Simultaneous Kinematic and Contact Force Modeling of a Human Finger Tendon System Using Bond Graphs and Robotic Validation

Abstract

This paper aims to use bond graph modeling to create the most comprehensive finger tendon model and simulation to date. Current models are limited to either free motion without external contact or fixed finger force transmission between tendons and fingertip. The forward dynamics model, presented in this work, simultaneously simulates the kinematics of tendon-finger motion and contact forces of a central finger given finger tendon inputs. The model equations derived from bond graphs are accompanied by nonlinear relationships modeling the anatomical complexities of moment arms, tendon slacking, and joint range of motion (ROM). The structure of the model is validated using a robotic testbed, Utah's Anatomically correct Robotic Testbed (UART) finger. Experimental motion of the UART finger during free motion (no external contact) and surface contact are simulated using the bond graph model. The contact forces during the surface contact experiments are also simulated. On average, the model was able to predict the steady-state pose of the finger with joint angle errors less than 6 deg across both free motion and surface contact experiments. The static contact forces were accurately predicted with an average of 11.5% force magnitude error and average direction error of 12 deg.

Introduction

The tendon system of the hand gives humans the capability of performing tasks that range from powerful grasping to fine manipulation. Modeling the human hand can lead to a better understanding of how humans are able to perform such a broad range of tasks. Clinicians and researchers use tendon models to estimate motion and forces in human hands during various tasks [1–4] while roboticists use tendon models to explore new ways to design tendon-based prosthetics, robotic hands, and exoskeletons [5–11]. Continued improvement of tendon models will help advance understanding in hand function and improve future devices that mimic this functionality.

Modeling any human biomechanical system is a challenging task due to the complexity of the human body, and the hands and fingers are no exception. Previous models of the hand have limited or inaccurate implementations of the underlying tendon system. Original modeling of the finger tendon system by Landsmeer developed moment arm relationships that are still used in many finger models today [12]. The first model to create a two-dimensional kinematic relationship between tendon excursion and finger motion (flexion–extension only) was Leijnse et al. [13]. This and other models identified different modeling complexities of the finger tendons regarding kinematics. These complexities include redundant tendons, tendon slacking, tendon interconnectivity, and variable moment arms. These complexities grow when considering external contact. Contact forces, joint stiffness, friction, and contact kinematics then become important.

Regarding these growing complexities, later models attempted to include more detail but all simplified the interconnections of the extensor mechanism (EM) [9,14–16]. The EM is a tendon aponeurosis made from a combination of multiple tendons located on the dorsal side of the finger [17]. The EM combines redundancy and interconnectivity to create an indeterminate and pose-dependent problem as described in Refs. [18] and [19]. A detailed description of the EM is discussed in the Muscles and Tendons section.

Modern models attempt to include the EM in more detail with varying degrees of complexity. Sancho-Bru et al. developed an inverse dynamics model utilizing muscle effort optimization to examine forces while grasping [2,10,18]. Valero-Cuevas et al. have developed a forward dynamics model that utilizes a relaxation algorithm to handle the EM force transfer [3]. Most recently, interconnected tendon networks, like the EM, have been modeled by Niehues et al. using a frame work for interconnected springs for both forward and inverse simulations [20]. Their work addresses the interconnection using optimization within the EM tendons. Each of these models needs a way to work with the indeterminate and pose-dependent force distribution in the EM utilizing an iterative approach. A current limitation is that none of these models have demonstrated simultaneous simulation of finger motion and external contact forces while also including tendon complexities discussed above. To the authors' knowledge, there are no models that present a complete and accurate tendon system of the whole hand, as most tendon models still strive for accurate modeling of only a single finger.

Biomechanical modeling is often investigated using the opensim environment. The opensim software is an open-source musculoskeletal modeling software, which includes a tendon-driven model of the forearm and hand [21,22]. Currently, opensim is unable to accurately represent muscle-tendon units that split and insert into different structures. opensim modeling of branching muscles or tendons is often replaced by multiple muscles, one muscle for each branch. This multiple muscle approach works well for muscle-tendon units that split or converge; however, the EM interconnects multiple branches of four different muscles. This was attempted for the index finger [23] for a static inverse simulation by creating multiple overlapping muscle routes for each muscle that contributes to the different interconnection of the EM. Minimization based on muscle effort was used to estimate force between muscles, but a sense of force distribution cannot be quantified.

This paper first describes a new single finger tendon model that simultaneously simulates kinematics and force transfer while in contact with the environment. For now, only quasi-static finger motion and forces are simulated, but as demonstrated with current models accounting for the EM, an iterative approach is required. Because of this, a dynamic forward simulation was chosen that uses numerical integration to handle the pose-dependent EM force transfer as well as finger motion. To accomplish this task, bond graph modeling techniques are used to assist in modeling the interconnected nature of the finger tendon system. Bond graphs are a powerful graphical description of energy transfer through a system and are often used in modeling the dynamics of complex systems [24]. Bond graphs are particularly advantageous in this application because of how naturally they facilitate the modeling of various subsystems and their complex interconnections. Previous researchers have used bond graphs to model only the extensor mechanism of a finger [25] with minimal experimental validation. We seek a model of the entire finger that can contact the environment.

Bond graphs are a dynamic modeling tool that facilitates derivation of closed-form dynamic state equations for the entire finger system. In general, only quasi-static parameters (e.g., stiffness, friction) may be necessary to simulate quasi-static motion and forces. However, by adding in dynamic parameters like inertia, the derivation of the bond graph's equations of motion becomes much easier by alleviating differential-algebraic loops that arise from the interconnections of the finger-tendon system. This results in first-order ordinary differential equations (ODEs) that can easily be numerically simulated.

The second goal of this paper is to validate the model using a biomimetic robotic finger. An earlier version of this model was preliminarily validated using the anatomically correct testbed (ACT) hand [26]. The ACT Hand is a proven tool for studying the human hand due to its unique mimicking of the hand's anatomical structure, while not having the disadvantages of in vivo or cadaver experimentation like specimen degradation under prolonged use [27–29]. These preliminary experiments looked at free finger motion or static force transfer with fixed joints, but never simultaneous simulation. The preliminary validation with the ACT hand proved difficult due to high joint friction. This joint friction was difficult to quantify and subsequently produced errors during model validation. To address this, a new finger was constructed with reduce joint friction, Utah's anatomically correct robotic testbed (UART) finger, in resemblance to the ACT hand. Using this new finger testbed, a set of validation experiments have been conducted to validate the model's ability to simultaneously simulate finger kinematics and force transmission under quasi-static loading and motion.

Finger Anatomy

The finger model described in this paper will be described in terms of the anatomical features and complexities that it will simulate. First, the anatomical features will be described. The human hand has five digits/fingers. The index, middle, and ring finger all have very similar tendon structures and are commonly referred to as the central fingers. The anatomical structures considered in our model of the human finger will be the joints, bones, and tendons. Figure 1 contains a description of the anatomical structures that will be discussed in terms of modeling.

Fig. 1.

Anatomy of a central finger including bones, tendon and muscles, and joints

Joints.

The three joints of a central finger are the metacarpophalangeal (MCP) joint, the proximal interphalangeal (PIP) joint, and the distal interphalangeal (DIP) joint (Fig. 1). The finger has four degrees-of-freedom (DOF), abduction–adduction (AB) at the MCP joint and flexion–extension (FE) at all joints. Each joint is contained within a low friction joint capsule formed by fine ligaments that determine its DOF and range of motion (ROM) [30]. The joint capsules also contribute to joint friction and joint stiffness. The joint capsules of the human have very low friction due to synovial fluid, and the stiffness of the joint can significantly vary depending on joint angle [31]. The nonlinearity of variable joint stiffness is ignored in previous tendon models since the data on joint stiffness variation are limited. Instead, a constant stiffness from literature is used [32].

Bones.

The four bones that make up the structure of the index finger are the metacarpal, proximal phalanx, middle phalanx, and distal phalanx, as shown in Fig. 1. The length of the bones is used in forward kinematics showing how joint angles relate to finger position. In addition to length, the shape of the bones is also an important factor in finger tendon modeling [28,33]. The shape and routing of the tendons on the bones influences the mechanical advantage of each tendon. The resulting tendon mechanical advantage, or moment arm, around each joint is dependent on the shape of the bone and pose of the finger. The nonlinearity of changing moment arms is often ignored in previous tendon models [14–16,34]. Our model will use variable moment arm models discussed in the Characterization section.

Muscles and Tendons.

