Patent Publication Number: US-2002003540-A1

Title: Method and apparatus for representing motion of multiple-jointed object, computer graphic apparatus, and robot controller

Description:
BACKGROUND OF THE INVENTION  
       [0001] The present invention relates to a method of representing a motion in which a motion of a multiple-articulated object such as a human or an animal is represented, and in particular, to a motion representation method, a motion representing apparatus, a computer graphic apparatus, and a robot controller suitable for facilitating an operation to set and to change a motion of the multiple-hinged object when a size thereof is altered and for enabling various kinds of motions to be represented.  
       [0002] In order to represent in computer graphics such motions as walking and running actions of a human and motions of a horse and a spanworm, a key frame method has been employed in general. According to the key frame method, to generate motions of a multiple-articulated object such as a human or a horse, the user first defines contours thereof at a point of time and at a subsequent point of time, respectively. Contours between these periods are determined based on an interpolation so that the respective contours or shapes thus attained are sequentially displayed in a time-series manner to resultantly produce a motion picture in which the multiple-jointed object seems to make a real action. However, the key frame method is attended with a problem that the contours thus determined in the time-series manner for the motion of the object takes a long period of time, which hence requires a considerably large amount of processing time and which imposes a heavy load on the operator.  
       [0003] Heretofore, to overcome this problem, as described in an article entitled “A Study of Computer Animation Composed of Animation Primitives by Trigonometric Motion Approximation” written in the Proceedings of IECE of Japan, January 1980, Vol. J63-D No. 1, an action of a human is shot by a camera to attain an animation picture thereof on a 16 millimeter (mm) film so as to measure movements of representative points of joints or articulations. For each joint portion above, horizontal and vertical positions X and Y thereof are obtained in centimeters relative to reference positions in a form of a function of time T, thereby determining a locus of the movement of each joint portion. Thereafter, the locus of the movement is approximated by a straight line and a trigonometric curve such that the computer system achieves computations on the approxiamted curve to attain respective contour data items in a time sequence, which are then sequentially displayed as a motion picture in the graphic system.  
       [0004] According to the prior art above, the movement of each articulation thus obtained through the shooting operation on a 16 mm film is analyzed to extract changes in the X-directional and Y-directional positions relative to the respective reference positions at each point of time, thereby determining the approximated curve of the motion of the human. Consequently, a satisfactory animation picture is developed when the action is to be expressed by the approximated curve. However, the approxiamted curve cannot be applied to a case, for example, where the size of the object is varied or where dimensional ratios between the respective joints are altered in the motion. In this case, there arises a problem that the shooting operation is required to be again actually achieved on the object with the pertinent size and/or with the associated ratio between the joints, which leads to a limited degree of freedom for representing the animation. That is, according to the conventional technology, when generating a motion picture of a multiple-jointed object in the computer graphic system, the image can be presented only as an analogy of the real object thus undergone the shooting operation. This means that various actions cannot be developed in computer graphics. For example, only an ordinary walking action of a human shot by the camera can be displayed in the graphic image. Namely, even when the ordinary or standard action is modified, a motion picture of, for example, a violent walking action or a joyful or pleasant walking motion cannot be obtained. In consequence, heretofore, to express such an action above, for example, a human model is required to actually walk with a violent feeling to be shot by a camera so as to attain an image of the violent walk, which is then analyzed to implement an objective animation picture. In other words, for example, when producing a motion picture of animals, insects, and imaginary objects of which various actions cannot be actually shot by a camera, various movements thereof cannot be easily presented in computer graphics.  
       [0005] In addition, it has been impossible in the conventional technology to produce an action with a human sentimental feature, which is usually expressed, for example, by a feature. That is, a characteristic action with a human emotion cannot be reflected on animation picture of the multiple-jointed object.  
       SUMMARY OF THE INVENTION  
       [0006] It is therefore an object of the present invention to provide a motion representation method and a computer graphic apparatus in which even when a size of a multiple-jointed object and/or a dimensional ratio between joints thereof are altered, a motion of the object can be easily changed in computer graphics.  
       [0007] Another object of the present invention is to provide a motion representation method and a computer graphic apparatus in which a multiple-articulated object can achieve various motions such as those having characteristics of actions of the object in computer graphics.  
       [0008] Still another object of the present invention is to provide a method of and an apparatus for controlling a robot in which an action of the robot can be determined independent of a size thereof and in which instructions of various motions can be issued to the robot, thereby solving a problem, similar to that described above, appearing when the robot is actually operated for various actions.  
       [0009] A first feature of the present invention resides in that in a case where a motion of a multiple-articulated object is presented on a display screen by controlling an action of each joint of the object, a bending angle of each articulation is expressed by a function independent of a length between articulations such that data of a contour of the multiple-jointed object in motion are obtained from the function, thereby displaying an image of the object according to a change in a size and/or a dimensional ratio between joints thereof.  
       [0010] A second feature of the present invention resides suitably in that in a case where a motion of a multiple-jointed object is presented on a display screen by controlling an action of each joint of the object, a bending angle of each articulation is expressed by the following function independent of a length between articulations.  
                 θ   m          (   t   )       =       D   m     +       ∑     n   =   1                A   mn     ·   sin                     (       n   ·   t     +     ψ   mn     -     φ     m   ·   n         )                   (   1   )                       
 
       [0011] D m : Direct-current component  
       [0012] A mn : Amplitude of each frequency component  
       [0013] Ψ mn : Phase  
       [0014] m: Joint number  
       [0015] n: Higher harmonic of n-th order  
       [0016] Φ m : Phase difference of 1st order higher harmonics between reference joint and m-th joint (Φ m =0 for reference joint)  
       [0017] Data of a contour of the multiple-jointed object in motion are obtained from values of the function θ m (t) for each joint.  
       [0018] A third feature of the present invention resides in that when a motion of a multiple-jointed object is presented on a display screen by controlling an action of each joint of the object, a bending angle of each articulation is expressed by a function independent of a length between articulations and components of the respective functions are modified when presenting the motion of the object.  
       [0019] A fourth feature of the present invention resides in that in the function (1), at least either one of the parameter values of D m , A mn , and Ψ mn  is changed.  
       [0020] A fifth feature of the present invention resides in that in a robot control operation in which instructions of operations are supplied to a multiple-jointed robot so as to instruct the robot to achieve a desired motion, a bending angle of each articulation is expressed by a function independent of a length between articulations such that positional data is computed for each joint of the robot based on the function, thereby producing the operational instructions.  
       [0021] A sixth feature of the present invention resides in that a bending angle of each joint of a multiple-jointed object is expressed by a function independent of a length between joints so as to reflect onto the function a feature of a motion such as one expressed by an element of a human emotion which may be linguistically represented by a feature.  
       [0022] A seventh feature of the present invention resides in that when presenting a motion of a multiple-jointed object on a display screen by controlling an action of each joint of the object, the action of the object is expressed by a function of time. Means for changing parameters disposed to develop various kinds of motions includes at least either one of means for obtaining a mean value of parameters of each function of time representing a plurality of actions, means for controlling a direction or an orientation of the multiple-jointed object, means for creating a route of a motion, means for changing a stride length between an inner side and an outer side of a curve, means for taking into consideration an influence of a centrifugal force, means for controlling a stride length in the motion, means for interpolating the function of time with respect to space, means for combining an action created in the key frame-method with an action generated from the function of time, means for producing a function of time from measured data, means for generating a function of time from the action created in the key frame method, and means for correcting the function of time.  
       [0023] As above, when a bending angle of each articulation of a multiple-hinged object is expressed by a function, the angle can be independent of a length between articulations. Consequently, in a case when a contour of the multiple-jointed object to be displayed is computed depending on the bending angle of each joint, the computation can be accomplished independently of the size of the object, namely, the function need not be prepared again for the computation. In consequence, even when the multi-articulated object is changed in its size, the computation of the contour thereof can be carried out by use of the functions beforehand prepared. Moreover, to change the representation of the motion, the user need only modify parameters of the functions to alter, for example, a change rate of the bending angle of each joint.  
       [0024] Furthermore, when computing mean values of the respective parameters of the functions of time representing a plurality of actions in the motion representation, the motion can be represented depending on a mean value of a plurality of functions of time.  
       [0025] The user may control the proceeding direction of the motion by adjusting the direction of the multiple-articulated object.  
       [0026] The object can be arbitrarily moved or displaced in a space based on a route of the motion beforehand prepared.  
       [0027] The inching width is varied between the inner and outer sides of a curve and hence the slip of a foot is prevented.  
       [0028] Owing to the influence of the centrifugal force taken into consideration, an inclination of a body can be presented in a circular motion.  
       [0029] The number of steps in an interval can be controlled depending on a stride width supervised in a movement.  
       [0030] Based on an interpolation of a function of time with respect to a space, the motion can be altered while the object is being moved. Depending on an interpolation of a function of time with respect to time, the motion can be varied depending on an elapsed time.  
       [0031] By combining a motion produced in the key frame method with one created from a function of time, there can be represented a motion which cannot be represented only from the function of time.  
       [0032] Measured data are processed to generate a function of time, which enables an actual motion to be produced depending on the function of time.  
       [0033] A function of time is created from a motion obtained in the key frame method and hence an imaginary action not actually existing in the accessible environment can be produced from the function of time.  
       [0034] A function of time can be corrected by interpolating the function of time. 
     
