How Do You Model θ in Segway Dynamics?

[saltine]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

The net torque acting on the body at A

τ_net = τ + Lmg sin(θ) /2 . . . (1)

From net torque to the angular acceleration:

α = τ_net/I . . . (2)

Integrals and derivatives:

θ = θ_o + ω_ot + 0.5α_ot² . . . (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 = τ_net/I = τ_net = (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

 

[anathema]

for the post! It looks like you have derived all the necessary equations to model θ. Just to clarify, the equations you have derived are for a simple pendulum with a constant torque acting on it. If you want to take into account other forces such as friction or air resistance, you would need to include those in your equations as well. Additionally, if you want to model the motion of the axle, you would need to include equations for that as well. Overall, your equations provide a good starting point for modeling θ, but you may need to modify them depending on the specific situation you are studying.

 

[thomate1]

for the question! It appears that you have derived all the necessary equations to model θ in this situation. Just to clarify, the equation for net torque (equation 1) should include the moment of inertia (I) in the denominator, as shown in equation 7. Other than that, your approach seems correct. Keep in mind that this is a simplified model and may not fully capture all the complexities of the Segway's motion, but it should give a good approximation for small values of θ.