3.1 Structural Configuration
The humanoid shoulder joint (HSJ) is based on a spherical 5R parallel mechanism, as shown in Figure 6(a). The initial unit axis vectors of all the revolute pairs in the HSJ are shown in Figure 6(b). Let O-XYZ be a base reference frame attached at the center O, where Y and OA2 axes are coincident. Let O-X1Y1Z1 be a moving reference frame of the HSJ attached at center O, where X1 and OC2 axes are coincident and Z1 and OC1 axes are coincident.
The humanoid elbow joint (HEJ) is based on a series 3-DOF kinematic chain RRR, as shown in Figure 7(a). The initial unit axis vectors of all the revolute pairs in the HEJ are shown in Figure 7(b). Let O2-X2Y2Z2 be a moving reference frame of the HEJ attached at the center O2, where Z2 and O2D axes are coincident.
The humanoid wrist joint (HWJ) is based on a spherical 3-RRP parallel mechanism, as shown in Figure 8(a). The initial unit axis vectors of all the motion pairs in the HWJ are shown in Figure 8(b). Let O3-X3Y3Z3 be a base reference frame of the HWJ attached at center O3, where Z3 and O2F axes are coincident. The moving reference frame attached at center O4, which coincides with point O3, is O4-X4Y4Z4.
The HRA can be equivalent to a series robotic arm, as shown in Figure 9. Specifically, β1 and γ1 denote the input of the equivalent series shoulder joint, and α4, β4, and γ4 denote the input of the equivalent series wrist joint.
3.2 Pre-processing of Parallel Joints
To clarify the kinematics analysis of the hybrid HRA, the Jacobian matrices of the parallel joints and corresponding equivalent series joints are calculated first.
3.2.1 Pre-processing of Humanoid Shoulder Joint
The HSJ exhibits only 2 degrees of rotational freedom. Hence, the Jacobian matrix of kinematics chain 1 can be obtained according to Eqs. (6) and (8) as follows:
$$\left\{ {\begin{array}{*{20}l} {\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]^{1} = \left[ {\begin{array}{*{20}l} {{\varvec{S}}_{{{ A}_{1} }} } & {{\varvec{S}_{{B_{1} }}^{\prime}} } & {{\varvec{S}_{{C_{1} }}^{\prime}} } \\ \end{array} } \right],} \\ {\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]^{1} { = }\left[ {\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]^{1} } \right]^{ - 1} ,} \\ \end{array} } \right.$$
(24)
where \({\varvec{S}_{i}^{\prime}}\) denotes the real-time unit axis vector of the ith motion pair.
However, for the kinematic chain 2, a virtual revolute pair D2 is added to make the Jacobian matrix a square matrix of the following form:
$$\left\{ {\begin{array}{*{20}l} {{{\varvec{\xi}}}_{{D_{{2}} }} = \left[ {\begin{array}{*{20}l} {{\varvec{S}}_{{D_{{2}} }} } & {{\varvec{S}}_{{D_{{2}} }}^{0} } \\ \end{array} } \right]^{{\text{T}}} ,} \\ {\theta_{{D_{{2}} }} = \dot{\theta }_{{D_{{2}} }} = \ddot{\theta }_{{D_{{2}} }} = 0,} \\ \end{array} } \right.$$
(25)
where \({\varvec{S}}_{{D_{{2}} }} = \left[ {\begin{array}{*{20}l} 0 & 0 & 1 \\ \end{array} } \right]^{{\text{T}}}\) and \({\varvec{S}}_{{D_{{2}} }}^{0} = \left[ {\begin{array}{*{20}l} 0 & 0 & 0 \\ \end{array} } \right]^{{\text{T}}}\).
