Constraint Study for a Hand Exoskeleton: Human Hand Kinematics and Dynamics

Abstract

In the last few years, the number of projects studying the human hand from the robotic point of view has increased rapidly, due to the growing interest in academic and industrial applications. Nevertheless, the complexity of the human hand given its large number of degrees of freedom (DoF) within a significantly reduced space requires an exhaustive analysis, before proposing any applications. The aim of this paper is to provide a complete summary of the kinematic and dynamic characteristics of the human hand as a preliminary step towards the development of hand devices such as prosthetic/robotic hands and exoskeletons imitating the human hand shape and functionality. A collection of data and constraints relevant to hand movements is presented, and the direct and inverse kinematics are solved for all the fingers as well as the dynamics; anthropometric data and dynamics equations allow performing simulations to understand the behavior of the finger.

1. Introduction

The human hand is a complex mechanism; it has a wide range of DoFs, allowing a great variety of movements. In recent years, as robotics has advanced, significant efforts have been devoted to the development of hand devices. The two main related application fields are prosthetic/robotic hands and exoskeletons. On one side, robotic hands are developed with the characteristics complying to those of the human hand, taking advantage of its variety of movements, thereby avoiding the use of a large number of end effectors when performing tasks with different objects (e.g., Eurobot [1], Robonaut [2]). On the other side, exoskeletons are designed to fit onto the human hand, aiming at enhancing performance in the carrying out of daily activities (e.g., improving astronauts’ hand performance during extravehicular activity [3]) or supporting the rehabilitation stage of hand injury recovery.

There are currently many different projects underway. Schabowsky et al. [4] introduced a newly developed Hand Exoskeleton Rehabilitation Robot (HEXORR), which was designed to provide a full range of motion for all fingers. NASA and General Motors presented a prototype of the Human Grasp Assist device [5] (K-Glove). Worsnopp et al. [6] introduced a finger exoskeleton for hand rehabilitation following strokes, to facilitate movement, especially pinch. Another project is being developed by Ho et al. [7]: their exoskeleton hand is EMG driven, again for rehabilitation, but working on all the fingers. All of these projects present a different number of DoFs and different structures, but in general they are developed with the objective of mimicking the main characteristics of the human hand. This implies a complete understanding of these characteristics, involving the anthropometric dimensions of the human hand, its kinematics, and its dynamics.

This paper aims at analyzing all the aforementioned aspects. It is organized as follows: Section 2 presents anthropometric data about the hand, such as dimensions of the hand and phalanges; Section 3 contains the main constraints of finger movements, explaining each type of constraint in the natural movement of the human hand; Section 4 describes the kinematical model of the hand; direct and inverse kinematics are developed step by step. Section 5 presents the dynamics of a single finger; Section 6 shows the implementation of the dynamic equations on a practical example. Finally, conclusions are presented in Section 7.

2. Human Hand Data

The human hand is composed of 5 digits: 4 fingers (index, ring, middle, and little fingers) and the thumb. The thumb is characterized by three articulations and three phalanges. The fingers also comprise three different articulations and four phalanges. Figure 1 shows names and acronyms of each articulation and phalanx. The wrist has two functional DoFs. The TMC joint of the thumb holds two DoFs (flexion/extension and adduction/abduction). A single DoF (flexion/extension) characterizes the MCP and IP joints of the thumb as well as the PIP and DIP joints of the fingers. Whereas the eight bones of the carpus articulate finely with each other producing small deformations, their representation in a single rigid segment is a consistent approximation [8].

Figure 1 

Anatomical details of the hand skeleton.

The analysis of kinematics and dynamics requires knowledge regarding the dimensions of the fingers and of the palm and their respective range of motion (RoM); those data are reported below. Tables 1 and 2 show the results of Garrett’s studies [9, 10] for finger lengths and palm dimensions, measurements that were taken from the right hands of 148 men and 211 women. In Table 1, crotch to tip is the distance along the axis of the digit from the midpoint of its tip to the level of the corresponding webbed crotch between two digit; wrist crease to tip is the distance along the axis of the digit from the midpoint of its tip to the wrist crease baseline.

Table 1 

Mean finger lengths and palm dimensions of USAF male (M)/female (F) flying personnel [9, 10] (cm).

Table 2 

Average hand dimensions of USAF male (M)/female (F) flying personnel [9, 10] (cm).

A few researchers have measured the length of each phalanx separately. A study with a variety of candidates is the one performed by Habib and Kamal [11]. The results of their study for each phalanx of index, middle, ring, and little fingers are in Table 3 (I1 means distal phalanx, I2 middle phalanx, and I3 proximal phalanx of the index finger. The same notation is used for other digits). A similar survey is also present in Jasuja and Singh’s study [12]. As shown in Table 3, the average dimensions of the hand are quite similar to the ones presented in Table 2, with a maximum difference of 2.18%. The data presented here is the sample distribution over geographic regions and Air Force Commands, which may be quite representative of possible EVA glove users.

Table 3 

Mean length of the hand and of the phalanges of index, middle, ring, and little fingers (cm).

As mentioned above, those data can give us an idea of the mean values of the length of each element that composes the human hand. With proper modifications, this data can also be used to compose a model to simulate its movements: in particular, it must be taken into account that the distance between a joint and a digit webbing is not representative of the equivalent link length. A simulation was performed, and the results are presented in Section 6.

3. Constraints Overview

Hand and digit motions are subject to several constraints that limit the range of the natural movements of human fingers. Constraints can be roughly divided into three types: static, intrafinger, and interfinger constraints. Intra- and inter-finger, constraints are often called dynamic constraints that in the fingers of the hand are essentially constraints between joint motions. However, this range of movement is somewhat ambiguous because the range depends on various factors involving human hand biomechanics; therefore they are difficult to be expressed in closed forms (i.e., equations) and how to model such constraints still needs further investigation. The significance of taking finger constraints into consideration is that this causes the DoFs of the human hand to decrease. It is important to underline the fact that different individuals can show significant variations regarding the following constraints.

3.1. Intrafinger Constraints

Intrafinger constraints are constraints between different joints in the same finger. Cobos et al. [14] presented several constraints for fingers and thumb. Equation (1) presents the relation between the joints of a finger, as first proposed by Rijpkema and Girard [15]: 𝜃_DIP≈23𝜃_PIP𝜃_PIP≈34𝜃_MCP𝑓/𝑒. (1)

In (2), the constraint relations between the joints when flexing or extending the thumb are 𝜃_IP≈12𝜃_MCP𝑓/𝑒,𝜃_MCP𝑓/𝑒≈54𝜃_TMC𝑓/𝑒. (2)

These constraints are not strict. In fact, there are individuals who are more able than others to control their DIP joint, and anybody can force a behavior slightly out of these equations for any joint. However, in normal conditions those constraints are respected quite faithfully.

It is important to underline that the constraints mentioned above are due to the physiologic nature of the human hand. Additionally, other constraints can be considered, due to ergonomics. In this case, the relations of the constraints are related to a specific task, such as grasping a circular or a prismatic object.

Table 4 shows the typical ergonomic constraints relative to a circular grasping.

