Soft Robots: Computational Design, Fabrication, and Position Control of a Novel 3-DOF Soft Robot

Abstract

This paper presents the computational design, fabrication, and control of a novel 3-degrees-of-freedom (DOF) soft parallel robot. The design is inspired by a delta robot structure. It is engineered to overcome the limitations of traditional soft serial robot arms, which are typically low in structural stiffness and blocking force. Soft robotic systems are becoming increasingly popular due to their inherent compliance match to that of human body, making them an efficient solution for applications requiring direct contact with humans. The proposed soft robot consists of three soft closed-loop kinematic chains, each of which includes a soft actuator and a compliant four-bar arm. The complex nonlinear dynamics of the soft robot are numerically modeled, and the model is validated experimentally using a 6-DOF electromagnetic position sensor. This research contributes to the growing body of literature in the field of soft robotics, providing insights into the computational design, fabrication, and control of soft parallel robots for use in a variety of complex applications.

1. Introduction

The emerging field of soft robotics derives much of its motivation from bioinspired designs of robotic systems that can safely interact with unknown environments [1,2,3,4]. Alongside the growth of soft robotics, the past decade has seen the development of numerous novel soft actuators and sensors [5]. These newly introduced transducer designs can function based on various actuation mechanisms such as pneumatic [6,7], hydraulic [8,9,10], thermal [11,12], electrical [13,14,15], and chemical [16,17,18].

Conventional rigid robotic systems heavily rely on applying force sensors and monitoring systems for interaction with humans. However, failure of such systems can be catastrophic, including harm and loss of life [19]. Inherent compliance match to the human body in soft robotic systems makes them among the best options for safe interaction with living systems. It is possible to engineer the structure of soft robotic systems such that the contact forces will always remain in a safe range in case of accidental collision with humans. Thus, it is possible to use soft robotic systems in a variety of applications, including medical [20,21,22,23], assistive technology [24,25,26,27], and search and rescue missions [28,29,30,31,32].

It is possible to break up safety compliance in robotic systems into two categories: active and passive. Active compliance in rigid robots utilizes sensory feedback force and torque data to control the position of the end-effector [33]. Feedback systems can be more complicated and require a rapid real-time response, limiting their application. On the other hand, passive compliance is achievable by integrating soft compliance joints and links in the structure of the robot. Passive systems can deform under external forces, without requiring any sensory data or control system, to a safe limit for human–robot interactions.

However, passive-compliant systems suffer from low accuracy and complex dynamics. Most soft robotic systems include serial or hybrid structures, which limits their blocking force and accuracy. Soft parallel robots can address these shortcomings [19,34]. The research area of soft parallel robots is relatively new. Amiri Moghadam et al., for the first time in 2015, used the term “soft parallel robot” in their work on developing a 2-DOF soft parallel robot equipped with electroactive polymer actuators [35].

Since the field of soft parallel robots is a newly evolving discipline, it is important to categorize existing soft parallel robots to compare and understand their working principles better. One effective way to categorize these robots is based on the application of soft joints, links, or both soft joints and links.

We have utilized both soft links, and joints in 2-, 3-, and 6-DOF soft parallel robots [19,35,36]. There are several contributions for soft-link-only parallel robots [37,38,39,40,41,42,43,44,45,46]. Yang et al. have demonstrated the design and modeling of a soft-joint-only parallel robot inspired by the structure of delta robots [47].

In the current work, we propose a novel 3-DOF soft parallel robot that can move in the x, y, and z directions. The proposed soft robot is analogous to the rigid delta robot consisting of three soft active links connected to passive links and the robot platforms through soft joints. However, the very small scale and mass of the robot prototype and its soft links make it highly unlikely to injure people during operation as a medical device.

Compliant mechanisms utilize flexible hinges or flexible members to induce relative motion between neighboring links. Such mechanisms offer several advantages over rigid joint connections, including minimal friction loss, superior performance, and no need for assembly [48,49,50]. Additionally, it is possible to substantially increase a robot’s workspace if compliant parts such as flexures or links are integrated into a soft robot’s design [19].

