Parabolic position and attitude interpolation for a robot hand

What is claimed is:

1. A control system for an industrial robot comprising a robot having a hand at an end of the robot; instruction means for providing a polygonal path defined by a plurality of straight lines which interconnect a plurality of point coordinates taught to said robot; arithmetical means for continuously interconnecting said straight lines by a parabola at predetermined locations in the vicinity of the centers of said straight lines, said parabola being generated by dividing a span between two predetermined positions through arithmetic linear interpolation technique in a manner in which points located between said predetermined positions are defined on a straight line interconnecting said two predetermined positions; and driving means for continuously moving said hand of the robot along said parabola generated by said arithemical means from a straight line to the next straight line.

2. A control system for an industrial robot comprising a robot having a hand at an end of the robot; instruction means for providing a path constituted by a plurality of first and second straight lines (AB, BC) which interconnect a plurality of point coordinates taught to the robot and providing a speed (v.sub.a) on said first straight line (AB) and a second speed (v.sub.b) on said second straight line (BC); arithmetical means for dividing said second straight line (BC) arithmetically in dependence on the second speed (v.sub.b) corresponding to the speed at which said hand is moved along said second straight line, determining a starting coordinate point (D) by a parabolic interpolation in correspondence with the speed (v.sub.a) on said first straight line (AB), and arithmetically dividing sequentially by a predetermined value the spans between said starting coordinate point (D) and coordinate points (E.sub.m) determined by said first arithmetic division for determining parabolic interpolating points (p.sub.m); and driving means for continuously moving said hand of the robot along the interpolating points (p.sub.m) determined by said arithmetical means from said first to said second straight line.

3. A control system for an industrial robot according to claim 2, wherein said arithmetical means comprises setting means for setting the parabolic interpolation starting point (D) so that a span (DB) between said starting point and the intermediate nodal point (B) is equal to n/n.sub.a .multidot.v.sub.a .multidot.T, and a parabolic interpolation ending point (E) on the second straight line so that a span (BE) between said parabolic interpolation ending point (E) and said intermediate nodal point (B) is equal to n'/n.sub.b .multidot.v.sub.b .multidot.T where v.sub.a represents speed of the hand on said first straight line, v.sub.b represents speed of the hand on said second straight line, T represents a sampling time, n.sub.a represents a natural number selected such that AB/v.sub.a T.ltoreq.n.sub.a <AB/v.sub.a T+1, n.sub.b represents a natural number selected such that BC/v.sub.b T .ltoreq.n.sub.b <BC/v.sub.b T+1, and n and n' represents predetermined natural numbers.

4. A control system for an industrial robot according to claim 3, wherein n is equal to n'.

5. A control system for an industrial robot according to claim 4, wherein coordinate points (E.sub.m) on said second straight line determined through arithmetic division divide equally a line segment (BE) by 2n, and line segments among said coordinate points (E.sub.m) are arithmetically divided by n+1-m/2 with said coordinate point (D) serving as the starting point for the parabolic interpolation, to thereby determine the parabolic interpolating points (p.sub.m).

6. A control system for an industrial robot according to claim 4 or 5, wherein n is so selected that BE.ltoreq.1/2BC, so that the parabolic interpolation can be effected successively.

7. A control system for an industrial robot comprising: a robot having a hand at an end of the robot; instruction means for providing two attitudes (f.sub.o, g.sub.o ; f.sub.n, g.sub.n) of the hand taught to the robot in terms of two unit vectors (f, g); arithmetical means for interpolating the the rotation angle of said hand about an axis (f.sub.o -f.sub.n).times.(g.sub.o -g.sub.n) for rotating from the attitude (f.sub.o, g.sub.o) to the attitude (f.sub.n, g.sub.n); and driving means for driving said robot's hand to change the attitude thereof in accordance with the rotation angle interpolated by said arithemical means.

8. A control system for an industrial robot according to claim 7, wherein said interpolation is a parabolic interpolation.

