From Simple to Complex Models: Gaits as Oscillations - RAM-Lab
From Simple to Complex Models: Gaits as Oscillations Yevgeniy Yesilevskiy? , Zhenyu Gan? , Weitao Xi? , and C. David Remy ? the authors contributed equally to this work
G AITS ARE VERY USEFUL ...
oscillates out of phase with the energy associated with the vertical motion (the sum of potential energy and kinetic energy in the vertical direction). For running, the two energy values oscillate in phase. These different energy oscillations are explained by an inverted pendulum model for walking, and a spring loaded inverted pendulum model for running [10]. This line of reasoning broadly sorts gaits into stiff and springy legged motions. As another example, Alexander [11] described bipedal gaits using their stride length, duty factor, and a shape factor which is defined as the relative value of coefficients in a Fourier analysis of the ground reaction forces (GRFs).
In nature, humans and animals move in ways that allow them to achieve energetically economical motion over a large range of velocities. In particular, researchers have shown that legged animals use multiple gaits in order to minimize their cost of transport (COT; the amount of energy used per distance traveled [1], [2]). For example, humans transition from walking to running [3], and horses transition from walking, to trotting, to galloping [4] as their current gait becomes uneconomical. In addition, Hoyt and Taylor found that when horses were allowed to self-select their velocities within a particular gait, they chose to move at the energetic minimum. Similar results have been obtained in a number of optimization studies that were conducted with conceptual dynamic models [5], [6], [7], showing that energetic efficiency can be achieved by using different modes of locomotion.
A reason why so many definitions exist, is that none of them are without shortcomings, and therefore, none are able to explain all gaits in a unified manner. Hildebrand’s footfall classification is unable to differentiate between gaits with identical footfall patterns, but different GRFs. For example, it cannot classify the grounded running that ostriches use at intermediate speeds. This gait has the same contact sequence as walking, but shows the vertical ‘single-hump’ GRFs characteristic of running [12]. The use of two models to differentiate gaits (as inspired by Cavagna) proves problematic as well. The inverted pendulum model is able to correctly explain part of the energy cycling in walking, but it is unable to reproduce the correct GRFs [13]. Quantitatively, only 65 % of energy fluctuations can be attributed to the energy exchange in an inverted pendulum [9]. 35 % remain unexplained. Moreover, for galloping, the vertical energy oscillates in phase with the horizontal energy at times, and out of phase at other times [14]. Therefore it does not fit cleanly into the two model paradigm. From the perspective
... BUT WHAT EXACTLY IS A G AIT ? Despite this obvious usefulness, it is not fully clear what a gait actually is. Many different definitions exist. Hildebrand [8], for example, classifies different gaits using the “percent of the stride interval that each hind foot is on the ground” along with the “percent of the stride interval that the footfall of a forefoot lags behind the footfall of the hind foot on the same side of the body”. From those two numbers, he was able to extract the footfall pattern, the support pattern, and the duration of the various combinations of support. Cavagna et al. [9], on the other hand, differentiated gaits from observations about energy cycling. For walking, the kinetic energy of the center of mass in the forward direction
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Fig. 1. The dependency of gait, motion, and morphology is illustrated in this analogy. Gaits of legged systems can be interpreted as different dynamic modes of the mechanical structure; similar to the different modes of a flexible beam (A, B). For both systems, a periodic motion can be sustained with the least amount of effort if the motion and the actuator inputs are matched to these natural dynamics (a-c). The modes can be shaped by adjusting the morphology of the system (I, II) which can be used to improve the performance at desired operating points.
support alternate with an air phase, no noticeable body pitch was observed. Consequently, the main body exhibited a much larger vertical motion than during walking or t¨olting. la dC O Mla
0
mo, jo
l
of a roboticist, it is probably most problematic that all of these definitions are purely descriptive. While they are able to classify different characteristics of locomotion, they give no explanation of where these gaits originate. Alexander’s shape factor is probably the best example for this. His Fourier coefficients can be used in an optimization problem to yield ‘natural looking’ GRFs [11], yet they do not give any insight into why these different force profiles enable efficient locomotion at different speeds. These definitions are not useful from a synthetic point of view. For a roboticist, this synthetic perspective still poses an open question: How can we build the ability to perform multiple efficient gaits into legged robotic systems?