There are six muscles per central finger; however, the index finger has an extra extensor, extensor indicis, that can be combined with the extensor digitorum (ED) for modeling purposes since they are equivalent during single finger analysis [28]. The six muscles presented for the finger are the flexor digitorum profundus (FDP), the flexor digitorum superficialis (FDS), the ED, the palmar interosseous (PI), the dorsal interosseous (DI) and the lumbrical (LUM) (Fig. 1) [30].

Tendons connect muscle to bone. It is common in tendon models to assume the tendons are inextensible, i.e., infinite tendon stiffness, and unchanging [14–16]. However, the passive stiffness of the finger has contributions from connective joint tissue, muscles, and tendon stiffness [35]. An important tendon structure is the EM. The EM combines the tendons of the ED, DI, PI, and LUM to provide extension at the DIP and PIP joints. Depending on pose and muscle contraction, the EM can cause flexion at the MCP while extending the distal joints. When contracted, the ED muscle transfers most of its energy through the EM to extend the MCP, PIP, and DIP joints; it can also cause a slight abduction or adduction of the MCP joint, depending on MCP-FE joint angle. For the index finger, the PI muscle causes adduction and the DI and LUM muscles cause abduction. Depending on MCP-FE joint angle, the PI and DI can also cause flexion or extension. The LUM muscle flexes the MCP and extends the PIP and DIP joints. The routing of the extensor tendons allows MCP to be independently controlled, whereas the PIP and DIP joints are coupled together through the extensor mechanism and cannot be extended separately [4].

Accurate representation of the extensor mechanism is important to retain, due to its unique role in tendon tension distribution and finger motion. Different interconnections of the extensor mechanism change the tendon tension distribution throughout the extensor mechanism [3,19,36]. Previous researchers have looked at the effects of different tendon interconnection arrangements of extensor mechanisms [3,36,37]. The unique function and nonlinearity of the extensor mechanism are often over simplified in previous tendon models, but Valero-Cuevas et al. and Lee et al. [3,19] have shown its complexity is essential for anatomical accuracy. The most common representation of the extensor mechanism is the Winslow's Rhombus extensively investigated in Ref. [3].

Bond Graph Modeling

In this section, the anatomical structures presented above are modeled, using bond graphs, to create a complete model of the tendon system of the finger. The model will be presented in generalized submodels. First, a single joint model will be presented followed by a tendon input model. Each joint and input model framework can be used multiple times for the number of joint axes (four) and the number of input tendons (six). Next, a model for the FDP tendon will be presented combining input and joint models to provide a frame of reference for combining these submodels. The more complex structure, the extensor mechanism, is then described. The last model description will be the transmission of an external force on the fingertip to the joint models. Then a complete combination of the submodels will be discussed for the whole finger. Finally, the implementation of the bond graph model in a numerical solver and additional nonlinear elements are discussed.

Bond Graph Basics.

The basics of bond graph modeling will be discussed. A full description of bond graphs can be found in Ref. [24]. Bond graph elements include energy sources, storage elements, dissipative elements, junctions, energy bonds, and energy transformers. In bond graphs, inputs are described by sources. For mechanical systems, an effort source, $S_e$, prescribes an input force or torque, while a flow source, $S_f$, prescribes a velocity. There are two kinds of energy storage elements that can be either capacitances, $C$, (spring) and inertance, $I$, (mass/inertia). There are also resistances, R, that model effects like viscous friction. Junctions either represent shared effort or flow between elements. 0-junctions, graphically shown as $0$, denote a shared effort and a summation of flow between the connected elements. 1-junctions, graphically shown as $1$, denote a shared flow and a summation of effort. Finally, there are transducer junctions that convert energy into different forms (i.e., translation to rotation). In this paper, transformers or modulated transformers transducers are used to transform tendon tensions to joint torques and tendon translations to joint rotations. All elements are connected to junctions by bonds. The direction of the arrow designates direction of energy flow and is used for sign convention during state equation derivation. Perpendicular strokes at one end of the bond, named causal strokes, denote the direction of effort enforcement versus flow enforcement. Causality can be assigned systematically based on the nature of the elements and junctions in the model [24].

Once a bond graph is constructed with proper elements, bonds, arrows, and causal strokes, the state equations can be systematically derived. The required state variables are defined by the energy storage elements that are in integral causality (see Ref. [24] for details). Choosing power variable formulation for the states, the effort of each capacitance in integral causality becomes a state, and the flow of each inertance in integral causality becomes a state. State equations begin with the derivative of each state, and are completed by following the causal paths through the bond graph (applying junction equations and constitutive laws of elements) until arriving back at inputs or states.

Notation.

Going forward, a subscript notation will be used. Subscripts $j$ correspond to the joint axes (MCP-AB, MCP-FE , PIP, or DIP) referenced as $j$ = 1, 2, 3, or 4, respectively. Subscripts $T$ represent tendons and tendon segments. For joint models, this refers to tendons that wraps around a joint.

Joints.

A bond graph framework for joints is shown in Fig. 2. Each anatomical joint is simplified to either a pin or universal joint. A single-DOF pin joint is used for PIP and DIP joints. A 2DOF universal joint is used for the MCP joint. Each DOF can have dynamic parameters including passive joint stiffness, $K_j$, joint inertia, $I_j$, and viscous friction, $b_j$. The bond graph uses a rotational spring to represent joint stiffness, inertia of the distal finger bones to represent the joint inertia, and rotational linear viscous damper to represent the small amount of joint friction. Each of these elements shares the same joint flow (1-junction) imposed by the rotary inertia. Energy is supplied to the joint from the tendons wrapping around each joint with a moment arm, $r_{T,j}$. A tendon force transforms into a joint torque equal to force times the moment arm. Multiple tendons wrapping around a joint create a net torque, ${\tau }_{\mathrm{net}}=\sum r_{T,j}{F}_T$. The moment arm also determines the relationship between tendon translation and joint rotation (${\dot{x}}_T=r_{T,j}{\dot{\theta }}_j).$ Finally, externally applied torques are also included, ${\tau }_{\mathrm{EXT},j}$. These torques can be applied by gravity or external contact forces, which will be discussed in the External Forces and Torques section.

Fig. 2.

Bond graph framework for a joint model actuated by tendons. The tensions of the tendons are transformed into torques using the moment arms for each tendon over the joint. These torques are summed with the torques generated by the joint stiffness, the viscous joint friction, and the external torques. These combined torques enforce effort on the inertia of the joint, which determines the flow (velocity) of the joint.

Using power variable formulation (force/velocity or torque/rotational velocity), the state variables for the joint model bond graph (Fig. 2) become: the torque of the torsion spring, ${\tau }_{K,j}$ and the rotational velocity of the joint inertia, ${\dot{\theta }}_j$. Their state equations are derived by taking derivatives and following causality through the bond graph (applying junction equations and constitutive laws) until arriving at inputs/states. The rate of change of the spring's torque is determined by the velocity of the spring multiplied by the stiffness of the joint. Since the spring and the inertia are connected to the same 1-junction, the velocity of the spring is equal to the velocity of the joint, which is itself a state. As such the final state equation for the joint stiffness becomes the following equation:

$$\frac{d{\tau }_{K,j}}{dt}=\hspace{0.17em}{K}_j{\dot{\theta }}_j$$ (1)

The rate of change of the joint velocity (acceleration) is determined by the sum of torques, denoted by the 1-junction, divided by the joint inertia. The sum of torques around each joint includes torques created by the tendon moments, joint stiffness, joint friction, and external forces. Assuming the tendon forces, $F_T$, are states (as shown in the Tendon Inputs section), the state equation for joint velocity becomes the following equation:

$$\frac{{d\dot{\theta }}_j}{dt}=\frac{1}{I_j}\left(\sum r_{T,j}{F}_T-{\tau }_{K,j}-b_j{\dot{\theta }}_j+{\tau }_{\mathrm{EXT},j}\right)$$ (2)

A third state can be defined as the position of the joint, ${\theta }_j$, with its state Eq. (3). While Eq. (3) is not completely necessary, it provides flexibility in determining the joint position if $K_j$ is not linear (i.e., human ligaments).

$$\frac{d{\theta }_j}{dt}={\dot{\theta }}_j$$ (3)

Bones.

The bones do not explicitly show up in the bond graph, as their inertia is lumped into the rotational inertia around the joints. The lengths of the bones are used in the model to convert from finger joint motion to whole finger motion using a D-H parameter kinematic model. The shape of the bones determines the moment arm relationships discussed previously.

Tendon Inputs.

In the current formulation of the model, the inputs are treated as tendon excursion velocities (flow sources). The dynamics of muscle activation will not be considered but could be added in the future. Each tendon is assumed to have some stiffness, $k_T$. Assuming no losses of energy in tendon translation, an input tendon's tension is determined by the stiffness of the tendon and the tendon displacement. The total displacement of the tendon is determined by the joint model dynamics and the dynamics of the EM. The bond graph framework of this model is shown in Fig. 3.

