Balance of Humanoid robot in Multi-contact and Sliding Scenarios

Saeid Samadi, Stephane Caron, Arnaud Tanguy, and Abderrahmane Kheddar

Abstract— This study deals with the balance of humanoidor multi-legged robots in a multi-contact setting where achosen subset of contacts is undergoing desired sliding-taskmotions. One method to keep balance is to hold the center-of-mass (CoM) within an admissible convex area. This areashould be calculated based on the contact positions and forces.We introduce a methodology to compute this CoM supportarea (CSA) for multiple fixed and sliding contacts. To selectthe most appropriate CoM position inside CSA, we accountfor (i) constraints of multiple fixed and sliding contacts, (ii)desired wrench distribution for contacts, and (iii) desiredposition of CoM (eventually dictated by other tasks). These areformulated as a quadratic programming optimization problem.We illustrate our approach with pushing against a wall andwiping and conducted experiments using the HRP-4 humanoidrobot.

Index Terms— Humanoid and multi-legged robots, balance,multi-contacts, sliding contacts.

I. INTRODUCTION

Humanoid robots are designed to interact with the envi-ronment and are supposed to be able to reproduce all sortsof physical actions that a human could do such as running,flipping, jumping, crawling, etc. In theory, state-of-the-arthumanoid robots have the hardware capability to achieve –to some extended and relative performances, some of thesecomplex behaviors. Yet, they lack efficient control strategieswith robust equilibrium conditions.

Sustaining equilibrium is more challenging in multi-contact settings. Balance in multi-contact conditions wasstudied theoretically in [1] but restricted to non-slidingcontacts. A recent study on the interaction of the robot withthe environment introduces a passivity-based whole bodybalancing framework [2]. In this latter work, gravito-Inertialwrench cone (GIWC) –introduced in [3], is used to keep thebalance in multi-contact, eventually under moving but notsliding contacts.

Sliding motion is one of the actions that humans master inachieving several tasks in their daily lives. Sliding contactshave been studied on several robots by considering dynamicfriction forces and planning and controlling objects by push-ing, see a recent example in [4].

Recent works on humanoid robots are mostly focused onavoiding [5], [6] or recovering [7], [8] slips. Keeping thebalance while sliding on purpose has been studied for twofeet contacts in both rotational and translational directions.Linear contact constraints have been used in [9] for the

The authors are with the Montpellier Laboratory of Informatics,Robotics and Microelectronics (LIRMM), CNRS-University of Montpel-lier, France. A. Kheddar is also with the CNRS-AIST Joint RoboticsLaboratory (JRL), UMI3218/RL, Tsukuba, Japan. Corresponding author:saeid.samadi@lirmm.fr

Fig. 1. Keeping balance of HRP-4 humanoid robot in pushing-against-a-wall scenario by adjusting position of the CoM.

controller to keep the dynamic balance while slipping inrotational directions on the feet. While rotational-slipping,considering the force distribution within the contact surfacesis the most challenging part of the study. Slip-turn motionfor the robot is produced in [10] by minimizing the floorfriction power. The works in [11], [12] are about generatingquick turning motion by rotational shuffling.

As a motion planning for translational shuffling, [13]implemented two-layer controller to satisfy of the stabilitycriteria of the robot using Zero-tilting Moment Point (ZMP)and achieve proper force distribution in two feet contacts.Total motion sequence of this method consists of both slide-and-stop phases and CoM trajectory is generated solvinga QP and related constraints. Also, keeping the dynamicbalance while shuffling is studied in [14] by formulatingcomplementarity constraints into a QP, and in [15] byuniform force distribution assumption on the sole for singlesupport phase.

Multiple sliding contacts for humanoid robots are still achallenge to solve. Our contribution to this field of interestis in introducing an applicable method to keep the balanceof the robot in multi-contact settings. Our method not onlyguarantees equilibrium criteria for fixed contacts, as shownin Figure (1), but also keeps the balance of the robot in thepresence of sliding contacts.

In the following, we structured our work in four sections.

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In Section II, we introduce a CoM support area for multiplefixed and sliding contacts. Holding the position of theCoM inside this area guarantees the balance of the robotwhile some chosen contacts slide and other not. Next, inSection III, we illustrate our methodology for calculatingthe position of CoM under constraints. Sections IV and Vshows the experimental results of proposed method and givesa conclusion of the whole work, respectively.

