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# Chains of rigid bodies and their numerical simulation by local frame methods

• * Corresponding author: Antonella Zanna
• We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of . In this framework, the dynamics of the $j$th body is described in a frame relative to the $(j-1)$th one. Starting from the Lagrangian formulation of the system on ${{\rm{SO}}}(3)^{N}$, the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.

Mathematics Subject Classification: Primary: 70F10, 70E55, 65P99; Secondary: 65K99, 65L05.

 Citation: • • Figure 1.  System of chained rigid bodies

Figure 2.  Single solid pendulum, consisting of a rectangular prism. Left: classical planar motion, described by the angle $\theta$. Right: planar motion, described in 3D by a quaternion and the MFM. At rest (vertical position), the inertial and frame axes are aligned

Figure 3.  Numerical solution for the angle $\theta$ for the solid pendulum. The MFM is solved with IMR and stepsize $h = 0.01$ and the exact solution is computed with RK45 imposing machine precision on the tolerances

Figure 4.  A simulation of the spinning top, idealized as a rod with a disk (left). Right: position of the center of mass in $[0,30]$ integrated with step size $h = 0.01$. See text for the values of the remaining parameters. The formulation of the equations is identical to that of the single pendulum

Figure 5.  4-body pendulum with torsional springs

Figure 6.  16-body

Figure 7.  Conservation of quaternion-unit length (left) and relative energy error (right) for the 16-body pendulum solved with GLRK scheme with a step length $h = 0.01$

Figure 8.  Spring loaded/damped chain. The red vector represents $n$, the unit vector in the direction of the displacement of the spring

Figure 9.  64-body pendulum

Table 1.  Comparison of the error in unit length (for quaternions), relative energy error and function evaulations at $T = 10$ for the RK45 and 2-stages GLRK method for a 3D pendulum with 4 links. See text for more details

 Link 1 Link 2 Link 3 Link 4 Relative energy error Function evaluations RK45 5.55e-15 2.47e-14 2.24e-14 2.50e-14 7.22e-14 15794 GL2 2.22e-15 5.55e-16 3.00e-15 1.66e-15 3.00e-13 11334

Table 2.  Computational times for systems with $N$ links. The computational cost grows quadratically with the number of links. See text for more details

 $N$ 4 8 16 32 48 64 Time 719ms$\pm$4.31ms 2.27s$\pm$15.8ms 10.7s$\pm$122ms 45.6$\pm$1.55s 1m43s$\pm$1.94s 3m24s$\pm$5.52s
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Tables(2)

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