An efficient online trajectory generating method for underactuated ...

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2014; 24:1653–1663 Published online 17 January 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.2953

An efficient online trajectory generating method for underactuated crane systems Ning Sun and Yongchun Fang* ,† Institute of Robotics and Automatic Information System, Nankai University, Tianjin 300071, China

SUMMARY In this paper, a trajectory planning approach is proposed for underactuated overhead cranes. Different from existing trajectory planning methods, the presented approach generates trajectory commands online without the necessity of iterative optimization, which is convenient for practical implementation. We demonstrate the performance of the proposed method, including swing elimination and precise trolley positioning, with rigorous Lyapunov-based mathematical analysis. Both numerical simulation and experimental results suggest that the presented method is feasible and efficient for practical applications. Copyright © 2013 John Wiley & Sons, Ltd. Received 25 June 2012; Revised 5 December 2012; Accepted 6 December 2012 KEY WORDS:

underactuated robots; overhead cranes; motion planning; coupling analysis

1. INTRODUCTION In practice, robots have been used in far-ranging fields, such as industrial manufacturing, packaging, machine assembly, home service, benchmark research [1], and caregiver assistance [2]. Underactuated robots refer to those with more degrees-of-freedom than control inputs [1, 3–6]. Traditionally, there are two ways to approach robot control: stabilization/regulation and trajectory tracking. For the latter one, we can conveniently incorporate requirements such as sufficiently high speed and acceleration when planning suitable trajectories and then design appropriate tracking controllers to make the robot track these trajectories. The importance of proper trajectory planning has been illustrated in the literature, such as [2, 7]. Overhead cranes can be seen as a special type of underactuated robots, and the control objective is to transport the unactuated payload to the desired location with no swing at the end. With regard to this topic, abundant efforts have been devoted to regulation schemes, and much excellent work has been reported [8–15]. Trajectory-based methods typically focus on combining efficient and smooth trajectories with anti-sway tracking controllers without regard for payload motion, and the control performance relies greatly on the tracking control laws [16–19]. Hence, these trajectorybased methods are not trajectory planning in the sense of simultaneous swing elimination and trolley positioning. Other methods including input shaping [20], time-optimal control [21], and phase plane analysis-based methods [22] have been successfully employed to plan anti-sway trolley trajectories based on analyzing the natural frequency of the crane system. Sun et al. propose a kinematic coupling-based motion planning method [23] that employs offline iterative learning to revise the trajectory parameters for precise trolley positioning.

*Correspondence to: Yongchun Fang, Institute of Robotics and Automatic Information System, Nankai University, Tianjin 300071, China. † E-mail: [email protected] Copyright © 2013 John Wiley & Sons, Ltd.

1654

N. SUN AND Y. FANG

The main idea in the work that follows is to incorporate swing-damping terms in a smooth trolley trajectory generated online in real time‡ that achieves the anti-sway objective without additional tracking controllers and does not affect trolley positioning performance. The proposed method consists of two parts: a swing-elimination component and a positioning reference component, where the first part can damp out the payload swing without affecting the positioning performance of the latter one. The constructed trajectory needs no iterative optimization, and it presents a straightforward structure. The central contribution is that the proposed method can be directly implemented online without the necessity of offline or advance planning as in [21–23]. The performance of the planned trajectory is guaranteed with some Lyapunov-like mathematical analysis. Analysis, simulation, and experiments indicate improved performance with respect to the kinematic coupling-based method in [23]. The paper is organized as follows. In Section 2, the crane trajectory planning problem is described. Next, we provide the trajectory generating procedure in Section 3. Simulation and experiments are presented in Section 4. Section 5 provides some concluding remarks. 2. PROBLEM STATEMENT Let us consider a two-dimensional overhead crane system described by [22, 23]§ .M C m/xR C ml R cos   ml P 2 sin  D Fx ,

(1)

ml 2 R C ml cos  xR C mgl sin  D 0,

(2)

wherein x.t / denotes the trolley displacement, .t / is the payload swing angle w.r.t. the vertical, M and m represent the trolley mass and the payload mass, respectively, l denotes the rope length, and Fx .t / denotes the resultant force imposed on the trolley, which is expressed as Fx .t / D Fa .t /  fr .t /,

(3)

with Fa .t / denoting the actuating force and fr .t / representing the girder friction. It is clear from (1) and (2) that the crane model includes two components of the actuated dynamics (1) and the unactuated kinematics (2), where the latter can be reduced, by dividing (2) with ml, to be l R C cos  xR C g sin  D 0.

