Input shaping control of a three‐cable suspension manipulator

A multi‐cable suspension manipulator moves a heavy payload through a large workspace by several spatially arranged cables with winches mounted on a crane system. All actuators are independently controllable to achieve a desired position and orientation of the payload with six degrees of freedom in space. As the payload is not fully constrained by the length of the cables, undesired load sway occurs. To transition the payload between predefined stationary equilibrium points, a feedforward control concept based on the input shaping method is presented. For the determination of the natural frequencies and damping ratios required for the design of an input shaper, a nonlinear dynamic model is linearised around a varying equilibrium point by an efficient procedure. The occurring kinematic actuator redundancy is resolved by optimisation. Experimental results from the prototype three‐cable suspension manipulator Cablev confirm the effectiveness of the approach.

shaper is most common.It consists of a sequence of two impulses and completely compensates self-induced oscillations under ideal conditions.Practical applications include cranes with single cable configuration [4,5] or flexible robots [6][7][8][9].
Since multi-cable suspension manipulators have multiple dynamically coupled vibrational degrees of freedom [10][11][12], an extended input shaper, the so-called multi-mode input shaper [3], is necessary to reduce undesired sway motions.Here, the original input signals are formed with multiple sequences of impulses based on the number of modes of the system.First studies for spatial payload positioning with a two-cable suspension manipulator have been presented in refs.[13,14].But only translational movements of the payload achieved through synchronous motions of the actuators are considered.The additional control of the rotational degrees of freedom of the payload requires differential motions of the actuators.Consequently, the resolution of the kinematic actuator redundancy by either task space augmentation or optimisation must be taken into account in the feedforward control design.Corresponding approaches are so far only known from the field of cable-driven parallel robots, where the cable winches are fixed in space [15][16][17].
The present contribution introduces a feedforward control concept based on the input shaping method for multicable suspension manipulators to transition a payload between predefined stationary equilibrium points with reduced sway motions with up to six degrees of freedom in space.For experimental validation, the topology of the three cablesuspension manipulator Cablev available at the University of Rostock [1,18] is considered.In Section 2 the control design model is formulated in the descriptor coordinates of the payload.Based on the linearised equations of motion derived analytically by a Taylor series expansion, the natural frequencies and the damping ratios are determined in Section 3 by solving the generalised eigenvalue problem.The input shaping based feedforward control concept is described in Section 4. Experimental results from the prototype system Cablev shown in Section 5 document the effect of the feedforward control.

MODELLING
For control design the nonlinear equations of motion of the considered cable suspension manipulator are formulated in descriptor form.They consist of the implicit algebraic constraint equations and the kinetic differential equations of the payload.The actuators are treated as kinematic inputs of the dynamic model under the assumption of individual drive controllers.

