DETC99/VIB-8223 FLATNESS-BASED CONTROL OF UNDERCONSTRAINED CABLE SUSPENSION MANIPULATORS

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1 Proceedings of DETC ASME Design Engineering Technical Conferences September -5, 999, Las Vegas, Nevada, USA DETC99/VIB-83 FLATNESS-BASED CONTROL OF UNDERCONSTRAINED CABLE SUSPENSION MANIPULATORS Thomas Maier Institute of Drive Systems and Mechatronics University of Rostock D-859 Rostock Germany Christoph Woernle Institute of Drive Systems and Mechatronics University of Rostock D-859 Rostock Germany ABSTRACT Underconstrained cable suspension manipulators support a load platform in space by less than six spatially arranged cables with independently controllable winches. To control the motion of the platform along desired trajectories in space, the classical inverse kinematic or computed-torque schemes are generalized using the concept of flat systems. For desired trajectories of the load platform and a sufficient number of their time derivatives, the control inputs can be algebraically calculated. Asymptotically stable tracking behaviour is achieved by cascaded feedback of state variables. The application of the method is shown for a planar overhead trolley crane as well as for a novel three-cable suspension manipulator. INTRODUCTION Cable suspension manipulators support a payload platform in space by several spatially arranged cables with computercontrolled winches. The winches are mounted either fixed or on movable trolleys. Compared to conventional cranes, it is possible to control not only the translational motion of the payload but also its orientation in order to perform, for example, assembly tasks. By this, cable suspension manipulators combine the ability of cranes to support heavy payloads in a large workspace with the dexterity of robot manipulators. Various configurations have been proposed (Dagalakis et al. 989, Arai and Osumi 99, Tadokoro et al. 996, Maier and Woernle 998). A cable suspension manipulator that supports a payload platform in space is underconstrained, if it has less than six cables (Ming and Higuchi 994). The payload platform of an underconstrained cable suspension manipulator may carry out sway motions even if the drives at the winches and trolleys are not moved. Obviously, the most simple underconstrained cable suspension manipulator is an overhead crane with a load mass. The application standing behind the present investigation is the novel cable suspension manipulator CABLEV being developed at University of Rostock (Figure ). Its load platform is supported by three cables with winches mounted on trolleys that move theirselves on a gantry. Applications are, for example, handling and assembling large and heavy components on construction sites or on shipyards. Since the position of the platform of an underconstrained cable suspension manipulator is not uniquely determined by the robot coordinates at the trolleys and winches, its motion cannot be controlled like the end-effector of an conventional robot by inverse kinematic control (at the kinematic level) or computedtorque control (at the dynamic level). Instead, the dynamics of the sway motion has to be taken into account. This can be achieved by means of the concept of flat systems that represents a relatively new approach for the analysis and control of a broad class of nonlinear systems (Fliess et al. 99). Flat systems may be regarded as a generalization of controllable linear systems. For flat systems many trajectory-tracking control problems can be systematically solved, both by open-loop or closed-loop control. Its application to underconstrained robotic systems represents a generalization of computed-torque control. Copyright 999 by ASME

