Soft Robot Kinematic Control Via Manipulability-Aware Redundancy Resolution Free

Soft robots have garnered increasing attention within the robotics community due to their potential in applications requiring distinctive attributes such as dexterity and safe interaction with the environment [1–3]. Despite these promising benefits, the control of soft robots remains challenging due to their continuum and elastic nature, a characteristic absent in rigid robots. Current methods, whether based on geometrically exact models or data-driven approaches, face significant limitations in achieving effective control of soft robots [4]. Specifically, model-based approaches, including Cosserat rod theory [5–8] and finite element methods [9], provide accurate models but are computationally intensive and not well-suited for real-time applications. On the other hand, data-driven approaches offer faster real-time control but require extensive data collection and significant training, and generally lack generalization and system stability.

Simplified computational models like piece-wise constant curvature (PCC) [10–12] and piece-wise variable curvature (PVC) [13] are able to balance computational efficiency and accuracy [4,14]. However, one of the primary challenges in PCC and PVC methods is the curve parameter representation, which has been criticized for its inherent kinematic singularities and discontinuities [15]. Parametrization singularities occur when certain parameters approach infinity (e.g., radius of curvature) or zero (e.g., curvature). This leads to incorrect values for other system parameters and eventually affect the system models, including forward kinematics, differential kinematics, and dynamic terms [4,16]. This parametrization singularity could cause critical problems in practical implementations, such as sudden motion jumps, oscillatory motion, and system instability [16]. While some parametrization methods have been proposed to resolve these issues [17,18], their focus has been limited to addressing expression singularities, i.e., mathematical inconsistencies in the kinematic model. However, these methods do not account for physical singularities that arise during control. Physical singularities occur when the robot encounters specific configurations (e.g., straight postures) where it is unable to generate the required movements to transition between poses. This gap underscores the need for approaches that address both parametrization and physical singularities in soft robotic control strategies.

In model-based kinematic control of soft robots, the resolved-rate algorithm is a common approach for mapping desired task space movements to actuator movements. Early implementations for position control relied on the PCC assumption [19]. To handle joint limits, later approaches incorporated null space projection within the resolved-rate algorithm using redundancy resolution [20]. However, these methods were evaluated only in simplified application scenarios [19] or open-loop simulations [13], and they did not address the issue of expression singularity in control. A more comprehensive study [21] extended inverse kinematic control from pure position control to pose control for a redundant soft robotic arm. This work replaced the use of null space projection for joint limit avoidance [13] with a weighted Jacobian approach, drawing inspiration from traditional methods in redundant manipulator control [22]. To address expression singularity issues, two heuristics were employed: (i) numerical calculation of the Jacobian using the central difference method and (ii) approximation of the singular Jacobian with a nearby nonsingular configuration [21]. However, the first method is computationally intensive, and the second suffers from a lack of accuracy. Although the approach was formulated as a combination of feedback (closed-loop) and feed-forward control, only the feed-forward component was implemented, and the evaluations of control success rates within the robot’s workspace were not detailed. Additionally, the orientation control was restricted to two rotational axes rather than achieving full orientation control [21].

To overcome the expression singularity and achieve full orientation control, Godage et al. [17] proposed the use of modal shape functions. In this approach, the curve parameters were approximated using Taylor series to address the expression singularity. In this study, the inverse kinematic formulation was examined through a global optimization approach, with the cost function defined as the  norm errors of position and orientation, utilizing matlab’s fmincon solver. One of the limitations of this method is its reliance on a global optimization-based control strategy, which significantly limits its real-time applicability due to high computational demands. Additionally, this approach cannot be implemented in closed-loop control because it does not address whether the robot can transition between poses in specific configurations. As we demonstrate later, the robot exhibits limited ability in certain configurations to generate movement and achieve specific target poses.

