Abstract:
A method for performing a convergence calculation using a projective transformation between images captured by two cameras to observe a flat part of an object in the images, wherein a computational load is reduced while securing a convergence property of the convergence calculation. Initial values (n 0 (i), d 0 (i)) are set to values satisfying a limiting condition that should be satisfied by the initial values (n 0 (i), d 0 (i)), where the limiting condition is that a plane πa(i) defined by the initial values (n 0 (i), d 0 (i)) of given types of parameters (n(i), d(i)) of a projective transformation matrix in the convergence calculation is inclined with respect to an actual plane including the flat part of the object to be observed.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to a projective transformation convergence calculation method for observing a flat part (planar region) of an object in images captured by two cameras by using a projective transformation between the captured images. 
         [0003]    2. Description of the Related Art 
         [0004]    If there is an object having a flat part in images captured by two imaging cameras and one captured image (reference image) is transformed by an appropriate projective transformation matrix, it is possible to match an image of the flat part of the object in the transformed image with an image of the flat part of the object in the other captured image (standard image). In this case, this type of projective transformation matrix capable of image transformation depends on the spatial posture (for example, a normal line direction) and position (for example, a distance from the optical center of the camera) of the plane including the flat part of the object. In other words, the projective transformation matrix defines the spatial position and posture of the plane including the flat part of the object in the captured images. 
         [0005]    As a technique for observing the flat part of the object in two captured images (measuring the posture or the like of the flat part) using the above, there is conventionally known a technique disclosed in, for example, a document titled “Extraction of Planar Region and Obstacle Detection Using Stereo Images” (Akihito Seki and Masatoshi Okutomi, IPSJ Transactions on Computer Vision and Image Media, Vol. 45, No. SIG13 (CVIM10), December 2004) (Nonpatent Document 1). In this technique, a plurality of appropriate templates are prepared for the projective transformation matrix between the images captured by the two vehicle-mounted cameras in order to extract a road plane on which the vehicle is running and the projective transformation matrix capable of the above image transformation is estimated by the convergence calculation with the templates used as the initial values of the projective transformation matrix. Preparing the plurality of templates as the initial values of the projective transformation matrix is, in other words, preparing a plurality of initial values of parameters representing the plane defined by the projective transformation matrix. 
         [0006]    The technique disclosed in Document 1 is intended to increase the robustness of the estimation of the projective transformation matrix for the plane to be measured by preparing the plurality of initial values of the projective transformation matrix in the convergence calculation or initial values of the plane to be measured and intended to prevent the dispersion of the estimation or to prevent an inappropriate projective transformation matrix from being estimated (a plane different from an actual plane from being estimated). 
         [0007]    The technique disclosed in Nonpatent Document 1, however, is based on a premise that an error in position between the plane defined by the initial values and the actual plane to be measured (an error in distance from the optical center of the camera) is sufficiently small. Therefore, if the error in position between them is relatively large, there is a possibility that the required number of iterative calculations is excessively increased until the projective transformation matrix is well converged or that it is impossible to converge the projective transformation matrix. 
         [0008]    Furthermore, the necessity of performing the convergence calculation for the plurality of initial values causes a problem that a computational load is easily excessive until the projective transformation matrix is conclusively determined. 
       SUMMARY OF THE INVENTION 
       [0009]    The present invention has been provided in view of the above background and therefore it is an object of the present invention to provide a projective transformation convergence calculation method for performing a convergence calculation using a projective transformation between images captured by two cameras in order to observe a flat part of an object in the images, wherein it is possible to reduce the computational load while securing convergence property of the convergence calculation. 
         [0010]    In order to achieve the above object, according to the present invention, there is provided a projective transformation convergence calculation method for performing a convergence calculation of updating values of given types of parameters of a projective transformation matrix between two cameras so as to converge a plane defined by the parameter values to a plane including a flat part of an object in images captured by the two cameras, while comparing a transformed image obtained by transforming an object image in the image captured by one of the two cameras by the projective transformation matrix with an object image in the image captured by the other camera in order to observe the flat part of the object in the images captured by the two cameras, wherein the initial values are set to values satisfying a limiting condition that should be satisfied by the initial values, where the limiting condition is that the plane defined by the initial values of the parameters in the convergence calculation is inclined with respect to an actual plane including the flat part of the object. 
         [0011]    The present inventor found the following as a result of various considerations on the projective transformation convergence calculation method. Although a specific example will be described later, specifically regarding the initial values of the parameters of the projective transformation matrix updated in the convergence calculation, the number of iterative operations in the convergence calculation tends to rapidly increase along with an increase in error in spatial position between a plane defined by the initial values of the parameters (hereinafter, referred to as the initial plane in some cases) and a plane including the actual flat part of the object (hereinafter, referred to as the actual plane in some cases) in the case where the initial plane is in parallel with the actual plane. Moreover, if the error in spatial position between the initial plane and the actual plane increases to some extent, the values of the parameters cannot be converged to certain values. In other words, in the case where the initial values of the parameters of the projective transformation matrix are set to values that allow the initial plane and the actual plane to be in parallel with each other, the computational load in the convergence calculation rapidly increases or the parameter values cannot be converged unless the initial values are set in such a way as to prevent a large error in spatial position between the initial plane and the actual plane. 
         [0012]    On the other hand, in the case where the initial values of the parameters of the projective transformation matrix are set in such a way that the initial plane is inclined to some extent with respect to the actual plane, the number of iterative operations in the convergence calculation does not rapidly increase even in the case of a relatively large error in spatial position between the initial plane and the actual plane, and it is possible to converge the values of the parameters with a relatively small number of iterative operations. 
         [0013]    Therefore, in the present invention, the initial values are set to values satisfying the limiting condition that should be satisfied by the initial values, where the limiting condition is that the plane defined by the initial values of the parameters in the convergence calculation (initial plane) is inclined with respect to the actual plane including the flat part of the object (actual plane). 
         [0014]    Since the initial plane defined by the initial values is inclined with respect to the actual plane by setting the initial values of the parameters as described above, it is possible to reduce the computational load while securing the convergence property of the convergence calculation even if the error in position between these planes is relatively large. 
         [0015]    In the above invention, the plane defined by the initial values of the parameters has a tilt angle with respect to a horizontal plane equal to a given angle, which has been previously determined so as to deviate from the range of values that can be used for a tilt angle with respect to the horizontal plane of the actual plane including the flat part of the object. 
         [0016]    More specifically, generally the values that can be used for the tilt angle of the plane including the flat part of the object to be observed (the tilt angle with respect to the horizontal plane) are within a certain range. Therefore, it is possible to set the initial values reliably satisfying the limiting condition by previously setting the given angle to a value deviating from the range and setting the initial values of the parameters that define the initial plane so that the tilt angle of the initial plane has the given angle. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0017]      FIG. 1  is an appearance perspective view of a bipedal walking robot, which is a legged locomotion robot in one embodiment of the present invention. 
           [0018]      FIG. 2  is a block diagram illustrating a schematic configuration of a control system mounted on the robot shown in  FIG. 1 . 
           [0019]      FIG. 3  is a flowchart illustrating an estimation of a floor shape in the embodiment. 
           [0020]      FIG. 4  is a flowchart illustrating the details of a process in step  5  shown in  FIG. 3 . 
           [0021]      FIG. 5  is a diagram for explaining a setting example of predetermined landing positions of a plurality of steps of robot&#39;s feet and image projection areas. 
           [0022]      FIG. 6(   a ) and  FIG. 6(   b ) are diagrams for explaining a setting example of predetermined landing positions of the plurality of steps of the robot&#39;s feet and the image projection areas. 
           [0023]      FIG. 7  is a diagram for explaining a setting example of predetermined landing positions of a plurality of steps of robot&#39;s feet and image projection areas. 
           [0024]      FIG. 8(   a ) and  FIG. 8(   b ) are diagrams for explaining a setting example of the predetermined landing positions of the plurality of steps of the robot&#39;s feet and the image projection areas. 
           [0025]      FIG. 9  is a diagram illustrating a relationship between the robot&#39;s feet and the image projection area. 
           [0026]      FIG. 10  is a diagram illustrating a state in which the image projection area is projected on a captured image. 
           [0027]      FIG. 11  is a diagram for explaining a setting example of initial values of shape parameters in a convergence calculation for estimating the shape parameters of an actual floor partial area. 
           [0028]      FIG. 12(   a ) and  FIG. 12(   b ) are graphs for explaining a setting approach for the initial values of the shape parameters in the convergence calculation for estimating the shape parameters of the actual floor partial area. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0029]    One embodiment of the present invention will be described with reference to  FIG. 1  to  FIG. 11 . This embodiment assumes that a projective transformation convergence calculation according to the present invention is applied to a process of estimating the shape of a floor where a legged locomotion robot is moved. Therefore, the floor corresponds to an object in the present invention. 
         [0030]    First, a schematic configuration of a legged locomotion robot in this embodiment will be described with reference to  FIG. 1  and  FIG. 2 .  FIG. 1  shows an appearance perspective view of a bipedal walking robot as a legged locomotion robot in this embodiment and  FIG. 2  shows a block diagram illustrating an outline of a control system of the robot. 
         [0031]    The bipedal walking robot  1  is a humanoid robot, having a body (base)  3 , a pair of left and right legs  5 L and  5 R, a pair of left and right arms  7 L and  7 R, and a head  9 . In the description of this embodiment, references “L” and “R” are used as references representing the left element and the right element of the robot  1  facing front, respectively. Note that, however, the references “L” and “R” are omitted if there is no particular need to distinguish between left and right. 
         [0032]    The legs  5 L and  5 R have the same structure, extending from a leftward portion of a lower end of the body  3  and a rightward portion thereof via hip joints  11 , respectively. Additionally, each leg  5  has a knee joint  13  in the middle portion in its longer direction. Furthermore, a foot  17  as a landing portion is attached to the end (lower end) of each leg  5  via an ankle joint  15 . In this embodiment, the hip joint  11 , the knee joint  13 , and the ankle joint  15  of each leg  5  have three degrees of freedom, one degree of freedom, and two degrees of freedom, respectively. Therefore, the foot  17  has six degrees of freedom with respect to the upper body  3 . 
