Optimisation of Body-ground Contact for Augmenting

Whole-Body Loco-manipulation of Quadruped Robots

Wouter J. Wolfslag, Christopher McGreavy, Guiyang Xin, Carlo Tiseo,Sethu Vijayakumar and Zhibin Li∗†

Abstract

Legged robots have great potential to perform loco-manipulation tasks, yet it is challenging to keep therobot balanced while it interacts with the environ-ment. In this paper we study the use of additionalcontact points for maximising the robustness of loco-manipulation motions. Specifically, body-ground con-tact is studied for enhancing robustness and manipu-lation capabilities of quadrupedal robots. We proposeto equip the robot with prongs: small legs rigidly at-tached to the body which ensure body-ground contactoccurs in controllable point-contacts. The effect ofthese prongs on robustness is quantified by computingthe Smallest Unrejectable Force (SUF), a measure ofrobustness related to Feasible Wrench Polytopes. Weapply the SUF to assess the robustness of the system,and propose an effective approximation of the SUFthat can be computed at near-real-time speed. Wedesign a hierarchical quadratic programming basedwhole-body controller that controls stable interactionwhen the prongs are in contact with the ground. Thisnovel concept of using prongs and the resulting con-trol framework are all implemented on hardware tovalidate the effectiveness of the increased robustnessand newly enabled loco-manipulation tasks, such asobstacle clearance and manipulation of a large object.

1 Introduction

Combined locomotion and manipulation tasks are akey competence for legged robots in applications suchas warehousing, search and rescue, and offshore in-spection and maintenance. To manipulate objects, arobot must exert forces onto the environment. To lo-comote, the robot must remain balanced and stableunder the load of the manipulation. The main chal-lenge of loco-manipulation is performing these taskssimultaneously by managing the limited resources re-quired to complete them: motor torques and tangen-tial contact forces [1,2]. Better management of theseresources will improve the robot’s workspace, payload,

∗This work was supported by EPSRC UK RAI Hub for Off-shore Robotics for Certification of Assets (ORCA, grant refer-ence EP/R026173/1), the EPSRC CDT in Robotics and Au-tonomous Systems (EP/L016834/1) and EPSRC UK RAI HubNCNR (EPR02572X/1)

†Authors are with the School of Informatics, University ofEdinburgh, UK.

Figure 1: Body-ground contact for enabling diverseloco-manipulation tasks: manoeuvring objects by legs.

robustness and stability. This paper investigates howto improve that management by adding contact pointsto a quadruped robot.

Previous work has shown that extra contact pointsreduce resource consumption and improve stability.Examples are found in humans or humanoid robotsusing their arms for balance and in multi-finger andarm manipulation, [3], and [4–6], and [7] respectively.Accurate force control at multiple contacts increasesrobustness of quadrupedal locomotion, especially inrough terrains and with disturbances [8].

Additional contacts, however, also produce chal-lenges in control, due to the uncertainty in estimatingexact contact locations, and dealing with non-trivialsurface geometries of the contacting body. Complexcontacts do not fit well into multi-contact frameworks,which rely on simple contact geometry, and often oncontacts only occurring at the end of the kinematicchain. This can limit the versatility of using extra con-tact points such as knee-ground contact [9], sliding [10]or rolling interactions in humanoid robots [11]. Recentmachine learning approaches address more complexcontact scenarios, such as in hand-manipulation andJenga [12, 13] but also have limited versatility due tochallenges of learning.

In contrast, humans and animals use variousparts of their bodies to increase movement stabil-ity. We are motivated to investigate how quadrupedalrobot movement might benefit from additional non-conventional body-ground contacts, which will beevaluated in this paper.

To enable versatile body-ground contact, we equip aquadruped robot with additional fixed limbs (see Fig.

arX

iv:2

002.

