Exploiting the Passive Dynamics of a Compliant Leg to Develop Gait Transitions
aa r X i v : . [ c s . R O ] A ug APS/123-QED
Exploiting the Passive Dynamics of a Compliant Leg to DevelopGait Transitions
Harold Roberto Martinez Salazar ∗ and Juan Pablo Carbajal † Artificial Intelligence Laboratory,Department of Informatics,University of ZurichAndreasstrasse 15 8050 Zurich Switzerland (Dated: June 4, 2018)
Abstract
Abstract:
In the area of bipedal locomotion, the spring loaded inverted pendulum (SLIP) modelhas been proposed as a unified framework to explain the dynamics of a wide variety of gaits. In thispaper, we present a novel analysis of the mathematical model and its dynamical properties. Weuse the perspective of hybrid dynamical systems to study the dynamics and define concepts suchas partial stability and viability. With this approach, on the one hand, we identified stable andunstable regions of locomotion. On the other hand, we found ways to exploit the unstable regionsof locomotion to induce gait transitions at a constant energy regime. Additionally, we show thatsimple non-constant angle of attack control policies can render the system almost always stable. ∗ http://ailab.ifi.uzh.ch/martinez/; martinez@ifi.uzh.ch † http://ailab.ifi.uzh.ch/carbajal/; carbajal@ifi.uzh.ch, both authors can be contacted regarding the content of the paper . INTRODUCTION One of the most accepted mathematical models for bipedal running is the spring loadedinverted pendulum (SLIP, for an extensive review see[1]). In a similar fashion, the rigidinverted pendulum has been extensively used to model bipedal walking[2]. In 2006, Geyeret al.[3] propose the SLIP model as a unifying framework to describe walking as well asrunning. The unified perspective proves useful for accurately explaining data from humanlocomotion[3]. Additionally, it allows describing both gaits (walking and running) in termsof dynamical entities observed in a discrete map, obtained by intersecting the trajectoriesof the system with a predefined section of lower dimension. Geyer associates these entitieswith limit cycles of the hybrid dynamical system [4, 5] and named their attracting behavioras self-stabilization . Though the nature of the observed dynamical properties is not yetclarified, those results emphasize that bipedal locomotion may be dictated solely by themechanics of the system. As a consequence, the control necessary for locomotion is thusreduced to the swing phase of the leg, showed in Fig. 1 between points A and B. The mostpopular control policy is to produce touchdowns at constant angle of attack α (CAAP ( α )),i.e. the angle spanned by the landing leg and the horizontal.In the last decade, many energy-efficient bipedal walking machines have been developed.Through careful design, they exploit the passive dynamics of their own body to move forward,requiring little control or none[6–10]. However, the construction of bipedal machines capableof exploiting passive dynamics in different gaits remains an unsolved engineering challenge.In this context, Geyer et al.[3] report that, in the SLIP model, it is not possible to havemultiple gaits at the same energy. The results are based on simulations that do not cover allpossible initial conditions of the system. In addition, Rummel et al.[11] prove that walkingand running is possible at the same energy level. They use a new map that allows comparingdifferent gaits with ease. The map is defined at the vertical plane crossing the landing pointof the foot (Fig. 1). In this way, they find the self-stable regions, but their intersection isempty. To concretize these ideas, let us describe this region for the running map R . E R ∞ = { x | x ∈ S ∧ ( ∃ α | x = R α ( x )) } , (1)where the subscript in R α denotes running using CAAP ( α ) and S denotes the section wherethe map is defined. Therefore, if for different gaits these stable regions do not