Soft robots hold significant potential in legged locomotion due to their inherent deformability, enabling enhanced adaptability to various environmental conditions and the generation of diverse locomotion gaits. While various soft robots have been proposed for terrestrial locomotion, research on dynamically-stable locomotion, such as trotting, with actuated soft bending limbs remains limited. We introduce a pneumatically-actuated soft quadruped featuring a soft body capable of a variety of dynamically-stable trotting locomotion. We utilize soft limb kinematics and parameterize fundamental limb locomotion to obtain quadrupedal locomotion trajectories for both linear and curvilinear motions. We also employ a physics-enabled dynamic model to optimize and evaluate trotting locomotion trajectories for dynamic stability. We further validate the stable locomotion trajectories through empirical experiments conducted on a soft quadruped prototype. The results demonstrate that the quadruped trots at a peak speed of 1.24 body lengths per second when traversing flat and uneven terrains, including slopes, cluttered areas, and naturalistic irregular surfaces. Furthermore, we compare the energy efficiency between trotting and crawling locomotion. The findings reveal that trotting is significantly more energy-efficient than crawling, with an average energy saving of up to 42%.Note to Practitioners—This paper was motivated by the challenge of achieving dynamically stable and efficient locomotion in soft quadrupeds. Many soft-legged robots are typically designed for statically stable, albeit inefficient and slow, locomotion gaits such as crawling. Our research aims to address this practical challenge of improving mobility in soft-legged robots. We develop a novel soft quadruped with pneumatically-actuated soft limbs that achieves efficient trotting that is 42% more energy-efficient than crawling. This work is particularly relevant for industries requiring adaptable and efficient navigation in environments, such as search and rescue, agricultural monitoring, and exploration. The development and optimization of trotting gaits through a physics-enabled dynamic model for dynamic stability provide a foundational framework for enhancing the adaptability and operational utility of soft robots. While our findings mark a significant step forward, challenges remain in deploying these locomotion strategies on autonomous untethered robots with onboard sensor feedback. Future research will focus on these areas, aiming to improve the practical deployment and robustness of soft robotic locomotive systems.

Continuum arms are more adaptable to their environments and inherently human-friendly compared to their rigid counterparts. Path planning of continuum arms is an active research area with many challenges. The hyper-redundancy of continuum arms, which renders them highly versatile, is their curse in path planning. This problem becomes even more challenging in dynamic environments in the presence of mobile obstacles. In this paper, we propose an anticipatory path planning approach for continuum arms in dynamic environments. Our approach is based on obstacle prediction coupled with temporal graphs to model the dynamic environment. We evaluate the proposed approach’s performance and compare it to prevailing path planning approaches for continuum arms in dynamic environments.

Continuum arms, with their mix of compliance, payload, safety, and manipulability, are perfectly suited to serve as co-robots, and their applications range from industry and manufacturing to human healthcare. Their hyper-redundancy serves as their most significant challenge for path planning and path planning approaches commonly used with rigid-link robots, such as inverse kinematics, that fail to provide reliable trajectories for continuum arms. We propose an Inverse Kinematics-based approach to address the limitations of previously-proposed Kinematics-based approaches. Using this new approach, we are able to efficiently generate very rich sets of configurations, which, in turn, lead to smooth path planning for such continuum manipulators. To validate the smoothness of the paths generated by our approach, we apply dynamics constraints to the generated trajectories. We show that, when tracked by a controller, the paths that are generated using the proposed approach are much smoother than previously-proposed Kinematics-based approaches: The proposed approach allows the continuum arm to traverse the trajectories very accurately and in time less than half of that taken by previous (reliable) path planning approaches.

