Fast trajectory planning by design of initial trajectory in overhead traveling crane with considering obstacle avoidance and load vibration suppression

This paper is concerned with an advanced transfer trajectory planning method of 2-Dimensional transfer machine with vibrational element such as an overhead traveling crane. In the 2-D transfer machine, it is required to reach the target position in a short time, avoid the obstacles, and suppress the vibration. In recent years, the authors have proposed the trajectory planning method using the optimization problem with considering input and state constraints in the transfer system, obstacle avoidance and vibration suppression. However in the previous approaches, it takes a long time to derive the reference trajectory because of many variables in the optimization. Therefore in this study, we propose the fast solution for optimizing the transfer trajectory by giving a feasible initial trajectory. And, it is discussed how to give the cost function in the trajectory optimization with reducing the fluctuating cart motion. Moreover, we discuss the practical case that the proposed approach is applied to the large transfer space. The effectiveness of the proposed transfer trajectory planning method is verified by simulations of the overhead traveling crane.


Introduction
In many industries, overhead traveling crane systems frequently have been used for efficiently transporting heavy loads. In the overhead traveling crane, it is required to reach the target position in a short time, avoid the obstacles, and suppress sway of the load [1] [4].
In order to fulfill the above requirements, the transfer control systems have been proposed in the previous studies. The vibration suppression control to load of the overhead crane using optimal control theory was proposed in [5]. In the crane system with varying the rope length, the load vibration with varying the natural frequency is suppressed by the gain scheduling control [6], [7]. Furthermore, natural frequency elements of vibration are eliminated by shaping the acceleration of the cart [8]. These control systems have specialized in the load sway suppression. In the 2-D transfer systems such as the overhead traveling crane, a path planning considering obstacle avoidance is required. In the studies of path planning, the path of transferred object has been derived by the potential method [9], the probabilistic road map method [10], and predicting the action of the moving obstacle [11]. In the study [12], the path in consideration of obstacles avoidance is derived by the potential method. Then, the path is reshaped to the trajectory with vibration suppression. In this approach, the reshaped trajectory is different from the path for avoiding the obstacle derived by the potential method. Therefore, the transfer object is in danger of collision with obstacles. In recent years, the trajectory planning methods in consideration of input and state constraints in the transfer system, suppressing the load sway with varying the rope length, and the obstacles avoidance in the transfer space have been proposed by authors [13], [14]. In these approaches, the trajectory planning problem is formulated as the quadratic problem with quadratic constraints in the optimization problem. Thus, the trajectory is optimized by a sequential quadratic programming approach.
However in the previous approaches, it takes a long time to derive the reference trajectory because of many variables in the optimization. Therefore in this study, we propose the fast solution for optimizing the transfer trajectory by giving a feasible initial trajectory to the trajectory optimization. And, it is discussed how to give the cost function in the trajectory optimization with reducing the fluctuating cart motion. Moreover, we discuss the practical case that the proposed approach is applied to the large transfer space. The effectiveness of the proposed transfer trajectory planning method is verified by simulations of the overhead traveling crane.

Model of overhead traveling crane system
In this study, the proposed control system is applied to the overhead traveling crane system as one of the 2-D transfer system with vibrational element. The overhead traveling crane used in this study is shown in Figure 1(a). And, schematic illustration of the load sway is shown in Figure 1(b). Specifications of the overhead traveling crane in Figure 1(a) are shown in Table 1. The model of load sway dynamics is linearized as θ ≪ 1, ϕ ≪ 1, and it is represented as where m is mass of the load, g is a gravity acceleration, l is the rope length, and θ and ϕ are the load sway angles, a x and a y are acceleration of the cart transfer of each axis. The cart transfer system consists of the position feedback control system. The feedback controllers on x-and y-axes have the proportional gain. The cart transfer systems on x-and y-axes are represented as where x cm , u m and r m are the cart position, the control input, and the reference trajectory of each axis, respectively. T m and K m are the time constant and the gain in the motor of each axis, respectively. K pm is the proportional gain in the position feedback controller of each axis. In this study, the time constant and the gain in each motor can be obtained as T x =0.008[s], T y =0.036 [s] and K x =K y =0.037[m/s/V] by the parameter identification, respectively. The proportional gains are given as K px =K py =100.

Transfer trajectory planning method
The transfer trajectory planning method using optimization method has been proposed in the author's studies [13], [14]. The detailed derivation of the method is referred to in [13], [14].

