JPH0659740A - Orbit controller - Google Patents

Abstract

(57) [Summary] [Purpose] Efficiently perform optimal vibration-damping trajectory control of structures involving motion. [Structure] A drive unit for moving and controlling the vibration-damped body,
A state equation described by the dynamic characteristics of the control target and the drive system, the state quantities corresponding to them, and the operation amount added to the control target, which includes a calculation unit that calculates and controls the drive pattern and a sensor unit that senses the state amount. On the basis of,
Calculate the feedback gain that maximizes or minimizes the value of the evaluation function described by the state quantity and the operation quantity,
The state equation rewritten by including this feedback gain in the previous state equation is used in the mathematical programming with some parameters such as the maximum allowable speed, maximum allowable acceleration, moving distance, and residual vibration set by the drive as constraints. The trajectory to be moved in the shortest time is obtained by solving using the equation, and the obtained trajectory is added to the controlled object by adding the controlled variable described by the multiplication of the feedback gain and the state quantity to the controlled object. Optimal vibration control trajectory control is performed.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a device for repeating start-up, movement and conveyance, and more particularly to a start-up control device which reduces vibrations of movable parts and enables high-speed movement and positioning of the device.

[0002]

2. Description of the Related Art In recent years, a transfer device such as an XY table, a transfer feeder, etc., which repeats a cycle of starting, moving and stopping, has been increased in size and speed. For this reason, the rigidity of both the servo system and the structural system is relatively reduced, vibration during movement and residual vibration after stoppage occur, and the positioning accuracy of the movable part of the transport device decreases. There is.

It has been found here that the natural frequency and damping of the system may be increased in order to reduce the system vibration that generally accompanies motion. Therefore, a method of relatively increasing the rigidity is adopted by increasing the rigidity of the servo system and the structural system, or by adding the damping by taking the state feedback of both the servo system and the structural system.

However, these techniques often reach their limits in practice due to restrictions on the increase in output of the servo system and the occurrence of oscillation problems. Therefore, the following methods are actually adopted.

1) Applying the optimal regulator theory of time-varying coefficient feedback including the structural system to the positioning problem at the time of moving and stopping, with the constraint that the input energy applied to the controlled object is minimized (Yamada, Nakagawa, "Measurement Automatic Control Papers", 18-1, (1982), 84).

2) A method of approximating the suppression of residual vibration at a stop as a feedforward problem in a moving process and approximating its optimal trajectory to a polynomial consisting of a minimum energy consumption and a constraint condition for residual vibration (Ono, "Mechanism") , 51-46
9, C (1985), 2233).

3) Searching for a trajectory that suppresses residual vibration by linear programming as the shortest time control under each constraint condition of the driving unit (Hamada et al., 6 persons, "Mechanism Theory", No. 870).
-3, (1987), 157), (Shinha, A .;
and Peng, H .; , ASME, 18-5, (19
89), 277. ).

[0008]

However, these 1),
As for 2) and 3), the optimal trajectory is determined by giving a command to follow this trajectory and performing vibration suppression. As various constraints become stricter, the optimal trajectory becomes a complicated curve. Are susceptible to modeling errors, mechanical system play and friction, and servo system non-linearity. Furthermore, there is a tendency that a sufficient effect cannot be obtained for a system whose characteristics change with time. Therefore, in order to solve these problems, a combination of feedforward control by an optimum trajectory and state feedback control of servo system and structural system is considered, and the following methods are also attempted.

4) A combination of minimum energy damping trajectory control and tracking control of a motor section to a target trajectory (Yamaura, Ono, "Mechanism", 54-508, C (1988), 299).
7. ). 5) Feedback control is used in combination with the optimal trajectory obtained by the linear programming method to correct the friction and dead zone of the DC motor (Yellow and 2 others, "Technical Journal", NLP85-
1, (1985), 1. ).

Since the methods 4) and 5) use the control system so that the trajectory of the actual structure follows the optimal damping trajectory obtained by mathematical programming, modeling error, etc. If there is not, the trajectory of the flexible structure will be the optimum one obtained above, and the desired vibration damping effect can be achieved. However, when there is a modeling error or the like, the desired trajectory to be moved is not optimal due to the error, and the desired damping effect cannot be achieved. Furthermore, in the methods 4) and 5), a feedback control system is added after obtaining the optimum trajectory, and the vibration caused by the optimum trajectory is not predicted in advance. Therefore, there is a tendency that the control system is not used efficiently.

The present invention has been made in view of these circumstances, and the mechanical system to be controlled has backlash and friction, and the servo system also has non-linear elements. Therefore, accurate modeling is performed. It is an object of the present invention to provide a trajectory control device capable of suppressing the vibration of a vibration-damped body regardless of these influences even if there is no such situation.

