Bipedal locomotion using variable stiffness actuation
aa r X i v : . [ c s . R O ] J un Bipedal locomotion using variable stiffness actuation
Ludo C. Visser, Stefano Stramigioli, and Raffaella Carloni
Abstract —Robust and energy-efficient bipedal locomotion inrobotics is still a challenging topic. In order to address issuesin this field, we can take inspiration from nature, by study-ing human locomotion. The Spring-Loaded Inverted Pendulum(SLIP) model has shown to be a good model for this purpose.However, the human musculoskeletal system enables us to activelymodulate leg stiffness, for example when walking in rough terrainwith irregular and unexpected height variations of the walkingsurface. This ability of varying leg stiffness is not consideredin conventional SLIP-based models, and therefore this paperexplores the potential role of active leg stiffness variation inbipedal locomotion. It is shown that the conceptual SLIP modelcan be iteratively extended to more closely resemble a realistic(i.e., non-ideal) walker, and that feedback control strategies canbe designed that reproduce the SLIP behavior in these extendedmodels. We show that these extended models realize a cost oftransport comparable to human walking, which indicates thatactive leg stiffness variation plays an important role in humanlocomotion that was previously not captured by the SLIP model.The results of this study show that active leg stiffness adaptationis a promising approach for realizing more energy-efficient androbust bipedal walking robots.
I. I
NTRODUCTION
Robust and energy-efficient bipedal locomotion in roboticsis an interesting research topic with many open questions.In particular, on one side of the spectrum, robust bipedalrobots are being developed, but without much considerationfor energy efficiency [1]. On the other side of the spectrum,extremely efficient bipedal locomotion is being achieved byexploiting passive robot dynamics [2]. However, the gaits ofsuch passive dynamic walkers lack robustness [3].In contrast, human walking is both robust and energyefficient, and, therefore, a better understanding of humanwalking could aid the design of robots achieving similarperformance levels. To develop this understanding, modelshave been proposed that capture the essential properties ofhuman gaits. One remarkably simple model is the bipedalspring-loaded inverted pendulum (SLIP) model, proposed in[4]. Despite its simplicity, the model accurately reproducesthe hip trajectory and ground reaction force profiles observedin human gaits. Furthermore, the model encodes a wide varietyof gaits, ranging from slow walking to running [5].It has been shown that the bipedal SLIP model can be usedto generate reference gaits for a fully actuated bipedal robot[6], [7]. However, the human musculoskeletal system enablesthe leg stiffness to be varied continuously, in order to adapt todifferent gaits and terrains. These stiffness variations also play
This work has been partly funded by the European Commission’s SeventhFramework Programme as part of the project VIACTORS under grant no.231554. { l.c.visser, s.stramigioli, r.carloni } @utwente.nl, MIRA Institute, Faculty ofElectrical Engineering, Mathematics and Computer Science, University ofTwente, The Netherlands. a role in disturbance rejection, for example in uneven terrainwith sudden and unexpected changes in the walking surface.Variable stiffness in the legs of the bipedal SLIP walker hasbeen shown to increase energy efficiency [8], [9] and improverobustness [10], but continuous leg stiffness adaptation hasnot yet been considered. Recent advances in the field ofvariable stiffness actuators, a class of actuators that allow theactuator output stiffness to be changed independently from theactuator output position [11], are enabling the realization ofrobotic walkers with physically variable leg stiffness. There-fore, further research into bipedal walking with variable legstiffness should be pursued, with the aim of getting closer torealizing bipedal walking robots with human-like performancecharacteristics.In this work, we explore modeling and control of bipedalwalking with controllable leg stiffness. The aim of this workis to show that SLIP-like walking behavior can be embeddedin more sophisticated models of bipedal walkers. This isachieved by the development of control strategies based onthe principles of leg stiffness variation, inspired by the capa-bilities of the human musculoskeletal system. Starting fromthe conventional bipedal SLIP model, we iteratively extendthis model, first by making the leg stiffness controllable, thenby adding a swing leg and its dynamics, and then further byincluding a knee in the swing leg. In parallel, we derive acontrol strategy, which is extended with each model iteration.For each iteration, we show the stabilizing properties of thecontroller, demonstrating that the controller derived for theSLIP model in the first iteration is sufficiently robust to handlethe increasingly complex dynamics in subsequent iterations.The final model and controller can serve as a template forbipedal robot control strategies.The paper is organized as follows. In Section II, we revisitthe bipedal SLIP model, as presented in [4], and analyze itsdynamics. Then, in Section III, we extend the bipedal SLIPmodel to have controllable stiffness (the V-SLIP model, forVariable SLIP), and derive a control strategy that rendersa natural gait of the SLIP model asymptotically stable. InSection IV, the controlled V-SLIP model is extended to includefeet, with the aim of introducing swing leg dynamics. The V-SLIP control strategy is extended to handle the extra degreesof freedom, and the stabilizing properties of the controllerare demonstrated. The swing leg model is further refinedin Section V by adding knees, with again further extensionof the control strategy. A comparison of the models andtheir controllers is presented in Section VI, and Section VIIconcludes the paper with a discussion and final remarks. Conventions in Notation
