Patent Publication Number: US-8532824-B2

Title: Control device for robot

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a control device for a robot, such as a legged mobile robot. 
     2. Description of the Related Art 
     As a technique for controlling a legged mobile robot, such as a bipedal walking robot, there has been generally known so-called position control proposed in, for example, Japanese Patent Application Laid-Open No. 5-305579 by the applicant of the present application. According to the disclosed technique, the desired displacement amount of each joint of a robot is sequentially determined on the basis of a desired motion of the robot (e.g., the trajectory of a desired position of the distal end of a leg of the robot), and then the actuator of each joint is drivably controlled so as to cause an actual displacement amount to coincide with the desired displacement amount. 
     According to the position control described above, the actual displacement amount of a joint is maintained to be constant as long as the desired displacement amount of the joint remains constant, thus inconveniently presenting poor flexibility in response to a change in an external force acting on the robot. 
     As a solution to the inconvenient inflexibility, there has been known a technique disclosed in, for example, Japanese Patent Application Laid-Open No. 2007-160446. According to the technique, a spring member is interposed between a joint and an actuator to impart flexibility to the joint, and a desired driving force for each joint to accomplish a desired motion of a robot is determined by inverse dynamics calculation so as to control the actual driving force of each joint to the desired driving force thereof by so-called force control. 
     According to the technique based on the force control, as disclosed in the aforesaid Japanese Patent Application Laid-Open No. 2007-160446, the desired rotational angle, angular velocity and angular acceleration of each joint for accomplishing a desired motion of the robot are determined, and then the inverse dynamics calculation is carried out on the basis of these desired values, thereby determining the feed-forward value of a desired driving torque to be imparted to each joint. Further, according to the technique disclosed in Japanese Patent Application Laid-Open No. 2007-160446, the desired driving torque for each joint is determined by adding a feed-back manipulation torque, which is for converging the rotational angle of each joint of the robot to a desired value, to the feed-forward value. 
     Thus, according to the technique disclosed in Japanese Patent Application Laid-Open No. 2007-160446, the actual motion of each joint of the entire robot will be significantly susceptible to restrictions based on the desired values of the rotational angle, the angular velocity and the angular acceleration of each joint. Hence, as long as the desired values remain constant, the desired driving torque for each joint will be inconveniently adjusted so as to maintain the motion of each joint to be constant even if an external force acting on the robot changes. 
     Therefore, according to the technique disclosed in Japanese Patent Application Laid-Open No. 2007-160446, the actual motion of each joint of the robot will be undesirably least flexible, possibly making it difficult to maintain a stable posture of the robot when, for example, an external force acting on the robot suddenly changes. 
     SUMMARY OF THE INVENTION 
     The present invention has been made with the background described above, and it is an object of the invention to provide a control device capable of controlling a driving force to be imparted to each joint of a robot to a driving force that is proper for causing the robot to perform a desired motion while enhancing the flexibility of the motion of joints of the robot in the case where the motion of the robot is controlled by force control. 
     To this end, a control device for a robot in accordance with the present invention is a control device configured to determine a desired driving force to be imparted to each joint of a robot, which has a plurality of links interconnected through joints and actuators which drive the joints, and to control the operation of each of the actuators on the basis of the determined desired driving force in the case where a motion of the robot is effected while causing at least one or more contact portion of the robot to come in contact with an external world of the robot, the control device including:
         a basic parameter group calculating unit which calculates a basic parameter group constituted of an inertia matrix for converting the second-order differential value of a generalized variable vector of the robot, which includes at least the displacement amount of each joint of the robot as a component, into a generalized force vector, and a gravity-dependent generalized force vector value, which is the value of a generalized force vector produced by the gravitational force acting on each link of the robot, or a basic parameter group constituted of the inertia matrix, the gravity-dependent generalized force vector, and a centrifugal force/Coriolis force-dependent generalized force vector value, which is the value of a generalized force vector generated by a centrifugal force and a Coriolis force acting on each link of the robot, on the basis of generalized variable observation information, which includes at least the observed value of an actual displacement amount of each joint of the robot;   a contact portion Jacobian matrix calculating unit which receives contact state information indicating the contact state of the one or more contact portions and the generalized variable observation information and calculates a contact portion Jacobian matrix, which is a Jacobian matrix expressing a relationship between the motion velocity of a predetermined contact portion representative element set as the element representing the motion of the one or more contact portions being in contact with the external world and a first-order differential value of the generalized variable vector on the basis of at least the received contact state information and generalized variable observation information;   a state amount Jacobian matrix calculating unit which receives the generalized variable observation information and calculates a state amount Jacobian matrix, which is a Jacobian matrix expressing a relationship between a predetermined type of state amount having a value dependent upon the value of the generalized variable vector and the first-order differential value of the generalized variable vector on the basis of at least the received generalized variable observation information;   a desired driving force determining unit which receives the basic parameter group, the contact portion Jacobian matrix and the state amount Jacobian matrix, which have been calculated, a desired value of a contact portion representative element motion acceleration, which is the desired value of the motion acceleration of the contact portion representative element, the generalized variable observation information, and the desired value of the first-order differential value of the predetermined type of state amount, uses the received data to calculate a component value corresponding to the displacement amount of each joint in a desired generalized force vector τcmd, which is a target of a generalized force vector that satisfies the relationship of expression 01 given below, and determines the calculated component value as a desired driving force to be imparted to the joint; and   an actuator control unit which controls the operation of the actuator on the basis of at least the determined desired driving force (a first aspect of the invention).
 
 S′ +( Js*M   −1   *Tc−Js ′)* q′ =( Js*M   −1   *Pc )*(τ cmd−τcmpn )  Expression 01
 
where
   q: generalized variable vector   q′: observed value of first-order differential value of q (=dq/dt)   S: state amount   S′: desired value of first-order differential value of S (=dS/dt)   M: inertia matrix   τcmpn: vector defined by τcmpn≡(N+G)−Pc −1 *Cc or τcmpn≡G−Pc −1 *Cc   G: gravity-dependent generalized force vector value   N: centrifugal force/Coriolis force-dependent generalized force vector value   Pc: matrix defined by Pc≡I−Jc T *(Jc#) T      I: unit matrix   Jc: contact portion Jacobian matrix   Jc#: matrix defined by Jc#≡(M −1 ) T *Jc T *Rc T      Rc: matrix defined by Rc≡((Jc*M −1 *Jc T ) −1 ) T      Cc: vector defined by Cc≡−Jc T *Rc*C   C: desired value of contact portion representative element motion acceleration (scalar or vector)   Tc: matrix defined by Tc≡−Jc T *Rc*Jc′   Jc′: matrix obtained by subjecting Jc to first-order differentiation (=dJc/dt)   Js: state amount Jacobian matrix   Js′: matrix obtained by subjecting Js to first-order differentiation (=dJs/dt)       

     In the present description, “*” denotes multiplication. 
     Supplementarily, in the present invention, the aforesaid robot may be an installed type robot having at least one link thereof fixed relative to a global coordinate system in an operating environment of the robot (fixed to a floor or the like in the operating environment) or a mobile robot whose entire body can be moved in relation to the global coordinate system. The global coordinate system means a coordinate system that can be regarded as an inertial coordinate system or an approximately inertial coordinate system. 
     Further, the aforesaid generalized variable vector is a vector having the displacement amounts (rotational angles or stroke amounts) of all joints of the robot as component values thereof in the case where the aforesaid robot is an installed robot. Meanwhile, in the case where the aforesaid robot is a mobile robot, the generalized variable vector is a vector having the position and posture of an arbitrary single representative link of the robot on the global coordinate system and the displacement amounts of all joints of the robot as component values thereof. 
     The term “position” of a link of the robot means a spatial position of a representative point fixed relative to the link, and the term “posture” of the link means a spatial orientation of the link. 
     Further, the aforesaid generalized force vector is a vector having the components thereof corresponding to the components of a generalized variable vector (a vector constituted of the same number of components as that of the generalized variable vector). To be more specific, the aforesaid generalized force vector is a vector whose components provide a force (a translational force or a moment) that causes temporal changes in the corresponding component values of the generalized variable vector. 
     The aforesaid generalized variable observation information means the information having, as its constituent element, an observed value (specifically, an actual instantaneous value of the component or the observed value of the temporal change rate thereof (first-order differential value)) related to an actual value of the generalized variable vector component (including the displacement amount of a joint). In this case, the observed value may be of course a value directly detected by a sensor. Alternatively, the observed value may be an estimated value that is estimated according to an appropriate model or a natural law from the detection value or values of one or more other physical quantities having correlativity with the observed value. 
     According to the first aspect of the invention, the aforesaid basic parameter group (the inertia matrix M, the gravity-dependent generalized force vector value G and the centrifugal force/Coriolis force-dependent generalized force vector value N, or M and G), the aforesaid contact portion Jacobian matrix Jc, and the aforesaid state amount Jacobian matrix Js are calculated by the basic parameter group calculating unit, the contact portion Jacobian matrix calculating unit, and the state amount Jacobian matrix calculating unit, respectively, as the parameters necessary to calculate the desired generalized force vector τcmd according to the aforesaid expression 01. 
     To supplementarily describe the basic parameter group, the total sum of the centrifugal force/Coriolis force-dependent generalized force vector value N and the gravity-dependent generalized force vector value G (=N+G) becomes nearly G (N+G≈G) in some cases. In such a case, only M and G may be calculated as the basic parameter group. 
     Further, the aforesaid contact portion representative element related to the contact portion Jacobian matrix Jc means an element that represents the motion of the aforesaid one or more contact portions of the robot that will be subjected to restrictions upon contact with an external world (the aforesaid one or more contact portions whose motion or motions (e.g., the positions or posture angles, or the temporal change rates thereof) are restricted according to the motion thereof). Incidentally, the term “posture angle” means a spatial orientation expressed using an angle as the unit thereof. The contact portion representative element may be, for example, an actual contact portion or a particular portion or a point included therein. Alternatively, the contact portion representative element may be a virtual element whose positional relationship or posture relationship with a contact portion has a predetermined relationship. 
     Further, the aforesaid contact state information is the information that indicates the state of the contact between the aforesaid one or more contact portions and an external world (e.g., the information on which contact portion is in contact with an external world or the state of an external force applied due to the contact of each contact portion with an external world). 
     Further, the desired driving force determining unit uses, as input data, the basic parameter group (M, G and N or M and G), the contact portion Jacobin Jc and the state amount Jacobian matrix Js, the contact portion representative element motion acceleration desired value C, the generalized variable observation information (specifically, q′ recognized from the generalized variable observation information), and a desired value S′ of the first-order differential value of the predetermined type of state amount so as to calculate the component value corresponding to the displacement amount of each joint in a desired generalized force vector τcmd, which is the target of a generalized force vector that satisfies the relationship of the aforesaid expression 01, and determines the calculated component value as the desired driving force to be imparted to the joint. 
     Thus, according to the first aspect of the invention, the desired driving force to be imparted to each joint is determined by using the desired value of the contact portion representative element motion acceleration and the desired value of the first-order differential value of the predetermined type of state amount as the targets restricting the motion of the robot. 
     Here, the motion of the contact portion is the motion of a part of the robot, so that the motion of each joint of the robot for achieving the desired value of the contact portion representative element motion acceleration is provided with redundancy. In other words, the degree of freedom of the motion acceleration of the contact portion representative element is smaller than the degree of freedom of the entire robot (the degree of freedom of the generalized variable vector). 
     Further, when causing the robot to accomplish a predetermined motion, it is generally not essential to make the motion itself of every joint coincide with a desired motion, whereas it is essential to control a certain state amount (e.g., the motion state of the overall center-of-gravity point of the robot), the value of which changes depending on the motion of the joint (expressed in the form of a function value of the motion of the joint) to a proper target. In general, the motion of each joint of the robot for achieving a target related to such a state amount is also provided with redundancy. In other words, the degree of freedom of the state amount is smaller than the degree of freedom of the entire robot (the degree of freedom of the generalized variable vector). 
     Thus, the arbitrariness of the motion of a joint of the robot is enhanced by determining a desired driving force to be imparted to each joint according the aforesaid expression 01, using the desired value of the contact portion representative element motion acceleration and the desired value of the first-order differential value of the predetermined type of state amount as the targets restricting the motion of the robot. 
     In addition, defining the motion of the contact portion representative element by using the desired value of the contact portion representative element motion acceleration makes it possible to determine the desired driving force of each joint that allows the desired value of the first-order differential value of the predetermined type of state amount to be achieved according to the aforesaid expression 01 without identifying an external force acting on the robot at the aforesaid contact portion (the external force mutually influencing the driving force to be imparted to each joint of the robot). 
     Thus, the first aspect of the invention makes it possible to control the driving force to be imparted to each joint to a driving force that is proper for causing a robot to accomplish a desired motion while at the same time enhancing the flexibility of the motion of each joint of the robot. 
     Supplementarily, in the first aspect of the invention, if the matrix of the right side (Js*M −1 *Pc) of the aforesaid expression 01 (hereinafter referred to as the matrix As in some cases) is a regular matrix, then the aforesaid desired generalized force vector τmcd may be calculated by carrying out the calculation of the right side of expression 5-22, which will be discussed later, by using a matrix Bs (i.e., the inverse matrix of As) determined according to expression 5-21a, which will be discussed later. If the matrix As is not a regular matrix, then the aforesaid desired generalized force vector τmcd can be calculated by carrying out the calculation of the right side of expression 5-22, which will be discussed later, by using, for example, the matrix Bs determined according to expression 5-21b, which will be discussed later. 
     In the first aspect of the invention, if the degree of freedom of the motion of the contact portion representative element is larger than the actual degree of freedom restricted by contact, then the motion of a joint of the robot tends to be easily subjected to the restrictions based on the desired value of the contact portion representative element motion acceleration. For this reason, to enhance the arbitrariness of the motion of a joint of the robot, the excess of the degree of freedom of the motion of the contact portion representative element over the actual degree of freedom is preferably minimized. Further preferably, the motion of the contact portion representative element does not develop a discontinuous change. 
     Considering the requirements described above, if, for example, the robot is a bipedal mobile robot that travels on a floor surface by carrying out a motion in which the distal portions of two leg links alternately come in contact with the floor surface (i.e., a robot in which the distal portions of two leg links function as the portions contacting with an external world), then the following construction is ideally adopted. 
     The contact portion representative element is an element, the position and posture of which exhibit springy displacements based on a total floor reaction force, which is the resultant force of floor reaction forces acting on the two leg links, and also an element in which the springy displacement amounts of the position and posture of the contact portion representative element are such that the springy displacement amounts of the position and posture of the distal portion of a first leg link caused by a floor reaction force acting on the first leg link out of the two leg links and the springy displacement amounts of the position and posture of the distal portion of a second leg link caused by a floor reaction force acting on the second leg link out of the two leg links have a relationship indicated by expression 02 given below, and the contact portion Jacobian matrix calculating unit calculates the contact portion Jacobian matrix Jc by calculating the right side of expression 03 given below (a second aspect of the invention).
 
 Xcc=r at*A 1 *X 1+(1 −r at )* A 2* X 2  Expression 02
 
 Jc=r at*A 1 *J 1+(1 −r at )* A 2* J 2  Expression 03
         where   Xcc: vector, the components of which are the springy displacement amounts of the position and posture of a contact portion representative element   r at: coefficient set according to contact state information such that r at=0 in a state wherein only the first leg link is in contact with the ground, r at=1 in a state wherein only the second leg link is in contact with the ground, or r at continuously changes in a range defined by 0≦r at≦1 in a state wherein both the first leg link and the second leg link are in contact with the ground   X1: vector, the components of which are the amounts of the springy displacements of the position and posture of the distal portion of the first leg link produced according to a floor reaction force acting on the first leg link   X2: vector, the components of which are the amounts of the springy displacements of the position and posture of the distal portion of the second leg link produced according to a floor reaction force acting on the second leg link   A1: matrix for converting a first floor reaction force vector constituted of a translational force component and a moment component of a floor reaction force acting on the first leg link into a floor reaction force vector acting on the contact portion representative element   A2: matrix for converting a second floor reaction force vector constituted of a translational force component and a moment component of a floor reaction force acting on the second leg link into a floor reaction force vector acting on the contact portion representative element   J1: Jacobian matrix expressing the relationship between a first-order differential value X1′ of X1 (=dX1/dt) and a first-order differential value q′ of a generalized variable vector q by X1′=J1*q′   J2: Jacobian matrix expressing the relationship between a first-order differential value X2′ of X2 (=dX2/dt) and a first-order differential value q′ of a generalized variable vector q by X2′=J2*q′       

