Type Synthesis of Self-Alignment Parallel Ankle Rehabilitation Robot with Suitable Passive Degrees of Freedom

Abstract

The current parallel ankle rehabilitation robot (ARR) suffers from the problem of difficult real-time alignment of the human-robot joint center of rotation, which may lead to secondary injuries to the patient. This study investigates type synthesis of a parallel self-alignment ankle rehabilitation robot (PSAARR) based on the kinematic characteristics of ankle joint rotation center drift from the perspective of introducing "suitable passive degrees of freedom (DOF)" with a suitable number and form. First, the self-alignment principle of parallel ARR was proposed by deriving conditions for transforming a human-robot closed chain (HRCC) formed by an ARR and human body into a kinematic suitable constrained system and introducing conditions of "decoupled" and "less limb". Second, the relationship between the self-alignment principle and actuation wrenches (twists) of PSAARR was analyzed with the velocity Jacobian matrix as a "bridge". Subsequently, the type synthesis conditions of PSAARR were proposed. Third, a PSAARR synthesis method was proposed based on the screw theory and type of PSAARR synthesis conducted. Finally, an HRCC kinematic model was established to verify the self-alignment capability of the PSAARR. In this study, 93 types of PSAARR limb structures were synthesized and the self-alignment capability of a human-robot joint axis was verified through kinematic analysis, which provides a theoretical basis for the design of such an ARR.

1 Introduction

The incidence of stroke gradually increases as aging progresses and many patients suffer from lower-extremity motor dysfunction owing to foot drop [1]. Medical research has indicated that effective rehabilitation training can accelerate the healing of damaged tissues, restore ligament elasticity, prevent joint adhesion, maintain joint flexibility, and accelerate joint rehabilitation, enabling patients to return to normal exercise performance quickly [2, 3]. Numerous studies have acknowledged the accurate work of rehabilitation physicians who perform a number of exercises significantly better than robots. However, the increasing number of patients and lack of medical personnel have driven a paradigm shift in rehabilitation. Thus, improved solutions are necessary to achieve faster recovery while simultaneously treating multiple patients [4]. To address this issue, ankle rehabilitation robots (ARRs) have been developed to reduce the labor intensity of rehabilitation physicians.

An ARR assists patients in rehabilitation exercise training and has many advantages compared to the traditional manual rehabilitation training modes, including: (i) ensuring the intensity, effect, and precision of rehabilitation training and better consistency of movement [5]; (ii) providing personalized training with different intensities and modes according to the degree of injury and rehabilitation of patients, and enhancing active participation of patients [6]; and (iii) quantitatively evaluating the rehabilitation progress of patients and providing a basis for improving the rehabilitation treatment scheme [7]. However, ARRs currently have problems, such as poor flexibility and low safety. Therefore, the research regarding ARRs has attracted increasing attention.

ARRs can be classified into two types according to their structures: platform and wearable robots [8]. Wearable robots have been used for gait rehabilitation training [9]. Platform robots can perform complex and specialized spatial motions, and are used to assist patients with physical disabilities or mobility difficulties during rehabilitation. They aid in preventing joint stiffness and improving joint range of motion and ankle proprioception [10]. Among platform ARRs, parallel ARRs have received wide attention because of their advantages, such as small inertia, high stiffness, compact structure, and high precision [11, 12].

Dai et al. [13] developed several ARRs using 3-SPS/S and 3-SPS/SP parallel mechanisms as the main structures. Saglia et al. [14] developed a two degrees of freedom (DOF) ARR controlled by three linear actuators. Ai et al. [15] developed a two DOF parallel-compliant ARR actuated by pneumatic muscles. Tsoi et al. [16, 17] developed an ARR that did not set a rotational axis to assist the rotation of the moving platform. Therefore, the subjects needed to control the lower limb as the rotation support and the ankle as the rotation center during rehabilitation training. Jamwal et al. [18, 19] proposed a lightweight and compliant parallel ARR in which pneumatic muscle actuators were placed parallel to the shinbone of the patient. Zou et al. [20] proposed a 3-RRS parallel ARR that could realize single and compound ankle rehabilitation training. Zhang et al. [21] and Qian et al. [22] developed ARRs driven by pneumatic muscle actuators that could realize changes in motion and dynamic performance by reconfiguring the actuator layouts. Li et al. [23, 24] and Dong et al. [25] presented novel parallel ARRs with the key characteristics of a simple configuration and no actuator redundancy. Liu et al. [26] designed a compliant ankle-rehabilitation robot redundantly driven by pneumatic muscles and cables to provide full range of motion and torque ability for the human ankle. Abu-Dakka et al. [27] developed a 3-PRS parallel ARR with three DOF that realized dorsiflexion (DO)/plantarflexion (PL), inversion (IN)/eversion (EV), and translation in the height direction. Zhang et al. [28] and Liu et al. [29] proposed compact generalized spherical parallel mechanisms suitable for ankle rehabilitation based on an ankle-motion fitting model with a high matching degree. Zeng et al. [30] proposed a new type of ankle-foot rehabilitation robot based on a decoupling series-parallel mechanism with one translational and three rotational DOF. Liu et al. [31] proposed a multi-locomotion-mode ARR based on a 2-UPU/RPU parallel mechanism that supported the training of muscle groups and ligaments related to ankle motion in addition to the rehabilitation training of the basic motion orientation of the ankle.

The aforementioned parallel ARR can be divided into two categories according to the relative positional relationship between the actuators and end effector: actuators below the end effector (AbEE) and actuators above the end effector (AaEE). The driving directions of ARRs with AaEE and the ankle were similar, providing a more suitable motion of the moving platform for the human ankle. However, a common problem between the two types of ARRs is that the mechanical axis of the rehabilitation robot is difficult to align or it coincides with the ankle joint axis/center in real time [32]. This, in turn, leads to human-robot motion incompatibility that results in secondary injury to the patient and is prevalent in joint rehabilitation robots. Several studies have been conducted to address this issue.

Niu et al. [33] proposed a compliant five-bar parallel mechanism that offered two mobilities in the sagittal plane. Additionally, the torsional springs mounted on this mechanism had the potential to automatically adjust the instantaneous center of rotation of the output link connected to the thigh with respect to the basis link connected to the shank. Sarkisian et al. [34, 35] proposed a novel powered knee exoskeleton with a self-alignment mechanism using a prismatic-revolute-revolute (PRR) configuration. Awad et al. [36] presented a novel approach for solving the self-axis alignment problem of an elbow exoskeleton using a planar parallel mechanism. Sun et al. [37] proposed a novel index finger exoskeleton with three motors to help post-stroke patients perform finger abduction (AB)/adduction (AD) and flexion/extension training. In addition, a spatial mechanism with passive DOF for the metacarpophalangeal joint was designed to realize human-robot axes self-alignment. Li et al. [38, 39] developed an upper-limb rehabilitation exoskeleton that achieved kinematic compatibility between the exoskeleton mechanism and human upper arm by introducing passive kinematic pairs. Trigili et al. [40] presented the design and experimental characterization of a four-DOF shoulder-elbow exoskeleton for upper-limb neurorehabilitation and treatment of spasticity that employed a self-alignment mechanism based on passive rotational joints to smoothly self-align the robot’s rotational axes to the axes of the user.

Rehabilitation robots for relatively simple self-aligning uni-/bi-axial joints (knee, elbow, and finger) have been thoroughly studied through analysis and comparison. Current research is mainly focused on series shoulder rehabilitation robots for a self-aligned multi-axis joint (shoulder). More complex joint rehabilitation robots based on parallel mechanisms have rarely been reported because the motion of the moving platform output feature point is realized by coupling more than two limbs, making the configuration design difficult.

To this end, the type synthesis of a self-alignment ankle rehabilitation robot (PSAARR) is researched, focusing on the issue of the human-robot joint axis of the current parallel ARR being difficult to align in real time. The "suitable DOF" is introduced to transform the human-robot closed chain (HRCC) formed by an ARR and human body into a kinematic suitable constrained system. The remainder of this paper is organized as follows. The self-alignment principle of the parallel ARR is described in Section 2, the type synthesis conditions of the PSAARR are presented in Section 3, type synthesis method and process of the PSAARR are described in Section 4, self-alignment capability analysis of the PSAARR is presented in Section 5, and conclusions are drawn in Section 6.