Table 4 

Intrafinger constraints for a circular grasp.

Therefore the thumb is defined by 2 DoFs, index finger by 2 DoFs, middle finger by 1 DoF, ring finger by 1 DoF, and little finger by 3 DoFs. The ring finger is calculated on the basis of the little and middle fingers joints. Therefore, the thumb, the index finger, and the little finger are the most important fingers when defining circular grasps [13].

Table 5 shows the ergonomic intrafinger constraints for a prismatic grasp. As a result, the thumb is defined by 2 DoFs, index finger by 2 DoFs, middle finger by 1 DoF, ring finger by 1 DoF, and little finger by 3 DoFs. Comparing the circular grasping and the prismatic one, the latter presents fewer constraints among fingers.

Table 5 

Intrafinger constraints for a prismatic grasp.

3.2. Interfinger Constraints

Interfinger constraints correlate two joints belonging to different fingers. For instance, with the hand in the open position (at rest), when one bends the index finger at the MCP joint, the MCP joint of the middle finger bends automatically as well, with respective certain proportionality.

As for intra-finger constraints, different individuals can show important differences. Moreover, also in this case one can forcefully overcome some of these constraints.

On the other hand, there are constraints that cannot be explicitly represented in equations. A few cases are explained below.

Some joints belonging to different fingers seem to be “naturally” interconnected. By that, we mean that the respective angles vary more or less proportionally, that is, unless a voluntary counter force is applied, imposing a different type of motion. However, trying to move a finger in such an unnatural way often results in oddly fatiguing efforts.

For instance, there is a coupled movement when the index finger and little finger are at rest. The flexion of the middle finger is equal to the flexion of the ring finger as described in 𝜃_{MCP𝑓/𝑒Middle}≈𝜃_{MCP𝑓/𝑒Ring}. (3)

Another coupled movement is produced when flexing the ring finger, thereby causing a slight flexion on the middle finger and on the little finger, that happens to be the same angle rotation as described in 𝜃_{MCP𝑓/𝑒Middle}≈𝜃_{MCP𝑓/𝑒Little}. (4)

There is also a coupled movement in abduction/adduction, which is generated among ring and little fingers. In most cases, the movement is similar to 𝜃_{MCP𝑎𝑏/𝑎𝑑Ring}≈𝜃_{MCP𝑎𝑏/𝑎𝑑Little}. (5)

Additionally, some joints can be tied by more intricate relations, divided into two types. The first type occurs when there is a unique motion, like the flexion in the MCP joint of the little finger. Equation (6) represents this type of relation between the fingers 𝜃_{MCP𝑓/𝑒Ring}≈7𝜃12^{MCP𝑓/𝑒Little},𝜃_{MCP𝑓/𝑒Middle}≈23𝜃_{MCP𝑓/𝑒Ring},𝜃_{MCP𝑓/𝑒Ring}−𝜃_{MCP𝑓/𝑒Middle}<60^∘,𝜃_{MCP𝑓/𝑒Little}−𝜃_{MCP𝑓/𝑒Ring}<50^∘. (6)

These equations indicate that when there is flexion, the ring and middle fingers will also flex in a certain range with respect to the little finger. For instance, when the little finger is flexed, the middle finger will flex 2/3 times the ring finger.

The second relationship is when there is a simple MCP flexion as in the case of the index finger, thereby causing some additional natural movement. Equation (7) represents this type of relation between the index and the middle fingers 𝜃_{MCP𝑓/𝑒Middle}≈15𝜃_{MCP𝑓/𝑒Index}. (7)

This relation is only present when there is a single flexion on the index finger MCP joint, so when it occurs, the middle finger MCP joint will flex naturally and passively with this proportionality.

These constraints are important because they express the natural relation between single joints. Thus devices designed to mimic the human hand must comply as precisely as possible with these constraints.

3.3. Static Constraints

The normal range of motion (ROM) of human hand joints corresponds to static constraints on joint angles in the model. These constraints are limits on the values that the 𝜃 parameters can assume. Main static constraints (Table 6) were collected by Cobos et al. [13]. By applying these constraints to the inverse kinematics presented later in Section 4.2, some DoFs can be neglected, and therefore the complexity of the system can be reduced.

Table 6 

Statics constraints [13].

4. Kinematic Model

The kinematic model proposed here is composed of 19 links corresponding to the human bones and 24 DoFs modeled by rotational joints. Two different kinematic configurations are considered for the fingers, one for the thumb modeled as 3 links and 4 joints and another for the other fingers (index, middle, ring, and little fingers). Each of them is modeled using 4 links and 5 joints, as in Figure 2. Note that the CMC joint represents the deformation of the palm, for instance, when the hand is grasping a ball, while the MCP abduction/adduction joint is defined before MCP flexion/extension.

Figure 2 

Kinematic configuration of the human hand. The thumb is defined by 3 links and 4 degrees of freedom whereas index, middle, ring, and little fingers are defined by 4 links and 5 DoFs.

4.1. Direct Kinematics

Direct kinematic equations are used to obtain the fingertip position and orientation according to the joint angles. The model equations are calculated by means of modified Denavit-Hartenberg (MDH) parameters, introduced by Craig [16]. The difference from the Denavit-Hartenberg convention is the fact that, on MDH, the 𝑧-axis of the reference frame {𝑗}, called 𝑧_𝑗, is coincident with the axis of joint 𝑗. In comparison to MD the 𝑧-axis of the frame {𝑗} is coincident with the axis of joint 𝑗+1. The advantage of using MDH is that there is no need for further transformation of the references in order to work with the dynamics of the rigid bodies.

4.1.1. Direct Kinematics of the Index, Middle, Ring, and Little Fingers

Each of the fingers contains four bones: metacarpal, proximal, middle, and distal (Figure 1). These bones correspond approximately to the links of the serial kinematic chain (Figure 2). Each articulation presented above for these four fingers corresponds to the joints: CMC, MCP, PIP, and DIP. The MCP joint can be split into 2 DoFs, which carry out the adduction/abduction and flexion/extension movements, respectively. All the other joints only allow flexion/extension movements. Having defined a numbering of the fingers, from 0 to 4 where finger 0 is the thumb and finger 4 is the little finger, Table 7 shows the MDH parameters for fingers 1 to 4.

Table 7 

Modified D–H parameters for fingers 1 to 4.