II. COM SUPPORT AREA

The dynamic model of any robot which can be describedby Newton-Euler equations is in the following form:∑

Wc = −Wm (1)

where Wc ∈ R6 and Wm ∈ R6 are the contact andthe gravity wrenches respectively. We follow the commonpractice to write the resultant linear term (force) first in thewrench followed by the moment term. So, W = [f τ ]T . Byseparating forces and torques of (1), we have:∑

i

fi = −mg (2)

∑i

pi × fi = −pG ×mg (3)

where m is the total mass of the robot, pi,fi ∈ R3 arethe position and force of the ith contact point respectively;g ∈ R3 is the gravity vector and is equal to

[0 0 −9.81

]Tand pG ∈ R3 is the position of the CoM. By introducing theunit vector ez as

[0 0 1

]T, we can rewrite these equations

by separating the vertical component of the gravity vector:∑i

fi = mgez (4)

∑i

pi × fi = mgpG × ez (5)

where gravity acceleration is g = −9.81 m/s2. Moreover, byapplying a cross-product to both sides of (5) by ez , we getthe following equation:

mgez × pG × ez =∑i

ez × (pi × fi) (6)

Next, we use two well-known properties of the cross-product that are:

ez × pG × ez =

pxG

pyG

0

(7)

a× (b× c) = (a.c)b− (a.b)c (8)

by applying equations 7 and 8 into (6) we have:

mgpSG =

∑i

fzi pi − pzi fi (9)

where pSG denotes the position of the CoM in the horizontal

plane and is equal to [pxG pyG 0]T .

Let’s consider we have a robot in a 3-contacts setting. Theposition of the CoM will be formulated as follows:

pSG =

fz3mgp3 +

fz2mgp2 +

fz1mgp1

− [pz3mgf3 +

pz2mgf2 +

pz1mgf1]

(10)

The equation 10 shows the general form of this formularegardless of orientation and positions of contacts. As aparticular case, we assume that both feet are on the groundso that pz1 = pz2 = 0. Consequently (10) becomes:

pSG =

fz1mgp1 +

fz2mgp2 +

fz3mgp3 −

pz3mgf3 (11)

In the following, we specify a region for the feasibleposition of CoM without losing the balance according toour sliding and multi-contact conditions. This region wasintroduced in [16] as a static equilibrium CoM area, and itsextension to 3D in [17]. The purpose of our present work isto be able to balance the robot in multi-contact configurationswith fixed or sliding contacts. Furthermore, this method willguarantee the stability of the robot for sliding contacts suchas wiping a board by hand or shuffling foot motions.

Let’s consider a humanoid having its feet on the groundand one of its arm (e.g. right one) wiping a board (non-coplanar with the other contacts). The wiping trajectorycould be anything and there is no limitation for the directionof wiping. Hence, the position of the sliding contact isa pre-defined parameter according to the designed wipingtrajectory. Therefore, the last two elements of (11) is notrelated to the main variables (p1 and p2) for introducing theCoM support Area (CSA), and we can define them as anindependent variable A:

pSG =

fz1mgp1 +

fz2mgp2 + A (12a)

A =fz3mgp3 −

pz3mgf3 (12b)

then:pSG −A =

fz1mgp1 +

fz2mgp2 (13)

On the other hand, by considering (2) in vertical direction,we have:

fz1mg

+fz2mg

+fz3mg

= 1 (14)

From (14), we introduce “Sc” as the sum of the coeffi-cients of (12a):

Scdef= 1− fz3

mg(15)

and (13) becomes:

pSG −A =

∑αi(Scpi) (16a)

αi =fi

Scmg(16b)

where, ∑αi = 1 (17)

X

YZ

(a) (b)

Fig. 2. Humanoid robot with sliding hand contact on the wall simulateddynamically by Choroenoid1(a) and prototyped in pymanoid2(b).

Equation (17) is a sufficient condition to show that the pointpSG − A should be inside the convex polygon constructed

by connecting pi points. There are two equivalent ways torepresent the wrench applied by the environment on the robotunder a surface contact:

1) Contact forces applied at the vertices of the contactarea [3];

2) A single contact wrench applied at a given point [17].We choose the latter method. To be able to use the surface

contact instead of single points, we replace each of thesepoints (pi) with four edges of the related foot. Coordinatesof new points are available by considering the dimensionof each foot. CSA is depicted in Fig. 2. The green area onthe ground of Fig. 2(a) is CSA for close foot contacts andFig. 2(b) emphasizes this area for far feet contacts in wipingmotion.

As a result, the CSA is constructed according to bothsliding and multi-contact conditions. To keep the balanceof the robot, the CoM should remain inside the CSA duringoperating motions. Setting the exact position for the CoMinside the CSA is the next contribution of our work. Thisis done by formulating this problem as a QP where desiredbehaviors are specified via a set of tasks and constraints.

