Mobile Robots -Towards New Applications 2008 Part 3 ppt

Modelling of Bipedal Robots Using Coupled Nonlinear Oscillators 71 Fig. 21. Trajectory in the phase space (limit cycle). Fig. 22. Phase portrait. 5.3 Coupled Oscillators System Oscillators are said coupled if they allow themselves to interact, in some way, one with the other, as for example, a neuron that can send a signal for another one in regular intervals. Mathematically speaking, the differential equations of the oscillators have coupling terms that represent as each oscillator interacts with the others. According to Kozlowski et al. (1995), since the types of oscillators, the type and topology of coupling, and the external disturbances can be different, exist a great variety of couplings. In relation to the type of coupling, considering a set of n oscillators, exists three possible basic schemes (Low & Reinhall, 2001): 1) coupling of each oscillator to the closest neighbours, forming a ring (with the n-th oscillator coupled to the first one): ; 1,1 1 21,1 1,1 , 1 ° ¯ ° ®  =− −=−+ =+ == nii niii ini jni (5) 72 Mobile Robots, Towards New Applications 2) coupling of each oscillator to the closest neighbours, forming a chain (with the n-th oscillator not coupled to the first one): ; 1 1 21,1 11 , 1 ° ¯ ° ®  =− −=−+ =+ == nii niii ii jni (6) 3) coupling of each one of the oscillators to all others (from there the term "mutually coupled"): ijnjni ≠== , 1, 1 (7) This last configuration of coupling will be used in the analyses, since it desires that each one of the oscillators have influence on the others. Figure 23 presents the three basic schemes of coupling. Fig. 23. Basic schemes of coupling: in ring (a), in chain (b) and mutually coupled (c). 5.4 Coupled Oscillators with the Same Frequency From the equation (3), considering a net of n-coupled Rayleigh oscillators, and adding a coupling term that relates the velocities of the oscillators, we have: () () () nișșcșșșqș n j jii,jioiiiiiii ,2,1,01 1 22 ==−−−Ω+−− ¦ =    θδ (8) where δ i , q i , Ω i and c i,j are the parameters of this system. For small values of parameters determining the model nonlinearity, we will assume that the response is approximated by low frequency components from full range of harmonic response. Therefore periodic solutions can be expected, which can be approximated by: () iiioi tA α ω θ θ ++= cos (9) In this case, all oscillators have the same frequency ω. Deriving the equation (9) and inserting the solutions in (8), by the method of harmonic balance (Nayfeh and Mook, 1979), the following system of nonlinear algebraic equations are obtained: () () () () ° ° ° ¯ ° ° ° ®  =−+ ¸ ¸ ¹ · ¨ ¨ © § −+Ω− =−+ ¸ ¸ ¹ · ¨ ¨ © § −+−Ω ¦ ¦ = = 0coscoscos 4 3 1sin 0sinsinsin 4 3 1cos 1 , 22 22 1 , 22 22 jjii n j jii ii iiiii jjii n j jii ii iiiii AAc qA AA AAc qA AA ααωα ω ωδαω ααωα ω ωδαω (10) With this system of equations, the parameters q i and Ω i can be calculated: () [] niAAc AA q jiji n j ji iii i ,2,1,cos 3 4 3 4 1 , 3222 =−−+= ¦ = αα δωω (11) Modelling of Bipedal Robots Using Coupled Nonlinear Oscillators 73 () nicA A ji n j jij i i ,2,1,sin 1 , 2 =−−=Ω ¦ = αα ω ω (12) Given the amplitude i A and j A , phase α i and α j , the frequency ω, and the chosen values of δ i and c i,j , the value of the parameters q i and Ω i can be calculated. 5.5 Coupled Oscillators with Integer Relation of Frequency Oscillators of a coupling system, with frequency ω, can be synchronised with other oscillators with frequency n ω, where n is an integer. In the study of human locomotion, we can observe that some degrees of freedom have twice the frequency of the others (n = 2). Therefore, a net of coupled Rayleigh oscillators can be described as: () () () [] () 01 1 , 1 , 22 =−−−−−Ω+−− ¦¦ == n k khkh m i ioiiihhohhhhhhh ccq θθθθθθθθθδθ  (13) where the term () [ ] ioiiih c θθθ −  , is responsible for the coupling between two oscillators with different frequencies, while the other term () khkh c θθ  − , makes the coupling between two oscillators with the same frequencies. If the model nonlinearity is determined for small values of parameters, periodic solutions can be expected which can be approximated by the harmonic functions: () hhhoh tA α ω θ θ ++= 2cos (14) () iiioi tA