VRM: A Unified Framework for Closed-Form Solutions of a Special Class of Serial Manipulators

Abstract

This paper proposes the virtual reconfiguration method (VRM) to construct the unified framework for closed-form solutions of a special class of serial manipulators. Central to the research is the inverse kinematics problem (KP) of 6- and 7-DOF serial manipulators, which contain either the Pieper's geometry or the Duffy's geometry. Given the desired end-effector pose of the manipulator, a virtual single chain (SLC) is developed by connecting the base and the end-effector with a hypothetical link. The equivalent single open chain (SOC) with different configurations can be obtained by cutting open the virtual SLC at one link between adjacent joints. Kinematic equivalence between the original manipulator and the new SOC is proven. Closed-form solutions of the original manipulator can be determined by solving the KP of the equivalent SOC. The VRM is further developed on the basis of the relationship between the manipulator and its equivalent SOC. In this paper, the KPs of 6-DOF manipulators with the spherical wrist and manipulators with the three-axis parallel shoulder joint are analysed. Principles and applications of the VRM are proposed. Finally, the validity and efficiency of the VRM are demonstrated by kinematics simulations of four different manipulators. Unlike traditional approaches, the VRM simplifies the computation of KPs and establishes a unified framework for closed-form solutions of the special class of 6- and 7-DOF serial manipulators, irrespective of the allocation of either the Pieper's geometry or the Duffy's geometry.

Keywords

Virtual Reconfiguration Method Closed-Form Solution Reconfigurable Robot Single Loop Chain Equivalent Single Open Chain

1. Introduction

Serial manipulators are robots in a sequence of links and joints that begins at the base and ends with an end-effector. They are widely used in industry, humanoid robots and space exploration. The inverse kinematics problem (KP) of serial manipulators is the most important aspect of robot analysis and motion planning. It concerns determination of joint variables corresponding to the relative pose between two members of the manipulator. In this paper, the KP of a special class of 6- and 7-DOF serial manipulators, which contain either the 3-DOF spherical joint (Pieper's geometry) or the three-axis parallel geometry (Duffy's geometry), is proposed. In order to obtain inverse kinematic solutions of manipulators,

many methods have been developed based on different representations of position and orientation and their derivatives with respect to time [1, 2]. In general, the IKP of the special classes of manipulators can be solved with universal methods and particular methods, respectively.

Numerical methods are based on the computation of nonlinear kinematic equations in the position domain. Approximate inverse kinematic solutions can be determined using numerical methods such as interval methods, continuation methods and iterative algorithms [3, 4], etc. Numerical methods are independent of robot morphology and are applicable to robots with any kinematic structure. Nevertheless, a trade-off exists between the computational complexity and the solutions' precision. Also, it is impossible to find all kinematic solutions corresponding to the specified end-effector pose. Besides, the IKP of manipulators can be solved on the basis of inverse instantaneous kinematics. Typical methods are the simple Jacobian-based method and other extended methods including closed algorithms, Jacobian transpose methods, and so on [2, 5]. The velocity-based methods are applicable to manipulators irrespective of their geometry. However, these methods have intrinsic inaccuracy because of the linear approximation of the Jacobian matrix, especially in the near region of singularity. These methods are also dependent on the initial conditions and do not consider the feasible region of solutions. In addition, artificial neural networks and numerous intelligent algorithms are also capable of computing inverse kinematic solutions [6, 7]. These methods have excellent capabilities of parallel computing, and do not need an explicit kinematic model. But they are usually required to learn online or off-line from kinematic samples. The precision of the results is inadequate for some sensitive tasks.

Particular methods for approaching the IKP of manipulators are based on direct mapping from the Cartesian space to the joint space. Inverse kinematic solutions can be expressed symbolically using algebraic or geometric methods [1, 8]. Closed-form solutions are more efficient than numerical algorithms and readily identify all possible exact solutions leading to the desired terminal position and orientation. This indicates the desirability of closed-form solutions above other universal approaches. However, closed-form solutions depend on the robot morphology, and they only exist for a few types of robots, including six-degree-of-freedom (DOF) serial manipulators with three adjacent revolute joint axes intersecting at a common point (Pieper's geometry or spherical joint), or manipulators with three consecutive revolute joint axes in parallel (Duffy's geometry) [9, 10]. Manipulators with the spherical wrist joint or the three-axis parallel shoulder joint are desirable because the IKP can be solved easily by kinematics decoupling. Nevertheless, for general manipulators including humanoid arms with the spherical shoulder joint or space robots with the Duffy's geometry locating in the middle part, it is difficult to derive with closed-form solutions by solving high-degree polynomial equations. The IKP of these manipulators simply resorts to numerical algorithms or intelligent approaches.

A few studies have been carried out for closed-form solutions for these types of manipulators. In [11], closed-form solutions of the 7-DOF humanoid arm are expressed by fixing one redundant joint. In [6], the approximate closed-form solutions of the inverse kinematic model are obtained using evolutionary algorithms and genetic programming technology. However, the establishment of the approximating models is time consuming, and the results are inadequate for fine manipulations. In [12], an analytical method is used for the 7-DOF humanoid arm with a spherical shoulder joint, and closed-form solutions are derived using the virtual joint method. However, the seventh joint angle needs to be given before other solutions are obtained, and the method is invalid for 6-DOF serial manipulators. The reverse-decoupling method is presented for humanoid robots with the spherical joint, and the IKP is approached using algebraic algorithm and matrix transformation [1]. However, there is no in-depth study on the principle of the reverse-decoupling method. In [13], the reverse-coordinates method is used for closed-form solutions of 6-DOF manipulators with the SRRR configuration. But this method is only applicable to manipulators with spherical shoulder joint. There is no unified framework for the IKP of manipulators satisfying the Pieper's condition and manipulators satisfying the Duffy's condition.

This research is motivated by closed-form solutions of a class of serial manipulators with the special geometry. This paper is organized as follows. The IKPs of two kinds of typical manipulators are analysed in Section 2. Principles and applications of the VRM are proposed in Section 3. The validity and efficiency of the VRM are demonstrated by case studies of typical manipulators in Section 4. Conclusions are presented in Section 5.

2. IKPs of Typical Manipulators

2.1 Manipulators with the Spherical Wrist

Pieper's geometry is equivalent to the spherical joint (S, which can achieve rotations in three different directions. Manipulators are usually designed with a spherical wrist to achieve better performances of the end-effector. The spherical wrist is widely used in industrial applications.

Suppose that the general 6-DOF manipulator contains the revolute joint (R) and the prismatic joint (P). The manipulator with the RPRS arrangement is shown in Figure 1.

Figure 1.

General RPRS manipulator with the spherical wrist

Let ^jA_i be the homogeneous transformation matrix describing the pose of Frame O_ixyz with respect to Frame O_jxyz. Point W is the intersection of the last three joint axes. The kinematic equation of the general 6-DOF serial manipulator with the spherical wrist can be expressed as

^{0} A_{6} = [\begin{matrix} f_{0R6} (θ, d) & f_{0P6} (v_{1}, v_{2}, v_{3}) \\ O & 1 \end{matrix}]

(1)

where $f_{0R6} (θ, d) \in SO (3)$ is the function of all joint variables, and $f_{0P6} (v_{1}, v_{2}, v_{3}) \in R^{3}$ is the function of the first three joint variables. O is the zero vector. The variable v_i depends on the type of joints. If joint i is revolute, v_i = θ_i. If joint i is prismatic, v_i = d_i.