III. CENTROIDAL QUADRATIC PROGRAM

A common solution for keeping the balance of the robotis to put a constraint on the position of the CoM and hold itinside the CSA. This constraint will ignore some parametersthat affect the performance of the robot such as keepingthe maximum distance of CoM from edges of the CSAor considering an appropriate wrench distribution on thecontacts.

A. Decision Variables

Collete et. al [1] introduced a quadratic program thattakes into account to the position of CoM, linearized contactforces and the gravity wrench in static equilibrium. Here weextend this QP to enable sliding contacts and apply slidingconstraints to the related contact. Our QP is designed withthe following decision variables:

Y =[pSG Wrf Wlf Wrh

]T1×21

(18)

1http://choreonoid.org/en/2https://github.com/stephane-caron/pymanoid/

where W denotes the wrench in the world frame andsubscripts “rf”, “lf” and “rh” correspond to the right-foot,left-foot and right-hand contacts, respectively. The solverdeals with position of the CoM. This centroidal QP computesdecision variables at each time step and sends them to therobot as a command. The static equilibrium of the system isgiven in (1). Accordingly, by considering three contacts (onehand contact and two feet contacts), Newton-Euler equationwrites:

Wm + Wrf + Wlf + Wrh = 0 (19)

where wrenches can be formulated as follows:

Wm =

0 0 00 0 00 0 00 −mg 0mg 0 00 0 0

pSG +

00−mg000

(20)

Wrf,lf =

I3×3 03×3

0 −pz pypz 0 −px I3×3

−py 0 px

wrf,lf (21)

where w shows the wrench in contact frame and is equal to[f ∈ R3 τ ∈ R4]T . Wrh is calculated in a same way withWrf,lf. In this way, we can shortly write these equations as:

Wm = Em1.pSG + Em2 (22a)

Wrf = ErfWrf (22b)

Wlf = ElfWlf (22c)

Wrh = ErhWrh (22d)

B. Sliding Condition

Consider a box in direct contact with the ground as shownin Fig. (3). An external force Fext is applied to this box. Thebox will not move as long as the contact force fc lies withinthe Coulomb friction cone C [18].

Fig. 3. Friction cone of a fixed box in presence of external force

Usual works in locomotion, multi-contact conditions ormotion generation for humanoids avoid slippage whereas weaim for it when needed. To avoid slipping, it is common touse inequality constraints to hold each contact force withinits associated friction cone. Instead, we are implementingconstraints to generate the sliding motion.

For generating controlled slipping motion, force vector ofthe sliding contact (fc) should remain at the edge of therelated contact’s friction cone, and hence expressed as equal-ity constraints. Hand torques are considered as inequalityconstraints.

We introduce two matrices named [S1] and [S2] such thatthe normal hand contact force and the related friction forcescould be given to the QP as a constrain:

[S1]6×6 =

[I3×3 03×3

03×3 03×3

](23)

so that:[S1]Wrh =

[f3×1

03×1

]rh

(24)

For the sliding contact calculations, f xrh shows the normal

force applied to the right hand in the world frame. Accordingto the wiping trajectory, the contact forces in y and zdirections could be calculated:

[f3×1

03×1

]rh=

0 0 . . . 0fyfx

0 . . . 0

fzfx

......

...0 0 . . . 0

6×6

wrh +

fx00000

(25)

where fyfx

and fzfx

are constant numbers and they are inde-pendent from the forces. From (24) and (25), we have:

[S1]wrh = [S2]wrh + k (26)

thereby:[S1 − S2]wrh = k (27)

C. Non-sliding Conditions

The sliding contacts are expressed with equality con-straints. However, to keep the rest of the contacts fixed onthe ground, we have to implement inequality constraints tomaintain the related contact force inside the friction cone.The general constraints that should be applied to contactwrenches in the QP solver are same as [19]:

| fx |6 µfz , | fy |6 µfz , fminz 6 fz 6 fmin

z

| τx |6 Yfz , | τy |6 Xfz , τminz 6 τz 6 τmin

z

(28)

where X and Y are dimentions of contact surfaces, so:

τminz = −µ(X+Y)fz+ | Yfx−µτx | + | Xfy−µτy | (29)

and

τmaxz = µ(X+ Y)fz− | Yfx + µτx | − | Xfy + µτy | (30)

The equations (28) are combined in the following form forfeet contacts:

±

fxfyfzτxτy

rf,lf

6

µfzµfzfmaxz

YfzXfz

rf,lf

(31)

Furthermore, for sliding contacts, we consider torquessimilarly to (31) in all directions inside the inequality con-straints. Because, sliding contact forces have been consideredin equality constraints to create the sliding motion. Notethat, there is no τz element inside the vectors and it willbe considered separately. We need to generate vectors in away that we can apply it as an inequality constraint to thesystem. For this reason, we re-write these equations usingG ∈ R6×21 and h ∈ R6 matrices and vectors:

