Computational details for : "Optimal Estimation of the Centroidal Dynamics of Legged Robots"

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Computational details for : ”Optimal Estimation of the Centroidal Dynamics of Legged Robots”

François Bailly, Justin Carpentier, Philippe Souères

To cite this version:

François Bailly, Justin Carpentier, Philippe Souères. Computational details for : ”Optimal Estimation

of the Centroidal Dynamics of Legged Robots”. [Research Report] Rapport LAAS n° 21072, LAAS-

CNRS; Université de Montréal. 2021. �hal-03180052v4�

Computational details for : “Optimal Estimation of the Centroidal Dynamics of Legged Robots”

Franc¸ois Bailly ^a,b,* , Justin Carpentier ^c and Philippe Sou`eres ^a

This document complements the paper entitled “Differential Dynamic Programming for Maximum a Posteriori Centroidal State Estimation of Legged Robots” [1]. The purpose of this work was to estimate the centroidal dynamics of legged robots by formulating a maximum a posteriori problem and solving it thanks to differential dynamic programming (DDP). In the following, the computations of the partial derivatives of the unoptimized value function (Q k ) are provided for the DDP algorithm. Then, the hypothesis about the 0−mean property of the stochastic part of the dynamics is validated by Fig. 1 (result of the simulation ) which demonstrates that the DDP minimization of Eq.(11) does keep ω k 0−mean.

A PPENDIX I

P ARTIAL DERIVATIVES OF Q _k

• Q xk = ∇ x

l k + ∇ x

V k+1 (f (x k , ω k )),

∇ x

l _k = 2 ∂(g(x _k ) − y _k ) ^T

∂x k

Σ ⁻¹ _η

k

(g(x _k ) − y _k ),

∂g(x k )

∂x _k = C(x k )

∂x _k x k + C(x k ) =

_{1 0 0 0 0}

0 0 2c

× 0 1 0 0 1 0 0 0 0 0 1 0

ˆ

= ˜ C(x k ), Q _xk = 2 ˜ C(x _k ) ^T Σ ⁻¹ _η

k

(g(x _k ) − y _k ) + A ^T V _x ⁰ .

• Q ωk = ∇ ω

l k + ∇ ω

V i+1 (f (x k , ω k )),

∇ ω

l _k =

∂||ω k || ² _Σ

∂ω i

= 2Σ ⁻¹ _ω

k

ω _k , Q ωk = 2Σ ⁻¹ _ω

ω k + B ^T V _x ⁰ .

• Q _xxk = ∇ ² _x

l _k + ∇ ² _x

V i+1 (f (x _k , ω _k )),

where, the element of ∇ ² _x

l _k at the i ^th row and j ^th

a

Laboratoire de Simulation et Mod´elisation du Mouvement, Facult´e de M´edecine, Universit´e de Montr´eal, Laval, QC, CanadaLAAS-CNRS, 7 Avenue du Colonel Roche, F-31400 Toulouse, France

b

LAAS-CNRS, 7 Avenue du Colonel Roche, F-31400 Toulouse, France

c

Inria, D´epartement d’informatique de l’ENS, ´ Ecole normale sup´erieure, CNRS, PSL Research University, Paris, France

*

corresponding author: [email protected]

column is denoted by:

[∇ ² _x

k

l _k ] _ij =

Y

X

m=1 Y

X

n=1

[Σ ⁻¹ _η

k

] _mn ∂ C ˜ _im ^T

∂x _j [g(x _k ) − y _k ] _n + ˜ C _im ^T ∂[g(x _k )] _n

∂x j

, [∇ ² _x

k

l k ] ij = [ ˜ C ^T Σ ⁻¹ _η

k

C] ˜ ij

+

Y

X

n=1 Y

X

m=1

∂ C ˜ _km ^T

∂x l

[Σ ⁻¹ _η

] mn [g(x k ) − y k ] n , [∇ ² _x

l k ] ij = [ ˜ C ^T Σ ⁻¹ _η

C] ˜ ij +

c ^T _ij Σ ⁻¹ _η

(g(x k ) − y k ),

where,

c _ij ∈ R ^Y s is the stacked vector of ∂ C ˜ _im ^T

∂x j

for m ∈ [1..Y ].

• Q ωωk = ∇ ² _ω

k

l k + ∇ ² _ω

k

V i+1 (f (x k ω k )),

∇ ² _ω

l k = 2Σ ⁻¹ _ω

∂ω k

∂ω k

= 2Σ ⁻¹ _ω

Q ωωk = 2Σ ⁻¹ _ω

+ B ^T V xx B

Fig. 1: Time evolution of the stochastic control inputs of the dynamics for the simulated walk of the HRP-2 robot.

R EFERENCES

[1] F. Bailly, J. Carpentier, and P. Sou`eres, “Optimal estimation of the

centroidal dynamics of legged robots,” in Internat. Conf. on Robotics

and Automation. IEEE, 2021.