         [0033]    The left and right arms  7 L and  7 R have the same structure, extending from the left side of the upper portion of the body  3  and the right side thereof via shoulder joints  19 , respectively. In addition, each arm  7  has an elbow joint  21  in its middle portion in the longer direction. Furthermore, a hand (palm and fingers)  25  is attached to the end of each arm  7  via a wrist joint  23 . In this embodiment, the shoulder joint  19 , the elbow joint  21 , and the wrist joint  23  have three degrees of freedom, one degree of freedom, and two degrees of freedom, respectively. Therefore, the hand  25  has six degrees of freedom for the upper body  3 . 
         [0034]    The head  9  is mounted on the top end of the upper body  3 . The head  9  includes a pair of left and right cameras  27 L and  27 R. These cameras  27 L and  27 R are each formed of, for example, a CCD camera. The cameras  27 L and  27 R are used to capture the surroundings of the front of the head  9  (generally ahead of the robot  1 ). Then, the cameras  27 L and  27 R are fixed to the head  9  so that their optical axes point substantially in the same direction toward the front of the head  9 . Furthermore, the head  9  includes a joint mechanism (not shown) that allows the head  9  to move with respect to the upper body  3 . The joint mechanism causes the head  9  to be movable with respect to the upper body  3  in such a way as to be integral with the cameras  27 L and  27 R. For example, the head  9  is capable of rotating motion around the trunk axis (generally a vertical axis) of the upper body  3  and of rotating motion around an axis in the right/left direction. Alternatively, the head  9  may be fixed to the upper body  3 . 
         [0035]    In addition, a box  29  is attached to the rear of the upper body  3  and the box  29  contains a controller  33  described later. 
         [0036]    The robot  1  includes a control system shown in  FIG. 2  to control its motions (walking and the like). The control system has a joint actuator  31  which drives the above joints of the legs  5 , the arms  7  and the head  9 , the controller  33  which controls the motions (movements) of the robot  1  via the joint actuator  31 , and a sensor  35  which detects various internal state quantities of the robot  1 . Although only one joint actuator  31  is shown in  FIG. 2 , the joint actuator  31  is provided for each of the above joints of the legs  5 , the arms  7 , and the head  9 . In addition, the joint actuator  31  is formed of an electric motor or the like. 
         [0037]    Although the details of the sensor  35  is not shown, the sensor  35  includes an acceleration sensor which detects 3-axis acceleration of a predetermined portion, for example, the upper body  3  of the robot  1 , a rate sensor which detects a 3-axis angular velocity of the upper body  3 , a tiltmeter which detects a tilt angle (the tilt angle with respect to the vertical direction) of the upper body  3 , and a force sensor (6-axis force sensor) which detects a floor reaction force (a 3-axis translational force and a moment around three axes) that acts on each foot  17  of the robot  1  from the floor side. 
         [0038]    The controller  33  is an electronic circuit unit including a CPU, a RAM, a ROM, an interface circuit, a drive circuit for the joint actuator  31 . The controller  33  accepts inputs of images captured by the cameras  27 L and  27 R and inputs of detection signals of the sensor  35 . Furthermore, the controller  33  accepts an input of a command that defines the moving speed and moving direction (traveling direction) of the robot  1  (the command is hereinafter referred to as the speed and direction command) transmitted from a remote control device (not shown) of the robot  1 . Moreover, the controller  33  generates a desired gait of the robot  1  by performing required arithmetic processing using the above input data. Furthermore, the controller  33  determines desired motions of the joints provided on the robot  1  such as the hip joints  11 , the knee joints  13 , and the ankle joints  15  of each leg  5  according to the generated desired gait. Thereafter, the controller  33  moves the robot  1  by controlling the joint actuators  31  according to the desired motions. 
         [0039]    Moreover, in this embodiment, the controller  33  sequentially recognizes the self-position of the robot  1  in the global coordinate system (world coordinate system), which is fixed with respect to the floor where the robot  1  is to be moved, on the basis of detection signals from the sensor  35  (for example, the acceleration sensor and the rate sensor). Alternatively, the self-position of the robot  1  may be recognized by using an appropriate external sensor. For example, it is also possible to recognize the self-position of the robot  1  on the basis of landmarks in the images captured by the cameras  27 L and  27 R with the landmarks previously provided on the floor where the robot  1  is to be moved. 
         [0040]    Moreover, in this embodiment, the controller  33  performs a process of estimating the shape of the floor where the robot  1  is to be moved by using the images captured by the cameras  27 L and  27 R. 
         [0041]    Although the speed and direction command of the robot  1  is entered from the remote control device to the controller  33  in this embodiment, alternatively it is possible to previously teach the controller  33  the speed and direction command or a movement plan of the robot  1  for specifying the speed and direction command. 
         [0042]    Among control processes performed by the controller  33  of the robot  1 , the control processes related to the present invention will be described in detail below with reference to  FIG. 3  to  FIG. 11 . The controller  33  sequentially estimates the shapes of the floor ahead in the traveling direction of the robot  1  on the basis of the images captured by the cameras  27 L and  27 R while performing a control process of moving the robot  1  according to the speed and direction command (a motion control process of the robot  1 ). 
         [0043]    Referring to  FIG. 3 , there is shown a flowchart showing an estimation process of the floor shape (hereinafter, referred to as the floor shape estimation). The floor shape estimation is sequentially performed at predetermined arithmetic processing intervals in parallel with the motion control of the robot  1 . 
         [0044]    The terms used in the following description will be supplementarily described here. Each arithmetic processing period in the floor shape estimation is referred to as the frame, the current arithmetic processing period is referred to as the current frame, and an arithmetic processing period immediately preceding the current frame is referred to as the previous frame. In addition, a floor that the controller  33  assumes at present for the motion control of the robot  1  (the floor whose area includes the current position of the robot  1 ) is referred to as the present assumed floor, the real floor is referred to as the actual floor, and the floor estimated in the floor shape estimation is referred to as the estimated floor. 
         [0045]    The present assumed floor is defined by information on the floor shape necessary for the motion control of the robot  1 . The information on the floor shape includes the position of an inclined plane or horizontal plane in the global coordinate system, the gradient of the inclined plane, or the like and the information is entered into the controller  33  before the start of the motion of the robot  1 . Basically, the controller  33  controls the motions of the robot  1  on the assumption that the floor where the robot  1  is to be moved matches the present assumed floor. In this process, the present assumed floor ahead in the traveling direction of the robot  1  is updated according to the estimated floor in the case where a difference between the present assumed floor and the estimated floor (for example, a difference in a tilt angle between these floors) is large to some extent in this embodiment. It is assumed that the present assumed floor in the vicinity of the ground contact point of the robot  1  is in a horizontal plane and approximately matches the actual floor when the robot  1  starts the motion. 
         [0046]    Among the feet  17 ,  17  of the robot  1 , the foot  17  lifted in the air (to leave the floor) during the movement of the robot  1  is referred to as the free leg&#39;s foot  17  and the other foot  17  (the foot  17  of the leg  5  that supports the empty weight of the robot  1  on the floor) is referred to as the supporting leg&#39;s foot  17 . In the case of continuously moving the robot  1 , the free leg&#39;s foot  17  and the supporting leg&#39;s foot  17  are alternately switched to each other. 
         [0047]    Moreover, it is assumed that the image captured by one of the cameras  27 L and  27 R is considered as a standard image in a stereo image and the image captured by the other camera is considered as a reference image in the stereo image in this embodiment. In the following description, the image captured by the camera  27 L may be considered as a standard image and the camera  27 L may be referred to as the standard camera  27 L for convenience in some cases. 
         [0048]    On the premise of the above, the floor shape estimation will be described below. Referring to  FIG. 3 , the controller  33  first obtains the left and right captured images (the standard image and the reference image) from the cameras  27 L and  27 R in step  1  in the floor shape estimation. 
         [0049]    Next in step  2 , the controller  33  determines future predetermined landing positions of a plurality of steps from the first to Nth steps on the basis of the entered speed and direction command, the shape of the present assumed floor, and a gait parameter used by the controller  33  to generate the desired gait of the robot  1 . The predetermined landing position of the first step is the predetermined landing position of the current free leg&#39;s foot  17  of the robot  1 . 
         [0050]    In this embodiment, the predetermined landing positions determined in step  2  are, for example, those of the first to seventh steps (N=7). The predetermined landing positions are in the global coordinate system. In addition, the predetermined landing positions mean the positions of the representative points of the foot  17 . 
         [0051]    In this embodiment, the gait parameter that the controller  33  uses to generate the desired gait of the robot  1  includes gait parameters that define the predetermined landing position of the free leg&#39;s foot  17  in the first-step gait (hereinafter, referred to as the current gait in some cases) (the predetermined landing position of the current free leg&#39;s foot  17 ), the predetermined landing position of the free leg&#39;s foot  17  in the second-step gait as the next gait (hereinafter, referred to as the next gait in some cases) (the predetermined landing position of the free leg&#39;s foot  17  subsequent to the current free leg&#39;s foot  17 ), and the landing position of the foot  17  for each step in a normal gait which is a virtual cyclic gait (a cyclic gait whose one cycle corresponds to two steps) which follows the current gait. The predetermined landing positions of the free leg&#39;s foot  17  of the current gait and the next gait and the landing position of the free leg&#39;s foot  17  in the normal gait defined by the above gait parameters are in the coordinate system fixed with respect to the robot  1  (for example, a coordinate system with the origin at a given representative point within the ground surface of the current supporting leg&#39;s foot  17  of the robot  1 ). 
         [0052]    The predetermined landing positions of the free leg&#39;s foot  17  of the current gait and the next gait defined by the gait parameters are determined according to a predetermined given rule on the basis of the speed and direction command entered into the controller  33  and the shape information on the present assumed floor (the present assumed floor in the vicinity of the current ground contact point of the robot  1 ). In the present embodiment, the predetermined landing positions of the free leg&#39;s foot  17  in the current gait and the next gait are set so that the free leg&#39;s foot  17  comes in contact with the present assumed floor in the predetermined landing positions. Furthermore, regarding the normal gait which is the virtual cyclic gait that follows the current gait, the landing position of the supporting leg&#39;s foot  17  and the landing position of the free leg&#39;s foot  17  in the first-step gait of each cycle (cycle of two steps) are determined in such a way that a relative positional relationship between them coincides with the relative positional relationship between the landing position of the supporting leg&#39;s foot  17  in the next gait (namely the predetermined landing position of the free leg&#39;s foot  17  in the current gait) and the predetermined landing position of the free leg&#39;s foot  17  in the next gait. Furthermore, the landing position of the supporting leg&#39;s foot  17  in the second-step gait of each cycle in the normal gait (namely the predetermined landing position of the free leg&#39;s foot  17  in the first-step gait of the normal gait) and the landing position of the free leg&#39;s foot  17  in the second-step gait of each cycle are determined in such a way that a relative positional relationship between them coincides with the relative positional relationship between the landing position of the supporting leg&#39;s foot  17  in the current gait and the predetermined landing position of the free leg&#39;s foot  17  in the current gait. 