9. A method of controlling the movement of a hand of an industrial robot comprising defining a polygonal path for movement of the robot hand by a plurality of straight lines which interconnect a plurality of point coordinates, continuously interconnecting said straight lines by a parabola at predetermined locations in the vicinity of the centers of said straight lines, said parabola being generated by dividing a span between two predetermined positions through arithmetic linear interpolation technique in a manner in which points located between said predetermined positions are defined on a straight line interconnecting said two predetermined positions, and continuously moving said hand of the robot along said generated parabola from a straight line to the next straight line.

10. A method of controlling the movement of a hand of an industrial robot comprising defining a path for movement of the robot hand constituted by a plurality of first and second straight lines (AB, BC) which interconnect a plurality of point coordinates, determining parabolic interpolating points (p.sub.m) wherein said second straight line (BC) is arithmetically divided in dependence on a second speed (v.sub.b) corresponding to the speed at which said hand is to be moved along said second straight line, and wherein a parabolic interpolation is effected starting from a coordinate point (d) determined in correspondence with a speed (v.sub.a) on said first straight line (AB) to thereby arithmetically divide sequentially by a predetermined value the spans between said starting coordinate point (d) and coordinate points (E.sub.m) determined by said first arithmetic division for determining said parabolic interpolating points (p.sub.m), and continuously moving said hand along the determined interpolating points (p.sub.m) continuously from said first to said second straight line.

11. A method of controlling an industrial robot according to claim 10, wherein said parabolic interpolation starting point (d) on said first straight line is so set that a span (DB) between said starting point and the intermediate nodal point (B) is equal to n/n.sub.a .multidot.v.sub.a .multidot.T, while a parabolic interpolation ending point (E) is so set on the second straight line so that a span (BE) between said parabolic interpolation ending point (E) and said intermediate nodal point (B) is equal to n'/n.sub.b .multidot.v.sub.b .multidot.T where v.sub.a represents speed of the hand on said first straight line, v.sub.b represents speed of the hand on said second straight line, T represents a sampling time, n.sub.a represents a natural number selected such that AB/v.sub.a T<n.sub.a <AB/v.sub.a T+1, n.sub.b represents a natural number selected such that BC/v.sub.b T.ltoreq.n.sub.b <BC/v.sub.b T+1, and n and n' represent predetermined natural numbers.

12. A method of controlling an industrial robot according to claim 11, wherein n is equal to n'.

13. A method of controlling an industrial robot according to claim 12, wherein coordinate points (E.sub.m) on said second straight line determined through arithmetic division divide equally a line segment (BE) by 2n, and line segments among said coordinate points (E.sub.m) are arithmetically divided by n+1-m/2 with said coordinate point (D) serving as the starting point for the parabolic interpolation, to thereby determine the parabolic interpolating points (p.sub.m).

14. A method of controlling an industrial robot according to claim 12 or 13, wherein n is so selected that BE.ltoreq.1/2BC, so that the parabolic interpolation can be effected successively.

15. A method of controlling the attitude of a hand of an industrial robot, comprising teaching attitudes of said hand to said robot in terms of two unit vectors (f, g), and wherein two attitudes (f.sub.o, g.sub.o ; f.sub.n, g.sub.n) are taught, parabolically interpolating the rotation angle for rotating said hand from the attitude (f.sub.o, g.sub.o) to the attitude (f.sub.n, g.sub.n) about an axis (f.sub.o -f.sub.n).times.(g.sub.o -g.sub.n) and rotating said hand through said rotation angle about said axis.

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BACKGROUND AND SUMMARY OF THE INVENTION

The present invention relates to a control system for controlling a hand of an industrial robot.

Many industrial robots have heretofore been developed for use in materials handling or spot welding and adapted to be driven under PTP control (i.e. point-to-point control). Recently, the robots tend to be increasingly used in various industrial fields for arc welding, assembling and others, and the PTP control tends to be replaced by a CP control (i.e. continuous path control) which assures higher performance.

The CP control involves a method of storing in a memory device data obtained at every sampling time of the control circuit incorporated in the robot. This method however requires an enormous storage capacity because a great deal of data must be stored for the particular arc welding, assembling or other operations to be performed. Such being the circumstances, the teaching is usually effected under the PTP control and a continuous path is generated through linear interpolation of spans between the points given by the teaching for driving the robot.

