Bipedal Robotic Running on Stochastic Discrete Terrain

Ayush Agrawal and Koushil Sreenath

Abstract— To navigate over discrete terrain with large dis-placements in the stepping locations, bipedal robots have to beable to perform agile and dynamic maneuvers such as jumpingor running while also satisfying strict constraints on footplacement and ground contact forces. In this paper, we analyzethe problem of bipedal running over stochastically varyingdiscrete terrain with large changes in step lengths. Specifically,our method is based on designing a library of running gaitsthat are two-step-periodic. We illustrate the capabilities of theproposed controller through numerical simulations of a five linkunderactuated robot RABBIT, running over discrete terrainwith step lengths that vary between 0.6m and 1.2m. This isabout 1.5 times the robot’s leg lengths and twice the step lengththat could have been achieved by walking.

I. INTRODUCTION

Legged robots have the promise of being able to servein applications such as in space and urban exploration, aspersonal robots in homes and in search and rescue operations.It is their inherent morphology and mechanical structurethat renders them as potentially superior candidates overtheir wheeled counterparts. A key task in such applicationsis the ability to locomote over discrete footholds such asin unstructured environments like wooded paths or overa flight of stairs in indoor environments. This, however,introduces several challenges and constraints including (a)Strict constraint on foot placement, (b) Friction constraintsand (c) Input constraints. Violation of any of these constraintswill render the system unstable.

A. Related Work

The problem of robotic legged bipedal walking overdiscrete terrain has been studied in the past, with a widevariety of techniques being used. Early methods for foot-step planning, such as in [11], relied on simple models likethe 3D inverted pendulum and cart-table models to generatewalking patterns to walk over randomly generated steppingstones. In [20], the authors present a method based on theconcept of capture points to control a bipedal robot to walkover discrete steps. The controller regulates the center-of-pressure on the stance foot so as to ‘guide’ the capturepoint to the desired stepping location. More recently, in [23],the authors present a centroidal momentum based controllerfor a high-dimensional robot ATLAS, to walk over partialfootholds including line and point contact surfaces. In [4], amixed integer quadratically constrained quadratic program ispresented for footstep planning of a humanoid robot to walk

Ayush Agrawal and Koushil Sreenath are with Department of MechanicalEngineering, University of California at Berkeley {ayush.agrawal,koushils}@berkeley.edu

This work was supported in party by NSF Grant IIS-1834557.

Fig. 1: Illustration of the stepping stones problem. The goalof the feedback control design is for the bipedal robot totraverse over a set of discrete terrain with wide gaps. Such aterrain can only be traversed by performing agile maneuverslike running or jumping.

on uneven terrain with obstacles. In [17], the authors presenta method based on Control Barrier Functions (CBFs) todesign feedback controllers for high dimensional 3D robotsto walk over stochastically generated discrete steps withchanging step heights and step lengths. In [14], the authorspropose a method that combines a one-step-periodic gaitlibrary approach with a CBF based feedback controller thatsignificantly improved the performance as in [18]. In [6], theauthors propose a bio-inspired controller based on CentralPattern Generators (CPGs) to achieve step length and stepheight modulations over a wide range for bipedal walking.

With regards to bipedal robotic running, while numerousstudies have been carried out, there have been limited studieson agile locomotion, like running, of legged systems overdiscrete terrain. One of the earliest works on running overrough terrain was by Hodgins and Raibert [9]. The authorspresented three intuitive control designs for regulating thestep length for a bipedal robot that could run over roughterrain including stairs with changing step heights. In [19],the authors propose a reinforcement learning based approachto control a physics-based legged character to navigate overterrain with wide gaps, steps and obstacles. The authors areable to translate their method to a wide variety of high-dimensional characters including a 21-link planar dog anda 7-link planar biped. In [5], the authors propose an intuitivedead-beat control strategy based on a point-mass modelfor running over 3D stepping stones. The authors performnumerical simulations on a point-mass model with masslesslegs for running over 3D stepping stones. In [7], the authorspresent a neuromuscular controller for bipedal running.