kF
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y αH
αF
x
(a)
A NEW PERSPECTIVE ON G AIT Recent work in passive dynamic walking suggests a new interpretation of gaits. Geyer et al. [15] were able to show that a bipedal model with two independent, massless legs with springs was able to accurately reproduce the energy cycling and GRFs in both walking and running. This discovery of multiple gaits within a single model suggests that gaits are different complex non-linear oscillation modes of a single oscillatory system. This concept can be understood by its analogy to the different modes of oscillation of a flexible beam (Fig. 1, left). Using similar model approximations as Geyer et al., we studied passive dynamic walking of quadrupeds (Fig. 2a). Our simple model was able to move in a variety of different modes while using the same set of model parameters [16]. In particular, we discovered passive motions that had the same footfall patterns as walking, trotting, t¨olting, pacing, bounding, and galloping of horses. These gaits are merely different dynamic modes of the same mechanical system. In addition to replicating a large number of footfall sequences, our models also accurately predicted vertical GRFs of horses at walk, trot, and t¨olt (Fig. 3a,b). Analyzing the detailed motion of these different locomotion modes, we clearly observed fundamentally different oscillatory patterns. Walking, for example, had phases of double support alternating with phases of triple support. The vertical GRFs exhibited a double hump with a mid-stance relief. These two force peaks coincided with a double compression of the leg springs. Also, in walking, the main body showed a clear pitching motion. The main body oscillated between -1.45 ◦ and 1.23 ◦ during a gait cycle. T¨olting has the same footfall sequence as walking. Yet, in this gait phases of double support alternate with phases of single support and there are fundamentally different oscillatory patterns. For t¨olting, the vertical GRFs exhibited only a single hump. This single peak was roughly twice the magnitude of the peaks in walking. A pitching motion is still present, but less pronounced than in walking (the main body oscillates between -0.47 ◦ and 0.48 ◦ ). Overall, t¨olting exhibited the least variability in main body vertical and horizontal velocity throughout the course of a stride. Trotting also showed a single hump in the vertical GRFs - corresponding to a single spring compression. Yet, since trotting is a two-beat gait in which phases of double-
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Fig. 2. Our work explored two different models of legged systems. Passive models with massless legs and no damping in the springs (a) and actuated models with distributed mass, series elastic actuators in the legs and hips, and damping in the springs (b). Both models exhibited a number of similar oscillation modes that yielded a variety of gaits.
While these results provide significant evidence towards interpreting gaits as different oscillatory modes of a single system, the simplicity of the underlying model limits possible conclusions with respect to actual robotic systems. The missing actuation, in particular, conceals issues of efficiency and other energetic benefits when exploiting different oscillations modes. Such benefits would be expected, given that exciting an oscillatory system in a natural mode would minimize the amount of required excitation power. To investigate this hypothesis, we additionally conducted optimal control studies with more complex models of legged robots (Fig. 2b). These models have fully distributed masses, which leads to complex swing leg dynamics and collision losses at touchdown. Additional losses are created by viscous friction in the springy legs. The models are not energetically conservative and are driven by detailed series elastic actuators. In order to not bias the model towards established gaits and to reduce the influence of optimization errors, we conducted two studies using different optimization methods. We used direct collocation in an optimization framework that could detect optimal footfall patterns automatically [17], and we used an established multiple shooting framework (MUSCOD [18]) for which we pre-selected a wide range of different footfall sequences. Both approaches yielded similar results. Despite the addition of actuation, collision losses, swing
leg dynamics, and damping losses, these complex models were found to perform optimally at the same oscillatory modes as the passive system. For low speeds, four-beat walking was the most economical gait, and for higher speeds, trotting was most economical (Fig. 3c). In addition, t¨olting proved to be only slightly more energetically expensive than trotting, indicating why it might be adopted by some horses.
