## 1 Introduction

## 2 Methods for Force Analysis of Passive Overconstrained PMs

_{ υ }, B

_{ υ }, C

_{ υ }, …, denote the joints of the υth (υ = 1, 2, …, t) supporting limb from the moving platform to the base in sequence. Assume that the friction in the kinematic joints is ignored, and the stiffness of the moving platform is much greater than that of the supporting limbs.

### 2.1 Traditional Method

_{ υ }(υ = 1, 2, …, t) and t

_{ υ }represent the three-dimensional reaction force and moment vectors of joint A

_{ υ }connecting the moving platform and the υth limb, respectively, which are expressed in the local coordinate system {o

_{ υ }} of the υ-th limb, \({}_{o\upsilon }^{O} {\varvec{R}}\) is the rotational transformation matrix of {o

_{ υ }} with respect to {O}, g is the gravity vector expressed in the global coordinate system, \({}_{g}^{O} {\varvec{R}}\) is the rotational transformation matrix of the global system with respect to {O}, m

_{ O }is the mass of the moving platform, r

_{ Oυ }is the position vector from origin O to the center of joint A

_{ υ }expressed in {O}, and h

_{ O }and n

_{ O }denote the inertia force and moment vectors of the moving platform expressed in {O}, respectively.

_{ υ }B

_{ υ }close to the moving platform in the υth limb can be built as

_{ Bυ }and t

_{ Bυ }represent the three-dimensional reaction force and moment vectors of joint B

_{ υ }, respectively, \({}_{g}^{o\upsilon } {\varvec{R}}\) is the rotational transformation matrix of the global system with respect to {o

_{ υ }}, m

_{ υ1}is the mass of the link A

_{ υ }B

_{ υ }, r

_{ oA }and r

_{ oB }are the position vectors from the origin o

_{ υ }to the centers of the joints A

_{ υ }and B

_{ υ }, respectively, and h

_{ υ1}and n

_{ υ1}denote the inertia force and moment vectors of the link A

_{ υ }B

_{ υ }, respectively. f

_{ Bυ }, t

_{ Bυ }, r

_{ oA }, r

_{ oB }, h

_{ υ1}and n

_{ υ1}are expressed in the local coordinate system {o

_{ υ }}.

_{ u,υ }and δ

_{ u,υ+1}denote the linear deformations generated at the ends of the υth and the (υ + 1)th limbs in the axis of the uth redundant constraint force, respectively, and ψ

_{ v,υ }and ψ

_{ v,υ+1}represent the angular deformations generated at the ends of the υth and (υ + 1)th limbs in the axis of the vth redundant constraint moment, respectively.

### 2.2 Method Based on the Judgment of Constraint Jacobian Matrix

_{1}, q

_{2}, …, q

_{ N }denote the N coordinates.

### 2.3 Method under the Condition of Decoupled Deformations

_{ υ }driving forces/torques and constraint forces/moments in total, as shown in Fig. 1, the elastic deformations generated at the end of the υth limb by the N

_{ υ }driving forces/torques and constraint forces/moments are considered to be decoupled to each other [53‐56]. In this case, the stiffness of each supporting limb can be expressed as a scalar quantity or a diagonal matrix. The steps of this method can be summarized as follows:

_{ j }be the stiffness between the jth driving force/torque or constraint force/moment and the elastic deformation generated at the end of the corresponding limb under the action of the jth driving force/torque or constraint force/moment. There exists

_{ j }denotes the magnitude of the jth driving force/torque or constraint force/moment, and δ

_{ j }represents the elastic deformation generated at the end of the corresponding limb by f

_{ j }.

### 2.4 Method Based on Resultant Constraint Wrenches

_{ i }, f

_{ k }, f

_{ p }and f

_{ q }represent the magnitudes of the ith actuation wrench, the kth independent constraint wrench, the resultant constraint wrench and the resultant constraint couple, respectively. All screws are expressed in the global system.