2 moment of inertia J) and a load (absolute position coordinates y, y, mass m), Figure. The cable is assumed to be massless and longitudinally stiff. The winch radius r is neglected against the cable length q. The crane is controlled by the trolley force u and the winch torque u. The driven robot coordinates q and the coordinates y of the load mass are, respectively, q = q q y = y y : () Figure. Cable suspension manipulator CABLEV A flat system has the property that the state variables and the control inputs can be algebraically expressed in terms of the control outputs and their time derivatives up to a certain order. Two benefits are: If the reference trajectories of the control outputs and a sufficient number of their time derivatives are specified, the motion of the system and the control inputs can be algebraically calculated. This property is useful for open-loop control design and for motion planning (Fliess et al. 995). Asymptotically stable tracking of precribed reference trajectories is achieved by a static, i.e. algebraic, feedback of state variables. This feedback is called quasi-static (Delaleau and Rudolph 995). The procedure is described for the motion control of the load mass of an overhead trolley crane because of its simple governing equations, refer also to Fliess et al. (993) and Lévine et al. (997). The three-cable suspension manipulator CABLEV with the same structure of governing equations is subsequently treated. FLATNESS-BASED CONTROL OF AN OVERHEAD TROLLEY CRANE. Dynamic Model A simple model of an overhead crane consists of a trolley (position q, mass m k ) with a winch (radius r, cable length q, Figure. Overhead trolley crane The control task considered here is to make the load mass track a desired trajectory in the vertical plane ŷ(t) = ŷ (t) ŷ (t) : () The coordinates y and q are constrained by ϕ(q;y) (y ; q ) + y ; q = : (3) The first- and second-order total time derivatives of the constraint (3) needed in the sequel are ẏ Φ q ϕ Φ y q with Φ y = y ; q = y Φ q = ; y ; q q (4) Copyright 999 by ASME

3 and ϕ Φ y with ϕ = Φ y ÿ Φ q + ϕ = q Φ q ẏ q The equations of motion for the load mass and the trolley/winch motion are coupled by the generalized cable force λ (Lagrangemultiplier) in direction of the cable, m m ÿ ÿ = mg + : y ; q y λ M y ÿ = F y + Φ T y λ (5) (6). Flatness of the System Flatness of the considered system means that the control inputs u as well as all internal variables, such as q and λ, can be expressed in terms of the control outputs y and time derivatives of the outputs. This can be done in two steps, refer also to Fliess et al. (993) and Lévine et al. (997). First the robot coordinates q and the generalized cable force λ are expressed. With (3) and (6) there are three equations available for q, q, and λ. Elimination of λ between the two equations in (6) yields the trolley position q, that can be inserted into (3) to express the cable length q, q (y ÿ) = y ; ÿ y ÿ ; g q () q (y ÿ) = [y ; q (y ÿ)] + y () : mk J r q q = r u y ; q ; λ u q M q q = B q u + Φ T q λ : (7) The generalized cable force λ is then expressed, for example, by the second equation of (6), λ(y ÿ) = m(ÿ ; g) y : () The differential-algebraic set of equations (6), (7), and (3) then describes the dynamics of the crane. Hereby, the constraint must be fulfilled at the position, velocity and acceleration levels, according to eqs. (3), (4), and (5), respectively. If the positions y, q and the velocities ẏ, q, which have to be consistent with the constraint (3) and their first-order time derivative (4), as well as the control forces u are given, eqs. (6), (7), and (5) together represent a set of five linear equations to determine uniquely the four accelerations ÿ and q, and the cable force λ: (6) : (7) : (5) : 64 M y Φ T y M q Φ T q Φ y Φ q ÿ q ;λ = 64 F y B q u ; ϕ 75 : (8) By eliminating the Lagrange-multiplier λ, the equations of motion can be reformulated in minimal form as ordinary differential equations in terms of three independent coordinates, e.g. q, q, y. Then y represents a coordinate of the sway motion, and the independent state variables of the system are In a second step the control inputs u are obtained by inserting the generalized cable force λ from () into the trolley/winch