(4)

As stated previously, the objective is to position the payload to the desired location with no swing at the end of the transportation. Because of the underactuated nature of the system, these objectives are usually accomplished in two steps: (i) moving the trolley, together with the payload, to the place right above the desired location and (ii) controlling the trolley motion to damp out the payload swing indirectly via their coupling relationship of (4). Motivated by the previous facts, we will carefully analyze both requirements of fast swing damping and precise trolley positioning to generate a smooth trolley trajectory. Before trajectory planning, we make the following assumption by taking the practical conditions of overhead cranes into consideration. Assumption 1 The payload is always beneath the trolley in the sense that =2 < .t / < =2, 8 t > 0 [9, 10, 20–23]. ‡

In this paper, we refer to the term ‘offline’ as calculating the trajectory commands in advance of implementation. By contrast, the term ‘online’ implies that the trajectory commands are generated and applied real time to the system. For instance, assume that the control period is 5 ms, and then the trajectory command for the next control action is calculated within the current 5 ms. § This paper considers two-dimensional overhead cranes for convenience of presenting the online trajectory generating method; hence, the payload swing out of the plane is not considered. Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

ONLINE TRAJECTORY GENERATING FOR CRANE SYSTEMS

1655

3. TRAJECTORY GENERATION In this section, the trajectory generating method will be detailed with performance analysis. 3.1. General framework for trajectory generation It is known that a trolley trajectory neglecting the swing effects, such as the S-shaped trajectory in [16–18], can only achieve precise trolley positioning without guaranteed swing damping [22, 23]. On the basis of the analysis implemented in Section 2, we need to appropriately plan a suitable trolley trajectory to minimize the payload swing while driving the trolley to the desired location. To this end, the to-be-planned trajectory consists of two parts: (i) a positioning reference trajectory component xR r .t / to guide the trolley to the desired location and (ii) a swing-eliminating component P to effectively eliminate the payload swing without affecting the trolley position. Thus, the ., / desired acceleration trajectory xR c .t / is expressed as P xR c .t / D xR r .t / C ., /,

(5)

Subsequently, we will construct ., P / from a swing damping-out point of view and then determine a suitable positioning reference trajectory xR r .t / and, finally, combine these two components to obtain the ultimate trajectory. The structure of the online trajectory generator is illustrated in Figure 1 and interpreted as follows. According to the previous analysis, the trolley positioning reference acceleration trajectory xR r .t / P is also derived beforehand. In addition, by is determined in advance, and the structure of ., / substituting (5) into (4), one can obtain P C g sin  D  cos  xR r . l R C cos   ., /

(6)

P we can calculate .t / and .t P / by solving (6) online Hence, given xR r .t / and the structure of ., /, and obtain the desired acceleration trajectory xR c .t / from (5). 3.2. Trajectory generation and performance analysis To illustrate the approach, consider only the swing-elimination component and develop an expression for ., P /. That is, temporarily set xR r .t / D 0. In this situation, consider the following nonnegative function: 1 V .t / D l P 2 C g.1  cos / > 0. 2 Taking the derivative of (7) w.r.t. time and substituting (4) into the resulting expression yields VP .t / D P .l R C g sin / D P cos  x. R

(7)

(8)

P will not influence the positioning capacity To render that VP .t / 6 0 and also to guarantee that ., / P of xR r .t /, ., / is determined as follows: P , xR e .t / D  P , ., /

(9)

Figure 1. Illustration for the trajectory generator. Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