The remainder of this paper is structured as follows. Section 2 details the design and fabrication of the proposed soft robot. Section 3 presents the modeling of the robot, encompassing both kinematics and dynamics. Finally, Section 4 provides the conclusions and future works.

2. Design and Fabrication

Rigid delta robots have wide adoption in systems requiring high-speed manipulation, such as pick and place tasks, using a pure axis of coordinate translation while preventing top plate rotation. Soft delta robots are designed by replacing the traditional revolute joints with their equivalent soft joints and replacing the active base links with soft actuators, as illustrated in Figure 1. thus allowing for increased workspace movement since the soft joints undergo large displacements when subjected to loading.

In contrast, revolute or pinned joints on the lower rigid links tend to exhibit limitations. Furthermore, designing and fabricating soft delta robots as a single piece is feasible, minimizing the number of parts and accessories while reducing manufacturing and assembly time.

The soft delta robot incorporates an active soft link connected to a passive upper compliant four-bar (CFB) arms through a flexure hinge called the single-soft link compliant four-bar arm (SL-CFB). The soft actuator links consist of a fin-like beam that allows for rotation and bending, emulating the motion of rigid bottom links. The single-piece CFB arm integrates flexure hinges, enabling side-to-side translation relative to the soft links. A servo motor with a pulley wheel that can pull a string in two directions actuates the tendon-driven soft links. The soft links bend inwards towards the center of the base with clockwise servo motor rotation. Conversely, rotating the servo contour-clockwise allows the soft links to bend outwards. Figure 2a demonstrates the bending configuration when the soft links are deflected outwards compared to the rotation on the rigid delta robot. Figure 2b shows the upper link lateral movement configuration with the upper link’s translation to the left versus the rotation to the left on the rigid delta robot upper links.

The main components of the soft delta robot are three SL-CFBs, a top plate, a bottom platform, three 20 kg servo motors (20 kg· cm of torque) to actuate the soft links through wires, wire string, and an embedded Arduino-based controller for the servo motor. Each soft link is encircled and separated by 120° on a triangular base, as depicted in Figure 3.

A single SL-CFB consists of an 85 mm long soft link and 163 mm long compliant four-bar arm with an overall length of 248 mm connected to the base and top plate. These dimensions are within the printing workspace of the 3D printer we used for this prototype. The upper compliant four-bar arms are 3D-printed using overture thermoplastic polyurethane (TPU) filament at a 40% infill density. The soft links utilize NinjaFlex 3D printing and TPU filaments at 100% infill. The combination of the TPU and NinjaFlex materials provide the required flexibility for this design. The selection of infills is a trial-and-error process, with a tradeoff between the required flexibility for the soft design and the solidity to make it controllable.

The top plate, base platform, and servo mounts are 3D-printed using polylactic acid (PLA) filament. PLA provides a firm basis for the support of flexible delta robot links. The soft delta robot is controlled and programmed using the embedded controller with pulse-width modulation to operate the servo motor given a set trajectory. Each servo motor rotates a pulley wheel with two strings attached to the soft links via a 3D-printed anchor piece for optimal attachment. The experimental data are collected from the prototype soft delta robot using various sensors, including a TrakSTAR 6-degree-of-freedom position sensor. Figure 4 depicts the experimental setup. Figure 5 shows the working principle of tendon-driven actuators. These actuators consist of a soft backbone that can bend both clockwise and counterclockwise by pulling the tendons on either side using a servomotor.

3. Modeling the Kinematics

We used two different modeling techniques for the kinematics of the soft robot. The first model is an analytical kinematic model. The second models is a numerical kinematic model that is developed in MATLAB Simscape. Since the robot incorporates soft base links and a compliant four-bar linkage designed as a single-piece arm, analytical modeling may not be accurate. MATLAB Simscape provides a platform for accurate modeling. Regulating the position of the top plate employs inverse kinematics for well-defined trajectories.