In brief, for the general 6-DOF serial manipulator with the spherical wrist, the position of the wrist point W depends only on the first three joint variables. The spherical wrist is used to adjust the terminal orientation. Given the desired end-effector pose, closed-form solutions can be obtained in two steps: the first three joint variables are derived from the position-based equation f_0P6(v₁, v₂, v₃), and then the spherical wrist angles can be solved based on the orientation-based equation f_0R6(θ, d).

2.2 Manipulators with the Duffy shoulder joint

Duffy's geometran achlies to the three-axis parallel planar robot, which can achieve two-dimensional motions. In order to simplify the inverse kinematics, Duffy's geometry is designed as the shoulder joint of manipulators. Similarly, the general 6-DOF manipulator with the 4RPR configuration is shown in Figure 2.

Figure 2.

General 4RPR manipulator with a Duffy shoulder joint

According to the forward kinematics, the end-effector pose ⁰A₆ can be expressed as

^{0} A_{6} = [\begin{matrix} * & * & * & * \\ * & * & * & * \\ f_{n} (v_{4}, v_{5}, v_{6}) & f_{o} (v_{4}, v_{5}, v_{6}) & f_{a} (v_{4}, v_{5}, v_{6}) & f_{p} (v_{4}, v_{5}, v_{6}) \\ 0 & 0 & 0 & 1 \end{matrix}]

where the elements denoted by ^* are omitted, which vary with all joint variables. f_n(v₄, v₅, v₆), f_o (v₄, v₅, v₆), f_a(v₄, v₅, v₆) and f_p (v_4, v_5, v₆) are functions related to the last three joint variables.

Therefore, for the general 6-DOF serial manipulator with the Duffy shoulder joint, the position and the orientation of Frame m O₆x yz along the z direction of Frame O₀xyz depend only on the last three joints. Given the desired end-effector pose, closed-form solutions can be obtained in two steps: the last three joint variables are derived from the third row of ⁰A₆, and then the shoulder joint angles can be computed from other rows of ⁰A₆.

3. Theory of the VRM

3. 1 Backgrounds

As mentioned above, for the general 6-DOF manipulator with the spherical wrist or the Duffy shoulder joint, closedform solutions can be directly obtained by decoupling certain elements of the end-effector pose. Furthermore, the general 7-DOF manipulator with these special geometries can be simplified as the 6-DOF robot by fixing one redundant joint, and the IKP can then be solved analytically [11]. Nevertheless, in order to meet special requirements of operations and environments, different types of serial manipulators are developed which satisfy the Pieper's or the Duffy's condition. However, these manipulators are not equipped with the spherical wrist or the Duffy shoulder joint. For instance, humanoid arms with the spherical shoulder joint and humanoid legs with the Duffy's geometry in the middle part can better emulate human functions [1]. Space manipulators are usually designed as symmetrical robots with the Duffy's geometry, so they can be folded for delivery to ISS [14–16]. Though closed-form solutions for these manipulators exist, it is difficult to derive the solutions directly since all elements of the position and orientation are coupled.

This paper presents a unified procedure for solving the IKP of a special class of the 6-DOF and the 7-DOF serial manipulators which contain either the Pieper's geometry or the Duffy's geometry.

3.2 Analysis of Kinematic Equivalence

Manipulators are kinematically equivalent if they have the same kinematic relationship between joints and links, inherently. Kinematic equivalence between the general serial manipulator and its reconfigured form is proven as follows.

1. Forward Kinematic Equivalence

In Figure 3(a), the general n-DOF serial manipulator with a closed is illustrated. Given the specified joint variables, the position and orientation of links are uniquely determined.

Figure 3.

Kinematic chain of the general manipulator: (a) the closed-loop manipulator; (b) Robot-A; (c) Robot-B

Based on the closed-robot, an n-DOF reconfigured manipulator “Robot-A” with an open loop is developed by cutting open the kinematic chain at Link n, as shown in Figure 3(b). For a certain serial manipulator, the position and orientation of links are determined by joint variables. Since all joint variables remain unchanged, there are no joint motions during the reconfiguration process. Thus, given the specified joint variables, the position and orientation of the corresponding elements in the Cartesian space are identical between the two manipulators. The original closed-loop manipulator is forward-kinematically equivalent to its reconfigured form with a closed loop.

Another reconfigured manipulator, “Robot-B”, can also be obtained by cutting open the kinematic chain at Link i, as shown in Figure 3(c). Similarly, given the specified joint variables, the poses of corresponding elements in the Cartesian space are the same between these closed-loop manipulators and Robot-B. This implies that forward-kinematic equivalence exists between the two serial manipulators.

According to the above analysis, the two different reconfigured manipulators with the open loop are forward-kinematically equivalent because of the identical pose of corresponding elements in the Cartesian space corresponding to the certain joint variables. On the other hand, Robot-B can be developed by connecting two ends of Robot-A with a hypothetical link and cutting open the kinematic chain at Link i. Robot-B can be deemed one of the reconfigured manipulators of Robot-A. Consequently, it can be concluded that the general serial manipulator is forward-kinematically equivalent to its reconfigured manipulator with an open loop.

In order to validate the forward kinematic equivalence, the kinematics of PUMA 560 is established. DH coordinate frames of the robot are established in Fig 4(a), DH parameters are listed in Table 1. Based on this, the kinematic equation yields

Table 1.

DH parameters of PUMA 560

link	a_i	α_i	d_i	θ_i
1	0	90°	L₁	θ₁
2	L₃	0°	− L₂	θ₂
3	L₄	90°	0	θ₃
4	0	−90°	L₅	θ₄
5	0	90°	0	θ₅
6	0	0°	L₆	θ₆

Figure 4.

DH frames of robots: (a) PUMA 560; (b) the reconfigured robot

^{0} A_{6} =^{0} A_{1}^{1} A_{2}^{2} A_{3}^{3} A_{4}^{4} A_{5}^{5} A_{6}

(2)

The reconfigured robot can be obtained by connecting two ends of the robot with a hypothetical link and cutting open the kinematic chain at Link 2. The virtual frames (Frame vb and Frame ve) are arbitrary and identical. DH coordinate frames are established as shown in Fig 4(b). The kinematic equation of the reconfigured robot is expressed as

^{v b} A_{v e} =^{v b} A_{2}^{2} A_{3}^{3} A_{4}^{4} A_{5}^{5} A_{6}^{6} A_{0}^{0} A_{1}^{1} A_{ve}

(3)

Based on (2) and (3), it can be seen that PUMA 560 and the reconfigured robot have the same kinematic equation. This demonstrates that the reconfigured robot is forward-kinematically equivalent to PUMA 560.

2. Inverse Kinematic Equivalence

Inverse kinematic equivalence between the original manipulator and the open-reconfigured manipulator is proven using proof by contradiction.

