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Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot 97 (27) Kll lql lIl ll lqlv c ftot c ftot ))cos()cos( )cos()cos( )cos(())sin( )sin()sin( )sin()sin(( 444333 22200 111444 333222 111004 OTZOTZ OTSZEZ OTZOTZ OTZOTSZ O T Z E Z && && && && & & & (28) Kqll ll qll Iqll ll lqlv bcmscm ftot bcmscm ftot ))2/cos()cos( )cos()cos( )cos()cos(( ))2/sin()sin( )sin()sin( )sin()sin(( .,5.5444 333222 00111 .,5.5444 333222 111005 EOSZOTZ OTZOTSZ EZOTZ EOSZOTZ OTZOTSZ O T Z E Z && && && && && & & & (29) Kll qll Ill lqlv tortortor ftot tortortor ftottor ))2/cos()cos( )cos()cos(( ))2/sin()sin( )sin()sin(( 222 00111 222 11100 OTSZOTSZ EZOTZ OTSZOTSZ O T Z E Z && && && && & Accordingly, the linear acceleration of the links can be calculated easily. After generation of the robot trajectory paths with the aid of interpolation process and with utilization of MATLAB commands, the simulation of the biped robot can be performed. Based on the all above expressed relations and the resulted parameters and subsequently with inserting the parameters into the program, the simulation of the robot are presented in simulation results. 3. Dynamic of the robot In similarity of human and the biped robots, the most important parameter of stability of the robot refers to ZMP. The ZMP (Zero moment point) is a point on the ground whose sum of all moments around this point is equal to zero. Totally, the ZMP mathematical formulation can be presented as below: (30) )cos( )sin()cos( 1 111 i n i i n i ii i ii n i iii n i i zmp zgm Izxgmxzgm x ¦ ¦¦¦ O TOO Where, i x and i z are horizontal and vertical acceleration of the link's mass center with respect to F.C.S where i T is the angular acceleration of the links calculated from the interpolation process. On the other hand, the stability of the robot is determined according to attitude of ZMP. This means that if the ZMP be within the convex hull of the robot, the stable movement of the robot will be obtained and there are no interruptions in kinematic parameters (Velocity of the links). The convex hull can be imagined as a projection of a Humanoid Robots, New Developments98 pyramid with its heads on support and swing foots and also on the hip joint. Generally, the ZMP can be classified as the following cases: 1) Moving ZMP 2) Fixed ZMP The moving type of the robot walking is similar to human gait. In the fixed type, the ZMP position is restricted through the support feet or the user's selected areas. Consequently, the significant torso's modified motion is required for stable walking of the robot. For the explained process, the program has been designed to find target angle of the torso for providing the fixed ZMP position automatically. In the designed program, torso q shows the deflection angle of the torso determined by the user or calculated by auto detector mood of the program. Note, in the mood of auto detector, the torso needed motion for obtaining the mentioned fixed ZMP will be extracted with respect to the desired ranges. The desired ranges include the defined support feet area by the users or automatically by the designed program. Note, the most affecting parameters for obtaining the robot's stable walking are the hip's height and position. By varying the parameters with iterative method for sded xx , [Huang and et. Al, 2001] and choosing the optimum hip height, the robot control process with respect to the torso's modified angles and the mentioned parameters can be performed. To obtain the