Introduction

Show All

Mechanism and Theory

A. Mechanism

1) Cross Spherical Gear (CS-Gear)

The CS-gear is formed by engraving two axisymmetric tooth structures on a spherical material. Specifically, the tooth structure is first engraved on the surface of the sphere around the x-axis like a lathe processing [Fig. 2 (b)]. This structure is based on the profile of an ordinary involute gear projected on the x–y plane formed by the x- and y-axes. Next, by additionally engraving around the y-axis, a quadrature spherical tooth structure around both axes is formed on the spherical surface [Fig. 2 (c)]. In this article, the gears in panels (b) and (c) of Fig. 2 are called basic spherical gears and CS-gears, respectively. The central axis of a tooth structure is called a structural axis. The mechanism of ABENICS uses CS-gears, but basic spherical gears are sometimes used for convenience when explaining the principle of the mechanism.

Fig. 2.

Formation process of a CS-gear. (a) Material of the sphere before engraving the teeth. The x–y plane showing the teeth profile includes the center point of the sphere. (b) Basic spherical gear, in which the tooth structure is formed around the x-axis (structural axis A). (c) Cross spherical gear, in which the tooth structure is formed around both structural axis A and the y-axis (structural axis B). (a) Material. (b) Basic spherical gear. (c) Cross spherical gear (CS-gear).

Show All

The module msph of the involute tooth profile of the CS-gear is unconstrained. In contrast, the number of teeth zsph is subjected to two constraints.

1. The number of teeth zsph must be even.

   (zsph∈EVEN).

2. The x and y-axes are located in the center of the valley or peak of the tooth.

These conditions are necessary to avoid obtaining incomplete teeth due to the interference of the two tooth structures and to mesh with the MP-gear, as described in Section II-A2.

The pitch circle diameter dsph mm, the addendum circle diameter da.sph mm, and the tooth depth hsph mm of a CS-gear are defined as follows, just like the typical spur gears:

dsphda.sphhsph=msphzsph=dsph+2msph=2.25msph.(1)(2)(3)

View Source

This idea of engraving tooth structures on a sphere has long been proposed as an improvement for universal joints [46]–[49]. Although multiple RDoF have been constructed from several planar [45] and basic spherical gear-like structures [40], [41], [44], an equivalent CS-gear engraved over the full surface of a sphere has not been previously reported.

2) Monopole Gear (MP-Gear)

The MP-gears have a unique tooth structure, the pole, which can mesh with the CS-gears (Fig. 3). When the z- and x-axes are aligned with the rotational axis and the pole, an MP-gear is symmetric in the x–y plane and its cross-section has a typical involute gear profile. The module mmpl and number of teeth zmpl of the profile are, respectively, given by

mmplzmpl=msph=zsph2.(4)(5)

View SourceTaking advantage of the repeated shape of CS-gear, MP-gear becomes smaller, i.e., one pole of the MP-gear can correspond to two poles of the CS-gear.

Fig. 3.

Shape of the MP-gear. The x-axis is aligned with the pole, and the z-axis is aligned with the pitch rotation axis.

Show All

To explain the tooth structure of the MP-gear, the spherical hobbing cutter (SH-cutter) (Fig. 4) is shaped similarly to a basic spherical gear [Fig. 2 (b)] and is formed by rotating a typical gear profile around a structural axis.1 Like a standard hobbing cutter, the SH-cutter engraves the tooth structure by pressing the pattern into the material. For simplicity, we describe the SH-cutter by its structural axis and gear profile hereafter.

Fig. 4.

SH-cutter used for sharpening. The SH-cutter is shaped like a rotating body with a gear profile around the structural axis.

Show All

The gear profiles of the MP-gear and SH-cutter are arranged on the same plane and mesh like a typical pair of spur gears (Fig. 5). If the distance between the centers is fixed at some appropriate length, the system behaves as a planetary gear mechanism. The length of the carrier Rcr mm, the revolution angle of the planetary gear ψcr rad around the sun gear, and the rotation angle of the planetary gear ψhb rad are, respectively, defined as follows:

Rcrψhb=msph(zsph+zmpl)2=msph23zsph2=34dsph=ψcr2(0≤ψcr<2π).(6)(7)

View SourceNote that in the initial state, the x-axis is assumed to be aligned with the structural axis.

Fig. 5.

Hobbing process of a MP-gear. The gear profiles of the MP-gear and SH-cutter mesh and collectively behave like a planetary gear mechanism with a sun gear, planetary gear, and carrier. The material placed in the center is continuously sharpened by the SH-cutter. The SH-cutter is symmetric about a structural axis (omitted here; see Fig. 4). Note that the angle of rotation of the planetary gear is expressed as a relative angle to the carrier.

Show All

Now, we suppose that the material of the MP-gear is placed at the center of the x–y plane. As the orbital revolution angle ψcr continuously increases from 0 rad to 2π rad, the SH-cutter rotates around the material like a planetary gear while also cutting the material. As the SH-cutter revolves around the material, the structural axis rotates as described by (7). This process is essential for meshing the MP-gear with the CS-gear.

As the two poles of the basic spherical gear are symmetrical, an MP-gear with half the number of teeth and only one pole can unlimitedly mesh with the basic spherical gear. This behavior appears to draw a polar orbit with respect to the CS-gear.2 Moreover, an MP-gear can mesh not only with the basic spherical gears but also with the CS-gears possessing two structural axes [Fig. 2 (c)].3

The large tooth width of the MP-gear effectively constrains the CS-gear, thus stabilizing “coupling motion.”4 The tooth width is determined by considering the size of the “Driving module” (see Section II-A3).

3) Driving Module

When assembled into a differential mechanism, the MP-gear can drive the CS-gear. In this article, the MP-gear driving system is defined as the driving module.

The driving module uses two actuators and outputs two RDoF, roll θr and pitch θp of the MP-gear. The skeleton of the equivalent linkage is shown in Fig. 6. The roll axis intersects with the centers of an MP-gear and CS-gear, and the pitch axis coincides with the z-axis of Fig. 3. The driving module is essentially a two-axis gimbal mechanism with its pitch axis adjacent to its roll axis. Our prototype model replaces this equivalent linkage with a differential mechanism.

Fig. 6.

Skeleton model of the driving module. The MP-gear is driven by two active joints, providing two RDoF. A passive slide between the CS-gear and the MP-gear (indicated by the arrow) provides another RDoF. This slide is generated when the pole of the MP-gear is directed to the right side of this figure.

Show All

ABENICS requires two driving modules. Most importantly, each MP-gear must mesh with one of the two tooth structures of the CS-gear. For example, Fig. 7 shows the two tooth structure based on the structural axis A and B of CS-gear meshing with the MP-gear of driving module A and B, respectively. This mechanism enables the CS-gear/output link to be driven with three RDoF.

Fig. 7.

View of the two tooth structures and their respective pairings with the two monopole gears (MP-gears). Each MP-gear is driven by a driving module. Note that the orientation of the pitch axis is affected by the rotation around the roll axis.

Show All

B. Theory

This section shows how the interaction between the components of the ABENICS mechanism can drive three RDoF.

1) Three Motions Invoked by Gear Interaction

The interaction between a CS-gear and an MP-gear generates gearing, coupling, and sliding motions. In Fig. 8, these three motions are invoked by a basic spherical gear [Fig. 2 (b)] and an MP-gear.

Fig. 8.

Three motions are invoked by the interaction between the MP-gear and the CS-gear.

Show All

As the MP-gear rotates around the pitch axis, its teeth mesh with those of the CS-gear, and the driving force is transferred from the MP-gear to the CS-gear (this behavior resembles that of a spur gear). The coupling motion transmits the driving force through the roll rotation of the wide-toothed MP-gear, similarly to Oldham's shaft coupling. The sliding motion caused by passive slippage of the tooth surface does not transfer the driving force.

