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Model-free control framework for multi-limb soft material robots

Soft body robots have potential for search and ex-
ploration during disaster relief operations. Soft
deformable materials promise versatility, adapt-
ability and impact resistance. However, control of
modular, multi-limbed soft robots for terrestrial
locomotion is challenging due to the continuum
nature of soft materials and robot-environment
interaction. Traditionally, robot control is usually
performed by modeling kinematics using exact
geometric equations and finite element analysis.
An alternative model-free control framework ad-
dresses following challenges
1. Generic - control exists in task space, not actu-
ator space, thus, independent of type of actua-
tor, soft material properties, etc.
2. Adaptability - allows learning for change in
surface of locomotion. Does not directly model
robot-environment interaction.
3. Modeling - indirect modeling by identifying
key factors that dominate robot control, e.g.
friction manipulation, and discretize them.
The model-free approach to control of soft robots
does not directly model the robot. It can be sum-
marized as a four-step process
1. Discretization: Discretizing the key factors
that dominate robot-environment interaction.
In process, defining finite robot states.
2. Visualization: Use graph theory for providing
mathematical representation of periodic con-
trol patterns and locomotion gaits.
3. Learning : Rewards are weighted displace-
ment and orientation change for robot state-
to-state transitions. Learn rewards specific to
locomotion surface.
4. Optimization: Optimization of reward depen-
dent locomotion task (translation or rotation)
cost function is an Integer Linear Program-
ming (ILP) problem.
Friction manipulation mechanisms dominate control by influencing the robot-
environment interactions.
Node NKState dec2bin(K 1)
Arc (Ai): transition from one state to another
Simple cycle (ci∈ R56): periodic cycles of state
transitions and act as linear basis for finding
locomotion gaits.
ci,j =1 if ciincludes arc Aj
xici, xi∈ {0,1,2, ..}
State transition reward (Rj∈ R3×1) : weighted result, some
translation (x, y) and rotation (θ), of transition from one
robot state to another. This is unique to surface of locomotion
and is learned.
Simple cycle cost J(ci) =
ci,j ·Rj
Locomotion (L): defined as circulation i.e. integer sum of sim-
ple cycles.
Locomotion cost: J(L) =
xiJ(ci)=[Jx, Jy, Jθ]T
For translation in +Xdirection
s.t. len(L)lmax and
Jy[y, y+], Jθ[θ, θ+]
Integer Linear Programming problem
with linear constraints!
No need for re-learning state transition rewards.
Graph theory analogy: Facilitates mathemati-
cal definition of periodic control (simple cycle)
which are instrumental in defining locomotion.
Speed of locomotion: Dependent on speed at
which a robot can transition from one state to
another as control exists in task space.
Optimization problem: Integer linear program-
ming problem with linear constraints. Can be
quickly solved using standard linear solvers for
small to medium size graphs.
-Complex locomotion gaits: Optimization for large
angular displacements and time constraint (ac-
tuator specific) will result in complex gaits.
-Way point navigation: Curvilinear path following
for way point navigation can be achieved by up-
dating the cost function.
-Intelligent learning: State transition rewards
(arc weights) can be learned intelligently for
new unstructured environment and dynami-
cally changing environment scenarios.
[1] R. Diestel. Graph Theory. Springer, 4th edition.
[2] D. B. Johnson. Finding all the elementary circuits of a directed graph. SIAM Journal on Computing, 4(1):77–84, 1975.
[3] G. Optimization et al. Gurobi optimizer reference manual. URL: http://www. gurobi. com, 2012.
... Another approach, introducing a feedback mechanism in order to apply adaptive control, is presented in [7,8]. A model-free control framework, based on the decomposition of possible gaits as paths between a finite number of basic states, has been proposed in [48]. ...
... . Dividing by h m the left-hand side of (64) and taking into account the above list of necessary conditions, one arrives to (48) and (49). The proof is concluded. ...
... Observe that, thanks to the Positive Linear-Independence Constraint-Qualification and to the complementarity conditions (48) we have also ...
We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger $W^{1,2}$ convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.
... Consequently, most locomotion control strategies for soft terrestrial robots rely on biomimetic, intuitive approaches, or trial-and-error [4], [10]. More recently, environment-centric, data-driven model-free approaches have been implemented to synthesize gaits [11], [12]. This paper utilizes the gaits synthesized by these approaches as briefly discussed later in the paper. ...
... The reader may refer to[11],[12] for detailed analysis, and to[24] for the graph theory terminology and concepts used. ...
