A Linear Multiport Network Approach for Elastically Coupled Underactuated Grippers

Robotic hands range from just a single degree-of-freedom [1] to 15 or more [2]. The most complicated and dexterous hands have never been used outside the research environment, like the anatomically correct testbed hand [3], or are designed for exceptional applications, e.g., Robonaut 1 and 2 [4,5] for manipulating tools during spaceflights. More degrees-of-freedom do not better accommodate uncertain real-world environments; as the number of degrees-of-freedom of an end effector increases, the number of implementations in everyday use tends to decrease [6]. Several researchers have incorporated compliance into hands and grippers [7–9] or constructed the hand from entirely soft materials [10,11], and this has improved performance in the presence of position inaccuracy or unknown object properties.

Underactuation lowers the cost and complexity of the device, usually making it easier to repair and simpler to operate. It can be traced back to the earliest robotic hands, both finger-to-finger (interfinger) [12] and between joints of an individual finger (intrafinger) [13]. They also passively adapt to objects to be grasped [7,14]. However, underactuated hands also have drawbacks; because they do not provide individual control of each joint, increased actuation effort may cause ejection of the object [15–17]. Underactuation has been physically implemented in most grippers using either bar linkages or pulley and tendon control [18].

Some methods of underactuation are illustrated in Fig. 1. “Branching” underactuation (Fig. 1(a)) simply has the tendon pulled by an actuator split into tendons to individual fingers. This means that the displacement of each tendon will be equal, and the actuator force is equal to the sum of each of the branching tendons. Due to its branching structure, the Toronto–Bloorview–MacMillan hand [8] was particularly vulnerable to object ejection. A cable “circuit,” which passes over fixed and movable pulleys and is anchored only at the ends [15], is shown in Fig. 1(b). The constant tension throughout the cable tends to distribute grasp forces equally. Some hands have multiple cable circuits for separate collections of fingers [19,20].

While these examples of underactuation illustrate some clever heuristics, they do not share much in common with the human hand, considered a benchmark for robotic hands [21]. Human fingers exhibit some enslavement, meaning they move together in some way [22,23]. Empirical studies on humans [24,25] indicate that the brain controls the human hand as a whole, rather than individually planning the joint space trajectory of each finger, a concept known as synergies [6,26–28].

This idea of synergies inspires an architecture where each actuator has some effect on all fingers, and each actuator corresponds to a specific synergy. This idea has been implemented on complex, high degree of actuation hands to simplify the software and communication channels [29]. Others have tried to implement synergies in hardware. The Pisa/IIT SoftHand [9,30] and others have found clever ways to implement a single “soft” or “adaptive” synergy mechanically in a grasping device. The implementation of two or more is elusive: the SoftHand+ [31] exhibits superposition of a clutching motion and preferential motion of the fingers on the lateral or medial side by actuating each end of the cable circuit and exploiting pulley friction properties. Differential mechanisms have been used in some specific instances [15,32,33]. These designs are based on clever ideas specific to the context; there are no systematic methods of producing this behavior known to the authors.

Compliance essentially converts uncertainty in position into variations in force. Passive compliance is inherent in different levels of a robotic system, such as in its links, actuators, and transmission systems like tendons, drive cables and joints [34], and exists between an end effector or tool when it makes contact with an object [6]. Active compliance notwithstanding, several researchers have spoken to the benefits when flexibility is built-in through integrated elastic elements, allowing the hand to conform to a variety of object shapes and sizes [1,18,35]. However, few works have paid attention to how deliberately varying these flexible properties can enforce desired hand behavior. This paper explores specifically structuring the compliance to create the desired mapping between actuator actions and finger motions, as illustrated in Fig. 1(c). The compliance allows us to superimpose multiple actuator motions onto the same set of fingers: Actuator motions individually produce deformations of an elastic material. In linearly elastic materials, these effects will be additive due to the principle of superposition.

One could envision combining underactuation, synergies, and compliance to produce a hand like the one in Fig. 2. All finger tendons originate from an elastic “web.” The University of Tulsa Anthro-pomorphic Robotic Hand [36] uses this idea with a single compliant mechanism, called the “mainspring,” but the design authority over its behavior is limited. If there were instead a library of compliant components with known properties, one could synthesize a web that produces the actuator–finger behavior desired, so long as it is built on a suitable framework relating the elastic transmission's physical design parameters to the actuator–finger behavior.

