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All content in this area was uploaded by Xuebo Zhang on Dec 06, 2022
Content may be subject to copyright.
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Minimum-time Trajectory Planning for
Underactuated Overhead Crane Systems with State
and Control Constraints
Xuebo Zhang, Member, IEEE, Yongchun Fang, Senior Member, IEEE, and Ning Sun, Student Member, IEEE
Abstract—In this paper, we propose a novel off-line minimum-
time trajectory planning approach for underactuated overhead
cranes. To the best of our knowledge, it is the first optimal
solution to the minimum-time trajectory planning problem for
overhead crane systems, which simultaneously takes into ac-
count various constraints including bounded swing angle for the
payload, bounded velocity, acceleration and even jerk for the
trolley. Different from existing approaches, by means of system
discretization and augmentation, the quasiconvex optimization
technique is successfully adopted to find the minimum-time solu-
tion while satisfying all the aforementioned constraints. Extensive
simulation and experiments with comparisons to previously
published methods are conducted to show the superior perfor-
mance of the proposed method. Note that, the results derived
in this paper also serve as promising guidance in engineering
applications, since it provides a performance limit, namely the
possible highest efficiency, for automatic or manual operation of
overhead cranes.
Index Terms—Minimum-time trajectory planning, underactu-
ated systems, overhead cranes.
I. INT ROD UC TI ON
UNDERACTUATED mechatronic systems have been
widely applied in modern industries [1]–[6]. As a typical
kind of underactuated systems, overhead crane systems play an
important role in many industrial areas to accomplish payload
transportation tasks [5]–[10]. For such tasks of overhead crane
systems, a well-known fact is that the transportation efficiency
and the payload anti-swing requirement are generally contra-
dictory with each other. If the trolley motion is too fast with
large acceleration and velocity, then undesirable payload swing
usually appears, even possibly leading to serious accidents
such as collisions and so on. On the other hand, the pursuit of
the highest efficiency under safety requirements and motor per-
formance limits, is always a persistent drive for manufacturers
Manuscript received April 19, 2005; revised January 11, 2007. This work
is supported in part by National Natural Science Foundation of China
under Grant 61203333 and 61325017, in part by Specialized Research
Fund for the Doctoral Program of Higher Education of China under Grant
20120031120040, in part by Tianjin Natural Science Foundation under Grant
13JCQNJC03200, in part by the Open Project of Chongqing Key Laboratory
of Computational Intelligence under Grant CQ-LCI-2013-03.
X. Zhang, Y. Fang and N. Sun are with the Institute of Robotics
and Automatic Information System (IRAIS) and also Tianjin Key
Laboratory of Intelligent Robotics, Nankai University, Tianjin 300071,
China (e-mail: zhangxuebo@nankai.edu.cn; yfang@robot.nankai.edu.cn;
sunn@robot.nankai.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier ****/TIE.201*.*******
to enhance their competitiveness in the markets. In this situa-
tion, a constrained minimum-time trajectory planning problem
naturally arises, that is, under constraints including bounded
swing angle, bounded velocity, bounded acceleration and even
bounded jerk, how to find the minimum-time trajectory to
transport the payload from the initial location to the desired
one? The study in this paper aims to provide a satisfactory
solution to this practical and important problem.
To reduce the oscillatory response during the transportation
process, many control techniques have been proposed in the
literature, such as input shaping [11], [12], energy-based con-
trol [13], saturation control [15], sliding-mode control [16]–
[18], genetic algorithm-based control [19], neural network-
based control [20], fuzzy control [21], [22], partial feedback
linearization-based control [23], and so on. Among these
techniques, input shaping is essentially an open-loop control
strategy, and hence it is easy to implement and has been
applied to many crane systems with great success [11], [12].
Fang et al. propose a series of energy coupling feedback
control methods to guarantee the system stability [7], [13].
Several different sliding-mode control methods are proposed
to help accomplish the anti-swing transportation task [16]–
[18]. Note that, these works mainly concentrate on how to
suppress the undesirable payload swing, while little effort has
been made to reveal time-optimal solutions under various other
constraints.
In addition to the anti-swing requirement, some other practi-
cal physical limits, such as maximum velocity and maximum
acceleration specified by the motor manufacturers or users,
should also be taken into account for rational control design
of underactuated crane systems. In the literature [28]–[31],
several time-optimal control methods for crane and other
flexible systems are presented to improve the transportation
efficiency. In [29], the overhead crane system is first linearized
by using the small angle approximation, based on which bang-
bang acceleration control is proposed to steer the system
from the initial position to the target one. Nevertheless, the
swing angle is constrained to be zero only at the boundary
states (i.e., the initial position and the target one), while there
is no guarantee for the swing angle and the velocity to be
in a tolerable interval during the transportation process. In
[30], several robust time-optimal control methods for flexible
systems, which present similar structures with crane systems,
are analyzed and it is shown that they are equivalent to the non-
robust time-optimal control of some different systems. Indeed,
these aforementioned works successfully yield several (robust
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or non-robust) minimum-time control techniques satisfying the
boundary conditions, however, they do not take into account
the safety or physical constraints such as the limits of the
swing angle and the trolley velocity during the motion process.
Some minimum-time control methods also incorporate state
constraints in the control design [27], [32]. The work in [32]
converts the problem to a parametric optimization problem to
obtain an optimal solution, however, the controller structure
and the switching time are assumed to be empirically known,
and hence the consequent optimality is local, i.e., it only
holds in a specified parametric space. Loock et al. utilize
the property of differential flatness and fourth-order B-splines
to optimize the trajectory in the parametric space [27]. In
their work, the bounded constraints of the swing angle and
the acceleration input during the evolution are considered in
the optimization. However, the velocity bounds are not taken
into consideration, and moreover, the optimality is also local
which only holds in a fourth-order B-spline parametric space.
Therefore, the minimum-time trajectory planning considering
all state and control constraints, is still a challenging problem.
