Soft Robotic Finger with Energy-Coupled Quadrastability

Abstract

The performance of the human finger is a significant inspiration for designing soft robotic fingers that can achieve high speed and high force or perform delicate and complex tasks. Existing soft grippers and actuators can be excellent in specific capabilities. However, it is still challenging for them to meet an all-around performance as the human finger, characterized by high actuation speed, wide grasping range, sensing ability, and gentle and high-load grasping capability. The proposed tendon pulley quadrastable (TPQ) finger has combined these qualities in the conducted gripping tasks. A pair of elastic tendons is utilized as the sole energy reservoir to create a novel energy distribution pattern: energy-coupled quadrastability. An energy model is built to analyze and predict the behaviors of the TPQ finger. Mechanical instability is utilized to enhance the actuation speed. The proposed soft lever mechanism endows the TPQ finger with sensing ability. The energy barrier adjusting plates control the energy barrier, adjusting the sensitivity of both active and passive actuation mechanisms. The transition of four stable states forms preplanned trajectories that are applied to create multiple grasping manners. Experiments show that it can respond to stimuli and finish a grasping task in merely 31 ms, and its payload can reach 33.25 kg. At the same time, it can also handle fragile objects such as a piece of rose and grasp a wide range of objects ranging from a thin nut (3.3 mm in height) or a thin card (0.76 mm thick) to a football (220 mm).

Introduction

The human hand has been the design paradigm of robotic soft grippers for long. Its dexterity and compliance enable it to handle complex tasks delicately, while it can also firmly lift heavy objects as well as be acute and fast in dynamic scenarios, such as catching a relay baton in a relay race. For robotic grippers, these qualities can be described as perception, dexterity, softness, actuation speed, and load tolerance. Progress has been made in specific areas of these qualities. For example, some grippers are designed with a gentle gripping capability, allowing them to manipulate extremely delicate objects such as sea creatures.^1–4 Some have highlighted reconfigurability as a typical feature.^5,6

Others focus on enhancing the manipulation strength through various means, such as fabrication method,⁷ material,⁸ or structures.^9–13 In terms of sensing ability, many researches use embedding sensors in soft actuators.^14–19 Recently, a passive sensing strategy leveraging mechanical stimuli to elicit motion has been explored,^20–23 resulting in a shorter response time compared with traditional sensors-based approaches.²⁰ Regarding adaptivity and dexterity, underactuated cable-driven hands have made significant progress, including the shape deposition manufacturing hand,²⁴ the iRobot-Harvard-Yale hand,²⁵ the Pisa/IIT soft hand,²⁶ and the commercialized shadow dexterous hand. Recent frontline research on bioinspired prosthetic hand had also made a significant progress toward the complexity and dexterity of human hand.^27–29

To enhance actuation speed of soft grippers, researchers have utilized mechanical instability³⁰ that can store and rapidly release energy when transforming between two stable states,^{20–22,31–33} and the minimal actuation time of these bistable grippers can reach an impressive 20 ms.²⁰ Nonetheless, grasping range and outputting force are still challenging for them. To pursue both high stiffness and safety of grasping, rigid soft coupled structure was applied by previous researchers;^10,13 a type of self-locking mechanism was applied by Guo et al.,¹⁰ reaching a payload of 9 kg. The hybrid design by Zhu et al.¹³ could reach a wide-range outputting force (payload 27 kg) with satisfactory tolerance of object dimension and deviation.

Due to the innate monostability of these grippers, actuation and response speed are still limited, and applications in high-speed scenarios are yet to be explored. Also, research on the grasping range in the height direction is still rare in the literature. Existing soft end effectors can be excellent in particular tasks, but there are still some gaps to fill to meet a considerably all-around performance of being soft, high-payload, fast, stimuli-perceptible, and with a wide grasping range in a single prototype. And this will be the goal throughout our design and experiments.

We proposed an energy distribution pattern for robotic finger prototypes with multistable feature: the energy-coupled quadrastability, which is formed by two bistable units that are coupled by the elastic potential energy of the system and determined by three geometrical parameters (the three finger joint angles). An energy model will be used to analyze and predict the behavior of the tendon pulley quadrastable (TPQ) finger. The transformation of the four stable states produces several preplanned trajectories, which the majority of grasping experiments are based on. The motion of the TPQ finger (joint angle, angular speed) will be analyzed with respect to stored elastic potential energy.

Controllable energy barrier will be realized by utilizing the proposed energy barrier adjusting (EBA) plates, and the effect of energy barrier on actuation mechanism sensitivity is statistically analyzed. Two types of working modes of the TPQ finger (quadrastable mode and cable-driven mode) will be explored in the experiments, with elastic tendons used in the quadrastable mode for high-speed and light-to-middle range outputting force scenarios and inextensible cables used in the cable-driven mode for high force scenarios. This article aims to achieve the following goals in the functionality of the robotic finger:

Grasping a broad range of objects in the dimensions of both width and height based on the preplanned trajectories.

Utilizing mechanical instability to pursue high actuation speed.

Endowing the TPQ finger with sensing ability with the proposed “soft lever” mechanism to create passive intelligence and enabling it to capture objects in dynamic scenarios on a millisecond scale.

Grasping extremely delicate objects based on an antagonistic effect of the tendon and soft unit.

Grasping a heavy object up to 33.25 kg using a triple-finger gripper in a cable-driven mode; in the quadrastable mode, reaching a payload of 6.25 kg.

Anatomical Inspiration of the TPQ Finger

The inspiration of TPQ finger stems from the anatomical structure of human finger: the flexor tendon pulley system, which provides fulcrums to elicit flexion and extension (Fig. 1a, b)^34,35 when flexor tendons are driven by muscles. Functional anatomical experiments show that among all the pulleys there are two most functionally important pulleys (A2 and A4 shown in Fig. 1) concerned with finger flexion,^36–38 and A2 was found to be the sole most important pulley of the two for flexor tendon function.³⁷

FIG. 1.