This work combines underactuation and compliance together in a systematic way using multiport network theory, a notion borrowed from electrical networks. Making force–current and displacement–voltage analogies, this provides a framework for connecting compliant mechanisms together to synthesize an elastic transmission mechanism that maps the actions of a small number of actuators to a larger number of finger tendons.

In multiport modeling of the hand, each constituent element used in the elastic web is modeled as an entity where the relationship between forces and displacements between nodes or “ports” on the mechanism are described by an immittance matrix [37]. Ports are points or nodes on a mechanism that are physically connected to other compliant mechanisms or the environment, with its interior behavior being a “black box,” as shown in Fig. 3. When multiple individual elements are connected together at their ports in series, parallel, and nested [38] configurations, they behave as and may be modeled as a single larger element. A multiport network is specified by a single one of its immittance relationships; the most convenient is in stiffness form: $f=Sδ$⁠, where f is a vector with each of its elements representing the tension in a particular tendon emerging from the elastic transmission mechanism. $δ$ is the excursions in each of the tendons. The square matrix S has units of stiffness and completely characterizes the behavior of an individual compliant component. It must be emphasized that this method (based on matrix multiplication) presupposes that whatever the physical manifestation is, it deforms with linear elastic behavior. Because S has units of stiffness, rather than a general impedance, the results only apply to grasping and slowly executed manipulations, not to extremely dynamic motions of fingers, jaws, or objects. S can be calculated analytically, as in the spring network example in Sec. 4, evaluated experimentally using a procedure similar to that performed by Schultz and Ueda [39], or determined using finite element methods [40].

Connecting individual ports means physically enforcing equal displacements (parallel connection) or equal forces (series connection) at these nodes. For instance, a parallel connection may be accomplished by a rigid linkage, while a series connection may be accomplished by a flexural joint or cable and pulley. Ports could be located anywhere on an elastic mechanism and may be arbitrarily numbered. The following argument shows that inter- and intranetwork port interconnections in parallel and series always yield connected stiffness matrices that are positive definite.

Consider two multiport networks, each possessing a positive definite stiffness matrix. The first of these is diagrammed in Fig. 3, where the mechanism itself is shown as a box and each port is given two terminals. In keeping with the electrical analogy, displacement $δ1k$ at the port is denoted with a pair “+” and “−” symbols. This indicates that as with voltage, displacement is measured “across the terminals” or between two points. Likewise, the force $f1k$ is given an arrow as when showing the direction of positive current. Tensile force is considered to be positive at a port. Multiport network 1 is an n-port network, and multiport network 2 (whose diagram is identical except for the total number of ports) is an m-port network. General expressions will be derived for the stiffness matrices resulting from the parallel and series connection of j ports of multiport network 1 to j ports of multiport network 2, where j ≤ n and j ≤ m.

Making a parallel connection between each of j pairs of ports between multiport networks 1 and 2 means that the new multiport model will leave the n + m − j unconnected ports as-is and replace each pair of connected ports with a new port, consolidating the port variables equations according to the connection conditions. The conditions for the parallel connection of j ports of multiport network 1 to j ports of multiport network 2, as shown in Fig. 4, are given in Eqs. (6) and (7), each representing j equations, where $(δparallel,1δ2,2δ2,fparallel,1f2,2f2,)∈ℝj$⁠. Combining these conditions with the individual multiport force expressions will eliminate displacements at the connected ports $1δ2$ and $2δ2$ in favor of $δparallel$ (the vector of displacements of the group of connected ports) and retain the displacement vectors of the unconnected ports $1δ1$ and $2δ1$

The system of equations for the parallel-connected multiport network described by Eqs. (8)–(10) can be combined into Eq. (11). The block matrix in Eq. (11) is the stiffness matrix for the network formed by connecting j ports of each network in parallel, denoted $Sparallel∈ℝ(n+m−j)×(n+m−j)$

Lemma 1. If two multiport networks each possesses a symmetric positive definite stiffness matrix, then the parallel connection between some number of ports of the first network and an equal number of ports of the second network will yield a multiport network with a symmetric positive definite stiffness matrix.