Suppose that $\bar{Ψ} = {{\bar{Θ}}_{1}, {\bar{Θ}}_{2}, \dots, {\bar{Θ}}_{m}}$ is the set of inverse kinematic solutions for the original manipulator corresponding to a specified relative pose of all links in the Cartesian space, and $\underline{Ψ} = {{\underline{Θ}}_{1}, {\underline{Θ}}_{2}, \dots, {\underline{Θ}}_{n}}$ is the set of inverse kinematic solutions for the reconfigured manipulator corresponding to the same relative pose in the Cartesian space. In addition, make an assumption that the original manipulator is not inverse-kinematically equivalent to the reconfigured manipulator. Then,

Case I: \exists {\bar{Θ}}_{i}, {\bar{Θ}}_{i} \notin \underline{Ψ} (i = 1, 2, \dots m)

Case II: \exists {\underline{Θ}}_{j}, {\underline{Θ}}_{j} \notin \bar{Ψ} (j = 1, 2, \dots n)

If Case I is true, the relative pose of corresponding elements of the original manipulator is different from that of the reconfigured manipulator corresponding to the joint variables ${\bar{Θ}}_{i}$ . This case contradicts the forward-kinematic equivalence of the two manipulators. Hence, the assumption (Case I) is false. It can be concluded that

\forall {\bar{Θ}}_{i} \in \bar{Ψ} \Rightarrow {\bar{Θ}}_{i} \in \underline{Ψ} (i = 1, 2, \dots m), and m \leq n .

Similarly, if Case II is true, the relative pose of corresponding elements of the reconfigured manipulator is different from that of the original manipulator corresponding to ${\underline{Θ}}_{j}$ . This case contradicts the forward-kinematic equivalence of the two manipulators. Hence, Case II is false. This implies that

\forall {\underline{Θ}}_{j} \in \underline{Ψ} \Rightarrow {\underline{Θ}}_{j} \in \bar{Ψ} (j = 1, 2, \dots n), and n \leq m .

(4)

Based on the above analysis, inverse-kinematic solutions $\underline{Ψ}$ of the reconfigured manipulator are the same as kinematic solutions $\bar{Ψ}$ of the original manipulator. That is, the inverse kinematic equivalence exists between the original manipulator and the open-reconfigured manipulator.

Therefore, it is proven that the original manipulator is kinematically equivalent to its reconfigured form.

The original manipulator and the reconfigured manipulators have identical kinematic singularity. The spherical joint at a singularity is illustrated in Figure 5(a). An arbitrary reconfigured form of the spherical joint is developed, as shown in Figure 5(b). It can be found that the two manipulators have identical characteristics at the singularity. Also, the equivalence of singularities can be validated by the kinematic equations from (2) and (3).

Figure 5.

Kinematic singularity of the 3-DOF manipulator: (a) the spherical joint at singularity; (b) the reconfigured manipulator at singularity

3.3 Principles of the VRM

The serial manipulator with an open kinematic chain is termed the single open chain (SOC), and the manipulator with a closed chain is termed the single chain (SLC). As mentioned above, given the specified joint variables, the end-effector pose with respect to the base frame is confirmed, and a stationary constraint exists between the base and the end-effector. Assuming that one virtual connection is built between the base frame and the end-effector frame (EEF), the virtual SLC is developed to be kinematically equivalent to the SOC. At this point, a new equivalent SOC with different configurations can be obtained by cutting open the virtual SLC at one link between adjacent joints. Various equivalent SOCs can be gained from the same manipulator by selecting different breaking points. The equivalent SOC also varies directly with different poses of the end-effector. Hence, the equivalent SOC can be deemed the variable-geometry robot.

With respect to the specified end-effector pose, the equivalent SOC has the same inner relationship between joints as the original serial manipulator. Hence, by cutting open the connection before the Pieper's geometry, manipulators with one internal spherical joint can be transformed into the equivalent SOC with the spherical wrist. Likewise, manipulators with the Duffy shoulder joint can also be derived from robots which contain the three-axis parallel structure. The theory to change the configuration of manipulators through the equivalent SOC is termed the virtual reconfiguration method (VRM). Based on this, reconfigurations of the manipulator with the Pieper's geometry and the manipulator with the Duffy's geometry are performed respectively.

A 6-DOF kinematic chain with the spherical joint is shown in Figure 6. The base frame and the EEF are established, and then the SLC is developed by virtually connecting these two frames. In order to obtain a new SOC with the spherical wrist, Link 4 is broken into two parts: Link 4' and Link 4″.

Figure 6.

Kinematic chain of the manipulator with the spherical joint: (a) the original SOC; (b) the equivalent reconfigured SOC

Link 4' connects with Joint 5, and Link 4” connects with Joint 4. The virtual base frame O_vbxyz is built arbitrarily at the breaking point on Link 4’, and the virtual EEF O_vexyz is fixed to the breaking point on Link 4”.

The pose of Frame O_vexyz with respect to Frame O_vbxyz can be expressed as

^{vb} A_{ve} =^{vb} A_{4'}^{4'} A_{5}^{5} A_{e}^{e} A_{b}^{b} A_{0}^{0} A_{1}^{1} A_{2}^{2} A_{3}^{3} A_{4″}^{4″} A_{ve}

(5)

The relationship between Frame O_4‘xyz and Frame O_4″xyz can be derived from (4). Because Frame O_vbxyz coincides with Frame O_vexyz, ^4’A_4” is simplified as

^{4'} A_{4″} =^{4'} A_{5}^{5} A_{e}^{e} A_{b}^{b} A_{0}^{0} A_{1} \underset{θ_{2,3,4}}{\underset{︸}{^{1} A_{2}^{2} A_{3}^{3} A_{4″}}}

(6)

According to (5), the new equivalent SOC is equipped with the spherical wrist. Since the homogenous transformation matrixes ^bA₀ and ^4‘A_4″ depend only on the geometrical structure of manipulators, given the desired end-effector pose, closed-form solutions of the manipulator can be solved directly with traditional methods.

The 6-DOF kinematic chain with the three-axis parallel structure is shown in Figure 7. Similarly, the SLC is developed by virtually connecting the base frame and the EEF. In order to obtain the new equivalent SOC with the Duffy shoulder joint, Link 1 is broken into two parts: Link 1' and Link 1′′. Link 1' connects with Joint 2, and Link 1′′ connects with Joint 1. The virtual frame O_vbxyz is built at the breaking point on Link 1', and the virtual EEF O_vexyz is fixed to the breaking point on Link 1″. Frame O_vexyz coincides with Frame O_vbxyz.

Figure 7.

Kinematic chain of the manipulator with the Duffy's geometry: (a) the original SOC; (b) the equivalent reconfigured SOC

The pose of Frame O_vexyz with respect to Frame O_vbxyz can be obtained by composing the homogeneous transformation, and the relationship between Frame O_1‘xyz and Frame O_1″xyz can be simplified as

^{1'} A_{1″} = \underset{θ_{2,3,4}}{\underset{︸}{^{1'} A_{2}^{2} A_{3}^{3} A_{4}}}^{4} A_{5}^{5} A_{e}^{e} A_{b}^{b} A_{0}^{0} A_{1″}

(7)

Therefore, the new equivalent SOC is equipped with the Duffy shoulder joint. Since the matrixes bA₀ and ^1‘A_1″ depend only on the geometrical structure of manipulators, given the desired end-effector pose, closed-form solutions can be obtained using traditional methods.