1055

2v1

[cs

.RO

] 2

4 Fe

b 20

20

1), which we call prongs. These prongs are rigidlyattached to the base of the robot and ensure point-contact at a known location. Such contact fits into thewhole-body force control pipeline shown to be versa-tile in other multi-contact scenarios [14]. Rigidly con-necting the prongs to the robot’s torso means they willreduce actuator loads by supporting the robot’s weightwhen they are in contact with the ground, thereby al-lowing the robot to perform additional tasks. Notethe contrast between prong-ground and belly-groundcontact: by using prongs, we know the exact contactlocations, which would be difficult to estimate whenusing the belly. Furthermore, the height of the prongsallows the body to be mobile while maintaining con-tact with the ground, which would be more difficultwith belly-ground contact.

Prong-like concepts are seen in wheeled platforms.Using outriggers, a wheeled robot can resist more dis-turbance force [15] [16], with an estimate of the ben-efits found in [17]. Legged robots can be augmentedwith wheels or skates at the feet to speed up locomo-tion in easy terrain [18,19], or with a tail to counteractinertial shifts during fast locomotion [20]. Augmenta-tions proposed in this paper can be used in parallelwith those mentioned above.

While the prongs provide controllable ground con-tact, three open questions remain: how to designprongs so they provide maximal benefit, how to dealwith the control challenges posed by body-groundeven with the simplified point contacts, and how toplan motions while deciding if and how to make con-tact with the prongs. The focus of this paper is proof-of-concept and analysis of enabling body-ground con-tact, therefor it considers neither planning, nor furthermechanical enhancement such as retractable prongs.

We first deal with prong design, which must con-sider placement, ground-clearance (length), manipu-lability and disturbance rejection capabilities. We de-fine the Smallest Unrejectable Force (SUF) as a met-ric to quantify disturbance rejection ability in loco-manipulation tasks under actuation limits and inter-action constraints, and provide a fast-to-compute ap-proximation, which are used optimise the design of theprongs. Our approach extends previous work show-ing the importance of optimizing posture for robust-ness [2, 21].

To control the robot with prongs, we use the es-tablished framework for modern quadruped robots:Quadratic Programming (QP) based inverse dynamiccontrollers [8, 14, 22, 23]. However, these controllerswere designed for contacts at the ends of the kinematicchain, not at the torso. The control of non-end-effectorlimb contact has been studied in manipulation [24]considering contacts with moving obstacles which donot kinematically constrain the contact limbs. Whenusing prongs, the torso will be constrained, and theabove controllers might become unstable. We proposea hierarchical QP controller that uses body-groundcontact constraints to minimise motor torques.

1.1 Contributions

Our paper studies the design of prongs for body-ground contact in quadrupedal robots. We vali-date their performance in three hardware experiments:push-rejection, obstacle clearance and object manip-ulation. The last two experiments use two conven-tional legs freed for manipulations by the support ofthe prongs. This provides the following contributions:

1. A proof-of-concept prong design for the ANYmalrobot which enables effective body-ground con-tact (Sec. 3).

2. A novel method to quickly compute an approxi-mation of the Smallest Unrejectable Force, a mea-sure for the robustness of the robot (Sec. 4).

3. Metrics for benchmarking the robustness and sta-bility of a robot with and without prongs (Sec.7).

4. A hierarchical QP controller that enables therobot to be operated with prongs by includingcontact constraints on base movement (Sec. 6).

Section 2 explains our notation for robot dynamics.Section 3 discusses the optimal design of the prongs.Our robustness measure, the SUF, and novel approx-imations of it are explained in Section 4. Section5 shows results from simulations and optimisations.Section 6 explains our controller for the hardware ex-periments. Hardware experiments highlighting the ef-ficacy of the prongs are shown in Sect. 7. Finally,discussion and conclusion are in Secs. 8 and 9.