where $v⊤Sparallelv>0$ because S₁ and S₂ are positive definite matrices by definition, and the result is proved. ◻

Remark1. It is worth pointing out that in the case of connecting all ports of two multiport networks of identical dimension in parallel, i.e., j = m = n, the dimensions of A, C, P, and R are zero, and thus $S1≡B, S2≡Q$⁠. In this special case, we recover the well-known result found in Ref. [37] that the immittance matrices add (S_parallel = S₁ + S₂), which is positive definite by inspection. Others may note that such a result would be expected based on passivity of multi-input, multi-output dynamic systems [42]. Our method holds even when an arbitrary number of ports are connected and when the two multiports are of different dimension.

Performing an analogous set of operations to those described in Sec. 2.2 to eliminate the variables associated with connected ports in favor of the new port formed by the connection results in an expression analogous to Eq. (11) for the new multiport formed by the series connection, with a symmetric stiffness matrix $Sseries∈ℝn+m−j×n+m−j$ in place of $Sparallel$⁠.

Unfortunately, the elements of S_series do not have the same clear relationship with the entries of S₁ and S₂ as S_parallel, making the path to a proof cumbersome at best, intractable at worst. However, because the series connection of ports is dual to the parallel connection, we can look at the series compliance matrix, C_series, the inverse of the stiffness matrix. The calculation of its elements from the elements of S₁ and S₂ is presented in the author's thesis [41].

Lemma 2. If two multiport networks each possesses a symmetric positive definite stiffness matrix, then a series connection between some number of ports of the first network and an equal number of ports of the second will yield a multiport network with a symmetric positive definite stiffness matrix.

The proof follows the procedure of the proof of Lemma 1 with C_series in place of S_parallel. This works because of the dualistic relationship between the stiffness and compliance matrices of series and parallel connections.

One of the most basic constitutive mechanisms with three ports or more may have both types of connections to other multiports in the transmission. We can get around this problem of mixed connections by making only the series connections in a combination to form an intermediate multiport with all the parallel-connected terminals left disconnected, then making parallel connections within the terminals of the intermediate multiport, and finally forming the multiport desired. The following lemmas provide for this case.

Lemma 3 (Lemma 4). If a multiport network possesses a positive definite stiffness matrix, then the parallel (series) connection of some number of ports to an equal number of other ports within the network will yield a multiport network with a positive definite stiffness matrix.

The proofs of Lemmas 3 and 4 (found in the author's thesis [41]) require the construction of a sparse matrix L, which represents the intranetwork connections.

Theorem 1.Any multiport network created through the interconnection of ports of multiport networks possessing positive definite stiffness matrices will have a positive definite stiffness matrix.

Proof. The multiport model of any elastic transmission composed of smaller constituent multiport elements can be created by connecting these elements in a sequence of steps involving either interport or intraport connections in parallel and series. The positive definiteness of interport connections is given in Lemmas 1 and 2, and the positive definiteness of intraport connections is proven by Lemmas 3 and 4 at each step. Each intermediate multiport in the sequence will, therefore, have a positive definite stiffness matrix.

Cutkosky and Kao [34] studied the stability of a grasp when small perturbations are applied to an object. Key points and tendon-driven hands are briefly summarized. These are then reexamined when the hand is underactuated using a compliant mechanism.

where $BPJ$ is a concatenation of the fingertip Jacobian that takes fingertip forces to body forces.

where $H$ is the Boolean contact selection matrix [43] (e.g., point contact, point contact with friction, and soft finger). $Jθ$ is a concatenated Jacobian of each of the fingers, sometimes known as the hand Jacobian. $Cs$ is the structural compliance (inverse of stiffness) of links and joints, which is always positive definite [34], and $Kθ$ is the stiffness in the axis of motion of the joint. Assuming a fully actuated hand, they interpret this as a servo gain on each joint. In a more realistic scenario, the fingers will be tendon-driven as described in the following.

