flâneur

Ch. 23 - Multi-Body Dynamics

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© Russ Tedrake, 2024 Last modified 2024-11-11. How to cite these notes, use annotations, and give feedback. Note: These are working notes used for a course being taught at MIT. They will be updated throughout the Spring 2024 semester. Lecture videos are available on YouTube. The equations of motion for a standard robot can be derived using the method of Lagrange. Using 𝑇 as the total kinetic energy of the system, and 𝑈 as the total potential energy of the system, 𝐿 = 𝑇 − 𝑈 , and 𝜏 𝑖 as the element of the generalized force vector corresponding to 𝑞 𝑖 , the Lagrangian dynamic equations are: (1) 𝑑 𝑑 𝑡 𝜕 𝐿 𝜕 𝑞 ˙ 𝑖 − 𝜕 𝐿 𝜕 𝑞 𝑖 = 𝜏 𝑖 . You can think of them as a generalization of Newton's equations. For a particle, we have 𝑇 = 1 2 𝑚 𝑥 ˙ 2 , so 𝑑 𝑑 𝑡 𝜕 𝐿 𝜕 𝑥 ˙ = 𝑚 𝑥 ¨ , and 𝜕 𝐿 𝜕 𝑥 = − 𝜕 𝑈 𝜕 𝑥 = 𝑓 amounting to 𝑓 = 𝑚 𝑎 . But the Lagrangian derivation works in generalized coordinate systems and for constrained motion. Consider the s

Deriving the equations of motion The equations of motion for a standard robot can be derived using the method of Lagrange. Using $T$ as the total kinetic energy of the system, and $U$ as the total potential energy of the system, $L = T-U$, and $\tau_i$ as the element of the generalized force vector corresponding to $q_i$, the Lagrangian dynamic equations are: \begin{equation} \frac{d}{dt}\pd{L}{\dot{q}_i} - \pd{L}{q_i} = \tau_i.\end{equation} You can think of them as a generalization of Newton's equations. For a particle, we have $T=\frac{1}{2}m \dot{x}^2,$ so $\frac{d}{dt}\pd{L}{\dot{x}} =…

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