Analysis of coordinated grasping kinematics and optimization of grasping force of a parallel hybrid hand

Abstract

It is important to obtain optimal grasping performance of a parallel hybrid hand in order to reduce the grasping impact force and power loss, to avoid the damage of object or human body during grasping operation or rescue missions. In this article, the coordinated grasping kinematics of the multi-fingers and the optimization of grasping force for a parallel hybrid hand are studied and analyzed systematically. First, the coordinated grasping condition is analyzed, and the formulae are derived for solving the displacement, the velocity, and the acceleration of the contact points between the object and the multi-fingertips of the parallel hybrid hand. Second, the formulae are derived for solving the contact forces at the contact points. Third, the frictional constraints between the multi-fingers and the object are analyzed during stable grasping, and the formulae are derived for solving the optimum contact forces. Finally, an analytic numerical example is given, the analytic solutions of the coordinated grasping kinematics and optimum contact forces are solved and verified by simulation solutions.

Introduction

A parallel hybrid hand is a parallel manipulator (PM) equipped with several fingers on the moving platform. It has played an important role in application of forging operation, rescue missions, manufacturing and fixture of parallel machine tool, Computer Testing (CT)-guided surgery, and micro-operation of biomedicine and assembly cells. ^{1 –8} In this aspect, Zhang and Gao ⁹ designed a biologically inspired 4 universal joint, prismatic joint and spherical joint (4UPS) + PU hybrid PM for mine rescue robot and studied its kinematics and structure optimization. Ramadan and Inoue ¹⁰ developed architecture of a hybrid two-fingered micro–nano manipulator hand. Liu et al. ¹¹ proposed a hybrid robot for CT-guided surgery. Wang et al. ¹² designed a reconfigurable robot for search/rescue. The fingers are key tools of the forging operator, rescue robots, and mediation robots. In this aspect, Mir-Nasiri ¹³ designed a four-axis parallel robotic arm for assembly operations. Yan et al. ¹⁴ established a coordinated kinematic modeling of gripper of the manipulator for heavy-duty manipulators in an integrated open-die forging center. Nuttall and Klein ¹⁵ analyzed mechanical compliance effects in a gripper with parallel jaws. Zhang et al. ¹⁶ classified grasping robotic hands into three categories based on the object connectivity. In various PMs, the 5- and 6-degree of freedom (DoF) PMs with composite spherical joint S_c should be selected as the parallel hybrid hand because the moving platform can provide more room for arranging more fingers and has more DoFs and higher stiffness. In the aspect of PMs with S_c , Zhang ⁷ synthesized various PMs with different limbs and different S_c . Huang et al. ⁸ analyzed the structure of a 3/6 Gough-Stewart PM with 3S_c . Ben-Horin and Shoham ¹⁷ synthesized a class of PMs with six limbs and several S_c . Lu and Hu studied kinematics and statics of a 5-DoF 4SPS + spherical joint-active prismatic joint-revolute joint (SPR) PM with 2S_c . ¹⁸ In aspect of coordinated grasping kinematics and the optimization of grasping force of robot hand, Lippiello et al. proposed a grasping force optimization algorithm for multiarm robots with multi-fingered hands ¹⁹ and studied multi-fingered grasp synthesis based on the object dynamic properties. ²⁰ Roa and Suárez ²¹ solved the independent contact regions for grasping 3-D objects. Qin et al. ²² optimized the grasping force of robotic multi-fingered hands based on the frictional constraints. Watanabe and Yoshikawa ²³ studied grasping optimization using a required external force set. Xia et al. ²⁴ optimized grasping force of multi-fingered robotic hands using a recurrent neural network. Zheng and Qian ²⁵ studied weaknesses existing in optimal grasp planning of robot hand. Song et al. ²⁶ proposed the dual-fingered stable grasping control for an optimal force angle. In fact, it is difficulty to manufacture S_c . Therefore, a 5-DoF PM with two equivalent composite universal joints U_e is proposed for constructing a parallel hybrid hand equipped with three fingers in this article. Up to now, the studies on the coordinated grasping kinematics and the optimization of grasping force of the parallel hybrid hand have not been found. Since this parallel hybrid hand is designed for high-speed and heavy-load coordinative operation or rescue missions, it is significant to obtain optimal grasping performance for the parallel hybrid hand in order to reduce the grasping impact force and power loss and to avoid the damage of object or the injury of human body during grasping and rescue missions. For this reason, this article focuses on the coordinated grasping kinematics of the multi-finger and the optimization of grasping force of a parallel hybrid hand in the light of its applications.

Prototype of hybrid hand and its structural characteristics

A 3-D prototype of the parallel hybrid hand is constructed as grasping different objects, see Figure 1.

Figure 1.

A 3-D prototype of parallel hybrid hand as grasping objects.

The parallel hybrid hand is composed of a 5-DoF PM with two equivalent composite universal joints U_e and three 1-DoF finger mechanisms. The 5-DoF PM with 2U_e includes a platform m, a base B, two composite limbs, and an SPR-type active leg r ₃. Here, m is a regular triangle Δb ₁ b ₄ b ₃ with three vertices, three sides s_m , and a central point o; B is an equilateral pentagon B ₁ B ₂ B ₃ B ₄ B ₅ with five vertices, five sides s_B , and a central point O. Each of the composite limbs includes an equivalent composite universal joint U_e and two SPR-type active limbs. U_e is composed of a beam g_i (i = 1, 2) with two collinear revolute joints and a universal joint U with two cross revolute joints R_vi and R_hi . The middle of g ₁ is connected with m by U joint at point b_j (j = 1, 2). Two ends of g ₁ are connected with the upper ends of the jth SPR active leg l_j (j = 1, 2) by revolute joints R_j (j = 1, 2) at a_j . The middle of g ₂ is connected with m by U joint at point b_j (j = 4, 5). Two ends of g ₂ are connected with the upper ends of the jth SPR active leg l_j (j = 4, 5) by revolute joints R_j (j = 4, 5) at a_j . The lower end of l_j (j = 1, 2, 4, 5) is connected with B by the spherical joint S at B_j . r ₃ connects B with m by a spherical joint S at B ₃, an active prismatic joint P and revolute joint R ₃ at b ₃, see Figure 2.

Figure 2.

Kinematics model and grasping force situation of parallel hybrid hand. (a) The kinematics/statics model of 5-DoF PM with 2U_e , (b) the kinematics/statics coordinate system object, and (c) contact force model of fingertip and object. DoF: degree of freedom; PM: parallel manipulator.

