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Edited by: Concepción A. Monje, Universidad Carlos III de Madrid, Spain

Reviewed by: Zheyuan Gong, University of Toronto, Canada; Yisheng Guan, Guangdong University of Technology, China

This article was submitted to Soft Robotics, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Modeling of soft robots is typically performed at the static level or at a second-order fully dynamic level. Controllers developed upon these models have several advantages and disadvantages. Static controllers, based on the kinematic relations tend to be the easiest to develop, but by sacrificing accuracy, efficiency and the natural dynamics. Controllers developed using second-order dynamic models tend to be computationally expensive, but allow optimal control. Here we propose that the dynamic model of a soft robot can be reduced to first-order dynamical equation owing to their high damping and low inertial properties, as typically observed in nature, with minimal loss in accuracy. This paper investigates the validity of this assumption and the advantages it provides to the modeling and control of soft robots. Our results demonstrate that this model approximation is a powerful tool for developing closed-loop task-space dynamic controllers for soft robots by simplifying the planning and sensory feedback process with minimal effects on the controller accuracy.

Soft robotic technologies are becoming increasingly prevalent in the design and development of robots (Kim et al.,

The most common modeling and control strategy for soft robots are based on steady-state models, which, under the steady-state assumption, can be equated to the kinematic model (George Thuruthel et al.,

A popular method for developing dynamic models for soft robots is based on the cosserat-rod mechanics. Such models have been extensively used for soft robotic manipulators driven by tendon actuation (Rucker and Webster,

Unlike static controllers, developing fully-dynamic controllers would involve a planning stage. Typically, this has to be performed using some optimization techniques irrespective of the modeling strategy. A good example is the use of trajectory optimization for the control of a soft robotic manipulator using a model-based (Marchese et al.,

This article investigates the viability of a first-order dynamic model for soft robots. It must be noted that unlike state-space dimensionality reduction methods (Thieffry et al.,

We investigate the viability of this simplifying assumption using extensive simulation studies. First, we present the theoretical reasoning behind the first-order assumption and its corresponding controller. Then we briefly introduce the fully dynamic simulation model that is used to verify the learned forward models and the dynamic controller. Finally, we present details on the learning architecture together with results of the model and the closed-loop task-space controller.

Given a soft robot that can be kinematically modeled by the configuration-space ^{n}, the task-space variable can be obtained by the kinematic transformation:

Where, ^{m} and

Here,

Soft robots typically have high damping values with low inertial properties. This is because they are commonly fabricated with viscoelastic materials with low material density. After the initial transient motion of a soft robot from rest (when

After discretizing the equation, the dynamic model can now be represented through the mapping : (_{i}, τ_{i}) → _{i+1}. Where, _{i}, _{i+1} are the current and the next configuration of the soft robot, respectively. Correspondingly, this implies that a closed-loop dynamic controller would require only the zero-order state feedback (

The obtained first-order dynamical Equation (3), the first-order configuration-space term can be replaced using the well known inverse kinematics mapping:

Where, ^{†} is any generalized inverse matrix. Note that we have ignored the null-space terms for brevity. Now the first-order dynamic equation can be reformulated as function of the task-space variables:

This functional mapping can now be directly learning by a machine learning architecture. We can now transform the mapping and introduce the target task-space variable ^{d} as:

The operational space controller mapping now becomes:

Here, (^{c}, ^{c}) are the current configuration-space coordinates and the current task-space coordinates, respectively. For the special case when the cardinality of the configuration-space coordinate is the same as the cardinality of the task-space coordinate (mapping from ^{c}, ^{d}) → τ. A simple feedforward neural network can be used to learn this mapping (

The dynamic model is based on the Piece-wise Constant Strain (PCS) approach for soft-rigid multibody system of Renda and Seneviratne (^{˜} represents the usual isomorphism between a vector in ℝ^{3} and its corresponding skew-symmetric matrix in 𝔰𝔬(3).

Schematic of the soft manipulator used for our studies. The manipulator is driven by three tendons arranged in the configuration shown above. Unless stated otherwise, the single section manipulator is used for the study.