For controller design of underactuated crane systems, the
recently proposed motion planning-based control approach
[24] first plans the motion trajectory of the trolley, and
then it utilizes tracking control techniques to accomplish the
transportation task. Since the motion trajectory is generated by
known planning techniques, the sketch of the state transition
process is then also known during the transportation as the
tracking error is generally small under feedback control. The
merit with generally known trajectories enhances the operation
safety, and the planning based control is thus widely used in
various areas such as industrial robots, unmanned vehicles,
and so on. However, the existing planning control techniques
in [24], [25] cannot handle all the state constraints such as
velocity and swing angle bounds. Although the previous work
in [26] can cope with most constraints for a crane system, it
yields only a feasible solution rather than a time-optimal one.
In this paper, a minimum-time trajectory planning method
is proposed to yield the highest transportation efficiency under
bounded state and control inputs including the swing angle,
trolley velocity, acceleration and even jerk. To the best of our
knowledge, it is the first minimum-time trajectory planning
method for such kind of underacutated systems subject to
both state and control constraints. Different from existing ap-
proaches, we use the theory from quasiconvex optimization to
guarantee that the aforementioned constraints can be satisfied.
Specifically, we first discretize the continuous-time system
to obtain a discrete-time acceleration-driven system model,
based on which the minimum-time criterion is shown to be
quasiconvex with respect to the optimal control sequence.
Subsequently, the constraints are analyzed to be convex and
then the trajectory planning problem is formulated into a
constrained quasiconvex optimization problem. Using the bi-
section method, the solution converges to the optimal one by
solving a convex feasibility problem at every step of search.
In addition, by augmenting the system state, the jerk-driven
system model is derived, and then the proposed method is
extended to further cope with the jerk constraint as well as the
constraints on the swing angle, the velocity and acceleration.
Extensive simulation and experimental results with compar-
isons to previously published methods are provided to show
the superior performance of the proposed method.
Compared with existing approaches, the main contribution
lies in that: (1) The optimal minimum-time solution success-
fully handles bounded state and control constraints including
the swing angle, trolley velocity, acceleration and even jerk;
(2) It should be emphasized that the proposed approach is
of significant importance in real industrial applications, since
it provides a performance limit, namely the possible highest
efficiency, for automatic or manual operation of overhead
cranes. This can be further utilized to evaluate other online
planning and control strategies or even manual operation
techniques of trained workers under safety and physical con-
straints; (3) Since no specific control or trajectory structures
are assumed known to convert the original problem into an
optimization problem over a corresponding parametric space,
the consequent planned trajectory is globally optimal.
The remainder of this paper is organized as follows. Section
II describes the overhead crane system and the continuous-time
model is given. In section III, the acceleration-driven optimal
planing technique is presented in detail. In section IV, we
extend the proposed method to the jerk-driven situation, which
yields a continuous acceleration profile that is easy for the
actuators to implement. Simulation and experimental results
are provided in section V. Finally, the conclusion is given in
section VI.
II. SY ST EM DESCRIPTION
Consider a planar overhead crane system as shown in Fig. 1.
Let x(t)∈Rand θ(t)∈Rdenote the trolley displacement and
the payload swing angle with respect to the vertical direction,
respectively, with lbeing the rope length.
Trolley
Payload
x
l
R
R
ail
mg
Fig. 1. Overhead crane system model
According to Newton’s second law, it is shown that the
motion of an underactuated crane system is governed by the
following equation [25]:
l¨
θ+ cos θ¨x+gsin θ= 0 (1)
where the constant g∈Ris the gravity acceleration.
Since the proposed method can guarantee that the swing
angle is always in a small range by using the quasiconvex op-
timization techniques during the whole transportation process,
we use the standard small angle approximations of sin θ≈θ
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and cos θ≈1as usually adopted in the literature [11], [26],
[27], [29], [32]. In this case, the system equation is rewritten
as follows:
¨
θ+g
lθ=−1
l¨x. (2)
From this equation, it is clear to see that the motion of the
trolley (the acceleration ¨x) directly influences the evolution
of the payload swing angle θ. Therefore, it is important
to elegantly design the trolley motion to obtain a highly
efficient trajectory under various constraints such as bounded
swing angle, bounded velocity, bounded acceleration and even
bounded jerk, and so on.
In industrial applications, a typical crane task is to transport
the payload horizontally from an initial position x(t0) = x0
to a target position x(tf) = xfunder safety and physical
constraints, with t0, tf∈Rbeing the initial time and final
arrival time, respectively. Without loss of generality, suppose
t0= 0 in the following analysis.
By considering the requirements for real transportation
tasks, the trajectory planning of overhead crane systems here
is defined as to find feasible acceleration (or jerk) trajectories
such that the resultant trolley position trajectory evolutes
smoothly from x(0) = x0to x(tf) = xf, and that the resultant
swing angle, velocity, acceleration and even jerk are kept
in specified (explicitly bounded) ranges during the transient
motion process.
Accordingly, the minimum-time trajectory planning aims to
find among all the feasible trajectories, the optimal one which
minimizes the final arriving time tf. Note that the minimum-
time optimal trajectory planning will definitely improve the
transportation efficiency for an automatic overhead crane
system, and on the other hand, it also provides a practical
performance limit of the manual operation which can be used
to evaluate the efficiency of human operators.
To further clarify the function of the minimum-time tra-
jectory planner, the overall block diagram for planning and
control of overhead crane systems is provided in Figure 2.
The whole system is composed of two modules: a trajectory
planning module and a tracking control module. The work
of this paper mainly focuses on the minimum-time off-line
trajectory planning module. The inputs of this module include
initial and desired configurations, an acceleration-based kine-
matics model or a jerk-based kinematics model(which will be
introduced by system augmentation in section IV), and various
state and control constraints. The outputs of the planner consist
of all the state and control trajectories, including the position
trajectory, the velocity trajectory, the swing angle trajectory,
the acceleration trajectory (and the jerk trajectory if using jerk-
based kinematics model). Yet, it should be remarked that,
in general cases, it is unnecessary to choose all of these
trajectories as the reference trajectories for the subsequent
controller to track, though they can be arbitrarily selected by
interested users. In this paper, we just choose the position
trajectory and the velocity trajectory as reference trajectories,
and the previously developed trajectory tracking controller in
literature [14], [25], [26] is directly adopted in the tracking
control module. Consequently, the torque control inputs are
computed in a real-time way, which are issued to the crane
system to track the reference trajectories. By combing the
minimum-time trajectory planner and the tracking controller,
the transportation task can be completed in a stable and highly
efficient way.
