Configuration and singularity analysis of a parallel hip joint simulator based on the forward kinematics

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Contents lists available at ScienceDirect

Applied

Mathematical

Modelling

journal homepage: www.elsevier.com/locate/apm

Conﬁguration

and

singularity

analysis

of

a

parallel

hip

joint

simulator

based

on

the

forward

kinematics

Gang

Cheng

a

_,

_Yang

_Li

a, ∗

_,

_Gabriel

_Lodewijks

b

_,

_Yusong

_Pang

b

_,

_Xianlei

_Shan

a a School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou, China

b Department of Maritime and Materials Engineering, Delft University of Technology, Delft, Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 9 December 2014 Revised 10 December 2015 Accepted 9 March 2016 Available online 25 March 2016

Keywords: Parallel manipulator Forward kinematics Singularity Conﬁguration spaces

a

b

s

t

r

a

c

t

Thesingularitiesofparallelmanipulatorsareusuallyidentifiedbygeometricalmethodsor bythekinematicprinciplesbasedontheposeparameters.Themethodshavelimitations inapplicationsthatinvolvesingularityavoidance,suchasmotionplanningfrominput pa-rameters.Toidentifyasingularityfromaninput parameterpoint ofview,whichwould makethesingularityavoidancestrategymoredirectandmoreeffectiveinpractical appli-cations,thispaperfocusesontherelationshipbetweenthesingularities,theconfiguration spaces and the inputparameters with a3SPS+1PS parallelhip joint simulatorselected toimplementthisapproach.Aunivariate-formpolynomialequationoftheforward kine-maticsisobtainedwiththeaidoftheSylvesterdialyticmethodofelimination,therefore provingthatthemanipulatorhasatmosteightconfigurationsforasingleinput.The con-figurationsarethendividedintotwotypesofspacesaccordingtotheirdistributions.Itis discoveredthatinpractice,weonlyneedconcernourselveswiththebasicconfiguration spaces,wherethesingularlocidegenerateintoasinglesurface.Finally,thesingular con-ditionisprovedtobeequivalenttothatwhentheunivariate-forminput–outputequation hasarepeatedrootintherealnumberfield.Therefore,thesingularconditionequationof theinputparametersandthesingularlociofthe inputparametersinthebasic configu-rationspacesareobtained.Thisstudyprovidesnewinsightintothesingularityavoidance ofaparallelmanipulator,especiallyforthesingularity-freedesigninthemotionplanning frominputparameters.

1. Introduction

A hip joint simulator using a 3SPS + 1PS spatial parallel manipulator as the core module was proposed [1]. This parallel simulator is used to replicate the kinematic and dynamic characteristics of a natural human hip joint to evaluate the friction and wear characteristics of the biomaterials in a hip joint prosthesis. While it overcomes the defects that arise in traditional serial simulators during complex motion simulations with large dynamic loading, it also introduces the inherent defect of all parallel manipulators (i.e., singularities), which results in uncontrolled motion and poor stiffness during certain poses.

The analysis of singularities in mechanisms is a very active research ﬁeld. A parallel manipulator may either lose or gain one or more degrees of freedom (DOF) in a singular conﬁguration [2], where the Jacobian matrix loses rank, and therefore, it loses the ability to counteract the external forces in certain directions. Researchers have performed a large amount of

∗ _{Corresponding author. Tel.: + 86 13225232379; fax: + 86 516 83591916.}

E-mail address: leeyang2012@163.com (Y. Li). http://dx.doi.org/10.1016/j.apm.2016.03.017

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work on the identiﬁcation and analysis of the singularities in parallel manipulators by means of the Jacobian matrix [3–6], the geometrical method [7–9], the kinematic method [10–14]and screw theory [15–17].

Gosselin [3]proposed a classical way to classify singularities according to the mathematical singularities of two Jacobian matrices, which deﬁne the relationship between the input and output velocities. Based on the cascaded expansion of the determinant of the Jacobian matrix, Li [4]presented an analytic form of the six-dimensional singularity locus of the general Gough–Stewart platform. Similar algebraic methods [5,6] would have diﬃculties dealing with complex mechanisms with multiple DOFs or with special mechanisms.

Merlet [7]introduced the Grassmann geometry for the singularity analysis of a parallel manipulator. The singular configurations were solved using line geometry by looking for possible actuator-line dependencies. This new method was accepted and applied by other researchers [8,9]. However, this method relies heavily on observation, and it is also hard to find all of the singular configurations or to obtain the singularity distribution.

Huang [10]proposed a kinematic method that is based on the relationship of the velocities of any three non-collinear points on the platform. All singularities of the Stewart parallel manipulator were classified into three different linear- complex singularities using this method [11]. Extending this work, Li [12]transformed the singularity analysis into a simpler position analysis of the singularity-equivalent-mechanism and obtained a simpler singularity locus equation. Similarly, Gre- gorio [13] presented a new expression of the singularity condition for the most general mechanism based on the mixed products of vectors, and he transformed the singularity condition into a ninth-degree polynomial equation that is cubic in the platform position parameters and is a sixth-degree polynomial in the platform orientation parameters. Park [14]first introduced the method of Riemannian geometry for a differential geometric analysis of the kinematic singularities for closed- chain mechanisms. The singularities were classified by configuration space, actuator and end-effector types according to the first-order properties of their kinematics.