The anatomical inspiration and structural design of the TPQ finger. (a) The anatomical structure of human hand. Image courtesy of Complete Anatomy. Annular pulleys from A1 to A5 are highlighted, and A2 and A4 labeled in red are the most important pulleys concerned with joint motion.^36–38 (b) QFP. Taking the mechanical advantage of bowstringing effect for amplified moment arm of finger joints. (i) Finger bowstringing with ruptured pulleys A3, A4, and A5. Human finger is unable to fully flex when certain crucial pulleys are ruptured due to limited shortening length of muscle. (ii) The QFP leverages bowstringing to increase the mechanical advantage of force source while keeping the flexing ability using artificial tendon with adjustable shortening length. (c) The structure of the four-chamber soft unit. (d) The TPQ finger with two bistable units linked by a bidirectional pulley A2. QFP, quadrastable finger phalanxes; TPQ, tendon pulley quadrastable.

Loss of tendon pulleys can lead to bowstringing of fingers (Fig. 1b); this rare hand injury will reduce the bending ability of finger (i.e., reduce angular rotation of finger joint per linear distance of muscle contraction³⁹), and despite one's best efforts, a pulley-ruptured finger, as shown in Figure 1b-i, will be unable to achieve full flexion due to the limited shortening range of the flexor digitorum profundus.⁴⁰ While tendon bowstringing reduces finger's flexing ability, it significantly increases the moment arm about the finger joint and hence the mechanical advantage of muscles.³⁹ The proposed pulley system of the TPQ finger keeps the most functionally important pulley A2 and eliminates pulleys A1, A3, A4, A5 to artificially create the hand injury “finger bowstringing” that enhances the bending moment and the mechanical advantage of force source.

Structure and Actuation Mechanism of TPQ Finger

Structural design of TPQ finger

The TPQ finger is composed of two parallel quadrastable finger phalanxes (QFPs) working in the same pattern and a soft unit composed of four independent air chambers (Fig. 1c, d). The rigid phalanxes and soft unit are connected by four shafts. The QFP contains four stable states formed by two bistable units shown in Figure 1d-i, that is, the triangular bistable unit (TBU) and the quadrangular bistable unit (QBU). Pulley A2 connects the two bistable units and is composed of two bidirectional pulleys placed on the proximal phalanx (Fig. 1b-ii). Pulley A2 can provide fulcrums for all three finger joints to rotate clockwise or counterclockwise, depending on the direction of torque of tendon force with respect to the corresponding joint center.

All joints have two ridges as the end position for all rotating phalanxes, forming two stable states for the two bistable units when tension in tendons exist. The independent pressurization of T1, T2 and Q1, Q2 (Fig. 1c) enables the stable-state transition. The EBA plates are mounted on proximal and middle phalanxes (Fig. 1b) and are used to adjust ridge angles of joints 2 and 3, hence adjust the work required for traversing snap-through point from one side of the ridges. In the QBU, A node pulley is set on the distal phalanx and flexes joint 3 and joint 2 in a coupled manner. The tendons start from the wrist pulley, go through pulley A2 and node pulley, and are fastened at the fixed end.

Actuation mechanisms of TPQ finger

Active actuation mechanism

As the active actuation mechanism, the four air chambers of soft unit can be controlled to switch the stable states of each bistable unit when tendons are prestressed. Without tendon pretension, the soft unit works independently in a gentle gripping manner.

Passive actuation mechanism

As shown in Figure 2 and Supplementary Movie S1, the deflection produced by external stimuli in the force-sensing region is transformed into the rotation of the soft lever about the pivot (joint 3/shaft 3), which enables the distal phalanx to rotate and switches the two stable states in QBU sharply when tendon prestores energy.

FIG. 2.

The actuation mechanisms of TPQ finger. The active actuation mechanism leverages air pressure to create torques at finger joints. The passive actuation mechanism is elicited by external force applied at the force sensing region, transforming deflection into rotation.

Energy Distribution Pattern of the TPQ Finger

Energy calculation of TPQ finger

The energy of the TPQ finger consists of two components: the energy stored in the tendons (dominant) and strain energy of soft unit (nondominant): $E_{T P Q} = E_{T e n d o n} + E_{S o f t} .$ (1)

Following assumptions are made about the energy of TPQ finger:

(1)

Small friction in the tendon pulleys: neglecting the friction between the tendon, and the pulleys are regarded as “ideal pulleys” so that the tensions of tendon on each side of the pulleys are identical, and the longitude deformation of the tendon is uniform.

(2)

Simplifying the tendon as a spring with nonlinear force–elongation function relation, and the tension force and elongation are one-to-one correspondence, that is, $F = F (Δ l)$ .

One single tendon in the QFP comprises of several segments (segment A to E, Fig. 3a and Supplementary Fig. S1), whose lengths vary with the finger joint angle $α_{10}, α_{21}, α_{32}$ as the finger morphology changes. The variable $α_{i, j}$ represents the relative angle of phalanx i about phalanx j and counterclockwise direction is chosen as the positive direction. Based on assumptions (1) and (2), the energy stored in the tendon is calculated as the work required to elongate it. For the purpose of calculation, we consider stable state 1 as the reference point. Any change in the length of the tendon segments (designated as A to E in Fig. 3a) leads to a change in the total elongation length $Δ L$ and consequently the tension force $F (Δ l)$ . The total energy in tendon for a random finger configuration is the integral of the tension $F (Δ l)$ with respect to $Δ L$ :

FIG. 3.