3.1. Analytic Kinematics

Kinematic analysis is necessary when designing robotic systems to provide a mapping that relates the end-effector position to the displacement of robot joints. Inverse and forward kinematics differ in the mapping direction. It is possible to obtain the soft robot’s kinematics model assuming constant curvature for the motion of the 3D-printed tendon-driven actuators [2,24]. Afterward, assigning the proper frames to the soft robot platforms is possible. Figure 6 explains the kinematics model of the soft delta robot derived as follows.

$$\begin{array}{c}\hfill \left\{B_i^B\right\}+\left\{L_i^B\right\}+\left\{l_i^B\right\}=\left\{P_P^B\right\}+\left[R_P^B\right]\left\{P_i^P\right\}\\ \hfill i=1,2,3.\end{array}$$

(1)

$B_i^B$ is the position of the vertex of the robot’s fixed platform, $L_i^B$ is the position of the soft actuators, $l_i^B$ is the position of the passive links, $P_P^B$ is the position of the end-effector, $\left[R_P^B\right]$ is the rotation matrix, and $P_i^P$ is the positions of the vertices of the moving platform. Since the robot has 3 DOFs (x, y, z), the rotation matrix R is a unit matrix. The following equations provide the positions of the soft actuators:

$$\begin{array}{c}\hfill \left\{L_1^B\right\}={\displaystyle \frac{L}{{\theta }_1}}{[0cos{\theta }_1-1sin{\theta }_1]}^T\\ \hfill \left\{L_2^B\right\}={\displaystyle \frac{L}{{\theta }_2}}[{\displaystyle \frac{\sqrt{3}(1-cos{\theta }_2)}{2}}{\displaystyle \frac{1-cos{\theta }_2}{2}}sin{\theta }_2{]}^T\\ \hfill \left\{L_3^B\right\}={\displaystyle \frac{L}{{\theta }_3}}[{\displaystyle \frac{\sqrt{3}(1-cos{\theta }_3)}{2}}{\displaystyle \frac{1-cos{\theta }_3}{2}}sin{\theta }_3{]}^T\end{array}$$

(2)

L is the length of the soft actuators and ${\theta }_i$ is their bending angles. It is possible to rewrite Equation (1) as follows when considering the constant length of the passive links:

$$\begin{array}{c}\hfill l_i=\parallel l_i^B\parallel =\parallel \left\{P_P^B\right\}+\left[R_P^B\right]\left\{P_i^P\right\}-\left\{B_i^B\right\}-\left\{L_i^B\right\}\parallel \\ \hfill i=1,2,3.\end{array}$$

(3)

It is beneficial to square the equation to avoid the square root in the norms of Equation (3).

$$\begin{array}{c}\hfill l_i^2=\parallel l_i^B{\parallel }^2=l_{ix}^2+l_{iy}^2+l_{iz}^2\\ \hfill i=1,2,3.\end{array}$$

(4)

Finally, solving Equation (4) numerically provides the value of the required bending angle in the active links for a given position of the robot end-effector.

We performed a workspace analysis using this developed kinematics model. A robot’s workspace defines all the reachable points of the space for the robot end-effector, which establishes the applicability of the robot for a given task. The workspace helps in optimizing a robot’s structure using the kinematic model to satisfy specific needs. Varying the robot’s end-effector height from 90 mm to 160 mm and recording the maximum circular trajectory reach of the soft robot at each step aided in acquiring the robot workspace.

Figure 7 shows the workspace of the soft robot. It should be noted that the robot can follow any trajectory within the identified workspace. The choice of a circular trajectory is an arbitrary choice to show this capability within the feasible workspace.

The soft robot simulation demonstrated the application of the kinematic model for horizontal and vertical trajectories before experimentally validating the results. Figure 8 shows the simulation results of the kinematics model for a horizontal trajectory and the required bending angles of the soft actuators.

Figure 9 depicts the simulation results for a vertical trajectory with the necessary bending angles of the soft actuators. The simulation results portray the kinematic model’s effectiveness in predicting the soft robot’s motion in different trajectories.