To avoid notational clutter, variable names will be reusedfor different models. However, this reuse is consistent, and q m h k k c c Fig. 1. The bipedal SLIP model—The model consists of a point mass m h , located in the hip joint, i.e. where two massless telescopic springs,with a constant spring stiffness k and rest length L , are connected. Theconfiguration variables ( q , q ) describe the position of the hip. variables with the same name indicate the same quantity inthe various models. For example, q i denotes configurationvariables, and p i denotes momentum variables, state vectorsare named x , and f ( x ) and g i ( x ) are drift and input vectorfields on the state manifold respectively. The Lie-derivativeof a function h along a vector field X is denoted by L X h .Function arguments are omitted where this is consideredpossible. II. T HE B IPEDAL
SLIP M
ODEL
In this Section, we revisit the bipedal SLIP model, aspresented in [4]. The model is depicted in Figure 1. It consistsof a point mass m h , located in the joint connecting the twolegs, i.e. the hip joint. The legs consist of massless telescopicsprings of stiffness k and rest length L . The configurationvariables ( q , q ) =: q describe the planar position of the pointmass, with ( p , p ) =: p the associated momentum variables.In the following, we derive the dynamic equations for thissystem, and analyze its dynamics. A. System Dynamics
The bipedal SLIP model shows, for appropriately choseninitial conditions [4], [5], a passive walking gait as illustratedin Figure 2. In order to derive the dynamic equations thatdescribe the gait of this model, two phases need to beconsidered: 1) two legs are in contact with the ground (i.e.the double support phase), and 2) one leg is in contact withthe ground (i.e. the single support phase). Furthermore, weconsider the parameter α , which is the angle at which themassless leg touches down at the end of the single supportphase, as indicated in Figure 2.The contact conditions are determined by the spring restlength L and angle of attack α , as shown in Figure 2. Inparticular, if the system is in the single support phase, thetouchdown event of the trailing leg occurs when q = L sin( α ) , (1)and at this moment the foot contact point c is calculated as c = q + L cos( α ) . L α q ( t ) L sin( α ) SS DS SS DS SSFig. 2. Passive gait of the bipedal SLIP model—The model alternates betweensingle support ( SS ) and double support ( DS ) phases, depending on the hipheight and the model parameters L and α . The gray shading will be usedthroughout this paper to indicate that the walker is in the double supportphase. Conversely, when the system is in the double support phase,the transition to the single support phase occurs when eitherof the two springs reaches its rest length with non-zero speed,and thus loses contact with the ground, i.e. when q ( q − c i ) + q = L , i = 1 , . (2)In nominal conditions, only the trailing leg is allowed to liftoff, after which the contact point c is relabeled as c tocorrespond to the notation used for the single support phase.In order to derive the dynamic equations, we define thekinetic energy function K = p T M − p , where M = diag( m h , m h ) (3)is the mass matrix and p := M ˙ q are the momentum variables.The potential energy function is defined as V = m h g q + 12 k ( L − L ) + 12 k ( L − L ) , where L i := p ( q − c i ) + q , and g is the gravitationalacceleration. During the single support phase, we set L ≡ L to eliminate the influence of this virtually swinging leg.The dynamic equations in Hamiltonian form are definedthrough the Hamiltonian energy function H = K + V andgiven by dd t (cid:20) qp (cid:21) = (cid:20) I − I (cid:21) " ∂H∂q∂H∂p . (4)From (4), it can be noted that the configuration variables q ( t ) are of class C . This is due to the fact that the ∂V∂q isnot differentiable at the moment of phase transition. This isbecause the massless second leg does not have a zero rate ofchange of length at the moment of touchdown, i.e.dd t L (cid:12)(cid:12)(cid:12) t = t +touchdown = 0 , where t +touchdown indicates that the time-derivative is taken onthe right of the discontinuity. It will be shown later that thishas consequences for the controller design. q m h k + u k + u c c Fig. 3. The V-SLIP model—In contrast to the bipedal SLIP model, the V-SLIP model has a controllable leg stiffness. This provides two control inputsduring the double support phase, but only one control input during the singlesupport phase, rendering the system underactuated.
III. T HE C ONTROLLED
V-SLIP M
ODEL
The passive bipedal SLIP model provides no control inputs,and therefore the only way to influence its behavior is bythe choice of initial conditions. Therefore, it is proposed toextend the bipedal SLIP model to have massless telescopicsprings with variable stiffness [12]. This bipedal V-SLIP (forVariable SLIP) model is depicted in Figure 3. The differencewith respect to the bipedal SLIP model is that the leg stiffnessnow has a controllable part, i.e. k i = k + u i , i = 1 , . In thisSection we give the dynamic equations for this system andpresent a stabilizing controller. A. System Dynamics
In deriving the dynamics of the V-SLIP model, we assumethat: • no slip or bouncing occurs in the foot contact points; • the springs are unilateral, meaning that we only allowthem to be compressed;The autonomous part of the dynamics of the bipedal V-SLIPmodel is the same as for the bipedal SLIP model. To includethe control inputs, (4) is extended, arriving at the dynamicsfor the V-SLIP model in port-Hamiltonian form:dd t (cid:20) qp (cid:21) = (cid:20) I − I (cid:21) " ∂H∂q∂H∂p + (cid:20) B (cid:21) uy = (cid:2) B T (cid:3) " ∂H∂q∂H∂p , (5)with u = ( u , u ) the controlled leg stiffness, and H is asdefined in Section II-A. The input matrix B is given by B = " ∂φ ∂q ∂φ ∂q ∂φ ∂q ∂φ ∂q , with φ = − ( L − L ) and φ = − ( L − L ) . The output y is dual to u , and it is readily verified that the dual product h u | y i has the units of power [13].As in Section II-A, we set L ≡ L during the singlesupport phase to eliminate the influence of the swing leg. It is emphasized that the control inputs u i , i = 1 , are restricted,such that the total leg stiffness is physically meaningful, i.e. u i ∈ R | < k + u i < ∞ . (6) B. Controller Design