[0012]

Therefore, according to the present invention,
It is equipped with a drive unit that moves and positions the vibration-damped body, a calculation unit that calculates and controls the drive pattern, and a sensor unit that senses the state quantity. The dynamic characteristics of the control target and drive system, and the state quantity corresponding to them , And a feedback gain that maximizes or minimizes the value of the evaluation function described by the state variable and the manipulated variable, based on the state equation described by the manipulated variable added to the controlled object, and this feedback gain is calculated as the previous state. The state equation rewritten by the equation is solved by using mathematical programming with some parameters such as the maximum allowable speed, maximum allowable acceleration, moving distance, and residual vibration defined by the drive as constraints. Control that is calculated by finding the trajectory to move in time and multiplying the obtained trajectory by the feedback gain and state quantity And controls the movement of the controlled object by applying to the control target not containing added. Since the equation of state used when obtaining the trajectory considers the feedback gain in advance, the control system can be effectively used.

[0013]

According to this structure, the optimum damping trajectory is obtained by using the state equation including the feedback gain on the assumption that the feedback control system is also used. Therefore, the obtained trajectory is the vibration characteristic of the control system. It is based on the change of. Therefore, the damping effect due to the improved trajectory and the damping effect due to the addition of the feedback control system can be efficiently obtained without contradicting each other. In addition, since each performance of the drive unit is described in the constraint condition, a reasonable trajectory can be obtained.

[0014]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of a control device according to the present invention will be described below with reference to the drawings.

FIG. 1 is a block diagram showing the apparatus configuration of the embodiment, and this experimental apparatus controls the vibration generated when a work is conveyed.

As shown in the figure, at the tip of the steel plate 1b,
The additional mass 1a is arranged, and the flexible structure 1 including the additional mass 1a and the steel plate 1b is dominated by the primary vibration mode. The other end of the steel plate 1b is attached to the table 2
The table 2 is moved in the X-axis direction by a hydraulic servo actuator (hydraulic cylinder) 3. The wheels of the table 2 are on rails (not shown),
The table 2 performs movements such as starting, moving, and stopping on the rail.

The controlled object is configured as described above.

The laser displacement meters 4 and 5 detect the state quantity of the controlled object, and the control unit 8 outputs the manipulated variable to the controlled object by performing a calculation described later based on the detected state quantity.

That is, the laser displacement meter 4 is a flexible structure 1.
Is detected in the x-axis direction, and the detected displacement x1 is applied to the A / D converter 9 of the control unit 8. Further, the detection signal x1 is applied to the differentiating circuit 6 to be converted into the speed x1.multidot. And applied to the A / D converter 9. It should be noted that “x ·” represents the first-order time derivative of x, and “x ··” represents the second-order time derivative of x.

In the laser displacement meter 5, the stroke amount x2 of the rod 3a of the hydraulic actuator 3 in the x-axis direction,
That is, the displacement x2 of the table 2 is detected. The detected displacement x2 is added to the A / D converter 9 and
The speed is set to x2 through the differentiating circuit 7 and added to the A / D converter 9. The A / D converter 9 converts the input displacements x1 and x2 and the velocities x1 and x2 into digital signals, respectively, and converts these into digital signals.
Add to. The operation amount calculation unit 10 calculates a drive signal uc for driving the hydraulic actuator 3 as described later, and the optimum damping trajectory command signal generation unit 12 calculates a drive signal for driving the hydraulic actuator 3. u
Are calculated, and these are added to the D / A converter 10. The drive signals, that is, the first manipulated variable uc and the second manipulated variable u are amplified as required by the amplifier 11, and these are added to the hydraulic actuator 3 as actuator driving signals. As a result, the stroke amount of the hydraulic actuator 3 is controlled according to the applied actuator drive signal, and the table 2 is moved according to the drive signal.

The contents of the calculation performed by the control unit 8 will be described below with reference to the equivalent models shown in FIGS.

As shown in FIG. 4, the flexible structure 1 has a mass m.
1, a spring constant k1, and a damping coefficient c1 are equivalent to a one degree of freedom mass-spring-viscous damping system, and the table 2 and the hydraulic actuator 3 have a mass m2, a spring constant k2, and a damping coefficient c.
2 is equivalent to the mass-spring-viscous damping system as the second-order lag element, and the signal input system supporting the hydraulic actuator 3 has mass m3, spring constant k3, and damping coefficient c3 mass-spring-viscous damping. The system is equivalent to the system and the device shown in FIG. 1 is represented by these connections. In FIG. 4, fc is the first
Is a force generated by the hydraulic actuator 3 in accordance with the operation amount uc of f, and f is a force generated by the hydraulic actuator 3 in accordance with the second operation amount u, which is a force acting on the mass of mass m3. expressed.

