Using Energy Shaping and Regulation for Limit Cycle

Using Energy Shaping and Regulation for LimitCycle Stabilization, Generation, and Transition in

Simple Locomotive Systems

Mark YeatmanStudent Member of ASME

Department of Mechanical EngineeringUniversity of Texas at Dallas

Robert D. Gregg ∗Department of Electrical Engineering and Computer Science

University of MichiganEmail: [email protected]

This paper explores new ways to use energy shaping and reg-ulation methods in walking systems to generate new passive-like gaits and dynamically transition between them. We re-capitulate a control framework for Lagrangian hybrid sys-tems, and show that regulating a state varying energy func-tion is equivalent to applying energy shaping and regulat-ing the system to a constant energy value. We then con-sider a simple 1-dimensional hopping robot and show howenergy shaping and regulation control can be used to gen-erate and transition between nearly globally stable hoppinglimit cycles. The principles from this example are then ap-plied on two canonical walking models, the spring loadedinverted pendulum (SLIP) and compass gait biped, to gener-ate and transition between locomotive gaits. These examplesshow that piecewise jumps in control parameters can be usedto achieve stable changes in desired gait characteristics dy-namically/online.

1 IntroductionResearch for creating periodic gaits in locomotive sys-

tems has been in development since the early 2000’s, with acelebrated example of Hybrid Zero Dynamics (HZD) [1, 2].This methodology revolves around designing state outputfunctions via Bezier polynomials such that when they aredriven to zero, a robot achieves a walking gait with pre-defined trajectories. More recent work has focused on ex-tending HZD to achieve dynamic motion transition throughmotion planning techniques [3–5]. A trajectory-free methodthat contrasts HZD is to mimic and stabilize “natural” orpassive dynamic walking gaits through “energy shaping”,a termed coined in works on Interconnection and Damp-

∗Corresponding Author

ing Assignment Passivity-Based Control (IDA-PBC) [6, 7]and Controlled Lagrangian [8, 9] techniques. The gen-eral equivalence between passivity-based energy shaping andother techniques has been demonstrated, such as “ControlledHamiltonian” [10] or “Immersion and Invariance” [11]. En-ergy shaping methods have a focus on using physicallymeaningful parameters to produce desired system proper-ties, which can provide intuition in the analysis and designof extremely nonlinear systems in ways other control meth-ods cannot, such as generalization of stability margins [12].As seen in [13–16], energy shaping can force a biped robotto emulate the dynamics of a target passive locomotive sys-tem, then exploit a passivity property between the input andenergy-based output to regulate the desired energy level as-sociated with a limit cycle without canceling the nonlineardynamics. Our paper is in this vein, but expands upon theseprevious works by dynamically transitioning between gaitsthrough online switching of both the target energy and thesystem parameters of the emulated locomotive system. Thisoffers a more simple (and arguably more natural) proce-dure for achieving dynamic gait transition than designing aplethora of gait trajectories and the transitions between them.

Inspired by [7] and [13] , our approach to generate limitcycles is as follows. First, identify a target Lagrangian sys-tem endowed with a continuum of periodic orbits where eachorbit is associated with a unique energy value, and then setthe control equal to the difference between the open-loopsystem dynamics and the desired system dynamics. Thisforces the closed-loop system to behave like the desired sys-tem. We refer to this step as energy shaping. Second, usean outer-loop controller to target a specific orbit by driv-ing the system to the associated energy level set, creatinga self-sustaining oscillator. We refer to this technique as en-

1 Copyright © by ASME

ergy regulation. Ideally, this would allow us to switch be-tween orbits by changing just the target system energy. Forwalking systems it is not quite that simple because hybriddynamics with dissipative impact maps can cause limit cy-cles, which by definition preclude other nearby periodic or-bits [17]. However, changes in the system parameters, likemass and gravity, can result in changes in the limit cycle tra-jectory. Thus, we will modify both the virtual system param-eters and target reference energy to generate and transitionbetween new limit cycles (essentially the idea of Lyapunovfunneling [18, 19]).

Our paper extends and connects the work on the com-pass gait biped in [13] by Spong, Holm, and Lee and thespring based models in [15, 20] by Garofalo and Ott. Thepaper by Spong et al. uses “Controlled Symmetries”, aform of energy shaping, to change the direction of the vir-tual gravity vector of the compass gait biped combined withan energy regulation technique to robustify walking gaits.The paper [20] by Garofalo et al. uses energy regulationtechniques to control the hopping height of a springy robot,while [15] uses energy regulation in conjunction with an em-bedded SLIP controller on a walker with knees and a torso.One of the main differences between the SLIP and compassgait models is that the compass gait has energy dissipationat impact while the SLIP model does not, which plays a keyrole in the stability of their passive limit cycles. The con-sideration of both of these models is important because theyare both fundamental to biped locomotion [21, 22]. We notethat this idea of causing systems to emulate self sustainingnonlinear oscillators has connections outside the domain ofwalking bipeds; [23] is an example in the area micro-electro-mechanical-systems (MEMS) that uses feedback control tocause a microbeam to behave like a van-der-Pol or Rayleighoscillator.