_{ b }is non-singular, the magnitudes of the actuation wrenches, the independent constraint wrenches, the resultant constraint wrench, and the resultant constraint couple can be solved from Eq. (13) as

_{ γ }, and that of the (λ + 1)th and the λth supporting limbs with coaxial constraint moments is η

_{ λ }. In view that the constraint forces and moments are in direct proportion to the stiffness of the corresponding limbs, the complementary equations can be given as

_{ p,γ }and f

_{ q,λ }are the magnitudes of the γth collinear constraint force and the λth collinear constraint moment, respectively.

### 2.5 Method Based on the Stiffness Matrix of Limb’s Overconstraint or Constraint Wrenches

_{a,i }, f

_{r,ε }, and \(f_{{{\text{r,}}\sigma }}^{\text{e}}\) are the magnitudes of the ith actuation wrench, εth non-overconstraint wrench, and σth equivalent constraint wrench, respectively. Details about the non-overconstraint wrenches, overconstraint wrenches, and equivalent ones of overconstraint wrenches are given in Ref. [58].

_{e}is the vector composed of the elastic deformations in the axes of equivalent constraint wrenches.

_{ υ }constraint wrenches (including actuation wrenches) to the moving platform, the force and moment equilibrium equations of the moving platform can be expressed as

### 2.6 Weighted Generalized Inverse Method

## 3 Methods for Force Analysis of Active Overconstrained PMs

_{ υ }links. Without loss of generality, each limb can possess more than one actuator.

### 3.1 Pseudo-inverse Method

_{act}consists of ζ driving forces/torques, G

_{act}is the coefficient matrix, and F

_{extr}is the generalized external force vector composed of inertia forces/moments, weight, and external loads encountered in the components of the mechanism.

_{act}is singular, the pseudo-inverse of G

_{act}is used to find the minimum norm of f

_{act}in some situations [27, 30, 72]:

### 3.2 Weighted Coefficient Method

_{non}is composed of the driving forces/torques of the generalized joints, i.e., the n non-redundant driving forces/torques, and τ

_{over}consists of the driving forces/torques of the non-generalized joints, i.e., the remaining (ζ–n) redundant driving forces/torques. \({\not\!{\varvec{S}}}_{F,h}^{\upsilon }\) and \({\not\!{\varvec{S}}}_{F}\) denote the resultant force vectors of the external loads, gravity, and inertia force/moment acted on the h-link of the υth limb and the moving platform, respectively. They are expressed in the corresponding local coordinates. \(\left( {{\varvec{G}}_{h}^{\text{non}} } \right)^{\upsilon }\), \({\varvec{G}}_{F}^{\text{non}}\), and \({\varvec{G}}_{\text{over}}^{\text{non}}\) represent the transformation matrices of \({\not\!{\varvec{S}}}_{F,h}^{\upsilon }\), \({\not\!{\varvec{S}}}_{F,h}^{\upsilon }\), and τ

_{over}from the corresponding local coordinates to the generalized coordinates, respectively.

_{ i }and W

_{ ξ }(ξ = n+1, n + 2, …, ζ) are weighted coefficients.

_{non}and W

_{over}, for example:

_{ i }and W

_{ ξ }be the velocities of the ith and ξth actuated joints, respectively. The driving forces/torques are distributed with the minimum input energy of the actuators [36, 66, 67].

### 3.3 Method Based on the Optimal Internal Forces

_{spcl}satisfies

_{homo}meets

_{act}, which are known as the internal forces. Hence, two kinds of methods are proposed for the statically indeterminate problem of active overconstrained PMs from the perspective of dealing with the internal forces. The first method is that the driving forces/torques are distributed without internal forces [18, 35, 70]. A number of studies have shown that the internal forces can be utilized to change the stiffness [79], improve the motion accuracy [32], increase the load-carrying capacity [19], and eliminate the backlash [34] of active overconstrained PMs, so the second method is that the driving forces/torques are distributed by utilizing the advantages of internal forces [19, 32, 34].