III. MINIMUM-TIME TRAJECTORY PLANNING WITH
CONTINUOUS VELOCITY
To achieve the global optimality, one way is to formulate
the problem as a convex or quasiconvex optimization problem.
By discretization of continuous-time system, we first derive a
discrete-time dynamic system, based on which it is shown
that the minimum-time optimal criterion is quasiconvex and
other state/control constraints are convex with respect to the
control sequences to be designed. Therefore, the minimum-
time trajectory planning problem is successfully converted to a
quasiconvex optimization problem. Then, the bisection method
is utilized to find the optimal solution. It should be noted
that, in this section, the control input of the system is the
acceleration, hence, the velocity is guaranteed to be continuous
since it is the integral of the acceleration profile.
A. Preliminaries
As stated earlier, the main approach is to formulate the
crane planning problem into a quasiconvex optimization prob-
lem by using system discretization and augmentation. Hence,
before proceeding to the concrete procedures, we first state
some basic concepts and related techniques on quasiconvex
optimization.
Definition 1: Quasiconvex function [34].
A scalar function
f(ρ) : Rn→Ris called quasiconvex (or unimodal) with
respect to ρ∈Rnif its domain and all its sublevel sets Sα
Sα={ρ∈dom f|f(ρ)≤α},(3)
for α∈R, are convex. The notation dom fdenotes the domain
for the function f.
Definition 2: quasiconvex optimization problem [34].
A qua-
siconvex optimization problem is one of the form:
minimize f0(ρ)(4)
subject to fi(ρ)<0, i = 1,2,· · · , m (5)
aT
iρ=bi, i = 1,2,· · · , p. (6)
where the objective function f0is a quasiconvex function,
and f1, f2,··· , fmare convex functions for a number of
mconvex constraints , with aiand bibeing constant known
vectors for a number of plinear constraints.
In addition to the aforementioned concepts, we should also
keep in mind that any quasiconvex optimization problem can
be converted into solving a sequence of convex feasibility
problems, and thus the globally optimal solution can be
obtained by using the bisection method as stated in [34].
Therefore, once the problem is successfully formulated as a
standard convex optimization algorithm, it can be solved using
the bisection method, wherein the users should carefully shrink
the range between the lower and upper bounds of the objective
function to increase the computing efficiency.
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( )
t
τ
×
Fig. 2. The overall block diagram for planning and control of overhead crane systems
B. System Discretization
In this subsection, we first express the overhead crane
system in the continuous-time state space form, based on
which the discretization can be easily carried out.
To facilitate the subsequent analysis, we rewrite (2) into the
state space representation as
˙
q=Aq+Bu, (7)
with
q= [x v θ ξ]T,(8)
A=
0 1 0 0
0 0 0 0
0 0 0 1
0 0 −g
l0
, B =
0
1
0
−1
l
(9)
where v(t)∈Rand ξ(t)∈Rdenote the trolley velocity
and the payload swing angular velocity, respectively, and the
control input u(t)∈Rrepresents the acceleration of the
trolley. q(t)∈R4is referred to as the system state vector,
with matrices A∈R4×4and B∈R4×1being the state matrix
and the input matrix, respectively.
Let T∈Rdenote the sampling period, then after some
calculations, we can obtain the corresponding exact discrete-
time model of the computer-based control system under the
zero order hold (ZOH) assumption. Accordingly, the control
input u(k)remains constant between kT and (k+ 1)T, and
hence the system (7) is discretized as follows:
q(k+ 1) = G(T)q(k) + H(T)u(k)(10)
with
G(T) = eAT , H(T) = ∫T
0
eAtdtB (11)
where G(T)∈R4×4and H(T)∈R4×1are the corresponding
discrete-time state matrix and input matrix for a fixed sampling
time T, respectively. It follows from (10) that
q(k) = Gkq0+Gk−1Hu(0) + · · · +Hu(k−1)
=Gkq0+
k
∑
i=1
Gk−iHu(i−1),(12)
which shows the state transition process, and it is clear that
the system state at a specified time kT is affine to the exerted
control sequence (u(0), u(1),· · · , u(k−1)). Hence, we
conclude that, by means of system discretization, the linear
constraints on the system state can be further transformed into
linear constraints with respect to the input control sequences
(u(0), u(1),· · · , u(k−1)).
C. Quasiconvex Optimization-based Minimum-time Planning
With the discrete-time system model, the aim of the
minimum-time trajectory planning now becomes to find the
optimal control input sequences, namely the acceleration pro-
file u(k), to steer the system state from an initial state q0=
[x0v0θ0ξ0]Tto the desired final state qf= [xf0 0 0]T
under state and control constraints including bounded swing
angle, velocity and acceleration. According to the derived tra-
jectory for the control input, those state trajectories, including
the position, velocity and the swing angle trajectories, can then
be obtained using the state transition equation (10). Note that
the proposed approach is able to handle the planning problem
from any initial state to the desired state with θ0being a small
swing angle. Nevertheless, consider the practical requirement
of general transportation tasks for overhead crane systems, it
is generally assumed q0= [x00 0 0]T[24], [26]. Namely,
the initial trolley velocity v0, the initial swing angle θ0and
the initial angular velocity ξ0are all zero.
1) Analysis of the minimum-time criterion function: in this
part, we will first show that the minimum-time criterion is
actually a quasiconvex function with respect to the acceleration
input u(k). This is an important result in this paper, which
makes it possible to formulate the trajectory planning task as
a quasiconvex optimization problem.