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
     [0035]FIG. 1 is a schematic diagram showing the configuration of a computer graphic apparatus in a first embodiment according to the present invention;  
     [0036]FIG. 2 is a diagram showing the display screen layout of the computer graphic apparatus;  
     [0037]FIG. 3 is a diagram showing an image of a human in a line drawing;  
     [0038]FIG. 4 is a graph presenting measured values of a change in the bending angles of primary joints of the human;  
     [0039]FIG. 5 is a graph showing a spectrum related to a curve  3   a  of FIG. 4;  
     [0040]FIG. 6 is a schematic diagram showing the configuration of a computer graphic apparatus in a second embodiment according to the present invention;  
     [0041]FIG. 7A is a graph, identical to the curve  3   a  of FIG. 4, presenting the bending angle change of a knee articulation in an ordinary walking action;  
     [0042]FIG. 7B is a graph showing the bending angle change of a knee articulation in a joyful walking action;  
     [0043]FIG. 8 is a graph showing a spectrum related to the curve of FIG. 7B;  
     [0044]FIG. 9 is a graph showing a difference between spectra of FIGS. 5 and 8, respectively;  
     [0045]FIG. 10 is a schematic diagram showing the configuration of a computer graphic apparatus in a third embodiment according to the present invention;  
     [0046]FIG. 11 is a schematic diagram showing the configuration of a computer graphic apparatus in a fourth embodiment according to the present invention;  
     [0047]FIG. 12 is a diagram showing the constitution of a control system of a multiple-jointed object in a fifth embodiment according to the present invention;  
     [0048]FIG. 13 is a diagram showing the configuration of a control system of a multiple-jointed object in a sixth embodiment according to the present invention;  
     [0049]FIG. 14 is a diagram showing the constitution of a control system of a multiple-jointed object in a seventh embodiment according to the present invention;  
     [0050]FIG. 15 is a diagram showing the configuration of a control system of a multiple-jointed object in a eighth embodiment according to the present invention;  
     [0051]FIG. 16 is a diagram illustratively showing a relationship between a transit point and a moving direction;  
     [0052]FIG. 17 is a diagram showing the configuration of a control system of a multiple-jointed object in a ninth embodiment according to the present invention;  
     [0053]FIG. 18 is a diagram showing relationships between transit points and a moving direction;  
     [0054]FIG. 19 is a diagram showing the configuration of a control system of a multiple-jointed object in a tenth embodiment according to the present invention;  
     [0055]FIG. 20 is a diagram showing relationship of strides in a movement along a curve;  
     [0056]FIG. 21 is a diagram showing the configuration of a control system of a multiple-jointed object in an 11th embodiment according to the present invention;  
     [0057]FIG. 22 is a diagram schematically showing components of force applied onto a human in a circular motion;  
     [0058]FIG. 23 is a diagram showing a correction of a human posture;  
     [0059]FIG. 24 is a diagram showing the configuration of a control system of a multiple-jointed object in a 12th embodiment according to the present invention;  
     [0060]FIG. 25 is a diagram showing the configuration of a control system of a multiple-jointed object in a 13th embodiment according to the present invention;  
     [0061]FIG. 26 is a diagram showing a relationship between a moving route and a stride length;  
     [0062]FIG. 27 is a diagram showing the constitution of a control system of a multiple-jointed object in a 14th embodiment according to the present invention;  
     [0063]FIG. 28 is a diagram schematically showing relationships between periods of time and stride lengths;  
     [0064]FIG. 29 is a diagram showing the configuration of a control system of a multiple-jointed object in a 15th embodiment according to the present invention;  
     [0065]FIG. 30 is a diagram showing the constitution of a control system of a multiple-jointed object in a 16th embodiment according to the present invention;  
     [0066]FIG. 31 is a diagram showing the configuration of a control system of a multiple-jointed object in a 17th embodiment according to the present invention;  
     [0067]FIG. 32 is a diagram illustratively showing an interpolation of an image during a displacement thereof;  
     [0068]FIG. 33 is a diagram showing the configuration of a control system of a multiple-jointed object in an 18th embodiment according to the present invention;  
     [0069]FIG. 34 is a diagram showing an interpolation of a motion with respect to time;  
     [0070]FIG. 35 is a diagram showing the constitution of a control system of a multiple-jointed object in a 19th embodiment according to the present invention;  
     [0071]FIG. 36 is a diagram showing the configuration of a control system of a multiple-jointed object in a 20th embodiment according to the present invention;  
     [0072]FIG. 37 is a diagram showing the constitution of a system in which a function of time is created from measured data;  
     [0073]FIG. 38 is a schematic diagram showing a method of generating a function of time representing a motion;  
     [0074]FIG. 39 is a diagram showing another system in which a function of time is produced from measured data;  
     [0075]FIG. 40 is a diagram showing still another system in which a function of time is generated from data measured in the key frame method;  
     [0076]FIG. 41 is a diagram showing the configuration of a system for correcting a data base;  
     [0077]FIG. 42 is a diagram showing the configuration of a control system of a multiple-jointed object in a 21st embodiment according to the present invention;  
     [0078]FIG. 43 is a diagram showing the operation of an editing section of the embodiments above in which various parameters are supplied from a display screen;  
     [0079]FIG. 44 is a diagram illustratively showing a screen example employed to specify expressions of a motion;  
     [0080]FIG. 45 is a diagram showing an example of a screen used to specify a route and a speed of an action in the motion editor;  
     [0081]FIG. 46 is a diagram showing a screen example adopted to correct a motion obtained by the key frame method in the motion editor;  
     [0082]FIG. 47 is a block diagram showing the configuration of a control system of a multiple-jointed object in a 22nd embodiment according to the present invention; and  
     [0083]FIG. 48 is a schematic perspective view showing the constitution of a robot. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS  
     [0084] Referring now to the drawings, description will be given of embodiments according to the present invention.  
     [0085]FIG. 1 shows the configuration of a computer graphic apparatus in a first embodiment according to the present invention. In this system, as shown in FIG. 2, an example of a multiple-jointed object i.e. a human is presented in an animation picture of a multiple-articulated object  1  on a display screen such as a CRT. In operation of the computer graphic apparatus, the multiple-hinged object  1  is represented as a multiple-jointed object in a linework as shown in FIG. 3. In this line drawing, a bending angle of each joint (a coxa  2 , a knee joint  3 , an elbow joint  4 , and a shoulder joint  5  in the example of FIG. 3) is controlled to attain various kinds of contours. Thereafter, the body portion is added to the line drawing to display a motion image of a human (FIG. 2).  
     [0086] The computer graphic apapratus of FIG. 1 includes a basic motion function storage  10  for storing therein for each basic motion a periodic function expressing a bending angle of each articulation, a basic motion selector  11  for selecting a desired function from the various functions loaded in the storage  10 , a joint angle computing unit  12  for receiving the function selected from the storage  10  by the selector  11  and for computing based thereon a bending angle of the pertinent joint in a time series manner, a speed controller  13  for controlling a speed at which the computing unit  12  achieves the computation of the joint angle in a time series fashion, namely, the speed of a motion conducted by the joint, a multiple-hinged object contour storage  15  for storing therein contours used to draw bodies related to lineworks of the various multiple-jointed objects, a contour selector  16  for selecting a contour of a multiple-articulated object, a rendering unit  14  for drawing a body for the motion of the multiple-jointed object, which is expressed only by joint bending angles computed by the computing unit  12 , based on the contour read from the storage  15 , thereby generating image information for displaying the picture in a two-dimensional manner, and a display  17  for presenting the picture depending on the image information.  
     [0087] The basic motion function storage  10  is loaded with functions associated with a motion of each joint for each basic action such as a walking action, a running action, or a sitting action. Under this condition, to select a basic motion, the operator picks a basic operation specification icon  30  presented on the display screen as shown in FIG. 2. In the following paragraph, a walking action of a human image is taken as an example to be expressed by a function representing a motion to be displayed.  
     [0088]FIG. 4 is a graph showing curves representing changes with respect to time of measured bending angles of primary joints of a walking person viewed from a horizontal direction (X-axis direction). Curves  2   a ,  3   a ,  4   a , and  5   a  designate changes with respect to time of measured bending angles of the coxa, the knee joint, the elbow joint, and the shoulder joint, respectively. The angle changes are measured for two periods (strides or steps). These curves show correlations existing between the actions of the respective articulations. FIG. 5 is a spectral diagram obtained by achieving a Fourier analysis on the measure values  3   a  of the knee joint of FIG. 4. Essential spectrum compounds can be expressed by use of a function of at most degree 5 or 6. Namely, there need not be employed a complicated expression including high-frequency components. In consequence, the change Θ m (t) with respect to time of the joint bending angle can be expressed by the following equation of a quite low degree. For example, as shown in FIG. 5, seven spectrum elements need only be required to express the angle change Θ m (t). The following function (1) is identical with the previously described above, but since the embodiment is clearly described, the function (1) is again used as follows;  
                 Θ   m          (   t   )       =       D   m     +       ∑     n   +   1                A   mn     ·   sin                     (       n   ·   t     +     ψ   mn     -     φ     m   ·   n         )                   (   1   )                       
 
