Available online at  Procedia Engineering 41 ( 2012 ) 1277  International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)  A Relevant Reduction Method for Dynamic Modeling of a Seven  Humanoid Robot in the Three  Amira Aloulou  a   National Institute of Applied Sciences and Technology, Centre Urbain Nord, BP 676  Abstract  In this paper, the dynamic modeling of a seven linked humanoid robot  Newton- Euler formalism. The aim of this study is to provide a clear and a systematic approach so that starting from generalized moti  equations of all rigid bodies of the humanoid robot one can establish a reduced dynamic  either for simulation propositions or implemented for any given control law. In addition, transformations and developments, p  here, can be exploited for modeling any other three  rigid bodies and degrees of freedom.  © 2012 The Authors.   Published by Elsevier Ltd. Selection and/or peer  Humanoid Robots and Bio- Sensor (HuRoBs), Faculty of  Keywords : Newton- Euler formalism, Three dimensional  1.   Introduction  The study and materialization of humanoid robots is a wild field of  role since all following stages of implementation are narrowly depending on it and especially the synthesis of an efficient  control law. Indeed, the dynamic model has to be accurate by taking account of  dynamic motion of the robot. Determining the degrees of freedom’s number and location and getting the kinematic model  are the two prerequisites needed to establish a dynamic modeling.  represent the main methodologies employed allowing the dynamic modeling of any given robotic system [1]. The  Lagrangian formalism gives a simple representation of a system having several joints by supplying at every instant  generaliz ed coordinates of all particles involved in the robotic system motion [2]. However, this method shows certain  limitations. For complex systems, choosing coordinates for the particles during motion is a difficult task. Moreover, this  approach doesn’t   provid e   direct   data   regarding the   required   torques needed   to   activate   each link.   Especially,   the  implementation of some complex links involving a friction as in the double  quite unrealizable in a purely Lagrangian contex  nonlinear and dense inertia operators thus generating a high calculation cost  has been chosen to lead our study. It is based on the obser  observation points. These velocities represent particles velocities when moving through the observation point at different  times [4]. Thus , observation points stand for current positions of partic  formalism involves taking account of more parameters than the Lagrangian one in the case of systems composed of many  rigid bodies, the determination and inversion of the inertia operators doesn’t imply a hig  * Corresponding author. Tel.:+0-216-1-703- 929; fax: +0  E-mail address:   olfa.boubaker@insat.rnu.tn.  Available online at   www.sciencedirect.com  Procedia Engineering 41 ( 2012 ) 1277   – 1284  International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)  A Relevant Reduction Method for Dynamic Modeling of a Seven  Humanoid Robot in the Three -dimensional   Space  Amira Aloulou a , Olfa Boubaker a, *  National Institute of Applied Sciences and Technology, Centre Urbain Nord, BP 676   -   1080 Tunis, Tunisia  In this paper, the dynamic modeling of a seven linked humanoid robot , is accurately developed, in the   three dimensional  Euler formalism. The aim of this study is to provide a clear and a systematic approach so that starting from generalized moti  equations of all rigid bodies of the humanoid robot one can establish a reduced dynamic al model. The resulting model can be expended  either for simulation propositions or implemented for any given control law. In addition, transformations and developments, p  here, can be exploited for modeling any other three -dimensional humanoid robo t with a different morphology and variable number of  Published by Elsevier Ltd. Selection and/or peer - review under responsibility  Sensor (HuRoBs), Faculty of   Mechanical Engineering, Universiti Teknologi  Euler formalism, Three dimensional   dynamic modeling, Model reduction, Seven linked humanoid rob ot  The study and materialization of humanoid robots is a wild field of   investigation where dynamic modeling plays a crucial  role since all following stages of implementation are narrowly depending on it and especially the synthesis of an efficient  control law. Indeed, the dynamic model has to be accurate by taking account of   all forces and torques involved in the  dynamic motion of the robot. Determining the degrees of freedom’s number and location and getting the kinematic model  are the two prerequisites needed to establish a dynamic modeling.   