or “how do morphological changes affect gait dynamics and their optimality?”. In terms of robotics, our in-depth analysis of the origin of gaits and their relation to the natural dynamics of a legged system will help hardware designers to systematically create systems that take advantage of natural motion. Roboticists can use this work to design the mechanical dynamics of legged systems in a way that requires minimal actuation, and that takes advantage of different ‘resonance’ cases that yield different gaits. The presented optimal control framework and the optimization of complex models can be used in a design process that synthesizes robotic gaits, motions, and morphologies at the same time. We expect that the present paper will spark a lively discussion at the conference, given that there is such a variety of definitions and explanations for gaits. R EFERENCES
Fig. 3. Model predicted ground reaction forces (solid lines) for walking (a) and t¨olting (b) are compared to data recorded from actual horses (dotted lines). Labels refer to the different legs (L-left, R-right, H-hind, F-front). While being based on the same model, t¨olting undergoes a fundamentally different oscillation pattern than walking. The graph in (c) shows the cost of transport obtained via optimal control for various gaits for an actuated model. At low speeds a four-beat walk is the most efficient gait, and at high speeds, trotting is most efficient.
The optimizers’ discovery of solutions that undergo the same fundamental motions as the passive model is a strong indication that we can interpret legged locomotion as a complex nonlinear oscillation which is merely excited and maintained through periodic actuator inputs. The underlying premise is that the mechanical dynamics of the legged system (driven by inertia, gravity, and elastic oscillations) can periodically store and return energy and are passively creating a substantial part of a desired motion. With this, we can consider gaits as a ‘resonant’ usage of a number of different oscillatory modes that optimally exploit the natural dynamics to improve energetic efficiency at different operating points. This interpretation of gait enables a synthetic perspective on legged systems since mass properties, stiffness values, and geometrical properties can be tuned to improve or alter the performance of such legged systems (Fig. 1, right). I. W HY IS THIS USEFUL ? The question of ‘what is a gait?’ has relevance for both the biological and robotic research communities. Our research gives new insight into why humans and animals move the way they do. Showing how animal locomotion can be represented by simple models helps to understand the underlying dynamics of their motion and adds another piece of information to the study of gaits. These simple models might additionally open the door to answering intricate questions whose answers can be lost in the complexity of real animals. For example, “what drives gait transitions?”
[1] G. Gabrielli and T. Von Karman, What price speed?: specific power required for propulsion of vehicles. 1950. [2] V. A. Tucker, “The energetic cost of moving about: Walking and running are extremely inefficient forms of locomotion. much greater efficiency is achieved by birds, fishand bicyclists,” American Scientist, vol. 63, no. 4, pp. 413–419, 1975. [3] P. Di Prampero, “The energy cost of human locomotion on land and in water.,” International journal of sports medicine, vol. 7, no. 2, pp. 55– 72, 1986. [4] D. F. Hoyt and C. R. Taylor, “Gait and the energetics of locomotion in horses.,” Nature, 1981. [5] C. Chevallereau, Y. Aoustin, et al., “Optimal reference trajectories for walking and running of a biped robot,” Robotica, vol. 19, no. 5, pp. 557–569, 2001. [6] M. Srinivasan and A. Ruina, “Computer optimization of a minimal biped