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A concise guide to modelling the physics of embodied intelligence in soft robotics


Embodied intelligence (intelligence that requires and leverages a physical body) is a well-known paradigm in soft robotics, but its mathematical description and consequent computational modelling remain elusive, with a need for models that can be used for design and control purposes. We argue that filling this gap will enable full uptake of embodied intelligence in soft robots. We provide a concise guide to the main mathematical modelling approaches, and consequent computational modelling strategies, that can be used to describe soft robots and their physical interactions with the surrounding environment, including fluid and solid media. We aim to convey the challenges and opportunities within the context of modelling the physical interactions underpinning embodied intelligence. We emphasize that interdisciplinary work is required, especially in the context of fully coupled robot–environment interaction modelling. Promoting this dialogue across disciplines is a necessary step to further advance the field of soft robotics.

Key points

  • Embodied intelligence is one of the main motivations for soft robotics.

  • Body compliance enables embodied intelligence and helps to simplify the control of robots for achieving the required tasks in complex environments.

  • A full mathematical description of the deformations of a soft robot, given by its internal interactions with actuators and by the external interactions with the surrounding environment, can be the tool for mastering body compliance and embodied intelligence.

  • Relevant modelling and computational techniques are within grasp, in an interdisciplinary effort.

  • Soft robotics can transition to a model-informed discipline, with embodied-intelligence-based design and control, that can pave the way towards soft-robot digital twins.

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Fig. 1: The three main components of a typical sensorimotor scheme, either for a robot or a biological system.
Fig. 2: Scope of this Technical Review in modelling the physics of embodied intelligence.
Fig. 3: Examples of practical simulations in the context of internal interactions.
Fig. 4: Examples of practical simulations in the context of external interactions.


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G.M. acknowledges NUS support through his start-up grant (R-265-000-A36-133). G.S.C. acknowledges MOE Tier 1 grant (R-265-000-655-114). C.L. acknowledges NUS support through her start-up grant (R-265-000-A31-133 and R-265-000-A31-731). Part of this research is supported by the National Research Foundation, Singapore, under its Medium Sized Centre Programme — Centre for Advanced Robotics Technology Innovation (CARTIN). G.M., G.S.C. and C.L. acknowledge MOE Tier 2 grant ‘REBOT’. This work was supported in part by the US Office of Naval Research Global under grant N62909-21-1-2033 and Khalifa University of Science and Technology under grants CIRA-2020-074 and RC1-2018-KUCARS.

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Correspondence to Gianmarco Mengaldo or Cecilia Laschi.

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Soft robotics

The RoboSoft Community defines soft robots as “soft robots/devices that can actively interact with the environment and can undergo ‘large’ deformations relying on inherent or structural compliance”.

Embodied intelligence

Part of control and intelligence contributed by the physical body and its interaction with the environment and the task.


Hyperelastic materials are models of material that have a nonlinear elastic response. They are commonly used to approximate soft tissues exhibiting large deformations, including rubber material and biological tissues.

Lagrange multipliers

A method to solve a constrained optimization problem, that is, to find the minima or maxima of a function under equality constraints. The method can be generalized to inequality constraints by the Karush–Kuhn–Tucker conditions.


The property of exhibiting different characteristics along different directions.

Finite element method

(FEM). A popular numerical method to solve partial differential equations. It reduces the original differential system into a solvable algebraic system by discretizing the domain of interest (time and space) into finite elements.

Lie group

A mathematical set that is differentiable and closed under a product operation. The Lie groups of 3D rotations (SO(3)) and 3D rigid motions (SE(3)) are particularly important here.

Euler angles

A triplet of angles that parametrize a 3D rotation. Each angle represents an elementary rotation around the x, y or z axis.

Non-Newtonian fluids

A fluid whose viscosity depends on stress, thereby breaking Newton’s law of viscosity.

Open-loop control

A control is called open-loop (or feedforward) when the controller operates independently from the output of the system. An example is a heating system controlled only by a timer to switch on or off. Conversely, control is closed-loop (or feedback) when the controller operates by using the output of the system as input to determine its behaviour. An example is a heating system using a controller that senses the temperature of the room.

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Mengaldo, G., Renda, F., Brunton, S.L. et al. A concise guide to modelling the physics of embodied intelligence in soft robotics. Nat Rev Phys 4, 595–610 (2022).

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