In our most recent work [15], [16], we presented a methodto design a feedback controller for an underactuated leggedrobot to walk over discrete terrain with stochastically varyingstep heights and step lengths. The method relied on pre-

computing a library of a small number of walking gaits(through an offline nonlinear program) that were two-step-periodic and parametrized by the step lengths and stepheights in the first and second steps. The controller thenperformed a bilinear interpolation between the different gaitsbased on the current and desired step lengths/heights duringrun-time. By switching among a set of two-step-periodicgaits, the controller was able to achieve aperiodic walking,with precise footstep placement over randomly generateddiscrete terrain. The control method was successfully im-plemented on an underactuated bipedal robot, ATRIAS, overa wide range of complex terrain.

B. Problem Statement and Approach

In this paper, we study the problem of bipedal roboticrunning over randomly varying discrete terrain with largechanges in step lengths (Figure 1). In particular, we develop amodel-based feedback controller for an underactuated leggedsystem that does not have knowledge of the entire terrainahead of time but only the position of the next steppinglocation is known. By only using a one-step preview ofthe upcoming stepping location, the method renders itselfcapable of being combined with vision sensors (like camerasand LiDAR) to estimate the stepping locations.

Motivated by the success of the ‘two-step-periodic’ gaitlibrary approach, we extend this approach to the case ofbipedal running. A primary advantage of the proposedmethod is that, unlike most other methods that rely onsimplifications of the system dynamics, it considers thefull nonlinear hybrid dynamics of the system, both, duringoffline gait generation and during the control phase. Giventhe current state-of-the-art in trajectory optimization fornonlinear hybrid systems and computational power, anotheradvantage of our method is that it is easy to implementon a physical system. A key reason for the success of thetwo-step-periodic gait approach in [16] was it allowed forsmooth transitions between walking gaits at different step-lengths/heights, thereby inherently preventing violations infriction cone and unilateral ground reaction force constraintsand by using only a small number of gaits in the gait library.The method, therefore, seems promising to use in richerlocomotion behaviors such as running and jumping.

The problem of robotic running over discrete footholds,however, places some additional challenges which stemsfrom the loss of control authority of the angular momentumduring flight phases (i.e. when the robot is completely in theair). Additional challenges also arise due to the increasednumber of possible hybrid modes of the system. These arepotential reasons for why the applicability of the ’two-step-periodic’ gait library approach might not be straightforward.In particular, we show that by reasoning about the dynamicsand by the proper choice of output variables to be controlled,the two-step-periodic gait library approach can be extendedto the case of running as well.

C. Contributions

The key contributions of this paper are as follows:

1) We extend the two-step-periodic gait library approach,initially presented in [15], [16] to develop a con-trol strategy for bipedal robotic running over discretefootholds that follows strict constraints on the statesand control inputs of the system;

2) By doing so, we are able to expand the capabilities ofthe robot to traverse over wider gaps in the discretefootholds that was not possible with only walking;

3) We show that, by a proper selection of output variablesto be controlled, we can significantly reduce the errorin the foot placement locations while running.

D. Organization

The rest of the paper is organized as follows: In SectionII, we present the hybrid model for running, followed by thetrajectory optimization and control design method in SectionIII. Finally, in Section IV, we present results from numericalsimulation of a five link underactuated bipedal robot.

II. DYNAMICAL MODEL FOR RUNNING

In this section, we present a brief overview of the dynam-ical model of a planar five-link two legged robot for runningbehaviors. These models will be used for generating optimaltrajectories as well as for control synthesis. The specificrobot under consideration is RABBIT. More details aboutthe dynamical model can be found in [2].

