stabilizing series-elastic point-foot bipeds using whole-body

IEEE TRANSACTIONS ON ROBOTICS, VOL. XX, NO. YY, JULY 16, 2016 1

Stabilizing Series-Elastic Point-Foot Bipeds usingWhole-Body Operational Space Control

D. Kim, Y. Zhao, G. Thomas, B. Fernandez, and L. Sentis

Abstract—Whole-Body Operational Space Controllers(WBOSC) are versatile and well-suited for simultaneouslycontrolling motion and force behaviors which can enablesophisticated modes of locomotion and balance. In this paper,we formulate a WBOSC for point-foot bipeds with SeriesElastic Actuators (SEA) and experiment with it using a teensize SEA biped robot. Our main contributions are on: 1)devising a WBOSC strategy for point-foot bipedal robots, 2)formulating planning algorithms for achieving unsupporteddynamic balancing on our point foot biped robot and testingthem using WBOSC, and 3) formulating force feedback controlof the internal forces – corresponding to the subset of contactforces that do not generate robot motions – to regulate contactinteractions with the complex environment. We experimentallyvalidate the efficacy of our new whole-body control andplanning strategies via balancing over a disjointed terrain andattaining dynamic balance through continuous stepping withouta mechanical support.

I. INTRODUCTION

During DARPA’s Robotics Challenge held in 2013 and2015, various humanoid robots were required to execute manypractical tasks [1]. Most of these robots implemented a certainclass of controllers called whole-body controllers (WBC),which are easily reconfigurable to accomplish multiple con-trol objectives under environmental constraints. A perceivedlimitation of these robots was that their motions were muchslower than their human counterparts. To remedy this issue,future WBC methods must significantly improve the speed andaccuracy of the executed motions without losing their naturalversatility. Additionally, humanoid robots with series elasticactuators are increasingly being sought for safe whole-bodyphysical interactions among people and to move in clutteredenvironments. However, controlling fast motion behaviors withhigh precision on robots with series elastic actuators is quitechallenging [2].

To demonstrate that WBCs can achieve highly dynamicand versatile behaviors on biped robots with series elasticactuators we devise here: 1) software and hardware controlmethods to achieve whole-body operational space behaviorsthat are stable and accurate, 2) feedback control strategies toaccurately regulate internal force behavior that yield pullingor pushing forces on complex terrains, 3) a pose estimator thatcan overcome the limitations of low-cost inertial measurementunits (IMU) and be well suited for highly dynamic behaviors,and 4) a locomotion planner and controller that can stabilizea point-foot robot on flat and rough terrains.

We consider WBCs to be very important for all typesof humanoid robots because of their versatility and intuitive

All authors are with the Department of Mechanical Engineering, TheUniversity of Texas at Austin, Austin TX 78712-0292

usage. Their versatility implies that WBCs designed for onetype of robotic structure could be transferred to a differenttype of robot with minimum supervision. Additionally, WBCshave the advantage of decoupling task dynamics. In thecase of WBOSC, it is possible to simultaneously controlmotion tracking behaviors and feedback-based internal forcebehaviors. Internal forces correspond to the subset of contactforces that do not generate robot motions. In a previouspaper we discussed the need to reduce joint torque controllergains to achieve better task motion tracking performance[2]. By combining task motion control with internal forcefeedback control, WBOSC achieves precise tracking of theinternal forces regardless of the tracking setpoint of the jointtorque controllers. This capability is unique to WBOSC andespecially important when moving in sophisticated types ofterrains that require the feet to apply tensions or compressionsto the terrain by precisely controlling the contact forces.

With no doubt, WBCs, in general, offer many advantagesfor versatility and contact interactions. And to get their highestperformance in terms of whole-body motion and force interac-tions they benefit by being implemented in torque controlledrobots such as those using series elastic actuators. Previously,WBOSC had not been formulated for and experimented withpoint-foot bipedal robots with series elastic actuators. There-fore, the main contribution of this paper is to design a WBOSCfor highly dynamic point-foot bipedal robots and assess itsperformance capabilities for various types of locomotion andbalance behaviors.

In most cases, robots with unstable dynamics, such as point-foot bipeds, use control structures that are customized to theparticular robot structure in hand. This customization allowsthem to perform optimally for the designed task but makes itdifficult to improvise controllers on demand or transfer them toother types of robotics structures. One of the most successfulpoint-foot biped robots is ATRIAS [3], which can continu-ously walk unsupported using small passive feet. However,its locomotion controller highly relies on the characteristicsof the hardware, whose natural dynamics is similar to thespring-mass model. The reliance on hardware characteristicsmakes the controller less versatile than WBCs. Additionally,ATRIAS has not been shown to regulate internal forces ondisjointed terrains for gaining balance. Compared to ATRIASwhich uses small passive feet to stabilize the yaw motion, ourbiped robot Hume does not have any feet. MABEL [4] andMARLO [5] also have clever controllers designed to follow theHybrid Zero Dynamics, the latter robot being able to achieveunsupported locomotion for several steps. But identifying theHybrid Zero Dynamics is a non-trivial process and difficult toadapt to other locomotion patterns. MARLO achieves 20 steps


of unsupported walk [5]. In comparison, our biped robot Humeachieves 18 steps of unsupported balance. Some quadrupedrobots, such as StarlETH [6] and HyQ [7], use WBCs forlocomotion and multi-contact tasks. They are also able tobalance on disjointed terrains. Compared to WBOSC they donot use feedback control of the internal forces. With respect tobalancing capabilities, quadrupedal robots are fundamentallydifferent compared to point-foot bipeds since the additionaltwo legs provide multiple support points to regain and maintainbalance. For instance, when a quadruped robot receives akick it can reposition two or three of its legs to create alarge support polygon for balance adaptation and center ofmass control. On the other hand point-foot bipeds do nothave a supporting polygon and therefore need to rely on therobustness of their locomotion planners and the precision ofthe foot placement controller.

Other inverted pendulum based algorithms such as the ZeroMoment Point [8], the Capture Point [9], or the DivergentComponent of Motion [10] have not been experimentallydemonstrated in unconstrained point-foot robots to date. Pre-viously, we introduced a new foot placement planner [11]which we extend for this paper for correctness and physicalimplementation via 1) formulating a stability analysis, 2)extending it to rough terrains, 3) extending it to unsupportedpoint-foot dynamic locomotion, and 4) explaining the effectof ground impacts on the robot’s trajectory. We use this loco-motion planner and controller on the physical biped systemfor flat terrain locomotion and we also present a walkingdemonstration on rough terrain in a simulation.

The paper is structured as follows. In Section II, we presentbackground of WBOSC and point-foot biped locomotion.Section III presents details of our experimental system includ-ing hardware and software. We explain gain scheduling andinternal force feedback control in Section IV. In Section V,we present our foot placement planner. Section VI presentsexperimental results. Finally, Section VII discusses interestingissues and future issues to be addressed. Additionally, the Ap-pendix briefly summarizes WBOSC, establishes the equationsthat we use for the prismatic inverted pendulum, and explainsour state estimation approach for estimating the robot’s bodyorientation.

II. RELATED WORK

A. Whole-Body Controllers

“Whole-Body Operational Space Control” (WBOSC) is atorque-level, prioritized, projection-based, multi-task, opera-tional space, feedback controller for floating based robotsoriginally outlined in [12] and extended to include internalforce control in [13]. It uses null-space projections to accom-plish task prioritization and computes contact and constraintconsistent reduced Jacobian matrices to project multiple taskimpedance controllers into joint torques. WBOSC also controlsinternal forces using the internal force matrix, which residesin the orthogonal space to joint accelerations and by extensionto operational space motion tasks.

Many experimental approaches for WBC rely onimpedance-based techniques or inverse dynamics and

contain optimization algorithms to solve for constraints andcontacts. One of them is the pioneering work by [14] whichformulates task space impedance controllers to fulfill contactand task constraints; the concept of impedance control wasfirst presented in [15]. [16] represents the first implementationthat we know of to achieve full dynamic model based taskcontrol with contact constraints on a humanoid robot. In[17] the implementation of hierarchical inverse dynamicalgorithms using a quadratic program is presented anddemonstrated on a Sarcos biped robot. Experiments includebalancing while withstanding external forces, balancing ona moving platform, and standing on a single foot. In [18] atorque-based WBC is presented for controlling the Atlas andValkyrie humanoid robots. A quadratic program solver is usedto minimize the error with respect to the desired momentumrate, contact forces, and task accelerations. In [19] WBC withinequality constraints via inverse dynamics and a quadraticprogram solver is proposed for the humanoid robot HRP-2.The algorithm is used offline to generate trajectories that arethen tracked by a real-time controller. In [20] a torque-basedWBC focused on optimizing multi-contact wrenches derivedfrom desired center of mass movements is presented. Thesestudies aim at controlling humanoid robots with actuatedankles, and thus do not consider the underactuation natureand fast locomotion dynamics of point-foot biped robots.

