Abstract — In this paper, a simple trajectory generation method for biped walking is proposed. The dynamic model of the five link bipedal robot is first reduced using several biologically inspired assumptions. A sinusoidal curve is then imposed to the ankle of the swing leg's trajectory. The reduced model is finally obtained and solved: it is an homogeneous 2 nd   order differential equations with   constant   coefficients.   The   algebraic   solution   obtained ensures   a   stable   rhythmic   gait   for   the   bipedal   robot.   It’s continuous in the defined time interval, easy to implement when the boundary conditions are well defined.  Index Terms —   Trajectory   Generation, Biped Locomotion, model reduction.  I.   I NTRODUCTION  Gait pattern generation [1] is one of key problems of research   devoted   to   bipedal   robots.   Two   kinds   of   works dedicated to the bipedal walking pattern generation can be distinguished:   studies   assimilating   robots   as   elementary models and works considering all morphological data of the robot, see [2] and references therein. For the first approach, the linear inverted pendulum model concept is the mostly used concept in order to generate the gait trajectory [3, 4, 5].   For the second group of works attention is paid on the generation of   a   trajectory   tracking   control   using   objective   function composed of one or more terms to minimize [2, 6, 7]. In this paper, a simple trajectory generation method is proposed. The dynamic model of the five link bipedal robot is first reduced using several biologically inspired assumptions. A sinusoidal curve is then imposed to the ankle of the swing leg's trajectory. The reduced model is finally obtained and solved: it is an homogeneous 2 nd   order differential equations with constant coefficients. The solution gives an algebraic solution for joint desired trajectories that ensures a stable rhythmic gait for the bipedal robot.  II.   T HE FIVE LINK BIPEDAL   R OBOT  The planar bipedal robot prototype is composed of five links associated to five DOF.   Fig.1 shows the involved rotations for each link.   All physical parameters involved in the kinematic and dynamic models are given by Table 1. They are inspired from [8]. The parameters   i i i   k , L , m   and   i I   , (   5 ,.., 1 = i   ), design mass, length, position of center of mass and inertia about center of mass, respectively.  Figure 1. The five link bipedal robot Table1. Physical parameters of the bipedal robot [8] link   Right leg   Right thigh   Pelvis   Left thigh   Left leg  joint   Right ankle  Right  knee  Right  hip  Left  hip  Left  knee  Link number   1   2   3   4   5  Mass (kg)   3.255   7.000   24.850   7.000   3.255  Lenght (m)   0.426   0.424   0.299   0.424   0.426  Center of  mass   (kg.m)   0.164   0.366   1.530   0.366   0.164  Inertia about centre of mass (Kgm 2 )  0.184   0.184   0.206   0.184   0.184  Gait trajectory generation for a five link bipedal robot based on a reduced dynamical model  1 Yosra Arous &   2 Olfa Boubaker  1,2   National Institute of Applied sciences and Technology, Tunis, Tunisia  2 olfa.boubaker@insat.rnu.tn