2 Self-Alignment Principle of Parallel ARR

The ankle is composed of tibia, fibula, talus, and calcaneus, which belong to the trochlear joint [41]. Based on the morphological structure and function of the ankle, the basic motion types include DO/PL, IN/EV, and AD/AB [42], as depicted in Figure 1. It is known from human anatomy that there is a coupling relationship between various movements of the ankle joint, and that the ankle joint rotation center drifts as its angle changes. Therefore, the ankle joint is equivalent to a three DOF spherical joint with the characteristics of a rotation center drift.

Figure 1

Morphological structure of the ankle joint and its HRCC model

From the perspective of the mechanism theory, the problem of human-robot kinematic incompatibility is caused by the insufficient DOF of the HRCC, making it difficult for the ARR to adapt to changes in the center of rotation of the ankle joint. Therefore, a suitable number and form of passive DOF must be introduced to transform the HRCC formed by the ARR and human body into a kinematic suitably constrained system. In this study, the passive DOF with a suitable number and form is defined as a "suitable DOF".

The concept of an "equivalent virtual series kinematic chain" for a parallel robot that was proposed by Kong et al. [43] is introduced to simplify the HRCC model. For any parallel robot, the corresponding virtual series kinematic chain can be determined by solving the kinematic space analytical expression for its moving platform. A schematic of the HRCC formed by the human body and equivalent virtual series kinematic chain of the ARR is shown in Figure 1.

The DOF of the HRCC can be determined using the modified Grübler–Kutzbach criterion [44] as:

$$M = \sum\limits_{i = 1}^{g} {f_{i} } + 6\left( {n - g - 1} \right) + \mu ,$$

(1)

where M represents the DOF of the HRCC, f_i represents the DOF of the ith kinematic pair of the HRCC, g represents the number of kinematic pairs of the HRCC, n represents the number of components (including the frame) of the HRCC, and μ represents the number of over constraints of the HRCC. According to "Euler's formula" in the graph theory, l = g + 1−n, and l represents the number of loops of the HRCC.

For the PSAARR, Eq. (1) can be expressed as:

$$M = f_{k} + f_{uk} - 6l + \mu ,$$

(2)

where f_k and f_uk represent the DOF of the known and unknown kinematic pairs of the HRCC, respectively.

According to the morphological structure and function of the PSAARR HRCC, the DOF of the HRCC is M = 3, that is, the DO/PL, IN/EV and AD/AB of the ankle joint. The DOF of the known kinematic pairs is f_k = 6, that is, three active kinematic pairs of the PSAARR and three DOF of the ankle joint. The number of loops is l = 1 and the number of the over constraint is μ = 0. Therefore, the DOF of unknown kinematic pairs f_uk can be solved using Eq. (2).

$$f_{uk} = M - f_{k} + 6l - \mu = 3 - 6 + 6 \times 1 - 0 = 3.$$

(3)

According to Eq. (3), three passive DOF should be introduced into the HRCC, that is, three translation DOF in mutually perpendicular directions. Therefore, the PSAARR should have six DOF, including three active DOF of rotation (moving platform orientation) and three passive DOF of translation (moving platform position). The three active DOF of rotation are controlled by the three driving pairs, and the three passive DOF of translation are controlled by the human body to achieve the self-alignment of the human-robot axis.

Simultaneously, active and passive DOF are required to be decoupled (be in a "decoupled" condition) to ensure that active rehabilitation training and passive self-alignment are independent of one other and to avoid mutual interference between them. In addition, the number of limbs should be equal to the number of active DOF to reduce the number of limbs, reduce mutual interference of limbs, and expand the workspace of the robot. This is known as the "less limb" condition. Based on this, a three-limb six-DOF rotation and translation decoupled parallel mechanism was proposed as the main structure of the PSAARR.

In summary, a self-alignment principle of parallel ARR was proposed that included:

• (i) Three "suitable passive DOF" of translation that are based on three active DOF of rotation, such that the HRCC formed by the rehabilitation robot and human body was transformed into a kinematic suitable constraint system.

• (ii) Active and passive DOF that are decoupled, and a number of limbs that was equal to the number of active DOF.

3 Type Synthesis Conditions of the PSAARR

The velocity Jacobian matrix of the parallel robot represents the velocity transfer relationship between the operational and joint spaces. The velocity Jacobian matrix of a PSAARR has a specific form according to the self-alignment principle. The relationship between the velocity Jacobian matrix of the PSAARR and its actuation wrenches (the twists) can be established based on the screw theory. Thus, the relationship between the self-alignment principle of the parallel ARR and the actuation wrenches (the twists) of the PSAARR can be established with the velocity Jacobian matrix as "the bridge".

3.1 Velocity Jacobian Matrix Form of PSAARR

The relationship between the joint space velocity ${\dot{\varvec{q}}}$, velocity Jacobian matrix J, and operating space velocity ${\dot{\varvec{X}}}$ for a parallel robot is ${\dot{\varvec{q}}} = {\varvec{J\dot{\varvec{X}}}}$.

For the PSAARR:

(i) Joint space velocity ${\dot{\varvec{q}}}$ of the limbs can be expressed as:

$$\dot{{\varvec{q}}}={\left(\begin{array}{cccccc}{\dot{q}}_{11}& {\dot{q}}_{12}& {\dot{q}}_{21}& {\dot{q}}_{22}& {\dot{q}}_{31}& {\dot{q}}_{32}\end{array}\right)}^{{\text{T}}},$$

(4)

where $\dot{q}_{ij}$ represents the velocity of the jth kinematic pair of ith limb. The active kinematic pairs q_i1 (main active) and q_i2 (subactive) control the rotation and translation of the moving platform, respectively.

(ii) The operating space velocity ${\dot{\varvec{X}}}$ is expressed as:

$${\dot{\varvec{X}}}{ = }\left( {\begin{array}{*{20}c} {v_{X} } & {v_{Y} } & {v_{Z} } & {\omega_{X} } & {\omega_{Y} } & {\omega_{Z} } \\ \end{array} } \right)^{{\text{T}}} ,$$

(5)

where v_X, v_Y, and v_Z represent the translational velocity components of the operating space. Similarly, ω_X, ω_Y, and ω_Z represent the rotational velocity components of the operating space.

(iii) The velocity Jacobian matrix J is expressed as:

$${\varvec{J}}{ = }\left( {\begin{array}{*{20}c} {{\varvec{J}}_{1} } & {{\varvec{J}}_{2} } & {{\varvec{J}}_{3} } \\ \end{array} } \right)^{{\text{T}}} ,$$

(6)

$${\varvec{J}}_{i} = \left( {\begin{array}{*{20}l} 0 & 0 & 0 & {J_{i14} } & {J_{i15} } & {J_{i16} } \\ {J_{i21} } & {J_{i22} } & {J_{i23} } & 0 & 0 & 0 \\ \end{array} } \right),\quad \left( {i = 1,\;2,\;3} \right).$$

Eq. (6) represents the velocity Jacobian matrix form of the PSAARR.

3.2 Velocity Jacobian Matrix Analysis of PSAARR

The velocity Jacobian matrix for parallel robots can be analyzed using the screw theory. A typical parallel robot consists of n limbs in which each limb typically has at least one active kinematic pair and the other kinematic pairs are passive. Multi-DOF kinematic pairs must be equated to a combined form of single-DOF kinematic pairs to facilitate characterization. This way, each limb can be regarded as an open-loop kinematic chain composed of several single-DOF kinematic pairs with its end connected to a moving platform.

The instantaneous output motion of a parallel robot moving platform can be represented by a linear combination of instantaneous screw motions in the limbs as:

$$\begin{gathered} \hfill {\varvec{v}} = \sum\limits_{j = 1}^{{m_{i} }} {\dot{q}_{ij} } \textbf{\textit{\$}}_{ij} = \left( {\textbf{\textit{\$}}_{i1} \;\;\textbf{\textit{\$}}_{i2} \;\; \cdots \;\;\textbf{\textit{\$}}_{im} } \right)\left( {\dot{q}_{i1} \;\;\dot{q}_{i2} \;\; \cdots \;\;\dot{q}_{i3} } \right)^{{\text{T}}} , \\ \hfill \left( {i = 1,\;2,\; \cdots ,\;n} \right), \\ \end{gathered}$$

(7)

where v represents the instantaneous output velocity vector at the end of the limb, $_ij represents the instantaneous twist of the jth single-DOF kinematic pair of the ith limb, and m_i represents the degrees of connection of the ith limb.