     According to the second aspect of the invention, the contact portion representative element is defined as an element, the position and the posture of which develop springy displacements based on a total floor reaction force, which is the resultant force of the floor reaction forces acting on the two individual leg links, and also as an element in which the springy displacement amounts of the position and posture of the contact portion representative element are such that the springy displacement amounts of the position and posture of the distal portion of a first leg link caused by a floor reaction force acting on the first leg link out of the two leg links and the springy displacement amounts of the position and posture of the distal portion of a second leg link caused by a floor reaction force acting on the second leg link out of the two leg links have a relationship indicated by the aforesaid expression 02. 
     According to the contact portion representative element defined as described above, either in the state wherein the distal portion of only one of the two leg links of the robot is in contact with the ground (or a floor) or in the state wherein the distal portions of both leg links are in contact with the ground, the degree of freedom of the motion thereof is only a total of six degrees of freedom composed of three degrees of freedom of a translational motion and three degrees of freedom of a rotational motion. This means that the degree of freedom of a motion of a contact portion representative element is the same degree of freedom as that of the distal portion of a single leg link. 
     Thus, if the motion of a contact portion representative element is restricted by the desired value of the contact portion representative element motion acceleration, then the resulting restricted degree of freedom of the robot will be only six degrees of freedom. This makes it possible to further enhance the arbitrariness of the motion of the robot, while minimizing the excess of the degree of freedom of the motion of the contact portion representative element over an actual degree of freedom. 
     Moreover, making the coefficient r at continuously change in the range of 0≦r at≦1 allows the continuity of the motion of the contact portion representative element to be achieved. This in turn makes it possible to enhance the stability of the motion of the robot while restraining a desired driving force for each joint of the robot from discontinuously changing. 
     Further, in the second aspect of the invention, the contact portion Jacobian matrix Jc related to the contact portion representative element defined as described above is calculated according to the aforesaid expression 03. In this case, the matrixes A1 and A2 and the Jacobian matrixes J1 and J2 need to be calculated. The matrixes A1 and A2 depend upon the contact state of a leg link (i.e., the state of contact with a floor), so that the matrixes A1 and A2 can be determined on the basis of, for example, the aforesaid contact state information. 
     The first-order differential value X1′ of X1 (=dX1/dt) may be considered to agree with the motion velocity (the translational velocity and the rotational velocity) of the distal portion of the first leg link, and the first-order differential value X2′ of X2 (=dX2/dt) may be considered to agree with the motion velocity (the translational velocity and the rotational velocity) of the distal portion of the second leg link. Hence, the Jacobian matrixes J1 and J2 of expression 03 can be calculated on the basis of the aforesaid generalized variable observation information. 
     Thus, the contact portion Jacobian matrix can be calculated by calculating the right side of the aforesaid expression 03. 
     In the first aspect or the second aspect of the invention, preferably, the actuator control unit includes a contact portion external force desired value determining unit which determines a contact portion external force desired value as an estimated value of an external force acting on the contact portion by carrying out processing of inverse dynamics calculation, using the desired driving force and the contact portion motion acceleration desired value, which have been determined, in the case where it is assumed that a driving force to be actually imparted to each joint of the robot coincides with the determined desired driving force and the actual motion acceleration of the contact portion representative element coincides with the desired value of the contact portion representative element motion acceleration, and a driving force correction amount calculating unit which determines a correction amount of the desired driving force according to a feedback control law such that a difference between the contact portion external force desired value and an observed value of an external force actually acting on the contact portion is converged to zero, wherein the operation of the actuator is controlled on the basis of the desired driving force that has been corrected using the correction amount (a third aspect of the invention). 
     According to the third aspect of the invention, the estimated value of an external force acting on the contact portion in the case, where it is assumed that the driving force actually imparted to each joint of the robot is made to coincide with the determined desired driving force and the actual motion acceleration of the contact portion representative element is made to coincide with the desired value of the contact portion representative element motion acceleration, is determined as the contact portion external force desired value. Further, the desired driving force is corrected on the basis of a correction amount determined according to a feedback control law such that the difference between the contact portion external force desired value and the observation value of an external force actually acting on the contact portion is converged to zero. Then, the operation of the actuator is controlled on the basis of the corrected desired driving force. 
     Thus, the third aspect of the invention makes it possible to compensate for an influence of an error or the like of a model on which the aforesaid expression 01 is based, thereby preventing a mismatch between a contact portion external force desired value corresponding to a desired driving force determined according to the aforesaid expression 01 and an external force actually acting on the contact portion. This permits enhanced stability of the motion of the robot. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram illustrating a skeleton framework of a robot to which an embodiment of the present invention is applied; 
         FIG. 2  is a schematic diagram illustrating a skeleton framework of a joint drive system provided in the robot shown in  FIG. 1 ; 
         FIG. 3  is a block diagram illustrating major functions of a control device of the robot shown in  FIG. 1 ; 
         FIG. 4  is a block diagram illustrating the functions of a desired joint driving torque determiner shown in  FIG. 3 ; 
         FIG. 5(   a ) through  FIG. 5(   c ) are diagrams for describing the processing in a first embodiment of a contact portion Jacobian matrix determiner shown in  FIG. 4 ; and 
         FIG. 6  is a diagram for describing the processing in a second embodiment of the contact portion Jacobian matrix determiner shown in  FIG. 4 . 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     First Embodiment 
     The following will describe a first embodiment of the present invention with reference to  FIG. 1  to  FIG. 5 . 
       FIG. 1  schematically illustrates the skeleton framework of a robot illustrated in the present embodiment. This robot  1  is a bipedal mobile robot as a legged mobile robot. The robot  1  is roughly constituted of a body link  2 , a pair of right and left leg links  3 R,  3 L, a pair of right and left arm links  4 R,  4 L, and a head  5 . 
     In the present description, a member on the right side observed facing to the front of the robot  1  or a variable indicating a quantity associated with the member will be suffixed by a character “R,” while a member on the left side observed facing to the front of the robot  1  or a variable indicating a quantity associated with the member will be suffixed by a character “L.” However, the characters “R” and “L” may be omitted unless it is particularly necessary to distinguish between the right side and the left side. 
     The body link  2  is constituted of a lower body  10 , which provides the base body of the robot  1 , and an upper body  11 , which is disposed above the lower body  10 . The upper body  11  is connected to the lower body  10  through a body rotating joint  12 , which has a freedom degree of rotation about a yaw-axis (about a Z-axis). 
     The pair of right and left leg links  3 R and  3 L share the same construction. To be more specific, each leg link  3  is provided with a thigh  14  connected to the lower body  10  through a hip joint  13 , a crus  16  connected to the thigh  14  through a knee joint  15 , and a foot  18  connected to the crus  16  through an ankle joint  17  as a plurality of links constituting the leg link  3 . 
     In this case, the hip joint  13  is constructed of three joints  19 ,  20 , and  21 , which have the freedom degrees of rotation about the yaw-axis, about a pitch-axis (about a Y-axis), and about a roll-axis (an X-axis), respectively. The knee joint  15  is constructed of a joint  22  having a freedom degree of rotation about the pitch-axis. The ankle joint  17  is constructed of two joints  23  and  24  having the freedom degrees of rotation about the pitch-axis and the roll-axis, respectively. 
     Thus, each of the leg links  3  in the present embodiment has a motion freedom of a six-degree-of-freedom relative to the lower body  10 . Incidentally, the rotational axes (the roll-axis, the pitch-axis, and the yaw-axis) of the joints  19  to  24  of each of the leg links  3  in the above description indicate the rotational axes in a state wherein the leg link  3  is stretched in the vertical direction. 
     The pair of right and left arm links  4 R and  4 L share the same construction. To be more specific, each of the arm links  4  is provided with an upper arm  26  connected to the upper body  11  through a shoulder joint  25 , a forearm  28  connected to the upper arm  26  through an elbow joint  27 , and a hand  30  connected to the forearm  28  through a wrist joint  29  as a plurality of links constituting the arm link  4 . 
     In this case, the shoulder joint  25  is constructed of three joints  31 ,  32 , and  33  having the freedom degrees of rotation about the pitch-axis, about the roll-axis, and about the yaw-axis, respectively. The elbow joint  27  is constructed of a joint  34  having a freedom degree of rotation about the pitch-axis. The wrist joint  29  is constructed of three joints  35 ,  36 , and  37  having the freedom degrees of rotation about the yaw-axis, about the pitch-axis, and about the roll-axis, respectively. 
     Thus, each of the arm links  4  in the present embodiment has a motion freedom of a seven-degree-of-freedom relative to the upper body  11 . Incidentally, the rotational axes (the roll-axis, the pitch-axis, and the yaw-axis) of the joints  31  to  37  of each of the arm links  4  in the above description indicate the rotational axes in a state wherein the arm links  4  are stretched in the vertical direction. 
     The head  5  is disposed above the upper body  11  and connected to the upper body  11  through a neck joint  38 . In this case, the neck joint  38  is composed of two joints  39  and  40  having the freedom degree of rotation about the pitch-axis and about the yaw-axis, respectively. Hence, the head  5  has a motion freedom of a two-degree-of-freedom relative to the upper body  11 . Incidentally, the rotational axes (the pitch-axis and the yaw-axis) of the joints  39  and  40  indicate the rotational axes in a state wherein the neck joint  38  is stretched in the vertical direction. 
     The robot  1  constructed as described above implements spatial motions of the leg links  3  by driving the six joints  19  to  24  of each of the leg links  3 , thus enabling the robot  1  to travel on a floor. For example, moving the leg links  3 R and  3 L in modes (gaits) similar to human walking or running motions enables the robot  1  to walk or run. In this case, each of the links of the robot  1  has six-degree-of-freedom for the spatial position and posture (spatial orientation) thereof. 
     Further, the seven joints  31  to  37  of each of the arm links  4  are driven to cause each of the arm links  4  to perform spatial motions, enabling a hand  30  of each of the arm links  4  to carry out work, such as touching an appropriate object. 
     In the present embodiment, since the lower body  10  and the upper body  11 , which constitute the body link  2 , are connected through the joint  12 , the body link  2  can be twisted by driving the joint  12 . 
     Further, the head  5  can be tilted (a rotational motion about the pitch-axis) or panned (a rotational motion about the yaw-axis) by driving the joints  39  and  40  of the neck joint  38 . 
     Supplementarily, the robot  1  in the present embodiment is a humanoid robot having the arm links  4 R,  4 L and the head  5 . Alternatively, however, the robot  1  may be a robot without the arm links  4 R,  4 L or the head  5 . Further alternatively, the joint  12  of the body link  2  may be omitted, thus combining the lower body  10  and the upper body  11  into one body. 
     Although not specifically illustrated in  FIG. 1 , the robot  1  has a joint drive system (a mechanism for driving the joints), including electric motors serving as actuators that output motive power for rotatively driving the aforesaid joints. According to the present embodiment, in order to impart flexibility to each joint, a motive power transmitting system between each joint and an electric motor is provided with a spring member that converts the motive power of the electric motor into elastic force and imparts the elastic force to a driven member of the joint as described below. 
       FIG. 2  schematically illustrates the skeleton framework of the joint drive system of each joint. The joint drive system includes a reduction gear unit  52 , which is constructed of a harmonic drive gear or the like, and a torsion bar  53 , which is a spring member, as the motive power transmitting element between an electric motor  50  and a driven member  51  of the joint (hereinafter referred to as the joint driven member  51 ). An output shaft  50   a  of the electric motor  50  is connected to the joint driven member  51  through the intermediary of the reduction gear unit  52  and the torsion bar  53  in this order. Thus, the joint drive system is constructed such that the motive power (torque) output by the electric motor  50  is transmitted to the joint driven member  51  via the reduction gear unit  52  and the torsion bar  53  in order. In this case, the motive power (torque) output by the electric motor  50  is converted into an elastic force (torsional force) by the torsion bar  53 , and the elastic force is imparted as a drive torque to the joint driven member  51 . 
     Supplementarily, the torsion bar  53  (spring member) here generically represents an elastic member added between the reduction gear unit  52  and the joint driven member  51  or an elastic member originally placed in the motive power transmitting system between the electric motor  50  and the joint driven member  51 . For this reason, if the reduction gear unit  52  originally has an elastic member in an output stage thereof, as in the harmonic drive gear, such as Harmonic Drive (a registered trademark), then the elastic member is also included in the torsion bar  53  (spring member). 
     The joint driven member  51  is one of a pair of members relatively and rotatively connected to each other by the joint having the pair of members. For example, in the joint drive system that drives the joint  22  of the knee joint  15 , one of the links of the thigh  14  and the crus  16  is the joint driven member  51 , while the housings of the electric motor  50  and the reduction gear unit  52  are secured to the other link. 
     As described above, according to the present embodiment, the joint drive system is provided with the torsion bar  53  serving as a spring member, permitting flexible motions of each joint. 
     The present embodiment is provided with a control device  60  composed of an electronic circuit unit, which includes a CPU, a RAM, a ROM and the like, and a variety of sensors to control the motions of the robot  1  having the construction described above. 
     In this case, as illustrated in  FIG. 1 , the sensors include a posture sensor  61  mounted on the lower body  10  to measure the angular velocity, the posture angle (the tilt angle relative to the vertical direction and the azimuth about the yaw-axis) and the like of the lower body  10 , which is the base body of the robot  1 , force sensors  62  ( 62 R and  62 L), each of which is interposed between the ankle joint  24  and the foot  18  of each of the leg links  3  to measure an external force (floor reaction force) applied to each of the feet  18  at the time of coming in contact with the ground, and force sensors  63  ( 63 R and  63 L), each of which is interposed between the wrist joint  29  and the hand  30  of each of the arm links  4  to measure an external force applied to each of the hands  30  from an object with which the hand  30  comes in contact. 
     The aforesaid posture sensor  61  is constituted of, for example, a gyro sensor which detects angular velocities about three axes and an acceleration sensor which detects the accelerations in the directions of three axes. Each of the force sensors  62  and  63  is constructed of, for example, six-axis force sensor which detects the translational forces in the directions of the three axes and moments about the three axes, respectively. The acceleration sensor constituting the posture sensor  61  is used also to measure the moving velocity of the lower body  10 . 
     Further, according to the present embodiment, as illustrated in  FIG. 2 , the joint drive system for each joint is provided with a rotary encoder  64  serving as an angle sensor for detecting the rotational angle of the output shaft  50   a  of the electric motor  50  (the rotational angle of a rotor) and a differential encoder  65  serving as an angle sensor which detects the rotational angle difference between both ends of the torsion bar  53 . 
     The differential encoder  65  in the present embodiment is a sensor which has a pair of discs  65   a  and  65   b  secured to both ends of the torsion bar  53  and outputs a signal based on a relative rotational angle of one of the discs  65   a  and  65   b  in relation to the other as a signal based on a rotational angle difference between both ends of the torsion bar  53 . 
     The rotary encoder  64  and the differential encoder  65  are sensors used to measure mainly the rotational angle of the output shaft  50   a  of the electric motor  50  (or the rotational angle of one end of the torsion bar  53  adjacent to the reduction gear unit  52  (=the rotational angle of the output shaft of the reduction gear unit  52 )) and the temporal change rate (rotational angular velocity) thereof, the rotational angle of the joint driven member  51  and the temporal change rate (rotational angular velocity) thereof, and the rotational angle difference between both ends of the torsion bar  53  (=the torsional angle of the torsion bar  53 ). 
     In place of the differential encoder  65 , an angle sensor, such as a rotary encoder which detects the rotational angle of the joint driven member  51 , may be used. Further, in place of the rotary encoder  64 , an angle sensor, such as a rotary encoder which detects the rotational angle of the output shaft of the reduction gear unit  52  may be used. 
     In the following description, the rotational angle of the joint driven member  51  of each joint of the robot  1  may be referred to as the rotational angle of a joint or a joint angle. 
     The control device  60  receives detection outputs of the aforesaid sensors (or filtering values obtained by passing the detection outputs through appropriate filters). The control device  60  then uses required measurement values (e.g., the measurement value of the posture angle of the lower body  10  and the temporal change rate (angular velocity) thereof, the measurement value of the moving velocity of the lower body  10 , the measurement value of an external force acing on each of the feet  18  or the hands  30 , and the measurement values of the rotational angle of each joint and the temporal change rate (angular velocity) thereof recognized from the received detection outputs so as to determine the desired value of the driving torque to be imparted to the joint driven member  51  of each joint in order to cause the robot  1  to accomplish a desired motion (hereinafter referred to as the desired joint driving torque). The control device  60  also carries out processing for controlling the output torque of the electric motor  50  of each joint drive system on the basis of the desired joint driving torque. 
     The following will explain in detail the control processing carried out by the control device  60 . In the present embodiment, the control processing in the case where the desired motion of the robot  1  is a motion for traveling on a floor by walking will be taken as an example. 
     As illustrated in  FIG. 3 , the control device  60  has a gait generator  71 , a desired joint driving torque determiner  72 , and a joint drive controller  73  as its major units for implementing functions by running an installed program or the like. 
     The gait generator  71  is a functional unit that carries out processing for generating a desired gait including at least a desired motion of the robot  1 . The desired motion generated by the gait generator  71  includes the time series of a desired joint angle, which indicates a desired value of the displacement amount (rotational angle) of each joint of the robot  1  and the time series of a desired position of the overall center-of-gravity point of the robot  1  (hereinafter referred to as the desired overall center-of-gravity point in some cases). 
     The aforesaid desired motion is generated by using, for example, the gait generating technique proposed by the applicant of the present application in Japanese Patent Laid-Open H5-305579 and the like. 
     The gait generating technique will be schematically explained. The expected landing position, the expected landing posture, and expected landing time aimed by each of both feet  18 R and  18 L of the robot  1  are determined mainly on the basis of the floor geometry information of an environment in which the robot  1  moves and the requirements on the directions in which the robot  1  travels. 
     Based on the expected landing position/posture and the expected land time, a foot position/posture trajectory, which is the trajectory (a time series pattern) of a desired position and a desired posture of each of the feet  18 , is determined. Further, based on the foot position/posture trajectory, a desired ZMP trajectory, which is the trajectory of a desired position of zero moment point (ZMP) (hereinafter referred to as the desired ZMP in some cases), is determined. 
     Further, the trajectories of desired positions and desired postures of links (the body link  2  and the like) other than the two leg links  3 R and  3 L of the robot  1  are determined such that the stability of the robot  1  in the future will be secured while at the same time satisfying a dynamic balance condition that the horizontal component of a moment generated about a desired ZMP by the resultant force of the inertial force generated due to a motion of the entire robot  1  and the gravitational force acting on the robot  1  becomes zero. 
     Then, based on the desired positions and the desired postures, the desired joint angles of individual joints are determined by using a geometric model (a rigid link model) of the robot  1 . 
     The desired gait may not necessarily be generated in real time while the robot  1  is actually in motion. Alternatively, a desired gait prepared beforehand (off-line) may be input to the control device  60  from outside by means of wireless communication or the like or may be stored and retained beforehand in a storage device (not shown) of the robot  1 . In this case, the gait generator  71  is not required. 
     A desired gait may include a desired value of a particular component (e.g., a vertical component of a translational force) of an overall floor reaction force (total floor reaction force) acting on one or both of feet  18 R and  18 L or the desired position of the acting point of the total floor reaction force (i.e., the desired ZMP), in addition to a desired motion of the robot  1 . In the present embodiment, a desired gait output by the gait generator  71  includes a desired ZMP. 
     The desired joint driving torque determiner  72  is a functional unit which sequentially carries out processing for determining a desired joint driving torque of each joint for accomplishing the aforesaid desired motion generated by the gait generator  71  (for making the actual motion of the robot  1  follow the desired motion). The desired joint driving torque determiner  72  is a functional unit that carries out core processing related to the present invention, which will be discussed in detail later. 
     The joint drive controller  73  is a functional unit which sequentially carries out processing for controlling the output torque of the electric motor  50  of each joint drive system so as to achieve a desired joint driving torque (specifically, a desired joint driving torque of each joint defined by a corrected desired joint driving torque ↑τtc_cmd, which will be discussed hereinafter) of each joint sequentially determined by the desired joint driving torque determiner  72  (so as to cause the driving torque actually imparted to the joint driven member  51  of each joint to follow a desired joint driving torque). 
     The joint drive controller  73  controls the output torque of the electric motor  50 , for example, as described below. The joint drive controller  73  sequentially estimates an elastic force torque actually generated by the torsion bar  53  (the driving torque actually imparted from the torsion bar  53  to the joint driven member  51 ) on the basis of a measurement value of the rotational angle difference between both ends of the torsion bar  53  (the torsional angle of the torsion bar  53 ). Further, the joint drive controller  73  sequentially sets the desired rotational angular velocity of the output shaft  50   a  of the electric motor  50  such that the difference between the estimated value of the elastic force torque and the desired joint driving torque is converged to zero. 
     Subsequently, the joint drive controller  73  determines the desired output torque of the electric motor  50  such that the difference between the desired rotational angular velocity and the measurement value of the rotational angular velocity of the output shaft  50   a  of the electric motor  50  is converged to zero. Then, the joint drive controller  73  converts the desired output torque into a desired value of energizing current of the electric motor  50 , and the actual energizing current is subjected to feedback control based on the desired value, thereby controlling the actual output torque of the electric motor  50  to the desired output torque. 
     In the control processing by the joint drive controller  73 , the output torque of the electric motor  50  may be controlled while compensating for the influences of vibrations of the torsion bar  53  or compensating for the influences of an external force acting on the joint driven member  51 . 
     The processing by the desired joint driving torque determiner  72 , the explanation thereof had been postponed, will now be explained in detail. 
     In the following description, a vector (a column of a plurality of components) will be denoted by adding “↑” to the beginning of a variable name like, for example, “↑A.” Unless otherwise specified, a vector will be a longitudinal vector in which components are longitudinally arranged (column vector). In the present description, “*” will be used as a multiplication symbol. 
     As illustrated in  FIG. 4 , the desired joint driving torque determiner  72  is provided with, as its major functions, a basic parameter group determiner  81 , a contact portion Jacobian matrix determiner  82 , a state amount Jacobian matrix determiner  83 , a state amount desired value determiner  84 , a control input determiner  85 , and a compliance controller  86 .  FIG. 4  illustrates major outputs of the functional units and the destinations to which the major outputs are input. 
     The desired joint driving torque determiner  72  sequentially carries out the processing by the functional units at a predetermined arithmetic processing cycle thereby to sequentially determine the desired joint driving torque of each joint. 
     The desired joint driving torque determiner  72  first carries out the processing by the basic parameter group determiner  81 , the processing by the contact portion Jacobian matrix determiner  82 , the processing by the state amount Jacobian matrix determiner  83 , and the processing by the state amount desired value determiner  84 . 
     The basic parameter group determiner  81  is a functional unit which sequentially carries out processing for calculating a parameter of a robot motion model, which is a model representing the relationship (motion equation) between a motion of the robot  1  and a force. 
     Here, the robot motion model is represented by the following expression 1-1 if an external force (a floor reaction force or the like) acting on the robot  1  from an external world is zero.
 