The primary contribution of this paper is an analyticalframework and simulation results for the use of energy shap-ing and regulation to dynamically transition between a rangeof walking gaits on hybrid locomotive systems by leveragingrelations between system parameters and desired limit cyclecharacteristics. Using this framework, we demonstrate anequivalence between regulation of a time/state-varying en-ergy function and a 2-step process of energy shaping thenenergy regulation of a constant reference value. We thenargue that regulating a constant energy value is the moremeaningful and clear method of control construction becausethe asymptotic limit cycle trajectory is an energy level set ofsome system. We consider a simple hopping robot that illus-trates the relationships between impact dissipation, systemparameters, energy, and limit cycle behavior. This leads usto create a novel discrete update law for the reference en-ergy that increases the robustness of a passive limit cycle inthe face of parameter uncertainty. We apply this law to thecompass gait biped in simulation, which is useful becausethe energy associated with a limit cycle cannot be analyti-cally computed in this model. The difference between thiswork and [13] is that we change the virtual mass instead ofgravity. Changing the virtual mass does not enjoy the same“Controlled Symmetry” property and is fundamentally more

difficult to analyze. A strict extension of [13] in this paperis that we dynamically transition between different walkinggaits on the compass gait biped using the energy regulationtechnique. In the same vein, our paper extends work on theSLIP model in [15, 20] by accomplishing dynamic gait tran-sitions through switches in the target energy and spring stiff-ness. Finally, we also explicitly demonstrate that energy reg-ulation can stabilize unstable limit cycles on the compass gaitbiped.

The organization of the paper is as follows: Section 2 ofthe paper gives a brief review of hybrid Lagrangian dynamicswith impacts and the application of energy shaping and regu-lation to this class of systems. Section 3 details the dynamicsand control of a hopping robot to illustrate concepts. Section4 presents the SLIP model dynamics, control, and simula-tions that transition between various running speeds. Section5 presents the compass gait biped model dynamics, control,and simulations that transition between walking speeds andstabilize previously unstable limit cycles of the passive sys-tem.

Notation: Given two matrices a and b of suitable dimen-sions, the matrix [a>, b>]> is denoted by [a;b] where> is thetranspose operator.

Remark 1. In this paper, it is important to distinguish be-tween the terms “passive” versus “passivity” and the con-cepts to which they refer. To be clear, the term “passive”refers to an uncontrolled mechanical system composed ofconnected masses, springs, and dampers. The term “pas-sivity” refers to the following mathematical definition.

Definition 1. Let S(q, .q) : R2n → R be a continuously dif-ferentiable, non-negative scalar function. A system [

.q; ..q] =f ([q; .q]) + g([q; .q])u, γ = η([q; .q]) has a passivity relation-ship between input u and output γ with storage functionS(q, .q) if

.S(q, .q)≤ u>γ.

2 Energy Shaping and Regulation with Lagrangian Dy-namicsThere are many papers on the idea of energy shaping

control for mechanical systems (which can be described byLagrangian or Hamiltonian dynamics); seminal work in thisarea includes “Controlled Lagrangians” by Bloch et al. in [8]and “Interconnection and Damping Assignment Passivity-Based Control” by Ortega et al. in [6]. Recently, there hasbeen a focus on using similar methods to stabilize periodicorbits, in both non-hybrid [7] and hybrid systems [16, 24].We review some methods and notation to apply these tech-niques on hybrid systems with Lagrangian dynamics and im-pacts. The general idea is to first use energy shaping to gen-erate desired virtual Lagrangian dynamics in the closed-loop,then use energy regulation to drive the associated virtual en-ergy function to a desired reference value associated with alimit cycle.


2.1 Hybrid Lagrangian DynamicsWe begin by defining the general notation of hybrid La-

grangian dynamics with impacts. The state of the system isgiven by the vector q ∈Rn. The continuous dynamics can bederived from the Euler-Lagrange (EL) equations which havethe following form:

ddt

∂L(q, .q)∂

.q− ∂L(q, .q)

∂q= B(q)u. (1)

The Lagrangian L = K (q, .q)− P (q) is the difference be-tween the system’s kinetic energy K and the potential en-ergy P . The external control forces u ∈ Rm are mapped intothe dynamics by the matrix B ∈ Rn×m. Kinetic energy canbe expressed as K = 1

2.q>M(q) .q, where M(q) is a positive

definite symmetric matrix that represents the mass/inertia ofthe system. The EL equations expressed in matrix form are

M(q) ..q+C(q, .q) .q+G(q) = B(q)u (2)

where the vector C .q represents Coriolis and centripetalforces and the vector G = ∂P (q)

∂q represents gravity. The ma-

trix C is constructed such that.

M−2C is skew symmetric.Discrete impact dynamics capture the effect of system

components coming in sudden contact with surfaces thatconstrain the motion of the system. The superscript nota-tion − and + denotes variables just before and after impact,respectively. We define a switching surface S as

S = (q, .q) |h(q−) = 0,.h < 0 (3)

where the function h gives the distance of the system com-ponent from the constraint surface. The inequality

.h < 0 en-

sures that the component is moving into the constraint beforeimpact. The impact triggers when the state of system entersthe switching surface. We use the impact model from [2]which causes instant, dissipative changes in the joint veloc-ities of the system, but not the joint positions. However, wedo allow the world frame to jump and the coordinates of thebiped to be relabeled, which can appear as jumps in the po-sition depending on the choice of coordinates. The impactmap is denoted by (q+, .q+) = ∆(q−, .q−). The combinationof the continuous and discrete dynamics results in hybrid La-grangian dynamics with impacts, expressed as

B(q)u = M(q) ..q+C(q, .q) .q+G(q) if(q, .q) /∈ S (4)