The relative position and orientation of a soft body _{i}(·) : _{i}]↦_{i}(

The continuous models of the position, velocity and acceleration of a soft body can be derived from the Cosserat rod theory, which gives (Boyer and Renda,

where

with

To model constrained rod, such as the Kirchhoff-Love case with angular strain only, the strain field is specified as:

where

Assuming piece-wise constant strains (Renda et al.,

where

Successive applications of the kinematics (Equation 9) for all the bodies of the system, yields to the definition of the geometric Jacobian _{i}(

where the block elements of the ^{th} Jacobian

Once a Jacobian is found, the generalized dynamics of the system can be obtained by projecting the free dynamics of each soft body by virtue of the D'Alembert's principle. The free dynamic equation, with its boundary conditions, of a soft body is given by (Renda et al.,

where _{·i}(_{i}(^{6×6} is the co-adjoint (respectively co-Adjoint) map of the Lie algebra (respectively Lie group) defined in Nomenclature. Regarding the internal elastic force, a linear viscoelastic constitutive model is usually chosen:

where

are the screw stiffness and viscosity matrix (_{i} being the young modulus, _{i} the shear modulus and ν_{i} the shear viscosity).

By Jacobian projection of the free dynamics (Equation 11), we obtain the generalized dynamics in its classical configuration-space form:

where ^{n×n} is the generalized mass matrix, ^{n×n} is the generalized Coriolis matrix, ^{n×n} is the block-diagonal generalized damping matrix, ^{n×n} is the block-diagonal generalized stiffness matrix, ^{n} is the vector of generalized position-dependent external forces and ^{n} is the vector of applied actuation forces. Note that the dynamic Equation (13) can be written in the form required by Equation (2), with

It is worth noting here the different structure of the components of the generalized dynamics Equation (13). Similarly to the minimal Lagrangian models of traditional rigid robots, inertial loads are characterized by full coefficient matrices, as can be see from Equations (14a) and (14b), while damping and stiffness loads are characterized by block-diagonal coefficient matrices, as for Equations (14c) and (14d). This is in contrast with other modeling approaches that use absolute coordinates, such as Finite Elements, for which the opposite holds. Inertial coefficient matrices are block-diagonal while damping and stiffness coefficient matrices are full. Thus, neglecting inertial terms will be well suited for minimal Lagrangian models for soft robotic manipulator, such as the PCS approach.

This section investigates two studies. First, we validate the accuracy of the learned first-order model with respect to the learned second-order model. Second, we perform simulated experiments to validate the accuracy of the proposed controller. All the tests are performed on the fully dynamic model described in section 3.

The learned models are derived using a kind of recurrent neural network called a nonlinear autoregressive exogenous (NARX) model (Billings,

Modeling of the forward dynamics and the inverse dynamics controller. Note that for the non-redundant cases, the configuration-space (

Parameters of the learned forward dynamic model.

Type | NARX network |

Hidden layer size | 40 |

No. of samples | 7000 |

Training algorithm | Levenberg-Marquardt backpropagation |

Training:Testing:Validation ratio | 70:15:15 |

Stopping criterion | Validation set error |

Maximum no. of epochs | 100 |

Random actuation of the tendons are performed (motor babbling) for 70 s to obtain the samples for learning the forward model (

Dynamic workspace of the manipulator obtained by motor babbling. This is obtained by recording end-effector position of the manipulator when actuated by random continuous actuation signals.

Step response of the learned models in comparison to the actual analytical model. The step signal is given to single actuator at time = 2 s.

Periodic response of the learned models in comparison to the actual analytical model.

In order to analyze the effects of the viscosity and the inertial effects on our modeling assumption, we further perform studies on the accuracy of the first-order model for varying material viscosity and density. As material density increases and the material viscosity decreases, the inertial effects become more and more dominant. Hence, one would expect the accuracy of the first-order model to decrease and the second-order model to remain constant. However, this is not necessarily the case as the training of second-order recurrent neural networks is more prone to instabilities (Pascanu et al.,

Root sum squared error of the learned model for varying dynamic properties of the manipulator.