Let ⊥, ||, and | be the perpendicular, parallel, and coinciding constraints. Let {s_i } be a coordinate frame o_i -x_iy_iz_i of the ith finger mechanism, {m} be a coordinate frame o-xyz fixed on m at o, and {B} be a coordinate frame O-XYZ fixed on B at O. This PM includes the following geometric conditions {b ₁|b ₂, b ₄| b ₅, z⊥m, R_vi ||z, R_hi ⊥R_vi , R ₁|R ₂, R ₄|R ₅, R_{h 1}⊥R ₁, R_{h 2}⊥R ₄, R_j ⊥l_j , b ₁ b ₄||x, R ₃||x, R ₃⊥r ₃, ob ₃|y, a_jb_j = b, ob_j = e, (i = 1, 2; j = 1, 2, 3, 4, 5)}. Each of the finger mechanisms has 1-DoF, and it is formed by a planar four bar mechanism which is composed of a frame, a claw, and an SPR-type active rod L_i . The three fingers can coordinately grasp object. The frame is fixed on m at a point o_i and distributed uniformly on m at the same circumference.

Coordinated grasping kinematics of parallel hybrid hand

Generally, the object is kept in a static state or moving, and the multi-fingertips are moved toward object during multi-fingertips grasping. Thus, the multi-fingertips apply the impact forces onto the object. Since the relative motion of object and the multi-fingertips must has a large influence on grasping performance, it is necessary to analyze the coordinated grasping kinematics of parallel hybrid hand and object.

Displacement of parallel hybrid hand

The kinematics model and the grasping force situation of the parallel hybrid hand are shown in Figure 2. A vector x in {s_i }, {m}, and {B} is represented by ^si x , ^m x , and x , respectively. Let δ _j be the unit vectors r_j from B_j to b_j , e _j be the vector from o to b_j , E _j be the vector from O to B_j , and e_j = e, E_i = E are satisfied. Let f _τ be a constrained force exerted on the SPR-type kinematic limb r ₃ at B ₃, τ be the unit vector of f _τ , and ς be the vector from o to f _τ . Since the constrained force do not produce any power, the constrained conditions τ ||R ₃ and τ ⊥l ₃ are satisfied. Let f _ci (i = 1, 2, 3) be a grasping force of the fingertip applied onto object, see Figure 1(b) and (c).

Let (X_o , Y_o , Z_o ) be the position components of o in {B}. Let (α, β, λ) be three Euler angles of m in {B} and E be the distance from O to B_i ; (x_l , x_m , x_n , y_l , y_m , y_n , z_l , z_m , z_n ) be nine orientation parameters of m. Let ϕ be one of (α, β, γ, φ, 2φ, θ, α_i , β_i , γ_i , θ_i ). Set s_ϕ = sinϕ, c_ϕ = cosϕ, and t_ϕ = tanϕ. The formulae for solving b _j , e _j , B _j , δ _j , c , ς in {B} and r_j (j = 1, …, 5) of the 5-DoF 4SPS + SPR PM have been derived by Lu and Hu, ¹⁸ see equation (1A) in Appendix 1. From the geometric relations of r_j (j = 1, 2, 4, 5) and l_j in Figure 2(a), it leads to

$l_j^{\text{2}}=r_j^{\text{2}}-b^{\text{2}}$

1

When given (α, β, γ, Y_o , Z_o ), l_j can be solved from equation (1) and equation (1A) in Appendix 1.

Displacement of the ith fingertip

Let c_i be the ith fingertip. As grasping object by multi-fingers, each of the fingertips contacts with object at contact point c_i without slide. Therefore, it is necessary to analyze the kinematics of the ith finger mechanism w_i (i = 1, 2, 3). The kinematics model of w_i (i = 1, 2, 3) and their distribution on m are shown in Figure 3. In {s_i }, (x_i ||z, y_i |oo_i ) are satisfied. Let L_i be an extension of the active leg of w_i , and W_ki (k = 1, 2, 3, 4) be the key points on w_i . Let d ₁ and d ₂ be the distances from o_i to W _1i and W _2i , respectively. Let d, d ₃, and d ₄ be the distances from W _1i to c_i , W _3i , and W _4i , respectively. Let d ₅ be the distance from W _2i to W _3i and d ₆ be the distance from W _4i to c_i . Let α_i be the angle between d ₃ and d ₄, β_i be the angle between d ₄ and d, γ_i be the angle between d ₁ and d ₃, and θ_i be the angle between x_i and d. Let ^m _si R be a rotation matrix from {s_i } to {m} and ^B _m R be a rotation matrix of PM from {m} to {B} in order XYX. ²⁷ The position vector of c_i (i = 1, 2, 3) in {s_i }, {m}, and {B} and the position vector o_i in {m} have been derived Lu et al., ²⁸ see equation (2A) in Appendix 2.

Figure 3.

Finger mechanism and its coordinate system.

Velocity and acceleration of the ith fingertip and parallel hybrid hand

Let v , ω , and V be the translational, angular, and general output velocities of the PM at o. Let a , ε , and A be the translational, angular, and general output accelerations of the PM at o. Let v_Li be the input velocity of the ith finger mechanism. Let v_lj and a_lj be the input velocity and acceleration of PM. Let V _l and A _l be the general input velocity and acceleration of the PM. They are represented ^18,27 as follows

$\begin{array}{c}\mathit{V}\text{=}\left(\begin{array}{c}\mathit{v}\\ \mathit{\omega }\end{array}\right),{\mathit{V}}_b\text{=}\left(\begin{array}{c}{\mathit{v}}_b\\ {\mathit{\omega }}_b\end{array}\right),\\ \mathit{A}\text{=}\left(\begin{array}{c}\mathit{a}\\ \mathit{\epsilon }\end{array}\right),{\mathit{A}}_b\text{=}\left(\begin{array}{c}{\mathit{a}}_b\\ {\mathit{\epsilon }}_b\end{array}\right),\end{array}{\mathit{V}}_l\text{=}\left(\begin{array}{c}{v}_{l1}\\ v_{l2}\\ v_{l3}\\ v_{l4}\\ v_{l5}\end{array}\right),{\mathit{A}}_l\text{=}\left(\begin{array}{c}{a}_{l1}\\ a_{l2}\\ a_{l3}\\ a_{l4}\\ a_{l5}\end{array}\right)$

2

The kinematic formulae between V and V _l and A and A _l of PM have been derived by Lu and Hu ²⁷ as follows

$\begin{array}{l}{\mathit{V}}_l={\mathit{J}}_l\mathit{V}, \mathit{V}\text{=}{\mathit{J}}_{\text{l}}^{-1}{\mathit{V}}_l,\\ {\mathit{A}}_l={\mathit{J}}_l\mathit{A}+{\mathit{V}}^T\mathit{H}\mathit{V}\end{array}$

3

Here, J _l and H _l are a 6 × 6 Jacobian matrix and a 6 × 6 × 6 Hessian matrix of the 5-DoF PM with 2U_e , respectively. The derivations of formulae for solving J _l , H _l , V , V _l , A , and A _l are given in Appendix 1.