Coordinates.
Figure 1A shows the mechanical model of the cable suspension manipulator.Three stiff and massless cables suspend a rigid payload (mass  p , inertia tensor  p ∈ ℝ 3×3 with respect to the reference point  p in the system  o , centre of mass  p , centroid  p and side lengths  p1 ,  p2 ,  p3 of the triangle defined by the attachment points  1 ,  2 ,  3 of the cables) in space.The corresponding cable winches are mounted on trolleys of an overhead crane system consisting of a bridge with three parallel rails (distance  o between the rails) and three trolleys moving along these rails.Bridge, trolleys and cable winches are independently controllable to achieve a desired payload position and orientation.Thus, the seven actuator coordinates  ∈ ℝ 7 include the position of the bridge  b , the positions of the trolleys  t1 ,  t2 ,  t3 and the lengths of the cables  c1 ,  c2 ,  c3 , The six load coordinates  ∈ ℝ 6 describe the spatial position and orientation of the payload-fixed coordinate system  p relative to the inertial-fixed coordinate system  o by the three cartesian coordinates   ,   ,   and three Cardan angles  1 ,  2 ,  3 , The six load velocities ẏ ∈ ℝ 6 depict the spatial velocity of  p relative to  o by the three translational velocities ṙ , ṙ , ṙ and the three angular velocities   ,   ,   in  o .The relation between the time derivatives of the load coordinates ẏ and the load velocities ẏ is given by kinematic differential equations of the form [19] For further considerations, the load coordinates  and velocities ẏ are partitioned into the minimal coordinates  and velocities q suitable for describing the sway motions of the payload and the remaining dependent coordinates  and velocities ̇, Nonlinear equations of motion.
Due to the stiff cables, three implicit constraint equations hold between the actuator coordinates  and the payload coordinates .According to Figure 1B they are as well as the transformation matrix op () from  p to  o and the unit vectors whereby the vector p   describes the position of  p with respect to  p and is arbitrary for positioning and the vectors p   1 , p   2 , p   3 represent the positions of  1 ,  2 ,  3 with respect to  p and result from the geometry of the triangle.
The nonlinear equations of motion of the payload in descriptor form with respect to  p consist of the six kinematic differential equations (3), the three kinematic constraint equations ( 5) and the six kinetic differential equations Here,  ȳ ∈ ℝ 6×6 is the mass matrix,  c ȳ ∈ ℝ 6 contains the generalised gyroscopic forces,  g ȳ ∈ ℝ 6 includes the generalised gravitational forces and the Jacobian matrix  ȳ ∈ ℝ 3×6 maps the cable force coordinates (Lagrange multipliers)  ∈ ℝ 3 in direction of the load coordinates .Moreover,  3 ∈ ℝ 3×3 denotes the identity matrix, the vector   ∈ ℝ 3 describes the position of  p relative to  p ,  is the gravitational acceleration, and the tilde operator represents the vector product.

MODAL ANALYSIS
To apply input shaping for generating trajectories that reduce undesired load sway, the natural frequencies and damping ratios of the cable suspension manipulator are required.The modal parameters can be determined by solving the generalised eigenvalue problem based on the minimal form of the equations of motion of the payload linearised around a stationary equilibrium point under the assumption of small sway motions.For the derivation it is advantageous to linearise the equations of motion in descriptor form and then transform them into minimal form using the linearised explicit constraint equations.

Linearised equations of motion.
Since only the homogeneous part of the equations of motion is relevant for the calculation of the modal parameters, the small variations of the actuator coordinates are not taken into consideration.Under this assumption the Taylor series expansion of the kinematic constraint equations ( 5) and the kinetic differential equations (7) up to the first order terms yield the autonomous linearised equations of motion of the payload in descriptor form whereby Δ ȳ ∈ ℝ 6 are the small variations of the load coordinates, and Δ ∈ ℝ 3 are the small variations of the cable force coordinates.The constant coefficient matrices are the mass matrix  Δ ȳ ∈ ℝ 6×6 , the stiffness matrix  Δ ȳ ∈ ℝ 6×6 and the Jacobian matrix  Δ ȳ ∈ ℝ 3×6 .The notation |  means that for the actuator coordinates , the load coordinates  and the cable force coordinates  the values of the respective stationary equilibrium point are to be inserted.Moreover, note that all other terms vanish, because for a stationary equilibrium point the kinematic constraint equations as well as the kinetic differential equations are fulfilled, and the coordinates ẏ, ÿ are zero.
To transform the linearised equations of motion (8) from descriptor form to minimal form, the linearised explicit constraint equations associated with the linearised implicit constraint equations are required whereby  Δ ȳ ∈ ℝ 6×3 ,  Δ q ∈ ℝ 3×3 (first, second and sixth column of  Δ ȳ ) and  Δs ∈ ℝ 3×3 (third, fourth, and fifth column of  Δ ȳ ) are the Jacobian matrices,  2 ∈ ℝ 2×2 denotes the identity matrix and Δ q ∈ ℝ 3 are the small variations of the minimal coordinates.Consequently, the first, second and sixth linearised explicit constraint equation are obtained by trivial assignment of Δ q to Δ ȳ and the third, fourth and fifth linearised explicit constraint equation are determined by rearranging the linearised implicit constraint equations in (8) according to the small variations of the dependent coordinates Δs.
The minimal form of the linearised equations of motion ( 8) is obtained by expressing the small variations of the descriptor coordinates Δ ȳ in terms of the small variations of the minimal coordinates Δ q using the linearised explicit constraint equations (9) and by eliminating the small variations of the cable force coordinates Δ using the orthogonality property  T Δ ȳ  T Δ ȳ = ,  Δ qΔ q +  Δ qΔ q =  with  Δ q =  T Δ ȳ  Δ ȳ  Δ ȳ ,  Δ q =  T Δ ȳ  Δ ȳ  Δ ȳ , whereby  Δ q ∈ ℝ 3×3 ,  Δ q ∈ ℝ 3×3 are the coefficient matrices calculated for a stationary equilibrium point.