         [0053]    The normal gait is a virtual gait that is generated to determine the condition for generating the current gait (a favorable condition for stabilizing the motion of the robot  1 ), but is not a gait used to control the actual motion control of the robot  1 . In addition, one cycle of the normal gait does not need to be composed of two steps, but may be composed of three or more steps. Since the normal gait is described in detail, for example, in PCT International Publication No. WO 02/40224 of the present applicat or the like, its detailed description will be omitted in this specification. 
         [0054]    Then, in step  2 , the controller  33  determines the predetermined landing positions of the steps up to the third step on the basis of the above-mentioned gait parameters used by the controller  33  to generate the desired gait (the current gait) for the robot  1 . More specifically, the predetermined landing position of the first step (the predetermined landing position of the free leg&#39;s foot  17  in the current gait) and the predetermined landing position of the second step (the predetermined landing position of the free leg&#39;s foot  17  in the next gait) defined by the gait parameters are transformed to the global coordinate system and are then directly determined as the predetermined landing positions of the feet  17  of the steps up to the second step. In addition, the landing position of the second step of the first cycle in the normal gait defined by the above gait parameter is transformed to the global coordinate system and is then determined as the predetermined landing position of the third step. 
         [0055]    Furthermore, the controller  33  determines the predetermined landing positions of the fourth to Nth steps (N=7 in this embodiment) according to a given rule on the basis of the entered speed and direction command and the shape information on the present assumed floor. The rule is the same as the rule for determining the predetermined landing positions of the first and second steps. In determining the predetermined landing positions, it is assumed that the current speed and direction command and the present assumed floor are held constant without change. 
         [0056]    Setting examples of the predetermined landing positions of the first to Nth steps in the process of step  2  will be described below with reference to  FIG. 5 ,  FIG. 6(   a ),  FIG. 6(   b ),  FIG. 7 ,  FIG. 8(   a ), and  FIG. 8(   b ).  FIG. 5  is an explanatory diagram illustrating the robot  1  and the actual floor in the sagittal plane (side view) of the robot  1 .  FIG. 6(   a ) and  FIG. 6(   b ) are diagrams of the actual floor and the like shown in  FIG. 5  viewed from the above in the vertical direction.  FIG. 7  is an explanatory diagram illustrating the robot  1  and the actual floor in the sagittal plane (side view) of the robot  1 .  FIG. 8(   a ) and  FIG. 8(   b ) are diagrams of the actual floor and the like shown in  FIG. 7 , viewed from the above in the vertical direction. In this instance,  FIG. 5  shows an example where the actual floor with which the robot  1  is in contact at the present moment is in a horizontal plane and where the actual floor on the distant side from a position P 1 , which is located ahead in the traveling direction of the robot  1 , changes from the horizontal plane to an inclined plane (a rising slope in  FIG. 5) . In addition,  FIG. 7  shows an example where the actual floor with which the robot  1  is in contact at the present moment is in an inclined plane (a rising slope in  FIG. 7 ) and where the actual floor on the distant side from a position P 2 , which is located ahead in the traveling direction of the robot  1 , changes from the inclined plane to a horizontal plane. The “inclined plane” means a floor inclined with respect to the horizontal plane. Moreover, in the description of this embodiment, the both ends of the inclined plane on the traveling path of the robot  1  are referred to as the inclined plane edges in the case where the floor (the actual floor, the present assumed floor, or the estimated floor) in one section on the traveling path of the robot  1  is in an inclined plane having substantially a certain tilt angle (a tilt angle with respect to the horizontal plane) within the section. Among these inclined plane edges, the inclined plane edge closer to the movement starting point of the robot  1  is referred to as the inclined plane start edge and the other inclined plane edge is referred to as the inclined plane end edge. For example, in the actual floor shown in  FIG. 5 , the inclined plane start edge is an intersection between the horizontal plane and the inclined plane in the position P 1  where the actual floor changes from the horizontal plane to the inclined plane. Also, in the actual floor shown in  FIG. 7 , the inclined plane end edge is an intersection between the inclined plane and the horizontal plane in the position P 2  where the actual floor changes from the inclined plane to the horizontal plane. 
         [0057]    It is assumed that the present assumed floor is in the horizontal plane (the horizontal plane continuous with the actual floor (horizontal plane) with which the robot  1  is in contact at the present moment) in the case where the actual floor is as shown in  FIG. 5 . In this situation, for example, as indicated by dashed lines in  FIG. 5  and  FIG. 6(   a ), the predetermined landing positions of the free legs&#39; feet  17 ( 1 ) to  17 (N) (N=7) are set for the first to Nth steps, respectively. In this instance, the step length of each step of the robot  1  is substantially constant. In the description of this embodiment, an index (i) (i=1, 2, . . . , N) denotes the predetermined landing position of the ith step. 
         [0058]    Moreover, it is assumed that the present assumed floor substantially matches the actual floor in the case where the actual floor is as shown in  FIG. 5 . In this situation, for example, as indicated by dashed lines in  FIG. 6(   b ), the predetermined landing positions of the free legs&#39; feet  17 ( 1 ) to  17 (N) (N=7) of the robot  1  are set for the first to Nth steps, respectively, in such a way that any of the free legs&#39; feet  17 ( 1 ) to  17 (N) of the predetermined landing position of the corresponding step is not placed on the inclined plane start edge and is not too close to the inclined plane start edge. 
         [0059]    Furthermore, it is assumed that the present assumed floor is in the inclined plane (the inclined plane continuous with the actual floor (inclined plane) with which the robot  1  is in contact at the present moment) in the case where the actual floor is as shown in  FIG. 7 . In this situation, for example, as indicated by dashed lines in  FIG. 7  and  FIG. 8(   a ), the predetermined landing positions of the free legs&#39; feet  17 ( 1 ) to  17 (N) (N=7) are set for the first to Nth steps, respectively. In this instance, the step length of each step of the robot  1  is substantially constant. The step length, however, is smaller than those in  FIG. 5  and  FIG. 6(   a ) (in the case where the present assumed floor is in the horizontal plane). 
         [0060]    Moreover, it is assumed that the present assumed floor substantially matches the actual floor in the case where the actual floor is as shown in  FIG. 7 . In this situation, for example, as indicated by dashed lines in  FIG. 8(   b ), the predetermined landing positions of the free legs&#39; feet  17 ( 1 ) to  17 (N) (N=7) of the robot  1  are set for the first to Nth steps, respectively, in such a way that any of the free legs&#39; feet  17 ( 1 ) to  17 (N) of the predetermined landing position of the corresponding step is not placed on the inclined plane end edge and is not too close to the inclined plane end edge. 
         [0061]    As described above, the predetermined landing positions of the steps up to the Nth step are set so as to prevent the free leg&#39;s foot  17  from being placed on the inclined plane start edge or inclined plane end edge. 
         [0062]    Additionally, while the predetermined landing position of the third step is determined in association with the gait parameter of the normal gait in this embodiment, the predetermined landing position of the third step may alternatively be determined in the same manner as for determining the predetermined landing positions of the fourth and subsequent steps. Basically, on the assumption that the speed and direction command and the present assumed floor are held constant as they are in the future, the predetermined landing positions of the steps up to the Nth step only have to be set so as to allow the robot  1  to move stably on the present assumed floor while an average moving speed and moving direction of the robot  1  of the steps up to the Nth step coincide with the speed and direction command and further geometric constraints on interference between the legs  5 ,  5  are satisfied. 
         [0063]    Until the current free leg&#39;s foot  17  of the robot  1  lands, the respective predetermined landing positions of the first to Nth steps set in step  2  are the same in all frames as long as the speed and direction command and the present assumed floor are held constant. On the other hand, in the case where the speed and direction command changes or the present assumed floor is updated before the current free leg&#39;s foot  17  lands, the predetermined landing positions set for the first to Nth steps set in step  2  change over time. In addition, after the current free leg&#39;s foot  17  lands, at least the Nth predetermined landing position is set anew. 
         [0064]    After performing the process of step  2  as described above, the controller  33  subsequently performs the process of step  3 . In the process of step  3 , the controller  33  sets image projection areas A(i) (i=1, 2, . . . , N) in the vicinity of the predetermined landing positions of the first to Nth steps (N=7) determined in step  2 . The image projection areas A(i) are projected on each of the images captured by the cameras  27 L and  27 R as described later. Also, the image projection areas A(i) are planar areas set on the present assumed floor. 
         [0065]    Referring to  FIG. 9 , there is shown a diagram illustrating a setting example of the image projection area A(i).  FIG. 9  shows the image projection area A(i) (i=1, 2, . . . , N) viewed in the direction perpendicular to the present assumed floor and the foot  17 (i) placed on the present assumed floor in the predetermined landing position of the ith step. As shown, the image projection area A(i) is set so as to include the ground surface of the foot  17 (i) (the right foot  17 R in the shown example) in the region of the ground surface in a situation where the entire planar surface of the bottom of the foot  17 (i) is placed on the present assumed floor in the corresponding predetermined landing position. 
         [0066]    More specifically, the image projection area A(i) is set, for example, in a rectangular form in this embodiment. In this instance, a length L 1  of the image projection area A(i) in the traveling direction of the robot  1  is slightly larger than a length L 3  of the ground surface of the foot  17 (i) in the traveling direction. Moreover, a length L 2  of the image projection area A(i) in the direction perpendicular to the traveling direction of the robot  1  is set to a length that the ground surfaces of the both feet  17 ,  17  are included in the image projection area A(i) (the length slightly larger than a distance L 4  between the left side edge of the left foot  17 L and the right side edge of the right foot  17 R viewed in the traveling direction of the robot  1 ), on the assumption that landing state of the foot  17 (i) on the present assumed floor is maintained while the other foot  17  (the left foot  17 L indicated by a two-dot chain line in the example shown in  FIG. 9 ) is placed on the present assumed floor in parallel with the foot  17 (i) in the horizontal direction. 