The actuation wrench represents the active-force screw applied to the moving platform by the driving pair of limbs. The actuation wrench does not work on the twist corresponding to the passive kinematic pair (referred to as a passive twist) in the same limb, and the reciprocal product of the two is zero. Assuming that there are g active kinematic pairs in each limb, there are at least g reciprocal screws (actuation wrench) that are reciprocal to all passive screws in each limb.

The unit screw of the actuation wrench can be expressed as $_rij (j=1, 2, …, g), and the passive twist in Eq. (7) can be eliminated using the reciprocal screw theory. The orthogonal operation on both sides of Eq. (7) with $_rij yields the following relationship:

$${\varvec{J}}_{Xi} {\dot{\varvec{X}}} = {\varvec{J}}_{qi} {\dot{\varvec{q}}}_{i} \quad \left( {i = 1,\;2,\; \cdots ,\;n} \right).$$

(8)

In Eq. (8),

$${\varvec{J}}_{Xi} = \left( {\begin{array}{*{20}c} {\textbf{\textit{\$}}_{ri1} } & {\textbf{\textit{\$}}_{ri2} } & \cdots & {\textbf{\textit{\$}}_{rig} } \\ \end{array} } \right)^{{\text{T}}} ,$$

$${\varvec{J}}_{qi} = \left( {\begin{array}{*{20}l} {\textbf{\textit{\$}}_{ri1} \circ \textbf{\textit{\$}}_{i1} } & {\textbf{\textit{\$}}_{ri1} \circ \textbf{\textit{\$}}_{i2} } & \cdots & {\textbf{\textit{\$}}_{ri1} \circ \textbf{\textit{\$}}_{ig} } \\ {\textbf{\textit{\$}}_{ri2} \circ \textbf{\textit{\$}}_{i1} } & {\textbf{\textit{\$}}_{ri2} \circ \textbf{\textit{\$}}_{i2} } & \cdots & {\textbf{\textit{\$}}_{ri2} \circ \textbf{\textit{\$}}_{ig} } \\ \vdots & \vdots & {} & \vdots \\ {\textbf{\textit{\$}}_{rig} \circ \textbf{\textit{\$}}_{i1} } & {\textbf{\textit{\$}}_{rig} \circ \textbf{\textit{\$}}_{i2} } & \cdots & {\textbf{\textit{\$}}_{rig} \circ \textbf{\textit{\$}}_{ig} } \\ \end{array} } \right),$$

$${\dot{\varvec{q}}}_{i} = \left( {\begin{array}{*{20}l} {\dot{q}_{i1} } & {\dot{q}_{i2} } & \cdots & {\dot{q}_{ig} } \\ \end{array} } \right)^{{\text{T}}} .$$

Eq. (8) contains n equations, written in matrix form as:

$${\varvec{J}}_{X} {\dot{\varvec{X}}} = {\varvec{J}}_{q} {\dot{\varvec{q}}},$$

(9)

$${\varvec{J}}_{X} = \left( {\begin{array}{*{20}c} {{\varvec{J}}_{{X{1}}} } & {{\varvec{J}}_{{X{2}}} } & \cdots & {{\varvec{J}}_{Xn} } \\ \end{array} } \right)^{{\text{T}}} ,$$

$${\varvec{J}}_{q} { = }\left( {\begin{array}{*{20}c} {{\varvec{J}}_{{q{1}}} } & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & {{\varvec{J}}_{{q{2}}} } & \cdots & \mathbf{0} \\ \vdots & \vdots & {} & \vdots \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & {{\varvec{J}}_{qn} } \\ \end{array} } \right),$$

where J_X and J_q represent the positive and inverse Jacobian matrices of the parallel robot, respectively.

According to Eq. (9), if the inverse Jacobian matrix J_q is a full-rank matrix, then:

$${\dot{\varvec{q}}}{ = }{\varvec{J}}_{q}^{ - 1} {\varvec{J}}_{X} {\dot{\varvec{X}}} = {\varvec{J\dot{\varvec{X}}}}.$$

(10)

The positive Jacobian matrix J_X of the three-limb six-DOF parallel robot is expressed as:

$${\varvec{J}}_{X} = \left( {\begin{array}{*{20}c} {{\varvec{J}}_{{X{1}}} } & {{\varvec{J}}_{{X{2}}} } & {{\varvec{J}}_{{X{3}}} } \\ \end{array} } \right)^{{\text{T}}} ,$$

(11)

$${\varvec{J}}_{Xi} { = }\left( {\begin{array}{*{20}c} {\textbf{\textit{\$}}_{{ri{1}}} } & {\textbf{\textit{\$}}_{{ri{2}}} } \\ \end{array} } \right)^{{\text{T}}} ,\quad \left( {i = 1,\;2,\;3} \right).$$

The inverse Jacobian matrix J_q of the three-limb six-DOF parallel robot is expressed as:

$${\varvec{J}}_{q} { = }\left( {\begin{array}{*{20}l} {{\varvec{J}}_{{q{1}}} } & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{J}}_{{q{2}}} } & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {{\varvec{J}}_{{q{3}}} } \\ \end{array} } \right),$$

(12)

$${\varvec{J}}_{qi} = \left( {\begin{array}{*{20}c} {J_{qi11} } & {J_{qi12} } \\ {J_{qi21} } & {J_{qi22} } \\ \end{array} } \right){ = }\left( {\begin{array}{*{20}c} {\textbf{\textit{\$}}_{{ri{1}}} \circ \textbf{\textit{\$}}_{i1} } & {\textbf{\textit{\$}}_{{ir{1}}} \circ \textbf{\textit{\$}}_{i2} } \\ {\textbf{\textit{\$}}_{{ri{2}}} \circ \textbf{\textit{\$}}_{i1} } & {\textbf{\textit{\$}}_{{ri{2}}} \circ \textbf{\textit{\$}}_{i2} } \\ \end{array} } \right),\quad \left( {i = 1,\;2,\;3} \right).$$

According to Eqs. (10), (11), and (12), the velocity Jacobian matrix J can be expressed as:

$${\varvec{J}}{ = }\left( {\begin{array}{*{20}c} {{\varvec{J}}_{1} } \\ {{\varvec{J}}_{2} } \\ {{\varvec{J}}_{3} } \\ \end{array} } \right){ = }\left( {\begin{array}{*{20}c} {{\varvec{J}}_{{q{1}}}^{ - 1} } & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & {{\varvec{J}}_{{q{2}}}^{ - 1} } & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & {{\varvec{J}}_{{q{3}}}^{ - 1} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {{\varvec{J}}_{{X{1}}} } \\ {{\varvec{J}}_{{X{2}}} } \\ {{\varvec{J}}_{{X{3}}} } \\ \end{array} } \right).$$

(13)

The instantaneous twist of the single-DOF kinematic pair $_ij can be expressed as:

$$\begin{gathered} \hfill \textbf{\textit{\$}}_{ij} = \left( {\begin{array}{*{20}c} {L_{ij} } & {M_{ij} } & {N_{ij} \;;} & {P_{ij} } & {Q_{ij} } & {R_{ij} } \\ \end{array} } \right), \\ \hfill \left( {i = 1,\;2,\;3} \right),\quad \left( {j = 1,\;2,\; \cdots ,\;6} \right), \\ \end{gathered}$$

(14)

where $_ij represents the active and passive kinematic pair twists when j = 1, 2 and j = 3, 4, 5, 6, respectively.

