from this formulation that the vehicle's coordinates are strongly perturbed by the motion of the manipulator; the manipulator-to-vehicle coupling effects dominate the vehicle-to-manipulator ones. This confirms the expected effect that the manipulator high bandwidth overcomes most of the nonlinear and coupling effects. The vehicle requires a more sophisticated controller that can cope with the dynamic coupling effects. The most important outcome of the above remark is that full decoupling and linearization may not be needed. The above approximated model can be used as a basis for designed simplified linearizable and partially decoupling feedback, as will be shown in the next two sections. IV. CONTROL DESIGN
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Two important cases will be considered here. 1) The control is based on a position dependent matrix , i.e., . . 2) The desired vehicle trajectory is constant, i.e., In the first case, the right-hand side (RHS) of (36) is zero and the inertia matrix appears as a left factor. The right factor is a second-order linear differential equation whose origin is exponentially stable. In the second case, the RHS of (36) is also zero but the matrix cannot be factorized. This makes it difficult to prove exponential stability. To obtain this property, the control law (33) is simplified to (37) whose approximated form is (38) . Note that Remark 4: This control law is simpler than (33) because it does not compensate the Coriolis terms associated with the vehicle. Remarks 2 and 3 still hold. Using either control law (37) or (38), the error equation becomes
The approximated model (31)–(32) suggests designing (39)
as (33)
which is, indeed, exponentially stable at the origin. This can be verified with the Lyapunov function
which has the approximated form (34) Remark 2: The equivalent control (34), can be obtained using various combinations of the arguments in (33). For example, if is used instead of the desired manipulator velocity vector the measured velocity (to avoid noise measurements), then the same expression for the equivalent control (34) is obtained. The approximation. The control difference is absorbed by the law (33) is refered to throughout this paper as partial singular perturbed model-based compensation. Remark 3: The structure of the control (33) is simpler and requires less information on the model equation than the control performing the exact compensation and decoupling, which would be designed without this approximation. For instance, the to be destandard computed torque control law will require fined as (35) Note that, in the context of our approximation, the term is replaced by . The control (35) is refered to throughout this article as partial model-based compensation. Either control (33) or (34) used in the approximated model -coordi(31)–(32) gives the following error equation in the nates: (36) describes the error between the approxiwhere mated solutions and the desired motion.
for a small positive (See [15]). As a consequence of the exponential stability of (36) and (39), we have the following. , has the following property 1) The difference :
Hence after some time , the vehicle error is of order of magnitude . [13, Th. 2) The above property can be extended to 9.4]. The following theorem summarizes our first result. be a smooth time function and consider Theorem 2: Let the following two cases. consider the con1) For the tracking case trol law
with and as in (6) and (7), with 2) For the regulation case control law:
. consider
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Fig. 3. Closed-loop errors of the VORTEX/PA10 system under: (a) Linear PD control (continuous line). (b) Partial singular perturbed model-based compensation (dotted line). (c) Partial model-based compensation (dashed line).
with and as in (6) and (7), where definite constant matrix. Then for a sufficiently small , a finite time that the solution of the closed-loop system following properties:
is a positive exists such , satisfies the
with bounded before the finite time is reached. Remark 5: From the continuity and existence of the closed-loop solutions (our system fulfills the Lipschitz local is bounded in the compact conditions), the evolution of . time interval A. Simulation Results Simulations have been performed with the 6-DOF planar VORTEX/PA10 simulator developed by LAG and IFREMER [14]. More detail on this simulations can also be found in [15]. The vehicle's 3-DOF are: the horizontal position, vertical posi. The maniption, and the vehicle orientation . ulator has three coplanar rotational angles The manipulator when fully extended is about 1 m long, from base to end effector. Its dry-weight is 40 kg. The vehicle is 1 m long and its dry-weight is 150 kg. The closed-loop bandwidth rad/s, while frequency for the VORTEX was set to rad/s. This gives a that of the PA10 was set to . Although smaller ratios can be bandwidth ratio of reached, we have selected the worst case here.