The bipedal SLIP model already shows stable walking gaits,with a relatively large basin of attraction [5]. As shown in ourprevious work, it is possible to tune the spring stiffness k tofurther increase the robustness of the gait, while minimallymodifying the natural dynamics of the walker [12].The control strategy uses a natural gait of the bipedal SLIPmodel as reference, i.e. trajectories ( q ◦ ( t ) , ˙ q ◦ ( t )) that are asolution of (4), where ˙ q is defined as ˙ q = M − p . However,the bipedal V-SLIP model is underactuated during the singlesupport phase (since there is only one leg in contact withthe ground), and thus it is not possible to track ( q ◦ ( t ) , ˙ q ◦ ( t )) exactly. To avoid that the walker lags behind the referenceduring the underactuated phase, we propose to instead de-fine a curve in the configuration space by parameterizing ( q ◦ ( t ) , ˙ q ◦ ( t )) by the horizontal position q , similar to theapproach presented in [14]. This is possible as long as ˙ q > .Then, the desired reference gait can be equivalently describedas ( q ∗ ( q ) , ˙ q ∗ ( q )) . The control objective is to have the hiptrajectory converge to an arbitrary small neighborhood of thisreference gait.In formulating the control strategy, we define the state x =( q, p ) and rewrite (5) in standard form as ˙ x = f ( x ) + X i g i ( x ) u i . (7)The following control strategy is proposed. Proposition 1:
Given parameterized reference state trajec-tories ( q ∗ , ˙ q ∗ ) , define the error functions h = q ∗ − q ,h = ˙ q ∗ − ˙ q . Then the following control strategy renders solutions of (5) asymptotically converging to ( q ∗ , ˙ q ∗ ) : • during the single support phase, u = − L g L f h (cid:0) L f h + κ d L f h + κ p h (cid:1) (8) and u ≡ ; • during the double support phase, when the leading leglength satisfies L − L e ≤ L < L (i.e. just after thetouchdown event), or when the trailing leg length satisfies L − L e ≤ L < L (just before the lift-off event), forsome small L e > : (cid:20) u u (cid:21) = − A ♯ (cid:0) L f h + κ d L f h + κ p h (cid:1) , (9) with A = (cid:2) L g L f h L g L f h (cid:3) , Exact parameterization is not possible, because q ( t ) is of class C only,as outlined in Section II-A. Approximating ( q ◦ ( t ) , ˙ q ◦ ( t )) by finite Fourierseries is an alternative that gives satisfactory results, as will be demonstrated. nd with ♯ denoting the Moore-Penrose pseudo inverse; • during the double support phase, when both leg lengthssatisfy L i < L − L e , (cid:20) u u (cid:21) = − A − (cid:20) L f h + κ d L f h + κ p h L f h + κ v h (cid:21) , (10) with A = (cid:20) L g L f h L g L f h L g h L g h (cid:21) . For any arbitrary small ε > , there exist constants κ p , κ d , κ v > for the control strategy (8) , (9) , (10) such that: lim t →∞ q ∗ ( q ( t )) − q ( t ) = 0 and lim t →∞ | ˙ q ∗ ( q ( t )) − ˙ q ( t ) | < ε. Remark 1:
The control input (9) is introduced, because thesystem (5) is no longer controllable when one of the legsreaches its rest length L . As such, the transition domaindefined by L e is necessary to comply with (6). ⊳ Remark 2:
As observed in Section II-A, the state trajectories q ( t ) are of class C only, and therefore it is not possible tomake the leg stiffness a state of the system, because higherorder Lie-derivatives do not exist. ⊳ Remark 3:
The control inputs, as given by Proposition 1,renders solutions of (5) asymptotically converging to ( q ∗ , ˙ q ∗ ) ,which are parametrized by the horizontal position q . Since q is strictly monotonically increasing, the control inputsasymptotically stabilize the system (5). ⊳ Proof 1: It is straightforward to show that, during the singlesupport phase, L g L f h in (8) is bounded away from zero if < L < L . Similarly, during the double support phase,the matrix A in (10) is invertible if < L i < L , i = 1 , and in addition c = c . These conditions are met through thedefinition of the phase transitions (1) and (2) .During the double support phase, (10) renders the systemstrongly input-output decoupled, i.e. h i is invariant under u j for i = j [15]. Therefore, and by (8) , (9) , during both thesingle and double support phases the error dynamics h ( t ) are described by ¨ h + κ d ˙ h + κ p = 0 . If κ p , κ d are chosen such that the zeros of the characteristicpolynomial are in the open left half-plane, then the dynamicsof h are asymptotically stable during each phase. The errorfunction h depends on the configuration q only, and q ( t ) is continuous and differentiable across the phase transitions.Therefore, lim t →∞ q ∗ ( q ( t )) − q ( t ) = 0 will be achieved.The dynamics of the error function h are, during the doublesupport phase, described by ˙ h + κ v h = 0 , which has as analytic solution h ( t ) = e − κ v ( t − t ds ) h ( t ds ) , t ≥ t ds , Since we are addressing a numerical issue, we are not concerned aboutderiving a proper invariant metric for defining the pseudo-inverse. Instead, weuse the Euclidian metric. TABLE IC
ONTROLLED
V-SLIP
