3D Underactuated Bipedal Walking via H-LIP based Gait Synthesis and Stepping Stabilization
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3D Underactuated Bipedal Walking via H-LIPbased Gait Synthesis and Stepping Stabilization
Xiaobin Xiong and Aaron Ames
Abstract —In this paper, we present a Hybrid-Linear InvertedPendulum (H-LIP) based approach for synthesizing and stabi-lizing 3D underactuated bipedal walking. The H-LIP model isproposed to capture the essential components of the underactu-ated part and actuated part of the robotic walking. The walkinggait of the robot is then synthesized based on the H-LIP. Wecomprehensively characterize the periodic orbits of the H-LIPand provably derive their stepping stabilization. The step-to-step(S2S) dynamics of the H-LIP is then utilized to approximate theS2S dynamics of the horizontal state of the center of mass (COM)of the robotic walking, which results in a H-LIP based steppingcontroller to provide desired step sizes to stabilize the roboticwalking. By realizing the desired step sizes, the robot achievesdynamic and stable walking. The approach is evaluated in bothsimulation and experiment on the 3D underactuated bipedalrobot Cassie, which demonstrate dynamic walking behaviors withboth versatility and robustness.
Index Terms —Bipedal Walking, Underactuation, Hybrid-LIP,Stepping Stabilization, Step-to-step Dynamics
I. INTRODUCTIONUnderactuation is prominent in bipedal locomotion [7], [8].The robot moves under switching support legs and internaljoint actuation. The actuation at the foot contact is significantlylimited due to practical actuation limits and finite reactionmoments (due to finite support regions) [9], [10]; the foot actu-ation can also be missing due to the lack of motor actuation atthe foot in the robot design [7], [11] for agility and simplicity.Moreover, the constant switching of support legs rendersthe dynamics to be hybrid [8]: cycling between continuousdynamics and discrete transitions. These conditions differ fromthose of the manipulation of a robot arm [12] where the base isfixed and the dynamics is typically continuous. Thus, generallyspeaking, it is challenging to control locomotion behaviors onthe high dimensional underactuated bipedal robots.Various approaches have been proposed to generate stableunderactuated bipedal walking. The hybrid zero dynamics(HZD) approach [13], [8], born in the control community,plans attractive periodic orbits with virtual constraints in thefull-dimensional state-space of the robot via large-scale trajec-tory/parameter optimizations [14] or numerical methods [15].Feedback controllers [16], [13] are employed to enforce thevirtual constraints, and the stability of the generated walkingis typically determined by the analysis on the numerically
The original conference version of the paper was presented in IROS 2019[1], and some results were also presented in Dynamic Walking 2019 [2].The work is supported by Amazon Fellowship in Artificial Intelligence.The authors are with the Department of Mechanical and Civil Engineering,California Institute of Technology. Corresponding author: Xiaobin Xiong( [email protected] ). The videos of the results can be seen in https://youtu.be/qEp1RUf6X-U as well as in [3], [4], [5], [6].
H‐LIP
Fig. 1. The H-LIP based approach on generating underactuated bipedalwalking: (left) the periodic orbits of the H-LIP and (right) 3D walking onCassie. derived Poincaré map. Practical realizations, however, chal-lenge the theoretical soundness. The optimization is highlynon-convex and difficult to solve in general. Furthermore,the stability of optimized 3D walking gaits cannot be easilydetermined in the optimization. Lastly, even if the optimizedgait is stable, the walking on the robot can be unstable if therobot deviates from the model that is used in the optimization.Another widely-studied approach uses the Spring LoadedInverted Pendulum (SLIP) [17], [7] model for generating com-pliant legged locomotion behaviors. The SLIP model sparkedwide interests in the legged locomotion community since itwas found to successfully capture both walking and runningof biological systems [18]. The controllers on the SLIP canalso be derived either by intuition [7] or based on its returnmap [19]. Several bipedal robots [20], [21], [11], [22] havebeen designed and built to resemble the SLIP model. For thosetype of robots, the SLIP inspired controllers are thus utilizedto render the corresponding locomotion behaviors. However,the SLIP based approach cannot be easily used for non-SLIPlike underactuated bipedal robots in practice. Furthermore,transitions between periodic walking behaviors on both theSLIP and the robot is not well studied.In this paper, we present a walking synthesis and controlbased on a model simplification, which focuses on the switchof support legs and neglects foot actuation. The simplifiedmodel is a variant of the Linear Inverted Pendulum (LIP)[23] with passive pivot contact and hybrid domain structure.Thus, we name it the Hybrid-LIP (H-LIP) [1]. The H-LIP ispassive in the continuous domains of walking, and the only"actuation" that changes the walking behavior is the step size.By formulating the dynamics at the step level, the step size a r X i v : . [ c s . R O ] F e b becomes the input to its linear step-to-step (S2S) dynamics.The H-LIP approximates hybrid walking of an underactu-ated bipedal robot assuming that the center of mass (COM)is approximately constant and the swing foot periodicallylifts off and strikes the ground. Then, the S2S of the H-LIP approximates the S2S of the walking of the robot. Bytreating the model difference as a disturbance to the linearS2S, state-feedback stepping controllers (i.e., H-LIP stepping [1], [24]) can be synthesized to control the walking of thehorizontal COM state of the robot at the pre-impact event; thedifference of the horizontal states between the robot and theH-LIP converges to disturbance invariant sets.To implement the H-LIP based approach on the 3D robot,we first realize desired walking behaviors on the H-LIP,by characterizing its periodic orbits and synthesizing theirstabilization. The desired H-LIP walking is then used in thestepping controller to find desired step sizes on the robot torealize desired walking behaviors. We realize our approach onthe 3D underactuated bipedal robot Cassie, shown in Fig. 1.The desired walking trajectories are constructed based on theH-LIP and its stepping controller and then stabilized via joint-level controllers. Various walking behaviors are thus realizedon the robot in both simulation and experiments.
A. Contributions
The main contributions of this paper are: • A low-dimensional model (H-LIP) with its compre-hensive orbit characterizations to approximate under-actuated bipedal walking.
We take the underactuatedversion of the canonical LIP and apply it to approximate the underactuated bipedal walking. We provably and geo-metrically characterize all the Period-1 (P1) and Period-2(P2) orbits of the H-LIP in its state space. • A versatile gait synthesis with stepping stabilizationbased on the H-LIP for realizing 3D underactuatedbipedal walking.
We present a walking synthesis methodthat directly maps the features of the H-LIP walking to therobotic walking. We also derive stepping stabilization ofthe robotic walking to the desired walking of the H-LIP;the stepping stabilization is based on the approximationof the S2S of the robot via the S2S of the H-LIP. • A computationally-efficient and robust realization onthe physical hardware of the complex 3D underac-tuated bipedal robot Cassie with passive compliance.
We realize the H-LIP based approach on the robot Cassiein both simulation and experiment. In the experiment, wepresent computationally-efficient and rigorous implemen-tations to solve the practical problems including contactdetection and COM velocity approximations, which areshown to be effective and robust to uncertainties of thehardware system and external disturbances.This paper builds on our previous conference paper [1], inwhich we formally presented the H-LIP, its orbit characteri-zation, and stepping stabilization. In [1], we also applied thestepping controller on an actuated SLIP (aSLIP) model and theunderactuated bipedal robot Cassie in simulation. This paperextends the previous results in the following aspects. First, the equivalent characterizations of the P1 and P2 orbits arepresented in this paper to complete the orbit identification.Second, the orbit stabilization is elegantly synthesized viathe step-to-step (S2S) dynamics of the H-LIP. We verify thatthe optimal stepping controllers in [1] are actually deadbeatcontrollers. Third, the gait synthesis and stabilization via theH-LIP on underactuated bipedal walking are more formallypresented here. Last but not least, we validate the approachon Cassie in experiments. The technical components of thehardware realization are detailed in this paper.The H-LIP based stepping has also been realized on theaSLIP model in [1] to show its application to a differentdynamical system of walking, and later [24], [25] extendedthese results to 3D and walking on rough terrains. Thus, wedo not present the aSLIP walking in this paper and mainlyfocus on the validation on the complex 3D robot Cassie.
B. Related Work
The H-LIP is a variant of the canonical LIP model [26] withfoot underactuation and hybrid dynamics structure. The LIPhas been extensively applied in the Zero Moment Point (ZMP)approach [9], [27] for realizing humanoid walking. The LIP iscontinuously actuated; the H-LIP is only discretely actuated byswapping support legs. One can view the ZMP-LIP approachesas using the ZMP of the LIP to approximate the ZMP ofthe robot with the LIP dynamics directly embedded on thehumanoid. Instead, in this paper, we use the H-LIP dynamicsto approximate the horizontal COM dynamics of underactu-ated bipedal robots, which do not have foot actuation and thuscan not strictly embed the pendulum dynamics. Additionally,compared to the LIP with foot-placement controllers [28],[27], [29] on humanoid walking, this approach focuses on theperiodic walking and stabilization on underactuated bipedalrobots.Compared to the periodic walking realized via HZD [8],[30], [31], the periodic orbits of the H-LIP are directlycontrolled via the step size on the S2S dynamics. Thus,the stability of the orbits and their transitions are solelydetermined by the stepping controller. Additionally, the robotis not necessarily controlled to evolve on a strict orbit in itsonly state-space. Instead, it converges closely to the walkingbehavior of the H-LIP. The H-LIP walking is pre-determinedbut the walking of the robot is not.The notation of the S2S in legged locomotion is an adap-tation of the Poincaré return map in nonlinear dynamics [32].The S2S has mostly appeared in controlling SLIP running[33], [34]. By investigating the evolution of the apex states,the S2S/return map of running can be easily obtained on theSLIP. Feedback controllers thus can be synthesized based onthe S2S to stabilize the running of the SLIP. However, theS2S of the walking of the SLIP (bipedal SLIP) has not beenshown to be obtained easily, possibly due to the complexityof the inclusion of the DSP dynamics. Similarly, the S2S of a3D bipedal walking robot cannot be obtained easily. By andlarge, the control based on the return map of walking hasbeen focused on the linearization at the fixpoint [35], [36],[37], [38] of a periodic solution on the return map (very few exceptions [33], [39] learned the S2S). This paper, instead,approximates the S2S of the robotic walking over a largeregion of the state-space at the Poincaré section. Additionally,the S2S approximation is linear and readily facilitates periodicwalking gaits to be characterized and feedback controllers tobe synthesized.
C. Paper Outline
We start by presenting some preliminaries of the underactu-ated bipedal walking in section II. Then section III introducesthe H-LIP and its gait synthesis and stepping stabilization.In section IV, we describe the orbit characterization andstabilization on the H-LIP. After that, we apply the H-LIPbased approach on the robot Cassie in section V, and presentthe results in simulation in section VI and the realization on thehardware in section VII. Finally, section VIII presents somediscussions, and section IX concludes this paper. Additionally,the simulated walking and experiment videos are listed inreferences as well as in the supplementary materials.II. P
RELIMINARY
In this section, we briefly introduce the hybrid model ofbipedal walking and underactuation of bipedal robots. Thosespecific characteristics provide correspondence to the lowdimensional model of walking at the high-level.