Figure 2 illustrates a schematic diagram ofthe robot. The configuration variables, defined asq :=

[pxhip p

zhip qT q1R q2R q1L q2L

]Tinclude

the world frame position of the hip[pxhip p

zhip

]T, world

frame orientation of the torso qT and the relative joint anglesof the thigh q1 and shin q2 links. The subscripts L and R referto the left and right links respectively. For future reference,we define the actuated joints qa :=

[q1R q2R q1L q2L

]The equations of motion are derived using the method ofLagrange and have the form during stance,

D(q)q̈ + C(q, q̇)q̇ +G(q) = Bu+ JTF, (1)

where u ∈ R4 are the control inputs that actuates each of thejoints qa, F ∈ R2 are the ground reaction forces at the stancefoot and J := ∂pst∂q ∈ R

2×7 is the Jacobian of the stance footposition pst. We note that during flight, since there are noexternal contact forces acting on the robot, F ≡ 0.

Like walking, the dynamics for running motions is alsohybrid. Specifically, running comprises of alternating phasesof single-support Σs and flight Σf phases as illustrated inFigure 3. The hybrid dynamics is written as

Σs :

{ẋ = fs(x) + gs(x)u, (x, u) /∈ Ss→fx+ = ∆s→f (x

−) , (x, u) ∈ Ss→f

Σf :

{ẋ = ff (x) + gf (x)u, x /∈ Sf→sx+ = ∆f→s (x

−) , x ∈ Sf→s(2)

Fig. 2: Generalized coordinates of the bipedal robot RAB-BIT.

Stance Flight

Fig. 3: Illustration of the domains in running.

Here Ss→f := {(x, u) | F z(x, u) = 0} is the switchingsurface corresponding to the transition between stance andflight domains and is defined as the set of states and controlinputs such that the vertical ground reaction force F z(x, u)is zero (this corresponds to the case when the stance footlifts off from the ground). Similarly, we define the switchingsurface Sf→s := {x | pzsw(x) = 0} corresponding to thetransition between flight to stance as the set of states suchthat the vertical component of the position of the swing footis zero.

In addition, ∆s→f = I is the identity operator and ∆f→sis obtained from rigid impact dynamics. The vector fieldsf(x) and g(x) are obtained for stance and flight phases usingLagrange’s equations of motion (1).

In the next sections, we present our control approach basedon the ‘two-step-periodic’ gait library.

III. HYBRID ZERO DYNAMICS BASED CONTROL

In this section, we briefly present the Hybrid Zero Dynam-ics (HZD) framework [22], [21] which uses the dynamicalmodel presented in Section II to generate periodic gaits anddesign feedback controllers for running. The HZD methodbegins by selecting a set of outputs ya for the hybriddynamical system as in (2). Driving these outputs to a set

of desired quantities yd define how the various links of therobot move. The HZD controller then implements an Input-Output (IO) Linearizing controller to drive the outputs ya toyd.

A. Output Selection

In this section we present our choice of outputs ya (alsoknown as virtual constraints). We note that several validchoices for the outputs exist. As in [15], a candidate forya is the set of actuated joint angles qa. However, sincewe are interested in achieving precise step-lengths, which isachieved through the flight phase, we choose the horizontalvelocity of the center of mass as one of the outputs duringthe stance phase. This choice of output is motivated by thefact that the horizontal distance achieved during flight isdependent on the exit-velocity of the stance phase (velocityof the center of mass at the instant before entering flightphase). We also note that this is a relative degree 1 output[10] [1]. The complete set of outputs during the stance phaseis chosen as

ysa =

vxcomqst1qsw1qsw2

. (3)During flight, we choose the following outputs to be

controlled

yfa :=

qTqst2qsw1qsw2

(4)The superscripts st and sw in ys denote stance and swing

legs respectively, and st = L/R (and sw = R/L), dependingon whether the Left or Right Leg is in stance respectively. Inyf , the superscripts st and sw denote the stance and swinglegs in the preceding stance phase.