joint’s actuator torques, the Lagrangian relation [Kraige, 1989] has been used at the single support phase as below: (31) )(),()( ii qGqqqCqqH W where, 6,2,0 i and GCH ,, are mass inertia, coriolis and gravitational matrices of the system which can be written as following: » » » » » » » » ¼ º « « « « « « « « ¬ ª 67 57 47 37 27 17 666564636261 565554535251 464544434241 363534333231 262524232221 161514131211 )( h h h h h h hhhhhh hhhhhh hhhhhh hhhhhh hhhhhh hhhhhh qH » » » » » » » » ¼ º « « « « « « « « ¬ ª 67666564636261 57565554535251 47464544434241 37363534333231 27262524232221 17161514131211 ),( ccccccc ccccccc ccccccc ccccccc ccccccc ccccccc qqC » » » » » » » » ¼ º « « « « « « « « ¬ ª tor G G G G G G qG 5 4 3 2 1 )( Obviously, the above expressed matrices show the double support phase of the movement of the robot where they are used for the single support phase of the movement. On the other hand, the relation (31) is used for the single support phase of the robot. Within the double support phase of the robot, due to the occurrence impact between the swing leg and the ground, the modified shape of relation (31) is used with respect to effects of the reaction forces of the ground [Lum and et. Al. 1999 and Westervelt, 2003, and Hon and et. Al., 1978]. For the explained process and in order to obtain the single support phase equations of the robot, the value of 0 q (as can be seen in figure (1.4)) must be put equal to zero. The calculation process of the above mentioned matrices components contain bulk mathematical relations. Here, for avoiding the aforesaid relations, just the simplified relations are presented: Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot 99 torso torsoctorsotorsoctorso torsoctorsoeeectorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing fswingfswingecfswinge eeecfswing ccc eceee ccc eceecc ecececc IIII IIqqllqqllqqll qllqllqlllllm qqllqqll qqllqqllqqll qqllqqllqqll qqllqqllqllqll qllqllqlllllllm qqllqqllq qllqqllqqll qqllqllqllqllqll llllmqqllqqllqqll qllqllqlllllmqqll qllqllllmqlllm 543 2122111221 2211 22 2 2 1 3344 3443222442 2332111441 1331122144 332211 22 4 2 3 2 2 2 15 43344242233241411331 122144332 211 2 4 2 3 2 2 2 14323231311221 332211 2 3 2 2 2 131221 2211 2 2 2 1211 2 1111 ))]2/(cos(2))2/(cos(2)cos(2 ))2/(cos()cos()cos(([ ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos(2)cos(2 )cos(2))2/(cos(2)cos(2 )cos(2)cos(2))2/(cos()cos( )cos()cos()cos(([ ))]cos(2)cos(2)cos(2)cos(2) cos(2 )cos(2)cos()cos()cos()cos( ([))]cos(2)cos(2)cos(2 )cos()cos()cos(([))]cos(2 )cos()cos(([))]cos(([h SS SMMM ESES ES ES ESMM MMM MMMM MMM MMM torsotorsoctorso torsoctorsoctorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing cfswing ccc ccc cccc IIIIIIqqll qqllqqlllllm qqllqqll qqllqqll qqllqqllqqll qqllqqllqqlllllllm qqllqqllqqllqqllqqll qqllllllmqqll qqll qqlllllmqqllllmlm 5432122 111221 22 2 2 1 3344 344322 2442233211 144113311221 22 4 2 3 2 2 2 15 43344242233241411331 1221 2 4 2 3 2 2 2 1432323131 1221 2 3 2 2 2 131221 2 2 2 12 2 1112 ))]2/(cos(2 ))2/(cos(2)cos(2([ ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos(2 )cos(2)cos(2))2/(cos(2 )cos(2)cos(2)cos(2([ ))]cos(2)cos(2)cos(2)cos(2)cos(2 )cos(2([))]cos(2)cos(2 )cos(2([))]cos(2([ )]([h S S ESES ES ES Humanoid Robots, New Developments100 torso torsoctorsotorsoctorsoctorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing cfswing ccc cc cccc IIIII qqllqqllqqllllm qqllqqll qqllqqll qqllqqllqqll qqllqqllqqllllllm qqllqqllqqllqqll qqllqqlllllmqqll qqll qqllllmqqlllm 5432 