(q+, .q+) = ∆(q−, .q−) if(q, .q) ∈ S (5)

2.2 Energy Shaping and Regulation ControlOur control approach is partitioned into two parts u =

us + ur so that us performs energy shaping and ur performsenergy regulation. The method of energy shaping we willuse is that of Controlled Lagrangians [8]. We define a target

mechanical system with Lagrangian L , derive target dynam-ics through the EL equations (1), and set the control equal tothe difference between the open loop dynamics and the targetdynamics. This yields the control law

us = (B>B)−1B>(C .q+G−MM−1(C .q+ G)), (6)

which will render the desired dynamics if and only if the so-called “matching condition”

B⊥(C .q+G−MM−1(C .q+ G)) = 0, (7)

is satisfied. Here, B⊥ is a full rank left annihilator of B, i.e.,B⊥B = 0. The matching condition basically ensures thereis never a component of the difference between the open-loop and desired dynamics in the nullspace of B. The newcontinuous dynamics are then

MM−1Bur = M ..q+C .q+ G. (8)

2.2.1 Regulating a Constant Reference EnergyThe energy of the new system can be regulated using

the passivity-based control method from [13,24] through theterm ur. Consider the following storage function (Definition1)

S =12(E−Eref)

2, (9)

where E(q, .q) is the closed-loop system energy and Eref is thereference energy. The time derivative of this storage functionis

.S = (

.E−

.Eref)(E−Eref). (10)

From the equation for the shaped dynamics (8),

.E =

.q>MM−1Bur (11)

=.q>Bur. (12)

If the reference energy is constant then.

Eref = 0 and

.S =

.q>Bur(E−Eref). (13)

Moreover,.S can be rendered negative semi-definite by

choosing

ur =−κ(E−Eref)ΩB> .q. (14)

Here, κ > 0 is a scalar gain while the matrix Ω ∈ Rn×n isa positive definite weighting matrix with each element less


than one. The storage function and its derivative are relatedby

.S =−κ(E−Eref)

2 .q>BΩB> .q (15)=−2κ|| .q||ΩS, (16)

where the term .q>BΩB> .q is a norm. The convergence of thevirtual energy to the target reference energy is guaranteed ifthe system state cannot enter some positively invariant setwhere || .q||Ω = 0 (LaSalle’s invariance principle can be ap-plied to validate this).

This controller differs from a similar method for Port-Hamiltonian systems in [7]; their control construction re-quires the ability to identify a state variable transformationthat decomposes a limit cycle into a 2 dimensional subman-ifold where periodic motion occurs and a n−2 dimensionalsubmanifold where the state is constant. The energy regu-lation controller then only operates on the energy of the pe-riodic submanifold. This is also the main idea behind thework of [25] for inducing limit cycles in springy robots. Ourcontrol method does not require the explicit identification ofthese submanifolds, which can be difficult for passive loco-motive systems. The disadvantage of assuming the implicitexistence of these submanifolds/limit cycles through the totalenergy is that multiple periodic orbits can exist in the sameenergy level set for a n > 2 dimensional system. This meansthat in general, convergence to a reference energy does notguarantee convergence to a desired limit cycle. However,there are several examples of previous work on hybrid lo-comotive systems [13, 15, 24, 26, 27] that demonstrate thatachieving the desired energy does achieve a desired limit cy-cle in practice.

2.2.2 Regulating a Varying Reference EnergyIt is possible to consider a non-constant reference en-

ergy Eref(q,.q, t) that varies with state and time. However,

achieving a control that ensures the convergence of the sys-tem to E = Eref(q,

.q, t) is essentially equivalent to the stepsof applying energy shaping then energy regulation as we didin the previous section. First, define an energy/Hamiltonianfunction E = E−Eref(q,

.q, t). Then, use the Legendre trans-formation to derive the associated Lagrangian L , assumingthat the transformation is well-defined. Finally, obtain thetarget dynamics through the EL equation and use the Con-trolled Lagrangians technique to arrive at an energy shap-ing control (because of this more general form of Eref, moregeneral matching conditions from [9] must be satisfied). Ifthe reference energy has some constant term C such thatEref = f (q, .q, t) +C, it will vanish after the EL equationsare applied. However, the desired convergence can be re-covered by applying an outer-loop energy regulating controlwith E = E − f (q, .q, t) and Eref = C. This two step proce-dure allows us to clearly interpret the effect of the controlas determining the shape of an energy level set through theshaping step and then stabilizing the set through energy regu-lation, similar to [7]. Directly regulating a time-state varying

reference energy results in a less clear effect.

2.3 Application to Hybrid Locomotive SystemsOur main interest in this paper is to use energy shaping

control to match the closed-loop dynamics to a passive loco-motive system. This is what allows us to bypass the issue ofexplicitly identifying a coordinate transformation or subman-ifold in the state space. These systems generally have limitcycles that depend on the value of their parameters [17, 21],meaning that we can generate new cycles by changing thoseparameters in the virtual system. The basin of attraction ofthese limit cycles are typically small, hence we will use en-ergy regulation to increase the basin in order to make gaittransitions more robust as shown in [13]. This method ofgait transition is basically the idea of Lyapunov funneling,see [18, 19]. However, the hybrid nature of the dynamicscan present conceptual challenges to the application of bothenergy shaping and regulation control methods.