The desired outputs yd := yd (τp, αp) are parametrized byBézier splines, where αp are the coefficients of the Bézierspline and τp is a phase variable that monotonically increasesfrom 0 to 1 and p ∈ {s, f}. Specifically, we will find theBézier parameters αp and phase variable parameters througha Nonlinear Program such that enforcing the outputs ya to thedesired outputs yd (τp, αp) through a feedback controller willresult in a two-step-periodic solution for the hybrid modelin (2). This is schematically illustrated in Figure 4. We notethat, there are several choices for the phase variable τp. Inparticular, we choose the normalized absolute stance leg-angle qstLA as the phase variable during stance and normalizedtime during flight,

τs :=qstLA − qstLA,max

qstLA,max − qstLA,max, (5)

τf :=t− tmin

tmax − tmin, (6)

Fig. 4: Illustration of two-step-periodic gait design. Weoptimize over two running steps with constraints on thestep lengths l0 and l1 in the first and second running stepsrespectively. The periodicity constraint enforces the states ofthe robot at the end of the second step x2 to return to thestates at the beginning of the first step x0.

with qstLA defined as

qstLA := qT + qst1 +

qst22− π

2; (7)

qstLA,max, qstLA,min, tmin, tmax are constants to be deter-

mined. For future reference, we collect the constant param-eters used in the gait phase variable definitions above intothe following vectors,

θs :=

[qstLA,maxqstLA,min

], (8)

θf :=

[tmaxtmin

]. (9)

B. Two-Step-Periodic Gait Design

Having presented the hybrid dynamical model for runningand the choice of outputs ya to be controlled, we now presenta method to find the parameters αs, αf , θs and θf , such thatthe resulting gaits are two-step-periodic.

The two-step-periodic gait design involves obtaining gaitparameters such that the post-impact states of the systemafter two running steps return to the initial states at the startof the first step. The gaits are parametrized by the step-lengths in the first and second second running step l0 and l1respectively and we define the set of parameters as

P (l0, l1) := {αs (l0, l1)αf (l0, l1) , θs (l0, l1) , θf (l0, l1)}.(10)

Subsequently, we find parameters P (l0, l1) for (l0, l1) ∈L× L to build a library of gaits,

G := {P (l0, l1) | (l0, l1) ∈ L× L}, (11)

Motor Torque |u| ≤ 10Nm

Friction Cone∣∣∣∣FhstFvst

∣∣∣∣ ≤ 0.6Vertical Ground Reaction Force during stance F vst ≥ 0N

Swing Foot Clearance during stance hf | ≥ 0.05m

TABLE I: Optimization constraints

where L is a predefined set of step lengths. Specifically, wechoose L = {0.6, 0.8, 1.0, 1.2}, with a total of 16 gaits inthe gait library.

The problem of obtaining the gait library G is cast as anonlinear program with the objective function taken as theintegral of squared torques over step length:

J =

∫ T0

||u(t)||22 dt. (12)

and constraints for the optimization are formulated as inTable I.

In addition to the above constraints, we also need toguarantee the periodicity of the gait through the periodictyconstraints:

1) The initial state at start of the first stance phase is givenby x = x+0 with corresponding (initial) step length l0.

2) Transition constraints between stance and flight: Thestate at the end of the first stance phase is equalto the state at the beginning of the first flight phase(corresponding to the step length l0).

3) Step Length constraint: The step-length constraint isenforced as the difference between the position of thestance foot at the beginning of the flight phase and theposition of the swing foot at the end of the flight phasebeing equal to the desired step-length l0.

4) The state at the end of the first flight phase (beforeimpact) is x = x−1 with (resulting) step length l0.

5) Impact constraints at the end of the first flight-phaseare enforced as x+1 = ∆(x

−1 ).

6) The initial state at start of the second stance phaseis given by x = x+1 with corresponding (initial) steplength of l1.

7) Transition between Stance and Flight phase: The con-straint is enforced as in 2 between the second stanceand flight phase.

8) The state at the end of the second flight phase (beforeimpact) is x = x−2 with (resulting) step length of l2.

9) Impact constraints at the end of the second step areenforced as x+2 = ∆(x

−2 ).

10) Periodic constraints are then enforced as x+2 = x+0 ,

resulting in l2 = l0.The generation of the two-step-periodic running gaits

using direct collocation with the specifications mentionedabove, involves discretization of each phase in time by aspecified number of nodes N ,

0 = t0 < t1 < t2 < · · · < tN = T, (13)

Fig. 5: Snapshots of the robot running over the discrete footholds using the control method presented in this paper.