 M*↑q″+↑N+↑G=↑τ   Expression 1-1
 
     ↑q″ in this expression 1-1 denotes a second-order differential value of a generalized variable vector ↑q of the robot  1  (=d 2 ↑q/dt 2 ), M denotes an inertia matrix, ↑N denotes a centrifugal force/Coriolis force-dependent generalized force vector, ↑G denotes a gravity-dependent generalized force vector, and ↑τ denotes a driving force-dependent generalized force vector. 
     The aforesaid generalized variable vector ↑q in the present embodiment is a vector comprised of an array of components, namely, the position and the posture (the position and the posture observed in a global coordinate system) of a single representative link of the robot  1  selected beforehand and the displacement amount (rotational angle) of each joint. The aforesaid representative link in the present embodiment is the lower body  10  serving as the base body of the robot  1 . Hereinafter, the lower body  10  will be referred to as the representative link  10  in some cases. 
     Here, in the description of the present embodiment, more specifically, the term “position” of each link, including the representative link  10  of the robot  1 , means the position in the global coordinate system of a representative point fixedly set relative to the link. Similarly, more specifically, the term “posture” of each link means the spatial orientation of the link observed in the global coordinate system (the orientation expressed in terms of the dimension of angle). In the following description, the posture will be referred to as the posture angle in some cases. 
     The aforesaid global coordinate system is an inertial coordinate system (including a coordinate system that can be approximately regarded as an inertial system), such as a coordinate system secured to a floor (including the ground surface) in an environment in which the robot  1  travels. 
     Further, each of the position and the posture (posture angle) of each link has three degrees of freedom in the present embodiment, allowing each of them to be expressed by means of three coordinate components. Hence, the number of the components of the generalized variable vector ↑q, i.e., the degree of freedom of ↑q, in the present embodiment is the total sum of the total degree of freedom of the position and the posture of the representative link  10  (=6) and the total degree of freedom of all joints of the robot  1 . In the robot  1  of the present embodiment illustrated in  FIG. 1 , the total degree of freedom of all joints is 29. Therefore, the degree of freedom of ↑q is 35. 
     The inertia matrix M is a matrix (symmetric matrix) for converting the second-order differential value ↑q″ of the generalized variable vector ↑q into a generalized force vector. i.e., a matrix in which the component value of a vector obtained by multiplying inertia matrix by ↑q″ (=M*↑q″) becomes a value having the dimension of a force corresponding to each component of ↑q″ (a translational force or a moment). 
     Here, each component of ↑q is a position or a posture angle or a rotational angle, so that each component of ↑q″ has the dimension of a translational acceleration or an angular acceleration. Therefore, each component value of a vector obtained by reversing the sign of M*↑q″ (−M*↑q″) means the inertial force generated by a motion of the robot  1 . In this case, in the vector (−M*↑q″), the inertial force of a component corresponding to the second-order differential value (translational acceleration) of the position of the representative link  10  is a translational inertial force (an inertial force turning into a translational force). The inertial force of a component corresponding to the second-order differential value (angular acceleration) of the posture of the representative link  10  and the inertial force of the component corresponding to the second-order differential value of the rotational angle (angular acceleration) of each joint are inertial force moments (inertial forces turning into moments). 
     The inverse matrix M −1  of the inertia matrix M is a diagonal matrix. 
     The generalized force vector is a vector whose components correspond to the components of the generalized variable vector ↑q (a vector constituted of the same number of components as that of ↑q). More specifically, the generalized force vector is a vector, each component of which provides a force (a translational force or a moment) that causes a change (acceleration) in a component value of ↑q corresponding thereto. In this case, each component of the generalized force vector is a force of the same type as that of the inertial force based on the second-order differential value of a component of ↑q corresponding thereto. 
     The centrifugal force/Coriolis force-dependent generalized force vector ↑N is a generalized force vector generated by a centrifugal force and a Coriolis force acting on each link of the robot  1 . Further, the gravity-dependent generalized force vector ↑G is a generalized force vector generated by a gravitational force acting on each link of the robot  1 . In the following description, the sum of ↑N and ↑G (=↑N+↑G) will be referred to as the nonlinear term generalized force vector in some cases. 
     The driving force-dependent generalized force vector ↑τ is a generalized force vector generated by a driving force of the actuator (the electric motor  50  in the present embodiment) that drives each joint of the robot  1 . In the robot  1  according to the present embodiment, the values of the components corresponding to the position and the posture of the representative link  10  in the driving force-dependent generalized force vector ↑τ are always zero. Further, in the ↑τ, the value of the component corresponding to the rotational angle of each joint will mean the driving torque to be imparted to the joint (hereinafter referred to as the joint driving torque in some cases). 
     The parameters calculated by the basic parameter group determiner  81  include the inertia matrix M, the centrifugal force/Coriolis force-dependent generalized force vector ↑N, and the gravity-dependent generalized force vector ↑G (or the nonlinear term generalized force vector (↑N+↑G)) in expression 1-1. 
     Supplementarily, if an external force (≠0), such as a floor reaction force, acts on the robot  1 , then the robot motion model is represented by an expression obtained by adding the term of the generalized force vector depending on the external force (external force term) to expression 1-1 (expression 5-1 to be discussed later). However, the aforesaid M, ↑N, and ↑G, which are the parameters calculated by the basic parameter group determiner  81  do not depend on the external force, and the values thereof depend on the motion state of the robot  1 , such as the rotational angle and the temporal change rate (angular velocity) thereof of each joint of the robot  1 . For this reason, the aforesaid expression 1-1 for describing the basic parameter group determiner  81  omits the external force term. 
     To calculate the parameters M, ↑N, and ↑G (or M and (↑N+↑G)), the basic parameter group determiner  81  sequentially receives the measurement value of the moving velocity of the representative link (the lower body)  10 , the measurement value of the posture angle of the representative link  10  and the measurement value of the temporal change rate (angular velocity) thereof, and the measurement value of the rotational angle of each joint and the measurement value of the temporal change rate (angular velocity) thereof. These measurement values are the measurement values recognized by the control device  60  on the basis of the detection outputs of the posture sensor  61 , the rotary encoder  64 , and the differential encoder  65 . 
     Further, the basic parameter group determiner  81  uses the input measurement values to sequentially calculate the parameters M, ↑N, and ↑G (or M and (↑N+↑G)) according to a publicly known method, such as the Recursive Newton Euler Algorithm (RNEA) or the unit vector method. 
     Supplementarily, there are some cases where the magnitude of a component value of ↑N becomes sufficiently minute in comparison to the magnitude of the corresponding component value of the gravity-dependent generalized force vector ↑G or becomes zero, leading to ↑N+↑G≈↑G. In such a case, ↑N out of the ↑N and ↑G may be ignored (regarding that the nonlinear term generalized force vector (↑N+↑G) agrees with ↑G), and only ↑G may be calculated. 
     The aforesaid contact portion Jacobian matrix determiner  82  is a functional unit which carries out processing for sequentially calculating Jacobian matrix (hereinafter referred to as the contact portion Jacobian matrix) that expresses the relationship between the motion speed of a contact portion representative element as an element representing a portion of the robot  1  in contact with the external world (more specifically, one or both of the translational velocity and the rotational velocity (angular velocity) of the contact portion representative element) and the first-order differential value (temporal change rate) of the generalized variable vector ↑q. In other words, the contact portion Jacobian matrix is a matrix that expresses the ratio (sensitivity) of a change in the spatial position/posture (more specifically, one or both of the position and the posture) of the contact portion representative element with respect to a change (minute change) of each component value of the generalized variable vector ↑q. 
     More specifically, the contact portion representative element serving as the element representing a portion of the robot  1  in contact with the external world means a substantive or virtual element, the motion of which is defined, depending on the state of contact of every contact portion in contact with the external world (the motion state of each contact portion at the time of contact or the state of an external force acting on each contact portion). 
     Here, a desired motion of the robot  1  described in the present embodiment is a motion for the robot  1  to perform a walking motion, as described above. In this case, the ground contact portion as an actual portion of the robot  1  in contact with the external world (the portion in contact with a floor surface) changes with time. More specifically, the ground contact state of the robot  1  during a walking motion is such that a one-leg ground contact state in which only one of the feet  18 R and  18 L is a portion in contact with the ground, a two-leg ground contact state in which both feet are portions in contact with the ground, and a one-leg ground contact state in which only the other one is a portion in contact with the ground are repeated in order. Hence, the position/posture of the contact portion representative element should change according to the ground contact states of the feet  18 R and  18 L. 
     Thus, the contact portion Jacobian matrix determiner  82  sequentially calculates the contact portion Jacobian matrix while sequentially determining the value of a parameter (the rate value r at, which will be discussed later) that defines the position/posture of the contact portion representative element of the robot  1  according to the ground contact state of the feet  18 R and  18 L. In this case, the position/posture of the contact portion representative element are determined such that the position/posture will not discontinuously change, i.e., the position/posture will always continuously change, at the time when the ground contact state of the feet  18 R and  18 L changes or during the periods of the one-leg ground contact state and the two-leg ground contact state, respectively. 
     To be more specific, in the one-leg ground contact state, the position/posture of the contact portion representative element are set as elements that depend upon the position and the posture of the foot  18 R or the foot  18 L, whichever is in contact with the ground. In the two-leg ground contact state, the position/posture of the contact portion representative element are set as elements that depend upon the position and the posture of both feet  18 R and  18 L and how a floor reaction force acts on both feet  18 R and  18 L. 
     The following will specifically describe the processing by the contact portion Jacobian matrix determiner  82 . 
     In the present embodiment, the contact portion representative element is regarded as an element formed by connecting a predetermined number of plural points subjected to motional restrictions when one or both of the feet  18 R and  18 L as the portions to be in contact with the external world come in contact with the ground (or comes in contact with a floor), and the motions (translational motions and rotational motions) of the contact portion representative element are expressed by the motions (translational motions) of the predetermined number of points. Hereinafter, the predetermined number of plural points will be referred to as the contact portion representative points. The number of the contact portion representative points will be maintained to be constant in any one of the ground contact states of the feet  18 R and  18 L. 
     More specifically, the contact portion representative element is constructed of four contact portion representative points a 1 , a 2 , a 3 , and a 4  in any one of the ground contact states of the feet  18 R and  18 L, as illustrated by the black dots in  FIGS. 5(   a ) to  5 ( c ). In  FIGS. 5(   a ) to  5 ( c ), one or both of the feet  18 R and  18 L in contact with the ground are indicated by solid lines, while the foot off the floor (moving in the air) is indicated by the two-dot chain line. 
     In this case, the positions of the four contact portion representative points a 1  to a 4  are specified to the positions based on the ground contact states of the feet  18 R and  18 L. 
     To be more specific, in the one-leg ground contact state in which the left foot  18 L is in contact with a floor, while the right foot  18 R is off the floor (the state wherein the left leg is in contact with the ground), four predetermined points a 1 L, a 2 L, a 3 L, and a 4 L set beforehand at predetermined positions on the bottom surface of the left foot  18 L (the positions secured relative to the left foot  18 L) are determined as the contact portion representative points a 1 , a 2 , a 3 , and a 4 , respectively, as illustrated in  FIG. 5(   a ). 
     The positions of the four predetermined points a 1 L to a 4 L in the present embodiment are the positions of four corners of the bottom surface of the left foot  18 L. More specifically, the positions of the predetermined points a 1 L to a 4 L are the front left corner of the bottom surface of the left foot  18 L, the front right corner thereof, the rear left corner thereof, and the rear right corner thereof, respectively, in order. Hence, the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  in the state wherein the left leg is in contact with the ground are defined on the basis of the position and the posture of the left foot  18 L. 
     Further, in the one-leg ground contact state in which the right foot  18 R is in contact with a floor, while the foot  18 L of the left leg link  3 L is off the floor (the state wherein the right leg is in contact with the ground), four predetermined points a 1 R, a 2 R, a 3 R, and a 4 R set beforehand at predetermined positions on the bottom surface of the right foot  18 R (the positions secured relative to the right foot  18 R) are determined as the contact portion representative points a 1 , a 2 , a 3 , and a 4 , respectively, as illustrated in  FIG. 5(   c ). 
     As with the four predetermined points a 1 L to a 4 L of the left foot  18 L, the positions of the four predetermined points a 1 R to a 4 R are the positions of four corners of the bottom surface of the right foot  18 R (the positions of the front left corner, the front right corner, the rear left corner, and the rear right corner). Hence, the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  in the state wherein the right leg is in contact with the ground are defined on the basis of the position and the posture of the right foot  18 R. 
     The mutual positional relationship among the four predetermined points a 1 R to a 4 R of the right foot  18 R is the same as the mutual positional relationship among the four predetermined points a 1 L to a 4 L of the left foot  18 L. 
     In the two-leg ground contact state in which both feet  18 R and  18 L are in contact with the ground, the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  are determined to be a position on a line segment connecting the predetermined points a 1 R and a 1 L of both feet  18 R and  18 L, a position on a line segment connecting the predetermined points a 2 R and a 2 L, a position on a line segment connecting the predetermined points a 3 R and a 3 L, and a position on a line segment connecting the predetermined points a 4 R and a 4 L in order, as illustrated in  FIG. 5(   b ). 
     Here, in the following description, a line segment connecting arbitrary two points A and B will be denoted by line segment A_B, and the length of the line segment A_B (line segment length) will be denoted by L(A_B). 
     In this case, the positions of the contact portion representative points a 1  to a 4  in the two-leg ground contact state are more specifically defined as follows. 
     The positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  are determined such that the ratio of a line segment length L(a 1 _a 1 L) to a line segment length L(a 1 R_a 1 L). i.e., L(a 1 _a 1 L)/L(a 1 R_a 1 L), the ratio of a line segment length L(a 2 _a 2 L) to a line segment length L(a 2 R_a 2 L), i.e., L(a 2 _a 2 L)/L(a 2 R_a 2 L), the ratio of a line segment length L(a 3 _a 3 L) to a line segment length L(a 3 R_a 3 L), i.e., L(a 3 _a 3 L)/L(a 3 R_a 3 L), and the ratio of a line segment length L(a 4 _a 4 L) to a line segment length L(a 4 R_a 4 L), i.e., L(a 4 _a 4 L)/L(a 4 R_a 4 L), present a mutually equal ratio value r at (0≦r at≦1) as indicated by expression 2-1a given below. 
     In other words, the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  are determined such that the ratio of a line segment length L(a 1 _a 1 R) to a line segment length L(a 1 R_a 1 L), i.e., L(a 1 _a 1 R)/L(a 1 R_a 1 L), the ratio of a line segment length L(a 2 _a 2 R) to a line segment length L(a 2 R_a 2 L), i.e., L(a 2 _a 2 R)/L(a 2 R_a 2 L), the ratio of a line segment length L(a 3 _a 3 R) to a line segment length L(a 3 R_a 3 L), i.e., L(a 3 _a 3 R)/L(a 3 R_a 3 L), and the ratio of a line segment length L(a 4 _a 4 R) to a line segment length L(a 4 R_a 4 L), i.e., L(a 4 _a 4 R)/L(a 4 R_a 4 L), present a mutually equal ratio value (1−r at) as indicated by expression 2-1b given below. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             L 
                             ⁡ 
                             
                               ( 
                               a1_a1L 
                               ) 
                             
                           
                           / 
                           
                             L 
                             ⁡ 
                             
                               ( 
                               a1R_a1L 
                               ) 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             L 
                             ⁡ 
                             
                               ( 
                               a2_a2L 
                               ) 
                             
                           
                           
                             L 
                             ⁡ 
                             
                               ( 
                               a2R_a2L 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             L 
                             ⁡ 
                             
                               ( 
                               a3_a3L 
                               ) 
                             
                           
                           
                             L 
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     Further, according to the present embodiment, the aforesaid ratio value r at, which defines the positions of the contact portion representative points a 1  to a 4  in the state wherein both legs are in contact with the ground, is determined on the basis of the ratio of a vertical component Fz_act_R of a translational floor reaction force (a translational force of a floor reaction force) actually acting on the right foot  18 R to a vertical component Fz_act_L of a translational floor reaction force actually acting on the left foot  18 L. More specifically, the contact portion Jacobian matrix determiner  82  determines the aforesaid ratio value r at according to expression 2-2a or 2-2b given below.
 