         [0067]    Although the entire bottom surface of the foot  17  is shown as if it were the ground surface (planar surface) for illustration purpose in the example shown in  FIG. 9 , the entire bottom surface of the foot  17  does not need to be the planar surface. For example, the front end and the rear end of the bottom surface of the foot  17  may be upwardly curved while the remaining part other than the front end and the rear end of the bottom surface is planar. Alternatively, a plurality of flush planar surfaces can be formed in the bottom surface of the foot  17 . In these cases, the ground surface of the foot  17  is a part of the bottom surface of the foot  17 . If it is the case, then the image projection area of the ith step may be set so that an area other than the ground surface of the foot  17 (i) of the area generated by projecting the entire foot  17 (i) on the present assumed floor protrudes from the image projection area. 
         [0068]    In addition, the image projection area A(i) does not always have to be set in a rectangular form. Moreover, for example, as shown by A(i)′ indicated by a virtual line in  FIG. 9 , it is also possible to set an area that includes the ground surface of the foot  17 (i) and has dimensions slightly larger than those of the ground surface (an area including only the ground surface of one foot  17 (i)) as the image projection area of the ith step. 
         [0069]    If the present assumed floor is in the horizontal plane, for example, in the situation shown in  FIG. 5  when the image projection area A(i) is set as described above, the image projection areas A( 1 ) to A(N) of the first to Nth steps (N=7) are set as shown in  FIG. 5  and  FIG. 6(   a ). Moreover, if the present assumed floor substantially matches the actual floor in  FIG. 5 , the image projection areas A( 1 ) to A(N) of the first to Nth steps are set as shown in  FIG. 6(   b ). 
         [0070]    On the other hand, if the present assumed floor is in the inclined plane, for example, in the situation shown in  FIG. 7 , the image projection areas A( 1 ) to A(N) of the first to Nth steps (N=7) are set as shown in  FIG. 7  and  FIG. 8(   a ). Moreover, if the present assumed floor substantially matches the actual floor in  FIG. 7 , the image projection areas A( 1 ) to A(N) of the first to Nth steps are set as shown in  FIG. 8(   b ). 
         [0071]    Subsequently, the controller  33  performs the process of step  4 . In the process of step  4 , the controller  33  projects the image projection areas A(i) (i=1, 2, . . . , N) set in step  3  on the images captured by the cameras  27 L and  27 R. In this instance, in order to project each image projection area A(i) on the image captured by each camera  27 , the controller  33  recognizes the position and posture of the camera  27 L or  27 R in the global coordinate system or the position and posture of the image projection area A(i) on the coordinate system fixed with respect to the camera  27 L or  27 R on the basis of the current position of the robot  1  in the global coordinate system and the displacement (detected value or desired value) of each joint of the robot  1 . The “posture” means a spatial orientation. 
         [0072]    The process of step  4  allows each captured image to include a partial image area Ac(i) generated by projecting each image projection area A(i) on the captured image, for example, as illustrated in  FIG. 10 . Note that the image in  FIG. 10  is captured by the right camera  27 R. 
         [0073]    The image in each partial image area Ac(i) is the image of a partial area of the actual floor existing ahead in the traveling direction of the robot  1 . Hereinafter, the partial area of the actual floor, which is captured as an image within each partial image area Ac(i), is referred to as the actual floor partial area Ar(i). The actual floor partial area Ar(i) corresponds to a flat part (the floor in this embodiment) of an object in the present invention. 
         [0074]    Additionally, if the present assumed floor in the image projection area A(i) matches the actual floor, the actual floor partial area Ar(i) corresponding to the image projection area A(i) matches the image projection area A(i). On the other hand, unless the present assumed floor in the image projection area A(i) matches the actual floor, the image in the partial image area Ac(i) corresponding to the image projection area A(i), namely the image of the actual floor partial area Ar(i) is an image of a partial area of the actual floor that is located on a straight line between each point in the image projection area A(i) and the optical center of each camera  27 . For example, if the present assumed floor is in the horizontal plane in the situation shown in  FIG. 5 , actual floor partial areas Ar( 6 ) and Ar( 7 ) corresponding to the image projection areas A( 6 ) and A( 7 ) of the sixth and seventh steps, respectively, are partial areas in the inclined plane of the actual floor as shown in  FIG. 5 . Then, the images of the actual floor partial areas Ar( 6 ) and Ar( 7 ) are the images within the partial image areas Ac( 6 ) and Ac( 7 ), respectively. Similarly, if the present assumed floor is in the inclined plane in the situation shown in  FIG. 7 , actual floor partial areas Ar( 6 ) and Ar( 7 ) corresponding to the image projection areas A( 6 ) and A( 7 ) of the sixth and seventh steps, respectively, are partial areas in the horizontal plane of the actual floor as shown in  FIG. 7 . 
         [0075]    Note that, however, the actual captured image by each camera  27  does not always include the partial image areas Ac(i) corresponding to all the image projection areas A(i) of the first to Nth steps (N=7). 
         [0076]    Subsequently, the controller  33  performs the process of step  5 . The process is to estimate shape parameters representing the shape of an actual floor captured within the partial image area Ac(i) corresponding to each image projection area A(i), namely the actual floor partial area Ar(i). 
         [0077]    The outline of the estimation process will be described below. In this embodiment, assuming that the actual floor partial area Ar(i) corresponding to each image projection area A(i) is an area included in one plane, a pair consisting of a floor posture parameter that represents the spatial (three-dimensional) posture of the actual floor partial area Ar(i) and a floor position parameter that represents the spatial position of the actual floor partial area Ar(i) in the normal line direction of the actual floor partial area Ar(i) is used as the shape parameters of the actual floor partial area Ar(i). 
         [0078]    In general, an arbitrary plane in a space is uniquely identified from its normal line direction and from the position of the plane in its normal line direction. Therefore, in this embodiment, a normal vector (unit vector) n(i) of the plane including the actual floor partial area Ar(i) is used as the floor posture parameter of the actual floor partial area Ar(i). Moreover, a distance d(i) between a representative point of the robot  1  and the plane including the actual floor partial area Ar(i) (hereinafter, referred to as the plane distance d(i)) is used as the floor position parameter in this embodiment. In this instance, for example, the optical center of the standard camera  27 L among the cameras  27 R and  27 L is used as the representative point of the robot  1  regarding the plane distance d(i). 
         [0079]    Therefore, the process of step  5  is to estimate the pair consisting of the normal vector n(i) that includes the actual floor partial area Ar(i) and the plane distance d(i) (n(i), d(i)) as the shape parameters of the actual floor partial area Ar(i) for each actual floor partial area Ar(i). In this embodiment, the normal vector n(i) estimated in step  5  is described as a pair of the coordinate axis components in the three-dimensional coordinate system fixed with respect to, for example, the camera  27 L or  27 R (hereinafter, referred to as the camera coordinate system) in each frame. The camera coordinate system is, for example, a three-dimensional orthogonal coordinate system with the origin as the optical center of the standard camera  27 L. In this instance, the plane distance d(i) is from the origin of the camera coordinate system to the plane including the actual floor partial area Ar(i). 
         [0080]      FIG. 5  and  FIG. 7  each show an example of the shape parameters defined as described above. References n( 2 ) and d( 2 ) in  FIG. 5  denote a normal vector n( 2 ) and a plane distance d( 2 ) as shape parameters of an actual floor partial area Ar( 2 ) corresponding to an image projection area A( 2 ) of the second step (in the shown example, Ar( 2 ) substantially matches A( 2 )), respectively. Similarly, references n( 6 ) and d( 6 ) in  FIG. 5  denote a normal vector n( 6 ) and a plane distance d( 6 ) as shape parameters of an actual floor partial area Ar( 6 ) corresponding to an image projection area A( 6 ) of the sixth step, respectively. Furthermore, references n( 3 ) and d( 3 ) in  FIG. 7  denote a normal vector n( 3 ) and a plane distance d( 3 ) as shape parameters of an actual floor partial area Ar( 3 ) corresponding to an image projection area A( 3 ) of the third step (in the shown example, Ar( 3 ) substantially matches A( 3 )), respectively. Similarly, references n( 6 ) and d( 6 ) in  FIG. 7  denote a normal vector n( 6 ) and a plane distance d( 6 ) as shape parameters of an actual floor partial area Ar( 6 ) corresponding to an image projection area A( 6 ) of the sixth step, respectively. 
         [0081]    Furthermore, in the process of step  5 , the shape parameters (n(i), d(i)) of each actual floor partial area Ar(i) are estimated by a convergence calculation using projective transformation between the images captured by the cameras  27 L and  27 R. In the convergence calculation, a comparison is made between an image, which is generated by transforming an image in the partial image area Ac(i) of the reference image (the image captured by the camera  27 R) using a projective transformation matrix which is determined according to the values of the shape parameters (n(i), d(i)), and the image in the partial image area Ac(i) of the reference image (the image captured by the camera  27 L), and in parallel with the comparison the values of (n(i), d(i)) are gradually updated from certain initial values so that these images substantially match. In other words, (n(i), d(i)) are assumed to be variable parameters, and the values of the variable parameters are updated from the certain initial values so that the plane defined by the variable parameter values is converged to a plane including the actual floor partial area Ar(i) (consequently, the projective transformation matrix determined by the variable parameter values is updated). The final convergence values of the variable parameters by the convergence calculation are obtained as estimated values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). The projective transformation matrix determined by the convergence values of the variable parameters has a meaning of the projective transformation matrix capable of converging the image in the partial image area Ac(i) of the reference image to an image that matches the image in the partial image area Ac(i) of the standard image. 
         [0082]      FIG. 4  is a flowchart showing the details of the process of step  5 . This process is performed for each image projection area A(i) set in step  3 . Hereinafter, in the description of the process of step  5 , a reference character i indicating the order of the step of the predetermined landing position is used as one denoting a single arbitrary value among 1 to N (N=7). Furthermore, the image projection areas A( 1 ) to A(N) set in the previous frame are referred to as previous image projection areas A_p( 1 ) to A_p(N). The suffix “_p” is used to mean that the area relates to the previous frame. 
