2.8.1 Computing the Motion of Free Rigid Bodies

Lagrange’s equations for the motion of a free rigid body in terms of Euler angles are quite disgusting, so we will not show them here.

Some configurations may have coordinate singularities when using Euler angles (e.g. gimbal lock). In the explicit Lagrange equations, the singularity arises when we try to find the expression for generalized accelerations. The expression for this involves the inverse of $\partial_2 \partial_2 L$. The determinant of this quantity may become zero when the Euler angle $\theta$ is zero (for 3-1-3 Euler angles).

(def Euler-state
(up 't
(up 'theta 'varphi 'psi)

(rendermd (determinant
(((square (partial 2)) (rigid/T-rigid-body 'A 'B 'C)) ;; rigid/T-rigid-body = T-body-Euler from book
Euler-state)))

$$A\,B\,C\,{\sin}^{2}\left(\theta\right)$$

So we cannot solve for the second derivatives when $\theta = 0$ and the Euler angles may change drastically when $\theta$ is small. This does not mean the actual motion of the rigid body is anything but well-behaved. The problem lies entirely in the representation of the motion using Euler angles. We may use another set of Euler angles when necessary to avoid this problem, but this tends to be cumbersome. In this section, we will be limiting our focus to trajectories that will not contain singularities for the chosen Euler angle set.

To obtain trjectories, we numerically integrate the Lagrange equations. The system derivative can be obtained directly from the Lagrangian using Lagrangian->state-derivative

TBD: Figure out plotting in Clojupyter

(defn rigid-sysder [A B C]
(Lagrangian->state-derivative (rigid/T-rigid-body A B C)))

(defn monitor-errors [win A B C L0 E0]
(fn [state]
(let [t (time state)
L ((rigid/Euler-state->L-body A B C) state)
E ((rigid/T-rigid-body A B C) state)]
(plot-point win t (relative-error (ref L 0) (ref L0 0)))
(plot-point win t (relative-error (ref L 1) (ref L0 1)))
(plot-point win t (relative-error (ref L 2) (ref L0 2)))
(plot-point win t (relative-error E E0)))))

Syntax error compiling at (REPL:9:5).
Unable to resolve symbol: plot-point in this context

Util.java:   221 clojure.lang.Util/runtimeException
core.clj:  3214 clojure.core$eval/invokeStatic core.clj: 3210 clojure.core$eval/invoke
main.clj:   437 clojure.main$repl$read_eval_print__9086$fn__9089/invoke main.clj: 458 clojure.main$repl$fn__9095/invoke main.clj: 368 clojure.main$repl/doInvoke
RestFn.java:  1523 clojure.lang.RestFn/invoke
AFn.java:    22 clojure.lang.AFn/run
AFn.java:    22 clojure.lang.AFn/run