If the impact map ∆ depends on the system parametersand we are limited only to continuous/non-impulsive control,then we are unable to completely emulate arbitrary virtualhybrid systems. This idea is related to work on energy shap-ing and Controlled Symmetries [28], of which a key compo-nent was demonstrating that the impact dynamics of a rigidbiped are invariant with respect to the ground slope param-eter. However, it is completely possible to change the vir-tual parameters of only the closed-loop continuous dynam-ics, while using the original open-loop impact dynamics. Weexpect that the qualitative relationship between gait charac-teristics (e.g., walking speed) and parameters will be similarbetween the true system and partially emulated virtual sys-tem, and we are not aware of any previous work that hasconsidered this approach.

The impact map also influences the construction of anenergy regulating control. If a passive walker has a con-served energy on its periodic orbit in the continuous dynam-ics, then the energy must be conserved across the discretedynamics as well. This implies an equilibrium between thekinetic energy lost from dissipative impact and the potentialenergy gained from shifting the world frame [29]. This equi-librium can be unstable, such that a small perturbation willcause the impact dynamics to drive the energy away from thelimit cycle [17]. However, it could be possible to use energyregulation to stabilize these passively unstable limit cycles.The idea is that over the flow of the continuous dynamics,the energy regulating controller compensates for the desta-bilizing effect of the impact dynamics. Consider the step tostep storage function

S−i+1 =∫ ti

0(

.q>Bur)(E−Eref)dτ+S+i (17)

=∫ ti

0(

.q>Bur)(E−Eref)dτ+∆S(q−,.q−)+S−i (18)

(19)

where ti is the time between impacts and ∆S is the change in


the storage function at impact. If

0≥∫ ti

0(

.q>Bur)(E−Eref)dτ+∆S(q−,.q−) (20)

then S−i+1 ≤ S−i , meaning the storage function is always de-creasing between impacts and the energy is converging to thetarget energy. This is basically the notion of a hybrid storagefunction and so-called “jump and flow passivity” [30], butapplied to orbital stabilization. For an unstable passive limitcycle, ∆S > 0, while a stable one corresponds to ∆S < 0.From equation (15), the amount of storage dissipated overthe continuous dynamics can be modulated with the gain κ.Again, even if the inequality (20) is satisfied and the refer-ence energy is asymptotically stable, this is only a necessarybut not sufficient condition for achieving a limit cycle in gen-eral. Furthermore, this leaves the method of finding the Erefassociated with the equilibrium point induced by the impactdynamics as an open question. At the end of the next section,we offer an adaptive discrete update law to accomplish thisonline for a hopping robot.

3 A Hopping RobotThis section considers a simple model of a hopping

robot with 1 degree-of-freedom (DOF). This makes it easierto offer analytical proofs of stability due to the linear contin-uous dynamics and low dimensionality. Some of the infor-mation on the passive dynamics is similar to other works onhoppers [20] and the rimless wheel [31]. The model servesas a simple non-abstract example to demonstrate and developour ideas for application to the SLIP and compass gait mod-els later in the paper. The novelty in this section is: 1) theanalytical procedure of using energy shaping and regulationto achieve desired limit cycle characteristics, 2) proof of sta-bilization of a range of limit cycles using energy shaping andregulation, and 3) a discrete update law for the reference en-ergy to deal with unknown environments.

3.1 Hopper DynamicsConsider an actuated mass-spring system hopping on a

static flat surface and constrained to move along the verticalaxis. The continuous dynamics has two phases/equations

Stance (ST) u = m ..y+ k(y− y0)+mg (21)Flight (FL) 0 = m ..y+mg (22)

where the mass of the point is m, the distance from the pointto the ground is y, the relaxed length of the spring is y0,the gravitational acceleration constant is g, and the actua-tion force is u. The discrete dynamics that govern the switch

between these phases are

if phase == FL and y≤ yc (23).y+ = e .y−

phase := STif phase == ST and y≥ y0 (24)

phase := FL

where the superscripts - and + indicate pre-impact and post-impact states, respectively. The spring length at impact is yc,and e is the coefficient of restitution. If e = 1, the impact iselastic, and if 0 < e < 1, the impact is plastic. The switchfrom stance to flight at y ≥ y0 occurs because the groundis a unilateral constraint that cannot pull the mass into theground, but rather can only push it away. For this reason, weonly consider a relaxed or pre-compressed spring yc ≤ y0 atimpact. We note that this exact model might be difficult tophysically realize; it serves as simple abstract template for ahybrid Lagrangian system with periodic motion.

3.2 Hopper Passive DynamicsPeriodic orbits in Lagrangian systems necessarily have

a conserved energy, which implies the system energy mustbe conserved at impact. For periodic orbits of the pas-sive hopper, this means the energy added by the spring pre-compression must equal the energy dissipated by the impactwith

12

k(y0− yc)2 =

12

m(1− e)2(.y−)2. (25)

If the impact is plastic, then the orbit can be shown to be lo-cally exponentially stable in the sense of Lyapunov by con-sidering the energy state from impact to impact. Before im-pact,

E−i =12

m(y−)2 +mgyc, (26)

and after

E+i =

12

m(ey−)2 +12

k(y0− yc)2 +mgyc. (27)

Since energy is conserved over the continuous dynamics,E+

i = E−i+1, which implies that

E−i+1 = e2E−i +12

k(y0− yc)2 +(1− e2)mgyc. (28)

Because 0 < e < 1, this discrete system is exponentially sta-ble and

E→ k(y0− yc)2

2(1− e2)+mgyc = Elim. (29)


This forms an analytical Poincare map [2], thus there is al-ways a locally exponentially stable hybrid limit cycle for thehopper.