When the poles of the CS-gear and MP-gear are coincident, the roll rotation of the MP-gear does not transmit any torque. However, the torque direction of the gearing motion can be freely chosen there, just like a steering motion in a car.

If the MP-gear in Fig. 8 is fixed, the interaction of the three motions constrains all motions except the rotation around the structural axis. In other words, a single MP-gear can constrain or drive two of the three RDoF of the CS-gear.

A CS-gear with two crossed-tooth structures [Fig. 2 (c)] can superimpose the interactions of two MP-gears. One MP-gear that meshes with the tooth structure of the structural axis A constrains the CS-gear's rotations except for the rotation around the structural axis A. Similarly, the other MP-gear constrains the rotations except around the structural axis B. Because structural axes A and B are orthogonal, the two MP-gears can constrain or drive all RDoF of the CS-gear. This fact is clarified in a mechanical analysis in the next section.

2) Equivalent Linkage and Number of Degrees of Freedom

Fig. 9.

Skeleton view of a MP-gear with a driving module expressed as an equivalent linkage. The output link has the equivalent capability as the mechanism of the driving module in Fig. 6. For a more accurate comparison between the figures, the axis of the passive joint should be perpendicular to the paper surface. The amount of rotation of the pitch axis is half that of Fig. 6 and the direction is opposite.

Show All

Fig. 10.

Skeleton model of a new ABENICS represented by an equivalent linkage. This parallel linkage mechanism is composed of the two mechanisms shown in Fig. 9. The output link achieves a driving torque with three RDoF.

Show All

Based on the interactions described in Section II-B1, Fig. 6 is replaced with the linkage mechanism shown in Fig. 9, i.e., the relationship between the CS-gear and the two driving modules can be expressed by the equivalent closed linkage shown in Fig. 10. This linkage satisfies the following two conditions.

1. The axes of rotation of all six joints intersect at the center O of the CS-gear.

2. The angle between the axes of any two joints (but without the angle between two joint axes on the ground) is 90 [deg]

This closed-linkage mechanism is a type of spatial mechanism that becomes an overconstrained spherical mechanism under the condition (1). Therefore, from Gruebler's equation, the number of DoF is given by

F=3(N−J−1)+∑i=1Jfi(8)

View Sourcewhere N is the number of links, J is the number of joints, and fi is the DoF of the ith joint. In Fig. 10, suppose that N\,=\,6, J\,=\,6, and f_{1 \ldots 6}\,=\,1. The number of DoF of this equivalent linkage is F\,=\,3, meaning that the output link in Fig. 10 has three RDoF.

We now focus on one of the driving modules in Fig. 10, i.e., the link consisting of A_1, A_2, and A_3. This module is an RR-type two-link serial arm with the passive joint A_3 as the end-effector. As the linkage is spherical, the end-effector A_3 has two DoF on the spherical surface centered at O. The same description holds for the other driving module. Therefore, the CS-gear can be considered driven in parallel by two serial arms. Since each arm is connected to the CS-gear via a passive joint and all joints are orthogonal, the output link acquires three RDoF.

Interestingly, this equivalent linkage suggests that the capability of the ABENICS mechanism is independent of the positions of joints A_1 and B_1, i.e., the relative angle between the driving modules need not be 90 [deg]. For example, the modules can oppose each other along a straight line.

As shown in Fig. 10, the possible orientation angles of the output link in a manufactured equivalent linkage are limited largely by mechanical interference between the links. Accordingly, one of the most significant achievements of this study is the realization of unlimited motion range by replacing this equivalent link with a gear-based mechanism of ABENICS.5

Control Algorithm

A. Definitions

Three coordinate axes are defined in this mechanism: The \Sigma _{\mathrm{B}} world coordinate system fixed at the holder, the \Sigma _{\mathrm{H}} local coordinate system fixed at the CS-gear, and the \Sigma _{\mathrm{M.}n}(n\,=\,\rm {A},\,\rm {B}) local coordinate system fixed at the driving module. The coordinate systems of the driving modules \rm A and \rm B are separately defined as \Sigma _{\mathrm{M.A}} and \Sigma _{\mathrm{M.B}}, respectively, giving four coordinate systems in total. As only rotation is considered here, the origin points O_{\mathrm{B}}, O_{\mathrm{H}}, O_{\mathrm{M.}n} of all coordinate systems coincide. These coordinate systems are shown in Fig. 11.

Fig. 11.

Definitions of the coordinate systems. (a) Cross spherical gear (CS-gear). (b) Holder/Base. (c) Driving module.

Show All

The derivation of the algorithm involves two types of coordinate transformations: A transformation ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{H}} showing the relative orientations of \Sigma _{\mathrm{B}} and \Sigma _{\mathrm{H}} and a transformation ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{M.}n} showing the relative orientations of \Sigma _{\mathrm{M.}n} and \Sigma _{\mathrm{B}}.

Because ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{H}} is the target orientation of the CS-gear, it is expressed as the following function of r, p, and y rad: \begin{align*} ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{H}} & = \boldsymbol{R} (r, p, y) \tag{9} \end{align*} View Sourcewhere \boldsymbol{R} is the rotation matrix.

Moreover, as ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{M.}n} is the orientation of the driving module with respect to \Sigma _{\mathrm{B}}, it is expressed as the following function with mechanical constants \alpha _n, \beta _n, \gamma _n rad: \begin{align*} ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{M.}n} & = \boldsymbol{R} (\alpha _n, \beta _n, \gamma _n). \tag{10} \end{align*} View SourceBased on these definitions, the control algorithm is derived as below.

B. Algorithm

To derive the inverse kinematics of this mechanism, we require two unit vectors \boldsymbol{j}_{\rm A} and \boldsymbol{j}_{\rm B} extending from the origin O_{\mathrm{H}} in \Sigma _{\mathrm{H}}. These vectors determine the directions of the two structural axes [Fig. 11 (a)] as follows: \begin{align*} \boldsymbol{j}_{\rm A} & = \left[ \begin{array}{c}j_{A.x}\\ j_{A.y}\\ j_{A.z} \end{array} \right] = \left[ \begin{array}{c}1\\ 0\\ 0 \end{array} \right]\tag{11}\\ \boldsymbol{j}_{\rm B} & = \left[ \begin{array}{c}j_{B.x}\\ j_{B.y}\\ j_{B.z} \end{array} \right] = \left[ \begin{array}{c}0\\ 1\\ 0 \end{array} \right]. \tag{12} \end{align*} View SourceNext, the coordinates of the unit vectors \boldsymbol{j}_n (n= \rm {A}, \rm {B}) are transformed by ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{H}} and the inverse matrix of ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{M.}n}, obtaining unit vectors \boldsymbol{J}_n in the coordinate systems \Sigma _{\mathrm{M.}n} of the driving modules \begin{align*} \boldsymbol{J}_n = (^{\mathrm{B}}\boldsymbol{q}_{\mathrm{M.}n})^{-1{\mathrm{B}}}\boldsymbol{q}_{\mathrm{H}}\boldsymbol{j}_n. \tag{13} \end{align*} View SourceHere, when the vector \boldsymbol{J}_n is projected onto the y–z plane in \Sigma _{\mathrm{M.}n}, the projected vector and the pitch axis of the driving module n must be directed orthogonally to each other around x axis, as shown in Fig. 12. Therefore, the roll angle \theta _{\mathrm{r.}n} is given by \begin{align*} \theta _{\mathrm{r.}n} & = \tan ^{-1} \left(\frac{J_{n.y}}{J_{n.z}}\right) \tag{14} \end{align*} View Sourcewhere \tan ^{-1} (J_{n.y} / J_{n.z}) is calculated by the inverse tangent function ATAN2(y,x) provided in most programming languages, specifically \mathrm{ATAN2} (J_{n.y}, J_{n.z}). This calculation obtains the angle between the position vector J_{n.y}, J_{n.z} and the z-axis. Note that when J_{n.y} and J_{n.z} are both zero, equation (14) does not provide the solution. In this case, exception handling is performed, such as reusing the previous angle value.