Full-text available
The adaptability of soft robots makes them ideal candidates to maneuver through unstructured environments. However, locomotion challenges arise due to complexities in modeling the body mechanics, actuation, and robot-environment dynamics. These factors contribute to the gap between their potential and actual autonomous field deployment. A closed-loop path planning framework for soft robot locomotion is critical to close the real-world realization gap. This paper presents a generic path planning framework applied to TerreSoRo (Tetra-Limb Terrestrial Soft Robot) with pose feedback. It employs a gait-based, lattice trajectory planner to facilitate navigation in the presence of obstacles. The locomotion gaits are synthesized using a data-driven optimization approach that allows for learning from the environment. The trajectory planner employs a greedy breadth-first search strategy to obtain a collision-free trajectory. The synthesized trajectory is a sequence of rotate-then-translate gait pairs. The control architecture integrates high-level and low-level controllers with real-time localization (using an overhead webcam). TerreSoRo successfully navigates environments with obstacles where path re-planning is performed. To best of our knowledge, this is the first instance of real-time, closed-loop path planning of a non-pneumatic soft robot.
... e authors of [23] used ILC to generate flexible impact behavior, and the authors of [24] reported an ILC-based method to learn the grasping task of a soft, fluid, and elastomeric manipulator. A graph-based, model-free flexible robot motion control framework was proposed in [25][26][27]. In literature [28,29], the authors have suggested a control strategy influenced by marine life. ...
... Procedure formula (25) can be obtained as follows: ...
Full-text available
Traditional and typical iterative learning control algorithm shows that the convergence rate of error is very low for a class of regular linear systems. A fast iterative learning control algorithm is designed to deal with this problem in this paper. The algorithm is based on the traditional P-type iterative learning control law, which increases the composition of adjacent two overlapping quantities, the tracking error of previous cycle difference signals, and the current error difference. Using convolution to promote Young inequalities proved strictly that, in terms of Lebesgue-p norm, when the number of iterations tends to infinity, the tracking error converges to zero in the system and presents the convergence condition of the algorithm. Compared with the traditional P-type iterative learning control algorithm, the proposed algorithm improves convergence speed and evades the defect using the norm metric’s tracking error. Finally, the validation of the effectiveness of the proposed algorithm is further proved by simulation results.
... 2.3. model based control CCM [21,82,83] NCCM [12,84] FEM [45-47, 85, 86] model free control PID [26,66,87,88] model-less [44,89,90] learn inverse kinematic equation [91,92] learn the control strategy [93,94] Most model-based control are proposed based on the CCM. In [21], the visual servoing approach is applied for position control of cable-driven soft robotic manipulator based on a CCM model. ...
... There are a number of work where machine learning is used to control soft robots. Some work tried to learn the inverse kinematic equations [91,92], some tried to learn the control strategies directly [93,94] using the reinforcement learning technique. Besides, for the control of soft actuators or the manipulator with a single actuator, PID type of control strategy is simple and effective [26,66,88]. ...
Soft robots can interact with the environment in a safe and compliant way because of their deformable structures. However, the modeling of soft robots which have, theoretically, infinite degrees of freedom, are extremely difficult especially when the robots have complex configurations. This difficulty of modeling leads to new challenges for the calibration and the control design of the robots, but also new opportunities with possible new force sensing strategies. This dissertation aims to provide new and general solutions using modeling and vision. The thesis at first presents a discrete-time kinematic model for soft robots based on the real-time Finite Element (FE) method. Then, a vision-based simultaneous calibration of sensor-robot system and actuators is investigated. Two closed-loop position controllers are designed and the robust stability of the closed-loop system is analyzed using Lyapunov stability theory. Besides, to deal with the problem of image feature loss, a switched control strategy is proposed by combining both the open-loop controller and the closed-loop controller. Using soft robot itself as a force sensor is available due to the deformable feature of soft structures. Two methods (marker-based and marker-free) of external force sensing for soft robots are proposed based on the fusion of vision-based measurements and FE model. Using both methods, not only the intensities but also the locations of the external forces can be estimated. The marker-based approach is proposed to find the correct locations of external forces among several possible ones. If there are no obvious feature points on the surface of the soft robot, the marker-free force sensing strategy is available using an RGB-D camera. As a specific application, a cable-driven continuum catheter robot through contacts is modeled based on FE method. Then, the robot is controlled by a decoupled control strategy which allows to control insertion and bending independently. Both the control inputs and the contact forces along the entire catheter can be computed by solving a quadratic programming (QP) problem with a linear complementarity constraint (QPCC). A simplified solution is proposed for the computation of QPCC by converting it into a standard QP problem.