Many of the previous WBCs formulate structures for con-trolling contact forces. There are many styles for regulatingthose forces. In [14], desired center of mass accelerations andtask wrenches are projected and summed up yielding a totalcenter of mass wrench. When there are multiple contacts,an optimization algorithm is applied to optimally distributethe contact wrenches before projecting them to the center ofmass. A center of mass Jacobian transpose projection is usedto achieve the desired global wrench. One of the problemswith this method is that the forward kinematics model usedfor controlling the center of mass does not account for contactconstraints which could result in less accurate motion trackingand force regulation. Nonetheless, the method allows theSarcos humanoid robot to balance fairly well but with visiblyslow center of mass motions.

One great controller is the Full Body Controller by [21]focusing on two stages consisting on first solving for the con-tact forces then followed by solving the full-body constrainedinverse dynamics problem. To deal with multiple contacts,a quadratic program is posed based on a simplified relationbetween center of mass and task wrenches to solved for thecontact forces that comply with friction constraints. Althoughthis work was one of the first to produce locomotion on aphysical humanoid robot using WBCs, its algorithm suffersfrom two problems: First task wrenches are projected intothe center of mass using a heuristic model. Second, it solvesthe inverse dynamics problem by artificially incorporatingfictitious task wrenches yielding a physically incorrect whole-body dynamic equation.

Another successful WBC is the Momentum-Based Con-troller by [22]. Variations of this controller were used to obtainthe second place for the DARPA Robotics Challenge. Thecontroller has a similar two-stage nature than the previous


Full Body Controller. But it has multiple advantages. First,for the contact solver, it uses the centroidal momentum matrix[23], [24] which correctly describes the relationship betweenexternal wrenches, i.e. contact wrenches and gravitation forces,and joint accelerations. Then it solves a quadratic problem thatincludes motion task constraints. In turn, the output reactionforces are dynamically correct with respect to center of massbehavior and motion tasks. Second, using the centroidal mo-mentum matrix it solves for joint accelerations, which is amore generic output than reaction forces alone. In its secondstage, it solves for the constrained inverse dynamics based onthe output joint accelerations and contact wrenches. In terms ofbeing dynamically consistent, we cannot think of advantagesor disadvantages between the Momentum-Based Controllerand our WBOSC. The biggest difference is that WBOSCuses internal forces as the control input for regulating contactforces. Because internal forces are fully controllable, WBOSCuniquely formulates a sensor-based feedback controller toprecisely regulate internal forces. In comparison, previouscontrollers rely on feedforward control of the internal forceswhich require high-bandwidth joint torque control. However,the high-bandwidth joint torque control impedes achievinghigh-bandwidth task control as discussed in Section IV-A.

It is necessary for us to convert desired contact forces tointernal forces. Fortunately, this is realizable by using thesimple projection operators from contact forces to internalforces described in Equation (41). As for determining the de-sired contact forces that comply with frictional constraints, wedescribed in [13] a search based method to do so. In the currentpaper we describe a simple multi-contact model to solve forthe contact forces that comply with frictional constraints. Itwould also be possible to use the contact solvers mentionedin [21], [22] as a first stage, then proceed to solve the inversetask dynamics problem using WBOSC as the second stage ofthe controller. The takeaway message is that we can use anycontact solver of our choice, convert contact forces to internalforces using simple projections as described in Equation(41), then proceed to solve for the whole-body torques usingWBOSC. Note that controlling task accelerations plus internalforces is equivalent to controlling reaction forces. For instance,the Momentum-Based Controller previously reviewed solvesfor the joint accelerations and reaction forces that complywith task accelerations and friction constraints. In our case,we directly solve for the task accelerations and compute theinternal forces that comply with friction constraints. The twocases are functionally equivalent. However, WBOSC allowsfor feedback control of internal forces which other controllersdo not.

There are several advantages of using WBOSC as theinverse task dynamics solver. First, we can formulate a sensor-based feedback controller to regulate internal forces. Usingfeedback control for the internal forces is beneficial sincewe intentionally reduce torque controller feedback gains infavor of position tracking performance as discussed in SectionIV-A. Employing internal force feedback control leads to moreaccurate tracking of the desired internal forces without re-ducing the task position tracking accuracy. Another advantageis that WBOSC exposes the task’s effective inertia matrices

which can be used to study dynamic properties such as theeffect of collisions on end effectors. Another difference is thatWBOSC provides a framework for task prioritization whichallows behaviors to automatically repress conflicting tasks thatare less critical for safety during runtime. Finally, WBOSCsupports the execution of overdetermined tasks, i.e. tasks thatattempt to control more degrees of freedom than there areavailable, given desired priorities. For instance in [25] wecontrol multiple tasks associated with a cloud of marker points.There are more markers than degrees of freedom in the robot.

The quadruped robot described in [6] uses a reaction forcemethod to balance a quadruped on a ramp with 40◦ inclination.Another quadruped described in [7] balances on a split terrainby regulating the contact forces. These types of controllers areof a similar nature than WBOSC. One difference is that theydo not implement feedback control of the internal forces.

Overall, there are many great WBCs out there supportingthe control of multiple tasks under contact constraints whichwe have compared to WBOSC. The main point of this paperis to demonstrate and analyze the performance of WBOSCon a bipedal point-foot robot which is characterized by fasterlocomotion dynamics than humanoid robots with actuatedankles and has no supporting polygon compared to quadrupedrobots.

B. Locomotion of Point-Foot Biped Robots

Point-foot biped robots similar to ours have often beenstudied [26]–[31] due to their mechanical simplicity and fastlocomotion capabilities. Only a few have achieved dynamicalbalancing without a constraint mechanism, the three mostnotable ones being the hydraulically actuated hopper from [32]and the bipeds from [28] and [3]. These last two biped robotsare based on the same mechanical architecture but differ inthe type of controller.

One of the most successful approaches to point-foot loco-motion comes from [33], which stabilizes the Hybrid ZeroDynamics of a high dimensional multi-body model of therobot. However, the algorithm is designed for robots supportedby a planarizing mechanism. Recently, the same group [28]has combined this full-body locomotion algorithm with theSIMBICON method [34] to achieve unsupported point-footlocomotion. Like ours, the robot accomplishes numerous stepsbefore falling.

In the works [3], [35], an unsupported point-foot robotaccomplishes continuous walking by using only simple rule-based algorithms to stabilize the walk. Our locomotion algo-rithm is different in that it searches adequate foot positionsto stabilize the robot around a balance point. The balancepoint can stay in place or track a desired trajectory for lo-comotion. In terms of walking, we cannot think of advantagesor disadvantages compared to that line of work. The higherperformance of their experimental locomotion results is dueto a number of factors which we believe are not related to thechoice of locomotion planner. First, the mechanical structureof their robot is more rigid than ours providing significantlyhigher foot positioning accuracy. Second, they use a highperformance IMU which contributes to precision and tracking


Fig. 1. Hume Robot Kinematics. Blue schematics describe the floating basejoints, while black describes the kinematics of the six leg joints. The locationsof the LED tracking markers identified by the PhaseSpace Impulse MotionCapture system are shown as red dots.

stability. Third, their robot uses small passive feet that preventit from drifting on the yaw direction. In contrast, our robotdoes not have passive feet. One key difference of WBOSC isthat it allows biped robots to use internal forces for balancingon highly irregular terrains, such as our example on balancingHume on a steep disjointed terrain.

Another successful locomotion approach also based onhybrid zero dynamics, utilizes human-inspired trajectories togenerate stable periodic locomotion and even handle someroughness on the terrain [31], [36]. However, the locomotionexperiments use a mechanical planarizing stage to constraintthe robot to a plane.