Soft robotic manipulators (SRM) are biologically inspired from animal appendages such as elephant trunks and octopuses’ arms. In contrast to traditional rigid robotic manipulators, SRMs are made of flexible material and generate motion through structural deformation. Thus, they more easily adapt to unstructured environments. Teleoperated SRMs can be used in spaces harmful or impractical to humans (i.e., nuclear radiation or minimally invasive surgical sites). Limited research is available on the factors that affect human performance during the teleoperation of SRMs. We present two Human-Robot Interfaces (HRI) and conduct user studies on an SRM in object handling tasks and assess the performance of teleoperators using spatial input devices and augmented reality-based input devices. The user interaction is quantified for two types of controllers (Direct and Indirect) in an immersive pick and place operation. A System Usability Scale (SUS) questionnaire is administered to assess the usability of each HRI. Results suggest that the users perform more effectively, make fewer errors using the Indirect Control HRI and participants rated the Indirect Control HRI as more usable regardless of the hardware device.

Tendon actuated multisection continuum arms have high potential for inspection applications in highly constrained spaces. They generate motion by axial and bending deformations. However, because of the high mechanical coupling between continuum sections, variable length-based kinematic models produce poor results. A new mechanics model for tendon actuated multisection continuum arms is proposed in this paper. The model combines the continuum arm curve parameter kinematics and concentric tube kinematics to correctly account for the large axial and bending deformations observed in the robot. Also, the model is computationally efficient and utilizes tendon tensions as the joint space variables thus eliminating the actuator length related problems such as slack and backlash. A recursive generalization of the model is also presented. Despite the high coupling between continuum sections, numerical results show that the model can be used for generating correct forward and inverse kinematic results. The model is then tested on a thin and long multisection continuum arm. The results show that the model can be used to successfully model the deformation.

In this paper, we propose and investigate a new approach to modeling variable curvature continuum robot sections, based on Euler spirals. Euler spirals, also termed Clothoids, or Cornu spirals, are those curves in which the curvature increases linearly with their arc length. In this work, Euler spirals are applied to the kinematic modeling of continuum robots for the first time. The approach was evaluated using the sections of numerous continuum robots, including two novel parallel continuum robots. Each robot consists of three parallel sections, each with three thin, long McKibben actuators. These sections are poorly modeled by the widely used constant curvature kinematic model. The constant curvature and Euler spiral models were compared and the Euler spiral method was seen to be a significantly better match for a wide range of configurations of the robot hardware.

Multisection continuum arms offer complementary characteristics to those of traditional rigid-bodied robots. Inspired by biological appendages, such as elephant trunks and octopus arms, these robots trade rigidity for compliance and accuracy for safety and, therefore, exhibit strong potential for applications in human-occupied spaces. Prior work has demonstrated their superiority in operation in congested spaces and manipulation of irregularly shaped objects. However, they are yet to be widely applied outside laboratory spaces. One key reason is that, due to compliance, they are difficult to control. Sophisticated and numerically efficient dynamic models are a necessity to implement dynamic control. In this paper, we propose a novel numerically stable center-of-gravity-based dynamic model for variable-length multisection continuum arms. The model can accommodate continuum robots having any number of sections with varying physical dimensions. The dynamic algorithm is of O (n2) complexity, runs at 9.5 kHz, simulates six to eight times faster than real time for a three-section continuum robot, and, therefore, is ideally suited for real-time control implementations. The model accuracy is validated numerically against an integral-dynamic model proposed by the authors and experimentally for a three-section pneumatically actuated variable-length multisection continuum arm. This is the first sub-real-time dynamic model based on a smooth continuous deformation model for variable-length multisection continuum arms.

Precise actuation of continuum surfaces in combination with continuum robotic arms that undergo large deformation is of high interest in soft robotics but of limited model-based study to date. This work develops this area towards enabling the robust design and control of large deformation continuum surfaces (LDCS) across multiple industrial applications in the healthcare, aerospace, manufacturing, and automotive domains. It introduces an actuation based dynamic model of LDCSs to accurately determine their deflection due to application of concentrated external forces while maintaining many physical characteristics and constraints on actuation elements and surface structure such as gravity, inertia, damping, elasticity, and interactive forces between actuators and LDCS. Using the lumped-mass methodology, a 3D integrated surface-arm model is developed, simulated and then validated experimentally where a pair of parallel arms are attached to the surface to actuate and deform it. The surface is then simultaneously subjected to a concentrated constant external force at its top center between the two arms. Comparing measured displacements between the experimental and modelling results over actuation time yielded the maximum error is less than 1% of the length of the surface's side at its final deflected profile despite the limited number of nodes (masses) used in the LDCS model while it is exposed to a significant external force.

