MODELOWANIE INŻYNIERSKIE ISNN 1896-771X 32, s. 53-60, Gliwice 2006
TRAJECTORY PLANNING AND CONTROL OF OVERHEAD CRANES IN THE WORK ENVIRONMENT WITH OBSTACLES
W
OJCIECHB
LAJERK
RZYSZTOFK
OŁODZIEJCZYKInstitute of Applied Mechanics, Technical University of Radom
Abstract. Manipulating payloads with cranes can be challenging due to undesirable load pendulations induced by the crane motion and external perturbations. The problem gets increasingly difficult when the work environment is cluttered with ob- stacles. Automatic control schemes offer a high potential of rationalization. In this paper a methodology of constructing appropriate load trajectories allowing for one coordinated maneuver that omits the obstacles is addressed. The developed formula- tion leads then to the analysis of crane dynamics and synthesis of control in the specified motion. Some results of numerical simulations are reported.
1. INTRODUCTION
In the industrial practice overhead gantry cranes are commonly operated manually – the op- erator actuates the trolley position and the rope length, by joysticks and/or buttons, so that to move the load from its initial position to the desired final destination along an operator- assessed trajectory. Even though almost the same paths are often repeated, which allows the operator to ‘learn’ the maneuver, the cycle time is usually relatively large since the operator is compelled to perform the maneuver slowly in order to avoid inertia-induced load oscillations [1,2]. The uncontrolled oscillations spoil the load positioning accuracy and are the main reason for the payloads cannot be manipulated quickly enough. The problem gets increasingly difficult when the work environment is cluttered with obstacles that must be omitted, resulting in the need for appropriate load trajectory planning and more demanding/cautious control commands [3]. Automatic control systems have a potential to pay back the desired routes much faster and more accurately than the repeated manual cycles. Specialized model-based control strategies need then to be developed for the precise manipulating payloads along specified curvilinear tra- jectories. A methodology of this type is addressed in this paper.
2. THE OVERHEAD CRANE MODEL
Cranes belong to a broader class of underactuated systems in which the number of control inputs is smaller than the number of degrees-of-freedom. For the overhead crane seen in Fig. 1, these numbers are m=3 and n=5, respectively. The control inputs are the forces F and 1 F 2 actuating the trolley positions s and 1 s , and the winch torque 2 M changing the rope length l, w
T b
w M
M
F ]
=[
u . The degrees of freedom are described by q=[s1 s2 l θ1 θ2]T , where the swing angles θ and 1 θ can be treated as uncontrolled variables. The dynamic equations of 2 the crane can be written in the following generic matrix form
u B q q f q q d q q
M( )&&+ ( ,&)= ( ,&)− T (1) where M is the n×n generalized mass matrix, d and f are n-vectors of generalized dynamic and applied forces, respectively, and B is the T n×m matrix of influence of control inputs u on the generalized actuating force vector fu =−BTu. The explicit form of the equations is re- ported in [4].
F2
x
m Mw
l mt
2 1
z
s2
s1
F1
mb y
Fig. 1. The overhead crane model
The crane performance goal is to execute a prescribed motion of the load, i.e. the m=3 control outputs are time-specified load coordinates, rd(t)=[xd(t) yd(t) zd(t)]T, the desired load trajectory in the inertial reference frame xyz. Expressed in terms of q, the task require- ments lead to m servo-constraints [4,5] on the system, i.e.
0 r
Φ q q
c( ,t)= ( )− d(t)= (2) After twice differentiating with respect to time, these initial servo-constraint equations can be transformed to the constraint conditions at the acceleration level
0 q ξ q, q q C
c&= ( )&&− ( &,t)=
& (3)
where C=∂Φ ∂q is the m×n program constraint matrix, and ξ=&r&d −C&q& is the m-vector of constraint induced accelerations. Explicit forms of Eqs. (2) and (3) are reported in [4].
