Payload motion control for a varying length flexible gantry crane

Cranes play a very important role in transporting heavy loads in various industries. However, because of its natural swinging characteristics, the control of crane needs to be considered carefully. This paper presents a control approach to a flexible cable crane system in consideration of both rope length varying and system constraints. At first, from Hamilton's extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry are obtained. The equations of motion consist of a system of partial differential equations. Then, a barrier Lyapunov function is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive experimental studies.


Introduction
Overhead cranes are widely used in many applications such as manufacturing factories, marine industries and harbour operations, due to their capability of transporting heavy loads or hazardous materials. However, as a typical under-actuated system, and in some situations, rope flexible deformation cannot be ignored, the crane load is frequently swinging during transportation processes, which affects the positioning accuracy of the load, and brings danger, damage, even accidents in working sites. Thus, the main problem in handling the crane system is to reduce the sway angle of the load and moving it to a desired position with a fast motion.
Recently, various studies have been done to solve the above-mentioned problems. Many control strategies have been applied to the crane system that can be divided into three categories including open loop (such as input shaping, filtering, command smoothing), closed loop (such as classical linear control, intelligent control, optimal control, adaptive control, sliding mode control and so on), and combined open and closed loop control, see [1] for more details. In addition, several efforts to use some other control algorithms have been investigated. In [2], time-optimal flatness-based control has been used to minimize the transition time. Model predictive control in combination with disturbance predictor is used in [3] for control of the crane system with strong disturbances and uncertainties. In [4], the hybrid partial feedback linearization and deadbeat control scheme is applied to control the crane. In this research, a deadbeat control is used to control and accelerate the position response, while partial feedback linearization is in charge of minimizing and stabilizing the sway angle. Moreover, to delivery high-performance control operation for overhead crane, four control schemes are combined in [5] to control a overhead crane. The discrete-time controller is formulated based on state feedback approach to provide servo control operation. The reference signal generator based on typical anti-swing trajectory performed by an expert crane operator is used to supply reference state trajectory profiles. The feedforward control that generates the designed output trajectory from system model to reduce nonlinear disturbance and improve the tracking accuracy. Simultaneously, the load swing control that uses a high-gain observer to damp the load swings. To avoid the dependence of controller design on the crane model, a model-independent control called proportional-derivative with sliding mode control is proposed in [6]. The controller is model free that makes it robust with uncertain/unknown system parameters. In addition, to overcome the sensitivity of measured signal for feedback control scheme, inverse dynamic that uses simulations of feedback control by machine learning has been proposed in [7]. In this research, artificial neural network that can act in realtime is used to learn inverse dynamic model from actual crane.
All of aforementioned works treat crane motion as a pendulum-like system and the crane cable is assumed as a rigid body. However, the crane cable is flexible in practice, that assumption is not valid, especially in case of light loaded situations or underwater operations. This leads to a requirement of considering the flexibility of crane cable while designing the crane controller.
In [8], the overhead crane with a flexible cable is modelled as a hybrid partial differential equationordinary differential equation system. And a feedback stabilization controller is proposed to asymptotically stabilize the system. To obtain the exponential stabilization for the flexible cable crane system, cascade approach [9] and back-stepping approach [10] are applied. Some researchers have also designed controller for flexible cable crane based on Lyapunov theory such as in [11][12][13]. Boundary control for stabilizing the inplane motion gantry crane system is introduced in [14,15], where the problem mentioned in [15] consisting of two payloads that is not very common in practice. An interesting control solution for the overhead crane can be found in [16] where the authors consider delayed boundary condition, the closed loop system is proven to be well-posed and asymptotic stable. The dynamics of flexible cable are also described by the wave equation and a finite-time stabilization controller is designed for the crane in [17]. Moreover, control of the flexible cable crane with variable cable length is also considered in [18][19][20]. In these works, the authors concentrate to the ultimate goal which is achieving the payload to desired position and minimizing swinging angle in steady state. For certain applications when working space is limited, it is necessary to have a hard constraint on the payload motion especially in transient period.
In some certain applications, such as safety-critical systems, or mechanical stoppages, the violation of space constraints will cause serious hazards. Thus, dealing with payload vibration constrains might be necessary. The problem of above control schemes for the flexible cable crane have not considered the constraints. In order to overcome this obstacle, the concept of barrier Lyapunov function has been applied [21][22][23]. However, these systems use an additional boundary control force at payload that might be problematic in practice. The barrier Lyapunov function is also used in [24] to control the flexible cable crane with constraints in the face of non-varying rope length. However, none of abovementioned works consider both variable cable length and payload constraints in controller design of flexible cable cranes.
The purpose of this research is to design a controller acting on trolley end for flexible cable with varying rope length crane that can precisely position the crane payload while minimize and retain the payload swinging motion in a predefined range for safety operation, this is also the main contribution of the paper. In order to obtain this purpose, the crane model with flexible and variable length cable is carefully derived. Then a control scheme is formulated from the proposed barrier Lyapunov function that stabilizes the system and takes the desired constraints in consideration. Experimental works are carried out to validate the effectiveness of the proposed controller.