The actuation wrench $_rij can be expressed as:

$$\begin{gathered} \hfill \textbf{\textit{\$}}_{rij} = \left( {\begin{array}{*{20}c} {L_{rij} } & {M_{rij} } & {N_{rij} \;;} & {P_{rij} } & {Q_{rij} } & {R_{rij} } \\ \end{array} } \right), \\ \hfill \left( {i = 1,\;2,\;3} \right),\quad \left( {j = 1,\;2} \right). \\ \end{gathered}$$

(15)

Combining Eqs. (13) – (15) results in:

$${\varvec{J}}_{i} = {\varvec{J}}_{i1} {\varvec{J}}_{i2} ,\quad \left( {i = 1,\;2,\;3} \right),$$

(16)

$${\varvec{J}}_{i1} = \frac{1}{{J_{qi11} J_{qi22} - J_{qi12} J_{qi21} }}\left( {\begin{array}{*{20}l} {J_{qi22} } & { - J_{qi21} } \\ { - J_{qi12} } & {J_{qi11} } \\ \end{array} } \right),$$

$${\varvec{J}}_{i2} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {L_{ri1} } & {M_{ri1} } & {N_{ri1} } & {P_{ri1} } & {Q_{ri1} } & {R_{ri1} } \\ \end{array} } \\ {\begin{array}{*{20}c} {L_{ri2} } & {M_{ri2} } & {N_{ri2} } & {P_{ri2} } & {Q_{ri2} } & {R_{ri2} } \\ \end{array} } \\ \end{array} } \right).$$

Eq. (16) represents the relationship between the velocity Jacobian matrix and limb actuation wrench (and twist) of the PSAARR.

3.3 Condition of Actuation Wrench and Twist

According to the screw theory, the actuation wrench in the same limb works on the corresponding actuated twist, and the reciprocal product of the two cannot be zero. Therefore, J_qi11 and J_qi22 in Eq. (16) were not zero. Subsequently, the main active q_i1 and subactive q_i2 kinematic pairs control the rotation and translation of the moving platform, respectively. Thus, the actuation wrenches $_ri1 and $_ri2 that correspond to the main active q_i1 and subactive q_i2 kinematic pairs were a zero (or infinite) pitch screw and a zero-pitch screw, respectively. This results in L_ri2, M_ri2, and N_ri2 in Eq. (16) not being zero at the same time.

Combining Eqs. (6) and (16) results in:

$$\left\{ \begin{gathered} J_{qi12} = J_{qi21} = 0, \hfill \\ L_{ri1} { = }M_{ri1} { = }N_{ri1} { = }0, \hfill \\ P_{ri2} { = }Q_{ri2} { = }R_{ri2} { = }0. \hfill \\ \end{gathered} \right.$$

(17)

According to Eq. (17), the actuation wrench can be expressed as:

$$\left\{ \begin{gathered} \textbf{\textit{\$}}_{ri1} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & {P_{ri1} } & {Q_{ri1} } & {R_{ri1} } \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{ri2} = \left( {\begin{array}{*{20}c} {L_{ri2} } & {M_{ri2} } & {N_{ri2} \;;} & 0 & 0 & 0 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(18)

According to Eq. (18), the actuation wrench $_ri1 is an infinite pitch screw and the actuation wrench $_ri2 is a zero pitch screw passing through the coordinate origin.

Combining the nature of the reciprocity between the actuation wrench and passive twist yields:

$$\left\{ \begin{gathered} \textbf{\textit{\$}}_{{ri{1}}} \circ \textbf{\textit{\$}}_{ij} = 0\quad \left( {j = 2,\;3,\;4,\;5,\;6} \right), \hfill \\ \textbf{\textit{\$}}_{ri2} \circ \textbf{\textit{\$}}_{ij} = 0\quad \left( {j = 1,\;3,\;4,\;5,\;6} \right). \hfill \\ \end{gathered} \right.$$

(19)

According to Eq. (19), the following condition for a PSAARR should be satisfied: In the same limb, the reciprocal product of the actuation wrench and all other twists is zero, except for the corresponding twist.

4 Type Synthesis Method and Process of the PSAARR

4.1 Type Synthesis Method of the Limb

The kinematic output characteristics of the parallel robot moving platform were determined using the kinematic pair type, number, order, and configuration of each limb, and the assembly form between the limbs. Limb-type synthesis is the key link, theoretical basis, and prerequisite for realizing parallel robot-type synthesis. The limb-type synthesis method and specific steps of the PSAARR were as follows:

• (i) All the possible forms of passive twist were determined based on the reciprocal product of the actuation wrench and passive twist in the same limb being zero.

• (ii) All possible forms of the active twist were determined based on the reciprocal product of the actuation wrench and corresponding active twist not being zero, and the reciprocal product of the actuation wrench and non-corresponding active twist being zero.

• (iii) The feasible number of various twists in the limb was determined based on the maximum linear irrelevance group condition of the twist system.

• (iv) The type, number, and configuration orientation of the kinematic pairs in each limb were determined according to the forms of the active and passive twists. Additionally, the existence of an inert pair was considered.

• (v) A limb-type synthesis was conducted according to the different degrees of limb connectivity, and all feasible limb structures were listed.

The limb type synthesis process of the PSAARR is shown in Figure 2.

Figure 2

Limb type synthesis process of the PSAARR

Only the basic kinematic pair was considered for the type synthesis of the limb to simplify the structure and limit space. For the case where the basic kinematic pair was replaced by a composite kinematic pair, refer to Ref. [45].

4.2 Type Synthesis of Passive Kinematic Pair Limb

All possible forms of the passive kinematic pair twist were determined based on the reciprocal product of the actuation wrench and all passive kinematic pair twists in the limb being equal to zero. According to Eq. (18), the passive kinematic pair twist can be expressed as:

$$\left\{ \begin{gathered} \textbf{\textit{\$}}_{i3} = \left( {\begin{array}{*{20}c} { - Q_{ri1} } & {P_{ri1} } & {0\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i4} = \left( {\begin{array}{*{20}c} { - R_{ri1} } & 0 & {P_{ri1} \;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i5} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & { - M_{ri2} } & {L_{ri2} } & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i6} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & {\; - N_{ri2} } & 0 & {L_{ri2} } \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(20)

Other forms of twisting can be obtained using a linear combination of the four twists in Eq. (20). However, the twists after the linear combination must be linearly independent. The case of redundant kinematics in the limb was not considered. That is, only the case in which the connectivity of the passive kinematic pair limb is four was considered. The passive kinematic limb pairs are listed in Table 1.

Table 1 Passive kinematic pairs of PSAARR

4.3 Type Synthesis of Active Kinematic Pairs Limb

4.3.1 Type Synthesis of Subactive and Passive Kinematic Pairs

The structures of the subactive and passive kinematic pairs were further determined based on the structure of the passive kinematic pair and according to the condition that the reciprocal product of the subactive kinematic pair twist $_i2 and main actuation wrench $_ri1 was zero.

According to Eqs. (18) and (20), the twisting system of the subactive and passive kinematic pairs can be expressed as:

$$\left\{ \begin{gathered} \textbf{\textit{\$}}_{i2} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 1 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i3} = \left( {\begin{array}{*{20}c} { - Q_{ri1} } & {P_{ri1} } & {0\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i4} = \left( {\begin{array}{*{20}c} { - R_{ri1} } & 0 & {P_{ri1} \;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i5} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & { - M_{ri2} } & {L_{ri2} } & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i6} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & {\; - N_{ri2} } & 0 & {L_{ri2} } \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(21)

Other forms of twisting can be obtained using a linear combination of the five twists in Eq. (21). However, the twists after the linear combination must be linearly independent. The case of redundant kinematics in the limb was not considered. That is, only the case where the connectivity of the subactive and passive kinematic pair was five was considered. The kinematic pair structures from the obtained twist system whose passive kinematic pairs were according to those listed in Table 1 were selected, as listed in Table 2.

Table 2 Subactive and passive kinematic pairs of PSAARR

The following conditions must be satisfied for the structure of the subactive and passive kinematic pairs, as listed in Table 2:

• (i) The axis direction of the revolute pair must be two.

• (ii) The number of revolute pairs in the same direction must be less than four.

• (iii) The number of prismatic pairs must be less than four.

4.3.2 Type Synthesis of Main Active and Passive Kinematic Pairs

The structure of the main active and passive kinematic pairs was based on determining the structure of the passive kinematic pair and was further determined according to the condition that the reciprocal product of the main active kinematic pair twist $_i1 and subactuation wrench $_ri2 was zero.