3.4 VRM Algorithm

Corresponding to the end-effector pose, the VRM can be used to transform manipulators into robots with different configurations. It is worth sketching the procedure to compute closed-form solutions for serial manipulators satisfying the Pieper's condition and serial manipulators satisfying the Duffy's condition. In general, the algorithm is implemented in six steps, as follows.

Step 1: Establish the DH coordinate frames for the base and the end-effector;

Step 2: Build the virtual SLC by connecting the base frame and the EEF;

Step 3: Determine the breaking point on one link according to the geometry of the manipulator. For instance, for manipulators with the spherical joint, the breaking point is located on the link after the spherical joint.; for manipulators with the Duffy's geometry, the breaking point is fixed to the link before;

Step 4: Establish the DH coordinate frames for the equivalent SOC;

Step 5: Compute the kinematics function by composing the homogenous transformations;

Step 6: Solve the IKP of the manipulator analytically using traditional methods.

3.5 Discussions

The VRM is developed based on the kinematic equivalence between the original manipulator and the equivalent reconfigured manipulator. The reconfigured manipulator is a subset of the reconfigurable robots because of the changeability of geometry. The comparisons between the equivalent reconfigured manipulator and the general reconfigurable robot are summarized as follows.

1. The general reconfigurable robot consists of a collection of interchangeable modules. According to the requirements of specified tasks, various robot geometries can be developed based on different selections and arrangements of the modules. Because the existence of closed-form solutions depends on the robot morphology, the IKP of reconfigurable robots is generally solved using numerical approaches [17–19].

2. The equivalent reconfigured manipulator concerns on a special class of serial manipulators which contain the Pieper's geometry or the Duffy's geometry. Once the manipulator has been reconfigured, the geometry and kinematic relationship are completely expressed in this manner. Besides, the reconfigured manipulator in this paper is kinematically equivalent to the original manipulator, and closed-form solutions can be obtained directly.

Another aspect to be considered for manipulators is the kinematic optimizations. Inverse kinematics involves one-to-many mapping from the Cartesian space to the joint space. For instance, given the specified terminal pose, there are at most eight solutions for the 6-DOF manipulator with either the Pieper's or the Duffy's geometry [20]. In order to determine the best of all feasible solutions, typical criteria are defined as follows.

Manipulability w₁(v)

w_{1} (v) = \sqrt{\det (J (v) J^{T} (v))}

Manipulators move away from singularities by maximizing w₁(v).

Distance from joint limits w₂(v)

w_{2} (v) = \frac{1}{n} \sum_{i = 1}^{n} {(\frac{v_{i} - {\bar{v}}_{i}}{v_{i, M a x} - v_{i, M i n}})}^{2}

where v_{i, Max} (v_{i, Min}) denotes the maximum (minimum) joint limit, and

{\bar{v}}_{i}

is, the middle value of the joint range. Manipulators move towards the centre of joint ranges by minimizing w₂(v).

Distance from previous solutions w₃(v)

w_{3} (v) = - \frac{1}{n} \sum_{i = 1}^{n} {(v_{i} - {v^{'}}_{i})}^{2}

where v′ denotes the previous joint variables. Manipulators move with little change in joint variables by minimizing w₄(v).

Distance from an obstacle w₄(v)

w_{4} (v) = \min_{p (v), p (θ)} ‖ p (v) - p (o) ‖

where p(o) is the obstacle position, and p(v) is the generic point position along the structure. Manipulators move away from the obstacle by maximizing w₄(v).

As mentioned above, the specified criteria are considered according to the requirement of the operations. The optimal joint displacements can be determined from all feasible solutions.

3.6 Applications

The topological structure describes a group of mechanisms which have the common characters between adjacent members. In this section, the topology structures of serial manipulators which contain either the Pieper's geometry or the Duffy's geometry are illustrated.

The topological structure of manipulators can be exactly expressed with three basic elements: the type of kinematics pair, connection relations between structural units, and the dimensional constraint type [21]. The prismatic joint and the revolute joint are the typical kinematics pairs. The basic dimensional constraint types are defined as follows.

SOC with two adjacent kinematics pair (R or P) axes intersecting at one common point is expressed as

SOC {\tilde{- R (or P) R (or P) -}};

SOC with two adjacent kinematics pair (R or P) axes being parallel to each other is expressed as

SOC {- R (or P) / / R (or P) -};

SOC with two adjacent kinematics pair (R or P) axes in the general configuration is expressed as

SOC {- R (or P) - R (or P) -} .

The topology structure of the Pieper's geometry and the Duffy's geometry can be represented as $SOC {\hat{- RRR -}}$ and SOC{- R / / R / / R -} respectively. The two types of manipulators with special geometry ensuring the existence of closed-form solutions are summarized as follows.

Type I: 6 DOFs open-manipulators containing P and R and satisfying the Pieper's condition

Suppose that the 6 DOFs SLC with the Pieper's geometry is denoted as $SLC {- R (or P) - \hat{RRR} - R (or P) - R (or P) -}$ , and the topology structure of the Type I manipulators can be obtained from the SLC by cutting open one linkage between any two joints except the Pieper's geometry.

Type II: 6-DOF open-loop manipulators containing P and R and satisfying the Duffy's condition The 6-DOF SLC which contains the Duffy's geometry is denoted as SLC{-R(orP) - R / / R / / R - R (orP) - R(orP) -}, and the topology structure of the Type II manipulators can be obtained from the SLC by cutting open one linkage between any two joints except the Duffy's geometry.

Based on the topological structures of these two types of manipulators, many manipulators can be developed due to different link parameters and joint allocations [1, 14–16]. The VRM is applicable to any manipulator belonging to Type I and Type II manipulators. It simplifies computations of the IKP and constructs the unified framework for closed-form solutions for one special class of serial manipulators.

4. Case Studies

4.1 IKPs of humanoid arms

The 6-DOF serial manipulator is generally used for humanoid robots to be able to imitate human motions. The humanoid arm of Hubo KHR-4, which has the spherical shoulder joint, is illustrated in Figure 8.

Figure 8.

The equivalent SOC and DH coordinate frames

Case 1:

The base frame and the EEF of the humanoid arm are established, and the virtual SLC is developed. Based on this, the equivalent SOC is obtained by cutting open the virtual SLC at Link 3. DH coordinate frames of the equivalent SOC are established, and DH parameters are as listed in Table 2.

Table 2.