2 Preliminaries/Robot Dynam-ics

The dynamics of a quadrupedal robot with a manipu-lator, and prongs attached, as shown in Figure 2, aregiven by:

M(q)q+h(q, q)=d(q, q, q)=Bτ +J>f λf +J>p λp +J>e F,(1)

where q are the generalised coordinates of the robotdescribing the position and orientation of the body,and the position of each joint, M(q) is a positive defi-nite mass matrix, h(q, q) is the dynamic bias contain-ing of centrifugal, Coriolis and gravitational effects, τare the joint torques, B is a selection matrix, Jf, Jpand Je are the Jacobians of the feet, prongs and end-effector of the arm respectively, and λf, λp and F areexternal (reaction) forces at those points.

These equation of motions are subject to furtherconstraints to ensure physically feasible ground inter-action and joint/motor torques. Ground interactionconstraints ensure the robot does not slip or pene-trate the ground, and are only considered when theassociated body part is in ground contact. For compu-tational efficiency, these conditions are approximated

2

as linear constraints for each contact point i:[0 0− 1

]λi ≤ 0 (2)[

1 0 − 12

√2µ]

abs(λi) ≤ 0 (3)[0 1 − 1

2

√2µ]

abs(λi) ≤ 0 (4)

Jiq + Jiq = 0 (5)

where µ is a friction coefficient, and the abs-operatorreturns the piece-wise absolute value. Additionally,the motor capabilities are reflected in bounds on thejoint torque for each joint index i and torque limit τi:

− τi ≤ τi ≤ τi. (6)

3 Optimal Prong Design

The prongs enlarge the buffer between the motortorque limits and the torques required to stand, whichcan then be used to reject disturbances or perform sec-ondary tasks. However, the magnitude of the benefitsdepend on how the prongs are placed and sized. To in-vestigate the effects of the prong design we focus on ascenario in which a force is applied at the end-effectorof an arm attached to the torso of a robot (Fig. 1).We find the effect of prong placement on the size ofthe disturbance the end-effector can sustain withoutmoving the robot. We then optimise the prong place-ment and the robot configuration for this robustnessmeasure.

By using a two prong configuration, the robot caneither fix torso pitch (by grounding both prongs) orallow pitching (by grounding one prong) depending ontask requirements. To ensure symmetry both prongshave equal length, and are placed on the x-axis of therobot frame. Furthermore, we enforce a symmetricposition of the feet. In this configuration, maximumrobustness is achieved when the prongs are furthestapart, so prongs are placed as far apart as possiblewithout interfering with the leg motion. As a result,the optimisation only requires three parameters:

max{xf ,yf ,bz}

FSUF(xf , yf , bz), (7)

where the SUF is a measure for the robustness (de-fined in the next section) and xf , yf and bz are thefeet x and y position and torso z position. These deci-sion variables are shown in Figure 2. When optimizingprong position, the prong length matches the heightof the torso. To solve the inverse kinematics, we usea standard iterative procedure with the transpose Ja-cobian, which requires no further regularization.

4 Smallest Unrejectable Force

A key element of he robustness of a robot is theamount of external force it can withstand while track-ing a target motion. Computing such forces and as-sociated robustness metrics can be done via Feasi-ble Wrench Polytopes, as discussed for legged robots

Figure 2: Rejectable Force Polytope and maximal re-jectable force for optimised robot configuration withprong (bottom) and without prongs(top). The end-effector is set to a world frame position: {0.8m, 0.2m,0.4m}.

in [25], and for manipulation in [26]. The FWP isthe set of wrenches applied to the robot, such thatthe ground reaction forces and joint torques requiredto execute the desired motion stay within the fric-tion cone and motor limits respectively. Here, we areinterested in a slight variation: the Rejectable ForcePolytope (RFP), the set of forces that can be appliedto the robot at the end-effector, such that it is able toperform the desired accelerations while satisfying theconstraints in Eqs. 1-6.