Fig. 12.

Schematic of the control theory. The variable \theta _{\mathrm{p.}n} in the MP-gear is derived from the angle \phi _{n} between J_{n} and the x-axis of \Sigma _{\mathrm{M.}n}. The variable \theta _{\mathrm{r.}n} is derived from the angle between J_{n} and the z-axis projected onto the y–z plane.

Show All

The angle between the vector \boldsymbol{J}_n and the x-axis in the ^{\mathrm{B}}\boldsymbol{q}_{\mathrm{M.}n} is obtained as follows: \begin{align*} \phi _{n} & = \cos ^{-1} (J_{n.x}). \tag{15} \end{align*} View SourceBy (5) giving the tooth ratio of the CS-gear and an MP-gear, the pitch angle \theta _{\mathrm{p.}n} is given by \begin{align*} \theta _{\mathrm{p.}n} & = 2 \phi _{n} = 2\cos ^{-1} (J_{n.x}). \tag{16} \end{align*} View SourceThese calculations generate the angles \theta _{\mathrm{r.}n} and \theta _{\mathrm{p.}n} of the two RDoF of an MP-gear. In practice, the rotation angles of the four motors are determined by the effect of the differential unit and their respective reduction ratios.

C. Actuation Redundancy

In this article, the mechanism principle is revealed using simple position-velocity control with MP-gear target angles. On the other hand, ABENICS is an overactuated mechanism, thus in practice, those joint angles and torques need to be distributed strategically.

In this mechanism, one of the four active joints can be regarded as following the others. This relational equation for joint angles is shown in Appendix C-A. Furthermore, the torques of the joints are also considered to affect each other. The relationship between these torques is described in Appendix C-C.

These increase the complexity of the control algorithm, but also bring the possibility to enhance the stiffness of the joint. By referring to the torque relationship and controlling the torque of the motors, this mechanism is expected to work more effectively.

Prototype

Show All

A. Design Concept and Requirements

The specifications of the CS-gear in the prototype are shown in Table I. In determining the mechanical parameters, it is assumed that a rod of length 300 mm and weight 0.5 kg is connected to the output link. The CS-gear's motion range allowed by the output link depends on the opening area of the holder. The contact surface area between the holder and the CS-gear is an important stabilizing factor in the mechanism of ABENICS. Considering this factor in our design, we widened the motion range of the CS-gear.

TABLE I Specifications of the Cross Spherical Gear in the Prototype

In this study, the angle between the two driving modules was varied as 90 [deg] and 180 [deg]. However, this angle can be arbitrarily set as described in Section II-B2. In addition, the CS-gear and MP-gear modules and the number of teeth can also be set arbitrarily as long as the conditions are met. In other words, the size and design values of this prototype are one of the possible forms of ABENICS.

B. Mechanical Design

Following the design concept, this study developed a prototype of the ABENICS mechanism. The computer-aided design model of the perpendicular-type prototype is shown in Fig. 14. The holder attached to the base plate holds the CS-gear like a ball joint. Both driving modules are connected to the holder. Each driving module has a differential unit for pitch and roll rotations of the MP-gear that meshes with the CS-gear. This unit is composed of an inner rotor, a differential inner worm gear, a differential pinion, and the MP-gear. The inner rotor is rotated around the roll axis by the roll driving motor. The differential inner worm gear that has teeth inside the cylinder is rotated around the same axis by the pitch driving motor. The differential pinion installed on the inner rotor transmits the force to the MP-gear caused by the rotational speed difference between the inner rotor and the inner worm. The MP-gear installed on the inner rotor obtains the driving force of the rotation around the roll axis and the pitch axis from the inner rotor and the differential pinion, respectively. Note that the orientation of the pitch axis is affected by the rotation around the roll axis. This driving module also has an ability to adjust the distance or backlash between the CS-gear and the MP-gear, which consequently corrects manufacturing and assembly errors.

Fig. 14.

Prototype of the new ABENICS. (b) Two driving modules are installed in (a) one prototype. The base supports the whole device and the output link provides the driving force. The dotted lines are auxiliary lines indicating that the parts are disassembled. In (b), part of the driving module is removed to show the internal structure. (a) Design of the ABENICS mechanism. (b) Design of the driving module.

Show All

The differential unit enables the attachment of all four motors to the base. However, the large difference between the reduction ratios of the roll and pitch rotations creates an asymmetric backlash on the MP-gear.6 Moreover, the driving module of this prototype is more elongated than necessary to verify multiple forms.

1) Gear Design and Motor Selections

Table II lists the specifications of each gear and Table III summarizes the performance of the selected motor. The CS- and MP-gears satisfy the conditions described in Section II-A1 and  II-A2. The shape of the MP-gear was generated by numerical calculation simulating the hobbing process shown in Fig. 5. The motors were brushless dc (BLDC) motors (Maxon Motor Inc., Switzerland) that are selected for their superior responsiveness.

TABLE II Specifications of the Spherical and Monopole Gears

TABLE III Specifications of Motors

2) Material Selections

The materials of the prototype are listed in Table IV. The CS- and MP-gears are constructed from polyetheretherketone (PEEK), which is a super engineering plastic with high lubricity, machinability, strength, and stability. As the holder is fabricated using polyoxymethylene (POM), the gear–holder contact surfaces require no oiling. The base and housing of the device are constructed from a lightweight and strong aluminum alloy (A2017).

TABLE IV Materials of the Main Parts

Many parts of the prototype were built from high-performance plastics with high machinability and stable properties because the components of ABENICS have complex shapes, which makes estimating problems during prototyping difficult. Construction from strong materials such as metals will also improve the capability of the device.

C. Controller Design

The flow of the prototype control system is shown in Fig. 15. Process (A) provides the trajectory function ^{\mathrm{B}}q_{\mathrm{H}}(t) using a trajectory generator, and process (B) generates the motor rotation angles \theta _{\mathrm{r.}n} and \theta _{\mathrm{p.}n} using inverse kinematics. Process (C) is the servo controller that is synchronized with (B). The processing frame composed of (B) and (C) operates at a sufficiently high frequency (1 kHz).

Fig. 15.

Schematic of the control system of the new ABENICS, showing the processes (blocks) and flow of information (arrows). The target angles r_t, p_t, y_t of the CS-gear are given by the user. The parameters indicated by the thin arrows are provided by the control system itself. SVPWM denotes the space vector pulsewidth modulation, a field-oriented control method that generates a rotating magnetic field with a three-phase ac voltage V_{\mathrm{r.}n}, V_{\mathrm{p.}n} via inverters.

Show All

Process (A) interpolates a path between the current orientation r_{\mathrm{c}}, p_{\mathrm{c}}, y_{\mathrm{c}} rad and the target orientation r_{\mathrm{t}}, p_{\mathrm{t}}, y_{\mathrm{t}} rad of the CS-gear with a great circle path and generates a trajectory function ^{\mathrm{B}}q_{\mathrm{H}}(t) that moves along the path. Process (B) calculates four motor rotation angles \theta _{\mathrm{r.A}}, \theta _{\mathrm{p.A}}, \theta _{\mathrm{r.B}}, \theta _{\mathrm{p.B}} from the target orientation ^{\mathrm{B}}q_{\mathrm{H}} (obtained by inputting the time t s to ^{\mathrm{B}}q_{\mathrm{H}}(t)) and the orientations ^{\mathrm{B}}q_{\mathrm{M.A}} and ^{\mathrm{B}}q_{\mathrm{M.B}} of the driving modules. Process (B) is the described algorithm in Section III and process (C) provides the closed loop control of the four BLDC motors, which is composed of an inverter, a motor, and a rotary encoder. It also provides a proportional-integral (PI) feedback. Note that because of the redundancy of this mechanism, the upper limit of the integral of the driving module B is set to small.