Let u and s( u ) be a vector and its skew-symmetric matrix, and I be a 3 × 3 unit matrix. They satisfy ^1,2

$\begin{array}{l}\mathit{u}\times =s\left(\mathit{u}\right)\text{, }s{\left(\mathit{u}\right)}^T=-s\left(\mathit{u}\right)\\ -s{\left(\mathit{u}\right)}^2=\mathit{I}-s\left(\mathit{u}\right)s{\left(\mathit{u}\right)}^T\end{array}$

4

Here, u may be one of ( ^si c _i , ^m _si R ^m c _i , ω , r _bi , i = 1, 2, 3).

Let v _ci , ω _ci , and V _ci be the translational, angular, and general velocities of the ith fingertip c_i . When ( V , ^si c _i , ω _ci , V _ci ) are given or solved, the formulae for solving v _ci , ω _ci , V _ci in {s_i }, {m}, and {B} have been derived by Lu et al., ²⁸ see equation (2B) in Appendix 2.

Let a _ci and ε _ci be the translational and angular acceleration of the ith fingertip c_i . When ( A , v _ci , ω _ci , V _ci ) are given or solved, the formulae for solving a _ci and ε _ci in {s_i }, {m}, and{B} have been derived by Lu et al., ²⁸ see equation (2C).

Let r _bi (i = 1, 2, 3) be the vector from o_b to c_i . Let v _b , ω _b , and V _b be the translational, angular, and general velocities of the object at o_b . Suppose that the fingertip contact with object at c_i without slide. The linear velocity v _ci of the contact point c_i (the ith fingertip) in {B} is represented as follows

$\begin{array}{l}{\mathit{v}}_{ci}={\mathit{v}}_b+{\mathit{\omega }}_b\times {\mathit{r}}_{bi}=\left(\begin{array}{c}{\mathit{I}}_{3\times 3}-s\left({\mathit{r}}_{bi}\right)\end{array}\right){\mathit{V}}_b\\ {\mathit{r}}_{bi}={\mathit{r}}_{ci}-{\mathit{r}}_{ob}\end{array}$

5

From equations (3) and (5), it leads to

${\mathit{J}}_{vci\_V_l}{\mathit{V}}_l+{\mathit{J}}_{vci\_v_{Li}}{v}_{Li}=\left(\begin{array}{c}{\mathit{I}}_{3\times 3}-s\left({\mathit{r}}_{bi}\right)\end{array}\right){\mathit{V}}_b$

6

When ( r _bi , V _b ) are solved or given, the formulae for solving the general input velocity V _l of PM and the input velocities of the finger mechanisms are derived as follows

$\begin{array}{l}\mathit{J}{\mathit{V}}_n={\mathit{J}}_c{\mathit{V}}_b\text{, }{\mathit{V}}_n={\mathit{J}}^{-1}{\mathit{J}}_c{\mathit{V}}_b\text{,}\\ {\mathit{V}}_n=\left(\begin{array}{c}{\mathit{V}}_l\\ v_{L1}\\ v_{L2}\\ v_{L3}\end{array}\right),{\mathit{J}}_{c9\times 6}=\left(\begin{array}{cc}\mathit{I}& -s\left({\mathit{r}}_{b1}\right)\\ \mathit{I}& -s\left({\mathit{r}}_{b2}\right)\\ \mathit{I}& -s\left({\mathit{r}}_{b3}\right)\end{array}\right)\end{array}$

7

$\underset{9\times 9}{\mathit{J}}=\left(\begin{array}{cccc}{\mathit{J}}_{vc1\_V_l}& {\mathit{J}}_{vc1\_v_{L1}}& {\mathit{0}}_{6\times 1}& {\mathit{0}}_{6\times 1}\\ {\mathit{J}}_{vc2\_V_l}& {\mathit{0}}_{6\times 1}& {\mathit{J}}_{vc2\_v_{L2}}& {\mathit{0}}_{6\times 1}\\ {\mathit{J}}_{vc3\_V_l}& {\mathit{0}}_{6\times 1}& {\mathit{0}}_{6\times 1}& {\mathit{J}}_{vc3\_vL3}\end{array}\right)$

8

Let a _b , ε _b , and A _b be the translational, angular, and general accelerations of the object at o_b . Suppose that the multi-fingertips contact with object at c_i without slide. When given ( r _bi , a _b , ε _b , A _b ), the translational acceleration a _ci (i = 1, 2, 3) of the contact point (the ith fingertip c_i ) in {B} can be solved as follows

$\begin{array}{c}{\mathit{a}}_{ci}={\mathit{a}}_b+{\mathit{\epsilon }}_b\times {\mathit{r}}_{bi}+{\mathit{\omega }}_b\times {\mathit{\omega }}_b\times {\mathit{r}}_{bi}\\ =\left(\begin{array}{c}{\mathit{I}}_{3\times 3}-s\left(r_{bi}\right)\end{array}\right){\mathit{A}}_b+s^2\left({\mathit{\omega }}_b\right){\mathit{r}}_{bi}\end{array}$

9

A relation between A _b and A is represented by equations (4) and (7) to (9) as follows

$\begin{array}{l}\left(\begin{array}{c}{\mathit{I}}_{3\times 3}-s\left({}_m{}^B{\mathit{R}}^m{c}_i\right)\end{array}\right)\mathit{A}+{}_m{}^B{\mathit{R}}^m{\mathit{a}}_{ci}\\ +2s\left(\mathit{\omega }\right){}_m{}^B{\mathit{R}}^m{\mathit{v}}_{ci}+s^2\left(\mathit{\omega }\right){}_m{}^B{\mathit{R}}^m{\mathit{c}}_i\\ \text{=}\left(\begin{array}{cc}{\mathit{I}}_{3\times 3}& -s({}_m{}^B{\mathit{R}}^m{c}_i)\end{array}\right){\mathit{A}}_b+s^2\left({\mathit{\omega }}_b\right){\mathit{r}}_{bi}\end{array}$

10

Analysis of contact force

Basic concept and condition of grasping

When grasping object or operating using multi-finger, a kinematic model is depended material and outline of grasped object. In this case, the fingertips apply grasped constraint forces onto object to balance the external workloads applied on object, and a statics balance of grasped object is kept. Thus, the sum of all force vectors should be 0. ²² However, as the multi-fingers apply very large grasping forces onto the object or human body and also satisfy the statics balance condition of multi-finger grasping object, the multi-fingers may damage the object or injury human body. Therefore, it is necessary to establish a statics model for solving reasonable contact force of the parallel hybrid hand for avoiding damage of the object or injury of human body.

Suppose that the contact between the multi-fingertips and the object is a friction contact without slide. When a arbitrarily workload is applied on object, if the contact force exerted onto object by fingertip is within a cone angle of friction, then the frictional force has not reached the maximum statics frictional force, and slide does not occur. ^{19 –22} In this case, contact forces resist the arbitrarily workload applied on object. This grasping is a force closed grasping. ^29,30 In order to complete grasping and operating task using the multi-fingers, the arbitrarily workload applied on object must be balanced by the contact forces between multi-fingertip and object. It is defined as a multi-fingertip stability balanced condition. Based on this condition, the contact force applied on object can be solved as given arbitrarily workload on object.