MODEL PARAMETER VALUES .Parameter Value Unit Description m h . kg Hip mass L . m Spring rest length L e . m Phase transition margin α . ◦ Angle of attack k N/m Nominal leg stiffness k min N/m Lower bound on leg stiffness k max N/m Upper bound on leg stiffness κ p Control parameter κ d Control parameter κ v Control parameter where t ds is the time instant of the last touchdown event. Forany κ v > , h ( t ) is asymptotically stable during the doublesupport phase.However, during the single support phase, the dynamics of h ( t ) are uncontrolled. During this phase, the control actionof u will result in a change of kinetic energy with respectto the constant energy level of the SLIP reference gait. Since u is bounded, as defined in (6) , the total increase in kineticenergy is also bounded. Let ∆ E ss denote the increase ofenergy during the single support phase due to u . There existsa constant C such that | ∆ E | < C , which implies, since h ( t ) is a function of the momentumvariable p , that | h ( t ds ) | < C < C . This in turn implies that there exists κ v < ∞ that brings h ( t ) in a neighborhood ε of zero within the duration of the doublesupport phase. With the parameters as listed in Table I, a numeric sim-ulation has been carried out using the PyDSTool softwarepackage [16]. The reference ( q ∗ , ˙ q ∗ ) has been obtained bysearching for a limit cycle of the uncontrolled SLIP modelwith the same parameters. As shown in Figure 4, the controllerindeed achieves the converges as claimed in Proposition 1. Theerror h is never exactly zero, because the solutions to (4) arenot analytical. Therefore the parameterized reference is notan exact representation of the natural dynamics, yielding themismatch in h during the single support phase.IV. T HE C ONTROLLED
V-SLIP M
ODEL WITH S WING L EG D YNAMICS
While the bipedal (V-)SLIP models do accurately reproducehip trajectories observed in human walking, and thus can giveinsights in human walking performance, the models are con-ceptual. In particular, all mass is assumed to be concentratedin a single point mass at the hip, and the legs are assumed tobe massless springs—assumptions that cannot be consideredvalid for a robotic system.In this Section we extend the V-SLIP model to incorporateswing leg dynamics. This is done by adding a foot mass, asshown in Figure 5 [17]. During the swing phase, the leg isassumed to have a fixed length L , while during the stancephase it is again assumed to be a massless spring connecting h
25 26 27 28 29 30Time [s]-0.010.000.01 h Fig. 4. Steady-state error functions for the controlled V-SLIP model—It canbe seen that for the control parameters listed in Table I the error functionsconverge like claimed in Proposition 1. q q q m h m f m f k + u L τ Fig. 5. The V-SLIP model with feet—By adding feet masses m f to the V-SLIP model, swing leg dynamics are introduced. The swing leg is assumedto be constraint at a length L during swing, and the stance foot is assumedto be fixed to the ground, i.e. no slip or bouncing in the contact point. the foot and the hip masses. In this Section we derive thedynamic equations that govern the system behavior, and extendthe controller from Section III-B to handle the swing legdynamics. A. System Dynamics
In deriving the dynamics of the V-SLIP model with feet,we assume that: • no slip or bouncing occurs in the foot contact points; • the springs are unilateral, meaning that we only allowthem to be compressed; • during the single support phase, the swing leg is con-straint to have length L .Under these assumptions, during the double support phase, wecan use the double support phase model used in Section III-A,and the model behavior is described accordingly by (5).During the single support phase, the model can be simplifiedas shown in Figure 6. The configuration of the system can bedescribed by ( q , q , q ) , where q ∈ [0 , π ) is the orientationof the swing leg. The total mass of the swing leg is m = m h + m f . Since the swing leg is assumed to be a rigid linkof length L , its center of mass is at a distance d com = m f L m h + m f q q q m, J com k + u L d c o m τ c Fig. 6. Model simplification—Under the assumptions of a rigid swing leg andno slip or bouncing in the foot contact point, the model depicted in Figure 5can be simplified during the single support phase. During the double supportphase, the model is reduced to the V-SLIP model, as shown in Figure 3. from the hip joint (as indicated in Figure 6). The moment ofinertia of the leg around its center of mass is given by J com = m h d + m f ( L − d com ) . In order to derive the dynamic equations of the system forthe single support phase, we let ( v , v , v ) =: v denote thehorizontal, vertical and rotational velocity of the (center ofmass of the) swing leg. These velocities are related to the rateof change of the configuration variables ˙ q by the Jacobianmatrix S ( q ) , defined as: S ( q ) = d com sin( q )0 1 − d com cos( q )0 0 1 , (11)such that v = S ( q ) ˙ q . This allows to have the configurationvariables q coincide with those used in the V-SLIP model ofSection III. In particular, by defining p := M v , with M = diag( m h + m f , m h + m f , J com ) (12)the mass matrix of the rigid body representing swing leg, thedynamics during the single support phase can be derived interms of ( q, p ) as follows.The kinetic energy is given by K = p T M − p , and wederive the potential energy function V as V = ( m h + m f ) g ( q − d com sin( q )) + 12 k ( L − L ) . Then, the Hamiltonian energy function is given by H = K + V , and we derive the dynamic equations in port-Hamiltonianform by using the Boltzmann-Hamel equations [18], yielding:dd t (cid:20) qp (cid:21) = J " ∂H∂q∂H∂p + (cid:20) B (cid:21) uy = (cid:2) B T (cid:3) " ∂H∂q∂H∂p , (13)where the skew-symmetric matrix J is given by J = " S − − S − T S − T (cid:16) ∂ T ( S T p ) ∂q − ∂ ( S T p ) ∂q (cid:17) S − . gain, the output y is dual to the input u , so that h u | y i defines a power flow. The control input u = ( u , τ ) , i.e. thecontrollable part of the stance leg stiffness, and the torqueapplied to the swing leg. The input matrix B is given by B = S − T ∂φ ∂q ∂φ ∂q
00 1 , with φ = − ( L − L ) . The mapping by S − T is necessarybecause the inputs are not collocated with v , but with ˙ q , ascan be seen in Figure 6. B. Phase Transitions
Unlike the V-SLIP model, where in both the double andsingle support phases the same configuration variables areused, this model uses two sets of configuration variables: in thedouble support phase only the position of the hip with respectto the foot contact points is relevant, while in the single supportphase also the swing leg orientation is required. Therefore,phase transition mappings need to be defined as follows.