G1rf,lf,rhY 6 h1

rf,lf,rh (32)

G2rf,lf,rhY 6 h2

rf,lf,rh (33)

where:

G1rf =

[06×3 Υ1

rf 06×12

](34a)

G2rf =

[06×3 Υ2

rf 06×12

](34b)

G1lf =

[06×9 Υ1

lf 06×6

](34c)

G2lf =

[06×9 Υ2

lf 06×6

](34d)

G1rh =

[06×15 Υ1

rh

](34e)

G2rh =

[06×15 Υ2

rh

](34f)

where Υ1rf,lf ∈ R6×6 is defined as:

Υ1rf,lf =

1 0 −µ 0 . . .

0 1 −µ...

... 0 1−Yrf,lf 1

−Xrf,lf 1...

0 0 . . . 0

Note that Υ2

rf,lf is same as Υ1rh,lf except the diagonal values

of the matrix which should be multiplied by −1. Also, forthe sliding contact, we have:

Υ1rh =

0 0 . . . 00...

. . ....

0 0 0...

Xrh 0 . . . 0 1 0Yrh 0 . . . 0 1

and for Υ2

rh, just replace two last elemets on the diagonal ofthe matrix with -1. For the other side of the inequality, h iszero vector for sliding contacts. For fixed contacts, we have:

h1rf,lf =

[0 0 fmax

z 0 0 0]T

(35a)

h2rf,lf =

[0 0 fmin

z 0 0 0]T

(35b)

h1,2rh = 06×1 (35c)

Besides, we should consider inequality constraints on τz .According to (28), we divide it to two types of boundarieswhich should be implemented for all contacts:

τz − τmaxz 6 0 (36)

−τz + τminz 6 0 (37)

consider that there is two absolute values inside each of τmaxz

and τminz based on (29) and (30) that each equation results

in four more rows inside the inequality matrix that should bemultiplied by Y. We introduce Gz ∈ R4×21 matrices whichcovers the inequality constraints on τz element of wrenches:

Gz1,2rf,lf,rhY 6 04×1 (38)

where:

Gz1rf =[04×3 Ψ1

rf 04×12

](39a)

Gz2rf =[04×3 Ψ2

rf 04×12

](39b)

Gz1lf =[04×9 Ψ1

lf 04×6

](39c)

Gz2lf =[04×9 Ψ2

lf 04×6

](39d)

Gz1rh =[04×15 Ψ1

rh

](39e)

Gz2rh =[04×15 Ψ2

rh

](39f)

where the Ψ ∈ R4×6 matrices are different for each contactand should be defined separately. They are calculated asfollows. Consider (36) for the right foot. By implementingthe amount of τmax

z from (30) and considering positive signfor the both of the absolute values, we get to the followingequation:

Yrhfx + Xrhfy + Crffz + µτx + µτy + τz 6 0

where Crf = −µ(X + Y) and is computed in the same wayfor other contacts.By considering negative sign for absolutevalues, the other three equations are available and Ψ matrixfor for upper bound of the right foot is calculated:

Ψ1rf =

Yrf Xrf Crf µ µ 1−Yrf Xrf Crf −µ µ 1Yrf −Xrf Crf µ µ 1−Yrf −Xrf Crf −µ µ 1

Notice that the first row of this matrix corresponds to

the above equation. Other matrices for the lower and upperbound of the contacts will be calculated in the same way.

D. QP Formulation

The goal of centroidal QP is to achieve Ydes as desiredamount of these decision variables as much as possible.Therefore, the minimization problem is:

‖Y −Ydes‖2 (40)

where the desired position for CoM is the ‘middle’ of theCSA and for feet contacts, desired quantity is equal to thewrench distribution for both.

According to (40) and sliding and non-sliding conditions,the QP formulation is in the following form:

minY

1

2YTPY + qTY (41a)

GY 6 h (41b)AY = b (41c)

where P = 2I21×21 and q = −2Ydes. By introducing GTrf as

transpose matrix of Grf and the same for the other contacts,matrices and vectors of the equality constraints in (41) are:

G =[G1,T

rf G2,Trf G1,T

lf G2,Tlf G1,T

rh G2,Trh

]T21×60

(42a)

h =[h1,T

rf h2,Trf h1,T

lf h2,Tlf h1,T

rh h2,Trh

]T21×1

(42b)

On the other hand, for equality constraints, (22) from sec-tion III-A and (27) from section III-B, should be used. Tocombine these constraints in one equation and be able to usethem in QP directly, matrix A ∈ R12×21 and vector b ∈ R12

are defined as follows:

A =

[Em1 Erf Elf Erh06×3 06×6 06×6 S1 − S2

](43a)

b =[−ET

m2 ∈ R6×1 −fx 0 . . . 0]T

(43b)

E. Controller Specification

Position of the CoM and wrenches of contacts is calculatedby centroidal QP. These values should be applied on thereal robot as commands. For this purpose, we use a whole-body dynamic controller based on another QP formulationintroduced in [20] using three following tasks.