The diagonal constant matrices used in the linear part of the control have been selected as the diagonal values of the inertia matrix when the vehicle orientation is zero and the arm kg kg kg m and is fully extended: kg m kg m kg m . Simulations were performed for the second case described above, when the desired vehicle position is constant. The references are set so that the vehicle must stay at an assigned fixed position while the manipulator performs a pendulum-like movement around the first joint. The first joint reference is a 90 amplitude with a 5-s period. The other joints are kept at rest. This movement is important enough to induce substantial hydrodynamic coupling (swimming). The following cases have been simulated: ; 1) Linear PD control: 2) Partial singular perturbed model-based compensation: (34) 3) Partial model-based compensation: (35). The results shown in Figs. 3 and 4 demonstrated that simple PD control does not suffice for good performance, while both partial singular perturbed model compensation and partial modelbased compensation provide better results. Note that our simplified partial singular perturbed model compensation obtained from the singular perturbation setup comes close to the partial model-based compensation after a few seconds. The difference in performance of these two controllers vanishes as the bandwidth ratio becomes smaller. Note also that the small deviation during the transient is in part due to the control saturation.
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Fig. 4. Control inputs corresponding to the simulation shown in Fig. 3. (a) Linear PD control (continuous line). (b) Partial singular perturbed model-based compensation (dotted line). (c) Partial model-based compensation (dashed line). Dot-dash line describes the actuator saturation levels.
Errors in the manipulator coordinates do not vary excessively in magnitude as it was expected.
trol to be designed and defined as follows:
is a state dependent disturbance
V. ROBUST NONLINEAR CONTROL In the last two sections, it has been shown that the vehicle/manipulator system can be decoupled partially with a linear feedback. In this case, the closed-loop system can be divided into two subsystems via approximated equations. One of these is fully independent of the other (manipulator) while the other is perturbed by the first subsystem (vehicle). It has been proved that the solutions of these approximated equations are close to the real ones in a time interval that can be extended to infinity under some conditions. In the last section, a control law was designed to compensate the perturbations on the vehicle due to the manipulator motion. Even if the closed-loop behavior of such a control law is acceptable, it continues to depend on the model knowledge of the system. In this section, a robust nonlinear control law will be proposed as an alternative. For this purpose, the approximated error equation resulting from the singular perturbation model will be used. The closed-loop error equation of system (23) can be written in the following state-space form: (40) where is the approximated error state vector [in the sense of the singular perturbation analysis, see (10)]. is a constant stable matrix whose eigenvalues are defined by the linear feedis the input matrix. is the auxiliary conback gains.
(41) The variables and are defined by (20)–(21). must be designed so that it can From this formulation, cope with the state dependent disturbance . Although the state disturbance may not be explicitly known, an upper bound can be obtained as follows: (42) and are some positive bounded scalar where is the bound of the quadratic terms: Coriolis functions. is the bound due to and the hydrodynamic damping. is the bound of the gravity the residual linear feedback. vector, the fluid velocity disturbance, and the desired velocity and acceleration of both the vehicle and the manipulator. The proposed robust control has the following form: (43) where is the real error variable defined in (10), , and is the unique solution of
,
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Fig. 6. Circle tracking with robust nonlinear control law where = 10 .
Fig. 5. Circle tracking with = 0. i.e. u = simple PD controller.
. In the context of the singular perturbation theory, the approximated control law would be (44) It is clear that if the control law (43) is used in the real system, the singular perturbation analysis would reduce it to (44). Therefore, the use of the reduced controller in the following analysis is consistent with the previous results. Theorem 3 [19]: Consider the error system (40) where the is bounded according to (42). state dependent disturbance is defined by Assume that the auxiliary control law (44), where and are defined as above. The error system (40) is globally exponentially stable if the is of the form scalar function
Fig. 7. Circle tracking with partial model-based compensation.
