IEEE Robotics & Automation Magazine - September 2023 - 112

TEST 3: DYNAMICS OF A FLEXIBLE FLYING ROD
In this comparison, we look at the dynamics of a freely flying
flexible rod (also known as the flying spaghetti problem)
which is a problem introduced by Simo and Vu-Quoc [22]
and replicated in [17]. We model a 10-m-long soft rod, with a
free lumped joint and initially at rest in the position shown
in Figure 6(a)(1). The position and orientation of the base of
the soft rod are parameterized by lumped DoF of the free
joint. We used the inextensible Kirchhoff model with a quadratic
polynomial basis to define the basis of the rod. Hence,
including the six DoF of the free joint, there are 15 DoF in
this system. Time-dependent point force F1 and moments
M2 and M3 are applied at the tip of the rod, as in Figure 6(a)
(1). The magnitude of M2 is defined as a triangular pulse
function that starts at time
t 0s= peaks at 200 N·m in 2.5 s,
,
and goes back to zero at 5 s. The numerical values of the
magnitude of F1 and M3 are 1/10 and half of M2, respectively.
The user can define such dynamic inputs as a function of
time (t) in the GUI.
20 kN
(1)
60
40
20
-20
-40
-60
50 kN
70 kN
5 kN
90 kN
F = 0 kN
100
50
110 kN
130 kN
050
X (m)
(1)
f = 50 kN
f = 20 kN
f = 70 kN
f = 5 kN
f = 90 kN
Vertical
Horizontal
f = 110 kN
f = 130 kN
-50
050
103 × TIP Load
(b)
FIGURE 4. (a) The toolbox simulation output. (b) The results obtained in [21]. (1) The clamped beam profile under varying follower
tip loads. (2) The horizontal and vertical tip displacement at different loads.
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IEEE ROBOTICS & AUTOMATION MAGAZINE SEPTEMBER 2023
100
100
(a)
-50
020406080
Load (kN)
100
120
(2)
10 kN
We perform the dynamic analysis of the system for the first
7 s. Figure 6(a)(2) and (3) show two views of superimposed
snapshots of the rod in midflight at different times, as solved
by the SoRoSim Toolbox. The rod's position, orientation, and
deformation match exactly with the published results in [17],
provided in Figure 6(b)(2) and (3). This example also demonstrates
the capabilities of the toolbox in modeling lumped and
distributed joints (soft body) within the same framework.
We use this example to highlight the computational efficiency
of the toolbox. The system uses 15 DoF to simulate a
complex dynamic motion in 3D. Boyer et al. report a computational
time of 4 h, 30 min for a 30-s simulation of the same
problem [17]. Hence, on average, the reported computational
time for a second of the simulation was 9 min. Using SoRoSim,
we were able to solve a 7-s simulation in less than 1 s of
computational time, which is three orders (, )
23 000# faster
than the previously reported computational time. Faster-thanreal-time
computation (7# faster in this case) will allow the use
of SoRoSim for real-world applications. To better appreciate
Vertical (z-Axis)
Horizontal (x-Axis)
100
50
(2)
Z (m)
Displacement
Displacement (m)
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