Energy modeling of the TPQ finger. (a) Parameters and simplified model of the TPQ finger. The dimensions of the soft unit and finger phalanxes are provided in Supplementary Fig. S6 and Supplementary Table S1, respectively. All parts for TPQ finger are shown in Supplementary Fig. S7. (b) Finger morphology inspired stable states. (c) The “energy hexahedron” of the QFP. A four-dimensional plot depicting the energy distribution with respect to working space defined by joint angles, based on the material property of tendon sample 1 (Supplementary Fig. S12 and Supplementary Table S4), and under a pretension of 300 mm $(δ = 83.3 %)$ . (d) Energy distribution pattern of the QBU, considering the resisting effect of soft unit. (i) Two energy decreasing paths detouring the local energy peak. Adopting either of the two paths, rather than attaining the energy summit by having joints 2 and 3 at 0° simultaneously, will result in a lower energy expenditure to reach S2 and is done by having either joint 2 or joint 3 rotate first. As revealed by the coordinates, for energy decreasing path 1, joint 2 rotates first, after overcoming the boundary energy barrier $E_{B a r r, p a t h 1}$ , rotation of joint 3 starts. For energy decreasing path 2, joint 3 rotates first. Rotation of joint 2 starts after overcoming the boundary energy barrier $E_{B a r r, p a t h 2}$ . (ii) Two real situations reflecting the two energy decreasing paths. Path 1 was created by pushing the joint tip (joint 2 rotate first), and a pulsed air pressure of 60 kPa was used to create path 2 (joint 3 rotate first). The coupled rotation effect happened after the first-rotating joint create moment arm for the subrotating joint. QBU, quadrangular bistable unit.

E (α_{10}, α_{21}, α_{32}) = \int_{0}^{p + L (α_{10}, α_{21}, α_{32}) - L_{S 1}} F (Δ l) d (Δ l),

(2)

where p represents the length of pretension at stable state 1, $L (α_{10}, α_{21}, α_{32})$ is a geometry function specifying the relation between the total tendon length in the QFP and the joint angles. $L_{S 1}$ is the tendon length in QFP at stable state 1. The length function $L (α_{10}, α_{21}, α_{32})$ is derived in Supplementary Section 1.1. The energy of the soft unit is derived based on a tendon excursion test, as a transform approach (see Supplementary Section 1.2 and Supplementary Figs. S1–S4). The force–displacement function $F (Δ l)$ that represents material property of the tendon is a ninth order polynomial after the curve fitting (R² > 0.999, Supplementary Table S4 and Supplementary Fig. S5). Substituting $F (Δ l)$ into Equation (2), the energy of the tendon is acquired.

The correspondence of energy hexahedron and geometrical configuration

All energy distribution patterns shown in Figures 3 and 4 are derived from the proposed energy model. There are four total stable states in the QFP, and they correspond to four finger morphologies that are the ultimate positions of flexion or extension of finger joints (Fig. 3b): full extension (stable state 1), extrinsic-plus position³⁹ (stable state 2), intrinsic-plus position³⁹ (stable state 3), and full flexion (stable state 4). The four-dimensional “energy hexahedron” (Fig. 3c) illustrates the energy distribution pattern of the QFP related to its geometrical configuration (the three joint angles) and demonstrates the feature of energy-coupled quadrastability: changes of any joint angle or stable state of a single bistable unit alter the system energy in an overall way.

FIG. 4.

Analysis of stable-state position. (a, b) Effect of tendon pretension on the flexion angle of the TBU. Under small tendon pretension region $(p \leq 70 m m)$ , the resisting effect of soft unit significantly influences finger flexion and the stable-state angle, while the flexion angles keep constant under a large-pretension situation. Note S1 and S3 refer to positions of stable states 1 and 3. (c, d) Effect of tendon pretension on the flexion of the QBU. Similar results are shown in the QBU, except that larger pretension is required to fully flex joint 2 than that needed in TBU. Energy data of soft unit are acquired from a tendon excursion test (see Supplementary Section 1.2, Supplementary Figs. S3, S4, and Supplementary Table S2) originally used in a functional finger anatomy experiment (Supplementary Information).³⁸ TBU, triangular bistable unit.

The end faces of hexahedron correspond to the ultimate angles controlled by finger joint ridges (Fig. 3a). The four stable states are situated at the four corners of the hexahedron, and transitions between these states require overcoming specific energy barriers. The position of snap-through points (unstable points, that is, the colored slices in Fig. 3c) is obtained when the derivatives of energy with respect to $α_{10}, α_{21}, α_{32}$ equal to 0: $\frac{d E}{d α_{10}} = 0, α_{21} = α_{21}^{S 1}, α_{32} = α_{32}^{S 1},$ (3) $\frac{d E}{d α_{21}} = 0, α_{10} = α_{10}^{S 1}, α_{32} = α_{32}^{S 1},$ (4)

\frac{d E}{d α_{32}} = 0, α_{10} = α_{10}^{S 1}, α_{21} = α_{21}^{S 1},

(5)

where $α_{10}^{S 1}, α_{21}^{S 1}, α_{32}^{S 1}$ are the joint ultimate angles for joints 1, 2, 3 at stable state 1, respectively. The positions of the snap-through points are solely determined by geometric configuration of QFP and are independent of tendon parameters since the energy function $E (α_{10}, α_{21}, α_{32})$ is the function of joint angles, in which only one term, that is, the geometrical function $L (α_{10}, α_{21}, α_{32})$ , contains joint angle variables.

