Lecture 11&12 - Rigid Body Dynamics 5&6

Lecture 11: Generalized Forces, and turning a set of second order equation, into 4 first order equation

Lecture 12: Constrained Lagrange and Differential Algebraic Equations

Sub-Topics

Lecture 11 State Space Formulations

Lecture 12 Constrained Lagrange

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We have

q = [q_{1} q_{2}]

Getting $v_{1}$

v_{1} = J \overset{q}{˙} = 100000 [\overset{q}{˙}_{1} \overset{q}{˙}_{2}] = \overset{q}{˙}_{1} 00

And getting $v_{2}$

r_{2} = L sin q_{2} + q_{1} 0 L cos q_{2} + h

Giving

v_{2} = J \overset{q}{˙} = 100 L cos q_{2} 0 - L sin q_{2} [\overset{q}{˙}_{1} \overset{q}{˙}_{2}]

Mass inertia matrix

W W = W_{1} + W_{2} = J_{1}^{T} J_{1} m_{1} + J_{2}^{T} J_{2} m_{2} = \dots = [m_{1} + m_{2} L m_{2} cos q_{2} L m_{2} cos q_{2} L^{2} m_{2}]

We now need to find $\frac{\partial L}{\partial q}$

T \frac{\partial T}{\partial q} = \frac{1}{2} \overset{q}{˙}^{- 1} W (q) \overset{q}{˙} = \frac{1}{2} [\overset{q}{˙}^{T} \frac{\partial w}{\partial q _{1}} \overset{q}{˙} \overset{q}{˙}^{T} \frac{\partial w}{\partial q _{2}} \overset{q}{˙}] = [0 - L m_{2} sin (q_{2}) \overset{q}{˙}_{2} \overset{q}{˙}_{1}]

We find the potential energy, $V$

V = \frac{1}{2} k_{1} q_{1}^{2} + \frac{1}{2} k_{2} q_{2}^{2} + m_{2} g (h + L cos q_{2}) 4

We have that $F_{d}$ acts on the position

p = q_{1} - \frac{l}{2} 0 \frac{h}{3}

And

\frac{\partial p}{\partial q _{1}} = 100 \frac{\partial p}{\partial q _{2}} = 000 F_{d} = - F_{d} 00

We then get

Q_{F_{d}} = [F_{d}^{T} \frac{\partial p}{\partial q _{1}} F_{d}^{T} \frac{\partial p}{\partial q _{2}}] = [- F_{d} 0] = [q_{1} q_{2}]

We have tjat $F_{a}$ acts on the position

p = q_{1} + L sin q_{2} 0 h + L cos q_{2}

Which gives

Q_{F_{a}} = [F_{a}^{T} \frac{\partial p}{\partial q _{1}} F_{a}^{T} \frac{\partial p}{\partial q _{2}}] = [F_{a} F_{a} L cos q_{2}] = [q_{1} q_{2}]

We also find the torque

T = d_{2} \overset{q}{˙}_{2} \to Q_{T} = [0 d_{2} \overset{q}{˙}_{2}]

Then the total generalized forces becomes

Q = [- F_{d} 0] + [F_{a} F_{a} L cos q_{2}] + [0 d_{2} \overset{q}{˙}_{2}]