A. Hybrid Dynamics Model of Bipedal Walking
The dynamics of bipedal walking can be described as ahybrid dynamical system [8] with continuous dynamics indifferent domains and in-between discrete transitions. Thecontinuous dynamics are affine control systems: ˙ x = f v ( x ) + g v ( x ) τ, (1)where x is the system state, τ is the vector of input torques,and the subscript v is the domain index. The discrete transi-tions between the consecutive domains can be described by: x + = ∆ v → v +1 ( x − ) , (2)where the superscripts − and + stand for the instants beforeand after the transition, respectively.The hybrid dynamics of bipedal walking are either com-posed of a single domain: single support phase (SSP) or by twodomains: a SSP and a double support phase (DSP). We referto the two as one-domain walking and two-domain walking. Fig. 2. The hybrid graphs of one-domain walking and two-domain walking. Fig. 3. The foot-underactuation of bipedal walking with the illustration byan inverted pendulum.
For one-domain walking, the transition ∆ SSP − → SSP + happensat the impact when the swing foot strikes the ground. Theimpact is modeled as plastic impact [8] where the velocityof the swing foot becomes zero after the impact, and thusthe state undergoes a discrete jump. As for the two-domainwalking, the transition ∆ SSP → DSP is also the impact event, andthe transition ∆ DSP → SSP is when one of the stance feet lifts offfrom the ground. The existence of the DSP typically happenswhen there is compliance in the leg, which prevents the stanceleg from instantaneously lifting off at the impact.The one-domain walking can be viewed as a special case ofa two-domain walking with an instantaneous DSP. The two-domain walking is then chosen as the general model that westudy for walking.
B. Foot Underactuation on Bipedal Robots
Underactuation on bipedal robots typically happens at thefoot-ground contact. The foot may not be designed withactuation to rotate (for reducing the leg inertia), or the footcontacts the ground with its edges. Both can be viewed asforms of underactuation since actuation is missing at thecontact location. This is in contrast to fully-actuated humanoidwalking, where the foot rotation typically remains actuatedfrom the control synthesis.The foot underactuation prevents the direct continuouscontrol on the center of mass (COM) of the robot to desiredtrajectories in the horizontal plane. A simple illustration is thatan inverted pendulum would roll passively without any actua-tion at the contact with the ground (see Fig. 3). Theoreticallyspeaking, the robot has rotational linkages, which can generaterotational momentum and thus indirectly affect the COM [28],[10], [40], e.g., the angle of a flywheel-inverted pendulumcan be controlled via the continuous rotation of the flywheel[41]. However, the joints on robots typically have limitedranges, control bandwidth, and torques in practice. Thus, itis not possible to purely depend on the angular momentumto continuously control the COM. Therefore, the horizontalCOM state of the underactuated robot is equivalently theunderactuated ("weakly actuated" [42]) states in practice.
Remark:
In this section, we mainly deliver two messages:the hybrid model with two domains is the general one todescribe walking; the horizontal COM state can be treatedas the underactuated state that highly relates to the footunderactuation. As a result, the reduced-order model in the
Fig. 4. The walking of the Hybrid-LIP model in SSP (a), at the preimpactstate (b), and during DSP (c). paper is proposed with these two aspects to approximate theunderactuated bipedal walking.III. H
YBRID L INEAR I NVERTED P ENDULUM M ODEL
In this section, we first present the model of the H-LIP,including the hybrid dynamics and the step-to-step (S2S)dynamics. We then present the H-LIP based gait synthesis andstepping stabilization for robotic bipedal walking.
A. Walking Dynamics of the H-LIP1) The Hybrid Dynamics:
The H-LIP is a point-massmodel with a constant center of mass (COM) height and twotelescopic legs with point-feet (see Fig. 4). The point-feetcorrespond to the underactuated feet of bipedal robots. Basedon the number of contacts with the ground, the walking iscomposed by a Single Support Phase (SSP) and a DoubleSupport Phase (DSP). In the SSP, the model is a passiveLIP with no actuation; in the DSP, we assume that the massvelocity is constant. The state of the system is composed ofthe position p and the velocity v of the mass. p is defined asthe position of the mass relative to its stance foot. In the DSP,the stance foot as the previous stance foot in the SSP. Thus,the dynamics are: ¨ p = λ p, (SSP) ¨ p = 0 , (DSP)where λ = (cid:113) gz and z is the height of the H-LIP. We assumethat the domain durations ( T SSP and T DSP ) are constant. Sincethe H-LIP is a point-mass model, the swing foot behaviorof lift-off and touch-down is not explicitly described. Thetransitions between domains are assumed to be smooth: ∆ SSP → DSP : (cid:26) v + = v − p + = p − ∆ DSP → SSP : (cid:26) v + = v − p + = p − − u where u is the step size, and the + / − indicate the states afterand before the transition, respectively. Since the dynamics arelinear and the transitions are in closed-form, the solutions are:SSP : (cid:26) p ( t ) = c e λt + c e − λt v ( t ) = λ ( c e λt − c e − λt ) (3)DSP : (cid:26) p ( t ) = p − SSP + v − SSP tv ( t ) = v − SSP (4)where c = ( p + SSP + λ v + SSP ) and c = ( p + SSP − λ v + SSP ) .
3D H-LIP:
The H-LIP is a planar model. Similar to LIP, theH-LIP can be presented in the 3-dimensional space. Since its dynamics are completely decoupled in each plane, a H-LIP in3D is equivalent to two orthogonally-coupled planar H-LIPs.
Equivalence to a One-Domain System:
The hybrid dynamicsof the H-LIP with two domains can be equivalently simplifiedto a single-domain hybrid system. This will simplify thedescriptions of periodic orbits. Since the closed-form solutionof the DSP is known, we virtually treat the DSP and itsassociated transitions as a single transition from the final stateof the SSP to the initial state of the next SSP. Thus, thetransition is defined as: ∆ SSP − → SSP + : (cid:26) v + = v − p + = p − + v − T DSP − u. (5)As a result, we have a hybrid dynamical system with acontinuous SSP dynamics and a virtual discrete transition.When T DSP = 0 , the dynamics becomes an actual one-domainsystem with only SSP, which is the passive LIP (LIP withpoint foot) model in the literature [43], [44], [45], [46].
Correspondence to robotic walking:
The assumptions on theH-LIP are designed to approximate the horizontal COM dy-namics on the underactuated bipedal robot. The contact isunactuated to match the foot-underactuation. We include theDSP in the model to make it general to represent both one-domain walking and two-domain walking on the robot. Theassumption of the constant COM height is to simplify thedynamics, which will be enforced on the robot.
2) The Step-to-step Dynamics:
The dynamics of the H-LIPare piecewise linear. As the durations are constant, the pre-impact states at consecutive steps can be related in closed-form. The state-space representation of the SSP dynamics is: ddt (cid:20) pv (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ˙ x SSP = (cid:20) λ (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) A SSP (cid:20) pv (cid:21)(cid:124)(cid:123)(cid:122)(cid:125) x SSP . (6)Thus, the final state of the SSP is calculated from the initialstate of the SSP: x − SSP
H-LIP = e A SSP T SSP x + SSP
H-LIP . (7)The transition in Eq. (5) can be written as: x + SSP k +1 = (cid:20) T DSP (cid:21) x − SSP k + (cid:20) − (cid:21) u k , (8)where k is the step index. Plugging Eq. (8) into Eq. (7) yields, x − SSP k +1 = e A SSP T SSP (cid:20) T DSP (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) A x − SSP k + e A SSP T SSP (cid:20) − (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) B u k . From now on, we treat the final state of the SSP as the discretestate of the hybrid dynamics of the H-LIP. Thus we drop somesubscripts and superscripts and rewrite the above equation as: x k +1 = A x k + Bu, (9)which is referred to as the step-to-step (S2S) dynamics of theH-LIP. The S2S is a discrete linear time-invariant system withthe step size being the input. 𝑢 (cid:3038) 𝑢 (cid:3038)(cid:2878)(cid:2869) 𝐱 (cid:3038) 𝐱 (cid:3038)(cid:2878)(cid:2869) S2S of H‐LIP
DSPSSP
H‐LIP
Gait Synthesis
Step Size
Timed verticalswing foot traj. vertical COM
Features
Stepping Stabilization
Approx.
Planar S2S
Sagittal Coronal 𝑢 (cid:3038) 𝑘 𝑘 (cid:3397) 1𝑢 (cid:3038)(cid:2878)(cid:2869) 𝑢 (cid:3038)(cid:2878)(cid:2869) 𝑢 (cid:3038) 𝑘 𝑘 (cid:3397) 1 Fig. 5. Illustration of the H-LIP based gait synthesis and stepping stabilization on the robot.
B. H-LIP based Gait Synthesis
Now we present the H-LIP based walking synthesis for 3Dunderactuated bipedal walking (Fig. 5). The H-LIP is set toapproximate the underactuated walking, and the walking itselfshould be specified to best match the H-LIP to reduce theapproximation error.Vertical Height of the COM: The vertical height of the COM z COM should remain approximately constant during walking.When it is possible, one can strictly enforce z COM to beconstant. For underactuated robots with passive compliance inthe leg (e.g. Cassie), strictly enforcing this condition is chal-lenging; hence, we only make sure that z COM is approximatelyconstant on the robot.Vertical Trajectory of the Swing Foot: The second componentis on the synthesis of the vertical trajectory of the swing foot z sw . As the step frequency on the H-LIP is assumed to beconstant, the swing foot on the robot is expected to periodicallylift-off and strike the ground with the same frequency. Thiscreates continuing hybrid execution on the dynamical systemand makes sure that the S2S dynamics of the robot exists. Asa result, z sw should evolve on a time-based trajectory, whichcreates the lift-off and touch-down behaviors based on time.Horizontal Trajectory of the Swing Foot: As the step size isthe control input on the H-LIP, the horizontal trajectory of theswing foot should be constructed to achieve a certain desiredstep size on the robot. Since the impact is time-based, thehorizontal trajectory of the swing foot is constructed to swingto the desired step location at the time of impact.3D Walking Decomposition: As the H-LIP is a planar model,the application to 3D robotic walking requires an orthogonalcomposition of two H-LIPs. We select the sagittal plane andcoronal plane of the robot as the decomposition of the roboticwalking. The horizontal COM state, swing foot position, andthe step size of the robot are decoupled into those in thesagittal and coronal plane, respectively. C. Stepping Stabilization via S2S Dynamics Approximation
We next present the H-LIP based stepping to generatethe desired step size on the robot for achieving the desiredwalking. The stepping in the sagittal plane is used as anexample; it is applied identically to the coronal plane. As therobot is controlled to periodically lift-off and touch-down thefoot, the hybrid dynamics of walking repeats a walking cycle. In other words, the pre-impact state exists, despite the numberof domains in the hybrid walking.Let { q − , ˙ q − } be the pre-impact state of the robot. Theevolution of the pre-impact states at the step level, i.e., thestep-to-step (S2S) dynamics of the robot, can be representedby: { q − k +1 , ˙ q − k +1 } = P ( q − k , ˙ q − k , τ ( t )) , (10)where k is the index of the step, and τ ( t ) represents the torqueswhich are applied during the step k . Each step starts with aDSP (if exists) and a following SSP. Let x R = [ p R , v R ] T be the horizontal COM state of the preimpact state of therobot in the sagittal plane. p R , v R are the horizontal positionand velocity of the COM of the robot, which are functions ofthe preimpact state { q − , ˙ q − } . Thus the S2S dynamics of thehorizontal COM state can be represented by: x Rk +1 = P x ( q − k , ˙ q − k , τ ( t )) . (11)In the latter, we directly refer Eq. (11) as the S2S of the robot.Due to the nonlinear dynamics of the robot, the exactexpression of the S2S dynamics can not be computed inclosed-form. Thus, synthesizing the controller based on theS2S dynamics is difficult in general. Since we design the gaitof the robot based on the H-LIP, the S2S of the robot shouldbe close to the S2S of the H-LIP. Therefore, we use the S2Sof the H-LIP in Eq. (9) to approximate the S2S of the robot.Eq. (11) can be rewritten as: x Rk +1 = A x Rk + Bu Rk + w (12) w := P x ( q − k , ˙ q − k , τ ( t )) − A x Rk − Bu Rk . (13)where u Rk is the step size of the robot, and w is the differenceof the S2S dynamics between the robot and the H-LIP. w is also the integration of the difference of the continuousdynamics over one step between the two systems. As the gaitof the robot is designed to match the walking of the H-LIP,the dynamics error should be small. Each step also happensin a finite time (determined by the vertical trajectory of theswing foot), thus the realizable walking velocity is bounded.Therefore, w , the integration of the continuous error dynamicsover a finite time, is assumed to belong to a bounded set, i.e., w ∈ W . w is treated as the disturbance to the linear system.Thus, on the robot, we apply the H-LIP based stepping : u R = u H-LIP + K ( x Rk − x H-LIP k ) (14) where u H-LIP is the step size on the H-LIP to realize a certainbehavior, and K is the feedback gain to make A + BK stable,i.e., eig ( A + BK ) < . Let e = x R − x H-LIP be the error state .Applying H-LIP based stepping yields the error dynamics : e k +1 = ( A + BK ) e k + w. (15)Since A + BK is stable, the error dynamics has a minimumdisturbance invariant set E . By definition, ( A + BK ) E ⊕ W ∈ E. (16)where ⊕ is the Minkowski sum. We call E the error invariantset, i.e., if e k ∈ E then e k +1 ∈ E . If W is small, then E issmall. Thus, the desired walking behavior (of the horizontalCOM state) can be first realized on the H-LIP, and thenapplying the H-LIP based stepping yields the behavior tobe approximately realized on the robot, with the error beingbounded by E . Note that the feasible step size on the robot isbounded ( u R ∈ U ). The desired behavior of the H-LIP thenshould satisfy u H-LIP ∈ U (cid:9) KE . In the latter, if possible, thesuperscripts R and H-LIP are omitted for conciseness.IV. O
RBIT C HARACTERIZATION , C
OMPOSITION AND S TABILIZATION ON THE
H-LIPIn this section, we describe the identification of the desiredperiodic walking behaviors on the H-LIP and their stabiliza-tion. We will briefly present the resulting theorems and leavethe proofs in the Appendix.