αωθθ ++= cos (15) () kkkok tA α ω θ θ ++= 2cos (16) Deriving the equation (14-16) and inserting the solutions in (13), by the method of harmonic balance (Nayfeh and Mook, 1979), the following system of nonlinear algebraic equations are obtained: () ( ) () () ( ) () ° ° ° ° ° ° ° ¯ ° ° ° ° ° ° ° ®  =− ++−+Ω− =− ++−+−Ω ¦ ¦ ¦ ¦ = = = = 0coscos2 2cos 2 cos312sin4 0sinsin2 2sin 2 sin312cos4 1 , 1 , 2 2222 1 , 1 , 2 2222 kkhh n k kh m i iih i hhhhhhhh kkhh n k kh m i iih i hhhhhhhh AAc c A qAAA AAc c A qAAA ααω αωαωωδαω ααω αωαωωδαω (17) With this system of equations, the parameters q k and Ω k can be obtained: () () [] ¦ ¦ = = −− +−+= n k khkhkh hh m i ihihi hhh h AAc A cA AA q 1 , 32 1 , 2 3222 cos 3 1 2cos 12 1 3 1 αα δω αα δωω (18) 74 Mobile Robots, Towards New Applications () () kh n k khk h m i ihihi h h cA A cA A αα ω αα ω ω −−−+=Ω ¦¦ == 1 , 1 , 22 sin 2 2sin 2 4 (19) Given the amplitude h A , i A and k A , phase α h , α i and α k , the frequency ω, and the chosen values of δ h , c h,i and c h,k , the value of the parameters q h and Ω h can be calculated. 6. Analysis and Results of the Coupling System To generate the motion of knee angles θ 3 and θ 12 , and the hip angle θ 9 , as a periodic attractor of a nonlinear network, a set of three coupled oscillators had been used. These oscillators are mutually coupled by terms that determine the influence of each oscillator on the others (Fig. 24). How much lesser the value of these coupling terms, more “weak” is the relation between the oscillators. Fig. 24. Structure of coupling between the oscillators. Considering Fig. 24, from the Equation (13) the coupling can be described for the equations: ( ) () () () 0][1 12312,39999,333 2 33 2 3333 =−−−−−Ω+−− θθθθθθθθθδθ  ccq oo (20) ( ) () () ( ) 0][][1 12121212,93333,999 2 99 2 9999 =−−−−−Ω+−− ooo ccq θθθθθθθθθθδθ  (21) ( ) ()() () 0][1 3123,129999,121212 2 1212 2 12121212 =−−−−−Ω+−− θθθθθθθθθδθ  ccq oo (22) The synchronised harmonic functions, corresponding to the desired movements, can be writing as: () 3333 2cos α ω θ θ ++= tA o (23) () 999 cos α ω θ += tA (24) () 12121212 2cos α ω θ θ ++= tA o (25) Considering α 3 = α 9 = α 12 = 0 and deriving the equation (23-25), inserting the solution into the differential equations (20-22), the necessary parameters of the oscillators (q i and Ω i , i ∈ {3, 9, 12}) can be determined. Then: () 3 3 3 2 9,3 2 93312312,3 3 12 44 δω δ A cAAAAc q ++− = (26) ω 2 3 =Ω (27) 2 9 2 9 3 4 A q ω = (28) Modelling of Bipedal Robots Using Coupled Nonlinear Oscillators 75 ω =Ω 9 (29) () 12 3 12 2 9,12 2 912123123,12 12 12 44 δω δ A cAAAAc q ++− = (30) ω 2 12 =Ω (31) From equations (20-22) and (26-31), and using the MATLAB ® , the graphs shown in Fig. 25 and 26 were generated, and present, respectively, the behaviour of the angles as function of the time and the stable limit cycles of the oscillators. These results were obtained by using the parameters showed in Table 1, as well as the initial values provided by Table 2. All values were experimentally determined. In the Fig. 26, the great merit of this system can be observed, if an impact occurs and the angle of one joint is not the correct or desired, it returns in a small number of periods to the desired trajectory. Considering, for example, a frequency equal to 1 s − 1 , with the locomotor leaving of the repose with arbitrary initial values: θ 3 = −3°, θ 9 = 40° and θ 12 = 3°, after some cycles we have: θ 3 = 3°, θ 9 = 50° and θ 12 = −3°. Fig. 25. Behaviour of θ 3 , θ 9 and θ 12 as function of the time. Fig. 26. Trajectories in the phase space (stable limit cycles). 76 Mobile Robots, Towards New Applications c 3,9 c 9,3 c 3,12 c 12,3 c 9,12 c 12,9 ε 3 ε 9 ε 12 0.001 0.001 0.1 0.1 0.001 0.001 0.01 0.1 0.01 Table 1. Parameters of Rayleigh oscillators. Cycle A 3 A 9 A 12 θ 3o θ 9o θ 12o 0 < ω t ≤π −29 50 10 32 0 −13 π < ω t ≤ 2π−10 50 29 13 0 −32 Table 2. Experimental initial values. Comparing Fig. 25 and 26 with the experimental results presented in Section 3 (Fig. 5, 6, 12, 13), it is verified that the coupling system supplies similar results, what confirms the possibility of use of mutually coupled Rayleigh oscillators in the modelling of the CPG. Figure 27 shows, with a stick figure, the gait with a step