In a multi-finger grasping system, let c_i (i = 1, 2, 3) be the three contact points between the fingertips and the object. Let {c_i } be a coordinate system (c_i -x_iy_iz_i ) attached onto c_i . Here, x_i and y_i are located in tangent plane of c_i , and z_i is normal of plane which includes c_i (i = 1, 2, 3). Let ^ci f _ci be the contact force of each fingertip applied onto object in {c_i }. Let ^c F _c be the general contact force of ^ci f _ci , and F _o be the general external workload applied onto object in {B}. They are represented as follows

$\begin{array}{c}{\mathit{F}}_o=\left(\begin{array}{c}{\mathit{f}}_o\\ t_o\end{array}\right),\\ \text{(}i\text{=}1,2,3)\end{array}{}^c\mathit{F}_c=\left(\begin{array}{c}{}^{c1}\mathit{f}_{c1}\\ {}^{c2}\mathit{f}_{c2}\\ {}^{c3}\mathit{f}_{c3}\end{array}\right),{}^{ci}\mathit{f}_{ci}=\left(\begin{array}{c}{f}_{cix}\\ f_{ciy}\\ f_{ciz}\end{array}\right)$

11

Based on the stability balanced condition, the frictional force between the multi-fingertip and the object must be less than the maximum statics friction force. Thus, a friction constraint equation is described as follows

$\begin{array}{l}\sqrt{f_{cix}+f_{ciy}}\le {\text{μ}}_i{f}_{ciz}\\ f_{ciz}\ge 0\text{,}(i\text{=}1,2,3)\end{array}$

12

Here, μ_i is the friction coefficient at point c_i .

Let ^B _ci R be rotation matrix from {c_i } to {B}. When transformed the contact force ^ci f _ci from {c_i } into {B}, the contact force f _ci in {B} can be derived as

${\mathit{f}}_{ci}={}_{ci}{}^B\mathit{R}^{ci}{\mathit{f}}_{ci}\text{,}(i\text{=}1,2,3)$

13

When f _ci satisfying equation (12), the statics balanced equation between f _ci (i = 1, 2, 3) and F _o can be established. Let ( f _c , t _c ) be a contact wrench. The general contact force F _c can be derived as follows

${\mathit{F}}_c=\left(\begin{array}{c}{\mathit{f}}_c\\ {\mathit{t}}_c\end{array}\right)\text{,}\begin{array}{c}\begin{array}{l}{\mathit{f}}_c={\mathit{f}}_{c1}+{\mathit{f}}_{c2}+{\mathit{f}}_{c3}\text{,}\\ {\mathit{t}}_c={\mathit{r}}_{b1}\times {\mathit{f}}_{c1}+{\mathit{r}}_{b2}\times {\mathit{f}}_{c2}+{\mathit{r}}_{b3}\times {\mathit{f}}_{c3}\end{array}\end{array}$

14

Here, r _bi is vector from o_b to c_i .

Substituting equation (13) into equation (14), a balance equation for the generalized forces applied to the object can be represented as follows

$\begin{array}{l}{\mathit{F}}_c={\mathit{G}}^c{\mathit{F}}_c\text{,}\\ \underset{6\times 9}{\mathit{G}}=\left(\begin{array}{ccc}{}_{c1}{}^B\mathit{R}& {}_{c2}{}^B\mathit{R}& {}_{c3}{}^B\mathit{R}\\ s\left({\mathit{r}}_{b1}\right){}_{c1}{}^B\mathit{R}& s\left({\mathit{r}}_{b2}\right){}_{c2}{}^B\mathit{R}& s\left({\mathit{r}}_{b3}\right){}_{c3}{}^B\mathit{R}\end{array}\right)\end{array}$

15

Here, G is a 6 ×9 grasp matrix, by which the relation of F _c and the external load can be discovered.

In order to balance object and obtain stability of grasping, a statics equation between the general external workload F _o and the general contact force F _c is derived from equation (15) as follows

${\mathit{F}}_o-{\mathit{F}}_c=0\Rightarrow {\mathit{F}}_o=-{\mathit{G}}^c{\mathit{F}}_c$

16

Thus, ^c F _c can be solved from equation (16) during stable grasping object.

Since G is a 6 × 9 matrix, it is a challenging issue to solve F _o by utilizing the theory of generalized inverse matrix. For this reason, it is necessary to explain the basic theory of equations set with generalized inverse matrix in next section.

Basic theory of equations set with generalized inverse matrix

Let P be a 1 × n vector, 0 be a 1 × m 0 vector, and G be a m × n (m<n) matrix. The generalized inverse matrix G ⁺ of G is a n × m matrix and satisfy ^30,31

${\mathit{G}}^{+}={\mathit{G}}^T{\left(\mathit{G}{\mathit{G}}^T\right)}^{-1}$

17

An equation set GP = 0 can be represented as follows

$\begin{array}{l}\mathit{G}\mathit{P}\text{=}\mathit{0}\text{,}\underset{m\times n}{\mathit{G}}\text{=}\left(\begin{array}{cccc}{g}_{11}& g_{12}& \cdots & g_{1n}\\ g_{21}& g_{22}& \cdots & g_{2n}\\ ⋮& ⋮& & ⋮\\ g_{m1}& g_{m2}& \cdots & g_{mn}\end{array}\right)\\ \underset{n\times 1}{\mathit{P}}\text{=}\left(\begin{array}{c}{p}_1\\ p_2\\ ⋮\\ p_n\end{array}\right)\text{, }\underset{m\times 1}{\mathit{0}}=\left(\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right)\end{array}$

18

Here, G is coefficient matrix, g_ij (i = 1, 2,…, m; j = 1, 2,…, n) is item of G , and p_j is item of P .

G can be transformed automatically into an equivalent matrix G _e ^30,31 by utilizing MATLAB as follows

$\begin{array}{l}\underset{m\times n}{\mathit{G}}=\left(\begin{array}{cccc}{g}_{11}& g_{12}& \cdots & g_{1n}\\ g_{21}& g_{22}& \cdots & g_{2n}\\ ⋮& ⋮& & ⋮\\ g_{m1}& g_{m2}& \cdots & g_{mn}\end{array}\right)\Rightarrow \\ \underset{m\times n}{{\mathit{G}}_e}=\left(\begin{array}{ccccccc}1& 0& \cdots & 0& b_{11}& \cdots & b_{1(n-m)}\\ 0& 1& \cdots & 0& b_{21}& \cdots & b_{2(n-m)}\\ ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 0& 0& \cdots & 1& b_{m1}& \cdots & b_{m(n-m)}\end{array}\right)\end{array}$

19

Here, b_ij (i = 1,2,…, m; j = 1, 2,…, n-m) is each simplified item of G _e by transforming G , and the result of b_ij can be solved.