Increasing the length of the manipulator is another way to increase the inertial properties and weaken the first-order assumption. For this, we test the same methodology on a 4 section manipulator, however, actuated only on the first section. Each section has approximately the same length, with the total length of the manipulator adding to 418 mm. We test two designs, one with a tapered morphology and the other with a cylindrical morphology (higher inertial properties). The radius of the tapered morphology linearly reduces from 30 to 10 mm while the cylindrical morphology has a fixed diameter of 30 mm. For this test and the following controller results, the default parameters of the manipulator is used (i.e., with the material density of 1,080 kg/m^{3} and viscosity of 300 Pas; Renda et al.,

Root sum squared error of the learned model for the 4-section underactuated manipulator.

The closed-loop task-space controller is derived by learning the operational-space dynamics mapping as described in section 2.1. As the mapping is not recursive, it can be learned using a simple feedforward neural network, as shown in ^{i}, ^{i+1}) → τ. When testing the controller, the next task-space coordinate, ^{i+1}, is replaced by the desired task-space variable ^{d}. The parameters of the neural network used for learning the first-order inverse dynamics mapping is given in

Parameters of the first-order inverse dynamics controller.

Type | Feedforward neural network |

Hidden layer size | 30 |

No. of samples | 7,000 |

Training algorithm | Levenberg-Marquardt backpropagation |

Training:Testing:Validation ratio | 70:15:15 |

Stopping criterion | Validation set error |

Maximum no. of epochs | 100 |

Path planning is usually a complex problem in inverse dynamics based controllers, but as our inverse model is developed with only zeroth order state feedback, the development of the task-space trajectory is greatly simplified. Acceptable paths can easily be generated using the data points obtained from the workspace of the manipulator, which is obtained during the motor babbling phase. The desired paths can be generated by picking reachable points from the workspace and routing a path through them, ensuing that there is sufficient time for the manipulator to reach adjacent points. This can be easily done by fixing a cap on the maximum distance between adjacent task-space variables.

To test our controller, we generate randomized linear paths for the end-effector of the manipulator to follow. This is done by picking two random points from the robot workspace and linearly interpolating a trajectory between them and from the initial position of the end-effector. If the initial position of the end-effector is ^{0} and the two random points are ^{1} and ^{2}, the generated path is from ^{0} → ^{1} → ^{2} → ^{1}. Note that the intermediate points are not necessarily reachable by this naive approach. Accurate trajectories can be generated by searching for adjacent points in the workspace or by projecting the trajectory onto the workspace surface.

The results of the trajectory tracking is shown in

Controller results for a continuous path. The motion of the end-effector is shown here.

The

Testing the first-order controller in open-loop, closed-loop, and with a late start.

Tracking error for the scenario shown in

This paper presents and verifies a model simplifying assumption for soft robots. The core idea of the assumption is that soft robots, by definition, tend to have low inertial and high viscoelastic properties. This leads to dynamic behaviors which are well approximated by a first-order system, as typically observed in nature. We verify this assumption using a simulated fully dynamic model of an Octopus-like manipulator and a type of recurrent neural network called NARX network. Finally, we develop easy-to-develop closed-loop task-space dynamic controllers based on this assumption. Our results indicate that controllers developed on this assumption can compensate the errors in modeling accuracy with the increased control frequency. Our method makes path planning simpler for non-redundant cases. Moreover, the sensory requirements for closed-loop dynamic control is reduced because of the simplification. This is because the state feedback required for the controller is only the zeroth-order component. Our work also indicates that the additional modeling complexity that soft elements introduce can, to some extent, be reduced by designing low inertial highly visco-elastic soft robot designs.

Although we use machine learning tools to test our modeling assumption and develop our controller, the approach is equally suited for analytical approaches. In fact, ignoring the inertial elements would greatly simply the modeling and parameter estimation process involved in model-based control of soft robots. Not only can we reduce the states of the dynamical system, but we also avoid the problem of estimating and inverting the full mass matrix (see section 3.2). However, it must be kept in mind that first-order systems present numerical challenges in their implementation. Such problems are not found in a learning based approach and hence desirable in that respect. Typically, the first-order model leads to a stiff differential equation and requires specialized techniques for solving them. Interested readers are suggested to refer to Strogatz (

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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