In the equivalent model shown in FIG. 4, the signal input system has a larger mass and smaller spring constants and damping constants than other flexible structures. Therefore, when these are not taken into consideration, the model shown in FIG. 4 is represented as an equivalent model with two degrees of freedom as shown in FIG. 5, and the gain of the state feedback in this equivalent model is calculated as follows.
First, the equation of motion of the controlled object is obtained as follows.

M1 x1 ··· + c1 (x1 · −x2 ·) + k1 (x1 −x2) = 0 (1) m2 x2 ··· + c1 (x2 · −x1 ·) + c2 x2 · + k1 (x2 −x1) + k2 x2 = fc (2) Here, the relative displacement between x1 and x2 is expressed as x
Let r be the state quantity Xg (t) and force fc (t) be defined as follows.

Xr = x1−x2 (3) Xg (t) = [xr.x2.xrx2] T (4) fc (t) =-Kcgucg (t) (5) When used, the relationship of the above equations (1) and (2) becomes as follows, and the state quantity Xg and the manipulated variable ucg
A state equation describing the dynamic characteristics of the controlled object can be obtained by.

Xg. (T) = Ag Xg (t) + bg ucg (t) (6) bg = [-Kcg / m2−Kcg / m2 0 0] T (8) Then, the following state feedback is performed.

Ucg (t) =-Kg Xg (t) (9) Kg = [K1 K2 K3 K4] (10) Here, the evaluation function J for evaluating the controlled object is obtained as follows.

J = ∫ (XT QX + uT ru) dt (11) where Q and r are weighting factors.

The manipulated variable u for performing the optimum control is a feedback gain Kg that minimizes (or maximizes) the value of the evaluation function J with the equation (6) as a constraint condition.
It is obtained by asking for. In this case, the known optimal regulator theory can be used.

Next, the equation of motion is established for the equivalent model of FIG. 4 as follows.

M1 x1 ··· + c1 (x1 · −x2 ·) + k1 (x1 −x2) = 0 (12) m2 x2 ·· + c1 (x2 · −x1 ·) + c2 (x2 · −x3 ·) + k1 (x2) -X1) + k2 (x2-x3) = fc (13) m3 x3 .. + c2 (x2 .-x3.) + C3 x3. + K2 (x2-x3) + k3 x3 = f (14) where the state quantity Letting X be X (t) = [x1.x2.x3.x1x2x3] T (15), the above equations (12), (13) and (14) can be expressed as X. (t) = AX (t) ) + Bu (t) + bc uc (t) (16), the state quantity X (t) and the first manipulated variable uc
It can be expressed by the state equation described by (t) and the second manipulated variable u (t).

Here, f (t) = Kfu u (t) (18) fc (t) = Kc uc (t) (19) b = [0 0 Kf / m3 0 0 0] T (20) bc = [0 Kc / m2 0 0 0 0] T (21).

Next, state feedback is applied to such a structural system. Here, the same calculation as the calculation of the equivalent model of FIG. 5 is performed. That is, the state quantity Xg ^ is expanded to Xg ^ = [x1.-x2.x2.-x3.x1-x2x2-x3] T (22) so as to match the dimension of the above equation (16). Further, the feedback gain K and the first manipulated variable uc are obtained by deforming so as to match the state quantity X as follows.

K = [K1 K2-K1 -K2 K3 K4-K3 -K4] (23) uc = -KX (t) (24) Therefore, the state equation of the above equation (16) is expressed as follows. To be done.

X · (t) = Ac X (t) + bu (t) (25) Ac = A−bc K (26) Now, the state equation of the above equation (26) is calculated from the continuous time system at the sampling time Δt. When converted to a discrete time system, the time (k
The state at +1) Δt is X (k + 1) = PX (k) + qu (k) (27), and the following equation is obtained as the solution.

X (k) = P <k> X (0) + ΣP <k-1-
i> qu (k) However, “P <n>” represents P to the n-th power. Also, Σ is from i = 0 to k-1 (28) where P = e <AcΔt> (29) q = (e <AcΔt> -I) Ac-1 b (30) Become. However, “e <n>” represents the n-th power of e.

