A.B=|α||β|Cos[θ]

A = {a_1, a_2, a_3} ;

B = {b_1, b_2, b_3} ;

A . B = a_1 b_1 + a_2 b_2 + a_3 b_3

A . B = | A | | B | Cos[θ]

υ={i,j,k}

i = {1, 0, 0} ;

j = {0, 1, 0} ;

k = {0, 0, 1} ;

υ = {i, j, k} ;

A = i a_1 + j a_2 + k a_3 ;

B = i b_1 + j b_2 + k b_3 ;

i . i = j . j = k . k = 1

i . j = j . i = k . i = i . k = j . k = k . j = 0

A//Norm

Norm[A] = (Abs[a_1]^2 + Abs[a_2]^2 + Abs[a_3]^2)^(1/2)

N={λ,μ,ν}

λ = Cos[α] ; μ = Cos[β] ; ν = Cos[γ] ;

N = {λ, μ, ν}

λ^2 + μ^2 + ν^2 = 1 ;

Cos[α]^2 + Cos[β]^2 + Cos[γ]^2 = 1 ;

r,v,a,t

v = r/t

a = v/t = ^2r/t^2

∫at

Δt = t_2 - t_1 ;

Δr = r_2 - r_1 ;

v = Δr/Δt = (r_2 - r_1)/(t_2 - t_1)

Δt = t_2 - t_1 ;

Δv = v_2 - v_1 ;

a_2 = Δv/Δt = (v_2 - v_1)/(t_2 - t_1) = (v_2 - v_1)/(t_2 - t_1)

v_1 = (r_1 - r_0)/(t_1 - t_0) ;

v_2 = (r_2 - r_1)/(t_2 - t_1) ;

a_2 = (v_2 - v_1)/(t_2 - t_1) = (v_2 - v_1)/Δt ;

a_2 = (r_2 - r_1)/(t_2 - t_1)^2 - (r_1 - r_0)/((t_1 - t_0) (t_2 - t_1))

= (r_2 - r_1)/Δt^2 - (r_1 - r_0)/Δt^2

= (r_0 - 2 r_1 + r_2)/Δt^2

a_2Δt = v_2 - v_1 = Δv ;

a_3Δt = v_3 - v_2 = Δv ;

a_4Δt = v_4 - v_3 = Δv ;

:

a_nΔt = v_n - v_ (n - 1) = Δv ;

v = ∫at = a t

∫_t_0^t_nat

w = (b - a)/Div ;

∫_a^bf[x] x = w ((f[a] + f[b])/2) = (b - a) ((f[a] + f[b])/2)

= w ((f[a] + f[a + w])/2) + w ((f[a + w] + f[a + 2w])/2) + … + w ((f[a + (Div - 1) w] + f[b])/2)

= w/2 (f[a] + 2f[a + w] + 2f[a + 2w] + … + 2f[a + (Div - 1) w] + f[b]) = w/2 (f[a] + f[b]) + w (f[a + w] + f[a + 2w] + … + f[a + (Div - 1) w])

= (b - a)/Div ((f[a] + f[b])/2 + (f[a + (b - a)/Div] + f[a + 2 (b - a)/Div] + … + f[a + (Div - 1) (b - a)/Div]))

∫_t_0^t_na[t] t

(t_n - t_0)/n = Δt

v_n = ∫_t_0^t_na[t] t

= (t_n - t_0)/n ((a[t_0] + a[t_n])/2 + (a[t_0 + Δt] + a[t_0 + 2Δt] + … + a[t_0 + (n - 1) Δt]))

= Δt ((a[t_0] + a[t_n])/2 + (a[t_0 + Δt] + a[t_0 + 2Δt] + … + a[t_0 + (n - 1) Δt]))

= Δt ((a[t_0] + a[t_n])/2 + Underoverscript[∑, i = 1, arg3] a[t_0 + i Δt])

v_n = Δt ((a_0 + a_n)/2 + (a_ (0 + Δt) + a_ (0 + 2Δt) + … + a_ (0 + n - 1Δt)))

= Δt ((a_0 + a_n)/2 + Underoverscript[∑, i = 1, arg3] a_ (0 + i Δt))

Encoder

r = C * Dist/Pulse = C * D/P

(C_2 * D/P - C_1 * D/P)/Δt = (C_2 - C_1)/Δt * D/P = (r_2 - r_1)/Δt = v_2

C_i≠0

v_n = (C_n - C_ (n - 1))/Δt * D/P = (C_n - C_ (n - 1))/(1s) * (1s)/Δt * D/P = (C_n - C_ (n - 1)) * 1/Δt * D/P

Total

v_n = (C_n - C_ (n - 1)) * D/P * 1/Δt ;

v_n = Δt ((a_0 + a_n)/2 + Underoverscript[∑, i = 1, arg3] a_ (0 + i Δt)) ;

(C_n - C_ (n - 1)) = Δt^2 * P/D * ((a_0 + a_n)/2 + Underoverscript[∑, i = 1, arg3] a_ (0 + i Δt))

If[C_i == 0, {n = 0, a_0}, {n ++; Δt^2 * P/D * ((a_0 + a_i)/2 + Underoverscript[∑, k = 1, arg3] a_k)}]