DH parameters of the humanoid arm

Link	a_i	α_i	d_i	θ_i
1	0	-90°	L₀	θ₁
2	0	90°	0	θ₂
3″	0	0°	0	θ₃
3′	0	-90°	−L₁	0°
4	0	90°	0	θ₄
5	0	−90°	− L₂	θ₅
6	L₃	0°	0	θ₆

The base frame coincides with Frame O₀xyz and ^3″A_3‘ is a constant transformation matrix which can be derived from Table 2. Consequently, computations of the kinematic equation as in (5) yield

^{3'} A_{3″} =^{3'} A_{4}^{4} A_{5}^{5} A_{e}^{e} A_{b}^{0} A_{1}^{1} A_{2}^{2} A_{3″}

(8)

Since the spherical joint is the wrist of the equivalent SOC, the position of Frame O₁xyz viewed from Frame O₃xyz depends only on the joint variable as θ_4,5,6. The equation about the wrist position can be derived from (7). Let [p_X p_Y p_Z]^T or be the end-effector position with respect to the base frame. If $P_{15, x}^{2} + P_{15, y}^{2} + P_{15, z}^{2} - {(L_{1} + L_{2})}^{2} \neq 0$ or $P_{15, x}^{2} + P_{15, y}^{2} + P_{15, z}^{2} - {(L_{1} - L_{2})}^{2} \neq 0$ yields

θ_{4} =atan2 (- g_{1}, g_{3})

(9)

θ_{5} =atan2 (- f_{3}, f_{1}) or θ_{5} =atan2 (f_{3}, - f_{1})

(10)

θ_{6} =atan2 (ρ, \pm \sqrt{1- ρ^{2}}) -atan2 (B, A)

(11)

The parameters g₁, g₃, f ₁, f ₃, ρ, A and B are computed in the Appendix.

Otherwise, the kinematic singularity occurs. In this case, sinθ₄ = 0, and θ₅ can be determined directly according to some roles.

Supposing that R=^bR_e ^eR₅ ⁵R₄ ⁴R_3‘ ^3’R_3″, substituting (8)–(10) into (7), solutions of the spherical joint can be derived. If sinθ₂ ≠ 0, the joint angles are given by

θ_{1} =atan2 (sign (\sin θ_{2}) R_{2,3}, sign (\sin θ_{2}) R_{1,3})

(12)

θ_{2} = \pm acos (R_{3,3})

(13)

θ_{3} =atan2 (sign (\sin θ_{2}) R_{3,2}, -sign (\sin θ_{2}) R_{3,1})

(14)

Otherwise, the spherical joint is at a singularity. In this case, θ₁ can be determined directly according to some roles. θ₃ is given by

\tan θ_{3} = \frac{B_{sc} (1)}{B_{sc} (2)}

(15)

where

B_{s c} = {[\begin{matrix} - sin θ_{1} & cos θ_{1} cos θ_{2} \\ - cos θ_{1} cos θ_{2} & - sin θ_{1} \end{matrix}]}^{-} [\begin{matrix} R_{1, 1} \\ R_{1, 2} \end{matrix}] .

Finally, closed-form solutions of the humanoid arm are obtained. Given the desired end-effector pose, the corresonding joint angles can be determined directly.

Case 2:

With respect to Figure 9, another spherical joint exists which consists the third joint, fourth joint and fifth joint. In order to validate the universality of the VRM, another equivalent reconfigured manipulator is developed based on the humanoid arm. Differently from Case 1, the equivalent SOC is obtained by cutting open the virtual SLC at Link 5. DH coordinate frames of the equivalent SOC are established, and DH parameters are listed in Table 3.

Table 3.

DH parameters of the humanoid arm

Link	a_i	α_i	d_i	θ_i
1	0	-90°	L₀	θ₁
2	0	90°	0	θ₂
3	0	-90°	−L₁	θ₃
4	0	90°	0	θ₄
5″	0	0°	0	θ₅
5′	0	−90°	−L₂	0°
6	L₃	0°	0	θ₆

Figure 9.

The equivalent SOC and DH coordinate frames

Similarly, ^5″A_5‘ is a constant transformation matrix which can be derived from Table 3. Computations of the kinematic equation as in (5) yield

^{5'} A_{5″} =^{5'} A_{e}^{e} A_{0}^{0} A_{1}^{1} A_{2}^{2} A_{3}^{3} A_{4}^{4} A_{5″}

(16)

Since the spherical joint is the wrist of the equivalent SOC, the position of Frame O₄xyz viewed from Frame O_5‘xyz depends only on the joint variable θ_1,2,6. With reference to the above analysis, the equation of the wrist position can be derived from (15). θ_1,2,6 can be derived using analytical methods. Finally, closed-form solutions of the new equivalent reconfigured manipulators are determined by kinematic decoupling. Due to the limitation of space, mathematical formulations of closed-form solutions are not expressed again.

In order to verify the validity of the VRM, kinematics simulations are performed. The joint range is from −180° to 180° without joint limits. The link parameters of the humanoid arm are obtained from its specifications as:

L₀=214.5(mm), L₁=179.14(mm), L₂=181.57(mm), L₃=80(mm). Let $^{b} X_{e}^{d} \in R^{3}$ and $^{b} R_{e}^{d} \in SO (3)$ be the desired position and orientation of the hand, respectively. Two sets of the hand pose are performed here. The first one (Pose A) is the general hand pose, while the second one (Pose B) is the pose at singularity.

\begin{array}{c} Pose A:^{b} X_{e}^{d} = [\begin{matrix} 0 .2949 & 0 .1659 & 0 .3498 \end{matrix}] (m) \\ ^{b} R_{e}^{d} = [\begin{matrix} 0 .7557 & - 0 .5755 & - 0 .3125 \\ 0 .5334 & 0 .2642 & 0 .8035 \\ - 0 .3799 & - 0 .7740 & 0 .5066 \end{matrix}] \\ Pose B:^{b} X_{e}^{d} = [\begin{matrix} - 0 .1962 & 0 .1133 & 0 .5244 \end{matrix}] (m) \\ ^{b} R_{e}^{d} = [\begin{matrix} - 0 .7500 & - 0 .0474 & 0 .6597 \\ 0 .4330 & - 0 .7891 & 0 .4356 \\ 0 .5000 & 0 .6124 & 0 .6124 \end{matrix}] \end{array}

Closed-form solutions are completed based on the analysis in Case 1 and Case 2. Eight sets of solutions are obtained for Pose A, and four sets of solutions for Pose B, as illustrated in Table 4. Based on this, optimal kinematic solutions can be selected based on the typical criteria mentioned in Section 3.5.

Table 4.