intersect, e.g.2 R ∞ ∩ E W ∞ = ∅ , we conclude that a transition between the two gaits cannot occur if thesystem is to remain in these regions. In other words, x ∈ E R ∞ ∧ y ∈ E W ∞ ⇒R α ( y ) / ∈ E R ∞ ∧ W β ( x ) / ∈ E W ∞ ∀ α, β. (2)In this study, we will show how transitions between gaits are found at points outsidethese stable regions. The transitions require the selection of the angle of attack; thereforeCAAP’s are not suitable for this task. We will also show evidence indicating that it ispossible to find an angle of attack θ that maps a point into a stable region, e.g. x / ∈ E R ∞ ∧ (cid:0) ∃ θ, y | y = x, y ∈ E R ∞ , y = R θ ( x ) (cid:1) . Additionally, we introduce the concepts of partialstability and viability that will be useful in the construction of the transitions presentedherein.This paper is organized as follows. In section II, we describe the models used for oursimulations, their representation in state variables and the definition of the discrete map.Next, in section III, we introduce the new concepts, and we show the regions where thetransitions between gaits exist. Later, in section IV, we discuss about the requirements of a FIG. 1. (Color online) Illustration of the evolution of the SLIP model for running and walking.The mass is represented with a filled circle. The color of the fill indicates touchdown event (black),takeoff event (white), and the crossing of the section (pink (grey)). The landing leg is picturedwith a thick solid line, and the leg at takeoff is represented with a blurred line. Due to the passiveproperties of these models, control is necessary only during the swing of the leg, i.e. during freefall while running and from point A to B while walking.
II. METHODS
As explained previously, we use the SLIP model to study bipedal gaits. We adopt theframework in [12], which is described in the language of hybrid dynamical systems. There-fore, we reintroduce some notation and definitions.To represent the different phases of a gait, the model is segmented into three sub-models.We will call these sub-models charts [4] or phases see Fig.1. Each chart represents the motionof a point mass under the influence of: only gravity (ff-chart or flight phase), gravity anda linear spring (s-chart or single stance phase), gravity and two linear springs (d-chart ordouble stance phase). The point mass represents the body of the agent and the masslesslinear springs model the forces from the legs (Fig.1). A trajectory switches from one chart toanother when some real valued functions evaluated on it cross zero ( event functions [4, 13]).We define a running gait as a trajectory that switches from the s-chart to the ff-chart andback to the s-chart. A walking gait is defined as a trajectory that switches from the s-chartto the d-chart and back again to the s-chart. Switches from the ff-chart to d-chart or viceversa are not included in this study.
A. Equations of motion in each chart
The motion in all the charts is governed by a system of ordinary differential equations:˙ ~X = ~F i (cid:16) ~X (cid:17) , (3)where ~X is the vector of state variables and ~F i is a force function characteristic of eachchart. Since all forces are conservative, the energy of the system is constant. For the ff-chartthe state is described by the Cartesian coordinates of the position of the point mass and its4elocity ~X ff = ( x, y, v x , v y ) T , ˙ ~X ff = v x v y − g , (4)where g is the acceleration due to gravity.The state in the s-chart is represented in polar coordinates ~X s = (cid:16) r, θ, ˙ r, ˙ θ (cid:17) T , where r isthe length of the spring and θ is the angle spanned by the leg and the horizontal, growingin clockwise direction. Thus, the equations of motion are:˙ ~X s = ˙ r ˙ θ km ( r − r ) + r ˙ θ − g sin θ − r (cid:16) r ˙ θ + g cos θ (cid:17) . (5)It is important to note that θ ( t T D ) = α , i.e. the angular state at the time of touchdown isequal to the