A. Orbit Characterization
The periodic orbits of the H-LIP that encode walking can begeometrically characterized in its state space. We categorizethe orbits of interest into two types, Period-1 (P1) and Period-2 (P2) orbits, depending on the number of steps that the orbitcontains. P1 orbits have a period of one step, and P2 orbitshave a period of two steps. There are also P N ( N > ) orbits,and we do not investigate them in this paper.The H-LIP is a two-dimensional system, thus we can presentthe periodic orbits explicitly in its state space with its phaseportraits. For the H-LIP in SSP, its phase portraits are identicalto that of the canonical passive LIP (Fig. 6 (a)). It is dividedinto four regions by the cross lines v = ± λp , based onthe orbital energy [26]: E o ( p, v ) = v − λ p . The physicalmeaning of E o > is that the H-LIP rotates over the stancefoot, i.e., the system passes through the states where p = 0 .In DSP, the phase portrait is simple, shown in Fig. 6 (b). Fig. 6. Phase portraits of the H-LIP walking in its (a) SSP and (b) DSP. 𝑝 (cid:4666)m(cid:4667) 𝑣 (cid:4666) m / s (cid:4667) Fig. 7. The Period-1 orbits illustrated in the phase portrait. The yellow crosslines are the orbital lines of P1 orbits. The red, blue and gray lines are walkingorbits with v − = 1 , . , − . m/s. The dashed lines indicate the transitions inEq. (5). The right-side sub-figures are the illustrations of the periodic walkingof each orbit. For conciseness, we use the equivalent one-domain sys-tem in Section III-A of the H-LIP. Then the orbits can berepresented only with a continuous trajectory in the SSPand a discrete transition. In the following, we present thegeometric characterization of P1 and P2 orbits in the phaseportrait of the SSP. The subscripts of
SSP on the states areomitted. Additionally, the pre-impact states and the step sizesof the orbits are presented explicitly from the desired walkingvelocity.
1) Period-1 Orbits:
We start with the geometriccharacterization of the P1 orbits. The velocity is thesame between the start and the end of the SSP of the P1orbits, i.e., v + SSP = v − SSP . Since the phase portrait is left-rightsymmetric, the orbits are left-right symmetric as well. Byinspection, all P1 orbits should only exist in the E o > regions and pass the vertical line of p = 0 . The comprehensivecharacterization is stated as follows: Theorem 1.
The initial and final states of the P1 orbits inSSP are on the orbital lines v = ± σ p , with v − p − = σ := λ coth ( T SSP λ ) , (17)being the orbital slope . Each state on v = σ p represents thefinal state of the SSP of a unique P1 orbit with the step size: u = 2 p − + T DSP v − . (18)Here, we defined the orbital lines and orbital slopes to locatethe boundary states of the orbits. Additionally, given a desirednet velocity v d , there is a unique P1 orbit for realization. It isobvious that the step size of the P1 orbit is: u ∗ = v d ( T DSP + T SSP ) := v d T. (19)with T being the entire step duration. Then the final states ofSSP of the P1 orbit are calculated from Eq. (17) and Eq. (18): [ p ∗ , v ∗ ] = [1 , σ ] v d T T DSP σ . (20)Fig. 7 shows three P1 orbits in the phase portrait of theSSP to illustrate the characterization of the P1 orbits with T SSP = 0 . s. A different T SSP would produce a different set 𝜎 (cid:2870) 𝑣 (cid:4666) m / s (cid:4667) 𝑝 (cid:4666)m(cid:4667)𝑣 (cid:3404) (cid:3398)𝜎 (cid:2870) 𝑝 (cid:3397) 𝑑 (cid:2870) 𝑣 (cid:3404) 𝜎 (cid:2870) 𝑝 (cid:3397) 𝑑 (cid:2870) Fig. 8. The Period-2 (P2) orbits illustrated in the phase portrait. The yellowcross lines and the green cross lines are the orbital lines of P2 orbits. The blueand gray orbits are the P2 orbits with net velocity being 0. The net velocityof the red orbit is . m/s. of orbital lines (the cross yellow lines). As T SSP → ∞ , theorbital lines converges to the black lines.
2) Period-2 Orbits:
P2 orbits take two steps to complete aperiodic walking. We differentiate the consecutive two stepsby its stance foot, indexed by L / R . Similar to the P1 orbits,we identify the orbital slope and orbital lines of P2 orbits,and therefore the P2 orbits are geometrically characterized: Theorem 2.
For P2 orbits, the initial and final states of SSPare located on the orbital lines defined as v = ± σ p + d with d being a constant and the orbital slope: σ := λ tanh ( T SSP λ ) . (21)Each state on the line v = σ p + d represents the final stateof the SSP of a P2 orbit with the step size being: u L/R = 2 p − L/R + T DSP v − L/R . (22)Geometrically, d shifts the set of orbital lines up or down.The magnitude of d determines the net velocity of the P2orbit. Given the desired velocity v d , d can be calculated: d = λ sech ( λ T SSP ) T v d λ T DSP + 2 σ , (23)which does not depend on the selection of the boundary states.This indicates that there are infinite number of P2 orbits torealize one desired net velocity. Another way to look at thisis through the fact that the step sizes are determined by v d : u ∗ L + u ∗ R = 2 v d T. There are infinite combinations of u ∗ L , u ∗ R to satisfy this, and therefore there are infinite P2 orbits torealize the desired velocity. Selecting one step size (e.g. u ∗ L )determines the other one and thus determines the P2 orbit. Thefinal states of the SSP can then be determined from Eq. (22), p ∗ L/R = u ∗ L/R − T DSP d T DSP σ , v ∗ L/R = σ p ∗ L/R + d . (24)Fig. 8 illustrates three P2 orbits. The blue and the gray orbitsare located on the same set of orbital lines (yellow lines with d = 0 ), thus they have a zero net velocity. For the red P2orbit, it has a non-zero net velocity. P1P2 𝑣 (cid:3404) (cid:3399)𝜎 (cid:2870) 𝑝 (cid:3397) 𝑑 (cid:2870) 𝑣 (cid:3404)(cid:3399) 𝜎 (cid:2869) (cid:4666) 𝑝 (cid:3398) 𝑑 (cid:2869) (cid:4667) 𝑣 (cid:3404)(cid:3399) 𝜎 (cid:2869) (cid:4666) 𝑝 (cid:3397) 𝑑 (cid:2869) (cid:4667) Fig. 9. Equivalent characterization of the periodic orbits. The dashed crosslines are the equivalent orbital lines.
3) Equivalent Characterization:
The P1 and P2 orbits arecharacterized by their orbital lines, respectively. We also findthat under certain conditions, the orbital lines of P1 orbitscan also characterize P2 orbits and vice versa. It is clear thatwhen u L = u R , a P2 orbit becomes an equivalent P1 orbit,which can be stated as: Proposition 3.
The orbital lines v = ± σ p + d characterizethe P1 orbits when u L = u R , which yields the final state ofthe SSP as p ∗ = d sinh ( T SSP λ )2 λ , v ∗ = σ p ∗ + d . Similarly, P1 orbital lines can characterize P2 orbits:
Proposition 4.
The extended P1 orbital lines v = ± σ ( p ± d ) characterize the P2 orbits: the initial states are on v = − σ ( p ± d ) and the final states are on v = σ ( p ± d ) . Thecorresponding step sizes are as stated in Eq. (22).The non-uniqueness of P2 orbits to realize the desiredvelocity comes from the non-uniqueness of d . Given a d ,the final states of the P2 orbits can thus be determined. Fig. 9illustrates the equivalent characterizations of the orbits in Fig.7 and 8. In the latter, we only use the results from Theorem1 and 2 to find the desired walking orbits.