Equation (8) shows the direct kinematics from index (𝑖=1) to the little finger (𝑖=4)𝑄_𝑖=⁰₀𝑇_𝑖⋅⁰₆𝑇_𝑖𝜃_𝑗=⁰₀𝑇_𝑖⋅⁵_𝑗𝑗=1^𝑗−1𝑇_𝑖𝜃_𝑗⋅⁵₆𝑇_𝑡𝑖,𝑄_𝑖=⁰₀𝑇_𝑖⋅⁰₁𝑇_𝑖𝜃_CMC𝑓/𝑒⋅¹₂𝑇_𝑖𝜃_{MCP𝑎𝑏/𝑎𝑑}⋅²₃𝑇_𝑖𝜃_MCP𝑓/𝑒⋅³₄𝑇_𝑖𝜃_PIP𝑓/𝑒⋅⁴₅𝑇_𝑖𝜃_DIP𝑓/𝑒⋅⁵₆𝑇_𝑡𝑖, (8) where(i)𝑄_𝑖 represents a matrix containing the position and orientation of the fingertip of each finger;(ii)⁰₀𝑇_𝑖 represents a rototranslation matrix taking into account the fact that the fingers are slightly fanned out and making it possible to pass from the initial base reference frame (𝑅₀) to the alignment of the 𝑖th finger first reference frame (𝑅_0𝑖);(iii)⁰₆𝑇_𝑖(𝜃_𝑗) is a matrix containing the geometrical transformation between the 𝑖th finger first reference frame and the 𝑖th fingertip (ft_𝑖). The matrix is composed of the concatenation of the transformation matrices of each finger link;(iv)_𝑗^(𝑗−1)𝑇_𝑖(𝜃_𝑗) is a matrix containing the geometrical transformation between the (𝑗−1)th reference frame and the 𝑗th reference frame of the 𝑖th finger;(v)⁵₆𝑇_𝑡𝑖 represents the position 𝑝_𝑡𝑖=𝑙_𝑡𝑖𝑥𝑙_𝑡𝑖𝑦𝑙_𝑡𝑖𝑧^𝑇 of the fingertip with respect to the distal (5th) reference frame.

𝑖 corresponds to index(1), middle(2), ring(3), and little(4) fingers. 𝑗 corresponds to each finger’s joint CMC_𝑓/𝑒, MCP_{𝑎𝑏/𝑎𝑑}, MCP_𝑓/𝑒, PIP_𝑓/𝑒, and DIP_𝑓/𝑒. 𝑡_𝑖 stands for the tip of the 𝑖th finger.

The coefficients 𝑄_𝑖𝑠𝑡, which are the elements of the 𝑠th row and 𝑡th column of the matrix expressed in (8), relative to the 𝑖th finger, are given in Appendix A.

4.1.2. Direct Kinematics of the Thumb

The thumb presents three bones (Figure 1): metacarpal, proximal, and distal. These bones correspond approximately to the length of each link. The respective joints are TMC, MCP, and IP. The TMC joint presents 2 DoFs, allowing adduction/abduction and flexion/extension. Table 8 shows the MDH parameters. Equation (9), with the same notation scheme as (8), shows the direct kinematics for the thumb 𝑄₀=⁰₀𝑇₀⋅⁰₅𝑇₀𝜃_𝑘,𝑄₀=⁰₀𝑇₀⋅⁰₁𝑇₀𝜃_{TMC𝑎𝑏/𝑎𝑑}⋅¹₂𝑇₀𝜃_TMC𝑓/𝑒⋅²₃𝑇₀𝜃_MCP𝑓/𝑒⋅³₄𝑇₀𝜃_IP𝑓/𝑒⋅⁴₅𝑇_𝑡0, (9) where 𝑘 corresponds to the thumb joint TMC_{𝑎𝑏/𝑎𝑑}, TMC_𝑓/𝑒, MCP_𝑓/𝑒, and IP_𝑓/𝑒. The position of the fingertip is 𝑝_𝑡0=𝑙_𝑡0𝑥𝑙_𝑡0𝑦𝑙_𝑡0𝑧^𝑇. The coefficients 𝑄_0𝑠𝑡 of the matrix expressed in (9) are given in Appendix A.

Table 8 

Modified D–H parameters of the thumb.

4.2. Inverse Kinematics

Inverse kinematics is used to obtain the joint angle values according to the fingertip position and orientation. The inverse kinematics will be solved separately for the thumb and for the other fingers. The model of the human hand is a redundant case; therefore several solutions to the inverse kinematic problem exist. In order to find a unique solution, it is necessary to take into account the constraints presented in Section 3.

4.2.1. Inverse Kinematics of a Finger (Index (𝑖=1) to Little Finger (𝑖=4))

The angles 𝜃_{𝑖_CMC𝑓/𝑒}, 𝜃_{𝑖_MCP𝑎𝑏/𝑎𝑑}, 𝜃_{𝑖_MCP𝑓/𝑒}, and 𝜃_{𝑖_PIP𝑓/𝑒}, 𝜃_{𝑖_DIP𝑓/𝑒} are obtained from (8), where the matrix 𝑄_𝑖, that is, the left member of the equation, is known. Hence, algebraically we solve the inverse kinematics for the CMC_𝑓/𝑒, MCP_{𝑎𝑏/𝑎𝑑}, and DIP_𝑓/𝑒 joints. The procedure here shown solves the kinematics of the index finger (𝑖=1); however, it is valid for every other finger except the thumb (𝑖=2,3,4). For sake of brevity in the following, the 𝑖 index is omitted.

The duplication of solutions, when solving an arctangent, for example, is eliminated thanks to the constraints of Section 3.3, because the ranges of all physiological angles are smaller than 180° and most of them are smaller than 90°. This fact is implicitly taken into account in the following. 𝜃_CMC𝑓/𝑒𝑄=atan³³𝑄₁₃,𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑄=atan¹³−𝑄₂₃cos𝜃_CMC𝑓/𝑒,𝜃_DIP𝑓/𝑒𝜖=atan¹𝜖₂, (10) where 𝜖₁ and 𝜖₂ are expressed as 𝜖₁=(−1)⋅𝑚𝑄₂₂+𝑛𝑄₂₁𝑚²−𝑛²,𝜖₂=Q₂₁𝑚+𝑛𝑚⋅(−1)⋅𝑚𝑄₂₂+𝑛𝑄₂₁𝑚²−𝑛²,𝑚=sin𝜃_{MCP𝑎𝑏/𝑎𝑑}cos𝜃_MCP𝑓/𝑒cos𝜃_PIP𝑓/𝑒−sin𝜃_{MCP𝑎𝑏/𝑎𝑑}sin𝜃_MCP𝑓/𝑒sin𝜃_PIP𝑓/𝑒,𝑛=sin𝜃_{MCP𝑎𝑏/𝑎𝑑}cos𝜃_MCP𝑓/𝑒sin𝜃_PIP𝑓/𝑒−sin𝜃_{MCP𝑎𝑏/𝑎𝑑}sin𝜃_MCP𝑓/𝑒cos𝜃_PIP𝑓/𝑒. (11)

The MCP_𝑓/𝑒, PIP_𝑓/𝑒 joints are solved through a geometric method; see Figure 3.

Figure 3 

Inverse kinematics for the index finger and thumb.