1656

N. SUN AND Y. FANG

where  2 RC is a positive gain. Hence, we have VP .t / D  P 2 cos  6 0 by substituting (9) P /, .t R / ! 0 by LaSalle’s invariance into (8). It can be further proved that xR e .t / ensures .t /, .t principle [24]. Yet, the swing-eliminating component xR e .t / cannot guarantee precise trolley positioning. Thus, we need to choose further a suitable positioning reference trajectory and combine it with xR e .t / to generate the ultimate trajectory. Out of concern for smoothness and precision, the positioning reference trajectory xr .t / should be selected such that the following conditions are satisfied [16, 23].  The positioning reference trajectory xr .t / should converge to the desired location pr , that is,

xr .t / ! pr , within finite time without overshoot. .3/  xP r .t /, xR r .t /, and xr .t / need to satisfy 0 6 xP r .t / 6 kv , jxR r .t /j 6 ka , jxr.3/ .t /j 6 kj , lim xP r .t / D 0, lim xR r .t / D 0,

t !1

t !1

(10) (11)

where kv , ka , kj 2 RC denote the corresponding upper bounds.  The initial conditions are chosen as xr .0/ D 0, xP r .0/ D 0.

(12)

Many trajectories, such as the one constructed in [16], achieve conditions (10)–(12). Combining xR r .t / with xR e .t / as in (5), we obtain the ultimate trolley acceleration trajectory as xR c .t / D xR r .t / C xR e .t / D xR r .t / C  P .t /,

(13)

wherein  in (9) is determined such that >

1 H)  cos .max /  1 > 0, cos .max /

(14)

with max being the maximum payload swing amplitude. Because cos  > cos .max / > 0, it is indicated from (14) that  cos   1 >  cos .max /  1 > 0.

(15)

Further, from (9) and (12), we can obtain the ultimate trolley velocity and displacement trajectories, respectively, as follows: ¶ xP c .t / D xP r .t / C xP e .t / D xP r .t / C .t /, Z xc .t / D xr .t / C xe .t / D xr .t / C 

(16)

t

d .

(17)

0

The main results of the paper are stated in the following theorem. Remark 1 In practice, the reference acceleration trajectories usually satisfy jxR r .t /j 6 ka  g to guarantee smooth transportation and safety [9], where g denotes gravitational acceleration. Under such circumstances, the swing angle is usually kept within max 6 10ı ; hence, we can conveniently choose  > 1.0154 D

¶

1 1 > ı cos.10 / cos.max /

(18)

P In this paper, we consider zero initial conditions as in [22, 23], that is, x.0/ D 0, x.0/ P D 0, .0/ D 0, .0/ D 0.

Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

ONLINE TRAJECTORY GENERATING FOR CRANE SYSTEMS

1657

to guarantee (14). Also, the following approximations hold [9, 21–23]: sin   , cos   1.

(19)

Theorem 1 The trajectory (xc .t /, xP c .t /, xR c .t /) is smooth and uniformly continuous with the following properties: (1) it guarantees that the payload swing angle, the angular velocity, and the angular acceleration converge to zero asymptotically in the sense that P /, .t R / D Œ0, 0, 0I lim Œ.t /, .t

t !1

(20)

(2) xc .t / ensures that the trolley arrives at the desired location while the corresponding velocity and acceleration converge to zero in the sense that lim Œxc .t /, xP c .t /, xR c .t / D Œpr , 0, 0.

t !1

(21)

Proof We start the analysis from the first part of the theorem. Considering again V .t / in (7) and differentiating it w.r.t. time, one has obtained the results of (8). Substituting the planned acceleration trajectory (13) into (8) and making some arrangements produces the following result: VP .t / D P xR r cos    P 2 cos .