Fig. 6: Illustration of gait interpolation. The desired gaitparameters are obtained based on the desired step lengthof the previous step ld0 and the desired step length of thecurrent step ld1 . Points marked by blue circles denote gaitparameters in the gait library. Red star denotes the gaitparameters based on the desired step lengths and obtainedusing bilinear interpolation of the existing gaits in the gaitlibrary.

where T represents the time to impact. In particular, weuse N = 10 for each phase and use the method of Directcollocation to solve the following trajectory optimizationproblem,

J = minu(t)

∫ T0

||u(t)||22 dt (14)

st. x(t) =

∫ t0

f(x(t)) + g(x(t))u(t)dt

c(x(t), u(t)) ≤ 0, 0 ≤ t ≤ T.

Here, c(x(t), u(t)) represent the physical constraints de-scribed in Table I as well as periodicity constraints. Thedesired gait parameters P (l0, l1) can be extracted from theoptimal state trajectories through a simple Bézier curve fit.We use the open-source optimization and simulation toolboxFROST [8] to perform the above optimization. We referthe reader to [12] for more details on the specifics of thetrajectory optimization scheme. We then generate the gaitlibrary G by obtaining the parameters P (l0, l1) for differentvalues of (l0, l1) ∈ L×L through the NLP described above.

C. Control Design

We use an input-output linearizing controller as in [22],[21]. We first define the outputs to be regulated to zero asthe difference between the actual and desired quantities yaand yd as:

yp := ypa − ypd, p ∈ {s, f}. (15)

We further differentiate between the relative degree oneand relative degree two outputs during stance as

ys1 :=[1 0 0 0

](ysa − ysd) , (16)

ys2 :=

0 1 0 00 0 1 00 0 0 1

(ysa − ysd) . (17)The IO Linearizing controller is then given by,

u = (Ap)−1

(−Bp (x, αp, θp) + vp) , p ∈ {s, f}, (18)

where Ap is the decoupling matrix and is defined as

As :=

[Lgy

s1

LgLfys2

](19)

Af := LgLfyf (20)

and Bp is defined as

Bs :=

[Lfy

s1

L2fys2

](21)

Bf := L2fyf . (22)

Here, Lfy and Lgy denote the Lie-derivatives of the outputy with respect to the vector fields f and g. Further, v isa feedback term, which could be, for example, a linearfeedback controller

vs :=

[−K1ys1

−K2ys2 −K3ẏs2

](23)

vf := −K4yf −K5ẏf , (24)

where Ki > 0, i = 1, 2, ...5 are appropriate gain matrices.The specific values of the parameters αp and θp depend on

the desired step-lengths of the preceding step ld0 and currentstep ld1 . The superscript d represents a desired quantity. Werestrict the desired step lengths ld1 to be within the rangeof L. This is schematically illustrated in Figure 6 We usebilinear interpolation of the gait parameters P (10) as in[15] to compute the gait parameters P (l0, l1) correspondingto the step lengths ld0 and l

d1 .

Remark: 1. In our method, we perform a linear interpolationof the Bézier parameters that parametrize the periodic gaitsrather than the time trajectories of the states. The inter-polated gait is therefore also a smooth Bézier curve. Themain motivation behind doing so is since the desired steplength is different in every step, planning for each of these

0.6 0.7 0.8 0.9 1 1.1 1.2desired step length (m)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

actu

al s

tep

leng

ht (m

)

Fig. 7: Resulting location of foot steps from 300 steps, over10 experiments (30 steps per experiment). Outer dashed linesindicate a 5cm deviation from the desired step length. Reddots denote the actual value of the step length obtained fromsimulation.

transients would cause an explosion of planned trajectories.Instead, we plan for periodic gaits - specifically a two-stepperiodic gait to build a library of gaits G as in (11). Duringimplementation, depending on the current step length l0 andthe desired step length l1 of the next step, we select the fourclosest gaits (in terms of step length) from G and performa bilinear interpolation (See Figure 6) resulting in P (l0, l1).Specifically, we only use the first step of the interpolatedtwo-step periodic gait and then switch to another interpolatedgait at the end of the first step. This makes the transientssmoother (as opposed to large jumps in the desired outputswhich generally causes a violation of unilateral constraintssuch as friction constraints and input constraints) as the exitstate at the end of the first step is close to the entry state ofthe periodic gait used for the second step.