Fig. 3.

Bond graph model of an input tendon assuming a tendon flow (velocity/displacement). The tendon is modeled as a series spring with stiffness, $k_t$. The spring enforces the tension in the tendon. The velocity of the tendon is spread across each joint it crosses. All tendons will have a flow component for the two MCP joint models. The ED, PI, DI, and LUM will connect to the EM model, which will affect the PIP and DIP joint model. The FDS and FDP tendons both directly affect the PIP joint. The FDP is the only input tendon that directly affects the DIP joint.

All six input tendons cross the MCP joint, so they all have some contribution to the MCP joint model for both flexion/extension and ab/adduction. The ED, PI, DI, and LUM tendons all insert into the EM, which contributes to extension in both the PIP and DIP joints as discussed below. Both the FDP and FDS directly cause flexion in the PIP joint, where only the FDP tendon causes flexion at the DIP joint.

Again, using power variables, the state variable in the input model is the force of the translational spring (4)–(6). For each tendon, the spring force state equation is determined by the stiffness of the translational spring, $k_T$, and the relative velocity of the string. For the two flexor tendons (FDP and FDS), the relative velocity is determined by the input tendon velocity, ${\dot{x}}_T$, and the sum of the change in tendon length due to joint rotations. The FDP tendon has changes in length from all four joint rotations (4). The FDS tendon only has three joint rotations (5). For the tendons inserting into the EM, only the contributions of joint motion of the MCP joints are directly used. They also have a change in relative velocity linked to intermediate masses, ${\dot{x}}_{T,m,1}$, of the EM inserts (6). These masses will be discussed in the Extensor Mechanism section.

$$\frac{d{F}_{\mathrm{FDP}}}{dt}=k_{\mathrm{FDP}}\left({\dot{x}}_{\mathrm{FDP}}-\sum _{j=1}^4{r}_{\mathrm{FDP},j}{\dot{\theta }}_j\right)$$ (4)

$$\frac{d{F}_{\mathrm{FDS}}}{dt}=k_{\mathrm{FDS}}\left({\dot{x}}_{\mathrm{FDS}}-\sum _{j=1}^3{r}_{\mathrm{FDS},j}{\dot{\theta }}_j\right)$$ (5)

$$\frac{d{F}_T}{dt}=k_T\left({\dot{x}}_T-\sum _{j=1}^2{r}_{T,j}{\dot{\theta }}_j-{\dot{x}}_{T,m1}\right)$$ (6)

In Eq. (6), T is either ED, DI, PI, or LUM, for each tendon in the EM.

Flexor Tendons.

All the energy generated by the contracting flexor muscles is transferred through the flexor tendons to flex the joints of the finger. It is assumed that no energy is lost due to sliding friction. For example, the energy transfer of the FDP is illustrated in a bond graph (Fig. 4), where the energy from the translational input of the FDP tendon is transferred to the rotational elements of the finger joints. When using a flow source (i.e., tendon excursion) as the input to the bond graph model, some energy is stored in the tendon compliance while the rest of the energy is transferred to each joint model (Fig. 3).

Fig. 4.

Example schematic model and bond graph of finger with only the FDP tendon that utilizes the tendon input and joint bond graphs without external forces. Each block represents an instance of a bond graph model. Four instances of the joint bond graph (Fig. 2) are used to represent each DOF of the finger. The input model (Fig. 3) of the FDP tendon connects to each joint. The connections represent the shared tension of the tendon and the individual tendon velocities per joint.

Extensor Mechanism.

The muscles connected to the EM have a different tendon transmission through the finger. The connections of the EM are commonly described by Winslow's Rhombus model [3]. The connections of the EM in the bond graph mimic Winslow's Rhombus with an added lumbrical segment. The bond graph model mimics the tendon routing of the ED, PI, DI, and LUM into the extensor mechanism by representing the tendon segments of the Winslow's Rhombus (Fig. 5(a)) as a series of interconnected springs (Fig. 5(b)).

Fig. 5.

(a) The EM described as the Winslow's rhombus with additional LUM. (b) Model of EM using interconnected translational springs in the interconnected Winslow's rhombus configuration. Intermediate masses are included at intersections of springs to elevate differential algebraic loops. (c) The bond graph representation of EM model using capacitive elements (springs) and intermediate inertial elements. 1-junctions on the left are connected to the outputs of the input tendon models for each tendon connecting to the EM. PIP and DIP represent the respective joint model bond graphs. The bond graph visually represents the Winslow's rhombus and demonstrates the intuitive application of bond graphs to interconnected systems.

The bond graph for the EM begins with the tendon inputs (Fig. 5(c)). The inputs of the finger model are transferred from the input models discussed above and into the EM. Where the tendon segments of each input tendon bifurcate, a “one” junction is used to signify a shared flow at this intersection. Tendon segments with subscript, 2, are tendons that only cross the PIP joint. They converge together at the proximal slip of the extensor mechanism and wrap around the PIP joint (1-junction in joint domain). The subscript, 3, tendon segments converge on the lateral bands of the extensor mechanism and wrap around the PIP joint and connect in the terminal slip that acts around the DIP joint.

The bond graph model's extensor mechanism also includes intermediate inertial masses at tendon interconnections (Fig. 5(c)). The masses are included to simplify equation derivation. The implications of using these inertias are discussed in the following Inertia section. Masses with subscript 1 will be referred to as level 1 masses that connect the input tendons to the EM. The masses with subscript 2 will be referred to as level 2 masses that connect the lateral band tendons with the first level of tendon segments.

The state equations from this model include the force of the tendon segment springs and the velocity of the intermediate masses. Again, spring-force state equations are determined by stiffness and a relative velocity. Segments with subscript 2 have relative velocities defined by the difference between the velocity of level 1 inertias and the change in velocity due to wrapping segments around the PIP joint (7). The segments with subscript 3 have a relative velocity defined by the difference between level 1 and 2 inertias (8). The segments with subscript 4 have a relative velocity defined by the difference between the connected level 2 mass and the change in velocity due to the lateral bands wrapping round the DIP and PIP joints (9). Inertial velocity state equations are defined by the sum of forces divided by the mass of the junction. For the level 1 masses, the input forces are the input tendon forces and the output forces are the forces of subscript 2 and 3 segments (10). The level 2 masses have input forces from the subscript 3 segments and output forces for the subscript 4 segments (11).

$$\frac{d{F}_{T2}}{dt}=k_{T3}\left({\dot{x}}_{m,1}-r_{T,2}{\dot{\theta }}_2\right)$$ (7)

$$\frac{d{F}_{T3}}{dt}=k_{T3}\left({\dot{x}}_{m,1}-{\dot{x}}_{m,2}\right)$$ (8)

$$\frac{d{F}_{T4}}{dt}=k_{T4}\left({\dot{x}}_{m,2}-r_{T,2}{\dot{\theta }}_2-r_{T,3}{\dot{\theta }}_3\right)$$ (9)

$$\frac{d{\dot{x}}_{m1}}{dt}=\frac{1}{m}\left(\sum F_{T,in}-\sum F_{T,2}-\sum F_{T,3}\right)$$ (10)

$$\frac{d{\dot{x}}_{m2}}{dt}=\frac{1}{m}\left(\sum F_{T,3}-\sum F_{T,4}\right)$$ (11)

The nonlinearity of rerouting tendons in the extensor mechanism is discussed in great detail in Ref. [36]. The unique structure and interconnection of the extensor mechanism leads to “somatic logic,” as described in Ref. [36], where input tendon tensions can preferentially propagate tension to the proximal and terminal slips by nonlinearly rerouting the distribution of tensions. This nonlinearity is not implemented into the bond graph model because there are no results in the literature that accurately describe how the tendons reroute under loading.

External Forces and Torques.

External forces (e.g., contact forces and pose dependent gravity) can be added to the bond graph modeling framework. The bond graph presented demonstrates the external forces and torque framework for applied fingertip and external gravity forces (Fig. 6).

Fig. 6.

Bond graph of external torques acting on the finger. A vector effort source, in the form of a wrench, represents the forces generated by a contact model. Those forces are transformed to the joint domain in the form of torques, using a transposed kinematic Jacobian. Those transformed torques are combined with torques from other sources (e.g., gravity) to create a total external torque at each joint. Gravity is calculated using pose dependent gravitational torque modeling [38].