Starting from the vector 𝐹₁ which contains the position of the fingertip, it is possible to obtain 𝐻₁ with the following expression: 𝐻₁=𝐹₁−𝐿₃₁∗𝑖̂. (12) Then, the vector 𝑃₁ is expressed as 𝑃₁=𝑃_1𝑥𝑃_1𝑦𝑃_1𝑧^𝑇, (13) where 𝑃_1𝑥=𝐿₀₁cos𝜃_CMC,𝑃_1𝑦=𝐿₀₁sin𝜃_CMC,𝑃_1𝑧=0. (14) Knowing 𝐻₁ and 𝑃₁ from (12) and (13), the vector 𝑢₁ is expressed as 𝑢₁=𝐻₁−𝑃₁. (15) Angles 𝜑₁ and 𝜑₂ are expressed as 𝜑₁𝑟=acos²²+𝐿²₀₁−𝑟²₁2𝑟₂𝑙₀₁,𝜑₂𝑟=acos²²+𝐿²₁₁−𝐿²₂₁2𝑟₂𝐿₁₁ (16) with variables 𝑟₁ and 𝑟₂ equal to 𝑟₁=𝐻₁,𝑟₂=𝑢₁. (17) So 𝜃_MCP𝑓/𝑒 is obtained as 𝜃_MCP𝑓/𝑒=𝜋−𝜑₁−𝜑₂. (18) And 𝜃_PIP𝑓/𝑒 is obtained as 𝜃_PIP𝑓/𝑒=𝜋−𝜑₃, (19) where 𝜑₃𝐿=acos²²¹+𝐿²₁₁−𝑟²₂2𝐿₂₁𝐿₁₁. (20)

4.2.2. Inverse Kinematics of the Thumb

A similar procedure can be applied to the thumb, in which 𝑄₀, that is, the left side of (9), is known; then the joint angle 𝜃_{TMC𝑎𝑏/𝑎𝑑} is obtained algebraically as follows: 𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑄=atan¹³−𝑄₂₃, (21) where the 𝑄_𝑠𝑡 term in (21) is the element of the 𝑠th row and 𝑡th column of the 𝑄₀ matrix. Then, using the geometrical method (Figure 3), the joint angles 𝜃_MCP𝑓/𝑒 and 𝜃_IP𝑓/𝑒 can be obtained: 𝜃_MCP𝑓/𝑒=𝜋−𝛾₁,𝜃_IP𝑓/𝑒=𝜋−𝛾₂−𝛾₃, (22) where 𝛾₁𝐿=acos²¹⁰+𝐿²₀₀−𝑟²₃2𝐿₁₀𝐿₀₀,𝛾₂𝐿=acos²²⁰+𝑠²₃−𝑟²₄2𝐿₂₀𝑟₃,𝛾₃𝑟=acos²³+𝐿²₁₀−𝐿²₀₀2𝑟_𝑠𝐿₁₀,𝑟₃=𝐻₀,𝑟₄=𝐹₀,𝐻₀=𝐹₀−𝐿₂₀∗𝑖̂. (23) The joint 𝜃_TMC𝑓/𝑒 is obtained algebraically as follows: 𝜃_TMC𝑓/𝑒𝜖=atan⁴𝜖₃, (24) where 𝜖₃=𝑄₃₂+𝑄₃₁𝜖cos𝜇,₄=𝐴₃₁−𝑄₃₁+𝑄₃₂cos𝜇⋅sin𝜇,cos𝜇𝜇=𝜃_MCP+𝜃_IP. (25)

5. Dynamics of a Single Finger

This section provides the dynamics equation system of a generic single finger, neglecting the MCP abduction/adduction motion. The 𝑖th index is again omitted. Based on the convention of Figure 4, the dynamic model is determined using Euler-Lagrange equations. It is important to highlight the fact that on the model the metacarpus is fixed, while only the finger phalanges are moving parts. Thus, 𝑅₂ is the base reference system, and all the equations are written with respect to 𝑅₂.

Figure 4 

Dynamic model of index(1).

The following equations are applicable to any 3R planar robot, after making the necessary adaptations to the variables.

The equations require the estimation of the kinetic energy and potential energy, as shown in the following.

In order to simplify the expressed vectors and further developments, the following notation is used: 𝜃₃=𝜃_MCP𝑓/𝑒,𝜃₄=𝜃_PIP𝑓/𝑒,𝜃₅=𝜃_DIP𝑓/𝑒. (26)

The kinetic energy is calculated starting from the position vectors of the center of mass of each phalanx with respect to the base reference frame 𝑅₂; in general, the position of the center of mass of the 𝑗th phalanx with respect to the 𝑗 reference frame is 𝐺_𝑔=𝑏_𝑔𝑥𝑏_𝑔𝑦0;𝑔=1,2,3. (27)

The mass of the 𝑗th phalanx is equal to 𝑚_𝑔, and the respective moment of inertia with respect to the axis 𝑧 is equal to 𝐼_𝑔.

Thus, each position vector with respect to the base reference frame 𝑅₂ is presented below, where 𝑐𝜃_𝑗 and 𝑠𝜃_𝑗 stand for cosine and sine of 𝜃_𝑗, respectively. Consider 𝐺₁=𝑏_1𝑥𝑐𝜃₃−𝑏_1𝑦𝑠𝜃₃𝑏_1𝑥𝑠𝜃₃+𝑏_1𝑦𝑐𝜃₃0,𝐺₂=𝐿₁𝑐𝜃₃𝐿₁𝑠𝜃₃0+𝑏_2𝑥𝑐𝜃₃+𝜃₄−𝑏_2𝑦𝑠𝜃₃+𝜃₄𝑏_2𝑥𝑠𝜃₃+𝜃₄+𝑏_2𝑦𝑐𝜃₃+𝜃₄0,𝐺₃=𝐿₁𝑐𝜃₃𝐿₁𝑠𝜃₃0+𝐿₂𝑐𝜃₃+𝜃₄𝐿₂𝑠𝜃₃+𝜃₄0+𝑏_3𝑥𝑐𝜃₃+𝜃₄+𝜃₅−𝑏_3𝑦𝑠𝜃₃+𝜃₄+𝜃₅𝑏_3𝑥𝑠𝜃₃+𝜃₄+𝜃₅+𝑏_3𝑦𝑐𝜃₃+𝜃₄+𝜃₅0. (28)

The velocities can be obtained by differentiating the position vectors with respect to time, taking into account the following notation of (29) and representing time derivatives through dot notation: 𝜃₃+𝜃₄=𝜃₃₄,𝜃̇₃+𝜃̇₄=𝜃̇₃₄,𝜃₃+𝜃₄+𝜃₅=𝜃₃₄₅,𝜃̇₃+𝜃̇₄+𝜃̇₅=𝜃̇₃₄₅. (29)

The velocities of each center of mass are obtained: 𝑣_𝐺1=−𝑏_1𝑥𝜃̇₃𝑠𝜃₃−𝑏_1𝑦𝜃̇₃𝑐𝜃₃𝑏_1𝑥𝜃̇₃𝑐𝜃₃−𝑏_1𝑦𝜃̇₃𝑠𝜃₃0,𝑣_𝐺2=−𝐿₁𝜃̇₃𝑠𝜃₃−𝑏_2𝑥𝜃̇₃₄𝑠𝜃₃₄−𝑏_2𝑦𝜃̇₃₄𝑐𝜃₃₄𝐿₁𝜃̇₃𝑐𝜃₃+𝑏_2𝑥𝜃̇₃₄𝑐𝜃₃₄−𝑏_2𝑦𝜃̇₃₄𝑠𝜃₃₄0,𝑣_𝐺3=−𝐿₁𝜃̇₃𝑠𝜃₃−𝐿₂𝜃̇₃₄𝑠𝜃₃₄−𝑏_3𝑥𝜃̇₃₄₅𝑠𝜃₃₄₅−𝑏_3𝑦𝜃̇₃₄₅𝑐𝜃₃₄₅𝐿₁𝜃̇₃𝑐𝜃₃+𝐿₂𝜃̇₃₄𝑐𝜃₃₄+𝑏_3𝑥𝜃̇₃₄₅𝑐𝜃₃₄₅−𝑏_3𝑦𝜃̇₃₄₅𝑠𝜃₃₄₅0. (30)