For all parameter cases of the passive hopper, the systemwill get stuck in the stance phase if the energy is too low toachieve liftoff. In addition, the mass can bottom out againstthe ground if the energy is too large, which we consider tobe a system failure. This implies that basin of attraction isbounded by these two energy level sets as

mgy0 < E <12

ky20. (30)

If e= 1 and yc = y0, the energy is always conserved across alldynamic regimes and there is a family of marginally stableperiodic orbits within these energy bounds.

3.3 Hopper ControlThe hopping height yapex is limited by the upper bound

on the energy with the expression

yapex <ky2

02mg

. (31)

The good news is that energy shaping can be used to changethe virtual spring stiffness k to increase the basin ceiling andachieve arbitrary hopping heights, while energy regulationcan be used to create energy for liftoff and minimize thebasin floor. The resulting control and closed loop stance dy-namics are

u = us +ur (32)

= (k− k)(y− y0)−κ(E−Eref).y (33)

0 = m ..y+κ(E−Eref).y+ k(y− y0)+mg (34)

where E = 12 m .y2 + 1

2 k(y− y0)2 +mgy. If there is a linear

damping term in the stance dynamics d .y, it acts as a shift onthe reference energy as Enew = Eref− d

κso the damped hop-

per system can be addressed by this example as well. Inter-estingly, the new stance dynamics correspond exactly to theharmonic Rayleigh-Van-der-Pol oscillator from [32] (whichwe elaborate on in the following section on the SLIP model).

In the case of e = 1 and yc = y0, we can choose any ref-erence energy that satisfies equation (30) to achieve a virtualpassive limit cycle. In the case of 0 < e < 1 and 0 < yc < y0,we must choose Eref to be exactly equal to equation (29) if wewant to ensure that the hopping limit cycle mimics a passivesystem. Consider the step-to-step storage function at impact,

S+i =12(E+−Eref)

2 (35)

=12(e2E−i +

12

k(y0− yc)2−Eref)

2. (36)

If 0 < e < 1 and Eref = Elim, then S+i = e4S−i and the storagefunction decreases after impact, i.e., S+i ≤ S−i . The energyregulation control causes the storage to decrease over thecontinuous dynamics, implying S−i+1 ≤ S+i . It follows thenthat S−i+1 ≤ S−i ; the post impact storage function monotoni-cally decreases from event to event. Thus, the hybrid limitcycle of the system under the energy regulation control isasymptotically stable.

One could expect a hopping robot to operate in differ-ent environments with varying coefficients of restitution thatcannot be estimated before hand. So it would be useful inpractice to have a method of updating Eref. Inspired by struc-ture of the impact dynamics, we propose the update policy

Erefi+1 = Erefi +λ(E+i −E−i ). (37)

where λ is a scaling gain. Essentially, if the impact dynam-ics cause a net gain in E then Eref is increased, and vice-versa. From the convergence of S in the continuous dynam-ics, |E−i+1−Erefi | ≤ |e2E−i −Erefi |. Though some manipula-tion, this can be converted into the form

[Erefi+1

E−i+1

]=

[1 λ(e2−1)c d

][Erefi

E−i

]+B (38)

= A[

Erefi

E−i

]+B (39)

where B is a constant, and |c+d|< 1. Then from the Perron-Frobenius theorem [33], the maximum absolute eigenvalueof A is less than one if |1+λ(e2− 1)| < 1. Therefore, thisupdate law is globally stable for −1 < λ < 0 and the refer-ence energy will converge to the energy of the passive limitcycle.

This update law can generalize to higher dimensionalsystems, which we show through numerical simulation onthe compass gait biped in a later section. We remark thatin practice, using a different constant value of Eref can stillresult in a limit cycle as in [20], but the asymptotic trajectorywill not emulate a virtual passive system and is dependent onthe value of the gain κ. Finally, the proof of stability for thehopper system relies on the stability properties of the passivelimit cycle, 0< e< 1. This leads us to believe that in general,the energy associated with an unstable limit cycle cannot bearrived at via (37).

4 Spring Loaded Inverted PendulumThe SLIP model can be considered as an extension of

the hopping robot that exhibits behaviors and properties sim-ilar to human walking and running [21]. It comprises apoint mass that moves via connecting the mass to the groundthrough ideal springs. The version we consider here has 2-DOF, can exhibit both walking and running behaviors, andhas additional control actuation along the spring axis. Thewalking behavior alternates between a single support phase


•-------

𝛼𝛼

𝑔𝑔

𝑚𝑚

𝐿𝐿0

𝑢𝑢 + 𝐹𝐹𝑘𝑘

𝑦𝑦𝑥𝑥

Fig. 1. Diagram of the spring loaded inverted pendulum.

and a double support phase, while the running behavior al-ternates between a single support phase and a flight phase.In this paper, we consider only the running behavior for sim-plicity. A diagram of the model is given in Fig. 1. Theprimary novelty of this section is using energy regulationbased methods to achieve dynamic gait transitions in theSLIP model.