and the scalars
and
satisfy for all time (45)
Proof: This can be proven with the Lyapunov function . Remark 6: If the scalar function , the control (43) also minimizes the cost function [19]
where
is as follows:
As a consequence, the control (43) would fulfill the return difference condition [19]. This means that the closed-loop energy
dissipation of the system (40) is higher with the controller (43) than without it. The control law (43) is said to be robust because it reduces the effects of the state disturbance . Even if gravity and buoyancy effects are compensated, will never be zero for the tracking , the robust control (43) leads to case. Therefore, if uniformly bounded stability of the error system. The bounds of and the smallest invariant set are functions of the size of the gain . In this case, exponential stability is lost but stability in the Lyapunov sense remains. The robust control (43) mainly copes with the quadratic term of the state dependent disturbance . The linear term is handled by the linear feedback. So the linear gains must be chosen in accordance with the desired transient perwhich is not state dependent, formance. Finally, the term cannot be completely handled by this controller. This term will induce errors in the tracking problem. To reduce them the gain needs to be large.
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Fig. 8. Closed-loop errors of the VORTEX-PA10 system, corresponding to the first 10 s of simulations shown in Figs. 5 –7. (a) = 0 (dashed line). (b) = 10 (continuous line). (c) Partial model-based compensation (35) (dotted line).
Fig. 9. Closed-loop control inputs corresponding to the first 10 s of the simulations shown in Figs. 5–7. (a) = 0 (dashed line). (b) = 10 (c) Partial model-based compensation (35) (dotted line).
(continuous line).
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A. Simulation Results for the Robust Nonlinear Control Simulations were performed for different values of the gain and the vehicle tracking case. In this paper only the most significant are reported. The vehicle/manipulator references are as follows: the desired vehicle trajectory is a 4-m diameter circle with an 80-s period. The vehicle orientation reference is set to zero degrees. The manipulator reference is still the pendulum-like movement described in Section IV-A. was calculated as the solution of the Lyapunov Matrix equality
Hence, the robust nonlinear control law (43) becomes (46) The reported simulations concern the following three cases: ; 1) Linear PD control: (46); 2) Robust nonlinear control where 3) Partial model-based compensation (35). The simulations were performed without actuator saturation. Figs. 5–7 show the system evolution in the plane for the three controllers. Figs. 8 and 9 show the tracking error and the control torques, respectively. The most significant result, is that the VORTEX orientation error droops from almost 20 amplitude to almost 0.1 when the robust control is applied (when changing the gain from zero to 10 ), see Fig. 8. Note that the apparently “large” magnitude of results from the fact that this gain weights the third-order to be efficient, it is polynomial controller. For the control necessary for the product of this gain with the high-order function shown in (46), to have an order of magnitude similar to the linear feedback (6)–(7). Again, it is important to stress that the proposed nonlinear law does not belong to the category of high-gain controllers which are, in general, locally not lipschitz (i.e., infinite local slope). Indeed, due to the high-order polynomial nature of this law, the first and second partial derivative evaluated at the origin are zero. This ensures low noise sensicomtivity near to the origin. The main contribution of the ponent is thus when the tracking error is large. Also note that the tracking performance and the control are torques obtained with the robust control and similar to those obtained by means of the partial model-based compensation, as shown in Figs. 8, and 9 (continuous and dotted lines are almost superimposed). The similarity in the control law evolution of these two approaches clearly shows that the proposed robust control strategy does not belong to the high-gain control class as discussed in the introduction. When the gain increases, the robust control approaches the partial model-based controller. Also note in the lower left plot of Fig. 9 that the VORTEX orientation torque for both controllers (the robust control and the partial model-based compensation) exhibits the saturation limits. Saturation would normally affect the performance of both controllers.
Fig. 10. (a) Circle tracking with robust nonlinear control law and actuator saturation. = 10 . (b) Circle tracking with partial model-based compensation.