This paper presents general modal dual Quaternion (DQ) kinematics for multisection continuum arms. DQ's produce higher accuracy than homogeneous transformation matrices (HTM) when transformed to modal shape functions (MSF) of similar order and are numerically stable. Thus, the model is compact, more accurate and computationally efficient than the modal kinematics proposed by the author based on HTM's. Also, DQ kinematics does not suffer from singularity related limitations of Euler angle based inverse orientation kinematics. Recursive schemes for deriving DQ's and DQ Jacobians are also presented and can be extended arbitrarily. Both modal HTM and modal DQ kinematics are then applied to solve illustrative spatial inverse position and orientation tracking problems. Based on the results, this paper quantitatively compares both methods and highlights the advantages of modal DQ kinematics. The proposed DQ kinematics are easily extensible to variable length multisection continuum arm with general actuator configurations.

In this paper, we present a new forward kinematic model for a novel class of long and thin continuum robots for operation in spatial workspace. Such robots are well suited for navigation through unstructured environments with superior reach using their flexible and thin profile, especially for inspection applications. This cable-like continuum robot design [1] has a concentric backbone arrangement but is spring-loaded and tendon-actuated, has improved compliance with a light and compact motor-encoder actuation mechanism. To account for the spring-loading, a compression factor is introduced on top of the established constant curvature continuum kinematics. The resulting continuum or shape variables are then estimated as a function of the measurable encoder variables. The effectiveness of the model is validated by performing experiments with the robot prototype.

Several robotic systems have been proposed for removing blood from the brain in patients who have undergone a hemorrhagic stroke. In this paper we explore the use of imagebased feedback to address tissue deformation when aspirating a hemorrhage in a phantom model. This is the first time intraoperative image feedback has been used with a concentric tube robot in this application. We describe a layer by layer approach to motion planning. Computed tomography (CT) images are collected periodically during hemorrhage removal. After each CT scan, the robot's tip path is re-planned to account for the tissue deformation that has occurred since the previous scan. We compare open loop hemorrhage removal to our sequential imaging-replanning approach, illustrating that the latter has the potential to enhance the safety and efficacy of the procedure.

Soft limb locomotion is a relatively new and challenging research field. However, soft limbs can not yet transition to practical application due to difficulties associated with control methods. Motivated by this research problem, in this paper, we investigate the performance of energy based control of underactuated soft limbed systems. We augment the previously reported energy shaping function for rigid bipeds with a new set of functions for the novel class of underactuated compass gait soft bipeds. We evaluate the controller performance and identify desired features and characteristics for better speed performance of such a biped. The proposed energy shaping functions are compared through controlled Lagrangian (CL) method for Euler-Lagrangian (EL) models and interconnection and damping assignment passivity-based control (IDA-PBC) methods for port-controlled Hamiltonian (PCH) models. Results for system stability, speed performance, and input torque profiles are compared. The IDA-PBC controllers are observed to produce better input torque and performance over the CL methods.The findings assist in extending and developing novel controllers to implement on soft limbed robots for practical control applications of soft multi-continuum limbed robots.

In this paper, we present a novel design for constructing multi-section continuum robots with a special focus on thin (less than 1 cm diameter) and relatively longer length (more than 100 cm), along with its new kinematic modeling and performance evaluation. This spring-loaded, tendon-actuated cable-like continuum robot incorporates key features of a concentric tube style design with added local compression to most sections and actively controllable bending along its entire backbone. It also avoids a complicated and large actuator system. Such robotic cables can be well suited for current space and terrestrial applications like exploration, teleoperation, surveillance, maintenance activities, etc., benefiting from these unique structural properties.