... Moreover, imitation learning from expert demonstrations may be used to obtain initial estimates of the policy parameters (Peters & Schaal, 2006). Over the last decade, model-free policies have demonstrated inspiring results in a wide range of challenging robotics applications, including control of multi-limb soft robots (Vikas, Grover, & Trimmer, 2015), flapping-wing micro-robots (Pérez-Arancibia, Duhamel, Ma, & Wood, 2015), visual servoing of elastic objects using robotic manipulators (Navarro-Alarcon, Liu, Romero, & Li, 2013), and robust active visual perching with quadrotors on inclined surfaces (Mao, Nogar, Kroninger, & Loianno, 2023). More recently, Kumar, Fu, Pathak, and Malik (2021) trained a robust model-free policy to enable legged-robot locomotion across many real-world terrains, such as sand, mud, hiking trails, tall grass and dirt pile. ...
... Wang et al. [26] proposed a MFC method considering external environmental disturbance, which eliminated the influence of environmental disturbance on the motion of space robots. Vikas et al. [27] proposed a control framework of a soft robot based on the model-free theory to solve the time-varying problem of the model in the operation process. Han et al. [28,29] applied the model-free control (MFC) method to the post-capture combined spacecraft for the first time. ...
Full-text available
A model-free control method is applied to the attitude and orbital operation of the post-capture combined spacecraft, which consists of a space robot and debris. The main contribution of this paper lies in the following three aspects. Firstly, the discrete dynamic linearization method of the motion equation for a post-capture combined spacecraft is proposed, and then, the standardized expression form of multiple input and multiple output system for the attitude and orbital dynamics motions of post-capture combined spacecraft are presented. Secondly, the data mapping model of the post-capture combined spacecraft is defined, and based on this, an initial value online optimization method for the data mapping model is provided, which is key for the convergence of model-free control. Finally, a test system based on the ground-based three-axis spacecraft simulator is built to simulate the attitude and orbital operation of post-capture combined spacecraft, and the experimental system is implemented to verify the validation of the model-free control method proposed in this paper. The results show that the model-free control has a good control effect on the attitude and orbit of the post-capture combined spacecraft, even if the configuration of the spacecraft is time-varying.
... The control performance is relevant to the accuracy of the model. Compared with model-based controllers, the model-free controllers require no model information from soft manipulators but require control structures based on real-time accurate feedback data (Vikas et al. 2015;George et al. 2018;Li et al. 2017;Jiang et al. 2020;Bruder et al. 2002. ...
Full-text available
Soft robotic manipulators have promising features for performing non-destructive underwater tasks. Nevertheless, soft robotic systems are sensitive to the inherent nonlinearity of soft materials, the underwater flow current disturbance, payload, etc. In this paper, we propose a prediction model-based guided reinforcement learning adaptive controller (GRLMAC) for a soft manipulator to perform spatial underwater grasping tasks. In the GRLMAC, a feed-forward prediction model (FPM) is established for describing the length/pressure hysteresis of a chamber in the soft manipulator. Then, the online adjustment for FPM is achieved by reinforcement learning. Introducing the human experience into the reinforcement learning method, we can choose an appropriate adjustment action for the FPM from the action space without the offline training phase, allowing online adjusting the inflation pressure. To demonstrate the effectiveness of the controller, we tested the soft manipulator in the pumped flow current and different gripping loads. The results show that GRLMAC acquires promising accuracy, robustness, and adaptivity. We envision that the soft manipulator with online learning would endow future underwater robotic manipulation under natural turbulent conditions.
... A numerical representation of the shape could be used to build a model of the compliant body. This description could be used to optimize locomotion of a soft robot, 9 actively deform a compliant object through visual inspection, 10 or in the case of morphological computation, be used to represent the intelligence inherent to the body of a robot. 11 Several approaches have been proposed to quantitatively describe shape. ...