Problem formulation
Before proceeding to derive mathematical model of the gantry crane, some important assumptions are specified as follows [20]: Assumption: (1) The gantry crane operates and deforms in one plane only. (2) Hooke's law is applied to the gantry cable elongation deformation. (3) The system friction is totally ignored. (4) Hook effect between the gantry cable and payload is not considered. (5) The payload is modelled as a point-mass i.e. payload geometry is not taken into account. (6) Deflection angle from vertical Z axis is very small.
Remark: Assumption 1 implies that single-beam gantry crane is considered. Assumption 2 indicates the gantry cable material is homogeneous, isotropic, and linearly elastic. Assumptions 3 and 4 emphasize the scope of the paper, system friction and hook effect will be our future concerns. Assumptions 4 and 6 generally hold in low capacity gantry cranes.
In order to formulate the dynamical model of a crane, we set the crane in a Cartesian coordinate system as shown in Figure 1. The crane includes a cart with weight m T run along Ox axis. At time t, the cart is at position x(t). The rope is with linear density ρ and a load with weight m P is mounted at the rope's end. At a time t, the rope length is l(t). There are two input forces, F x , to move the cart, and F l to lift the load. In practice, the forces are generated by torque controlled electric motors through mechanical gearing systems. In the paper scope, it is assumed that the electro-mechanical system is ideal. Hence, it is straightforward to consider acting forces F x and F l as control inputs. A point P in the rope at the time t can be expressed by it position along Oz axis z(t), and the difference from the cart along Ox axis, w(z(t), t). The point P can be described as r p and is calculated as follows: where i and k are unit vectors of Ox and Oz axes respectively. Then, velocity vector v p at P can be calculated as (2) where we have used the following notations Total kinetic energy KE of the system includes kinetic energy of the cart, rope and load, and can be calculated as follows: Total potential energy PE of the system includes potential energy of the cart, the rope, the load, and can be calculated as follows: with g is gravity acceleration. According to Hamilton's principle with W is the work done by external forces and δW = F x δ x + F l δ l . Substitute (3) and (4) into (5), and set where T = m p g + ρg(l − z) is the cable tension. Equation (5) can be rewritten in a compact form as The variation of L c can be calculated as whereL c = L c | z=l(t) . In addition, it is straight forward to yield and Moreover using integration by parts, it can be shown that From (11) and (12), we can obtain For the sake of simplicity in the presentation, let us calculate each single term in (13) Similarly, we also have (16) Substitute (14)-(16) into 17 we obtain We also have then, by calculating as same as with L c , we can obtain t2 t1 Because of w 2 z << 1, δz = δl, substituting (17) and (19) Since t 1 and t 2 are arbitrarily, (20) implies and and and ∂L c ∂w z z=0 δw(0, t) = 0 and Equation (21) represents the motion of the cable, a simple operation give a more detailed form of the equation of motions as ρ(x tt + w tt + z tt w z + z t (2w zt + w zz )) = (Tw z ) z (26) It is noted that we have used the following notation and w zt = ∂ 2 ∂z∂r .

Similar approach yields from (22)
ρ(x tt + w tt + z tt w z + z t (2w zt + z t w zz ))dz and from (23), performing integration by parts shows that Equations (27) and (28) characterize the relation between control forces and the transverse, hoisting motions, respectively. The relation will be used to further in control design step. Equation (24) simply becomes Carefully applying time and spatial derivatives, (25) simply turns into Equations (26)-(30) describe the dynamical model of a crane system with varying cable length. Based on the dynamics of the cable and boundary conditions, a variant of traditional Lyapunov function will be employed to develop the position and vibration controller for the system.