Looking at Eq. (16), one sees that it is nothing more than a series of congruence transforms on the matrices $Cs$ and $Kθ−1$⁠. Congruence transformations retain the signs of eigenvalues [44]. In practice, few robotic hands have servomotors at the joints. Murray et al. [45] introduce the notion of “tendon excursion space,” a Euclidean space where each coordinate corresponds to the excursion of an individual tendon. They point out that the principle of virtual work holds between joint space and tendon excursion space with its own “tendon Jacobian” in a manner analogous to the kinetostatic duality between joint space and task space. Following this line of reasoning, and assuming tendon moment arms do not change substantially under the perturbation, this does nothing more than replace $Kθ$ in Eq. (16) by $PKtP⊤$⁠, adding yet another congruence transform. (⁠$Kt$ is the stiffness in tendon excursion space.) This applies to fully actuated hands where the joints are driven by tendons. We can now combine this with the results of Sec. 2, where the stiffness is not a software stiffness, but a physical stiffness, and the hand is underactuated by virtue of the compliant mechanism.

Once the grasp is achieved, f and $δ$ are taken to mean differential displacements [46]. If the actuators are fixed while the object is perturbed, then $δn+1=δn+2=⋯=δn+m≡0$⁠. This is equivalent to deleting the last m columns of S. Likewise, the stability of the grasp is not concerned with the actuator tendon forces, so the final m rows can be disregarded. This means that $Kt$ is exactly S₁₁. Since S₁₁ is one of the principal minors of S, which is positive definite, by Theorem 1 it too is positive definite. Consequently, it also produces a positive definite joint-dependent stiffness K_b.²

Does this mean that an underactuated hand with a compliant web transmission will always produce a grasp that is stable to external perturbations of the grasped object? No—but the elastic transmission helps. The previous argument cannot say anything about the second term in Eq. (15), K_J, the configuration dependent stiffness. A definitive statement can only be made by analyzing the specific morphology and grasp pose. The inter- and intrafinger coupling can introduce off-diagonal terms in K_b that can counteract destabilizing influences in K_J. The off-diagonal coupling terms of S₁₁ can be used to the same effect. If certain “risky” poses are identified where K_J has negative eigenvalues, design choices could be made in the elastic web which produce a counteracting effect in K_b at that pose. So while an elastic transmission mechanism like the one described in this paper is not a guarantee against ejection of objects, it does produce a stabilizing effect.

The congruence transform argument does not address the situation when the actuators execute volitional actions, e.g., attempting to clench the object more tightly. The multiport framework can also be used to understand this situation, as will be illustrated in the example in Sec. 4.

A simple three-digit underactuated grasping device was developed to evaluate how faithful the multiport model is when the device is constructed from real-world components, and to demonstrate how the multiport model can be used to predict intuitive behaviors encountered in real-world situations. Its simple structure and kinematics were chosen to minimize the importance of rolling contact (K_J-related) effects and clearly display the behavior of the compliant mechanism when changes are made to the actuator end. If it were to have articulated fingers where rolling contact is significant, the behaviors described in this section would still be present, but would be more difficult to discern from the experimental data. In a five-fingered anthropomorphic hand with two actuators, the multiport approach will describe the behavior equally well, although differences in behavior between one transmission network and another may be more subtle than the contrast between the two cases presented here. The analytical development in this section was presented in Ref. [47], but the experimental validation presented in Sec. 4.3 appears here for the first time.

The gripper, pictured in Fig. 6, has a single actuator and two movable fingers. It has one actuator, two fingers, and a stationary thumb. Motion of the individual fingers is unconstrained kinematically, but their force–displacement behavior is linked by a three-port elastic transmission mechanism. The device can be configured to effectuate two distinct compliant couplings: a branch connection (parallel-connected two-port models) pictured in Fig. 6(a), and a cable circuit connection (series-connected two-port models) pictured in Fig. 6(b), where the digit systems are connected by a system of cables and pulleys. Note that this different from branching and cable circuit connections directly to the fingers; this is branching and cable circuit connections of the two-port models, forming a different three-port model in each case.

The gripper will be operated in a displacement-control mode, although it is possible to also use this in a force-controlled mode [46]. For each test, the experimenter manually displaced the actuator drawbar to complete a grasp. Each digit is connected to mechanical ground by one spring and is connected by the other spring to the actuator. As shown in Figs. 6(a) and 6(b), the displacements of the digits and the actuator are measured by HEDS-9200 linear optical encoders (Broadcom Limited, Singapore) with a US Digital LIN-300-24-N (Vancouver, WA) strip with a resolution of 300 counts/in, and the compressive force experienced by each semicylindrical digit is measured by a Futek LSB-200 uniaxial force sensor (Futek Advanced Sensor Technology, Inc., Irvine, CA).