4) 3D Composition:
Full 3D walking can be encoded bytwo orthogonally composed planar orbits. The desired 3Dwalking behavior is first described via the desired walkingvelocities v dx,y in the sagittal and coronal plane. The orbitthat realizes the walking velocity is then identified in eachplane. A typical composition is choosing a P1 orbit in thesagittal plane and a P2 orbit in the coronal plane (sP1-cP2).The non-uniqueness of the P2 orbit can prevent foot collisionsby selecting step sizes to have opposite signs. This will be themain composition we realize on the 3D robot. B. Orbit Stabilization
We now derive the stepping stabilization on the periodicorbits. It can be viewed as generating a controller on u suchthat the S2S state is controlled to the desired final states inEq. (20) for P1 orbits and in Eq. (24) for P2 orbits. In [1], thestabilization was formulated based on the hybrid dynamics,and the proof was on the contraction on the distance betweenthe state and the target state of the orbit. Hence, the ranges of the gain and the optimal gains in terms of contraction ratewere derived from the contraction. One can also derive thestabilization via the S2S dynamics. It becomes a canonicallinear control problem: controlling the state to the desired onebased on the linear dynamics. We do not present this approachhere. Instead, we directly apply the H-LIP based stepping inEq. (14) to stabilize the orbits of the H-LIP, which yields thesimplest derivation. The stepping stabilization for P1 orbits is: u = u ∗ + K ( x − x ∗ ) , (25)where x = [ p − , v − ] T is the current pre-impact state of theH-LIP, u ∗ is the step size of the desired P1 orbit, and x ∗ =[ p ∗ , v ∗ ] T is the pre-impact state of the desired P1 orbit. Theerror state is e = x − x ∗ , and the error dynamics becomes: e k +1 = ( A + BK ) e k . (26)This is Eq. (15) with w = 0 since the H-LIP stepping isapplied on the H-LIP itself. To drive the H-LIP to its orbit,i.e., e → , we only need to find K to make A + BK stable.The deadbeat control can be applied. Since the system hastwo states and one input, it requires to two steps to make e → for all e ∈ R . The deadbeat gain is calculated from: ( A + BK deadbeat ) = 0 , which yields, K deadbeat = (cid:2) T DSP + λ coth ( T SSP λ ) (cid:3) . (27)Plugging the deadbeat gain into Eq. (25) yields, u k = p + p ∗ + T DSP v + λ coth ( T SSP λ )( v − v ∗ ) . (28)This is verified to be equal to the optimal stepping controllerin Theorem 2.1 in [1], which globally stabilizes the system tothe desired P1 orbit with two steps.Similarly, the stepping stabilization for P2 orbits is: u L = u ∗ L + K ( x L − x ∗ L ) , u R = u ∗ R + K ( x R − x ∗ R ) , (29)which yields the same error dynamics in Eq. (26). When thedeadbeat gain in Eq. (27) is chosen, the controller becomesidentical to the optimal controller in Theorem 2.2 in [1].This application of the "H-LIP stepping" on the H-LIPshould not be confused to that on the robot. In the applicationto the robot, we initialize the state of the H-LIP to be identicalto the robot, stabilize the H-LIP to the desired behavior, andthen stabilize the robot to the walking of the H-LIP. Remark:
The S2S formulation of H-LIP stepping for itsorbit stabilization also enables the use of many linear con-trollers. For instance, the Linear Quadratic Regulator (LQR)controller [47], [24] can be applied to provide the optimalgain K subject to a quadratic cost on the states and inputs.Model predictive controllers [48] can be easily synthesizedon the linear S2S dynamics to directly stabilize the state tothe desired one on the periodic orbits. In this paper, we donot present those results or their comparison. Instead, we onlydemonstrate the deadbeat controller on the H-LIP and then onthe robot due to its simplicity. Comparison to Capture Point:
The deadbeat stepping con-troller on the H-LIP is similar to the capture point controller[28]. In the capture point controller, the step location isdetermined by the passive LIP model so that the robot can come to a stop, i.e., v → as t → ∞ . In comparison, the H-LIP with zero velocity is a P1 orbit with v ∗ = 0 . Additionally,if we assume T SSP → ∞ and T DSP → , the step size controllerin Eq. (28) becomes identical to the instantaneous capturepoint controller: u = p + (cid:0)(cid:0)(cid:18) p ∗ + (cid:8)(cid:8)(cid:8)(cid:42) T DSP v + λ (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58) coth ( T SSP λ )( v − (cid:26)(cid:26)(cid:62) v ∗ ) = p + vλ . Thus, the capture point controller on the H-LIP is a specialcase of the deadbeat stepping controller on this model. Moreimportantly, the capture point controller based on the LIPmodel is typically directly applied on the robot; whereas thestepping controller on the H-LIP, e.g. in Eq. (28), is notdirectly applied on the robot but is used at the nominal stepsize u H-LIP in Eq. (14).V. A
PPLICATION TO
3D U
NDERACTUATED B IPEDAL R OBOT C ASSIE
In this section, we apply the H-LIP based gait synthesisand stepping stabilization on the 3D underactuated bipedalrobot Cassie. We start by presenting the mathematical modelof the robot. Then, we apply the H-LIP based approach todesign the desired output trajectories for realizing 3D walking.Lastly, low-level optimization-based controllers are presentedfor trajectory tracking on the output.
A. The Robot Model
The robot Cassie, shown in Fig. 10, is a 3D underactuatedbipedal robot with compliance. It is designed and built byAgility Robotics [49] to resemble the Spring-loaded InvertedPendulum (SLIP) model [18], [50] for locomotion. It has aconcentrated upper body and light-weight springy legs. Here,we describe the mathematical model that best captures itsdynamics with certain simplifications.Each leg on the robot can be modeled with seven degreesof freedom, including five motor joints and two leaf springs.The two leaf springs can be modeled as two rotational jointswith torsional springs [22], [51]. As shown in Fig. 10, thefour-bar linkage on the lower part of the leg transmits adistant motor actuation to the pitch of the foot. The closed-loop linkage on the upper part of the leg can be viewed as totranslate the rotation of the knee joint to the tarsus joint, which hip linkknee linkshin linkshin springtarsus linkachilles rod toe/foot linkheel springplantar rod toe motor deflection axis motor spring passive joints hip yawhip pitch hip rollshin pitchknee pitchtoe pitch tarsus pitch
Fig. 10. The underactuated bipedal robot Cassie, its joints, and linkages. extends and retracts the foot. Since the push-rods are light-weighted, we neglect their inertia and their associated degreesof freedom (dofs) for simplifications. The foot is then assumedto be directly actuated from the toe motor. The achilles rod isreplaced by a holonomic constraint on the distance betweenthe end-points of the rod.We use the floating-base coordinate to describe the config-uration of the robot: q = [ q pelvis , q Lleg , q
Rleg ] T , where q pelvis =[ q x,y,z pelvis , q rpy pelvis ] ∈ SE (3) is the pelvis configuration, and q Lleg , q
Rleg are the configuration of the left and right leg, re-spectively. q L/Rleg = [ q rollhip , q yawhip , q pitchhip , q knee , q shin , q tarsus , q heel , q toe ] ,where the individual element is the joint angle. The motorjoints are q motor = [ q rollhip , q yawhip , q pitchhip , q knee , q toe ] T , and the springjoints are q spring = [ q shin , q heel ] T . The continuous dynamics are: M ( q )¨ q + C ( q, ˙ q ) + G ( q ) = Bτ + J Th F h , (30) J h ¨ q + ˙ J h ˙ q = 0 , (31)where M ( q ) is the mass matrix, C ( q, ˙ q ) contains the Cori-olis and centrifugal forces, G ( q ) is the gravitational vector, τ = [ τ Tm , τ Ts ] T is the actuation vector, τ m represents themotor torque vector, τ s is the vector of the torsional forcesof the spring joints, B = [ B m , B s ] is the actuation matrix, J h = [ J T rod , J T Foot ] T represents the Jacobian of the holonomicconstraints, and F h = [ F T rod , F T GRF ] T contains the holonomicforces, including the forces on the push-rods and groundreaction forces (GRF). The spring forces are calculated by: τ shin/heel = K s shin/heel q shin/heel + D s shin/heel ˙ q shin/heel , where K s shin/heel and D s shin/heel are the stiffness and damping of the springs. Theholonomic constraints h ( q ) include the distance constraints onthe achilles push-rods and the ground contact constraints. Forthe dynamics in the SSP, the ground contact can be describedvia 5 holonomic constraints and the dimension of h ( q ) is 7. InDSP, the dimension of h ( q ) becomes 12. Note that, Eq. (30)and Eq. (31) provide an affine mapping from the input torquesto the holonomic forces: F h = A h τ + b h . (32)The exact expressions of A h , b h are omitted here. Hybrid Model of Underactuated Walking:
The walking ofCassie is modeled as a two-domain hybrid system. Due to thecompliance in the legs, the transition from the current SSP tothe next SSP is not likely to instantaneously happen right afterthe impact event. In other words, the DSP typically exists inwalking. The impact between the swing foot and the groundis assumed to be plastic [8], where the velocity of the stateundergoes a discrete transition. Note that the foot is smalland its rotation is not actuated in the lateral direction. Thus,the walking in the coronal plane is underactuated at the foot.The toe actuation on the stance foot is virtually removed bysetting the torque to 0 to render the underactuated walking inthe sagittal plane as well.
B. H-LIP based Gait Design and Stepping on Cassie
Now we apply the H-LIP based approach on Cassie. Theoutput is designed to satisfy the requirement of the H-LIPbased gait synthesis: the vertical center of mass (COM) position z COM (w.r.t. the stance foot) should be (approximately)constant, and the vertical position of the swing foot z sw isconstructed to periodically lift off and strike the ground. Thehorizontal position of the swing foot { x, y } sw (w.r.t. the stancefoot) is controlled to achieve the desired step size u dx,y fromthe H-LIP based stepping. Additionally, the orientation of thepelvis q rpy pelvis and the swing foot φ sw py should be controlled tofully constrain the walking. The output in SSP (illustrated inFig. 11) is then defined as: Y = z COM { x, y, z } sw q rpy pelvis φ sw py − z d COM { x, y, z } d sw q rpy d pelvis φ sw pyd . (33)
1) Accommodations for Compliance:
The output definitionin Eq. (33) is sufficient for robots without evident compliantelements. However, the passive compliance on Cassie createschallenges on precise control on the vertical COM and swingfoot positions. If the output contains the compliant degreesof freedom, the spring can create undesired resonance, whichthen destabilizes the output, especially in the vertical direction.Thus, accommodations have to be made for the compliance.In [1] [52], the uncompressed leg length (i.e. the leglength with zero spring deflections) was used as the outputto indirectly control the vertical position of the COM and theswing foot. Here we select the uncompressed vertical COMand swing foot positions as the approximation to the actualones. By definition, the vertical position of the COM w.r.t. thestance foot is a function of { q rpy pelvis , q motor , q tarsus , q spring } . TheCOM height with uncompressed springs is: ˜ z COM = z COM ( q rpy pelvis , q motor , q tarsus → q rigidtarsus , q spring → .q rigidtarsus is the uncompressed tarsus angle under the holonomicconstraint of the push-rod: q rigidtarsus = Root ( h rod ( q knee , q shin → , q heel → , q tarsus ) = 0) , which is solved via Newton-Raphson method. z COM in Eq.(33) is thus approximated by ˜ z COM . Similarly, the position ofthe swing foot w.r.t. the stance foot are approximated in thesame way by { ˜ x, ˜ y, ˜ z } sw . Since the springs on the stance legare expected to oscillate less, we only set the springs on theswing legs to 0 in { ˜ x, ˜ y, ˜ z } sw for better approximations. 𝜙 (cid:2935)(cid:2911)(cid:2933)(cid:2929)(cid:2933)(cid:2919)(cid:2924)(cid:2917) (cid:4670)𝑥(cid:3556) (cid:2929)(cid:2933) , 𝑦(cid:3556) (cid:2929)(cid:2933) (cid:4671)𝜙 (cid:2926)(cid:2919)(cid:2930)(cid:2913)(cid:2918)(cid:2929)(cid:2933)(cid:2919)(cid:2924)(cid:2917) 𝑧̃ (cid:2913)(cid:2925)(cid:2923) 𝑧̃ (cid:2929)(cid:2933) 𝑞 (cid:2926)(cid:2915)(cid:2922)(cid:2932)(cid:2919)(cid:2929)(cid:3045)(cid:3043)(cid:3052) 𝑧 (cid:2868) 𝑧̃ (cid:2913)(cid:2925)(cid:2923) (cid:4666)𝑡 (cid:3404) 0(cid:4667) 𝑧 (cid:2929)(cid:2933)(cid:2923)(cid:2911)(cid:2934) 𝑢 (cid:3051)(cid:3031) COM 𝑧 (cid:2887) (cid:2899) (cid:2897) (cid:3031) 𝑧 (cid:2929) (cid:2933) (cid:3031) 𝑥 (cid:2929) (cid:2933) (cid:3031) Fig. 11. Illustrations of the definition and desired trajectories of the output.