Since G _e is equivalent to G , the solutions of G _e are the same as that of G . ³¹ p_j (j = 1, 2,…, m) in equation (18) can be represented by equation (19) and solved from p_j (j = m+1,…, n) as follows

$\begin{array}{l}\mathit{G}\mathit{P}=\mathit{0}\Rightarrow \\ \{\begin{array}{l}{p}_1=-b_{11}{p}_{m+1}-b_{12}{p}_{m+2}\cdots -b_{1(n-m)}{p}_n\\ p_2=-b_{21}{p}_{m+1}-b_{22}{p}_{m+2}\cdots -b_{2(n-m)}{p}_n\\ \text{………………………………}\\ p_m=-b_{m1}{p}_{m+1}-b_{m2}{p}_{m+2}\cdots -b_{m(n-m)}{p}_n\end{array}\end{array}$

20

Here, p_j (j = m+1,…, n) is an arbitrarily constant.

Thus, a general solution of equation set GP = 0 in equation (20) can be solved as follows

$\begin{array}{l}\mathit{P}\text{=}\left(\begin{array}{c}{p}_1\\ ⋮\\ p_m\\ p_{m+1}\\ ⋮\\ p_n\end{array}\right)\text{=}{p}_{m+1}\left(\begin{array}{c}-b_{11}\\ ⋮\\ -b_{m1}\\ 1\\ ⋮\\ 0\end{array}\right)+\cdots p_n\left(\begin{array}{c}-b_{1(n-m)}\\ ⋮\\ -b_{m(n-m)}\\ 0\\ ⋮\\ 1\end{array}\right)\\ {\mathit{P}}_m=\left(\begin{array}{c}{p}_{m+1}\\ ⋮\\ p_n\end{array}\right)\end{array}$

21

A simple formula of general solution of GP = 0 is represented as follows

$\begin{array}{l}\mathit{P}\text{=}\mathit{W}{\mathit{P}}_m\text{, }\mathit{W}=\left(\begin{array}{cccc}{\mathit{W}}_1& {\mathit{W}}_2& \cdots & {\mathit{W}}_{n-m}\end{array}\right),\\ {\mathit{W}}_1=\left(\begin{array}{c}-b_{11}\\ ⋮\\ -b_{m1}\\ 1\\ ⋮\\ 0\end{array}\right),\cdots \text{,}{\mathit{W}}_{n-m}=\left(\begin{array}{c}-b_{1(n-m)}\\ ⋮\\ -b_{m(n-m)}\\ 0\\ ⋮\\ 1\end{array}\right)\end{array}$

22

Here, W is a mapping matrix from P _m to P .

When replacing vector 0 in equation (20) by a vector b (b ₁,…, b_m ), GP = 0 is transformed into GP = b . A matrix [ G b ] is derived from equation (22). Similarly, [ G b ] can be transformed into an equivalent matrix [ G b ] _e by utilizing MATLAB as follows

$\left(\begin{array}{cc}\mathit{G}& \mathit{b}\end{array}\right)\text{=}\left(\begin{array}{ccccc}{g}_{11}& g_{12}& \cdots & g_{1n}& b_1\\ g_{21}& g_{22}& \cdots & g_{2n}& b_2\\ ⋮& ⋮& & ⋮& ⋮\\ g_{m1}& g_{m2}& \cdots & g_{mn}& b_m\end{array}\right)\Rightarrow {\left(\begin{array}{cc}\mathit{G}& \mathit{b}\end{array}\right)}_e\text{=}\left(\begin{array}{cccccccc}1& 0& \cdots & 0& b_{11}& \cdots & b_{1(n-m)}& d_1\\ 0& 1& \cdots & 0& b_{21}& \cdots & b_{2(n-m)}& d_2\\ ⋮& ⋮& & ⋮& ⋮& & ⋮& ⋮\\ 0& 0& \cdots & 1& b_{m1}& \cdots & b_{m(n-m)}& d_m\end{array}\right)$

23

Here, d_i (i = 1, 2,…, m) are the simplified item of [ G b ] _e by transforming b_i of [ G b ].

Since ( G b ) _e is equivalent to ( G b ), the solutions of ( G b ) _e are the same as that of ( G b ) in the study by Lay. ³¹

Substituting equation (23) into GP = b , p_j (j = 1, 2,…, m) can be represented and solved as follows

$\begin{array}{l}\mathit{G}\mathit{P}=\mathit{b}\Rightarrow \\ \{\begin{array}{l}{p}_1=-b_{11}{p}_{m+1}-b_{12}{p}_{m+2}\cdots -b_{1(n-m)}{p}_n+d_1\\ p_2=-b_{21}{p}_{m+1}-b_{22}{p}_{m+2}\cdots -b_{2(n-m)}{p}_n+d_2\\ \text{……………………………………}\\ p_m=-b_{m1}{p}_{m+1}-b_{m2}{p}_{m+2}\cdots -b_{m(n-m)}{p}_n+d_m\end{array}\end{array}$

24

The solutions P _b of GP = b can be derived from equation (24) and can be represented as follows

$\begin{array}{l}{\mathit{P}}_b\text{=}\mathit{D}\text{+}\mathit{P}=\left(\begin{array}{c}{d}_1+p_1\\ ⋮\\ p_n\end{array}\right)\text{, }\\ \mathit{D}={\left(\begin{array}{cccccc}{d}_1& \cdots & d_m& 0& \cdots & 0\end{array}\right)}^T\end{array}$

25

Here, D is a special solution of GP = b corresponding to b , and P is a general solution of GP = 0 .

P _b is represented as follows

${\mathit{P}}_b\text{=}{\mathit{G}}^{+}\mathit{b}+\mathit{\lambda }\mathit{W}{\mathit{P}}_m$

26

Here, λ is an arbitrarily constant for optimizing workload.

Since P _m (p_{m+ 1}, p_{m+ 2},…, p_n-m ) is (n-m) × 1 vector formed from (n-m) arbitrarily constants, λP _m don’t change general solution model of GP = b .