III.   K INEMATIC AND   D YNAMIC   M ODELING  The five-link bipedal robotic system with five degrees of freedom can be described by the following direct kinematic model:  ) ( θ h X   =   (1) where   [   ]   5 5 4 3 2 1   ℜ ∈ =   T  θ θ θ θ θ θ   is   the   joint displacement   vector,   [   ]   2  ℜ ∈ =   T  y x X   is   the   Cartesian position   vector   and   2  ) (   ℜ ∈ θ h   is   a   nonlinear   function described by:  ( )          + + + − − + =  5 5 4 4 2 2 1 1 5 5 4 4 2 2 1 1  sin sin sin sin cos cos cos cos  θ l θ l θ l θ l  θ l θ l θ l θ l  θ h  The time derivative of the direct kinematic model (1) yields the following differential kinematic model:  θ θ J X   & &   ) ( =   (2) where   [   ] T  y x X   & & &   =   is   the   Cartesian   velocity   vector,  [   ] T  θ θ θ θ θ θ   5 4 3 2 1   & & & & & &   =   is the vector of joint velocities and  ) ( θ J   is the so-called analytical Jacobian matrix given by:  ( )             − − =  5 5 4 4 2 2 1 1 5 5 4 4 2 2 1 1  cos cos 0 cos cos sin sin 0 sin sin  θ l θ l θ l θ l  θ l θ l θ l θ l  θ J  A typical walking cycle may include three phases [1]: the single support phase (SSP), the impact phase (IP) and the double support phase (DSP), (see Fig.2). In this paper we focus only on SSP. Using Lagrange approach [9], the dynamical model of the bipedal robot in SSP phase is described by:  DU θ G θ θ H θ θ M   = + +   ) ( ) , ( ) (   & & &   (3) where   [   ] T  θ θ θ θ θ θ   5 4 3 2 1   & & & & & & & & & & & &   =   is   the   vector   of   joint accelerations,   [   ] T  U U U U U U   5 4 3 2 1 =   is the   vector of torque inputs,   ) ( θ M   is the symmetric positive definite inertia matrix,   ) , (   θ θ H   &   is the vector of centripetal and Coriolis torques and   ) ( θ G   is the   vector of gravitational torques   given by  [   ]   5 1 5 1 ) (   ≤ ≤ ≤ ≤ =   j i ij M θ M   ,   [   ]   5 1 5 1 ) , (   ≤ ≤ ≤ ≤ =   j i ij h θ θ H   &   ,   [   ]   5 1 ) (   ≤ ≤ =   i i G θ G  where:  ) cos( ) ( ) cos( )) ( ( ) cos( ) cos( ) ( ) (  5 1 5 5 1 5 15 5 1 5 1 5 4 4 1 4 14 3 1 3 1 3 13 2 1 2 1 5 2 1 4 2 1 3 2 1 2 12 2 1 5 4 3 2 1 2 1 1 11  θ θ k l l m M  θ θ l l m k l l m M  θ θ k l m M  θ θ l l m l l m l l m k l m M l m m m m I k m M  + − = + + − = − = − + + + = + + + + + =  ) cos( ) ( ) cos( ) ( ( ) cos( ) ( ) cos( ) ) ( (  5 2 5 5 2 5 25 4 2 4 2 5 4 4 2 4 24 3 2 3 2 3 23 2 2 5 4 3 2 2 2 2 22 2 1 2 1 5 4 3 2 1 2 21  θ θ k l l m M  θ θ l l m k l l m M  θ θ k l m M l m m m I k m M  θ θ l l m m m k l m M  + − = + + − = − = + + + + = − + + + =  Figure 2.The three phases of a typical walking cycle  0 ) cos( ) cos(  35 34 3 2 3 3 33 3 2 3 2 3 32 3 1 3 1 3 31  = = + = − = − =  J M I k m M  θ θ k l m M  θ θ k l m M  ) cos( )) ( ( ) ( 0 ) cos( ) ) ( ( ) cos( ) ) ( (  5 4 5 5 4 5 45 4 2 4 5 2 4 4 4 44 43 4 2 4 2 5 4 4 2 4 42 4 1 4 1 5 4 4 1 4 41  θ θ k l l m M I l m k l m M M  θ θ l l m k l l m M  θ θ l l m k l l m M  − − = + + − = = + + − = + + − =  5 2 5 5 5 55 5 4 5 5 4 5 54 53 5 2 5 5 2 5 52 5 1 5 5 1 5 51  ) ( ) cos( ) ( 0 ) cos( ) ( ) cos( ) (  I k l m M  θ θ k l l m M M  θ θ k l l m M  θ θ k l l m M  + − = − − = = + − = + − =  ) sin( )) ( ( ) sin( ) ) ( ( ) sin( ) ( ) sin( ) ( 0  5 1 5 5 1 5 15 4 1 4 1 5 4 