When saturation is included, the overall performance of the VORTEX was slightly reduced for the robust control as well as for the partial model-based compensation as can be seen in Figs. 10(a) and (b). The VORTEX orientation error for both cases (with and without saturation) is shown in Fig. 11. This figure shows that both controllers achieve a similar performance even in the presence of saturation. VI. CONCLUSION We have presented, on the one hand, an alternative control design for the UVM composite system with substantially different bandwidth characteristics. For this purpose we have resorted to a singular perturbed model that allows reduction of control complexity. We have presented a formal study of the asymptotic properties of the closed-loop system solution. As a result of this study, we have shown that the error variables are of the order of magnitude of the bandwidth ratio, and that this bound holds for a time horizon that can be extended to
De WIT et al.: NONLINEAR CONTROL OF AN UNDERWATER VEHICLE/MANIPULATOR WITH COMPOSITE DYNAMICS
Fig. 11.
VORTEX orientation error and control errors. (a) robust control where = 10
infinity. The more meaningful result concerns the case where the vehicle reference is constant and the manipulator reference is time varying. Extension to the complete tracking case may be possible if the desired vehicle acceleration can be scaled down to slow motions of magnitude . Simulations have shown that a good precision is obtained when compared to partial model-based compensation. Moreover, we have presented a robust nonlinear feedback control for the UVM system with composite dynamics. Singular perturbed model was used as a basis for this study. The approximation results of this analysis are only valid in an in. The extension of terval of time can be obtained if , i.e., if and the regulation case is studied. However, these results are only sufficient, and it seems possible that this extension can be achived via a different analysis, using the same control law. Simulations have , the performance obtained with shown that even if this nonmodel-based control is similar to the one resulting from model-based feedback. It was also shown by simulation that this is still true in the case of saturation. In this case, the robust nonlinear control proposed in this paper is a good compromise between control complexity and closed-loop performance.
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(continuous line). (b) Partial model-based compensation (dotted line).
REFERENCES [1] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 1994. [2] C. C. de Wit, D. Williamson, and R. Bachmayer, “Performance-oriented robust control for a class of mechanical systems: A study case,” presented at the Proc. Int. Conf. Syst., Man, Cybern., Le Touquet, France, Oct. 17–20, 1993. [3] A. Healey and D. Lienard, “Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles,” IEEE J. Ocean. Eng., vol. 18, 1993. [4] M. Perrier and C. C. de Wit, “Experimental comparison of PID versus PID plus nonlinear controller for subsea robots,” J. Autonomous Robots (Special Issue on Autonomous Underwater Robots), 1996. [5] M. Perrier, V. Rigaud, C. C. de Wit, and R. Bachmayer, “Performanceoriented robust nonlinear control for subsea robots,” presented at the Proc. IEEE Int. Conf. Robot. Automat., San Diego, CA, 1994. [6] C. C. de Wit, E. O. Díaz, and M. Perrier, “Control of underwater vehicle/manipulator system with composite dynamics,” presented at the Proc. Amer. Contr. Conf., Philadelphia, PA, 1998. [7] , “Robust nonlinear control of an underwater vehicle/manipulator system with composite dynamics,” presented at the Proc. Int. Conf. Robot. Automat., Leuven, Belgium, 1998. [8] H. Mahesh, J. Yuh, and R. Lakshmi, “A coordinated control of underwater vehicle and robot manipulator,” J. Robot. Syst., vol. 8, no. 3, pp. 339–370, June 1991. [9] T. W. McLain, S. M. Rock, and M. J. Lee, “Coordinated control of an underwater robotic system,” presented at the Proc. IEEE Int. Conf. Robot. Automat. (Video Submission), Minneapolis, MN, 1996.