 r at=Fz _act —   R/ ( Fz _act —   R+Fz _act —   L )  Expression 2-2a
 
 r at= 1 −Fz _act —   L /( Fz _act —   R+Fz _act —   L )  Expression 2-2b
 
     Thus, the positions of the contact portion representative points a 1  to a 4  in the state wherein both legs are in contact with the ground are determined on the basis of the positions and the postures of both feet  18 R and  18 L and how a floor reaction force acts on both feet  18 R and  18 L. 
     In expression 2-2a or 2-2b, if Fz_act_R is zero, then r at will be zero. In this case, therefore, the contact portion representative points a 1 , a 2 , a 3 , and a 4  will coincide with the aforesaid predetermined points a 1 L, a 2 L, a 3 L, and a 4 L, respectively, of the left foot  18 L. In expression 2-2a or 2-2b, if Fz_act_L is zero, then r at will be 1. In this case, therefore, the contact portion representative points a 1 , a 2 , a 3 , and a 4  will coincide with the aforesaid predetermined points a 1 R, a 2 R, a 3 R, and a 4 R, respectively, of the right foot  18 R. Further, Fz_act_R will be zero in the state wherein the left leg is in contact with the ground, while Fz_act_L will be zero in the state wherein the right leg is in contact with the ground. 
     Hence, not only in the state wherein both legs are in contact with the ground but also in the state wherein the left leg is in contact with the ground and in the state wherein the right leg is in contact with the ground, determining the ratio value r at according to expression 2-2a or 2-2b and defining the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  on the basis of the determined ratio value r at such that the aforesaid expression 2-1a or 2-1b is satisfied will consequently specify the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  to be the aforesaid positions in the state wherein the left leg is in contact with the ground and the state wherein the right leg is in contact with the ground. 
     In the present embodiment, therefore, the contact portion Jacobian matrix determiner  82  carries out processing for determining the aforesaid ratio value r at as a parameter defining the positions of the contact portion representative points a 1 , a 2 , a 3 , and a 4  (consequently the position and the posture of the contact portion representative element). The following will specifically describe how the processing is carried out. 
     The measurement values of the actual translational floor reaction force vertical components Fz_act_R and Fz_act_L acting on the feet  18 R and  18 L are input to the contact portion Jacobian matrix determiner  82  to determine the aforesaid ratio value r at (hereinafter referred to as the contact portion representative element defining ratio r at). The measurement values of Fz_act_R and Fz_act_L are the measurement values recognized by the control device  60  on the basis of mainly the detection outputs of the force sensors  63 R and  63 L. 
     Then, based on the received measurement values of the translational floor reaction force vertical components Fz_act_R and Fz_act_L, the contact portion Jacobian matrix determiner  82  calculates the contact portion representative element defining ratio r at according to the aforesaid expression 2-2a or expression 2-2b. 
     Determining the contact portion representative element defining ratio r at as described above consequently determines the positions of the four contact portion representative points a 1  to a 4  such that these positions continuously change. In turn, the position and the posture of the contact portion representative element formed by connecting the four contact portion representative points a 1  to a 4  are determined such that the position and the posture will continuously change. 
     Supplementarily, according to the present embodiment, the positions of the four contact portion representative points a 1  to a 4  are determined as described above, so that the foot  18 R or  18 L itself in contact with the ground corresponds to the contact portion representative element in the state wherein one leg of the robot  1  is in contact with the ground. Meanwhile, in the state wherein both legs of the robot  1  are in contact with the ground, the contact portion representative element corresponds to a virtual element existing between the two feet  18 R and  18 L. 
     Further, the contact portion Jacobian matrix determiner  82  determines a contact portion Jacobian matrix by using the contact portion representative element defining ratio r at determined as described above. 
     In this case, according to the present embodiment, the motion of a contact portion representative element is expressed by the motions of the four contact portion representative points a 1  to a 4 , so that the contact portion Jacobian matrix is indicated by the matrix Jc which represents the relationship between a temporal change rate (a first-order differential value) ↑Xc′ of a vector ↑Xc constructed by an array of the positions of the contact portion representative points a 1  to a 4  as components (=d↑Xc/dt), i.e., the vector ↑Xc′ constructed using the motion velocities of the individual contact portion representative points a 1  to a 4  as the components thereof, and a first-order differential value ↑q′ of the aforesaid generalized variable vector ↑q by expression 2-3 given below.
 
↑ Xc′=Jc*↑q′   Expression 2-3
 
     The aforesaid vector ↑Xc means the position and the posture of a contact portion representative element constructed of the contact portion representative points a 1  to a 4 , and the temporal change rate ↑Xc′ of ↑Xc means the motion velocity of the contact portion representative element. Hereinafter, the ↑Xc will be referred to as the contact portion representative element position vector. The motion velocity (translational velocity) of each of the contact portion representative points a 1  to a 4  has three degrees of freedom, so that ↑Xc′ is expressed as a vector formed of twelve components (3*4=12). 
     Here, the contact portion representative element position vector ↑Xc in the case where the contact portion representative points a 1  to a 4  are set to coincide with the aforesaid predetermined points a 1 R to a 4 R of the right foot  18 R is denoted by ↑Xc_R, while the contact portion representative element position vector ↑Xc in the case where the contact portion representative points a 1  to a 4  are set to coincide with the aforesaid predetermined points a 1 L to a 4 L of the left foot  18 L is denoted by ↑Xc_L. Further, the Jacobian matrix that indicates the relationship between a first-order differential value ↑Xc_R′ of the ↑Xc_R (=d↑Xc_R/dt) and the first-order differential value ↑q′ of the generalized variable vector ↑q is denoted by Jc_R, while the Jacobian matrix that indicates the relationship between a first-order differential value ↑Xc_L′ of the ↑Xc_L (=d↑Xc_L/dt) and the first-order differential value ↑q′ of the generalized variable vector ↑q is denoted by Jc_L. In other words, ↑Xc_L′=Jc_L*↑q′ and ↑Xc_R′=Jc_R*↑q′ apply. 
     At this time, the contact portion Jacobian matrix Jc is given by the following expression 2-4.
 
 Jc=r at*Jc   —   R +(1− r at )* Jc   —   L   Expression 2-4
 
     According to the present embodiment, therefore, the contact portion Jacobian matrix determiner  82  calculates the Jacobian matrixes Jc_R and Jc_L of the right side of expression 2-4, and the right side of expression 2-4 is calculated from the calculated values of the Jacobian matrixes Jc_R and Jc_L and the contact portion representative element defining ratio r at calculated as described above, thereby calculating the contact portion Jacobian matrix Jc. 
     In this case, the contact portion Jacobian matrix determiner  82  receives the moving velocity of the lower body  10 , the measurement values of the posture angle and the temporal change rate (angular velocity) of the lower body  10 , and the measurement values of the rotational angles and the temporal change rates (angular velocities) of the joints of each of the leg links  3 R and  3 L in order to calculate the aforesaid Jacobian matrixes Jc_R and Jc_L. Then, the contact portion Jacobian matrix determiner  82  calculates the aforesaid Jacobian matrixes Jc_R and Jc_L from the received measurement values. 
     Thus, in the processing carried out by the contact portion Jacobian matrix determiner  82 , the contact portion representative element defining ratio r at is determined on the basis of the measurement values of the translational floor reaction force vertical components Fz_act_R and Fz_act_L acting on the robot  1  (by extension, on the basis of the ground contact states of both feet  18 R and  18 L). Further, the Jacobian matrixes Jc_R and Jc_L are calculated on the basis of the measurement values of predetermined components of the generalized variable vector ↑q and the first-order differential value ↑q′ thereof. Then, the contact portion Jacobian matrix Jc is calculated according to the aforesaid expression 2-4 from the calculated values of the contact portion representative element defining ratio r at and the Jacobian matrixes Jc_R and Jc_L. 
     Supplementarily, if the denominator of the right side of the aforesaid expression 2-2a or 2-2b can be regarded to be maintained substantially constant, then the contact portion representative element defining ratio r at may be determined according to an expression derived by replacing the denominator of the right side of expression 2-2a or 2-2b by the gravitational force acting on the entire robot  1  (=overall mass of the robot  1 *gravitational force acceleration constant). 
     If the aforesaid desired gait generated and output by the gait generator  71  (or a desired gait supplied to the control device  60  from outside) includes desired values Fz_cmd_R and Fz_cmd_L of the translational floor reaction force vertical components of the feet  18 R and  18 L, respectively, then the contact portion representative element defining ratio r at may be determined according to an expression derived by replacing Fz_act_R and Fz_act_L of the aforesaid expression 2-2a or 2-2b by Fz_cmd_R and Fz_cmd_L, respectively. 
     Alternatively, for example, the contact portion representative element defining ratio r at may be determined by combining (adding up) a filtering value obtained by subjecting the value of r at, which has been calculated according an expression obtained by replacing Fz_act_R and Fz_act_L of expression 2-2a or 2-2b by Fz_cmd_R and Fz_cmd_L, respectively, to filtering of a low-pass characteristic and a filtering value obtained by subjecting the value of r at, which has been calculated by the computation of expression 2-2a or 2-2b, to filtering of a low-cut characteristic (a high-pass characteristic). 
     In other words, a component on a low frequency side of the contact portion representative element defining ratio r at may be determined on the basis of the ratio of the desired values Fz_cmd_R and Fz_cmd_L and a component on a high frequency side of the contact portion representative element defining ratio r at may be determined on the basis of the ratio of the measurement values of Fz_act_R and Fz_act_L, and then these determined components may be combined (added up), thereby determining the contact portion representative element defining ratio r at. 
     In this case, for the filtering of a low-pass characteristic and the filtering of a low-cut characteristic, filters whose transfer functions are expressed by, for example, 1/(Tc*s+1) and Tc*s/(Tc*s+1), respectively (Tc: predetermined time constant) may be used. 
     In the present embodiment, the four points a 1  to a 4  have been used as the contact portion representative points to express the motion of a contact portion representative element. Alternatively, however, three, or five or more points may be used as the contact portion representative points. 
     The aforesaid state amount Jacobian matrix determiner  83  is a functional unit which sequentially carries out processing for calculating a Jacobian matrix expressing the relationship between a state amount predetermined as a desired state amount for controlling the robot  1  to a desired value proper to ensure a stable walking motion (hereinafter referred to as the state amount to be controlled in some cases) and the first-order differential value ↑q′ of the generalized variable vector ↑q (the Jacobian matrix will be hereinafter referred to as the state amount Jacobian matrix). 
     The state amount to be controlled is a state amount whose value changes as the value of any one component of the generalized variable vector ↑q changes. To be more specific, the state amount to be controlled is a state amount whose value (instantaneous value) is obtained by multiplying a certain state amount Jacobian matrix by the first-order differential value ↑q′ of the generalized variable vector ↑q. 
     In the present embodiment, as the state amount to be controlled, a set of a total translational momentum ↑Ptotal and a total angular momentum ↑Ltotal of the robot  1  observed in the global coordinate system (more specifically, a total angular momentum ↑Ltotal about a predetermined reference point in the global coordinate system) is used. The global coordinate system in the present embodiment uses, for example, a coordinate system (a so-called supporting leg coordinate system) having the origin thereof set at a position based on the ground contact position of the foot of a supporting leg out of the feet  18 R and  18 L of the robot  1  (e.g., a position in a ground contact surface). 
     More specifically, the total translational momentum ↑Ptotal is a translational momentum resulting from adding up the momentum of the center-of-gravity point of every link of the robot  1  (=the mass of each link*the moving velocity of the center-of-gravity point of the link). In other words, ↑Ptotal denotes the translational momentum of the overall center-of-gravity of the robot  1 . The ↑Ptotal in the present embodiment denotes a vector having three degrees of freedom (a vector constituted of components in the directions of three axes, namely, the roll axis, the pitch axis, and the yaw axis). 
     In the following description, the roll axis means a horizontal axis in the longitudinal direction of the robot  1 , the pitch axis means a horizontal axis in the lateral direction of the robot  1 , and the yaw axis means an axis in the vertical direction (gravitational direction). 
     The total angular momentum ↑Ltotal denotes an angular momentum obtained by adding up the angular momentum about the aforesaid reference point attributable to the translational motion of the center-of-gravity point of every link of the robot  1  (=the vector product of the positional vector of the center-of-gravity point of each link relative to the reference point and the translational momentum of the center-of-gravity point) and the angular moment attributable to a rotational motion of every link about the center-of-gravity point (=the inertial moment of each link*the angular velocity about the center-of-gravity point of the link). The total angular momentum ↑Ltotal in the present embodiment is a vector having two degrees of freedom, excluding a component about the yaw axis (i.e., a vector constituted of components about two axes, namely, the roll axis and the pitch axis). The aforesaid reference point is a point preset on the aforesaid global coordinate system (e.g., the origin of the supporting leg coordinate system). 
     The state amount Jacobian matrix determiner  83  determines the state amount Jacobian matrix corresponding to the aforesaid state amount to be controlled. In this case, the state amount Jacobi an matrix is a matrix Js1 which represents the relationship between a vector ↑S1 (=[↑P, ↑L] T ; hereinafter referred to as the state amount vector ↑S1) constructed of the total translational momentum ↑Ptotal and the total angular momentum ↑Ltotal arranged as the components thereof, and the first-order differential value ↑q′ of the generalized variable vector ↑q according to expression 3-1 given below.
 
↑ S 1 =Js 1* ↑q′   Expression 3-1
 
     In the present embodiment, the total translational momentum ↑Ptotal is a vector having three degrees of freedom and the total angular momentum ↑Ltotal is a vector having two degrees of freedom. Hence, ↑S1 is a vector constituted of 5 components (3+2=5). 
     To calculate the state amount Jacobian matrix Js1, the state amount Jacobian matrix determiner  83  receives the measurement values of the moving velocity of the representative link (the lower body)  10  and the posture angle and the temporal change rate (angular velocity) thereof, and the measurement values of the rotational angle of each joint and the temporal change rate (angular velocity) thereof. Then, the state amount Jacobian matrix determiner  83  calculates the state amount Jacobian matrix Js1 from the received measurement values. 
     The state amount desired value determiner  84  is a functional unit which carries out processing for determining a desired value related to the state amount to be controlled. 
     In the present embodiment, the desired value related to the state amount to be controlled is a desired value ↑S1′_cmd of the temporal change rate (first-order differential value) ↑S1′ of the state amount vector ↑S1 (=d↑S1/dt). 
     In this case, ↑S1′ is a vector having the first-order differential value ↑Ptotal′ of the total translational momentum ↑Ptotal of the robot  1  (=d↑Ptotal/dt) and the first-order differential value ↑Ltotal′ of the total angular momentum ↑Ltotal (=d↑Ltotal/dt) as the components thereof (=[↑Ptotal′, ↑Ltotal′] T ). Hence, the desired value ↑S1′_cmd of ↑S1′ (hereinafter referred to as the state amount desired value vector ↑S1′_cmd) corresponds to the desired value of the total inertial force (excluding a moment component about the yaw axis) generated by a motion of the entire robot  1 . 
     More specifically, the state amount desired value vector ↑S1′_cmd is a vector having the desired value of ↑Ptotal′ and the desired value of ↑Ltotal′ as the components thereof. Further, the desired value of ↑Ptotal′ of ↑S1′_cmd corresponds to the desired value of the total translational inertial force generated by a motion of the entire robot  1  (the desired value of the translational inertial force of the overall center-of-gravity of the robot  1 ), while the desired value of ↑Ltotal′ corresponds to the desired value of the total moment inertial force (excluding the component about the yaw axis) generated by a motion of the entire robot  1 . 
     Further, according to the present embodiment, the state amount desired value determiner  84  determines the aforesaid state amount desired value vector ↑S1′_cmd as described below. 
     The state amount desired value determiner  84  calculates the desired value ↑Ptotal′_cmd of the temporal change rate ↑Ptotal′ of the total translational momentum ↑Ptotal and the desired value ↑Ltotal′_cmd of the temporal change rate ↑Ltotal′ of the total angular momentum ↑Ltotal in the state amount desired value vector ↑S1′_cmd by carrying out separate arithmetic processing. 
     To be more specific, regarding the processing for calculating ↑Ptotal′_cmd, the state amount desired value determiner  84  calculates a difference ↑Δr gc between a desired position ↑r gc_cmd of the overall center-of-gravity point of the robot  1  in the global coordinate system and the measurement value of an actual position ↑r gc_act (=↑r gc_cmd−↑r gc_act), and a first-order differential value ↑Δr gc′ of the difference ↑Δr gc (=d↑Δr gc/dt). 
     In this case, the desired position ↑r gc_cmd of the overall center-of-gravity point of the robot  1  is input to the state amount desired value determiner  84  from the aforesaid gait generator  71 . The measurement value of the actual position ↑r gc_act of the overall center-of-gravity point is sequentially calculated by the control device  60  according to a publicly know method (e.g., a method using a Kalman filter) primarily from the measurement value of the rotational angle of each joint, and the calculated measurement value is input to the state amount desired value determiner  84 . 
     Incidentally, each of ↑r gc_cmd and ↑r gc_act denotes a vector constituted of positions in the directions of three axes, namely, the direction of the roll axis, the direction of the pitch axis, and the direction of the yaw axis (i.e., a position vector having three degrees of freedom). 
     Then, the state amount desired value determiner  84  calculates a feedback manipulated variable for converging the aforesaid difference ↑Δr gc to zero according to a predetermined feedback control law and determines the calculated feedback manipulated variable as the desired value ↑Ptotal′_cmd of ↑Ptotal′. 
     In this case, specifically, the state amount desired value determiner  84  determines ↑Ptotal′_cmd according to a proportional-differential law (PD law) such that the aforesaid difference ↑Δr gc is converged to zero. In other words, the state amount desired value determiner  84  determines ↑Ptotal′_cmd according to the following expression 4-1 from the difference ↑Δr gc and a first-order differential value ↑Δr gc′ thereof.
 
↑ P total′ —   cmd=Kp 1 *↑Δr gc+Kv 1 *↑Δr gc′   Expression 4-1
 
     In expression 4-1, Kp1 and Kv1 denote a proportional gain and a differential gain, respectively, of specified values set beforehand. These gains Kp1 and Kv1 are expressed by diagonal matrixes (third-order diagonal matrixes in the present embodiment). 
     Regarding the processing for calculating Ltotal′_cmd, the state amount desired value determiner  84  calculates a difference ↑Δrg_xy between the horizontal position of a desired ZMP of the robot  1  on the global coordinate system (the position in the roll-axis direction and the position in the pitch-axis direction) and the measurement value of the an actual horizontal position ↑r gc_act_xy of the overall center-of-gravity point of the robot  1  (=the component of the aforesaid ↑r gc_act in the roll-axis direction and the component thereof in the pitch-axis direction). In this case, the horizontal position of the desired ZMP of the robot  1  is input to the state amount desired value determiner  84  from the gait generator  71 . 
     Further, the state amount desired value determiner  84  calculates an inclination angle ↑θpend (an inclination angle about two axes, namely, the roll axis and the pitch axis) relative to the vertical direction of an inverted pendulum in the case where the overall center-of-gravity point of the robot  1  is regarded as the mass point of the inverted pendulum which uses a desired ZMP as its supporting point, and also calculates the first-order differential value ↑θpend′ thereof (=d↑θpend/dt) from the aforesaid difference ↑Δr g_xy. More specifically, if the component of ↑θpend about the roll axis and the component thereof about the pitch axis are denoted by θpend_x and θpend_y, respectively, then θpend_x and θpend_y are calculated from the component ↑Δr g_y of the aforesaid difference ↑Δr g_xy in the pitch-axis direction and the component ↑Δr g_x thereof in the roll-axis direction, respectively, according to expressions 4-2a and 4-2b given below.
 