4 1 4 14 3 1 3 1 5 13 2 1 2 1 5 2 1 4 2 1 3 2 1 2 12 11  θ θ k l l m h  θ θ l l m k l l m h  θ θ k l m h  θ θ l l m l l m l l m k l m h h  + − − = + + − − = − = − + + + = =  ) sin( ) ( ) sin( ) ) ( ( ) sin( 0 ) sin( ) ) ( (  5 2 5 5 2 5 25 4 2 4 2 5 4 4 2 4 24 3 2 3 2 3 23 22 2 1 2 1 5 4 3 2 1 2 21  θ θ k l l m h  θ θ l l m k l l m h  θ θ k l m h h  θ θ l l m m m k l m h  + − − = + + − − = − = = − + + + − =  0 0 ) sin( ) sin(  35 34 33 3 2 2 2 3 32 3 1 3 1 3 31  = = = − − = − − =  h h h  θ θ k l m h  θ θ k l m h  ) sin( )) ( ( 0 0 ) sin( ) ) ( ( ) sin( ) ) ( (  5 4 5 5 4 5 45 44 43 4 2 4 2 5 4 4 2 4 42 4 1 4 1 5 4 4 1 4 41  θ θ k l l m h h h  θ θ l l m k l l m h  θ θ l l m k l l m h  − − = = = + + − − = + + − − =  ) sin( )) ( ( 0 ) sin( )) ( ( ) sin( )) ( (  5 4 5 5 4 5 54 55 53 5 2 5 5 2 5 52 5 1 5 5 1 5 51  θ θ k l l m h h h  θ θ k l l m h  θ θ k l l m h  − − − = = = + − − = + − − =
5 5 5 5 5 4 4 5 4 4 4 4 3 3 3 3 2 2 5 4 3 2 2 2 1 1 5 4 3 2 1 1 1  cos )) ( ( cos ) ) ( ( cos cos ) ) ( ( cos ) ) ( (  θ k l m g G  θ l m k l m g G  θ gk m G  θ l m m m k m g G  θ l m m m m k m g G  − = + − = = + + + = + + + + =                   − − − − =  1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1  D  IV.   M ODEL   R EDUCTION AND   G AIT   T RAJECTORY   G ENERATION  The purpose of this section is to reduce the nonlinear dynamic model (3) into a solvable differential system in order to give algebraic solutions of gait trajectories. The reduced model will be obtained as follows: the robotic system (3) is first written around an equilibrium point   eq θ   as [9]:  v B x A x   + = &   (4)           ∂ ∂ − =   × − × ×  5 5 1 5 5 5 5  0 0  eq θ  θ  G J I A          =   − ×  D J B   1 5 5 0  [   ] T eq U U v   − =   × 5 1 0  [   ] T eq eq   θ θ θ θ x   & &   − − =  T o generate joint desired joint trajectory vector:  (   )   (   )   (   )   (   )   (   )   (   ) [   ] T d d d d d d   t θ t θ t θ t θ t θ t θ   , 5 , 4 , 3 , 2 , 1 =  we assume, for the bipedal robot, the following biologically inspired assumptions (see Fig.3):  Assumption   1: ZMP stability [10] is imposed. This is can be ensured by assuming that:  0 1   = U   (5)  Assumption   2: the supporting leg is kept straight as:  (   )   (   ) t θ t θ   d d   , 2 , 1   =   (6)  Assumption   3: The bipedal robot must have an upright posture that is to say, it keeps its back straight such that:  ( )   2  , 3  π  t θ   d   =   (7)  Assumption   4: the relationship between the right ankle joint and the left hip joint is imposed such that:  (   )   (   )   α t θ t θ   d d   + =   , 1 , 4   (8)  where   α   is a given constant angle to be chosen.  Assumption   5:   To generate a rhythmic stable movement, the relationship between the right ankle joint and the left knee joint is given by [11]:  ( )   ( )   ) ( sin 2 , 1 , 5   t  T  π  t θ t θ   d d   − =   (9)  From the assumptions (6)-(9), we can then easily deduce the desired velocity vector and the desired acceleration vector defined, respectively, by:  Figure 3. Biologically inspired assumptions  ( )   ( )   ( )   ( )   ( )   ( ) [   ] T d d d d d d   t θ t θ t θ t θ t θ t θ   , 5 , 4 , 3 , 2 , 1   & & & & & &   =  ( )   ( )   ( )   ( )   ( )   ( ) [   ] T d d d d d d   t θ t θ t θ t θ t θ t θ   , 5 , 4 , 3 , 2 , 1   & & & & & & & & & & & &   =  Particularly the following relations are deduced:  (   )   (   ) t θ t θ   d d   , 2 , 1   & & & &   =   (10)  (   )   0 , 3   = t θ   d  & &   (11)  (   )   (   ) t θ t θ   d d   , 1 , 4   & & & &   =   (12)               + =   t T  π  T  π θ θ   d d   2 cos 2  2 , 1 , 5   & & & &   (13)  By summing all the lines of the system (4) and using the relation (5), (10)-(13), the following reduced homogeneous 2 nd  order   differential   equations   with   constant   coefficients   is obtained as:  (   )   (   )   ) ( ) ( , 1 1 , 1 1   t s t h K t θ H t θ M   d d   − − − = + & &   (14)  where:  (   )  eq eq  θ θ  M M M M M M M M M M M  ) ( 2   45 25 15 14 12 23 55 44 22 11 1  + + + + + + + + + =  eq  θ  θ  G  θ  G  θ  G  θ  G H            ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ =  5 5 4 4 2 2 1 1 1  eq  θ θ  U  θ  G  θ  G π θ  G  α θ θ  G  θ θ  G  θ θ  G  θ θ  G  θ θ  G K  eq eq  1 5 5 3 3 4 4 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1  2 1 2   +          ∂ ∂ + ∂ ∂ + ∂ ∂ +           ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ − =         ∂ ∂ − =   t T  π θ  G t h   2 cos 2 ) (  5 5               =   t T  π  M T  π  t s   2 cos 2 ) (   2 2  55 45 25 15 2   M M M M M   + + + =  Solving (14) for the boundary conditions:  10 0 , 1   ) (   θ t θ   d   =  f f d   θ t θ   1 , 1   ) (   =
and for the equilibrium point   :  T eq  π π π π π θ          =   2 2 2 2 2  the analytical solution of the differential equation (14) is given by [12]:  1 3 2 1 , 1   2 cos ) (   2 1  H K t T  π  C e C e C t θ   t r t r d   −        + + =   (15) where :  10 3 1 2 1   θ C H K C C   − + − =  f f f  t r t r f t r f  e e  H K t T  π  C e H K  θ θ C C   2 1 1  1 3 1 10 1 3 2  2 cos  + +        −           − − + =            −              − ∂ ∂ =  1 1 2 2 2 5 5 3  4 2  H M T  π  T  π  M  θ  G C  1 1 1 2 , 1   M H M r   ⋅ − ± =   ,   0 1 1   〈 H M  To verify the algebraic solution (15) that asked a rather tedious calculation two approaches are used: a symbolic computation of the solution (15) using the symbolic toolbox of Matlab software   and   a   numerical   integration   of   the   differential equation (14) using the ode45 function of the same software. The three solutions were found superimposed for the physical parameters given by Table 1. Using the relations (15) and (6)-(9), the analytical expressions of the joint trajectories are then deduced. We impose then to the robotic model (3) the following second order linear input- output behavior [8]:  (   )   (   ) (   )   (   )   (   ) (   )   (   )   (   ) (   )   0 = − + − + −   t θ t θ K t θ t θ K t θ t θ   d p d v d   & & & & & &   (16)  where   5 5 ×  ℜ ∈ v K   and   5 5 ×  ℜ ∈ p K   are   two   positive   definite diagonal matrices chosen to guarantee global stability, desired performances and decoupling proprieties for the controlled system and such that the desired trajectories   d θ   ,   d θ &   and are derived from the solution (15) and the relations (6)-(9). The control law deduced from (3) and (16) is then given by:  (   )   (   )   (   ) (   )   (   )   (   ) (   ) (   ) [  )] ( ) , ( ) ( 1  θ G θ θ H t θ t θ K t θ t θ K t θ θ M D U   d p d v d  + + − − − − =   −  & & & & &  (17) Using   the   physical   parameters   given   in   table1,   the   joint trajectories of the controlled biped robot are generated as see by Fig. 4 where the desired joint position is designed by  (   )   (   )   (   )   (   )   (   )   (   ) [   ] T d   t teta t teta t tata t teta t teta t   5 4 3 2 1 θ   =   .   