At this point, the Lagrange and N  represent the main methodologies employed allowing the dynamic modeling of any given robotic system [1]. The  Lagrangian formalism gives a simple representation of a system having several joints by supplying at every instant  ed coordinates of all particles involved in the robotic system motion [2]. However, this method shows certain  limitations. For complex systems, choosing coordinates for the particles during motion is a difficult task. Moreover, this  e   direct   data   regarding the   required   torques needed   to   activate   each link.   Especially,   the  implementation of some complex links involving a friction as in the double -support- phase (DSP) walking cycle becomes  quite unrealizable in a purely Lagrangian contex t. Finally, the estimation and inversion of certain Jacobians may produce  nonlinear and dense inertia operators thus generating a high calculation cost   [3 ]. Therefore, the Newton  has been chosen to lead our study. It is based on the obser vation and determination of velocities in one or more given  observation points. These velocities represent particles velocities when moving through the observation point at different  , observation points stand for current positions of partic les at a given instant. Even if the Newton  formalism involves taking account of more parameters than the Lagrangian one in the case of systems composed of many  rigid bodies, the determination and inversion of the inertia operators doesn’t imply a hig h calculation cost.  929; fax: +0 -216-1-704-929.  International Symposium on Robotics and Intelligent Sensors 2012 (IRIS 2012)  A Relevant Reduction Method for Dynamic Modeling of a Seven -linked  Space  1080 Tunis, Tunisia  three dimensional   space using the  Euler formalism. The aim of this study is to provide a clear and a systematic approach so that starting from generalized moti on  al model. The resulting model can be expended  either for simulation propositions or implemented for any given control law. In addition, transformations and developments, p roposed  t with a different morphology and variable number of  review under responsibility   of the Centre of  Mechanical Engineering, Universiti Teknologi   MARA.  ot .  investigation where dynamic modeling plays a crucial  role since all following stages of implementation are narrowly depending on it and especially the synthesis of an efficient  all forces and torques involved in the  dynamic motion of the robot. Determining the degrees of freedom’s number and location and getting the kinematic model  At this point, the Lagrange and N ewton–Euler approaches  represent the main methodologies employed allowing the dynamic modeling of any given robotic system [1]. The  Lagrangian formalism gives a simple representation of a system having several joints by supplying at every instant  ed coordinates of all particles involved in the robotic system motion [2]. However, this method shows certain  limitations. For complex systems, choosing coordinates for the particles during motion is a difficult task. Moreover, this  e   direct   data   regarding the   required   torques needed   to   activate   each link.   Especially,   the  phase (DSP) walking cycle becomes  t. Finally, the estimation and inversion of certain Jacobians may produce  ]. Therefore, the Newton -Euler methodology  vation and determination of velocities in one or more given  observation points. These velocities represent particles velocities when moving through the observation point at different  les at a given instant. Even if the Newton -Euler  formalism involves taking account of more parameters than the Lagrangian one in the case of systems composed of many  h calculation cost.