θpend —   x =tan−1(Δ r g   —   y/h )≈Δ r g   —   y/h   Expression 4-2a
 
θpend —   y =tan−1(Δ r g   —   x/h )≈Δ r g   —   x/h   Expression 4-2b
 
     Then, the first-order differential values (temporal change rates) of these θpend_x and θpend_y are calculated, thereby calculating each component of ↑θpend_xy′. 
     In expressions 4-2a and 4-2b, “h” denotes a specified value set beforehand as a mean height of the overall center-of-gravity point of the robot  1 . 
     Subsequently, according to a predetermined feedback control law, the state amount desired value determiner  84  calculates the feedback manipulated variable (control input) for converging the inclination angle ↑θpend (=[θpend_x, θpend_y] T ) of the inverted pendulum calculated as described above to zero, and determines the calculated feedback manipulated variable as the desired value ↑Ltotal′_cmd of ↑Ltotal′. 
     In this case, specifically, the state amount desired value determiner  84  determines ↑Ltotal′_cmd such that the aforesaid inclination angle ↑θpend is converged to zero according to the PD law (proportional-differential law). More specifically, the state amount desired value determiner  84  determines ↑Ltotal′_cmd according to the following expression 4-3 from the inclination angle ↑θpend and the first-order differential value ↑θpend′ thereof.
 
↑ L total′ —   cmd=Kp 2*↑θpend+ Kv 2*↑θpend′  Expression 4-3
 
     In expression 4-3, Kp2 and Kv2 denote a proportional gain and a differential gain, respectively, of specified values set beforehand. These gains Kp2 and Kv2 are expressed by diagonal matrixes (second-order diagonal matrixes in the present embodiment). 
     Thus, the desired value of ↑Ptotal′_cmd of ↑Ptotal′ and the desired value ↑Ltotal′_cmd of ↑Ltotal′ are determined by the processing described above, thereby determining the state amount desired value vector ↑S1′_cmd (=[↑Ptotal′_cmd, ↑Ltotal_cmd] T ). 
     In this case, according to the present embodiment, ↑Ptotal′_cmd of ↑S1′_cmd is determined such that the position of the overall center-of-gravity point ↑r gc_act of the robot  1  converges to the desired position ↑r gc_cmd. 
     Further, ↑Ltotal′_cmd of ↑S1′_cmd is determined such that, in the case where the actual overall center-of-gravity point of the robot  1  is regarded as the mass point of an inverted pendulum, the inclination angle ↑θpend of the inverted pendulum converges to zero (in other words, such that the actual overall center-of-gravity point of the robot  1  converges to a position above a desired ZMP in the vertical direction). 
     Supplementarily, θpend_x and θpend_y are approximately proportional to Δr g_y and Δr g_x. Hence, the processing for calculating θpend_x and θpend_y may be omitted, and ↑Ltotal′_cmd_xy may be determined such that Δr g_y and Δr g_x converge to zero according to a feedback control law (e.g., the PD law). For example, ↑Ltotal′_cmd_xy may be calculated according to an expression in which ↑θpend of the right side of expression 4-3 has been replaced by a vector [Δr g_y, Δr g_x] T . 
     Thus, after the processing by the basic parameter group determiner  81 , the contact portion Jacobian matrix determiner  82 , the state amount Jacobian matrix determiner  83 , and the state amount desired value determiner  84  is carried out, the desired joint driving torque determiner  72  carries out the processing by the control input determiner  85 . 
     The control input determiner  85  is a functional unit which sequentially carries out the processing for determining a desired joint driving torque of each joint of the robot  1  required to achieve a desired value related to a state amount to be controlled (the aforesaid state amount desired value vector ↑S1′_cmd). Hereinafter, a vector formed of desired joint driving torques of joints, which are determined by the control input determiner  85 , arranged as the components thereof will be denoted by the desired joint driving torque ↑τ1_cmd. 
     In the description of the present embodiment, ↑τ1_cmd is a vector of the same order as the aforesaid driving force dependent generalized force vector ↑τ (the same order as the generalized variable vector ↑q). In this case, of the components of ↑τ1_cmd, the components corresponding to the position and the posture of the representative link (the lower body  10 ) of the robot  1  are both zero. 
     Here, the basic expression used for the processing by the control input determiner  85  will be described below. 
     In the case where a portion of the robot  1  in contact with the external world is subjected to an external force, a robot motion model expressing the relationship between a motion of the robot  1  and a force (a motional equation) is represented by the following expression 5-1 derived by adding an equivalent of the influence of the external force to the aforesaid expression 1-1 representing a robot motion model in the case where the external force is zero.
 
 M*↑q″+↑N+↑G+Jc   T   *↑Fc=↑τ   Expression 5-1
 
     In the expression 5-1, ↑Fc denotes an external force acting on the contact portion representative element caused by an external force acting on the portion of the robot  1  in contact with the external world, that is, the foot  18 R and/or the foot  18 L in the present embodiment (specifically, a total external force acting on the robot  1  through the intermediary of the foot  18 R and/or the foot  18  when the contact portion representative element is expressed as the portion on which the external force acts). Jc T  denotes a transposed matrix of the contact portion Jacobian matrix Jc. 
     As described below, the aforesaid expression 5-1 can be transformed to expression 5-11, which will be discussed later, by defining the motion of a contact portion of the robot  1  subjected to an external force, i.e., the motion of the aforesaid contact portion representative element (the motion is supposed to match a predetermined motion). 
     Multiplying both sides of expression 5-1 by a matrix Jc*M −1  results in the following expression 5-2.
 
 Jc*↑q″+Jc*M   −1 *(↑ N+↑G )+ Jc*M   −1   *Jc   T   *↑Fc=Jc*M   −1 *↑τ  Expression 5-2
 
     Meanwhile, the following expression 5-3 is derived by subjecting both sides of the aforesaid expression 2-3, which indicates the relationship between a first-order differential value ↑Xc′ of the contact portion representative element position vector ↑Xc as the motion velocity of the contact portion representative element and the first-order differential value ↑q′ of the generalized variable vector ↑q, to first-order differentiation and then by carrying out expression formation.
 
 Jc*↑q″=↑Xc″−Jc′*↑q′   Expression 5-3
 
     In this expression 5-3, Jc′ denotes a first-order differential value of the aforesaid contact portion Jacobian matrix Jc (=dJc/dt) and ↑q″ denotes a second-order differential value of ↑q (=d 2 ↑q/dt 2 ). 
     Further, the following expression 5-4 is derived by substituting expression 5-3 into expression 5-2.
 
↑ Xc″−Jc′*↑q′+Jc*M   −1 *(↑ N+↑G )+ Jc*M   −1   *Jc   T   *↑Fc=Jc*M   −1 *↑τ  Expression 5-4
 
     Here, a weight matrix Wat (diagonal matrix) is used to introduce a matrix Jc# and a matrix Rc defined by the following expressions 5-5 and 5-6, respectively.
 
 Jc#=Wat   T   *Jc   T   *Rc   T   Expression 5-5
 
 Rc ≡(( Jc*Wat*Jc   T ) −1 ) T   Expression 5-6
 
     The matrix Jc# defined by expression 5-5 means a generalized weighted pseudo inverse matrix of the contact portion Jacobian matrix Jc. 
     In this case, as the aforesaid weight matrix Wat may use, for example, the inverse matrix of the aforesaid inertia matrix M. If Wat=M −1 , then (Jc#) T =Rc*Jc*Wat=Rc*Jc*M −1 . Hence, the following expression 5-7 is derived by multiplying both sides of the aforesaid expression 5-4 by the matrix Rc (provided that Wat=M −1 ). 
     More specifically, the aforesaid expression 5-1 is transformed to the following expression 5-7 by using matrixes Jc# and Rc defined by the following expressions 5-8 and 5-9, respectively, obtained as Wat=M −1  according to expressions 5-5 and 5-6.
 
 Rc*↑Xc″−Rc*Jc′*↑q′+ ( Jc# ) T *(↑ N+↑G )+ ↑Fc =( Jc# ) T *↑τ  Expression 5-7
 
where
 
 Jc# ≡( M   −1 ) T   *Jc   T   *Rc   T   Expression 5-8
 
 Rc ≡(( Jc*M   −1   *Jc   T ) −1 ) T   Expression 5-9
 
     M −1  denotes a diagonal matrixeso that (M −1 ) T =M −1 . 
     Here, a case is assumed where a motion of the contact portion representative element is defined and the second-order differential value ↑Xc″ of the contact portion representative element position vector ↑Xc (=d 2 ↑Xc/dt 2 ) is set to coincide with a predetermined desired value ↑C (a case where the motion accelerations of the contact portion representative points a 1  to a 4  are respectively set to coincide with certain predetermined desired values). In other words, a case is assumed where the motions of the feet  18 R and  18 L, which are contact portions, are defined such that ↑Xc″ equals ↑C. 
     In this case, the following expression 5-10 is derived from the above expression 5-7.
 
↑ Fc =( Jc# ) T *(↑τ−↑ N−↑G )− Rc*↑C−Rc*Jc′*↑q′   Expression 5-10
 
     This expression 5-10 is substituted into the above 5-1, and further, a matrix Tc, a vector ↑Cc, a matrix Pc, and a generalized force vector ↑τcmpn, which are defined by the following expressions 5-12, 5-13, 5-14, and 5-15, respectively, are used thereby to provide the following expression 5-11.
 
 M*↑q″+Tc*↑q′=Pc *(↑τ−↑τ cmpn )  Expression 5-11
 
where
 
 Tc≡−Jc   T   *Rc*Jc′   Expression 5-12
 
↑ Cc≡−Jc   T   *Rc*↑C   Expression 5-13
 
 Pc≡I−Jc   T *( Jc #) T   Expression 5-14
 
↑τ cmpn ≡(↑ N+↑G )− Pc   −1   *↑Cc   Expression 5-15
 
     The above expression 5-11 denotes the motion equation of the robot  1 , which applies when the motion of a contact portion representative element is specified, in a form excluding an external force ↑Fc. 
     An arbitrary state amount ↑S having a value that depends on one or more components of the aforesaid generalized variable vector ↑q of the robot  1  is assumed, and it is also assumed that the relationship between the ↑S and the first-order differential value ↑q′ of the generalized variable vector ↑q is denoted by the following expression 5-16 by using a certain Jacobian matrix Js.
 
↑ S=Js*↑q′   Expression 5-16
 
     This expression 5-16 is a generalized version of the above expression 3-1. Although ↑S in expression 5-16 does not have to be a vector (may be a scalar), it is denoted as a vector for the convenience of consistency with the above expression 3-1 in this description. 
     Here, the above expression 5-11 is transformed to the following expression 5-20 by using the relational expression of the following expression 5-17 derived from the above expression 5-16.
 
 Js*↑q″=↑S′−Js′*↑q′   Expression 5-17
 
↑ S ′+( Js*M   −1   *Tc−Js′ )*↑ q′ =( Js*M   −1   *Pc )*(↑τ−↑τ cmpn )  Expression 5-20
 
     Thus, determining the values of the matrixes Js, M, Tc, Js′, and Pc in expression 5-20 and the values of ↑S′, ↑q′, and ↑τcmpn makes it possible to determine the driving-force-dependent generalized force vector ↑τ (by extension, the joint driving torque of each joint) according to the above expression 5-20. 
     Here, the matrix of the right side of expression 5-20 (Js*M −1 *Pc) (hereinafter defined as As≡Js*M −1 *Pc) is not necessarily a regular matrix (i.e., an inverse matrix As −1  of As does not necessarily exist). 
     Therefore, in the present embodiment, ↑τ is calculated according to expression 5-22 given below by using a matrix Bs determined from the matrix As according to an expression 5-21a or 5-21b given below. 
     In the case where As(≡Js*M −1 *Pc) is a regular matrix;
 
 Bs=As   −1   Expression 5-21a
 
     In the case where As (≡Js*M −1 *Pc) is not a regular matrix;
 
 Bs=As   T *( k*I+As*As   T ) −1   Expression 5-21b
         where   k: positive real number determined such that (k*I+As*As T ) becomes a regular matrix
 
↑τ= Bs *(↑ S′ +( Js*M   −1   *Tc−Js′ )*↑ q′ )+↑τ cmpn   Expression 5-22
       

     This expression 5-22 is the basic expression used to calculate the desired joint driving torque ↑τ1_cmd in the present embodiment. In this case, the matrix Bs determined by the above expression 5-21b means a pseudo inverse matrix of the matrix As. 
     In the following description, for an arbitrary matrix A, “A −1 ” will mean a matrix matching a matrix B determined by the following expression 5-23 (a pseudo inverse matrix) if the matrix A is not a regular matrix. On the other hand, if the matrix A is a regular matrix, then A −1  will mean an inverse matrix of A in its original meaning. Further, whether A is a regular matrix or not, A −1  will be referred to as the inverse matrix of A for the sake of convenience.
 
 B=A   T *( k*I+A*A   T ) −1   Expression 5-23
         where   k: positive real number determined such that (k*I+A*A T ) becomes a regular matrix       

     In the above expression 5-22, the matrix Bs denotes the inverse matrix of As (=As −1 ) in the aforesaid meaning. 
     Supplementarily, both sides of the above expression 5-20 may be multiplied by an appropriate matrix to constitute an expression for calculating ↑τ. For example, the expression for calculating ↑τ may be constituted in the same manner as described above on the basis of an expression obtained by multiplying both sides of expression 5-20 from the left by a matrix Rsc defined by Rsc≡(Js*Pc T *M −1 *Js T ) −1 ) T . In this case, multiplying both sides of expression 5-20 from the left by the matrix Rsc causes both sides of expression 5-20 to be converted to the dimension of a generalized force vector. Further, the expression for calculating ↑τ will be an expression obtained by replacing the matrix Bs of expression 5-22 by a matrix ((Rsc*Js*M −1 *Pc)*Rsc). 
     Based on the above description, the following will specifically describe the processing by the control input determiner  85 . 
     To calculate the desired joint driving torque ↑τ1_cmd, the control input determiner  85  sequentially receives the inertia matrix M, the centrifugal force/Coriolis force-dependent generalized force vector ↑N and the gravity-dependent generalized force vector ↑G (or the nonlinear term generalized force vector (↑N+↑G)) determined by the basic parameter group determiner  81 , the contact portion Jacobian matrix Jc determined by the contact portion Jacobian matrix determiner  82 , the state amount Jacobian matrix Js1 determined by the state amount Jacobian matrix determiner  83 , and the state amount desired value vector ↑S1′_cmd determined by the state amount desired value determiner  84 . In addition, the control input determiner  85  sequentially receives the measurement values of the temporal change rates (first-order differential values) ↑q′ of the generalized variable vector ↑q (the measurement values of the moving velocity of the lower body  10  and the temporal change rate (angular velocity) of the posture angle and the measurement value of the temporal change rate (angular velocity) of the rotational angle of each joint), and also receives a contact portion representative element desired acceleration ↑C, which will be discussed later. 
     Then, the control input determiner  85  determines the value of each parameter used for calculating the right side of expression 5-22 by using a predetermined input value, and calculates the right side of expression 5-22 by using the determined parameter, thereby calculating the desired joint driving torque ↑τ1_cmd. 
     More specifically, the control input determiner  85  carries out processing for calculating the matrixes Tc and Pc and the generalized force vector ↑τcmpn necessary to calculate expression 5-22 by using a predetermined input value. 
     In this case, the control input determiner  85  uses the inertia matrix M, the centrifugal force/Coriolis force-dependent generalized force vector ↑N and the gravity-dependent generalized force vector ↑G (or the nonlinear term generalized force vector (↑N+↑G)) and the contact portion Jacobian matrix Jc, which have been received, to calculate Tc, Pc, and ↑τcmpn according to the definitional equations of the above expressions 5-8, 5-9, 5-12, 5-13, 5-14, and 5-15. 
     To be more specific, the control input determiner  85  calculates the matrixes Jc# and Rc according to the above definitional equations 5-8 and 5-9 from the received inertia matrix M and the contact portion Jacobian matrix Jc. Further, from these matrixes Jc# and Rc and the received Jc, the control input determiner  85  calculates the matrixes Tc and Pc according to the above definitional equations 5-12 and 5-14. In this case, Jc′, i.e., the temporal change rate (a first-order differential value) of Jc, is calculated from the time series of the received Jc. 
     Further, the control input determiner  85  calculates the vector Cc according to the definitional equation of the above expression 5-13 from the received Jc and the calculated Re, and the desired value ↑C of the second-order differential value ↑Xc″ of the contact portion representative element position vector ↑Xc (the motion accelerations of the contact portion representative points a 1  to a 4 ). Hereinafter, the desired value ↑C will be referred to as the contact portion representative element desired acceleration ↑C. 
     Here, according to the present embodiment, the desired accelerations of the four contact portion representative points a 1  to a 4 , which are the components of the contact portion representative element desired acceleration ↑C, are preset to zero for all the contact portion representative points a 1  to a 4 . In other words, according to the present embodiment, the contact portion representative points a 1  to a 4 , which constitute a contact portion representative element, remaining stationary relative to a floor or moving at a constant velocity is established as a target for the motions of the feet  18 R and  18 L serving as the contact portions. 
     Then, the control input determiner  85  calculates the vector ↑Cc according to the definitional equation of the above expression 5-13 by using the contact portion representative element desired acceleration ↑C. In the present embodiment, the contact portion representative element desired acceleration ↑C is a zero vector, so that all components of the vector ↑Cc are zero. Hence, in actuality, the processing for calculating ↑Cc is unnecessary, and the components of ↑Cc may simply be set to zero. 
     Further, the control input determiner  85  calculates ↑τcmpn according to the definitional equation of the above expression 5-15 from the above ↑Cc, the calculated Pc, and the received ↑N and ↑G. 
     The processing for calculating Tc, Pc, and ↑τcmpn is carried out as described above. 
     Supplementarily, in the case where ↑N+↑G≈↑G applies, the ↑τcmpn may be calculated according to an expression derived from the above expression 5-15 without ↑N. 
     The control input determiner  85  carries out the processing for calculating the matrix Bs used for the calculation of expression 5-22. In the following description, the matrix Bs corresponding to the state amount to be controlled will be denoted by Bs1. 
     In this case, the control input determiner  85  calculates the matrix Bs1 from the received inertia matrix M and the state amount Jacobian matrix Js1 and the aforesaid calculated value of Pc. More specifically, the inverse matrix As −1  of a matrix (Js1*M −1 *Pc)(≡As) is determined as Bs1. As described above, if As is not a regular matrix, then As −1  is a pseudo inverse matrix calculated according to the above expression 5-23. 
     Subsequently, the control input determiner  85  calculates ↑τ by carrying out the computation of the right side of the above expression 5-22 (more specifically, the computation of an expression derived by replacing Bs, ↑S′, Js, Js′, and ↑q′ of the right side of expression 5-22 by the measurement value ↑q′, Bs1, ↑S1′cmd, Js1, and Js1′, respectively) by using the calculated values of Bs1, Tc and ↑τcmpn, the received state amount desired value vector ↑S1′_cmd, state amount Jacobian matrix Js1 and inertia matrix M, and the measurement value of ↑q′, and determines the calculated value of ↑τ as the desired joint driving torque ↑τ1_cmd. In this case, Js1′ (=the first-order differential value of Js1) is calculated from the time series of the state amount Jacobian matrix Js1. 
     In the present embodiment, the desired joint driving torque ↑τ1_cmd is determined by the processing carried out by the control input determiner  85  described above on the basis of the calculated values of M, ↑N, ↑G, Jc, and Js1 (or the calculated values of M, ↑G, Jc, and Js1), the state amount desired value vector ↑S1′_cmd, the measurement value of ↑q′, and the contact portion representative element desired acceleration ↑C. 
     In this case, ↑τ1_cmd is determined such that the temporal change rate ↑S1′ of the state amount vector ↑S1 coincides with the state amount desired value vector ↑S1′_cmd while setting the motion accelerations of the contact portion representative points a 1  to a 4  to coincide with the acceleration defined by the contact portion representative element desired acceleration ↑C. 
     At this time, as described above, ↑Ptotal′_cmd of ↑S1′_cmd is determined such that the position of the overall center-of-gravity point ↑r gc_act of the robot  1  converges to the desired position ↑r gc_cmd, and ↑Ltotal_cmd of ↑S1′_cmd is determined such that, in the case where the actual overall center-of-gravity point of the robot  1  is regarded as the mass point of an inverted pendulum, the inclination angle ↑θpend of the inverted pendulum converges to zero. Hence, ↑τ1_cmd is consequently determined such that the overall center-of-gravity point of the robot  1  can be maintained at a proper position. 
     After carrying out the processing by the control input determiner  85  as described above, the desired joint driving torque determiner  72  carries out the processing by the compliance controller  86 . 
     The compliance controller  86  is a functional unit which determines the desired value of the total floor reaction force (overall floor reaction force) acting on the robot  1  on the basis of the desired joint driving torque ↑τ1_cmd determined by the control input determiner  85 , and then sequentially carries out processing for correcting the desired joint driving torque ↑τ1_cmd to bring the difference between the determined desired value and the measurement value of the actual total floor reaction force close to zero. Hereinafter, the torque obtained by correcting the desired joint driving torque ↑τ1_cmd by the compliance controller  86  will be referred to as the corrected desired joint driving torque ↑τ1c_cmd. In the present embodiment, the corrected desired joint driving torque ↑τ1c_cmd is used as the final desired joint driving torque for controlling the electric motor  50  (the desired joint driving torque to be input to the joint drive controller  73 ) at each arithmetic processing cycle of the control device  60 . 
     In order to calculate the corrected desired joint driving torque ↑τ1c_cmd, the compliance controller  86  sequentially receives the centrifugal force/Coriolis force-dependent generalized force vector ↑N and the gravity-dependent generalized force vector ↑G (or the nonlinear term generalized force vector (↑N+↑G)) determined by the basic parameter group determiner  81 , the desired joint driving torque ↑τ1_cmd determined by the control input determiner  85 , and the contact portion representative element defining ratio r at determined by the contact portion Jacobian matrix determiner  82  (or the contact portion representative element position vector ↑Xc defined thereby). 
     In addition, the compliance controller  86  sequentially receives the matrixes Rc, Jc′ and Jc# calculated by the control input determiner  85 , the contact portion representative element desired acceleration ↑C (zero vector in the present embodiment), the measurement values of actual floor reaction forces ↑FM_act_R and ↑FM_act_L acting on the feet  18 R and  18 L, respectively, the measurement value of the moving velocity of the representative link  10 , the measurement value of the posture angle of the representative link (the lower body)  10  of the robot  1  and the measurement value of the temporal change rate (angular velocity) thereof, and the measurement value of the rotational angle of each joint and the measurement value of the temporal change rate (angular velocity) thereof. 
     The measurement values of the floor reaction forces ↑FM_act_R and ↑FM_act_L are the measurement values recognized by the control device  60  on the basis of the detection outputs of the aforesaid force sensors  63 R and  63 L. 
     The compliance controller  86  first calculates the right side of the above expression 5-10 from the received ↑N, ↑G (or ↑N+↑G), Re, Jc′, Jc#, and ↑τ1_cmd, and the measurement value of the first-order differential value ↑q′ of the generalized variable vector ↑q, thereby calculating a contact portion representative element desired acting force ↑Fc_cmd as a vector formed of an array of the desired values of forces (translational forces) acting on the contact portion representative points a 1  to a 4 . 
     More specifically, ↑Fc_cmd is calculated according to expression 7-1 given below. The contact portion representative element desired acting force ↑Fc_cmd means an estimated value of a force (translational force) to be applied to the robot  1  at the contact portion representative points a 1  to a 4  in the case where the motion acceleration of the contact portion representative element is set to coincide with the contact portion representative element desired acceleration ↑C and the actual joint driving torque of each joint of the robot  1  is set to coincide with the desired joint driving torque defined by ↑τ1_cmd.
 