The branch interconnection is accomplished by connecting the digit systems by a rigid drawbar, which imposes the (parallel) condition $1δ2=2δ2=δact$⁠. Using s_i,j from S_parallel gives the immittance parameters in column 2 of Table 1.

The cable circuit interconnection is accomplished by connecting the digit systems with a pulley system, which imposes the (series) conditions $1f2=2f2=fact$ (because the tension in the cable must be constant throughout its length). This follows from equilibrium on the pulleys. The no-slack condition in the cable imposes the other multiport series connection condition: $δact=1δ2+1δ2$⁠. Using the series connection relationships (omitted from Sec. 2 for brevity, but found using a process similar to the construction of S_parallel [41]), one arrives at the immittances in column 3 of Table 1.

The three-port model derived will predict how each gripper type will behave. The gripper begins with all compliant elements at their resting length. As the actuator drawbar is pulled, the fingers will begin to close on the object.

Once the digits make contact with an object, the contact forces on the digits are nonzero, while the displacements are governed by the geometry of the object. In this stage, the actuator's and two digits' displacements are measured relative to their respective positions when contact was first established. To predict if an object will rotate or not, it is helpful to analyze how digit forces would change with respect to increased actuator displacement, δ_act, by assuming a fictitious moment is applied which precisely counteracts any difference in digit forces predicted by the multiport and prevents the object from rotating. If the forces on digits 1 and 2 scale equally, then the fictitious moment would be equal to zero. If the two digit forces scale unequally, the fictitious moment would need to be nonzero to equilibrate the difference in applied forces. This implies that the object will rotate.

Note that f₁ scales with $−1k2$⁠, and f₂ scales with $−2k2$⁠, which would cause a net moment except in the special case that $1k2=2k2$⁠. This indicates that as the actuator displaces beyond the point of contact, the two digit forces will scale differently. This leads to rotation of the object that depends on the magnitude of the force. This is referred to as a “spurious squeeze” by Gabiccini et al. [48].

In the series case, both digit forces scale by the same coefficient of δ_act. The net moment on the object will remain at zero, and the object will not rotate; this is referred to as a “pure squeeze” by Gabiccini et al. [48].

A set of 19 objects, consisting of 7 geometric shapes, 11 household/workshop objects, and a cutout of the U.S. state of Oklahoma were placed in the gripper. The gripper was closed until contact was made, and then actuator displacement was increased until a mechanical limit was reached, a sensor saturated, or object ejection occurred, whichever came first. Fingertip force, finger displacement, and actuator displacement were recorded during forward and reverse actuator motions. The test was performed for both the series and parallel three-port mechanism. The video footage of these trials is available on the Biological Robotics At Tulsa research group's website. Data from the full test suite appear in Ref. [41].

In this section, representative results demonstrating the difference between the parallel and series configurations are shown for a highly asymmetric shape, an acrylic cutout of the U.S. state of Oklahoma. Spring constant values for the trial shown are in Table 2. Figure 7(a) shows the forces measured at each finger versus the actuator displacement after contact was made (solid lines). The negative values indicate compression. Fingertip forces predicted by numerically evaluating the three-port model with the measured finger displacements (Eqs. (20) and (21)) are shown by the dashed and dotted lines. Pictures of this object at different stages of the parallel-configured experiment are shown in Fig. 7(b) with the object's edge emphasized in red. As predicted by the model, in the parallel configuration, the shape rotated clockwise as the actuator displacement increased.

An identical test was conducted in the series-configured gripper. The digit forces with the model prediction are shown in Fig. 7(c). Note that the lines for fingers 1 and 2 overlap one another as predicted by the model. In contrast to the example in Fig. 7(a), the forces for each digit scale nearly equally as predicted by Eq. (26). Figure 7(d) shows the object at two different stages of actuator displacement, with the edge of the object emphasized in red. Unlike the parallel case, the object did not rotate significantly with increased actuator displacement after contact.