Frameworks such as the Capture Point [9] and the DivergentComponent of Motion [10] based on the linear point masspendulum model are very practical and widely used forcontrolling humanoid robots with actuated feet. However, theyhave not been used to control point-foot robots to date.

III. SYSTEM CHARACTERIZATION

A. Hardware Setup

Our robot, Hume, is a biped robot that is 1.5 meters talland 20 kg in weight. The leg kinematics resemble simplifiedhuman kinematics and contain an adduction/abduction hipjoint followed by a flexion/extension hip joint followed byflexion/extension knee joint as shown in Fig. 1. The lackof ankle joints allows the shank to be a lightweight carbonfiber tube. The tip of the shank contains contact sensorsbased on limit switches. The series elastic actuators on allsix joints are based on a sliding spring carriage connected tothe output by steel ropes. The deformation in the springs isdirectly measured within the carriage assembly. The concept,kinematics, and specifications of the robot were proposed byour team at UT Austin, and the manufacturing was donemostly by Meka Robotics with some of our help. For fall

safety the robot is attached to a trolley system with a block andtackle system allowing easy lifting and locking at a particularheight.

Hume is controlled with distributed digital signal processorsconnected by an EtherCAT network to a centralized PCrunning a real-time RTAI Linux kernel. This communicationsystem introduces a 0.4 ms delay from the Linux machineto the actuator DSPs and back. Each DSP controls a singleactuator and they do not communicate directly with each other.Power is delivered through a tether.

Similar to [37], Hume is electrically powered and hassix actuated joints. Our poses estimation setup combines anoverhead motion capture system with a low cost IMU. Thissensing modality contrasts the setup found in [3] which takesadvantage of its highly accurate IMU sensor. A relatively inex-pensive Microstrain 3DM-GX3-25-OEM inertial measurementunit on our robot’s torso measures angular velocity and linearacceleration, which is used in the state estimator. Additionally,the robot has a full overhead PhaseSpace Impulse motion cap-ture system that gives it global coordinate information aboutseven uniquely identifiable LED tracking devices mountedrigidly to the torso. The PhaseSpace system produces a datastream at 480 Hz and communicates to the Linux Control PCvia a custom UDP protocol. There is approximately 5 ms ofdelay in the feedback data. It accomplishes this using a systemof eight high speed sensors mounted on the ceiling above therobot, and a proprietary software package to fuse their readingsinto a single estimate for the three dimensional position of eachmarker. On each update, the system reports the location of asmany of the uniquely identifiable LED markers as it can seein Fig. 1.

B. End-to-End Controller Architecture

The feedback control system is split into six joint level con-trollers and a centralized high level controller (see Fig. 2). Thisforms a distributed control system where the joint controllersfocus on high speed actuator dynamics while the centralizedcontroller focuses on overall system dynamics. Yet the high-level feedback is necessary to create the coupling betweenjoints implied by operational space impedance tasks as well asregulating the internal forces between multi-contact supports.In SEA control, we kept Meka’s joint torque controller, whichis based on passivity as described in [38] (shown on the lowerright corner of Fig. 2).

C. Whole-Body Operational Space Control Algorithm

WBOSC [13] is a feedback control strategy based onOperational Space Control [39], which extends it to floatingbase robots in contact with the environment. It allows theuser to specify multiple task objectives and their impedance inoperational space. It also subdivides the torques applied to therobot into orthogonal spaces that affect either robot motion orinternal forces. When the user specifies these internal forces,WBOSC achieves them using feedback. The full details ofWBOSC are explained in Appendix A.

At the implementation level, WBOSC works well providedthat communication latencies are sufficiently small. Achieving


Fig. 2. Overall Control Diagram. This figure illustrates WBOSC and the joint-level torque controllers. One of our main contributions comes from thefeedback control of internal forces. Note that the gains for the controllers are treated as additional input parameters to represent the gain scheduling forachieving the best possible performance of each task.

a 0.667 ms latency required significant software optimiza-tion. We modified Meka’s firmware to ensure our controlleroperated within a real-time context, and to incorporate itinto a hierarchical chain structure that ensures minimumlatency for stacked control systems. We also reduced thebasic computational cost of our WBOSC implementation bybypassing recursive dynamics software and instead using aclosed-form expression to calculate the mass, gravity, andCoriolis matrices. To reduce the tracking error, we added anintegral term to all position tasks.

D. Contact Switching Transitions

To reduce the high speed behavior caused by a suddenchange in joint torques, we devise a strategy that smoothensout the torque commands when the robot transitions betweensingle and dual support. The sudden change in torque com-mand is due to the instantaneous switch between constraintsets within WBOSC. When the controller uses a single contactconstraint, it returns τsingle. When it uses a dual contact con-straint, it returns τdual. To ensure a smooth torque trajectory,we employ τtrans, which is a special instance of WBOSC thatuses a single contact constraint but adds an artificial transitionforce associated with the contact behavior of the transitioningfoot. The transitioning foot corresponds to the foot that justlanded on the ground or the foot that is about to lift up from theground. In either case, we do not instantaneously switch thecontroller between single and dual contact phases, but insteadtransition smoothly from single contact to dual contact or vice

versa. And the manner that we transition is by adding theartificial transition force to the single contact controller andslowly increasing or decreasing its value until it matches thefinal torques of the next phase. That is, for the single contacttransitioning controller, we adjust Equation (36) to include anartificial transitional force, ftrans, as follows:

Ftask = Λ∗taskutask + µ∗task + p∗task + ftrans. (1)

Based on the dual contact hypothesis, τdual, we find thereaction force that would be generated on the foot that justlanded on the ground or that is generated on the foot about totake off. Using the second row of the matrix Equation (30),and based on the forces corresponding to the task set in singlecontact phase resulting from the dual contact phase reactionforces, i.e. fdual→single , Sdual→singleFr, we get

fdual→single =

Sdual→single

(JT

s

[UT τdual − b− g

]+ ΛsJsq

), (2)

where Sdual→single is a projection operator that transformsthe dual contact reaction forces to equivalent task forces ofthe single contact controller. Let us take the case of theswinging foot that just landed on the ground. Rather thanswitching instantaneously to the dual contact controller, wecontinue using the single contact controller with the artificialtransition force. That force is smoothly increased until itmatches the dual contact reaction forces projected to thesingle contact controller. Therefore, as the controller transitionapproaches to dual support we increase the artificial force


from (ftrans = 0) → (ftrans = fdual→single). Conversely,after the controller switches from dual support phase to singlesupport phase prior to lifting the foot from the ground, wesmoothen the transition by gradually removing the artificialforce, (ftrans = fdual→single) → (ftrans = 0). To achievethis gradual transition, we define ftrans , w · fdual→single,w ∈ [0, 1], where w varies linearly with time from one tozero or from zero to one over the course of the transition asappropriate.

To avoid running the controller twice every servo cycleduring transitions, we reuse previous values for τdual andfdual→single. When a foot lifts, the old value of τdual is simplythe last controller action before the start of the transition, andfdual→single is calculated once based on that controller state.When a foot lands, we run the dual support controller onceat the start of the transition for the sole purpose of acquiringfdual→single. This process is similar to [6] except that theyapply it to a quadratic-programming based controller whereaswe apply it to a WBOSC-based controller.

IV. ENHANCEMENT OF WHOLE-BODY OPERATIONALSPACE CONTROL

A. Online Feedback Gain Adjustment

To enhance the bandwidth of the proposed WBOSC con-troller (shown in the upper portion of Fig. 2), we loweredthe gains of the torque feedback controllers (shown in thelower portion of the same figure). A detailed study on stabilityand bandwidth trade-off between high-level position controland low-level torque control is presented in [2]. In particular,reducing the low-level torque controller bandwidth allowsus to increase the high-level position controller bandwidthwithout compromising stability. To achieve a higher bandwidthwith position control, we increase the PID feedback gains inWBOSC, reduce the joint-level torque feedback gains, and setthe integral gains on the low-level torque controllers. Detuningtorque gains would reduce the accuracy of internal forcetracking performance if we would solely rely on feedforwardmodels – see discussion in Section IV-B. For this reason wedesign internal force feedback controllers to precisely trackthat subset of the contact forces – this technique was alsodiscussed in Section IV-B.