↑ Fc   —   cmd =( Jc# ) T *(↑τ1 —   cmd−↑N−↑G )− Rc*↑C−Rc*Jc′*↑q′   Expression 7-1
 
     If ↑N+↑G≈↑G applies, then ↑N in expression 7-1 may be omitted. 
     Subsequently, the compliance controller  86  converts the contact portion representative element desired acting force ↑Fc_cmd to a desired total floor reaction force ↑FMt_cmd, which is the desired value of a force to be applied to a predetermined overall floor reaction force acting point (e.g., a desired ZMP) serving as the acting point of an overall floor reaction force relative to the robot  1 . 
     In the present embodiment, the desired total floor reaction force ↑FMt_cmd is a vector formed of translational force components in the directions of three axes and moment components about three axes (a vector having six degrees of freedom). In this case, the translational force component of ↑FMt_cmd will be the result of adding up the desired translational forces acting on the individual contact portion representative points a 1  to a 4 . Further, the moment component of ↑FMt_cmd will be the result of adding up the moments generated about the total floor reaction force acting point by the desired translational forces acting on the contact portion representative points a 1  to a 4 . 
     The conversion of ↑Fc_cmd to ↑FMt_cmd is effected by multiplying a conversion matrix, which has a component determined on the basis of the positional relationship between each of the contact portion representative points a 1  to a 4  and the aforesaid floor reaction force acting point, by ↑Fc_cmd. 
     Further, the compliance controller  86  calculates the measurement value of the total floor reaction force ↑FMt_act actually acting on the aforesaid overall floor reaction force acting point from the measurement values of the actual floor reaction forces ↑FM_act_R and ↑FM_act_L acting on the feet  18 R and  18 L, respectively. 
     In this case, according to the present embodiment, the measurement value of ↑FM_act_R is a vector formed of the translational force components in the directions of three axes and the moment components about three axes, which act on a predetermined floor reaction force acting point corresponding to the right foot  18 R. Similarly, the measurement value of ↑FM_act_L is a vector formed of the translational force components in the directions of three axes and the moment components about three axes, which act on a predetermined floor reaction force acting point corresponding to the left foot  18 L. 
     Further, the compliance controller  86  adds up the result obtained by converting the measurement value of ↑FM_act_R to a force (a set of a translational force component and a moment component) acting on an overall floor reaction force acting point by using a conversion matrix, which has a component determined on the basis of the positional relationship between the overall floor reaction force acting point and the floor reaction force acting point of the right foot  18 R and the result obtained by converting the measurement value of ↑FM_act_L to a force (a set of a translational force component and a moment component) acting on an overall floor reaction force acting point by using a conversion matrix, which has a component determined on the basis of the positional relationship between the overall floor reaction force acting point and the floor reaction force acting point of the left foot  18 L. Thus, the measurement value of the total floor reaction force ↑FMt_act is calculated. 
     The translational force component of the measurement value of ↑FMt_act provides a translational force of the total sum (vector sum) of the translational force component of ↑FM_act_R and the translational force component of ↑FM_act_L. Further, the moment component of ↑FMt_act provides a moment of the total sum (vector sum) of the moment generated about the overall floor reaction force acting point by the moment component of ↑FM_act_R, the moment component of ↑FM_act_L and the translational force component of ↑FM_act_R, and the moment generated about the overall floor reaction force acting point by the translational force component of ↑FM_act_L. 
     After calculating the measurement values of the desired total floor reaction force ↑FMt_cmd and the total floor reaction force ↑FMt_act as described above, the compliance controller  86  calculates a difference ↑ΔFMt therebetween (=↑FMt_cmd−↑FMt_act). Then, the compliance controller  86  calculates, according to a predetermined feedback control law, a correction amount ↑Δτt of the desired joint driving torque ↑τ1_cmd serving as the manipulated variable for converging the difference ↑ΔFMt to zero. 
     To be more specific, according to the present embodiment, the compliance controller  86  converts a vector Gk*↑ΔFMt, which is obtained by multiplying the difference ↑ΔFMt by a proportional gain Gk of a predetermined value, into an error ↑ΔFc relative to the desired value (the contact portion representative element desired acting force ↑Fc_cmd) of an actual force (translational force) acting on the contact portion representative points a 1  to a 4 . 
     The conversion is effected by multiplying the vector Gk*↑ΔFMt by an inverse matrix of a conversion matrix for converting the aforesaid ↑Fc_cmd into ↑FMt_cmd. In the present embodiment, the conversion matrix for converting ↑Fc_cmd into ↑FMt_cmd is not a regular matrixeso that the inverse matrix of the conversion matrix is a pseudo inverse matrix determined according to the above expression 5-23. 
     Further, the compliance controller  86  multiplies the error ↑ΔFc by an inverse matrix (Jc# T ) −1  of a matrix Jc# T , as indicated by the following expression 7-2, thereby calculating a correction amount ↑Δτt of a resultant desired joint driving torque ↑τt_cmd.
 
↑Δτ t =( Jc#   T ) −1   *↑ΔFc   Expression 7-2
 
     Thus, the correction amount ↑Δτt is determined so as to converge the difference ↑ΔFMt to zero according to a proportional law serving as a feedback control law. 
     Then, the compliance controller  86  adds the correction amount ↑Δτt to the desired joint driving torque ↑τ1_cmd as indicated by expression 7-3 given below thereby to determine a corrected desired joint driving torque ↑τ1c_cmd.
 
↑τ1 c   —   cmd=↑τ 1 —   cmd+↑Δτt   Expression 7-3
 
     The processing by the compliance controller  86  described above corrects the desired joint driving torque ↑τ1_cmd so as to bring the difference between the desired total floor reaction force ↑FMt_cmd, which is the total floor reaction force corresponding to the desired joint driving torque ↑τ1_cmd, and the measurement value of an actual total floor reaction force ↑FMt_act close to zero. 
     According to the present embodiment, the corrected desired joint driving torque ↑τ1c_cmd determined as described above (more specifically, the component of ↑τ1c_cmd corresponding to the rotational angle of each joint) is input to the joint drive controller  73 . 
     This completes the detailed explanation of the processing carried out by the desired joint drive torque determiner  72 . 
     According to the present embodiment described above, the desired value of the first-order differential value ↑Ptotal′ of the total translational momentum of the state amount desired value vector ↑S1′_cmd is determined such that the position of the overall center-of-gravity point of the robot  1  including the position in the vertical direction converges to the desired position of the overall center-of-gravity point of a desired gait generated by the gait generator  71 . In other words, the desired value of ↑Ptotal′ corresponding to the desired value of the translational inertial force generated by a motion of the robot  1  is determined such that the actual position of the overall center-of-gravity point of the robot  1  converges to a proper desired position. 
     Further, the desired value of the first-order differential value ↑Ltotal′ of the total angular momentum in ↑S1′_cmd is determined such that the horizontal position of the overall center-of-gravity point of the robot  1  converges to a position in the vertical direction above a desire ZMP, which serves as the supporting point of a virtual inverted pendulum having its mass point at the overall center-of-gravity point. In other words, the desired value of ↑Ltotal′ corresponding to the desired value of a moment inertial force generated by a motion of the robot  1  is determined such that the actual horizontal position of the overall center-of-gravity point of the robot  1  is converged to a position that makes it possible to prevent the aforesaid virtual inverted pendulum from tilting. 
     Further, in the present embodiment, the desired joint driving torque ↑τ1_cmd is determined so as to allow the desired value of ↑Ptotal′ and the desired value of ↑Ltotal′ of the state amount desired value vector ↑S1′_cmd to be achieved. 
     Here, in the robot  1  according to the present embodiment, the joint drive system of each joint is provided with the torsion bar  53 , which is a spring member, thus making each joint flexible. This tends to cause the rotational angle of each joint to easily change due to the influence of an external force acting on the robot  1 , thus causing the position of the overall center-of-gravity point (especially the position thereof in the vertical direction) to easily change. 
     However, the state amount desired value vector ↑S1′_cmd is determined as described above, so that, if the actual position of the overall center-of-gravity point of the robot  1  deviates from a desired position, then the desired joint driving torque ↑τ1_cmd is determined such that the deviation is corrected. 
     This arrangement makes it possible to restrain the actual position of the overall center-of-gravity point of the robot  1  from deviating from a proper desired position. As a result, even though the joints of the robot  1  are flexible, it is possible to prevent the posture of the robot  1  from becoming unstable, thus permitting higher stability of the behaviors of the robot  1 . 
     In the present embodiment, the desired joint driving torque ↑τ1_cmd is determined according to the above expression 5-20. Therefore, the motions of the contact portion representative points a 1  to a 4  of the contact portion representative element are defined by the contact portion representative element desired acceleration ↑C, and then ↑τ1_cmd is determined on the basis of the state amount desired value vector ↑S1_cmd, which is a desired value related to the state amount to be controlled of the robot  1 . 
     With this arrangement, an appropriate ↑τ1_cmd that allows ↑C and ↑S1′_cmd to be achieved can be determined without identifying the value of an external force acting on the robot  1 , which is susceptible to the influence of a disturbance or the like. In this case, the total degree of freedom of ↑C and ↑S1′_cmd is smaller than the total degree of freedom of the entire robot  1  (the degree of freedom of the aforesaid generalized force vector ↑q), so that the joint driving torque ↑τ1_cmd can be determined without the need for identifying the rotational motions of all joints of the robot  1 . This makes it possible to enhance the flexibility of the motions of the joints of the robot  1  in response to, for example, a change in an external force acting on the robot  1 . 
     Here, the relationship of correspondence between the embodiment described above and the present invention will be supplementarily described. 
     In the present embodiment, the basic parameter group determining unit in the present invention is implemented by the aforesaid basic parameter group determiner  81 . The measurement values of the moving velocity of the lower body  10 , the posture angle and the temporal change rate (angular velocity) thereof, and the measurement values of the rotational angle of each joint and the temporal change rate (angular velocity) thereof correspond to the generalized variable observation information in the present invention. 
     Further, the contact portion Jacobian matrix calculating unit in the present invention is implemented by the aforesaid contact portion Jacobian matrix determiner  82 . In this case, the feet  18 R and  18 L of the robot  1  correspond to the contact portions in the present invention, and the contact portion representative element in the present invention is set as the element constituted by connecting the aforesaid contact portion representative points a 1  to a 4 . In the present embodiment, the measurement values of the aforesaid translational floor reaction force vertical components Fz_act_R and Fz_act_L are used as the contact state information in the present invention. 
     The state amount Jacobian matrix calculating unit in the present invention is implemented by the state amount Jacobian matrix determiner  83 . In this case, according to the present embodiment, the state amount vector ↑S constituted of the set of the total translational momentum ↑Ptotal and the total angular momentum ↑Ltotal corresponds to a predetermined type of state amount in the present invention. 
     Further, the desired driving force determining unit in the present invention is implemented by the control input determiner  85 . In this case, the desired joint driving torque ↑τ1_cmd corresponds to the desired generalized force vector τcmd in the present invention, and the component of this ↑τ1_cmd that corresponds to the rotational angle of each joint corresponds to the desired driving force in the present invention. Further, the aforesaid contact portion representative element desired acceleration ↑C corresponds to the desired value of the contact portion motion acceleration in the present invention, and the state amount desired value vector ↑S1′_cmd corresponds to the desired value of the first-order differential value of the predetermined type of state amount in the present invention. 
     The compliance controller  86  and the joint drive controller  73  implement the actuator control unit in the present invention. The processing carried out by the compliance controller  86  implements the contact portion external force desired value determining unit and the driving force correction amount calculating unit in the present invention. More specifically, the contact portion representative element desired acting force ↑Fc_cmd corresponds to the contact portion external force desired value in the present invention, and the contact portion external force desired value determining unit is implemented by the processing for calculating the ↑Fc_cmd according to the above expression 7-1. Further, the aforesaid correction amount ↑Δτt corresponds to the correction amount of a desired driving force in the present invention and the processing for calculating the correction amount ↑Δτt as described above implements the driving force correction amount calculating unit in the present invention. 
     Second Embodiment 
     A second embodiment of the present invention will now be described with reference mainly to  FIG. 6 . 
     The present embodiment differs from the first embodiment in the method for setting a contact portion representative element in the processing by a desired joint driving torque determiner  72  of a control device  60 . Accordingly, in the processing by a desired joint driving torque determiner  72 , the processing by a contact portion Jacobian matrix determiner  82 , a control input determiner  85 , and a compliance controller  86  is partly different from that of the first embodiment. The rest of the processing by the desired joint driving torque determiner  72  (the processing by a basic parameter group determiner  81 , a state amount Jacobian matrix determiner  83 , and a state amount desired value determiner  84 ), and the processing by a gait generator  71  and a joint drive controller  73  is the same as that in the first embodiment. 
     Therefore, the description of the present embodiment will be focused mainly on the different aspects from the first embodiment, and the same matters as those of the first embodiment will not be described again. 
     In the first embodiment, the motion of a contact portion representative element is defined by the motions of the four contact portion representative points a 1  to a 4 . For this reason, if the motion of the contact portion representative element is restrained, then a restriction equivalent to twelve degrees of freedom (3*4=12) out of the total degree of freedom of an entire robot  1  (the degree of freedom of the generalized force vector ↑q) will result. The present embodiment sets the contact portion representative element such that the degree of freedom subjected to a restriction when the motion of a contact portion representative element is restrained will be less than that in the first embodiment. 
     According to the present embodiment, the contact portion representative element is defined as an element subjected to the resultant force of external forces (the resultant force on all contact portions) applied from an external world to each portion of the robot  1  in contact with the external world. Further, the external force acting from the external world to each portion of the robot  1  in contact with the external world (a translational force and a moment) is regarded as an external force acting on a contact portion of the robot  1  as an elastic force attributable to the springy displacements of the position and the posture of the surface of contact between the contact portion and the external world (the displacement caused by the elastic deformation of the contact portion or a portion in the external world). Similarly, the resultant force of external forces acting on a contact portion representative element is regarded as an external force acting on a contact portion representative element as an elastic force attributable to the springy displacements (elastic deformation) of the position and the posture of the surface of contact (virtual or substantive surface of contact) between the contact portion representative element and the external world. 
     More specifically, according to the present embodiment, when the robot  1  travels on a floor surface, the contact portion representative elements in, for example, a walking motion, are defined as the elements to which the resultant force of floor reaction forces (the total floor reaction force) acting on both feet  18 R and  18 L are applied. 
     Hence, in the present embodiment, during a walking motion of the robot  1 , if a floor reaction force vector (a vector having a translational force and a moment as its components) applied to an acting point existing in the ground contact surface (the surface of contact with the floor) of the right foot  18 R is denoted by ↑FM_R=[↑F_R, ↑M_R] T , a floor reaction force vector (a vector having a translational force and a moment as its components) applied to an acting point existing in the ground contact surface (the surface of contact with the floor) of the left foot  18 L is denoted by ↑FM_L=[↑F_L, ↑M_L] T , and the resultant force vector acting on a contact portion representative element (the resultant vector of the floor reaction force vectors [↑F_R, ↑M_R] T  and [↑F_L, ↑M_L] T ) is denoted by ↑FMt=[↑Ft, ↑Mt] T , then the relationship between [↑Ft, ↑Mt] T  and [↑F_R, ↑M_R] T  and [↑F_L, ↑M_L] T  is indicated by the following expression 8-1. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ↑ 
                           FMt 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             A_R 
                             * 
                             