Computed solutions of the humanoid arm (unit: degree)

Pose	Solutions	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Pose A	solution 1	45.0000	-150.0000	30.0000	60.0000	-45.0000	120.0000
	solution 2	45.0000	-150.0000	-150.0000	-60.0000	135.0000	120.0000
	solution 3	-135.0000	150.0000	-150.0000	60.0000	-45.0000	120.0000
	solution 4	-135.0000	150.0000	30.0000	-60.0000	135.0000	120.0000
	solution 5	56.0101	-106.0380	127.0661	60.0000	-135.0000	165.9475
	solution 6	56.0101	-106.0380	-52.9339	-60.0000	45.0000	165.9475
	solution 7	-123.9899	106.0380	-52.9339	60.0000	-135.0000	165.9475
	solution 8	-123.9899	106.0380	127.0661	-60.0000	45.0000	165.9475
Pose B	solution 1	0	180.0000	124.4395	60.0000	-45.0000	45.9475
	solution 2	0	180.0000	-150.0000	60.0000	-135.0000	90.0000
	solution 3	0	180.0000	30.0000	-60.0000	45.0000	90.0000
	solution 4	0	180.0000	-55.5605	60.0000	135.0000	45.9475

According to Table 4, the computed pose corresponding to each solution can be derived. Errors between specified pose and computed pose are computed which are all less than 1e-14. The verification process is illustrated in Figure 10, which shows that closed-form solutions accurately correspond to either the general terminal pose or the special pose at kinematics singularities. Hence, the VRM is valid to solve the IKP of manipulators satisfying the Pieper's condition.

Figure 10.

The validation process for kinematic solutions

4.2 IKPs of SFA

Duffy's geometry is the typical structure used in space robots, which can achieve a larger motion range and can be folded with minimal volume. Small Fine Arm (SFA) is one manipulator among the Japanese Experiment Modules [16]. The 6-DOF space robot with the three-axis parallel structure is shown in Figure 11.

Figure 11.

The space robot and DH coordinate frames

Similarly, the equivalent SOC is obtained by cutting open the virtual SLC at Link 1. The DH coordinate frames of all links are established. DH parameters are listed in Table 5.

Table 5.

DH parameters of the 6-DOF robot

link	a_i	α_i	d_i	θ_i
1″	0	0°	0	θ₁
1′	0	90°	L₀	180°
2	L₁	0°	0	θ₂
3	L₂	0°	0	θ₃
4	0	−90°	−L₃	θ₄
5	0	90°	L₄	θ₅
6	0	0°	L₅	θ₆

It is obvious that ^bA₀ is the identity matrix and ^1″A_1‘ is a constant transformation matrix which can be derived from Table 5. The computation of kinematic equations is specified as

^{1'} A_{1″} =^{1'} A_{2}^{2} A_{3}^{3} A_{4}^{4} A_{5}^{5} A_{e}^{e} A_{b}^{0} A_{1″}

(17)

Since the Duffy's geometry is the shoulder joint of the equivalent SOC, the wrist pose in z direction of the base frame depends only on θ_1,5,6. The first joint angle is expressed as

θ_{1} =atan2 (L_{3}, \pm \sqrt{a^{2} + b^{2} - L_{3}^{2}}) -atan2 (b, a)

(18)

θ_{5} = \pm acos (a_{y} \cos θ_{1} - a_{x} \sin θ_{1})

(19)

\begin{array}{l} θ_{6} =atan2 (sign(sin θ_{5}) (o_{y} \cos θ_{1} - o_{x} \sin θ_{1}), \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} -sign(sin θ_{5}) (n_{y} \cos θ_{1} - n_{x} \sin θ_{1})) \end{array}

(20)

The parameters a and b are computed in the Appendix.

Otherwise, sinθ₅=0, and θ₆ can be determined directly according to some roles.

The space robot is now simplified as a planar robot including Joint 2, Joint 3 and Joint 4. Closed-form solutions of the 3-DOF planar robot are given by

θ_{2} =atan2 (H_{sc2} (1), H_{sc2} (2))

(21)

\sin θ_{3} = \frac{L_{SW}^{2} - L_{1}^{2} - L_{2}^{2}}{2 L_{1} L_{2}}

(22)

θ_{4} =atan2 (H_{sc4} (1), H_{sc4} (2))

(23)

The parameters L_SW, H_sc2 and H_sc4 are illustrated in the Appendix.

Finally, closed-form solutions of the 6-DOF space robot are obtained from (17)–(22).

In order to demonstrate the validity of the VRM, kinematics simulations are performed. The joint range is from −180° to 180° without joint limits. Link parameters of the robot are expressed as follows: L₀=200 (mm), L₁=435 (mm), L₂=435(mm), L₃=180(mm), L₄=180(mm), L₅=441(mm).

Similarly, two sets of the desired end-effector poses are examined. The first one (Pose A) is the general hand pose, and the second one (Pose B) is the pose at singularity.

\begin{array}{c} Pose A:^{b} X_{e}^{d} = [\begin{matrix} 0 .5067 & -0 .2880 & 0 .8801 \end{matrix}] (m) \\ ^{b} R_{e}^{d} = [\begin{matrix} -0 .0388 & 0 .2977 & 0 .9539 \\ -0 .5388 & 0 .7977 & -0 .2709 \\ -0 .8415 & -0 .5245 & 0 .1294 \end{matrix}] \\ Pose B:^{b} X_{e}^{d} = [\begin{matrix} - 0 .4966 & 0 .2610 & 0 .6667 \end{matrix}] (m) \\ ^{b} R_{e}^{d} = [\begin{matrix} - 0 .5000 & 0 .8660 & 0 \\ 0 .0000 & - 0 .0000 & 1 \\ 0 .8660 & 0 .5000 & 0 \end{matrix}] \end{array}

According to (17)–(22), eight sets of solutions are obtained for Pose A, and six sets of solutions for Pose B, as illustrated in Table 6. Optimal kinematic solutions can be selected based on the typical criteria mentioned in Section 3.5. With reference to Figure 10, the errors between specified pose and computed pose are obtained, which are less than 1e-14. Closed-form solutions are accurate corresponding to either the general terminal pose or the special pose at kinematics singularities. It is verified that the VRM is valid for the IKP of manipulators satisfying the Duffy's condition.

Table 6.

Computed solutions of the SFA (unit: degree)

Pose	Solutions	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6
Pose A	solution 1	45.0000	60.0000	45.0000	60.0000	150.0000	45.0000
	solution 2	45.0000	105.4302	-45.0000	105.0000	150.0000	45.0000
	solution 3	45.0000	29.5979	117.8100	-162.4079	-150.0000	-135.0000
	solution 4	45.0000	147.4079	-117.8100	-44.5979	-150.0000	-135.0000
	solution 5	9.0917	69.5090	45.6538	56.6466	114.7204	54.6197
	solution 6	9.0917	115.1629	-45.6538	102.3005	114.7204	54.6197
	solution 7	9.0917	42.0462	117.2856	-167.5223	-114.7204	-125.3803
	solution 8	9.0917	159.3318	-117.2856	-50.2367	-114.7204	-125.3803
Pose B	solution 1	-140.1535	101.2335	97.5331	161.2335	140.1535	60.0000
	solution 2	-140.1535	-161.2335	-97.5331	-101.2335	140.1535	60.0000
	solution 3	-140.1535	107.1136	40.8138	32.0725	-140.1535	-120.0000
	solution 4	-140.1535	147.9275	-40.8138	72.8864	-140.1535	-120.0000
	solution 5	0	-8.7393	73.9530	-25.2136	0	20
	solution 6	0	65.2136	-73.9530	48.7393	0	20

4.3 IKPs of 7-DOF manipulators

7DOF manipulators are redundant manipulators. There are infinite inverse kinematic solutions corresponding to one specified end-effector pose. The non-redundant manipulator can be developed by fixing one of the joints. In this section, the 7-DOF humanoid arm with the offset wrist and the space manipulator SSRMS is examined.