4.1 SLIP Dynamics and ControlIn general, the energy of the system is

E = K (.q)+P (q) (40)

=12

m(.x2 +

.y2)+12

k(L(x,y)−L0)2 +mgy (41)

The configuration vector of the model is q= [x,y]>, the pointmass is m, and the length and stiffness of the springs are Land k, respectively. The gravitational acceleration constant isg, and is along the vertical coordinate y. At the relaxed springlength L= L0 the system releases the spring from the ground,and it engages the spring again at the contact angle α wheny = Lsin(α). The rules for releasing and engaging the springensure that the spring energy is zero at phase transitions, thusthe energy of the open-loop system is conserved across allregimes. The switching rules and the EL equation lead to thefollowing equations of motion:

Stance (ST) Ju = M ..q+G (42)

Flight (FL) 0 = M ..q+[

0mg

](43)

if phase == FL and y≤ L0 sin(α) (44)phase := ST

if phase == ST and L≥ L0 (45)phase := FL

The matrix J maps the control input u (collinear with thespring forces) into the stance coordinates. The system is un-deractuated with degree 1 during the single support phase

and has no continuous actuation during the flight phase,though we claim control authority over the touchdown con-tact angle α.

In [21], it is shown that for a constant spring stiffness kand touchdown angle α, there is a compact set of energiesthat correspond to walking or running periodic orbits in theSLIP model. They also show that these energy sets exist andchange for a range of stiffnesses and touchdown angles. Thisis our motivation to use energy shaping to change the springstiffness. Because the energy of the open-loop system is con-served, any periodic orbit is only marginally stable. Thus, itis reasonable to use energy regulation to stabilize the orbit.However, because the SLIP model has a 4 dimensional statespace instead of the 2 dimensional space of the hopper, a sin-gle energy value does not uniquely define a trajectory of theopen-loop system. Additionally, work in both [21] and [34]indicates that controlling the contact angle is critical to thestability of this model. Inspired by [34], we use the policy

αi+1 =12(αi +π−θi) (46)

to update the contact angle, where θi is the take off angle.This policy reaches an equilibrium when the touchdown andliftoff positions are symmetric about the y axis.

The control is partitioned into u = us +ur. The stiffnesschange is accomplished with us = (k− k)(L−Lo). Using thestorage function of equation (9), the time derivative under theSLIP dynamics is

.S = (E−Eref)

.q>Jur (47)

which means we should choose

ur =−κ.L(E−Eref) (48)

to ensure that.S is negative semi-definite.

In section 3.3 we mentioned that the energy regulationcontrol caused the closed-loop hopper system to take theform of a harmonic Rayleigh-Van-der-Pol oscillator. By in-serting these closed loop dynamics into the SLIP model, wecan examine the effect of the “unmodeled” rotational dynam-ics around the spring contact point. The motivation is thatthis will be suggestive of qualitative behavior of embeddingthis energy regulated SLIP model into higher order bipedmodels as in [15] without aggressive compensation of dy-namics transverse to the spring action. The recycling of the1-DOF hopper controller for the SLIP model is accomplishedby using only the energy of the SLIP model along the springaxis as

ur =−κ.L(

12

m.L2 +

12

k(L−L0)2−Eref). (49)

The potential benefit of controller (49) over (48) is that itrequires less state information, but with the drawback thatthe limit cycle trajectory will certainly not emulate a passivesystem.


4.2 SLIP SimulationsThis section offers simulation results that demonstrate

the ability to use energy shaping regulation methods toachieve different running speeds on the SLIP model. Thecontrol in equation (48) is termed the regulation controlwhile equation (49) is termed the oscillator control. Westarted with an initial known running gait from [34], withm = 70 kg, Lo = 1 m, α = 55, k = k = 8200 N

m , and E =1860J (marked as a red dot in Fig. 2-6). For all cases, κ = 1.For the oscillator, we heuristically found a reference energyvalue E = 583J that resulted in an average speed similar tothe known gait (≈ 6 m

s ). We created a grid of target energiesand stiffnesses around this configuration, used the flight apexof the known gait as the initial condition for every grid point,and allowed the system to converge to a new limit cycle. Thismeans that every stable grid point is a stable transition fromthe initial limit cycle to a new limit cycle due to a singlechange in reference energy and/or virtual stiffness. The exis-tence and stability of the limit cycle were confirmed via thenumerical linearization of the Poincare return map via themethod from [2].

The results for the stable average running speed of themodel under the energy shaping and regulation control andthe embedded harmonic Rayleigh-Van-der-Pol oscillator aregiven in Fig. 2 and 4. The edge of each surface indicatesthe edge of the sampling grid, or a case where the model fellthrough the floor or went backwards. For the regulation con-trol in Fig. 2, the speed level set projections indicate thatstiffness does not determine the average speed which agreeswith the results of [21] that show a range of walking speedsfor a given spring stiffness. However, the oscillator controlcauses a qualitative change in this relationship so that the av-erage speed does depend on stiffness and reference energyas seen in Fig. 4. The two methods give a similar range ofachievable walking speeds. We emphasize again that thesenew gaits are all the result of stable transitions from the pas-sive known gait. A sample trajectory for the energy regula-tion controller is given in Fig. 3, where the known gait is runfor 3 steps then the parameters are switched and the trajec-tory and contact angle converge to a new gait.

Plots of the equilibrium contact angle as a function ofstiffness and energy are given in Fig. 5 and 6 for direct com-parison to the results in [21]. Both methods have a simi-lar range of contact angles; the difference between them islargely that the embedded oscillator surface in Fig. 6 seemsto be flatter than the surface in Fig. 5. This indicates thatthere might be some constant normal vector in this space as-sociated with stability for the system under the embeddedoscillator.