22111221 22 2 3344 344322 2442233211 144113311221 22 4 2 3 2 25 4334424223324141 13311221 2 4 2 3 2 243232 31311221 2 3 2 231221 2 2213 ))]2/(cos(2))2/(cos()cos(([ ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos(2 )cos(2)cos(2))2/(cos( )cos()cos()cos(([ ))]cos(2)cos(2)cos(2)cos( )cos()cos(([))]cos(2 )cos()cos(([))]cos(([h SS ESES ES ES 5433344 34432224422332 1114411331 2 2 4 2 354334424223324141 1331 2 4 2 3432323131 2 3314 ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos()cos()cos( ))2/(cos()cos()cos( ([))]cos(2)cos()cos()cos( )cos(([))]cos()cos(([h IIIqqllqqll qqllqqllqqllqqll qqllqqllqqlll ll mqqllqqllqqllqqll qqllllmqqllqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswingcfswing ccc cccc ESES ES ES 543344 344322 2442111441 2 2 45433442424141 2 4415 ))])2/(cos())2/(cos(2 )cos())2/(cos( )cos())2/(cos()cos( ([))]cos()cos()cos(([h IIqqllqqll qqllqqll qqllqqllqqlll lmqqllqqllqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswingcfswing cccc ESES ES ES 53344 2211 2 516 ))])2/(cos())2/(cos( ))2/(cos())2/(cos(([h Iqqllqqll qqllqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingfswingfswingcfswingcfswing ESES ESES torosotorsoctorsotorsoctorsoctorsotorso Iqqllqqlllm ))])2/(cos())2/(cos(([h 2211 2 17 SS Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot 101 torsotorsoctorsotorsoctorso torsoctorsoeectorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing fswingfswingecfswing eeecfswing ccc ecee cccec eccecc IIIIIqqllqqll qqllqllqllllm qqllqqll qqllqqllqqll qqllqqllqqll qqllqqllqll qllqllqllllllm qqllqqllqqllqqll qq llqqllqllqllqll lllmqqllqqllqqllqll qllllmqqllqlllm 54322211 122122 22 2 3344 3443222442 2332111441 13311221 443322 22 4 2 3 2 25 4334424223324141 13311221443322 2 4 2 3 2 243 2323131122133 22 2 3 2 23122122 2 2221 ))]2/(cos(2))2/(cos( )cos())2/(cos()cos(([ ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos(2)cos(2 )cos(2))2/(cos()cos( )cos()cos())2/(cos( )cos()cos()cos(([ ))]cos(2)cos(2)cos(2)cos( )cos()cos()cos()cos()cos( ([ ))]cos(2)cos()cos()cos( )cos(([))]cos()cos(([h SS SMM ESES ES ES ESM MMM MMM M MM torso torsoctorsotorsoctorsoctorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing cfswingcc ccc cccc IIIII qqllqqllqqllllm qqllqqll qqllqqllqqll qqllqqllqqllqqllqqll llllmqqllqqllqqll qqllqqllqqlllllmqqll qqll qqllllmqqlllm 5432 22111221 22 2 3344 3443222442 233211144113311221 22 4 2 3 2 25433442422332 414113311221 2 4 2 3 2 243232 31311221 2 3 2 231221 2 2222 ))]2/(cos(2))2/(cos()cos(([ ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos(2)cos(2 )cos(2))2/(cos()cos()cos()cos( ([))]cos(2)cos(2)cos(2 )cos()cos()cos(([))]cos(2 )cos()cos(([))]cos(([h SS ESES ES ES torso torsoctorsoctorsotorfswingfswingcfswing fswingfswingcfswingfswingfswingcfswing cfswingcc cccc IIIII qqllllmqqll qqllqqllqqll qqllqqllllllmqqllqqll qqlllllmqqllllmlm 5432 22 22 233 44344322 24422332 22 4 2 3 2 2543344242 2332 2 4 2 3 2 243232 2 3 2 23 2 2223 ))]2/(cos(2([))])2/(cos(2 ))2/(cos(2)cos(2))2/(cos(2 )cos(2)cos(2([))]cos(2)cos(2 )cos(2([))]cos(2([)]([h SES ESES Humanoid Robots, New Developments102 543 33443443 2224422332 22 4 2 35 433442422332 2 4 2 343232 2 3324 ))])2/(cos(2))2/(cos(2)cos(2 ))2/(cos()cos()cos(([ ))]cos(2)cos()cos(([))]cos(([h III qqllqqllqqll qqllqqllqqlllllm qqllqqllqqllllmqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingcfswing ccccc ESES ES 5433 44344322 2442 22 4543344242 2 4425 ))])2/(cos( ))2/(cos(2)cos())2/(cos( )cos(([))]cos()cos(([h IIqqll qqllqqllqqll qqllllmqqllqqlllm fswingfswingcfswing