Since the system is controllable, there must exist some
feasible control sequences which can drive the system state
from q0to qf. Let K∈Rbe a sufficiently large integer such
that the goal configuration can be reached at time KT , then,
for every feasible control sequence (u(0), u(1),· · · , u(K)),
we define the planning time ft(·)as follows:
ft(u(0), u(1),· · · , u(K)) =
min{kp|q(k) = qffor kp≤k≤K}(13)
which indicates that the overhead crane keeps still after the
planning time. To show that the function ft(·)is quasiconvex
with respect to variables (u(0), u(1),··· , u(K)), given an
arbitarty constant fs∈Z+, the sublevel set Saderived by
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ft(·)≤fsis defined as follows:
Sa={(u(0),· · · , u(K)) |q(k) = qf
for any fs≤k≤K}.(14)
Since the trolley remains stationary at the desired position,
it can be seen that once the final configuration is arrived, the
acceleration input should be kept as zero from then. By further
using the state transition equation (12), we have
Sa={(u(0),· · · , u(K)) |u(k) = 0 for fs≤k≤K
Gfsq0+
fs
∑
i=1
Gfs−iHu(i−1) = qf}.(15)
Based on the fact that the solution set of linear equations
is always affine, we know from (15) that the sublevel set
Sais an affine set with respect to the control sequence.
Since every affine set is also convex, it is concluded that the
sublevel set Sais convex, which indicates that the function
ft(·)is quasiconvex with respect to the control sequence to be
optimized (u(0), u(1),··· , u(K)) based on the definition
of the quasiconvex function in section III-A.
2) Analysis of state and control constraints: to ensure
safety transportation, the swing angle θ(k)should be sup-
pressed into a small range during the motion process. Hence,
the following constraint C1 should be satisfied:
C1 : |θ(k)| ≤ θmax (16)
where θmax is the maximum allowable swing angle. In ad-
dition, since the motor attached to the trolley has its own
performance limits, which are generally provided by the
manufacturers, the velocity v(k)and the acceleration u(k)
should meet the following requirements:
C2 : |v(k)| ≤ vmax,(17)
C3 : |u(k)| ≤ umax (18)
where vmax and umax denote the maximum velocity and
acceleration. Using the result obtained by the system dis-
cretization in (12), we know that these constraints can be
further transformed into linear constraints with respect to
the input control sequences (u(0), u(1),· · · , u(k−1)).
Specifically, since the position, velocity and the swing angle
are all entries of the system state q(k), we have
v(k) = IT
vq(k), θ(k) = IT
θq(k)(19)
with Ivand Iθbeing unit vectors as follows
Iv= [0,1,0,0]T,Iθ= [0,0,1,0]T.(20)
By further substituting (12) into (19), we know that
v(k) = IT
vGkq0+IT
v
k
∑
i=1
Gk−iHu(i−1) (21)
θ(k) = IT
θGkq0+IT
θ
k
∑
i=1
Gk−iHu(i−1) (22)
Hence,the aforementioned state and control constraints in (16),
(17) and (18) can be converted into linear constraints with
respect to (u(0), u(1),· · · , u(k−1)) as follows:
IT
θGkq0+IT
θ
k
∑
i=1
Gk−iHu(i−1) ≤θmax (23)
−IT
θGkq0−IT
θ
k
∑
i=1
Gk−iHu(i−1) ≤θmax (24)
IT
vGkq0+IT
v
k
∑
i=1
Gk−iHu(i−1) ≤vmax (25)
−IT
vGkq0−IT
v
k
∑
i=1
Gk−iHu(i−1) ≤vmax (26)
u(k)≤umax,−u(k)≤umax .(27)
This transformation makes it possible to consider these con-
straints in a convex optimization or even linear programming
framework.
3) Quasiconvex optimization formulation: now, we can
formulate the minimum-time trajectory planning problem
as finding the optimal control sequence uop(k)(k=
0,1,2,· · · , K )among all feasible sequences to minimize
the planning time ft(u(0), u(1),· · · , u(K)):
minimize ft(u(0), u(1),· · · , u(K)) (28)
s.t. q(0) = [x0v0θ0ξ0]T;
For ft≤k≤K, q(k) = qf;(29)
For 0≤k≤K,
∥θ(k)∥ ≤ θmax,∥v(k)∥ ≤ vmax ,∥u(k)∥ ≤ umax.(30)
By using the fact that the trolley will stop and keep still at
the desired position, after some mathematical analysis, the
previous optimization problem (28)-(30) can be reformulated
as follows:
minimize ft(u(0), u(1),· · · , u(K)) (31)
s.t. q(0) = [x0v0θ0ξ0]T;
For ft≤k≤K, u(k) = 0;
Gftq0+
ft
∑
i=1
Gft−iHu(i−1) = qf;(32)
For 0≤k≤K,
linear constraints for u(k), v(k), θ(k)in (23)–(27).
So far, the minimum-time constrained trajectory planning
problem is converted into a constrained optimization problem
formulated by (31), (32) and (23)–(27). In the following,
we will show that this problem is essentially a quasiconvex
optimization problem. By the definition of the quasiconvex
function in section III-A, we have shown that the objective
function ft(·)in (31) is quasiconvex with respect to the
control sequence (u(0), u(1),· · · , u(K)). In addition, the
constraints for the swing angle θ(k), the velocity v(k)and
the control input u(k)are all linear constraints of the control
sequence u(k)(k= 0,1,· · · , K )as shown in (23)–(27).
Therefore, the constraints (32) and (23)–(27) are all standard
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Algorithm 1 Bisection-based quasiconvex optimization for
minimum-time trajectory planning of overhead cranes
Input: q0,qf,kl, kup, G(T), H (T), vmax , θmax, umax
Output: k⋆, uop(k)(k= 0,1,· · · , k⋆)
1: repeat
2: set kmid = (kl+kup)/2
3: solve the convex feasibility problem:
Find u(k)(k= 0,1,··· , kmid)(33)
s.t. Gkmid q0+
kmid
∑
i=1
Gkmid−iH u(i−1) = qf;(34)
For 0≤k≤kmid,
IT
θGkq0+IT
θ
k
∑
i=1
Gk−iHu(i−1) ≤θmax (35)
−IT
θGkq0−IT
θ
k
∑
i=1
Gk−iHu(i−1) ≤θmax (36)
IT
vGkq0+IT
v
k
∑
i=1
Gk−iHu(i−1) ≤vmax (37)
−IT
vGkq0−IT
v
k
∑
i=1
Gk−iHu(i−1) ≤vmax (38)
u(k)≤umax,−u(k)≤umax .(39)
4: if the problem is feasible then
5: kup =kmid
6: else {the problem is infeasible}
7: kl=kmid
8: end if
9: until kl−kup ≥0
10: k⋆=kl, and uop(k) = u(k)for k= 0,1,··· , k⋆.
convex constraints. To sum up, the objective function is quasi-
convex and other constraints are convex, hence it is concluded
that the problem formulated by (31), (32), together with
(23)–(27), is essentially a quasiconvex optimization problem
according to definition 2 provided in section III-A.
