Perform nonlinear regression with nlsLM within a function in r

Question

I want to add a modification factor to an existing equation to fit data. The original equation is defined through a function because the variable N_l is a vector of numbers and the function is selecting the largest outcome of the equation by going through all possible values in the vector N_l. The original function is defined as:

library(utils)

R <- function(x){
  N_b <- x[1]
  N_l <- x[2]
  A <- x[3]
  x.sqr <- x[4]
  S <- x[10]
  e <- x[grepl("e_\\d",names(x))]
  f <- sapply(seq(N_l),function(k) max(Multi.Presence$m[k] * ((k/N_b) + 
                                       (A * combn(e,k,sum) / x.sqr))))
  c(val = max(f), pos = which.max(f))
}

DATA.GIRDER1 <- cbind(DATA.GIRDER1, vars = t(apply(DATA.GIRDER1, 1, R)))
colnames(DATA.GIRDER1)[12:13] <- c("Proposed.Girder1","Lanes")

The equation defines in the function is:

The first 5 rows of the dataframe DATA.GIRDER1 and dataframe Multi.Presence are provided:

> dput(DATA.GIRDER1[(1:5),]
structure(list(N_b = c(5, 5, 5, 5, 5), N_l = c(4, 4, 4, 4, 4), 
    A = c(-12, -12, -12, -12, -12), x.sqr = c(1440, 1440, 
    1440, 1440, 1440), e_1 = c(21.8, 21.8, 21.8, 21.8, 21.8), 
    e_2 = c(9.8, 9.8, 9.8, 9.8, 9.8), e_3 = c(-2.2, -2.2, -2.2, 
    -2.2, -2.2), e_4 = c(-14.2, -14.2, -14.2, -14.2, -14.2), 
    e_5 = c(0, 0, 0, 0, 0), S = c(12, 12, 12, 12, 12), 
    R = c(0.59189685884369, 0.583646426252063, 
    0.556293941275237, 0.576160481501275, 0.597435112708129)), 
    row.names = c(NA, 5L), class = "data.frame")

> dput(Multi.Presence)
structure(list(N_l = c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10), m = c(1.2, 
1, 0.85, 0.65, 0.65, 0.65, 0.65, 0.65, 0.65, 0.65)), row.names = c(NA, 
-10L), class = "data.frame")

The theoretical data to fit the equation to is CSi.Girder1. At the moment, the way the function is set up, it calculates the maximum R for each row of dataframe DATA.GIRDER1.

I want to add a regression term based on variable S in dataframe DATA.GIRDER1to the second part of the equation to find parameters a and b to best fit the data in CSi.Girder1. The desired output would implement the equation below:

To use nlsLM I need to define a function for the equation such as:

library(minpack.lm)

Prposed.Girder1 <- function(N_b, N_l,A,x.sqr,e_1,e_2,e_3,e_4,e_5,S,a,b) {
  R <- function(x){
    N_b <- x[1]
    N_l <- x[2]
    A <- x[3]
    x.sqr <- x[4]
    e <- x[grepl("e_\\d",names(x))]
    f <- sapply(seq(N_l),function(k) max(Multi.Presence$m[k] * ((k/N_b) + 
                (A * combn(e,k,sum) / x.sqr) * (b*S^a))))
    c(val = max(f), pos = which.max(f))
  }
  DATA.GIRDER1 <- cbind(DATA.GIRDER1, vars = t(apply(DATA.GIRDER1, 1, R)))
  colnames(DATA.GIRDER1)[12:13] <- c("Proposed.Girder1","Lanes")

  return(R)
}

Girder1_nlsLM <- nlsLM(R ~ Prposed.Girder1(N_b, N_l,A,x.sqr,e_1,e_2,e_3,e_4,e_5,S,a,b), 
                 data = DATA.GIRDER1, 
                 start = c(a = 0.01, b = 0.01))

summary(Girder1_nlsLM)

But this is not successful and I get the error:

Error in model.frame.default(formula = ~R + N_b + N_l + A + x.sqr + e_1 + : object is not a matrix

How can I add this modification factor in terms of variable S to solve for the parameters a and b.