1) Posture task: is a regularization task based on degreesof freedom q that brings the robot to a reference joint-angle half-sitting configuration qhalf-sit and its derivatives byconsidering task stiffness K via:

q = K(qhalf-sit − q)− 2√

Kq (44)

2) End-effector admittance task: takes a desired positionpd ∈ R6 as target in world frame, desired wrench wd ∈ R6

in sensor frame and admittance gains A ∈ R6. For a givendegree of freedom i, the task is a position task if Ai = 0and a force task if Ai 6= 0 and in this case, force feedbackis applied by pi = Ai(w

di −wi). In experiments, we apply

this task to the right hand of the robot which is in contactwith the wall.

3) CoM task: takes as target a desired position cd ∈ R3,velocity cd ∈ R3 and acceleration cd ∈ R3 in world frame.This is a standard second-order task. Internally, it will realize:

c = K(cd − c) +B(cd − c) + cd (45)

where B is the task damping (usually 2√K) and c = Jcomq

is the CoM velocity where J shows Jacobian matrix.Finally, we regulate the weight distribution between the

two feet on the ground while the end effector of the robotacts on the contact surface. For this purpose, we use:

4) Foot force difference control [21]: applies to two end-effectors whose targets are in the same plane, in our case theleft foot and right foot. The force difference (fz,lf− fz,rf) isregulated to the value (fdz,lf−fdz,rf) provided by the centroidalQP by applying damping control to a virtual offset z:

z = Az((fz,lf − fz,rf)− (fdz,lf − fdz,rf)) (46)

The velocity −z is then applied to the left foot, while +z isapplied to the right foot.

Fig. 4. Normal force tracking while Pushing the wall using fixed CoM.

Fig. 5. Normal force tracking while Pushing the wall using proposedstrategy

IV. EXPERIMENTS AND RESULTS

We implemented our new methodology for keeping thebalance and did several experiments with HRP-4 humanoidrobot. In this section, we discuss three of these experimentsand show the performance of our controller in pushing andwiping scenarios. The baseline for our study is consideringa fixed position for the robot’s CoM. The Figure (4) showsthe robot scrabbling to increase the normal force on the handcontact and achieve the target force of 60 N but fails.

Force tracking of admittance control shows that the failureoccurs in less than 20 N normal force and the robot is notable to achieve the target force with this posture anymore.On the other hand, we did the same experiment with ourproposed controller. The Figure (5) shows successful trackingof the normal force with the generated posture of the robotdue to the position of CoM and kinematics of the robot.

Also, the trajectory of CoM while increasing and decreas-ing the right-hand force is shown in Figure (6). Position ofthe CoM moves forward by increasing the target normal forceand lays back by decreasing the force. The last experimentdeals with sliding contact and shows the performance of thecontroller while wiping a vertical surface with a normal forceof 30 N on sliding contact. Figure (7) shows the normal forcetracking of sliding contact. In this experiment, the normal

Fig. 6. Position of CoM while pushing the wall.

Fig. 7. Normal force tracking while wiping the wall using proposedstrategy.

force is set to 30 N by admittance task of the controllerand starts wiping from 16th second. This is the reason oferror occurred at this moment but then the normal forceconverges to the target value. Experiments are available inaccompanying video 3.

V. CONCLUSION

Stability criteria for humanoid robots based on the positionof the CoM have been studied in the current work. A method-ology for constructing CSA in multi-contact conditions isintroduced. This method, covers multiple fixed and slidingcontacts. For setting CoM inside this the CSA, we use cen-troidal QP by considering constraints. The robot achieves tothe desired configuration by implementing inverse kinematicsof the system according to the position of the CoM.

Simulations and experiments show that proposed balancecontrol is valid for both fixed and sliding contacts in practice.Also, by using this method, the robot is able to achieve aproper body configuration in order to reach target forces incontacts. Online estimation of the contact friction should beaddressed as part of future work. The transition between fixedand sliding contact modes, which is still a challenge to solve,causes errors in target force tracking.

3https://www.youtube.com/watch?v=Wai-Lp4e5FE

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