Modal parameters.
Using the ansatz Δ q = Δ qe  with the eigenvector Δ q and the eigenvalue  for the linearised dynamic equations (10), the three natural frequencies  ∈ ℝ 3 are obtained by numerically solving the generalised eigenvalue problem where the -th natural frequency   corresponds to the -th solution  2  of the characteristic polynomial ( 2 ),  2  = − 2  ,  = 1, 2, 3.The three damping ratios  ∈ ℝ 3 of the kinematically indeterminate system are zero for the assumptions made.

INPUT SHAPING BASED FEEDFORWARD CONTROL
The objective of the feedforward control is to transition the payload of the cable suspension manipulator between stationary equilibrium points without residual load sway with up to six degrees of freedom in space.Utilising the input shaping method, the required actuator coordinates are obtained by using a multi-mode zero-vibration input shaper and by calculating the so-called generalised inverse kinematics.

Multi-mode input shaper.
To transition the payload between stationary equilibrium points, trajectories are generated for the load coordinates y.The task of the input shaper is to reshape these trajectories of the load coordinates y in such a way that the shaped trajectories of the load coordinates y self-cancel undesired sway motions of the payload.Since the three-degree-of-freedom sway motions of the payload are dynamically coupled due to the kinematic parallel structure of the three-cable suspension and the modal parameters cannot be unambiguously assigned to individual load coordinates, a multi-mode input shaping algorithm is used.Here, the trajectories of the load coordinates y are convolved with three sequences of impulses whereby the symbol * represents convolution, and the number of sequences corresponds to the number of modes of the cable suspension manipulator [20,21].The required impulse intensities  , and excitation times  , of the Dirac delta functions are mathematically calculated from the system's natural frequencies  and damping ratios  using the formulas of the zero-vibration input shaper for each sequence where   = 1 for the assumed case   = 0. Note that other input shapers can be used to obtain the impulse intensities  , and the excitation times  , of each sequence.For example, the zero-vibration-and-derivative input shaper or the extrainsensitive input shaper achieve a higher robustness against uncertainties at the expense of an increased computational effort [22,23].

Generalised inverse kinematics.
The generalised inverse kinematics problem is to determine the actuator coordinates p in terms of the load coordinates y at each time step.With the three kinematic constraint equations ( 5) and the six equilibrium conditions (7) for the seven unknown actuator coordinates p and the three unknown cable force coordinates λ, the problem is kinematically redundant with one redundant degree of freedom.The resolution is done through optimisation by minimising the Euclidean norm of the Jacobian matrix  q ∈ ℝ 3×3 (first, second, and sixth column of  ȳ ).The selected criterion leads to solutions where the inclination angles of the cables and thus the variations of the natural frequencies  and the damping ratios  are kept as small as possible during a payload transition between stationary equilibrium points.The corresponding constrained optimisation problem reads whereby the nonlinear constraint equations ensure that for a desired pose the payload is in a kinematically compatible and static equilibrium configuration.In comparison to the flatness-based approach described in refs.[2,24], the inertial forces and the generalised gyroscopic forces are neglected, since the sway dynamics of the payload are compensated by the multi-mode input shaper.