length of 0.63 m. Figure 28 shows the gait with a step length of 0.38 m. Dimensions adopted for the model can be seen in Table 3. More details about the application of coupled nonlinear oscillators in the locomotion of a bipedal robot can be seen in Pina Filho (2005). Toes Foot Leg (below the knee) Thigh Length [m] 0.03 0.11 0.37 0.37 Table 3. Model dimensions. Fig. 27. Stick figure showing the gait with a step length of 0.63 m. Fig. 28. Stick figure showing the gait with a step length of 0.38 m. Modelling of Bipedal Robots Using Coupled Nonlinear Oscillators 77 7. Conclusion From presented results and their analysis and discussion, we come to the following conclusions about the modelling of a bipedal locomotor using mutually coupled oscillators: 1) The use of mutually coupled Rayleigh oscillators can represent an excellent way to signal generation, allowing their application for feedback control of a walking machine by synchronisation and coordination of the lower extremities. 2) The model is able to characterise three of the six most important determinants of human gait. 3) By changing a few parameters in the oscillators, modification of the step length and the frequency of the gait can be obtained. The gait frequency can be modified by means of the equations (23-25), by choosing a new value for ω. The step length can be modified by changing the angles θ 9 and θ 12 , being the parameters q i and Ω i , i ∈ {3, 9, 12}, responsible for the gait transitions. In future works, it is intended to study the behaviour of the ankles, as well as simulate the behaviour of the hip and knees in the other anatomical planes, thus increasing the network of coupled oscillators, looking for to characterise all determinants of gait, and consequently simulate with more details the central pattern generator of the human locomotion. 8. Acknowledgments The authors would like to express their gratitude to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazilian governmental entity promoter of the scientific and technological development, for the financial support provided during the course of this present research. 9. References Bay, J.S. & Hemami, H. (1987). Modelling of a neural pattern generator with coupled nonlinear oscillators. IEEE Trans. Biomed. Eng. 34, pp. 297-306 Brandão, M.L. (2004). As Bases Biológicas do Comportamento: Introdução à Neurociência, Editora Pedagógica Universitária (EPU), 225 p., Brazil Calancie, B.; Needham-Shropshire, B.; Jacobs, P.; Willer, K.; Zych, G. & Green, B.A. (1994). Involuntary stepping after chronic spinal cord injury. Evidence for a central rhythm generator for locomotion in man. Brain 117(Pt 5), pp. 1143-1159 Collins, J.J. & Richmond. S.A. (1994). Hard-wired central pattern generators for quadrupedal locomotion. Biological Cybernetics 71, pp. 375-385 Collins, J.J. & Stewart, I. (1993). Hexapodal gaits and coupled nonlinear oscillators models. Biological Cybernetics 68, pp. 287-298 Cordeiro, J.M.C. (1996). Exame neurológico de pequenos animais, EDUCAT, 270 p., Brazil Dietz, V. (2003). Spinal cord pattern generators for locomotion. Clinical Neurophysiology 114, pp. 1379-1389 Dimitrijevic, M.R.; Gerasimenko, Y. & Pinter, M.M. (1998). Evidence for a spinal central pattern generator in humans. Annals of the New York Academy of Sciences 860, pp. 360-376 Dutra, M.S.; Pina Filho, A.C.de & Romano, V.F. (2003). Modeling of a Bipedal Locomotor Using Coupled Nonlinear Oscillators of Van der Pol. Biological Cybernetics 88(4), pp. 286-292 78 Mobile Robots, Towards New Applications Dutra, M.S. (1995). Bewegungskoordination und Steuerung einer zweibeinigen Gehmaschine, Shaker Verlag, Germany Eberhart, H.D. (1976). Physical principles for locomotion, In: Neural Control of Locomotion, R. M. Herman et al., Plenum, USA Grillner, S. (1985). Neurobiological bases of rhythmic motor acts in vertebrates. Science 228, pp. 143-149 Horak, F.B. & Nashner, L.M. (1986). Central programming of postural movements adaptation to altered support-surface configurations. Journal of Neurophysiology 55, pp. 1369-1381 Jackson, E.A. (1990). Perspectives of Nonlinear Dynamics, Cambridge University Press Kapandji, I.A. (1980). Fisiologia articular: esquemas comentados de mecânica humana, Ed. 