In this connection, it is noted that in case the hand of the robot is caused to move at a certain speed along the path defined by two line segments so as to pass the points determined through the linear interpolation, acceleration would become infinite at a nodal point of the path, making the control impossible, unless the path is so modified in the vicinity of the nodal point that one line segment passes over to the other succeeding line segment continuously along a curved path. To this end, there have been proposed methods of interconnecting the two polygonal line segments by an arc or a curve of multi-degree. However, these prior art methods are disadvantageous in that difficulty is encountered in arithmetically determining the path on a real time base upon driving the robot because the amount of calculations for determining the path will be significantly increased and that the teaching becomes extremely complicated. For these reasons, it is difficult to adopt the prior art methods mentioned above for practical applications.

The robot's hand thus has to be stopped at the taught point, providing a major cause for elongating the time required for the operation of the robot.

Further, for controlling the attitude of the robot's hand by resorting to interpolation, a rotation angle or deviation in contraposition of the robot mechanism is divided equally. However, since the rotation angle mentioned above is inherently in a non-linear relation with the attitude, there will be a remarkable difference in data for different types of robots. This is disadvantageous because when the robot is to be driven under a direct numerical control (DNC) or the like, data can not be used in common for different types of robots, while linear interpolation which allows a uniform rotation speed is difficult and, therefore, also disadvantageous.

Further, there is known a method of controlling the attitude of the robot's hand through interpolation by making use of a Eulerian angle. However, representation by Eulerian angle includes a singular point, as the result of a which an attitude which can not be represented in terms of a Eulerian angle is inevitably present. Besides, the linear interpolation which makes the rotation speed uniform is difficult, as is the case of the control performed on the basis of the rotation angle described above.

With a view to solving the problems arising in conjunction with the control of the robot's hand described above, the present invention provides a method of interpolating the path by continuously interconnecting a plurality of straight line segments with parabola by resorting only to the arithmetic interpolating functions inherently incorporated in the conventional robot. Since the inventive method can be carried out solely by the operation of the linear interpolation, the addition of a long program is not necessary, whereby the amount of calculations may remain small, providing the operation of a real time base to drive the robot. As to the attitude interpolation of the robot's hand, there is proposed according to the invention a method of driving the robot through interpolation of the rotation angle with the attitude being represented by two unit vectors. According to the inventive method, interpolation of the attitude of the robot's hand can be accomplished with an improved accuracy.

Accordingly, it is an object of the present invention to provide a control system for a robot's hand which is capable of determining the path for the movement of a robot's hand through interpolation based on simplified arithmetic processing on a real time base.

It is another object of the present invention to provide a robot hand controlling system which is capable of varying the speed of the hand at a uniform acceleration through simplified arithmetic processing.

According to a general aspect of the present invention, arithmetic interpolation processing is employed to divide spans among individual positions set for determining a path along which the robot's hand is to be moved, whereby the path for the robot's hand originally defined by a polygon or line segments is made continuous by a parabola in the vicinity of a nodal point so that the hand moves at a uniform acceleration in the vicinity of the nodal point.

According to another general feature of the invention, the attitude of the hand is represented by two unit vectors f and g, wherein a rotation axis for motion of the robot's hand on the basis of two sets of given data (f.sub.o, g.sub.o) and (f.sub.n, g.sub.n) representative of attitudes of the hand, respectively, and the rotation angle required for the robot's hand to move from the one attitude given by (f.sub.o, g.sub.o) to the other (f.sub.n, g.sub.n) about the rotation axis defined by (f.sub.o -f.sub.n).times.(g.sub.o -g.sub.n) is interpolated for driving the robot's hand.