     [0089] where,  
     [0090] D m : Direct current component  
     [0091] A mn : Amplitude of each frequency component  
     [0092] ψ mn : Phase  
     [0093] m: Joint number  
     [0094] n: Higher harmonic of n-th order  
     [0095] Φ n : Phase difference of 1st order higher harmonic between reference joint and m-th joint (Φ m =0 for reference joint)  
     [0096] In this connection, the harmonization between the actions of the respective joints is expressed by use of a difference between phases of the joints. The phase differences are kept retained through the motion.  
     [0097] The joint angle computing unit  12  of FIG. 1 achieves computations based on the function (1) to obtain an action of the knee joint while changing the value of the variable  t  such that the position of the joint is sequentially displayed at the point of time  t , thereby presenting a motion picture of the knee. Incidentally, to set the action speed, the operator uses a motion specification icon  33  of FIG. 2. Namely, when the display item of a vertical bar  40  in the icon  33  is horizontally shifted by a mouse cursor or the like, the speed controller  13  of FIG. 1 is activated to develop its operation. The joint angle computing unit  12  computing based on the function (1) the values of Θ m (t 1 ), Θ m (t 2 ), Θ m (t 3 ), etc. in a time series fashion increases the value of t 2 −t 1 =t 3 −t 2 =. . . =Δt before achieving the computations when the operator specifies a higher motion speed.  
     [0098] Since the function (1) above does not include any parameter denoting a length, even when a size and/or a dimensional ratio between joints of the object are/is varied, the function (1) need not be modified. In consequence, also when it is desired to alter a size of a multi-hinged object to be displayed, the load imposed on the operator is not increased; moreover, after the size is changed, the animation picture of the multi-articulated object can be displayed at a high speed. That is, according to the embodiment above, the operator need only select a kind of each basic motion, a motion speed thereof, and a contour of each joint for a motion picture. Namely, a motion of a multi-jointed object can be developed with a quite small amount of information and through a small number of operation steps.  
     [0099] Referring now to the motion representing apparatus of FIG. 1, the operation of the first embodiment will be described. First, the contour storage  15  is loaded, for example, with contour data related to a state of a person image standing in an upright style and contour data of a basic posture of a flying butterfly. The contour selecting unit  16  selects desired contour data therefrom. For example, when the contour data of a person image are selected, the selected data are fed to the rendering unit  14 . On the other hand, the basic motion function storage  10  is loaded, for example, with functions for which parameters of basic actions of a person such as a walking action, a running action, and a sitting action are respectively specified. One of the basic motions is selected by the basic motion selector  11 . For example, when a running action is selected, the function associated with the running action is transferred to the joint angle computing unit  12 , which in turn achieves a computation of the function (1) to produce data of angles. The computation result is transferred together with a speed change rate indicated by the speed controller  13  to the rendering unit  14 . Based on the contour data from the contour storage  15  and the angle data thus received, the rendering unit  14  creates image data to be sent to the display  17 , which resultantly presents thereon a picture of the image data.  
     [0100]FIG. 6 shows the constitution of a computer graphic apparatus in a second embodiment according to the present invention. As compared with the configuration of the first embodiment of FIG. 1, the second embodiment includes a feature component storage  20  and a feature component selector  21 . The joint angle computing unit  12  is configured to accomplish computations based on a selected basic operation function and a selected feature component. The feature component is here associated with a feature and is used to qualify an action or a motion. That is, the feature component is a feature such as joyfully, sadly, or delightfully. In this embodiment, the feature components are expressed in the form of functions so that the operator specifies a feature component to be selected. As can be seen from FIG. 2, the operator need only pick a feature selection icon  31  in the screen to choose a characteristic component. For example, with the mouse cursor or the like arranged over the icon  31 , each time the left-side button is actuated features such as “joyfully”, “merrily”, “like a drunkard”, and “calmly” are sequentially presented. When a desired feature is displayed on the screen, the operator actuates the right-side button to select the feature. In a case where “walk” is selected as the basic motion and “joyfully” is chosen as the feature element, the object of a human image  1  displayed achieves an indicated action, namely, the object  1  walks joyfully or merrily. Next, a description will be given of a control operation of the characteristic element.  
     [0101]FIG. 7A is a graph presenting a change with respect to time of the measured bending angle of a knee joint in an ordinary walking action, whereas FIG. 7B is a graph showing a change with respect to time of the measured bending angle of a knee joint in a joyful walking action (i.e. the person walks joyfully). FIG. 8 shows a result of a spectral analysis achieved on the data of the joyful walking motion of FIG. 7B. In this connection, a result of a spectral analysis conducted on the measured values of the ordinary walking action of FIG. 7A is presented in the graph of FIG. 5. Consequently, a difference between the spectral analyses denotes the component associated with the feature “joyfully” as shown in FIG. 9. Teh graphs of FIGS. 5, 8 and  9  present only the power spectrum components having respectively different phases. As can be seen from the spectral graph of FIG. 9, the primary spectrum representing the expression “joyfully” as a feature element can be expressed with a function of at most degree 5 or 6. With the characteristic “joyfully” taking into consideration, the knee bending angle is represented by a function Θ m (t) as follows.  
                       θ   m          (   t   )       =                  (       D   m     +     d   m       )     +       ∑     n   =   1            [         (       A   mn     +     a   mn       )     ·   sin          {       n   ·   t     +                                            (       Ψ   mn     +     ψ   mn       )     -     φ     mn   ·   n         }     ]                 (   2   )                       
 
     [0102] where,  
     [0103] d m : Direct-current compound of “joyfully” 
     [0104] a mn : Component of “joyfully” in each frequency component  
     [0105] ψmn: Phase component of “joyfully” 
     [0106] When the operator selects “joyfully” as the feature component by means of the selector  21  (FIG. 6), values of d m , a m , and ψ mn  are read from the feature component storage  20  to be substituted in the function (1), thereby attaining a function (2) to be fed to the computing unit  12 . As a result, the system displays on the screen an animation picture of a joyfully walking person.  
     [0107] In accordance with the second embodiment, only the feature selection is added to the operations of the first embodiment so that the multiple-articulated object can conduct an action qualified by the selected feature.  
     [0108] Since the operation of the motion representation apparatus of the second embodiment shown in FIG. 6 can be understood from the descriptions of the first embodiment of FIG. 1 and the second embodiment above, a redundant explanation thereof will be here omitted.  
     [0109]FIG. 10 shows the configuration of a computer graphic apparatus in a third embodiment according to the present invention. When compared with the second embodiment of FIG. 6, this constitution additionally includes a feature controller  22  and a feature magnitude designating unit  23 . In this embodiment, the joint angle computing unit  12  achieves computations based on the following function (3).  
                       θ   m          (   T   )       =                  (       A   m     +       α   m     ·     d   m         )     +       β   m     ·       ∑     n   =   1            [       (       A   mn     +       α   m     ·     a   mn         )     ·                                      sin        {       n   ·   t     +     (       Ψ   mn     +       α   m     ·     ψ   mn         )     -     φ     mn   ·   n         }       ]                 (   3   )                       
 
