Torque Controlled Locomotion of a Biped Robot with Link Flexibility Nahuel A. Villa1 Pierre Fernbach2 Maximilien Naveau12 Guilhem Saurel1 Ewen Dantec1 Nicolas Mansard13 Olivier Stasse13

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Torque Controlled Locomotion of a Biped Robot with Link Flexibility

Nahuel A. Villa1, Pierre Fernbach2, Maximilien Naveau1,2, Guilhem Saurel1,

Ewen Dantec1, Nicolas Mansard1,3, Olivier Stasse1,3

Abstract— When a big and heavy robot moves, it exerts

large forces on the environment and on its own structure,

its angular momentum can vary substantially, and even the

robot’s structure can deform if there is a mechanical weakness.

Under these conditions, standard locomotion controllers can fail

easily. In this article, we propose a complete control scheme to

work with heavy robots in torque control. The full centroidal

dynamics is used to generate walking gaits online, link deﬂec-

tions are taken into account to estimate the robot posture and

all postural instructions are designed to avoid conﬂicting with

each other, improving balance. These choices reduce model and

control errors, allowing our centroidal stabilizer to compensate

for the remaining residual errors. The stabilizer and motion

generator are designed together to ensure feasibility under

the assumption of bounded errors. We deploy this scheme to

control the locomotion of the humanoid robot Talos, whose hip

links ﬂex when walking. It allows us to reach steps of 35 cm,

for an average speed of 25 cm/sec, which is among the best

performances so far for torque-controlled electric robots.

I. INTRODUCTION

Legged robots are normally modeled and controlled as a

chain of rigid bodies with actuated joints connecting them

[1]. This simpliﬁcation of the structural material properties

is specially accurate to deal with robots that are light or

have multiple legs [2]. Nevertheless, heavy biped robots such

as Talos or Walkman (≈100 kg) can present small but

meaningful deﬂections of their structure. These unmodeled

deﬂections produce a bad estimation of contact points as

well as a slow transference of forces through the kinematic

chain, resulting, therefore, in wrong contact forces and a bad

tracking of the desired robot motion. Due to the unstable

dynamics of legged robots, the tracking error tends to grow,

ending up with a control failure.

Flexible components are the subject of several studies in

robotics in general [3] and humanoids in particular:

Flexible joints based on Series Elastic Actuators (SEA)

[4], [5] have been used and studied on humanoid robots

such as Walkman [6], Coman [7] or Valkirie [8] which,

thanks to joint sensors, take advantage of the ﬂexibility

for safe environment interaction, disturbance rejection and

dissipation of walking impact energy. In our case, however,

deﬂections are not directly measurable as they are produced

on the robot links.

* For this work N. A. Villa, O. Stasse were supported by the cooper-

ation agreement ROB4FAM. M. Naveau, G. Saurel and N. Mansard were

supported by the H2020 Memmo project. P. Fernbach by the cooperation

agreement DynamoGrade, E. Dantec was supporter by ANITI.

1Gepetto Team, LAAS-CNRS, Universit´

e de Toulouse, France.

2TOWARD, Toulouse, France.

3Artiﬁcial and Natural Intelligence Toulouse Institute, France.

e-mails: nahuel.villa@laas.fr,pierre.fernbach@toward.fr,

firstName.lastName@laas.fr

Fig. 1. Snapshots of Talos walking dynamically, in torque-control, with a

calmed velocity of 15 cm/s.

Flexible bodies use to be incorporated to the end effectors

of position-controlled robots to measure interaction forces

from their deﬂection and to damp walking impacts, such as

in the HRP series [9], [10], [11]. The locomotion of these

robots is normally controlled with approaches derived from

[12], where the deﬂection is estimated based on the desired

contact forces.

Similar to the later case, in this article, we estimate the

link deﬂections based on the commanded joint torques to

avoid the noise and delay introduced with the measurement

of torques. Using a rigid-robot model, we incorporate such

deﬂections in the closest joints to obtain a better posture

estimation. We also reduce typical approximation errors

by previewing all centroidal non-linear behaviors in our

motion planning scheme. We obtain even further reduction

of the control errors by making all references of the inverse

dynamics consistent with the full centroidal motion and with

each other.

The remaining (much smaller) model error, as well as all

internal and external disturbances, produce tracking errors

that grow with the robot dynamics. We use state feedback to

stabilize the behavior of the Center of Mass (CoM) of the

robot and, based on a reachability analysis of the resulting

closed-loop system [13], we deploy a tube-based MPC [14]

scheme that guarantees robust feasibility when disturbances

are bounded.

In particular, we use the robot Talos, shown in Fig. 1, as

an experimental platform for this work. Talos is a commer-

cial humanoid robot equipped with powerful actuators and

arXiv:2210.15205v1 [eess.SY] 27 Oct 2022

precise sensors in a strong structure [15]. Nevertheless, this

platform has shown high difﬁculties to develop repeatable

walking gaits in torque control. As an outcome of this work,

we can claim that such difﬁculties are largely explained by

the unforeseen ﬂexibility that the model Talos exhibit on its

hip links.

We propose a simple method to identify the stiffness on

the robot hips and using the identiﬁed values on the proposed

control scheme, we have obtained fast and dynamic walking

gaits on torque control with velocities up to 25 cm/s, the

fastest walking velocity reached on Talos up to date.