Transition Conditions:
Similar to the (V-)SLIP models, thetouchdown event occurs when the foot of the swing leg haspassed in front of the hip, and in addition, recalling that theswing leg is constraint to have length L during the swingphase, q = L sin( q ) . At the time instance that both of these conditions are met, thenew foot contact point c is computed as c = q − L cos( q ) . The lift-off event occurs when the trailing leg reaches itsrest length L with non-zero speed, since we do not allow thesprings to pull. Momentum Variable Mapping:
To complete the phase tran-sitions, the momentum variables of the double support phaseneed to be mapped to the momentum variables for the singlesupport phase and vice versa. This mapping also needs toensure that the constraints on the foot contact points aremaintained. In particular, upon touchdown, the foot of theformer swing leg needs to be instantaneously constraint tofulfill the no-slip condition. This can be realized by applyinga momentum reset at the instant of touchdown [19]. It wasshown in our previous work that, despite the energy lossassociated with the impact, energy-efficient locomotion canbe realized [17]. However, in this work, we will focus onthe added benefit of the compliant legs, and thus assume acompliant impact. This implies that, upon impact, the footmass m f will instantaneously dissipate its kinetic energy,while the hip mass m h remains unaffected by the impact dueto the compliant leg. Essentially the swing leg is allowed to swing through the floor. This willbe addressed in the next model iteration in Section V. To be accurate, the transition occurs when the foot starts to accelerateaway from the floor. However, this is practically equivalent to assuming thatthe transition occurs at the moment the spring length becomes equal to its restlength and assuming that the leg instantly becomes rigid at the same moment.
To map the momentum variables between the phases, weneed to account for the disappearing and reappearing of thefoot mass. For this purpose, we define new coordinates z = ( q , q , q ) and z = ( q , q , c i ) , where c i denotes the contact point that is subject to change dueto the transition. During both the touchdown and the lift-offevent, the leg length is assumed to be L , so that we obtain z ( z ) = q q q − L cos( q ) . We define the Jacobian matrix Z := ∂z /∂z accordingly.For the transition from single support to double support,using the subscripts “old” and “new” for post- and pre-transition values, we have: ˙ z , new = Z ˙ z , old , where ˙ z , old is defined by the momentum variables p old justbefore the phase transition: ˙ z , old = S − ( q ) M − p old , with S ( q ) defined in (11) and M ss the mass matrix defined in(12). Note that the expression for ˙ c is irrelevant in this phasetransition, since we assume that the foot is instantly constraint.The post-transition momentum variables for the double supportphase p new are then given by p new = M ds (cid:20) ˙ q ˙ q (cid:21)| {z } ∈ ˙ z , new , with M ds the mass matrix defined in (3).Similarly, for the transition from double support to singlesupport, we have ˙ z , new = Z − ˙ z , old , where ˙ z , old is defined through the momentum variables p old just before the phase transition: ˙ z , old = M − p old , with M ds the mass matrix defined in (3), and setting ˙ c = 0 ,since the foot is stationary at the moment of lift-off. The post-transition momentum variables p new for the single supportphase are then calculated as p new = M ss S ( q ) ˙ z , new , with S ( q ) defined in (11). C. Controller Design
During the double support phase, the model is equivalentto the V-SLIP model, and therefore, during this phase, thecontrol strategy proposed in Proposition 1 can be used. For thesingle support phase, the control strategy has to be extendedto regulate the swing leg trajectory q ( t ) . For this purpose, wedefine a reference trajectory q ∗ ( t ) as a polynomial: q ∗ ( t ) = X i =0 a i ( t − t lo ) i , t lo ≤ t < t lo + T swing , here T swing is the desired swing time, e.g. obtained from thenominal single support phase time of the SLIP model referencegait, and t lo is the time instant of the last lift-off event. Thecoefficients a i are such that q ∗ ( t ) is a minimum-jerk trajectorywith boundary conditions q ∗ ( t lo )˙ q ∗ ( t lo )¨ q ∗ ( t lo ) = q ( t lo )00 and q ∗ ( t lo + T swing )˙ q ∗ ( t lo + T swing )¨ q ∗ ( t lo + T swing ) = π − α . In formulating the control strategy, we will define for boththe double support and single support phases a state vectorof the form x = ( q, p ) and write the respective differentialequations (5) and (13) in the standard form (7). The followingcontrol strategy is proposed, extending the V-SLIP controlstrategy formulated in Proposition 1. Proposition 2:
Given reference state trajectories ( q ∗ , ˙ q ∗ , q ∗ ) , define the error functions h = q ∗ − q ,h = ˙ q ∗ − ˙ q ,h = q ∗ − q . During the double support phase, the corresponding controlstrategy formulated in Proposition 1 renders solutions of (5) asymptotically converging to ( q ∗ , ˙ q ∗ ) .During the single support phase, the following controlstrategy renders solutions of (13) asymptotically convergingto ( q ∗ , q ∗ ) : (cid:20) u τ (cid:21) = − A − (cid:20) L f h + κ d L f h + κ p h L f h + κ w L f h + κ a h (cid:21) , (14) with A = (cid:20) L g L f h L g L f h L g L f h L g L f h (cid:21) . For any arbitrary small ε , ε , δ > , there exist constants κ p , κ d , κ v , κ a , κ w > for the control strategy (9) , (10) , (14) such that lim t →∞ | q ∗ ( t ) − q ( t ) | < ε , and lim t →∞ | ˙ q ∗ ( t ) − ˙ q ( t ) | < ε , and, during t lo ≤ t < t lo + T swing , | q ∗ ( t lo + T swing ) − q ( t lo + T swing ) | < δ. Proof 2: The control strategy is such that the system isstrongly input-output decoupled. Therefore, the dynamics of h ( t ) are given by ¨ h + κ d ˙ h + κ p h = e , where e ( t ) is a disturbance due to the phase transitions. Asa result, q ( t ) is continuous, but not differentiable. However, The velocity ˙ q ( t lo ) and the acceleration ¨ q ( t lo ) are not matched bythe reference trajectory q ∗ ( t ) , because these quantities are in practice verydifficult to measure accurately. -0.010.000.01 h -0.200.000.200.400.600.801.00 h