         [0083]    In the process of step  5 , the controller  33 , first in step  501 , searches N previous image projection areas, A_p( 1 ) to A_p(N), for the previous image projection area A_px (A_px is one of A_p( 1 ) to A_p(N)) that is closest to the image projection area A(i) corresponding to the predetermined landing position of the ith step in the current frame. Note that the term “closest to” means “closest in spatial position.” More specifically, the controller  33  calculates a distance between the center of the image projection area A(i) and the center of each of N previous image projection areas, A_p( 1 ) to A_p(N), on the basis of the positions in the global coordinate system of the centers and then considers one previous image projection area having the minimum distance as the previous image projection area A_px closest to the image projection area A(i). Hereinafter, the previous image projection area A_px is referred to as the close previous image projection area Ap_x. 
         [0084]    Next, in step  502 , the controller  33  determines whether a difference in position between the image projection area A(i) and the close previous image projection area Ap_x is equal to or greater than a given threshold value. The difference in position between the image projection area A(i) and the close previous image projection area Ap_x corresponds to a distance between the center of the image projection area A(i) and the center of the close previous image projection area Ap_x calculated in step  501 . If the determination result is NO, the controller  33  performs the process of steps  503  and  504 . If the determination result is YES, it performs the process of steps  505  to  510 . 
         [0085]    Additionally, the process of steps  501  to  510  is performed to set the initial values of (n(i), d(i)) in the convergence calculation described later. 
         [0086]    The situation where the determination result of step  502  is NO means that the image projection area A(i) overlaps the close previous image projection area A_px and it is predicted that the plane including the actual floor partial area Ar(i) corresponding to the image projection area A(i) coincides or substantially coincides with the plane including the actual floor partial area Ar_px corresponding to the close previous image projection area A_px with high accuracy. On the other hand, the situation where the determination result of step  502  is YES means that the image projection area A(i) does not overlap the close previous image projection area A_px (in other words, it has no relevance to the actual floor partial area Ar_px whose shape parameters have already been estimated in the previous frame) and it is difficult to predict the spatial posture and position of the plane including the actual floor partial area Ar(i) corresponding to the image projection area A(i) with high accuracy at the present moment. The situation where the determination result of step  502  is YES occurs in the case of setting of an image projection area corresponding to the predetermined landing position of a new step in the current frame (the step subsequent to the Nth step in the previous frame) or in the case of updating the present assumed floor ahead in the traveling direction of the robot  1  between the previous frame and the current frame. 
         [0087]    In the process of steps  503  and  504  performed when the determination result of step  502  is NO, initial values (n 0 (i), d 0 (i)) are set to estimate the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i) corresponding to the image projection area A(i) by the convergence calculation. More specifically, in step  503 , the controller  33  transforms the shape parameters (n_px, d_px) estimated in the previous frame (the shape parameters are described in the camera coordinate system in the previous frame) to the camera coordinate system in the current frame in correspondence with the close previous image projection area Ap_x and defines the transformed shape parameters as (n_px′, d_px′). The relationship between the camera coordinate system in the previous frame and the camera coordinate system in the current frame is recognized on the basis of the displacements of the respective joints of the robot  1  in the period between the previous frame and the current frame. 
         [0088]    Next, in step  504 , the controller  33  sets the shape parameters (n_px′, d_px′) obtained as described above as initial values (n 0 (i), d 0 (i)) of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). Setting the initial values (n 0 (i), d 0 (i)) in this manner means that the initial values of the plane including the actual floor partial area Ar(i) are matched with the plane including the actual floor partial area Ar_px corresponding to the close previous image projection area A_px. 
         [0089]    On the other hand, the initial values (n 0 (i), d 0 (i)) of (n(i), d(i)) are set as described below in the process of steps  505  to  510  if the determination result of step  502  is YES. In other words, in step  505 , the controller  33  obtains a plane πx defined by the shape parameters (n_px, d_px) estimated in the previous frame in association with the close previous image projection area Ap_x (obtains the position and posture of the plane πx in the global coordinate system). More specifically, for example, the controller  33  transforms the shape parameters (n_px, d_px) described in the camera coordinate system in the previous frame to the camera coordinate system in the current frame by the same process of step  503 . It then describes the plane πx in the global coordinate system on the basis of the shape parameters (n_px′, d_px′) after the transformation and the relationship between the camera coordinate system and the global coordinate system in the current frame. Alternatively, the controller  33  describes the plane πx in the global coordinate system on the basis of the shape parameters (n_px, d_px) described in the camera coordinate system in the previous frame and the relationship between the camera coordinate system and the global coordinate system in the previous frame. The relationship between the camera coordinate system and the global coordinate system in each frame can be recognized on the basis of the position in the global coordinate system of the robot  1  in each frame and the displacement of each joint of the robot  1  in the frame. 
         [0090]    Next, in step  506 , the controller  33  obtains a straight line L(i), which passes through the center of the image projection area A(i) and the optical center of the standard camera  27 L, in the global coordinate system. In other words, the straight line L(i) is described in the global coordinate system. 
         [0091]    Next, in step  507 , the controller  33  obtains an intersection point P(i) between the plane πx and the straight line L(i). In other words, it obtains the position in the global coordinate system. Alternatively, for example, it is possible to obtain the position of the intersection point P(i) in the camera coordinate system in the current frame and then transform it to the global coordinate system. 
         [0092]    Subsequently, the controller  33  proceeds to step  508  to obtain a plane πa(i), which passes through the intersection point P(i) obtained as described above and is inclined at a given angle θ from the horizontal plane, in the global coordinate system. The predetermined angle θ is, for example, 45° in this embodiment. In this case, there are countless planes, which pass through the intersection point P(i) and are inclined at the given angle θ, and any of the planes can be obtained as pa(i). By way of example, the controller  33  obtains a plane as pa(i) that passes through the intersection point P(i), includes a horizontal straight line parallel to the Y-axis direction of the global coordinate system or to the right/left direction of the robot  1 , and is inclined at the predetermined angle θ from the horizontal plane (a plane forming a rising slope or falling slope, viewed from the robot  1 ) in this embodiment. Note that the point P(i) (existing on the plane πa(i)) through which the plane πa(i) passes does not always have to be set as described above, for example, an arbitrary point in the image projection area A(i) may be set as P(i). Moreover, for example, d_px′ of (n_px′, d_px′), which is obtained by transforming the shape parameters (n_px, d_px) estimated in the previous frame in association with the close previous image projection area Ap_x to the camera coordinate system in the current frame, may be set to the plane distance to the plane πa(i). In this case, there is no need to define the point P(i) through which the plane πa(i) passes. 
         [0093]    Subsequently, the controller  33  proceeds to step S 509  to obtain the shape parameters (πa(i), da(i)) representing the plane πa(i) in the camera coordinate system in the current frame. The shape parameter πa(i) is a normal vector (unit vector) in the camera coordinate system of pa(i), and the shape parameter da(i) is a distance between the optical center of the standard camera  27 L and the plane πa(i) in the current frame. 
         [0094]    Subsequently, the controller  33  proceeds to step  510  to set (na(i), da(i)) as the initial values (n 0 (i), d 0 (i)) of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). 
         [0095]    If the determination result of step  502  is YES, in other words, if it is difficult in that situation to predict the spatial posture and position of the plane including the actual floor partial area Ar(i) corresponding to the image projection area A(i) at present with high accuracy, the initial values (n 0 (i), d 0 (i)) of the shape parameters (n(i), d(i)) are set so that the plane defined by the shape parameters is inclined at a given angle θ to the horizontal plane. 
         [0096]    A setting example of the initial values (n 0 (i), d 0 (i)) in the above will be described with reference to  FIG. 11 .  FIG. 11  shows an explanatory diagram therefor, in which the robot  1  and the actual floor are shown in the sagittal plane (side view). In this instance, it is assumed that the close previous image projection area A_px that is closest to the image projection area A( 6 ) of the sixth step in the current frame is the image projection area A_p( 5 ) of the fifth step in the previous frame. Moreover, in this example, it is assumed that the actual floor in the vicinity of the predetermined landing position of the fifth step in the previous frame substantially coincides with the present assumed floor and consequently that the actual floor partial area Ar_p( 5 ) corresponding to the close previous image projection area A_px (=A_p( 5 )) substantially coincides with the close previous image projection area A_px. 
         [0097]    In this case, the plane πx shown in  FIG. 11  (the plane including the actual floor partial area Ar_p( 5 )) is obtained in step  505  regarding the image projection area A( 6 ) of the sixth step in the current frame. Thereafter, a straight line L( 6 ) shown in  FIG. 11  is determined in step  506 . Furthermore, a point P( 6 ) in  FIG. 11  is determined in step  507 . In this example, the point P( 6 ) coincides with the center of the image projection area A( 6 ) of the sixth step. Then, a plane πa( 6 ) shown in  FIG. 11  is determined in step  508 . The plane πa( 6 ) passes through the point P( 6 ) and is inclined at the given angle θ (for example, 45°) to the horizontal plane. Furthermore, in step  509  the controller  33  obtains the shape parameters πa( 6 ), da( 6 )) representing the plane πa( 6 ), namely a pair consisting of the normal vector πa( 6 ) of the plane πa( 6 ) and the plane distance da( 6 ) from the optical center of the standard camera  27 L to the plane πa( 6 ) in the current frame. Then, the shape parameters (πa( 6 ), da( 6 )) are set as the initial values (n 0 ( 6 ), d 0 ( 6 )) of the shape parameters (n( 6 ), d( 6 )) of the actual floor partial area Ar( 6 ) corresponding to the image projection area A( 6 ) of the sixth step. 
         [0098]    The process of steps  505  to  510 , which is performed when the determination result is YES in step  502 , corresponds to the main technique of the present invention. It will be described later why this process is performed to set the initial values (n 0 (i), d 0 (i)) of the shape parameters (n(i), d(i)) as described above. 
         [0099]    The process of steps  501  to  510  described hereinabove is performed to set the initial values (n 0 (i), d 0 (i)) of the shape parameters (n(i), d(i)). 
         [0100]    After setting the initial values (n 0 (i), d 0 (i)) as described above, the controller  33  proceeds to step  512  to set the initial values (n 0 (i), d 0 (i)) as temporary values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). The process from step  512  corresponds to the convergence calculation for estimating the shape parameters (n(i), d(i)). In this embodiment, the convergence of the Gauss-Newton method is employed as the convergence calculation. 