The coupled rotation phenomenon of finger joints in QBU

The TPQ robotic finger shows similar kinesiological pattern in flexion and extension as human finger, that is, the distal interphalangeal joint (Fig. 3b) and the proximal interphalangeal joint of a healthy hand rotate in a coupled manner and reach hyperflexion or hyperextension position simultaneously. The energy hexahedron (Fig. 3c) also demonstrates the coupled rotation phenomenon in the flexion process of QBU (from stable states 1 to 2 or from stable states 3 to 4). As one of the energy barriers in QBU is broken (the colored slices in Fig. 3c), that is, joint 2 or joint 3 crosses its snap-through point, and the other energy barrier in QBU becomes “barrier-free.” Its corresponding finger joint would cross its snap-through position automatically, which enables the system energy to decrease to stable state S2 or S4.

More intuitively, Figure 3d presents the system energy by setting $α_{10}$ to −15°. From S1 to S2, there are two energy decreasing paths go around the local energy peak. This consumes less energy for stable-state transition process compared with reaching the energy peak. We verified the existence of the two energy paths in two real situations. Energy decreasing path 1 was created by pushing the tip of QBU and energy decreasing path 2 was created by a pulsed air pressure of 60 kPa, as shown in Figure 3d-ii.

Analysis of stable-state position

The total energy of the TPQ finger is the superposition of the tendon elastic potential energy and the strain energy of soft unit (Fig. 4). The elongation of tendon leads the total energy to increase in a step-by-step manner, and the stable-state positions S2 and S3 are solely related to tendon pretension length. For all finger joints, the torques produced by tendon and crumpled soft unit are antagonistic.

Under small tendon deformation (tendon sample 1; $δ < 25 %$ ; initial length = 360 mm; select stable state 1 as the pretension reference point), the addition of soft unit shifts stable points S2 and S3 (local energy minimum) of the QFP significantly (Fig. 4a, c). Under large deformation $(δ > 40 %)$ , the impact of soft unit on the positions of stable states is virtually negligible. For the TBU, analysis result in Figure 4b shows that tendon under a strain of 42% (150 mm) can fully flex the finger joint 1 (QBU maintains extension state). With this pretension length, the soft unit's maximum energy only accounts for about 10% of the total energy. For the QBU, results in Figure 4d show that tendons with pretension length of 150 mm can fully flex joint 3 and reach a flexing rate of 83% (100°/120°) for joint 2 (TBU maintains extension state). The resisting effect of soft unit on changing stable point position can be completely neglected when the tendon deformation reaches 400 mm $(δ = 111.1 %)$ .

Analysis and Statistical Experiments

Trajectory, motion analysis, and morphology control

Internal and external trajectories

The profiles of two types of tip trajectories were captured by a high-speed camera, shown in Figure 5a and b (Supplementary Movie S2). The internal trajectory produced by actuating chamber Q1 sweeps three stable states in sequence: stable states 1, 2, and 4. Under a pretension length of 450 mm, it takes 32 ms to reach a tip velocity of more than 4 m/s (a position near stable state 2) and 98 ms to reach the fully flexing morphology (stable state 4). The external trajectory with a broader profile, produced by actuating chamber T1, sweeps stable states 1, 3, and 4 in sequence. Its acceleration process takes longer than the internal trajectory, with 163 ms to reach stable state 3 and 203 ms for full flexion. However, a larger tip velocity was acquired in the external trajectory, which is more than 6 m/s. The angles of EBA plates were all set at −1° for fast actuation, and the pneumatic signals for internal and external trajectories were 90 kPa with a time interval of 0.2 and 0.5 s, respectively. Residual oscillations of tip trajectory were chopped.

FIG. 5.

The preplanned basic trajectories, motion analysis, and morphology control. (a, b) Demonstration of the two basic trajectories in TPQ finger during flexion process. The internal trajectory is obtained by actuating the Q1 chamber with a 90 kPa pulsed pressure of 0.2 s, for external trajectory, the same magnitude of pulsed pressure was applied to actuate chamber T1, with a time interval of 0.5 s. (c, d) Obtaining tunable transition speed by tuning tendon elongation. Two independent motion analyses of the TBU and QBU are conducted under different tendon elongation; curves of tip-to-joint angle and angular speed about time are obtained by a high-speed camera after processing by video analysis tool (tracker 6.0.8). Theoretical energy decrease $δ E_{12}$ (S1 to S2), $δ E_{13}$ with respect to tendon pretension is put on (c, d) as inset plots. (e) Morphology control and trajectory preplanning by selectively actuating the chambers of soft unit.

Independent motion analysis of the TBU and the QBU

In the proposed motion analysis experiment, the flexing speed of TPQ finger during stable-state transition is solely controlled by tendon pretension length. The model value of energy decrease that is majorly transformed into dynamic energy (denoted as $δ E_{12}$ [S1 to S2] and $δ E_{13}$ in the inset of Fig. 5c, d) during stable-state transition process has a monotonic incremental function relation with tendon pretension length p. Theoretically, the motion of TPQ finger would be faster due to higher released energy as tendon pretension length increases.

This experiment took the angle and angular speed of fingertip about finger joint 1 (for TBU analysis) or 2 (for QBU analysis) as the variables representing finger motion, that is, $θ_{T J i}$ , $ω_{T J i}, i = 1, 2$ . Results in Figure 5c and d show that for both QBU and TBU the increases of tendon elongation length p lead to higher transition angular speed and shorter transition time. The tip-to-joint angle and speed were captured by a high-speed camera and analyzed using a video analysis and modeling tool Tracker (version 6.0.8); the whole experiment process and slow-motion clip are demonstrated in Supplementary Movie S2.

Morphology control and trajectory preplanning

As demonstrated in Figure 5e, the transition of finger morphologies can be directed toward preplanned tip trajectories during a unidirectional transformation process by selectively pressurizing one or two of the four air chambers. These trajectories are the segments of the two basic trajectories, that is, internal and external trajectories, except the transition process between stable states 2 and 3, which is defined as the internal–external trajectory transition. All the transition process between two stable states can be inversed. Figure 5e shows that the TPQ finger can totally produce 13 transformation ways between 2 of the 4 stable states (including direction) by inflating the soft unit.