2) Desired Output Trajectories:
The desired orientations ofthe pelvis and swing foot are chosen to be constant. The rest ofthe desired trajectories are designed with Bézier polynomialsto satisfy the requirements of the H-LIP based approach. Theexact Bézier coefficients are listed in the Appendix.First, the desired step sizes in the sagittal and coronal planeare constantly decided from the H-LIP based stepping: u dx/y = u H-LIP x/y + K ( x / y − x / y H-LIP ) , (34)where x / y is the horizontal COM state of the robot in thesagittal or coronal plane. The desired horizontal trajectoriesof the swing foot are designed as: x/y d sw = (1 − b h ( t )) x/y + sw + b h ( t ) u dx/y , (35)where x/y + sw is the horizontal position of the swing foot w.r.t.the stance foot in the beginning of the current SSP. b h ( t ) is aBézier polynomial that transits from 0 ( t = 0 ) to 1 ( t = T SSP ),where the clock of the gait t is reset to 0 after each step.The vertical COM position should be controlled to z , whichis also the constant height of the H-LIP. At swapping supportlegs, ˜ z COM has a small discrete jump. The desired trajectoryof the vertical COM position is then constructed as: z d COM = (1 − b h ( t ))˜ z + COM + b h ( t ) z , (36)where ˜ z + COM is the uncompressed COM height in the beginningof the SSP. Lastly, the vertical position of the swing foot z d sw ( t ) is constructed as: z d sw ( t ) = b v ( t, z maxsw , z negsw ) , (37)where b v is another Bézier polynomial to create lift-off andtouch-down behaviors. It is designed to transit from 0 ( t = 0 )to z maxsw (e.g., t = T SSP ) and back to z negsw ( t = T ). z maxsw is aconstant to determine the foot-ground clearance, and z negsw is asmall negative value to ensure foot-strike at the end.
3) Desired DSP Output:
In DSP, two feet contact theground at all times. With more holonomic constraints on thesystem, the dimension of the outputs decreases. Instead of re-formulating a different set of DSP outputs, we directly usethe SSP outputs and set the desired values of the outputs onthe swing foot to be the actual ones (including the horizontalpositions and orientation), which preserves the holonomicconstraints in the DSP and also simplifies the gait design.
Versatility of the Gait Design:
The output construction di-rectly allows the COM height z , step frequency (inverse ofthe walking period T ) and the swing foot clearance z maxsw tobe individually chosen. Different combinations of the param-eters render different walking behaviors. The desired walkingvelocity in each plane is individually stabilized via the H-LIPstepping, which is independent of the chosen gait parameters.Additionally, for P2 orbits on the robot, there are infiniteorbits for realizing the same desired walking velocity. Thecombination of the gait parameters and orbit selections rendersversatile walking behaviors on the robot. Remark:
In [1], a stepping-in-place gait was optimizedon the actuated SLIP model and its periodic trajectories ofthe leg length were then applied on Cassie. Various walkingbehaviors were realized via perturbing the stepping-in-place gait by changing the step size based on the H-LIP. The periodictrajectories of the leg length indirectly realized the lift-offand touch-down behaviors on the swing foot and renderedan approximately constant height on the vertical COM. Here,the output is constructed in a more direct and general fashion.The use of the aSLIP is not necessary, and the desired outputtrajectories are directly constructed from the gait parameters.
C. Joint-Level Optimization-based Controller
Nonlinear controllers can be applied to drive the output Y in Eq. (33) to zero. In particular, we consider using QuadraticProgram (QP) based controllers [53], [16] for stabilization,where the contact constraints and torque limits can be includedas the inequality constraints in the QP. When the foot contactsthe ground, the resultant ground reaction forces should satisfythe friction cone constraints and the non-negativity constraintson the normal forces. This can be encoded as: A F GRF ≤ , where A is a constant matrix. The motors can only providecertain amount of torques at certain speeds. Thus, we set: τ lb ( ˙ q ) ≤ τ m ( ˙ q ) ≤ τ ub ( ˙ q ) , where τ lb/ub is the lower or upperbounds on the motor torques.The control objective is to drive the output Y → . Herewe illustrate two prominent approaches for realization: oneis in the task space control (TSC) formulation [53] throughminimizing the difference between the actual acceleration anda desired acceleration, which yields stable linear dynamics onthe output; the other one is in the control Lyapunov function(CLF) formulation [16] via an inequality condition on thederivative of the Lyapunov function of the output to yieldexponential convergence.For the TSC, the desired acceleration ¨ Y d is chosen as: ¨ Y d = − K p Y − K d ˙ Y , (38)with K p , K d being the feedback gains. An optimization prob-lem is formulated to minimize || ¨ Y − ¨ Y d || subject to thephysical constraints and the robot dynamics. Then the actualoutput dynamics evolves closely to the desired linear stabledynamics in Eq. (38), which thus realizes the control objective.Since the acceleration ¨ Y is affine w.r.t. the input torque, theoptimization problem is a quadratic program (QP).For the CLF, a quadratic Lyapunov function V ( Y , ˙ Y ) isconstructed on the output Y and ˙ Y , thus V ( Y , ˙ Y ) → ifand only if [ Y , ˙ Y ] → . The convergence condition of Y isenforced the derivative of V , i.e., ˙ V ≤ − γV, (39)with γ > , which yields V (and thus Y ) to decrease at leastat an exponential rate. As ˙ V is affine w.r.t. the input torque,a QP can be formulated on minimizing the norm of the inputtorque subject to the inequality constraint in Eq. (39) andadditional physical constraints. The two QP based controllersare summarized as follows. TSC-QP CLF-QP [ τ ∗ m , ∼ ] = argmin τ m ,F h , ¨ q || ¨ Y − ¨ Y d || [ τ ∗ m , ∼ ] = argmin τ m ,F h τ Tm τ m s.t. Eq.(30) , (31) , A F GRF ≤ τ lb ( ˙ q ) ≤ τ m ( ˙ q ) ≤ τ ub ( ˙ q ) s.t. Eq.(32) , (39) , A F GRF ≤ τ lb ( ˙ q ) ≤ τ m ( ˙ q ) ≤ τ ub ( ˙ q ) Remark:
We do not intend to compare the two controllersor propose any other variants of the QP based controllers inthis paper. They are merely used as tools to stabilize the outputto demonstrate the H-LIP based approach. With proper gain-tuning, both controllers can perform equivalently.VI. S
IMULATION E VALUATION
In this section, we realize and evaluate the approach onCassie in simulation. The simulation environment allows thor-ough evaluations on the robot model at the stage beforehardware realization. In the simulation, we have full access toall the states of the system. Thus, the information of contactsand the horizontal velocity of the robot are exactly known,which provides a rigorous analysis of the proposed approach.
A. Setup
The robot starts from a static standing configuration. Thedynamics of the robot are numerically integrated using theODE 45 function in Matlab with event-based functions fortriggering domain switching. Target final velocities v tx and v ty are given with the goal of controlling the robot to realize thesetarget velocities. We first select an orbit composition and thenconstruct continuous desired velocity profiles v dx/y ( t ) to reachthe target velocities. For simplicity, we use piecewise lineartrajectories to design v dx/y ( t ) . For P2 orbits, the desired stepsize should be specified. The desired output trajectories areconstructed via the H-LIP based gait synthesis and stepping.The gait parameters such as the swing foot clearance andstep frequency are specified in the beginning. The low-leveloptimization-based controller is constructed and solved at1kHz using qpOASES [54]. The video of the simulation resultscan be seen in [3]. B. Results1) Forward Walking:
We first present forward walking onCassie as the basic realization of the proposed H-LIP basedapproach. The orbit composition is chosen as having a P1orbit in its sagittal plane and a P2 orbit in its coronal plane.The velocities are chosen to be v tx = 1 m/s, and v ty = 0 m/s,thus the robot only progresses in its sagittal plane. We choose T = 0 . s, z maxsw = 0 . m, z negsw = − . m, and the orbit-determining step width of the P2 orbit is u y ∗ L = − . m. Thedesired walking velocity v dx ( t ) is chosen from 0 to ramp up to1m/s within 3s. Fig. 12 shows the plots of the forward walkingvia the H-LIP based approach. The output trajectories arewell-tracked via the optimization-based controller. The actualvelocities converge to the desired ones with negligible errors.The horizontal COM states of the robot converge closely tothe desired orbits of the H-LIP that realize the target velocitiesin each plane.Then we change the target velocity in the sagittal planefrom -1.5m/s to 1.5m/s with a 0.5m/s increment. Fig. 13shows the results. For clarity, in the phase portraits, we onlyplot the steady walking behavior where the desired walkingvelocity becomes constant (after 5s). We also demonstrate thatthe evolution of the error states is within the error invariant set. The error states in each plane can be directly calculatedfrom the pre-impact states of the robot and the desired statesof the H-LIPs. To calculate the error invariant set, we firstnumerically calculate the dynamics error w from the evolu-tion of the horizontal COM states in each realized walkingbehavior. Since W cannot be calculated analytically, we useall w to construct a polytope to numerically approximate W in each plane. As K is chosen from the deadbeat controller,i.e., ( A + BK ) = 0 , the invariant set E = ( A + BK ) W ⊕ W .The set operation is calculated using the MPT [55] toolbox.Fig. 13 (d) shows that the error states are indeed inside theerror invariant set.
2) Lateral and Diagonal Walking:
The approach can alsorealize walking to different directions by selecting differentdesired velocities in each plane of the robot. Here we presentwalking in the lateral and diagonal direction. Fig. 14 illustratesthe converged walking behaviors with different choices ofthe target velocities. The gait parameters are identical to theprevious examples. By tracking the desired velocity in eachplane, the robot walks in the desired direction. The convergedorbits of the horizontal COM states are also relatively closeto the desired orbits of the H-LIP in both cases. Moreover, byselecting different desired step width u y ∗ L , different P2 orbitsare realized in the coronal plane with the same desired velocity v dy (Fig. 14 (l-y) and (d-y)).
3) Variable Orbit Compositions:
The two types of orbitsof the H-LIP provide four kinds of orbit compositions in 3D.If the kinematic constraints are neglected, all four types oforbit compositions can be realized to achieve the same desiredwalking velocity. Fig. 15 illustrate the four realizations toachieve the lateral walking with v ty = 0 . m/s. Each realizationis abbreviated by the type of orbit in each plane, e.g., sP1-cP1 indicates both P1 orbits are selected in the sagittal andcoronal plane. For certain compositions, kinematic collisionscan happen between the legs. E.g., the sP1-cP1 gait with v tx,y = 0 clearly creates foot overlaps. The complex leg designon Cassie further increases the possibilities of kinematiccollisions between the legs. Although it is still possible torealize those compositions with certain orbits, we only focuson the realization of the sP1-cP2 orbits on the hardware.VII. E XPERIMENT E VALUATION ON C ASSIE
In this section, we realize the H-LIP based approach onthe physical hardware of Cassie. Unlike in simulation, therobot state is no longer completely and exactly known on thehardware. Moreover, the computation capacity of the on-boardcomputer has to be taken into consideration for realization.Therefore, we first present the control realization with practicalconsiderations on the hardware, and then present the resultsof the experiment.