The above and other objects, features and advantages of the present invention will become more apparent by reading the following description of the preferred embodiments of the invention made with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a side view showing schematically an industrial robot to which the present invention may be applied;

FIG. 2 is a view to show a driving mechanism for the robot;

FIG. 3 is a view to show the same along the line X.sub.1 --X.sub.1 in FIG. 2;

FIG. 4 is a view showing a polygonal path;

FIG. 5 is a view showing a general arrangement of a control system for the robot;

FIG. 6 is a view showing a manipulating panel of a teaching box;

FIG. 7 is a view to illustrate linear interpolation of two line segments;

FIG. 8 is a view to illustrate a method of making the path continuous by a parabola according to the teaching of the invention;

FIG. 9 is a view to illustrate a parabolic interpolation according to the invention;

FIGS. 10 and 11 are views showing results of a parabolic interpolation;

FIG. 12 is a view to illustrate a parabolic interpolation according to the invention applied to a case where the speed of the robot's hand is varied when moving along a straight line segment;

FIG. 13 shows graphically the result of the interpolation illustrated in FIG. 12;

FIG. 14 is a view to illustrate a parabolic interpolation according to the invention applied to a case where the speeds of the robot's hand are different between line segments;

FIG. 15 is a view to illustrate a parabolic interpolation for the motion of the robot's hand in the three-dimensional coordinate system;

FIGS. 16 and 17 show flow charts to illustrate controlling processes for controlling the motion of the robot's hand according to the invention;

FIG. 18 is a vector diagram to illustrate vectors representative of attitude of the robot's hand;

FIG. 19 shows schematically in a perspective view a hand of the robot controlled according to the present invention; and

FIG. 20 is a view similar to FIG. 19 and showing two attitudes (f.sub.o, g.sub.o) and (f.sub.n, g.sub.n) of the robot's hand represented as vectors .

DETAILED DESCRIPTION

In the following, the present invention will be described in concrete terms by referring to the accompanying drawings.

Referring to FIGS. 1, 2 and 3, there is shown an example of an articulated robot having six degrees of freedom to which this invention is applied and which is disclosed in the copending U.S. patent application Ser. No. 285637 filed July 21, 1981 and assigned to the present assignee (the corresponding European Patent Application No. 81105693.6 filed July 10, 1981). FIG. 1 is a side view of an articulated industrial robot to which the principle of the invention can be applied. The articulated industrial robot has a base 1, a turret 2, an upper arm 3, a forearm 4 and a wrist 6. The robot has 6 (six) kinds or degrees of freedom: namely a rotation around a Z-axis, rotation of the upper arm 3 around Y.sub.1 -axis, rotation of the forearm 4 around a Y.sub.2 -axis, rotation of a lower half part 5b of the forearm around an X.sub.1 -axis, rotation of the wrist 6 around a Y.sub.3 -axis, and rotation of tool mount or hand 7 around an X.sub.2 -axis.

FIG. 2 illustrates a driving system for each of the motions mentioned above, in the state in which the upper arm 3 and the forearm 4 of the robot shown in FIG. 1 are stretched, while FIG. 3 is a sectional view taken along the line X.sub.1 --X.sub.1 of FIG. 2. More specifically, the turret 2 is rotatably supported on the base 1 through rotary bearings 13 and 13'. An output shaft 10 of a rotary drive source 9a (such as a D.C. servo motor) fixed to the base 1 is connected to a reduction gear 12 (such as a harmonic reduction gear) also fixed to the base 1, via a safety brake 11 (such as an ON/OFF type of solenoid brake) which is also fixed to the base 1. The low-speed side, i.e. the output side of the reduction gear 12 is fixed to the turret 2, so that a rotary drive source 9a imparts to the turret 2 rotation around the Z-axis as shown in FIGS. 1 and 2.

A rotary driving source 14a (such as a D.C. servo motor) for driving the upper arm 3 is mounted in the turret 2. The output shaft 15 of the rotary driving source is connected through a safety brake 16 and a pair of bevel gears 17 and 17' to a reduction gear 18 fixed to an upper part of the turret 2. The pair of upper arms 3 and 3' are rotatably secured to an upper portion of the turret 2 through respective rotary bearings 19 and 19'. The low speed output of the reduction gear 18 is connected to one of the upper arms 3 so that the rotary driving source 14a imparts to the upper arms 3 and 3', the rotation around the Y.sub.1 -axis shown in FIG. 2.