The kinetic energy is expressed as 1𝑇=2𝑚₁𝑣_𝐺1²+12𝑚₂𝑣_𝐺2²+12𝑚₃𝑣_𝐺3²+12𝐼₁𝜃̇₃²+12𝐼₂𝜃̇₃₄²+12𝐼₃𝜃̇₃₄₅². (31)

There are two forms of potential energy considered in this study: the gravitational potential energy and the elastic potential energy. Regarding the elastic potential energy, the value of the stiffness is considered as an average value (constant value 𝑘_𝑔) in this study, as a close approximation of the nonlinear and anisotropic finger stiffness (i.e., it varies with the direction). 𝑘₁, 𝑘₂, and 𝑘₃ are the stiffness values for the MCP, DIP, and PIP joints, respectively [17].

The potential energy for the finger is expressed as 𝑈=𝑚₁𝑔𝑏_1𝑥𝑠𝜃₃+𝑏_1𝑦𝑐𝜃₃+𝑚₂𝑔𝐿₁𝑠𝜃₃+𝑏_2𝑥𝑠𝜃₃₄+𝑏_2𝑦𝑐𝜃₃₄+𝑚₃𝑔𝐿₁𝑠𝜃₃+𝐿₂𝑠𝜃₃₄+𝑏_3𝑥𝑠𝜃₃₄₅+𝑏_3𝑦𝑐𝜃₃₄₅+12𝑘₁𝜃₃²+12𝑘₂𝜃₃₄−𝜃₃²+12𝑘₃𝜃₃₄₅−𝜃₃₄². (32)

Moreover, introducing a function of the generalized velocities, usually referred to as the Rayleigh dissipation function 𝐹, for the damping forces, this function is expressed as 1𝐹=2𝑐₁𝜃̇₃²+12𝑐₂𝜃̇₃₄−𝜃̇₃²+12𝑐₃𝜃̇₃₄₅−𝜃̇₃₄², (33) where the damping constant (𝑐_𝑔) stands for the nonconservative contribution caused by the muscles, actuating the finger. Nonconservative forces contributed less than 15% to the total force response to static displacement. Muscle viscosity is dissipative and, hence, non-conservative, resulting in a force field with nonzero curl [17]. To be more precise, values 𝑐₁, 𝑐₂, and 𝑐₃ are the damping values for the MCP, DIP, and PIP joints, respectively.

The Euler-Lagrange equations thus become, considering the three generalized coordinates 𝜃₃, 𝜃₃₄, and 𝜃₃₄₅, 𝑑𝜕()𝑑𝑡𝑇−𝑈𝜕𝜃̇₃−𝜕(𝑇−𝑈)𝜕𝜃₃+𝜕𝐹𝜕𝜃̇₃=𝜏₃,𝑑𝜕()𝑑𝑡𝑇−𝑈𝜕𝜃̇₃₄−𝜕(𝑇−𝑈)𝜕𝜃₃₄+𝜕𝐹𝜕𝜃̇₃₄=𝜏₄,𝑑𝜕()𝑑𝑡𝑇−𝑈𝜕𝜃̇₃₄₅−𝜕(𝑇−𝑈)𝜕𝜃₃₄₅+𝜕𝐹𝜕𝜃̇₃₄₅=𝜏₅ (34) in which the 𝜏_𝑗(𝑗=3,4,5) terms contain the forces applied through the muscles in order to actuate the phalanges and the contact forces, shown in Figure 5.

Figure 5 

Contact forces.

According to the virtual work principle, the equation to calculate the generalized force can be expressed as 𝜏_𝑗=∑⁵_𝑗=3𝛿𝑊_𝑗𝛿𝜃_𝑗;𝑗=3,4,5, (35) where 𝛿𝑊_𝑗 is the virtual work done by the force applied to the system. In the current case, it is 𝜏₃=𝐹_3𝑦𝑒_3𝑥−𝐹_3𝑥𝑒_3𝑦+−𝐿₁𝑠𝜃₃𝐿₁𝑐𝜃₃0⋅𝐹_4𝑥c𝜃₃₄−𝐹_4𝑦s𝜃₃₄𝐹_4𝑥s𝜃₃₄+𝐹_4𝑦c𝜃₃₄0+−𝐿₁𝑠𝜃₃𝐿₁𝑐𝜃₃0⋅𝐹_5𝑥c𝜃₃₄₅−𝐹_5𝑦s𝜃₃₄₅𝐹_5𝑥s𝜃₃₄₅+𝐹_5𝑦c𝜃₃₄₅0+𝐶_𝑚1−𝐶_𝑚2,𝜏₄=𝐹_4𝑦𝑒_4𝑥−𝐹_4𝑥𝑒_4𝑦+−𝐿₂𝑠𝜃₃₄𝐿₂𝑐𝜃₃₄0⋅𝐹_5𝑥c𝜃₃₄₅−𝐹_5𝑦s𝜃₃₄₅𝐹_5𝑥s𝜃₃₄₅+𝐹_5𝑦c𝜃₃₄₅0+𝐶_𝑚2−𝐶_𝑚3,𝜏₅=𝐹_5𝑦𝑒_5𝑥−𝐹_5𝑥𝑒_5𝑦+𝐶_𝑚3, (36) where the term 𝐶_𝑚𝑔 refers to the torque produced by the muscles on the 𝑗th joint, and 𝐹_𝑗=𝐹_𝑗𝑥𝐹_𝑗𝑦0^𝑇 is the contact force applied to the 𝑗th phalanx at the point defined by the position vector 𝑒_𝑗=𝑒_𝑗𝑥𝑒_𝑖𝑦0^𝑇.

Calculating each element of the Euler-Lagrange equations, the dynamical system (33) becomes 𝐴₁𝜃̈₃+𝐴₂𝜃̈₃₄+𝐴₃𝜃̈₃₄₅=𝐴₄,𝐵₁𝜃̈₃+𝐵₂𝜃̈₃₄+𝐵₃𝜃̈₃₄₅=𝐵₄,𝐶₁𝜃̈₃+𝐶₂𝜃̈₃₄+𝐶₃𝜃̈₃₄₅=𝐶₄. (37)

The terms 𝐴₁, 𝐴₂, and 𝐴₃ contain the coefficients of the accelerations and 𝐴₄ the remaining terms; the same goes for the other Euler-Lagrange equations.

Equation (37) allows the direct dynamics of the finger to be solved, where, given the torques exerted by the muscles on each phalange, the movement of the finger can be calculated. If, on the other side, an inverse dynamics problem is set, it is simple to rearrange the equation system (33) to obtain the trend of the unknown muscle torques from the phalanges motion laws.