A. Control Realization on Hardware1) Contact Detection:
The robot Cassie is not equippedwith contact sensors to detect foot-ground contact. It is pos-sible to measure the deflection of the spring joints and seta threshold for contact detection. However, the springs canstill have non-trivial deflections in the swing phase due to the (v-x)(v-y) (p-x)(p-y) 𝑣 (cid:3051) 𝑣 (cid:3052) (u-x)(u-y) (o-z) (o-s)(o-x) (o-y) 𝑣 (cid:3051)(cid:3031) 𝑣 (cid:3051)(cid:2892)(cid:2879)(cid:2896)(cid:2893)(cid:2900) 𝑣 (cid:3051) 𝑢 (cid:3051)(cid:2892)(cid:2879)(cid:2896)(cid:2893)(cid:2900) 𝑢 (cid:3051) 𝑣 (cid:3052)(cid:3031) 𝑣 (cid:3052)(cid:2892)(cid:2879)(cid:2896)(cid:2893)(cid:2900) 𝑣 (cid:3052) 𝑢 (cid:3052) 𝑢 (cid:3052)(cid:2892)(cid:2879)(cid:2896)(cid:2893)(cid:2900) 𝑧 (cid:2887)(cid:2899)(cid:2897) 𝑧 (cid:3031) 𝑧̃ (cid:2887)(cid:2899)(cid:2897) 𝑧 (cid:2929)(cid:2933) 𝑧 (cid:2929)(cid:2933)(cid:3031) 𝑧̃ (cid:2929)(cid:2933) 𝑥 (cid:2929)(cid:2933)(cid:3031) 𝑥(cid:3556) (cid:2929)(cid:2933) 𝑦 (cid:2929)(cid:2933)(cid:3031) 𝑦(cid:3556) (cid:2929)(cid:2933) Fig. 12. Simulation results on a forward walking with v tx,y = [1 , m/s, u y L ∗ = − . m: the trajectories of the horizontal velocities of the COM (red and bluelines) in the sagittal plane (v-x) and the coronal plane (v-y) compared with the desired velocities v dx,y ( t ) (black lines) and the corresponding velocities ofthe H-LIP (green circles); the phase trajectories of the horizontal states of the COM in the sagittal plane (p-x) and the coronal plane (p-y) compared with theH-LIP orbits (black) at the target velocities; comparisons of the step sizes (u-x, u-y) between the robot (red circles in the sagittal plane and blue circles in thecoronal plane) and the H-LIP (green circles). (o) Output tracking with the red dashed lines indicating the desired output trajectories and the blue continuouslines indicating the actual one: (o-z) the vertical COM trajectory ˜ z COM (the black line is the actual vertical COM position of the robot z COM ); (o-s) the verticalswing foot trajectory ˜ z sw (the black lines are z sw ); the horizontal trajectories of the swing foot in the sagittal plane (o-x) and the coronal plane (o-y). (a) (b)(c) (d) 𝐸 (cid:3051) 𝐞 (cid:3038) (cid:4666)𝑣 (cid:3051)(cid:3047) (cid:3404) (cid:3398)1.5(cid:4667)(cid:4666)𝑣 (cid:3051)(cid:3047) (cid:3404) 1.5(cid:4667) 𝐞 (cid:3038) 𝑣 (cid:3051)(cid:3047) (cid:3404) 1.5𝑣 (cid:3051)(cid:3047) (cid:3404) (cid:3398)1.5 𝑣 (cid:3051)(cid:3047) (cid:3404) (cid:3398)1.5𝑣 (cid:3051)(cid:3047) (cid:3404) 1.5 Fig. 13. Comparison on forward walking with different target velocities v tx : (a) forward velocities of the COM (continuous lines) compared with thedesired velocity profiles v dx ( t ) (dashed lines); (b) the converged orbits (red andblue lines) of the sagittal COM states compared with the desired target orbitsof the H-LIP (black lines). (c) the converged orbits (the red is with v tx = 1 . and the blue is with v tx = − . ) of the coronal COM states compared withthe target orbit of the H-LIP (black); (d) the error state trajectories (circles)and the error invariant set E x (the blue transparent box) in the sagittal plane. inertia forces in the leg. Therefore, the threshold has to beset large enough to avoid false detection of contact. However,this can cause significant late-detection of impacts and early-detection of lift-offs. Instead, we use the measured torque fromthe input current (similar to the proprioceptive sensing [56])along with the spring deflections to approximately calculatethe contact forces at the feet. A threshold is then set on the 𝑣 (cid:3051)(cid:3047) (cid:3404) 0.5, 𝑣 (cid:3052)(cid:3047) (cid:3404) 0.5𝑣 (cid:3051)(cid:3047) (cid:3404) 0, 𝑣 (cid:3052)(cid:3047) (cid:3404) 0.5 (l-x) (d-x)(l-y) (d-y) Fig. 14. Lateral walking ( v tx,y = [0 , . m/s, u y L ∗ = − . m) and diagonalwalking ( v tx,y = [0 . , . m/s, u y L ∗ = − . m) with their converged orbitsin the sagittal plane (l-x, d-x) and the coronal plane (l-y, d-y). magnitude of the forces to detect contact.Since the vertical COM of the robot will be controlledapproximately constant, we neglect the dynamics contributionto the contact forces. The EOM in Eq. (30) becomes: G ( q ) = Bτ + J Th F h , (40)where F h = [ F T rod , F L T GRF , F R T GRF ] T and τ = [ τ Tm , τ Ts ] T . τ m is Fig. 15. Simulated walking via different orbit compositions with the sametarget velocity ( v tx,y = [0 , . m/s). The trajectories of the swing foot areindicated by the red (left foot) and the blue (right foot) lines, with therectangles indicating the contact locations.Fig. 16. Contact detection via the GRF calculation: The transparent red andblue lines are the norm of the actual GRF on the robot in simulation, thedashed red and blue lines are the calculated GRF for contact detection. Thedetected DSP is close to the actual DSP in simulation. measured from the motor current and τ s is calculated from thespring deflections. This equation is invariant w.r.t. the pelvisposition q x,y,z pelvis , which is not known. Thus, we set q x,y,z pelvis to 0.The rest of q are measured via the IMU, joint encoders and legkinematics. F h can be directly solved via the pseudo-inverseof J Th : F h = pinv ( J Th )( G ( q ) − Bτ ) . The calculated F L/RGRF are then low-pass filtered with a cutoff frequency of 100Hz.A threshold is then set on the norm of F L/RGRF to determineif the foot is in contact with the ground. Fig. 16 shows thecontact detection via the GRF compared with the actual GRFin a simulated walking. The threshold is set to 100N, whichprovides precise contact detections.
2) H-LIP based Velocity Approximation on COM:
Thetransitional position and velocity of the floating-base q x,y,z pelvis and ˙ q x,y,z pelvis cannot be directly measured. ˙ q x,y,z pelvis is requiredfor calculating the COM velocity for realizing the walking.We implemented the extended Kalman filter in [57] forstate estimation by utilizing the inertia measurement unit(IMU). This state estimation required significant computation (a similar estimation scheme in [58] has to be implementedon a secondary computer on the robot). Additionally, themagnetometer drift inside the IMU also creates errors on theestimated velocities under certain circumstances. Due to thoseconcerns, we instead approximate the COM velocity based onthe H-LIP dynamics in the SSP.We use the walking in the sagittal plane to illustrate theapproximation. Let p and v be the horizontal position andvelocity of the COM in the beginning of the SSP. The dynam-ics of the horizontal COM in the SSP can be approximated bythe SSP dynamics of the H-LIP. Thus the current COM stateof the robot [ p t , v t ] in the SSP can be approximated by: (cid:20) p t v t (cid:21) ≈ e A SSP t (cid:20) p v (cid:21) , (41)where A SSP is defined in Eq. (6). Let A t := e A SSP t . Given themeasured positions p and p t and the current time t from thebeginning of the SSP, the velocity approximations are: (cid:20) ˜ v ˜ v t (cid:21) = (cid:20) − A (1 , t / A (1 , t / A (1 , t A (2 , t − A (1 , t A (2 , t / A (1 , t A (2 , t / A (1 , t (cid:21) (cid:20) p p t (cid:21) (42)where the superscripts indicate the elements of the matrix A t .Thus the continuous velocity approximation ˜ v t is obtained.The prediction of the pre-impact velocity ˜ v t = T SSP can alsobe continuously approximated by the H-LIP dynamics in theSSP based on the current state [ p t , ˜ v t ] T and the time-to-impact T SSP − t . The velocity approximation is solely basedon the position of the COM w.r.t. the stance foot, which onlyuses joint encoders and orientation readings of the IMU andthus is robust to sensor noises. Moreover, we show that thisapproximation is valid for applying the H-LIP based stepping,only with generating a different error invariant set.Let ˜ v − be the approximated velocity of the COM of therobot at the pre-impact. ˜ v − = ˜ v t = T SSP is calculated from Eq.(42). Let ˜ x = [ p − , ˜ v − ] T represent the approximated COMstate at the pre-impact. Assuming the COM position of therobot is measured with a negligible error, ˜ x − x = [0 , ˜ v − − v − ] T := δ x , where v − is the actual COM velocity of the robotat pre-impact event. Note that δ x is bounded: the velocity error ˜ v − − v − is the integration of the dynamics difference betweenthe H-LIP and the robot in the SSP. The approximated stateis used in the H-LIP based stepping, i.e., u = u H-LIP + K (˜ x − x H-LIP ) . Therefore, the error dynamics becomes: e k +1 = A x k + Bu k + w − A x H-LIP k − Bu H-LIP k = ( A + BK ) e k + w + BKδ x k (cid:124) (cid:123)(cid:122) (cid:125) ˜ w . ˜ w ∈ ˜ W is bounded since w and δ x are both bounded. Thisconsequently creates a new error invariant set ˜ E .To validate this, we use the H-LIP based velocity approxi-mations to replace the actual horizontal velocities of the COMin the controller in simulation. The performance is comparablewith that with true COM velocity in the previous section. Fig.17 shows the results on a simulated forward walking as aproof. As the horizontal COM dynamics of the robot is closeto the H-LIP dynamics, the velocity approximation works well(Fig. 17 (v-x) and (v-y)). Although the new disturbance ˜ w is (v-y)(v-x)(e-x) (e-y) 𝐸 (cid:3051) 𝐸(cid:3560) (cid:3051) 𝐸 (cid:3052) 𝐸(cid:3560) (cid:3052) 𝐞 (cid:3038)𝐲 𝐞 (cid:3038)𝐱 𝑣 (cid:3051) (cid:4666)𝑡(cid:4667)𝑣(cid:3556) (cid:3051) (cid:4666)𝑡(cid:4667) 𝑣 (cid:3052) (cid:4666)𝑡(cid:4667) 𝑣(cid:3556) (cid:3052) (cid:4666)𝑡(cid:4667) Fig. 17. Validation on the H-LIP based velocity approximation on a simulatedwalking with v tx,y = [1 , m/s, u y L ∗ = − . m: (v-x, v-y) the approximatedhorizontal velocities of the COM ˜ v x ( t ) (red line) and ˜ v y ( t ) (blue line) inthe SSP compared with the actual velocities (dashed black lines); (e-x, e-y)the error state trajectories e x , y and the new error invariant sets ˜ E x,y (bluetransparent polytopes) in each plane compared with the error invariant sets E x,y (white polytopes with black continuous bounding lines) from Fig. 12. w plus another term, it does not necessarily mean that the sizeof ˜ W and the resultant ˜ E are larger. Here we get a smaller setin the sagittal plane (Fig. 17 (e-x)), and the sets in the coronalplane are of similar sizes (Fig. 17 (e-y)).