One drawback of detuning torque controllers is that frictionand stiction cause torque errors. This effect results in jointsnot moving when torque commands are small compared to thestiction threshold. To tackle this problem, we simultaneouslyadjust the torque controller gains and the WBOSC feedbackcontroller gains based on each robot joint’s effective load: injoints belonging to the stance leg, we turn off the integral gainof the torque controller. In joints belonging to the swing leg,we turn on the integral gain of the same controller to reducethe effects of friction in the actuators.

This can be counter-intuitive since larger effective massimplies larger feedback gain. Intuitively, the natural frequencyof joint force output in SEA actuators decreases as the joint’seffective load increases [40]. When one of the robot’s legs is incontact with the ground, its effective mass increases and as aresult its natural force frequency decreases. Applying integral

Fig. 3. Illustration of Internal Forces for Various Robots. Internal forcesin point-foot robots correspond to tensions or compressions between pairs ofsupporting contacts.

torque gain in a controller with a small natural frequency canreduce the phase margin due to an increase in bandwidth and adrop in phase by −90◦. After various trials, we concluded thata proportional feedback control with motor velocity feedbackin the torque controller and a PID control in WBOSC givesthe best performance for the stance leg.

B. Sensor-Based Internal Force Feedback Control

Internal forces are associated with joint torques that produceno net motion. As such, internal forces correspond to mutuallycanceling forces and moments between pairs or groups of con-tact points, i.e. tensions, compressions and reaction moments.For instance, a triped point-foot robot has three internal forcedimensions while a biped point-foot robot has a single internalforce dimension as shown in Figure 3.

Internal forces are fully controllable since they are or-thogonal to the robot’s motion. As such, both the robot’smovements and its internal forces can be simultaneouslycontrolled. Moreover, in many types of contact poses, internalforces are easily identifiable using some physical intuition.For instance, in the triped pose of Figure 3 the three feet cangenerate three virtual tensions between the points of contact.The physics of tension forces were analyzed in greater detailusing a virtual linkage model in [13].

Internal forces are part of the core WBOSC. In Appendix A,we describe the model-based control structures enabling directcontrol of internal forces. In particular, the basic torquestructure derived in Equation (43) is copied here:

τint = J ∗Ti|l

(Fint,ref − Fint,{t} + µ∗i + p∗i

), (3)

where Fint,ref is the vector of desired internal forces andFint,{t} corresponds to the mapping of task torques into theinternal force manifold. The above equation would be suffi-cient for feedforward internal force control if the commandedtorques were equal to the actual torques, and if the kinematicand dynamic models were exact. However, as we mentioned inthe Introduction and will further explore in detail in SectionIV-A, we intentionally lower the bandwidth of joint torquecontrollers in order to increase the bandwidth of WBOSC’stask motion controllers. As a result, we cannot solely rely


on feedforward control of internal forces. Formulating in-ternal force feedback control is a significant advantage overincreasing the bandwidth of joint torque controller. It enablesgood force tracking accuracy but only in the dimensions thatare rendered of interest to the behaviors, which in our casecorrespond to internal forces. Because internal forces do notaffect task motion, the internal force feedback controller doesnot negatively affect the accuracy of the motion tasks as thejoint torque controllers do. Overall, it is best to formulatefeedback control only in the dimensions that are practical,and in our case they are the motion tasks in one hand, andthe internal force task on the other. To our knowledge, thisis the first successful use of sensor-based feedback control ofinternal forces in a real robot.

Since internal forces are fully controllable, we can achievebetter internal force tracking accuracy by incorporating asimple proportional controller of the measured internal forceerror via Equation (43),

τint = J ∗Ti|l


+KF (Fint,ref − Fint,act)), (4)

where KF is a proportional control gain and Fint,act are theactual sensor-based internal forces. To obtain these sensor-based forces, we use the torque sensors on the series elasticactuators to find the reaction forces as per Equation (30) andapply a projection Wint to find internal forces,

Fint,act ,Wint

[JT

s (UT τsensor − b− g) + ΛsJ q], (5)

where τsensors corresponds to the vector of torques sensed bythe spring element in each series elastic actuator (see Figure 2).

The above internal force mapping is distinguished fromprevious work due to its sensor-based force feedback nature,and its mapping is valid due to the physical fact of robotredundancy in the multi-contact case. The induced contactclosed loop causes the number of controlled motion tasksto be smaller than that of actuated joints. Correspondingly,additional DOFs are available to be controlled for more forcetasks, such as internal forces in Equation (5). This mappedinternal forces are consistent with contact constraints andcancellation of accelerations on the robot’s base or on theactuated joints [13]. More details can be found in Appendix A.

To calculate internal forces for Hume we need to de-fine the mapping given in Equation (41) in Appendix A,where Wint is the matrix representing the map from reac-tion forces to internal forces. In our case, Hume controlsthe internal forces between the two feet during dual con-tact phases. In dual support mode, the reaction forces are(fRx, fRy, fRz, fLx, fLy, fLz)T , where R and L meanthe robot’s right and left foot, respectively. According to [13],Wint consists of St, a selection matrix of tensions, Rt, arotation matrix from global frame to the direction parallel tothe line between two contact points, and ∆t, a differentialoperator matrix, i.e.

Wint = StRt ∆t, (6)

with

St =(1 0 0

), (7)

Rt =

xT

yT

zT

,

x = PR−PL

||PR−PL||

y = (−x(2), x(1), 0)T

z = x× y

, (8)

∆t =(I3×3 −I3×3

). (9)

where PR and PL are the position of the right and left feet,respectively.

There are two ways to compute desired internal forces: 1)by approximating them via a simple force statics problem andchoosing values that comply with friction cones, and 2) by firstsolving an optimization problem including task accelerationsand friction cones to obtain desired reaction forces, Fr,ref , andthen projecting them into internal forces using Equation (41),i.e.

Fint,ref = Wint Fr,ref . (10)

In this paper we use method 1). If one wants to use method2), she/he can use existing methods such as the QP-basedcontact solver defined in [16] or the centroidal-momentum-based contact solver defined in [22]. Then the equation abovecan be used to solve for the desired internal forces to besubsequently used as inputs to WBOSC.

V. FOOT PLACEMENT PLANNER

The foot placement planner determines footstep locationsbased on a prismatic inverted pendulum model (PIP) thatis representative of the robot in single contact phases. Thefootstep locations are selected such that the PIP model is stabi-lized. This should result in a stable balance behavior assumingthe trajectory generators and WBOSC, 1) successfully placethe feet at the desired locations, 2) achieve the desired heightof the center of mass, and 3) fix the orientation of the robot’sbody.

We previously devised a footstep location algorithm calledthe “phase space constant time to velocity reversal plan-ner” [11]. The swing motion is separated into two phases,lifting and landing the foot. The landing location is computedbefore the landing phase starts. In every step, when the liftingphase reaches 70% completion, the planner computes thenext footstep location. The value of 70% was empiricallydetermined to ensure the planner completes before the landingphase starts. It is a processor intensive task that must berun outside of the realtime thread. The operational space set-point trajectory for the swing foot is then defined based on apolynomial function and the desired landing position, with thetrajectory ending once ground contact is sensed. If the groundis at the expected height and the position tracking is ideal, thefootstep will land after the nominal swing time. If the plannedstep is outside the mechanical limits of the robot, the plannerchooses the closest reachable step.


Fig. 4. Constant Time to Velocity Reversal Planner. As shown in (a), we approximate the dynamics of the robot with the prismatic inverted pendulummodel, shown in (b). This model predicts the dynamics of the horizontal center of mass position x, y given the stance foot location (xp, yp) = and theheight surface z = h(x, y). This can be integrated forward in time via the numerical integration procedure shown in (c). When the planner starts operating itrecords the initial state and integrates this state forward to determine the switching state . As shown in timelines (d) and (e), the “Estimated sequence” ofthe planner has an analogue in the “Robot states” of the state machine. In particular, the switching state roughly corresponds to the dual support phase ofthe walking state machine. This state represents the planner’s guess at the time and state (x, x, y, y) values immediately after the switch. The goal of theplanner is to stabilize the robot, and this is achieved by choosing the next footstep such that x and y velocity equal zero t′x and t′y seconds, respectively,after the foot switch every step. For sufficiently smooth height surfaces, the relationship between the next footstep location and the velocity is monotonic, soonly a single solution exists. We use Newton’s method with numerical differentiation to identify this solution. There exist two velocity reversal states: x

and y as shown in (c).