An expansion of the coefficients of equation (37) can be found in Appendix B, equation (B.1).

6. No-Load Joint Velocities

A test campaign was carried out, in order to evaluate the maximum angular velocities of the finger joints. Ten male and ten female subjects were asked to perform the following operation as fast as possible during 10 seconds (i.e., before any fatigue phenomenon arises to deteriorate the subject performance): starting from the completely extended hand, they have to flex the fingers to a fist and then to extend them again. The maximum flexion angles for the index finger, corresponding to the fist configuration on each cycle, have been statically measured on the subjects as 85 deg for the MCP joint, 105 deg for the PIP joint, and 70 deg for the DIP one. In the following, those angles will be considered negative in accordance with the notations of Section 4.1. Ten male and ten female subjects were involved in the investigation: the statistical output of the test is reported in Table 9.

Table 9 

Joint velocity tests.

There are no significant differences between the male and the female subjects. The results obtained by our test campaign show velocities in accordance with previous tests, for instance, in [18].

The trend of the phalange angles was supposed to follow a sinusoidal law, in the form 𝜃_𝑗=(𝐷_𝑗/2)(cos(𝜋𝑡/𝑡₁)−1), where 𝑗=3,4,5. The term 𝐷_𝑗 is the range of the movement on the 𝑗th joint, while 𝑡₁=10s/num_cycles is the period of each cycle. The time differentiation of this law permits obtaining the trend of the velocities, as well as the maximum angular velocity of each joint, equal to 𝜃̇_{𝑗_max}=|−𝜋𝐷_𝑗/2𝑡₁|.

Here, such experimental motion law has been implemented in the dynamic modeling of the finger. The simulation deals with an inverse dynamics case study: given the motion law of the system (i.e., the kinematic angles of the phalanges and their time derivatives), the torques exerted by the muscles on each phalanx are calculated. In order to evaluate correctly such torques in absence of load, it is necessary to insert also the viscoelastic term that can be preponderant with respect to the dynamical term [19]. The values of the stiffness and the damping coefficients for the MCP joint are obtained through an extrapolation of the values of MCP stiffness and damping, reported in a series of tests in [20], for the no-load case, where the applied external force is null. Then, given the fact that the tests in [20] compute translational stiffness and damping coefficients, they are transformed into rotational ones, to be coherent with the dynamic modeling. The resulting coefficients are shown in Table 11, reported in the appendix, together with the anthropometric data and numerical constants imposed in the model to simulate the behavior of a human index finger.

The trends of the muscle torques 𝐶_𝑚1, 𝐶_𝑚2, and 𝐶_𝑚3 are reported in Figure 6. First of all, these values can be compared with the maximum torque capability of each phalange: Hasser [18] reports these maximum values as 𝐶_𝑚max=46302280775^𝑇. The current test, without external load, correctly produces a set of torques that are quite lower than these limits. Moreover, the results shown in [19] state that the average total torque for a complete flexion-extension movement of the MCP joint is 100.8 Nmm for the female subjects and 194.9 Nmm for the male ones: these values are quite near to the ones produced here, where the joint velocity is higher. The same paper states that the contribute of the viscoelastic terms is preponderant with respect to the dynamical one: the same is true for the current analysis, where the coefficients 𝑓_𝑘𝑗=|𝐶_𝑘𝑗|/|𝐶_𝑚𝑗|, 𝑓_𝑐𝑗=|𝐶_𝑐𝑗|/|𝐶_𝑚𝑗| and 𝑓_𝑑𝑗=|𝐶_𝑑𝑗|/|𝐶_𝑚𝑗|, that are, respectively, the torque portion due to the stiffness term, the damping term, and the dynamic term for the 𝑗th phalange, are reported in Table 10. However, if [19] shows that the stiffness term is about the double of the damping one, in the current analysis the situation is reversed: it is presumably due to the higher velocity investigated here.

Table 10 

Percentages of stiffness, damping, and inertial torques with respect to the total torque.

Table 11 

Anthropometric data and numerical constants used for simulation.

Figure 6 

Trend of the muscle torques.

In conclusion, the results of the current simulation are in good accordance with the literature data. More precise results could be obtained as a consequence of a more accurate estimation of the dynamical parameters inserted into the model, in particular the stiffness and the damping coefficients. For example, due to lack of data, in the current simulation these parameters have been set as equal for the MCP, PIP, and DIP joints: in reality, the PIP and DIP joints have different characteristics, which should not be neglected for a deeper investigation.

Finally, given the previously described test and the related muscle torque, shown in Figure 6, then the mean power associated with the opening and closing of the index finger at maximum velocity in the absence of load can be calculated: it is equal to 1.3 W. The maximum power for the MCP joint in the absence of load is 1.24 W, the maximum power for the PIP joint is 1.94 W, and for the DIP joint it is 0.86 W. This value constitutes a correction of the maximum power estimation for the finger joints, reported by Hasser [18], which states that the absence of external load corresponds to zero muscle torque.

7. Conclusions

An exhaustive study of the human hand was performed, dealing with the kinematics of the human hand and showing the matrices with the MDH parameters and detailed equations for both the direct and inverse kinematics of the hand. In addition, the dynamics of a single finger was analyzed and finally written using the same reference systems as per the kinematics. A case study was proposed, and a simulation was completed using the provided anthropometric data, in order to investigate the capabilities of the proposed analytical system.

The results of the current study can be exploited to conceive future hand devices. On one side, direct and inverse kinematics constitute preliminary stages for the development of any structure similar to the human hand, for instance, in robotic or rehabilitation hands projects. On the other side, dynamics equations allow the behavior of fingers to be simulated. Hence, the value of these equations is twofold: to model the finger itself, for the use in a control scheme model of a human-machine interface, such as a hand exoskeleton, or to model a finger-like architecture, such as a robotic hand.