Next, the output equation that defines the constraint condition is obtained. When the relative displacements x1 -x2 of the flexible structure 1 with respect to the table 2, the speed x2 of the table 2, and the acceleration x2 of the flexible structure 1 are introduced, and the state quantity y (k) representing these is introduced, the above equation (28) is obtained. Y (k) = [x1−x2x2.x2 ..] T = CcX (k) = CcP <k> X (0) +. SIGMA.CcP <k-1-i> qu (k) (where Σ Is from i = 0 to k-1) (31). here, dc = [0 0 Kc / m 2] T (34) Therefore, assuming that the constraint condition of the table 2 during movement is “maximum allowable value R”, the state quantity y (k) is −R ≦ y (k) ≦ R (35). In addition, when the maximum allowable value of the constraint condition is different between the positive side and the negative side, R and R ′ are set to different values, and −R ′ ≦ y
(K) ≦ R may be satisfied. Therefore, by substituting the above equation (31) into this equation (35), the following equation is obtained for the second manipulated variable u (k).

-R-Cc P <k> X (0) ≦ ΣCc P <k-1-i> qu (k) ≦ R−Cc P <k> X (0) (where Σ is from i = 0 k-1) (36) The value of k during the movement of the table 2 is in the range of 0 to N.

Further, the state quantity X at the time of starting the movement of the table 2
Initial condition of X (0) = [0 0 0 0 0 0] T (37), and when the distance le is moved with time Te and stopped (Te = N × Δt), the residual vibration becomes zero. From the condition, X (N) = [0 0 0 le le le] T (38) and the above equation (28), the following equation is obtained for the second manipulated variable u (k).

X (N) -P <N> X (0) ≦ ΣP <k-1-i> qu (k) ≦ X (N) -P <N> X (0) (where Σ is i = 0 to k−1) (39) Thus, if linear inequalities based on the constraint conditions during movement and at rest, which are shown in the equations (36) and (39), are obtained, the solution of these inequalities is obtained. The second manipulated variable u can be obtained.

Therefore, first, the moving time is fixed (N fixed).
The processing procedure for obtaining the solution u that minimizes the residual vibration level will be described with reference to the flowchart shown in FIG. First, the termination condition X (N) is initially set to the value shown in the equation (38) (step 101). Next, the constraint condition during movement shown in the above equation (36) (step 10
2) A linear programming method, for example, the simplex method, is searched for whether or not there is a solution u (k) satisfying the constraint condition (step 103) at the time of the stop shown in the equation (39) (step 104). As a result, if the solution u (k) exists (YES at step 105), (36),
A second manipulated variable u (k) that satisfies the expression (39), that is, an optimum damping trajectory command signal that minimizes the residual vibration level is output to the amplifier 11 (step 107).

On the other hand, in step 105 (36),
If it is determined that there is no solution u (k) that satisfies both expressions (39), then the termination condition X (N) is relaxed and reset. This is done by successively incrementing the current x1, x1.
06). If the second manipulated variable u (k) satisfying the expressions (36) and (39) is eventually obtained by repeating the processing of steps 102 to 106 again, the residual vibration level becomes the minimum. The signal is output to the amplifier 11 as an optimum damping trajectory command signal (step 107).

In this way, the second manipulated variable u obtained in order to minimize the residual vibration level is added to the hydraulic actuator 3 from the amplifier 11 together with the above-described first manipulated variable uc according to the state feedback, and the table 2 is changed. It moves for a predetermined distance le and stops according to the applied operation amount. As a result, during the movement, the flexible structure 1, the table 2 and the like satisfy the constraint condition R, the vibration level and the like move within a predetermined range, and the residual vibration level becomes the minimum when the soft structure 1 moves by a predetermined distance le and stops. Will be things.

The above processing is considered to be a fixed moving time. Now, a processing procedure for obtaining a solution u that fixes the residual vibration level and minimizes the moving time will be described with reference to the flowchart shown in FIG.

First, N is initially set to 1 (step 201). Next, a solution u (k) that satisfies the constraint condition during movement (step 202) shown in the above equation (36) and the constraint condition at stop (step 203) shown in the above equation (39).
Is checked by linear programming, for example, simplex method (step 204). As a result, if the solution u (k) exists (determination Y in step 205)
ES), a second manipulated variable u (k) satisfying the equations (36), (39), that is, an optimal damping trajectory command signal that minimizes the moving time Te at N = 1 is output to the amplifier 11 (step 207).

On the other hand, in step 205 (36),
If it is determined that there is no solution u (k) that satisfies both equations (39), then the minimum time condition is relaxed by incrementing N by +1 (step 20).
6). In this way, the processes of steps 202 to 206 are repeatedly executed again, and if the second manipulated variable u (k) satisfying the expressions (36) and (39) is eventually obtained, the moving time Te becomes the shortest. The signal is output to the amplifier 11 as an optimum damping trajectory command signal (step 207).