FRFP(q, q, qd) = {F ∈ R3|Eqs. 1-6 hold for some

values of(λf , λp and τ),

q = qd}, (8)

where qd is the desired acceleration. The RFP is apolytope, as the constraints are linear in the free vari-ables.

In practice, it is desirable to summarise the RFPinto a single robustness metric. For this we proposethe Smallest Unrejectable Force: the smallest distur-bance force the robot cannot withstand while perform-ing its desired motion. This is the same as finding thethe Chebyshev radius of the RFP, but with the centreof the circle fixed to the origin.

The scheme from [27] can compute the exact RFP,but its computation time does not scale well withthe number of contacts and joints, as it requires atransition between vertex-representation of the FWPto its half-space-representation. Computing the SUF

3

from the RFP is also computationally expensive, so asimplified metric that finds the Smallest UnrejectableForce in a single predetermined direction was proposedin [25]. Here we propose and investigate three approx-imations of the SUF: Fibonacci, affine, and quadratic.

The Fibonacci approximation is based on an inner(conservative) approximation for the RFP. Points onthe boundary of the RFP are found by solving theoptimisation problem:

maxf,λf ,λp,τ

f s.t. Eqs. 1-6, F = fF

where the resulting f is the maximum feasible scal-ing factor for force in the direction F , considering thedynamic equations and leg joint torque limits τ .

The approximation of the RFP is the polytopespanned by vertices found by solving the above optimi-sation for a set of approximately uniformly distributedforce directions according to the Fibonacci-sphere.

To determine the size, ρ of the Smallest Unre-jectable Force, we check for each halfspace that deter-mines the polytope to check if the worst case force di-rection would violate the associated constraint at thatvalue of ρ. Given half-space representations aiF ≤ bi,where i indices the halfspaces of the RFP, we knowthe worst case force is in the direction of ai (see [28]).Hence we solve the following optimisation problem us-ing enumeration:

maxρρ s.t. ρ||ai|| ≤ bi ∀i = max

i(bi||ai||−1)

(9)The Fibonnacci approximation still requires a con-version from vertex to halfspace representation. Thecomputation time depends on the number of verticessampled. We use 1024 samples for a high quality ap-proximation.

The affine and quadratic approximations of the SUFdo not compute the RFP explicitly. Instead, they findthe worst case disturbance force (similar as above),while simultaneously solving for an optimal controllaw determining how the joint-torques and ground re-action forces change with the disturbance force. Asthe true (nonlinear) optimal control law cannot becomputed efficiently, the two approximations assumean affine and quadratic control law respectively [29].The optimisation problems deviate slightly from thosein [29] to simplify handling of the equality constraintsfor this specific scenario, and to search for the largestsphere centred around the origin, rather than aroundan arbitrary point.

First reparametrise the control equations:[F τ λ

]>=

[I 0

−W+Je> N

] [FδQ

]+

[0

W+d

]where δQ are combinations of joint-torques andground reaction forces in the null-space of the dy-namics equation, which are solved using the matrixW =

[B J>f J>p

], the matrix N is a basis for the

nullspace of W and the Moore-Penrose pseudo-inverseis indicated by a +.

The affine approximation optimises an affine controllaw from disturbance to reaction forces and torques:

δQ = δQ0 + V F (10)

where δQ0 are nominal joint torques and ground reac-tion forces, and V is a gain matrix. These parametersare optimised along with the size of the SUF (ρ), viathe conical quadratic program:

maxρ,δQ0,V

ρ s.t.[0 δQ0

]ai +

∣∣[ρI V >]ai∣∣ ≤ bi ∀i

(11)the constraint coefficients in ai and bi are takenfrom Eqs. 5-6. The quadratic approximation usesa quadratic control law. The resulting semi-definiteprogram is included in the Appendix.

5 Simulation Experiments

This section first compares the proposed approxima-tions on computational efficiency and accuracy. Thenthe prong optimisation problem from Section 3 issolved using the affine approximation of the SUF.

