Combining aerial robots with soft continuum manipulators promises safe, compliant interaction with the environment, but the coupled dynamics of a free-flying base and a highly deformable, underactuated arm are difficult to study without a suitable simulator. This paper describes an open-source simulation framework, built on the MuJoCo physics engine, for a quadrotor that carries a tendon-driven soft continuum arm underneath its body. The arm is modelled as a pseudo-rigid-body chain of twelve segments joined by pairs of orthogonal compliant hinges. Three tendons, spaced \SI{120}{\degree} apart and routed through every segment, bend it in any direction. A geometric tracking controller on SE(3) flies the vehicle; its rotor allocation matrix is generated directly from the model description. A closed-form tendon-tension allocation keeps every tension within its limits and gives the same bending moment in every direction. We report the full set of modelling choices and parameters, and we evaluate the framework in four simulated scenarios: a \SI{36}{\second} mission with a figure-eight and a tendon-driven tip sweep, an attitude-gain study, a quasi-static characterization of the bending workspace, and an impulsive disturbance applied to the arm tip. With the nominal gains the vehicle tracks the mission with \SI{3.1}{\centi\meter} RMS position error while the arm tip swings up to \SI{37}{\centi\meter} from the body axis. The gain study shows that a low-bandwidth attitude loop lets the arm excite sustained oscillations. The simulation runs about 15× faster than real time on a laptop CPU. All models, controllers and scripts that reproduce the figures are released.
Figures & tables
Fig. 1: Rendered frames from the demonstration mission. (a) Hover with the arm hanging under gravity. (b) Figure-eight flight with the arm deflected by inertial loads. (c)–(f) Hover while the tendons sweep the bending direction through a full revolution. The arm curls into a hook toward the tensioned side.
Parameter
Value
Quadrotor
Mass (frame, rotors, flange)
1.06kg
Principal inertia
(6.7,6.7,12.9)×10−3kgm2
Arm (rotor) offset (±xr,±yr)
(±0.177,±0.177)m
Thrust limit per rotor
0N8N
Drag-to-thrust ratio κ
0.016m
Hover thrust per rotor
3.09N
TABLE I: Model parameters
Fig. 2: Demonstration mission. Top: quadrotor position (solid) and reference (dashed). Middle: position error norm (log scale). Bottom: tendon tensions from ( 8 ). Dotted lines mark the phase boundaries.
Fig. 3: Top view of the figure-eight phase. The arm tip lags the vehicle and swings outward at the turns.
Phase
∥ep∥ [cm]
Tilt
Tip lat.
Thrust
RMS
max
max [ ∘ ]
max [cm]
range [N]
Hover
< 0.01
< 0.01
0.0
0.0
3.09–3.09
Climb
0.03
0.05
0.0
0.0
2.97–3.21
Figure-eight
3.90
8.87
23.5
36.8
2.95–3.53
Tendon sweep
3.34
5.16
4.7
21.8
2.73–3.42
Return
0.50
1.50
0.6
2.1
3.02–3.16
TABLE II: Mission statistics per phase (nominal gains)
Set
KR,xy
Kω,xy
RMS
max
RMS t>24s
Tilt max
[cm]
[cm]
[cm]
[ ∘ ]
Soft
0.35
0.06
22.67
52.82
33.59
40.7
Medium
1.0
0.15
7.99
16.70
11.04
28.6
Stiff
2.0
0.25
4.80
11.97
5.56
25.4
Nominal
4.0
0.40
3.14
8.87
2.85
23.5
TABLE III: Mission performance versus roll/pitch attitude gains
Fig. 4: Quasi-static bending in hover. Left: tip lateral deflection and depth versus the bending command μ for β=0 . Right: tip workspace in the body xy -plane when β is swept, for μ=0.5 and μ=1 . Arrows show the tendon directions ϕi .
Fig. 5: Response to a 0.4Ns lateral impulse on the arm tip (shaded) during hover: vehicle position error and tip lateral displacement relative to the vehicle.