In this way, from the amplifier 11, the second operation amount u obtained in order to minimize the moving time Te is added to the hydraulic actuator 3 together with the above-described first operation amount uc according to the state feedback, and the table 2 is changed. It moves for a predetermined distance le in the shortest time according to the applied operation amount and stops. As a result, during the movement, the flexible structure 1, the table 2 and the like satisfy the constraint condition R while the vibration level and the like move within a predetermined range, and when the movable structure moves for a predetermined distance le in the shortest time and stops, the residual vibration level and the like. Will fall within a predetermined level range.

FIG. 6 is a graph for explaining the effect of the embodiment. In the figure, the upper graph shows the time change of the absolute displacement of the flexible structure 1, and the lower graph shows the time change of the relative displacement of the flexible structure 1 with respect to the table 2. The figure (a) shows a case where the trapecoid trajectory is said to be excellent in suppressing the cam motion residual vibration, and the figure (b) shows a case where the trapecloid trajectory control and feedback control are used together. FIG. 6 (c) shows a case of moving by the LP optimum trajectory obtained by using the linear programming (LP), FIG. 7 (d) shows a case of moving by using the control by the LP optimum trajectory and the feedback control in combination, and FIG. 6E shows the case where the movement is performed by the control of the embodiment.

As is clear from these figures, in the embodiment, the residual vibration is sufficiently suppressed as compared with the conventional cases (a) to (d). Moreover, since the experimental device has non-linear elements such as backlash and friction, it is considered that modeling is not performed accurately. It can be seen that even in such a case, the residual vibration is suppressed and the robustness is excellent.

Although the relative displacement during movement is not restricted in the embodiment, when restricted, it is robust against changes in the natural frequency of the system and changes in motion speed, and the control efficiency is high. It has been clarified from the result of the simulation that the above is good.

[0051]

As described above, according to the present invention, the mechanical system to be controlled has backlash, friction, etc., and the servo system also has non-linear elements, so that modeling is not performed accurately. Even in such a case, the vibration can be effectively suppressed regardless of these influences, and the vibration can be sufficiently reduced.

[Brief description of drawings]

FIG. 1 is a block diagram showing a configuration of an embodiment of a control device according to the present invention, and is a diagram showing a configuration of a device for controlling vibration.

FIG. 2 is a flowchart showing a procedure of processing for minimizing a vibration level performed by the control unit shown in FIG.

FIG. 3 is a flowchart showing a procedure of processing for minimizing a moving time, which is performed by the control unit shown in FIG.

FIG. 4 is a diagram showing an equivalent model of the apparatus shown in FIG. 1 in a 3-degree-of-freedom system.

FIG. 5 is a diagram showing an equivalent model of the device shown in FIG. 1 in a two-degree-of-freedom system.

FIG. 6 is a graph showing comparison between experimental results of the conventional control and experimental results of the example in order to explain the effect of the example.

[Explanation of symbols]

 1 Flexible structure 2 Table 3 Hydraulic servo actuator 8 Control unit

Claims (4)

[Claims] 1. A dynamic part of a controlled object and a drive system, which includes a drive part for moving and positioning a vibration-damped body, a calculation part for calculating and controlling a drive pattern, and a sensor part for sensing a state quantity, Based on the state quantity corresponding to the state quantity and the state equation described by the manipulated variable added to the controlled object, a feedback gain that maximizes or minimizes the value of the evaluation function described by the state quantity and the manipulated quantity is calculated, and this feedback is calculated. Mathematical programming is applied to the state equation in which the gain is included in the previous state equation and rewritten, with some parameters such as the maximum allowable speed, maximum allowable acceleration, moving distance, and residual vibration defined by the drive being constraints. The trajectory to move in the shortest time is found by solving the above, and the trajectory obtained is multiplied by the feedback gain and the state quantity. A trajectory control device, characterized in that the motion of a controlled object is controlled by adding to the controlled object a control amount described above. 2. The trajectory control device according to claim 1, wherein the trajectory of the minimum residual vibration is determined by mathematical programming by determining the travel time and relaxing the constraint condition of the residual vibration. 3. A mathematical programming method is used to obtain an orbit for moving in the shortest time or an orbit for moving with minimum vibration at an arbitrary time by including the vibration level of the vibration-damped body as a constraint in the constraint. The orbit control device according to claim 1, wherein: 4. The trajectory control device according to claim 3, wherein the constraint conditions for obtaining the shortest vibration trajectory and the minimum vibration trajectory include a maximum allowable control force and a maximum allowable control power.