dynamics (7), u u = mk J r q q + y ; q m(ÿ ; g) : (3) q r y Expressions for q and q are obtained by differentiation of () and (). Since q and q in (), () depend on the secondorder time derivative of the outputs y, the accelerations q and q are obtained in terms of the fourth-order time derivative of y. Altogether, the control inputs u can be algebraically expressed in terms of the outputs y and their time derivatives ẏ, ÿ, y (3), and y (4). Thus, the system is flat..3 Overall Control Structure The procedure described in the previous subsection represents an open-loop control structure to calculate algebraically control forces u for given output functions ŷ(t), T x = y q T T ẏ q : (9) The minimal form of the equations of motion in terms of x,however, is not needed in the sequel. In the following subsections a control strategy is described that exploits the flatness of the system. ŷ(t) ŷ(t) ŷ(t) ŷ (3) (t) ŷ (4) (t) + () () ˆq(t) ˆq(t) ˆq(t) + (3) u(t) : (4) 3 Copyright 999 by ASME

4 However, to counteract disturbances like initial conditions that are not consistent with the desired trajectory, wind forces, or incertainties of parameters, it is necessary to use a closed-loop control strategy with feedback of the actual errors. According to the open-loop calculation sequence according to (4), two cascaded controllers are provided (Figure 3): a) Tracking and anti-sway controller. A feedback of the sway variables y, ẏ generates robot coordinates ˆq(t) that make the load mass track the desired trajectory ŷ(t) and provides damping of undesired sway oscillations. For the layout of this controller it is assumed that the actual robot coordinates q(t) track ˆq(t). b) Trolley/winch motion controller. A feedback of the state vector x from (9) makes the robot coordinates q(t) track the functions ˆq(t) generated by the anti-sway controller. This cascaded structure is based on the assumption that the controlled trolley/winch dynamics can be made fast against the sway dynamics by means of sufficiently powerful trolley/winch motors. In the following two subsections, the controllers are described in more detail..4 Tracking and Anti-Sway Controller The control law is obtained from () and () by assigning new inputs according to ÿ! w y! w : (5) Pure tracking control (without anti-sway control) is achieved by w = ŷ w = ŷ : (6) Note that the new input w is only assigned to the second-order derivative of y, since for y and ẏ the actual values of the state variables are used. With (5) the control laws () and () read ˆq (y w w ẅ ) = y ; w w ẅ ; g q (7) ˆq (w w ẅ ) = (y ; ˆq ) + w (8) with ˆq from (7) : the sway coordinate y, whereby (6) is replaced by w = ŷ + α V ( ŷ ; ẏ )+α P (ŷ ; y ) (9) yˆ : () w = The feedback gains α P, α V are assigned by pole-placement for the dynamics of the position error of the sway motion ( ŷ ; ÿ )+α V ( ŷ ; ẏ )+α P (ŷ ; y )= : () According to (4), the robot coordinates ˆq(t) are needed not only at the position level, as given by (7) and (8), but also at the velocity and acceleration levels. Differentiating the expressions in (7) and (8) with respect to the time gives the robot coordinates at the velocity level ˆq = ẏ ; e w (3) ; ẇ w ; w ẇ () ẅ ; g ˆq ˆq = e ė + w ẇ with ˆq from () (3) and the abbreviations e = y ; ˆq ė = ẏ ; ˆq : The robot coordinates at the acceleration level are with the assignment ÿ! w from (5) ˆq = w ; e w (4) + ė w (3) ; ẅ w ; ẇ ẇ ; w ẅ ẅ ; g (4) ˆq = ˆq e (w ; ˆq )+ė + w ẅ + ẇ ; ˆq with ˆq from (4) : (5) The time derivatives of w and w needed in () to (5) are obtained by differentiating (9) under consideration of (5), ẇ = ŷ (3) + α V ( ŷ ; w ) + α P ( ŷ ; ẏ ) (6) ẅ = ŷ (4) + α V (ŷ (3) ; ẇ )+α P ( ŷ ; w ) (7) With (7) and (8) the load tracks the desired trajectory, i.e. y(t) ŷ(t) if the initial conditions are consistent with the reference trajectory, i.e. y ()=ŷ () and ẏ ()= ŷ (), and if there are no disturbances such as wind forces. Asymptotic stabilization of the sway motion is achieved by adding to w the actual position and velocity errors in terms of and ẇ = ŷ ẅ = ŷ w (3) = ŷ (3) w (4) = ŷ (4) : (8) Again it is seen that the desired trajectory ŷ(t) of the load mass from () has to be defined at least up to the fourth-order time derivative. 4 Copyright 999 by ASME