         [0101]    Subsequently, the controller  33  proceeds to step  513  to initialize an error before update e(k−1). The error before update e(k−1) initialized here is for use in comparison with an error e(k), which is obtained in the first process of step  516  in a loop process described later, and is set to an excessive error that is surely greater than the error e(k). 
         [0102]    Subsequently, the controller  33  performs the loop process for repeating the process of step  514  to step  522  until a required condition is satisfied. In this loop process, the shape parameters (n(i), d(i)) represent variable parameters and their values are updated appropriately. Specifically, the shape parameters (n(i), d(i)) correspond to parameters in the present invention. Hereinafter, the process of steps  514  to  522  in each round of the loop process is referred to as the in-loop calculation; the currently running in-loop calculation is referred to as the current in-loop calculation; and the immediately preceding in-loop calculation is referred to as the previous in-loop calculation. Moreover, in order to distinguish a value obtained by the current in-loop calculation from a value obtained by the previous in-loop calculation, (k) is appended to the former value and (k−1) is appended to the latter value, if necessary. Note that (n(i), d(i)) are referred to as the variable parameters (n(i), d(i)) until the values are determined. 
         [0103]    In step  514 , the controller  33  calculates a projective transformation matrix H(i) between the cameras  27 L and  27 R from the current values of the variable parameters (n(i), d(i)). The projective transformation matrix H(i) is given by equation (1) described below. Note that H(i) is represented in the normalized camera coordinate system. 
         [0000]        H ( i )= R+t·m ( i ) T   (1) 
         [0000]    where m(i)=n(i)/d(i) 
         [0104]    In the above, R is a rotation matrix between the cameras  27 L and  27 R and t is a translation vector (here, a column vector) between the respective optical centers of the cameras  27 L and  27 R. They depend on the internal parameters such as the relative position and posture relationship between the cameras  27 L and  27 R and are previously determined by a calibration. Additionally, m(i) is a vector (here, a column vector) obtained by dividing the normal vector n(i) by a plane distance d(i) as defined by the provision of equation (1). A reference character “T” in equation (1) means a transposition. 
         [0105]    As apparent from equation (1), the projective transformation matrix H(i) is defined by the values of the variable parameters (n(i), d(i)). 
         [0106]    Subsequently, the controller  33  proceeds to step S 515  to transform the projected image of the image projection area A(i) in the reference image (the image captured by the camera  27 R) of the images captured by the cameras  27 L and  27 R, namely the image in the partial image area Ac(i) in the reference image by using the projective transformation matrix H(i) and to obtain the image generated by the transformation as a transformed image. 
         [0107]    Subsequently, the controller  33  proceeds to step  516  to obtain the error e(k) between the projected image of the image projection area A(i) in the standard image (the image captured by the camera  27 L), namely the image in the partial image area Ac(i) in the standard image and the transformed image obtained in step  515 . The error e(k) is an amount that depends on a difference in luminance distribution or a difference in shape between the image in the partial image area Ac(i) in the standard image and the transformed image obtained in step  515 . 
         [0108]    If the plane defined by the current values of the variable parameters ((n(i), d(i)) coincide with the plane including the actual floor partial area Ar(i) with high accuracy, the transformed image obtained in step  515  matches or substantially matches the image in the partial image area Ac(i) in the standard image. In this case, the error e(k) becomes zero or very small. Therefore, the error e(k) is an index representing a degree of difference between the plane defined by the current values of the variable parameters (n(i), d(i)) (in other words, the plane defined by the projective transformation matrix H(i) determined by the current values of the variable parameters (n(i), d(i))) and the plane including the actual floor partial area Ar(i). In other words, the error e(k) is an index representing the error in current values of the variable parameters (n(i), d(i)) to the true values. 
         [0109]    Then, the controller  33  proceeds to step  517  to determine whether the current error e(k) is smaller than the error before update e(k−1), which is the error obtained in step  516  in the previous in-loop calculation. If the current in-loop calculation is the first-round in-loop calculation, the error before update e(k−1) to be compared with the error e(k) is a value initialized in step  513  described above. Therefore, in this case, the determination result of step  517  is always YES. 
         [0110]    Additionally, if the current in-loop calculation is the first-round in-loop calculation, it is also possible to omit the determination of step  517  and to perform the process from step  518 . In that case, the process of step  513  is unnecessary. 
         [0111]    If the determination result of step  517  is YES, the controller  33  proceeds to step  518  to obtain respective correction values Δn(i) and Δd(i) of n(i) and d(i). These correction values Δn(i) and Δd(i) are determined according to the error e(k). 
         [0112]    Subsequently, the controller  33  proceeds to step  519  to determine whether the current error e(k) is smaller than a given amount, in other words, whether the error e(k) is sufficiently small. 
         [0113]    If the determination result is NO, the controller  33  proceeds to step  520  to determine whether the correction values Δn(i) and Δd(i) obtained in step  518  are smaller than given amounts, respectively. 
         [0114]    If the determination result is NO, the controller  33  proceeds to step  521  to set (store) the current values of the variable parameters (n(i), d(i)) as the values of (ns(i), ds(i)) and to assign the current error e(k) to the error before update e(k−1) (set the current error e(k) as the error before update of the next in-loop calculation). 
         [0115]    Furthermore, the controller  33  proceeds to step  522  to update the current values of the variable parameters (n(i), d(i)) using the correction values Δn(i) and Δd(i) obtained in step  518 . Thereby, the current in-loop calculation ends. Then, the controller  33  restarts the in-loop calculation from step  514 . 
         [0116]    On the other hand, if the determination result of step  517  is NO, the determination result of step  519  is YES, or the determination result of step  520  is YES, the in-loop calculation ends (exits from the loop processing of steps  514  to  522 ) and the estimated values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i) are determined. In other words, the controller  33  determines the values of the variable parameters (n(i), d(i)) which define the projective transformation matrix H(i) that allows a transformed image generated by transforming the image of the actual floor partial area Ar(i) in the reference image (the image in the partial image area Ac(i)) using the projective transformation matrix H(i) to coincide with the image of the actual floor partial area Ar(i) in the standard image (the image in the partial image area Ac(i)). 
         [0117]    More specifically, the variable parameters (n(i), d(i)) are substantially converged to certain values in the situation where the determination result of step  520  in the current in-loop calculation is YES, and therefore in this situation, it is thought that the values of (n(i), d(i)) will not change almost at all even if the update of the values of (n(i), d(i)) is further repeated. In this case, the controller  33  proceeds to step  523  to update the current values of the variable parameters (n(i), d(i)) using the correction values Δn(i) and Δd(i) obtained in step  518  of the current in-loop calculation. Thereafter, in step  524 , the controller  33  determines the values of the variable parameters (n(i), d(i)) obtained after the update as the estimated values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). In this case, alternatively it is possible to omit the process of step  523  and to use the values of the variable parameters (n(i), d(i)) used in step  514  of the current in-loop calculation directly as the estimated values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). 
         [0118]    Moreover, in the situation where the determination result of step  519  in the current in-loop calculation is YES, it is thought that the plane defined by the variable parameters (n(i), d(i)) used in step  514  in the current in-loop calculation sufficiently matches the plane including the actual floor partial area Ar(i). If this is the case, the controller  33  proceeds to step  524  to determine the current values of the variable parameters (n(i), d(i)), namely (n(i), d(i)) used in step  514  of the current in-loop calculation as the estimated values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). 
         [0119]    If the determination result of step  517  in the current in-loop calculation is NO, the variable parameters (n(i), d(i)) possibly diverge even if the in-loop calculation is further repeated. Therefore, in this case, the controller  33  proceeds to step  525  to determine the values of (ns(i), ds(i)) set in step  521  in the previous in-loop calculation, namely variable parameters (n(i), d(i)) used in step  514  in the previous in-loop calculation as the estimated values of the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i). 
         [0120]    The above is the details of the process of step  5 . The process of step  5  allows the acquisition of the estimated values of the shape parameters (n(i), d(i)) of the actual floor, which is captured as an image in the partial image area Ac(i) corresponding to each image projection area A(i), namely the actual floor partial area Ar(i). 
         [0121]    Then, the following describes the reason why the plane πa(i) corresponding to the initial values (n 0 (i), d 0 (i)) of (n(i), d(i)) is set as described above in the case where the determination result of step  502  is YES. 
         [0122]    The present inventor performed an experiment as described below. Specifically, the inventor performed the convergence calculation of steps  512  to  524  with the value of d 0 (i) set to a plurality of values for a case where n 0 (i) of the initial values ((n(i), d(i)) is matched with the actual normal vector n(i) of the actual floor partial area Ar(i) for the actual floor partial area Ar(i) (the area in a certain position ahead of the robot  1 ) whose shape parameters (n(i), d(i)) are known (a case where the plane πa(i) is in parallel with the plane of the actual floor partial area Ar(i) of the plane πa(i)) and for a case where n 0 (i) is inclined with respect to the actual normal vector n(i) of the actual floor partial area Ar(i) (a case where the plane πa(i) is inclined with respect to the plane of the actual floor partial area Ar(i)). Then, the inventor measured the number of executions of the in-loop calculation until the completion of the convergence calculation for each case. 
         [0123]    The examples of the measurement result are shown in  FIG. 12(   a ) and  FIG. 12(   b ).  FIG. 12(   a ) is a graph illustrating a relationship between an error in set value of d 0 (i) (an actual error of d(i)) and the number of executions of the in-loop calculation in the case where the plane πa(i) corresponding to the initial values (n 0 (i), d 0 (i)) are in parallel with the plane of the actual floor partial area Ar(i), and  FIG. 12(   b ) is a graph illustrating a relationship between an error in set value of d 0 (i) (an actual error of d(i)) and the number of executions of the in-loop calculation in the case where the plane πa(i) corresponding to the initial values (n 0 (i), d 0 (i)) are inclined 45° to the actual floor partial area Ar(i). In these examples, the actual floor partial area Ar(i) is in the horizontal plane. In either diagram, the abscissa axis represents the error in set value of d 0 (i) and the ordinate axis represents the number of executions of the in-loop calculation. 