Solving contact force

When P (p ₁, p ₂,…, p ₉) is a 9 × 1 vector and G is a 6 × 9 matrix, G and its equivalent matrix G _e can be represented based on above theory of equations set with generalized inverse matrix ³⁰ as follows

$\begin{array}{l}\mathit{G}=\left(\begin{array}{ccc}{}_{c1}{}^B\mathit{R}& {}_{c2}{}^B\mathit{R}& {}_{c3}{}^B\mathit{R}\\ s\text{(}{\mathit{r}}_{b1}){}_{c1}{}^B\mathit{R}& s\text{(}{\mathit{r}}_{b2}){}_{c2}{}^B\mathit{R}& s\text{(}{\mathit{r}}_{b3}){}_{c3}{}^B\mathit{R}\end{array}\right)\Rightarrow \\ {\mathit{G}}_{e6\times 9}=\left(\begin{array}{ccccccc}1& 0& \cdots & 0& b_{11}& b_{12}& b_{13}\\ 0& 1& \cdots & 0& b_{21}& b_{22}& b_{23}\\ ⋮& ⋮& & ⋮& ⋮& ⋮& ⋮\\ 0& 0& \cdots & 1& b_{61}& b_{62}& b_{63}\end{array}\right)\end{array}$

27

Each of items in G _e can be solved by utilizing MATLAB. The general solution of GP = 0 is formed by a set of vectors W _i (i = 1, 2, 3) as follows

$\begin{array}{l}\mathit{G}\mathit{P}=\mathit{0}\text{, }\mathit{P}=\mathit{W}{\mathit{P}}_m\text{,}\\ \mathit{W}=\left(\begin{array}{ccc}{\mathit{W}}_1& {\mathit{W}}_2& {\mathit{W}}_3\end{array}\right)\text{,}\\ {\mathit{P}}_m=\left(\begin{array}{c}{p}_7\\ p_8\\ p_9\end{array}\right),\mathit{P}=\left(\begin{array}{c}{p}_1\\ ⋮\\ p_6\\ {\mathit{P}}_m\end{array}\right)\text{, }\\ {\mathit{W}}_1=\left(\begin{array}{c}-b_{11}\\ ⋮\\ -b_{61}\\ 1\\ 0\\ 0\end{array}\right)\text{, }{\mathit{W}}_2=\left(\begin{array}{c}-b_{12}\\ ⋮\\ -b_{62}\\ 0\\ 1\\ 0\end{array}\right)\text{,}{\mathit{W}}_3=\left(\begin{array}{c}-b_{13}\\ ⋮\\ -b_{63}\\ 0\\ 0\\ 1\end{array}\right)\end{array}$

28

Here, p ₇, p ₈, and p ₉ are three arbitrarily constants.

Replacing P _b and b by ^c F _c and − F _o , respectively, the general solution of ^c F _c in equation (14) can be solved using the generalized inverse matrix G ⁺ of G as follows

$\begin{array}{l}{\mathit{F}}_o=-{\mathit{G}}^c{\mathit{F}}_c\Rightarrow \\ {}^c\mathit{F}_c=-{\mathit{G}}^{+}{\mathit{F}}_o+\mathit{\lambda }\mathit{W}{\mathit{P}}_m\text{,}{\mathit{G}}^{+}={\mathit{G}}^T{\left(\mathit{G}{\mathit{G}}^T\right)}^{-1}\end{array}$

29

Here, λ is an arbitrarily constant for optimizing workload. ^c F _c is represented ³² as follows

${}^c\mathit{F}_c={}^c\mathit{F}_p+{}^c\mathit{F}_q$

30

Here, ^c F _p is the general operating force which is used to balance external force F _o applied on object; ^c F _q is the internal force which is the sum of contact forces. ³³

^c F _q has no influence on movement of object, but it influences on the grasping stability and slide of object and fingertip. ³³ Therefore, a key issue is to determine reasonable grasping force for completing coordinated operation. Suppose that each of multi-fingers always contacts object during grasping. Based on equation (30) and GP = 0 , ^c F _p and ^c F _q are solved as follows

${}^c\mathit{F}_p=-{\mathit{G}}^{+}{\mathit{F}}_o$

31

$\begin{array}{l}{}^c\mathit{F}_q={\mathit{\lambda }}^c{\mathit{F}}_g\text{,}{}^{ci}\mathit{f}_{qi}={\mathit{\lambda }}_i{}^c\mathit{f}_g\text{,}\\ \mathit{\lambda }=\text{max}({\mathit{\lambda }}_1\text{, }{\mathit{\lambda }}_2\text{, }{\mathit{\lambda }}_3\text{),}\\ {}^c\mathit{F}_g=\mathit{W}{\mathit{P}}_m\text{,}{}^{ci}\mathit{f}_{gi}={\mathit{W}}_i{\mathit{P}}_m\text{, }i\text{=}1,2,3\\ {}^{ci}\mathit{f}_{gi}=\left(\begin{array}{c}{f}_{gix}\\ f_{giy}\\ f_{giz}\end{array}\right)\text{, }\underset{9\times 3}{\mathit{W}}=\left(\begin{array}{c}{\mathit{W}}_1\\ {\mathit{W}}_2\\ {\mathit{W}}_3\end{array}\right)\text{, }\underset{3\times 3}{{\mathit{W}}_i}=\left(\begin{array}{c}{\mathit{W}}_{ix}^T\\ {\mathit{W}}_{iy}^T\\ {\mathit{W}}_{iz}^T\end{array}\right)\end{array}$

32

In equation (32), the magnitude and orientation of ^ci f _qi are determined by λ_i and WP _m , respectively. Here, λ_i is an arbitrarily constant for optimizing grasping force. Therefore, the optimization of contact force includes optimization of magnitude and orientation of contact force. In the optimization processes, the first solves optimum orientation of ^ci f _qi . After that, solves the optimum magnitude λ_i of ^ci f _qi along its optimum orientation based on the condition of friction cone. Since each of λ_i (i = 1, 2, 3) satisfies the condition of friction cone, select maximum one of λ_i (i = 1, 2, 3). When λ = max(λ ₁, λ ₂, λ ₃) satisfies the condition of friction cone, the optimum magnitude of ^c F _c can be solved.

If the contact forces ^ci f _ci (i = 1, 2, 3) satisfy the friction constraint equation (12), then ^ci f _gi must also satisfy the friction constraint equation. Thus, there is

$f_{gix}^2+f_{giy}^2\le {\mu }_i^2{f}_{giz}^2\text{, }{f}_{giz}\ge 0,(i\text{=}1,2,3)\text{.}$

33

Substituting ^ci f _gi in equation (32) into equation (33), it leads to

$\{\begin{array}{l}{\left({\mathit{W}}_{ix}^T{\mathit{P}}_m\right)}^T\cdot \left({\mathit{W}}_{ix}^T{\mathit{P}}_m\right)+{\left({\mathit{W}}_{iy}^T{\mathit{P}}_m\right)}^T\cdot \left({\mathit{W}}_{iy}^T{\mathit{P}}_m\right)\\ \le {\mu }_i^2\left({\mathit{W}}_{iz}^T{\mathit{P}}_m\right)T\cdot \left({\mathit{W}}_{iz}^T{\mathit{P}}_m\right)\\ -{\mathit{W}}_{iz}^T{\mathit{P}}_m\le 0\text{,}(i\text{=}1,2,3)\end{array}$

34

Next, an equivalent constraint equation is derived from equation (34) as follows

$\{\begin{array}{l}{\mathit{P}}_m^T\left({\mathit{W}}_{ix}{\mathit{W}}_{ix}^T+{\mathit{W}}_{iy}{\mathit{W}}_{iy}^T-{\mu }_i^2{\mathit{W}}_{iz}{\mathit{W}}_{iz}^T\right){\mathit{P}}_m\le 0\\ -{\mathit{W}}_{iz}^T{\mathit{P}}_m\le 0\text{, (}i=1\text{, }2\text{, }3).\end{array}$

35

In fact, equation (35) is a constraint condition of solving contact force in the light of friction cone constraint of multi-finger grasping system. Thus, the challenging issue of solving grasping force is transformed into to determine whether constraint condition (equation (35)) is satisfied. If constraint condition (equation (35)) is satisfied, then grasping is stable. After that, the optimum grasping force can be solved.