The most important take away from these simulation re-sults is that a single change in parameters using energy shap-ing and regulation can achieve a stable transition betweenfast and slow running. Also, the principles of energy shap-ing and regulation control can be applied to a subcomponentof the system to achieve qualitatively similar behavior. Weexpect that this could be extremely useful in the applicationof these methods to wearable devices to assist locomotion,like a powered prosthesis [35] or orthosis [36], where mea-

Fig. 2. Speed-Energy-Stiffness surface for the SLIP model underthe regulation control. The minimum speed achieved was 2.82 m

s ,the maximum 7.95 m

s .

0 5 10 150.75

0.8

0.85

0.9

0.95

Fig. 3. Sample trajectory of transition from 6 ms to 3 m

s under theregulation control. The stiffness and reference energy were switchedat the horizontal dashed line.

Fig. 4. Speed-Energy-Stiffness surface for the SLIP model underthe oscillator control. The minimum speed achieved was 3.65 m

s , themaximum 8.05 m

s .

suring the total energy of the combined human-robot systemis infeasible.


Fig. 5. Contact Angle-Energy-Stiffness surface for the SLIP modelunder regulation control.

Fig. 6. Contact Angle-Energy-Stiffness surface for the SLIP modelunder the oscillator control.

5 Compass Gait Biped

The compass gait biped can walk down a shallow slopeunder the power of gravity alone by reaching an energy equi-librium between the potential energy gained and the kineticenergy lost at each impact [17]. Much of the research in thecontrol of dynamic biped locomotion uses this model as atestbed, to the point some have titled it “The Simplest Walk-ing Model” [22]. Similar to the SLIP model, different walk-ing speeds emerge for a given set of system parameters [17],which motivates the application of the energy shaping andregulation control methods. A diagram of the model is givenin Fig. 7.

5.1 Compass Gait Biped Dynamics and Control

The biped undergoes an instantaneous rigid impactwhen the swing leg hits the ground. The impact modelfrom [2] and the EL equation gives the following hybrid dy-

Fig. 7. Diagram of the compass gait biped. The legs are symmetric.

namics for the compass gait biped:

M(q) ..q+C(q, .q) .q+G(q) = u (50)

[ .q+

FI

]= R

[M −A>

A 02x2

]−1 [ M02x2

].q− if h(q,Ψ)≤ 0

q+ = Rq−. (51)

The continuous dynamics are described by (41), where q =[θ1,θ2]

>, M is the mass matrix, C is the corriolis/centrifugalmatrix, G = ∂P (q)

∂q is the gravity vector, and u is the vector ofcontrol torques applied at the joints. The discrete dynamicsare described by (51). They have a plastic impact map thatdepends on M, a matrix A that describes the constraint ofthe impact foot to the ground, and a relabeling matrix R thatswaps the swing and stance legs. The impact maps a pre-impact velocity vector .q− to a post-impact velocity .q+ andan impact force FI . The vertical distance from the swing legto the ground is h. For additional modeling details, see [13]and [17].

Using the framework from Section 2.2, the energy shap-ing control is

us =−MM−1(C .q+ G)+C .q+G (52)

while the energy regulation control is

ut =−κ(E−Eref)MM−1Ω

.q (53)

The closed-loop dynamics are then

M ..q+(C+κ(E−Eref)Ω).q+ G = 0. (54)

The parameters available for shaping in the energy func-tion of the system are: mh, ms, a, b and g. The question then


becomes: how should we choose these values? In [37], theeffect of gravity shaping on the compass gait biped is thor-oughly explored, indicating that average walking speed isproportional to

√g. In [17], it is shown that the dynam-

ics can be normalized to depend on the mass ratio µ = mhms

and the length ratio β = ba , which both influence the average

speed. Because the impact dynamics depend only on M, wecan exactly emulate a target g but we cannot do the same forβ and µ. This means we could exactly reproduce the resultsfrom [37] in the closed-loop hybrid dynamics, but we can-not change µ and β to reproduce similar results from [17].The primary novelty of this section is the exploration of therelationship between the walking speed and a virtual changein µ and β during the continuous dynamics while using thenatural values in the impact dynamics. In addition, we showthat energy regulation can be used to stabilize period-1 gaitsthat otherwise would go unstable and bifurcate as mass pa-rameters vary [17].

In the hopper with plastic impacts, we were able to an-alytically compute the Eref for a given passive limit cycleusing equation (29). In general, it is not possible to knowthe energy of a passive biped limit cycle without simulatingit numerically. This poses a challenge to dynamically chang-ing the virtual parameters and reference energy to achievenew passive limit cycles without making a library of pre-computed gaits. In [26], a discrete step-by-step update lawfor Eref in an energy regulation control is proposed, as

Erefi+1 = Erefi +λ(vref− vi). (55)

This law achieves a desired average walking speed vref,where λ is a scaling gain and vi is the average walking speedfor the ith step. However this law simply shifts the problemto picking the vref associated with a natural limit cycle beforehand, instead of Eref. Instead, we can reuse the update lawin equation (37) from the 1-DOF hopper and apply it to thecompass gait biped. A comparison between these strategieson a simulated biped is given in the following section. Weuse unat to denote the control under equation (37) and uvel

for equation (55).

