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.
Learning contact-rich, whole-body manipulation for soft continuum robots is held back by the lack of simulation infrastructure that has accelerated rigid-robot manipulation. Existing soft robot simulators are physically grounded but lack the contact handling, actuation support, or learning integration needed for contact-rich manipulation; rigid-body approximations offer these capabilities but sacrifice physical grounding. We bridge this gap for tendon-driven continuum robots (TDCRs) by deriving a continuum-mechanics-informed discretization that places the soft robot natively inside MuJoCo, unifying tendon forces, body contact, and dynamics in a single physics pipeline. We validate the simulator against a Cosserat rod reference (static and dynamic) and real TDCR hardware. We then train state-based imitation learning policies via teleoperation in simulation and deploy them zero-shot to a physical 3-segment TDCR on a 7-DoF Franka arm across two contact-rich manipulation tasks. To our knowledge, this is the first demonstration of sim-to-real transfer for contact-rich manipulation with continuum robots.
Chengnan Shentu, Nicholas Baldassini, Tongjia Zheng +2
This article presents a novel SOFA based finite element method for the soft body modeling and the corresponding dynamic simulation and control of a pneumatic morphing soft quadrotor. The proposed modeling preserves the physical interpretability and control structure of traditional quadrotor dynamics, while capturing the complex, time-varying behavior of pneumatically actuated soft arms. In SOFA, the soft pneumatically actuated arms are discretized as a tetrahedral mesh following an elastic material law that produces internal forces adequate to the real dynamic behavior of the body. Pneumatic actuation governed by both periodic and error-based control signals is applied within the internal cavities to analyze the morphing capability. Finally, a proportional-integral controller is proposed to study the controlled dynamic behavior and morphing capabilities of the pneumatic arm, wherein the pneumatic actuation to the soft arm is controlled to achieve the desired target position. The simulation results show the effectiveness of the proposed novel modeling framework and the related controller design.
F. Labra Caso, V. Sumathy, P. Ferrentino +3
Robotics & AI Luleå University of Technology · IMEC Center VUB Brussels University
Tendon transmissions reduce distal inertia and add compliance, yet routing, slack, and friction govern motion and force transfer. Mainstream rigid-body robotics simulators such as MuJoCo do not jointly resolve moving noncircular contact, unilateral tension, and segment friction. We present MuJoCable, which adds a reduced-order, configuration-dependent cable transmission to MuJoCo. Its routing algorithm jointly optimizes an ordered path across moving analytic and mesh surfaces. A unilateral axial law, directional Capstan propagation, and nodal virtual work map this path to segment tensions and body forces. The warm-started engine plugin applies these forces during simulation and exposes route and load states for design. Pulley benchmarks recover analytical transmission relations with a Capstan-ratio error below 0.5%. On the underactuated 18-joint SpiRobs, MuJoCable reveals friction-driven load growth and proximal redistribution of joint rotation that the native tendon does not represent. Hardware tests on SpiRobs and a tendon-route-coupled finger reproduce observed motion sequences. By making physical threading executable, MuJoCable brings transmission sources of the simulation-to-reality gap into route, cable, and actuator design before fabrication.