     [0110] where, α m  and β m  denote a magnitude of a feature component and a magnitude or value of an amplitude, respectively. In this system, the feature magnitude α m  is changed to specify motions with expressions in a range including, for example, “walk ordinarily”, “walk joyfully”, and “walk quite joyfully”. Alternatively, as a feature “sadly” opposite to “joyfully”, there may be specified motions with expressions such as “quite sadly”, “ordinarily”, and “quite joyfully”. In this operation, when the display item of the vertical bar  40  is horizontally shifted in the icon  34  denoting the feature magnitude of FIG. 2, the feature magnitude designating unit  23  specifies the desired magnitude of the feature component. The amplitude value β m  is related to a stride length, which takes the larger value when the indication of the vertical bar  40  approches to the right end of the icon  32  of FIG. 2. Namely, the multiple-jointed object walks with a larger stride in the obtained animation picture.  
     [0111] According to the third embodiment, only by adding the magnitude specifying operation to the operations of the second embodiment, the degree of a feature component and a stride length can be altered; moreover, the feature magnitude and the stride length can be changed in a continuous manner.  
     [0112]FIG. 11 shows the construction of a computer graphic apparatus in a fourth embodiment according to the present invention. As compared with the third embodiment in which only one kind of the feature of an action can be specified, the user can specify a plurality of features of an action in this embodiment. For this purpose, there are adopted a feature controller  24  and a feature magnitude designating unit  25 . According to the fourth embodiment, a walking action may be accomplished with specifications of, for example, “joyfully” and “slowly”. In this case, the computations are carried out with the following function (4). It is assumed in this function that the value of each parameter can be changed.  
                       Θ   m          (   t   )       =                  A   mo     +       ∑   i              α     m                 i       ·     d     m                 i                  )         +       β   m     ·       ∑     n   =   1            [       (       A   mn     +       ∑   i            α     m                 i       ·     a     m                 i             )     ·                                      sin        {       n   ·   t     +     (       Ψ   mn     +       ∑   i            α     m                 i       ·     ψ   mni           )     -     φ     mn   ·   n         }       ]                 (   4   )                       
 
     [0113] Since a plurality of characteristic components can be specified for an action, the objective action is conducted with a wider variety of emotional expressions.  
     [0114] As above, according to the fourth embodiment, the values of parameters of the function (4) can be altered in a successive manner; moreover, the expressions of the action conducted by an image of a person can be changed depending on the modified parameters in a realtime manner and through an interactive operation.  
     [0115] In the description of the fourth embodiment, the apparatus controls the functions modified for a motion of a computer graphic image. However, when a computation result attained from the joint angle computation unit  12  is employed as an operational instruction to control an operation of a multiple-articulated robot having a real size of the pertinent object, there can be implemented a robot controller operating independently of the size of the robot. Furthermore, since the actions to be sent as instructions to the robot can be qualified with the features, the robot can perform various kinds of actions with desired functions.  
     [0116] According to the embodiments 1 to 4 described above, since the contour of a multiple-jointed object and/or a robot at an intermediate point of a motion can be computed by use of a function independent of the size thereof, multiple-hinged objects and/or robots of various sizes can be actuated in a realtime manner. In addition, the action can be qualified by a feature of the action; moreover, the degree of the feature can be varied. Consequently, the motion can be accomplished with quite a large number of functions.  
     [0117] Although a human model has been described in conjunction with the embodiments 1 to 4 above, an effect similar to those of the embodiments can naturally be developed also for an object such as a measuring worm.  
     [0118] Referring next to FIG. 12, a description will be given of a computer graphic apparatus in a fifth embodiment according to the present invention. The constitution of FIG. 12 includes a data base  51  for storing therein functions of time for expressions of motions, a joint angle computing unit  52 , a rendering unit  53 , a display  54 , and a controller  55 . Let us assume here that in order to actuate a person image, a bending angle of each joint is varied with respect to time based on the following function.  
                 θ   m          (   t   )       =       A   0     +       ∑     n   =   1     l          {       A   n     ·     sin        (     nt   +     ψ   n       )         }                 (   5   )                       
 
     [0119] This function expressing a joint bending angle with respect to time is obtained from a Fourier series expansion of a period function in which a letter  m  denotes a joint number. To represent a motion of the entire human body, there are required a function θ m (t) for each of the joints. In the function,  i  stands for the maximum degree of the series expansion, A 0  designates a mean value of the bending angle, A n  indicates a spectral intensity of an n-th order higher harmonic, and Ψ n  designates a phase of the n-th order higher harmonic.  
     [0120] The data base  51  is loaded, for each kind of motion, with coefficients A 0 , A 1 , . . . , A i , Ψ 1 , Ψ 2 , . . . , and Ψ i  for the function θ m (t) of time associated with each joint of a person image. The joint angle computing unit  52  computes, based on the coefficients A 0 , A 1 , . . . , A i , Ψ 1 , Ψ 2 , . . . , and Ψ 1  of an objective motion, a bending angle of each human articulation at a point of time. The rendering unit  53  receives the computated results from the joint angle computing unit  52  to further compute based thereon a position and a posture of the person image in a three-dimensional manner so as to project the attained image data onto a two-dimensional area. The display  54  presents the resultant picture on its screen. The controller  55  selects functions of time from the data base  51 , modifies the coefficients A 0 , A 1 , . . . , A i , Ψ 1 , Ψ 2 , . . . , and Ψ 1  of each selected function of time θ m (t), and controls a variable  t  of time.  
     [0121] As a result, according to the fifth embodiment, a desired action can be selected, expressions of operations other than those stored in the data base  51  can be developed, and the action speed can be controlled depending on the variable  t  of time.  
     [0122]FIG. 13 shows the configuration of a computer graphic apparatus in a sixth embodiment according to the present invention. This system includes a data base  51  for storing therein functions of time for expressions of motions, a time function selector  61 , storages  62 ,  63 , and  64  for temporarily storing therein the selected functions, a mean value computing unit  65  for computing a mean value of functions of time, a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . After the function selector  61  choses motions, the components of the functions of time representing the motions are stored in the temporary storage  62  thereof for each motion. Let us assume here that the number of the selected motions is  j  and the amplitudes and phase differences of the respective functions are as follows. Namely, A 10 , A 11 , . . . A 1i , Ψ 11 , Ψ 12 , . . . , Ψ 1i  for motion  1 ; A 20 , A 21 , . . . A 2i , Ψ 21 , Ψ 22 , . . . , Ψ 2i  for motion  2 ; and A j0 , A j1 , . . . A ji , Ψ j1 , Ψ j2 , . . . , Ψ j1  for motion j. The amplitudes and phases of the selected functions of time are processed by the function mean value computing unit  65 , which achieves computations thereon to attain a mean value of each frequency component as follows.  
               A   n   *     =         ∑     k   =   0     j          A   kn       j             (   6   )                 Ψ   n   *     =         ∑     k   =   1     j          Ψ   kn       j             (   7   )                       
 
     [0123] Using the following function (8), the joint angle computing unit  52  computes, for each angle of the person image, a bending angle at a point of time based on the amplitude and the phase resultant from the functions (6) and (7), respectively.  
                 θ   m   *          (   t   )       =       A   0   *     +       ∑     n   =   1     i          {       A   n   *     ·     sin        (     nt   +     Ψ   n   *       )         }                 (   8   )                       
 
     [0124] The rendering unit  53  processes the data resultant from the computing unit  52  to obtain information in a three-dimensional representation of a position and a posture of the person image so as to project the information onto a two-dimensional space. Based on the projected result, the display  54  presents a person image on a screen thereof.  
     [0125] In short, the system is capable of creating a motion other than those loaded in the function data base  51  as follows. Namely, the data base  51  is accessed to obtain therefrom a plurality of functions of time for expressions of motions such that the functions are subjected to the computations above, thereby producing a desired motion.  
     [0126]FIG. 14 shows the constitution of a computer graphic apparatus in a seventh embodiment according to the present invention. This configuration includes a data base  51  for storing therein functions of time for expressions of motions, a function selector  61 , temporary storages  62  to  64  for temporarily storing the respective components of the selected functions of time, a unit  71  for computing a weighted mean value of the functions of time, a joint angle computing unit  52 , a rendering unit  53 , a display  54 , and a controller  55 . For the functions of time representing the motions thus selected by the function selector  61 , the components thereof are stored in the temporary storage  62  associated therewith for each motion. It is assumed here that the number of the selected motions is  j  and the amplitudes and phase differences of the respective functions are as follows. That is, A 10 , A 11 , . . . A 1i , Ψ 11 , Ψ 12 , . . . , Ψ 1i  for motion  1 ; A 20 , A 21 , . . . A 2i , Ψ 21 , Ψ 22 , . . . , Ψ 2i  for motion  2 ; and A j0 , A j1 , . . . A j1 , Ψ j1 , Ψ j2 , . . . , Ψ ji  for motion  j . The amplitudes and phases of the selected functions of time are delivered to the function mean value computing unit  65 , which achieves based thereon computations to attain a mean value of each frequency component as follows.  
               A   n   **     =         ∑     k   =   0     j          (       A   kn     ·     α   k       )       j             (   9   )                 Ψ   n   **     =         ∑     k   =   1     j          (       Ψ   kn     ·     α   k       )       j             (   10   )                       
 