Similarly, the Jacobian matrix of the kinematic chain 2 can be obtained according to Eqs. (6) and (8):
$$\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]^{2} = \left[ {\begin{array}{*{20}l} {{\varvec{S}}_{{{ A}_{2} }} } & {{\varvec{S}_{{C_{2} }}^{\prime}} } & {{\varvec{S}_{{D_{2} }}^{\prime}} } \\ \end{array} } \right],} \\ {\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]^{2} = \left[ {\begin{array}{*{20}l} {{\varvec{S}}_{{{ A}_{2} }} } & {{\varvec{S}_{{C_{2} }}^{\prime}} } \\ \end{array} } \right],} \\ \end{array} } \right.} \\ {\left\{ {\begin{array}{*{20}l} {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]^{2} = \left[ {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]^{2} } \right]^{ - 1} ,} \\ {\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]^{2} = \left[ {\begin{array}{*{20}l} {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]_{{1{\varvec{ :}}}}^{{2}} } & {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]_{{{2}{\varvec{ :}}}}^{2} } \\ \end{array} } \right]^{{\text{T}}} ,} \\ \end{array} } \right.} \\ \end{array} } \right.$$
(26)
where (:1) denotes the first column of matrices.
According to Eqs. (13) and (14), the Jacobian matrix of the HSJ based on the active pairs can be obtained as follows:
$$\left\{ {\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} = \left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\phi_{{{\text{SJ}}}} }} } \right]^{ - 1} ,} \\ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} = \left[ {\begin{array}{*{20}l} {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]_{:1} } & {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]_{:2} } \\ \end{array} } \right],} \\ \end{array} } \right.} \\ {\left\{ {\begin{array}{*{20}l} {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\phi_{{{\text{SJ}}}} }} = \left[ {\begin{array}{*{20}l} {\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]_{1:}^{1} } & {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]_{1:}^{2} } & {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]_{3:}^{2} } \\ \end{array} } \right]^{{\text{T}}} ,} \\ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\phi_{{{\text{SJ}}}} }} = \left[ {\begin{array}{*{20}l} {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\phi_{{{\text{SJ}}}} }} } \right]_{1:} } & {\left[ {\left[ {\overline{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\phi_{{{\text{SJ}}}} }} } \right]_{2:} } \\ \end{array} } \right]^{{\text{T}}} .} \\ \end{array} } \right.} \\ \end{array} } \right.$$
(27)
For the corresponding equivalent series joint, the Jacobian matrix can be obtained according to Eqs. (6) and (8) as follows:
$$\left\{ {\begin{array}{*{20}l} {\left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} = \left[ {\begin{array}{*{20}l} {{\varvec{S}}_{{Y_{1} }} } & {{\varvec{S}_{{X_{1} }}^{\prime}} } \\ \end{array} } \right],} \\ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\psi_{{{\text{SJ}}}} }} { = }\left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} } \right]^{{\text{T}}} .} \\ \end{array} } \right.$$
(28)
3.2.2 Pre-processing of the Humanoid Wrist Joint
The HWJ has 3 degrees of rotational freedom. Hence, the Jacobian matrix of the kinematic chains can be obtained according to Eqs. (6) and (8) as follows:
$$\left\{ {\begin{array}{*{20}l} {\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} } \right]^{b} = \left[ {\begin{array}{*{20}l} {{}^{3}{\varvec{S}}_{{{ H}_{b} }} } & {{}^{3}{\varvec{S}_{{K_{b} }}^{\prime}} } & {{}^{3}{\varvec{S}_{{P_{b} }}^{\prime}} } \\ \end{array} } \right],} \\ {\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} } \right]^{b} { = }\left[ {\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} } \right]^{b} } \right]^{ - 1} ,} \\ \end{array} } \right.$$
(29)
where label 3 in the upper left corner indicates that the calculation is performed in the O3-X3Y3Z3 coordinate system.