Unlike expression singularities, which can be identified through explicit mathematical representations, determining when a redundant robot, especially a soft redundant robot, approaches physical or near-singular configurations is far more challenging. This difficulty persists even when sophisticated parametrization are employed to circumvent expression singularities. To address this issue, the concept of manipulability was introduced in robotics [23]. In the rigid robot community, this measure has often been integrated as a secondary objective within redundancy resolution frameworks [24], aiming to locally maximize dexterity and avoid singularities. However, simply using manipulability in redundancy resolution does not guarantee a monotonic improvement, and the robot may still encounter singularities [25]. In response to these limitations, alternative approaches have been explored, including dynamic neural networks [26], Riemannian metrics [27], and control barrier functions [28]. Yet, these methods often require computing higher-order derivatives, such as Hessians, which significantly increase computational complexity. This challenge is particularly pronounced in soft robotic systems, where the complex nature of the models makes Hessian calculations computationally expensive and impractical for real-time control. Moreover, despite extensive work on kinematic modeling, the use of manipulability concepts to guide control in soft robotics remains largely under-explored. Existing efforts, such as Ref. [29], have primarily focused on characterizing manipulability, while recent studies, like Ref. [30], have examined manipulability in relation to workspace reachability and dexterity. However, these studies did not utilize manipulability for the control of soft robots.

In this article, we propose an effective pose kinematic control framework for real-time implementation based on the PCC assumption. We extend the kinematic model from our previous work, transitioning from a one-section, 3-degree-of-freedom (DoF) robot to a three-section, 9-DoF robot [31,32]. This model incorporates both Taylor series approximations and the recursive differential kinematics from Ref. [17], resulting in a singularity-free expression and computationally efficient formulation. Building on this foundation, we introduce a novel resolved-rate control approach with the following contributions:

• We provide a deeper understanding of coupled position and orientation control from the perspective of manipulability, highlighting the challenges in soft robotics where the robot’s unique structure and motion constraints make the full pose control difficult, particularly near-singular configurations.

• We propose a redundancy resolution algorithm that prioritizes position control over orientation control. This approach emphasizes the importance of decoupling position and orientation to improve the success rate and motion smoothness of soft robotic manipulators.

• To address the physical singularities (e.g., inability to twist in straight postures), we leverage manipulability measures, enabling the controller to predict and avoid these near-singular configurations through a gain-scheduled algorithm. This ensures not only a mathematically singularity-free model but also a physically manipulability-aware control strategy capable of providing stable closed-loop behavior in real-time.

The remainder of the article is organized as follows. Section 2 presents an overview of the system hardware and mechatronics. Section 3 reviews the kinematic modeling, modal shape representation, and recursive formulation. In Sec. 4, we highlight our proposed control algorithm. Sections 5 and 6 present the simulation and experimental results, respectively. In Sec. 7, we discuss our simulation and experimental results in detail, highlighting the novelty, performance, and advantages of our approach. Finally, the article concludes in Sec. 8.

The soft robotic design featured in this study, consisting of three sections, is an advancement of our previous work on a single-section design [32]. In this part, we describe its mechanical design and mechatronic components for feedback control.

The soft robotic arm in this article is divided into proximal, medial, and distal sections, as depicted in Fig. 1(a). Each section is equipped with three evenly spaced actuation units around its circumference. For a larger stiffness, dual actuators are employed in each actuation unit of the proximal and medial sections. The length of these actuation units changes with respect to internal pressure. All actuators consist of a 15 cm silicone tube that has an inner radius of 6 mm and a wall thickness of 2.5 mm (4G-60518-latex tube, Feelers), housed within a 45 cm braided sleeve with a 5/8 in. width (B00ZCNTIRQ, Electriduct). The ends are connected to 3D-printed pneumatic adapters and plugs with clamps. Sacrificial tubing is placed between the silicone and clamps to prevent damage to the silicone from clamping. To efficiently place the actuators’ inlets in our robot, each section is oriented at a  deg angle to its adjacent sections. The design also includes intermediate plates to facilitate a constant curvature. Each actuator’s elongation is limited by  cm, as exceeding this length under high pressure will cause the silicone to tear.