     [0127] where, α k  denotes a weight of a function of time.  
     [0128] According to the following function (11), the joint angle computing unit  52  computes for each angle of the person a bending angle at a point of time by use of the amplitude and the phase attained from the functions (9) and (10), respectively.  
                 θ   m   **          (   t   )       =       A   0   **     +       ∑     n   =   1     i          {       A   n   **     ·     sin        (     nt   +     Ψ   n   **       )         }                 (   11   )                       
 
     [0129] Thereafter, the rendering unit  53  processes the data resultant from the computing unit  52  to attain information of a position and a posture of the person image in a three-dimensional manner so as to project the information onto a two-dimensional area. Based on the projected result, the display  54  presents a human image on a screen thereof.  
     [0130] As a result of the processing above, the apparatus is capable of generating a motion other than those loaded in the function data base  51  as follows. The data base  51  is accessed to obtain therefrom a plurality of functions of time for expressions of motions such that based on the functions, the computations above are executed with the weights for the selected motions taken into consideration, thereby producing a desired motion.  
     [0131]FIG. 15 shows the configuration of a computer graphic apparatus in an eighth embodiment according to the present invention. This system includes a data base  51  for storing therein functions of time for expressions of motions, a transit point specifying unit  81 , a moving direction controller  82 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 16 shows a relationship between a transit point and a moving direction of a multiple-jointed object  1 . It is assumed in this diagram that a plane defined by the x and y axes is a ground surface where a human as the multiple-articulated object  1  stands. In this graphic image, the person stands on the ground in a vertical direction i.e. along the z-axis. First, the transit point specifying unit  81  designates a transit point  401  of the person in FIG. 16. The moving direction controller  82  then rotates the object  1  about the y axis so that the front side thereof faces to the transit point  401 . For the displacement of the object  1 , the user selects expressions of the motion during the movement from the data base  51 . Thereafter, the joint angle computing unit  52  is operated to actuate joints of the multiple-jointed object  1 . Since the object  1  is facing to the passage point  401  as a result of the operation conducted by the moving direction controller  82 , the object  1  is moved or displaced toward the transit point  401 . The rendering unit  53  processes the data from the computing unit  52  to generate information of a position and a posture of the object  1  in a three-dimensional manner so as to project the information onto a two-dimensional space. Depending on the projected result, the display  54  presents a picture of the multiple-articulated object  1  on a screen thereof.  
     [0132] With the provision above, the system can control the moving direction of a human whose motions are represented by the functions of time.  
     [0133]FIG. 17 shows the constitution of a computer graphic apparatus in a ninth embodiment according to the present invention. This system comprises a data base  51  for storing therein functions of time for expressions of motions, a transit point specifying unit  81 , a moving direction controller  82 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 18 shows an example of a display screen presenting a relationship between a transit point and a moving direction of a multiple-jointed object  1 . Let us assume in this diagram that a plane defined by the x and y axes is a ground surface where the multiple-articulated object  1 , namely, the person stands. In this graphic image, the object  1  takes a stand posture on the ground in a vertical direction i.e. along the z-axis of the coordinate system. First, the transit point specifying unit  81  specifies transit points  601  to  605  of the object  1  on the plane. The specified transit points are connected to each other with a curve such as a free curve so as to create a moving route designated by a position  601  and a curve  606 . The moving direction controller  82  rotates the object  1  about the y axis so that the front side thereof is oriented to a direction of a tangent of the curve  606  at a position of the object  1  moving on the curve  606 . For the displacement of the object  1 , the user selects expressions of the motion during the movement from the data base  51 . Based on the selected data, the joint angle computing unit  52  actuates joints of the multiple-jointed object  1 . Since the object  1  is directed along the tangent direction at the current passage point on the curve  606  as a result of the operation conducted by the moving direction controller  82 , the object  1  is displaced along the generated curve  606 . The rendering unit  53  processes data received from the computing unit  52  to produce information of a position and a posture of the object  1  in a three-dimensional manner so as to project the information onto a two-dimensional space. Based on the projected result, the display  54  presents an image of the object  1  on a screen thereof.  
     [0134] Through the operation above, the apparatus can arbitrarily move an image of the human on the plane by controlling the functions of time representing expressions of motions of the human.  
     [0135] There may also utilize still another control method of controlling the motions of a multiple-hinged object as follows. Namely, the user supplies the system, from input means (not shown), with a position  601  denoting a starting point of a movement of the multiple-articulated object  1 , a tangent direction (vector information)  607  designating at least one of a speed of the object  1  at the starting point of the movement and a direction of the movement of the object  1  thereat, a position  603  indicating an ending point of the movement of the object  1 , and vector information (not shown) denoting at least one of a speed of the object  1  at the ending point of the movement and a direction of the movement of the object  1  thereat. The inputted information items are memorized and displayed on the screen. Based on these data items, the system accomplishes computations to obtain information denoting a route of the movement of the object  1  from the initial position  601  to the terminal position  603 . Thereafter, the moving passage portions are similarly computed between the positions  603  and  604 ,  604  to  605 , and  605  to  606  in a sequential manner.  
     [0136] Moreover, according to a still another motion control method, the system is supplied with a position  601  related to a position of the multiple-jointed object  1  in motion and a tangent direction  607  indicating a direction of the movement of the object  1 . The inputted information items are then stored in a memory and are displayed on a screen. Depending on these data items, the apparatus conducts computations to attain information designating a path of the movement of the multiple-articulated object  1 . In this connection, the information item indicating the direction of the moving object  1  is either one of information denoting a direction thereof with respect to a desired point in the pertinent coordinate system or to another multiple-hinged object, information designating a direction thereof with respect to a desired line in the pertinent coordinate system, information indicating a direction thereof with respect to a desired plane in the pertinent coordinate system, and information denoting a direction thereof with respect to the current direction of the movement of the object  1 .  
     [0137]FIG. 19 shows the construction of a computer graphic apparatus in a tenth embodiment according to the present invention. This system comprises a data base  51  for storing therein functions of time for expressions of motions, a transit point specifying unit  81 , a moving route generator  91 , a moving direction controller  82 , a controller  101  for adjusting a stride width in a movement along a curve, a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 20 shows a relationship between a stride width of a foot of the object  1  on a center side of the radius of curvature and a stride width of a foot on a side opposite thereto of the radius of curvature when the object  1  as the multiple-jointed object moves along a curve  701 . Let us assume here that a small interval ΔL or  702  at a point P or  709  on the curve  701  where the object  1  is moving has a center of curvature  0  or  703  and a radius of curvature R or  704 , the distance between the foot on the center side of the curvature and the center of curvature  0  is denoted as Ri or by a reference numeral  705 , and the distance between the foot on the side opposite to the center side of the curvature and the center of curvature  0  is denoted as Ro or by a reference numeral  706 . The passage route or curve  701  of the object  1  is then produced by the transit point specifying unit  81  and the moving route generator  91  of FIG. 19. The moving direction controller  82  adjusts the posture of the object  1  such that the front side thereof is oriented along a tangent direction at the point P. The stride controller  101  produces, based on the following functions (12) and (13), a stride length S i  or  707  on the center side of the curvature and a stride length S o  or  708  on a side opposite thereof when the multiple-articulated object  1  moves along the curve. In this connection, a letter S denotes a stride length of the object  1  moving along a straight line.  
               S   i     =         R   i     R     ·   S             (   12   )                 S   o     =         R   o     R     ·   S             (   13   )                       
 