4, Vol. 2, Editora Manole, Brazil Kozlowski, J.; Parlitz, U. & Lauterborn, W. (1995). Bifurcation analysis of two coupled periodically driven Duffing oscillators. Physical Review E 51(3), pp. 1861-1867 Low, L.A. & Reinhall, P.G. (2001). An Investigation of Global Coordination of Gaits in Animals with Simple Neurological Systems Using Coupled Oscillators. UW Biomechanics Symposium Mackay-Lyons, M. (2002). Central Pattern Generation of Locomotion: A Review of the Evidence. Physical Therapy 82(1) McMahon, T.A. (1984). Muscles, Reflexes and Locomotion, Princeton University Press Meirovitch, L. (1975). Elements of Vibration Analysis, McGraw-Hill, USA Moraes, I.A. (1999). Sistema Nervoso, Departamento de Fisiologia e Farmacologia da Universidade Federal Fluminense, Brazil Nayfeh, A.H. & Mook, D.T. (1979). Nonlinear oscillations, John Wiley & Sons, Inc. Pearson, K.G. (1993). Common principles of motor control in vertebrates and invertebrates. Annu. Rev. Neurosci. 16, pp. 265-297 Penfield, W. (1955). The Role of the Temporal Cortex in Certain Psychical Phenomena. Journal of Mental Science 101, pp. 451-465 Pina Filho, A.C.de (2005). Study of Mutually Coupled Nonlinear Oscillators Applied in the Locomotion of a Bipedal Robot, D.Sc. Thesis, PEM/COPPE/UFRJ, Brazil Pina Filho, A.C.de; Dutra, M.S. & Raptopoulos, L.S.C. (2005). Modeling of a Bipedal Robot Using Mutually Coupled Rayleigh Oscillators. Biological Cybernetics 92(1), pp. 1-7 Pinter, M.M. & Dimitrijevic, M.R. (1999). Gait after spinal cord injury and the central pattern generator for locomotion. Spinal Cord 37(8), pp. 531-537 Raptopoulos, L.S.C. (2003). Estudo e Desenvolvimento de Equipamento de Baixo Custo para Análise da Marcha de Amputados, D.Sc. Thesis, PEM/COPPE/UFRJ, Brazil Saunders, J.B.; Inman, V. & Eberhart, H.D. (1953). The Major Determinants in Normal and Pathological Gait. J. Bone Jt. Surgery 35A Strogatz, S. (1994). Nonlinear Dynamics and Chaos, Addison-Wesley Winter, D. (1983). Biomechanical motor patterns in normal walking. J. Motor Behav. 15(4), pp. 302-330 Zielinska, T. (1996). Coupled oscillators utilised as gait rhythm generators of a two-legged walking machine. Biological Cybernetics 74, pp. 263-273 5 Ground Reference Points in Legged Locomotion: Definitions, Biological Trajectories and Control Implications Marko B. Popovic The Media Laboratory Massachusetts Institute of Technology U.S.A. Hugh Herr The MIT Media Laboratory MIT-Harvard Division of Health Sciences and Technology Spaulding Rehabilitation Hospital, Harvard Medical School U.S.A. The Zero Moment Point (ZMP) and Centroidal Moment Pivot (CMP) are important ground reference points used for motion identification and control in biomechanics and legged robotics. Using a consistent mathematical notation, we define and compare the ground reference points. We outline the various methodologies that can be employed in their estimation. Subsequently, we analyze the ZMP and CMP trajectories for level-ground, steady-state human walking. We conclude the chapter with a discussion of the significance of the ground reference points to legged robotic control systems. In the Appendix, we prove the equivalence of the ZMP and the center of pressure for horizontal ground surfaces, and their uniqueness for more complex contact topologies. Since spin angular momentum has been shown to remain small throughout the walking cycle, we hypothesize that the CMP will never leave the ground support base throughout the entire gait cycle, closely tracking the ZMP. We test this hypothesis using a morphologically realistic human model and kinetic and kinematic gait data measured from ten human subjects walking at self-selected speeds. We find that the CMP never leaves the ground support base, and the mean separation distance between the CMP and ZMP is small (14% of foot length), highlighting how closely the human body regulates spin angular momentum in level ground walking. KEY WORDS Legged Locomotion, Control, Biomechanics, Human, Zero Moment Point, Center of Pressure, Centroidal Moment Pivot 1. Introduction Legged robotics has witnessed many impressive advances in the last several decades from animal-like, hopping robots in the eighties (Raibert 1986) to walking humanoid robots at