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[10] S. McMillan, D. E. Orin, and R. McGhee, “Efficient dynamic simulation of an underwater vehicle with a robotic manipulator,” IEEE Trans. Syst., Man., Cybern., vol. 25, pp. 1194–1206, Aug. 1995. [11] I. Schjølberg and T. I. Fossen, “Modeling and control of underwater vehicle-manipulator systems,” presented at the Proc. 3rd Conf. Marine Craft Maneuvering Contr. (MCMC'94), Southampton, U.K., 1994. [12] P. Kokotovic, H. K. Khalil, and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design. New York: Academic, 1986. [13] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [14] E. O. Díaz, “Underwater Mobile Manipulator Modeling,”, Int. Rep. 1-9LJ2/1-95-3-321285 LAG, Oct. 1996. [15] E. O. Díaz, C. C. de Wit, and M. Perrier, “Nonlinear Control of an Underwater Vehicle/Manipulator with Composite Dynamics: Appendices,”, Int. Rep. 99-100 LAG, Feb. 1999. [16] J. B. Marion and S. P. Thornton, Classical Dynamics of Particles and Systems, 4th ed. Orlando, FL: Harcourt Brace, 1995. [17] K. J. Åström, “Limitations on control systems performance,” presented at the IEEE Proc. Eur. Contr. Conf., Bruxels, Belgium, June 1997. [18] C. C. de Wit, B. Siciliano, and G. Bastin, Eds., Theory of Robot Control. New York: Springer-Verlag, 1996. [19] D. Williamson and C. C. de Wit, “Performance-oriented robust control for a class of nonlinear systems,” presented at the Proc. Eur. Contr. Conf., Rome, Italy, 1995.
Carlos Canudas de Wit was born in Villahermosa, Tabasco, Mexico, in 1958. He received the B.Sc. degree in electronics and communications from the Technological Institute of Monterrey, Mexico, in 1980. He received the M.Sc. degree from the Polytechnic of Grenoble, Department of Automatic Control, Grenoble, France, in 1984. He received the Ph.D. degree in automatic control from the Polytechnic of Grenoble in 1987. He was Visiting Researcher in 1985 at Lund Institute of Technology, Lund, Sweden. Since then he has been working at the same department as “Directeur de recherche at the CNRS,” where he teaches and conducts research in the area of adaptive and robot control. He teaches undergraduate and graduated courses in robot control and stability of nonlinear systems. He is also responsible for a team on Ccontrol of electromechanical systems and robotics. His research interests include adaptive control, identification, robot control, nonlinear observers, control of systems with friction, AC and CD drives, Automotive control, nonholonomic systems. He has written a book on Adaptive Control of Partially Known Systems: Theory and Applications (Amsterdam, The Netherlands: Elsevier). He also edited a book on Advanced Robot Control (New York: Springer-Verlag, 1991), and a joint 12–author book Theory of Robot Control (New York: Springer-Verlag, 1997). He has been Associate Editor of Automatica since March 1999. Dr. Canudas de Wit was Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, from January 1992 to December 1997.
Ernesto Olguín-Díaz (M’00) was born in Mexico City, Mexico, in 1968. He received the B.S. degree in mechanical and electrical engineering from the Technological Institute of Monterrey (ITESM), Mexico, in December 1991. He received the the M.S. degree in automation and the Ph.D. degree from the National Polytechnic Institute of Grenoble (INPG), Grenoble, France, in June 1995 and January 1999, respectively. His research includes mobile manipulation, underwater vehicles, nonlinear systems, nonlinear control theory, and robust control.
Michel Perrier (A’93) received the Ph.D. degree in robotics and automation in 1991 from the University of Montpellier II, France. In 1992, he spent one year as a Postdoctoral Researcher in the Underwater Vehicles Laboratory, Massachusetts Institute of Technology (MIT) Sea Grant College Program, Boston, MA, where he worked on the design and implementation of the low and high-level contol of the Autonomous Underwater Vehicle “Odyssey I.” Since December 1992, he had been a full-time Research Engineer in the Subsea Robotics and Artificial Intelligence Laboratory at the French Research Institute for the Exploitation of the Sea (IFREMER), La Seyne-sur-Mer, France. His interests include trajectory planning and path following for autonomous land mobile robots, using artificial intelligence technics for fusing sensor data and world model information, underwater vehicle systems including underwater vehicle control, coupled underwater vehicle-manipulator system control, telemanipulation systems, mission programming and planning, and control architecture.