Reconfigurable energy barrier and its effect on actuation mechanism

To quantify the effect of the EBA plates on adjusting the energy barrier, and to verify the validity of the energy model, the “path-one” (Fig. 3d) energy barrier of QBU is measured. The comparison between model value and experimental value is shown in Figure 6a. Three groups of EBA plates (angle of EBA plates set at [−1°, −1°], [−8°, −8°], and [−15°, −15°]) are tested. Energy barrier is obtained by the integral of force with respect to displacement and is measured by a force sensor mounted on a screw guide (Fig. 6a and Supplementary Fig. S8), which pushes the tip of TPQ finger in the direction that breaks the energy barrier. The experimental value fits the model well in a smaller tendon pretension range between 0 and 150 mm. While with larger tendon elongation from 150 to 300 mm, experimental values are larger than model values for all EBA plate angles, which can be attributed to the increasing friction torque at finger joints as tension of tendon increases.

FIG. 6.

The reconfigurable energy barrier and its effect on actuation mechanism. (a) The comparison between model and experimental value of energy barrier. The quasi-decoupled effect of the two parameters is seen when EBA plate angles are both set at 1°. Error band (with standard deviation) is used in the statistics of the measured data; each data point is repeatedly collected for 10 times. (b, c) The effect of the EBA plate angles on critical actuation force of the passive sensing mechanism and on critical actuation pressure from S1 to S2. The complete data see Supplementary Tables S5 and S6. By using EBA plates with the smallest angle (1°–1°), high sensitivity can be obtained in the passive sensing mode, with the minimal actuation force of 0.14 N when p = 50 mm $(δ = 13.8 %)$ . Similar regularity is seen in the critical actuation pressure (on chamber Q1, Fig. 4e) as the actuation force and energy barrier. Three independent groups are also tested; the critical actuation pressure is solely influenced by tendon pretension. EBA, energy barrier adjusting.

Energy barrier of the tested QBU is controlled by two parameters: the tendon pretension length and angle of EBA plates. Minimizing the angle of EBA plates can significantly reduce energy barrier. Data point at p = 300 mm shows that the mean energy barrier for 1°–1° EBA plates is 23.8 times smaller than the 15°–15° group; increasing tendon elongation length enhances not only the total system energy of the TPQ finger but also the value of energy barrier. However, by using 1°–1° EBA plates, this increasing effect of energy barrier can be negligible.

The experimental value shows that the difference of energy barrier for 1°–1° EBA plate at p = 0 and 300 mm is only 0.75 mJ; this enables the stable-state transition of QBU (S1 to S2) to be actuated easily. In this situation, the effect of pretension and EBA plate angle on energy barrier can be regarded as quasi-decoupled. The significance of decoupling the two parameters is that the minimization of energy barrier and high-density energy storage can be realized concurrently. Accordingly, the system can transform larger amount of elastic potential energy into dynamic energy by consuming less energy for triggering actuation mechanism.

Energy barrier could affect the sensitivity of the proposed actuation mechanisms. Figure 6b demonstrates the influence of energy barrier (controlled by the EBA configurations) on the critical actuation force of the passive actuation mechanism (Supplementary Fig. S9). The deduction of energy barrier could lower the force required to trigger passive sensing (increasing the passive sensitivity). However, the “quasi-decoupling” effect for 1°–1° EBA plate angle is less obvious compared with energy barrier (Fig. 6a). Similar effects of energy barrier are also seen in the critical actuation pressure of the QBU, during the S1 to S2 stable-state transition (Fig. 6c). The critical actuation pressures of the TBU in both positive and negative directions (positive: chamber T1 actuated; negative: chamber T2 actuated) and the reverse pressure of the QBU (chamber Q2 actuated) are independent of the energy barrier controlled by EBA plates. These values are solely related to tendon pretension length (see independent groups in Fig. 6c). The experiment setup is the same to the one used in the motion capture experiment (see Supplementary Fig. S10). All devices used in the tendon excursion, energy barrier measuring and critical actuation force test are shown in Supplementary Fig. S14.

Grasping Experiments and Analysis

Dynamic sensing scenario and its stable-state stability inspection

To fully explore the function of the preplanned trajectory in TPQ finger, a morphology mimicking the pre-capturing posture of human hand based on the passive sensing mechanism is studied. Similar to the manner of human hands to prey a moving object in space with a half-opened morphology and envelop it by quickly flexing fingers, a single TPQ finger can capture objects using its embedded passive sensing mechanism with similar morphology (stable state 3). And once the objects are sensed, the TPQ finger will release the energy stored in tendon and envelop the objects in tens of milliseconds. Figure 7a shows a comparison of TPQ finger and human hand on catching a screw driver (Supplementary Movie S3).

FIG. 7.

Dynamic sensing scenario and the analysis of its timescale and repeatability. (a) Comparison between human hand and TPQ finger on capture behavior of moving objects. (b) Proposed experiment of obtaining the time for passive grasping. (i) A catapult device (spring stiffness 190 N/m) used to control initial kinetic energy and direction of grasped objects. (ii) Proposed method of acquiring the time for passive grasping. A flex sensor (adjustable resistor that reacts to curvature, 2.2 inches; Spectra Symbol) was embedded into the QBU region of the TPQ finger. 10 V of voltage was applied to the flex sensor and a resistor (33 kΩ) in series. The changing of grasping state alters the resistance of the flex sensor, and its electric potential difference captured by an oscilloscope (Tektronix TBS 1102B). The time for passive grasping is defined by the rise time (from stable value of the lower voltage to initially reach the stable value of higher voltage) of the electric potential difference of the flex sensor. (c) Three groups of variables that affect the passive grasping time: (i) kinetic energy, (ii) energy barrier, and (iii) object dimension. Each variable was independently analyzed by controlling variates method. (d) Effect of inclination angle on grasping success rate. Both positive and negative inclination angle lower the success rate of dynamic catching of objects. High success rate (100%; over 30 trials) is seen in the region from −20° to 20°. The effective grasping region (−40° to −20° and 20° to 60°) has success rate higher than 60%. Success rate dropped to zero for the groups with inclination angle −50° and 70°. Supplementary Fig. S15 shows the experimental setup of the dynamic capturing experiment.