Optimization of contact force

When the contact force of multi-fingers applied onto object becomes too large, the finger mechanism or the parallel mechanism may be damaged. At the same time, more power is required for the unnecessary larger input active forces of finger mechanism and parallel mechanism. Therefore, the contact force of multi-fingers applied onto object must be limited within reasonable value. Thus, an optimum object is that the contact force of the multi-fingers applied onto the object should be the lowest when the stable grasping condition is satisfied.

The condition of internal force ^c F _q satisfying equation (35) is not unique because many solutions of contact force ^c F _q satisfy this condition. Therefore, a key issue is to select a reasonable contact force from them. In fact, an optimum task is to minimize the maximum contact force. It is equivalent to be away from the boundary of friction cone as far as possible using the mathematic model of minimizing the maximum value of linear object in MATLAB/Optimum Box.

Suppose that the constant η is an objective function, equation (35) is a constrained boundary condition for constructing optimum model. An optimum objective function is represented as follows

$\{\begin{array}{l}\text{min}\eta \\ {\mathit{P}}_m^T({\mathit{W}}_{ix}{\mathit{W}}_{ix}^T+{\mathit{W}}_{iy}{\mathit{W}}_{iy}^T-{\mu }_i^2{\mathit{W}}_{iz}{\mathit{W}}_{iz}^T){\mathit{P}}_m\le \eta \\ -{\mathit{W}}_{iz}^T{\mathit{P}}_m\le \eta ,\text{ (}i=1\text{, }2\text{, }3)\end{array}$

36

Each of elements P _m is limited in order to obtain the optimum solution of η. Let (η_o , P _o ) be the optimum solution corresponding to η and P _m . When η_o <0, grasping is stable grasping.

Substituting P _o solved from equation (36) into the formula of each contact force, the contact force ^ci f _qi can be represented and solved as follows

$\begin{array}{l}{}^c\mathit{F}_q\text{=}\left(\begin{array}{c}{}^{c1}\mathit{f}_{q1}\\ {}^{c2}\mathit{f}_{q2}\\ {}^{c3}\mathit{f}_{q3}\end{array}\right)\text{,}{}^{ci}\mathit{f}_{qi}\text{=}\left(\begin{array}{c}{f}_{qix}\\ f_{qiy}\\ f_{qiz}\end{array}\right)\text{=}{\mathit{\lambda }}_i{\mathit{W}}_i{\mathit{P}}_o\text{,}\\ \underset{3\times 3}{{\mathit{W}}_i}\text{=}\left(\begin{array}{c}{\mathit{W}}_{ix}^T\\ {\mathit{W}}_{iy}^T\\ {\mathit{W}}_{iz}^T\end{array}\right)\text{,}(i\text{=}1,2,3)\end{array}$

37

The general contact force ^c F _c is represented and solved as follows

$\begin{array}{l}\begin{array}{c}\begin{array}{l}{}^c\mathit{F}_c\text{=}{}^c\mathit{F}_p\text{+}{}^c\mathit{F}_q\text{,}\\ i\text{=1, 2, 3,}\end{array}\end{array}{}^c\mathit{F}_c\text{=}\left(\begin{array}{c}{}^{c1}\mathit{f}_{c1}\\ {}^{c2}\mathit{f}_{c2}\\ {}^{c3}\mathit{f}_{c3}\end{array}\right)\text{,}\\ {}^c\mathit{F}_p\text{=}\left(\begin{array}{c}{}^{c1}\mathit{f}_{p1}\\ {}^{c2}\mathit{f}_{p2}\\ {}^{c3}\mathit{f}_{p3}\end{array}\right)\text{,}{}^{ci}\mathit{f}_{ci}\text{=}\left(\begin{array}{c}{f}_{cix}\\ f_{ciy}\\ f_{ciz}\end{array}\right)\text{,}{}^{c\mathrm{i}}\mathit{f}_{p\mathrm{i}}\text{=}\left(\begin{array}{c}{f}_{pix}\\ f_{piy}\\ f_{piz}\end{array}\right)\end{array}$

38

The contact force ^ci f _ci satisfies equation (33); therefore, there is

$\begin{array}{l}{\left(f_{pix}+f_{qix}\right)}^2+{\left(f_{piy}+f_{qiy}\right)}^2\le {\mu }_i^2{\left(f_{piz}+f_{qiz}\right)}^2\\ f_{piz}+f_{qiz}\ge 0\end{array}$

39

After expending equation (39), it leads to

$\{\begin{array}{l}{\mu }_i^2{f}_{qiz}^2-f_{qix}^2-f_{qiy}^2\\ +2\left({\mu }_i^2{f}_{piz}{f}_{qiz}-f_{pix}{f}_{qix}-f_{piy}{f}_{qiy}\right)\\ +{\mu }_i^2{f}_{piz}^2-f_{pix}^2-f_{piy}^2\ge 0\\ f_{piz}+f_{qiz}\ge 0\text{,}(i\text{=}1,2,3)\end{array}$

40

Substituting ^ci f _qi in equation (37) into equation (40), it leads to

$\{\begin{array}{l}{C}_{ci}{\mathit{\lambda }}_i^2+2D_{ci}{\mathit{\lambda }}_i+E_{ci}\ge 0\\ F_{piz}+{\mathit{\lambda }}_i{\mathit{W}}_{iz}^T{\mathit{P}}_{\text{ο}}\ge 0\\ (i=1,2,3)\end{array}$

41

$\begin{array}{l}{C}_{ci}={\mathit{P}}_{\text{ο}}^T({\text{μ}}_i^2{\mathit{W}}_{iz}{\mathit{W}}_{iz}^T-{\mathit{W}}_{ix}{\mathit{W}}_{ix}^T-{\mathit{W}}_{iy}{\mathit{W}}_{iy}^T){\mathit{P}}_{\text{ο}}\text{,}\\ D_{ci}=({\text{μ}}_i^2{f}_{piz}{\mathit{W}}_{iz}^T-f_{pix}{\mathit{W}}_{ix}^T-f_{piy}{\mathit{W}}_{iy}^T){\mathit{P}}_{\text{ο}}\text{,}\\ E_{ci}={\text{μ}}_i^2{f}_{piz}^2-f_{pix}^2-f_{piy}^2\end{array}$

As λ_i satisfying equation (36), λ_i and λ are solved from equation (41) as follows