fswingfswingcfswingfswingfswingcfswing cfswingccc ES ESES 533 4422 2 526 ))])2/(cos( ))2/(cos())2/(cos(([h Iqqll qqllqqlllm fswingfswingcfswing fswingfswingcfswingfswingfswingcfswingcfswing ES ESES torsotorsoctorsoctorsotorso Iqqlllm )])2/(cos(([h 22 2 27 S 5433344 3443222442 2332111441 133144 33 22 4 2 3543344242 2332414113314433 2 4 2 343232313133 2 3331 ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos()cos( )cos())2/(cos()cos( )cos())2/(cos()cos( )cos(([))]cos(2)cos( )cos()cos()cos()cos()cos( ([))]cos()cos()cos(([h IIIqqllqqll qqllqqllqqll qql lqqllqqll qqllqllqll qlllllmqqllqqll qqllqqllqqllqllqll llmqqllqqllqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing fswingfswingecfswinge ecfswingcc cece cccecc ESES ES ES ESMM M MM M Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot 103 543 3344 34432224422332 1114411331 2 2 4 2 354334424223324141 1331 2 4 2 3432323131 2 3332 ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos()cos()cos( ))2/(cos()cos()cos( ([))]cos(2)cos()cos()cos( )cos(([))]cos()cos(([h III qqllqqll qqllqqllqqllqqll qqllqqllqqlll ll mqqllqqllqqllqqll qqllllmqqllqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswingcfswing ccc cccc ESES ES ES 543 33443443 2224422332 22 4 2 35 433442422332 2 4 2 343232 2 3333 ))])2/(cos(2))2/(cos(2)cos(2 ))2/(cos()cos()cos(([ ))]cos(2)cos()cos(([))]cos(([h III qqllqqllqqll qqllqqllqqlllllm qqllqqllqqllllmqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingcfswing ccccc ESES ES 543 3344 3443 22 4 2 354334 2 4 2 34 2 3334 ))])2/(cos(2))2/(cos(2 )cos(2([))]cos(2([)]([h III qqllqqll qqlllllmqqllllmlm fswingfswingcfswingfswingfswingcfswing cfswingccc ESES 543344 3443 22 454334 2 4435 ))])2/(cos())2/(cos(2 )cos(([))]cos(([h IIqqllqqll qqllllmqqlllm fswingfswingcfswingfswingfswingcfswing cfswingcc ESES 53344 2 536 ))])2/(cos())2/(cos(([h Iqqllqqlllm fswingfswingcfswingfswingfswingcfswingcfswing ESES 0h 37 Humanoid Robots, New Developments104 54 33443443 22244211 144144 22 45 43344242414144 2 4441 ))])2/(cos())2/(cos(2)cos( ))2/(cos()cos())2/(cos( )cos())2/(cos()cos(([ ))]cos()cos()cos()cos(([h II qqllqqllqqll qqllqqllqqll qqllqllqllllm qqllqqllqq llqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingfswingfswingcfswing fswingfswingecfswingecfswing cccecc ESES ESES ESMM M 5433 44344322 2442111441 22 45433442424141 2 4442 ))])2/(cos( ))2/(cos(2)cos())2/(cos( )cos())2/(cos()cos( ([))]cos()cos()cos(([h IIqqll qqllqqllqqll qqllqqllqqll llmqqllqqllqqlllm fswingfswingcfswing fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing cfswingcccc ES ESES ES 5433 44344322 2442 22 4543344242 2 4443 ))])2/(cos( ))2/(cos(2)cos())2/(cos( )cos(([))]cos()cos(([h IIqqll qqllqqllqqll qqllllmqqllqqlllm fswingfswingcfswing fswingfswingcfswingfswingfswingcfswing cfswingccc ES ESES 543344 34432442 22 454334 2 4444 ))])2/(cos())2/(cos(2 )cos()cos(([))]cos(([h IIqqllqqll qqllqqllllmqqlllm fswingfswingcfswingfswingfswingcfswing cfswingcc ESES 5444 22 45 2 4445 ))])2/(cos(2([)]([h IIqqllllmlm fswingfswingcfswingcfswingc ES 544 2 546 ))])2/(cos(([h Iqqlllm fswingfswingcfswingcfswing ES 0h 47 Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot 105 533 4422 11 2 551 ))])2/(cos( ))2/(cos())2/(cos( ))2/(cos())2/(cos(([h Iqqll qqllqqll qqllqlllm fswingfswingcfswing fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingfswingfswingecfswingcfswing ES ESES ESESM 53344 2211 2 552 ))])2/(cos())2/(cos( ))2/(cos())2/(cos(([h Iqqllqqll qqllqqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingfswingfswingcfswingcfswing ESES ESES 53344 22 2 553 ))])2/(cos())2/(cos( ))2/(cos(([h Iqqllqqll qqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingcfswing ESES ES 53344 2 554 ))])2/(cos())2/(cos(([h Iqqllqqlllm fswingfswingcfswingfswingfswingcfswingcfswing ESES 544 2 555 ))])2/(cos(([h Iqqlllm fswingfswingcfswingcfswing ES 5 2 556 )]([h Ilm cfswing 0h 57 torsotorsotor torsotortorsotoretor Iqqll qqllqlllmasstorsoh ))])2/(cos( ))2/(cos())2/(cos(([ 22 11 2 61 S SMS Humanoid Robots, New Developments106 torsotorsotortorsotortor Iqqllqqlllmasstorsoh ))])2/(cos())2/(cos(([ 2211 2 62 SS torsotorsotortor Iqqlllmasstorsoh ))])2/(cos(([ 22 2 63 S torsotor Ilmasstorsoh )]([ 2 67 , 0 66,65,64 h In the all of above mentioned components, the following parameters have been used and they can be seen in figure (1.6) : (32) ,., .00. cmcm qq E O M E I (a) (b) Fig. (1.6). (a) The foot’s and (b) The support link’s geometrical configurations. [...]... Control of a New Biped Walking Robot”, International Journal of Humanoid Robotics, Vol 3, Num.1, March 2006, pp 10 5-1 26 7 Analytical Criterions for the Generation of Highly Dynamic Gaits for Humanoid Robots: Dynamic Propulsion Criterion and Dynamic Propulsion Potential Olivier Bruneau Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire d’Ingénierie des Systèmes de Versailles (LISV, EA 40 48) France... Fc1 + D2 p Fc 2 Thus, we obtain the close-loop constraint as function of the joint torques: 0 = f (Γ, q p , q p , q1 , q2 , q3 , q4 , q p , q1 , q2 , q3 , q4 ) (9) 4 Dynamic criterions Based on the previous equations, two dynamic criterions are proposed : the dynamic propulsion criterion and the dynamic propulsion potential 126 Humanoid Robots, New Developments 4. 1 Dynamic Propulsion Criterion To start,... gaits of robots In this research field, four kinds of approaches are used The first one uses pragmatic rules based on qualitative studies of human walking gaits (Pratt et al., 2001) (Sabourin et al., 20 04) The second one focuses on the mechanical design of the robot in order to obtain natural passive dynamic gaits (Collins et al., 2005) The third one deals with theoretical 122 Humanoid Robots, New Developments. .. walking anthropomorphic robots able to move in environments well-adapted to human beings and able to cooperate with them Among the more advanced projects of humanoid robots, one can mention : the Honda robots P2, P3 (Hirai, 1997) (Hirai et al., 1998) and Asimo (Sakagami et al., 2002), the HRP series developed by AIST (Kaneko et al., 1998) (Kajita et al., 2005) (Kaneko et al., 20 04) (Morisawa et al., 2005),... joint limit q imax , if we apply Γimax In order to perform that, we 128 Humanoid Robots, New Developments calculate, with (17) and (18), the new joint configuration that we would have obtained if we had applied Γimax Then, in order to stop the joint motion, we apply the minimum torque that the joint actuator can generate in this new joint configuration We continue this loop as long as the joint velocity... angular acceleration, : Tight angular acceleration (j) : Shank M.C inertial force, : Tight M.C inertial force (h) : Ankle joint torque, : Hip joint torque, …: Shank joint torque 113 Humanoid Robots, New Developments 1 14 (a) Stick Diagram (b) ZMP (c) Velocity (d) Acceleration Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot (e) Angular Vel... (Morisawa et al., 2005), the small robot QRIO proposed by Sony (Nagasaka et al., 20 04) , the KHR series developed by KAIST (Kim et al., 20 04) (Kim et al., 2005), the last robot