3) Joint-level Controller:
The optimization-based con-troller in section V can be potentially implemented on thehardware by utilizing the secondary computer on the robot.Here we apply a
PD + Gravitation Compensation (PD+G)controller, which in practice provides an equivalent trackingperformance and a much-lower computational effort. ThePD+G controller is directly implemented on the main com-puter on the robot, which is written as: τ m = τ PD + τ G , where τ PD represents the PD component and τ G representsthe gravitation compensation part.For the PD component, we directly map the desired accel-eration of the output ¨ Y d to the joint torques. ¨ Y d is identicallychosen to that in Eq. (38). Y and ˙ Y are measured on thehardware. Note that the output selections (e.g. Eq. (33)) aremainly functions of the motor joints. The actual accelerationof the output is assumed to be: ¨ Y = J Y ¨ q m + ˙ J Y ˙ q m , where J Y = ∂ Y ∂q m . The desired accelerations of the motor joints areapplied as the motor torques: τ PD = ¨ q dm = J − Y ( ¨ Y d − ˙ J Y ˙ q ) . (43)For the gravity compensation, we need to find joint torquesto cancel the gravitational terms in Eq. (40) based on thecurrent configuration and contact. The problem is inverseto the contact detection. Given q , we find τ G to minimize: (cid:107) B m τ G + B s τ s + J Th F h − G ( q ) (cid:107) . Note that the foot contactof the robot is underactuated and thus there does not exist anyset of joint torques to completely cancel out the gravitationalterm, unless the foot is fully-actuated. This yields a leastsquare problem of min : (cid:107) AX − b (cid:107) where A = [ B m , J Th ] , X = [ τ TG , F Th ] T , b = G ( q ) − B s τ s . Similarly, this problem canbe solved via the pseudo-inverse of A , i.e., X = pinv ( A ) b ,which yields the gravity compensation term τ G . B. Hardware Implementation Scheme
The robot is controlled via a remote controller that sendsradio commands to the robot. The on-board computer isprogrammed to interpret the radio signals, read all the sensorson the robot, and send torque commands to the robot.The implementation is illustrated in Fig. 18. The remotecommands are processed to get the desired walking behaviors.The H-LIP based gait synthesis and stepping stabilizationcalculate the output based on the gait parameters, contact,and COM states. The joint-level controller then calculates themotor torques and sends them to the motor modules to stabilizethe outputs. Based on the computation capacity of the mainon-board computer, the control loop is set at 1kHz.
C. Directional Walking
We demonstrate directional walking behaviors on the robotby using the joysticks on the remote to steer the robot to itsforward, backward, and lateral directions. The joystick valueson the remote are used as the desired walking velocities v dx,y .We use low-pass filters to smooth the reading of the joysticks.Thus the desired velocities between consecutive steps do notvary significantly. Fig. 21 demonstrates the snapshots of therobot walking in its sagittal and coronal plane.In order to analyze the generated walking behaviors onthe hardware, we use the extended Kalman filter (EKF) [57]offline to get a continuous estimation of the horizontal velocityof the COM on the robot. The estimated velocities are usedas references rather than the ground truth. This is because theestimation has non-neglectable errors that in nature come fromthe imperfections of the dynamics models and sensors.Fig. 20 shows the horizontal COM states of a forwardwalking [4]. The estimated velocities are compared withthe approximated velocities from the H-LIP and the desiredvelocities. The desired walking velocities are tracked withinreasonable errors. The error invariant sets are approximated inthe same way (in section VI), and the error states are all insidethe invariant sets. Note that the error invariant sets are largerthan those in Fig. 17. This is because the dynamics difference w is calculated using the estimated velocities; the estimationerrors directly yield larger sets W x,y and thus larger sets E x,y .Additionally, the translational dynamics and transversal dy-namics can be controlled separately. Therefore, we implementa turning controller that only changes the hip yaw angles andkeep the stepping controller intact. With turning, the robot canbe joystick-controlled easily in confined environments [4]. D. Versatile Walking
To demonstrate the versatility in the gait design, we utilizethe potentiometers on the remote to vary the gait parameters inreal-time. As indicated previously, we select the step duration,swing foot clearance, step width in the P2 orbit, and the COMheight. There are four potentiometers on the remote controller, Desired Walking • DesirStep Frequency: • Desired Velocities: 𝑣 (cid:3051)(cid:3031) , 𝑣 (cid:3052)(cid:3031) • Lateral Step Width (P2): 𝑢 (cid:2896)∗ • Nominal Height 𝑧 (cid:2868) • Swing foot clearance 𝑧 (cid:2929)(cid:2933)(cid:2923)(cid:2911)(cid:2934) H‐LIP based Gait Synthesis and Stepping Stabilization Joint‐level Controller 𝜏 (cid:3040) EncodersIMU
Remote Controller
H‐LIP based Velocity Approximation Contact Detection
Fig. 18. Illustration of the hardware realization of the controller on Cassie. (a) (b)(c) (d)(e)
DSP
Fig. 19. Illustrations of the output tracking and contact detection on thehardware: the desired output trajectories (the blue dashed lines) and the actualoutput trajectories (the red lines) of (a) the vertical COM position, and (b,c,d)the vertical, forward and lateral positions of the swing foot; (e) the contactdetection via the GRF, where the boxed regions indicate the DSP.TABLE IV
ERSATILE W ALKING P ARAMETERS
Variables Definition Range
Step Duration T . − . sDesired Swing Foot Clearance z maxsw . − . mDesired Step Width u ∗ L . − . mDesired COM Height z . − m and each potentiometer corresponds to one parameter. Thereading on the potentiometers can oscillate, and we do notlow-pass filter the values to show the robustness of thecontrol implementation. Fig. 22 demonstrates a stepping-in-place walking with varying the four parameters. The rangesof the parameters are listed in Table I. All the parameterscan be varied continuously, and the H-LIP based stepping stillcan stabilize the walking. Fig. 23 demonstrates the continuouschanges of the values in the experiment [5]. Additionally, [5]shows forward walking behaviors with different COM heights.For parameters outside of the range, the walking can beinfeasible or destabilized. For instance, if the step width istoo small or too larger, the walking becomes kinematicallyinfeasible. If the desired COM is too tall or too low, the leg canextend or retract to its kinematic limits. If the foot clearance (v-x)(v-y)(s-x) (p-y)(s-y) 𝑣 (cid:3051)(cid:3031) 𝑣(cid:3556) (cid:3051) 𝑣̂ (cid:3051) 𝑣 (cid:3052)(cid:3031) 𝑣(cid:3556) (cid:3052)(cid:2879) 𝑣̂ (cid:3052) (p-x) Fig. 20. Trajectories of a forward walking with varying target velocities.(v-x, v-y) plot the horizontal COM velocities including the desired velocities v dx,y ( t ) (the black dashed lines), the velocity in the SSP from the H-LIPbased approximation (the red lines), the predicted pre-impact velocity (theblue lines), and the estimated velocities (the gray lines). (p-x, p-y) plot thehorizontal states in the sagittal plane (in different time segments) and thecoronal plane, respectively. The black orbits are the desired orbits of the H-LIP. (s-x, s-y) plot the error state trajectories (red and blue circles) inside thecalculated error invariant sets (transparent blue polytopes) in each plane. is extremely low, the robot then has a trivial SSP and cannotstabilize its walking via stepping; if it is too high, the verticalswing trajectory then requires large accelerations to lift-offand touch-down and thus exceeds the joint actuation limits.Similarly, the actuation limits prevent the walking durationfrom being too small to track the swing trajectories. If theduration is too long, the robot can fall over before the swingleg strikes the ground to stabilize it.The changes of the COM height and the step durationchange the S2S approximation of the H-LIP (e.g. Eq. (9)).The implementation of the H-LIP stepping directly respondsto the new S2S dynamics. Note that the vertical COM heightis assumed constant on the H-LIP and that on the robot iscontrolled approximately constant. The height, however, canchange between steps, as long as the vertical dynamics is not Fig. 21. Walking tiles of the robot: the walking in the sagittal plane demonstrated P1 orbits and that in the coronal plane illustrated a P2 orbit.
FasterSlower Higher Lower (v1) (v2) (v3) (v4)
Fig. 22. Illustration of the versatility of the realized walking by varying stepfrequency (v1), step clearance (v2), step width (v3), and COM height (v4). causing significant disturbance to the horizontal dynamics. Thechange of the swing foot clearance can change the impactvelocity and potentially change w . Similarly, the variationon the step frequency also changes the integration of thedynamics error in the continuous domains, which then change w . In the experiment, the qualitative and quantitative effectsof these parameters on w and then E are not analyzed dueto the existence of the horizontal velocity error (from the H-LIP based velocity approximation in the control or the stateestimation in the analysis). Instead, the experiment showsthat versatile walking behaviors are stably generated with theparameter variations on the fly. E. Disturbance Rejection
Lastly, we demonstrate the robustness of the walking withdisturbance rejection on the hardware [6]. Since the H-LIPstepping provides COM state-dependent step size planning(Eq. (34)), the robot instantaneously reacts to external distur-bances. We consider two types of disturbances: external pushesand ground variations. The external pushes directly disturbthe S2S dynamics of the robot; the ground variations change (a)(b)(c)
840 860 880 900 9200.30.40.5 (d)
Fig. 23. The trajectories of the (a) COM height and (b) vertical height of theswing foot in terms of the actual outputs, (c) the desired step width comparedwith the target step size u ∗ L/R (black lines), and (d) the duration of the walking. the domain durations, impact, and vertical COM behaviors,which indirectly disturb the S2S dynamics. We demonstratewalking with lateral pushes from a human operator (Fig. 24)and walking on grassy and uncertain terrain (Fig. 25).With the push disturbances, the error state e y can temporar-ily go outside of the invariant set E y . The stepping controllerthen brings e y back in E y . In terms of the horizontal velocity,the robot is pushed to have large velocities and then thestepping controller drives the robot back to its nominal walkingbehavior. Since the kinematically-feasible step width is verylimited, the robot may take several steps to recover. If the pushis excessive, the robot can fall over due to infeasible u dy .When walking on grassy terrain, the horizontal dynamicsof the robot are disturbed as the soil deforms and the heightof the terrain varies. The continuous horizontal velocities thushave large variations as shown in Fig. 25; the error states,however, still lie inside the invariant sets. (v)(p) (e)nominaldisturbed Fig. 24. Push disturbance rejection on Cassie: (v) the estimated horizontalvelocity in the coronal plane, (p) the horizontal state trajectory (the blue) inthe coronal plane compared with the desired orbit of the H-LIP (the black),and (e) the error state trajectory compared with the error invariant set.
VIII. D
ISCUSSION
After demonstrating our approach in simulation and experi-ment, we now discuss the implications, limitations, extensions,and future directions of the approach in this section.
A. Implications
The H-LIP based gait synthesis and stepping stabilization issuccessfully realized on the complex robot Cassie, demonstrat-ing both robustness and versatility on the walking behaviors.The realization is extremely simple in computation. There areno non-convex optimizations to be solved offline or online.The periodic orbits of the H-LIP and its stepping are allin closed-form. Besides these benefits, there are also severalimplications of the approach as follows.