According to Eqs. (13) and (14), the Jacobian matrix of the HWJ is based on the active pairs and can be obtained as follows:
$$\left\{ {\begin{array}{*{20}l} {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} = \left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\phi_{{{\text{WJ}}}} }} } \right]^{ - 1} ,} \\ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\phi_{{{\text{WJ}}}} }} { = }\left[ {\begin{array}{*{20}l} {\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} } \right]_{{1{\varvec{ :}}}}^{1} } & {\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} } \right]_{{1{\varvec{ :}}}}^{2} } & {\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} } \right]_{{1{\varvec{ :}}}}^{3} } \\ \end{array} } \right]^{{\text{T}}} .} \\ \end{array} } \right.$$
(30)
With respect to the corresponding equivalent series joint, the Jacobian matrix can be obtained according to Eqs. (6) and (8) as follows:
$$\left\{ {\begin{array}{*{20}l} {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} = \left[ {\begin{array}{*{20}l} {{}^{3}{\varvec{S}}_{{{ Z}_{4} }} } & {{}^{3}{\varvec{S}_{{Y_{4} }}^{\prime}} } & {{}^{3}{\varvec{S}_{{X_{4} }}^{\prime}} } \\ \end{array} } \right],} \\ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\psi_{{{\text{WJ}}}} }} { = }\left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} } \right]^{ - 1} .} \\ \end{array} } \right.$$
(31)
3.3 Forward Kinematics Analysis of the Humanoid Robotic Arm
According to Eq. (22), the velocity vector of motion pairs of the corresponding equivalent series joint can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\dot{\varvec{\psi }}}_{{{\text{SJ}}}} = \left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\psi_{{{\text{SJ}}}} }} \left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} \dot{\varvec{\phi }}_{{\text{ SJ}}} ,} \\ {{\dot{\varvec{\psi }}}_{{{\text{WJ}}}} = {}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\psi_{{{\text{WJ}}}} }} {}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} \dot{\varvec{\phi }}_{{\text{ WJ}}} .} \\ \end{array} } \right.$$
(32)
Meanwhile, according to Eq. (11), the velocity vectors of passive motion pairs of the corresponding parallel joint can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\dot{\varvec{\theta }}}_{{{\text{SJ}}}}^{\left( b \right)} = \left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]^{\left( b \right)} \left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} \dot{\varvec{\phi }}_{{\text{ SJ}}} ,} \\ {{\dot{\varvec{\theta }}}_{{{\text{WJ}}}}^{\left( b \right)} = \left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} } \right]^{\left( b \right)} {}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} \dot{\varvec{\phi }}_{{\text{ WJ}}} .} \\ \end{array} } \right.$$
(33)
Thus, according to Eqs. (18) and (19), \(\left[ {{\varvec{H}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}}\) and \({}^{3}\left[ {{\varvec{H}}_{\omega } } \right]_{{\psi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}}\) can be calculated. Then, according to Eq. (23), the acceleration vector of the corresponding equivalent series joint can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\varvec{\ddot{\psi }}}_{{{\text{SJ}}}} = \left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\psi_{{{\text{SJ}}}} }} \left( {\left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} \varvec{\ddot{\phi } }{ + }\dot{\varvec{\phi }}_{{\text{ SJ}}}^{{\text{ T}}} \left[ {{\varvec{H}}_{\omega } } \right]_{{\phi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} \dot{\varvec{\phi }}_{{\text{ SJ}}} } \right.