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  With regard to other works dealing with the dynamic modeling of bipedal robots in the three dimensional space, we may notice the lack of papers introducing a clear and sequential methodology to follow in order to establish a reduced dynamic model so that it can be directly expended for simulations and implemented for any given control law [5]. Throughout this article, our aim is then to propose a plain and exhaustive method based on the Newton-Euler formalism to obtain a satisfactory three-dimensional dynamic modeling whatever number of rigid bodies or degrees of freedom is considered. Nevertheless, the lower body morphology of a female humanoid robot prototype introduced in a previous paper [6] will be adopted to realize this demonstration. This robot prototype is composed of seven links associated to 12 DOF. In previous works [6], all physical parameters corresponding to each link have been determined and a kinematic model has been established. In this paper, generalized three dimensional motion equations for the rotation and the translation of each link are reached. A first dynamic modeling corresponding to the free robotic system when achieving the single support phase is proposed via some rigorously explained transformations and developments. We show how to finally get reduced free or constraint three dimensional dynamic models. Both free and constraint resulting dynamic models are directly expendable for reaching a whole walking cycle including the single support, impact and double support phases.  2.   Dynamic Modeling  The bipedal robot prototype [7] is composed of seven links associated to 12 DOF. Each rigid body Ci of the humanoid robot is characterized by its own physical parameters determined using Winter statistical model and detailed in [6]. As a prerequisite to dynamic modeling in the three dimensional space, the three-dimensional kinematic model basically founded on Euler’s transformation matrix has been established and then the dynamic model will be computed using the Newton- Euler method.  2.1.   Newton-Euler Principle for Dynamic Modeling  For each link Ci, generalized motion equations for the rotation and the translation are used such as [7]:  I  w       = f    + F    + F    + G    + G    + τ    + τ    (1)  M  X    = M  g + Γ    – Γ    (2) where each term of (1) and (2) is described in the nomenclature given below.  Nomenclature  W  ,   W     Angular position and velocity of the link Ci  X    ,   X  , X    Linear position, velocity and acceleration of the link Ci  F    , F    Torques due to the holonom force applied respectively to the proximal and distal articulation of  the link Ci expressed in the body coordinate system  G   ,   G    Non-holonom torques applied respectively to the proximal and distal articulation of the link Ci  expressed in the body coordinate system  τ  ,   τ    Muscular torques applied respectively to the proximal and distal articulation of the link Ci  expressed in the body coordinate system  Γ    ,   Γ    Holonom forces applied respectively to the proximal and distal articulation of the link Ci  expressed in the inertial coordinate system  f    Intrinsic torque of the link Ci expressed in the body coordinates system ( x  , y  , z  )   and relating  angular velocity to the link inertia  Λ    Non-holonom torque applied to the proximal articulation of the link Ci expressed in the body  coordinate system of the previous link Ci-1  Λ    Non-holonom torque applied to the proximal articulation of the link Ci expressed in the body  coordinate system of the previous link Ci-1  R    ,   R    Non holonom torque applied to the distal articulation of the link Ci expressed in the body  coordinate system  T    Torque applied to the proximal articulation of the link Ci expressed in the body coordinate  system of the previous link Ci-1  T    Torque applied to the distal articulation of the link Ci expressed in the body coordinate system  Human body’s balance of forces and torques reveals that humanoid limbs are subject to three kinds of forces: holonom, non