5 β - β - Figure 3. Cascaded control structure.5 Trolley/Winch Motion Controller The trolley/winch motion controller makes the actual robot coordinates q(t) track the functions ˆq(t) that are generated by the tracking and anti-sway controller according to (7) and (8), and () to (5). If the system exactly follows the desired trajectory ŷ(t), the needed control forces can be calculated by (3). However, to counteract non-consistent initial conditions or other disturbances, a feedback of the actual position and velocity errors of the robot coordinates q is provided. If the deviations of the robot coordinates q(t) from the nominal trajectory ˆq(t) are not too large, a linear feedback of the errors may be sufficient. In practice this should be seeked for in order to avoid too complex control structures. From a theoretical point of view it is also possible to design a nonlinear controller that deals with large deviations from the nominal trajectory, too. This controller is obtained by the method of exact inputoutput linearization (IO linearization), refer, for example, to Isidori (995). Here, the nonlinearities of the mechanical system are compensated by an inverse dynamic system and new control inputs are applied that act independently on the actual robot coordinates q. The governing equations for the inverse system are the dynamic equations of the mechanical system (6) and (7), and the constraint (5). Assigning new inputs v to the accelerations of the robot coordinates q according to q v! or q! v (9) q v these equations lead to the linear set of equations (6) : (7) : (5) : 64 M y Φ T y Φ T q B q Φ y ÿ ;λ ;u = F y ;M q v ;Φ q v ; ϕ 3 75 : (3) The control forces u obtained from (3) generate actual accelerations q that are equal to the inputs v, q v = or q = v : (3) q v 5 Copyright 999 by ASME

6 β - β - Figure 4. Structure of the IO-linearized mechanical system (equivalent to Figure 3) The auxiliary variables ÿ and λ in the solution vector of (3) are not further used. The system matrix and the right-hand side of (3) depend on the independent state variables x from (9). This calculation of the linearizing feedback directly from the differential-algebraic equations of motion avoids their reformulation in minimal form, i.e. in terms of x (Woernle 998). Asymptotic stabilization of the linearized and decoupled IO channels (3) is achieved by feedback of the actual position and velocity errors of the robot coordinates q (feedback gains β Pi, β Vi ), v = ˆq + β V ( ˆq ; q )+β P ( ˆq ; q ) (3) v = ˆq + β V ( ˆq ; q )+β P ( ˆq ; q ) : (33) The structure of the IO-linearized mechanical system is represented by Figure 4. It shows the decoupled IO channels (3) stabilized by (3) and (33). Note that the sway motion is still nonlinear. The sway motion can be described in terms of the sway coordinate y. It has the property that it is not observable from the outputs q of the IO channels (3). By this, it is the zero dynamics of the IO-linearized system (Isidori 995). The zero dynamics can be expressed as a second-order differential equation in terms of y, ÿ = r(y q ẏ q)+r(y q)v : (34) However, the zero dynamics is here not derived, since it is not further exploited..6 Example Tracking a straight-line trajectory between the rest positions (y y ) =(m 5m) at t = s and (y y ) E =(m 3m) at t E = s is considered as an example in Figure 5. To meet the smoothness requirements, the velocity along the trajectory (length L) is prescribed by v(t) =L 3ω 4 sin3 ωt with ω = π t E : 6 Copyright 999 by ASME