An external fingertip force was implemented into the bond graph model to simulate contact with an environmental surface. The external fingertip force is implemented in the bond graph model using the Jacobian transpose as a multiport transformer. The Jacobian transpose, ${\mathit{J}}^T$, generated from the Denavit–Hartenberg (D–H) parameters of the finger as in Ref. [38], is used to relate the external fingertip force/torque wrench, ${\mathit{W}}_{\mathbf{FT}}$, to joint torques, ${\mathit{\tau }}_{\mathbf{FT}}$ (12)

$${\mathit{\tau }}_{\mathit{FT}}={\mathit{J}}^T{\mathit{W}}_{\mathrm{FT}}$$ (12)

The fingertip force is calculated by specifying an environmental impedance contact model. As the finger moves through the environment, a force is generated by stiffness and friction models. In this work, a specific surface contact model will be discussed, but any impedance contact model could be used. This can be replicated for any external contact force. A different Jacobian would be required for each contact point.

The model can handle external torques as well. Gravity could be implemented with additional Jacobians for each center of mass of the bones, but for kinematics such as the finger, gravity is often computed as torques using classical robotics methods for serial manipulators [38]. To do this in the bond graph, a torque (effort) source can be added at each joint that is modulated by the joint angles of the finger, ${\mathit{\tau }}_{\mathit{G}}=f\left(\mathit{\theta }\right)$. A total external torque created by fingertip forces and gravity can be combined such that a total external torque can be generated, thus ${\mathit{\tau }}_{\mathbf{EXT}}={\mathit{\tau }}_{\mathbf{FT}}+{\mathit{\tau }}_{\mathit{G}}$. This relationship is shown graphically in Fig. 6 by 1-junctions adding the torques generated by the fingertip forces and the modulated gravity sources.

Complete Bond Graph.

To create a complete bond graph for a single central finger, the joint, input tendon, extensor mechanism, and external force models were combined. Four instances of the joint model (Fig. 2) are used to model the MCP flexion/extension, MCP ab/adduction, PIP, and DIP joints. Six instances of the input tendon model (Fig. 3) are used to model the ED, FDP, FDS, DI, PI, and LUM tendons. A single EM model (Fig. 5(c)) and a single fingertip contact model (Fig. 6) are used to complete the single finger model. A complete bond graph is shown in Fig. 7, connecting the multiple submodels together.

Fig. 7.

Complete bond graph model of the central finger combining multiple instances of the joint and tendon input models, the extensor mechanism bond graph, and the external force bond graph. Six instances of the joint tendon input model are used for each input tendon. The PI, ED, DI, and LUM connect to the two MCP joint bond graphs and the EM bond graph. The FDP connects to all four joint bond graphs. The FDS connects to all but the DIP joint bond graphs. The EM then affects the PIP and DIP joints. The external forces bond graph is connected to each joint bond graph.

Because of the model formulation, causal strokes are placed at the joints. This demonstrates two things. First, the displacement of the input tendon causes a force that is transmitted through compliances to the joints. Second, the joint motion is caused by imbalance of effort imposed by tendon inputs, external forces, and intrinsic joint dynamics.

The bond graph model has seventeen translational springs, four rotation springs, six translational masses, and four rotational inertias. This means that a minimum of 31 ODE state equations are required to simulate forces and velocities. For ease of joint angle calculation, four additional equations are used in the form of (3). A summary of the required state equations is shown in Table 1. Additionally, if the fingertip contact model utilizes differential state equations, more ODEs may be required.

Table 1.

Summary of instances of the equations used in complete bond graph using equations of motion describing the joint, input tendon, and extensor mechanism model

Equations	Instances	Purpose	Model
(1)	4	Rotary spring torques	Joint model
(2)	4	Rotary inertia velocities
(3)	4	Rotary inertia angles
(4)	1	Force in FDP input spring	Input tendon model
(5)	1	Force in FDS input spring
(6)	4	Force in ED, DI, PI, and LUM input spring
(7)	4	Force in e2, d2, p2, and L2 springs	Extensor mechanism model
(8)	5	Force in ed3, ep3, d3, p3, and L3 springs
(9)	2	Force in d4 and p4 springs
(10)	4	Translational inertia velocity for mass e1, d1, p1, and L1
(11)	2	Translational inertial velocity for mass ep2 and ed2

Inertia.

The primary purpose of including inertial elements in the joint and EM models is to facilitate the derivation of explicit state equations. The details of this are tied to how causality is assigned in the bond graph method. By adding an inertial element to a 1 junction, the common flow at that junction becomes a state and provides a termination point during state equation derivation, which prevents the formation of algebraic loops that lead to differential algebraic equations. In the case where the differential algebraic equations include nonlinear constitutive laws that are noninvertible (e.g., static friction), we are left with a set of implicit state equations that can only be solved numerically at great computational expense. By including inertias, the resulting state equations are a set of explicit ODEs that are easy to derive and computationally inexpensive to simulate.

The inclusion of these inertial parameters brings up the issue of numerical values. For the joints, the inertias are physically analogous to the inertia of the skeleton and the values are included in many biomechanical models. For the EM, the inertias at the intersections of tendon segments are more enigmatic. They do not have a value that is easily measured, and an arbitrary value cannot be chosen. If an inertial value is too small, numerical issues arise during numerical integration. If inertia is too large, the dynamics of the simulation can be impeded. To address this problem, the impedance of the translational inertias in the EM was matched to the physically measured joint inertia. The impedance matching for the finger is discussed in Ref. [39]. This method ties the value of the translational inertia of the EM masses to the rotational inertia of the joint using the moment arms.

Implementation.

The bond graph model is used to determine the relationship between tendon excursion or tension (input) to final finger position in joint angles (output). A fourth-order Runge–Kutta solver is used to solve the set of ODEs derived from the bond graph model. Issues encountered when implementing the solver were assessed with the goal of maintaining ease of computation and anatomical accuracy; these issues include selecting a moment arm model, implementing tendon slack, and implementing joint ROM limits.

Very few tendon models attempt to implement the nonlinearities of tendon slacking and maximum joint ROM [13,14,34], These modeling complexities are inherently nonlinear, and there are no bond graph elements that innately model these nonlinearities; instead, nonlinearities are introduced into the constitutive laws of existing bond graph elements. The following two nonlinearities are defined in the system. Tendon tension values cannot go below 0 N (as a negative tension would imply compression of the tendon). In the simulation of the model, this is enforced for all tendon segments, $T$, by the following equation:

$$\frac{d{F}_T}{dt}=\left\{\hspace{0.17em}\begin{array}{cc}0,& \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{d{F}_T}{dt}<0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{F}_T\le 0\\ \mathrm{E}\mathrm{q}.\hspace{0.17em}\left(\left(4\right)-\left(9\right)\right),& \mathrm{else}\end{array}\right\}$$ (13)

Joint angles cannot exceed values beyond maximum ROM of the index finger ROM. The ROM is enforced by the following equation:

$$\frac{d{\theta }_j}{dt}=\left\{\hspace{0.17em}\begin{array}{cc}0,& \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{d{\theta }_j}{dt}<0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_j\le {\theta }_{\min }\\ 0,& \mathrm{else}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{d{\theta }_j}{dt}>0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_j\ge {\theta }_{\max }\\ \mathrm{E}\mathrm{q}.\hspace{0.17em}\left(2\right),& \mathrm{else}\end{array}\right\}$$ (14)

Another important nonlinearity of note is regarding the moment arms. The moment arms for the human finger system are variable over the fingers workspace [33]. This means that the moment arms depend on anthropomorphic data. In our model, we assume they are only dependent on joint angles. $r_{T,j}=f\left({\theta }_j\right)$. The specifics on how these parameters are found are discussed in the Characterization section of the experimental testbed.