A rotary driving source 20a (such as a D.C. servo motor) for driving the forearm 4 is attached to the rear end of the upper arm 3. The output shaft 22 of the driving source 20a is connected through a safety brake 21 and a pair of bevel gears 23 and 23' to a reduction gear 24 fixed to the lower end of the upper arm 3. An upper half portion 5a of the forearm 4 is rotatably clamped between the upper arms 3 and 3' through respective rotary bearings 25 and 25' and connected to the low speed output shaft of the reduction gear 24. In consequence, the rotary driving source 20a attached to the rear end of the upper arm 3 imparts to the upper half part 5a of the forearm the rotation around the Y.sub.2 -axis as shown in FIGS. 1 and 2.

A lower half part 5b of the forearm 4 is rotatably secured to the end of the upper half part 5a of the forearm through bearings 29 and 29' for free rotation around the X.sub.1 -axis as shown in FIGS. 1 and 2. A gear 27 is attached to the output shaft of a rotary driving source 26a (such as a D.C. servo motor) fixed to the upper half part 5a of the forearm. Another gear 28 is fixed to the input shaft of a reduction gear 30 which is fixed to the upper half part 5a and having a low speed output connected to the lower half part 5b of the forearm. These gears 27 and 28 mesh with each other so that the rotary driving source 26a causes the lower half part 5b of the forearm to rotate around the X.sub.1 -axis.

The driving system constituted by the rotary driving source 20a for driving the upper half part 5a of the forearm, the output shaft 22, the bevel gears 23 and 23' and so forth is mounted only on one of the two upper arms 3. Fixed to the rear end of the other upper arm 3' is a rotary driving source 31a (such as a D.C. servo motor) for driving the wrist 6. The output shaft 32 of this rotary driving source 31a is coupled through a pair of bevel gears 33 and 33' to the upper half part 5a and further to the end portion of the lower half part 5b of the forearm through another pair of bevel gears 34 and 34' and a rotary shaft 35. The wrist 6 is rotatably secured to the lower half part 5b of the forearm through a rotary bearing 38. A reduction gear 37 fixed to the lower half part 5b has a low-speed output coupled to the wrist 6. The rotary shaft 35 is connected at its end to the input shaft of the reduction gear 37 through bevel gears 36 and 36' so that the rotary driving source 31a fixed to the rear end of the upper arm 3' imparts to the wrist 6 a rotation around the Y.sub.3 -axis shown in FIG. 1 and 2. The transmission of power at the inside of the forearm 4 is achieved by the rotary shaft 35 which is coaxial with the axis of rotation of the lower half part 5b of the forearm and extending through a cavity or bore formed in the gear 28 and the reduction gear 30 for driving the lower half part 5b. Therefore, the rotary motion of the lower half part 5b can be achieved without substantially hindering the operation of the driving system.

A rotary driving source 39a (such as a D.C. servo motor) is mounted within the wrist 6 and has an output shaft 40 connected to the input shaft of the reduction gear 41 fixed to the wrist 6. The low-speed output end of the reduction gear 41 is connected to the tool mount or hand 7 which is rotatably supported by the wrist 6 through rotary bearings 42 and 42' so as to impart to the hand the rotation around the X.sub.2 -axis shown in FIGS. 1 and 2.

The articulated industrial robot R imparted with six varieties of freedom as elucidated above is illustrated in a form of a schematic model in FIG. 4.

Next, by referring to FIGS. 5 and 6, description will be made of a control unit 70 for controlling the articulated robot R having six types of freedom and a teaching unit for teaching information about a previously programmed path and speeds for causing the robot R to be operated or moved along the predetermined path at the programmed speeds on a point-to-point base.