turn of the century (Hirai 1997; Hirai et al. 1998; Yamaguchi et al. 1999; Chew, Pratt, and 80 Mobile Robots, Towards New Applications Pratt 1999; Kagami et al. 2000). Although the field has witnessed tremendous progress, legged machines that demonstrate biologically realistic movement patterns and behaviors have not yet been offered due in part to limitations in control technique (Schaal 1999; Pratt 2002). An example is the Honda Robot, a remarkable autonomous humanoid that walks across level surfaces and ascends and descends stairs (Hirai 1997; Hirai et al.1998). The stability of the robot is obtained using a control design that requires the robot to accurately track precisely calculated joint trajectories. In distinction, for many movement tasks, animals and humans control limb impedance, allowing for a more robust handling of unexpected disturbances (Pratt 2002). The development of animal-like and human-like robots that mimic the kinematics and kinetics of their biological counterparts, quantitatively or qualitatively, is indeed a formidable task. Humans, for example, are capable of performing numerous dynamical movements in a wide variety of complex and novel environments while robustly rejecting a large spectrum of disturbances. Given limitations on computational capacity, real-time trajectory planning in joint space does not seem feasible using optimization strategies with moderately-long future time horizons. Subsequently, for the diversity of biological motor tasks to be represented in a robot’s movement repertoire, the control problem has to be restated using a lower dimensional representation (Full and Koditschek 1999). However, independent of the specific architecture that achieves that reduction in dimension, biomechanical motion characteristics have to be identified and appropriately addressed. There are two ground reference points used for motion identification and control in biomechanics and legged robotics. The locations of these reference points relative to each other, and relative to the ground support area, provide important local and sometimes global characteristics of whole-body movement, serving as benchmarks for either physical or desired movement patterns. The Zero Moment Point (ZMP), first discussed by Elftman 1 (1938) for the study of human biomechanics, has only more recently been used in the context of legged machine control (Vukobratovic and Juricic 1969; Vukobratovic and Stepanenko 1972; Takanishi et al. 1985; Yamaguchi, Takanishi and Kato 1993; Hirai 1997; Hirai et al. 1998). The Centroidal Moment Pivot (CMP) is yet another ground reference point recently introduced in the literature (Herr, Hofmann, and Popovic 2003; Hofmann, 2003; Popovic, Hofmann, and Herr 2004a; Goswami and Kallem 2004). When the CMP corresponds with the ZMP, the ground reaction force passes directly through the CM of the body, satisfying a zero moment or rotational equilibrium condition. Hence, the departure of the CMP from the ZMP is an indication of non-zero CM body moments, causing variations in whole-body, spin angular momentum. In addition to these two standard reference points, Goswami (1999) introduced the Foot Rotation Indicator (FRI), a ground reference point that provides information on stance-foot angular accelerations when only one foot is on the ground. However, recent study (Popovic, Goswami and Herr 2005) find that the mean separation distance between the FRI and ZMP during the powered plantar flexion period of single support is within the accuracy of their 1 Although Borelli (1680) discussed the concept of the ZMP for the case of static equilibrium, it was Elftman (1938) who introduced the point for the more general dynamic case. Elftman named the specified point the “position of the force” and built the first ground force plate for its measurement. [...]