1. Field of the Invention
The present invention relates to a system having a plurality of axes oriented in different directions such as a machine tool. More particularly, the present invention is concerned with a position control method which is used in a system of the type described and which can minimize the path error while attaining high response velocity and control stability.
2. Related Art
A typical conventional position control method in a system of the type described relies upon a so-called soft servo system shown in FIG. 8 conducted on each of a plurality of axes. FIG. 7 shows an example of a position control system for controlling the position of an object along a circular path. The system illustrated in FIG. 7 basically employs position controls along two axes. Namely, the system shown in FIG. 7 has a position command generating means 1 capable of outputting a shape (position) command signal (R.sub.x, R.sub.y) and control means 2X, 2Y which receive the command values R.sub.x, R.sub.y of the components of the respective axes and delilver control inputs U.sub.x, U.sub.y to the control objects 3X, 3Y having the respective driving means. In operation, the control means 2X, 2Y of the respective axes, upon receipt of the respective command values R.sub.x, R.sub.y from the position command generating means 1, operate to simultaneously control the corresponding control objects 3X, 3Y, whereby a circular path is formed.
FIG. 8 shows, for the purpose of clarification and simplicity of explanation, the detail of the position control system for uni-axial position control. This system employs an outer control loop of a comparatively low level gain (.omega..sub.o) intended for controlling the position and an inner control loop of a comparatively high level gain (.omega..sub.c) intended for controlling velocity.
The velocity control loop, which has a higher level of gain (.omega..sub.c) exhibits a greater stability against any disturbance and fluctuation of parameters. In addition, since the position control loop has a comparatively low level of gain (.omega..sub.o), there is no risk of imparting excessive impact (acceleration) to the mechanical system. For these reasons, the known system shown in FIGS. 7 and 8 offers an advantage in that no specific consideration is needed in the formation of the machining or processing program when this sytem is applied to, for example, a numerical control machine tool.
This known position control system however, still suffers from the following problems.
Namely, this known system does not enable the response speed and the stability of the system to be adjusted independently. Thus, the system is often required to operate under such a compromise that either one or both of the response speed and the stability do not reach the required level. Even if such a compromise could be obtained, the known system still encounters a problem in that much labor and time are required for the purpose of comparison of the instant value with the command value. Furthermore, a critical problem encountered with this known system is that the precision of the position control is undesirably limited by the maximum value of the gain (.omega..sub.o) of the position control loop.
This critical problem, which is serious particularly from the view point of current demand for higher accuracy or precision, will be described in more detail.
The characteristic (G(s)) of the position control loop in the position control system shown in FIG. 8 can be handled as a servo having a first-order lag, because the gain (.omega..sub.c) of the velocity control loop is greater than the gain (.omega..sub.o) of the position control loop. For example, the gain (.omega..sub.c) of the velocity control loop is usually 4 to 20 times greater than the gain (.omega..sub.o) of the position control loop. The characteristic (G(s)) therefore is represented as follows. ##EQU1##
On the other hand, the response characteristic V(t) to a stepped velocity command (V.sub.0 /S) is represented as follows. EQU V.sub.(t) =V.sub.o (1-e.sup.p) (2)
where, P represents -.omega..sub.o t. PA1 =( .sub.x, .sub.y) . . . increment of position command vector PA1 = - =(e.sub.x, e.sub.y) . . . position error vector PA1 =(U.sub.x, U.sub.y) . . . control input vector PA1 =(C.sub.x, C.sub.y) . . . position vector PA1 = + PA1 = PA1 : state variable PA1 : coefficient matrix of control object PA1 : input matrix
Therefore, the acceleration a.sub.(t) required can be obtained as follows, by conducting a first order differentiation of the formula (2) as follows. EQU a.sub.(t) =V.sub.o .multidot..omega..sub.o .multidot.e.sup.p ( 3)
A machine system which may be a machine tool is assumed here to have a maximum cutting feed velocity V.sub.0 max. In such a system, the maximum acceleration a max is given by the following formula (4). EQU a.sub.max =V.sub.o max .multidot..omega..sub.o ( 4)
On the other hand, the machine system itself has an allowable maximum acceleration A max which is determined by the construction thereof. Obviously, the condition of a max .ltoreq.A max has to be met. Therefore, the maximum value .omega..sub.0 max was of the gain of the position control loop is limited by the following condition. EQU .omega..sub.o max =A.sub.max /V.sub.o max (5)
The shape precision during cutting of a circular work is represented here in terms of a radius reduction rate .delta. of the circle. The radius reduction rate .delta. is represented as follows. ##EQU2##
Where, R represents the radius (mm) of the diameter, while .DELTA.R represents the decrement (mm) of the radius. V.sub.0 represents the cutting velocity (mm/sec).
When a circle of a radius R is scribed at a constant cutting velocity V.sub.0, the radius reduction rate .delta. is proportional to 1/.omega..sub.o.sup.2, i.e., varies in inverse proportion to .omega..sub.o.sup.2. On the other hand, the maximum value .omega..sub.o max of the gain of the position control loop is limited due to the reason concerning the machine system, as explained before in connection with formula (5). Thus, in a position control system which is approximated by the formula (1), the upper limit of the maximum value .omega..sub.o max of the gain is determined by the allowable maximum acceleration Amax and the maximum cutting feed velocity V.sub.o max which are determined by the machine system. Therefore, when the factors (Amax, V.sub.0 max) of the machine system are definitely determined, the maximum gain .omega..sub.o max is automatically limited, thus making it impossible to obtain a higher gain. In consequence, the precision 8 is limited by the maximum value .omega..sub.o max of the gain, thus preventing attainment of any higher degree of precision.
The known position control method explained before in connection with FIG. 7 involves another problem. Namely, in this system, a single position command generating means 1 is adapted for conducting a machining along a circular path by conducting bi-axial position control, i.e., a position control in the direction of a first axis and a position control in the direction of a second axis. To this end, the position command generating means outputs command values R.sub.x and R.sub.y for the respective directions or axes, so that the respective control means 2X and 2Y operate so as to control the conrol systems 3X and 3Y on the basis of the commands R.sub.x and R.sub.y. Therefore, the position control by the system shown in FIG. 7 essentially requires that two control groups each consisting of the control means 2X or 2Y and the control system 3X or 3Y, have exactly the same mechanical and electrical characteristics, otherwise the control may be effected such that one of the shafts moves ahead while the other runs aback in response to the commands, resulting in an error in the determination of the path.