To analyze the factors that affect the reliability and capturing time of the dynamic sensing scenario, three groups of variables were independently studied by controlling variates method: kinetical energy, dimensions of grasped objects, and energy barrier of the TPQ finger. The success rates under these variables were also recorded. A catapult device (Fig. 7b) was applied to control the initial kinetic energy and the speed direction of grasped objects. To record the grasping time, a flex sensor (adjustable resistor that reacts to curvature) was embedded into the soft unit, and 10 V of voltage was applied to it, with a resistor (33 kΩ) in series (Fig. 7b-ii). The dynamic catching time is defined by the rise time (from stable value of lower voltage to initially reaching the stable value of higher voltage, Fig. 7b) of the electric potential difference of the flex sensor and was captured by an oscilloscope. The test was repeatedly recorded for 10 times with error bars $(m e a n \pm S D)$ for each variable (Fig. 7c, all individual results are shown in Supplementary Figs. S16–S18). $Δ L_{T e n d o n}$ (sample 2) was set to 500 mm for all studied variables.

Kinetic energy is negatively related to the grasping time. Result shows that higher kinetic energy could cause a more intense acceleration of the TPQ finger (Supplementary Movie S4) and reduce the average passive grasping time (Fig. 7c-i). Under kinetic energy of 1.15 J, the average grasping time can reach 31.6 ms. However, the negative impact force would be enlarged with higher kinetic energy applied, occasionally leading to failure of grasping as objects bounce out of the capturing profile (i.e., the external trajectory). The success rates are all 100% (30 trials) for the first three groups and drop to 83.3% and 53.3% for spring elongation length of 95 and 110 mm, respectively.

Energy barrier is another factor affecting passive grasping time and repeatability. Three groups of EBA plate configuration (“1°–1°,” “8°–8°,”and “15°–15°”) were used to adjust the energy barrier of the passive sensing mechanism. Result shows that longer time interval is needed for dynamic catching with higher energy barrier (Fig. 7c-ii). And higher energy barrier could lessen the residual kinetic energy of the grasped object, and the acceleration of the closing process is lowered (Supplementary Movie S4). Failed grasps were caused when object kinetic energy could not overcome the energy barrier. The success rate for “1°–1°,” “8°–8°,” and “15°–15°” EBA plate configurations is 100% (30 trials), 90%, and 70%, respectively.

The influence of object diameter on passive grasping time is shown in Figure 7c-iii. Result demonstrates that larger object dimension can result in a shorter time of gripping, leaving less space for finger to flex, the minimum mean time required for a whole passive sensing, and grasping is 31.1 ms, which is seen in the dynamic gripping task of the PVC tube of 63 mm in diameter.

The influence of catching angle on the repeatability of the dynamic capturing scenario was studied using a bar-shape sample with uniform mass distribution (PVC tube, 150 g, D = 32 mm). The initial kinetic energy and speed direction were controlled by the catapult device (Fig. 7d). Result shows that the success rate for inclination angles from −20° to 20° can all achieve 100% (the high repeatability region); the effective grasping region shown in Figure 7d can achieve success rate higher than 60%. The increasing or decreasing of inclination angle reduces the probability and the normal force of the object that hits the force-sensing region of the soft unit, leading to lower success rate. Supplementary Movie S5 demonstrates the dynamic capturing tasks of TPQ finger under different inclination angles.

Based on the passive sensing strategy, a series of experiments were designed to demonstrate its possible use in dynamic interactive tasks. The pipe-in-pipe task (Fig. 8a) requires TPQ finger to continuously captured three tubes of different diameters and insert the smaller tubes into the larger ones. Leveraging both the passive sensing and morphology control ability, it can also finish a continuous job of catching, placing, active grasping, and returning an object (the PVC tube shown in Fig. 8b). By tuning the tendon pretension length to a larger value, it can capture heavier objects such as an umbrella, a vacuum cup, or a spanner weighing more than 0.6 kg (Fig. 8c and Supplementary Movie S3).

FIG. 8.

Featured grasping tasks and stable-state stability test of the dynamic sensing scenario. (a) “Pipe-in-pipe” task. (b) Combination task of dynamic sensing and active actuation. (c) High-force capturing. (d) Stable-state stability test for the passive sensing morphology by creating large inertial acceleration with a rotation platform. The studied stable and fail states are stable states 3 and 4. (e) Inertial acceleration data obtained from an accelerometer. Data of the constant-speed phase are demonstrated. Three groups of EBA chips were used to study their influence on stability.

In dynamic capturing scenarios, the capturing of targets would require the gripper to instantly move an estimated location and actuate the passive sensing mechanism; the process would produce large inertial force that could cause a sudden collapse of the sensing state and reduce its reliability. Considering this issue, we studied the influence of energy barrier on stable-state stability. The test was conducted using a rotation platform to create inertial force (Fig. 8d and Supplementary Fig. S11), and it was applied along the direction that breaks the energy barrier of the QBU (the most unstable direction $a_{I, x}$ shown in Fig. 8d). Acceleration of the constant-speed phase of the platform is shown in Figure 8e; the maximum inertial acceleration that the pre-gripping morphology (S3) can stand is 5.75 times of gravity, as the EBA plate angles were set to 15°. This number is 4.2 and 2 times of gravity for 8°–8° and 1°–1° EBA plates, respectively (Supplementary Movie S6).