The final nonlinearity is that joint friction is not usually just viscous. The current bond graph formulation has a viscous friction component, but static friction is not quantified. A simple Coulombic joint friction model was added to the derived state equations. The constant Coulombic friction threshold torque for each joint, ${\tau }_{c,j}$, was determined by a coefficient of friction, ${\mu }_c$, times the magnitude of the resultant compressive forces. The compressive forces are a combination of the sum of tendon tensions crossing each joint, $\sum F_{T,j}$, gravitational reaction forces, ${\mathit{F}}_{G,j}$, and external fingertip forces, ${\mathit{F}}_{\mathit{FT}}$ (15). The parameter value used will be discussed in the Characterization section. The Coulombic relationship now modifies (2) to include a Coulombic friction term (16). The Coulombic friction term, ${\tau }_{f,j}$, is calculated using (17) described in detail in Ref. [40]. When the joint is not moving, the static friction term is equal to either the Coulombic threshold, ${\tau }_{c,j}$, or the net torque acting on the joint, ${\tau }_{\mathrm{net},j}$. The net torque is calculated as the sum of the torques from tendons, joint stiffness, and external torques (18). This forces the joint to be static unless ${\tau }_{\mathrm{net},j}$ is greater than the threshold. Once the joint can move, the Coulombic friction is only the threshold torque acting against motion

$${\tau }_{c,j}={\mu }_c||\sum {\mathit{F}}_{T,j}+{\mathit{F}}_{G,j}+{\mathit{F}}_{\mathrm{FT}}||$$ (15)

$$\frac{{d\dot{\theta }}_j}{dt}=\frac{1}{I_j}\left(\sum r_{T,j}{F}_{T,j}-{\tau }_{K,j}-b_j{\dot{\theta }}_j+{\tau }_{\mathrm{EXT},j}+{\tau }_{f,j}\right)$$ (16)

$${\tau }_{f,j}=\left\{\begin{array}{cc}-\min \left(\left|{\tau }_{\mathrm{net},j}\right|,{\tau }_{c,j}\right)\hspace{0.17em}\mathrm{sign}\left({\tau }_{\mathrm{net},j}\right)& \mathrm{i}\mathrm{f}\hspace{0.17em}{\dot{\theta }}_j=0\\ -{\tau }_{c,j}\hspace{0.17em}\mathrm{sign}\left({\dot{\theta }}_j\right)& \mathrm{else}\end{array}\right\}$$ (17)

$${\tau }_{\mathrm{net},j}=\hspace{0.17em}\sum r_{T,j}{F}_{T,j}-{\tau }_{j,K}+{\tau }_{\mathrm{EXT},j}$$ (18)

Robotic Testbed

The robotic finger construction and parameters will be described in this section along with the experimental setup.

Mechanisms.

To experimentally validate the model, a robotic testbed was used. A robotic testbed provides a repeatable platform to conduct experiments. In contrast, a cadaver can degrade during experimentation and can be difficult to characterize. The precursor of this model's kinematics and static contact simulation were separately validated using the ACT hand [26,41].

In order to validate the model's simultaneous simulation, a new finger was developed, UART finger (Fig. 8). This finger was developed with joint friction in mind. In previous validation experiments, joint friction was difficult to characterize and caused issues with kinematics and force transmission [26]. In this new finger, imbedded ball bearings were used to reduce this joint friction. The bearings are standard unshielded ABEC-5 stainless steel ball bearings (57155K341 McMaster-Carr, Santa Fe Springs, CA). To accommodate these bearings, the finger bones were scaled 1.5 times the average size [42]. Since, the primary purpose of the testbed is to validate the model framework, the size of the bones and resulting moment arms do not change the framework of the model, but rather are parameters in the model to be calibrated. This calibration will be discussed in the testbed Characterization section.

Fig. 8.

UART finger

Following the ACT's success, the mechanisms of the human finger were emulated robotically [7]. The finger has four pin joint axes with near human range of motion. Like the ACT hand the MCP ab/adduction joint axis is pitched 60 deg with respect to the metacarpal bone [43]. The bones of the finger are made of two sections. The external bone surface is three-dimensional (3D) printed acrylonitrile butadiene styrene (ABS) plastic, printed in two halves. The internal structure is supported by a plastic frame. The ball bearings are imbedded in sockets of the 3D printed bone. A visual description of the bones internal structure and the corresponding D–H parameter variables are shown in Fig. 9. The D–H parameters values used for kinematics are described in Table 2. An offset for the MCP-FE angle was chosen to make the zero-angle pose of the finger correspond to the finger bones being colinear. Negative joint angles correspond to flexion. The distal phalanx, $a_4$, is longer than the scaled anatomical value to account for the added fingertip.

Fig. 9.

Detailed view of the UART finger's mechanisms and D-H parameters. Axes are fixed on distal joint connected to each bone. The X axes represent the length dimension of the bones. The Z axes represent the joint axes, $a_i$ represents the distance between each joint, and ${\theta }_j$ represents the joint angle between each X axes. The MCP joint is modeled with two perpendicular joints. There is a 30$\hspace{0.17em}\mathrm{deg}$ offset on the MCP flexion angle. This makes the zero angle of the finger to be when the four bones are inline. The offset in the MCP ab/adduction angle is consistent with Ref. [43].

Table 2.

D-H parameters of the UART finger used for kinematic representation

$i$	$a_i$	$d_i$	${\alpha }_i$	${\theta }_i$	Joint
1	0	0	$90\hspace{0.17em}\mathrm{deg}$	${\theta }_1$	$\mathrm{MC}{\mathrm{P}}_{\mathrm{AB}}$
2	$60.7$ mm	0	0	${\theta }_2-30\hspace{0.17em}\mathrm{deg}$	$\mathrm{MC}{\mathrm{P}}_{\mathrm{FE}}$
3	30 mm	0	0	${\theta }_3$	$\mathrm{PIP}$
4	40 mm	0	0	${\theta }_4$	$\mathrm{DIP}$

The tendons of the UART finger are made from a 50 lb. test strength braided Spectra kite string (Spectra50100, Goodwinds LLC, Mount Vernon, WA). The EM utilizes crocheting to connect the different segments to form the interconnected structure that follows the Winslow's rhombus model. While the hood of the EM is not reproduced, routing eyelets in the finger are used to ensure the motion of tendons follows near human tendon routing. Specific eyelets are used at the PIP joint to ensure the routing of the lateral bands during full range of motion. Tendons are secured to embedded pins in the middle and distal phalanges.

A soft fingertip silicon rubber fingertip is also included on the finger to emulate a soft finger pad and increase the static equilibrium space of the finger under many different tendon loading configurations [44]. This fingertip is a silicone (Ecoflex 0030, Smooth-on, Macungie, PA).

Actuation and Sensing.

For this research, six series elastic actuators (SEAs) were used to provide the input tendon excursions for each input tendon (Fig. 10). A servo motor was selected (Dynamixel MX-64, Robotis, Lake Forest, CA) for position control. A custom-ordered linear stiffness torsion spring with gapped coils was used as the elastic element for the SEA. A potentiometer was used to measure the spring deflection, which can be used to calculate the tendon force. The primary purpose of the SEA actuator is to provide force measurement capabilities and ease of use. A six-axis force/torque transducer (F/T Nano17 SI-12-0.12, ATI Industrial Automation, Garner, NC) was used to measure the fingertip force applied at the surface. A USB-DAQ 6210 (National Instruments, Austin, TX) was used to collect the SEA and the F/T transducer signals. Finally, the pose of the finger was measured using an optical motion capture (MOCAP) system with cameras (FLEX V100:R2, Optitrack, Corvallis, OR) and evaluation software (AMASS, C-Motion, Inc., Germantown, MD).

Fig. 10.

The UART's SEA used to actuate the finger and measure the forces of the input tendons

Characterization.

The bond graph contains many parameters for joint, tendon, moment arm, and contact models. The many parameters of the model were calibrated in Ref. [39] by conducting various static and dynamic characterization experiments on the UART finger. The following Static and Dynamic Characterization sections will briefly describe the estimation of the various parameters, but the full characterization of values is discussed in Ref. [39]. A complete summary of the characterized parameters values is shown in Table 3.

Table 3.

Summary of parameters values used in the bond graph simulation of the UART's experimental motion and contact

Parameter	Value	Units	Description
$K_j$	0	N mm/rad	Joint stiffness
$b_j$	0.01	N mm s/rad	Joint viscous friction
$k_t$	15	N mm	EM stiffnesses
$k_{t,\mathrm{IN}}$	0.3	N mm	SEA input stiffnesses
$m_t$	0.129	kg	Impedance matching
${\mu }_c$	0.97	mm	Columbic coefficient
$a$	0.258		Normal contact force coefficients
$b$	1.72
$K_x$	0.104	N/mm	Shear contact stiffnesses
$K_y$	0.138	N/mm
${\mu }_x$	0.481	mm	Shear contact slip coefficients
${\mu }_y$	0.553	mm

Static.

The static parameters are the stiffnesses, Coulomb friction, moment arms, and contact model. The joints have two main static components. The first is joint stiffness. For this application, the joints of the robotic finger do not have a spring and the bones are assumed as rigid; therefore, the joint stiffnesses were set to zero. However, the joints do have a Coulomb friction model. The Coulombic coefficient was experimentally estimated [39].

The translational part of the model has seventeen translational springs. Six of these are the input springs for the six input tendons. On the robotic testbed, the input tendons are Spectra kite string connected to the SEAs. The kite string has a much higher stiffness than the elastic element and can be assumed as rigid compared to the SEA. Therefore, the value of the input tendon stiffness used for simulation corresponds to the elastic elements of the SEAs, 0.3 $\pm$ 0.03 N mm. The other eleven springs correspond to the segments of the EM. Based on simple force–displacement characterization experiments on sample crocheted segments, the stiffnesses of the EM tendons segments are all estimated to be 15 N/mm for all segments.