4) Solving the quasiconvex optimization problem using bi-
section: the solution of a quasiconvex optimization problem
can be obtained by using the bisection method, together
with solving a sequence of convex feasibility problems. To
describe the approach in a self-contained way, we provide
the detailed computation process in Algorithm 1. In this
algorithm, k⋆∈Z+denote the minimum time index, with
top =k⋆Tbeing the minimum time for the presented con-
strained trajectory planning. Given the lower bound kl∈Z+
and the upper bound kup ∈Z+for k⋆, the minimum time
index k⋆and the corresponding optimal control sequence
uop(k)(k= 1,2,· · · , k⋆)with uop(k) = 0(k⋆< k < K)
can be obtained using the bisection method.
The “Find” operation in Algorithm 1, together with the state
and control constraints (34)–(39), constitutes a feasibility prob-
lem for convex optimization. Note that the objective function
is empty, and the objective of the undergoing computation
is reduced to find a feasible solution for variables u(k)(k=
1,2,· · · , kmid )subject to the constraints from (34)–(39).
In fact, this convex feasibility problem is essentially a linear
programming (LP) problem with an empty objective function,
because the constraints from (34)–(39) are all linear with
respect to the decision variables u(k)(k= 1,2,· · · , kmid ).
Many linear programming or more general convex program-
ming methods, such as interior-point methods, can be utilized
to solve this feasibility problem. These methods are included
in many available free or commercial optimization softwares.
To increase the search efficiency, it is generally a good way
to shrink the search range by providing a lower bound and
upper bound of the arrival time k⋆Tthat are close to the
optimal solution. In the algorithm, an initial estimation of
the lower bound for the time index klis given as
kl=xf
vmaxT,(40)
while the upper bound kup is regarded as the arrival time
determined by the online trajectory generating method [14].
It is worthwhile to point out that once the optimal trajectory
for the control input is obtained, we can use the state transition
equation (12) to compute all the state trajectories including the
position trajectory, the velocity trajectory and the swing angle
trajectory. Any of these trajectories can be utilized as reference
trajectories to be followed by a feedback tracking controller
as shown in Figure 2.
IV. EXTENSION TO MINIMUM-TIME PLANNING WITH
CONTINUOUS ACCELERATION
In this section, we will show that the proposed methodology
can be extended to ensure that the trolley acceleration is
continuous in discrete-time sense (i.e., the jerk is bounded),
which is beneficial to the actuator to avoid undesired vibration.
To address this problem, we will first derive an augmentation
system by adding the acceleration as a state variable. In
this case, the crane system is jerk-driven and thus we can
consider all state and control constraints including bounded
jerk, bounded acceleration, bounded velocity and bounded
swing angle in a unified framework.
Specifically, define the augmented system state q′∈R5as
follows:
q′= [q, a]T(41)
where a(t)∈Rdenotes the trolley acceleration (a(t)is
equivalent to u(t)in (7)). In addition, we define the control
input u′(t)∈Ras the jerk of the trolley motion. Subsequently,
the jerk-driven system model is derived as
˙
q′=A′q′+B′u′(42)
with
A′=
0 1 0 0 0
0 0 0 0 1
0 0 0 1 0
0 0 −g/l 0−1/l
0 0 0 0 0
, B′=
0
0
0
0
1
.(43)
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According to the continuous-time jerk-driven model (42),
the proposed approach can be applied by first obtaining its
discrete-time system model and then formulating the minimum
planning problem as a quasiconvex optimization problem. In
addition, the bounded jerk constraint can also be incorporated:
C4 : |u′(k)| ≤ u′
max.(44)
By using a formulation similar to (31), (32) and (23)–(27) with
extra jerk constraint (44), the bisection-based optimization
algorithm can be further applied to obtain the minimum time
index k∗and the corresponding control sequence u′(k)(k=
0,1,· · · , K ), as well as the generated trajectory.
Remark 1: By augmenting the system state, the proposed
approach has the potential capacity to generate minimum-time
trajectory satisfying high order smoothness of the resulting
practical control input such as motor speed/torque.
Remark 2: In addition to the typical constraints considered
in this paper, some transient performance index, such as the re-
quirement of no overshoot for x(t)(namely x(k)≤xffor any
k), can also be considered in the quasiconvex optimization-
based framework since it is a convex constraint with respect
to the control sequence.
V. SI MU LATION AND EXPE RI ME NTA L RES ULTS
A. Simulation
In this section, we present three groups of numerical simula-
tion results for the proposed minimum-time trajectory planning
method (MTTP) in comparison with existing approaches,
including bang-bang optimal control (BBOC) [29], the phase
plane-based trajectory planning method (PPBM) [26] and the
newly developed coupling analysis-based trajectory generating
method (CABM) [14]. To facilitate subsequent analysis, the
phase plane-based planning method (PPBM) that does not take
into account the bounded jerk constraint, is terms as “PPBM1”,
while the PPBM that guarantees the bounded jerk constraint
is abbreviated as “PPBM2”.
In the simulation, the rope length and the gravity constant
are set as
l= 1.2 m, g = 9.8 m/s2.(45)
The initial and desired configuration are selected as
q0= [0 0 0 0],qf= [10 (m) 0 0 0],(46)
while the state and control constraints are as follows:
θmax = 0.1 rad, vmax = 2 m/s, umax = 1 m/s2.(47)
It should be noted that the maximum swing angle is chosen
to satisfy the requirement of most general applications, while
other parameters are set close to those of real industrial
overhead cranes.