Position and vibration control design
Since the input forces are applied at the trolley end of the system, the control design process has to guarantee the location of the inputs, minimize and retain the payload swinging motion in a predefined range. In order to achieve the control objective, the direct Lyapunov method is employed. Considering the following Lyapunov candidate function given as where k b is a positive constant denoting a distance defined range that constraining the payload motion and k c is a positive constant. We assume that at the initial condition w(z, 0) < k b , which generally holds in practice. Compare to the conventional Lyapunov approach which includes system energy and tracking errors, the natural logarithm term log(•) is embedded to tackle the the requirement of maintaining the payload in a certain distance from the equilibrium position. The first derivative of V 1 (t) with time t is calculated as follows: × (x tt + w tt + z tt w z + z t (2w zt z t w zz )) + 2z t z tt + 2Tw z w zz z t + 2Tw zt w z dz Fundamental operations show thaṫ ρx t x tt + w tt + z tt w z + z t (2w zt + z t w zz ) dz + k 1 m T x t x tt + k 1 m P x t x tt +w tt + l ttwz + l t (2w zt + l t w tt ) ρw z z t x tt + w tt + z tt w z + z t (2w zt + z t w zz ) + z t z tt + Tw z w zz z t + Tw zt w z dz + m P l t w z x tt +w tt ρw t x tt + w tt + z tt w z + z t (2w zt + z t w wt ) dz + m Pwt x tt +w tt + l ttwz + l t (2w zt + l twzz ) − t x tt + w tt + z tt w z + z t (2w zt + z t w zz ) dz Using dynamic model (26)-(30) and integration by parts show thaṫ Tw zt w z dz Sincew t + l twz ≤ |w t + l twz |, it can be deduced thaṫ Tw zt w z dz There exists a positive constant k 0 such that |w t | ≤ k 0 x 2 t , this implies that |w t + l twz | ≤ |w t | + |l twz | + sign(l t )l t |w z | Since |w| < k b , [25], we have Inequality (37) suggests that (34) can be rewritten aṡ Tw zt w z dz + ρw t (x t +w t + l tw z)l t − m p gw twz Applying integration by parts on tension related terms leads to w zt Tw z dz (39) and Substituting (39), (40) into (38), and rearranging (38) lead tȯ At this step, recall the relation between input forces and system dynamics given in (27) and (28), the control can be selected as and The control input renderV(t) aṡ SinceV(t) ≤ 0 and V(t) is a function ofw, we conclude that V(w) is bounded. Due to the assumption w(z, 0) < k b it implies that |w| < k b ∀t. It is noted that the designed controls require cable ends information that are available for feedback. The controls are totally applicable when embedding in actuators such as electric motors in torque control mode.

Experimental results
The ability of maintaining payload in a certain gap is illustrated in this section via a set of experiments.
The gantry crane scaled model is depicted in Figure 2.
The crane is designed to operate in three-dimensional space and its motions are actuated by three servo motors working in torque control mode combined with gearbox/rack and pinion transmissions Figure 2(a). The proposed controls are calculated and transformed into required torque at the motor ends. Torque reference values are set through analog inputs of the servo drives in form of voltage. Payload swinging motion is detected by a special mechanism and converted into voltage signal that is proportional to payload fluctuation Figure 2(b).
Firstly, we only apply position control of the trolley to the system, payload fluctuation control is not activated. The PI position controller in tuned in such a way that desired trolley position is tracked when payload dynamic is removed. In this circumstance, system responses with different rope lengths and payload     mass (l = 0.5 m, m p = 3 kg and l = 0.7 m, m p = 5 kg) are given in Figures 3 and 4, respectively. It can be seen that, without payload vibration suppression, payload fluctuation angle can reach approximately 25 degree when l = 0.5 cm, m p = 3 kg. Lower vibration amplitude is witnessed in the case of m p = 5 kg due to higher payload inertia. The vibration affects trolley responses as can be observed in Figure 3(a) and 4(a).
In the second experimental scenario, the integrated trolley position and payload vibration suppression proposed in the paper is activated. The control parameters are k l = 3, k 4 =, k 2 = 5, k 3 = 10, k x = 8, k b = 0.15, this implies that the payload is to be maintained in a distance of 0.15 m away from the trolley vertical axis, and k c = 7. At first, the payload is positioned at (0 m, 0.5 m) and then proceeds to (0.5, 0.5 m) and to (0.5, 0.7 m). Figure 5 and 6 present the system performances with l = 0.5 m, m p = 3 kg and l = 0.7 m, m p = 5 kg, respectively.
It can be observed from Figure 5 and 6 that under control action, payload vibration is considerably reduced. Payload maximum swinging angles for l = 0.5 m, m p = 3 kg, and l = 0.7 m, m p = 5 kg are at about 13 degree and 12 degree, respectively. Assuming pendulum-like motion of the crane system and applying simple trigonometric operations, it is can be shown that the payload motion in well maintained in a range defined by k b which is corresponding to payload swinging angles of 17 degree and 12 degree, respectively. Control inputs are illustrated in Figures 5(c-d) and

Conclusions
Due to the tremendous applications in different fields, gantry crane dynamics and control draw researchers attention. In this paper, the problem of position and vibration suppression in the gantry crane system with variable length flexible cable and payload motion constraints are considered. The system was represented by partial differential equation model from Hamilton's extended principle. Based on the novel barrier Lyapunov function, the control scheme that stabilize the crane system has been derived. Moreover, the payload has been successfully moved to the desired point, vibrations of the variable flexible cable have been greatly suppressed, and the boundary payload motion constraint has been satisfied. Experimental results have been provided to verify the performance of the proposed control. However, the well-posed problem of the designed controller is not proven. The crane system consider is placed on a solid foundation, our future works will look at dynamics and control aspects of the system mounted on moving foundation in ship-to-ship and ship-to-shore operations.

Disclosure statement
No potential conflict of interest was reported by the author(s).