Figure 12 shows the equivalent reconfigured manipulator of the humanoid arm. By parameterizing the seventh joint of the humanoid arm, closed-form solutions can be derived using the VRM. The equivalent SOC is obtained by cutting open the virtual SLC at Link 3. The computation of kinematic equations is specified as

Figure 12.

The equivalent SOC of the 7-DOF humanoid arm

^{3'} A_{3″} =^{3'} A_{4}^{4} A_{5}^{5} A_{6}^{6} A_{e}^{e} A_{b}^{0} A_{1}^{1} A_{2}^{2} A_{3″}

(24)

Based on (23), parameterized solutions can be obtained. Let [p_X p_Y p_Z]^T be the EEF position with respect to the base frame. If $^{b} P_{15} {(1)}^{2} +^{b} P_{15} {(2)}^{2} +^{b} P_{15} {(3)}^{2} - {(L_{se} + L_{ew})}^{2} \neq 0$ or $^{b} P_{15} {(1)}^{2} +^{b} P_{15} {(2)}^{2} +^{b} P_{15} {(3)}^{2} - {(L_{se} + L_{e w})}^{2} \neq 0, θ_{4, 5, 6}$ yields

θ_{4} =atan2 (- g_{1}, g_{3})

(25)

θ_{5} =atan2 (f_{1}, f_{3}) or θ_{5} =atan2 (- f_{1},- f_{3})

(26)

θ_{6} =atan2 (ρ, \pm \sqrt{1- ρ^{2}}) +atan2 (C, A)

(27)

The parameters g₁, g₃, f₁, f₃, ρ, A and C are computed in the Appendix.

Otherwise, the kinematic singularity occurs. In this case, sinθ₄=0, and θ₅ can be determined directly according to certain principle.

Supposed that R=^bR_e ^eR ₆⁶R₅ ⁵R₄⁴R_3‘ ^3’R_3″, the solutions of the spherical joint can be derived. If sinθ₂≠0, the joint angles are given by

θ_{1} =atan2 (sign (\sin θ_{2}) R_{2,3}, sign (\sin θ_{2}) R_{1,3})

(28)

θ_{2} = \pm acos (- R_{3,3})

(29)

θ_{3} =atan2 (-sign (\sin θ_{2}) R_{3,2}, sign (\sin θ_{2}) R_{3,1})

(30)

Otherwise, the spherical joint is at the singularity. In this case, θ₁ can be determined directly according to certain principle. θ₃ is given by

θ_{3} =atan2 (B_{sc} (1), B_{sc} (2))

(31)

where

B_{sc} = {[\begin{matrix} \sin θ_{1} & \cos θ_{1} \cos θ_{2} \\ -cos θ_{1} & \sin θ_{1} \cos θ_{2} \end{matrix}]}^{-} [\begin{matrix} R_{1,1} \\ R_{2,1} \end{matrix}] .

Finally, closed-form solutions of the 7-DOF humanoid arm are obtained. Given the desired end-effector pose, the corresponding joint angles can be determined directly.

In order to demonstrate the efficiency of the VRM, closed-form solutions of the parameterized humanoid arm are carried out in Matlab 7.1 environment running on an Intel(R) Core(TM) 2 Duo CPU 1.83GHz PC. At the same time, the IKP is also solved by the well-known Newton-Raphson algorithm and the simple Jacobian-based method. The position error between the desired end-effector pose and the computed pose is defined as 0.001(m). Given 20 sets of arbitrary end-effector pose in the workspace, the efficiency of each method is illustrated in Figure 13. The comparisons demonstrate that the VRM is valid and efficient to solve the IKP of the manipulator analytically.

Figure 13.

The efficiency of three different methods

Similarly, the equivalent SOC of SSRMS is developed by cutting open the virtual SLC at Link 2, as shown in Figure 14. The computation of kinematic equations is specified as

Figure 14.

The equivalent SOC of the SSRMS

^{2'} A_{2″} =^{2'} A_{3}^{3} A_{4}^{4} A_{5}^{5} A_{6}^{6} A_{e}^{e} A_{b}^{0} A_{1}^{1} A_{2″}

(32)

By parameterizing the first joint of SSRMS, closed-form solutions are derived using the VRM. The second joint angle is expressed as

θ_{2} =atan2 (ρ, \pm \sqrt{1- ρ^{2}}) +atan2 (B, A)

(33)

If $\sin θ_{2} \neq 0$ are given by

θ_{6} = \pm acos (a_{z} \cos θ_{2} - (a_{x} \cos θ_{1} + a_{y} \sin θ_{1}) \sin θ_{2})

(34)

θ_{7} =atan2 (-sign(sin θ_{6}) D_{1},sign(sin θ_{6}) D_{2})

(35)

The parameters ρ, A, B, D₁ and D₂ are computed in the Appendix.

Otherwise, sinθ₆=0, and θ₇ can be determined directly according to some roles.

The space robot is then simplified as a planar robot including Joint 3, Joint 4 and Joint 5. Closed-form solutions of the 3-DOF planar robot are given by

θ_{3} =atan2 (H_{sc3} (1), H_{sc3} (2))

(36)

\sin θ_{4} = \frac{L_{SW}^{2} - L_{4}^{2} - L_{5}^{2}}{2 L_{4} L_{5}}

(37)

θ_{5} =atan2 (H_{sc5} (1), H_{sc5} (2))

(38)

The parameters L_SW, H_sc3 and H_sc5 are illustrated in the Appendix.

Finally, parameterized closed-form solutions of the 7-DOF space robot are obtained from (32)–(37).

In order to highlight the efficiency of the VRM, computations of the IKP of the parameterized manipulator are completed using different methods. Twenty sets of arbitrary end-effector poses in the workspace are given. Figure 15 shows the computation time using three different methods. It shows that the VRM is faster in determining kinematic solutions than the other two methods.

Figure 15.

The efficiency of three different methods

In conclusion, the VRM is a valid and efficient approach to solve the IKP of manipulators analytically. It is applicable to serial manipulators containing either the Pieper′ or the Duffy's geometry.

5. Conclusions

In this paper, a unified framework has been proposed for closed-form solutions of a special class of 6- and 7-DOF serial manipulators containing the Pieper's geometry or the Duffy's geometry. Given a specified end-effector pose of the general manipulator, an equivalent reconfigured form can be developed. The reconfigured manipulator is kinematically equivalent to the original manipulator. The virtual reconfiguration method (VRM) is proposed on the basis of the kinematic equivalence between the two manipulators. Hence, for any serial manipulator satisfying either the Pieper's condition or the Duffy's condition, the equivalent SOC with the spherical wrist or the Duffy shoulder joint can be developed using the VRM. Closed-form solutions can also be obtained by solving the IKP of the equivalent SOC using traditional methods. The validity and efficiency of the VRM are verified by kinematic simulations of humanoid arms and space robots. Unlike traditional approaches, the VRM simplifies the computation of the IKP and establishes a unified framework for closed-form solutions of the special class of 6- and 7-DOF serial manipulators irrespective of the allocation of the Pieper's geometry or the Duffy's geometry.