Based on Eqs. (18) and (20), the twist system of main active and passive kinematic pairs can be expressed as:

$$\left\{ \begin{gathered} \textbf{\textit{\$}}_{i1} = \left( {\begin{array}{*{20}c} 1 & 0 & {0\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i3} = \left( {\begin{array}{*{20}c} { - Q_{ri1} } & {P_{ri1} } & {0\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i4} = \left( {\begin{array}{*{20}c} { - R_{ri1} } & 0 & {P_{ri1} \;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i5} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & { - M_{ri2} } & {L_{ri2} } & 0 \\ \end{array} } \right), \hfill \\ \textbf{\textit{\$}}_{i6} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & {\; - N_{ri2} } & 0 & {L_{ri2} } \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(22)

Other forms of twisting were obtained using a linear combination of the five twists in Eq. (22). However, the twists after the linear combination must be linearly independent. The case of redundant kinematics in the limb was not considered. That is, only the case where the connectivity of the main active and passive kinematic pairs was five was considered. The kinematic pair structures from the obtained twist system whose passive kinematic pairs were according to those listed in Table 1 were selected, as listed in Table 3.

Table 3 Main active and passive kinematic pairs of PSAARR

The following conditions must be satisfied for the structure of the main active and passive kinematic pairs, as listed in Table 3:

• (i) The revolute pair must be divided into two groups, the axis of the first group must intersect the rotation center of the moving platform, the axis of the second group must be parallel to each other, and all revolute pairs must belong to one of the two groups.

• (ii) The number of revolute pairs belonging to the same group must be less than four.

• (iii) The number of revolute pairs belonging to the first group must be greater than one.

• (iv) When two revolute pairs belong to the first group, the axis of the prismatic pair must be perpendicular to that of the revolute pair belonging to the second group.

• (v) The number of prismatic pairs must be less than three.

4.4 Type Synthesis of the PSAARR

The kinematic pair structures in accordance with those listed in Table 2 and Table 3 were selected based on the determination of the passive and active kinematic pair structures, which was the limb structure of the PSAARR. The case of redundant kinematics in the limb was not considered. That is, only the case where the connectivity of the limb kinematic pairs was six was considered. The limb structure of the PSAARR is listed in Table 4 and a portion of the limb structure model is shown in Figure 3.

Table 4 Limb kinematic pairs of PSAARR

Figure 3

Limb structures of the PSAARR

As shown in Table 4, 93 types of PSAARR limb structures were obtained. A PSAARR was obtained by arbitrarily selecting three limb structures from the type synthesis results in Table 4 and connecting them to the moving and static platforms. This was done because the six-DOF limbs did not constrain the moving platform.

The $3{\text{ - R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ PSAARR belonging to category 3R3P is shown in Figure 4. For each limb: The first kinematic pair (revolute pair) is perpendicular to the second and third kinematic pairs (prismatic pairs), and not perpendicular to the fourth kinematic pair (prismatic pair); the two revolute pairs closest to the moving platform intersect the rotation center of the moving platform.

Figure 4

$3{ - }{\text{R}}_{{1{\text{ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ (3R3P) SPSAARR

The $3{\text{ - R}}_{{{\text{1ma}}}} \overline{\text{P}}{\tilde{\text{P}}}{\text{R}}_{{2{\text{sa}}}} \hat{\text{R}}_{2} \hat{\text{R}}$ PSAARR belonging to category 4R2P is shown in Figure 5. For each limb: The first kinematic pair (revolute pair) is perpendicular to the second and third kinematic pairs (prismatic pairs); the fourth kinematic pair (revolute pair) and the fifth kinematic pair (revolute pair) are parallel to each other; the two revolute pairs closest to the moving platform intersect the rotation center of the moving platform.

Figure 5

$3{\text{ - R}}_{{{\text{1ma}}}} \overline{\text{P}}{\tilde{\text{P}}}{\text{R}}_{{2{\text{sa}}}} \hat{\text{R}}_{2} \hat{\text{R}}$ (4R2P) PSAARR

The $3{\text{ - R}}_{{1}} {\text{R}}_{{1}} {\text{R}}_{{1}} \dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}_{{{\text{ma}}}} \hat{\text{R}}$ PSAARR belonging to category 5R1P is shown in Figure 6. For each limb: The three kinematic pairs closest to the static platform are parallel to each other, and not perpendicular to the fourth kinematic pair (prismatic pair); the two revolute pairs closest to the moving platform intersect the rotation center of the moving platform.

Figure 6

$3{\text{ - R}}_{{1}} {\text{R}}_{{1}} {\text{R}}_{{1}} \dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}_{{{\text{ma}}}} \hat{\text{R}}$ (5R1P) PSAARR

The $3{\text{ - R}}_{{1}} {\text{R}}_{{1}} {\text{R}}_{{1}} {\text{R}}_{{{\text{2sa}}}} \hat{\text{R}}_{2} \hat{\text{R}}_{{{\text{ma}}}}$ PSAARR belonging to category 6R is shown in Figure 7. For each limb: The three kinematic pairs closest to the static platform are parallel to each other; the fourth kinematic pair (revolute pair) and the fifth kinematic pair (revolute pair) are parallel to each other; the two revolute pairs closest to the moving platform intersect the rotation center of the moving platform.

Figure 7

$3{\text{ - R}}_{{1}} {\text{R}}_{{1}} {\text{R}}_{{1}} {\text{R}}_{{{\text{2sa}}}} \hat{\text{R}}_{2} \hat{\text{R}}_{{{\text{ma}}}}$ (6R) PSAARR

The main active kinematic pair should be a frame pair to improve the performance of the robot. The subactive kinematic pair is controlled by the human body. Therefore, it should be as close as possible to the moving platform. As shown in Table 4, the two-limb structures ${\text{R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ and ${\text{R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\text{R}_{{2{\text{sa}}}} \hat{\text{R}}_{2} \hat{\text{R}}$ satisfied this requirement. In comparison, ${\text{R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ structure was simpler. Therefore, the ${\text{3 - R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ symmetric parallel robot was selected as the main structure of the PSAARR.

5 Self-Alignment Capability Analysis of the PSAARR

5.1 Description of HRCC Structure

The x-, y-, and z-axes of the static platform coordinate system O-xyz for the rehabilitation robot coincided with the main active kinematic pair axes of the first, second, and third limbs, respectively. In addition, the origin O was the intersection point of the three axes. The origin O' of the moving platform coordinate system O'-x'y'z' was the intersection point of the axis of the revolute pair directly connected to the moving platform. The z'-axis pointed from point O' to the form center C of the moving platform. The y'-axis was located on the plane O'CO₁₆, which was opposite to the direction of the vector O'O_16. The x'-axis was determined by the right-hand rule, as shown in Figure 8.

Figure 8

Local coordinate system of the first limb for each kinematic pair

The local coordinate system defined for each kinematic pair of the second and third limbs was the same as that of the first limb owing to the symmetrical structure. Therefore, the local coordinate system of each kinematic pair was defined considering the first limb as an example. In addition, the coordinate origin O_ij coincided with the center point of each kinematic pair, as shown in Figure 8. The local coordinate systems O_i5-x_i5y_i5z_i5 and O_i6-x_i6y_i6z_i6 were established at the kinematic pairs O_i5 and O_i6, respectively. Points O_i5 and O_i6 coincided with point O', and O_i5 and O_i6 were drawn at the kinematic pair for convenience. Simultaneously, only the z_ij-axis was marked to express the local coordinate system, and the x_ij- and y_ij-axes were determined according to the principle of coordinate system establishment. The axes of each kinematic pair coincided with their corresponding local coordinate systems, which is expressed as:

$$\left\{ \begin{gathered} {}^{{O_{ij} }}{\hat{\varvec{\omega }}}_{ij} = \left( {\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} } \right)^{{\text{T}}} \quad \left( {i = 1,\;2,\;3;\;j = 1,\;5,\;6} \right), \hfill \\ {}^{{O_{ij} }}{\hat{\varvec{v}}}_{ij} = \left( {\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} } \right)^{{\text{T}}} \quad \left( {i = 1,\;2,\;3;\;j = 2,\;3,\;4} \right). \hfill \\ \end{gathered} \right.$$

(23)

The structural parameters of the limb are as follows: the distances between points O and O_i1, O_i1 and O_i3, O_i3 and O_i4, and O_i4 and O_i2 are L₁, L₂, L₃, and L₄ at the initial position, respectively. The distance between point O' and O_i6 is L₅. The side length of equilateral triangle O₁₆O₂₆O₃₆ is L₆, the angle between kinematic pairs O_i3 and O_i4 is α, the angle between kinematic pairs O_i5 and O_i6 is β, and the angle between axes O_i6O and OC is γ. The homogeneous transformation matrixes of each adjacent kinematic pair for the initial pose are presented in the Appendix.