The control unit 70 and the robot mechanism R constitute a position control system in which output values produced by pulse encoders PE coupled to respective actuators M are fed back to the control unit 70 through a counter 75, whereby differences between the target or desired coordinate values determined previously by a microprocessor (A) 72 and the corresponding transformed values of the encoder outputs are converted into analog values by means of a digital/analog (D/A) converter 74 for driving the actuators M. In accordance with the assumption that the robot R is imparted with 6 types of freedom, there are provided six actuators M, six tachometer generators TG and six pulse encoders PE. By the way, FIG. 6 illustrates in a plan view a teaching box 78. This teaching box incorporates a basic teaching job executing manipulator unit for storing position and attitude data or information in a storage by manually operating the conventional robot. With the terminology "position and attitude data or information", it is intended to mean nine data which are represented by orthogonal coordinate values of the robot including position data of the tool mount 7 on hand positions p in an XYZ-coordinate system (i.e. X-coordinate value, Y-coordinate value and Z-coordinate value) and two unit vectors {f (.theta..sub.x, .theta..sub.y, .theta..sub.z), g (.theta..sub.x, .theta..sub.y, .theta..sub.z)}.

There are provided on the teaching box 78, those switches 79g, 79h, 79i, . . . , 79r which indicate translational movements of the hand 7 along the coordinate axes and rotational movements thereof around the coordinate axes for driving in the orthogonal mode. Signs (+) and (-) attached to these switches indicate the directions, i.e. forward and backward directions, respectively, of the corresponding motion or movements. Also in the articulation mode, the same switches are used for driving separately and singularly the articulations attached with the corresponding reference numbers (i.e. #1, #2, . . . , #6). When execution of a teaching program by the microprocessor (A) 72 (stored in a read-only memory or ROM 94) is started, interpretation of commands inputted by actuating input keys provided on the teaching box 78 is performed after initialization processing of the relevant data. More specifically, when certain keys of the basic function teaching manipulator field are pressed, the corresponding input data is fetched by a microprocessor 97 through a keyboard display interface 86 periodically under the timing of a clock signal produced by a clock generator 81. Subsequently, in response to a data transfer request signal issued from the control unit 70, the code data to be supplied to the microprocessor (A) 72 through a bus line 85, an asynchronous communication interface adapter (ACIA) 83, level changer 82 and hence a serial interface (I/F) 76 and a bus line 73 is analytically determined by the microprocessor (B) 97 in dependence on which type of processing is to be executed by the microprocessor (A) 72, whereby the data for the relevant processing routine is transferred to the microprocessor (A) 72, whereupon the execution of the program proceeds to the next phase.

In this connection, detailed description will be omitted as to the teaching processing for loading the positional coordinate values of the robot in the random access memory or RAM 77 (triggered by the switch 79a), a mode alternating processing for selectively determining whether the robot is to be operated in the articulation mode independent of the individual axes or the hand 7 is to be operated along the axes of the orthogonal coordinate system in the orthogonal mode when the robot is manually operated (activated by the switches 79e and 79f), a processing for opening and closing of the hand 7 (activated by the switches 79c and 79d), and a processing for lighting those of light emission diodes of LED's 80a, 80b, . . . , 80f located above the key switches 79a, 79b, . . . , 79f, respectively, which correspond to these switches which have been pressed or actuated. The last mentioned processing is executed by the microprocessor (B) 87 for indicating the pressed ones of the above mentioned processing key switches 79a, 79b, . . . , 79f.

In the case of the processing for the motion of the hand as brought about by pressing the keys 79g, . . . , 79r, target or desired values for the selected operation mode determined by the mode change-over processing are arithmetically determined, whereupon they are stored in the RAM 77 as the target values represented by the orthogonal coordinate values or the articulation coordinate values. More specifically, the coordinate values in concern are so determined that the corresponding displacements are progressively increased within a range not exceeding the maximum displacement during a period beginning with the first pressing of the corresponding keys and ending with the release of them, while the displacements are progressively decreased to zero after the release of the keys. The coordinate values thus determined are added to or subtracted from the orthogonal coordinate values or the articulation coordinate values determined in the preceding cycle in dependence on the directions of the movements, to thereby determine the target coordinate values for the instant cycle. These target coordinate values thus obtained are transformed to the corresponding encoder output values for the respective axes of the robot and compared with the actual values read out from the pulse encoders PE through the counter 75. The resulting differences, if present, are supplied to the D/A converter 74 in terms of numbers of pulses to thereby feed speed commands to servo-drivers 96 provided for the axes of the robot, respectively. Thus, the robot R is operated.