Appendices

A. Rototranslation Matrix Elements

The 𝑖 subscript of the angles is omitted. First of all, the terms for the fingers (𝑖=1to4) are given. Consider 𝑄_𝑖11=𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒−𝑠𝜃_CMC𝑓/𝑒𝑠𝜃_MCP𝑓/𝑒×𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒+𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒×𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒,𝑄_𝑖12=𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒−𝑠𝜃_CMC𝑓/𝑒𝑠𝜃_MCP𝑓/𝑒×−𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒+𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒×−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒,𝑄_𝑖13=𝑐𝜃_CMC𝑓/𝑒𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑},𝑄_𝑖14=𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒−𝑠𝜃_CMC𝑓/𝑒𝑠𝜃_MCP𝑓/𝑒×𝐿_𝑡𝑖𝑥𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝐿_𝑡𝑖𝑦−𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝐿_1𝑖+𝐿_2𝑖𝑐𝜃_PIP𝑓/𝑒−𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒+𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒×𝐿_𝑡𝑖𝑥𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝐿_𝑡𝑖𝑦−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝐿_2𝑖𝑠𝜃_PIP𝑓/𝑒+𝑐𝜃_CMC𝑓/𝑒𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝐿_𝑡𝑖𝑧+𝐿_0𝑖𝑐𝜃_CMC𝑓/𝑒,𝑄_𝑖21=𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒×𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒×𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒,𝑄_𝑖22=𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒×−𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒×−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒,𝑄_𝑖23=−𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑},𝑄_𝑖24=𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒×𝐿_𝑡𝑖𝑥𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝐿_𝑡𝑖𝑦−𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝐿_1𝑖+𝐿_2𝑖𝑐𝜃_PIP𝑓/𝑒−𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒×𝐿_𝑡𝑖𝑥𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝐿_𝑡𝑖𝑦−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝐿_2𝑖𝑠𝜃_PIP𝑓/𝑒+−𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝐿_𝑡𝑖𝑧,𝑄_𝑖31=𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒+𝑐𝜃_CMC𝑓/𝑒𝑠𝜃_MCP𝑓/𝑒×𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+−𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒+𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒×𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒,𝑄_𝑖32=𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒+𝑐𝜃_CMC𝑓/𝑒𝑠𝜃_MCP𝑓/𝑒×−𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+−𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒+𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒×−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒,𝑄_𝑖33=𝑠𝜃_CMC𝑓/𝑒𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑},𝑄_𝑖34=𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑐𝜃_MCP𝑓/𝑒+𝑐𝜃_{𝐶𝑀𝐶𝑓/𝑒}𝑠𝜃_{𝑀𝐶𝑃𝑓/𝑒}×𝐿_𝑡𝑖𝑥𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝐿_𝑡𝑖𝑦−𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒−𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝐿_1𝑖+𝐿_2𝑖𝑐𝜃_PIP𝑓/𝑒+−𝑠𝜃_CMC𝑓/𝑒𝑐𝜃_{MCP𝑎𝑏/𝑎𝑑}𝑠𝜃_MCP𝑓/𝑒+𝑐𝜃_CMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒×𝐿_𝑡𝑖𝑥𝑠𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝐿_𝑡𝑖𝑦−𝑠𝜃_PIP𝑓/𝑒𝑠𝜃_DIP𝑓/𝑒+𝑐𝜃_PIP𝑓/𝑒𝑐𝜃_DIP𝑓/𝑒+𝐿_2𝑖𝑠𝜃_PIP𝑓/𝑒+𝑠𝜃_CMC𝑓/𝑒𝑠𝜃_{MCP𝑎𝑏/𝑎𝑑}𝐿_𝑡𝑖𝑧+𝐿_0𝑖𝑠𝜃_CMC𝑓/𝑒. (A.1)

Now, the terms for the rototranslation matrix of the thumb (𝑖=0) are reported. Take 𝑄₀₁₁=𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑐𝜃_TMC𝑓/𝑒×𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+−𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑠𝜃_TMC𝑓/𝑒×𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒,𝑄₀₁₂=𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑐𝜃_TMC𝑓/𝑒×−𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+−𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑠𝜃_TMC𝑓/𝑒×−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒,𝑄₀₁₃=𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑},𝑄₀₁₄=𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑐𝜃_TMC𝑓/𝑒×𝐿_{𝑡0_𝑥}𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝐿_{𝑡0_𝑦}−𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝐿₁₀𝑐𝜃_MCP𝑓/𝑒+𝐿₀₀+−𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑠𝜃_TMC𝑓/𝑒×𝐿_{𝑡0_𝑥}𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝐿_{𝑡0_𝑦}−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝐿₁₀𝑠𝜃_MCP𝑓/𝑒+𝐿_{𝑡0_𝑧}𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑},𝑄₀₂₁=𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑐𝜃_TMC𝑓/𝑒×𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+−𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑠𝜃_TMC𝑓/𝑒×𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒,𝑄₀₂₂=𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑐𝜃_TMC𝑓/𝑒×−𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+−𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑠𝜃_TMC𝑓/𝑒×−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒,𝑄₀₂₃=−𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑},𝑄₀₂₄=𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑐𝜃_TMC𝑓/𝑒×𝐿_{𝑡0_𝑥}𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝐿_{𝑡0_𝑦}−𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝐿₁₀𝑐𝜃_MCP𝑓/𝑒+𝐿₀₀+−𝑠𝜃_{TMC𝑎𝑏/𝑎𝑑}𝑠𝜃_TMC𝑓/𝑒×𝐿_{𝑡0_𝑥}𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝐿_{𝑡0_𝑦}−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝐿₁₀𝑠𝜃_MCP𝑓/𝑒−𝑐𝜃_{TMC𝑎𝑏/𝑎𝑑}𝐿_{𝑡0_𝑧},𝑄₀₃₁=𝑠𝜃_TMC𝑓/𝑒𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_TMC𝑓/𝑒𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒,𝑄₀₃₂=𝑠𝜃_TMC𝑓/𝑒−𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_TMC𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒,𝑄₀₃₃𝑄=0,₀₃₄=𝑠𝜃_TMC𝑓/𝑒×𝐿_{𝑡0_𝑥}𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝐿_{𝑡0_𝑦}−𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒−𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝐿₁₀𝑐𝜃_MCP𝑓/𝑒+𝐿₀₀+𝑐𝜃_TMC𝑓/𝑒×𝐿_{𝑡0_𝑥}𝑠𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝐿_{𝑡0_𝑦}−𝑠𝜃_MCP𝑓/𝑒𝑠𝜃_IP𝑓/𝑒+𝑐𝜃_MCP𝑓/𝑒𝑐𝜃_IP𝑓/𝑒+𝐿₁₀𝑠𝜃_MCP𝑓/𝑒. (A.2)