This paper presents an Octopus inspired walking robot with pneumatic muscle actuator (PMA) driven continuum arms. Each arm is made up of 4 longitudinally arranged PMAs, consistent with octopus arm anatomy. We first present the design and construction of a single continuum arm followed by its modeling and experimental validation. The design of the walking robot is then presented followed by details of extended dynamic model describing the full walking robot with four arms. Basic control architecture is introduced for the robot to achieve walking motion and experimental results analyzed. Initial results show good agreement between the experimental results and simulation results.

This paper presents the results of a multidisciplinary project where biologists, mechanical engineers and electronic engineers worked together to develop bio-inspired soft continuum arms, whose design captures and takes advantage of key features of the octopus anatomy and control. The cross-integration of such diverse expertise was channelled towards the design of soft continuum arms whose characteristics were inspired by nature, but with a focus on readily available engineering technologies and their effective integration from a system viewpoint. On one side the mechanical structure and the control was designed looking at the animal, in particular at the coupling between its anatomy and control system that allows the animal to survive in its ecosystem. On the other side engineering issues and constraints were carefully accounted for, namely material softness, intrinsic safety, energy efficiency, cost effectiveness and manufacturing aspects. The design evolution is presented through three different generations of prototypes where both bio-inspiration and engineering requirements are appropriately blended.

This paper presents an improved mode shape function-based 3D dynamic model for pneumatic muscle actuated continuum arms, and validates the model and simulation results through experimental testing. The model also facilitates the direct control of pneumatic muscle actuated continuum arms through the use of input pressure. This is achieved without additional intermediary transformations and does not have singularity problems present in previous models. The proposed arm model uses a new pneumatic muscle actuator (PMA) dynamic model with hysteresis that is capable of modelling both extending and contracting PMAs. The proposed hysteric model is simple, easily adaptable, and validated experimentally. The PMA model can be applied to dynamically model any PMA based system as well as PMA actuated continuum arms utilizing different actuator configurations.

Continuum structures with a very high or infinite number of degrees of freedom (DOF) are very interesting structures in nature. Mimicking this kind of structures artificially is challenging due to the high number of required DOF. This paper presents a kinematic and dynamic model for an underwater robotic manipulator inspired by Octopus vulgaris. Then, a prototype arm inspired by live octopus is presented and the model validated experimentally. Initial comparisons of simulated and experimental results show good agreement.

This paper presents the kinematics, dynamics, and experimental results for a novel quadruped robot using continuum limbs. We propose soft continuum limbs as a new paradigm for robotic locomotion in unstructured environments due to their potential to generate a wide array of locomotion behaviors ranging from walking, trotting, crawling, and propelling to whole arm grasping as a means of negotiating difficult obstacles. A straightforward method to derive the kinematics and dynamics for the proposed quadruped has been demonstrated through numerical simulations. Initial experiments on a prototype continuum quadruped demonstrate the ability to stand up from a flat-belly stance, absorb external disturbances such as maintaining stability after dropping from a height and after being perturbed by a collision, and crawling on flat and cluttered environments. Experiment results provide evidence that locomotion with soft continuum limbs are feasible and usable in unstructured environments for variety of applications.

This paper presents a new three dimensional kinematic and dynamic model for variable length continuum arm robotic structures using a novel shape function-based approach. The model incorporates geometrically constrained structure of the arm to derive its deformation shape function. It is able to simulate spatial bending, pure elongation, and incorporates a new stiffness control feature. The model is validated through numerical simulations, based on a prototype continuum arm, that yields physically accurate results.

This paper presents a new three dimensional (3D) kinematic model based on mode shape functions (MSF) for multisection continuum arms. It solves the singularity problems associated with previous models and introduces a novel approach for intuitively deriving exact, singularity-free MSFs, thus avoiding mode switching schemes and simplifying error models. The model is able to simulate spatial bending, pure elongation/contraction, and introduces inverse orientation kinematics for the first time to multisection continuum arms. Also, it carefully accounts for physical constraints in the joint space to provide enhanced insight into practical mechanics, and produces correct results for both forward and inverse kinematics. The model is validated through simulations, based on a prototype continuum robotic arm. Proposed approach is applicable to a broad spectrum of continuum robotic arm designs.