A. Velocity Reversal

Our planner attempts to stabilize the robot by making itscenter of mass reverse direction every step. In its simplifiedmodel, the feet instantly change between swing and supportmodes and the CoM reverses its velocity a time t′ after theprevious contact switch.

In contrast to approaches based on linear models withanalytically solvable dynamics such as the Capture Point orthe Divergent Component of Motion methods, our approachis different because the PIP model is not analytically solvableand numerical search is used to plan the steps. This leads tothe primary computational element in the planning procedure:a shooting method over the possible footstep locations to finda center of mass trajectory that accomplishes the CoM velocityreversal goal. In a previous paper [11], which only consideredconstant elevation of the center of mass, we used bisectionsearch to find footstep locations. In this new work, whichconsiders variable elevations, we use Newton’s method basedon numerical derivative approximations to find the footsteps.

As illustrated in Fig. 4, the planner begins calculating thelanding location when 70% of the lifting phase is reached. Assuch, the planner continuously re-plans to correct for trajectorydeviations. Using the current estimate of the CoM velocity andposition, it numerically integrates forward until the predefinedswing time ends to predict its CoM position and velocity whenits stance foot and swing foot will switch roles, in Fig. 4.The equation used in the numerical integration is presented inAppendix B.

The implementation of the planner enforces the choice of

t′. This time value remains constant for every step, thus theuser only needs to specify a single parameter. The methodfor selecting t′ is explained in Section V-B. The planner thenfinds and returns the footstep location that causes the robot’sCoM velocity to reach zero t′ seconds after the foot switch.For each potential footstep location considered, the plannerintegrates forward in time starting from the post-impact stateas suggested in Fig. 4, ensuring the velocity is zero after t′

seconds. This integration can be viewed as a function mappingfootstep location to a future velocity. Newton’s method usesthis function to find a foot placement that results in zero CoMvelocity. The number of integrations performed is very lowsince the process relies on the monotonicity of the relationshipbetween footstep location, xp and yp, and the velocity after t′

seconds, x(t′x|xp) and y(t′y|yp) [11].

As shown in Fig. 5, our planner allows Hume to step overa 7 cm tall platform and using a CoM height surface thatconforms to the terrain. Specifically, this height surface isdefined, piece-wise, as a function of a global x coordinate,with three constant height pieces connected by two sinusoidalsegments. The middle constant height piece is 7 cm abovethe others to account for the obstacle’s height. The surfacemaintains first order continuity. In this simulation the robot’splanner follows a moving goal location. As this goal locationpasses over the platform the robot ascends and descends itwhile constantly stepping. Although the planner usually findsthe proper foot placement, the robot shows variable forwardand backward swinging motions. This variability arises whenthe robot performs large motions such as stepping down


Fig. 5. Simulation of Unsupported Point Foot Walking Over Rough Terrain. Subfig. (a) shows the sagittal CoM phase portrait of three steps of walkingover a rough terrain, while (b) shows the lateral CoM phase portrait. The smaller plots correspond to individual steps. The corresponding three steps areshown in (c) using the SrLib Multi-Body Dynamic Simulation Environment. Subfig. (d) shows more steps of the dynamic walking simulation over the roughterrain.

from a platform. Since the simplified PIP model used forplanning does not account for the multi-limb robot dynamics,predicting the CoM path becomes increasingly more difficultwhen estimating large motions. Despite this discrepancy, theplanner and controller successfully achieve a stable walk overthe challenging terrain.

In the simulation, we assume zero time delay, perfectsensor data, perfect torque tracking, and correct dynamic andkinematic models. These assumptions, which are difficult toobtain in the real world, make it easier to check the basicfunctionality of the planner.

B. Stability Analysis

It can be shown that t′ is stable for a limited range ofvalues assuming linear height surfaces. A natural property ofour planner is that the footsteps converge towards the locationof the CoM projected to the ground. In practice, this is not adesirable behavior since the feet would move too close to eachother making the robot more sensitive to small disturbances.To account for this potential problem, we create a hybridbehavior—if the planned velocity at the next transition willbe too small we artificially extend the swing time, whichkeeps the dynamics away from converging to the origin. Inthis section, we only consider the system’s stability when thisconvergence effect is not present.

We begin by formulating the well known linear invertedpendulum model,

x =g

h(x− xp), (11)

where g, h, and xp are gravitational acceleration, the constantCoM height, and the stance foot location, respectively. Thesolution of this ODE is,

x(t) = xp + (x0 − xp) cosh(ωt) +x0ω

sinh(ωt), (12)

where ω =√g/h, and can be expressed as a discrete time

state-space system with a constant step duration T :

X((k + 1)T ) = AX(kT ) +Bxp,k k ∈ Z, (13)

where,

A =

[cosh(ωT ) ω−1 sinh(ωT )ω sinh(ωT ) cosh(ωT )

], (14)

B =

[1− cosh(ωT )

−ω sinh(ωT )

]. (15)

Here, X(kT ) =[xkT xkT

]Trepresents the state at the

instant when the input changes—the instant the stance footand swing foot change roles, and a new foot position xp isput into place.

As explained previously, the planner chooses xp such thatthe CoM velocity becomes zero at time t′. This can beexpressed as a function of the state of the discrete system,

0 = xkT+t′ =[ω sinh(ωt′) cosh(ωt′)

]X(kT ) (16)

− ω sinh(ωt′)xp,k,

xp,k =[1 ω−1 coth(ωt′)

]X(kT ). (17)

However we include an additional linear bias term in thecontrol law above,

xp,k = xdκp +[(1− κp) ω−1 coth(ωt′)

]X(kT ), (18)

to move the robot towards a goal location xd, effectivelycreating a locomotion behavior.

Without loss of generality, we assume the goal is the originfor the stability analysis. The closed loop system under the


Fig. 6. Unsupported Dynamic Balancing Experiment: This figure shows a portion of an experiment in which our point foot biped robot, Hume, accomplishes18 steps of unsupported dynamic balancing. (a) Motion within the balancing experiment is divided into three states: dual support, swing leg lift, and swingleg landing. Additionally, a transition state exists at the beginning of the lifting phase and the end of the landing phase to avoid the jerk caused by a suddenchange in controller constraints. (b) and (c) show the x and y-directional trajectory of the CoM. The blue line represents actual data while the red linesindicate the trajectories estimated by the planner. (d) Position and orientation task tracking is plotted over a representative portion of the stepping experiment,with sensor data in red and desired values, dotted, in blue. Height refers to CoM height. (e) The snapshots and data of steps 3-6 are shown. The phase pathsof steps 3-6 are expanded into individual step planning plots. For each step, a red line marks the actual CoM path up to the switching state, and a greenline continues the trajectory after the switch. The robot’s initial stance foot in the step planning plot is denoted with a , the planned second footstep witha black circle, and the achieved second stance foot location with a blue cross. Therefore, a green line in the ith step plot is the same path as the red line inthe (i+ 1)th step plot. A black line and a dark green dotted line are, respectively, the PIP model’s predicted paths before and after the switching state.

proposed feedback law is:

X((k + 1)T ) = (A+BK)X(kT ) (19)

K =[(1− κp) ω−1 coth(ωt′)

](20)

A+BK = A′ (21)A′11 = 1− κp + κp cosh(ωT ) (22)

A′12 = ω−1(sinh(ωT ) + (1− cosh(ωT )) coth(ωt′)) (23)A′21 = κpω sinh(ωT ) (24)A′22 = cosh(ωT )− sinh(ωT ) coth(ωt′) (25)

Using the eigenvalues of the matrix A′, we determine thestability of the closed loop system. To be robust to modeluncertainties, we chose parameters κp and t′ such that theyproduce eigenvalues with magnitudes close to 0.8 for thedesired h and T values. One interesting fact of t′ is that aninfinitely large t′ stops the robot’s CoM exactly at the stancefoot position, which is equivalent to the Capture Point [9].However, this is not a desired behavior because when theCoM is near the stance leg, crossing of the legs might occur.Thus, we design the planner parameters to gradually decrease


Position Gain

Dual Support Single Support

CoMz qRz qRy qRx CoMz qRy qRx footx footy footz

Kx(s−2) 270.0 100.0 200.0 210 270.0 200.0 210.0 220.0 237.0 400.0

Ix(s−3) 20.0 0.0 20.0 35.0 20.0 20.0 35.0 35.0 40.0 35.0

Dx(s−1) 10.0 0.0 15.0 10.0 10.0 15.0 10.0 35.0 40.0 60.0

Torque Gain

Stance Leg Swing Leg

Both Abduction Both Hip Both Knee Both Abduction Right Hip Left Hip Right Knee Left Knee

KP,τ (N m rad s−1) 2.84 43 22 3.09 58.57 32.70 49.75 26.1

KI,τ (N m rad s−1) 0.15 0 0 0.19 5.86 3.48 6.85 4.15

TABLE IGain Set for the Unsupported Dynamic Balancing Experiment

Transition Lifting Landing Dual Support κx, κy t′x, t′y λx λy

0.024 (s) 0.145 (s) 0.15 (s) 0.016 (s) 0.4, 0.4 0.185, 0.18 0.7, -0.88 0.69, -0.81

TABLE IIPlanner Parameters for the Unsupported Dynamic Balancing Experiment

Fig. 7. Knee Joint Torque: The plot shows torque trajectories of right andleft knee joints. Blue dotted lines are commanded torque and red solid linesare measured torque. Color indication of the state machine in Fig. 6 is used.

the robot’s CoM velocity at the moment when the stance legswitches to the other leg. We achieve this decrease in velocityby setting the magnitudes of the eigenvalues of A′ close to0.8.