When the hand 7 of the robot R has been moved to the required position and/or attitude in this way, the operator can now press the teaching key switch 79a. Then, the count outputs .theta..sub.1, .theta..sub.2, . . . , .theta..sub.6 of the counter 75 representative of the counted pulses output from the pulse encoders PE coupled to the actuators M, respectively, are fetched by the microprocessor (A) 72. On the basis of these values .theta..sub.1, .theta..sub.2, . . . , and .theta..sub.6, the microprocessor (A) 72 determines arithmetically the position/attitude quantities p, f and g of the hand 7 with the aid of an arithmetic processor 95. The values thus determined are stored as the position quantity p.sub.a (X.sub.a, Y.sub.a, Z.sub.a) of a point A (FIG. 7) and the attitude quantities f.sub.a (f.sub.xa, f.sub.ya, f.sub.za) and g.sub.a (f.sub.xz, f.sub.ya, f.sub.za) of the hand at the point A, where p.sub.a (X.sub.a, Y.sub.a, Z.sub.a)=p (.theta..sub.1, . . . , .theta..sub.6), f.sub.a (.theta..sub.xa, .theta..sub.ya, f.sub.za)=f (.theta..sub.1, . . . , .theta..sub.6) and g.sub.a (g.sub.xa, g.sub.ya, g.sub.za)=g (.theta..sub.1, . . . , .theta..sub.6). The speed v.sub.a at which displacement from the point A to a next point is to occur is commanded by a speed switch 88 and ten keys 90. The value of the commanded speed is always displayed by the display 87. In the similar manner, teaching is made as to the position p.sub.b (X.sub.b, Y.sub.b, Z.sub.b) of the point B and the corresponding information f.sub.b (f.sub.xb, f.sub.yb, f.sub.zb), g.sub.b (g.sub.xb, g.sub.yb, g.sub.zb) of the hand as well as the speed information v.sub.b. Further, information of the point C, p.sub.c (X.sub.c, Y.sub.c, Z.sub.c), f.sub.c (f.sub.xc , f.sub.yc, f.sub.zc), g.sub.c (g.sub.xc, g.sub.yc, g.sub.zc) and v.sub.c are taught. By the way, when a parabolic interpolation is to be effected for the point B, the switch 89 is closed in the point (B) teaching mode to designate the parabolic interpolation.

Next, description will be made of the parabolic interpolation effected on the basis of the taught information about the points A, B and C (inclusive of the case where these points A, B and C lie on a straight line). FIG. 7 shows an example of an equidistant interpolation mode along two line segments. Referring to FIG. 7, the three points A, B and C represent the positions p.sub.a (X.sub.a, Y.sub.a, Z.sub.a), p.sub.b (X.sub.b, Y.sub.b, Z.sub.b) and p.sub.c (X.sub.c, Y.sub.c, Z.sub.c) of the hand 7 taught in the robot R, and the hand is to be moved at the speed v.sub.a along the line segment AB and at the speed v.sub.b along the line segment BC. In the course of the motion, the microprocessor (A) 73 determines the positions passed by the hand of the robot R on these line segments at every sampling time T, to control the robot with the thus determined positions being employed as the target or desired positions. The number of samplings effected during the movement of the hand 7 of the robot R along the line segment AB is given by the following expression:

n.sub.a =[AB/v.sub.a T] (1)

In the similar manner, the number of samplings made during the passage along the line segment BC is given by

n.sub.b =[BC/v.sub.b T] (2)

In the above expressions (1) and (2), the blacket [ ] represents that the sampling number n.sub.a is a natural number falling within a range given by AB/v.sub.a T.ltoreq.n.sub.a .ltoreq.AB/v.sub.a T+1 and that n.sub.b is also a natural number in a range where AB/v.sub.b T.ltoreq.n.sub.b <AB/v.sub.b T+1. Accordingly, the target or desired points at the respective sampling time points correspond to those derived by dividing equally the line segments AB and BC by n.sub.a and n.sub.b, respectively. However, it should be noted that driving of the robot's hand in this way would require that the acceleration of the hand at the nodal point B be infinitely increased, thus making it practically impossible to drive the robot's hand. With a view to evading this difficulty, it is taught according to the present invention that the movement of the robot's hand is caused to depart from the direction of the line segment AB at the sampling point which precedes the nodal point B by a predetermined sampling number n so that the robot's hand will follow a parabolic path to reach the line segment BC by skipping the n sampling points leading to the nodal point B.