$\begin{array}{l}{\mathit{\lambda }}_i=\text{max}\left(\frac{-D_{ci}+\sqrt{D_{ci}^2-C_{ci}{E}_{ci}}}{C_{ci}},\frac{F_{piz}}{{\mathit{W}}_{iz}^T{\mathit{P}}_o}\right)\\ \begin{array}{c}\mathit{\lambda }=\text{max(}{\mathit{\lambda }}_1\text{, }{\mathit{\lambda }}_2\text{, }{\mathit{\lambda }}_3\text{), (}i\text{=1, 2, 3) }\end{array}\end{array}$

43

Next, the optimum grasping force ^c F _qo and the optimum contact force ^c F _co are solved from equations (36) and (42) as follows

$\begin{array}{l}{}^c\mathit{F}_{qo}=\mathit{\lambda }\mathit{W}{\mathit{P}}_o\\ {}^c\mathit{F}_{co}={}^c\mathit{F}_p+{}^c\mathit{F}_{qo}=-{\mathit{G}}^{+}{\mathit{F}}_o+\mathit{\lambda }\mathit{W}{\mathit{P}}_o\end{array}$

44

After that, the optimum statics formulae of parallel hybrid hand can be derived based on the principle of virtual work. ¹ Finally, the optimum active forces of multi-finger mechanism and the optimum active forces of PM can be solved based on the derived optimum statics formulae of parallel hybrid hand.

Solved example and verification

The geometric parameters and the input kinematics of PM and the fingers are given, see Table 1.

Table 1.

Given pose parameters, input velocity of PM, geometric parameters of finger mechanism, and the workloads exerted on fingers.

Symbol	Value (unit)	Symbol	Value (unit)
e, E	0.095, 0.136 (m)	ϕ 2	60° angle rb 2 and yb
p, r	0.1106, 0.05 (m)	ϕ 3	60° angle rb 3 and yb
d 0, d 1	0.1214, 0.039 (m)	μi	0.3
d 2, d 3	0.036, 0.082 (m)	f o	[0 0-110] T (N)
d 5, d 6	0.0169, 0.0546 (m)	t o	[0 0 0] T (N·mm)
φ, ρ	18°, 15.14°	vlj (j = 1–5)	(8, 6, 5, 4, 10)t (mm/s)
φ 1, φ 2, φ 3	180°, 60°, 60°	L 1, L 2, L 3	0.137 (m)
mg	10 (kg)	vL1, vL2, vL3	0

PM: parallel manipulator.

Based on the parameters in Table 1 and relative analytic formulae, the positions of contact point c_i (i = 1, 2, 3) in {B} are solved, see Figure 4(a). The velocities v_li of l_i in PM are solved and varied within 0–0.1 s, see Figure 4(b). The accelerations a_li of l_i in PM are solved and varied largely within 0–0.1 s, see Figure 4(b) and (c).

Figure 4.

Analytical kinematics solutions of contact point in {B} and l_j of PM. PM: parallel manipulator.

Let k be the number of iteration in optimum history. Based on analytical formulae in “Optimization of contact force” section, the optimum history of the contact force f _ci (i = 1, 2, 3) in the case k = 11 and the optimum results of contact force are solved, see Figure 5(a) and (c). The optimum results of f _ci (i = 1, 2, 3) are obtained, see Figure 5(d).

Figure 5.

Optimum history of contact force versus iterations k = 11 and optimum results of contact forces.

Let f_ciz be the normal component of the contact force f _ci . It is known from Figure 5(a) to (c) that

f_ciz (i = 1, 2, 3) are reduced as increasing k.

f_ciz satisfy constraint condition as k = 4.

As increasing k, f_ciz are reduced largely until k = 11.

As k = 11, optimum values of f_ciz are reached.

The optimum results of three components of contact force f _ci are solved, see Figure 5(d).

When the optimum results of three components of contact force f _ci are reached, the grasping impact force and power loss of parallel hybrid hand can be reduced and the damage of object or the injury of human body can be avoided.

By utilizing Solidwork, the positions of the contact point in {B} are obtained, see Figure 6(a). The velocities v_li and accelerations a_li of l_i (i = 1, 2, 3) in PM are obtained, see Figure 6(b) and (c). The contact forces before optimization are obtained, see Figure 6(d).

Figure 6.

Simulation solutions of kinematics and contact forces of contact point in {B} and l_j of PM by utilizing Solidwork. PM: parallel manipulator.

In fact, in the whole grasping processes, the multi-fingertip move, contact, and grape the object, the object is subjected a rigid collision force and is moved from static state to the moving state of the multi-finger. In order to simulate the whole grasping processes, a translational actuator is added onto the active limb of finger mechanism, and set v_lj = 0, thus the extension of linear actuator is kept as constant. Thus, it ensures that the fingertip is the same as contact point. Set object is cylinder with radius 0.05 m. The three fingertips c_i (i = 1, 2, 3) are distributed uniformly on the same circumference of the cylinder object and locate the same plane.

The absolute errors between Figures 4(a) and 6(a) are shown in Figure 7(a). The absolute errors between Figures 4(b) and (c) and 6(b) and (c) are shown in Figure 7(b) and (c). The absolute differences between the optimization solutions in Figure 5(d) and before optimization solutions in Figure 6(d) are shown in Figure 7(d).

Figure 7.

Absolute errors between analytic solutions and simulation solutions of kinematics and contact forces.

It is known from Figure 7(a)to (c) that the analytical solutions are coincident with the simulation solutions. It is known from Figure 7(d) that optimization solutions of the contact forces are less than that before optimization solutions. Therefore, the derived formulae in this article are correct.

Conclusions

A proposed parallel hybrid hand is composed of a 5-DoF PM with two equivalent composite universal joints and three 1-DoF finger mechanisms.

The coordinated grasping kinematics model of the multi-finger is established, and the formulae are derived for solving the displacement, the velocity, and the acceleration of the contact points of the object with the multi-fingertip of the parallel hybrid hand.

The optimum contact force model of the parallel hybrid hand is established, and the formulae for solving the optimum contact forces are derived.

The solved input velocity of parallel hybrid hand is varied within 0–0.1 s, and the input acceleration of parallel hybrid hand is varied largely within 0–0.1 s. The analytic solutions of the coordinated grasping kinematics and the optimum contact forces are verified by simulation solutions.

When the optimum results of contact force are reached, the grasping impact force and power loss of parallel hybrid hand can be reduced and the damage of object or the injury of human body can be avoided.

This parallel hybrid hand has potential applications for forging operation, rescue missions, industry pipe inspection, manufacturing and fixture of parallel machine tool, CT-guided surgery, health recover and training of human neck or waist, and micro–operation of biomedicine and assembly cells. The theoretical formulae and the solved results of the coordinated grasping kinematics and the optimum contact forces provide a foundation for its structure optimization, control, manufacturing, and applications.