3.2. Numerical Kinematics

Simscape modeling is a low-fidelity technique that is beneficial due to its ease of design through the Simulink library blocks and CAD files. The SDOF soft link is created by alternating extruded solid blocks and revolute joint blocks oriented to allow inwards and outwards link curling, as seen in Figure 10a.

Figure 10b shows the compliant four-bar arm made of extruded solids for the sides, revolute joints for the corners, and a gimbal joint that connects the rectangular piece to the tip. A pulley system and a motor provide the required tension in either direction for soft-link actuation.

The wire originates from a revolving cylindrical solid acting as the motor connected to each element of the flexible link using cylindrical solids and pulley blocks. Finally, the design ends at the last segment with a belt cable end block, illustrated in Figure 10a. Figure 10c depicts a single arm. Each cylindrical solid connects to a non-actuated revolute joint. The design includes three SL-CFB arms and motors connected to the base at the corresponding positions. The revolute joints representing the motors record the position or rotation angle as a time function, acting as sensors. Figure 11 illustrates the sensor data exported to the workspace in Simscape.

Actuating the soft robot is possible in two ways in the simulation. Like the physical robot, it is possible to actuate the motors by actuating the revolute joints connected to each soft link. Separately, it is possible to actuate the 6-DOF block attached to the center of the top plate. The torque is “automatically computed”, and the actuation is “provided by input”. A ramp signal provides the rotation angle, where a positive number moves the soft-link curl outward, and a negative number pulls the link inward. The simulation environment records the coordinates of the top plate in Simscape for experimental comparison.

Alternatively, to actuate via the 6-DOF joint connected to the top plate, the path of the top plate is generated for any desired trajectory and supplied as an input for the corresponding x, y, and z prismatic primitives. Through the simulation, obtaining the necessary angular displacements of the motor angles is essential to create the desired trajectory for the top plate.

We performed a validation and trajectory control to test the developed models. Experimental validation for the simulated soft robot confirmed the contributed findings. Figure 12 shows the validation of the Simscape model through experimental data.

Figure 13 and Figure 14 provide the comparison results of the Simscape and experimental system responses for circular and spatial motion, respectively. The experimental setup includes the 3D-printed prototype and an NDI Aurora brand 6-DOF position sensor attached to the center of the top plate to record the position. This means that the rigid body, object, or system can move and rotate independently along all six axes in 3D space. This high degree of freedom enables tracking of complex movements and orientations, which is crucial for navigation inside the body. For instance, a robotic arm with six degrees of freedom can achieve a broader range of positions and orientations compared to systems with fewer degrees of freedom. The collected data processing utilizes MATLAB 9.13 with the NDI (ndigital.com) software to compare to the simulation model. Before running the experimental setup, sensor calibration was necessary to capture more accurate transmitter locations.

The validation results include bending the soft links outward by 68° in the same direction, ensuring the plate moves up and down while staying parallel to the surface without tilting. The same actuation input was applied to the Simscape model and recorded for top-plate-coordinate comparison. The Simscape model responded the same as the prototype.

After the numerical validation, planar circular and 3D helical motion trajectories are created based on Equation (5). The coordinates are provided as input displacements to the top plate’s center to extract the base links’ corresponding angular deflections from the Simscape model.

$$\begin{array}{c}\hfill x=r\times cos\left(\theta \right),y=r\times sin\left(\theta \right),z=z_{initial}\\ \hfill x\left(t\right)=sin\left(t\right)\times cos\left(\omega t\right)\\ \hfill y\left(t\right)=sin\left(t\right)\times cos\left(\omega t\right)\\ \hfill z\left(t\right)=cos\left(t\right)+z_{initial}\end{array}$$

(5)

The top plate can follow any desired trajectory through the deflections of the flexure hinges of the compliant four-bar arm only, without the requirement of bending on the soft links in Simscape. However, the physical system actuation is only possible using the motors to bend the soft links outwards or inwards. Overcoming this limitation is possible by initial lowering of the top platform in the simulation to permit further desired motion via the deformation of the soft links, specifically for circular motion, since a helical trajectory already possesses vertical displacement.