The mechatronic setup, as depicted in Fig. 2, comprises nine proportional pressure regulators (ITV 1031-21N2BL4, SMC Corporation) with built-in pressure sensors. These regulators are connected to a compressor that generates an output pressure of up to 70 psi to pressurize the actuators. The control algorithm is implemented on a Simulink xPC Target machine, which is connected to an analog output board (DAC6703, National Instruments) that provides a 0–5 V signal to control the pressure regulators. The encoders’ outputs are measured with a 32-bit counter PCI (CNT32-8 M, Contec), and the system operates at 100 Hz sampling rate for control and data acquisition. The actuators’ elongation is measured by custom-designed rotary encoders, as detailed in Ref. [32]. Independent measurements for each actuation unit are achieved by decoupling the deformations using Bowden cables, which isolate elongation readings from movements in other sections. Additionally, the end-effector’s pose is measured using an optical camera (MicronTracker, Claronav, Toronto, Canada), operating at a frequency of 16 Hz. This measurement is sent to the xPC Target machine using User Datagram Protocol (UDP) communication and is integrated into the closed-loop control to provide precise feedback for position and orientation, compensating for uncertainties in the forward kinematics.

This section briefly revisits the forward and differential kinematics of the soft arm based on the PCC assumption [31,32]. We then review the recursive formulation to enhance the numerical calculation speed. Readers are encouraged to read Ref. [17] for a detailed derivation. Here we refer to the three sections as section ⁠, ⁠, and ⁠, from the proximal end to the distal tip, as depicted in Fig. 1(a). A local coordinate frame is assigned to section  by aligning axis  toward the first actuation unit and axis  with the tangent direction of the section’s centerline intersecting the base plate of that section. The local frame of section  is used as the robot’s base frame.

To describe the soft robot arm pose in the task space, two independent mappings are required: (i) from actuator space to configuration space and (ii) from configuration space to task space. Actuator space variables include the length changes, denoted as  for the th actuation unit in section ⁠. The initial length of all actuators is ⁠. The actuation units within each section are positioned  deg apart from each other at a radius of  from the section’s centerline. The actuation vector is defined as ⁠, where  denotes the actuation vector of section ⁠, defined as ⁠. Additionally,  denotes all actuator variables from section 1 to section  (e.g., ⁠).

Note that the radius of curvature  in (1) is undefined when section  becomes straight. This issue causes numerical instability in the computation of kinematics and Jacobians around straight configurations [17]. To address this challenge, we will employ the Taylor series expansion with an order of  to approximate the transformation matrix of each section [17].

The forward and differential kinematics for a multisection soft robotic arm involve a substantial computational load. To address this, we draw inspiration from Ref. [17] and will reformulate the above equations into a recursive form.

Having established the differential kinematics, our next step is to develop the kinematic control for the desired end-effector pose. In Sec. 4.1, we begin by reviewing an optimization formulation that finds the minimal joint velocities guiding the robot toward the desired pose, leading to a standard resolved-rate control approach. This is followed by a discussion on defining a weight matrix to handle the robot’s joint limits [22]. Then, in Sec. 4.2.1, we highlight the challenges of coupled position and orientation control using conventional resolved-rate control for the first time, primarily due to the unique serial–parallel structure and the absence of a wrist in soft robotic arms. Subsequently, in Sec. 4.2.2, we formulate a new optimization problem to address this challenge by decoupling position and orientation control utilizing a redundancy resolution algorithm [20]. Next, in Sec. 4.3, we discuss the potential failures of decoupled position-orientation control and the motivation for incorporating manipulability awareness. We conclude our formulation in Sec. 4.4 by introducing a manipulability-aware, gain-scheduled redundancy resolution algorithm, and we provide a summary in Sec. 4.5.

Selecting  as the identity matrix simplifies this to the general pseudo-inverse kinematic solution ⁠, where ⁠. By determining the necessary ⁠, we can find the  for moving the robot to the desired configuration.

In this part, we will first highlight the challenges associated with coupled position and orientation control of the soft robot arm from the standpoint of robot manipulability. We will then detail our proposed control algorithm formulation to address this challenge.

The independent measures for position (⁠⁠) and orientation manipulability (⁠⁠), illustrated separately in Fig. 4 using  and ⁠, display more detailed characteristics. Although the distribution of the position manipulability measure remains heterogeneous, the range of manipulability variation is considerably narrower than that of the coupled and orientation manipulability. For instance, a comparison between the extremes (maximum over minimum) of coupled and position manipulability reveals a significant disparity, ranging from  to ⁠. This indicates that simultaneously controlling the position and orientation is more challenging than controlling the position only, as the robot may be subject to singularity. This observation suggests a reasonable explanation for why the pose control of the soft robot arm is significantly more difficult and prone to instability compared to that of traditional robots with wrist joints that provide direct orientation control.