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  holonom and mechanic forces. Holonom forces and torques result of the interaction between limbs whereas non-holonom torques consist in the effort a limb is subject to in order to remain aligned with the previous limb. Finally, mechanical torques are muscular torques applied by human body’s muscles to move limbs. Fig.1 presents the lower body joints and links whereas Fig.2 shows the applied forces and torques to the considered humanoid robot lower body.  Fig.1: Links and Joints of the Bipedal Robot   Fig.2: Applied forces and torques to the Bipedal Robot  2.2.   Constraints  A humanoid robot is always subject to two kinds of constraints: non holonomic constraints and holonomic ones. For the case study of the seven-linked robot, we get the following constraints:  Non holonomic constraints:   We have enumerated four non holonomic constraints that are to be found at the knees and the ankles. The first constraint occurs when the bipedal robot is standing on its right foot. In that posture, the right leg can only rotate in the sagittal plane at the right ankle level. Therefore, there is only one degree of freedom at the right ankle and two degrees of connection. In the same way and for the second constraint, there is a similar configuration at the level of the left ankle. For each rigid body subject to a non holonomic constraint, the Ri matrix represents the degrees of connection involved while the Qi matrix contains the degrees of freedom. Regarding these two constraints, Ri and Qi are the following:  R    = R    =   1   0   0 0   0   1# $  a nd   Q    = Q    = (0   1   0) $  The third constraint is to be found in the region of the right knee. In that location, there is only one degree of freedom and two degrees of connection. Similarly for the fourth constraint, there is an identical configuration at the level of the left knee. Ri and Qi for these two last constraints are given by:  R *   = R +   =   1   0   0 0   0   1# $  and   Q *   = Q +   = (0   1   0) $  When translating the previous non holonomic constraints into equations, we get the following system:  R  $ (A $ A  W    − W   ) = 0  R * $ (A  $ A * W *   − W  ) = 0   (3)  R + $ (A 0 $ A + W +   − W 0 ) = 0  R  $ (A + $ A  W    − W + ) = 0  Holonomic constraints:   Twenty-one holonomic constraints directly arise from the bipedal robot kinematic modeling given in [6]. They may be described as constraints of connection between segments in order to maintain a rigid body linked to both the previous and next link. These constraints are enumerated in the system of equations given by (4):
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  X   = A   K  X    = A  K    − A   L   + X  X *   = A * K *   − A  L    + X   X 3   = A 3 K 3   − A * L *   + X *   (4)  X 0   = A 0 K 0   − A 3 L 3   + X 3  X +   = A + K +   − A 0 L 0   + X 0  X    = A  K    − A + L +   + X +  2.3.   A method for Dynamic Modeling  Holonom forces and muscular torques are applied to every articulation whereas non holonom forces are only to be found in the articulations of ankles J 1   and J 7   and knees J 2   and J 6 . Table 1 (resp. 2) is a summary of the forces and torques applied to the seven rigid bodies composing the bipedal robot for the rotation (resp. translation):  Table 1: Forces and torques applied to the bipedal robot for the equation of rotation  Body i   f i   F i   F i+1   G i   G i+1   τ i   τ i+1  1   f1   K 1 xA 1T Γ 1   -L 1 xA 1T Γ 2   0   -R 2 Λ 2   A 1T T 1   -T 2  2   f2   K 2 xA 2T Γ 2   -L 2 xA 2T Γ 3   A 2T A 1 R 2 Λ 2   -R 3 Λ 3   A 2T A 1 T 2   -T 3  3   f3   K 3 xA 3T Γ 3   -L 3 xA 3T Γ 4   A 3T A 2 R 3 Λ 3   0   A 3T A 2 T 3   -T 4  4   f4   K 4 xA 4T Γ 4   L 4 xA 4T Γ 5   0   0   A 4T A 3 T 4   -T 5  5   f5   -K 5 xA 5T Γ 5   L 5 xA 5T Γ 6   0   R 6 Λ 6   -A 5T A 4 T 5   T 6  6   f6   -K 6 xA 6T Γ 6   L 6 xA 6T Γ 7   -A 6T A 5 R 6 Λ 6   R 7 Λ 7   -A 6T A 5 T 6   T 7  7   f7   -K 7 xA 7T Γ 7   0   -A 7T A 6 R 7 Λ 7   0   -A 7T A 6 T 7   0  Table 2: Forces and torques applied to the bipedal robot for the equation of translation Body i   M i .g   Γ i   Γ   i+1  1   M 1. g   Γ 1   -   Γ 2  2   M 2. g   Γ 2   -   Γ 3  3   M 3. g   Γ 3   -   Γ 4  4   M 4. g   Γ 4   Γ 5  5   M 5. g   -   Γ 5   Γ 6  6   M 6. g   -   Γ 6   Γ 7  7   M 7. g   -   Γ 7   0  Based on generalized motion equations (1) and (2), results provided by Tables 1 and 2 can be written after simplification as follows:  I   W    = f   + KK   A $ Γ   − LL   A $ Γ    − R  Λ    + A $ T   − T   I  W      = f    + KK  A  $ Γ    − LL  A  $ Γ *   − R * Λ *   + A  $ A   R  Λ    + A  $ A   T    − T *  I * W *     = f *   + KK * A * $ Γ *   − LL * A * $ Γ 3   + A * $ A  R * Λ *   + A * $ A  T *   − T 3  I 3 W 3     = f 3   + KK 3 A 3 $ Γ 3   + LL 3 A 3 $ Γ 0   + A 3 $ A * T 3   − T 0   (5.a)  I 0 W 0     = f 0   − KK 0 A 0 $ Γ 0   + LL 0 A 0 $ Γ +   + R + Λ +   − A 0 $ A 3 T 0   + T +  I + W +     = f +   − KK + A + $ Γ +   + LL + A + $ Γ    + R  Λ    − A + $ A 0 R + Λ +   − A + $ A 0 T +   + T   I  W      = f    − KK  A  $ Γ    − A  $ A + R  Λ    − A  $ A + T    + T 