The walking cycle is shown by Fig.5. Compared to previous works [8], this paper gives an algebraic solution for desired joint trajectories that must be followed by the robotic system. The result is a stable rhythmic movement for the planar bipedal system.  Figure 4. Joint trajectories of the bipedal robot in the swing phase  Figure 5. The resulting walking cycle  V.   C ONCLUSION  A simple trajectory generation method is proposed for biped al gait by imposing a sinusoidal curve to the ankle of the swing leg's trajectory. R EFERENCES  [1]   M. Vukobratovic, B. Borovac, D. Surla, D. Stokic “Biped locomotion: dynamics, stability, control and application,” Springer-Verlag, 1989. [2]   A. Aloulou, O. Boubaker, “Minimum jerk-based control for a three dimensional bipedal robot,” Lecture notes in Computer Science, Vol. 7102, 2011, pp. 251-262.  [3]   K. Erbatur, U. Seven, “An inverted pendulum based approach to biped trajectory   generation   with   swing   leg   dynamics,”   Proc.   IEEE International Conference on Humanoid Robots, November 2007, pp. 216-221. [4]   N. Motoi, T. Suzuki, K. Ohnishi, “A bipedal locomotion planning based on virtual linear inverted pendulum model,” IEEE Transactions on Industrial Electronics, vol. 56, January 2009, pp. 54 – 61. [5]   S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, H. Hirukawa, “The 3D linear   inverted   pendulum   mode:   A   Simple   Modeling   for   a   Biped Walking Pattern Generation,” Proc. IEEE International Conference on Intelligent Robots and Systems, vol. 1, 2001, pp. 239-246. [6]   Y. Hurmuzlu, “Dynamics of bipedal gait: Part I-objective functions and the contact event of a planar five-link biped,” Journal of Applied Mechanics, vol. 60, June 1993, 331-336. [7]   C. Chevallereau, Y. Aoustin, “Optimal reference trajectories for walking and running of a biped robot,”   Robotica, vol. 19,   August   2001, 557- 569. [8]   A. Aloulou, O. Boubaker, “Control of a step walking combined to arms swinging for a three dimensional humanoid prototype,” Journal of Computer Science, vol. 6, August 2010, 886-895. [9]   M. Spong and M. Vidyasagar, “Robot dynamics and control,” John Wiley & Sons, New York, 1989. [10]   M. Vukobratovic, AA. Frank, D. Juricic, 1970, “On the stability of biped locomotion,” IEEE Transactions on Biomedical Engineering, vol. BME- 17, 1970, pp. 25-36. [11]  T.   Furuta,   T.   Tawara,   Y.   Okumura   &   M.   Shimizu.   “Design   and construction of series of compact humanoid robots and development of biped walk control strategies,” Robotics and autonomous Systems, vol. 34, November 2001, pp. 81-100. [12]   E. Hairer, G. Wanner, “Solving ordinary differential equations II: Stiff and differential-algebraic problems,” Springer-Verlag, 2010.