Representation of controlled object
In this study, the cart motion with input and state constraints as shown in Figure 2 can be controlled according to the trajectory planning. Therefore, the cart motion as the controlled object is represented as where x dm is the state variable vector, z m is the controlled variable vector. The vector z m consists of the cart position, velocity and acceleration, z um is the control input, y m is the cart position as the observed variable. In order to formulate the trajectory planning to optimization problem, the cart motion is represented with the discrete-time system as t = k∆T . Here, t is the time, ∆T is the sampling time and k is the sample number. As shown in Figure 2, the control input z um and the controlled variables z m are subject to the input and state constraints.

Cost function
The cost function is defined for the transfer trajectory optimization. The transfer trajectory derived by the previous approaches [13], [14] has the fluctuating motion. Because, the energy saving is not considered in the cost function. In order to reduce the fluctuating motion, the cost function with energy saving is proposed as min rx,ry where r 0x and r 0y are the target positions on x-and y -axes, respectively. v is the frequency. v 1 and v 2 are the minimum and maximum frequencies. And, w 1 ≥ 0, w 2 ≥ 0 and w 3 ≥ 0 are the weight coefficients.
Here, the vectors of the input and controlled variables are respectively defined as We also define vectors R m , R 0m and Y m (m = x, y) whose elements are r m (k), r 0m (k) and y m (k), (k = 0, 1, · · · , n − 1), respectively. Therefore, the control input and the controlled and observation variables can be compactly expressed as where M zm and M ym are given by Excluding the weight coefficient w 1 , the first term J 1 and second term J 2 in the cost function (4) becomes where since R T 0x R 0x and R T 0y R 0y are constants, they have been omitted from the cost function. We also express the third term J 3 and fourth term J 4 in the cost function (4) without the weight coefficient w 2 to the matrix representation as Excluding the weight coefficient w 3 , the fifth term J 5 in the cost function (4) can be discrete Fourier transformed as The sixth term J 6 in the cost function (4) can be represented similarly to J 5 . Therefore, the matrix representation of fifth term J 5 and sixth term J 6 in the cost function (4) can be expressed as where M e is given by Collecting (9), (10) and (12), the cost function (4) becomes min Rx,Ry

Input and state constraints on the transfer system
The input and state constraints on the transfer system can be expressed into the matrix representation as where Z xmc and Z umc are the input and state constraints, respectively given by where z mc and z umc are the boundary parameters of the constraints. In this study, the trajectory is planned over a finite time interval. To transfer the object to the goal position (r 0x , r 0y ) and ensure it is stationary at the final time, we impose the following equality constraints.
where M f m and Z f m are given by

Positional constraints on obstacle avoidance
In the 2-D transfer, it is necessary to avoid obstacles in the transfer space. The obstacle is defined as the ellipsoidal no-penetration area as shown in Figure 3. The interior of the ellipse is the no-penetration area that must be avoided in trajectory planning. Therefore, we impose the following inequality constraint on the position of the transferred object.
where (x 0 , y 0 ) is the center of the ellipse. a r and b r are respectively the lengths of axes of the ellipsoidal no-penetration area, and are given as where a and b are the corresponding axis lengths of the ellipse enclosing the obstacle. And, r is represented as where r e is the radius of the circle enclosing the transferred object. d max is the maximum moving distance of the transferred object which is caused by leaned the rope due to inertia. The maximum rope angle ϕ max leaned by inertia can be derived from (1) as where |a x | max is the maximum acceleration of cart transfer. Therefore, the maximum moving distance d max can be represented as where l max is the maximum rope length. The matrix representation of the inequality (19) can be represented as where E = [e T 1 e 1 · · · e T k e k · · · e T n e n ] T , η = [e 1 e T 1 · · · e k e T k · · · e n e T n ] T , e 1 = [1, 0, 0, 0, · · · , 0] ∈ R 1×n , e i = [0, · · · , 0, 1, 0, · · · , 0] ∈ R 1×n , X 0 = [x 0 , x 0 , · · · , x 0 ] T ∈ R 1×n and Y 0 = [y 0 , y 0 , · · · , y 0 ] T ∈ R 1×n .
By (14), (15) and (24), The trajectory planning of the overhead traveling crane can be resulted to quadratic programming problem with quadratic constraints. This quadratic programming problem is solved by a sequential quadratic programming method.