C. Impact Model

In many cases, the hybrid dynamics of a robot impactingthe ground are significant. This warrants a model that includesa discrete map to represent the sudden velocity changesthat occur during impact [41]. However, since our planningalgorithm focuses exclusively on the CoM behavior, and sincethe body mass of our robot is much larger than the leg mass,we used a model that predicts no change of the CoM velocityduring impact. Here we use the approximations that 1) theseries elastic actuators of our robot decouple the reflected rotor

inertias, that 2) the actuators are frictionless, that 3) the robot’supper body mass is fairly decoupled from the foot due to theleg’s kinematic chain, and that 4) the knee of the landing legis not entirely stretched.

VI. EXPERIMENTAL RESULTS AND ASSESSMENT

A. Unsupported Dynamic Balancing

To start the experiment, Hume is briefly supported while itrises to the desired height for balancing. Once it reaches itsstarting height, the experimenter balances the robot carefullyand lets it go as it takes its first step. Once free, the robotcontinuously steps until it falls over. There is a harness rope,slack when the robot is at its starting height, which catches itif it falls to prevent major damage. The power and Ethernettether hang slack from another rope.

The motion follows a time scripted state machine, shown inFig. 6(a). Since the states are symmetric with respect to thesupporting leg being either right or left, states are categorizedin two different compound tasks with left and right singlesupport having symmetric structures. The WBOSC compoundtask, xtask, differs between dual support and single supportphases of the stepping state machine. In dual support, thecompound task coordinates are [CoMz, qRz, qRy, qRx]T ,where CoMz represents the height of the center of mass. qRz ,qRy and qRx are body yaw, pitch and roll angles, respectively.Those coordinates are controlled via the acceleration inputof WBOSC, utask, shown in Equation (36) and via PD orPID control laws. In single support, the compound task is[CoMz, qRy, qRx, footx, footy, footz]T . The desiredheight is set to the initial height when Hume begins to stepand the body orientation is set to be straight up. The desiredfoot trajectory for the lifting phase consists on first reaching apredefined z height while keeping x and y constant. Then, a


Fig. 8. Trajectory Tracking and Human Disturbance Rejection on Disjointed Terrain: Subfigure (a) shows Hume standing between two inclined woodenpanels and tracking a position task with its center of mass. This position set-point follows a constant velocity trajectory along an elliptical path shown, alongwith the measured CoM path, in Subfigure (b). In Subfigure (c) the CoM has different impedances in the horizontal and vertical directions. When the robotis pushed backwards it moves as though the center of mass were connected to a low-spring-constant spring, whereas when the robot is pushed downwardsit reacts as though connected to its set point by a far stiffer spring. Due to the feedback regulation of internal forces, the biped does not fall down whendisturbed with large external forces.

3rd order polynomial is used to generate the desired (x, y, z)landing trajectory. Since Hume’s feet are points, in singlesupport phase it is not possible to control the yaw motion,qRz . To compensate for this deficiency, we use the brief timethat the robot spends in dual support to correct for it. In thebalancing experiment we do not use internal force feedbackcontrol to reduce complexity of the sequencing process. Sincethe standing surfaces are flat, internal force regulation is notneeded to keep the contacts stable. All control parameters forthe single and dual support phases, and for the stance andswing legs are shown in Table I. The parameters of the plannerused in this experiment are shown in Table II.

In Fig. 6(b) and (c), The x and y directional CoM tra-jectories (red) are superimposed on the one-step predictedpath by our planner (blue). Although predicting CoM path formultiple steps is difficult, computing the CoM path for a singlestep using the PIP model closely approximates the actualCoM motion. The orientation error is bounded by 0.05 rad(Fig. 6(d)). Given the model disturbances and the impacts,this error is reasonable small to validate the controller’sperformance. Fig. 6(e) shows snapshots and phase paths forthis experiment. The phase space data is corrupted by highfrequency noise from the IMU sensor signal and the jointencoders that combined to compute CoM velocity.

Transition and gain scheduling techniques also play animportant role by smoothing the motion and tracking the tasks.In Fig. 7, the commanded torque (blue) changes from 0 to−60 N m without significant jerk due to our contact transitiontechnique. When the right leg switches to a stance leg (greenbackground), around 20 N m of torque tracking error appears.This is expected because we detune the low-level torquecontroller to achieve stiffer position control by the WBOSCcontroller. The controller corrects yaw error during the dualsupport phase. Additionally, Hume incurs a significant bendingof the stance leg which results on uncertainty of the positionof the CoM with respect to the stance foot. Despite all these

sources of error our robot was able to dynamically balanceunsupported for 18 steps using its point contacts.

B. Balance on a Disjointed Terrain

In this experiment, Hume balances on a high pitch terraincomposed of two 45◦ wedges angled in towards the robotto create a convex floor profile. Because there is no way tocontrol lateral motion, the planarizer is used to constrain themotion of Hume to the sagittal plane.

Note that we do not include the 5 kg sliding linkage in thedynamic model and regard it as unmodeled disturbance. Therobot’s tasks were to maintain a 100 N internal force pushingoutwards against the two contact points, a desired impedancetask for the center of mass, and a desired impedance taskfor the body orientation. The reference of 100 N is roughlycalculated with the assumption that the 200 N body weight isequally distributed between each foot and a 100 N horizontalforce generates a reaction force normal to the 45◦ angledsurfaces.

By controlling the internal force, Hume does not slipwhile it tracks the reference CoM path within a 2 cm error.This experiment is divided into two sub-experiments: Humewas controlled to track a time-varying CoM trajectory whichfollowed an elliptical path in the sagittal plane, as shownin Fig. 8(b); and Hume was controlled to hold a Cartesianimpedance task on the CoM which had low stiffness horizon-tally and high stiffness vertically as shown in Fig. 8(c). In thetest, xtask = [CoMx, CoMz, qRy, qRx]T , Ftask,ref = [0]4×1and Fint,ref = 100 N, using the notation of Fig. 2. We do notcontrol CoMy and qRz since the planarizer rigidly constrainsthose directions of motion. However, we do control the robot’sroll, qRx, since the planarizer is slightly flexible in the rolldirection. Thus, controlling the roll and pitch helps to sustainthe body pose.

CoM tracking performance is shown in Fig. 8(b). Althoughtracking errors exist they are bounded within 2 cm. We believe


Fig. 9. Internal Force Control Without Feedback: Without feedback loop,internal force control shows larger than 50% error.

this error occurs because of various reasons: 1) the legs ofHume are flexible and that flexibility is not modeled, 2) theplanarizer that holds Hume on the back is unmodeled, and 3)the low cost IMU that we use suffers from quick orientationdrift. The second sub experiment shows that feedback controlof internal forces allows Hume to stay balanced despiteexternal forces exerted by a human. Due to the WBOSC’sinternal force feedback controller, the disturbances we exerton the robot do not produce large deviations in the internalforce tracking performance.