The most important of the parameters in the model are the moment arms. They determine the relationship between joint rotation and finger movement in space as well as tendon tension and joint torsion. This subsequently becomes important for the position of the finger during surface contact. As previously described, the moment arms are assumed to be dependent on the posture of the finger. An effective method developed by the ACT group [28,45] is used to characterize the moment arms of the UART finger experimentally. Following their neural network method in Ref. [45], the moment arms for all joint and tendons were determined. Moment arms were assumed to be only the function of the joint angles it crossed. For the MCP joint, all moment arms were a function of both flexion/extension and ab/adduction angles.

The methods described in Refs. [28] and [45] worked for all of the moment arm relationships except the lateral band's moment arms about the PIP joint. During flexion, the lateral bands of the EM, anatomically and on the UART, wrap around the end of the proximal phalanx. The moment arms for the lateral bands passing over the PIP joint, however, were not captured using ACT group's method. To estimate this moment arm, a simple two-dimensional model was used, described in Ref. [39].

Finally, the contact model for the fingertip was characterized in terms of a simple static impedance model. The surface contact was assumed to take the form of Fig. 11, having three translational springs. If $\mathrm{\Delta }Z$ is positive (away from surface), then all forces are zeros. The normal force, $F_{N,z}$, was a function of displacement ($\mathrm{\Delta }Z)$ of the fingertip in the normal direction ($Z$). The contact point (CP) of the fingertip was assumed to be static unless a static threshold was reached. The shear forces were assumed to be elastic as the finger bone displaces inside the silicone fingertip. $\mathrm{\Delta }X$ and $\mathrm{\Delta }Y$ are the displacements in the shear directions with respect to the orientation of the surface. Rotational affects inside the fingertip were neglected.

Fig. 11.

Fingertip contact model utilizing translational springs in the normal, $n_z$, and shear, $s_x$ and $s_y$, directions. The left of the figure shows a side view of the finger as the as it displaces ($\mathrm{\Delta }Z$ and $\mathrm{\Delta }X$) in the normal and shear ($s_x)$ directions. The right shows a front view of the finger as it displaces ($\mathrm{\Delta }Z$ and $\mathrm{\Delta }Y$) in the normal and shear ($s_y)$ directions. The CP is the initial point that touches the surface. The IP is the joint the kinematic Jacobian is mapped to. As the finger displaces the relative position between the IP and CP changes and generates forces using an impedance contact force model.

From experimental characterization, the normal force–displacement model was found to be a power relationship (19). As the fingertip compressed further, the surface area increased consistent with other artificial fingertip characterization [46]. The shear forces were found to be elastic with a slip threshold, (20) and (21). This threshold was approximately linear with respect to normal force (22) and (23). The contact forces were mapped back to the Jacobian reference frame at the interaction point (IP) using an equivalent wrench, ${\mathit{W}}_{\mathrm{FT}}\hspace{0.17em}$ (24).

$$F_{N,z}=a{\left|\mathrm{\Delta }Z\right|}^b$$ (19)

$$F_{s,x}=-\min \left\{K_x\Delta X,\hspace{0.17em}{F}_{\mathrm{slip},x}\mathrm{sign}\left(\mathrm{\Delta }\mathrm{X}\right)\right\}$$ (20)

$$F_{s,y}=-\min \left\{K_y\Delta Y,F_{\mathrm{slip},y}\mathrm{sign}\left(\mathrm{\Delta }\mathrm{Y}\right)\right\}$$ (21)

$$F_{\mathrm{slip},x}={\mu }_x{F}_{N,z}$$ (22)

$$F_{\mathrm{slip},y}={\mu }_y{F}_{N,z}$$ (23)

$${\mathit{W}}_{\mathbf{FT}}=\left[\begin{array}{c}{\mathit{F}}_{\mathbf{FT}}\\ \mathit{r}\times {\mathit{F}}_{\mathbf{FT}}\end{array}\right]$$ (24)

$${\mathit{F}}_{\mathrm{FT}}=\left[\hspace{0.17em}\begin{array}{c}{F}_{s,x}\\ F_{s,y}\\ F_{N,z}\end{array}\right]$$

$$\mathit{r}=\left[\begin{array}{c}-\Delta X\\ -\Delta Y\\ r_{ft}-\Delta Z\end{array}\right]$$

The contact model parameter values along with a summary of all constant value parameters are shown in Table 3.

Dynamic.

The dynamic parameters include viscous damping, translational masses, and joint inertias. While this model will be used for quasi-static simulations in this paper, the dynamic parameters can also influence steady-state response due to the complex nonlinearities in the model. The viscous friction at the joints is minimal but was estimated in Ref. [39] using a simplified pendulum characterization. As discussed before, the values for the translational masses are calculated using impedance matching [39].

The configuration of the skeletal structure leads to specific inertial properties. The distal bones are supported by bones proximal to it. This skeletal structure of the finger follows the kinematics of a serial manipulator. As with serial manipulators, the finger has a pose-dependent mass matrix. Since the finger skeletal configuration is comparable to a three-link serial manipulator, textbook robotics techniques [38] can be applied to model these configuration-dependent inertias. This requires knowledge of the inertia of each link about its center of gravity and the location of the center of gravity for each link. These values were estimated from mass properties of a 3D CAD model (Solidworks, Dassault Systems, Velizy-Villacoumblay, France) of the UART finger. Inertial coupling between each joint, Coriolis, and centripetal effects on joint acceleration were not included in this model, assuming low joint velocities and accelerations [38]. In the context of our quasi-static simulations, these inertial effects would be negligible, but could be included if the model is used for simulating dynamic trajectories in the future.

Experimental Setup.

The model was validated with two types of experiments. First, the model was validated in terms of kinematic simulation, free of external contact forces. The second experiment involved quasi-static loading and motion of the finger while in contact with the environment. The testbed setup for both types of experiments is shown in Fig. 12.

Fig. 12.

Experimental setup of the finger used in the free motion and static contact experiments. The free motion experiments began at the flex pose of the finger and moved as a single tendon was pulled. The finger did not contact the environment. The static finger experiments began at two different finger poses, Hook and Curl. The surface angle was varied for each pose and primary flexor tendon combination. The global Cartesian frame (X, Y, Z) represents the Cartesian frame of the finger. The surface frame ($n_z,s_y,s_x$) represents the normal and two shear forces that are relative to the surface angle.

Free Motion.

First, the kinematic simulation capabilities of the model were validated. The UART testbed was placed in a flexed (Flex) position. With all tendons connected to their respective SEAs, one of the tendons was pulled 15 mm (at servo output). For reference, the flexor and extensor tendons of a normal human hand move approximately 15 mm per 100 deg of MCP flexion [33]. The tendons were actuated over a four-second time span using a sigmoid trajectory to minimize the sharp dynamic response of steps and ramp inputs. This was repeated five times for each input tendon. The resulting finger motion was recorded using MOCAP and used for comparison with the simulated finger motion.

Surface Contact.

To simultaneously simulate contact forces and finger pose, the fingertip must be able to move through space without hard constraints. In the past, static force simulations have been done by pinning or fixing the fingertip to the environment [20,26,45,47,48]. However, this is not a natural constraint for humans interacting with the environment. In this research, unconstrained surface contact will be modeled and reproduced experimentally. In this case, the finger can move across the surface if static friction limits are exceeded. This surface contact simulation will also be more valuable when applying the model to human biomechanical simulations since humans interact with the world by touching or sliding their fingers across objects.

The surface contact experiments were setup by placing the finger in an initial orientation touching the surface, with initially small tensions in all the input tendons (Fig. 12). Then two tendons were pulled. To ensure that the finger remained in contact with the surface, a flexor tendon (either FDP or FDS) was pulled 15 mm (at servo output). A second tendon that connects to the EM (ED, DI, PI, or LUM) was pulled 7.5 mm. The tendons were actuated over a four-second time span using a sigmoid trajectory to minimize the dynamic response of steps and ramp inputs. This resulted in a quasi-static pose and force trajectory. The other four tendons had a passive elastic connection to their SEA.

Two different initial poses were explored: an extended hook pose (Hook) and a curled pose (Curl) as seen in Fig. 12. For each pose, every combination of a single flexor (FDP or FDS) and EM tendons (ED, DI, PI, or LUM) were actuated, resulting in 16 different tendon loading trials. This was repeated five times for each combination. The surface angle varied between flexor-pose combination pairs to keep the motion quasi-static. The surface angles for the FDP-Hook and FDP-Curl trials were 45 deg and 67.5 deg, respectively. The surface angles for the FDS-Hook and FDS-Curl trials were 22.5 deg and 45 deg, respectively.