angle of attack. The parameter r defines the natural length of the spring.In the d-chart the state is also represented in polar coordinates ~X d = (cid:16) r, θ, ˙ r, ˙ θ (cid:17) T , withthe origin of coordinates in the new touchdown point. The motion is described by:˙ ~X d = ˙ r ˙ θkm (cid:20) ( r − r ) + (cid:18) − r r ♂ (cid:19) ( x ♂ cos θ − r ) (cid:21) + r ˙ θ − g sin θ − r (cid:20) km (cid:18) − r r ♂ (cid:19) x ♂ sin θ + 2 ˙ r ˙ θ + g cos θ (cid:21) (6) r ♂ = p r + x ♂ − rx ♂ cos θ, (7)where x ♂ is the horizontal distance between the two contact points and r ♂ is the length ofthe back leg. B. Event functions
Event functions are functions on the phase space of the system. An event occurs whenthe trajectory of the system intersects a level curve of the event function. At the time of the5vent, the current state of the system is mapped to the state of another chart. Some eventfunctions are parameterized with the angle of attack and the natural length of the springs.Switches from the ff-chart to the s-chart are defined by: F ff → s (cid:16) ~X ff , α, r (cid:17) : y − r cos α = 0 v y < , (8)which means that the mass is falling and the leg can be placed at its natural length withangle of attack α . Therefore, the motion is now defined in the s-chart. The switch in theother directions is simply: F s → ff (cid:16) ~X s , r (cid:17) : r − r = 0 . (9)These are the only two event functions involved in the running gait. The map from onechart to the other is defined by: x = − r cos θ y = r sin θ. (10)It is important to have in mind that the origin of the s-chart is always at the touchdownpoint.For the walking gait, we have to consider switches between single and double stancephases. From the s-chart to the d-chart, we have: F s → d (cid:16) ~X s , α, r (cid:17) : r sin θ − r cos α = 0 θ > π , (11)which is similar to (8) with the additional condition that the mass is tilted forward. Addi-tionally, if we consider the sign of the radial speed, we differentiate between walking gait W with ˙ r < GR , with ˙ r > F d → s (cid:16) ~X d , r (cid:17) : r ♂ − r = 0 , (12)with r ♂ as defined in (7). The map from the d-chart to the s-chart is the identity. In theother direction we have: r d = r θ d = α, (13) x ♂ = r cos α − r s cos θ s , (14)6here the subscripts indicate the corresponding chart.If the system falls to the ground ( y ≤ v x < C. Simulation of the dynamics
The state of the model is observed when the trajectory of the system intersects the sectiondefined by S : θ = π / . In this way, the map R α : S → S transforms points through theevolution of the system from the s-chart to the ff-chart and back again to the s-chart using anangle of attack α . Similarly, the map W α : S → S transforms points through the evolutionof the system from the s-chart to the d-chart and back again to the s-chart using an angleof attack α .All initial conditions are given in the S section and in the s-chart, i.e. only one legtouching the ground and oriented vertically. Moreover, all the initial conditions are given atthe same total energy. The results are visualized using the values of the length of the spring r and the radial component of the velocity which, in S , equals the vertical speed ˙ r = v y ( v x is obtained from these values and the equation of constant energy). It is important to notethat all possible values of r , v y and v x , for a given value of the total energy E , lay on anellipsoid. Besides, there is a transformation that maps the ellipsoid to a sphere. This canbe shown as follows: the total energy in the section is, E = 12 k ( r − r ) + 12 m (cid:0) v x + v y (cid:1) + mgr (15)Defining the parameters L = r k h E − mg (cid:16) r − mg k (cid:17)i , (16) ω = r km , (17)the new variables ˆ v x = v x ω , (18)ˆ v y = v y ω , (19)ˆ r = r − (cid:16) r − mgk (cid:17) , (20)7 ABLE I. Values used for the simulations presented in this paper.Description Name ValueMass m