The hardware of processes (A), (B), and (C) is listed below.

1. A personal computer (Java Programs).

2. NXP-MK64, 32-bit 120-MHz ARM Cortex-M4.

3. NXP-MK20, 32-bit 72-MHz ARM Cortex-M4.

Table V gives the values of the mechanical constants \alpha _n, \beta _n, and \gamma _n in process (B). The perpendicular-type driving modules in Fig. 14 are perpendicularly aligned. In the opposite-type configuration, driving modules A and B are arranged along a straight line.

TABLE V Mechanical Constants

Experiment and Discussion

Show All

A. Positioning Experiment

Fig. 16 shows the experimental setup to confirm whether the theoretical motion range of the system is achievable in practice. First, the discrete target points were regularly arranged in the configuration space based on the XYZ-Euler angles in \Sigma _{\mathrm{B}}, within the motion range of the output link. Next, the prototype was operated to reach those points. Finally, the orientation of the output link was measured by a motion-capture system that originally had been coordinated8 to match the XYZ-Euler angles in \Sigma _{\mathrm{B}} of the prototype. If the prototype device can operate in three RDoF, the measured results plotted in the three-dimensional configuration space of r, p, and y should be arranged in a grid.

Both perpendicular- and opposite-type prototypes were employed in this experiment. Based on Table I, the measurement area was set as follows: Roll = \pm 90, pitch = \pm 45 (perpendicular type) or \pm 30 (opposite type), and yaw = 0 to 270. The target points were set in the area at equal intervals: Roll and pitch = 15 and yaw = 90. To reveal the continuity of positioning, the position measurements were made at each target point, approaching the roll, pitch, and yaw from both the positive and negative sides (\pm 15 [deg]) taking the six measurements at each point.

Positions of the tracking marker attached to the output link were measured by the motion-capture system (OptiTrack V120 Trio; Natural Point Inc., Oregon, USA). To obtain the results of one measurement, an average of 120 measurement data during one second was used. The measurement accuracy of this instrument is supplemented by Appendix A.

B. Trajectory Tracking Experiment

This experiment verifies the transient response of the mechanism's prototype as it moves from one orientation to the other: The CS-gear's orientation is measured by following the trajectory of a straight line set in the r, p, y [deg] three-dimensional configuration space. If the tracking of the prototype is good, the measurement points plotted in three-dimensional space should be well linear.

The perpendicular-type prototype was used in this experiment. The measurement method, experimental equipment, and coordinate system were the same as Section V-A.

To set up the measurement trajectory, we prepared reference points (Table VI). The trajectory runs in sequence from P0 to P8. In one straight movement, r, p, and y are each at a constant speed with the same travel time. The fastest rotation is set to 90

TABLE VI Reference Points

[deg/s], while the other two rotation speeds are adjusted to synchronize. Each reference point has a stopping time of 1 s.

The CS-gear orientation was measured with a period of 120 Hz and the results were given by a single continuous tracking test.

C. Experimental Results and Discussion

The experimental results are shown in Figs. 17, 18, and 20. The horizontal, depth, and vertical directions depict the r, p, and y angles, respectively. Fig. 19 displays the serial pictures showing the operation of the perpendicular-type prototype.

Fig. 17.

Scatter diagrams of the results of the positioning experiments. The XYZ-Euler angles (roll r, pitch p, and yaw y degrees) are plotted on the orthogonal coordinate system. The measurement markers reduced the motion range of the opposite type. The colors of the plotted points indicate the magnitudes of the errors between the true and measured values on the orthogonal coordinates. The six measurements at each target point are plotted without averaging each other. All results are plotted in the diagrams and no plots are eliminated. The regular alignment of the plot points shows the correctness of the theory of the proposed mechanism. (a) Perpendicular type (90). (b) Opposite type (180).

Show All

Fig. 18.

Average and standard deviation of the results of the positioning experiment. (a-1) and (b-1) are scatter diagrams based on the r–p plane, and (a-2) and (b-2) are based on the r–y plane. Each plotted point is the measured value at the target point of the intersection of the ticks connected by the red lines. These errors and standard deviations are exaggerated, and one square of the graph grid is 3 when reading the values.

Show All

Fig. 19.

Movement of the prototype of the new ABENICS in the tracking experiment. To clarify the movement, two infrared reflection markers were characterized with simple red and green circles in these pictures.

Show All

Fig. 20.

Scatterplot of the tracking experiment results. XYZ-Euler angles (roll r, pitch p, and yaw y [deg]) are plotted in the Cartesian coordinate system. The color of the plotted points illustrates the angle of y [deg] and the linearity of the plotted points indicates the transient response of this mechanism. (a) Perspective view. (b) Three-view.

Show All

1) Quantitative Discussion of Measurement Results

From Fig. 17, in both prototype configurations, most of the plotted measurement points were aligned evenly on the orthogonal coordinate system and coincided with their target points. This indicates that the prototypes reached any orientation from any direction. In addition, the plots of the perpendicular and opposite types were similar, suggesting that the capabilities of the ABENICS mechanism were independent of the angle between the two driving modules. Moreover, Fig. 20 shows that the orientation transition of the CS-gear is linear and suggests that the mechanism can follow the trajectory continuously.

Fig. 21.

Schematic diagram of the CS-gear and MP-gears of the perpendicular type prototype at the CS-gear orientation r=90 [deg] p=30 [deg], and y=90 [deg]. Note that the coordinate axes in this figure are in the right-hand coordinate system, i.e., this view is from the minus side of the Z-coordinate. In this case, the roll rotation r of the CS-gear is performed by the gearing motion of both the MP-gears.

Show All

A more detailed analysis shows that the error in the repetitive positioning depended on its orientation (e.g., see the error around r=90 [deg], p=30 [deg], and y=90 [deg] in Figs. 17 (a) and 21). This orientation dependence probably results from combining different driving principles, gearing motions, and coupling motions.9 The gearing motion uses a differential unit in the driving module, which introduces more error. On the other hand, the coupling motion is a torsional rotation caused by the tooth width of the MP-gear, so its error depends on the tooth width and backlash with CS-gear.10 The friction caused by the sliding motion depends on the meshing position of the CS-gear and MP-gear. Consequently, the total error due to the composition of the three motions is expected to depend on the relative orientations of CS-gear and MP-gear. According to dial gauge backlash measurements, there was approx. 0.7–0.9 of backlash in gearing motion and approx. 0.8 of backlash in coupling motion. The gearing motion included a backlash of 0.6 caused by the driving module. Note that there are two types of backlash here: 1) Backlash in the driving module and 2) backlash between the CS-gear and MP-gear. In the graphs that show the trajectory tracking, the vibrations in the yaw rotation are noticeable. At r = 90,-90 [deg], the output link is horizontal and is affected by gravity, which is also thought to be caused by the backlash. This problem will get better with improvements of design and manufacturing technology. In the near future, we will establish a method that accurately evaluates these effects by modeling and formulating the underlying mechanism.

D. Discussion on Design Issues

ABENICS has many unique components, such as CS-gear and MP-gear. Although these brought new possibilities, some difficulties are foreseen. Here, the design challenges of this mechanism are discussed.

2) Manufacturing of MP-Gear and CS-Gear

Establishing a manufacturing method for a CS-gear and MP-gear is one of the biggest challenges. 5-axis machining centers are a viable solution for both metallic and resin materials. On the other hand, although the machining procedure needs to be tested, it may also be possible to lathe-like machining using the CS-gear's linear symmetry. In addition, the MP-gear could be processed such as that in Fig. 5.

Using a 3D printer is a good way to fabricate this mechanism. Fig. 22 is an early prototype of this project and MP-gears, which were manufactured by stereolithography (SLA) 3D printer for theory validation and are confirmed to work. Also, the MP-gear's shape generation algorithm is already established and the module and number of teeth can be easily changed.