} \\ { \, \left. { - {\dot{\varvec{\psi }}}_{{{\text{SJ}}}}^{{\text{ T}}} \left[ {{\varvec{H}}_{\omega } } \right]_{{\psi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\dot{\varvec{\psi }}}_{{{\text{SJ}}}} } \right),} \\ {{\varvec{\ddot{\psi }}}_{{{\text{WJ}}}} = {}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\psi_{{{\text{WJ}}}} }} \left( {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\phi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} \varvec{\ddot{\phi } }{ + }\dot{\varvec{\phi }}_{{\text{ WJ}}}^{{\text{ T}}} {}^{3}\left[ {{\varvec{H}}_{\omega } } \right]_{{\phi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} \dot{\varvec{\phi }}_{{\text{ WJ}}} } \right.} \\ { \, \left. { - {\dot{\varvec{\psi }}}_{{{\text{WJ}}}}^{{\text{ T}}} {}^{3}\left[ {{\varvec{H}}_{\omega } } \right]_{{\psi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\dot{\varvec{\psi }}}_{{{\text{WJ}}}} } \right).} \\ \end{array} } \right.$$
(34)
Finally, according to Eqs. (6) and (7), the forward kinematics analysis of the HRA can be obtained as follows:
$$\left\{ {\begin{array}{*{20}l} {{\varvec{V}}_{{{\text{HRA}}}} = {\varvec{J}}_{\psi }^{{{\text{HRA}}}} {\dot{\varvec{\psi }}}_{{{\text{HRA}}}} ,} \\ {{{\varvec{\varepsilon}}}_{{{\text{HRA}}}} = {\varvec{J}}_{\psi }^{{{\text{HRA}}}} {\varvec{\ddot{\psi }}}_{{{\text{HRA}}}} { + }{\dot{\varvec{\psi }}}_{{{\text{HRA}}}}^{{\text{T}}} {\varvec{H}}_{\psi }^{{{\text{HRA}}}} {\dot{\varvec{\psi }}}_{{{\text{HRA}}}} { + }{\varvec{c}}_{{{\text{HRA}}}} ,} \\ \end{array} } \right.$$
(35)
where \({\varvec{J}}_{\psi }^{{{\text{HRA}}}} = \left[ {\begin{array}{*{20}l} {{{\varvec{\xi}}}_{{Y_{1} }} }{{\varvec{\xi}_{{X_{1} }}^{\prime}} }{{\varvec{\xi}_{D}^{\prime}} }{{\varvec{\xi}_{E}^{\prime}} }{{\varvec{\xi}_{F}^{\prime}} }{{\varvec{\xi}_{{Z_{4} }}^{\prime}} }{{\varvec{\xi}_{{Y_{4} }}^{\prime}} }{{\varvec{\xi}_{{X_{4} }}^{\prime}} } \\ \end{array} } \right]\), \({{\varvec{\psi}}}_{{{\text{HRA}}}} = \left[ {\begin{array}{*{20}l} {\psi_{{Y_{1} }} }{\psi_{{X_{1} }} }{\theta_{D} }{\theta_{E} }{\theta_{F} }{\psi_{{Z_{4} }} }{\psi_{{Y_{4} }} }{\psi_{{X_{4} }} } \\ \end{array} } \right]^{{\text{T}}}\), \({\varvec{H}}_{\psi }^{{{\text{HRA}}}} = \left\{ {\begin{array}{*{20}l} {{\varvec{\xi}_{m}^{\prime}} \times {\varvec{\xi}_{n}^{\prime}} ,} & {m < n,} \\ {0,} & {\text{other cases,}} \\ \end{array} } \right.\) \(m,n = Y_{1} ,X_{1} ,D,E,F,Z_{4} ,Y_{4} ,X_{4}\).
3.4 Inverse Kinematics Analysis of the Humanoid Robotic Arm
3.4.1 The First Method
According to Eqs. (8) and (9), the velocity and acceleration vectors of the equivalent series manipulator can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\dot{\varvec{\psi }}}_{{{\text{HRA}}}} = \left[ {{\varvec{J}}_{\psi }^{{{\text{HRA}}}} } \right]^{ - 1} {\varvec{V}}_{{{\text{HRA}}}} ,} \\ {{\varvec{\ddot{\psi }}}_{{{\text{HRA}}}} = \left[ {{\varvec{J}}_{\psi }^{{{\text{HRA}}}} } \right]^{ - 1} \left[ {{{\varvec{\varepsilon}}}_{{{\text{HRA}}}} - {\dot{\varvec{\psi }}}_{{{\text{HRA}}}}^{{\text{T}}} {\varvec{H}}_{\psi }^{{{\text{HRA}}}} {\dot{\varvec{\psi }}}_{{{\text{HRA}}}} - {\varvec{c}}_{{\text{M}}} } \right].} \\ \end{array} } \right.$$