                               ↑ 
                               FM_R 
                             
                           
                           + 
                           
                             A_L 
                             * 
                             
                               ↑ 
                               FM_L 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ↑ 
                                     F_R 
                                   
                                   + 
                                   
                                     ↑ 
                                     F_L 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     V_R 
                                     * 
                                     
                                       ↑ 
                                       F_R 
                                     
                                   
                                   + 
                                   
                                     ↑ 
                                     M_R 
                                   
                                   + 
                                   
                                     V_L 
                                     * 
                                     
                                       ↑ 
                                       F_L 
                                     
                                   
                                   + 
                                   
                                     ↑ 
                                     M_L 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   1 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   where 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ↑ 
                         FMt 
                       
                       ≡ 
                       
                         [ 
                         
                           
                             
                               
                                 ↑ 
                                 Ft 
                               
                             
                           
                           
                             
                               
                                 
                                   ↑ 
                                   M 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
                         ↑ 
                         FM_R 
                       
                       ≡ 
                       
                         [ 
                         
                           
                             
                               
                                 ↑ 
                                 F_R 
                               
                             
                           
                           
                             
                               
                                 ↑ 
                                 M_R 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ↑ 
                         FM_L 
                       
                       ≡ 
                       
                         [ 
                         
                           
                             
                               
                                 ↑ 
                                 F_L 
                               
                             
                           
                           
                             
                               
                                 ↑ 
                                 M_L 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       A_R 
                       ≡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               0 
                             
                           
                           
                             
                               V_R 
                             
                             
                               I 
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       A_L 
                       ≡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               0 
                             
                           
                           
                             
                               V_L 
                             
                             
                               I 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Here, ↑Ft in expression 8-1 denotes a translational force vector, which is constituted of translational force components in the directions of three axes, of the resultant vector ↑FMt acting on a contact portion representative element (hereinafter referred to as the total floor reaction force vector ↑FMt), ↑Mt denotes a moment vector, which is constituted of moment components about three axes, of the total floor reaction force vector ↑FMt, ↑F_R denotes a translational force vector, which is constituted of translational force components in the directions of three axes, of the floor reaction force vector ↑FM_R acting on the right foot  18 R, ↑M_R denotes a moment vector, which is constituted of moment components about three axes, of the floor reaction force vector ↑FM_R, ↑F_L denotes a translational force vector, which is constituted of translational force components in the directions of three axes, of the floor reaction force vector ↑FM_L acting on the left foot  18 L, and ↑M_L denotes a moment vector, which is constituted of moment components about three axes, of the floor reaction force vector ↑FM_L. 
     Further, the components of the matrixes I and 0 of the matrixes A_R and A_L in expression 8-1 denote a third-order unit matrix and a third-order zero matrix, respectively, the component V_R of the matrix A_R denotes a third-order matrixeset such that V_R*↑F_R coincides with a vector product of the positional vector of the ground contact position of the right foot  18 R observed from the ground contact position of a ground contact portion representative element (hereinafter denoted by ↑Vp_R) and ↑F_R (=↑Vp_R×↑F_R), and the component V_L of the matrix A_L denotes a third-order matrixeset such that V_L*↑F_R coincides with a vector product of the positional vector of the ground contact position of the left foot  18 L observed from the position of a ground contact portion representative element (hereinafter denoted by ↑Vp_L) and ↑F_L (=↑Vp_L×↑F_L). 
     More specifically, the ground contact position of the ground contact portion representative element is the position of the acting point of the total floor reaction force vector ↑FMt relative to the ground contact portion representative element. The ground contact position of the right foot  18 R is, to be more specific, the position of the acting point of the floor reaction force vector ↑FM_R relative to the right foot  18 R, and the ground contact position of the left foot  18 L is, to be more specific, the position of the acting point of the floor reaction force vector ↑FM_L relative to the left foot  18 L. 
     Further, according to the present embodiment, the floor reaction force vectors ↑FM_R and ↑FM_L as the external forces acting on the feet  18  in contact with the floor surface as the portions of the robot  1  in contact with the external world are regarded to act on the feet  18  as an elastic force attributable to the springy displacements of the positions and the postures of the contact surfaces between the feet  18  and the floor (the displacement caused by the elastic deformation of the feet  18  or the floor). 
     Similarly, the total floor reaction force vector ↑FMt as an external force acting on a contact portion representative element in the present embodiment is regarded as an external force acting on the contact portion representative element in the form of an elastic force attributable to the springy displacements of the position and the posture (the displacements caused by the elastic deformation of the contact portion representative element or a floor) of the contact surface (substantive or virtual contact surface) between the contact portion representative element and a floor. 
     Therefore, if the vector whose component is the springy displacement amount of the position and the posture of the ground contact surface (the surface in contact with the floor) when the right foot  18 R comes in contact with the ground (the amount of the translational displacement of the ground contact surface from a reference position and the amount of the angular displacement of the ground contact surface from a reference posture) is denoted by ↑X_R≡[↑Xorg_R, ↑Xrot_R] T , the vector whose component is the springy displacement amount of the position and the posture of the ground contact surface (the surface in contact with the floor) when the left foot  18 L comes in contact with the ground (the amount of the translational displacement of the ground contact surface from a reference position and the amount of the angular displacement of the ground contact surface from a reference posture) is denoted by ↑X_L≡[↑Xorg_L, ↑Xrot_L] T , and the vector whose component is the amount of the springy displacement of the position and the posture of the ground contact surface (the surface in contact with the floor) of the contact portion representative element (the amount of the translational displacement of the ground contact surface from a reference position and the amount of the angular displacement of the ground contact surface from a reference posture) is denoted by ↑Xcc≡[↑Xcc_org, ↑Xcc_rot] T , then the relationship between ↑FM_R and ↑X_R, the relationship between ↑FM_L and ↑X_L, and the relationship between ↑FMt and ↑Xcc are indicated by the following expressions 8-2, 8-3, and 8-4, respectively. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ↑ 
                         FM_R 
                       
                       ≡ 
                       
                         [ 
                         
                           
                             
                               
                                 ↑ 
                                 F_R 
                               
                             
                           
                           
                             
                               
                                 ↑ 
                                 M_R 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               kf_R 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               km_R 
                             
                           
                         
                         ] 
                       
                       * 
                       
                         ↑ 
                         X_R 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                         
 
                       
                       ↑ 
                       X_R 
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ↑ 
                               Xorg_R 
                             
                           
                         
                         
                           
                             
                               ↑ 
                               Xrot_R 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   2 
                 
               
             
             
               
                 
                   
                     
                       
                         ↑ 
                         FM_L 
                       
                       ≡ 
                       
                         [ 
                         
                           
                             
                               
                                 ↑ 
                                 F_L 
                               
                             
                           
                           
                             
                               
                                 ↑ 
                                 M_L 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               kf_L 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               km_L 
                             
                           
                         
                         ] 
                       
                       * 
                       
                         ↑ 
                         X_L 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                         
 
                       
                       ↑ 
                       X_L 
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ↑ 
                               Xorg_L 
                             
                           
                         
                         
                           
                             
                               ↑ 
                               Xrot_L 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   3 
                 
               
             
             
               
                 
                   
                     
                       
                         ↑ 
                         FMt 
                       
                       ≡ 
                       
                         [ 
                         
                           
                             
                               
                                 ↑ 
                                 Ft 
                               
                             
                           
                           
                             
                               
                                 
                                   ↑ 
                                   M 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               kfcc 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               kmcc 
                             
                           
                         
                         ] 
                       
                       * 
                       
                         ↑ 
                         Xcc 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                         
 
                       
                       ↑ 
                       Xcc 
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ↑ 
                               Xcc_org 
                             
                           
                         
                         
                           
                             
                               ↑ 
                               Xcc_rot 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     Here, ↑Xorg_R in expression 8-2 denotes a translational displacement amount vector of the springy displacement amount ↑X_R in the directions of three axes, ↑Xrot_R denotes the angular displacement amount vector about three axes in the springy displacement amount ↑X_R, kf_R denotes a third-order diagonal matrix indicating a spring constant (spring constant related to translational displacement) that expresses the relationship between each component of ↑F_R and each component of ↑Xorg_R by a relationship indicated by ↑F_R=kf_R*↑Xorg_R, and km_R denotes a third-order diagonal matrix indicating a spring constant (spring constant related to rotational angle displacement) that expresses the relationship between each component of ↑M_R and each component of ↑Xrot_R by a relationship indicated by ↑M_R=km_R*↑Xrot_R. 
     Further, ↑Xorg_L in expression 8-3 denotes a translational displacement amount vector of the springy displacement amount ↑X_L in the directions of three axes, ↑Xrot_L denotes the angular displacement amount vector about three axes in the springy displacement amount ↑X_L, kf_L denotes a third-order diagonal matrix indicating a spring constant (spring constant related to translational displacement) that expresses the relationship between each component of ↑F_L and each component of ↑Xorg_L by a relationship indicated by ↑F_L=kf_L*↑Xorg_L, and km_L denotes a third-order diagonal matrix indicating a spring constant (spring constant related to rotational angle displacement) that expresses the relationship between each component of ↑M_L and each component of ↑Xrot_L by a relationship indicated by ↑M_L=km_L*↑Xrot_L. 
     Further, ↑Xcc_org in expression 8-4 denotes a translational displacement amount vector of the springy displacement amount ↑Xcc in the directions of three axes, ↑Xcc_rot denotes the angular displacement amount vector about three axes in the springy displacement amount ↑Xcc, kfcc denotes a third-order diagonal matrix indicating a spring constant (spring constant related to translational displacement) that expresses the relationship between each component of ↑Ft and each component of ↑Xcc_org by a relationship indicated by ↑Ft=kfcc*↑Xcc_org, and kmcc denotes a third-order diagonal matrix indicating a spring constant (spring constant related to rotational angle displacement) that expresses the relationship between each component of ↑Mt and each component of ↑Xcc_rot by a relationship indicated by ↑Mt=kmcc*↑Xcc_rot. 
     Further, according to the present embodiment, the ground contact position (the acting point of ↑FMt) of a contact portion representative element coincides with the ground contact position of the right foot  18 R (the acting point of ↑FM_R) in a one-leg ground contact state wherein only the right foot  18 R comes in contact with the ground. In a one-leg ground contact state wherein only the left foot  18 L comes in contact with the ground, the ground contact position (the acting point of ↑FMt) of a contact portion representative element coincides with the ground contact position of the left foot  18 L (the acting point of ↑FM_L). In a two-leg ground contact state wherein both feet  18 R and  18 L come in contact with the ground, the ground contact position (the acting point of ↑FMt) of a contact portion representative element is at an internally dividing point of a line segment connecting the ground contact positions of the two feet  18 R and  18 L. 
     More specifically, referring to  FIG. 6 , the relationship between a position vector ↑Vp_R of the acting point of the floor reaction force vector ↑FM_R applied to the right foot  18 R observed from the acting point of the total floor reaction force vector ↑FMt acting on the contact portion representative element and a position vector ↑Vp_L of the acting point of the floor reaction force vector ↑FM_L applied to the left foot  18 L observed from the acting point of ↑FMcc is given by the following expression 8-5 on the basis of the contact portion representative element defining ratio r at described in the first embodiment.
 
 r at*↑Vp   —   R=− (1 −r at )*↑ Vp   —   L   Expression 8-5
 
     In other words, the ground contact position of the contact portion representative element (the acting point of ↑FMcc) is a position expressed by |↑Vp_R|:|↑Vp_L|=(1−r at):r at on a line segment connecting the ground contact positions of both feet  18 R and  18 L in the two-leg ground contact state (refer to  FIG. 6 ). 
     In the present embodiment, the ground contact position (the acting point of ↑FMt) of a contact portion representative element, the ground contact position (the acting point of ↑FM_R) of the right foot  18 R, and the ground contact position (the acting point of ↑FM_L) of the left foot  18 L are respectively set to, for example, the positions of points where the horizontal components (components about the horizontal axis) of the floor reaction force moments at the acting point, namely, ↑Mt, ↑M_R, and ↑M_L, are zero, i.e., the ZMP. Setting each acting point in this manner leads to the relationship indicated by expression 8-5. 
     Further, according to the present embodiment, the springy displacement amount ↑Xcc related to a contact portion representative element (hereinafter referred to as the contact portion representative element springy displacement amount ↑Xcc) coincides with the springy displacement amount ↑X_R related to the right foot  18 R in a one-leg ground contact state in which only the right foot  18 R is in contact with the ground. Similarly, the contact portion representative element springy displacement amount ↑Xcc coincides with the springy displacement amount ↑X_L related to the left foot  18 L in a one-leg state wherein only the left foot  18 L is in contact with the ground. In a two-leg state wherein both feet  18 R and  18 L come in contact with the ground, the contact portion representative element springy displacement amount ↑Xcc will be a weighted mean displacement amount of the springy displacement amounts ↑X_R and ↑X_L related to the two feet  18 R and  18 L, respectively. 
     To be more specific, the relationship between the contact portion representative element springy displacement amount ↑Xcc and the springy displacement amounts ↑X_R and ↑X_L related to the feet  18 R and  18 L, respectively, is indicated by expression 8-6 given below. r at and (1−r at) being the weighted coefficients, as with the relationship between the ground contact position of a contact portion representative element (the acting point of ↑FMt) and the ground contact positions of the feet  18 R and  18 L (the acting points of ↑FM_R and ↑FM_L).
 
↑ Xcc=r at*↑X   —   R +(1 −r at )*↑ X   —   L   Expression 8-6
 
     Further, in the processing by the contact portion Jacobian matrix determiner  82  according to the present embodiment, the contact portion representative element springy displacement amount ↑Xcc corresponds to the position and the posture of the contact portion representative element. A matrix Jc which indicates the relationship between the motion velocity of the contact portion representative element, that is, the temporal change rate (a first-order differential value) ↑Xcc′ of ↑Xcc (=↑Xcc/dt), and a first-order differential value ↑q′ of a generalized variable vector ↑q by expression 8-7 given below is defined as the contact portion Jacobian matrix in the present embodiment.
 
↑ Xcc′=Jc*↑q′   Expression 8-7
 
     The contact portion representative element springy displacement amount ↑Xcc is a vector constituted of six components, namely, the components of the translational displacement amounts in the directions of three axes and the components of the angular displacement amounts about three axes. For the sake of convenience, in the description of the present embodiment, the same reference character Jc as that in the first embodiment will be used for the contact portion Jacobian matrix. In the following description of the present embodiment, unless otherwise specified, the contact portion Jacobian matrix Jc will mean the Jacobian matrix that indicates the relationship between ↑Xcc′ and ↑q′ by expression 8-7. 
     Based on the above, the method for calculating the contact portion Jacobian matrix Jc in the present embodiment will be described. 
     First, the following expression 8-8 is derived by applying expressions 8-2, 8-3 and 8-4 to the above expression 8-1. 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               kfcc 
                               * 
                               
                                 ↑ 
                                 Xcc_org 
                               
                             
                           
                         
                         
                           
                             
                               kmcc 
                               * 
                               
                                 ↑ 
                                 Xcc_rot 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         A_R 
                         * 
                         
                           [ 
                           
                             
                               
                                 
                                   kf_R 
                                   * 
                                   
                                     ↑ 
                                     Xorg_R 
                                   
                                 
                               
                             
                             
                               
                                 
                                   km_R 
                                   * 
                                   
                                     ↑ 
                                     Xrot_R 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         A_L 
                         * 
                         
                           [ 
                           
                             
                               
                                 
                                   kf_L 
                                   * 
                                   
                                     ↑ 
                                     Xorg_L 
                                   
                                 
                               
                             
                             
                               
                                 
                                   km_L 
                                   * 
                                   
                                     ↑ 
                                     Xrot_L 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   where 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       A_R 
                       ≡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               0 
                             
                           
                           
                             
                               V_R 
                             
                             
                               I 
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       A_L 
                       ≡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               0 
                             
                           
                           
                             
                               V_L 
                             
                             
                               I 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     The equation of the first line of this expression 8-8 is denoted by the following expression 8-9.
 
 kfcc*↑Xcc _org= kf   —   R*↑X org —   R+kf   —   L*↑X org —   L   Expression 8-9
 
     Meanwhile, the following expression 8-10 is derived from the above expression 8-6.
 
↑ Xcc _org= r at*↑X org —   R +(1− r at )*↑ X org —   L   Expression 8-10
 
     Further, the following expressions 8-11a and 8-11b are derived by comparing the above expressions 8-9 and 8-10.
 
 kf   —   R=wcc*kfcc   Expression 8-11a
 
 kf   —   L =(1− wcc )* kfcc   Expression 8-11b
 
     The equation of the second line of the above expression 8-8 is represented by the following expression 8-12.
 
 kmcc*↑Xcc _rot=↑ Vp   —   R ×( kf   —   R*↑X org —   R )+ km   —   R*↑X rot —   R+↑Vp   —   L ×( kf   —   L*↑X org —   L )+ km   —   L*↑X rot —   L   Expression 8-12
 
where
 
↑ Vp   —   R ×( kf   —   R*↑X org —   R )= V   —   R *( kf   —   R*↑X org —   R )
 
↑ Vp   —   L ×( kf   —   L*↑X org —   L )= V   —   L *( kf   —   L*↑X org —   L )
 
     Here, referring to  FIG. 6 , if the positional vector of the ground contact position of the left foot  18 L (the acting point of ↑FM_L) observed from the ground contact position of the right foot  18 R (the acting point of ↑FM_R) is denoted by ↑V_RL (=↑Vp_L−↑Vp_R), then the following expressions 8-13a and 8-13b can be obtained from the above expression 8-5.
 