Fig. 22.

3D-printed prototypes and monopole gear (MP-gear) in various modules (m:1, 2, and 4 mm).

Show All

In the case of metal 3D printers, a presence of a rough surface due to the sintering of the metal powder is not suitable for CS-/MP-gear. On the other hand, it is remarkable that combined machining centers and metal 3D printers have emerged in recent years. The problem may be solved by cutting only the tooth contact surface thin in postprocessing.

3) Holder and Range of Motion

The CS-gear of the prototype must be held by a holder. Therefore, if there is an output axis, the range of motion is limited, i.e., there is a tradeoff between the range of motion and stability. Such systems can be found in a friction wheel, a spherical motor, and an ultrasonic system. On the other hand, a linkage or differential mechanism has no such limitations. A comparison of these movable ranges is shown in Table VII [33]. Note that the values of the working ranges in the table depend on the coordinate system of each literature and are not uniform.

TABLE VII Comparison of Range of Motion

One way to increase the range of motion is to remove the bottom part of the opposite type prototype's holder. This reduces the contact area between the CS-gear and the holder. Thus, the stiffness is one of the challenges, but it allows for a range of motion of 360, 90, 360 [deg]. Another way is to design the angle between the driving modules to be small. The prototype's angles are 90 and 180 [deg] but can be placed at a minimum angle of 60 [deg]. If a mechanism to prevent the CS-gear dropout is installed and the overhang is removed from the holder, the range of motion in this mechanism will be extended to approx. 180, 180, 360 [deg]. We expect that the linkages, magnets, or mobile spherical shells provide auxiliary holding for the CS-gear. Holding the CS-gear in a passive gimbal mechanism is one valid method.12 However, the singularity of the linkage mechanism will be an issue. On the other hand, a combination of a CS-gear containing a ferromagnetic material and a holder with permanent magnets is an alternative.13 In this case, we need to consider the friction caused by magnetic forces and the permissible load of the hand. However, the configuration of the mechanism is relatively simple. In addition, there is one option to hold the CS-gear with a movable spherical shell structure (e.g., Fig. 23). Although many challenges are foreseen in this design, it is possible to hold the CS-gear with surface contact. There are many challenges in designing the holder, but due to the flexibility of its design, we believe ABENICS has a high potential for the near future.

Fig. 23.

Illustration of the spherical shell mechanism to hold the CS-gear. This spherical shell is connected to the shell guide by a passive joint. The holder contains two driving modules that have optimized designs. The shell guide is held in place by passive rollers in the holder, which allow the spherical shell to move with the movement of the CS-gear output link, thus allowing for a wide range of motion.

Show All

Conclusion

Show All

Show All

Show All

Show All

Define the detailed coordinate system for Fig. 10 as shown in Fig. 28. Because translational movement is not considered here, all coordinate transformations are represented by rotation matrices. The relationship with the rotation angle of the driving module is as follows: \begin{align*} \theta _{\mathrm{r.A}} = &\; \theta _{\rm A1} & \theta _{\mathrm{r.B}} = &\; \theta _{\rm B1} \tag{17} \\ \theta _{\mathrm{p.A}} = &\; - 2 \theta _{\rm A2} & \theta _{\mathrm{p.B}} = &\; - 2 \theta _{\rm B2}. \tag{18} \end{align*} View Source

Fig. 28.

Detailed skeleton model of the new active ball joint mechanism (ABENICS). The coordinate system of the link \Sigma _{\rm a0}, \Sigma _{\rm a1}, \Sigma _{\rm a2}, \Sigma _{\rm b0}, \Sigma _{\rm b1}, \Sigma _{\rm b2}, the hand (CS-gear) coordinate system \Sigma _{\rm H}, and the holder coordinate system \Sigma _{\rm B}. The \theta is the rotation angle of each joint and \beta is the angle formed by the two driving modules. \boldsymbol R is the coordinate transformation. Since it is a spherical linkage, each coordinate system is actually in the same position as \Sigma _{\rm H}.

Show All

Expressing the rotation matrices around the x, y, and z axes as \boldsymbol R_{x}(\theta), \boldsymbol R_{y}(\theta), and \boldsymbol R_{z}(\theta), respectively, the coordinate transformation in Fig. 10 is as follows: \begin{align*} ^{\rm a0}{\boldsymbol R}_{\rm a1} & = {\boldsymbol R}_{x}(\theta _{\rm A1}) \tag{19} \\ ^{\rm a1}{\boldsymbol R}_{\rm a2} & = {\boldsymbol R}_{y}(\theta _{\rm A2}) \tag{20} \\ ^{\rm a2}{\boldsymbol R}_{\rm H} & = {\boldsymbol R}_{x}(\theta _{\rm A3}) \tag{21} \\ ^{\rm b0}{\boldsymbol R}_{\rm b1} & = {\boldsymbol R}_{x}(\theta _{\rm B1}) \tag{22} \\ ^{\rm b1}{\boldsymbol R}_{\rm b2} & = {\boldsymbol R}_{y}(\theta _{\rm B2}) \tag{23} \\ ^{\rm b2}{\boldsymbol R}_{\rm H} & = {\boldsymbol R}_{x}(\theta _{\rm B3}) {\boldsymbol R}_{z}(-\pi /2)\tag{24} \\ ^{\rm a0}{\boldsymbol R}_{\rm b0} & = {\boldsymbol R}_{z}(\beta). \tag{25} \end{align*} View SourceHere, for simplicity of expression, it is assumed that ^{\rm B}{\boldsymbol R}_{\rm a0} is the unit matrix by aligning \Sigma _{\rm B} with the coordinate system \Sigma _{\rm a0} of the driving module A.

This closed-link mechanism is a redundant mechanism that realizes three DoF of the CS-gear with four active joints. Thus, they can be considered as three independent active joints and one dependent active joint. Here, we assume that \theta _{\rm B2} is the dependent active joint.