Due to feedback control, the errors between desired (blue)and actual (red) internal forces are small enough to keep Humebalance on the difficult terrain. In contrast, in Fig. 9 we turnedoff the feedback control of internal forces. The experimentwas conducted on a 20◦ pitch terrain due to the robot beingunable to stay up on the 45◦ pitch terrain when turning off thefeedback. We can see that the error to command ratio is muchlarger when using feedforward control only (more than 50 %)relative to the experiment with the feedback enabled (less than20 %). These experiments ultimately show the effectiveness ofclosed loop internal force control as applied to maintaining africtional contact.

VII. DISCUSSION AND CONCLUSIONS

The central focus of our paper has been on formulating aWBOSC and a new balance planner to achieve unsupporteddynamic balancing of a point foot biped robot without passivefeet. The underactuated nature of point foot bipeds forces usto “give up” on the x and y components of CoM motion,leaving those aspects to evolve naturally while the WBOSCcontrols the remaining degrees of freedom. These liberateddegrees of freedom are still controlled, but must be controlledindirectly, on a step by step basis, by choosing the footsteplocations. By formulating a WBOSC and a phase-space-basedfoot placement planner we make Hume, our 6-DoF point-foot biped robot with series elastic actuators, balance onrough terrain when constrained to move in the robot’s sagittalplane and take 18 steps on flat terrain when unsupported. Inaddition, our framework also demonstrates its ability to stepover a 7cm obstacle in simulation. Achieving these capabilitiesrequired advancements in WBOSC that push its performanceboundaries to the next level by stabilizing highly dynamicbiped robots and showing the benefits of employing feedbackcontrol of internal forces.

WBOSC uses the robot’s dynamic model to simultaneouslycontrol motion and force behaviors in the task dimensionsthat are rendered practical for the operations at hand. Forexample, WBOSC on a torque controlled robot can precisely

control the position of the robot’s center of mass while alsoprecisely controlling the internal forces - in the sense of thetensions and compressions performed with the entirety of therobot’s body - between disjointed surfaces. And only whenjoint torque sensing is available, these type of sophisticatedbehaviors can be achieved without reliance on force sensorslocated on the end effectors. More importantly, using the sameidea, WBOSC on torque controlled robots could control theinternal forces between any two points in the robot’s structure,provided that they are distant enough from each other. In ourcase, we employed this unique force / motion feedback controlcapability, to balance our biped robot on a high pitch splitsurface. Beyond contacts on the feet or end-effectors, WBOSCaims at achieving contact awareness and command over theentire robot’s body structure. And as such, it may one day bethe ideal type of controller for walking or moving in directcontact with objects and people.

An important issue of this paper has been the integrationof WBOSC with joint level torque controllers. To obtaingood performance, torque control at the joint level has beensought. The reason is that joint torque control reduces theeffect of the natural dynamics of the actuator, i.e. stiction,dynamic friction, non-linearities, thus yielding excellent forcebehaviors for the actuator. However, using torque-feedbackcontrol nested within a position feedback controller needscareful understanding of its stability as a function of looplatencies [2]. Torque controllers substantially decrease theeffect of natural friction of the actuators, forcing the taskposition controllers to provide damping via velocity feedback.However, velocity feedback is very sensitive to loop latenciesas was analyzed in detail in [42]. In order to achieve stiff taskposition control, which is desirable for some tasks to rejectmodeling disturbances, system latencies both due to the WBCand to communications, need to be minimized. In the case ofHume, we have greatly optimized all control computations andembedded communications to decrease the servo round trip,including communications, to 0.667 ms.

Additionally, when there is a need to deliver stiff positiontracking control, it is desirable to detune or reduce torquegains. This effect was reported in [43] and more recentlyanalyzed in depth in our paper [2]. In essence, we prioritizeposition accuracy over torque accuracy when position trackingis essential. To compensate for the detuned torque controllers,when our biped balanced on the split terrain, we added aforce feedback loop that regulated the internal force fromjoint torque sensor. This feedback loop mitigated the effectsof trading off motion accuracy versus torque accuracy. Thekey advantage is that increasing internal force control gainsdoes not affect the task position control performance whereasincreasing torque gains reduces task position performance.

Finally, exact state measurement has been crucial to maneu-vering with the highest possible accuracy given the hardwarelimitations of our low cost IMU. However, many successfullegged robots, such as Atlas, ATRIAS, and the MIT Cheetah 2robot, use an optical IMU called KVH Industries Model 1750,which has a minuscule bias error, 0.000 55◦/s. This is not thecase in our experimental setup, as we used a low cost and muchless accurate MEMS IMU, i.e. the Microstrain 3DM-GX3-25-


OEM, with a bias error 500 times worse than the previousone, i.e. 0.25 ◦/s. In other words, our IMU’s orientationestimate drifts away quickly. In practice, the drift appearsto be much faster than the specification. To compensate forthis discrepancy, we devised a sensor fusion approach bycombining the IMU with visual data from our motion capturesystem. Nonetheless, the overall pose sensing system is farless accurate and speedy than using the high end IMU. In thefuture we plan on replacing our current IMU with the high endversion KVH IMU and also redesigning part of the leg jointsto increase mechanical rigidity. With those improvements webelieve that we will substantially increase the accuracy of thefoot positioning, enabling faster and more robust locomotionbehaviors.

APPENDIX AWHOLE-BODY OPERATIONAL SPACE CONTROL

WBOSC was first laid out in [12] with a further explorationof internal forces that was published later, in [13]. Interestedreaders should refer to these sources for a description of thetheoretical background complete with proofs for the conceptsbelow, as space limits us to a cursory overview of WBOSCas applied to bipedal robots.

Modeling biped robots entails representing not only the stateof each joint q1, . . . , qnjoints , but also the states of the robotas an object in space. We choose to parameterize the robot’sbase as a 6-dimensional floating joint between the world andthe robot’s hip coordinate system with state vector qb ∈ R6.Combining the robot and base states into a single vector wearrive at q ∈ R6 ⊕ Rnjoints = Rndofs , the generalized jointvector. The joint torques can only directly affect the jointsthemselves, and not the floating base dynamics, so we define

an underactuation matrix U ∈ R(ndofs−6

)×ndofs that maps the

global joint vector to the subspace of actuated joints.

qact = U q, (26)

with qact ∈ Rnjoints being the robot’s actuated joints. Thedynamics of the robot’s generalized joints can be describedby a single, linear, second order differential equation

Aq + b+ g = UT τcontrol (27)

where {A, b, g} represent the inertia matrix, Coriolis / cen-trifugal forces, and gravitational forces, respectively whileτcontrol is the desired control command applied to the outputjoints of the robot. Without considering contacts, or the subtlenon-holonomic effect of angular momentum, joint torqueswould have no effect on the geometrically uncontrollable sixdimensional subspace of the generalized joints: {z ∈ Rndofs :zTA−1UT = 0}. However we can sometimes gain the abilityto control more of this space due to contact constraints.

We consider two contact cases for point-foot biped robots:single support in which one foot is in contact, and dualsupport where the robot is supported by both feet. In the singlesupport phase we describe the contact via a support JacobianJs ∈ R3×ndofs , which maps from generalized joint velocityto the velocity of the constrained foot point in Cartesianspace. When considering dual contact, our support Jacobian

represents twice as many constraints. Since this generalizedpoint, either the single foot point in R3 or the dual foot pointin R6, is constrained, we know its acceleration must be zero.Substituting the constraint Jsq + Jsq = xfoot(or feet) = 0and adding the associated co-state of constraint space reactionforces Fr, the dynamics become

Aq + b+ g + JTs Fr = UT τcontrol, (28)

and we can find q and Fr by solving the matrix equation(A JT

s

Js 0

)(qFr

)=

(UT τcontrol − b− g

−Jsq

). (29)

Converting to upper diagonal form via Gaussian eliminationwe find(A JT

s

0 I

)(qFr

)=

UT τcontrol − b− g

JT

s

[UT τcontrol − b− g

]+ ΛsJsq

,

(30)with Λs , [JsA

−1JTs ]−1 and Js , A−1JT

s Λs. SubstitutingNT

s , I − JTs ΛsJsA

−1 to more conveniently express theresulting constrained dynamic equation,

A q +NTs (b+ g) + JT

s ΛsJsq = (UNs)Tτcontrol. (31)

This can be viewed as constraining the dynamics to thedynamically consistent null-space of the constraint by definingthe dynamically consistent pseudo inversion operator

X , A−1XT [XA−1XT ]−1 (32)

and observing that

Ns = I − JsJs (33)

is the null space projector of Js under dynamically consistentinversion such that JsNs = 0, NsA

−1JTs = A−1NT

s JTs = 0,

and NsNs = Ns.WBOSC for an operational task representation, ptask, is

defined by the differential kinematic equation

ptask = J∗task qact, (34)

where J∗task , JtaskUNs ∈ Rntask×nact is the contactconsistent task Jacobian and Jtask ∈ Rntask×ndofs is theunconstrained task Jacobian. The basic control structure forthe single support phase of the biped is thus

τcontrol = J∗TtaskFtask, (35)

with Ftask being the entry point for feedback control laws togovern trajectories, applied forces, or combinations of the twoin the operational space. For instance, when controlling anoperational space position trajectory, we use the model-basedcontrol law

Ftask = Λ∗taskutask + µ∗task + p∗task, (36)

with utask being a desired acceleration for the operational ref-erence. {Λ∗task, µ∗task, p∗task} are inertia, velocity-based forces,and gravity based forces in the operational space that can befound in the previous references.