(22)

Applying Young’s inequality, one can rewrite (22) into 1 1 VP .t / 6 xR r2 cos2  C P 2   P 2 cos  D xR r2 cos2   . cos   1/P 2 4 4 1 6 xR r2 cos2   Œ cos .max /  1P 2 . 4 Integrating VP .t / w.r.t. time, we obtain Z t Z 1 t 2 P 2 d C V .0/. xR r cos2 d  Œ cos .max /  1 V .t / 6 4 0 0

(23)

(24)

Together with (10) and (12), the first term on the right-hand side of the inequality (24) can be further calculated as Z t Z t Z t Z 1 t 2 xR r cos2 d 6 xR r2 d D ŒxR r xP r t0  xP r xr.3/ d 6 xR r .t /xP r .t / C kj xP r d 4 0 0 0 0 (25) D xR r .t /xP r .t / C kj xr .t / 2 L1 . In addition, it follows from (15) that Z

t

 Πcos .max /  1

P 2 d 6 0.

(26)

0

Hence, we obtain from (24), (25), (26), and further from (7) and (22) that P / 2 L1 H) VP .t / 2 L1 . V .t / 2 L1 H) .t In view of (24), (25), and (27), one derives Z t Z 1 t 2 P / 2 L2 . Œ cos .max /  1 xR cos2 d C V .0/  V .t / 2 L1 H) .t P 2 d 6 4 0 r 0 Copyright © 2013 John Wiley & Sons, Ltd.

(27)

(28)

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

1658

N. SUN AND Y. FANG

Moreover, substituting (13) into (4) and rearranging the resulting formula yields  1 1 R .t / D  g sin   cos  xR r  cos  P , l l l which indicates from the facts of P .t / 2 L1 and xR r .t / 2 L1 (see (10)) that R / 2 L1 . .t

(29)

(30)

Therefore, it is known from (27), (28), and (30) that P .t / 2 L2 \ L1 and R .t / 2 L1 . Barbalat’s lemma [25] can then be employed to conclude the following result: P / D 0. lim .t

(31)

t !1

Next, we will further analyze the convergence of .t / and R .t /. Specifically, (29) can be decomposed into the following two parts:   1 1  R .t / D  g sin   (32) cos  xR r C cos  P . l „ l ƒ‚ … „ l ƒ‚ … '1 .t /

'2 .t /

Together with (11), (27), and (31), it is clear that 1 'P1 .t / D  g cos  P 2 L1 , lim '2 .t / D 0. t !1 l

(33)

In view of (31) and invoking extended Barbalat’s lemma|| , the following results are obtained: lim '1 .t / D  lim

t !1

t !1

1 R / D 0. g sin  D 0, lim .t t !1 l

(34)

Because of Assumption 1, we have sin .t / D 0 ) .t / D 0. It is then concluded from (34) that lim .t / D 0.

t !1

(35)

Thus, (31), (34), and (35) confirm the first part of the theorem. Subsequently, we will prove the smoothness and uniform continuity of xc .t /, xR c .t /, and xR c .t /. Taking the time derivative of (13) produces xc.3/ .t / D xr.3/ .t / C  R .t /,

(36)

which, together with (13) and (16), indicates that xc .t /, xR c .t /, and xR c .t / are smooth and differentiable. Moreover, (10) implies xP r .t /, xR r .t /, xr.3/ .t / 2 L1 , and one can conclude from P /, .t R / 2 L1 , which further indicates from (13), (16), Assumption 1, (27), and (30) that .t /, .t and (36) that xP c .t /, xR c .t /, xc.3/ .t / 2 L1 .

(37)

Hence, xc .t /, xP c .t /, and xR c .t / are uniformly continuous. Finally, we will prove the second part of the theorem regarding trolley positioning. In view of (19), (4) can be linearized as [20–23] l R C xR C g D 0.

(38)

l R C xR c C g D 0.

(39)

Substituting (13) into (38) yields

||

Extended Barbalat’s Lemma [25, 26]: If a differentiable function f .t/ W R>0 ! R has a finite limit as t ! 1, and its time derivative can be expressed as fP .t/ D g1 .t/ C g2 .t/, where g1 .t/ is uniformly continuous (or, gP 1 .t/ 2 L1 ) and limt!1 g2 .t/ D 0, then limt!1 g1 .t/ D 0, limt!1 fP .t/ D 0.

Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

ONLINE TRAJECTORY GENERATING FOR CRANE SYSTEMS

1659

By taking the limit of (39) w.r.t. time and employing the conclusions of (34) and (35), we have R /  lim g.t / D 0. lim xR c .t / D  lim l .t

t !1

t !1

t !1

(40)

Similarly, applying (11) and (35) to (16) produces lim xP c .t / D lim ŒxP r .t / C .t / D 0.

t !1

t !1

(41)

To further analyze the convergence of xc .t /, we integrate (39) w.r.t. time, substitute (16) into the derived results, and make some mathematical arrangements to obtain Z t ± i 1° P 1h P d D  (42) lŒ .t /  P .0/ C ŒxP c .t /  xP c .0/ D  l .t / C xP r .t / C .t / g g 0 It then follows, after taking the time limit of (42) and using the conclusions of (31) and (41), that Z 1 Z t i h 1 P / C xP r .t / C .t / D 0, lim d D d D  lim l .t (43) t !1 0 g t !1 0 which indicates that the swing-eliminating component has no influence on trolley positioning. On the basis of (43), we can obtain the following result by taking the time limit of (17): Z 1 lim xc .t / D lim xr .t / C  d  D pr . (44) t !1

t !1

0

Thus, the proof for the second part of the theorem is completed.



Remark 2 The designed swing-eliminating component xR e .t / of (9) in this paper takes a more straightforward structure than the anti-swing mechanism in [23]. Moreover, there is only one to-be-tuned parameter  for the proposed approach, whereas four parameters must be adjusted for the method in [23]. The planned trajectory in [23] needs offline iterative optimization to guarantee precise trolley positioning performance. In contrast, the swing-eliminating component xR e .t / does not influence trolley positioning. In addition, when using the method in [23], once the desired location is changed, the trajectory parameters must be recalculated by iteration, whereas in the proposed method, pr may be reset to be consistent with the modified trolley location without any additional calculations. Hence, the proposed trajectory generating approach needs no iterations, and it is suitable for real-time implementation. Remark 3 The proposed method can be extended to three-dimensional (3D) overhead cranes without much difficulty. To illustrate this point, we will undertake some brief analysis. The kinematics for 3D cranes are given as [27] Cx Cy xR C lCy2 Rx  2lSy Cy Px Py C gSx Cy D 0, Sx Sy xR  Cy yR  l Ry  lSy Cy Px2  gCx Sy D 0, where x.t / and y.t / denote the trolley displacements in two orthogonal directions X and Y , respectively, x .t / and y .t / are used to describe the spatial payload swing, and Sx , sin x , Sy , sin y , Cx , cos x , Cy , cos y . For clarity, we only discuss the anti-sway performance of the extended swing-elimination components, which are designed as xR e .t / D x .Sx Sy Py  Cx Cy Px /, yRe .t / D y Py

(45)

with x , y 2 RC being damping gains, and the corresponding planned acceleration trajectories are obtained similar to (13). Consider the following positive definite function: 1 1 V3d .t / D lCy2 Px2 C l Py2 C g.1  Cx Cy /. 2 2 Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

1660

N. SUN AND Y. FANG

By taking its derivative w.r.t. time and substituting (45) into the resulting expression, we are led to VP3d .t / D x .Sx Sy Py  Cx Cy Px /2  y Cy Py2 6 0, where the fact of x .t /, y .t / 2 .=2, =2/ H) Cy > 0 has been used (Assumption 1). On the basis of this result, some invariant set-based analysis can be performed to obtain the conclusions of x .t /, y .t /, Px .t /, Py .t /, Rx .t /, Ry .t / ! 0. The positioning performance of the corresponding planned trajectories can also be proved by carrying out similar analysis to Theorem 1, which is, however, omitted here because it is beyond the scope of the paper. Remark 4 It is noted that the crane kinematics are not related to the payload mass m, and thus from the trajectory planning point of view, the performance of the planned trajectory is not influenced by different values of m. For different rope lengths, as long as they are known (note that this is a common precondition needed for all crane trajectory planning methods [20–23]), the conclusions of Theorem 1 always hold, despite the fact that the swing damping speeds may be different. This problem can be conveniently addressed by parameter tuning because there is only one to-be-tuned parameter . 4. NUMERICAL SIMULATION AND EXPERIMENTAL VERIFICATION We present both numerical simulation and experimental results to demonstrate the practical control performance of the proposed method. 4.1. Simulation results and analysis In the simulation study, we validate the proposed approach from the kinematics (motion planning) viewpoint, that is, we substitute the planned trajectory (13) into the crane kinematics (4) without considering the corresponding driving force to observe the trolley positioning and swing damping performance, as in [23]. The method in [23] is also simulated for comparison. The simulation is implemented in M ATLAB /S IMULINK, where the trolley and payload masses, the rope length, and the gravitational constant are M D 7 kg, m D 1.025 kg, l D 0.75 m, g D 9.8 m/s2 .