         [0124]    As shown in  FIG. 12(   a ), in the case where the plane πa(i) is in parallel with the plane of the actual floor partial area Ar(i), the number of executions of the in-loop calculation rapidly increases with increasing error in set value of d 0 (i). Moreover, if the error in set value of d 0 (i) increases to some extent, the variable parameters (n(i), d(i)) cannot be converged to certain values. On the other hand, as shown in  FIG. 12(   b ), in the case where the plane πa(i) is inclined with respect to the plane of the actual floor partial area Ar(i), the number of executions of the in-loop calculation does not increase so much even if the error in set value of d 0 (i) increases and it is possible to converge the variable parameters (n(i), d(i)) to certain values. From the above result, in order to estimate the actual variable parameters (n(i), d(i)) efficiently with the smallest possible number of executions of the in-loop calculation, it is desirable to incline the plane πa(i) corresponding to the initial values (n 0 (i), d 0 (i)) with respect to the plane of the actual floor partial area Ar(i). In addition, if that is performed, the set value of d 0 (i) does not need to be a value sufficiently close to the actual value of d(i), but can be a relatively rough value. 
         [0125]    On the other hand, in a legged locomotion robot like the robot  1  in this embodiment, generally the tilt angle (the tilt angle with respect to the horizontal plane) of the floor where the robot is to be moved is limited to a tilt angle within a certain allowable range by the mechanical constraints such as the joints of the robot or the constraints of the specifications. In addition, if the operating environment of the robot is limited to a predetermined environment, the range of the tilt angle of the floor in the environment is previously determined. A value that can be employed for the tilt angle of the floor where the robot is to be moved is a value within a certain range and the range is known in general. Moreover, the tilt angle of the floor where the robot is to be moved is generally on the order of 10° at most. 
         [0126]    Therefore, it is practically possible to always incline the plane πa(i) corresponding to the initial values (n 0 (i), d 0 (i)) with respect to the plane of the actual floor partial area Ar(i) by inclining the plane πa(i) at an angle deviating from the range of the values that can be employed for the tilt angle of the floor where the robot is to be moved. For example, it is practically possible to always incline the plane πa(i) with respect to the plane of the actual floor partial area Ar(i) by inclining the plane πa(i) 45° with respect to the horizontal plane. 
         [0127]    From the above considerations, if the determination result of step  502  is YES, it is determined that the initial values (n 0 (i), d 0 (i)) should satisfy a limiting condition that the plane corresponding to the initial values (n 0 (i), d 0 (i)) is inclined with respect to the plane of the actual floor partial area Ar(i) and that the plane πa(i) is inclined at a given angle θ with respect to the horizontal plane so as to satisfy the limiting condition in this embodiment. Then, 45° is employed as an example of the given angle θ. 
         [0128]    Although θ only has to deviate from the range of values allowed for the floor tilt angle as described above, it is more desirable that the absolute value of an angular difference from an arbitrary tilt angle within the range is equal to or higher than a given value. Since θ is 45° in this embodiment, the difference between the tilt angle of the plane πa(i) and the tilt angle of the plane of the actual floor partial area Ar(i) is 35° or greater on the assumption that the tilt angle of the actual floor is 10° at most. Note that, however, the difference between the tilt angle of the plane πa(i) and the tilt angle of the plane of the actual floor partial area Ar(i) does not need to be always 35° or greater, but may be 20° or greater. 
         [0129]    By performing the process of step  5  described hereinabove for each actual floor partial area Ar(i), it is possible to estimate the shape parameters (n(i), d(i)) of each actual floor partial area Ar(i) efficiently in a short period of time. 
         [0130]    Although the initial values (n 0 (i), d 0 (i)) are set (pa(i) is set) by the process of steps  505  to  510  only in the case where the determination result of step  502  is YES in this embodiment, alternatively it is possible to set the initial values (n 0 (i), d 0 (i)) by performing the same process of steps  505  to  510  also in the case where the determination result of step  502  is NO. In other words, the process of steps  502  to  504  may be omitted. 
         [0131]    Returning to the description of  FIG. 3 , the controller  33  subsequently performs the process of step  6 . In step  6 , the controller  33  removes noise components by filtering the estimated values of the shape parameters (n(i), d(i)) of each actual floor partial area Ar(i) (by the first-order lag filter or the like). This prevents the values of the shape parameters (n(i), d(i)) from excessively fluctuating under the influence of noise. In the subsequent description, it is assumed that the shape parameters (n(i), d(i)) for each actual floor partial area Ar(i) represent the estimated values filtered in step  6 . 
         [0132]    Subsequently, the controller  33  performs the process of steps  7  to  16  and then terminates the process of the current frame. In the process of steps  7  to  16 , the controller  33  identifies the inclined plane edge (the inclined plane start edge or the inclined plane end edge) of the actual floor ahead in the traveling direction of the robot  1  and the like. 
         [0133]    More specifically, first in step  7 , the controller  33  divides the actual floor partial areas Ar(i) (i=1, 2, . . . , N) into two groups, the front-side group and the back-side group, so that an evaluation function described below is minimized. The front-side group means a group on the side nearer the robot  1  and the back-side group means a group on the side farther from the robot  1  than the front-side group. In other words, the process of step  7  is to determine a certain step (the Kth step) from the first to Nth steps whose predetermined landing positions are determined in step  2  in such a way that an evaluation function described below is minimized and to divide the actual floor partial areas Ar(i) (i=1, 2, . . . , N) into the front-side group consisting of the actual floor partial areas Ar( 1 ) to Ar(K) corresponding to the predetermined landing positions of the first to Kth steps and the back-side group consisting of the actual floor partial areas Ar(K+1) to Ar(N) corresponding to the predetermined landing positions of the (K+1)th to Nth steps. 
         [0134]    As the evaluation function used for the above division, the evaluation function represented by the following equation (2) is used in this embodiment: 
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         [0135]    In the equation (2), mavf is more accurately a vector obtained by dividing a center vector (a unit vector having an average direction of the normal vectors n( 1 ) to n(K)), which is obtained by spherical linear interpolation (slerp) from the normal vectors n( 1 ) to n(K) of the shape parameters (n(i), d(i)) (i=1 to K) of the actual floor partial areas of the first to Kth steps, by the weighted mean of plane distances d( 1 ) to d(K). Similarly, mavb is a vector obtained by dividing a center vector (a unit vector having an average direction of the normal vectors n(K+1) to n(N)), which is obtained by the spherical linear interpolation (slerp) from the normal vectors n(K+1) to n(N) of the shape parameters (n(i), d(i)) (i=K to N) of the actual floor partial areas of the Kth to Nth steps, by the weighted mean of plane distances d(K+1) to d(N). 
         [0136]    The first term on the right-hand side in the above equation (2) has a meaning of distribution (the degree of variation) of the shape parameters of the Ar( 1 ) to Ar(K) and the second term on the right-hand side has a meaning of distribution (the degree of variation) of the shape parameters of Ar(K+1) to Ar (N) In the process of step  7 , the value of K is determined so that the value of the evaluation function is minimized and the actual floor partial areas Ar(i) (i=1, 2, . . . , N) are divided into two groups, the front-side group and the back-side group as described above. By the division of the actual floor partial areas Ar( 1 ) to Ar(N) into the front-side group and the back-side group, the actual floor partial areas Ar( 1 ) to Ar(N) are divided so that the distribution of the shape parameters in each group is minimized, in other words, so that the highest mutual approximation degree is achieved in the shape parameters in each group. 
         [0137]    When the actual floor partial areas Ar(i) (i=1, 2, N) are divided into the front-side group and the back-side group in this manner, for example, in the situation shown in  FIG. 5 , the value of K becomes “5” for the minimum evaluation function of equation (2) and the actual floor partial areas are thus divided in such a way that the actual floor partial areas Ar( 1 ) to Ar( 5 ) of the first to fifth steps belong to the front-side group and the actual floor partial areas Ar( 6 ) and Ar( 7 ) of the sixth to seventh steps belong to the back-side group. In addition, for example, in the situation shown in  FIG. 7 , there is an inclined plane edge (an inclined plane end edge) within the actual floor partial area Ar( 5 ) corresponding to the image projection area A( 5 ) of the fifth step and therefore the value of K becomes “4” or “5” in the minimum evaluation function of equation (2). In this case, the actual floor partial area Ar( 5 ) of the fifth step may belong to the front-side group in some cases or to the back-side group in other cases. Note, however, that the actual floor partial areas Ar( 1 ) to Ar( 4 ) of the first to fourth steps belong to the front-side group and the actual floor partial areas Ar( 6 ) and Ar( 7 ) of the sixth to seventh steps belong to the back-side group. 
         [0138]    Subsequently, the controller  33  proceeds to step  8  to obtain representative values of the shape parameters of the actual floor partial areas belonging to the respective groups divided as described above and calculates an angular difference between the plane corresponding to the front-side group (hereinafter, referred to as the front-side representative plane) and the plane corresponding to the back-side group (hereinafter, referred to as the back-side representative plane) from the representative values. The front-side representative plane has a meaning of a plane typically approximating the plane including Ar( 1 ) to Ar(K) and the back-side representative plane has a meaning of a plane typically approximating the plane including Ar(K+1) to Ar(N). 
         [0139]    More specifically, in step  8 , a pair (navf, davf) consisting of the center vector navf of the normal vectors n( 1 ) to n(K) and the weighted mean davf of the plane distances d( 1 ) to d(K) in the shape parameters of the actual floor partial areas Ar( 1 ) to Ar(K) belonging to the front-side group is obtained as the representative values of the shape parameters of the front-side group. In this instance, navf/davf is the same as mavf in equation (2). 
         [0140]    Similarly, a pair (navf, davb) consisting of the center vector navb of the normal vectors n(K+1) to n(N) and the weighted mean davb of the plane distances d(K+1) to d(N) in the shape parameters of the actual floor partial areas Ar(K+1) to Ar(N) belonging to the back-side group is obtained as the representative values of the shape parameters of the back-side group. In this instance, navb/davb is the same as mavb in equation (2). 
         [0141]    Then, the controller  33  considers the planes defined by (navf, davf) and (navb, davb) obtained as described above as the front-side representative plane and the back-side representative plane, respectively, and calculates an angular difference between the two planes. 
         [0142]    Additionally, the angular difference obtained in step  8  has a meaning of an index representing the degree of difference between the front-side representative plane and the back-side representative plane. Besides the angular difference between the planes, the index may be a difference between the above-mentioned weighted average efficiencies davf and davb or a distance in the vector space between mavf and mavb described in the above equation (2). 