Footnotes

6. Appendix

Parameters of the IKP for the 6-DOF humanoid arm are given by

\begin{array}{c} ^{b} A_{e} = [\begin{matrix} n_{r, x} & o_{r, x} & a_{r, x} & p_{r, x} \\ n_{r, y} & o_{r, x} & a_{r, x} & p_{r, x} \\ n_{r, z} & o_{r, x} & a_{r, x} & p_{r, x} \\ 0 & 0 & 0 & 1 \end{matrix}],^{e} A_{b} = [\begin{matrix} n_{x} & o_{x} & a_{x} & p_{x} \\ n_{y} & o_{x} & a_{x} & p_{x} \\ n_{z} & o_{x} & a_{x} & p_{x} \\ 0 & 0 & 0 & 1 \end{matrix}] . \\ P_{15, x} = p_{r,x} - L_{3} n_{x}, P_{15, y} = p_{r,y} - L_{3} n_{y}, P_{15, z} = p_{r,z} - L_{3} n_{z} - L_{0} \\ A= p_{x} + L_{0} a_{x} + L_{3}, B= p_{y} + L_{0} a_{y}, C= {(p_{z} + L_{0} a_{z})}^{2} + L_{2}^{2} \\ ρ= (L_{1}^{2} - A^{2} - B^{2} -C) / (2 L_{2} \sqrt{A^{2} + B^{2}}) \\ f_{1} =Acos θ_{6} -Bsin θ_{6}, f_{2} =Asin θ_{6} +Bcos θ_{6} \\ f_{3} = p_{z} + L_{0} a_{z}, g_{1} = f_{1} \cos θ_{5} - f_{3} \sin θ_{5}, g_{3} =- L_{2} - f_{2} \end{array}

Parameters of the IKP for SFA are given by

\begin{array}{c} a= p_{x} - L_{5}^{0} A_{6} (1,3) \\ b= L_{5}^{0} A_{6} (2,3) -^{0} A_{6} (2,4) \\ L_{SW} = \sqrt{^{1'} T_{4} {(1,4)}^{2} +^{1'} T_{4} {(2,4)}^{2}} \\ H_{sc2} = {[\begin{matrix} - L_{2} sin θ_{3} & L_{1} + L_{2} cos θ_{3} \\ L_{1} + L_{2} cos θ_{3} & L_{2} sin θ_{3} \end{matrix}]}^{-} [\begin{matrix} 1^{'} T_{4} (1, 4) \\ 1^{'} T_{4} (2, 4) \end{matrix}] \\ H_{sc4} = {[\begin{matrix} -sin (θ_{2} + θ_{3}) & cos (θ_{2} + θ_{3}) \\ cos (θ_{2} + θ_{3}) & sin (θ_{2} + θ_{3}) \end{matrix}]}^{-} [\begin{matrix} 1^{'} T_{4} (1, 1) \\ 1^{'} T_{4} (2, 1) \end{matrix}] \end{array}

Parameters of the IKP for the 7-DOF humanoid arm are given by

\begin{array}{c} ^{b} A_{e} = [\begin{matrix} n_{r, x} & o_{r, x} & a_{r, x} & p_{r, x} \\ n_{r, y} & o_{r, x} & a_{r, x} & p_{r, x} \\ n_{r, z} & o_{r, x} & a_{r, x} & p_{r, x} \\ 0 & 0 & 0 & 1 \end{matrix}],^{e} A_{b} = [\begin{matrix} n_{x} & o_{x} & a_{x} & p_{x} \\ n_{y} & o_{x} & a_{x} & p_{x} \\ n_{z} & o_{x} & a_{x} & p_{x} \\ 0 & 0 & 0 & 1 \end{matrix}], \\ A= (p_{r,x} - L_{w1t} - L_{bs} a_{r,x}) \cos θ_{7} + (L_{bs} a_{r,y} - p_{r, y}) \sin θ_{7} - L_{ww1}, \\ B= (p_{r,x} - L_{w1t} - L_{bs} a_{r,x}) \sin θ_{7} - (L_{bs} a_{r,y} - p_{r,y}) \cos θ_{7}, \\ C= p_{r,z} - L_{bs} a_{r,z} \\ ^{b} P_{15} =^{b} P_{6} +^{b} R_{6} \cdot {[\begin{matrix} L_{w w 1} & 0 & 0 \end{matrix}]}^{T} + {[\begin{matrix} 0 & 0 & - L_{b s} \end{matrix}]}^{T}, \\ ρ= (L_{se}^{2} - A^{2} - B^{2} - C^{2} - L_{e w}^{2}) / (2 L_{e w} \sqrt{A^{2} + C^{2}}) \\ f_{1} =Acos θ_{6} +Csin θ_{6}, f_{2} =Asin θ_{6} -Ccos θ_{6}, f_{3} =B, \\ g_{1} = f_{1} \cos θ_{5} + f_{3} \sin θ_{5}, g_{3} = f_{2} + L_{ew} \end{array}

Parameters of the IKP for the 7-DOF SSRMS are given by

\begin{array}{c} A= (L_{8} a_{x} - p_{x}) \cos θ_{1} + (L_{8} a_{y} - p_{y}) \sin θ_{1}, \\ B= L_{8} a_{z} + p_{z} - L_{1}, C= L_{3} + L_{6}, \\ D_{1} = (o_{x} \cos θ_{1} + o_{y} \sin θ_{1}) \sin θ_{2} - o_{z} \cos θ_{2}, \\ D_{2} = (n_{x} \cos θ_{1} + n_{y} \sin θ_{1}) \sin θ_{2} - n_{z} \cos θ_{2}, \\ L_{SW} = \sqrt{P_{M}^{2} (1) + P_{M}^{2} (2)}, \end{array}

P_{M} =^{2^{'}} P_{5} -^{2^{'}} R_{5} \cdot {[\begin{array}{l} L_{7} & 0 & 0 \end{array}]}^{T}

(where ^2′P₅ and ^2′P₅ are the position and the orientation of Frame 5 with respect to Frame 2′ respectively),

\begin{array}{c} H_{sc3} = {[\begin{matrix} - L_{5} sin θ_{4} & L_{4} + L_{5} cos θ_{4} \\ L_{4} + L_{5} cos θ_{4} & L_{5} sin θ_{4} \end{matrix}]}^{-} [\begin{matrix} P_{M} (1) \\ P_{M} (2) \end{matrix}] \\ H_{sc5} = {[\begin{matrix} -sin (θ_{3} + θ_{4}) & cos (θ_{3} + θ_{4}) \\ cos (θ_{3} + θ_{4}) & sin (θ_{3} + θ_{4}) \end{matrix}]}^{-} [\begin{matrix} \begin{matrix} ^{2'} R_{5} (1, 1) \\ ^{2'} R_{5} (2, 1) \end{matrix}] \end{matrix} \end{array}

7. Acknowledgements

This work is supported by the National 973 Programme under Grant No. 2011CB013306 and the National Natural Science Foundation of China under Grant No. 51175106.

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