(46)

We utilize the S-shaped trajectory in [16] as the positioning reference trajectory, which is given by   pr 1 cosh.k1 t  "/ , (47) xr .t / D ln C 2 2k2 cosh.k1 t  "  k2 pr / where k1 D

2ka kv

D 1.2, k2 D

2ka kv2

D 0.48, " D 3.5 are trajectory parameters determined in

accordance with [23]. The desired trolley location is pr D 0.6 m. The trajectory parameter in (13) is determined, after a few trials, as  D 8. The corresponding parameters in [23] are chosen as ˛ D ˇ D 50,  D 0.015, D 3.0, pr .1/ D 0.6 by iterative optimization [23]. The simulation results are illustrated in Figures 2 and 3, where, for comparison, we have replotted the trolley displacement curve of the positioning reference trajectory (47) in Figure 3 (dashed line). It is clear from Figure 2 that the payload swing presents simple-pendulum oscillations when the trolley reaches the desired location, indicating that the positioning reference trajectory, without considering the trolley/payload coupling, cannot eliminate the payload swing. By contrast, one can find in Figure 3 that the proposed online generated trajectory (13), without iterative optimization, achieves similar control performance as the offline optimized trajectory in [23], in terms of trolley positioning and swing elimination. 4.2. Experimental results and analysis Experiments for the proposed approach and the kinetic coupling-based method in [23] have been carried out on a prototype overhead crane (Figure 4) [14, 16, 23]. The physical parameters, Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

ONLINE TRAJECTORY GENERATING FOR CRANE SYSTEMS

1661

0.8

x [m]

0.6 0.4 0.2 0 0

5

10

15

10

15

Time [sec]

θ [deg]

5 0 −5 0

5

Time [sec]

Figure 2. Simulation results: the positioning reference trajectory (47) and its corresponding payload swing. 0.8

x [m]

0.6 0.4 0.2 0 0

5

10

15

10

15

Time [sec]

θ [deg]

2 1 0 −1 −2 0

5

Time [sec]

Figure 3. Simulation results: the planned trajectories (solid line, the proposed approach; dotted-dashed line, the method in [23]; dashed line, the positioning reference trajectory (47)).

trolley

rope

payload

Figure 4. The overhead crane experiment testbed.

control objective, and trajectory parameters are the same as those in the simulation. To make the trolley follow the planned trajectories, the following PD controller with feedforward friction compensation is utilized [22, 23, 28]: Fa .t / D kp e.t /  kd e.t P / C fr0 tanh.x= / P  kr jxj P x, P Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

1662

x [m]

N. SUN AND Y. FANG

0.5 0 0

5

10

15

10

15

10

15

θ [deg]

Time [sec]

2 0 −2 0

5 Time [sec]

F [N]