Wide grasping range

Leveraging the multiple preplanned trajectories during stable-state transition, the TPQ finger could grasp a broad range of objects in both width and height using simple control strategy.

Gripping of tiny objects

The TPQ finger could grasp an object that is 50 times smaller than its longitudinal dimension using the external trajectory (from S1 to S3) within 1 s (Fig. 8a and Supplementary Movie S7).

Flipping strategy for the stable grasping of thin objects

To achieve stable grasping of thin objects such as cards from flat ground, a flipping strategy is often used by human fingers. The general manipulation process used is to select one of the fingers as the fixed finger, which stops one side of the grasped thin object from sliding, while the other finger provides rotation motion that changes the object's inclination from horizontal to vertical. This method increases the contact area between object and finger, from approximately two points to two faces (Fig. 9b). To ensure a high success rate, another innate feature of human fingers, that is, fingernail, is often applied to handle ultra-thin objects with heights significantly lower than finger pad.

FIG. 9.

Demonstration of grasping range of TPQ finger based on the preplanned trajectories. (a) Manipulation of tiny objects. The gripping tasks of three tiny objects are shown; the pinching morphology is generated by external trajectory segment from S1 to S3. (b, c) Flipping strategy for stable grasping process (i) to (iv) refer to the four stages of the human and the TPQ finger flipping strategy: (i) Object location estimation before applying grasping force; (ii) Using fingernails to clamp grasped thin objects; (iii) The flipping process; (iv) Reaching the stable grasping position. The right finger works as the fixed finger that provides a downward force, creating a quasi-fixed hinge at the right edge of the thin object. The resultant force of the fixed finger and the ground goes along the tangent line of the thin object. The left rotating finger provides an upward force, the component force normal to the thin object is used for flipping, and the component along the object counteracts the resultant force from the fixed finger side. With flipping strategy, the contact area between fingers and the object is significantly enlarged. To maintain a high success rate, human fingers often utilize fingernails to ensure adequate contact with the grasped thin objects. Inspired by this feature, two silicone fingernails were added to the TPQ finger. (d) Success rate of fast flipping of the eight grasped objects. (e) Grasping range comprising dimensions of all tested objects. (f) Grasping a series of objects with incremental dimensions based on passive sensing strategy.

Inspired by this feature and the flipping grasping strategy, we used a two-finger gripper with silicone fingernails (fingernail tangent line at about 45°) at the fingertips. The right finger in Figure 9b was selected as the fixed finger (without active actuation), creating the fixed hinge (Fig. 9b-iii) during rotation. The inflated left finger provides upward force at the left contact point, enabling the object to rotate around the fixed hinge.

Eight objects were tested in the grasping experiment (Fig. 9d and Supplementary Movie S8). The two-finger TPQ gripper (with silicone fingernails) can grasp a thin card of 0.76 mm in 0.133 s, with a success rate of 83.3% (out of 30 trials). Six out of the eight objects are under 2 mm, and the corresponding grasping success rates are above 60%.

Two-finger grasping based on passive sensing

Beside tiny objects and thin-plate objects, the two-finger configuration could grasp a series of objects of larger dimension by tuning the distance and angle of wrist joint (Supplementary Table S3). Nine objects ranging from a 40-mm table tennis ball to a 220-mm football were successfully grasped using the passive sensing strategy (Fig. 9f and Supplementary Movie S9). Passive sensing, which eliminates the need to inflate the soft unit, saves energy compared with the active actuation method. The dimensions of all the objects constitute a wide grasping range; the grasping width ranges from 5.3 mm (a bolt) to 220 mm (a football); the grasping height ranges from 0.76 mm (a card) to 230 mm (a bottle), as shown in Figure 9e.

Gentle picking, high-force manipulation, and a speed competition with human response mechanism

Grasping of delicate objects

Gripping of two fragile targets: a rose and a hydrogel bead are demonstrated in Figure 10a and Supplementary Movie S10. The TPQ finger could gently grasp the fragile targets without damage, using two finger-mimic picking morphologies. The two antagonistic pairs (Fig. 10b) composed of two adverse torques produced by tendon and air chamber were applied to adjust the fingertip position, therefore adjusting the opening range of the gripper according to objects' sizes. Three trajectory segments were used in this task; trajectory segments 1 and 2 were controlled by antagonistic pair A and B, respectively; trajectory segment 3 was created by the slow pressurization of chamber T1.

FIG. 10.

Demonstration of the TPQ finger ability of being gentle, powerful, and fast. (a) Gently picking two delicate objects based on two finger-mimic morphologies. (b) Strategies used in the gentle picking task. Two antagonistic pairs were used to control the tip trajectory segments (segment 1 and segment 2) to adapt to the sizes of objects, by tuning the air pressure of chamber T2 (for antagonistic pair A) or chamber Q2 (for antagonistic pair B). The tip trajectory segment 3 is formed by the sole pressurization of chamber T1. (c) Lifting heavy objects based on the quadrastable mode using elastic tendon (sample 2). (d) In the cable-driven mode, firmly grasping a pair of pliers to cut an electric wire of 3 mm in diameter. (e) Payload of the triple-finger gripper in the cable-driven mode. The griper could powerfully lift an object up to 33.25 kg by flexing phalanxes in the bowstringing region. (f) A response time test based on a ruler falling-and-catching experiment originally used in clinical test.⁴¹ A similar “eye-to-muscle” system was constructed to sense and catch the falling ruler, the average response time of five tests is 105 ms, approximately half of the response time of human (average 202 ms). (g) The actuation time for the finger is 33 ms after chopping off the time interval for signal input (obtained by a high-speed camera). (h) A comparison of payload and actuation time of other soft actuators and this work. Note: The ones rounded by dotted circles have no specific payload but under 1 kg according to the heaviest object in corresponding experiments. For monostable grippers, only payload is compared.