(36)
According to Eqs. (20) and (21), the velocity and acceleration vectors of moving platforms of the parallel joints can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\varvec{V}}_{{{\text{SJ}}}} = \left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\dot{\varvec{\psi }}}_{{{\text{SJ}}}} ,} \\ {{{\varvec{\varepsilon}}}_{{{\text{SJ}}}} = \left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\varvec{\ddot{\psi }}}_{{{\text{SJ}}}} { + }{\dot{\varvec{\psi }}}_{{{\text{SJ}}}}^{{\text{ T}}} \left[ {{\varvec{H}}_{\omega } } \right]_{{\psi_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\dot{\varvec{\psi }}}_{{{\text{SJ}}}} { + }{\varvec{c}}_{{{\text{SJ}}}} ,} \\ \end{array} } \right.$$
(37)
$$\left\{ {\begin{array}{*{20}l} {{}^{3}{\varvec{V}}_{{{\text{WJ}}}} = {}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\psi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\dot{\varvec{\psi }}}_{{{\text{WJ}}}} ,} \\ {{}^{3}{{\varvec{\varepsilon}}}_{{{\text{WJ}}}} = {}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\psi_{{{\text{WJ}}}} }} {\varvec{\ddot{\psi }}}_{{{\text{WJ}}}} { + }{\dot{\varvec{\psi }}}_{{{\text{WJ}}}}^{{\text{ T}}} {}^{3}\left[ {{\varvec{H}}_{\omega } } \right]_{{\psi_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\dot{\varvec{\psi }}}_{{{\text{WJ}}}} { + }{}^{3}{\varvec{c}}_{{{\text{WJ}}}} .} \\ \end{array} } \right.$$
(38)
According to Eqs. (11) and (16), the velocity and acceleration vectors of all the motion pairs of the parallel joints can be obtained as follows:
$$\left\{ {\begin{array}{*{20}l} {{\dot{\varvec{\theta }}}_{{{\text{SJ}}}}^{\left( b \right)} = \left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} } \right]^{\left( b \right)} {\varvec{V}}_{{{\text{SJ}}}} ,} \\ {{\dot{\varvec{\theta }}}_{{{\text{WJ}}}}^{\left( b \right)} = \left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} } \right]^{\left( b \right)} {}^{3}{\varvec{V}}_{{{\text{WJ}}}} ,} \\ \end{array} } \right.$$
(39)
$$\left\{ {\begin{array}{*{20}l} {{\varvec{\ddot{\theta }}}_{{{\text{SJ}}}}^{\left( b \right)} = \left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{SJ}}}}^{{\theta_{{{\text{SJ}}}} }} \left( {{{\varvec{\varepsilon}}}_{{{\text{SJ}}}} - {\dot{\varvec{\theta }}}_{{{\text{SJ}}}}^{{\text{ T}}} \left[ {{\varvec{H}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\dot{\varvec{\theta }}}_{{{\text{SJ}}}} - {\varvec{c}}_{{{\text{SJ}}}} } \right)} \right]^{\left( b \right)} ,} \\ {{\varvec{\ddot{\theta }}}_{{{\text{WJ}}}}^{\left( b \right)} = \left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{{\text{WJ}}}}^{{\theta_{{{\text{WJ}}}} }} \left( {{}^{3}{{\varvec{\varepsilon}}}_{{{\text{WJ}}}} - {\dot{\varvec{\theta }}}_{{{\text{WJ}}}}^{{\text{ T}}} {}^{3}\left[ {{\varvec{H}}_{\omega } } \right]_{{\theta_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\dot{\varvec{\theta }}}_{{{\text{WJ}}}} - {}^{3}{\varvec{c}}_{{{\text{WJ}}}} } \right)} \right]^{\left( b \right)} .} \\ \end{array} } \right.$$
(40)
3.4.2 The Second Method
According to the mobility analysis of the hybrid HRA, kinematic chain 2 of the HSJ and kinematic chain 1 (or 2, or 3) of the HWJ are selected to form the corresponding branch series of the robotic arm.