25 26 27 28 29 30Time [s]-0.050.000.050.100.150.20 h Fig. 7. Steady-state error functions for the controlled V-SLIP model withswing leg—It can be seen that the error functions converge like claimed inProposition 2. Note that h ≡ during the double support phase. e ( t ) is bounded and impulsive, and therefore there exists con-stants κ p , κ d > such that h ( t ) converges to a neighborhood ε of zero.Similarly, the dynamics dynamics of h ( t ) are given by ˙ h + κ v h = e , where e ( t ) is also a disturbance due to the phase transitions.As a result of these disturbances, ˙ q ( t ) is not continuous.However, since e ( t ) is bounded and impulsive, there exista κ v > such that h ( t ) converges to a neighborhood ε ofzero.During the single support phase, the dynamics h ( t ) aregiven by ¨ h + κ w ˙ h + κ a h = 0 . For suitably chosen constant κ a , κ w > , such that the zero arein the open left half-plane, the error function h ( t ) convergesto a neighborhood δ of zero in finite time. The proposed control strategy has been validated throughnumeric simulations. The same parameters were used as forthe V-SLIP model as listed in Table I, and m f = 2 . kg.Furthermore, κ a = 1000 and κ w = 40 . It can be observedthat the error functions converge as claimed in Proposition 2.In particular, the influence of the swing leg can be clearlyobserved when the plots are compared to Figure 4. Specif-ically, we can see the influence of the swing leg motion inthe error function h at the onset of the single support phases(the unshaded areas of the plot). The error function h alsoshows a significant increase in amplitude during the swing.We can also observe in h the lift-off of the swing leg inthe form of discontinuities at the moment of transition fromthe double support phase (shaded areas) to the single supportphase (unshaded areas). The error function h shows that theswing leg motion is controlled as claimed by the proposed q θ θ m h m f m f k + u τ ℓ ℓ Fig. 8. The V-SLIP model with feet and knees—By adding an actuatedknee joint to the model of Section IV, the leg can be retracted during thesingle support phase. This allows the leg to swing forward without scuffingthe ground. It is assumed that no slip or bouncing occurs in the foot contactpoint of the stance leg. control law. Note that the degree of freedom q is not definedduring the double support phase, and therefore h ≡ duringthis phase.V. T HE C ONTROLLED
V-SLIP
MODEL WITH R ETRACTING S WING L EG D YNAMICS
In this Section, we further refine the model presented inSection IV by adding a knee, as shown in Figure 8. Thisallows the swing leg to be retracted, so that it can be swungforward without scuffing the ground. We derive in this Sectionthe dynamic equations for this model, and further extend thecontroller.
A. System Dynamics
Similar to the model presented in Section IV, we willassume that: • no slip or bouncing occurs in the foot contact points; • the spring are unilateral.These assumptions allow to again use the double support phasemodel used in Section III-A.To avoid notational clutter due to goniometric relations, thesimplified model depicted in Figure 9 is used. The simplifi-cation is possible, because the introduction of the knee jointintroduces only a kinematic relation between the hip mass andthe foot mass, since these are located at the extremities of theswing leg.In deriving the dynamic equations for the single supportphase of this model, we define new coordinates as z = ( q , q , q , q ) and z = ( q , q , s , s ) (15)where s = q − q cos( q ) ,s = q − q sin( q ) , (16)i.e. the position of the foot of the swing leg. We furthermoredefine the tangent map Z = ∂z /∂z . Using this relation and q q q q , τ τ m h m f k + u c Fig. 9. Model simplification—The configuration of the swing leg can beequivalently described by a linear degree of freedom q , corresponding to thedistance between the hip and the foot, and the orientation q , analogous tothe model of Figure 5. During the double support phase, the model is reducedto the V-SLIP model, as shown in Figure 3. noting that z = q and thus that ˙ z = Z ˙ q , we can derive themass matrix M ( q ) from the energy equality
12 ˙ q T M ( q ) ˙ q = 12 ˙ z T M ˙ z = 12 ˙ q T Z T M Z ˙ q, (17)with M = diag( m h , m h , m f , m f ) .By defining the momentum variables p := M ( q ) ˙ q , thekinetic energy K = p T M − ( q ) p , and the potential energyfunction V is found to be: V = m h g q + m f g ( q − q sin( q )) + 12 k ( L − L ) . Then, the Hamiltonian energy function H = K + V and thedynamic equations in port-Hamiltonian form are given bydd t (cid:20) qp (cid:21) = (cid:20) I − I (cid:21) " ∂H∂q∂H∂p + (cid:20) B (cid:21) uy = (cid:2) B T (cid:3) " ∂H∂q∂H∂p , (18)where u = ( u , τ , τ ) , i.e. the controllable parts of the stanceleg stiffness, and the torques collocated with q and q . Theinput matrix B is given by B = ∂φ ∂q ∂φ ∂q , with φ = − ( L − L ) . B. Phase Transitions
Just as in the model described in Section IV, also in thismodel we need to consider the different sets of configurationvariables in the single and double support phases. Therefore,in the following the phase transition mappings are defined. ransition Conditions:
The touchdown event occurs whenthe swing leg foot hits the ground, i.e. when, using (16), q = q sin( q ) . At this time instant, the new foot contact point c (seeFigure 3) is computed as c = q − q cos( q ) . The lift-off event is defined the same as in Section IV-B,since both models are reduced to the V-SLIP model duringthe double support phase.