7 5 m m 3 m y [m] y [m] displacement [m] force [N] / torque [Nm] time [s] time [s] Figure 5. Tracking a straight-line trajectory. To show the effects of the control, the initial conditions are not consistent with the reference trajectory, q (t )=m q (t )=6m (load mass below the desired trajectory), y (t )=m ẏ (t )=;ms ; (initial sway motion in negative y -direction). Mechanical parameters are r = :m m = kg m k = 3 kg J = :6m kg: The poles of the anti-sway controller according to () are s = ; :75i : The poles of the trolley/winch motion controller according to (3) with (9) are s = ; :5i : The time histories of y (t) and y (t) in Figure 5 that have been obtained by a MATLAB program show the asymptotically stable tracking behaviour. 3 CABLE SUSPENSION MANIPULATOR CABLEV 3. Dynamic Model To manipulate a load in space, the cable suspension manipulator CABLEV (CABle LEVitation) is being developed at University of Rostock. Its load platform is suspended by three spatially arranged cables. Based on earlier investigations (Maier and Woernle 997), a configuration was found, where the winches are mounted on trolleys moving along parallel tracks on a gantry that is itself movable (Figure 6). A laboratory system is under construction. The workspace will have a base area of about m and a height of about.5 m, the payload will be about 3 kg. The independently controllable robot coordinates q IR 7 are the gantry/trolley displacements q :::q 3 and the three cable lengths q 4 q 5 q 6. The platform coordinates y IR 6 are, for example, three cartesian coordinates x, y, z along the axis of an inertial frame, and three Cardan angles α, α, α 3. Thus, the 7 Copyright 999 by ASME

8 3. Flatness-Based Control The control goal is to make the platform track a desired trajectory ŷ(t) in space. The redundancy is treated in a first approach by introducing a control constraint between the three trolley coordinates q q, and q 3 of the form 3 ϕ (q) q + (q + q 3 ) ; c = with c = const : (38) According to Subsection.4, the tracking and anti-sway controller calculates reference functions of the robot coordinates ˆq(t) IR 7 for given reference functions ŷ(t). The following nonlinear set of ten scalar equations is solved to determine the unknowns ˆq(t) IR 7 and λ IR 3 for given ŷ, ŷ, ŷ: (36) : Φ T y (ŷ ˆq) ˆλ = M y (ŷ) ŷ + k y (ŷ ŷ) ; F y (ŷ ŷ) (35) : ϕ(ŷ ˆq) = (38) : ϕ (ˆq) = 9 >= > : (39) Figure 6. Robot coordinates of CABLEV. system is kinematically redundant. With three cables, there are three constraints between q and y analogous to (3), q ϕ(q y) =! ϕ Φ q Φ y = : (35) ẏ Analogous to (6) and (7), respectively, dynamic equations are formulated for the load platform in terms of the platform coordinates y, M y (y)ÿ + k y (y ẏ) =F y (y ẏ)+φ T y (y q)λ (36) and for the assembly of the gantry, the trolleys and the winches in terms of the robot coordinates q IR 7, M q (q) q + k q (q q) =F q (y ẏ)+b q (y)u + Φ T q (y q)λ : (37) Here, M y IR 6 6 and M q IR 7 7 are mass-matrices, k y IR 6 and k q IR 7 are centrifugal and Coriolis forces, F y IR 6 and F q IR 7 include all applied forces except for the control forces u IR 7 along the robot coordinates q IR 7, and λ IR 3 are the generalized cable forces. The time derivative of (39) leads a linear set of equations to determine ˆq in terms of ŷ :::ŷ (3), Φ T y ˆλ ΦT y ˆq Φ q ŷ Φ ˆq ˆλ 3 5 = My ŷ + k y ; F y ŷ + ŷ [k y ; F y ] ŷ + M y ŷ (3) ;Φ y ŷ 3 75 : (4) Symbolic calculation of the matrices of (4) is recommendable in order to exploit simplifications due to the particular structure of the expressions in (39). Differentiation of (4) finally gives a linear set of equations for ˆq in terms of ŷ :::ŷ (4). Investigations showed that the terms stemming from the time derivatives of the coefficient matrices of (4) can be neglected without loosing too much accuracy in calculating ˆq. According to (3), a feedforward tracking and trolley/winch controller is obtained by solving (37) with respect to u. For small deviations from the reference trajectory ˆq(t) linear error feedback may suffice. According to (3) exact IO linearization is possible that requires, however, a large computational amount. 3.3 Example To illustrate the idea of manipulating an object in space by the CABLEV manipulator, simulations of rotations of the load 8 Copyright 999 by ASME