80 kgElastic constant of linear springs k
15 kNmRest length of linear springs r E
820 JAcceleration due to gravity g .
81 m / s Angle of Attack α from 55 ◦ to 90 ◦ transform equation (15) into, L = ˆ v x + ˆ v y + ˆ r (21)which defines a sphere. Therefore, all initial conditions of ˆ r and ˆ v y with constant energy,are defined inside a circle. A Delaunay triangular mesh was created in the circle with 65896initial conditions as vertices (131245 triangles). Each vertex was transformed using R α , GR α and W α with 400 values of α ∈ [55 ◦ , ◦ ]. To compute the evolution of an arbitraryinitial condition, we used bilinear interpolation in the triangles of the mesh.The model implementation and data analysis were carried out in MATLAB(2009, TheMathWorks), GNU Octave[14] and Matplotlib[15]. Simulations were run for constant energy,using the step variable integrator ode45 (relative tolerance: 1 × − and absolute tolerance:1 × − ). Table I shows the values of the parameters used. III. RESULTS
In this section, we present the results of the analysis on the data collected from themodels as described in section II C. Aiming to define a controller, we introduce some im-portant properties of the dynamics of each gait, namely finite stability for a given CAAPand viability. 8 . Finite stability and Viability
Finite stability describes the set of initial conditions where the system can do a maxi-mum amount of steps (sequential applications of the map) before failing, using CAAP. Forexample, we can define for W E Wn = { x | x ∈ S ∧ ( ∃ α | y = W nα ( x ) , n ≥ , y ∈ S ) } . (22)That is, at a given state x = ( r, v y ) in S there is a CAAP ( α ) such that the system cando at most n steps before failing. The region E W are all the points in the section whereapplying W produces a failure. The existence of E Wn implies that a controller of the systemmay not need to take a decision at each step. In addition, the controller may exploit thisalleviation by planning future angles of attack. Viability describes how easy is to choose thefuture angle of attack. The level of ease is measured in terms of the size of the interval ofangles that can be chosen to avoid a failure of the system. For the running gait this regionis defined as: V R (∆ α ) = { x | x ∈ S∧ ( ∃ α ∈ I α , k I α k ≥ ∆ α | y = R α ( x ) , y ∈ S ) } , (23)where I α denotes a real interval and k · k measures its length. In a real system, it is requiredthat a viable angle of attack exists for a definite interval, since real sensors and actuatorshave a finite resolution and are affected by noise.Fig. 2 shows the finite stability regions for each gait. The stable region of R , as reportedin [12] ( v y = 0) is not visible. Although E R ∞ may have some area of attraction, due to theresolution we used for the angles of attack (described in section II C) we do not see it in ourresults. Based on results not presented here, we estimate that the resolution in the angle ofattack to detect such basin for the current energy is ∼ − . In despite of the low resolutionin the angles, the system can perform an average of 10 steps in R , and at least 25 steps(maximum calculated) in GR and W . This means that running is more difficult at thisenergy level than the other two gaits. Particularly for GR and W , we see that there is aplateau with the maximum number of steps. This is the evidence of the self-stable regionsof these gaits, and the plateau is related to the basing of attraction of that region.9 IG. 2. (Color online) Finite stability regions. The figures show initial conditions for R , GR and W that can do multiple steps under CAAP before failing. A region in white corresponds to E i forgait i . Fig. 3 shows the V i (∆ α ) regions for each gait i . Comparing with Fig. 2, we see thatin general long partial stability implies wider options for the angle of attack. Particularly,transitions are found near these regions of high viability and long partial stability, as willbe described in the next section. FIG. 3. (Color online) Viability regions for each gait. The figures show the range of angles ofattack that can be selected in each initial condition that allows the system give at least one morestep. Colors indicate the size of the window, spanning from 0 ◦ to 10 ◦ . Fig. 4 shows one of the strongest results presented here. For each gait i , there is at leastone angle of attack