Momentum variable mapping:
Similarly to the approachtaken in Section IV-B, we start from the new set of coordinatesdefined in (15) and the corresponding Jacobian matrix Z .Thus, for the transition from single support to double support: ˙ z , new = Z ˙ z , old , where ˙ z , old is defined by the pre-transition momentum vari-ables p old through ˙ z , old = M − p old . Here, M ss is the mass matrix defined in (17). As in Sec-tion IV-A, the post-transition momentum variables for thedouble support phase p new are given by p new = M ds (cid:20) ˙ q ˙ q (cid:21)| {z } ∈ ˙ z , new , with M ds the mass matrix defined in (3).For the transition from double support to single support, weagain have ˙ z , new = Z − ˙ z , old , where ˙ z , old is defined through the momentum variables p old just before the phase transition: ˙ z , old = M − p old , with M ds the mass matrix defined in (3), and setting ˙ s =˙ s = 0 , since the foot is stationary at the moment of lift-off.The post-transition momentum variables p new for the singlesupport phase are then calculated as p new = M ss ˙ z , new . C. Controller Design
During the double support phase, the control strategy pro-posed in Proposition 1 can again be used because of themodel correspondence during this phase. For the single supportphase, the control strategy has to be extended with respect tothe control strategy presented in Proposition 2. In particular, inaddition to the control of the swing leg orientation, the swingleg length has to be regulated as well. For this purpose, wedefine a reference trajectory q ∗ ( t ) of the form q ∗ ( t ) = b + b t + b t , t lo ≤ t ≤ t lo + T swing . The coefficients b i are such that the trajectory q ∗ ( t ) satisfiesthe following conditions: q ∗ ( t lo ) q ∗ ( t lo + T swing ) q ∗ ( t lo + T swing ) = q ( t lo ) L − ∆ L , with ∆ > the amount of retraction of the swing leg. Thistrajectory ensures that at the moment of lift-off the swingleg is immediately accelerating away from the floor, reachingthe maximum retraction during the predicted mid-stance. Attouchdown, the swing leg will have length L , correspondingto the (V-)SLIP model.Defining the state vector of the form x = ( q, p ) and writing(5) and (18) in the standard form (7), the following controlstrategy is proposed, extending the strategy formulated inProposition 2. Proposition 3:
Given reference state trajectories ( q ∗ , ˙ q ∗ , q ∗ , q ∗ ) , define the error functions h = q ∗ − q ,h = ˙ q ∗ − ˙ q ,h = q ∗ − q ,h = q ∗ − q . During the double support phase, the corresponding controlstrategy formulated in Proposition 1 renders solutions of (5) asymptotically converging to ( q ∗ , ˙ q ∗ ) .During the single support phase, the following controlstrategy renders solutions of (18) asymptotically convergingto ( q ∗ , q ∗ , q ∗ ) : (cid:20) u τ (cid:21) = − A − L f h + κ d L f h + κ p h L f h + κ w L f h + κ a h L f h + κ n L f h + κ ℓ h , (19) with A = L g L f h L g L f h L g L f h L g L f h L g L f h L g L f h L g L f h L g L f h L g L f h . For any arbitrary small ε , ε , δ , δ > , there exist con-stants κ p , κ d , κ v , κ a , κ w , κ ℓ , κ n > for the control strategy (9) , (10) , (19) such that lim t →∞ | q ∗ ( t ) − q ( t ) | < ε , and lim t →∞ | ˙ q ∗ ( t ) − ˙ q ( t ) | < ε , and, during t lo ≤ t < t lo + T swing , | q ∗ ( t lo + T swing ) − q ( t lo + T swing ) | < δ | q ∗ ( t lo + T swing ) − q ( t lo + T swing ) | < δ The proof is analogous to the proof given in Section IV-Cand is omitted for brevity. The control strategy is validatedthrough numeric simulations, with the same model parametersas used in Section IV-C, and in addition κ ℓ = 1000 and κ n = 40 , with a leg retraction ∆ = 7 . cm. The resulting errorfunction plots are shown in Figure 10, and it can be seen thatthey converge as claimed in Proposition 3. The error functionsshow now significantly bigger influences of the swing legdynamics when compared to Figure 7. h -0.200.000.200.400.600.801.00 h -0.050.000.050.100.150.20 h
25 26 27 28 29 30Time [s]-0.06-0.04-0.020.000.02 h Fig. 10. Steady-state error functions for the controlled V-SLIP model withleg retraction—It can be seen that the error functions converge like claimedin Proposition 3. Note that h ≡ and h ≡ during the double supportphase. VI. C
OMPARISON BY N UMERICAL S IMULATION
Starting from the bipedal SLIP model, three iterations ofmodel refinement have been presented in Section III, Sec-tion IV, and Section V. Also, in the first iteration, a robustcontroller for leg stiffness has been presented in Section III-B,which has been extended in subsequent iterations. In thissection, a comparison of the performance of these controllersis presented.