 ↑Vp   —   R =−(1 −r at )*↑ V   —   RL   Expression 8-13a
 
↑ Vp   —   L=r at*↑V   —   RL   Expression 8-13b
 
     Applying the above expressions 8-13a and 8-13b and the above expressions 8-11a and 8-11b to the above expression 8-12 leads to the following expression 8-14. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           kmcc 
                           * 
                           
                             ↑ 
                             Xcc_rot 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             wcc 
                           
                           * 
                           
                             ( 
                             
                               1 
                               - 
                               wcc 
                             
                             ) 
                           
                           * 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               ↑ 
                               V_RL 
                             
                             × 
                             
                               ↑ 
                               Xorg_R 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             km_R 
                             * 
                             
                               ↑ 
                               Xrot_R 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           wcc 
                           * 
                           
                             ( 
                             
                               1 
                               - 
                               wcc 
                             
                             ) 
                           
                           * 
                           
                             ↑ 
                             V_RL 
                           
                           × 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ↑ 
                             Xorg_L 
                           
                           + 
                           
                             km_L 
                             * 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           ↑ 
                           Xrot_L 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           wcc 
                           * 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 wcc 
                               
                               ) 
                             
                             ↑ 
                             V_RL 
                           
                           × 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ↑ 
                                 Xorg_L 
                               
                               - 
                               
                                 ↑ 
                                 Xorg_R 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             km_R 
                             * 
                             
                               ↑ 
                               Xrot_R 
                             
                           
                           + 
                           
                             km_L 
                             * 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           ↑ 
                           Xrot_L 
                         
                       
                     
                   
                 
               
               
                 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   14 
                 
               
             
           
         
       
     
     In this case, ↑V_RL and (↑Xorg_L−↑Xorg_R) can be regarded as vectors that are parallel to each other, so that the value of the first term of the right side of expression 8-14 will be zero. Therefore, expression 8-14 can be rewritten to expression 8-15 given below.
 
 kmcc*↑Xcc   —   rot=km   —   R*↑X rot —   R+km   —   L*↑X rot —   L   Expression 8-15
 
     Further, the following expression 8-16 is derived from the above expression 8-6.
 
↑ Xcc _rot= r at*↑X rot —   R +(1− r at )*↑ X rot —   L   Expression 8-16
 
     Further, the following expressions 8-17a and 8-17b are obtained by comparing the above expressions 8-15 and 8-16.
 
 km   —   R=r at*kmcc   Expression 8-17a
 
 km   —   L =(1− r at )* kmcc   Expression 8-17b
 
     Further, applying the above expressions 8-11a and 8-11b and the above expressions 8-17a and 8-17b to the above expression 8-8 leads to the following expression 8-18.
 
↑ Xcc=r at*A   —   R*↑X   —   R +(1− r at )* A   —   L*↑X   —   L   Expression 8-18
 
     Then, the following expression 8-19 is obtained by differentiating both sides of the above expression 8-18.
 
↑ Xcc′=r at*A   —   R*↑X   —   R′ +(1− r at )* A   —   L*↑X   —   L′   Expression 8-19
         where ↑Xcc′=d↑Xcc/dt, ↑X_R′=d↑X_R/dt, ↑X_L′=d↑X_L/dt       

     Here, the temporal change rates of r at*AR and (1−r at)*A_L will be sufficiently small, and r at*A_R and (1−r at)*A_L will be maintained substantially constant. 
     In this case, ↑X_R′ is considered to coincide with the temporal change rate of the ground contact position (the acting point of ↑FM_R) of the right foot  18 R, while ↑X_L′ is considered to agree with the temporal change rate of the ground contact position (the acting point of ↑FM_L) of the left foot  18 L. 
     Hence, if a Jacobian matrix denoting the relationship between the temporal change rate of the ground contact position of the right foot  18 R (the acting point of ↑FM_R) and the first-order differential value ↑q′ of the generalized variable vector ↑q is denoted by Jcc_R, and a Jacobian matrix denoting the relationship between the temporal change rate of the ground contact position of the left foot  18 L (the acting point of ↑FM_L) and the first-order differential value ↑q′ of the generalized variable vector ↑q is denoted by Jcc_L, then the following expression 8-20 is obtained from the above expressions 8-7 and 8-19.
 
 Jc=r at*A   —   R*Jcc   —   R +(1− r at )* A   —   L*Jcc   —   L   Expression 8-20
 
     According to the present embodiment, therefore, the contact portion Jacobian matrix determiner  82  calculates Jacobian matrixes, namely, Jcc_R and Jcc_L, of the right side of expression 8-20 and matrixes A_R and A_L. Then, the contact portion Jacobian matrix determiner  82  calculates the right side of expression 8-20 from the calculated values of the Jacobian matrixes Jcc_R, Jcc_L and the matrixes A_R, A_L and the contact portion representative element defining ratio r at calculated as described in the first embodiment, thereby calculating the contact portion Jacobian matrix Jc. 
     The following will describe in more detail the processing by the contact portion Jacobian matrix determiner  82  in the present embodiment. 
     In order to calculate the contact portion representative element defining ratio r at, the Jacobian matrixes Jcc_R, Jcc_L, and the matrixes A_R, A_L used to calculate the right side of the above expression 8-20, the contact portion Jacobian matrix determiner  82  receives the detection outputs of the force sensors  63 R and  63 L (the measurement values of floor reaction forces acting on the feet  18 R and  18 L), the measurement values of the moving velocity of the lower body  10  and the posture angle and the temporal change rate (angular velocity) thereof, and the measurement values of the rotational angles and the temporal change rates (angular velocities) of the joints of leg links  3 R and  3 L. 
     Subsequently, the contact portion Jacobian matrix determiner  82  computes the above expression 2-2a or 2-2b in the same manner as that in the first embodiment from the measurement values of the translational floor reaction force vertical components Fz_act_R and Fz_act_L so as to calculate the contact portion representative element defining ratio r at. As described in the first embodiment, the r at may be calculated by using the desired values of the translational floor reaction force vertical components Fz_act_R and Fz_act_L. 
     The contact portion Jacobian matrix determiner  82  also calculates the positions of the acting points (ZMPs in the present embodiment) of the floor reaction force vectors ↑FM_R and ↑FM_L relative to the feet  18 R and  18 L, respectively, and the position of the acting point (ZMP in the present embodiment) of a total floor reaction force vector ↑FMt, which is a resultant vector of the floor reaction force vectors, on the basis of the measurement values of the floor reaction forces acting on the individual feet  18 R and  18 L. Then, based on these calculated values of the positions of the acting points, the matrixes A_R and A_L defined by “where” of the above expression 8-1 are calculated. 
     If the desired gait generated and output by the aforesaid gait generator  71  (or the desired gait supplied to the control device  60  from an external source) includes the desired values of the floor reaction force vectors ↑FM_R and ↑FM_L of the feet  18 R and  18 L, respectively, then the desired values may be used in place of the measurement values of ↑FM_R and ↑FM_L to calculate the matrixes A_R and A_L. Alternatively, if the desired gait includes the desired positions of ZMPs related to the floor reaction force vectors ↑FM_R and ↑FM_L, respectively, and the desired position of a ZMP related to the total floor reaction force vector ↑FMt, then these desired positions may be used to calculate the matrixes A_R and A_L. 
     Further, the contact portion Jacobian matrix determiner  82  calculates Jacobian matrixes Jcc_R and Jcc_L on the basis of the measurement values of the moving velocity of the lower body  10  and the posture angle and the temporal change rate (angular velocity) thereof, the measurement values of the rotational angles and the temporal change rates (angular velocities) of the joints of the leg links  3 R and  3 L, and the positions (or the desired positions) of the acting points (the ZMPs in the present embodiment) of the floor reaction force vectors ↑FM_R and ↑FM_L calculated as described above. 
     Then, the contact portion Jacobian matrix determiner  82  uses the contact portion representative element defining ratio r at, the Jacobian matrixes Jcc_R and Jcc_L, and the matrixes A_R and A_L, which have been calculated as described above, to compute the right side of the above expression 8-20, thereby calculating the contact portion Jacobian matrix Jc. Thus, the contact portion Jacobian matrix Jc as the Jacobian matrix indicating the relationship between ↑Xcc′ and ↑q′ by the above expression 8-7 is calculated. 
     This completes the detailed explanation of the processing by the contact portion Jacobian matrix determiner  82  in the present embodiment. 
     The processing by the control input determiner  85  in the present embodiment will now be described. 
     As with the first embodiment, the control input determiner  85  in the present embodiment sequentially receives the inertia matrix M, the centrifugal force/Coriolis force-dependent generalized force vector ↑N and the gravity-dependent generalized force vector ↑G (or the nonlinear term generalized force vector (↑N+↑G)) determined by the basic parameter group determiner  81 , the contact portion Jacobian matrix Jc determined by the contact portion Jacobian matrix determiner  82 , the state amount Jacobian matrix Js1 determined by the state amount Jacobian matrix determiner  83 , and the state amount desired value vector ↑S1′_cmd determined by the state amount desired value determiner  84 . In addition, the control input determiner  85  sequentially receives the measurement values of the temporal change rates (first-order differential values) ↑q′ of the generalized variable vector ↑q (the measurement values of the moving velocity of the lower body  10  and the temporal change rate (angular velocity) of the posture angle and the measurement value of the temporal change rate (angular velocity) of the rotational angle of each joint), and also a contact portion representative element desired acceleration ↑C (the desired value ↑C of the motion acceleration of the contact portion representative element). 
     Then, as with the first embodiment, the control input determiner  85  calculates the right side of expression 5-22 by using the received values thereby to calculate the desired joint driving torque ↑τ1_cmd. 
     In the present embodiment, however, the contact portion Jacobian matrix Jc required to calculate the matrixes Jc#, Rc, Tc, the vector ↑Cc, and the matrix Pc defined by the above expressions 5-8, 5-9, 5-12, 5-13 and 5-14, respectively, is the Jacobian matrix calculated by the contact portion Jacobian matrix determiner  82  according to the above expression 8-20. 
     Further, according to the present embodiment, the motion acceleration of a contact portion representative element takes the second-order differential value ↑Xcc″ of the contact portion representative element springy displacement amount ↑Xcc (=d 2 ↑Xcc/dt2). Hence, the contact portion representative element desired acceleration ↑C required to calculate the vector ↑Cc defined by expression 5-13 is the desired value of ↑Xcc″. In this case, in the present embodiment, the desired value ↑C of ↑Xcc″ is preset to a zero vector whose component values are zero (however, a zero vector of six components in the present embodiment), as with the one in the first embodiment. According to the present embodiment, therefore, maintaining the contact portion representative element springy displacement amount ↑Xcc to be constant is the target related to the motions of the feet  18 R and  18 L serving as the contact portions. 
     The processing by the control input determiner  85  in the present embodiment is the same as that in the first embodiment except that the contact portion Jacobian matrix Jc and the desired value ↑C used to compute the right side of the above expression 5-22 are different from those in the first embodiment. 
     The processing by the compliance controller  86  in the present embodiment will now be described. 
     As with the first embodiment, the compliance controller  86  in the present embodiment sequentially receives the centrifugal force/Coriolis force-dependent generalized force vector ↑N and the gravity-dependent generalized force vector ↑G (or the nonlinear term generalized force vector (↑N+↑G)) determined by the basic parameter group determiner  81 , the desired joint driving torque ↑τ1_cmd determined by the control input determiner  85 , and the contact portion representative element defining ratio r at determined by the contact portion Jacobian matrix determiner  82 . Further, the compliance controller  86  sequentially receives the matrixes Rc, Jc′, and Jc# calculated by the control input determiner  85 , the contact portion representative element desired acceleration ↑C (zero vector in the present embodiment), the measurement values of the actual floor reaction forces ↑FM_act_R and ↑FM_act_L acting on the feet  18 R and  18 L, respectively, the measurement value of the moving velocity of the representative link  10 , the measurement value of the posture angle of the representative link (lower body)  10  of the robot  1  and the measurement value of the temporal change rate (first-order differential values) thereof, and the measurement value of the rotational angle of each joint and the measurement value of the temporal change rate (angular velocity) thereof. 
     Then, as with the first embodiment, the compliance controller  86  calculates a contact portion representative element desired acting force ↑Fc_cmd according to the above expression 7-1. In this case, the contact portion representative element desired acting force ↑Fc_cmd in the present embodiment has a meaning as an estimated value of the external force to be applied to a contact portion representative element, i.e., the estimated value of the total floor reaction force vector ↑FMt in the case where the motion acceleration ↑Xcc′ of a contact portion representative element is set to coincide with ↑C and the actual joint driving torque of each joint of the robot  1  is set to coincide with a desired joint driving torque defined by ↑τ1_cmd. 
     In the present embodiment, therefore, the compliance controller  86  sets the contact portion representative element desired acting force ↑Fc_cmd calculated according to the above 7-1 directly as a desired total floor reaction force ↑FMt_cmd as the desired value of a total floor reaction force. 
     Further, as with the first embodiment, the compliance controller  86  calculates the measurement value of an actual total floor reaction force ↑FMt_act from the measurement values of the actual floor reaction forces ↑FM_act_R and ↑FM_act_L acting on the feet  18 R and  18 L, respectively. 
     Then, the compliance controller  86  calculates a difference ↑ΔFMt between the calculated desired total floor reaction force ↑FMt_cmd and the measurement value of the total floor reaction force ↑FMt_act (=↑FMt_cmd−↑FMt_act), and calculates a correction amount ↑Δτt of the desired joint driving torque ↑τ1_cmd, which serves as the manipulated variable for converging the difference ↑ΔFMt to zero, according to the following expression 7-4 having the same form as that of the above expression 7-2.
 
↑Δτ t =( Jc#   T ) −1   *↑ΔFMt   Expression 7-4
 
     In this case, according to the present embodiment, the total floor reaction force ↑FMt directly acts on a contact portion representative element, so that the correction amount ↑Δτt is calculated by expression 7-4 in which ↑ΔFc of the right side of the above expression 7-2 has been replaced by ↑ΔFMt. Incidentally, Jc# of expression 7-4 is calculated according to the definitional equation of the above expression 5-8 by using the contact portion Jacobian matrix Jc in the present embodiment. 
     Thus, ↑Δτt is determined by a proportional law serving as a feedback control law such that the difference ↑ΔFMt is converged to zero. 
     Subsequently, as with the first embodiment, the compliance controller  86  determines a corrected desired joint driving torque ↑τ1c_cmd according to the above expression 7-3. 
     Thus, the desired joint driving torque ↑τ1_cmd is corrected such that the difference between the desired total floor reaction force ↑FMt_cmd, which is the total floor reaction force corresponding to the desired joint driving torque ↑τ1_cmd, and the measurement value of the actual total floor reaction force ↑FMt_act is brought close to zero. 
     This completes the description of the processing by the compliance controller  86  in the present embodiment. 
     The present embodiment is the same as the first embodiment except for the aspects described above. According to the present embodiment, the method for determining the state amount desired value vector ↑S1′_cmd is the same as that of the first embodiment, thus making it possible to restrain the actual position of the overall center-of-gravity point of the robot  1  from deviating from a proper desired position, as with the first embodiment. This in turn makes it possible to restrain the posture of the robot  1  from becoming unstable, thereby enhancing the stability of the behaviors of the robot  1  despite the flexible joints of the robot  1 . 
     Further, according to the present embodiment, the motion acceleration ↑Xcc″ of a contact portion representative element is defined by the contact portion representative element desired acceleration ↑C (zero vector in the present embodiment), and then ↑τ1_cmd is determined according to the above expression 5-20 on the basis of the state amount desired value vector ↑S1′_cmd as the desired value related to the state amount to be controlled of the robot  1 . 
     Therefore, as with the first embodiment, a proper ↑τ1_cmd that allows ↑C and ↑S1′_cmd to be realized can be determined without identifying the value of an external force acting on the robot  1 . In this case, the total degree of freedom of ↑C and ↑S1′_cmd is smaller than the total degree of freedom of the entire robot  1  (the degree of freedom of the aforesaid generalized force vector ↑q) also in the present embodiment, so that the flexibility of the motions of the joints of the robot  1  to accommodate changes or the like in an external force acting on the robot  1  can be enhanced, as with the first embodiment. 
     In addition, according to the present embodiment, the degree of freedom of the motion acceleration ↑Xcc″ of a contact portion representative element and the contact portion representative element desired acceleration ↑C is smaller than that in the first embodiment (six degrees of freedom in the present embodiment). Hence, the restrictions on the degree of freedom of each joint of the robot  1  when each of the feet  18 R and  18 L comes in contact with the ground (or a floor) can be reduced from those in the first embodiment. As a result, the flexibility of the motions of the joints of the robot  1  can be further enhanced. 
     Here, the relationship of correspondence between the present embodiment and the present invention will be supplementarily described. 
     In the present embodiment, the contact portion representative element springy displacement amount ↑Xcc corresponds to the springy displacement amount Xcc of the position and posture of the contact portion representative element in the present invention. 
     Further, the right leg link  3 R corresponds to the first leg link in the present invention, while the right foot  18 R corresponds to the distal end of the first leg link. The springy displacement amount ↑X_R related to the right foot  18 R corresponds to the springy displacement amount X1 of the position and posture of the distal end of the first leg link in the present invention. 
     Further, the left leg link  3 L corresponds to the second leg link in the present invention, while the left foot  18 L corresponds to the distal end of the second leg link. The springy displacement amount ↑X_L related to the left foot  18 L corresponds to the springy displacement amount X2 of the position and posture of the distal end of the second leg link in the present invention. 
     The above expression 8-18 corresponds to the above expression 02 in the present invention. In this case, matrixes A_R and A_L of the right side of expression 8-18 correspond to the matrixes A1 and A2, respectively, in the present invention. 
     Further, the above expression 8-20 corresponds to the above expression 03 in the present invention. In this case, Jacobian matrixes Jcc_R and Jcc_L of the right side of expression 8-20 correspond to the Jacobian matrixes J1 and J2, respectively, in the present invention. 
     The relationship of correspondence between the present embodiment and the present invention is the same as the relationship of correspondence between the first embodiment and the present invention except for the aspects described above. 
     The following will describe some modifications of the embodiments described above. 
     In each of the aforesaid embodiments, the robot  1  has been a bipedal walking robot, which is a legged mobile robot. However, the robot in the present invention may be a mobile robot other than a legged mobile robot (e.g., a robot that travels on wheels). Alternatively, the robot in the present invention may be an installed robot, any one of links of which is fixed to a floor or the like. 
     Further, the present invention can be applied also to a case where the robot  1  runs or travels and performs an operation with the hands  30 R and/or  30 L at the same time rather than being limited to the case where the robot  1  performs walking. 
     Further, the predetermined type of state amount may be another type of state amount rather than being limited to the total translational momentum ↑Ptotal and the total angular momentum ↑Ltotal. 
     Further, in the first embodiment, the contact portion representative element desired acceleration ↑C has been set to zero. Alternatively, however, the target of another type of motion of the contact portion representative points a 1  to a 4  may be set as ↑C. For example, the motion patterns of the contact portion representative points a 1  to a 4  may be set to sinusoidal patterns. In addition, in the case where, for example, the robot  1  exists in a system, such as an elevator, wherein acceleration is generated, the acceleration of the system (the acceleration observed from an inertial coordinate system) may be added to the contact portion representative element desired acceleration ↑C. 
     Similarly, a desired value other than the zero vector may be set as the contact portion representative element desired acceleration ↑C in the aforesaid second embodiment. 
     Further, the compliance controller  86  may be omitted. More specifically, the desired joint driving torque ↑τ1_cmd determined by the control input determiner  85  may be directly used to control the driving torque of each joint of the robot  1 .