9 [ ] Figure 7. Rotation of the load platform around an horizontal space-fixed axis α [ ] Figure 8. Rotation of the load platform around a vertical space-fixed axis platform around a horizontal and a vertical fixed axis in space are shown in Figures 7 and 8, respectively. The representations of the robot coordinates q i over the corresponding rotation angles show that all robot coordinates have to be adjusted in order to execute these platform motions. The simulation results have been obtained by means of the object-oriented programming environment M a abile (Kecskeméthy a a 995). 4 CONCLUSIONS The trajectory tracking problem of underconstrained cable suspension manipulators can be solved by a generalization of computed-torque control schemes, whereby the property of flatness of the describing dynamic equations is exploited. A cascaded feedback structure leads to asymptotically stable tracking behaviour. An objective of ongoing investigations is to find simplifications of the control structure. Examples are the neglection of terms with minor influence in the tracking and anti-sway controller or the investigation of simplified trolley/winch controllers. A central problem is the measurement of the actual position of the load platform. The design of the winches and the cable guidances allows measurements of the cable length via the rotation angle of the drum and of the inclination of each cable in one vertical plane. In combination with the trolley/gantry positions these six measurements are used to calcuate the absolute platform position and orientation. 9 Copyright 999 by ASME

10 REFERENCES Arai, T. and Osumi, H., 99, Three Wire Suspension Robot, Industrial Robot, Vol. 9, pp. 7. Dakalakis, N.G., Albus, J.S., Wang, B.L., Unger, J., and Lee, J.D., 989, Stiffness Study of a Parallel Link Robot Crane for Shipbuilding Applications, ASME Journal on Offshore Mechanics and Arctic Engineering, Vol., pp Delaleau, E. and Rudolph, J., 995, Decoupling and Linearization by Quasi-Static Feedback of Generalized States, Proc. 3rd European Control Conference ECC 95, Rome, pp Fliess, M., Lévine, J., Martin, P., and Rouchon, P., 99, On Differentially Flat Nonlinear Systems, In: Fliess, M. (Ed.) Nonlinear Control System Design, Pergamon Press, pp Fliess, M., Lévine, J., and Rouchon, P., 993, Generalized State Variable Description for a Simplified Crane Description, Int. Journal of Control, Vol. 85, pp Fliess, M., Lévine, J., Martin, P., and Rouchon, P., 995, Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples, Int. Journal of Control, Vol. 6, pp Isidori, A., 995, Nonlinear Control Systems, Springer, Berlin. Kecskeméthy, A., (995), Object-Oriented Modeling of Mechanical Systems, In: Angeles, J. and Kecskeméthy, A. (Eds.) Kinematics and Dynamics of Multibody Systems, Springer, Berlin, pp Lévine, J., Rouchon, P., Yuan, G., Grebogi, C., Hunt, B.R., Kostelich, E., Ott, E., and Yorke, J.A., 997, On the Control of US Navy Cranes, Proc. European Control Conference ECC 97, Brussels, Belgium, Paper No. 77. Maier, T. and Woernle, C., 998, Inverse Kinematics for an Underconstrained Cable Suspension Manipulator. In: Lenar ci c, J. and Husty, M. (Eds.) Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, Dordrecht, pp Ming, A. and Higuchi, T. (994), Study on Multiple Degree-of-Freedom Positioning Mechanism Using Wires, Int. Journal of the Japanese Society for Precision Engineering, Vol. 8, pp and Tadokoro, S., Nishioka, S., Kimura, T., Hattori, M., Takamori, T., and Maeda, K., 996, On Fundamental Design of Wire Configurations of Wire-Driven Parallel Manipulators with Redundancy, Proc. 996 Japan USA Symposium on Flexible Automation, Boston (Mass.), Vol., pp Woernle, C., 998, Control of Robotic Systems by Exact Linearization. In: Angeles, J. and Zakhariev, E. (Eds.) Computational Methods in Mechanical Systems, Springer, Berlin, pp Copyright 999 by ASME