Control structure.
The overall structure of the input shaping based feedforward control for the cable suspension manipulator is shown in Figure 2. Based on the actual payload coordinates  0 of the initial equilibrium point and the desired load coordinates  1 of the final equilibrium point, the trajectory generator creates suitable trajectories for the load coordinates y.The trajectories of the load coordinates y are then reshaped by the multi-mode input shaper (12) in order to reduce undesired load sway.Finally, the trajectories of actuator coordinates p that serve as the kinematic inputs of the cable suspension manipulator are calculated by solving the generalised inverse kinematics (14) using the shaped trajectories of the load coordinates y.
The natural frequencies  and the damping ratios  required for the design of the multi-mode input shaper (12) are determined by solving the generalised eigenvalue problem (11).To calculate the coefficient matrices  Δ q,  Δ q, the actual actuator coordinates  0 , the actual load coordinates  0 and the actual cable force coordinates  0 of the initial equilibrium point are used.Consequently, changes in the modal parameters ,  that may occur during a transition between two equilibrium points are not taken into consideration in the feedforward control.The error leading to undesired load sway can be minimised by the choice of the optimisation criterion in the generalised inverse kinematics (14) and by the separate execution of controlled movements of the payload in the directions of the minimal coordinates  and in the directions of the dependent coordinates .The measures are based on observations that large changes of the cable inclination angles or the cable lengths correlate with large changes in the modal parameters ,  .

EXPERIMENTAL RESULTS
The effectiveness of the input shaping based feedforward control is demonstrated by an experiment on the three-cable suspension manipulator Cablev.The prototype system is available at the University of Rostock for testing various control algorithms.Due to lack of space, please refer to the literature [1,18] whereby a motion in   -direction is studiously avoided to prevent large changes in the modal parameters ,  .To protect the test rig, the trajectories of the load coordinates y() between the initial and final equilibrium point are interpolated by continuous functions that take into account the kinematic limitations of the actuators.The experimental results are shown in Figure 3. Figures 3A-B respectively represent the trajectories of the bridge and trolley displacements ṕg () =  g () −  g0 as well as the cable length displacements ṕc () =  c () −  c0 obtained from the generalised inverse kinematics and from measurements.Figures 3C-D display the prescribed and measured trajectories of the payload displacements ŕ() = () −  0 and the payload rotations θ() = () −  0 .The occurring deviations during the transition are caused by the design constraint of the zero-vibration input shaper that the sum of the impulse amplitudes must equal one in order to ensure stationary accuracy [3].As a result, unlike the flatness-based approach, the gradient of the shaped signal cannot be greater than the gradient of the original signal.Finally, Figures 3E-F show the tracking errors   ,   of the load coordinates in the time interval 12 s ≤  ≤ 20 s and illustrate how the input shaping based feedforward control moves the payload to a desired stationary equilibrium point with small residual oscillations.The occurring stationary deviations are due to calibration inaccuracies.

CONCLUSION
For a three-cable suspension manipulator with kinematically indeterminate payload position, a feedforward control based on the input shaping method has been presented.Experimental results from the prototype system Cablev confirm the effectiveness of the approach.Ongoing research is devoted to the design of a suitable feedback controller that reduces tracking errors and actively dampens load sway due to non-modelled effects.

F I G U R E 1
Mechanical model of the three-cable suspension manipulator.(A) Coordinates.(B) Geometric constraints between  and .

F I G U R E 2
Modular structure of the input shaping based feedforward control for the three-cable suspension manipulator.

F I G U R E 3
Experimental results of the input shaping based feedforward control for a general movement of the payload in space.

2 .
for a detailed description of the structure.The non-zero physical parameters of Cablev used for the control design are given by  o = 0.45 m,  p1 =  p2 =  p3 = 0.90 m,  p = 12.5 kg and p  = diag{0.75,0.75, 1.50}kg m 2 .In the experiment, the payload is moved between two stationary equilibrium points with the initial and final conditions of the load coordinates  T 0 = [ 0.50 m 0.50 m 2.00 m 0.00 deg 0.00 deg 0.00 deg ] 10 m 1.30 m 2.00 m 10.00 deg 20.00 deg 30.00 deg ] , Open access funding enabled and organized by Projekt DEAL.O R C I D Erik Hildebrandt https://orcid.org/0009-0005-0644-044XR E F E R E N C E S