Soft materials are driving the development of a new generation of robots that are intelligent, versatile, and adept at overcoming uncertainties in their everyday operation. The resulting soft robots are compliant and deform readily to change shape. In contrast to rigid-bodied robots, the shape of soft robots cannot be described easily. A numerical description is needed to enable the understanding of key features of shape and how they change as the soft body deforms. It can also quantify similarity between shapes. In this article, we use a method based on elliptic Fourier descriptors to describe soft deformable morphologies. We perform eigenshape analysis on the descriptors to extract key features that change during the motion of soft robots, showing the first analysis of this type on dynamic systems. We apply the method to both biological and soft robotic systems, which include the movement of a passive tentacle, the crawling movement of two species of caterpillar (Manduca sexta and Sphacelodes sp.), the motion of body segments in the M. sexta, and a comparison of the motion of a soft robot with that of a microorganism (euglenoid, Eutreptiella sp.). In the case of the tentacle, we show that the method captures differences in movement in varied media. In the caterpillars, the method illuminates a prominent feature of crawling, the extension of the terminal proleg. In the comparison between the robot and euglenoids, our method quantifies the similarity in shape to ∼85%. Furthermore, we present a possible method of extending the analysis to three-dimensional shapes.
... Indeed, energy is absorbed by the soft and compressible foam material, making fast motions difficult. It may be that the best niche for soft foam robots is with slow, quasi-static motions to form configurations. 92 In this way, our robot can assume a particular shape in order to accomplish a task. For example, a crawling gait in a terrestrial robot could be created by alternating between flat and folded configurations. ...
A design and manufacturing method is described for creating a motor tendon–actuated soft foam robot. The method uses a castable, light, and easily compressible open-cell polyurethane foam, producing a structure capable of large (~70% strain) deformations while requiring low torques to operate ( < 0.2 N·m). The soft robot can change shape, by compressing and folding, allowing for complex locomotion with only two actuators. Achievable motions include forward locomotion at 13 mm/s (4.3% of body length per second), turning at 9◦/s, and end-over-end flipping. Hard components, such as motors, are loosely sutured into cavities after molding. This reduces unwanted stiffening of the soft body. This work is the first demonstration of a soft open-cell foam robot locomoting with motor tendon actuators. The manufacturing method is rapid (~30 min per mold), inexpensive (under $3 per robot for the structural foam), and flexible, and will allow a variety of soft foam robotic devices to be produced.
... Soft modules are manufactured of soft materials, such as silica gel or rubber, which have characteristics of flexibility and continuous deformation. Therefore, many researchers have developed on soft modular robots, including the FEAs soft robots [12], the SMS robots [13], the soft modular assistance robots [14], large and continuum deformation robot [15], soft actuator for robot [16], magnet connection of modular units for soft robot [17], artificial neural networks(ANN) soft robot [18], multi-limb soft robot [19], the common modular soft robot [20], modular soft robotic gripper [21]. ...
This conference was one of a series of Oberwolfach conferences, held every two years or so, with focus on graph structure, decomposition, and representation. There were 49 participants, including over a dozen graduate students and postdocs. At the request of the Oberwolfach Director, the conference schedule was designed to promote informal collaboration. In particular, there were fewer formal talks than usual, and instead there were a number of discussion groups or ``workshops'. Also, the first day (except for one plenary talk) was devoted to having the participants introduce themselves -- we asked all participants to give a five-minute presentation of their current interests. We were fortunate in that several of the plenary talks described major new results. For instance, Ron Aharoni and Eli Berger have just solved the ErdH{o}s-Menger conjecture; Bertrand Guenin has proved a major extension of the four-colour theorem; and Stephan Brandt and St'ephan Thomass'{e} have settled a long-standing question about the chromatic number of dense graphs. But probably the most distinctive feature of the meeting were the workshops. Some of these were planned before the conference, and others were held spontaneously. They were each on a topic with a chairman, but made as informal as possible. Some were more or less a sequence of talks on the topic, some were monologues, and some were genuine discussions. There were several different topics: infinite graphs and Ramsey theory, matroid theory, connectivity, graph minors and width, and topological methods. Three topics in particular gave rise to particularly active and long-running workshops: the proof of the ErdH{o}s-Menger conjecture, the prospects of extending the graph minors project to matroids, and the use of topological methods for combinatorial problems. Our thanks to the organizers of the workshops for making them run successfully, to the Director for encouraging us to try out new ways of informal collaboration, and to all the participants for making this a highly stimulating meeting.
An algorithm is presented which finds all the elementary circuits-of a directed graph in time bounded by O((n + e)(c + 1)) and space bounded by O(n + e), where there are n vertices, e edges and c elementary circuits in the graph. The algorithm resembles algorithms by Tiernan and Tarjan, but is faster because it considers each edge at most twice between any one circuit and the next in the output sequence.
Gurobi optimizer reference manual
  • G Optimization
G. Optimization et al. Gurobi optimizer reference manual. URL: http://www. gurobi. com, 2012.