To address the challenges mentioned above, we propose prioritizing position over orientation control. We leverage a redundancy resolution algorithm [20,35] to utilize the null space projection of the primary task (position control) to facilitate a secondary task (orientation control). Additionally, we integrate the weighted Jacobian concept from (16) into our formulation to handle joint limits.

While we have established a priority-based control approach, there exists a failure scenario when  approaches a singular state. This situation causes the singular values of  to decrease significantly, which leads to the divergence of its pseudo-inverse and  in (31). This singularity often occurs when the robot end-effector’s  axis aligns with the base frame’s  axis, as reflected by the orientation manipulability measure plot in Fig. 4, where most straight poses exhibit the lowest manipulability.

To improve stability, we integrate the manipulability measures into our algorithm. This enhancement allows for the early detection of potential singular configurations and facilitates a proactive adjustment. Predominantly affected by an orientation singularity, our strategy will pause position control when nearing singular configurations, focusing on orientation control excluding the  axis. Once the risk of singularity is reduced, we can revert to priority-based control. Considering the orientation manipulability distribution illustrated in Fig. 4(b), there is a significantly reduced likelihood of the robot encountering a singularity.

While the singularity issues have been previously addressed by employing manipulability awareness, the convergence rates between position and orientation may remain imbalanced, leading to a slow overall convergence rate. To accelerate convergence speed while maintaining numerical stability, we propose a gain-scheduled algorithm driven by four essential metrics: (i) position error, (ii) orientation error, (iii) position manipulability, and (iv) orientation manipulability. The algorithm switches the control mode and adjusts the orientation control gain based on these values.

Although it has been observed that singularities often arise with a reduction in orientation manipulability, our results indicate that incorporating position manipulability into our gain-scheduled algorithm can lead to implementations that are not only more efficient but also have a higher success rate. In the case of low position or orientation manipulability, we change our demands for the controller by switching to the two-axis orientation control mode, previously discussed in Sec. 4.3. The control mode is switched back to the priority-based control formulation in (31) once the robot is far away from the low manipulability configurations. The position and orientation errors are used so that if the robot approaches the desired pose, we can increase the effect of orientation control by amplifying ⁠. The detailed algorithm, along with the gains and thresholds used in our experiments, is presented in Algorithm 1. Note that  in line  is a user-defined parameter to tune the convergence speed and motion stability.

Our approach continuously monitors the robot’s manipulability—both in terms of position and orientation—to identify potential singularities or near-singular conditions. When low manipulability values are detected, the algorithm proactively switches from a full priority-based redundancy resolution to a restricted mode that excludes position control and focuses solely on orientation control around the  and  axes of the base frame. This adjustment is motivated by the robot’s inability to perform twisting motions due to the absence of a dedicated wrist joint.

The other key aspect of the algorithm involves adjusting convergence speed. The orientation control gain is dynamically modulated based on the position and orientation errors, ensuring that as the end-effector approaches the target pose, orientation control is incrementally intensified to accelerate convergence. Without this adjustment, even if the position converged to the target, the orientation would take significantly longer time to converge. Moreover, increasing the orientation control gain near singular configurations leads to more substantial changes in the robot’s configuration, thus improving manipulability and allowing the system to revert to the full priority-based redundancy resolution sooner. By integrating manipulability awareness, selective axis control informed by the robot’s structural limitations, and gain scheduling, our method provides a stable, generalizable, and efficient redundancy resolution strategy suitable for a wide range of kinematically redundany soft robotic systems.

In this section, we evaluate the performance of the proposed algorithm in comparison with other methods. We assess the algorithm’s effectiveness by measuring the success rate in obtaining feasible inverse kinematic solutions and the smoothness of the motion. The evaluation of smoothness includes analyzing the deviation from the intended path in both position and orientation upon reaching the desired pose. Path deviation is the difference between the actual path taken and the linear path. Similarly, orientation deviation is determined by the summation of the norms of incremental orientation changes, calculated using (15), and then subtracted from the ideal change, which is the norm of orientation error from initial to final pose as defined in (15).