Results

Using the model and parameters described above, the motion and contact forces of the finger were simulated for both the free motion and surface contact experiments. Allowing for noise in measurements, the initial conditions of the simulation were estimated from the average measured initial joint angles, input tendon forces, and contact forces. The internal EM tendon segment tensions were initialized based on balancing forces at the tendon junctions. All distal segments were assumed to have equal tensions summing to the tension of the proximal tendon. All velocities were assumed to be initially zero. For the contact experiments, the position of the surface was assumed to be located at the fingertip with initial fingertip deflection imposed.

The inputs to the model were the displacements of the input tendons. Following the sigmoid actuation trajectory of the experimental trials, the input trajectories for each tendon loading and pose combination were replicated. Each simulation was finalized at steady-state. Steady-state was assumed when all joint angle velocities were less than 0.1 deg/s. The resulting steady-state finger joint angles and contact forces were compared to the experimental measurements.

Free Motion.

A summary of steady-state poses of the free motion experiments is shown in Table 4. The table contains the steady-state mean absolute joint angle error for each tendon loading trial and the standard deviation (SD) of the error for each joint in parenthesis. The resulting finger pose is also shown graphically in Fig. 13. The average experimental resulting steady-state pose for each free motion loading trial is shown in solid lines. The simulated steady-state pose is shown in dashed lines. The mean absolute error across all trials for the joints are 5 deg, 3.9 deg, 5.6 deg, and 2.2 deg for the DIP, PIP, MCP-FE, and MCP-AB joints, respectively. Many of the joint angle errors were near or within the SD of the measured steady-state pose. This suggests good agreement between simulated and experimental finger motion.

Table 4.

Steady-state free motion pose comparison

Free motion	Mean absolute steady-state joint angle error, deg ($\mathrm{SD}$)
Trial	DIP	PIP	$\mathrm{MC}{\mathrm{P}}_{\mathrm{FE}}$	$\mathrm{MC}{\mathrm{P}}_{\mathrm{AB}}$
ED	4.65 (1.35)	4.53 (3.47)	4.85 (1.00)	1.08 (1.02)
FDP	3.36 (4.38)	7.35 (6.40)	8.38 (8.57)	5.27 (3.68)
FDS	10.35 (7.78)	1.81 (1.92)	7.20 (7.55)	3.00 (2.71)
DI	0.88 (0.86)	4.59 (3.89)	4.67 (4.18)	2.39 (2.27)
PI	2.39 (2.06)	3.23 (4.20)	2.33 (2.79)	0.52 (0.60)
LUM	8.79 (7.03)	2.01 (1.18)	6.08 (1.35)	1.00 (1.26)

Fig. 13.

Average experimental (solid) and the simulated (dashed) steady-state free motion poses for each free motion trial labeled at the fingertip. The Flex label signifies the starting position for all trials. The left plot is the finger pose in the X-Y global plane; the right plot is in the Z-Y global plane as described in Fig. 12.

Surface Contact.

A summary of steady-state pose comparisons for all sixteen tendon/pose trials is shown in Table 5. The table contains the steady-state mean absolute error of the finger joint angles for each trial. They are accompanied by the SD of each experimental trial for comparison in parenthesis. The mean absolute error across all trials for the joints are 3.1 deg, 2.9 deg, 1.5 deg, and 4.5 deg for the DIP, PIP, MCP-FE and MCP-AB joints, respectively. As with the free motion experiments, many of the experimental joint angles were within the SD of the measured static pose. The surface contact motion has smaller errors due to both the smaller total motion and the added position constraint of the surface.

Table 5.

Steady-state surface contact pose comparison

Tendon input trial		Mean absolute steady-state joint angle error, deg ($\mathrm{SD}$)
Flexor $({\theta }_s$)	EM	DIP	PIP	$\mathrm{MC}{\mathrm{P}}_{\mathrm{FE}}$	$\mathrm{MC}{\mathrm{P}}_{\mathrm{AB}}$
Hook
FDP $(45\hspace{0.17em}\mathrm{deg}$)	ED	6.1 (1.3)	5.6 (1.0)	2.3 (0.5)	1.4 (1.9)
	DI	2.1 (1.2)	4.3 (1.3)	1.9 (1.0)	1.8 (0.8)
	PI	8.1 (3.5)	8.8 (2.2)	3.0 (0.6)	1.4 (1.6)
	LUM	6.2 (4.6)	4.5 (2.4)	1.6 (0.2)	0.7 (0.9)
FDS $(22.5\hspace{0.17em}\mathrm{deg}$)	ED	3.4 (3.9)	2.5 (2.5)	1.0 (1.0)	0.9 (0.9)
	DI	2.0 (2.3)	1.5 (1.7)	1.4 (1.0)	9.6 (0.8)
	PI	1.2 (1.5)	2.1 (2.0)	2.0 (2.0)	1.6 (1.3)
	LUM	6.0 (6.6)	3.2 (3.0)	1.6 (0.7)	10.8 (1.3)
Curl
FDP $(67.5\hspace{0.17em}\mathrm{deg}$)	ED a	1.1 (1.5)	0.8 (0.8)	0.6 (0.7)	2.0 (1.5)
	DI	1.4 (1.7)	1.3 (1.3)	1.1 (1.0)	9.6 (0.9)
	PI	2.2 (2.5)	1.7 (1.4)	1.5 (1.2)	6.3 (0.5)
	LUM	2.1 (2.5)	0.9 (1.1)	1.2 (1.4)	11.0 (1.3)
FDS $(45\hspace{0.17em}\mathrm{deg}$)	ED	3.9 (2.4)	2.8 (1.2)	0.9 (0.5)	0.9 (0.7)
	DI	3.1 (1.3)	4.8 (1.2)	2.1 (0.8)	6.8 (0.3)
	PI	1.1 (1.2)	1.8 (2.1)	1.1 (1.4)	6.6 (0.5)
	LUM	0.8 (0.6)	1.3 (1.1)	0.8 (0.7)	1.0 (0.9)

^aTrial dataset shown in dynamic plots.

A summary of the contact forces is shown in Table 6. This table contains the steady-state error between the average measured fingertip force and the simulated forces in the global Cartesian frame (Fig. 12). The vector of the simulated fingertip force is also examined in terms of error in magnitude and direction compared to the experimental data. The average force magnitude error across all trials is 7.6% and the average direction error is 11 deg. There are a few outliers, but the result shows the model successfully estimates the contact forces at steady-state.

Table 6.

Steady-state contact force comparison

Tendon input trial		$F_{\exp }-F_{\mathrm{sim}},$ N Mean (SD)			Force vector error
Flexor $({\theta }_s$)	EM	X	Y	Z	${‖F‖}_{\mathrm{err}\hspace{0.17em}}\hspace{0.17em}(\%)$ b	${\varphi }_{\mathrm{err}}\hspace{0.17em}\left(\mathrm{deg}\right])$ c
Hook
FDP $(45\hspace{0.17em}\mathrm{deg}$)	ED	0.09 (0.07)	−0.17 (0.03)	0.00 (0.01)	8.9	17.4
	DI	0.01 (0.08)	−0.05 (0.04)	0.13 (0.02)	8.1	11.6
	PI	0.13 (0.08)	−0.10 (0.06)	0.06 (0.03)	4.8	15.1
	LUM	0.07 (0.01)	−0.07 (0.05)	0.15 (0.02)	4.3	14.9
FDS $(22.5\hspace{0.17em}\mathrm{deg}$)	ED	−0.01 (0.00)	0.25 (0.05)	0.09 (0.02)	38.3	11.9
	DI	−0.04 (0.01)	0.14 (0.03)	0.23 (0.03)	9.9	22.4
	PI	−0.03 (0.01)	0.27 (0.03)	−0.04 (0.01)	35.5	8.0
	LUM	−0.03 (0.01)	0.14 (0.04)	0.25 (0.02)	9.4	23.3
Curl
FDP $(67.5\hspace{0.17em}\mathrm{deg}$)	ED a	0.03 (0.04)	−0.07 (0.04)	−0.04 (0.01)	5.0	10.1
	DI	−0.07 (0.01)	−0.07 (0.01)	−0.03 (0.01)	11.5	5.9
	PI	0.10 (0.01)	−0.01 (0.03)	−0.23 (0.01)	9.0	21.3
	LUM	−0.04 (0.02)	−0.04 (0.02)	−0.04 (0.02)	7.1	4.8
FDS $(45\hspace{0.17em}\mathrm{deg})$	ED	−0.01 (0.01)	0.00 (0.02)	−0.01 (0.02)	1.8	4.4
	DI	0.02 (0.01)	−0.02 (0.03)	0.01 (0.01)	1.2	4.0
	PI	0.12 (0.01)	0.15 (0.01)	−0.17 (0.01)	22.2	18.2
	LUM	0.07 (0.01)	0.02 (0.01)	0.00 (0.01)	7.3	3.9

^aTrial dataset shown in dynamic plots.