B. Dynamic Equation Terms

Consider 𝐴₁=𝑚₁−𝑏_1𝑥𝑠𝜃₁−𝑏_1𝑦𝑐𝜃₁−𝑏_1𝑥s𝜃₁−𝑏_1𝑦c𝜃₁+𝑏_1𝑥𝑐𝜃₁−𝑏_1𝑦𝑠𝜃₁𝑏_1𝑥c𝜃₁−𝑏_1𝑦s𝜃₁+𝑚₂−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁+𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁+𝑚₃−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁+𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁×𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁+𝐼₁,𝐴₂=𝑚₂−𝑏_2𝑥𝑠𝜃₁₂−𝑏_2𝑦𝑐𝜃₁₂−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁++𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁+𝑚₃−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁++𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁,𝐴₃=𝑚₃−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁++𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁,𝐴₄=𝑚₁−𝑏_1𝑥𝜃₁̇𝑠𝜃₁−𝑏_1𝑦𝜃₁̇𝑐𝜃₁−𝑏_1𝑥𝜃₁̇c𝜃₁+𝑏_1𝑦𝜃₁̇𝑠𝜃₁+𝑏_1𝑥𝑐𝜃₁−𝑏_1𝑦𝑠𝜃₁−𝑏_1𝑥s𝜃₁−𝑏_1𝑦𝑐𝜃₁𝜃₁̇+𝑚₂−−𝐿_1𝑥𝜃̇²₁𝑐𝜃₁+𝐿_1𝑦𝜃̇²₁𝑠𝜃₁−𝑏_2𝑥𝜃̇²₁₂𝑐𝜃₁₂+𝑏_2𝑦𝜃̇²₁₂𝑠𝜃₁₂×−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−−𝐿_1𝑥𝜃̇²₁𝑠𝜃₁−𝐿_1𝑦𝜃̇²₁𝑐𝜃₁−𝑏_2𝑥𝜃̇²₁₂𝑠𝜃₁₂−𝑏_2𝑦𝜃̇²₁₂𝑐𝜃₁₂𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁+𝑚₃−−𝐿_1𝑥𝜃̇²₁𝑐𝜃₁+𝐿_1𝑦𝜃̇²₁𝑠𝜃₁−𝐿_2𝑥𝜃̇²₁₂𝑐𝜃₁₂+𝐿_2𝑦𝜃̇²₁₂𝑠𝜃₁₂−𝑏_3𝑥𝜃̇²₁₂₃𝑐𝜃₁₂₃+𝑏_3𝑦𝜃̇²₁₂₃𝑠𝜃₁₂₃×−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−−𝐿_1𝑥𝜃̇²₁𝑠𝜃₁−𝐿_1𝑦𝜃̇²₁𝑐𝜃₁−𝐿_2𝑥𝜃̇²₁₂𝑠𝜃₁₂−𝐿_2𝑦𝜃̇²₁₂𝑐𝜃₁₂−𝑏_3𝑥𝜃̇²₁₂₃𝑠𝜃₁₂₃−𝑏_3𝑦𝜃̇²₁₂₃𝑐𝜃₁₂₃×𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁−𝑚₁𝑔𝑏_1𝑥𝑐𝜃₁−𝑏_1𝑦𝑠𝜃₁+𝑚₂+𝑚₃𝑔×𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁+𝑘₁𝜃₁−𝑘₂𝜃₁₂−𝜃₁−𝑐₁𝜃̇₁+𝑐₂𝜃̇₁₂−𝜃̇₁,𝐵₁=𝑚₂−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−𝑏_2𝑥𝑠𝜃₁₂−𝑏_2𝑦𝑐𝜃₁₂+𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂+𝑚₃−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂+𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂,𝐵₂=𝑚₂−𝑏_2𝑥𝑠𝜃₁₂−𝑏_2𝑦𝑐𝜃₁₂−𝑏_2𝑥𝑠𝜃₁₂−𝑏_2𝑦𝑐𝜃₁₂++𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂+𝑚₃−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂++𝐿_2𝑥𝑐𝜃₁₂𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂+𝐼₂,𝐵₃=𝑚₃−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂++𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂,𝐵₄=𝑚₂−−𝐿_1𝑥𝜃̇²₁𝑐𝜃₁+𝐿_1𝑦𝜃̇²₁𝑠𝜃₁−𝑏_2𝑥𝜃̇²₁₂𝑐𝜃₁₂+𝑏_2𝑦𝜃̇²₁₂𝑠𝜃₁₂−𝑏_2𝑥𝑠𝜃₁₂−𝑏_2𝑦𝑐𝜃₁₂−−𝐿_1𝑥𝜃̇²₁𝑠𝜃₁−𝐿_1𝑦𝜃̇²₁𝑐𝜃₁−𝑏_2𝑥𝜃̇²₁₂𝑠𝜃₁₂−𝑏_2𝑦𝜃̇²₁₂𝑐𝜃₁₂𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂+𝑚₃−−𝐿_1𝑥𝜃̇²₁𝑐𝜃₁+𝐿_1𝑦𝜃̇²₁𝑠𝜃₁−𝐿_2𝑥𝜃̇²₁₂𝑐𝜃₁₂+𝐿_2𝑦𝜃̇²₁₂𝑠𝜃₁₂−𝑏_3𝑥𝜃̇²₁₂₃𝑐𝜃₁₂₃+𝑏_3𝑦𝜃̇²₁₂₃𝑠𝜃₁₂₃−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂−−𝐿_1𝑥𝜃̇²₁𝑠𝜃₁−𝐿_1𝑦𝜃̇²₁𝑐𝜃₁−𝐿_2𝑥𝜃̇²₁₂𝑠𝜃₁₂−𝐿_2𝑦𝜃̇²₁₂𝑐𝜃₁₂−𝑏_3𝑥𝜃̇²₁₂₃𝑠𝜃₁₂₃−𝑏_3𝑦𝜃̇²₁₂₃𝑐𝜃₁₂₃𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂−𝑚₂𝑔𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂+𝑚₃𝑔𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂+𝑘₂𝜃₁₂−𝜃₁−𝑘₃𝜃₁₂₃−𝜃₁₂−𝑐₂𝜃̇₁₂−𝜃̇₁+𝑐₃𝜃̇₁₂₃−𝜃̇₁₂,𝐶₁=𝑚₃−𝐿_1𝑥𝑠𝜃₁−𝐿_1𝑦𝑐𝜃₁−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃+𝐿_1𝑥𝑐𝜃₁−𝐿_1𝑦𝑠𝜃₁𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃,𝐶₂=𝑚₃−𝐿_2𝑥𝑠𝜃₁₂−𝐿_2𝑦𝑐𝜃₁₂−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃×+𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃,𝐶₃=𝑚₃−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃++𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃×𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃+𝐼₃,𝐶₄=𝑚₃−−𝐿_1𝑥𝜃̇²₁𝑐𝜃₁+𝐿_1𝑦𝜃̇²₁𝑠𝜃₁−𝐿_2𝑥𝜃̇²₁₂𝑐𝜃₁₂+𝐿_2𝑦𝜃̇²₁₂𝑠𝜃₁₂−𝑏_3𝑥𝜃̇²₁₂₃𝑐𝜃₁₂₃+𝑏_3𝑦𝜃̇²₁₂₃𝑠𝜃₁₂₃−𝑏_3𝑥𝑠𝜃₁₂₃−𝑏_3𝑦𝑐𝜃₁₂₃−−𝐿_1𝑥𝜃̇²₁𝑠𝜃₁−𝐿_1𝑦𝜃̇²₁𝑐𝜃₁−𝐿_2𝑥𝜃̇²₁₂𝑠𝜃₁₂−𝐿_2𝑦𝜃̈₁₂𝑠𝜃₁₂−𝐿_2𝑦𝜃̇²₁₂𝑐𝜃₁₂+𝑏_3𝑥𝜃̈₁₂₃𝑐𝜃₁₂₃−𝑏_3𝑥𝜃̇²₁₂₃𝑠𝜃₁₂₃−𝑏_3𝑦𝜃̇²₁₂₃𝑐𝜃₁₂₃×𝑏_3𝑥𝑐𝜃₁₂₃−𝑏_3𝑦𝑠𝜃₁₂₃−𝑚₂𝑔𝑏_2𝑥𝑐𝜃₁₂−𝑏_2𝑦𝑠𝜃₁₂+𝑚₃𝑔𝐿_2𝑥𝑐𝜃₁₂−𝐿_2𝑦𝑠𝜃₁₂+𝑘₂𝜃₁₂−𝜃₁−𝑘₃𝜃₁₂₃−𝜃₁₂−𝑐₃𝜃̇₁₂₃−𝜃̇₁₂. (B.1)