1) Approximated Analytical Continuous "Gait Library":
The orbit characterization of the H-LIP can be viewed asproviding an approximated analytical "gait library" for thehorizontal COM states of the bipedal robot. The "gait library"of the H-LIP is continuous, i.e., filling the state-space ofthe horizontal COM. Although the horizontal COM of therobot does not necessarily behave identically to the orbit, itconverges closely to the orbit under the H-LIP stepping. Moreimportantly, transitions between "gaits" or non-periodic walk-ing behaviors can be easily realized via the H-LIP stepping(e.g. the case of tracking a varying desired velocity). (v)(p) (e)
Fig. 25. Walking on grassy and uncertain terrain: (v) the estimated forwardvelocity (gray line) and the desired velocity (black dashed line), (p) thehorizontal state trajectory (the red) in the sagittal plane compared withthe desired orbit of the H-LIP (the black) for a walking segment withan approximately constant v dx of . m/s, and (e) the error state trajectorycompared with the error invariant set.
2) Gait Synthesis and Characterization:
The 3D com-position of planar orbits offers a way of synthesizing andcharacterizing 3D bipedal walking gaits. The gait synthesisand characterization via composition of planar orbits canpotentially be extended to other multi-legged systems, e.g.,the bounding behavior on quadrupedal locomotion [59] canbe viewed as producing a P2 orbit in its sagittal plane. Theextension appears to be non-trivial but possible.
3) Model-free Planning:
The H-LIP based approach can beviewed as a "model-free" approach, where the robot modelis not used in the planning. The walking of the H-LIP isshown to approximate the general hybrid nature of alternatingsupport legs in bipedal walking. The planning on the hybriddynamics of all the degrees of freedom (dofs) is encapsulatedinto the control on the horizontal dynamics of the COM; theindividual dynamics of each dof is not specifically described.As a result, the approach can tolerate the imperfections of therobot modeling in the planning of walking.
4) Interpretation of Stability:
The stability of underactuatedbipedal walking is typically understood and analyzed on theperiodic orbit of the robot [8]. The S2S dynamics formulationprovides a different perspective towards understanding the sta-bility of walking. Assuming that the strongly-actuated dynam-ics (the outputs) can be stabilized, the underactuated/weakly-actuated dynamics (the horizontal COM states) are shown tobe directly controlled by the step sizes in the S2S at the steplevel. Stabilization on the underactuated dynamics can thusbe directly synthesized. The stability of the walking is no longer on the periodic orbits but on the discrete horizontalCOM states. B. Limitations and Potential Solutions
We also identify the limitations of our approach along withcertain potential solutions.
1) Vertical COM Behavior:
The vertical COM height iscontrolled approximately constant in each step. As we haveshown, it permits gradual variations of the COM height be-tween steps. It is not yet known if it is possible to dramaticallychange the COM height within a step. One possible solutionto enable this is to employ a model (e.g. a height-varyingpendulum model [60], [61]) that captures both the verticaland horizontal COM behaviors.
2) Pelvis Orientation and Swing Foot Trajectory:
In ourapproach, the pelvis/upper-body orientation is fixed, and theswing foot trajectory is designed in the simplest way possible.Both are not optimized in terms of any criteria, e.g., energyconsumption. It is possible to apply data-driven approaches tofind a low dimensional representation of the energy consump-tion in terms of parameterized trajectories of the swing footor the pelvis. Optimal trajectories can then be constructed onthe swing foot and the pelvis.
3) Performance Accuracy:
The error state e directly de-scribes the performance of the stepping controller which drivesthe robot to a desired walking of the H-LIP. The error isnot controlled to zero but in the error invariant set E . Thereare two ways to further improve the performance in terms ofreducing e . The first is to develop a better approximation of theS2S so that the model difference w is smaller, which will befurther discussed later. The second is to employ a controllerthat can directly reduce the error e ; e.g., integral control ispotentially able to mitigate the error.
4) Kinematic Feasibility:
The robot joints are designed withlimited ranges of motion. Thus, the ranges of the availablestep sizes are also bounded, which then limits the behaviors(i.e. the walking speeds and orbit compositions) on the robot.Additionally, the legs can internally collide with each otherwithin their ranges of motion. This is more evident on Cassiedue to its complex design. The stepping controller presented inthis paper does not systematically take this into consideration.Instead, the kinematic feasibility is reflected on the choices ofthe desired walking of the H-LIP. In practice, this is sufficientto produce safe (despite conservative) walking on the robot. Amore theoretically sound approach should involve a systematicidentification of the kinematic feasibility. Moreover, advancedcontrollers [62] can be explored to include the disturbance andinput bounds, which will be one of our future work.
5) Dynamic Feasibility:
The realized walking behavior isassumed to be dynamically feasible. In other words, thedesired trajectories of the outputs are assumed to be trackablegiven the limitation of the motor design. In the optimization-based controller, the torque bounds are included. However,theoretically, it does not guarantee the trajectories (especiallythe swing foot trajectories) to be well-tracked, e.g., when thewalking duration is chosen too small, the motor joints maynot be able to move fast enough to drive the swing foot to the desired location. In practice, this can be identified empiricallyon the hardware despite the loss of theoretical soundness.
C. Extensions and Future Directions
Besides the potential methods to address the limitations,there are several extensions of the proposed approach.
1) Global Position Control:
The H-LIP based stepping canalso be used for controlling the global position [63] of theunderactuated bipedal robot [64], [65] by including the globalposition in the S2S dynamics. Then the H-LIP stepping canbe used to approximately control the global position of therobot, where the feedback is on the error in terms of the globalhorizontal position, local horizontal position (w.r.t. the stancefoot), and the horizontal velocity of the COM.
2) Walking over Rough Terrain:
The H-LIP model is as-sumed to walk on flat terrain in this paper, so is the robot.The walking synthesis on flat terrain is shown to stabilize therobot walking on grassy terrain with mild height variations.The H-LIP based approach can also be rigorously extended towalk on stairs, slopes and general rough terrains [25]. A linearapproximation of the S2S dynamics can be obtained if thevertical COM position is controlled to keep an approximatelyconstant height from the ground. However, it is challenging tocontrol the vertical COM state on a compliant robot to followcertain desired trajectories, which will be one of our focusesin the future.
3) Improving S2S Approximation:
The S2S of the H-LIPis a linear model-free approximation and renders closed-form controllers for stabilization. The S2S approximation canpotentially be improved, e.g., different dynamics quantitiessuch as the angular momentum [44] can also be explored forimprovement. It is also possible to investigate model-basedapproximations or data-driven approaches (e.g. [33], [39]),which could potentially offer better approximations and thusimprove the performances on the stepping stabilization.
4) On Fully-actuated Humanoid Walking:
The H-LIP basedapproach can also be potentially applied towards walking onfully-actuated humanoids. The foot is then actuated but withlimited controls, which comes from the ankle actuation andzero moment point (ZMP) constraint on the support polygon.The foot actuation helps to control the robot. Therefore, inthe future, we will explore the integration with H-LIP basedapproach and the foot actuation on humanoids for generatinghighly dynamic and versatile behaviors.IX. C
ONCLUSION
This paper presents a Hybrid Linear Inverted Pendulum(H-LIP) based approach to synthesize and stabilize 3D un-deractuated bipedal walking. Periodic orbits of the H-LIP aregeometrically characterized in its state space and then orthog-onally composed for 3D walking. The walking behaviors ofthe H-LIP are then approximately realized on the robot viathe H-LIP based gait synthesis and stepping stabilization. Theproposed approach is successfully realized on the 3D underac-tuated bipedal robot Cassie in both simulation and experiment.The implementation is straight-forward and computationally-efficient. The realized walking behaviors are demonstrated tobe both versatile and robust. X. A
PPENDIX : PROOFS ON THE
H-LIP
Proof of Thm. 1:
Combining Eq. (7) and Eq. (5) yields therelation between the boundary position and velocity (theorbital slope), and p + = − p − . Plugging this into Eq. (5) yields u = 2 p − + T DSP v − . Proof of Thm. 2:
We can first show that, any state on the line v = − σ p + d in the beginning of the SSP will flow tothe line v = σ p + d at the end of the SSP, with d beinga constant. This is easily proven by using Eq. (7). Then anarbitrary state is chosen, and it is easy to show the step sizesmust be u L/R = 2 p − L/R + T DSP v − L/R to get a two-step orbit.To derive d from the desired velocity, we first select anarbitrary state [ p , − σ p + d ] as the initial state of the P2orbit. The rest of the boundary states can be calculated asfunctions of p and d . Then the sum of the step sizes is u ∗ L + u ∗ R = d ( T DSP + T DSP cosh ( T SSP λ )+ λ sinh ( T SSP λ )) whichis equal to v d T . Solving this for d yields Eq. (23). Proof of Prop. 3:
This proof follows the previous paragraphby starting an arbitrary state [ p , − σ p + d ] as the initialSSP state of the P2 orbit. Letting u L = u R and solving for p yields p = − d sinh ( T SSP λ )2 λ . The rest follows immediately. Proof of Prop. 4:
This proof is similar to the proof of Them.2. First, it is easy to show that any initial state on the line v = − σ ( p + d ) will flow to the line v = σ ( p + d ) after T SSP .Similarly, any initial state on the line v = − σ ( p − d ) willflow to the line v = σ ( p − d ) after T SSP . Then an arbitrarystate is chosen on the line v = σ ( p + d ) (or equivalently v = σ ( p − d ) ), and it is easy to show the step sizes must be u L/R = 2 p − L/R + T DSP v − L/R to get a two-step orbit.XI. A
PPENDIX : B
ÉZIER P OLYNOMIALS
The Bézier polynomials are used to design the desiredoutput trajectories. The Bézier polynomials are defined as: b ( t ) = β (¯ t ( t ) := tT ) := (cid:80) Mk =0 β k M ! k !( M − k )! ¯ t k (1 − ¯ t ) M − k , where ¯ t ∈ [0 , and β k are the coefficients of the Bézierpolynomial. The following table lists the coefficients of thehand-designed Bézier polynomials, where N indicates a rowvector of size N with all elements being 1. On Notation Bézier Coefficients β Horizontal swing foot b h [0 , , ] Vertical swing foot b v [0 , z maxsw , , z negsw ] Vertical COM height b h [0 , , ] R EFERENCES[1] X. Xiong and A. D. Ames, “Orbit characterization, stabilization andcomposition on 3d underactuated bipedal walking via hybrid passivelinear inverted pendulum model,” in . IEEE, 2019, pp. 4644–4651.[2] ——, “Continuous gait generation for 3d underactuated bipedal walkingvia hybrid linear inverted pendulum model,” in
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Xiaobin Xiong is currently a PhD candidate inMechanical Engineering at the California Institute ofTechnology (Caltech). He received his B.S. degreefrom Tongji University, Shanghai, China in 2013,and his M.S. degree from Northwestern University,Evanston, Illinois in 2015. At Northwestern, heworked with Dr. Kevin M. Lynch and Dr. PaulUmbanhowar on nonprehensile manipulation. HisPhD study mainly focuses on legged locomotion,and his interest lies in controls for robotic dynamicalsystems. He is the Amazon Fellow in AI at Caltech.He is also the recipient of the IROS-Robocup Best Paper Award in 2019.