As mentioned in Sec. 4.2, the crucial difference between connecting the two-port transmission for each finger in a branched or cable circuit configuration is that the branched interconnection causes an imbalance in grasp forces in the two digits, which increases with actuator effort. In practice, this causes the object to rotate so that the forces trend towards equilibration (spurious squeeze). Using a strictly linear prediction, where the contact points translate directly in the actuation direction, one might assume that the measured force curves would overlay one another in the parallel case as well, because the object would rotate until any imbalance in forces was taken up. However, because there is friction between a grasped object and the platform, the object is unable to rotate freely enough to achieve complete equilibrium. Another artifact of this is that object rotation in the parallel configuration can upset predictions of finger trajectories if not all fingers close at the same time. This is why the model and experiment curves are offset from one another slightly in Fig. 7(a). In contrast, the cable circuit connection causes grasp forces to scale equally with respect to δ_act so that the grasp forces on each digit scale approximately equally, and the object does not rotate by any discernible amount.

In Figs. 7(a) and 7(c), the measured and predicted force curves from the displacement agree well, meaning the three-port model does a good job of capturing the elastic transmission's force–displacement behavior. A hysteresis-like curve is evident on both fingers in Fig. 7(c). This is because the fingers did not follow the exact same path (have the exact same displacement for the same value of actuator displacement) upon squeezing and releasing the grasp. This is likely due to Coulomb friction. It is worth noting that this deviation appears most pronounced on finger 1 (the finger that moves the most during the spurious squeeze) at low levels of contact force, when frictional effects would be most significant. The spring constants in Table 2 are based on the manufacturer-provided values and may exhibit some manufacturing variation that would affect the slope of the curves. The linear bearing on which the fingers ran was made from two tracks mounted end to end. Although the slide transitioned smoothly from one to the other, a small detent could be felt when moving it manually. With lower spring constants, this may have caused minor deviations there.

The multiport models do not include rolling contact at the fingertips. They only predict force components in a single spatial dimension (corresponding to tendon excursion). Transverse components are small when the digits make contact with the object at approximately the same time, the object does not rotate, and the line of action of the force is close to normal with the surface of an object. However, the transverse force components increase in relative magnitude if an object rotates significantly. In one case, the orientation of the Oklahoma cutout was reversed, accentuating the spurious squeeze effect, until the transverse components were so large that the object was ejected.

From this experiment, the implications for design choices in compliantly coupled industrial three-jaw grippers are clear—for asymmetric objects, the choice of stiffnesses and how they are connected will determine how the grasped object will behave. For complex, multidegrees-of-freedom hands, the framework described in Sec. 2 applies equally well. There will be more constitutive elements making up the transmission than a pair of two ports, and as a result, there will be more interconnections. The final multiport will have more than three ports, with more than two “finger” and one “actuator” ports. While we expect that the multiport description of the compliant transmission will also give insight into how these devices will behave, the nuances of the behavior may be more subtle than that displayed starkly by this experiment.

This paper has presented a framework for underactuated gripping devices based on multiport network theory. Under this framework, a small number of actuators can each influence the motion of all fingers by connecting through an elastic transmission or web. It has been shown that if the web is composed of smaller compliant components, each of which has a multiport model with a positive definite stiffness matrix, any combination of series and parallel interconnections between them will result in a web with a positive definite stiffness matrix. Because the stiffness matrix is positive definite, this architecture tends to promote stable grasps, although changes in finger contact kinematics due to the external wrench can still overcome this effect to dislodge the object. Experiments on a single-actuator gripper whose fingers are coupled by a multiport spring network show good model fidelity, and that the model can be used to identify real-world behaviors that result from design choices.

Future work will apply this framework to the design of more complex multifingered hands with multiple joints and biologically inspired finger kinematics once the platform hardware development (underway in the authors' research group) is complete. The multiport framework can be extended to dynamic motions of fingers and objects by creating multiport models of the constituent parts whose elements are transfer functions. While this framework provides a method for evaluating combinations of constituent components and transmission candidates, it does not produce the elastic web in response to design specifications. A systematic method to synthesize the elastic transmission suitable for a range of tasks is a meaningful problem in its own right and will be the subject of a future work.

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