^bForce magnitude error: ${‖F‖}_{\mathrm{err}}=\left|\frac{‖F_{\exp }‖-‖F_{\mathrm{sim}}‖}{‖F_{\exp }‖}\right|$.

^cForce direction error: ${\varphi }_{\mathrm{err}}={\text{cos}}^{-1}\left(\frac{F_{\mathrm{sim}}·F_{\exp }}{‖F_{\exp }‖‖F_{\mathrm{sim}}‖}\right)$ .

While the focus of this work was quasi-static pose and contact force estimation, the transient response can also be examined to discuss dynamic behavior. The transient response of the surface contact trial where FDP flexor and ED extensor were pulled in the curled pose will be examined to discuss the dynamic functionality of the model.

The input tendon tension response for the simulated and experimental systems is shown in Fig. 14. The simulated response of the two active tendons matches the experimental response with minor differences. The passive tendons have minor differences.

Fig. 14.

Average experimental (solid) and the simulated (dashed) input tendon tension for trial 3 with ED tendon over time. Actively actuated tendons are shown on the top axis, and the passively connected tendon tensions are shown on the bottom axis.

The joint angle responses of the simulated and experimental systems are compared in Fig. 15. Comparing the pose response shows that the simulation correctly estimates the static pose of the finger with minor differences. The simulated MCP's ab-adduction angle did not move compared to the experimental, while the DIP flexion angle moved further than experimental.

Fig. 15.

Average experimental (solid) and the simulated (dashed) joint angles for trial 3 with ED tendon over time

The contact forces of the fingertip, shown in the global Cartesian frame, are shown over time in Fig. 16. While the simulated forces begin at the measured initial forces, the finger immediately diverges from the initial condition by a small amount due to minor errors in modeling, resulting in a different static equilibrium. During actuation, they follow similar force trajectories and reach a steady-state near the experimental forces.

Fig. 16.

Average/SD experimental (solid/blur) and simulated (dashed) contact forces in the global reference frame for Trial 3 with ED tendon over time

Discussion

Experimental and simulated motion of a human like finger was compared in both free motion and surface contact. Results show good agreement for both the free motion and surface contact experiments in terms of quasi-static pose and fingertip force estimation. Some trials show more errors than others. The transient responses of the surface contact simulation show general agreement with the measured finger response with some evidence of error, most notably, in the fingertip force and tendon tension results. These discrepancies between the model and the UART finger may be caused by our limitations and assumptions.

One limitation between the model and the robotic finger is the possible inaccuracies in the variable moment arm functions. Both static pose and contact forces are heavily dependent on the accuracy of the variable moment arms. Any inaccuracies in the neural net fitting or geometric modeling, as discussed in the Characterization section, could have significant effect on the agreement between the two systems. It is unknown how much this limitation contributed to the error; however, the authors believe these errors to be minimal based on qualitative comparisons between the variable moment arm functions and the UART finger. This error is also suspected to be minimal given that the free motion experiments show minimal error. The error from initial tension estimation in the EM and the simplicity of the joint friction and contact force models are more likely causes of simulation error.

During the initialization of the dynamic simulation, the state variables are estimated as the average initial experimental measurements. While the input tendon force, joint angles, and fingertip contact are easily measured, the forces of the interconnected EM are not measurable. It was assumed that each tendon segment of the EM shared the load from the input tendon. This, however, cannot be verified. Whether these segments are slack or not is also not verifiable. This error in initial conditions could affect how the finger begins motion and subsequently reaches steady-state.

The joint friction model utilizes a simple static friction and viscous damping model. The static friction coefficient is both constant (invariable to joint position or velocity) and is estimated the same for all joints. Errors in this static friction model could contribute to how the finger moves in response to tendon tension and displacements. This would have larger implications for the transient response of the finger. This would subsequently affect how the impedance fingertip forces are generated. Both the free motion and the surface contact models have some evidence of error, but the contribution of joint friction modeling is unknown. This is suspected to be the largest contribution to the outliners in the surface contact force results (Table 6).

The largest suspected source of error for the surface contact experiments is from the simplicity of the contact model. This model only assumes translational forces with simple springs and force thresholds. The contact model neglects viscous effects that could be very important for a dynamic simulation and rotational effects that could be significant as the contact surface of the fingertip increases due to larger normal forces. In this static evaluation, the effects seem to be minimal; however, those trials with larger force direction error have larger force error in the Z direction. This force error largely corresponds to error in the MCP-AB joint estimation. The ab/adduction angle also has the most error on average. This resulting error in joint angles and force estimation suggests that the finger is not being impeded correctly as it tries to move and pivot across the surface.

The transient performance of the simulation results demonstrates the potential applications to more dynamic movement and contact situations. The general trajectories of the state variables match the measured values but do vary in the transient portion of the response. This could be due to the inaccuracies of the dynamic parameters like inertia and friction. More accurate parameters will be needed for more dynamic applications. However, the general agreement between the simulation and UART finger suggests that the model could be used in future dynamic applications.

Given the presented limitations, the bond graph tendon model can still accurately simulate tendon-driven finger motion during both free motion and external contact. The model was able to accurately predict the static equilibrium pose and contact forces while touching a surface in a curled and extended hooked posture. Additionally, the model has been shown to roughly simulate the transient response of a human like finger tendon system. These results demonstrate the forward simulation capabilities of the model for both free motion and surface contact application.

Conclusion

This works presents a modeling framework applied to a human central finger. The bond graph modeling method provided generalized models for tendon-driven finger inputs, joints, and fingertip contact. A specific bond graph was created for the interconnections of the extensor mechanism. Combining these models with nonlinear constitutive relationships provides a forward dynamics simulation of the human finger-tendon system. This model was validated using a biomimetic robotic testbed, the UART finger. Quasi-static experiments and simulations demonstrated the model's ability to simultaneously simulate kinematics and fingertip contact forces given tendon inputs, which has not been previously achieved by any finger tendon model. While the goal for now is to simulate quasi-static trajectories of the finger while in contact with the environment, the model could in the future be calibrated and validated for more dynamic trajectories. The current validation of the model is limited to human-like robotic tendon fingers, but the model could also be calibrated to match cadaver experiments given correct biological parameter identification of the human hand. In the future, we plan to use this model (along with both robotic and cadaveric validation) to simulate outcomes of tendon reconstructive surgeries, by including more digits and muscle activation in the model. This model could also be used as a tool for design of bioinspired robotic fingers and hands. This work represents a first step in combining tendon kinematics, statics, and complexities into one complete model, which could lead to a complete interconnected model for the human hand.

Funding Data

• Funding Incentive Seed Grant from the University of Utah and in part by the Nation Institute of Arthritis and Musculoskeletal and Skin Diseases (NIH Award No. R21AR076269; Funder ID: 10.13039/100000069). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Nomenclature

$b_j$ =

viscous damping coefficient for joint $j$

$F_{N,z}$ =

fingertip force normal to surface

$F_{s,x}$ =

shear fingertip force in the x direction

$F_{s,\mathrm{y}}$ =

shear fingertip force in the y direction

$F_{\mathrm{slip},x}$ =

fingertip static friction threshold in x directions

$F_{\mathrm{slip},y}$ =

fingertip static friction threshold in y direction

$F_T$ =

force in tendon segment, $T$

$I_j$ =

rotational inertia of joint $j$

${\mathit{J}}^T$ =

kinematic Jacobian of the finger skeleton referenced to the fingertip

$k_T$ =

translational stiffness for tendon segment, $T$

$K_j$ =

rotational joint stiffness for joint, $j$

$m$ =

intermediate mass of tendon intersections

$r_{T,j}$ =

moment arm for tendon segment, $T$, that passes over joint, $j$

${\mathit{W}}_{\mathbf{FT}}$ =

wrench, force and moment, matrix applied at the fingertip

${\dot{x}}_T$ =

translational velocity of tendon segment, $T$

${\theta }_j$ =

joint angle for joint $j$

${\dot{\theta }}_j$ =

joint rotational velocity for joint $j$

${\mu }_c$ =

Columbic joint friction coefficient

${\tau }_c$ =

static joint friction threshold torque

${\tau }_{\mathrm{EXT},j}$ =

external torque applied at joint, $j$

${\mathit{\tau }}_{\mathbf{FT}}$ =

joint torque vector generated by the fingertip contact

${\mathit{\tau }}_{\mathit{G}}$ =

joint torque vector generated by the gravitational forces on the finger

${\tau }_{K,j}$ =

torque supplied by the rotational joint stiffness, $K$, at joint, $j$