20 0 −20 0

5 Time [sec]

Figure 5. Experimental results: the planned trajectories (solid line, the proposed approach; dotted-dashed line, the method in [23]). Table I. Quantified experimental results. Controllers Proposed method Method in [23]

xf .m/

max .ı /

ts .s/

Fa max .N/

0.598 0.596

1.63 2.12

7.5 6.1

11.56 15.08

R 15 0

Fa2 .t /dt .N2  s/ 118.60 279.26

where kp , kd 2 RC are positive control gains; fr0 tanh.x= / P  kr jxj P xP is the feedforward part used to compensate for the girder friction fr .t / with parameters fr0 D 4.4, D 0.01, kr D 0.5 obtained via experimental studies on the testbed [22, 23]; e.t / D x.t /  xp .t / denotes tracking errors, with xp .t / representing the planned trajectories (both the newly proposed trajectory of xc .t / and the one in [23]). The control gains for the proposed approach are chosen as kp D 250, kd D 30, and those for the one in [23] are kp D 160, kd D 50. The real-time control period is set as 5 ms. Figure 5 and Table I illustrate the experimental results and some quantified indexes. In Table I, xf denotes the final position of the trolley; max denotes the maximum swing amplitude; ts D max¹tsp , tst º is the settling time, where tsp denotes the moment when .t / enters j.t /j 6 0.5ı , 8 t > tsp , and tst is the moment when the trolley reaches and stays at xf ; Famax denotes the largest R 15 control input amplitude; 0 Fa2 .t /dt is used to represent the consumed energy [6]. We can see that the experimental results in Figure 5 are very close to the simulation results in Figure 3. In addition, it is clearly seen from Figure 5 that the newly proposed trajectory (17) suppresses the payload swing into a smaller range with less control effort, although the transferring time is a bit longer when compared with the kinematic coupling-based method in [23]. The proposed approach is more practical because it generates trajectory commands online for the trolley to track, whereas offline iterations are unavoidable when using the method in [23]. 5. CONCLUDING REMARKS We have designed a new online trajectory generating method for overhead cranes. Compared with existing trajectory planning methods, the proposed approach has a straightforward structure without iterative optimization and is convenient for online implementation. Both the swing-eliminating performance and the trolley positioning performance are proved by Lyapunov-like analysis and (extended) Barbalat’s lemma. Numerical simulation and experimental results confirm the effectiveness and feasibility of the proposed approach. In future efforts, the payload hoisting and lowering motion will be incorporated in a more complete trajectory planning process. ACKNOWLEDGEMENTS

The authors would like to thank the reviewers for their valuable comments and suggestions that have improved the quality of the paper. They also acknowledge the financial support from the National Natural Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc

ONLINE TRAJECTORY GENERATING FOR CRANE SYSTEMS

1663

Science Foundation of China (Grant No. 61127006) and the National Science and Technology Pillar Program of China (Grant No. 2013BAF07B03). REFERENCES 1. Aguilar-Ibañez GF, Frias OOG. A simple model matching for the stabilization of an inverted pendulum cart system. International Journal of Robust and Nonlinear Control 2008; 18(6):688–699. 2. Kim D-J, Wang Z, Behal A. Motion segmentation and control design for UCF-MANU-An intelligent assistive robotic manipulator. IEEE/ASME Transactions on Mechatronics 2012; 17(5):936–948. 3. Dupree K, Liang C-H, Hu G, Dixon WE. Adaptive Lyapunov-based control of a robot and mass-spring system undergoing an impact collision. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 2008; 38(4):1050–1061. 4. Xin X, She JH, Yamasaki T, Liu Y. Swing-up control based on virtual composite links for n-link underactuated robot with passive first joint. Automatica 2009; 45(9):1986–1994. 5. Xin X, Kaneda M. Analysis of the energy-based swing-up control of the Acrobot. International Journal of Robust and Nonlinear Control 2007; 17(16):1503–1524. 6. Hu G, Makkar C, Dixon WE. Energy-based nonlinear control of underactuated Euler-Lagrange systems subject to impacts. IEEE Transactions on Automatic Control 2007; 52(9):1742–1748. 7. You C, Han J. Trajectory planning of manipulator for a hitting task with autonomous incremental learning. Proceedings of the 2007 IEEE International Conference on Robotics and Biomimetics, Sanya, China, 2007; 1446–1450. 8. Burg T, Dawson D, Rahn C, Rhodes W. Nonlinear control of an overhead crane via the saturating control approach of Teel. The 1996 IEEE International Conference on Robotics and Automation, Minneapolis, Minnesota, USA, 1996; 3155–3160. 9. Yu W, Moreno-Armendariz MA, Rodriguez FO. Stable adaptive compensation with fuzzy CMAC for an overhead crane. Information Sciences 2011; 181(21):4895–4907. 10. Le TA, Kim G-H, Kim MY, Lee SG. Partial feedback linearization control of overhead cranes with varying cable lengths. International Journal of Precision Engineering and Manufacturing 2012; 13(4):501–507. 11. Fang Y, Dixon WE, Dawson DM, Zergeroglu E. Nonlinear coupling control laws for an underactuated overhead crane system. IEEE/ASME Transactions on Mechatronics 2003; 8(3):418–423. 12. Chang DE. Pseudo-energy shaping for the stabilization of a class of second-order systems. International Journal of Robust and Nonlinear Control 2012; 22(18):1999–2013. 13. Liu D, Yi J, Zhao D, Wang W. Adaptive sliding mode fuzzy control for a two-dimensional overhead crane. Mechatronics 2005; 15(5):505–522. 14. Ma B, Fang Y, Zhang Y. Switching-based emergency braking control for an overhead crane system. IET Control Theory and Applications 2010; 4(9):1739–1747. 15. Liu R, Li S, Ding S. Nested saturation control for overhead crane systems. Transactions of the Institute of Measurement and Control 2012; 34(7):862–875. 16. Fang Y, Ma B, Wang P, Zhang X. A motion planning-based adaptive control method for an underactuated crane system. IEEE Transactions on Control Systems Technology 2012; 20(1):241–248. 17. Chwa D. Nonlinear tracking control of 3-D overhead cranes against the initial swing angle and the variation of payload weight. IEEE Transactions on Control Systems Technology 2009; 17(4):876–883. 18. Hua YJ, Shine YK. Adaptive coupling control for overhead crane systems. Mechatronics 2007; 17(2-3):143–152. 19. Chang C, Lie HW. Real-time visual tracking and measurement to control fast dynamics of overhead cranes. IEEE Transactions on Industrial Electronics 2012; 59(3):1640–1649. 20. Sorensen KL, Singhose W, Dickerson S. A controller enabling precise positioning and sway reduction in bridge and gantry cranes. Control Engineering Practice 2007; 15(7):825–837. 21. Yoshida Y, Tabata H. Visual feedback control of an overhead crane and its combination with time-optimal control. Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Xi’an, China, 2008; 1114–1119. 22. Sun N, Fang Y, Zhang X, Yuan Y. Transportation task-oriented trajectory planning for underactuated overhead cranes using geometric analysis. IET Control Theory and Applications 2012; 6(10):1410–1423. 23. Sun N, Fang Y, Zhang Y, Ma B. A novel kinematic coupling-based trajectory planning method for overhead cranes. IEEE/ASME Transactions on Mechatronics 2012; 17(1):166–173. 24. Khalil HK. Nonlinear Systems, (3rd edn). Prentice-Hall: New Jersey, 2002. 25. Dixon WE, Dawson DM, Zergeroglu E, Behal A. Nonlinear Control of Wheeled Mobile Robots. Springer-verlag: Berlin, 2001. 26. Fang Y, Dixon WE, Dawson DM, Chawda P. Homography-based visual servo regulation of mobile robots. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 2005; 35(5):1041–1050. 27. Ma B, Fang Y, Zhang X, Wang X. Modeling and Simulation for a 3D Overhead Crane (in Chinese with an English abstract). Proceedings of the World Congress on Intelligent Control and Automation, Chongqing, China, 2008; 2564–2569. 28. Makkar C, Hu G, Sawyer WG, Dixon WE. Lyapunov-based tracking control in the presence of uncertain nonlinear parameterizable friction. IEEE Transactions on Automatic Control 2007; 52(10):1988–1994. Copyright © 2013 John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control 2014; 24:1653–1663 DOI: 10.1002/rnc