I have an equation like ( 𝑓(𝑞,𝑞̇,𝑠)=𝑀(𝑞,𝑠)(𝑞𝑑̈+𝐾𝑝𝑒 ̇+𝐾𝑑𝑒)+𝑁(𝑞,𝑞,𝑠̇) )

where M(q, s) is the n × n inertia matrix of the entire system, q denotes the n × 1 column

matrix of joint variables (joint/internal coordinates), s represents system parameters such as

mass and the characteristic lengths of the bodies, and f (q, ˙ q, s) is the n×1 columnmatrix of

generalized driving forces which might be functions of the system’s generalized coordinates,

and/or speeds, and/or system parameters. The term N(q, ˙ q, s) includes inertia-related loads

such as Coriolis and centripetal generalized forces, as well as gravity terms.Defining

the n×1 column matrix of the desired joint trajectory as qd (t), one can express the tracking

error as (e(t) = qd (t) − q(t)).

I have 2 uncertanity parameters s=(s1,s2) in mass of two links , i wanna integrated uncertanity to find mean of 𝑓 for every joint of my two link sereise robot .

distribution of 2 non-deterministic parameters is uniform and basis function is legendre ,

by using galerkin projection can obtain this equation at end : 𝑓𝑗𝑙=(1/𝑐𝑙2)

l = 0;....;Nt ; and j = 1;....;n: that Nt is the coefficient number and n is size of f vector .

my question is : how can i calculate inner product of upper equation to find for example f10 from orthogonal equation that mentioned upper ( ) i know the 𝜑𝑙(𝑠) but what should i consider for (f j) for my example in this Integral :

∬𝑓𝑗(𝑠1,𝑠2).𝜑𝑙(𝑠1,𝑠2).𝑝𝑑𝑓(𝑠1).𝑝𝑑𝑓(𝑠2)𝑑𝑠1 𝑑𝑠2=𝑓𝑗𝑙