of Waseda University having seven degrees of freedom per leg (Ogura et al., 20 04) , Johnnie (Lohmeier et al., 20 04) and H7 (Kagami et al., 2005) These robots are namely able to climb stairs and to carry their own power supply during... joint torque, : Hip joint torque, …: Shank joint torque 117 Humanoid Robots, New Developments 118 (a) Stick Diagram (b) ZMP (c) Velocity (d) Acceleration Dynamic Simulation of Single and Combined Trajectory Path Generation and Control of A Seven Link Biped Robot e) Angular Vel (f) Angular Acc (j) Inertial Forces (h) Driver Torques Fig 1. 14 (a) The robot’s stick diagram on 8 , Fixed ZMP, H min 0.60m,... parameters given in fig.2 The reference’s point Rp is selected at the hip of the robot for the definition of the position, velocity and acceleration of the trunk Fig 2 Model parameters of Rabbit 1 24 Humanoid Robots, New Developments 3 Analytical dynamics of Rabbit The approach is based on the dynamic equations with the lagrangian form for the robot Rabbit at the reference point Rp 3.1 Lagrangian Dynamic Equations... Shank joint torque 119 Humanoid Robots, New Developments 120 (a) Stick Diagram (b) ZMP (c) Inertial Forces (d) Driver Torques (e) Driver Torques Fig 1.15 (a) The robot’s stick diagram on combined surface, nominal motion, Moving ZMP, 8 (b) The moving ZMP diagram in nominal gait which satisfies stability criteria (c) Inertial forces: : Supp tight, : Supp shank, …: Swing tight, -: Swing shank (d) Joint’s . Iqqllqqlllm fswingfswingcfswingfswingfswingcfswingcfswing ESES 0h 37 Humanoid Robots, New Developments1 04 54 3 344 344 3 22 244 211 144 144 22 45 43 344 242 4 141 44 2 44 41 ))])2/(cos())2/(cos(2)cos( ))2/(cos()cos())2/(cos( )cos())2/(cos()cos(([ ))]cos()cos()cos()cos(([h II qqllqqllqqll qqllqqllqqll qqllqllqllllm qqllqqllqq llqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswingfswingfswingcfswing fswingfswingecfswingecfswing cccecc . torsotorsoctorsotorsoctorso torsoctorsoeectorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing fswingfswingecfswing eeecfswing ccc ecee cccec eccecc IIIIIqqllqqll qqllqllqllllm qqllqqll qqllqqllqqll qqllqqllqqll qqllqqllqll qllqllqllllllm qqllqqllqqllqqll qq llqqllqllqllqll lllmqqllqqllqqllqll qllllmqqllqlllm 543 22211 122122 22 2 3 344 344 322 244 2 233211 144 1 13311221 44 3322 22 4 2 3 2 25 43 344 242 233 241 41 1331122 144 3322 2 4 2 3 2 243 2323131122133 22 2 3 2 23122122 2 2221 ))]2/(cos(2))2/(cos( )cos())2/(cos()cos(([ ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos(2)cos(2 )cos(2))2/(cos()cos( )cos()cos())2/(cos( )cos()cos()cos(([ ))]cos(2)cos(2)cos(2)cos( )cos()cos()cos()cos()cos( ([ ))]cos(2)cos()cos()cos( )cos(([))]cos()cos(([h SS SMM ESES ES ES ESM MMM MMM M MM torso torsoctorsotorsoctorsoctorsotor fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing cfswingcc ccc cccc IIIII qqllqqllqqllllm qqllqqll qqllqqllqqll qqllqqllqqllqqllqqll llllmqqllqqllqqll qqllqqllqqlllllmqqll qqll qqllllmqqlllm . ES ESES torsotorsoctorsoctorsotorso Iqqlllm )])2/(cos(([h 22 2 27 S 543 3 344 344 322 244 2 233211 144 1 133 144 33 22 4 2 3 543 344 242 233 241 41133 144 33 2 4 2 343 232313133 2 3331 ))])2/(cos(2))2/(cos(2 )cos(2))2/(cos()cos( )cos())2/(cos()cos( )cos())2/(cos()cos( )cos(([))]cos(2)cos( )cos()cos()cos()cos()cos( ([))]cos()cos()cos(([h IIIqqllqqll qqllqqllqqll qql lqqllqqll qqllqllqll qlllllmqqllqqll qqllqqllqqllqllqll llmqqllqqllqlllm fswingfswingcfswingfswingfswingcfswing fswingfswingcfswing fswingfswingcfswing fswingfswingecfswinge ecfswingcc cece cccecc