High-force manipulation

The experiment shows that the gripper can reach a payload of 6.33 kg (payload-to-weight ratio: 15.8) working in the quadrastable mode (Fig. 10c and Supplementary Movie S11). While working in the cable-driven mode, a single TPQ finger could firmly hold a pair of pliers and cut an electrical wire of 3 mm in diameter (Fig. 10d and Supplementary Movie S11). The payload in cable-driven mode was tested using a triple-finger gripper; it could grasp a heavy object up to 33.25 kg without damage to both the soft unit and finger phalanxes (Fig. 10e and Supplementary Movie S11).

Response speed comparison with human hand

The response time of the double-finger gripper was tested using a classical clinical test originally designed for testing the reaction time of human.⁴¹ A optoelectronic switch was applied to sense the falling ruler and transmit signal to a solenoid valve; the millimeter-scaled ruler was suspended closely above the laser detecting line. The optoelectronic switch functions as the “human eye” that processes information and gives signal to “the muscle”, that is, the two fingers in the designed system. Five repeated tests were conducted, and the average falling distance of the ruler is 5.38 cm, indicating that the average response time of the designed system is 105 ms, according to free falling body formula. This is roughly a half of the average reaction time of a healthy young adult (202 ms).⁴² By deducting the time interval for signal inputting (i.e., from the start of detection to the opening of solenoid valve, which is obtained from a high-speed camera, for details, see Supplementary Fig. S13), the time for actuation is only 33 ms (Fig. 10g and Supplementary Movie S12).

Conclusion and Discussion

We have proposed a novel prototype for the design of tendon-driven finger that works in the quadrastable mode, and it was embedded with a series of capabilities similar to those of human finger, including sensing, wide grasping range, softness, high-load tolerance, and quickness to be actuated. Variables influencing key features of TPQ finger were studied, and based on them, we acquired tunable transition speed (solely control of tendon pretension) and tunable passive and active sensitivity (based on the reconfigurable energy barrier, by tuning EBA plate or tendon pretension). These variables were utilized to optimize the performance during various grasping tasks.

Previous stable-state transition-type fast actuators or grippers have maximally reduced energy barrier to acquire short response time.^20,21,23 Still, no experimental investigations have analyzed the stability of the stable state or pre-grasping morphology following the reduction of the energy barrier, a crucial element in ensuring the dependability of the grasping methodology. Considering this, we combined a stability test with the conducted dynamic interactive task that can frequently encounter uncertain accelerations in real situations.

The TPQ finger takes the advantages of both cable-driven finger and the stable-state transition strategy. Cable-driven robotic hand in the literature already shows excellent adaptivity and operability in various tasks,^24,25 but their suitability for high-speed scenarios that occur on a millisecond timescale remains largely unexplored. Bistable grippers based on one-dimensional beam or two-dimensional plate structures often sacrifice grasping adaptivity and strength to improve speed by reducing inertial force (mass) to a minimal level, as noted in the previous literature.^{22,32,43–45} The embedded multistability of TPQ finger significantly enhances actuation speed, and multiple grasping morphologies help the finger to grasp a wide range of objects. The two driven modes can be either used in a dynamic task that happens in tens of milliseconds or in a slower situation but need high output force, or even slower, to gently grasp fragile objects. Figure 10h compares the actuation speed and payload of the TPQ finger with some high-speed bistable grippers/actuators^{19–21,27–29,40} and monostable grippers/actuators,^{7–10,12,13,24,46–50} respectively.

Our next prototype will focus on designing a comprehensive TPQ hand that can fulfill all the proposed functions in a single unit. Also, we are seeking a solution that could combine the two gripping modes, that is, the quadrastable mode and the cable-driven mode. This could be realized by using two types of tendons (both elastic and inextensible tendons) and lining them into the same pulley. However, actuating them would need two separate motors. For simplification, another method by altering the energy reservoir can be applied, using a single motor to drive a coil spring to store energy, and replacing the elastic tendons with inextensible cable (e.g., nylon cable).

In addition, the estimation of the preplanned trajectory is needed to work in harmony with the wrist joint angle and the tendon pretension, to ensure a higher success rate in gripping multidimensional objects. The rigid phalanxes of TPQ finger made by photosensitive resin have satisfactory static strength, but breakage could occur as colliding with tougher materials or working in periodic high-force scenarios, and possible damages could be posed to fragile objects when being impacted by the phalanxes. Employing materials with better impact resistance, such as nylon, and embedding the phalanxes into the soft unit would be our goals of iteration on these issues.

Footnotes

Authors' Contributions

P.J. supervised and sponsored the whole project. Y.Y. co-supervised this project. Z.S. conceived the whole work, designed, manufactured, iterated the prototypes, and formulated the quadrastable energy model. T.J., Z.S., and Z.W. constructed all the experimental devices and setup, and completed all experiments and data analysis section. Z.S., P.J., Y.Y., T.J., and Z.W. wrote the article. Z.S., T.J., Z.W., and H.L. shot the pictures and videos. T.M. and J.L. edited the figures and supplementary movies.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This project is partly funded by the National Natural Science Foundation of China (Grant Nos. 51705050, 52005269) and by the Natural Science Foundation of Chongqing (Grant No. 2022NSCQ-MSX1629).

Supplementary Material

Supplementary Information

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Supplementary Material

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