Afterward, validation is possible by applying the angular displacements obtained from the Simscape as inputs for the motors actuating in the physical testbed. Figure 10 and Figure 11 provide the tendon results. The prototype follows the same trajectories extracted from the Simscape model with a slight deviation. One of the reasons might be how the 6-DOF position sensor was attached to the plate. This is due to the fact that the attachment of the sensor to the plate is subject to some minor misalignment from the ideal location and orientation used for experiments.

4. Soft Robot Position Control

This section addresses the challenges of controlling soft robots in the real world and experimental environments to gather results. It is difficult to design a controller subject to uncertainties such as materials, geometry, and input loading [51,52,53,54]. The soft parallel manipulator in this study has an uncertain Jacobian that makes traditional control methods challenging to implement. Furthermore, simulations often need to be faster to run in real time, limiting their usefulness for control purposes.

To overcome these challenges, the experimental testbed utilizes a data-driven approach to perform inverse kinematics. Instead of relying on a Jacobian, the joint changes based on the inverse kinematics of the real-world position are estimated compared to the desired position. The experiments implement two test methods: a neural network approach and a k-nearest neighbors (KNN) regression approach.

4.1. Data-Driven Approaches in Control

Data-driven approaches have a wide range of applications in control systems engineering [55,56,57,58]. In robotic systems, inverse kinematics is a critical problem, particularly for soft manipulators subject to nonlinearity, elasticity, and other complex factors. Recent research has explored using data-driven methods to learn accurate and nonlinearity-compensated inverse kinematics models for these manipulators. One such approach is k-nearest neighbors regression (KNNR), which can be an efficient and effective alternative to numerical solutions [59,60,61].

In a study by Zhang et al., KNNR was applied to the problem of learning inverse kinematics for a tendon-driven serpentine manipulator (TSM) [62]. The high nonlinearities caused by cable tensioning, elastic elements, and interaction of adjacent joints in the serpentine structure make TSM a particularly challenging platform for inverse kinematics modeling. Zhang et al. found that KNNR outperformed other methods, including Gaussian mixture regression and extreme learning machine, with the lowest root mean square error in tracking experiments [63].

The use of KNNR in robotics has been studied in multiple studies in the past. For example, Fuppeteer, a wearable 6-DOF display for delivering fingertip tactile cues, required inverse kinematics interpolation to achieve positions and orientations in the continuous workspace of the device [64]. Different interpolation methods, including KNNR, were evaluated on 1000 random poses. The KNNR-method-estimated leg lengths as a weighted average of k-nearest neighbors demonstrated its potential for data-driven kinematics of robots.

Neural network-based approaches have become increasingly popular for a wide range of engineering applications [65,66,67], including robotic control, by compensating for unknown variables [68,69,70,71]. Zhang et al. employed a neural network model to enhance robot position control using force information [63]. The approach used multiple factors such as movement trajectory, speed, shape, length, and material as input data for the neural network model.

Shao et al. proposed an improved neural network adaptive control (INNAC) method to regulate the bending angle of each finger of a hand rehabilitation robot driven by flexible pneumatic muscles [72]. The application exhibited a better control effect and stability than single neuron network adaptive control (SNNAC). Bamgbose et al. developed a data-driven approach to controller design for autonomous systems using a neural network-optimized control scheme based on samples from test navigation, which did not require precise models of the plant [73]. The implementation was effective in terms of faster transient response and overall error minimization in two case studies, one involving mobile robot motion control and the other involving the pitch and yaw angle control of a 2-DOF helicopter.

Majumdar et al. synthesized a neural network-based gain-scheduled proportional–integral–derivative (PID) controller to regulate the position of a pneumatic artificial muscle [74]. The proposed controller exhibited superior performance to classical PID controllers, with fast convergence, minimum oscillations, and reduced steady-state error. Ding et al. presented a neural network-based hybrid position and force-tracking control strategy for a robot model [75]. The proposed control scheme guaranteed both position-tracking accuracy and force-tracking capability, making it suitable for industrial scenarios that require precise force control and frequent force switching.

