# Segway physics question

1. Dec 8, 2008

### saltine

1. The problem statement, all variables and given/known data
Derive all equations necessary to model θ.

In the figure, ω is the angular velocity of the body about the axle. v is the linear velocity of the axle. The body supplies a clockwise torque τ to the axle.

2. Relevant equations

3. The attempt at a solution

The net torque acting on the body at A

τnet = τ + Lmg sin(θ) /2 . . . (1)

From net torque to the angular acceleration:

α = τnet/I . . . (2)

Integrals and derivatives:

θ = θo + ωot + 0.5αot2 . . . (3)

ω = dθ/dt . . . (4)

α = dω/dt . . . (5)

Linearize: If the device is initially vertical and at rest, and θ is always small:

dθ/dt = ω . . . (6)

dω/dt = α = τnet/I = τnet = (Lmg/2I) θ + (1/I) τ . . . (7)

Let x := [ θ ; ω ], u = τ :

x' = Ax + Bu = [ 0 1 ; Lmg/2I 0 ] x + (1/I) τ . . . (8)

y = Cx + Du = [ 1 0 ] x . . . (9)

Is this all I need to consider for θ?

- Thanks