5.2 Compass Gait Biped SimulationsWe now present simulation results for the compass gait

biped under the energy shaping and regulation controls. Thebaseline parameters we use are mh = 10kg,ms = 5kg,a =0.5m,b = 0.5m,Ψ = 3.7 with a known initial condition fora stable limit cycle from [17]. We compare changes in thetrue ratios β against changes in the virtual ratios β during thecontinuous dynamics. We omit results for µ and µ for brevity.The energy regulation controls uvel

t and unatt are applied to

the system with the virtual ratios, using the control parame-ters λ = 0.5, κ = 100, and Ω = diag([1,0]). Each referencevelocity vref for a given ratio is taken from the correspond-ing physical ratio limit cycle. The ratios that we sample areon a uniform grid from 0.5 to 1, and we do not display datapoints in the grid range that either bifurcated or were unsta-

0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

Fig. 8. Length ratio versus average speed for physical and virtual

dynamics. β is the physical length ratio, β is the virtual length ratio,uvel

t converges to a desired walking speed, unatt converges to the

energy equilibrium induced by the discrete dynamics. Data pointsthat bifurcated or were unstable are not displayed.

0.5 0.6 0.7 0.8 0.9 1145

150

155

Fig. 9. Energy versus length ratio for the adaptive energy regula-tion controllers. Data points that bifurcated or were unstable are notdisplayed.

ble within our search tolerance. We confirm the stability ofthe limit cycles using the linearized Poincare return map.

The results are shown in Figs. 8, 9, and 10. We cansee that shaping the continuous dynamics alone through β

does not reproduce the same limit cycle as physically chang-ing the parameters. In Fig. 8, the virtual length ratio β hasthe same general trend between ratio and speed as β, but itcauses a larger increase in walking speed. The introductionof unat

t enables Eref to converge to Enat, as evidenced wherethe circle and cross data points overlap in the figures. It alsoincreases the range of achievable speeds, as seen by the circledata points that do not overlap the cross points. The velocityupdate law uvel

t causes the shaped system to converge to thetargeted walking speed of the associated physical system, asseen by the overlap of the squares and stars. Fig. 9 showsthe energies that the update laws converge to, indicating thereal parameters shift energy down more than the virtual ones.In Fig. 10, we can see that uvel

t causes limit cycles that areless efficient compared to those from unat

t , in the sense thatthey require more torque output from the control to achievethe same walking speed. These inefficient cycles are due tothe fact that they are unnatural and must compensate for theenergy mismatch between Eref and Enat.

Finally, we offer a simulation example of using energyregulation to stabilize an unstable limit cycle. In this case,the terrain is changed from a slope to stairs of a similar ge-


0.7 0.8 0.9 1 1.1 1.20

2000

4000

6000

8000

Fig. 10. Time integral of torque squared versus average speed forphysical and virtual dynamics length ratio changes. Data points thatbifurcated or were unstable are not displayed.

-40 -20 0 20 40-100

-50

0

50

100

150

200

Fig. 11. The slope period-1 passive limit cycle versus the stairsperiod-2 passive limit cycle.

ometry. Thus, the impact map ∆(q, .q) remains the same butthe switching surface S and distance function h are changed.The stair impact map still admits the energy equilibrium fromthe slope dynamics, however as seen in Fig. 11 this is not as-sociated with a stable period-1 limit cycle for the passive sys-tem. In Fig. 12, we present a stair walking simulation wherewe switch on the energy regulation control after 4 steps andrun it for 10 more steps. This causes the biped to convergeto the period 1 slope limit cycle while walking on the stairsterrain, indicating that we have stabilized this previously un-stable gait.

6 ConclusionIn this article we have explored some principles and ex-

tensions of energy shaping and regulation control for gen-erating and transitioning between limit cycles in simple lo-comotive systems. We analytically and numerically demon-strated that energy regulation can increase the basin of attrac-tion of the limit cycle so that parameter switches in an energyshaping control lead to stable gait transitions in a trajectory-free manner. Our contributions are: 1) the demonstration ofthe equivalence between the 2-step process of energy shap-ing then energy regulation, and regulation of time and statevarying energy functions, 2) a discrete update law for Eref toallow convergence to a passive limit cycle online, 3) transi-tioning between running gaits on the SLIP model, 4) a novelexamination of energy shaping on the compass gait biped, 5)

0 2 4 6 8 10-40

-20

0

20

40

Fig. 12. Transition from passive period-2 limit cycle to an energyregulated period-1 limit cycle. The energy regulation control is turnedon after 4 steps, at the green vertical line.

a theoretical explanation for using energy regulation to stabi-lize unstable limit cycles and demonstration on the compassgait biped. This paper also serves to connect and stream-line previous works in the areas of orbital stabilization andhybrid locomotive systems. The outcome of our work hereis a model with gaits based on continuously varying springstiffness and nonlinear damping in the form of energy reg-ulation and is similar in spirit to previous work on variableimpedance control for a powered prosthesis [38]. As such,we plan to utilize the theory and results in this work to shapethe dynamics of a powered prosthetic leg to mirror the SLIPmodel to aid locomotion with task variation like changingwalking speeds and slopes.

AcknowledgementsThis work was supported by NSF Award 1652514 /

1949869. Robert D. Gregg holds a Career Award at the Sci-entific Interface from the Burroughs Wellcome Fund.

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Using Energy Shaping and Regulation for Limit Cycle