A static coordinate system was established for the human body kinematic chain at the fixed connection part between the PSAARR and human lower leg, which coincided with the static platform coordinate system of the PSAARR. In addition, a moving coordinate system was established at the center of the human talus, which coincided with the moving platform coordinate system of the PSAARR. The homogeneous transformation matrix of the moving coordinate system O'-x'y'z' with respect to the static coordinate system O-xyz is expressed as:

$${}_{{O^{\prime}}}^{O} {\varvec{ T}}\left( {\varvec{q}} \right) = \left( {\begin{array}{*{20}l} {{}_{{O^{\prime}}}^{O} {\varvec{R}}} & {{}^{O}{\varvec{p}}_{{O^{\prime}}} } \\ {\mathbf{0}} & 1 \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {r_{13} } & {p_{x} } \\ {r_{21} } & {r_{22} } & {r_{23} } & {p_{y} } \\ {r_{31} } & {r_{32} } & {r_{33} } & {p_{z} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(24)

where (r₁₁, r₂₁, r₃₁)^T, (r₁₂, r₂₂, r₃₂)^T, and (r₁₃, r₂₃, r₃₃)^T represent the direction vectors of the x'-, y'-, and z'- axes, respectively. Additionally, r_ij (i = 1, 2, 3; j = 1, 2, 3) denotes known numbers when the rehabilitation locomotion mode is given. (p_x, p_y, p_z)^T represents the position vector of point O', and p_x, p_y, and p_z are known numbers according to the position of the human talus center.

There are two paths from the origin O of the static coordinate system O-xyz to the origin O' of the moving coordinate system O'-x'y'z' for the HRCC of the PSAARR, that is, the PSAARR and human body. Subsequently, the homogeneous transformation matrix between the two coordinate systems of the human body path is known. Therefore, all displacement or rotation angles of the PSAARR active kinematic pair can be obtained according to the kinematic constraint equation of the HRCC.

From the previous analysis, the position and orientation of the PSAARR moving platform were decoupled, such that they could be calculated from the human talar center position and rehabilitation locomotion mode, respectively.

5.2 Kinematic Position Analysis of the HRCC

From the previous analysis, point O_i5 coincided with point O', that is, ^Op_O' = ^Op_Oi5. Therefore, it was only necessary to know the homogeneous transformation matrix ${}_{{O_{i5} }}^{O} {\varvec{T}}\left( {\varvec{q}} \right)$ (the coordinate system O_i5-x_i5y_i5z_i5 with respect to the static platform coordinate system O-xyz) to obtain ^Op_O'.

According to the exponential product equation, ${}_{{O_{i5} }}^{O} {\varvec{T}}\left( {\varvec{q}} \right)$ can be expressed as:

$${}_{{O_{i5} }}^{O} {\varvec{T}}\left( {\varvec{q}} \right) = {{e}}^{{\theta_{i1} \left[ {{}^{O}{\hat{\textbf{\textit{\$}}}}_{i1} } \right]}} {{e}}^{{d_{i3} \left[ {{}^{O}{\hat{\textbf{\textit{\$}}}}_{i3} } \right]}} {{e}}^{{d_{i4} \left[ {{}^{O}{\hat{\textbf{\textit{\$}}}}_{i4} } \right]}} {{e}}^{{d_{i2} \left[ {{}^{O}{\hat{\textbf{\textit{\$}}}}_{i2} } \right]}} {}_{{O_{i5} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right),$$

(25)

where θ_i1 represents the rotation angle of the main active kinematic pair in the ith limb, d_ij (j = 2, 3, 4) represents the displacement of the jth prismatic pair in the ith limb, and ${}^{O}{\hat{\textbf{\textit{\$}}}}_{ij}$ represents the unit twist of the jth kinematic pair relative to the static platform coordinate system O-xyz when all kinematic pairs of the ith limb (except the jth kinematic pair) are fixed in the initial configuration.

For the first limb, the unit twist ${}^{O}{\hat{\textbf{\textit{\$}}}}_{ij}$ is expressed as:

$$\left\{ \begin{gathered} {}^{O}{\hat{\textbf{\textit{\$}}}}_{11} { = }\left( {\begin{array}{*{20}c} 1 & 0 & {0\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{13} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 0 & 0 & 1 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{14} = \left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 0 & { - {\text{s}} \alpha } & {{\text{c}} \alpha } \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{12} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 1 & 0 & 0 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(26)

where sα represents sinα and cα represents cosα.

For the second limb, the unit twist ${}^{O}{\hat{\textbf{\textit{\$}}}}_{ij}$ is expressed as:

$$\left\{ \begin{gathered} {}^{O}{\hat{\textbf{\textit{\$}}}}_{21} { = }\left( {\begin{array}{*{20}c} 0 & 1 & {0\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{23} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 1 & 0 & 0 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{24} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & {{\text{c}}\alpha } & 0 & { - {\text{s}}\alpha } \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{22} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 0 & 1 & 0 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(27)

For the third limb, the unit twist ${}^{O}{\hat{\textbf{\textit{\$}}}}_{ij}$ is expressed as:

$$\left\{ \begin{gathered} {}^{O}{\hat{\textbf{\textit{\$}}}}_{31} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {1\;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{33} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 0 & 1 & 0 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{34} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & { - {\text{s}}\alpha } & {{\text{c}}\alpha } & 0 \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\textbf{\textit{\$}}}}_{32} { = }\left( {\begin{array}{*{20}c} 0 & 0 & {0\;;} & 0 & 0 & 1 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(28)

For the revolute pair, the exponential product is expressed as:

$$\left\{ \begin{gathered} {\hat{\textbf{\textit{\$}}}}_{ij} = \left( {\begin{array}{*{20}c} {{\hat{\varvec{\omega }}}_{ij} } \\ {{{\varvec{\upsilon}}}_{ij} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\hat{\varvec{\omega }}}_{ij} } \\ {{\varvec{r}}_{ij} \times {\hat{\varvec{\omega }}}_{ij} } \\ \end{array} } \right)\quad \left( {{{\varvec{\omega}}}_{ij} \ne {\mathbf{0}}} \right), \hfill \\ {{e}}^{{\theta_{ij} \left[ {{\hat{\textbf{\textit{\$}}}}_{ij} } \right]}} = \left( {\begin{array}{*{20}c} {{{e}}^{{\theta_{ij} \left[ {{\hat{\varvec{\omega }}}_{ij} } \right]}} } & {\left( {{\varvec{E}} - {{e}}^{{\theta_{ij} \left[ {{\hat{\varvec{\omega }}}_{ij} } \right]}} } \right)\left( {{\hat{\varvec{\omega }}}_{ij} \times {{\varvec{\upsilon}}}_{ij} } \right) + \theta_{ij} {\hat{\varvec{\omega }}}_{ij} {\hat{\varvec{\omega }}}_{ij}^{{\text{T}}} {{\varvec{\upsilon}}}_{ij} } \\ {\mathbf{0}} & 1 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(29)

The exponential product for the prismatic pair is expressed as:

$${\hat{\textbf{\textit{\$}}}}_{ij} = \left( {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\hat{\varvec{\upsilon }}}_{ij} } \\ \end{array} } \right){,}\;{{e}}^{{d_{ij} \left[ {{\hat{\textbf{\textit{\$}}}}_{ij} } \right]}} = \left( {\begin{array}{*{20}c} {\varvec{E}} & {d_{ij} {\hat{\varvec{\upsilon }}}_{ij} } \\ {\mathbf{0}} & 1 \\ \end{array} } \right),$$

(30)

where ${\hat{\varvec{\omega }}}_{ij}$ represents the unit vector of the jth revolute pair axis direction in the ith limb, r_ij represents any point vector diameter of the jth revolute pair axis in the ith limb, ${\widehat{\varvec{\upsilon }}}_{ij}$ represents the unit vector of the jth prismatic pair axis in the ith limb, and $\left[ {{\hat{\varvec{\omega }}}_{ij} } \right]$ represents the antisymmetric matrix of ${\hat{\varvec{\omega }}}_{ij}$.