3. MODELING OF THE LOAD TRAJECTORY
In an obstacle-free workspace, the usual practice in manipulating payloads with overhead cranes is to perform a straight line maneuver [2,6] (Fig. 2a). The load trajectory can then be defined as rd(t)=rA+(rB −rA)s(t), where rA =[xA yA zA]T and rB =[xB yB zB]T are the
initial and target load positions at time t and 0 t , and f s(t) is an appropriately smooth rest-to- rest reference function ensuring that r&d(t0)=r&d(tf)=0 and &r&d(t0)=&r&d(tf)=0. For longer traveling distances the maneuver is often divided into the acceleration (I), steady velocity (II), and deceleration (III) phases. A reasonable proposition for s(t) is [6]
trajectory obstacle
obstacle workspace
obstacle
obstacle workspace
workspace
A
B
A
B
trajectory
trajectory
A
B
a) b) c)
Fig. 2. Load trajectories in the free (a) and cluttered work environment (b,c)
− − − + − − −
+ −
=
−
= −
− + − +
= −
4 0
5
5 0
6
6 0
7
7 0
8
0 0 0
4 0 5
5 0 6
6 0 7
7 0 8
0
) ( 7 ) ( 14 ) ( 10 2
) ( ) 5
( ) 2 (
7 14 10
2 ) 5
(
τ τ τ
τ τ
τ τ
τ τ τ
τ τ
τ
τ τ τ
τ τ
τ
t t
t t
S S t s
S t t
s
t t t
t t S
s
III II I
(4)
where τ =tf −t0, and τ is the acceleration/deceleration time. In the straight line trajectory 0 formulation the parameter S should be taken as S =1 and dimensionless. Given τ =20s,
s
0 =5
τ , and S =1, the above defined reference function s(t) and its first and second time de- rivatives are illustrated in Figure 3.
0 0.04 0.08
-0.04 0 0.04
0 5 10 15 20
0 0.5 1 s [1]
s [s-1] s
[s-2] . ..
s s.
s..
s..
t [s]
Fig. 3. The reference function and its time derivatives
Manipulating payloads in the work environment cluttered with obstacles is more challeng- ing. Most often the whole maneuver is divided into a sequence of straight line maneuvers as illustrated in Fig. 2b. The consequence is that the load needs to be accelerated, decelerated and then stop each time the straight line maneuver is executed, which causes that the total cycle time is relatively large. The proposition of this paper is to perform one rest-to-rest maneuver along an appropriately constructed curvilinear trajectory in a strictly specified time (Fig. 2c).
The trajectory modeling developed in this paper is divided into the following steps.
Step 1. The trajectory is roughly sketched by a succession of points P1,K,PN so that to omit the obstacles, with P and 1 P coinciding with the initial and target load posi-N tions, respectively.
Step 2. The trajectory function is determined numerically by approximating/interpolating the sketched route by spline functions of third order, using the algorithm described in [7]. Firstly, the length h of the broken line that passes the points P1,K,PN is used as abscissa of the smooth spline to get rd(h)=[xd(h) yd(h) zd(h)]T, which is then recalculated to rˆd(s)=[xˆd(s) yˆd(s) zˆd(s)]T following the procedure described in [8], where s is the arc length parameter of the smoothed trajectory.
Step 3. The rest-to-rest maneuver along the constructed trajectory is modeled using s(t) as introduced in Eq. (4), with S being the total length of the planned trajectory.
Step 4. The required system output used in Eq. (2) is obtained as rd(t)=rˆd[s(t)], and its second time derivative involved in Eq. (3) is &r&d =rˆ′′s&2 +rˆ′&s&, where )( ′ denotes dif- ferentiation with respect to s.