... in walking Human Biology 11: 529- 535 Full, B and Koditschek, D 1999 Templates and Anchors: Neural Mechanical Hypotheses of Legged Locomotion on Land J Exp Biol 202: 33 25 -33 32 102 Mobile Robots, Towards New Applications Goswami, A 1999 Postural stability of biped robots and the foot-rotation indicator (FRI) point International Journal of Robotics Research 18(6): 5 23- 533 Goswami, A and Kallem, V 2004... Walking Stabilized by Trunk Motion on a Sagitally Uneven Surface Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp 32 3 -33 0 104 Mobile Robots, Towards New Applications Tilley A R and Dreyfuss H 19 93 The measure of man and woman New York, Whitney Library of Design, an imprint of Watson-Guptill Publications Vukobratovic, M and Juricic, D 1969 Contributions to the... Man and Cybernetics, Part A, 34 (5): 630 - 637 Schaal, S 1999 Is imitation learning the route to humanoid robots? Trends in Cognitive Sciences 3: 233 -242 Shih, C L., Li, Y Z., Churng, S., Lee, T T and Gruver W A 1990 Trajectory synthesis and physical admissibility for a biped robot during the single-support phase Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp... angular momentum and balance maintenance of biped robots Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, LA, U.S.A., pp 37 85 -37 90 Gu, W 20 03 The Regulation of Angular Momentum During Human Walking Undergraduate Thesis, Massachusetts Institute of Technology, Physics Department Herr, H., Whiteley, G.P and Childress, D 20 03 Cyborg Technology Biomimetic Orthotic and... middle of a single support phase to the middle of the next single support phase of the opposite limb Subjects 1 2 3 4 5 6 7 8 9 10 Mean±STD A% 16 14 13 17 16 10 12 11 15 15 14 ± 2 B% 15 13 10 15 12 9.0 14 15 15 14 13 ± 2 C% 16 13 12 16 15 10 12 12 15 15 14 ± 2 Table 1 For ten healthy test participants walking steadily at their self-selected speeds, listed are the mean distances, normalized by foot length,... statistical analysis, significance was determined using p < 0.05 3. 2 Results Representative trajectories of the ZMP and CMP are shown in Figure 5 for a healthy female participant (age 21, mass 50.1 kg, height 158 cm, speed ~1 .3 m/s) For each study participant, Table 1 lists the mean normalized distances between the CMP and the ZMP For all participants and for all walking trials, the ZMP was always well... length (see Figure 5) Finally, for all participants and for all walking trials, the CMP remained within ground support base throughout the entire gait cycle The mean of the normalized distance between the CMP and the ZMP for the single support phase (14 ± 2%) was not significantly different from that computed for the double support phase ( 13 ± 2%) (p=0 .35 ) 3. 3 Discussion Since spin angular momentum... Animals, Borelli (1680) discussed a biomechanical point that he called the support point, a ground reference location where the resultant 82 Mobile Robots, Towards New Applications ground reaction force acts in the case of static equilibrium Much later, Elftman (1 938 ) defined a more general “position of the force” for both static and dynamic cases, and he built the first ground force plate for its measurement... attached to the ground surface (see equation (15)) (Popovic and Herr 20 03; Hofmann et al 2004, Popovic, Hofmann and Herr 2004b) 6 Here stability refers to the capacity of the system to restore the CM to a location vertically above the center of the ground support envelope ( x ZMP 0 ) after a perturbation 94 Mobile Robots, Towards New Applications Although controlling ZMP position is one strategy for stabilizing... definitions of the ZMP (see Section 2.1 for ZMP definitions, equations (1) and (2)) 98 Mobile Robots, Towards New Applications Given the definition of the CP (equation (A.1)), we can prove that the CP is identical to the ZMP by noting from equation (A.2) that G R (rCP ) | horizontal G R (0) | horizontal g g FG R Z rCP 0, (A .3) therefore satisfying one definition of the ZMP defined in equation (1), Section . Equation ( 13) the coupling can be described for the equations: ( ) () () () 0][1 1 231 2 ,39 999 ,33 3 2 33 2 33 33 =−−−−−Ω+−− θθθθθθθθθδθ  ccq oo (20) ( ) () () ( ) 0][][1 12121212, 933 33, 999 2 99 2 9999 =−−−−−Ω+−− ooo ccq θθθθθθθθθθδθ  . and Ω i , i ∈ {3, 9, 12}) can be determined. Then: () 3 3 3 2 9 ,3 2 933 1 231 2 ,3 3 12 44 δω δ A cAAAAc q ++− = (26) ω 2 3 =Ω (27) 2 9 2 9 3 4 A q ω = (28) Modelling of Bipedal Robots Using. Trajectories in the phase space (stable limit cycles). 76 Mobile Robots, Towards New Applications c 3, 9 c 9 ,3 c 3, 12 c 12 ,3 c 9,12 c 12,9 ε 3 ε 9 ε 12 0.001 0.001 0.1 0.1 0.001 0.001 0.01 0.1 0.01