Fast solution by using feasible initial trajectory
The transfer trajectory can be obtained by the optimization problem introduced in the previous section. However, it takes a long time to derive the optimized trajectory because of many variables in the optimization. In the previous study [14], the initial trajectory R ini ∈ R 1×n in the optimization has been given as where r 0m is the target position. Thus, the initial trajectory is no-feasible solution to be subject to (15), (17) and (24). In this case, it takes a long time to obtain the optimized trajectory. By giving the feasible initial trajectory to the optimization problem, the optimizing time can be shortened. The feasible initial trajectory is designed while satisfying the constraint conditions defined by (15), (17) and (24). And, the feasible initial trajectory is designed based on the designer's subjective view.
As an example, the feasible initial trajectory is shown in Figure 4 and Figure 5.  Table 1.  The feasible initial trajectory shown in Figure 4 and Figure 5 is given to the trajectory optimization.

Simulation Verification
In this study, the fast solution of trajectory optimization by using the feasible initial trajectory and the cost function with the energy saving for reducing the fluctuating motion of the cart are proposed. The proposed approaches are verified by simulation of the overhead traveling crane system which has the specifications as shown in Table 1. The transfer space with obstacle is shown in Figure 5, and the feasible initial trajectory shown in Figure 5        The weight coefficients in the proposed cost function are given as w 1 =10, w 2 =0.1 and w 3 =1. The trajectory optimization with the proposed cost function is performed by using the proposed fast solution. As the comparisons, the weight coefficients, which are same as the cost function in [14], are given as w 1 =10, w 2 =0 and w 3 =1. And, the trajectory is optimized using the conventional initial trajectory which has the same elements as the target position r 0 . Moreover, the trajectory is optimized using the cost function with the weight coefficients, w 1 =10, w 2 =0.1, w 3 =0, which the vibration suppression is not considered in the trajectory optimization.
The simulation results are shown in Figure 6 to Figure 10. Figure 6(a) and (b) show the time series of the cart transfer on x-and y-axes in simulation, respectively. In Figure 6, the upper left graphs show the acceleration of cart transfer, and the upper right graphs show the velocity. The lower left and lower right graphs show the cart position and control input, respectively. The solid line are the simulation results of the proposed approach, the broken lines are the conventional approach and the chain lines are the trajectory without vibration suppression. Figure 7 shows the cart trajectory in the transfer space. Figure 8(a) and (b) show the power spectrum of the acceleration of the cart transfer on x-and y-axes, respectively. The line types in Figure 8 are same as those in Figure 6. Figure 9 show the swaying angle of the rope whose length is l=0. 3[m]. Figure 10 show the swaying angle of the rope whose length is l=0. 6[m]. In Figure 9 and Figure  10, (a) and (b) show the simulation results on x-and y-axes, respectively. The line types in Figure 9 and Figure 10 are same as those in Figure 6. As seen from these simulation results, the load swaying can be suppression by the proposed approach, and the load is transferred in a short time while avoiding the obstacle and satisfying the constraints in the crane system. Moreover, fluctuating motion of the cart can be reduced by applying the proposed cost function. Table 2 shows the calculating time of the trajectory optimization in the proposed fast solution and the conventional solution. By using the proposed fast solution, the trajectory can be optimized in a short time.

Application to the large transfer space
The proposed approach is applied to the large transfer space. Here, the simulation conditions of the transfer trajectory are shown as follows.
Target position is located at (x, y) = (2.0, 0.   The simulation results are shown in Figure 11 to Figure 15. The layouts of the figures and the line types are same as those of Figure 6 to Figure 10. As the results, the cart is transferred in a short time by the proposed approach, even if the transfer space is expanded. And, the load sway can be suppressed. The calculation time of the trajectory optimization is 2224[s]. The calculation time can be increased with expanding the transfer space.

Conclusions
We proposed transfer trajectory planning method of 2-D transfer machine with vibrational elements such as an overhead traveling crane. In this study, we especially proposed the fast solution for optimizing the transfer trajectory by giving a feasible initial trajectory. And, the cost function with energy saving was proposed for reducing the fluctuating motion of the cart. The effectiveness of the proposed transfer trajectory planning method is verified by simulations of the overhead traveling crane. The computation time of the proposed trajectory optimization is faster than that of the conventional approach. In the future works, the experimental verification using the actual crane system should be confirmed for usefulness of the proposed method, and it will be required to develop the fast solution of the trajectory optimization for large transfer space.