that maps the current state of the system into E i ∞ , and this angle existsfor an extense region of S . This implies that if we consider control policies with variableangle of attack, almost any point in the section can be rendered stable. For this region the10ptimal control policy requires two angles: the first one maps the point to E i ∞ ; the secondangle, keeps the system in this region. FIG. 4. (Color online) Points that can be mapped to stable regions in one step. The figuresshow the initial conditions that can be mapped to a small neighborhood of the stable region E i ∞ , | v y | < × − ( v y = 0, dashed horizontal lines). Color indicates the angle chosen. Regions V i (2 ◦ )are marked with solid lines. B. Transition regions
As it was shown in the previous section, the only way of producing transitions betweengaits is to put the system in a region with finite stability (due to the empty intersection ofthe E i ∞ regions reported in [12], see Fig 4). In Fig. 5 we show transitions starting at E in andarriving at V j (2 ◦ ) for i = j and ( i → j ) = { ( R → GR ) , ( GR → W ) , ( W → GR ) , ( W → R ) } .We show the transitions that will be used in the next example, however transitions betweentwo any gaits are possible. It shall be noticed that wherever two regions of different gaitsintersect, the transition is trivial.Finally, Fig. 6 and Fig. 7 show one example of three transitions for a given initial condi-tion. The trajectory has a total of 26 steps and the angle sequence is α = (cid:0) . , . , . , . , . , . , . , . , . , . , . , . , . (cid:1) (24)where the exponent indicates how many times the angle was used. The path of the centerof mass in the Cartesian plane is also shown in the figures.11 IG. 5. (Color online) Transitions regions landing in ∆ α ≥ ◦ . All the initial conditions that havea future inside the region with ∆ α ≥ ◦ of the objective gait are plotted with black dots. The sameregion of the starting gait is given as a reference and appears shaded. Colors in the objective regionindicate the angle of attack used to perform the transition. Wherever two regions of different gaitsintersect, the transition is automatically given.FIG. 6. (Color online) Transition sequence. The plot shows a trajectory with three transitions.The Regions V i (2 ◦ ) are shown shaded with self-stable regions in dotted line. The arrows indicatethe order of the sequence and the step number is given. The angle of attack sequence is given in(24). All together we have shown that the SLIP model can be easily controlled to presenttransitions between gaits. To find transitions we must search for an intersection betweenthe future of the starting region and the desired objective region. Depending how theseregions are defined, it may be the case that multiple steps are required to achieve a successful12
IG. 7. Transition time series. The figure shows the motion of the point mass a in the plane isshown together with the crossing of the section (filled circles 6). Transition points are indicatedwith a vertical line. a transition. IV. DISCUSSION
There are two important aspects regarding the viability regions. First, it is importantto notice that V i (∆ α ) enclose the E i ∞ region, and the points that can be mapped to stableregions in one step (Fig. 4) . Second, as it can be seen in Fig. 3, the bigger the range ofthe angle of attack is, the smaller the viability region is. We can take advantage of theseproperties to stabilize the system more easily. The selection of an appropriate ∆ α e.g. 2 ◦ defines a set of V i (∆ α ) inside the section S , where the controller has at least a range of 2 ◦ to select an appropriate angle of attack. Moreover, the agent can select conservative angles,step by step, to bring itself to the E i ∞ region (Fig. 5).Despite the relief to the controller induced by the viability region, the selection of the∆ α can generate regions that do not intersect; e.g. in Fig. 4 we can see that V i (2 ◦ ) doesnot intersect any other region, which makes the gait transition more difficult to carry out.In order to cope with this situation, we look at the future of all the initial conditions in E in .As it is presented in Fig. 5, we found that there are some initial conditions, that under a setof angles of attack, are mapped from E in to E jn (e.g. E Rn to E GRn ). What is also importantis that the region where