In the case of dual support, there appear closed loop effectsbetween the legs of the robot in contact with the environment.


Our previous work has thoroughly addressed this problem bycreating structures to control the internal forces. Internal forcesare defined as those that are orthogonal to joint accelerations.As such, internal forces do not produce any movement andonly contribute to force generation within the closed-loopformed by the contacts. Analyzing the right hand side ofEquation (31), those forces correspond to the manifold(

U Ns

)Tτcontrol = 0, (37)

which reflect the cancellation of acceleration effects. There-fore, the torques that fulfill the above constraint belong to thenull space of (U Ns), which is defined by the projection

L∗ ,(I − U Ns U Ns). (38)

The torques associated with internal forces are those that donot contribute to net movement, i.e.,

τ ′int = L∗T τint, (39)

where τint is the set-point for the internal forces. Thus,when simultaneously controlling operational space tasks andinternal forces we superimpose the orthogonal structures ofEquation (35) and (39) yielding the WBOSC command

τcontrol = J∗TtaskFtask + L∗T τint. (40)

Internal forces can be physically defined as linear forcesand mutually canceling reaction moments between pairs ofsupporting contacts. As explained in our previous works, suchforces are expressed via the equation,

Fint = Wint Fr, (41)

where Fr is the set of all reaction forces on the environmentand Wint is a matrix containing geometric transformations andselection criteria to extract the internal forces (Equation (6)).Using this mapping, we demonstrated that the dynamics ofinternal forces correspond to

Fint =(J∗i|l

)Tτint + Fint,{t} − µ∗i − p∗i , (42)

where J∗i|l ,

(L∗UJsW

Tint

). Additionally, Fint,{t} are in-

ternal forces induced by task behavior, i.e. Fint,{t} ,

WintJT

s J∗TtaskFtask, and µ∗i and p∗i are Coriolis / centrifugal

and gravitational effects on the internal forces. Inverting theabove equation, we derive the torques needed to accomplisha desired internal force,

τint = J ∗Ti|l


), (43)

where J∗i|l is the Moore-Penrose left-pseudoinverse of J∗i|l,

also referred to as the reduced Jacobian of internal forcesacting on the contact closed loops, and Fint,ref is the vectorof desired internal forces which we use as an entry point tocontrol internal forces.

APPENDIX BPRISMATIC INVERTED PENDULUM MODEL

In the proposed planner, numerical integration starts withz(x(n), , y(n)) as described in Fig. 4. The PIP model can beexpressed as the differential equation

x =g + z

z(x− xp),

y =g + z

z(y − yp). (44)

Accounting for z being a function of x, y, the height surface,

dz

dt=∂z

∂xx+

∂z

∂yy, (45)

d2z

dt2=

d

dt

(∂z∂x

)x+

∂z

∂xx+

d

dt

(∂z∂y

)y, (46)

z =∂2z

∂x2x2 +

∂z

∂xx+

∂2z

∂y2y2 +

∂z

∂yy. (47)

By plugging Equation (44) into Equation (47), we obtain,

z =∂2z

∂x2x2 +

∂z

∂x

g + z

z(x− xp) +

∂2z

∂y2y2 +

∂z

∂y

g + z

z(y − yp),

(48)(1− g

z(∂z

∂x(x− xp) +

∂z

∂y(y − yp))

)z = (49)

∂2z

∂x2x2 +

∂z

∂x

g

z(x− xp) +

∂2z

∂y2y2 +

∂z

∂y

g

z(y − yp), (50)

z =

∂2z∂x2 x

2 + ∂z∂x

gz (x− xp) + ∂2z

∂y2 y2 + ∂z

∂ygz (y − yp)

1− gz ( ∂z

∂x (x− xp) + ∂z∂y (y − yp))

(51)

APPENDIX CSTATE ESTIMATION

The controller needs a body orientation estimate every servocycle, 0.667 ms, yet the motion capture system updates at only480 Hz, occasionally fails to track a subset of the markers,and has a processing delay. An IMU is used to mitigate thisproblem. When the motion capture position update arrives,the new best estimate of the orientation at the instant in thepast corresponding to the delayed sensor data is found. Wemaintain a list of recent IMU measurements, and calculate anew estimate of the current robots orientation by integratingthe angular velocities.

We use least squares to minimize the distance betweenmotion capture LED positions yki ∈ R3 and predicted motioncapture LED position yki ∈ R3 for i = 1, . . . , n, where n istypically 7, but decreases when LEDs are blocked, and wherek represents the time in the past associated with the delayedmotion capture data. Our model predicts motion capture LEDlocations based on an affine transformation of a default patternyki = xk + Akzki where T k = {xk ∈ R3, Ak ∈ R3×3} is theaffine transform at time k and the default pattern, zi ∈ R3,represents the LEDs in a known frame. The pattern originis the geometric center of the LED position, Σ7

i=1ejzi = 0for j = x, y, z. Each affine transform includes both a linear


translation and rotation term. Our estimation problem is linearin the individual components of this affine transform, which iswhy we use it. However, this linearity comes at a disadvantage:an affine transform can represent both physically realisticrotation and non-physical skewing and scaling of the pattern.Since the physical reality will always bias the estimationproblem towards valid rigid body transforms, we can safelyassume that these transforms can be converted to a physicalone later. We find the best fit affine transform as follows:

θk =

(xk

vec(Ak)

), yk =

yk1...yk7

= Rθk, (52)

R =

I3×3 exz

ᵀ1 eyz

ᵀ1 ezz

ᵀ1

I3×3 exzᵀ2 eyz

ᵀ2 ezz

ᵀ2

......

......

I3×3 exzᵀ7 eyz

ᵀ7 ezz

ᵀ7

, (53)

Ko ∈ Rn×7 =

eᵀ0 if LED 1 was foundeᵀ1 if LED 2 was found

...eᵀ6 if LED 7 was found

, (54)

W =

(I3n×3n 0

0 λI9×9

), (55)

Rr ,

((Ko ⊗ I3×3)R

0 I9×9

), (56)

θk , (RᵀrWRr)−1Rᵀ

rW

((Ko ⊗ I3×3)yk

θ(k|k − p)

). (57)

Here (52) describes affine transforms in vector form, anddemonstrates the linearity of prediction. The base regressor(53) incorporates the default pattern data, but assumes all theLEDs are visible. We use a knockout vector (54) of variablesize to define which LEDs are to be used in each update.We also employ a regularization term, and a weighting matrix(55) parameterized by a variable λ that controls the tradeoffbetween new rotational information and old data. This resultsin the full, regularized regressor (56), and the best affinetransform estimate (57).

Note that the regularization to θ(k|k−p) in (57) accounts fordynamics using the IMU data history, starting p steps beforek at k− p—the index of the last LED position sensor update,

θ(k|k − p) = θk−p +

k∑t=k−p

(03

vec(ωtIMU×)

)∆t. (58)

Finally, to attain a valid rigid body transform θk from thegeneral affine θk, we convert the direction cosine matrix Ak

into the closest fit quaternion using the method of [44], andthen use that quaternion to generate a new direction cosinematrix Ak which is guaranteed to be a rigid body rotation.The original xk and this new Ak form the rigid body transformθk. The algorithm returns θ(t, k) as the best estimate of the

orientation until a new motion capture LED position messageis received.

ACKNOWLEDGMENT

We are very grateful for the financial support of this projectfrom the Office of Naval Research and NASA Johnson SpaceCenter.

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