Following this introduction, Section II describes the robot

dynamics and ﬂexibility. Section III details our approach

to incorporate deﬂections in the posture estimation. Our

centroidal controller is exposed in Section IV. The stable

centroidal motion is mapped into joint torques by the whole-

body controller, as explained in Section V. We deploy all the

previous control infrastructure on the robot Talos to perform

the experiments discussed in Section VI, and Section VII

summarizes our main conclusions.

II. MODELING

A. Whole-body dynamics

Walking robots are normally represented as a kinematic

chain of njoints connecting n+1links, in which no link is

attached to the inertial world frame [16]. The robot conﬁgu-

ration q= [q>

wq>

j]>can be described by the position and

orientation of the base link (robot waist) qw∈SE (3), and

the posture given by all joint angles qj∈Rn.

Joint motors produce the torques τa∈Rn, required for

the robot motion, following the dynamics [1]:

M(q)¨q+h(q, ˙q) = Sjτa+X

Jk(q)>fk,(1)

where M(q)∈R(n+6)×(n+6) is the generalized inertia matrix,

h(q, ˙q)∈Rn+6stands for Coriolis, centrifugal and gravity

forces, Sjselects all directly actuated joints, and for the k-

th contact, fk∈R3is a force exerted by the environment on

the point associated to the Jacobian matrix J(q)>

k∈Rn+6×3.

Joint angles must lie on collision-free ranges, and joint

torques are limited by the employed motors and materials:

qmin

j≤qj≤qmax

j,(2)

τmin

a≤τa≤τmax

a,(3)

where the inequalities with lower and upper limit vectors

hold element-wise.

We assume that feet sdo not slide during ground contacts:

˙s=Js˙q, (4)

¨s=Js¨q+˙

Js˙q= 0,(5)

and that ground contact forces are unilateral, constrained to

friction cones of the form [1]:

kfp

kk ≤ µfn

k∀fkin the ground-feet contact,(6)

where the friction forces fp

kparallel to the contact surface are

limited by the normal force fn

kwith some friction coefﬁcient

µ > 0.

B. Centroidal dynamics

Balance and locomotion dynamics corresponds to the

under actuated part of (1) and can be isolated from the

posture dynamics without additional assumptions [17], [18].

Let us consider a Cartesian coordinate system with the origin

on some ground contact surface, and the axis zaligned with

the gravity. So, in the lateral coordinates xy , this dynamics

relates the motion of the Center of Mass (CoM) cof the

robot to the Center of Pressure (CoP) pof the ground contact

forces [19, Chapter 2] as

pxy =cxy −mcz¨cxy −S˙

Lxy

m(¨cz+gz)+Pkrz

kfxy

Pkfz

,(7)

where gis the vertical acceleration due to gravity, mis the

total robot mass, ˙

Lis the variation of the angular momentum,

rkare the ground contact points and S=0-1

1 0 is a π

rotation matrix. Due to unilaterality of ground contact forces

(6), the CoP is bound to the support polygon P[1]:

p∈ P(s)(8)

that varies depending on the current foot positions s.

C. Hip ﬂexibility

Deﬂections on Talos’ hips are concentrated on its waist-

leg connection, where the link cross-section narrows. This

introduces extra degrees of freedom that can be represented

with passive virtual joints [20]. As deformations appear on

vertical linkages, we only model pitch and roll deﬂections,

which produce the main impact on foot placement. We obtain

in this form a model for the robot Talos with 42 degrees of

freedom composed by 32 actuated joints, 4 elastic passive

joints and the global position and orientation of the robot.

We built a simulator for this model using pinocchio

[21] for the computation of rigid body dynamics and the

gepetto-viewer for visualizations, as in the companion

video1, where we have included massless collision-free plates

on the virtual joints to display deﬂections.

We simulate the elastic deformation of virtual joints as a

spring damper

τf=−kfθ−df˙

θ, (9)

with stiffness kfand damping dfcoefﬁcients that relate

deﬂections θto the ﬂexing torque τf. As the identiﬁed values

of stiffness (see Section VI-B) are relatively big, simulations

require short integration periods (≈0.1ms) for numerical

convergence.

III. POSTURE ESTIMATION WITH DEFLECTIONS

Hip conﬁgurations, outlined in Fig. 2, are composed by

3 measured joint rotations qhip ∈R3and 2 unmeasurable

elastic deﬂections θ∈R2that can be estimated. We condense

1video available in: https://gepettoweb.laas.fr/

articles/talos_centroidal_mpc_torque_control.html

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TorqueControlledLocomotionofaBipedRobotwithLinkFlexibilityNahuelA.Villa1,PierreFernbach2,MaximilienNaveau1;2,GuilhemSaurel1,EwenDantec1,NicolasMansard1;3,OlivierStasse1;3AbstractWhenabigandheavyrobotmoves,itexertslargeforcesontheenvironmentandonitsownstructure,itsangularmomentumcanvarysubstantially...

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Torque Controlled Locomotion of a Biped Robot with Link Flexibility Nahuel A. Villa1 Pierre Fernbach2 Maximilien Naveau12 Guilhem Saurel1 Ewen Dantec1 Nicolas Mansard13 Olivier Stasse13

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