According to the homogeneous transformation rule, ${}_{{O_{i5} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right)$ can be expressed as:

$${}_{{O_{i5} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right) = {}_{{O_{i1} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right){}_{{O_{i3} }}^{{O_{i1} }} {\varvec{T}}\left( {\mathbf{0}} \right){}_{{O_{i4} }}^{{O_{i3} }} {\varvec{T}}\left( {\mathbf{0}} \right){}_{{O_{i2} }}^{{O_{i4} }} {\varvec{T}}\left( {\mathbf{0}} \right){}_{{O_{i5} }}^{{O_{i2} }} {\varvec{T}}\left( {\mathbf{0}} \right).$$

(31)

Combining Eqs. (25)–(31) results in:

$${}^{O}{\varvec{p}}_{{O^{\prime}}} = \left( {\begin{array}{*{20}c} {I_{1} } \\ {f_{12} \left( {d_{13} ,\;d_{14} } \right)} \\ {f_{13} \left( {d_{13} ,\;d_{14} } \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {f_{21} \left( {d_{23} ,\;d_{24} } \right)} \\ {I_{2} } \\ {f_{23} \left( {d_{23} ,\;d_{24} } \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {f_{31} \left( {d_{33} ,\;d_{34} } \right)} \\ {f_{32} \left( {d_{33} ,\;d_{34} } \right)} \\ {I_{3} } \\ \end{array} } \right),$$

(32)

where I₁ = d₁₂ + p_45z + ( L₁ + L₄ ), I₂ = d₂₂ + p_45z + (L₁ + L₄ ), and I₃ = d₃₂ + p_45z + ( L₁ + L₄ ).

Combining Eqs. (24)–(32) results in:

$$\left\{ \begin{gathered} d_{12} = p_{x} - p_{45z} - \left( {L_{1} + L_{4} } \right), \hfill \\ d_{22} = p_{y} - p_{45z} - \left( {L_{1} + L_{4} } \right), \hfill \\ d_{32} = p_{z} - p_{45z} - \left( {L_{1} + L_{4} } \right). \hfill \\ \end{gathered} \right.$$

(33)

The central position of the talus was only related to the position of the subactive kinematic pair, not the main active kinematic pair, as seen in Eq. (33). Moreover, the different directional positions of the talus were only related to the position of the limb subactive kinematic pair. Therefore, the moving platform positions were completely decoupled.

5.3 Kinematic Orientation Analysis of the HRCC

According to the included angle between the kinematic pair, the O_i5 and O_i6 axis is β, and the relationship between ${}^{O}{\hat{\varvec{\omega }}}_{{i{5}}}$, ${}^{O}{\hat{\varvec{\omega }}}_{i6}$, and β is expressed as:

$${}^{O}{\hat{\varvec{\omega }}}_{{i{5}}} \cdot {}^{O}{\hat{\varvec{\omega }}}_{i6} = \cos \beta .$$

(34)

Using homogeneous transformation, the homogeneous coordinates of ${}^{O}{\hat{\varvec{\omega }}}_{{i{5}}}$ and ${}^{O}{\hat{\varvec{\omega }}}_{i6}$ is expressed as:

$$\left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {{}^{O}{\hat{\varvec{\omega }}}_{{i{5}}} } \\ 0 \\ \end{array} } \right) = {}_{{O_{i5} }}^{O} {\varvec{ T}}\left( {\varvec{q}} \right)\left( {\begin{array}{*{20}c} {{}^{{O_{i5} }}{\hat{\varvec{\omega }}}_{{i{5}}} } \\ 0 \\ \end{array} } \right), \hfill \\ \left( {\begin{array}{*{20}c} {{}^{O}{\hat{\varvec{\omega }}}_{i6} } \\ 0 \\ \end{array} } \right) = {}_{{O^{\prime}}}^{O} {\varvec{ T}}\left( {\varvec{q}} \right){}_{{O_{i6} }}^{{O^{\prime}}} {\varvec{ T}}\left( {\varvec{q}} \right)\left( {\begin{array}{*{20}c} {{}^{{O_{i6} }}{\hat{\varvec{\omega }}}_{i6} } \\ 0 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(35)

Combining Eqs. (25) and (35) results in:

$$\left\{ \begin{gathered} {}^{O}{\hat{\varvec{\omega }}}_{15} = \left( {\begin{array}{*{20}c} {a_{33} } \\ { - a_{13} {\text{s}}\theta_{11} - a_{23} {\text{c}}\theta_{11} } \\ {a_{13} {\text{c}}\theta_{11} - a_{23} {\text{s}} \theta_{11} } \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\varvec{\omega }}}_{25} = \left( {\begin{array}{*{20}c} {a_{13} {\text{c}}\theta_{21} - a_{23} {\text{s}}\theta_{21} } \\ {a_{33} } \\ { - a_{13} {\text{s}}\theta_{21} - a_{23} {\text{c}}\theta_{21} } \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\varvec{\omega }}}_{35} = \left( {\begin{array}{*{20}c} { - a_{13} {\text{s}}\theta_{31} - a_{23} {\text{c}}\theta_{31} } \\ {a_{13} {\text{c}}\theta_{31} - a_{23} {\text{s}}\theta_{31} } \\ {a_{33} } \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(36)

According to the exponential product formula, ${}_{{O_{i6} }}^{{O^{\prime}}} {\varvec{ T}}\left( {\varvec{q}} \right)$ can be expressed as:

$${}_{{O_{i6} }}^{{O^{\prime}}} {\varvec{ T}}\left( {\varvec{q}} \right) = {{e}}^{{\theta_{i6} \left[ {{}^{{O^{\prime}}}{\hat{\textbf{\textit{\$}}}}_{i6} } \right]}} {}_{{O_{i6} }}^{{O^{\prime}}} {\varvec{T}}\left( {\mathbf{0}} \right).$$

(37)

According to the positional relationship between the kinematic pair, the twist ${}^{{O^{\prime}}}{\hat{\textbf{\textit{\$}}}}_{i6}$ can be expressed as:

$$\left\{ \begin{gathered} {}^{{O^{\prime}}}{\hat{\textbf{\textit{\$}}}}_{26} = \left( {\begin{array}{*{20}c} 0 & { - {\text{s}} \gamma } & {{\text{c}} \gamma \;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ {}^{{O^{\prime}}}{\hat{\textbf{\textit{\$}}}}_{26} = \left( {\begin{array}{*{20}c} {{{\left( {\sqrt 3 {\text{s}} \gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 {\text{s}} \gamma } \right)} 2}} \right. \kern-0pt} 2}} & {{{\left( {{\text{s}} \gamma } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{s}} \gamma } \right)} 2}} \right. \kern-0pt} 2}} & {{\text{c}} \gamma \;;} & 0 & 0 & 0 \\ \end{array} } \right), \hfill \\ {}^{{O^{\prime}}}{\hat{\textbf{\textit{\$}}}}_{36} = \left( {\begin{array}{*{20}c} { - {{\left( {\sqrt 3 {\text{s}} \gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 {\text{s}} \gamma } \right)} 2}} \right. \kern-0pt} 2}} & {{{\left( {{\text{s}} \gamma } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{s}} \gamma } \right)} 2}} \right. \kern-0pt} 2}} & {{\text{c}} \gamma \;;} & 0 & 0 & 0 \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(38)