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  M   X   = M   g + Γ   − Γ   M  X    = M  g + Γ    − Γ *  M * X *   = M * g + Γ *   − Γ 3  M 3 X 3   = M 3 g + Γ 3   + Γ 0   (5.b)  M 0 X 0   = M 0 g − Γ 0   + Γ +  M + X +   = M + g − Γ +   + Γ   M  X    = M  g − Γ   In Table 1, it may be noticed that most involved forces include in their expressions a vector product. By using antisymmetric matrices to eliminate vector products, every equation for rotation is reduced as shown in (5). Indeed, the role of an antisymmetric matrix [8]   is to conceal the vector products by replacing them with simpler products. Assuming that  z   ∈ ℜ 36   is the state vector of the system composed of angular and linear accelerations and   Γ ∈ ℜ  6   , Λ   ∈ ℜ 76   and  T ∈ ℜ  6   are the holonom, non-holonom and muscular forces and torques respectively, the system (5) may be written as:  P   z   = P    + P * Γ + P 3 Λ + P 0 T   (6) where   P   , P  , P * , P 3   and   P 0   are matrices of appropriate dimensions including, on the one hand, the different products between proximal/distal distances and Euler’s transformation matrices and on the other hand inertias and masses of the rigid body whose forces and torques are at play. To exploit the results of non holonomic constraints, we will differentiate (3) with respect to time which gives:  R  $ (−W    − WW   A $ A  W    + A $ A  W   ) = 0  R * $ (−W     − WW  A  $ A * W *   + A  $ A * W  * ) = 0  R + $ (−W  0   − WW 0 A 0 $ A + W +   + A 0 $ A + W  + ) = 0  R  $ (−W  +   − WW + A + $ A  W    + A + $ A  W   ) = 0  When arranging the above system of equations into a more compact form, the following equation is obtained:  P 3 $ z   = P +   with   P +   =  9  :  ;  R  $ WW   A $ A  W   R * $ WW  A  $ A * W *  R + $ WW 0 A 0 $ A + W +  R  $ WW + A + $ A  W    <  =  >   ,   P +   ∈ ℜ 76   (7)  Let’s come-back to the system (4) that summarizes all holonomic constraints applied to the humanoid biped and study the case of the first rigid body C1. When differentiating the Euler’s transformation matrix   A   with respect to time, the resulting equation is (8):  A   =   ?@ A  ?B   = A   WW   (8) Using the result given by (8) and differentiating the first equation of (5) with respect to time, the result is:  X   − A   WW   K   = 0  If the first equation of (5) is differentiated with respect to time twice, the following equation is obtained:  X   −   ?(@ A CC A D A ) ?B   = X   − A   (WW   )  K   + A   KK   W    = 0  In order to apply the same reasoning to the six other rigid bodies, one has to calculate time derivatives of Euler’s transformation matrices   A  ,   A * , A 3 , A 0 , A +   and A    and then implement results into time derivatives of system (5) so that:
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  X   + A   KK   W    = A   (WW   )  K  X    + A  KK  W     − A   LL   W    − X   = A  (WW  )  K    − A   (WW   )  L  X *   + A * KK * W  *   − A  LL  W     − X    = A * (WW * )  K * − A  (WW  )  L   X 3   + A 3 KK 3 W  3   − A * LL * W  *   − X *   = A 3 (WW 3 )  K 3   − A * (WW * )  L *   (9)  −X    0   − A 0 KK 0 W  0   + A 3 LL 3 W  3   + X 3   = −A 0 (WW 0 )  K 0   + A 3 (WW 3 )  L 3  −X    +   − A + KK + W  +   + A 0 LL 0 W  0   + X 0   = −A + (WW + )  K +   + A 0 (WW 0 )  L 0  −X       − A  KK  W     + A + LL + W  +   + X +   = −A  (WW  )  K    + A + (WW + )  L +  Equation (10) is obtained by arranging the system (9) into a more compact form:  P * $ z   = P    with   P    =  9  :  :  :  :  ;  A   (WW   )  K  A  (WW  )  K    − A   (WW   )  L  A * (WW * )  K * − A  (WW  )  L   A 3 (WW 3 )  K 3   − A * (WW * )  L *  −A 0 (WW 0 )  K 0   + A 3 (WW 3 )  L 3  −A + (WW + )  K +   + A 0 (WW 0 )  L 0  −A  (WW  )  K    + A + (WW + )  L 3 <  =  =  =  =  >  ,   P    ∈ ℜ  6   (10) When combining equations (7) and (10), the following equality is reached:  (P *   P 3 ) $ E   z   = (P  $   P + $ ) $ E   (11)  P   is an inversible matrix,   z   can then be expressed according to other terms of the equality (6) :  z   = P   F   (P    + P * Γ + P 3 Λ + P 0 T )   (12) Replacing   z   in (11) by its expression in (12) and then isolating it in the left side of the equation, (13) is reached:  Γ Λ # =   (P *   P 3 ) $ E   P  F   (P *   P 3 )   F   E   ( P  $   P + $   ) $   −   (P *   P 3 ) F   E   (P    + P 0 T)   ( 13) In (13), holonomic force   Γ   and non holonomic torque   Λ   are expressed as state variables. Knowing that   Γ ∈ ℜ  6   and  Λ ∈ ℜ 76   , the formulation (13) presents a system composed of 29 state variables instead of the initial 42 ones. This already represents a reduction of the DOF involved. However, some transformations are still needed to be proceeded to get a system with a minimal number of state variables. Assuming that   P * $   = (P 7   P G ) , the following equality is obtained:  P * $ z   = (P 7   P G )z   = P 7 W    + P G X  If P 3   is replaced by its expression in equation (10), the following equation is reached:  P 7 W    + P G X = P    (14) To reach a state modeling from previous equation (11), vector   X   is isolated and expressed according to the remaining elements of the equation so that:  X = −P G F   P 7 W    + P G F   P   which can be written as:  HW  X I = H   I   −P G F   P 7  I W    + H o   P G F   P  I   where   o    ∈ ℜ  6   is the null vector and   I    ∈ ℜ  6   is the identity matrix. Let’s assume that:  P   K   = H   I   −P G F   P 7  I   ,   P   K   ∈ ℜ 36   and   P   = H o   P G F   P   I   ,   P   ∈ ℜ 36
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  It can be therefore written that:  z   = P   K W    + P   (15) Inserting the expression of   z   given in (15) into the equation of system (6) and then isolating the first left side term in the resulting equation, one gets:  P   P   K W    = −P   P   + P    + P * Γ + P 3 Λ + P 0 T  Let’s multiply the previous equality by   P   K$   so that:  P   K$ P   P   K W    = −P   K$ P   P   + P   K$ P    + P   K$ P * Γ + P   K$ P 3 Λ   (16) Verifying that   P   K $   P * Γ = 0 , equation (16) becomes:  P   K$ P   P   K W    = −P   K$ P   P   + P   K$ P    + P   K$ P 3 Λ + P   K$ P 0 T   (17) Angular accelerations   W    to W     defined in [6] may be written under a matricial form as:  W    = P    Φ    + P   *   (18) Where   Φ    is the second time derivative of the vector   Φ   that contains positions of only active DOFs. Then, using expression (18) in equation (17), the following equality is obtained:  P   K$ P   P   K NP    Φ    + P   * O = −P   K$ P   P   + P   K$ P    + P   K$ P 3 Λ + P   K$ P 0 T   (19) Let’s isolate the first left side term of the previous equality and multiply the whole equality by   P   $ :  P   $ P   K $ P   P   K P    Φ    = −P   $ P   K $ P   P   + P   $ P   K$ P    + P   $ P   K $ P 3 Λ + P   $ P   K $ P 0 T − P   $ P   K$ P   P   K P   *  It is finally shown that   P    $   P   K $   P 3 Λ = 0   consequently the below equation may be written as follows:  P   $ P   K $ P   P   K P    Φ    = −P   $ P   K $ P   P   + P   $ P   K$ P    + P   $ P   K $ P 0 T − P   $ P   K$ P   P   K P   *   (20) In order to get closer to the motion equation general form, all terms depending on the state variable   Φ   and its first derivative  Φ    and second derivative   Φ    are placed on one side of the equality and all remaining terms on the other side. Equation (20) becomes then:  P   $ P   K $ P   P   K P    Φ    + P   $ P   K $ P   (P   + P   K P   * ) − P   $ P   K$ P    = P   $ P   K $ P 0 T   (21)  The final reached equation (21) stands for the dynamic equation of a free robotic system. This last expression is quite conformed to the general form of a robotic dynamic modeling given in (22):  J(Φ)Φ    + HN Φ, Φ  O + G(Φ) = D. τ   (22)  where   Φ   is the state vector describing the dynamics of the humanoid robot system, J( Φ ) is the positive definite inertia matrix,   HN Φ, Φ  O is the vector of the Coriolis and centripetal torques, G( Φ ) is the positive definite gravity vector, D is a nonsingular input map matrix and   τ   is the vector of control inputs. Making correspondence between equations (21) and (22), the following system is obtained:  J(Φ) =   P    $   P   K $   P   P   K P    ,   J(Φ) ∈ ℜ   ∗     HN Φ, Φ    O = P    $   P   K $   (P   P   + P   P   K P   * )   ,   HN Φ, Φ  O ∈ ℜ   ∗  G(Φ) = − P    $   P   K $   P    ,   G(Φ) ∈ ℜ   ∗   (23)  D =   P    $   P   K $   P 0   ,   D ∈ ℜ   ∗ 