we can find these initial conditions are inside the viability region13Fig. 5).In these terms, the controller has two purposes. First, based on the state on the S section, it has to select the gait, and the angle of attack to keep the agent stable. Thus, thecontroller needs to have the knowledge of all the V R (∆ α ), and the desired ∆ α to identifywhich gait has to be selected; the angle of attack can be selected based on the gait model.Second, the controller has to be able to produce gait transition when it is needed. Hence,the transition regions should be known by the controller and with a model of the gait, theangle of attack required can be selected. We expect that this approach can be used to handleuneven terrain, given that these irregularities can be modeled (under certain restrictions) asa change in energy.All these results are conditioned to the selection of the S section. This means that weare analyzing the system in only one point in the whole trajectory. From what we see inthese results, in some regions the trajectories are very close. It would not be a surprise thatthese trajectories of R , W , and GR cross each other in another point along their continuousevolution, but given that we are looking just at the S section, this cannot be anticipated.Nevertheless, the selection of this section establishes the angle of attack as a natural controlaction to stabilize the system and to generate the transitions. V. CONCLUSION
In the present study we have taken advantage of the perspective of hybrid dynamicalsystems to represent locomotion as a process generated by several charts. Although, thisview makes evident a bigger set of connections among the charts, in this paper we takeinto account a small subset (s-chart to ff-chart, and s-chart to d-chart) which allow us todiscover new alternatives to perform gait transitions. The development of the maps W α , GR α , R α is fundamental to identify important regions in the S section that bring the systemto stable locomotion and to a gait transition. The present results bring new ideas aboutplausible mechanisms that biped creatures could use to carry out gait transitions and stablelocomotion. These mechanisms exploit the passive dynamics of the system, which reducesthe amount of energy needed to control the system. These features are also present in bipedmachines with compliant legs, and as suggested in this paper, these mechanisms can beexploited to develop stable gaits and gait transitions.14 CKNOWLEDGMENTS
Funding for this work has been supplied by SNSF project no. 122279 (From locomotionto cognition), and by the European project no. ICT-2007.2.2 (ECCEROBOT). Addition-ally, the research leading to these results has received funding from the European Com-munity’s Seventh Framework Programme FP7/2007-2013-Challenge 2-Cognitive Systems,Interaction, Robotics- under grant agreement No 248311-AMARSi. [1] P. Holmes, R. J. Full, D. Koditschek, and J. Guckenheimer, SIAM Rev. , 207 (2006), ISSN0036-1445.[2] S. Mochon and T. A. McMahon, J. Biomech. , 49 (1980), ISSN 00219290.[3] H. Geyer, A. Seyfarth, and R. Blickhan, P. Roy. Soc. B - Biol. Sci. , 2861 (Nov. 2006),ISSN 0962-8452.[4] J. Guckenheimer and S. Johnson, in Hybrid Systems II (Springer-Verlag, London, UK, 1995)pp. 202–225, ISBN 3-540-60472-3.[5] J. Cortes, IEEE Contr. Sys. Mag. , 36 (Jun. 2008), ISSN 0272-1708.[6] T. McGeer, Int. J. Robot. Res. , 62 (Apr. 1990), ISSN 0278-3649.[7] S. H. Collins, Int. J. Robot. Res. , 607 (Jul. 2001), ISSN 0278-3649.[8] M. Wisse and J. V. Frankenhuyzen, in Adaptive Motion of Animals and Machines (Springer-Verlag, Tokyo, 2006) pp. 143–154, ISBN 4-431-24164-7.[9] S. H. Collins, A. Ruina, R. Tedrake, and M. Wisse, Science , 1082 (Feb. 2005), ISSN1095-9203.[10] T. Geng, B. Porr, and F. Worgotter, Neural Comput. , 1156 (May 2006), ISSN 0899-7667.[11] J. Rummel, Y. Blum, H. M. Maus, C. Rode, and A. Seyfarth, in IEEE Int. Conf. Robot.(ICRA) (IEEE, 2010) pp. 5250–5255, ISBN 978-1-4244-5038-1.[12] J. Rummel, Y. Blum, and A. Seyfarth, in
Autonome Mobile Systeme (Springer, Berlin, Hei-delberg, 2009) pp. 89–96, ISBN 978-3-642-10283-7.[13] P. T. Piiroinen and Y. A. Kuznetsov, ACM T. Math. Software , 1 (May 2008), ISSN00983500.
14] J. W. Eaton,
GNU Octave Manual .[15] J. D. Hunter, Computing in Science and Engineering , 90 (2007), ISSN 1521-9615., 90 (2007), ISSN 1521-9615.