         [0143]    Subsequently, the controller  33  proceeds to step  9  to determine whether or not the angular difference obtained in step  8  is equal to or greater than a given threshold value. If the determination result is YES, the controller  33  proceeds to step  10  to calculate a temporary inclined plane edge on the assumption that an inclined plane edge exists at the intersection between the front-side representative plane and the back-side representative plane. More specifically, the controller  33  calculates the intersection between the planes from the representative values (navf, davf) of the shape parameters defining the front-side representative plane and the representative values (navb, davb) of the shape parameters defining the back-side representative plane and then considers the intersection as a temporary inclined plane edge. 
         [0144]    The temporary inclined plane edge corresponds to a temporary inclined change portion in the present invention. 
         [0145]    Subsequently, the controller  33  proceeds to step  11  and recalculates the representative values of the shape parameters of the front-side representative plane and the back-side representative plane and the inclined plane edge. More specifically, the controller  33  removes the actual floor partial area whose distance from the temporary inclined plane edge is shorter than a given distance (the actual floor partial area close to the temporary inclined plane edge) from the actual floor partial areas Ar( 1 ) to Ar(K) belonging to the front-side group and considers those other than the removed actual floor partial area as the front-side group anew. Similarly, the controller  33  removes the actual floor partial area whose distance from the temporary inclined plane edge is shorter than a given distance from the actual floor partial areas Ar(K+1) to Ar(N) belonging to the back-side group and considers those other than the removed actual floor partial area as the back-side group anew. This is performed because the actual floor partial area close to the temporary inclined plane edge may possibly extend over two planes different in tilt angle of the actual floor. Then, the controller  33  obtains the representative values (navf, davf) of the shape parameters of the actual floor partial areas belonging to the front-side group determined anew as described above by the same method as in the above step  8  from the shape parameters and then considers them as the representative values of the shape parameters of the front-side representative plane anew. Similarly, the controller  33  obtains the representative values (navb, davb) of the shape parameters of the actual floor partial areas belonging to the back-side group determined anew by the same method as in the above step  8  from the shape parameters and then considers them as the representative values of the shape parameters of the back-side representative plane anew. Thereafter, the controller  33  determines the intersection between the front-side representative plane and the back-side representative plane defined by (navf, davf) and (navb, davb) obtained anew as an inclined plane edge. 
         [0146]    The front-side representative plane and the back-side representative plane obtained as described above each have a meaning of an estimated floor. More specifically, the estimated floor on the front side of the inclined plane edge viewed from the robot  1  is represented by the front-side representative plane and the estimated floor on the back side of the inclined plane edge is represented by the back-side representative plane. In this case, the controller  33  obtains the shape parameters (navf, davf) and (navb, davb) that define the front-side representative plane and the back-side representative plane, respectively, anew after removing the actual floor partial area close to the temporary inclined plane edge in the process of step  8 , which thereby increases the accuracy of the estimated floor on the front side and the back side of the inclined plane edge and which allows the inclined plane edge to be identified with higher accuracy. 
         [0147]    Subsequently, the controller  33  proceeds to step  12  to update inclined plane parameters. The inclined plane parameters represent a macroscopic shape of the estimated floor along the traveling path of the robot  1 . In step  12 , more specifically, the inclined plane parameters representing the end position (the end position ahead in the traveling direction of the robot  1 ) of the estimated floor (the planar floor which is an estimated floor in the vicinity of the current ground contact point of the robot  1 . Hereinafter, the estimated floor is referred to as the current ground contact estimated floor) in the current position of the robot  1  are updated according to the inclined plane edge obtained in step  11 . 
         [0148]    Furthermore, the inclined plane parameters representing the start position and the degree of inclination of the estimated floor (the floor different in the degree of inclination from the current ground contact estimated floor) on the back side of the end position of the current ground contact estimated floor, viewed from the robot  1 , are updated according to the shape parameters (navb, davb) defining the back-side representative plane obtained in step  11 . 
         [0149]    On the other hand, if the determination result of step  9  is NO, the process of step  13  to step  16  is performed. Note that the situation where the determination result of step  9  is NO means that the actual floor partial areas Ar(i) (i=1, 2, . . . , N) exist substantially on the same plane. In the case where the degree of inclination of the actual floor ahead in the traveling direction of the robot  1  gradually changes, however, the determination result of step  9  may be NO in some cases. To cope with such a situation, the controller  33  performs the process described below if the determination result of step  9  is NO. 
         [0150]    Thus, first in step  13 , the controller  33  divides the actual floor partial areas Ar(i) (i=1, 2, . . . , N) into the front-side group and the back-side group at a given position. More specifically, the controller  33  considers the actual floor partial areas Ar( 1 ) to Ar(Kx) of the first step to a predetermined step (the Kxth step such as, for example, the third step) as the front-side group and the actual floor partial areas Ar(Kx+1) to Ar(N) of the (Kx+1)th step to the Nth step as the back-side group, for example. Alternatively, the controller  33  can consider the actual floor partial areas corresponding to the image projection areas A( 1 ) to A(Kx) whose distance from the current position of the robot  1  is equal to or less than a given distance in the traveling direction of the robot  1  as the front-side group and other actual floor partial areas Ar(Kx+1) to Ar(N) as the back-side group, instead of dividing the actual floor partial areas in units of steps. 
         [0151]    Subsequently, the controller  33  proceeds to step  14  to calculate an angular difference between the present assumed floor in the vicinity of the current landing point of the robot  1  used for the current motion control of the robot  1  (hereinafter, the present assumed floor is referred to as the present assumed contact floor) and the plane (the back-side representative plane) corresponding to the back-side group determined in step  13 . More specifically, in step  14 , the controller  33  obtains the plane, which is defined by a pair (navb, davb) consisting of the center vector navb of the normal vectors n(Kx+1) to n(N) in the shape parameters of the actual floor partial areas Ar(Kx+1) to Ar(N) belonging to the back-side group and the weighted mean davb of the plane distances d(Kx+1) to d(N), as the back-side representative plane. Then, the controller  33  calculates the angular difference between the back-side representative plane and the current ground contact assumed floor. 
         [0152]    Additionally, the angular difference obtained in step  14  has a meaning of an index representing the degree of difference between the current ground contact assumed floor and the back-side representative plane. The index can be other than the angular difference similarly to step  8 . 
         [0153]    Moreover, in the case where the shape of the floor in the vicinity of the ground contact point of the robot is estimated based on a detected value of a floor reaction force or the like as suggested by the applicant of the present application in Japanese Patent Application laid-Open No. 2001-322076, it is also possible to use the floor represented by the estimated floor shape as the current ground contact assumed floor in the process of step  14 . 
         [0154]    Subsequently, the controller  33  proceeds to step  15  to determine whether or not the absolute value of the angular difference obtained in step  14  is equal to or greater than a given threshold value similarly to step  9 . 
         [0155]    If the determination result is NO in this step, the back-side representative plane has substantially the same degree of inclination as the current ground contact assumed floor and is considered to be a plane continuous to the current ground contact assumed floor. Then, in this case, the controller  33  terminates the process of the current frame without updating the inclined plane parameters of the estimated floor. 
         [0156]    If the determination result of step  15  is YES, the controller  33  considers the back-side representative plane to be inclined with respect to the current ground contact assumed floor and calculates the inclined plane edge. In this case, the inclined plane edge is, for example, a line perpendicular to the traveling direction of the robot  1 , passing through the midpoint between the position of the heel of the foot  17  in the predetermined landing position of the Kxth step and the position of the heel of the foot  17  in the predetermined landing position of the (Kx+1)th step. 
         [0157]    Subsequently, the controller  33  proceeds to step  12  to update the inclined plane parameters of the estimated floor. More specifically, the controller  33  updates the inclined plane parameters representing the end position of the current ground contact estimated floor (the end position ahead in the traveling direction of the robot  1 ) according to the inclined plane edge obtained in step  16 . In addition, the controller  33  updates the inclined plane parameters, which represent the start position and the degree of inclination of the estimated floor (the floor different in the degree of inclination from the current ground contact estimated floor) on the back side of the end position of the current ground contact estimated floor viewed from the robot  1 , according to the shape parameters (navb, davb) that define the back-side representative plane obtained in step  14 . In this case, the back-side representative plane has a meaning of an estimated floor on the back side of the inclined plane edge, viewed from the robot  1 . 
         [0158]    The above is the details of the floor shape estimation in this embodiment. 
         [0159]    In this embodiment, the present assumed floor used for the motion control of the robot  1  is appropriately updated according to the inclined plane parameters of the estimated floor determined as described above. In this case, the parameters related to the degree of inclination of the current ground contact assumed floor are maintained as they are in order to increase the stability of the motion control of the robot  1 . Regarding the present assumed floor on the back side of the current ground contact assumed floor (on the back side of the end position of the current ground contact estimated floor), the parameters related to the inclined plane edge position and to the degree of inclination of the back-side present assumed floor is updated so as to coincide with the estimated floor if the difference in the degree of inclination between the present assumed floor and the estimated floor is equal to or greater than a given value (for example, 1.5 degrees or more). Note that it is not indispensable to maintain the inclined plane parameters related to the degree of inclination of the current ground contact assumed floor as they are, but alternatively it is possible to update the inclined plane parameters in conformity to the degree of inclination of the current ground contact estimated floor. 
         [0160]    In the robot  1  according to this embodiment described hereinabove, the shape parameters (n(i), d(i)) of the actual floor partial area Ar(i) are estimated by using the images captured by the cameras  27 L and  27 R for each actual floor partial area Ar(i) corresponding to the predetermined landing positions of the plurality of steps of the robot  1 . More specifically, the shape of the actual floor is estimated only within the partial area, necessary for the moving motions of the robot  1 , of the actual floor in the images captured by the cameras  27 L and  27 R. Thus, it becomes possible to perform appropriate movement control of the robot  1  while reducing arithmetic processing for recognizing the shape of the floor using the images captured by the cameras  27 L and  27 R. 
         [0161]    While the above embodiment has been described by taking the bipedal walking robot  1  as an example, naturally the present invention is also applicable to a legged locomotion robot having three or more legs. 
         [0162]    Although the above embodiment has been described with respect to an example in which the object is the floor on which the robot equipped with cameras moves, the present invention is not limited to such applications. For example, the present invention is also applicable to a case where the object is a road on which a vehicle equipped with cameras is running and the terrain shape of the road is estimated.