     [0138] For a movement of the object  1 , the user accesses the data base  51  to acquire therefrom functions for desired expressions of motions to be achieved during the movement of the object  1 . Resultantly, the joint angle computing unit  52  actuates joints of the object  1 . The difference between the strides on the inner and outer sides is supervised by the stride controller  101  and then the function correcting unit  102  corrects functions of time representing the motions. Since the image of the object  1  is oriented along a tangent direction at the current point of the passage as a result of the operation achieved by the moving direction controller  82 , the object  1  moves along the curved line  701  without causing any foot slip on the ground. The rendering unit  53  computes, based on the results of the computation of the joint angle computing unit  52 , information of a position and a posture of the object  1  in the three-dimensional manner so as to project the information onto a two-dimensional area. The resultant information is then presented on the display  54 .  
     [0139] With the provisions above, the apparatus can move the human image whose motions are represented by functions of time freely on a plane without causing any foot slip on the ground.  
     [0140]FIG. 21 shows the configuration of a computer graphic apparatus in an 11th emodiment according to the present invention. This system comprises a data base  51  for storing therein functions of time designating expressions of motions, a transit point specifying unit  81 , a moving route generator  91 , a moving direction controller  82 , a unit  201  for detecting a radius of curvature, a moving speed detector  202 , a centrifugal force computing unit  203 , a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . The moving path of a multiple-jointed object e.g. a human is created by the transit point specifying unit  81  and the moving path generator  91 . The moving direction controller  82  adjusts a posture of the person image such that the front side thereof is oriented along a tangent direction of a curve of the moving path. FIG. 22 shows a relationship between forces applied to a person image when the state of the person image is changed from an upright posture with respect to the ground surface to a circular motion. This diagram includes a multiple-hinged object  1 , a center of curvature  0 , a radius of curvature R, a gravity force g, a centrifugal force  ar , and a resultant force F of the centrifugal force  ar  and the gravity force g. In this state, the object  1  falls down outwardly due to the centrifugal force ar. More concretely, for the observer, the object  1  seems to fall down toward the outside. In this situation, the posture of the object  1  is controlled as follows. In the system, a radius of curvature of the curve where the object  1  is just passing is sensed by the detector  201  and the current moving speed is detected by the moving speed detector  202 . Using the radius of curvature and the moving speed, the centrifugal force computing unit  203  achieves computations to attain a centrifugal force  ar  applied to the object  1 . Thereafter, in order arrange the person posture to be parallel to the resultant force F of the centrifugal force  ar  and the gravity g, the function correcting unit  102  corrects the functions of time representing expressions of motions to incline the posture of the object  1  by an angle θ f . Resultantly, an equilibrium state is established between the centrifugal force  ar  and the gravity force g applied to the multiple-articulated object  1 , which hence does not fall down onto the ground. Actions of the related joints are then generated by the joint angle computing unit  52 . The rendering unit  53  computes, depending on the results of the computation of the joint angle computing unit  52 , information of a position and a posture of the object  1  in the three-dimensional manner so as to project the information onto a two-dimensional area. The display  54  then presents the obtained information on its screen.  
     [0141] In short, when moving along a curved line an image of a person whose motions are represented by functions of time, the apparatus takes the influence on the centrifugal force into consideration. Consequently, there can be prevented an unnatural action in which, for example, the person image seems to fall down onto the ground.  
     [0142]FIG. 24 shows the configuration of a computer graphic apparatus in a 12th embodiment according to the present invention. This system comprises a data base  51  for storing therein functions of time representing expressions of motions, a transit point specifying unit  81 , a moving route generator  91 , a moving direction controller  82 , a unit  201  for detecting a radius of curvature, a moving speed detector  202 , a gravity correcting unit  301 , a centrifugal force computing unit  203 , a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . The moving path of a human image is produced by the transit point specifying unit  81  and the moving route generator  91 . The moving direction controller  82  arranges a posture of the person image such that the front side thereof is oriented along a tangent direction of a curve of the moving path. Thereafter, a radius of curvature of the curve where the object  1  is just moving is detected by the detector  201  and the current moving speed is sensed by the moving speed detector  202 . Based on the radius of curvature and the moving speed, the centrifugal force computing unit  203  achieves computations to attain a centrifugal force  ar  applied to the person. In order to establish an equilibrium state between the centrifugal force  ar  and the gravity force g applied to the multiple-articulated object, the posture of the person image is corrected by means of the function correcting unit  102 . The gravity correcting unit  301  is disposed to correct the magnitude of information associated with the gravity force. When information of the gravity is varied, for example, when the gravity applied to the person image is reduced, the posture thereof is greatly inclined; conversely, when the gravity is increased, the person image becomes to be stable, thereby presenting actions of the person image in an arbitrary manner. Thereafter, motions of the related joints are then generated by the joint angle computing unit  52 . The rendering unit  53  computes, based on the results obtained from the joint angle computing unit  52 , information of a position and a posture of the person in the three-dimensional fashion so as to project the information onto a two-dimensional space. The display  54  presents the resultant information on its screen.  
     [0143] In short, when moving along a curve an image of a person whose motions are represented by functions of time, the apparatus can alter the magnitude of the gravity applied to the person image such that the change in the posture of the person image passing along the curved line is represented with an emphasized expression or a moderate expression.  
     [0144]FIG. 25 shows the construction of a computer graphic apparatus in a  13 th embodiment according to the present invention. This constitution includes a data base  51  for storing therein functions of time representing expressions of motions, a position specifying unit  501 , a distance computing unit  502 , a stride information input device  503 , a stride controller  504 , a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 26 shows a display example of a relationship between a moving route and a stride of a person image. This diagram includes a curved line  1001  denoting a moving route of the person image. The position specifying unit  501  is employed to specify two points x 1  and x 2  on the curve  1001  and then the distance computing unit  502  determines a distance L of a portion x 1 x 2  of the curve  1001 . The stride input device  503  is disposed to input therefrom a stride count n required when the person image moves along the curve  1001  from the point x 1  to the point x 2 . The stride controller  504  obtains the stride length S based on the following equation to send the value S to the function correcting unit  102 .  
             S   =     L   n             (   14   )                       
 
     [0145] The function correcting unit  102  corrests functions of time chosen from the data base  51  such that the stride width becomes to be S. Using the corrected functions of time, the joint angle computing unit  52  generates actions of the respective joints. Based on the computation results, the rendering unit  53  attains information of a position and a posture of the person image in the three-dimension fashion. The information is then projected onto a two-dimensional space to be presented on a screen by the display  54 .  
     [0146] In summary, the apparatus can move an image of a person along a preset interval on a curve with a predetermined number of strides.  
     [0147]FIG. 27 shows the configuration of a computer graphic apparatus in a 14th embodiment according to the present invention. This structure comprises a data base  51  for storing therein functions of time representing expressions of motions, a time specifying unit  511 , a time computing unit  512 , a stride count input device  503 , a controller  514  for supervising a period of time required for a stride or step, a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 28 shows a display example of the display  54  representing a relationship of a stride with respect to time (on a horizontal line or an abscissa) in a movement of the person image. The time specifying unit  511  is employed to specify two points of time t 1  and t 2  on the line  1101  designating elapsed time. The time computing unit  512  obtains an interval of time between the points of time t 1  and t 2  on the line  1101 . The stride count input device  503  is disposed to input therefrom a stride count n required when the person moves during the interval of time from t 1  to t 2 . The stride period controller  514  obtains the period of T S  required for a step based on the following equation, thereby sending the attained value T S  to the function correcting unit  102 .  
               T   S     =     T   n             (   15   )                       
 