model discovers walking and running,” Nature, vol. 439, no. 7072, pp. 72–75, 2006. [7] K. Tsujita, M. Kawakami, and K. Tsuchiya, “A study on optimal gait pattern of a quadruped locomotion robot,” in Systems, Man and Cybernetics, 2004 IEEE International Conference on, vol. 1, pp. 745– 749, IEEE, 2004. [8] M. Hildebrand, “Symmetrical gaits of horses,” Science, vol. 150, no. 3697, pp. 701–708, 1965. [9] G. A. Cavagna, H. Thys, and A. Zamboni, “The sources of external work in level walking and running.,” The Journal of physiology, vol. 262, no. 3, pp. 639–657, 1976. [10] R. Blickhan, “The spring-mass model for running and hopping,” Journal of biomechanics, vol. 22, no. 11, pp. 1217–1227, 1989. [11] R. M. Alexander, “A model of bipedal locomotion on compliant legs,” Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, vol. 338, no. 1284, pp. 189–198, 1992. [12] J. Rubenson, D. B. Heliams, D. G. Lloyd, and P. A. Fournier, “Gait selection in the ostrich: mechanical and metabolic characteristics of walking and running with and without an aerial phase,” Proceedings of the Royal Society of London-B, vol. 271, no. 1543, p. 1091, 2004. [13] M. G. Pandy, “Simple and complex models for studying muscle function in walking,” Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, vol. 358, no. 1437, pp. 1501– 1509, 2003. [14] A. E. Minetti, “The biomechanics of skipping gaits: a third locomotion paradigm?,” Proceedings of the Royal Society of London. Series B: Biological Sciences, vol. 265, no. 1402, pp. 1227–1233, 1998. [15] H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant leg behaviour explains basic dynamics of walking and running,” Proceedings of the Royal Society B, vol. 273, no. 1603, pp. 2861–2867, 2006. [16] Z. Gan and C. D. Remy, “A passive dynamic quadruped that moves in a large variety of gaits,” in Intelligent Robots and Systems (IROS), 2014 IEEE/RSJ International Conference on, p. Accepted for publication, IEEE, 2014. [17] W. Xi and C. D. Remy, “Optimal gaits and motions for legged robots,” in Intelligent Robots and Systems (IROS), 2014 IEEE/RSJ International Conference on, p. Accepted for publication, IEEE, 2014. [18] M. Diehl, D. B. Leineweber, and A. A. Sch¨afer, MUSCOD-II User’s Manual. IWR, 2001.
G AITS ARE VERY USEFUL ...
oscillates out of phase with the energy associated with the vertical motion (the sum of potential energy and kinetic energy in the vertical direction). For running, the two energy values oscillate in phase. These different energy oscillations are explained by an inverted pendulum model for walking, and a spring loaded inverted pendulum model for running [10]. This line of reasoning broadly sorts gaits into stiff and springy legged motions. As another example, Alexander [11] described bipedal gaits using their stride length, duty factor, and a shape factor which is defined as the relative value of coefficients in a Fourier analysis of the ground reaction forces (GRFs).
In nature, humans and animals move in ways that allow them to achieve energetically economical motion over a large range of velocities. In particular, researchers have shown that legged animals use multiple gaits in order to minimize their cost of transport (COT; the amount of energy used per distance traveled [1], [2]). For example, humans transition from walking to running [3], and horses transition from walking, to trotting, to galloping [4] as their current gait becomes uneconomical. In addition, Hoyt and Taylor found that when horses were allowed to self-select their velocities within a particular gait, they chose to move at the energetic minimum. Similar results have been obtained in a number of optimization studies that were conducted with conceptual dynamic models [5], [6], [7], showing that energetic efficiency can be achieved by using different modes of locomotion.