A. Comparison of Gait Control
Figure 11 shows the horizontal hip trajectory q ( t ) , anda detail of the corresponding vertical hip trajectory q ( t ) .The most notable difference between the three models istheir average forward velocity, which is . m/s for the V-SLIP model, but only . m/s and . m/s for the modelsincluding the swing leg dynamics and the knee. Looking at thehip trajectories, it can be seen that this is due to the inclusionof the swing leg dynamics, which introduces a lag in forwardmotion with respect to the V-SLIP dynamics.The leg stiffness trajectories k i = k + u i are shown inFigure 12. Since the reference trajectory is the natural gaitof the bipedal SLIP model, it is not surprising that the V-SLIP model requires very little control action to track the q [ m ] V-SLIP+ swing leg+ retraction25 26 27 28 29 30Time [s]0.840.860.880.900.920.940.960.98 q [ m ] Fig. 11. Hip trajectories q ( t ) and q ( t ) —It can be observed that swingingand retracting the leg has a negative influence on the forward velocity. Thisinfluence is particularly apparent in the vertical hip position trajectories. k + u [ N / m ]
25 26 27 28 29 30Time [s]01000200030004000500060007000 k + u [ N / m ] Fig. 12. Control inputs—See Figure 11 for the legend. The controlled V-SLIPmodel hardly requires any control input to track the reference, which is bydesign of the control strategy. Adding the swing leg and retracting introducesa significant disturbance in the dynamics, and thus more control input. Notethat u ≡ during the single support phase. reference. The small amount that is required is due to the smallmismatch between the parameterized reference trajectory andthe true dynamics, as pointed out already in Section III-B. Theintroduction of the swing leg dynamics introduces a significantdisturbance to the V-SLIP dynamics, exemplified by the largermagnitude of the control inputs. It is interesting to note thatthe V-SLIP model with swing leg, as introduced in Section IV,requires an impulse-like control input during the single supportphase to counter the acceleration and deceleration of the swingleg. In contrast, including the leg retraction (Section V) resultsin a smaller moment of inertia, resulting in a smoother control.However, it is noted that the swing leg retraction does resultin larger deceleration of the hip, which manifests itself inlarger control inputs during the double support phase (bottomplot), actually reaching the lower bound of zero leg stiffness k = k + u for short periods of time. B. Energy Balance
The natural gait of the bipedal SLIP model is associatedto a constant energy level [4], [5]. Since the V-SLIP modelpresented in Section III matches the bipedal SLIP model, itsenergy balance is the same if the reference trajectory for the V-SLIP controller exactly matches the solutions of (4). However, E n e r g y [ J ] Total energyKinetic energyPotential energyElastic energy
Fig. 13. Energy balance for the controlled V-SLIP model with swing leg—Thedashed line indicates the constant energy level of the bipedal SLIP model. Theinfluence of the swing leg is clearly seen in the bulges in the kinetic energy.
25 26 27 28 29 30Time [s]020406080100120140160180 E n e r g y [ J ] Total energyKinetic energyPotential energyElastic energy
Fig. 14. Energy balance for the controlled V-SLIP model with leg retraction—The dashed line indicates the constant energy level of the bipedal SLIP model.Also here the influence of swinging and retracting the swing leg is clearlyseen. as already noted before, the solutions of (4) are not analytical,and therefore a small mismatch between the natural dynamicsof the V-SLIP model and the reference is inevitable, resultingin small control action even in nominal conditions, as shownin Figure 12.The energy balance for the model including the swing leg(Section IV) is presented in Figure 13. It can be seen that theintroduction of the swing leg dynamics introduces a significantdeviation of the constant energy level of the bipedal SLIPmodel, indicated by the dashed line. The bulges in the kineticenergy plot clearly show the accelerating and the deceleratingof the swing leg.The energy balance for the swing leg model with legretraction (Section V) is shown in Figure 14. It clearly showsthat the leg retraction further slows down the system, asexemplified by the lower total energy level when compared toFigure 13. However, it is also noted that the bulges observedin Figure 13 have been reduced in amplitude in Figure 14.This is because the leg retraction results in a lower swing leginertia, mitigating the influence of the swing of the leg.Both Figure 13 and Figure 14 show relatively small varia-tions in the total energy level. This signifies that only smallamounts of energy are exchanged with the environment andvia the control action.
C. Cost of Transport
Cost of transport (also known as specific resistance) is ameasure of energy efficiency, as it measures the energy that asystem uses to travel a specified distance [20], [21]. Using thedefinition proposed in [21], the cost of transport is obtainedby exploiting the port-Hamiltonian formulation of the dynamicequations (5), (13), (18): C = 1 m total g ∆ x Z T |h u | y i| d t, (20)where m total denotes the total mass and ∆ x the distancetraveled during the time T . The cost C captures the amountof energy required for walking the distance ∆ x , taking intoaccount that, in general, actuators dissipate energy whennegative work is done, rather than storing it.Using (20), we find C = 3 · − for the controlled V-SLIPmodel. The cost is not exactly zero due to the aforementionedmismatch between the reference trajectory and the true naturaldynamics: if the reference had been exact, we would have C ≡ . For the model with swing leg (Section IV), weobtain C = 0 . . For the model with the retracting swingleg (Section V), we obtain C = 0 . . While Figure 14 hintedto lower energy expenditure when compared to Figure 13, thisgain in efficiency is offset by the lower average velocity. Thecost of transport C = 0 . is in the same range as of humanwalking [20], and thus the proposed control strategy allowsfor a theoretical performance that s approaches that of humanwalking. VII. C ONCLUSIONS
In this paper, we started from the bipedal SLIP model, andshowed how active leg stiffness variation can render gaitsof this model more robust. In particular, it was shown thatthe model could be extended to include swing leg dynamics,while still embedding the SLIP-like walking behavior byemploying feedback control strategies. These control strategiesare inspired by the capabilities of the human musculoskeletalcapability of varying leg stiffness. It was shown that this ap-proach yields a theoretical cost of transport that is comparableto human performance. This shows that active leg stiffnessvariation can be an important concept in human walking,which has not yet been transferred to the domain of realizingperformant robotic walkers.The starting point for the analysis presented in this workwas the bipedal SLIP model, extended by variable leg stiffnessto provide control inputs. Subsequent modeling iterationsextended the model to include full swing leg dynamics, whileat the same time the control strategy was extended to handlethe refined models. The final result is a template model of abipedal walker, based on the principles of the bipedal SLIPmodel, and a stabilizing controller that realizes an energeticperformance level comparable to human walking. This modelplus controller can serve as a basis for control of bipedalrobots. R
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