0 4 8 12 16 20
0 4 8 12 16 20 24
l l l
l l l l l l l
l l
obstacle
obstacle
x [m]
y [m]
initial position
target position
−
−
−
−
−
−
−
−
−
−
−
−
=
11 23 18
10 24 13
9 21 10
8 19 10
7 17 10
6 14 14
5 11 18
4 9 18
3 7 18
2 4 15
1 3 8
1 3 2
rd
Fig. 4. The sketching points and the resulted load trajectory
A trajectory modeled using the proposed procedure is illustrated in Fig. 4. Twelve points where used to sketch the load motion path so that to omit the obstacles. A mixed interpolation and approximation of the sketched path is possible due to the error estimates declared for each of the points P1,K,PN. Larger/smaller values of the estimates cause a looser/tighter fit (ap- proximation), and setting them all equal to zero results in an exact fit (interpolation); see [7]
for more details. By appropriately adjusting the error estimates, the spline can then be made tight at critical points and loose at others. In order to obtain as smooth fit as possible we used interpolation at the initial and target points, and approximation at the others. The trajectory smoothness can further be improved by using the procedure twice, i.e. using a bigger number of points from the first-fit trajectory as input data for the second approximation.
4. GOVERNING EQUATIONS AND CONTROL SYNTHESIS
By introducing v=q&, the dynamic equations (1) and servo-constraint conditions (3) form a set of 2n+m=13 differential-algebraic equations (DAEs) in the same number of 2n=10 state variables q and v, and m=3 controls u. The inconvenience in dealing with such DAEs is that their index is equal to five [4], which is a measure of complexity of a DAE system and usually causes difficulty in numerical treatment. The index is thus reduced to three by project- ing the dynamic equations into the constrained and unconstrained subspaces, defined by the
n
m× matrix C introduced in Eq. (3) and the n×k (k=n−m) matrix D – an orthogonal complement to C such that CD=0 ⇔ DTCT =0. The final governing equations are [4]:
d
T T T T
T
r Φ q 0
ξ u B M C d f M C 0
u B D d f D v M D
v q
−
=
−
−
−
=
−
−
=
=
−
−
) (
) (
) (
1 1
&
&
⇔
) , (
) , , , (
) , , , ( ) (
t t t
q c 0
u q q b 0
u q q h v q H
v q
=
=
=
=
&
&
&
&
(5)
and the DAE index is now equal to three. A solution to the DAEs are variations in time of state variables of the crane executing the load prescribed trajectory, qd(t) and vd(t), and the control ud(t) that ensures the realization of the motion. The solution covers thus both the dy- namic analysis and synthesis of control of the crane in the partly specified motion [4].
A very simple algorithm, based on Euler backward difference approximation scheme, was used to solve DAEs (5). Following this scheme (refer to [4] for more details), given q and n v at time n t , the values n qn+1, vn+1 and un+1 at time tn+1=tn +∆t are found as a solution to the following nonlinear algebraic equations:
0 q
c
0 u
q v b
0 u
q v v h
q v H
0 q v
q
=
=
=
∆ −
−
=
∆ −
−
+ +
+ + + +
+ + + + +
+
+ +
) , (
) , , , (
) , , , ( )
(
1 1
1 1 1 1
1 1 1 1 1
1
1 1
n n
n n n n
n n n n n
n n
n n n
t t t t
t
(6)
and in this way the solution is advanced from t to n tn+1, where t∆ is the time integration step.
The advantage of this scheme lies in its simplicity and numerical efficiency. Moreover, due to the original servo-constraint equations are involved in DAEs (5), the solution is free from the constraint violation problem, and the truncation errors do not accumulate in time. The code assures thus stable solutions of appropriate accuracy.