5.1 Comparing Approximation Meth-ods

This section compares the computation time and accu-racy of all computation methods: exact, single direc-tion [25], Fibonacci, affine and quadratic. The simula-tions were implemented with Julia’s libraries for rigidbody dynamics [30] and optimisation [31, 32]. Thecode ran on a PC with Intel Core i7-7830x processorand 32Gb of memory.

To compare the approximations the SUF was com-puted for random robot configurations from two sce-narios: 1. tele-operation scenario, similar to [27], inwhich there is no arm attached to the robot, threelegs are on the ground, and the remaining leg is usedas end-effector 2. a scenario with an arm attached tothe robot functioning as end-effector, and all four legsof the robot in contact with the ground (see Fig 2).

The results, shown in Table 1, affirm the slowcomputation of the exact method. The affine andquadratic approximations are faster than the Fi-bonacci approach. The quadratic approximation

0.2 0.3 0.4 0.5

70

75

80

85

90

95

Torso height [m]

Disturban

ceRejection

(SUF)[N

]

prongno prong

¸Approaching Max

Base Height

Figure 3: The SUF for each height of the base. Thefoot locations are optimized for each height.

4

affine fibonacci quadratic single0

1

2

3

Approximation

Nor

mal

ized

SU

Fundesirable

desirable

Figure 4: Boxplot of relative SUF size in each ap-proximation. The background color indicates the de-sirability: accurate approximations and underestima-tions are more desirable.

scales less well to the arm-attached scenario, due tothe number of parameters in the quadratic term of thecontrol law. The single direction approach is clearlyfastest.

Figure 4 conveys the quality of the approximations,by comparing the sizes of their SUFs relative to ex-act computation. Only the arm-attached scenario isshown to highlight the potential differences in quality,as those are larger in that scenario. The quadraticand Fibonacci approximations are very close to exact,with the affine approach resulting in slight underap-proximations. The affine approximation failed to con-verge once due to numerical issues. Note these threeapproximations are conservative. The single-directionapproximation is shown to have poor accuracy. Fur-thermore, this approach overestimates of the SUF,which is undesirable for robustness analysis. Due tothe favourable trade-off between conservativity, accu-racy and computation time, the affine approximationis used in the remainder of this paper.

5.2 Optimizing Robust Body-GroundContact

We optimised the prong length and placement, in or-der to maximise the SUF for a fixed end-effector po-sition, see Eq. 7. The results of this optimisation areshown for scenarios with and without prong in Fig-ure 2. Shown are the optimal configuration of therobot, the resulting rejectable force polytope, and asphere with the SUF as radius. The forces are scaledusing a ‘stiffness’ of 1000 N m−1. The polytopes andspheres are translated, such that their origin (0 dis-turbance force) is at the end-effector. The minimalnon-rejectable forces are 88 N and 96 N respectively.

To show the efficacy of prongs, we also found theSUF given a body height, optimising only the foot lo-cations, see Figure 3. Slight noise is caused by approx-imations in the IK algorithm. When the prongs are

Figure 5: The push recovery experimental setup. Therobot is pushed using a stick with a force/torque sen-sor.

attached, the SUF is pointed upwards, and is limitedby the unilaterality conditions. The prong’s lengthhas little effect in this direction, so does not effectthe SUF. Therefore, the prong length can be decidedby other considerations: ground clearance and a mini-mum height from the base. The prongs also have littleeffect on the SUF when the torso height is larger, asthe robot legs are then close to their singular position,which limits joint torques. However, as this singularitycomes with mobility and control issues, such heightsare undesirable. For more practical torso heights wesee that the prongs provide a benefit of up to 35%.

6 Controller Design

The whole-body-control of ANYmal uses the well es-tablished hierarchical QP paradigm [33]. To ensurethe robustness of this control paradigm, some ad-ditions have been proposed recently. For example,the techniques from [22] aim to improve robustnessagainst joint tracking errors. In this paper, we followthe hierarchical QP framework by combining the footcontact constraints and the prong contact constraintsinto a single augmented contact Jacobian. As such,the prong and foot contacts are considered in the sameway and their forces are optimised simultaneously.