In this section, we present a comparative analysis of kinematic control formulations for soft robot arms. Specifically, we discuss the standard weighted coupled control (WCC) method introduced in (12), as well as the coupled redundancy resolution (CRR) algorithm and the decoupled redundancy resolution (DRR) method.

We generated 100,000 random data points within the actuator space and computed the forward kinematics to establish a set of desired poses. We established the robot “home configuration,” where the actuators are close to their midpoint range (4 cm) with a small deviation. This is to avoid straight configurations, where the robot has low manipulability in orientation. The “home configuration,” distribution of desired poses, along with 10 random configurations, is depicted in Fig. 5. An attempt at inverse kinematic control was deemed successful if the robot achieved the desired position with an error of less than 1 mm and an orientation error of less than  deg within 4,000 iterations, equivalent to 40 s in real-time given a 0.01 s sampling rate. The results are summarized in Table 1.

From Table 1, it is evident that our proposed method achieves the highest success rate, with the least deviation from the intended path and orientation. To elucidate the superior behavior of our algorithm, we examine the significance of each component in various formulations by comparing them to other algorithms, specifically the significance of weighted Jacobian and priority-based control formulations. This comparison not only highlights the strengths and weaknesses of these methods but also demonstrates why our approach is superior.

Focusing first on the weighted approach, the use of a weighted Jacobian in the WCC algorithm for joint limit avoidance is more effective than in the CRR algorithm. This is because WCC’s weighted approach reduces the impact of actuators that approach their limits. In contrast, CRR addresses joint limits through null space projection, and only when this does not interfere with higher priority tasks. Thus, the weighted Jacobian consistently outperforms the null space projection method in terms of success rate performance. Turning to priority-based formulation, we analyze both CRR and DRR algorithms. In these approaches, joint limit avoidance is designated as the lowest priority task. The distinction lies in how each algorithm manages the control of position and orientation. DRR employs priority-based (or decoupled) control, which significantly enhances performance, resulting in better success rates and reduced path and orientation deviations.

Our approach synthesizes these insights, integrating both the weighted method for managing joint limits and priority-based control to significantly enhance the accuracy and smoothness of our algorithm. This strategy ensures that we reach the desired configuration while minimizing deviations from both the path and the target orientation, making our algorithm better suited for real-time applications. Moreover, it incorporates manipulability awareness, significantly improving the robot’s performance when the robot is close to singular configurations, highlighting the superiority of our approach.

To further investigate the performance of our proposed algorithm versus comparative approaches, we considered 10,000 additional targets with initial position errors  mm and orientation errors  deg. Additionally, the initial configuration for each desired target was randomly chosen. Based on the results, which are presented in Table 2, it is evident that our approach further underscores its superior performance and effectiveness in this challenging study.

In this section, we implement proposed and comparative algorithms on our soft robot hardware to investigate their performance in real-time implementations. In Sec. 6.1, we compare the performance of our proposed method against comparative approaches, highlighting the detrimental effects of high path and orientation deviations (as detailed in Tables 1 and 2) on convergence behavior and accuracy. The pose errors are calculated through a forward kinematics formulation with real-time measurements of actuator variables, isolating our implementation from kinematic uncertainties.

Next, in Sec. 6.2, we address the high deviations observed in comparative approaches by retuning their parameters for the experimental study. We enhance our implementation by incorporating real-time feedback from an external camera and continue calculating the Jacobians based on actuator variable measurements. To assess the performance of our algorithm, we compare the final pose errors to highlight accuracy and use the  norm of the pose error to demonstrate convergence speed. This experimental study further investigates the effects of kinematic uncertainty on the accuracy and convergence behavior of our system. In both studies, we reset the robot to a home configuration by applying predefined pressures to all actuators, followed by a three-second rest period to stabilize the robot before activating the closed-loop control. To ensure fairness, the gains for each algorithm were fixed after the initial tuning, as summarized in Table 3.