5. NUMERICAL SIMULATIONS
0 5 10 15 20 25
0 5 10 15 20
t [s]
s1 [m]
0 5 10 15 20 25
-100 0 100 200
t [s]
F1 [N]
0 5 10 15 20 25
0 5 10 15 20 25
t [s]
s2 [m]
0 5 10 15 20 25
-80 -40 0 40 80
F2 [N]
t [s]
0 5 10 15 20 25
0 4 8 12
t [s]
l [m]
0 5 10 15 20 25
-100 -98 -96
t [s]
Mw [Nm]
0 5 10 15 20 25
-8 -4 0 4 8
t [s]
θ1 [o]
0 5 10 15 20 25
-6 -3 0 3 6
t [s]
θ2 [o]
Fig. 5. Motion and control of the crane executing the specified trajectory
For the crane model seen in Fig. 1, the data used in computations were the following: the load and trolley masses – m=100kg and mt =10kg, respectively; the winch moment of iner- tia – Jw =0.1kgm2; the winch radius – rw =0.1m, and we neglected for simplicity any resis- tance in the crane motions. The performance aim was to move the load along the trajectory defined in Fig. 4, while imposing the rest-to-rest maneuver of Eq. (4) on the motion along the trajectory with τ =25s and τ0 =5s. The time integration step was ∆t=0.1s. Some essential results of the numerical simulations are reported in Fig. 5. The (nominal) control obtained from
the inverse simulation was then used as an input in the direct simulation. The crane motion pat- tern and the load trajectory were repeated with a numerical accuracy.
6. CONCLUSIONS AND DISCUSSION
A methodology for constructing desired trajectories for payloads maneuvered by overhead gantry cranes in the work environment cluttered with obstacles was developed. The trajectory is first sketched by a set of successive points in the work space so that to omit the obstacles, and then interpolated/approximated by appropriately smooth spline functions. After imposing a rest-to-rest reference function that governs the load motion along the curvilinear trajectory, one coordinated and fast maneuver is performed which is much faster and effective compared to the usual practice of traveling the load by means of a sequence of rest-to-rest maneuvers along straight lines. In this contribution the obstacles were assumed to be fixed. The approach can then easily be extended to the case of movable obstacles – the trajectory should be rede- fined each time the new position of obstacles is identified.
The prescribed load trajectory and the assumed rest-to-rest motion of the load along the tra- jectory lead to strict specifications in time of load coordinates in the crane workspace, which are then rearranged to servo-constraint conditions on the system. Involved the crane dynamics, a formulation is developed which allows for the dynamic analysis of the crane executing the load specified motion and the determination of control that enssures the motion realization. In this paper only the results of inverse dynamics analysis were presented, which lead to the syn- thesis of nominal (feedforward) control law. Some guidelines for constructing the control law enhanced by a closed-loop control strategy in order to provide a stable tracking of the required reference load trajectory in presence of perturbations are proposed in [4].
Acknowledgments. The research was supported by the Ministry of Education and Science under the project No. 4 T12C 062 30.
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PROJEKTOWANIE TRAJEKTORII I STEROWANIE SUWNIC W PRZESTRZENI ROBOCZEJ Z PRZESZKODAMI
Streszczenie. Realizacja założonego ruchu ładunku za pomocą suwnicy jest zadaniem złożonym ze względu na trudne do wyeliminowania wahania ładunku wywołane zarówno ruchem dźwignicy jak i zaburzeniami zewnętrznymi. Zadanie to jest szczególnie skomplikowane dla przestrzeni roboczych z przeszkodami. Systemy sterowania automatycznego mogą pozwolić na realizację tych zadań szybciej, bezpieczniej i z większą precyzją. W niniejszej pracy zaproponowano metodę efektywnego modelowania odpowiedniej trajektorii ruchu ładunku tak, by omijane były występujące w przestrzeni roboczej przeszkody. Trajektoria ta szkicowana jest najpierw za pomocą ciągu punktów, a następnie interpolowana/aproksymowana z użyciem funkcji sklejanych (splajnów). Nakładając odpowiedni warunek na ruch wzdłuż trajektorii, modelowany jest jeden dynamiczny manewr przeniesienia ładunku z położenia początkowego do końcowego w narzuconym czasie. Opisane w pracy różniczkowo-algebraiczne równania ruchu programowego pozwalają na analizę dynamiczną suwnicy i jej sterowania wymaganego dla realizacji narzuconego ruchu ładunku. Demonstrowane są wyniki odpowiednich symulacji numerycznych.
