Methodology under consideration

Property 2.3 The left-hand member of the RDM (equation (2.2)) and the FDM (equation (2.9))

can be rewritten in a linear form with respect to the physical parameters of the robot defined at the beginning of Section 2.3.1.

In the rigid case, this leads to:

φr(q, ˙q, ¨q) χr = τ + τext (2.20)

where φr(q, ˙q, ¨q) ∈ Rn,Nr is the rigid regression matrix and χr ∈ RNr is the vector of the rigid

base parameters. As described in [Gautier 1990,Khalil 2004], the base parameters constitute the minimum set of parameters that completely characterize the robot dynamic behavior. They are obtained from the dynamic model parameters ζr described previously by eliminating those

that have no effect on the dynamic model and by linearly regrouping some others. In the flexible case, it yields:

φf(q, ˙q, ¨q, θ, ˙θ, ¨θ) χf =

τext

τ !

(2.21) with φf(q, ˙q, ¨q, θ, ˙θ, ¨θ) ∈ R2n,Nf the flexible regression matrix and χf ∈ RNf the flexible base

parameters.

More explicitly, it can be observed from equation (2.21) and from the reduced form of the FDM in equation (2.9) that:    f φr(q, ˙q, ¨q) Dq−θ −I 0 0 0 0 _−Dq−θ I Dθ¨ Dθ˙ Dsign(_θ˙₎    | {z } φf(q, ˙q,¨q,θ, ˙θ,¨θ)           f χ_r χ_K χ_τ k0 χ_J m χ_F vm χ_F sm           | {z } χ_f = τext τ ! (2.22) where

− the motor-side contributions have been removed from φr(q, ˙q, ¨q) and χrin the tilde notations f

φr(q, ˙q, ¨q) and χfr,

− matrices Dq−θ, Dθ¨, Dθ˙ and Dsign(θ˙_{) are defined by D}var = diag(var1var2 . . . varn) ∈ Rn,n for var ∈ {q − θ, ¨θ, ˙θ,sign( ˙θ)},

− the vectors of the associated parameters χK, χτk0, χJm, χFvm and χFsm are defined by

χ_{par= [par1par2 . . . parn]}T _{for par ∈ {K, τ}

Single elastic joint example

If we consider a single link rotating in the horizontal plane (hence without gravity) and actuated with a motor through an elastic joint coupling, and if we assume that Coriolis and centrifugal terms are negligible, then the FDM is reduced to:

M ¨q + Fva ˙q + Fsasign( ˙q) + K (q − θ) − τk0 = τext (2.23a) Jmθ + F¨ vm ˙θ + Fsmsign( ˙θ) − K (q − θ) + τk0 = τ (2.23b)

where the same notation as inFigure 2.2are used. Then the terms φf ∈ R2,8 and χf ∈ R8

of equation (2.21) are expressed as follows:

φf =  ¨q ˙q sign( ˙q) q− θ −1 0 0 0 0 0 0 −(q − θ) 1 θ¨ ˙θ sign( ˙θ)   (2.24) χ_{f =} _{M Fva Fsa K τk0 Jm Fvm Fsm}T _(2.25)

Note that other models than the dynamic model can be used for the identification phase, as long as they can be rewritten in a linear form with respect to the base parameters. For instance, we can mention the energy (or integral) model [Gautier 1988] that does not require the computation of the joint accelerations or the power model [Gautier 2013a] that avoids large errors for low- frequency varying parameters such as offsets.

In the following, indexes are removed for clearness reasons but developments are applicable to both the rigid and flexible cases. For the identification procedure, the linear form with respect to the base parameters in equation (2.20) (rigid case) or (2.21) (flexible case) is evaluated without any external disturbance at a sufficient number of points on exciting trajectories. The motor currents and positions (and joint positions if available) are measured, and the velocities and accelerations are computed offline by numerical derivation using a central difference algorithm to avoid any phase shift. This leads to the following system of overdetermined linear equations:

W χ+ ρ = Y (2.26)

where, for r time samples along all the DOF (r ≫ N), W ∈ Rr,N _{is the observation matrix,}

ρ_{∈ R}r is the vector of errors due to noisy measurements and modeling errors and Y ∈ Rr is

the vector of torque measurements.

Assuming that ρ is a zero-mean additive independent noise of standard deviation σρ, the

covariance matrix of ρ is denoted Σρ∈ Rr,r and is defined by:

Σρ:= E[ρ ρT] = σρ2Ir (2.27)

2.3. Parameter uncertainties 41

An unbiased estimation of σρcan be calculated using the following equation:

ˆσρ2 = k

Y _{− W ˆ}χk2

r− N (2.28)

where ˆχ is the Least-Squares (LS) solution of equation (2.26) which is the most widespread resolution method. Nonetheless, alternative solutions have been proposed, such as maximum- likelihood parameter estimation [Swevers 1997], linear matrix inequalities [Sousa 2014], extended Kalman filtering [Gautier 2001] or machine learning [Tu 2018].

The determination of the optimal trajectories, also called persistently exciting trajectories, aims at increasing the parameters excitation to improve the convergence rate and the noise immunity of the estimation. Two procedures can be applied:

− Determining sequential sets of trajectories that stimulate specific dynamic parameters. For instance, the dynamic parameters can be identified link by link by moving some joints while blocking the other ones, starting from the last link to avoid cumulative errors due to the block-triangular form of W . In a similar approach, when shoulder and wrist contributions can be decoupled, associated parameters can be identified in two stages to address the difference in the order of magnitude. Another technique consists in dividing the base parameters in groups (e. g. inertial, gravity, friction) and defining characteristic trajectories that stimulate only one group of parameters at a time [Vandanjon 1995].

− Calculating exciting trajectories by minimizing a criterion to define. Since the sensitivity of the solution with respect to modeling errors and noise can be measured by the condition number of the observation matrix, a widely used optimization criterion is the minimization of the conditioning of W [Presse 1993]. Other approaches rely on the minimization of the uncertainties on the parameter estimates as in [Swevers 1997] with a maximum-likelihood estimator.

In order to deal with the problem of heterogeneous measurements between the different axes and experiments, a Weighted Least-Squares (WLS) formulation of the system (2.26) can be used [Gautier 2001]. For this purpose, for each experiment and each joint, we consider the subsystem Wsχ + ρs = Ys composed only of the associated equations, where s refers to the

subsystem number. Based on this subsystem, the standard deviation ˆσρs of the corresponding

error is calculated separately using equation (2.28). Then the rs equations of the subsystem s

are weighted by a factor 1/ˆσρs. The global system of weighted equations is written as follows:

Wwχ + ρw = Yw (2.29)

where the weighted matrices Ww, ρw and Yw are obtained with:

Ww = G W , ρw = G ρ , Yw = G Y and G = diag  ₁ ˆσρ1 Ir1 . . . 1 ˆσρns Ir_ns  (2.30) with G ∈ Rr,r _{the diagonal matrix containing the weights for each subsystem and n}

s the total

The solution χ⋆of equation (2.29) is obtained by LS minimization of the 2-norm of the weighted

errors vector ρw. Assuming that Ww is full column rank, the explicit solution χ⋆ is given by:

χ_{⋆ := min}

χ kρwk

2 _{= W}

w+Yw (2.31)

with Ww+ = (WwTWw)−1WwT the pseudo-inverse matrix of Ww.