According to Eqs. (8) and (9), the velocity and acceleration vectors of the branch series manipulator can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\dot{\varvec{\psi }}}_{{{\text{B}} - {\text{HRA}}}} = \left[ {{\varvec{J}}_{\psi }^{{{\text{B}} - {\text{HRA}}}} } \right]^{ - 1} {\varvec{V}}_{{{\text{HRA}}}} ,} \\ {{\varvec{\ddot{\psi }}}_{{{\text{B}} - {\text{HRA}}}} = \left[ {{\varvec{J}}_{\psi }^{{{\text{B}} - {\text{HRA}}}} } \right]^{ - 1} \left[ {{{\varvec{\varepsilon}}}_{{{\text{HRA}}}} - {\dot{\varvec{\psi }}}_{{{\text{B}} - {\text{HRA}}}}^{{\text{T}}} {\varvec{H}}_{\psi }^{{{\text{B}} - {\text{HRA}}}} {\dot{\varvec{\psi }}}_{{{\text{B}} - {\text{HRA}}}} - {\varvec{c}}_{{{\text{HRA}}}} } \right],} \\ \end{array} } \right.$$
(41)
where \({\varvec{J}}_{\psi }^{{{\text{B}} - {\text{HRA}}}} = \left[ {\begin{array}{*{20}l} {{{\varvec{\xi}}}_{{A_{2} }} }{{\varvec{\xi}^{{\prime}}_{{C_{2} }}} }{{\varvec{\xi}^{{\prime}}_{D}} }{{\varvec{\xi}^{{\prime}}_{E}} }{{\varvec{\xi}^{{\prime}}_{F}} }{{\varvec{\xi}^{{\prime}}_{{H_{1}} }} }{{\varvec{\xi}^{{\prime}}_{{K_{1} }}} }{{\varvec{\xi}^{{\prime}}_{{P_{1}} }} } \\ \end{array} } \right]\), \({{\varvec{\psi}}}_{{{\text{B}} - {\text{HRAHRA}}}} = \left[ {\begin{array}{*{20}l} {\theta_{{A_{2} }} }{\theta_{{C_{2} }} }{\theta_{D} }{\theta_{E} }{\theta_{F} }{\theta_{{H_{1} }} }{\theta_{{K_{1} }} }{\theta_{{P_{1} }} } \\ \end{array} } \right]^{{\text{T}}}\), \({\varvec{H}}_{\psi }^{{{\text{B}} - {\text{HRA}}}} = \left\{ {\begin{array}{*{20}l} {{\varvec{\xi}^{{\prime}}_{m}} \times {\varvec{\xi}^{{\prime}}_{n}} ,} & {m < n,} \\ {0,} & {\text{other cases,}} \\ \end{array} } \right.\) \(m,n = A_{2} ,C_{2} ,D,E,F,H_{1} ,K_{1} ,P_{1} .\)
According to Eqs. (20) and (21), the velocity and acceleration vectors of moving platforms of the parallel joints can be calculated as follows:
$$\left\{ {\begin{array}{*{20}l} {{\varvec{V}}_{{{\text{SJ}}}} = \left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\dot{\varvec{\theta }}}} \right]^{2} ,} \\ {{{\varvec{\varepsilon}}}_{{{\text{SJ}}}} = \left[ {\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\varvec{\ddot{\theta }}}{ + }{\dot{\varvec{\theta }}}^{{\text{ T}}} \left[ {{\varvec{H}}_{\omega } } \right]_{{\theta_{{{\text{SJ}}}} }}^{{{\text{SJ}}}} {\dot{\varvec{\theta }}}{ + }{\varvec{c}}_{{{\text{SJ}}}} } \right]^{2} ,} \\ \end{array} } \right.$$
(42)
$$\left\{ {\begin{array}{*{20}l} {{}^{3}{\varvec{V}}_{{{\text{WJ}}}} = \left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\dot{\varvec{\theta }}}} \right]^{1} ,} \\ {{}^{3}{{\varvec{\varepsilon}}}_{{{\text{WJ}}}} = \left[ {{}^{3}\left[ {{\varvec{J}}_{\omega } } \right]_{{\theta_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\varvec{\ddot{\theta }}}{ + }{\dot{\varvec{\theta }}}^{{\text{ T}}} \left[ {{\varvec{H}}_{\omega } } \right]_{{\theta_{{{\text{WJ}}}} }}^{{{\text{WJ}}}} {\dot{\varvec{\theta }}}{ + }{}^{3}{\varvec{c}}_{{{\text{WJ}}}} } \right]^{1} .} \\ \end{array} } \right.$$
(43)
Similarly, according to Eqs. (39) and (40), the velocity and acceleration vectors of all the motion pairs of the parallel joints can be obtained.