In the first simulation, we suppose that the control input
is the acceleration u(t)without the bounded jerk constraint,
and hence the trolley velocity is naturally continuous. Figure
3 depicts the simulation results in a comparative way by
utilizing various methods including BBOC, PPBM1 and the
proposed MTTP method. It is shown that, though the bang-
bang time optimal control presents the fastest convergent
0 5 10 15
0
5
10
15
x [m]
0 5 10 15
−1
0
1
2
3
v [m/s]
0 5 10 15
−1
−0.5
0
0.5
1
acc [m/s2]
0 5 10 15
−10
−5
0
5
10
θ [deg]
time [s]
PPBM1
BBOC
MTTP
PPBM1
BBOC
MTTP
PPBM1
BBOC
MTTP
PPBM1
BBOC
MTTP
Fig. 3. Comparison of the proposed method and existing approaches without
bounded jerk constraint. The solid lines (blue) denote the results using the
proposed MTTP method, the dashed lines (red) represent the results by BBOC
[29], and the dotted-dashed lines (green) denote the results by the PPBM1
[26].
speed to the desired configuration, the state constraints (C1
and C2), including the prescribed maximum swing angle
and the maximum velocity, are violated during the planning
process. Between the rest two methods that obey the state
and control constraints, it is clear that the proposed method
is much more efficient than the phase plane-based method
without the bounded jerk constraint (PPBM1) [26]. This fact
is further illustrated by the arrival time listed in Tab. I, from
which we can see that the resulting efficiency is improved by
(11.48 −7.68)/11.48 = 33.1% when compared with PPBM1.
Hence, the proposed method presents superior performance
over existing approaches in the presence of the state and
control constraints, and the resulting arrival time can be used
as the reference performance limit for automatic or manual
transportation using crane systems.
In the second simulation, we suppose that the jerk should be
bounded during the planning process. Hence, the acceleration
is continuous and it can be easily implemented by the actuators
without abrupt switchings of the acceleration in the bang-bang
control. The maximum jerk constraint C4 in (44) is set as
u′
max = 2 m/s3.(48)
To ensure that the C4 constraint is satisfied, we use the
jerk-driven system model described in section IV. Since the
acceleration-driven bang-bang optimal control leads to infinite
jerks at the switching point, the corresponding simulation
results are not included in this group of simulation. In the
following, we will show comparative simulation results using
the proposed MMTP, PPBM2 [26], and CABM [14]. As shown
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TABLE I
TRAJECTORY PROPE RTIE S
Methods C1 satisfied? C2 satisfied? C3 satisfied? Arrival time top (s)
BBOC [29] × × √6.51
PPBM1 [26] √ √ √ 11.48
MTTP √ √ √ 7.68
0 5 10 15
0
5
10
15
x [m]
0 5 10 15
−1
0
1
2
v [m/s]
0 5 10 15
−1
−0.5
0
0.5
1
acc [m/s2]
0 5 10 15
−10
−5
0
5
10
θ [deg]
PPBM2
CABM
MTTP
PPBM2
CABM
MTTP
PPBM2
CABM
MTTP
PPBM2
CABM
MTTP
0 5 10 15
−4
−2
0
2
4
jerk [m/s3]
time [s]
PPBM2
CABM
MTTP
Fig. 4. Comparison of the proposed method and existing approaches with
the bounded jerk constraint. The solid lines (blue) denote the results using
the proposed MTTP method, the dashed lines (red) represent the results by
the CABM [14], and the dotted-dashed lines (green) denote the results by
PPBM2 [26].
in Fig. 4, the constraints (C1-C3) on the velocity, the acceler-
ation and the swing angle can all be satisfied using different
methods. However, the maximum allowable jerk constraint
(C4) is violated by PPBM2 [26]. The reason is that, though
the resulting jerk by PPBM2 can be proven to be bounded in
[26], the specified upper bound cannot be arbitrarily enforced
as an explicit constraint. In addition, it is obvious that the
proposed method uses less time and it is thus more efficient
than the other two approaches. An underlying reason is that
the resulting acceleration and velocity are attractive since the
capacity of the actuators is employed as much as possible
under safety and physical requirements. The corresponding
arrival time using these three methods are also presented in
Tab. II, which further confirms the effectiveness of the pro-
0 2 4 6 8 10
0
5
10
x [m]
0 2 4 6 8 10
−1
0
1
2
v [m/s]
0 2 4 6 8 10
−1
0
1
acc [m/s2]
0 2 4 6 8 10
−10
0
10
θ [deg]
time [s]
Fig. 5. The results of MTTP with nonzero initial velocity and swing angle
posed method. It is shown that the efficiency of the proposed
method is improved by (12.58 −8.20)/12.58 = 34.82% with
respect to CABM [14], and (11.59 −8.20)/11.59 = 29.25%
with respect to PPBM2 [26].
To further verify that the proposed method works well with
any initial state, we conduct the third simulation with the
initial velocity and the swing angle being nonzero. In this
case, to the best of our knowledge, most existing methods
cannot be directly applied, thus we only provide the results of
the proposed approach. Without loss of generality, we use the
acceleration u(t)as control input, and the initial and desired
configurations are selected as
q0= [0 1(m/s) 0.1(rad) 0],qf= [10 (m) 0 0 0].
with other constraints being the same as those in (47).
Figure 5 depicts the trajectory results for position, velocity,
acceleration and the swing angle. It can be shown that the
system state is successfully planned to the target configuration
under differential constraints of the kinematic equation. Since
the initial velocity is greater than zero (1 m/s), the resul-
tant trajectories become much faster than those in the first
simulation. This simulation demonstrates that the proposed
approach can potentially act as an online trajectory planner, for
which the initial state can be any non-static system state. With
the fast development of computing hardware and optimization
software, it is promising that real-time online implementation
of the proposed planner can be realized in the future.
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TABLE II
TRAJECTORY PROPE RTIE S
Methods C1 satisfied? C2 satisfied? C3 satisfied? C4 satisfied? Arrival time top (s)
CABM [14] √ √ √ √ 12.58
PPBM2 [26] √ √ √ ×11.59
MTTP √ √ √ √ 8.20
x
Rail
Trolley
Payload
Motor
Rope
Fig. 6. Testbed
B. Experiments
In this subsection, comparative experimental results are
conducted to show the superior performance of the proposed
approach. The testbed used in the experiments is shown in Fig.