     [0148] The function correcting unit  102  achieves a correction on functions of time chosen from the data base  51  such that the stride period becomes to be T S . Depending on the corrected functions of time, the joint angle computing unit  52  generates motions of the respective joints. On receiving the computation results, the rendering unit  53  computes based thereon information of a position and a posture of the person in the three-dimensional fashion. The information is then projected onto a two-dimensional space so as to be presented on a screen by the display  54 .  
     [0149] As a result, the system can move an image of a person along a line during a preset interval of time with a predetermined number of steps.  
     [0150]FIG. 29 shows the constition of a computer graphic apparatus in a 15th embodiment according to the present invention. This structure comprises a data base  51  for storing therein functions of time representing expressions of motions, a posture specifying unit  522 , a position specifying unit  521 , a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . The posture specifying unit  522  is adopted to designate a desired posture (in a stationary state) of a person image. The position specifying unit  521  is used to denote a position in the space where the human takes the posture. In this situation, the person image is moving with motions associated with functions of time selected from the data base  51 . When the person image approaches the position specified by the position specifying unit  521 , in order to develop the posture denoted by the posture specifying unit  522 , the function correcting unit  102  corrects the functions of time. At the specified position, the person image takes the denoted posture. Depending on the corrected functions of time, the joint angle computing unit  52  creates motions of the respective joints. Based on the computation results, the rendering unit  53  obtains information of a position and a posture of the person image in the three-dimensional fashion. The information is then mapped onto a two-dimensional area to be presented on a screen by the display  54 .  
     [0151] Resultantly, the constitution can present an image of a multiple-articulated object in a specified posture at a predetermined position while the object is acting based on functions of time.  
     [0152]FIG. 30 shows the construction of a computer graphic apparatus in a 16th embodiment according to the present invention. This structure comprises a data base  51  for storing therein functions of time representing expressions of motions, a posture specifying unit  522 , a time specifying unit  531 , a function correcting unit  102 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . The posture specifying unit  522  is adopted to designate a desired posture (in a stationary state) of a person image. The time specifying unit  531  is used to denote a point of time when the person image takes the posture. The person image is moving with motions presented by functions of time selected from the data base  51 . At a point of time in the vicinity of the point of time specified by the time specifying unit  531 , in order to develop the posture designated by the posture specifying unit  522 , the function correcting unit  102  corrects the functions of time. At the specified point of time, the person image takes the denoted posture. Using the corrected functions of time, the joint angle computing unit  52  creates motions of the respective joints. Based on the computation results, the rendering unit  53  obtains information of a position and a posture of the person image in the three-dimensional fashion, which is then projected onto a two-dimensional area to be displayed on a screen by the display  54 .  
     [0153] With the operations above, the system can present an image of a multiple-articulated object in a specified posture at a predetermined point of time while the object is moving based on functions of time.  
     [0154]FIG. 31 shows the configuration of a computer graphic apparatus in a 17th embodiment according to the present invention. This constitution comprises a data base  51  for storing therein functions of time representing expressions of motions, a function selector (A)  542 , a position and function storage (A)  545 , a position specifying unit (B)  543 , a function selector (B)  544 , a position and function storage (B)  546 , a distance computing unit  547 , a function interpolating unit  548 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 32 show a screen display example of the display  54  useful to explain a method of interpolating expressions of motions of a person image while the person image is moving. The person image moves from a left-hand side to the right-hand side along a straight line  1401 . The position specifying unit (A)  541  is employed to specify a point Xm or  1402  on the straight line  1401  and then the function selector (A)  542  is initiated to select from the data base  51  functions of time related to an expression of the motion at the point  1402 . Let us assume here, that “the person walks in an ordinary manner” has been selected ( 1403  in FIG. 32) and that each frequency component of the function of time has a spectral intensity A mn  and a phase Ψ mn . The position X m  on the straight line  1401  and the spectral intensity A mn  and the phase Ψ mn  of the function of time representing the action are loaded in the position and function storage (A)  545 . Next, the position specifying unit (B)  543  is used to specify a point X m+1  or  1404  and then the function selector (B)  544  selects from the data base  51  functions of time related to an expression of the motion at the point  1404 . It is assumed here that “the person walks cheerfully” has been selected ( 1405  in FIG. 32) and that each frequency component of the function of time has a spectral intensity A m+1n  and a phase Ψ m+1n . The specified position X m+   1  on the straight line  1401  and the spectral intensity A m+1n  and the phase Ψ m+1n  of the function of time representing the motion are stored in the position and function storage (B)  546 . The distance computing unit  547  is adopted to determine a current position of the person image. Let us assume here that the current position is denoted as  x  or by a reference numeral  1406 . The function interpoalting unit  548  processes the functions of time related to the two points and the current position to obtain a spectral intensity A n (x) and a phase Ψ n (x) of the function of time at the present position  x  based on the following equations.  
                 A   n          (   x   )       =       A   mn     +           A     m   +   ln       -     A   mn           X     m   +   1       -     X   m              (     x   -     X   m       )                 (   16   )                   Ψ   n          (   x   )       =       Ψ   mn     +           Ψ     m   +   ln       -     Ψ   mn           X     m   +   1       -     X   m              (     x   -     X   m       )                 (   17   )                       
 
     [0155] Using the spectral intensity A n (x) and a phase Ψ n (x) of the function of time determined from the equations (16) and (17), the joint angle computing unit  52  solves the following function to attain the joint angles.  
                 θ   n          (   t   )       =         A   o          (   x   )       +       ∑     n   =   l     l          {         A   n          (   x   )       ·     sin        (     nt   +       Ψ   n          (   x   )         )         }                 (   18   )                       
 
     [0156] Based on interporated functions of time, the joint angle computing unit  52  creates motions of the respective joints. In this example, there is produced an intermediate action  1403  between the ordinary walk and the cheerful walk. Based on the computation results, the rendering unit  53  generates information of a position and a posture of the person image in the three-dimensional fashion. The information is then projected onto a two-dimensional space, which is then presented on a screen by the display  54 .  
     [0157] As a result, the system can present an image of a multiple-articulated object in which the image moves with an expression developed by an interpolation of motions between two specified positions.  
     [0158]FIG. 33 shows the configuration of a computer graphic apparatus in an 18th embodiment according to the present invention. This structure includes a data base  51  for storing therein functions of time representing expressions of motions, a time specifying unit (A)  551 , a function selector (A)  552 , a time and function storage (A)  555 , a time specifying unit (B)  553 , a time and function selector (B)  554 , a time and function storage (B)  556 , a time computing unit  557 , a function interpolating unit  548 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . FIG. 34 shows a screen display example produced by the display  54  for explaining a method of interpolating expressions of motions of a person image being displaced. In this diagram, a straight line  1501  stands for an axis of time. The time specifying unit (A)  551  is used to specify a point of time T m  or  1502  on the time axis  1501  and then the function selector (A)  552  selects from the data base  51  functions of time associated with an expression of the motion at the point of time  1502 . Let us assume here, that “the person walks in an ordinary manner” has been selected ( 1503  in FIG. 34) and that each frequency component of the function of time has a spectral intensity A mn  and a phase Ψ mn . The point of time T m  specified on the time axis  1501  and the spectral intensity A mn  and the phase Ψ mn  of the function of time representing the action are memorized in the time and function storage (A)  555 . Subsequently, the time specifying unit (B)  553  is adopted to specify a point of time T m+1  or  1504  and then the function selector (B)  554  selects from the data base  51  functions of time related to an expression of the motion at the point of time  1504 . Let us assume here that a motion “the person walks cheerfully” has been chosen ( 1505  in FIG. 34) and that each frequency component of the function of time has a spectral intensity A m+1  a phase Ψ m+1n . The specified position T m+1  on the time axis  1501  and the spectral intensity A m+1n  and the phase Ψ m+1n  of the function of time representing the motion are memorized in the time and function storage (B)  556 . The time computing unit  577  is then activated to determine a current point of time, which is assumed here to be denoted as  t  or by a reference numeral  1506 . The function interpolating unit  558  processes the functions of time associated with the two points and the current point of time to attain a spectral intensity A n (t) and a phase Ψ n (t) of the function of time at the present point of time  t  based on the following equations.  
                 A   n          (   t   )       =       A   mn     +           A     m   +   ln       -     A   mn           T     m   +   l       -     T   m              (     t   -     T   m       )                 (   19   )                   Ψ   n          (   t   )       =       Ψ   mn     +           Ψ     m   +   ln       -     Ψ   mn           T     m   +   l       -     T   m                (     t   -     T   m       )     .                 (   20   )                       
 
     [0159] Using the spectral intensity A n (t) and the phase Ψ n (t) of the function of time determined from the equations (16) and (17), the joint angle computing unit  52  solves the following function to obtain the joint angles.  
                 θ   n          (   t   )       =         A   o          (   t   )       +       ∑     n   =   1     i          {         A   n          (   t   )       ·     sin        (     nt   +       Ψ   n          (   t   )         )         }                 (   21   )                       
 