Wang et al. studied a control strategy for a planar underactuated manipulator system based on a wavelet neural network (WNN) model to achieve the position control objective of the system [76]. The study found that the proposed control scheme effectively achieved the position control objective of the system.

Overall, these studies highlight the potential of KNNR and neural networks for solving inverse kinematics problems in robotics, particularly for soft manipulators with high nonlinearities. By leveraging data-driven methods, KNNR and NN offer efficient and effective alternatives to traditional analytical kinematics models, enabling more precise and accurate motion control for these complex systems.

The present study employed a fundamental neural network control strategy to achieve position control. The network we employed, shown in Figure 15, consisted of a three-input and three-output model with three hidden layers, containing 9, 27, and 9 nodes, respectively.

The inputs are the Cartesian coordinates (x, y, z) of the moving platform with respect to the home position and the outputs are the three motor angles. To train the network, we utilized a comprehensive dataset comprising 226,983 observations of the correlation between motor degrees and position. Additionally, we employed the k-nearest neighbors (KNN) algorithm using the same dataset. Although the KNN algorithm does not require explicit training, we still had to tune the k-value, i.e., the number of neighbors involved in the estimation of motor positions. Our KNN algorithm uses a k-value of 15.

4.2. Closed-Loop Position Control

In traditional control loops, the Jacobian matrix helps relate the end-effector’s velocities to the joints’ velocities [77]. However, the uncertain Jacobian makes traditional control methods challenging for soft parallel manipulators like the one studied in this research [78,79]. Moreover, the analytical model used for simulations lacks real-time performance, rendering it unsuitable for the control loop. Hence, a data-driven approach to perform inverse kinematics (IK) and estimate joint changes facilitates the implementation.

A control loop implementation aids in achieving real-time control for the soft parallel manipulator, as shown in Figure 16. The control loop takes in the target position and feeds it into an inverse kinematics (IK) solver to calculate the desired angles for the servos.

The desired angles are then compared to the expected angles based on the real-world position obtained from the 6-DOF sensor, producing an angle error. The angle error is updated with a proportional–integral–derivative (PID) controller to estimate the required servo movements to compensate for the position error. This approach overcomes the challenges associated with the uncertain Jacobian of the manipulator while also providing real-time performance.

4.3. Experimental Position Control Results

This article presents a series of open- and closed-loop control experiments to evaluate the soft parallel manipulator’s performance. The open-loop tests implement three approaches: the kinematic model assuming circular curvature, KNN regression, and neural network. Figure 17a–c depict the results, where the red trajectory represents the target trajectory, and the blue trajectory represents the actual position recorded by the 6-DOF sensor.

It is important to note that all measurements utilizing the kinematic model had the poorest performance among the open-loop tests, with a root-mean-square error (RMSE) of 1.46. Optimizing the parameters for the specific trajectory can improve the performance of the kinematic model. In contrast, the neural network controller and the KNN regression controller performed similarly, producing RMSE values of 0.16 and 0.19, respectively.

Figure 17d,e show the KNN regression and neural network results in the closed-loop (PID controller) experiment tests. Both controllers produced an RMSE of 0.08, showcasing the effectiveness of the data-driven approach in compensating for the uncertain Jacobian and improving the control of the soft parallel manipulator.

5. Conclusions

This paper reports on the design, modeling, and fabrication of a novel 3-DOF soft parallel robot analogous to a rigid delta robot. The robot consists of three closed kinematic chains connecting the top moving platform to the bottom through soft active/passive links and joints. The kinematics model of the robot is derived based on the constant curvature assumption and used to obtain its typical workspace. The robot’s structure was 3D-printed using TPU and NinjaFlex and actuated through three tendon-driven soft actuators.

The kinematic models showed that the robot could move in arbitrary 3D trajectories within its workspace. The model of the soft robot is simulated using Simscape. Simulating the motion of the soft links includes using discrete sets of elements connected using rotary joints and springs in the modeling. Comparing the experimental data and simulation results indicates the effectiveness of the design and Simscape model. Future work includes robust position control of the soft robot and more optimal soft robots with six DOFs.