Combining Eqs. (35), (37), and (38) results in:

$$\left\{ \begin{gathered} {}^{O}{\hat{\varvec{\omega }}}_{16} = \left( {\begin{array}{*{20}c} {r_{13} {\text{c}} \gamma - r_{12} {\text{s}} \gamma } \\ {r_{23} {\text{c}} \gamma - r_{22} {\text{s}} \gamma } \\ {r_{33} {\text{c}} \gamma - r_{32} {\text{s}} \gamma } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {A_{11} } \\ {A_{12} } \\ {A_{13} } \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\varvec{\omega }}}_{26} = \left( {\begin{array}{*{20}c} {{{\left( {\sqrt 3 r_{11} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 r_{11} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + {{\left( {r_{12} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{12} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + r_{13} c\gamma } \\ {{{\left( {\sqrt 3 r_{21} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 r_{21} s\gamma } \right)} {2 + {{\left( {r_{22} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{22} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + r_{23} c\gamma }}} \right. \kern-0pt} {2 + {{\left( {r_{22} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{22} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + r_{23} c\gamma }}} \\ {{{\left( {\sqrt 3 r_{31} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 r_{31} s\gamma } \right)} {2 + {{\left( {r_{32} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{32} s\gamma } \right)} 2}} \right. \kern-0pt} 2}}}} \right. \kern-0pt} {2 + {{\left( {r_{32} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{32} s\gamma } \right)} 2}} \right. \kern-0pt} 2}}} + r_{33} c\gamma } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {A_{21} } \\ {A_{22} } \\ {A_{23} } \\ \end{array} } \right), \hfill \\ {}^{O}{\hat{\varvec{\omega }}}_{36} = \left( \begin{gathered} {{\left( {r_{12} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{12} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + r_{13} c\gamma - {{\left( {\sqrt 3 r_{11} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 r_{11} s\gamma } \right)} 2}} \right. \kern-0pt} 2} \hfill \\ {{\left( {r_{22} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{22} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + r_{23} c\gamma - {{\left( {\sqrt 3 r_{21} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 r_{21} s\gamma } \right)} 2}} \right. \kern-0pt} 2} \hfill \\ {{\left( {r_{32} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {r_{32} s\gamma } \right)} 2}} \right. \kern-0pt} 2} + r_{33} c\gamma - {{\left( {\sqrt 3 r_{31} s\gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 3 r_{31} s\gamma } \right)} 2}} \right. \kern-0pt} 2} \hfill \\ \end{gathered} \right) = \left( {\begin{array}{*{20}c} {A_{31} } \\ {A_{32} } \\ {A_{33} } \\ \end{array} } \right). \hfill \\ \end{gathered} \right.$$

(39)

Combining Eqs. (34), (36) and (39) results in:

$$B_{i1} {\text{s}}\theta_{i1} + B_{i2} {\text{c}}\theta_{i1} + B_{i3} = 0\quad \left( {i = 1,\;2,\;3} \right),$$

(40)

$$\left\{ \begin{gathered} B_{11} = - a_{13} A_{12} - a_{23} A_{13} , \hfill \\ B_{12} = a_{13} A_{13} - a_{23} A_{12} , \hfill \\ B_{13} = a_{33} A_{11} - \cos \beta , \hfill \\ \end{gathered} \right.$$

$$\left\{ \begin{gathered} B_{21} = - a_{13} A_{23} - a_{23} A_{21} , \hfill \\ B_{22} = a_{13} A_{21} - a_{23} A_{23} , \hfill \\ B_{23} = a_{33} A_{22} - \cos \beta , \hfill \\ \end{gathered} \right.\quad \left\{ \begin{gathered} B_{31} = - a_{13} A_{31} - a_{23} A_{32} , \hfill \\ B_{32} = a_{13} A_{32} - a_{23} A_{31} , \hfill \\ B_{33} = a_{33} A_{33} - \cos \beta . \hfill \\ \end{gathered} \right.$$

According to Eq. (40), the rotational angle of the main active kinematic pair can be expressed as:

$$\theta_{i1} = 2a{\text{tan}}\frac{{B_{i1} \pm \sqrt {B_{i1}^{2} + B_{i2}^{2} - B_{i3}^{2} } }}{{B_{i2} - B_{i3} }}.$$

(41)

The rehabilitation locomotion mode was only related to the rotational angle of the main active kinematic pair and not to the subactive kinematic pair, as seen in Eq. (41).

According to the inverse kinematics of position and orientation, the rehabilitation locomotion mode was only related to the rotation angle of the main active kinematic pair, and the central position of the talus was only related to the displacement of the subactive kinematic pair. Therefore, the ${\text{3-R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ parallel mechanism met the requirements of the PSAARR, and the kinematic analysis results provided a theoretical basis for subsequent design, analysis, and optimization.

6 Conclusions

• (1) A self-alignment principle of parallel ARR was proposed according to the conditions of "suitable passive DOF", "decoupled", and "less limb": (i) Three "suitable passive DOF" of translation were introduced on the basis of three active DOF of rotation to transform the HRCC formed by the ARR and human body into a kinematic suitable constrained system and (ii) the active and passive DOF were decoupled, and the number of limbs was equal to the number of active DOF.

• (2) The type synthesis condition was obtained according to the self-alignment principle of the PSAARR. The reciprocal product of the actuation wrench and all other twists in the same limb was zero, except for the corresponding twist. The type synthesis method of the PSAARR limb was proposed based on the screw theory and type synthesis conditions, obtaining 93 types of PSAARR limb structures.

• (3) A ${\text{3-R}}_{{{\text{1ma}}}} \overline{\text{P}}\tilde{\text{P}}\dot{\text{P}}_{{{\text{sa}}}} \hat{\text{R}}\hat{\text{R}}$ symmetric parallel robot was selected as the main structure of the PSAARR. The results of the kinematic analysis showed that the rotational angle of the main active kinematic pair was only related to the rehabilitation locomotion mode and the displacement of the subactive kinematic pair was only related to the central position of the talus, which met the requirements of the PSAARR.

Appendix

$${}_{{O_{11} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right){ = }\left( {\begin{array}{*{20}c} 0 & 0 & 1 & {L_{1} } \\ 0 & { - 1} & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A1)

$${}_{{O_{21} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right){ = }\left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & {L_{1} } \\ 0 & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A2)

$${}_{{O_{31} }}^{O} {\varvec{T}}\left( {\mathbf{0}} \right){ = }\left( {\begin{array}{*{20}c} 0 & { - 1} & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & {L_{1} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A3)

$${}_{{O_{i3} }}^{{O_{i1} }} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} 0 & 0 & 1 & {L_{2} } \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A4)

$${}_{{O_{i4} }}^{{O_{i3} }} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} { - {\text{c}} \alpha } & 0 & {{\text{s}} \alpha } & {L_{3} {\text{s}} \alpha } \\ 0 & { - 1} & 0 & 0 \\ {{\text{s}} \alpha } & 0 & {{\text{c}} \alpha } & {L_{3} {\text{c}} \alpha } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A5)

$${}_{{O_{i2} }}^{{O_{i4} }} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} {{\text{s}} \alpha } & { - {\text{c}} \alpha } & 0 & 0 \\ 0 & 0 & { - 1} & { - L_{4} } \\ {{\text{c}} \alpha } & {{\text{s}} \alpha } & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A6)

$${}_{{O_{i5} }}^{{O_{i2} }} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & {a_{13} } & {p_{45x} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {p_{45y} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {p_{45z} } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A7)

$${}_{{O_{i6} }}^{{O_{i5} }} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} {{\text{c}} \beta } & 0 & {{\text{s}} \beta } & 0 \\ 0 & 1 & 0 & 0 \\ { - {\text{s}} \beta } & 0 & {{\text{c}} \beta } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A8)

$${}_{{O_{16} }}^{{O^{\prime}}} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} 0 & { - 1} & 0 & 0 \\ {{\text{c}} \gamma } & 0 & { - {\text{s}} \gamma } & 0 \\ {{\text{s}} \gamma } & 0 & {{\text{c}} \gamma } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A9)

$${}_{{O_{26} }}^{{O^{\prime}}} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} { - \frac{\sqrt 3 }{2}{\text{c}} \gamma } & \frac{1}{2} & {\frac{\sqrt 3 }{2}{\text{s}} \gamma } & 0 \\ { - \frac{1}{2}{\text{c}} \gamma } & { - \frac{\sqrt 3 }{2}} & {\frac{1}{2}{\text{s}} \gamma } & 0 \\ {{\text{s}} \gamma } & 0 & {{\text{c}} \gamma } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$

(A10)

$${}_{{O_{36} }}^{{O^{\prime}}} {\varvec{T}}\left( {\mathbf{0}} \right) = \left( {\begin{array}{*{20}c} {\frac{\sqrt 3 }{2}{\text{c}} \gamma } & \frac{1}{2} & { - \frac{\sqrt 3 }{2}{\text{s}} \gamma } & 0 \\ { - \frac{1}{2}{\text{c}} \gamma } & {\frac{\sqrt 3 }{2}} & {\frac{1}{2}{\text{s}} \gamma } & 0 \\ {{\text{s}} \gamma } & 0 & {{\text{c}} \gamma } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right).$$

(A11)