Remark: 2. The proposed method performs a linear inter-polation as opposed to a nonlinear interpolation. There arecertainly several ways to represent this nonlinear model.For example, in [3], the authors propose Support VectorMachines (SVMs) and neural network model to interpolatebetween the different gaits.

IV. NUMERICAL VALIDATION

In this section, we present our numerical results fromsimulation of 5-link underactuated robot RABBIT. We per-formed multiple simulations with randomly varying desiredstep lengths. The desired step lengths were sampled from auniform distribution between 0.6m and 1.2m which is abouttwice the desired step length reported in [18] and about 1.5times the robot’s leg length.. Figure 5 presents snapshots of

Fig. 8: Motor Torques from one of the simulations. The topand bottom red lines indicate the maximum and minimumallowable control inputs.

0 2 4 6 8 10 12 14time (s)

-100

0

100

200

300

400

500

600

700

800

norm

al g

roun

d re

actio

n fo

rce

(N)

Fig. 9: Vertical Ground Reaction Forces from one of thesimulations.

the robot running over one realization of the discrete terrain.Figure 7 illustrates the foot step locations along with thedesired stepping locations from ten such simulations with30 steps per simulation. We note that the average step-lengtherror increases as the desired step-length increases.

In all our simulations, the foot placement was accurate to±5cm of the desired step lengths while all other constraintssuch as input limits (Figure 8) and unilateral vertical groundreaction force (Figure 9) constraints were met. The averagerunning velocity was 1.8m/s.

Remark: 3. As mentioned in Section III-A, a candidatechoice for the outputs during the stance phase are theactuated joint angles qa (all outputs have relative degree two).However, the results we obtained using these outputs werevery different from those obtained using the outputs definedin (3) with one relative degree one output. In particular, weobserve a significantly poor performance in the placement offootsteps with a maximum error of 39cm. We attribute thesuccess of the outputs in (3) to the fact that the horizontaldisplacement of the center of mass depends solely on its exitvelocity during stance. Regulating the exit velocity directly

0.6 0.7 0.8 0.9 1 1.1 1.2

desired step length (m)

0.6

0.7

0.8

0.9

1

1.1

1.2

actu

al s

tep

leng

th (

m)

Fig. 10: Resulting location of foot steps from 50 steps froma single simulation using qa (vector relative degree 2) as theoutputs during stance phase. Outer dashed lines indicate a39cm deviation from the desired step length. Red dots denotethe actual value of the step length obtained from simulation.

during stance will potentially lead to an accurate step-lengthat the end of the flight phase and hence smaller errors in steplength. Figure 10 illustrates the performance of the controllerusing the actuated joints qa as the outputs.

V. CONCLUSION

In conclusion, we have presented a control strategy forbipedal robotic running over stochastically varying discreteterrain and potentially increased the range of step lengthsto twice that could have been achieved only by walking.The controller maintains the stability of the robot whilerespecting critical safety constraints such as constraints onfoot placement and ground reaction forces. With properchoice of the outputs to be controlled, the resulting steplength from simulation is accurate to 5cm of the desired steplength. While we are yet to formally prove any theoreticalguarantees on switching between the different periodic gaits,we provide some intuitive explanation about the success ofthe method as in Remark 1. A potential direction to addressthis are the stability conditions for switching controllersbetween different exponentially stable periodic orbits asprovided in [13]. Moreover, the method here is simple toimplement and requires a small number of two-step periodicgaits to run over a terrain with a wide range of step lengths.

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