At each time-step, as part of the QP, the controllerminimises an error between the desired and actualtask-space accelerations space: ||x − xd||2w, with thedesired accelerations based on the error e in the taskspace position. If for x = xd the kinematic constraintsdo not hold, i.e. the system is overconstrained withrespect to its desired movement, minimising the ac-celeration error might lead to unstable behaviour. Toensure a solution that stabilises the robot, we use awhole-body controller consisting of the following fivehierarchical layers, each solving a QP. Each layer en-sures that the optimality conditions of the previouslayers are satisfied, i.e., it optimises in the nullspaceof the previous layers.

1. Dynamic feasibility: finds any feasible solutionfor the dynamic constraints, Eqs. 1-6.

2. Torso angular acceleration tracking: minimisesthe error between the desired angular accelera-tion and the executed angular acceleration.

5

Table 1: Computation time of SUF-approximations in ms

Task Exact Fibonacci Affine Quadratic Single

teleoperation 4301 588.8 13.8 84.169 4.65manipulation 37747 727.0 17.49 709.1 4.93

5 10 15 20

0.0

20.0

40.0

Push

Force

[N]

prongno prong

5 10 15 200.00

0.02

0.04

Displacement[m

]

5 10 15 205.0

15.0

25.0

Time[s]

Motor

Torque[N

m]

¶Torque Reduction

Figure 6: Comparison of torso displacement, maxi-mum push force and required motor torque (left frontknee) when pushing the robot with (red) and without(blue) prongs.

3. Torso translational acceleration tracking: whenthe prongs contact the ground, this layer has noeffect on the outcome, as translational accelera-tion is not in the available nullspace. This pre-vents unstable behaviour.

4. Swing foot acceleration tracking

5. Torque minimisation: minimises the sum ofsquared motor torques, in order to reduce energyconsumption.

The implementation also reduces computational loadvia the trick from [34] to avoid directly computing thetorques.

7 Hardware Experiments

To validate the use of prongs, we perform three ex-periments. 1. Pushing the robot to assess the joint–torques. 2. Clearing out an obstacle with the robots’conventional legs, freed for this task by the prongs.3. Lifting a box with two side legs, to establish theversatility of the controller.

7.1 Push Recovery

To verify that our controller is able to use the prongsto enhance the robustness of the robot, we push the

robot with a rod equipped with a force-torque sensor.The experiment is shown in Figure 5. In the experi-ment, we push the robot horizontally on the middle ofthe base. The robot is in the same configuration forboth experiments, which is the default standing con-figuration of the robot-platform with the torso heightlowered to the height of the prongs. The force is gradu-ally increased up to approximately 30 N. After holdingthis force for 5 seconds, the force is reduced to 0.

Figure 6 shows the base displacement, disturbanceforce and motor torque during the experiment. Thekey result is found by comparing the effective compli-ance and the amplification factor between disturbanceforce and motor torque during the period of maximumpush force. The motor torque is significantly lowerwith prong, despite a slightly stiffer torso behaviour.These reduced knee-joint torques result in a capacityto reject larger disturbance forces.

7.2 Obstacle Clearance

The second experiment shows robot’s ability to per-form basic manipulation using the prongs. Standardquadrupeds would be unable to perform manipulationwith more than one leg, as they are required for stand-ing. The prongs take over responsibility for standing,freeing the legs for manipulation. Using the free-gaitmotion description library [35], we generate a sequenceof body and end-effector targets, such that the robotspushes an obstacle away.

The resulting motion can be seen in Figures 7 and8. Note that, by necessity, the legs of a quadruped areequipped with relatively strong motors, which makesthem well suited for obstacle clearing tasks such asthis. Performing such a task is only possible whenrelying on body-ground contact. Enhancing the capa-bilities of the robot to allow obstacle clearing makesthem more versatile in rough terrain.