A reason why so many definitions exist, is that none of them are without shortcomings, and therefore, none are able to explain all gaits in a unified manner. Hildebrand’s footfall classification is unable to differentiate between gaits with identical footfall patterns, but different GRFs. For example, it cannot classify the grounded running that ostriches use at intermediate speeds. This gait has the same contact sequence as walking, but shows the vertical ‘single-hump’ GRFs characteristic of running [12]. The use of two models to differentiate gaits (as inspired by Cavagna) proves problematic as well. The inverted pendulum model is able to correctly explain part of the energy cycling in walking, but it is unable to reproduce the correct GRFs [13]. Quantitatively, only 65 % of energy fluctuations can be attributed to the energy exchange in an inverted pendulum [9]. 35 % remain unexplained. Moreover, for galloping, the vertical energy oscillates in phase with the horizontal energy at times, and out of phase at other times [14]. Therefore it does not fit cleanly into the two model paradigm. From the perspective
... BUT WHAT EXACTLY IS A G AIT ? Despite this obvious usefulness, it is not fully clear what a gait actually is. Many different definitions exist. Hildebrand [8], for example, classifies different gaits using the “percent of the stride interval that each hind foot is on the ground” along with the “percent of the stride interval that the footfall of a forefoot lags behind the footfall of the hind foot on the same side of the body”. From those two numbers, he was able to extract the footfall pattern, the support pattern, and the duration of the various combinations of support. Cavagna et al. [9], on the other hand, differentiated gaits from observations about energy cycling. For walking, the kinetic energy of the center of mass in the forward direction
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Fig. 1. The dependency of gait, motion, and morphology is illustrated in this analogy. Gaits of legged systems can be interpreted as different dynamic modes of the mechanical structure; similar to the different modes of a flexible beam (A, B). For both systems, a periodic motion can be sustained with the least amount of effort if the motion and the actuator inputs are matched to these natural dynamics (a-c). The modes can be shaped by adjusting the morphology of the system (I, II) which can be used to improve the performance at desired operating points.
support alternate with an air phase, no noticeable body pitch was observed. Consequently, the main body exhibited a much larger vertical motion than during walking or t¨olting. la dC O Mla
0
mo, jo
l
of a roboticist, it is probably most problematic that all of these definitions are purely descriptive. While they are able to classify different characteristics of locomotion, they give no explanation of where these gaits originate. Alexander’s shape factor is probably the best example for this. His Fourier coefficients can be used in an optimization problem to yield ‘natural looking’ GRFs [11], yet they do not give any insight into why these different force profiles enable efficient locomotion at different speeds. These definitions are not useful from a synthetic point of view. For a roboticist, this synthetic perspective still poses an open question: How can we build the ability to perform multiple efficient gaits into legged robotic systems?
kF
kH
y αH
αF
x
(a)
A NEW PERSPECTIVE ON G AIT Recent work in passive dynamic walking suggests a new interpretation of gaits. Geyer et al. [15] were able to show that a bipedal model with two independent, massless legs with springs was able to accurately reproduce the energy cycling and GRFs in both walking and running. This discovery of multiple gaits within a single model suggests that gaits are different complex non-linear oscillation modes of a single oscillatory system. This concept can be understood by its analogy to the different modes of oscillation of a flexible beam (Fig. 1, left). Using similar model approximations as Geyer et al., we studied passive dynamic walking of quadrupeds (Fig. 2a). Our simple model was able to move in a variety of different modes while using the same set of model parameters [16]. In particular, we discovered passive motions that had the same footfall patterns as walking, trotting, t¨olting, pacing, bounding, and galloping of horses. These gaits are merely different dynamic modes of the same mechanical system. In addition to replicating a large number of footfall sequences, our models also accurately predicted vertical GRFs of horses at walk, trot, and t¨olt (Fig. 3a,b). Analyzing the detailed motion of these different locomotion modes, we clearly observed fundamentally different oscillatory patterns. Walking, for example, had phases of double support alternating with phases of triple support. The vertical GRFs exhibited a double hump with a mid-stance relief. These two force peaks coincided with a double compression of the leg springs. Also, in walking, the main body showed a clear pitching motion. The main body oscillated between -1.45 ◦ and 1.23 ◦ during a gait cycle. T¨olting has the same footfall sequence as walking. Yet, in this gait phases of double support alternate with phases of single support and there are fundamentally different oscillatory patterns. For t¨olting, the vertical GRFs exhibited only a single hump. This single peak was roughly twice the magnitude of the peaks in walking. A pitching motion is still present, but less pronounced than in walking (the main body oscillates between -0.47 ◦ and 0.48 ◦ ). Overall, t¨olting exhibited the least variability in main body vertical and horizontal velocity throughout the course of a stride. Trotting also showed a single hump in the vertical GRFs - corresponding to a single spring compression. Yet, since trotting is a two-beat gait in which phases of double-
l1
m1, j1
RAMlab
m2, j 2
lRH
ταR F uαL H
τlR H αLH k
ulL H
y
l3
m3, j3
(b) x
Fig. 2. Our work explored two different models of legged systems. Passive models with massless legs and no damping in the springs (a) and actuated models with distributed mass, series elastic actuators in the legs and hips, and damping in the springs (b). Both models exhibited a number of similar oscillation modes that yielded a variety of gaits.