The coordinate transformation ^{\boldsymbol B}{\boldsymbol R}_{\boldsymbol H} to the hand (CS-gear) in the link chain A is expressed by the following equation, (26) shown at bottom of this page. \begin{align*} &^{\rm B}{\boldsymbol R}_{\rm H} = ^{\rm B}{\boldsymbol R}_{\rm a0} \; ^{\rm a0}{\boldsymbol R}_{\rm a1} \; ^{\rm a1}{\boldsymbol R}_{\rm a2} \; ^{\rm a2}{\boldsymbol R}_{\rm H} = \\ & \left[\begin{matrix}C_{\rm A2} & S_{\rm A2} S_{\rm A3} & C_{\rm A3} S_{\rm A2}\\ S_{\rm A1} S_{\rm A2} & C_{\rm A1} C_{\rm A3} - C_{\rm A2} S_{\rm A1} S_{\rm A3} & - C_{\rm A1} S_{\rm A3} - C_{\rm A2} C_{\rm A3} S_{\rm A1}\\ - C_{\rm A1} S_{\rm A2} & C_{\rm A1} C_{\rm A2} S_{\rm A3} + C_{\rm A3} S_{\rm A1} & C_{\rm A1} C_{\rm A2} C_{\rm A3} - S_{\rm A1} S_{\rm A3}\end{matrix}\right]\tag{26} \end{align*} View SourceThe S_{\rm m} and C_{\rm m} in the equation mean \sin {\theta _{\rm m}} and \cos {\theta _{\rm m}}. Here, to obtain the unknown variable, \theta _{\rm A3}, we use the following circularity of closed links: \begin{align*} ^{\rm a0}{\boldsymbol R}_{\rm a1} \; ^{\rm a1}{\boldsymbol R}_{\rm a2} \; ^{\rm a2}{\boldsymbol R}_{\rm H} = ^{\rm a0}{\boldsymbol R}_{\rm b0} \; ^{\rm b0}{\boldsymbol R}_{\rm b1} \; ^{\rm b1}{\boldsymbol R}_{\rm b2} \; ^{\rm b2}{\boldsymbol R}_{\rm H}. \tag{27} \end{align*} View SourceSolving this equation for the passive joint {\boldsymbol R}_{x}(\theta _{\rm B3}), we obtain the following equation, (28) shown at bottom of this page. \begin{align*} & {\boldsymbol R}_{x}(\theta _{\rm B3}) = \\ & (^{\rm b1}{\boldsymbol R}_{\rm b2})^{-1} \, (^{\rm b0}{\boldsymbol R}_{\rm b1})^{-1} \, (^{\rm a0}{\boldsymbol R}_{\rm b0})^{-1} \, {}^{\rm a0}{\boldsymbol R}_{\rm a1} \; ^{\rm a1}{\boldsymbol R}_{\rm a2} \; ^{\rm a2}{\boldsymbol R}_{\rm H} \; ({\boldsymbol R}_{\rm z}(-\pi /2))^{-1}\tag{28} \end{align*} View SourceExpanding this equation, we obtain \begin{align*} &\left[ \begin{matrix}1 & 0 & 0\\ 0 & C_{B3} & - S_{B3}\\ 0 & S_{B3} & C_{B3} \end{matrix} \right] = \left[ \begin{matrix}r_{11} & r_{12} & r_{13}\\ r_{21} & r_{22} & r_{23}\\ r_{31} & r_{32} & r_{33} \end{matrix} \right] \tag{29} \\ &r_{11} = 1 = C_{A3} F - S_{A3} (C_{A2} G - S_{A2} H) \tag{30} \\ &F = C_{A1} C_{B2} S_{\beta } + C_{A1} C_{\beta } S_{B1} S_{B2} - C_{B1} S_{A1} S_{B2} \\ &G = C_{A1} C_{B1} S_{B2} + C_{B2} S_{A1} S_{\beta } + C_{\beta } S_{A1} S_{B1} S_{B2} \\ &H = C_{B2} C_{\beta } - S_{B2} S_{B1} S_{\beta } \\ &r_{12} = 0 = - C_{B2} I - S_{B2} J \tag{31} \\ &I = C_{A2} C_{\beta } + S_{A1} S_{A2} S_{\beta } \\ &J = C_{A1} C_{B1} S_{A2} - C_{A2} S_{B1} S_{\beta } + C_{\beta } S_{A1} S_{A2} S_{B1} \\ &r_{13} = 0 = - S_{A3} F - C_{A3} (C_{A2} G - S_{A2} H) \tag{32} \\ &r_{21} = 0 = C_{A3} K - S_{A3} L \tag{33} \\ &K = C_{A1} C_{B1} C_{\beta } + S_{A1} S_{B1} \\ &L = S_{A1} C_{A2} C_{B1} C_{\beta } - C_{A1} C_{A2} S_{B1} + S_{A2} C_{B1} S_{\beta } \\ &r_{22} = C_{B3} = C_{A1} S_{A2} S_{B1} + C_{A2} C_{B1} S_{\beta } - C_{B1} C_{\beta } S_{A1} S_{A2} \tag{34} \\ &r_{23} = -S_{B3} = - C_{A3} L - S_{A3} K \tag{35} \\ &r_{31} = 0 = C_{A3} M - S_{A3} \left(C_{A2} N - S_{A2} O \right) \tag{36} \\ &M = - C_{A1} C_{B2} C_{\beta } S_{B1} + C_{A1} S_{B2} S_{\beta } + S_{A1} C_{B1} C_{B2} \\ &N = - S_{A1} C_{B2} C_{\beta } S_{B1} + S_{A1} S_{B2} S_{\beta } - C_{A1} C_{B1} C_{B2} \\ &O = S_{B2} C_{\beta } + C_{B2} S_{B1} S_{\beta } \\ &r_{32} = S_{B3} = - C_{A2} O - S_{A2} N \tag{37} \\ &r_{33} = C_{B3} = - C_{A3} N - S_{A3} M. \tag{38} \end{align*} View SourceHere, r_{21} leads to \theta _{\rm A3} based on \theta _{\rm A1}, \theta _{\rm A2}, and \theta _{\rm B1} \begin{align*} &\theta _{\rm A3} \\ &= \operatorname{atan}{\left(\frac{S_{A1} S_{B1} + C_{A1} C_{B1} C_{\beta }}{- C_{A1} C_{A2} S_{B1} + S_{A2} C_{B1} S_{\beta } + S_{A1} C_{A2} C_{B1} C_{\beta }} \right)}. \end{align*} View SourceThe forward kinematics can be derived by substituting \theta _{\rm A1}, \theta _{\rm A2}, and \theta _{\rm A3} into (26). Also, r_{12} represents the constraint between \theta _{\rm A1}, \theta _{\rm A2}, \theta _{\rm B1}, and \theta _{\rm B2}. Solving this equation for the driven active joint \theta _{\rm B2}, we obtain the following equation: \begin{align*} &\theta _{\rm B2} \\ &= - \operatorname{atan}{\left(\frac{C_{A2} C_{\beta } + S_{A1} S_{A2} S_{\beta }}{ C_{A1} S_{A2} C_{B1} - C_{A2} S_{B1} S_{\beta } + S_{A1} S_{A2} S_{B1} C_{\beta }} \right)} \end{align*} View Sourcewhere the CS-gear orientation is expressed in terms of XYZ-Euler angles \begin{align*} &\theta _{r} = - \operatorname{atan}{\left(\frac{ S_{B1} C_{A1} S_{A1} S_{A2}^{2} + C_{B1} U }{ C_{B1} C_{A1} S_{A2} (C_{\beta } S_{A1} S_{A2} - S_{\beta } C_{A2}) + S_{B1} V} \!\right) } \tag{39} \\ &\theta _{p} \\ &\!=\! \operatorname{asin}{\left(S_{A2} \left(\!- C_{A1} C_{A2} S_{B1} \!+\! C_{A2} C_{B1} S_{A1} C_{\beta } \!+\! S_{A2} C_{B1} S_{\beta } \right) \!\right) }\! \tag{40} \\ &\theta _{y} = - \operatorname{atan}{ \left(\frac{ S_{A2} }{C_{A2}} \left(C_{A1} C_{B1} C_{\beta } + S_{A1} S_{B1} \right) \right) } \tag{41} \\ &U = S_{\beta } C_{A2} S_{A1} S_{A2} - C_{\beta } (S_{A1}^{2} S_{A2}^{2} + 1) \tag{42} \\ &V = S_{A1}^{2} S_{A2}^{2} - S_{A2}^{2} + 1. \tag{43} \end{align*} View Source

Partial differentiation of the above equation at each joint angle yields the Jacobian of the link chain A. Also, the Jacobian of link chain B can be derived by the same procedure. \begin{align*} {\boldsymbol J_A} = & \left[ \begin{matrix}\frac{\partial \theta _r}{\partial \theta _{\rm A1}} & \frac{\partial \theta _r}{\partial \theta _{\rm A2}} & \frac{\partial \theta _r}{\partial \theta _{\rm B1}} \\ \frac{\partial \theta _p}{\partial \theta _{\rm A1}} & \frac{\partial \theta _p}{\partial \theta _{\rm A2}} & \frac{\partial \theta _p}{\partial \theta _{\rm B1}} \\ \frac{\partial \theta _y}{\partial \theta _{\rm A1}} & \frac{\partial \theta _y}{\partial \theta _{\rm A2}} & \frac{\partial \theta _y}{\partial \theta _{\rm B1}} \end{matrix} \right] = \left[ \begin{matrix}j_{11} & j_{12} & j_{13} \\ j_{21} & j_{22} & j_{23} \\ j_{31} & j_{32} & j_{33} \end{matrix} \right]. \tag{44} \end{align*} View Source