7.3 Lifting Box

Experiment three shows controlled torso mobilitywhile the prongs contact the ground, and further ma-nipulation with the legs. The robot lowers on itsprongs, and leans to its right side, freeing the left sidelegs for manipulation. The legs are controlled in task-space with low-gains, allowing basic dual arm manip-ulation: picking up a box. Figures 9 and 10 showssnapshots of this motion, lifting two different boxes.This and the previous experiment are captured in theaccompanying video.

6

Figure 7: The robot leaning on both prongs to push a box away. From top left: From a standing position (1),the robot lowers itself onto the prongs (2), which enables the front legs to be lifted from the ground. Thesefront legs are used to push the box (3), which is pushed out of the way (4-6), clearing space for the robot tomove into.

Figure 8: The robot leaning on both prongs to push a box away. From top left: From a standing position (1),the robot lowers itself onto the prongs (2), which enables the front legs to be lifted from the ground. Thesefront legs are used to push the box (3), which is pushed out of the way (4-6), clearing space for the robot tomove into.

When lifting the left-side legs, the desired ground-contact force changes quickly, which can result in jerkymotions. To counteract such motions, we enforce asmooth contact force. Future work on control shouldfocus on the capability of the robot to track more rapidtransitions.

8 Discussion

The improvement in robustness and manipulationcapabilities introduced by the body-ground contactopens several applications, improving the interven-tion capabilities in robots deployed for exploration andmonitoring. However, body-ground contact needs fur-ther research before quadrupeds are ready for suchapplications.

There are three main areas of improvement for fu-ture work. First is extending the affine and quadraticapproximations of the SUF. Knowledge about the dis-tribution of contact forces can be used to bias therobustness measure towards more likely disturbances.With further improvements to the computational effi-ciency, these approximations could be incorporated inreal-time planning and control. Second is incorporat-ing body-ground contact in motion planning. Makingdecisions about when to use the prongs (for robust-ness) and when not (for speed), maximising the util-ity of the prongs is a challenging problem due to theintermittent nature of the contact. Third is applyingprongs to body-environment contact with other robotmorphologies. Two examples are: a quadruped in-

creasing its robustness by leaning sideways on a wallwith its body, and a robot arm increasing its accuracyby contacting a table with a prong on its elbow.

9 Conclusion

This paper studied the use of prongs to enable body-ground contact in quadrupedal robots. We showedthat using prongs increases the robustness of therobot, as measured by its ability to reject forces ap-plied to the end-effector, by up to 35 %, largely inde-pendent on prong length.

We applied an optimisation-based whole-body con-troller that handles the constrained body motion re-sulting from body-ground contact. On the hardware,we verified the increased robustness in the form ofpush resistance with limited motor torques. We alsoshowed obstacle clearance and basic object manipula-tion, two capacities added by the prongs freeing thelegs from their body-support task.

7

Figure 9: The support provided by the prongs is used to free two of the legs for a manipulation task (i.e.,lifting a box)

Figure 10: Motion stills of a second box lifting experiment, using a differently sized box.

A Quadratic Inverse DynamicsRule

To compute the SUF, it is also possible to use aquadratic inverse dynamics law. Optimising this lawfor maximal force rejection, is a semi-definite program.The program is detailed below, adapted from [29].

maxρ,δQ0,V,W,ζ,ξ

ρ s.t. ζ ≤ b (12)

ξ ≥ 0 (13)ζi − ξi − a>i

(0δQ0

)− 1

2a>i

(ρIV

)− 1

2

(ρIV

)>

ai ξi −∑n2

j aij+n1Wj

� 0 ∀ i

(14)

Here the quadratic term of the inverse dynamics lawis W , and we introduced two helper variables ξ and ζ.

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