While these results provide significant evidence towards interpreting gaits as different oscillatory modes of a single system, the simplicity of the underlying model limits possible conclusions with respect to actual robotic systems. The missing actuation, in particular, conceals issues of efficiency and other energetic benefits when exploiting different oscillations modes. Such benefits would be expected, given that exciting an oscillatory system in a natural mode would minimize the amount of required excitation power. To investigate this hypothesis, we additionally conducted optimal control studies with more complex models of legged robots (Fig. 2b). These models have fully distributed masses, which leads to complex swing leg dynamics and collision losses at touchdown. Additional losses are created by viscous friction in the springy legs. The models are not energetically conservative and are driven by detailed series elastic actuators. In order to not bias the model towards established gaits and to reduce the influence of optimization errors, we conducted two studies using different optimization methods. We used direct collocation in an optimization framework that could detect optimal footfall patterns automatically [17], and we used an established multiple shooting framework (MUSCOD [18]) for which we pre-selected a wide range of different footfall sequences. Both approaches yielded similar results. Despite the addition of actuation, collision losses, swing
leg dynamics, and damping losses, these complex models were found to perform optimally at the same oscillatory modes as the passive system. For low speeds, four-beat walking was the most economical gait, and for higher speeds, trotting was most economical (Fig. 3c). In addition, t¨olting proved to be only slightly more energetically expensive than trotting, indicating why it might be adopted by some horses.
or “how do morphological changes affect gait dynamics and their optimality?”. In terms of robotics, our in-depth analysis of the origin of gaits and their relation to the natural dynamics of a legged system will help hardware designers to systematically create systems that take advantage of natural motion. Roboticists can use this work to design the mechanical dynamics of legged systems in a way that requires minimal actuation, and that takes advantage of different ‘resonance’ cases that yield different gaits. The presented optimal control framework and the optimization of complex models can be used in a design process that synthesizes robotic gaits, motions, and morphologies at the same time. We expect that the present paper will spark a lively discussion at the conference, given that there is such a variety of definitions and explanations for gaits. R EFERENCES
Fig. 3. Model predicted ground reaction forces (solid lines) for walking (a) and t¨olting (b) are compared to data recorded from actual horses (dotted lines). Labels refer to the different legs (L-left, R-right, H-hind, F-front). While being based on the same model, t¨olting undergoes a fundamentally different oscillation pattern than walking. The graph in (c) shows the cost of transport obtained via optimal control for various gaits for an actuated model. At low speeds a four-beat walk is the most efficient gait, and at high speeds, trotting is most efficient.
The optimizers’ discovery of solutions that undergo the same fundamental motions as the passive model is a strong indication that we can interpret legged locomotion as a complex nonlinear oscillation which is merely excited and maintained through periodic actuator inputs. The underlying premise is that the mechanical dynamics of the legged system (driven by inertia, gravity, and elastic oscillations) can periodically store and return energy and are passively creating a substantial part of a desired motion. With this, we can consider gaits as a ‘resonant’ usage of a number of different oscillatory modes that optimally exploit the natural dynamics to improve energetic efficiency at different operating points. This interpretation of gait enables a synthetic perspective on legged systems since mass properties, stiffness values, and geometrical properties can be tuned to improve or alter the performance of such legged systems (Fig. 1, right). I. W HY IS THIS USEFUL ? The question of ‘what is a gait?’ has relevance for both the biological and robotic research communities. Our research gives new insight into why humans and animals move the way they do. Showing how animal locomotion can be represented by simple models helps to understand the underlying dynamics of their motion and adds another piece of information to the study of gaits. These simple models might additionally open the door to answering intricate questions whose answers can be lost in the complexity of real animals. For example, “what drives gait transitions?”