The relationship from the coordinate system of the base (holder) to the orientation of the hand (CS-gear) can be expressed in each of the link chains A and B as follows. Here, ^{\rm B}{\boldsymbol R}_{\rm H} is a known value \begin{align*} ^{\rm B}{\boldsymbol R}_{\rm H} = & ^{\rm B}{\boldsymbol R}_{\rm a0} \; ^{\rm a0}{\boldsymbol R}_{\rm a1} \; ^{\rm a1}{\boldsymbol R}_{\rm a2} \; ^{\rm a2}{\boldsymbol R}_{\rm H} \tag{45} \\ ^{\rm B}{\boldsymbol R}_{\rm H} = & ^{\rm B}{\boldsymbol R}_{\rm a0} \; ^{\rm a0}{\boldsymbol R}_{\rm b0} \; ^{\rm b0}{\boldsymbol R}_{\rm b1} \; ^{\rm b1}{\boldsymbol R}_{\rm b2} \; ^{\rm b2}{\boldsymbol R}_{\rm H}. \tag{46} \end{align*} View SourceUsing (19)–(25) and arranging the unknown components on the left-hand side, each equation can be expressed as follows: \begin{align*} {\boldsymbol R}_{\rm x}(\theta _{\rm A1}) \; {\boldsymbol R}_{\rm y}(\theta _{\rm A2}) \; {\boldsymbol R}_{\rm x}(\theta _{\rm A3}) = & (^{\rm B}{\boldsymbol R}_{\rm A0})^{-1} \; {}^{\rm B}{\boldsymbol R}_{\rm H} \tag{47} \\ {\boldsymbol R}_{\rm x}(\theta _{\rm B1}) \; {\boldsymbol R}_{\rm y}(\theta _{\rm B2}) \; {\boldsymbol R}_{\rm x}(\theta _{\rm B3}) = \\ ({\boldsymbol R}_{\rm z}(\beta))^{-1} \; (^{\rm B}{\boldsymbol R}_{\rm A0})^{-1} & \; ^{\rm B}{\boldsymbol R}_{\rm H} \; ({\boldsymbol R}_{\rm z}(-\pi / 2))^{-1}. \tag{48} \end{align*} View SourceBy expanding both sides of each equation, the following equations are obtained. However, since the right-hand side is a known component, its detailed derivation is omitted. \begin{align*} \left[\begin{matrix}C_{A2} |^{*2} & S_{A2} S_{A3} |^{*3} & C_{A3} S_{A2} |^{*3}\\ S_{A1} S_{A2} |^{*1} & C_{A1} C_{A3} - C_{A2} S_{A1} S_{A3} & - C_{A1} S_{A3} - C_{A2} C_{A3} S_{A1}\\ - C_{A1} S_{A2} |^{*1} & C_{A1} C_{A2} S_{A3} + C_{A3} S_{A1} & C_{A1} C_{A2} C_{A3} - S_{A1} S_{A3} \end{matrix}\right] \\ = \left[ \begin{matrix}r_{\rm a11} |^{*2} & r_{\rm a12} |^{*3} & r_{\rm a13} |^{*3} \\ r_{\rm a21} |^{*1} & r_{\rm a22} & r_{\rm a23}\\ r_{\rm a31} |^{*1} & r_{\rm a32} & r_{\rm a33} \end{matrix} \right] \tag{49} \\ \left[\begin{matrix}C_{B2} |^{*2} & S_{B2} S_{B3} |^{*3} & C_{B3} S_{B2} |^{*3} \\ S_{B1} S_{B2} |^{*1} & C_{B1} C_{B3} - C_{B2} S_{B1} S_{B3} & - C_{B1} S_{B3} - C_{B2} C_{B3} S_{B1}\\ - C_{B1} S_{B2} |^{*1} & C_{B1} C_{B2} S_{B3} + C_{B3} S_{B1} & C_{B1} C_{B2} C_{B3} - S_{B1} S_{B3} \end{matrix}\right] \\ = \left[ \begin{matrix}r_{\rm b11} |^{*2} & r_{\rm b12} |^{*3} & r_{\rm b13} |^{*3} \\ r_{\rm b21} |^{*1} & r_{\rm b22} & r_{\rm b23}\\ r_{\rm b31} |^{*1} & r_{\rm b32} & r_{\rm b33} \end{matrix} \right] \tag{50} \end{align*} View SourceNow, focusing on the *1, *2, and *3 components of (49) and (50) shown at bottom of this page, we obtain the following equations: \begin{align*} \theta _{\rm A1} = & \operatorname{atan}{ \left(-\frac{r_{\rm a21}}{r_{\rm a31}} \right) } & \theta _{\rm B1} = & \operatorname{atan}{ \left(-\frac{r_{\rm b21}}{r_{\rm b31}} \right) }\tag{51} \\ \theta _{\rm A2} = & \operatorname{acos}{ \left(r_{\rm a11} \right) } & \theta _{\rm B2} = & \operatorname{acos}{ \left(r_{\rm b11} \right) }\tag{52} \\ \theta _{\rm A3} = & \operatorname{atan}{ \left(\frac{r_{\rm a12}}{r_{\rm a13}} \right) } & \theta _{\rm B3} = & \operatorname{atan}{ \left(\frac{r_{\rm b12}}{r_{\rm b13}} \right) }. \tag{53} \end{align*} View SourceIf the CS-gear orientation is defined by the XYZ-Euler angles, then ^{\rm B}{\boldsymbol R}_{\rm H} = R_{x}(r)R_{y}(p)R_{z}(y). Therefore, the specific angle for each joint is given by \begin{align*} \theta _{\rm A1} = & \operatorname{atan}{\left(\frac{C_{r} S_{y} + C_{y} S_{p} S_{r}}{C_{r} C_{y} S_{p} - S_{r} S_{y}} \right) } \tag{54} \\ \theta _{\rm A2} = & \operatorname{acos}{ \left(C_{p} C_{y} \right) } \tag{55} \\ \theta _{\rm A3} = & - \operatorname{atan}{ \left(\frac{C_{p} S_{y}}{S_{p}} \right) } \tag{56} \\ \theta _{\rm B1} = & - \operatorname{atan}{\left(\frac{ C_{y} C_{\beta } C_{r} + S_{y} \left(S_{\beta } C_{p} - C_{\beta } S_{p} S_{r} \right) }{C_{r} S_{p} S_{y} + C_{y} S_{r}} \right)} \tag{57} \\ \theta _{\rm B2} = & \operatorname{acos}{\left(C_{y} C_{r} S_{\beta } - S_{y} \left(C_{\beta } C_{p} + S_{\beta } S_{p} S_{r} \right) \right)} \tag{58} \\ \theta _{\rm B3} = & \operatorname{atan}{\left(\frac{C_{y} \left(C_{\beta } C_{p} + S_{\beta } S_{p} S_{r} \right) + S_{y} S_{\beta } C_{r}}{- C_{\beta } S_{p} + S_{\beta } C_{p} S_{r}} \right) }. \tag{59} \end{align*} View Source

Since this mechanism is a redundant closed-link mechanism, antagonistic or superimposed torques are expected to exist inside the mechanism. In this section, the relational equations are first derived from hand velocity to joint angular velocity. Next, the torque equation is derived using the principle of virtual work. To simplify the equation, \beta = 90

is used here.