Now, the principle of the present invention will be elucidated in detail on the illustrative assumption that the robot's hand is moved on a two-dimensional plane (i.e. the plane defined by the axes x and y). Referring to FIG. 8, it is assumed that the speed v.sub.a ' of the robot's hand 7 at which the latter is caused to move along the line segment A'B' is equal to the speed v.sub.b ' of the robot hand moving along the segment B'C'. Further, positions of the three points A', B' and C' are assumed to be so selected with respect to the x-axis and y-axis of the orthogonal plane coordinate system that the point B' coincides with the origin of the coordinate and the internal angle<A'B'C' is bisected by the y-axis, with the x-axis extending perpendicularly to the y-axis. A parabolic equation is expressed as follows:

Y=ax.sup.2 +b (3)

where a and b represent constants, respectively. Tangent to the parabola is then given by the following differential equation: ##EQU1## where c and d represent constants, respectively. Differentiation of the equation (4) by x results in the following singular solution: ##EQU2## Substituting this solution to the equation (4), the latter can be rewritten in a simplified form as follows: ##EQU3## From the equations (3) and (6)

a=-1/4c (7)

and

b=d (8)

Here, the point which precedes the nodal point or origin B' by the predetermined sampling number n is represented by D', while the point which follows the point B' by the number n samplings is represented by E'. Assume that the length of the line segments D'B' and B'E' is equal to each other and represented by l (=n.multidot.v.sub.a 'T and n.multidot.v.sub.b 'T). On these conditions, the tangents to the parabola at the points D' and E' pass through the origin of the x-y coordinate. Accordingly, when the angle<A'B'C' is represented by .alpha., the coordinates of the point D' are given by ##EQU4## and ##EQU5## respectively, while the inclination of the tangent at that point D' is given by ##EQU6## Similarly, the coordinates of the point E' are given by ##EQU7## and ##EQU8## respectively, while inclination of the tangent at this point E' is given by ##EQU9##

Further, on the assumption that the tangents pass the origin, following relation is derived from the equation (4). Namely, ##EQU10## Further, from the equation (3), a can be determined as follows: ##EQU11## From the equations (7), (8), (9) and (10), ##EQU12## and ##EQU13## are determined and put into the equation (6), whereby the equation for the parabola is expressed as follows: ##EQU14##

The control for the robot is, so to say, a sampling control. Accordingly, the motion from the point D' to the point E' must be effected on the basis of data obtained through an integral number of samplings. When the position of the robot's hand on the parabolic path at m-th sampling counted from the point D' is represented by coordinates x.sub.m and y.sub.m, then the coordinate x.sub.m is given by a below mentioned expression (12) because the motion in the direction of the x-axis is effected at a uniform or constant speed. ##EQU15## By putting the above equation (12) into the equation (11), the latter can be rewritten as follows: ##EQU16## Further, inclination of the tangent to the parabola at the point (x.sub.m, y.sub.m) is given by -(m/n-1).multidot.cos (.alpha./2). The coordinates (x.sub.m ', y.sub.m ') of the intersection between this tangent and the line segment B'C' are given by ##EQU17## The points corresponding to these coordinates (x.sub.m, y.sub.m) have tangents which intersect equidistantly on the line segment B'C' when plotted by successively substituting 1, 2, 3, . . . , for m in the above equation (14).

From the equations (12) and (14), the ratio of the distance between the points (x.sub.m-1, y.sub.m-1) and (x.sub.m, y.sub.m) to the distance between the points (x.sub.m-1, y.sub.m-1) and (x.sub.m ', y.sub.m ') is written as follows: ##EQU18## In other words, the component x.sub.m is determined by dividing the line segment connecting the point (x.sub.m-1, y.sub.m-1) and (x.sub.m ', y.sub.m ') by the factor (n+1-m/2). The values of the component y.sub.m lie approximately on the same line segment when the