6, where the rope length is l= 0.75 m, while the initial and
desired locations are set as x0= 0 m and xf= 0.6 m. During
the planning process, the maximum swing angle, velocity and
acceleration are selected as
θmax = 3.0 deg, vmax = 0.1 m/s, amax = 0.2 m/s2,(49)
while the maximum allowable jerk is set as
u′
max = 2 m/s3.(50)
Due to the small motion range of our experimental testbed, the
values for the maximum acceleration,velocity and the swing
angle are set smaller than those in the simulation such that
these constraints can really have functions during the planning
process. By taking into account the aforementioned state and
control constraints, various trajectories can be obtained by
using different planning methods. In this section, two groups
of comparative experimental results are presented, wherein the
jerk constraint is not considered during the planning process in
the first group while it is considered in the second group. Note
that these trajectory planning methods are employed to gener-
ate various reference trajectories, and in this experiment, the
planned position trajectory and the planned velocity trajectory
are utilized as the reference signal for the subsequent controller
to track. As previously shown in Figure. 2, a proportional-
derivative (PD) feedback torque controller is applied to track
the generated reference trajectories [14], [25], [26] to achieve
highly efficient transportation tasks.
1) Group 1–experimental results without the jerk con-
straint: In this group, the bounded jerk constraint is not
enforced during the planned process. For comparative analysis,
the proposed MTTP method based on the acceleration-driven
model in section III and PPBM1 [26], are adopted to generate
the reference trajectories under the constraints in (49), and then
the corresponding PD tracking controllers are implemented on
the testbed to track the trajectories.
Figure 7 depicts the experimental results by using MTTP,
PPBM2 and BBOC. It is shown that, the reference position
and velocity trajectories generated by using the MMTP and
PPBM2 approaches, are both tracked very well using PD
controllers and the swing angle is less than the prescribed
maximum allowable value of 3 degrees. However, it should
be noted that, though the arrival time of the BBOC approach
is the shortest, the maximum velocity is greater than 0.35m/s
for both reference and practical velocity trajectories, which is
obviously more than the specified maximum velocity 0.1m/s
in (49). The violation of the maximum velocity constraints
in the experiment, is consistent with the theoretic results that
the BBOC method cannot guarantee the velocity and swing
angle constraints to be satisfied. In addition, it is seen that,
in the presence of the same state and control constraints,
the proposed MTTP method is more efficient than PPBM2.
This fact is also illustrated in Tab. III, wherein the ideal
computed arrival time of the reference trajectories and the real
arrival time in the experiments are listed for comparisons. It
is seen that the efficiency of the proposed MTTP method is
improved by (7.910 −6.935)/7.910 = 12.33% with respect
to the PPBM1. It should be note that some residual swing or
chattering of the velocity may arise since the jerk is very large
at the bang-bang-like switching points, which usually leads to
physical vibration of motor with respect to its base.
2) Group 2–experimental results with the jerk constraint:
To ensure that the jerk lies in a specified interval, we adopt
the proposed MTTP method based on the jerk-driven system
model presented in section IV. For comparative analysis, other
two approaches, including CABM in [14] and phase plane-
based method with the jerk constraint (PPBM2) in [26], are
also implemented in real experiments.
Figure 8 depicts the comparative experimental results, from
which it is shown that the proposed MTTP method is the most
efficient approach under the same state and control constraints.
From Tab. IV, we know that the efficiency of the proposed
MTTP method is improved by (7.845 −6.945)/7.845 =
11.47% with respect to PPBM2, and (8.700−6.945)/8.700 =
20.17% with respect to CABM.
Additionally, it is worthwhile to point out that the proposed
MTTP method can guarantee the jerk to be less than the spec-
ified maximum value (50), however, the other two approaches
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0 2 4 6 8 10
0
0.2
0.4
0.6
x [m]
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
v [m/s]
0 2 4 6 8 10
−2
0
2
4
time [s]
θ [deg]
References
MTTP
PPBM1
BBOC
References
MTTP
PPBM1
BBOC
References
MTTP
PPBM1
BBOC
Fig. 7. Experimental comparison of the proposed MTTP method, PPBM1
[26] and BBOC. The dotted lines (red) represent the corresponding reference
trajectories computed by MTTP, PPBM1 and BBOC. The solid lines (blue)
denote the trajectory tracking experimental results based on the proposed
MTTP, the dotted-dashed lines (green) denote the experimental results by
PPBM1, and the dash lines (black) denote the results by BBOC.
TABLE III
TRA JEC TORY PR OPE RTI ES
Methods Computed time (s) Real time (s)
MTTP 6.900 6.935
PPBM1 [26] 7.738 7.910
TABLE IV
TRA JEC TORY PR OPE RTI ES
Methods Computed time (s) Real time (s)
MTTP 6.980 6.945
PPBM2 [26] 7.847 7.845
CABM [14] 9.235 8.700
including PPBM2 and CABM, only ensure that the jerk is
bounded, yet they cannot guarantee that the jerk is less than
the specified explicit maximum value. This fact is indicated
by the chattering of the velocity for the PPBM2 and CABM,
as shown in Fig. 8. For the proposed MTTP method, since
the jerk is explicitly limited to a range by using a jerk-driven
system model, it is seen that the vibration of the motor is less
than that for an acceleration-driven model in the first group of
experiments, and hence the residual swing is almost negligible.
Remark 3:
Note that the acceleration profile and jerk profile
are not given in the experiments due to the lack of sensors for
these variables on our testbed. However, it can be seen that
the position and velocity are tracked very well, which further
indicates that the real acceleration and jerk are generally close
to the reference ones when the noise and disturbance are small.
0 2 4 6 8 10
0
0.2
0.4
0.6
x [m]
0 2 4 6 8 10
−0.05
0
0.05
0.1
0.15
v [m/s]
0 2 4 6 8 10
−2
0
2
4
time [s]
θ [deg]
References
MTTP
PPBM2
CABM
References
MTTP
PPBM2
CABM
References
MTTP
PPBM2
CABM
Fig. 8. Experimental comparison of the proposed method and existing
approaches with the bounded jerk constraint. The dotted lines (red) represents
the corresponding reference trajectories computed by MTTP, PPBM2 and
CABM. The solid lines (blue) denote the results using the proposed MTTP, the
dotted-dashed lines (green) represent the results by PPBM2, and the dashed
lines (black) denote the results by CABM.