The inverse Jacobian {\boldsymbol J_A}^{-1} and {\boldsymbol J_B}^{-1} in the link chain A and B are calculated by substituting (54)–(59) into the following equations: \begin{align*} &{\boldsymbol J_A}^{-1} = \left[ \begin{matrix}\frac{\partial \theta _{\rm A1}}{\partial \theta _r} & \frac{\partial \theta _{\rm A1}}{\partial \theta _p} & \frac{\partial \theta _{\rm A1}}{\partial \theta _y} \\ \frac{\partial \theta _{\rm A2}}{\partial \theta _r} & \frac{\partial \theta _{\rm A2}}{\partial \theta _p} & \frac{\partial \theta _{\rm A2}}{\partial \theta _y} \\ \frac{\partial \theta _{\rm A3}}{\partial \theta _r} & \frac{\partial \theta _{\rm A3}}{\partial \theta _p} & \frac{\partial \theta _{\rm A3}}{\partial \theta _y} \end{matrix} \right] = \left[ \begin{matrix}ja_{11} & ja_{12} & ja_{13} \\ ja_{21} & ja_{22} & ja_{23} \\ ja_{31} & ja_{32} & ja_{33} \end{matrix} \right] \\ &ja_{11} = 1, \quad ja_{12} = \frac{C_{p} C_{y} S_{y}}{C_{p}^{2} C_{y}^{2} - 1} \quad ja_{13} = \frac{S_{p}}{- C_{p}^{2} C_{y}^{2} + 1} \\ &ja_{21} = 0, \quad ja_{22} = \frac{C_{y} S_{p}}{\sqrt{- C_{p}^{2} C_{y}^{2} + 1}} \quad ja_{23} = \frac{C_{p} S_{y}}{\sqrt{- C_{p}^{2} C_{y}^{2} + 1}} \\ &ja_{31} = 0,\quad ja_{32} = - \frac{S_{y}}{C_{p}^{2} C_{y}^{2} - 1} \quad ja_{33} = - \frac{C_{y} \tan {\left(\theta _{p} \right) }}{S_{y}^{2} + \tan ^{2}{\left(\theta _{p} \right) }} \tag{60}\\ &{\boldsymbol J_B}^{-1} = \left[ \begin{matrix}\frac{\partial \theta _{\rm B1}}{\partial \theta _r} & \frac{\partial \theta _{\rm B1}}{\partial \theta _p} & \frac{\partial \theta _{\rm B1}}{\partial \theta _y} \\ \frac{\partial \theta _{\rm B2}}{\partial \theta _r} & \frac{\partial \theta _{\rm B2}}{\partial \theta _p} & \frac{\partial \theta _{\rm B2}}{\partial \theta _y} \\ \frac{\partial \theta _{\rm B3}}{\partial \theta _r} & \frac{\partial \theta _{\rm B3}}{\partial \theta _p} & \frac{\partial \theta _{\rm B3}}{\partial \theta _y} \end{matrix} \right] = \left[ \begin{matrix}jb_{11} & jb_{12} & jb_{13} \\ jb_{21} & jb_{22} & jb_{23} \\ jb_{31} & jb_{32} & jb_{33} \end{matrix} \right] \\ &jb_{11} = \frac{C_{p} S_{y} \left(C_{r} C_{y} S_{y} - S_{p} S_{r} S_{y}\right)}{F}\\ & jb_{12} = \frac{S_{y} \left(C_{r} S_{y} + C_{y} S_{p} S_{r}\right) }{F} \\ &jb_{13} = - \frac{C_{p} S_{r}}{F} \\ &jb_{21} = \frac{\left(C_{r} S_{p} S_{y} + C_{y} S_{r}\right)}{G} \\ & jb_{22} = \frac{C_{p} S_{r} S_{y}}{G} \\ &jb_{23} = \frac{ C_{r} S_{y} + C_{y} S_{p} S_{r} }{G} \\ &jb_{31} = - \frac{C_{p} S_{y}}{H} \quad jb_{32} = \frac{S_{r} \left(C_{r} S_{p} S_{y} + C_{y} S_{r}\right) }{H} \\ &jb_{33} = \frac{C_{p} S_{r} \left(C_{r} C_{y} - S_{p} S_{r} S_{y}\right)}{H} \tag{61} \end{align*} View Source \begin{align*} F =& C_{p}^{2} S_{y}^{2} + \left(C_{r} S_{p} S_{y} + C_{y} S_{r}\right)^{2} \\ H =& C_{p}^{2} S_{r}^{2} + \left(C_{r} S_{y} + C_{y} S_{p} S_{r}\right)^{2} \\ G =& \sqrt{- \left(- C_{r} C_{y} + S_{p} S_{r} S_{y}\right)^{2} + 1}. \end{align*} View Source

Based on the principle of virtual work and using { (\boldsymbol J_A^{-1}) }^{T} and { (\boldsymbol J_A^{-1}) }^{T}, an expression is obtained to derive the torque \boldsymbol \tau _{\text{out}.A}, \boldsymbol \tau _{\text{out}.B} at the hand end in each link chain from the torque \boldsymbol \tau _{in.A}, \boldsymbol \tau _{in.B} at the joints \begin{align*} {\boldsymbol \tau _{\text{out}.A}} = & { (\boldsymbol J_A^{-1}) }^{T} \cdot {\boldsymbol \tau _{in.A}} \tag{62}\\ {\boldsymbol \tau _{ {out}.B}} = & { (\boldsymbol J_B^{-1}) }^{T} \cdot {\boldsymbol \tau _{in.B}}. \tag{63} \end{align*} View SourceHere, since the torque at the hand tip is {\boldsymbol \tau _{\text{out}}}={\boldsymbol \tau _{\text{out}.A}}+{\boldsymbol \tau _{\text{out}.B}}, the formula for the relationship between the torque at the hand tip and at the joint is derived as follows: \begin{align*} {\boldsymbol \tau _{\text{out}}} = & { (\boldsymbol J_A^{-1}) }^{T} \cdot {\boldsymbol \tau _{in.A}} + { (\boldsymbol J_B^{-1}) }^{T} \cdot {\boldsymbol \tau _{in.B}}\tag{64}\\ \left[ \begin{matrix}\tau _{\text{out}.r} \\ \tau _{\text{out}.p} \\ \tau _{\text{out}.y} \\ \end{matrix} \right] = & { (\boldsymbol J_A^{-1}) }^{T} \cdot \left[ \begin{matrix}\tau _{in.A1} \\ \tau _{in.A2} \\ \tau _{in.A3} \\ \end{matrix} \right] + { (\boldsymbol J_B^{-1}) }^{T} \cdot \left[ \begin{matrix}\tau _{in.B1} \\ \tau _{in.B2} \\ \tau _{in.B3} \\ \end{matrix} \right]. \tag{65} \end{align*} View SourceAssuming that the passive joint torques are \tau _{in.A3} = 0 and \tau _{in.B3} = 0, the following three more specific torque equations can be obtained: \begin{align*} \tau _{\text{out}.r} = & ja_{11} \cdot \tau _{in.A1} + jb_{11} \cdot \tau _{in.B1} + jb_{21} \cdot \tau _{in.B2} \tag{66}\\ \tau _{\text{out}.p} = & ja_{12} \cdot \tau _{in.A1} \tag{67}\\ & + ja_{22} \cdot \tau _{in.A2} + jb_{12} \cdot \tau _{in.B1} + jb_{22} \cdot \tau _{in.B2} \\ \tau _{\text{out}.y} = & ja_{13} \cdot \tau _{in.A1} \\ & + ja_{23} \cdot \tau _{in.A2} + jb_{13} \cdot \tau _{in.B1} + jb_{23} \cdot \tau _{in.B2}. \tag{68} \end{align*} View SourceBy solving this system of simultaneous equations, we can obtain \tau _{in.A1}, \tau _{in.A2}, \tau _{in.B1}, and \tau _{in.B2} that satisfy the target hand (CS-gear) torque. This means that antagonistic torques can be caused between the active joints depending on the orientation, but on the other hand, the torques can be superimposed on each other to increase the stiffness. In addition, using the passive joint torques \tau _{in.A3} and \tau _{in.B3} as the effect of friction in the sliding motion, a more accurate calculation is possible.