[1] G. Gabrielli and T. Von Karman, What price speed?: specific power required for propulsion of vehicles. 1950. [2] V. A. Tucker, “The energetic cost of moving about: Walking and running are extremely inefficient forms of locomotion. much greater efficiency is achieved by birds, fishand bicyclists,” American Scientist, vol. 63, no. 4, pp. 413–419, 1975. [3] P. Di Prampero, “The energy cost of human locomotion on land and in water.,” International journal of sports medicine, vol. 7, no. 2, pp. 55– 72, 1986. [4] D. F. Hoyt and C. R. Taylor, “Gait and the energetics of locomotion in horses.,” Nature, 1981. [5] C. Chevallereau, Y. Aoustin, et al., “Optimal reference trajectories for walking and running of a biped robot,” Robotica, vol. 19, no. 5, pp. 557–569, 2001. [6] M. Srinivasan and A. Ruina, “Computer optimization of a minimal biped model discovers walking and running,” Nature, vol. 439, no. 7072, pp. 72–75, 2006. [7] K. Tsujita, M. Kawakami, and K. Tsuchiya, “A study on optimal gait pattern of a quadruped locomotion robot,” in Systems, Man and Cybernetics, 2004 IEEE International Conference on, vol. 1, pp. 745– 749, IEEE, 2004. [8] M. Hildebrand, “Symmetrical gaits of horses,” Science, vol. 150, no. 3697, pp. 701–708, 1965. [9] G. A. Cavagna, H. Thys, and A. Zamboni, “The sources of external work in level walking and running.,” The Journal of physiology, vol. 262, no. 3, pp. 639–657, 1976. [10] R. Blickhan, “The spring-mass model for running and hopping,” Journal of biomechanics, vol. 22, no. 11, pp. 1217–1227, 1989. [11] R. M. Alexander, “A model of bipedal locomotion on compliant legs,” Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, vol. 338, no. 1284, pp. 189–198, 1992. [12] J. Rubenson, D. B. Heliams, D. G. Lloyd, and P. A. Fournier, “Gait selection in the ostrich: mechanical and metabolic characteristics of walking and running with and without an aerial phase,” Proceedings of the Royal Society of London-B, vol. 271, no. 1543, p. 1091, 2004. [13] M. G. Pandy, “Simple and complex models for studying muscle function in walking,” Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, vol. 358, no. 1437, pp. 1501– 1509, 2003. [14] A. E. Minetti, “The biomechanics of skipping gaits: a third locomotion paradigm?,” Proceedings of the Royal Society of London. Series B: Biological Sciences, vol. 265, no. 1402, pp. 1227–1233, 1998. [15] H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant leg behaviour explains basic dynamics of walking and running,” Proceedings of the Royal Society B, vol. 273, no. 1603, pp. 2861–2867, 2006. [16] Z. Gan and C. D. Remy, “A passive dynamic quadruped that moves in a large variety of gaits,” in Intelligent Robots and Systems (IROS), 2014 IEEE/RSJ International Conference on, p. Accepted for publication, IEEE, 2014. [17] W. Xi and C. D. Remy, “Optimal gaits and motions for legged robots,” in Intelligent Robots and Systems (IROS), 2014 IEEE/RSJ International Conference on, p. Accepted for publication, IEEE, 2014. [18] M. Diehl, D. B. Leineweber, and A. A. Sch¨afer, MUSCOD-II User’s Manual. IWR, 2001.