Remark 4:
Due to the limited motion distance of our testbed,
the improvements of the transportation efficiency by using
the proposed MTTP in the experiments, are less than that in
the simulation wherein the target distance is set as 10 meters
like the real industrial cranes. Hence, note that the proposed
MTTP is of vital importance for efficiency improvement if
it is implemented in industrial applications with long motion
distances.
C. Some Notes for Users
To successfully apply the proposed trajectory planner in
practice, many factors should be taken into account for a
realistic transportation task. Parameter variation, including the
changing of rope length and payload weight, usually happens.
In addition, both external and internal disturbances, such as the
wind and highly nonlinear friction, also appear in the motion
process. The possibility of implementing the algorithm in an
online manner should also be discussed. In order to deal with
these problems, we provide some notes and suggestions to
serve as guidance for potential users:
•Rope length variation: when the rope length changes, the
trajectory needs to be replanned by running the algorithm
again. An existing issue is that the current proposed
approach is off-line. To cope with this issue, a possible
way is to use lookup tables or neural networks to save
the planned trajectories in the memory, though this issue
will be ultimately solved in the future with the fast
development of computing hardware and optimization
software.
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•Payload weight variation: since the payload weight does
not appear in the system kinematics equation (1), the
planned trajectory still works. In this case, the motor and
its drive should be carefully selected and the feedback tra-
jectory tracking controllers should be elegantly designed
to gain more robustness in the presence of payload weight
variation [24], [25].
•Disturbances: the proposed approach is mainly focused
on the trajectory planning, which is actually an open-
loop optimal planner. The output of the planner will be
regarded as the reference trajectory to be tracked by using
a feedback tracking controller. Though the open-loop
control cannot deal with the disturbances, the rejection of
disturbances can be conducted in the feedback tracking
controller design. Again, similar with the payload vari-
ation case, interested readers can refer to the literatures
for feedback controller design.
•Off-line v.s. online planning: the algorithm is off-line in
its current form. Since the algorithm works well with re-
spect to any initial state, the planner can potentially act as
an online planner, provided that the computation process
can be implemented in a real-time manner. In fact, the
computing cost is increasing when the sampling time is
smaller, because in this case the discretized control input
sequences becomes a large set of variables. For typical
crane applications, by using CVX [35], [36] and GUROBI
[37] under the Matlab environment, the computing time
is generally less than 3 minutes given a sampling period
of 0.02s, which is acceptable for off-line planning but
cannot be directly implemented for online planning. For
industrial applications, other than directly implementing
the proposed approach, a possible way is to use the
lookup tables to memorize a set of useful trajectories,
or to use neural network-based methods to learn from a
huge set of off-line trajectories generated by the proposed
approach. When the time-consuming training is over, we
can then implement these tables or neural networks on an
industrial portable device in a real-time online manner.
Compared with direct implementation of optimization
algorithms, this solution is practical and can potentially
make the current planner an online planner, under limited
computing ability for portable devices.
VI. CONCLUSION
Most of current researches on the motion planning of
overhead cranes focus on the derivation of feasible trajectories
with the bounded swing angle constraint. Few results have
yet been presented for minimum-time optimal planning of
overhead cranes under constraints on the swing angle, veloc-
ity, acceleration and even jerk. This paper proposes an off-
line minimum-time trajectory planning approach for overhead
crane systems. An advantage of the proposed approach is
that both state and control constraints are successfully taken
into consideration during the planning process. Different from
existing approaches, the presented method adopts quasiconvex
optimization theory to guarantee the state and control con-
straints to be satisfied. Consequently, the proposed approach
gives a performance limit of the transportation efficiency
under safety requirements, which is of significant importance
in engineering applications to evaluate the performance of
automatic or manual operation of cranes. Both simulation and
experimental results with comparisons to existing mainstream
approaches are presented to show the superior performance of
the proposed approach.
Currently, collaborated with a local company, we are build-
ing a real industrial-level, automatic, large-scale overhead
crane systems, for which the transportation distance is more
than 10 meters and the rated payload around 20 tons. The
whole project will be completed before December 2015. At
that time, we will try to apply the proposed method in this
paper to real large-scale cranes by implementing the algorithm
on an industrial portable device with lookup tables or learning
techniques to record the necessary off-line planning data. In
this way, more extensive experimental study will be conducted
to improve the operation efficiency.
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Xuebo Zhang (M’12) received the B.Eng. degree in
Automation from Tianjin University in 2002, China,
and the Ph.D. degree in Control Theory and Con-
trol Engineering from Nankai University in 2011,
China. He is currently an assistant professor with
the Institute of Robotics and Automatic Information
System(IRAIS) and also Tianjin Key Laboratory of
Intelligent Robotics, Nankai University, China.
His research interests include mobile robotics,
motion planning, visual servoing, and visual sensor
network.
Yongchun Fang (S’00−M’02−SM’08) received the
B.S. degree in electrical engineering and the M.S.
degree in control theory and applications from Zhe-
jiang University, Hangzhou, China, in 1996 and
1999, respectively, and the Ph.D. degree in electrical
engineering from Clemson University, Clemson, SC,
in 2002.
From 2002 to 2003, he was a Postdoctoral Fellow
with the Sibley School of Mechanical and Aerospace
Engineering, Cornell University, Ithaca, NY. He is
currently a Professor with the Institute of Robotics
and Automatic Information System (IRAIS), Nankai University, Tianjin,
China. His research interests include AFM-based Nano-systems, visual ser-
voing, and control of underactuated systems including overhead cranes.
Ning Sun (S’12) received the B.S. degree in mea-
surement and control technology and instruments
(with honors) and the B.S. degree in marketing
from Wuhan University, Wuhan, China, in 2009.
He is currently working toward the Ph.D. degree
in control theory and control engineering in the
Institute of Robotics and Automatic Information
System, Nankai University, Tianjin, China.
His research interests include motion planning,
nonlinear control, and control of underactuated
mechatronic/robotic systems including cranes.