     [0160] Based on the functions of time determined through the interpolation, the joint angle computing unit  52  produces actions of the respective joints. In this example, there is generated an intermediate action  1503  between the ordinary walk and the cheerful walk. Depending on the computation results, the rendering unit  53  generates information of a position and a posture of the person image in the three-dimensional fashion, which is then projected onto a two-dimensional space to be displayed on a screen by the display  54 .  
     [0161] With the provision above, the apparatus can present an image of a multiple-articulated object in which the image moves with an interpolated expression developed by an interpolation of motions between two specified positions.  
     [0162]FIG. 35 shows the structure of a computer graphic apparatus in a 19th embodiment according to the present invention. This configuration includes a data base  51  for storing therein functions of time representing expressions of motions, an angle specifying unit  561 , a function expression separator  562 , a key-frame motion generator  563 , a motion combining unit  564 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . In some cases, all actions of a person image cannot be represented by use of functions of time stored in the data base  51 . For example, in a case where a motion “a waling person waves his or her hand” is desired to be produced, even when an action “a person walks” is already loaded in the data base  51 , if a motion of “wave a hand” is missing therein, the desired action cannot be obtained. Next, a description will be given of a method of generating a motion, for example, “a walking person waves his or her hand” in the computer graphic apparatus of the  19 th embodiment. First, functions of time representing an action “walk” are selected from the data base  51 . Let us assume here that the person image waves the left hand. In this situation, the joints ranging from the left shoulder joint to the joint of the tip of the hand are required to be separated from those to be represented with the functions of time above. This operation is accomplished by the joint specifying unit  561 . The function expression separator  562  accordingly separates the specified joints from the function expression. Actions of the remaining joints are then generated by the joint angle computing unit  52 . For the spearated joints, motions are produced by the key-frame motion generator  563  creating motions in the key frame method. The motion combining unit  564  combines the motions generated in the key frame method with those prepared depending on the functions of time. Using the combined results, the rendering unit  53  generates information of a position and a posture of the person image in the three-dimensional fashion. The information is then projected onto a two-dimensional space to be presented on a screen by the display  54 .  
     [0163] As a result, the apparatus can present an image of a multiple-articulated object in which the image conducts an action not registered to the data base in advance.  
     [0164]FIG. 36 shows the construction of a computer graphic apparatus in a 20th embodiment according to the present invention. This constitution comprises a data base  51  for storing therein functions of time representing expressions of motions, an angle specifying unit  561 , a function expression separator  562 , a key-frame motion generator  563 , a function transformer  571 , a motion combining unit  564 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . The structure of this embodiment is implemented by additionally disposing a function transformer  571  preceding to the key-frame motion generator  563  of the apparatus of the 19th embodiment shown in FIG. 35. An action generated by the key-frame motion generator  563  is transformed by the function transformer  571  into a function of time such as one represented by the function (5). The motion represented by the transformed result is registered to the data base  51  so as to be used again in another processing later.  
     [0165] In short, according to the 20th embodiment, a motion which has not been registered to the data base is generated in the key frame method and is then transformed into a function of time, which is registered to the data base so as to be employed again in an operation later.  
     [0166]FIG. 37 shows an apparatus for creating a function of time representing an action. The constitution includes a motion measuring unit  581 , a function transformer  582 , and a data base  51  storing functions of time for expressions of motions. FIG. 38 shows an example of a procedure used to create a function of time representing an expression of a motion. First, the motion measuring unit  581  measures an action (angle) of each joint of a person image as a multiple-articulated object  1 . In the example of FIG. 38, a video camera  1802  is adopted to shoot an image of the object  1  in motion such that based on an image  1803  presented on a screen for each frame, the motion of each articulation is measured, which is obtained, for example, as shown in a graph  1804 . The function transformer  582  then accomplishes a Fourier series expansion on the measured data to obtain a function (representing an expression of a motion) similar to the function (5). The resultant function of time is then loaded in the data base  51 .  
     [0167] Through the operation above, the apparatus can process an actual action of a person image to create a function of time representing an expression of the action.  
     [0168]FIG. 39 shows an apparatus for generating a function of time representing an expression of a motion. The configuration includes a motion measuring unit  581 , a measured data correcting unit  591 , a function transformer  582 , and a data base  51  for storing therein functions of time for expressions of motions. This system is materialized by adding the measured data correcting unit  591  to the apparatus of FIG. 37. The measured data has a difference with respect to actual data because of a measuring error and an inappropriate measurement. In order to minimize the discrepancy therebetween, the measured data correcting unit  591  accomplishes a filtering operation and a correction on the measured data.  
     [0169] As a result of these operations, according to the apparatus of FIG. 39, when an actual motion of a person image is measured, any error appearing in the measuring operation can be removed to appropriately create a function of time representing an expression of the motion.  
     [0170]FIG. 40 shows a structure of an alternative apparatus for generating a function of time representing an expression of a motion. The configuration includes a key-frame motion generator  801 , a function transformer  582 , and a data base  51  storing functions of time for expressions of motions. In this system, a motion of a person image is first generated by the motion generator  801  in the key frame method. The generated motion (a change with respect to time in the bending angle of each joint) is transformed into a function of time representing the motion, which is then registered to the data base  51 .  
     [0171] As a result, the apparatus can produce, based on the motion prepared according to the key frame method, a function of time representing an expression of the motion.  
     [0172]FIG. 41 shows the constitution of a still another apparatus for creating a function of time representing an expression of a motion. The configuration includes a data base  51  loaded with functions of time for expressions of motions and a function correcting unit  811 . In this system, the function correcting unit  811  is disposed to modify a function of time representing the motion selected from the data base  51 . The modifying operation here includes a filtering operation of a function representing an action, an interpolation on a motion function, and an operation to obtain a mean value of a plurality of motion functions. The modified functions are stored in the data base  51  so as to be used again in an operation later.  
     [0173] With this provision, the apparatus of FIG. 41 modifies a function of time representing the motion selected from the data base  51  and then stores the modified function in the data base, thereby enabling the resultant function to be employed again later.  
     [0174]FIG. 42 shows a constitution of a computer graphic apparatus in a 21st embodiment according to the present invention. The structure of this embodiment includes a data base  51  storing functions of time for expressions of motions, a function selector  821  for selecting a function of time for a joint constituting a body of a multiple-articulated object, a temporary storages  822  to  824  for temporarily storing therein selected functions of time for the associated joints, a function composing unit  825 , a joint angle computing unit  52 , a rendering unit  53 , and a display  54 . The system of this embodiment is adopted to present an action of a person image as a multiple-hinged object in which the upper-half body of the person image conducts a walking action and the lower-half body thereof achieves a running action. In this example, the system accomplishes operations as follows. The function selector  821  selects from the data base  51  functions of time for the walking action of the upper-half of the body, which are stored in the temporary storage  822 . Subsequently, the function selector  821  similarly selects functions of time for the running action of the lower-half of the body, which are stored in the temporary storage  823 . The function composing unit  825  then combines the functions of the upper-half of the body with those of the lower-half thereof. Based on the resultant function of time, the joint angle computing unit  52  achieves computations to determine actions of the respective joints. The rendering unit  53  processes the resultant data to obtain a position and a posture of the person image in the three-dimensional fashion. Information of the position and the posture is then projected onto a two-dimensional space so as to be displayed on a screen of the display  54 .  
     [0175] Resultantly, according to the 21st embodiment, a function of time can be selected for each articulation of the person image such that actions of the respective joints are combined with each other to achieve a motion of the person image.  
     [0176]FIG. 43 shows a motion editor  305  available as an editing section of the configurations of FIGS.  12  to  42 . The motion editor  305  is adopted to input various parameters and the like of the respective emobiment, for example, from a screen. The configuration of the editor  305  includes a motion expression specifying part  951 , a path and speed specifying part  952 , and a key-frame motion correcting part  953 .  
     [0177] In the screen example of FIG. 44, the motion expression specifying part  951  is used, for example, to select a function representing an expression of a motion in the embodiments above and to adjust weights applied to the selected functions. The motion expression specifying part  951  outputs a function (related to a spectral intensity and a phase angle for a motion of each joint) representing a motion desired by the user.  
     [0178]FIG. 45 shows a screen display example of the path and speed specifying part  952  in which a moving path, a moving speed, and an expression of a motion is specified for an image of a person in the embodiments above. This part  952  produces data indicating a bending angle of each joint of the person image for each frame of the picture.  
     [0179] In a screen example of the key-frame motion correcting part  953  of FIG. 46, a portion of an expression of a motion represented by functions of time is separated from the function expressing so as to be generated according to the key frame method.  
     [0180] In the embodiments described above, description has been given of examples in which a computer graphic system is adopted to represent an image of a person so as to control an action of the person image. However, the results obtained from the joint angle computing unit  12  may be used as instructions for actions, namely, action control signals of a multiple-articulated robot having a real size of the associated object, thereby implementing a robot control system driving the robot.  
     [0181] A description will now be given of such an example. FIG. 47 shows the constitution of an apapratus controlling a robot in a 22nd embodiment according to the present invention. As compared with the control apparatus of FIG. 1, the object contour storage  15  is dispensed with. Namely, the contour storage  15  is replaced with a robot  2000  shown in FIG. 48; moreover, the rendering unit  14  is substituted for actuators A. Each actuator A is disposed to bend an associated joint of the robot  2000  depending on a joint bending angle computed by the joint angle computing unit  12 . In consequence, like in the case of the embodiments, the respective articulations are actuated in a harmonized manner based on a basic function for an action such as “walk” or “run” selected by the basic motion selector  11 . Moreover, as already described in conjunction with the embodiments, when the basic motion function storage  10  is supplied with, in addition to the actions such as “walk” and “run”, feature components, for example, a characteristic element representing an emotional expression of a person, the robot  2000  can be actuated in a motion such as “walk merrily” or “run sadly”.