Amira Aloulou and Olfa Boubaker / Procedia Engineering 41 ( 2012 ) 1277 – 1284  2.4.   Resulting Constraint Dynamic Model  Whether considering the impact phase or the double support phase, the general robotic constraint dynamic model of the bipedal robot is described by [5]:  J(Φ)Φ    + H( Φ, Φ  ) + G(Φ) = Dτ +   UV W  UX   Γ Y   (24) where E is the contact point defined such as   E = E(Φ) = (E 6   E [   E \ ) $   and   Γ Y   is the contact force with the ground. During the ground contact, the free end of the biped, at the completion of the step, comes into contact with the ground. This phase is assumed to take place in an infinitesimal time interval. The impact force is described by:  Γ Y   ]^   = −   UV UX   |   X `ab   JNΦ ]^ OΦ    ( t K F )   (25)  where   t K F   is the instant that directly occurs before the impact. During the double support phase the reaction force is described by:  Γ Y YcdB   =   e UV UX   J F   (Φ)   UV W  UX   f e UV UX   J F   (Φ) gHN Φ, Φ  O + G(Φ) − Dτh −   U UX   e   UV UX   Φ  f Φ  f#  F  (26) Various solutions may be used to solve the position/ force control problem [9, 10, 11] .   For our case two control laws have been already applied to the reduced model: a feedback linearizing control [7] and a Minimum Jerk-based one [12]. Satisfactory simulation results have been obtained for the both cases.  3.   Conclusion  In this paper, the three dimensional dynamic modeling of a seven linked humanoid robot has been accurately developed using the Newton-Euler formalism. A clear and sequential methodology is provided to establish a reduced and expendable model. The reduced model allows a direct use for control law implementation.  References  [1 ] Wisama K, Ouarda I. General solution for the dynamic modeling of parallel robots. J. of Intelligent and Robotic Systems; 2007, Vol. 49, p. 19-37. [2] Abdellatif H, Heimann B. Computational efficient inverse dynamics of 6-DOF fully parallel manipulators by using the Lagrangian formalism. Mechanism and Machine Theory; 2009, Vol. 44, p.192–207. [3] Høifødt H. Dynamic modeling and simulation of robot manipulators: the Newton-Euler formulation. Master of Science in Engineering Cybernetics. Norwegian University of Science and Technology; 2011. [4] Dumas R, Aissaouib R, de Guise J.A. A 3D Generic Inverse Dynamic Method using Wrench Notation and Quaternion Algebra. Computer Methods in Biomechanics and Biomedical Engineering; 2004, Vol. 7, p.159-166. [5] Hemami H. A general framework for rigid body dynamics, stability and control. J. Dynamic Systems, Measurement, and Control; 2002, Vol.124, p. 241-252. [6] Aloulou A, Boubaker O. Control of a step walking combined to arms swinging for a three dimensional humanoid prototype. J. of Computer Science; 2010, Vol. 6, p.886-895. [7] Aloulou A, Boubaker O. Modeling and controlling a humanoid robot in the three dimentional space. IEEE International Symposium on Robotics and Intelligent Sensors; 2010, p.188-193. [8] Hemami H. Some aspects of Euler-Newton equations of motion. Archive of Applied Mechanics;1982, vol.52, p. 167-176. [9]   Mehdi H, Boubaker O. Impedance controller tuned by particle swarm optimization for robotic arms. International Journal of Advanced Robotic Systems;   2011; p. 93-103. [10] Mehdi H, Boubaker O. Stiffness and impedance control using Lyapunov theory for robot-aided rehabilitation. International Journal of Social Robotics; DOI: 10.1007/s12369-011-0128-5. [11] Mehdi H, Boubaker O. Rehabilitation of a human arm supported by a robotic manipulator: A position/force cooperative control. J. of Computer Science; 2010, Vol. 6, p. 912-919. [12] Aloulou A, Boubaker O. Minimum Jerk-based Control for a Three Dimensional Bipedal Robot. Intelligent Robotics and Applications , Lecture Notes in Computer Science; 2011, Vol. 7102, p. 251-262.