Another important tool that has served in the analysis of singularities is screw theory. Hunt [15] laid down the general framework for applying screw theory to singularity analysis, and introduced the notion of stationary and uncertainty conﬁgurations. This method is widely used by other researchers, and similar works can be found in [16,17].

Presently, few works have dealt with the relationship between the singularities, the input parameters and the configuration spaces. However, the multiplicity of forward kinematic solutions has been identified as the most significant reason for the singularities that occur in parallel manipulators [18].

Generally, the singularity avoidance is used under three circumstances: (1) online monitoring. In this case, the input parameters are known. However, the pose parameters rely on the detection system, so in this way the singularity avoidance methods based on the input parameters are obviously more direct; (2) motion planning for the pose of a moving platform. In this case, the singularity avoidance methods based on pose parameters are more direct. However, it should be noted that for most parallel manipulators, the inverse kinematics are very simple and have only one solution. Therefore, the input parameters could easily be obtained according to the pose parameters; (3) motion planning for the input parameters (such as returning to zero). Because the forward kinematics of parallel manipulators are extremely complex, and more importantly, the solution is non-unique, which means that the actual output pose is uncertain. This is where those methods based on the pose parameters have limitations whereas the ones based on input parameters would work.

In summary, the singularity avoidance methods that are based on the input parameters would be more direct and more effective than the traditional ones that are based on the pose parameters. Moreover, the analysis of the conﬁguration spaces requires that all possible solutions of the forward position analysis should be found, and the analysis of the singular conditions based on the input parameters also relies on a closed-form model of the forward kinematics. Therefore, an analytic model of the forward kinematics is necessary to conduct the analysis of the relationship between the singularities, the conﬁguration spaces and the input parameters.

This paper is organized as follows. In Section2, the forward kinematic model is built and verified with a single higher- degree equation obtained. In Section3, the singular loci of the pose parameters, the distribution of the configuration spaces and their singular status are presented and analyzed. Section4gives the singular condition equation for the input parameters and its singular loci in the basic configuration spaces that are more meaningful in practice. Finally, Section 5follows with the conclusions of this paper.

2. Forwardkinematics

The 3SPS +1PS spatial parallel manipulator consists of a moving platform and a base connected by three surrounding SPS-type driving legs and one intermediate PS-type leg, as shown in Fig.1(a). The driving legs use electric cylinders to drive the moving platform. The intermediate leg is fixed on the base and connected to the center of the moving platform with a thrust bearing. It is used to install the artificial hip joint and balance the loading force of the hydraulic cylinder. The height is determined by the initial adjustments and remains unchanged during the working process. A_iand B_iare the fixed points of the spherical hinges on the moving platform and the base, and they are distributed as two regular triangles, with radiuses

e and E, respectively.

The ﬁxed coordinate system { b} is established on the base at its center point O with the line OB2 as the Y-axis and

the normal line of the base as the Z-axis. The moving coordinate system { m} is established on the moving platform at its center point o with the line oA2 as the y-axis and the normal line of the moving platform as the z-axis, as shown in

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Fig. 1. Prototype and topology structure of the hip joint simulator.

The driving leg length liand the intermediate leg height h can completely determine the pose of the moving platform,

which is denoted as X= ( h, q). The orientation of the moving platform is q(

α

,

β

,

γ

) and is described by RPY-type Euler Angles and is deﬁned as follows: initially { m} has the same orientation as { b}, then { m} rotates around the X-axis, the

Y-axis and the Z-axis of { b} in turn with the rotation angles

α

,

β

and

γ

, respectively.

2.1. Solutionofthemotionconstraintequation

The traditional way to build the kinematics model of a parallel manipulator is based on the position and orientation parameters of a moving platform, which is a direct and simple method for building the model. However, the solution for this type of model would be too diﬃcult due to the massive trigonometric operations. Therefore, the coordinates of the hinge points on the moving platform are used instead. Though the number of variables is increased, a more constructed equation set, which is easy for elimination, can be achieved. The procedures are as follows:

According to the geometry and motion constrains of the manipulator, we can obtain:

⎧

⎪

⎨

⎪

⎩

G1= 3 i=1 → oAi=0, G2=

|

→ oAi

|

− e=0, G3=

|

→ AiBi

|

− li=0. i=1,2,3, (1)

We substitute with the coordinates in { b} of each point and eliminate x3, y3, and z3 from G1. We then eliminate the

quadratic terms in G3from G3−G2 and make the variable substitution as follows:

⎧

⎨

⎩

m1=z1− h, m2=z2− h, n=h2₊_e2₊_E2_. (2)

Then, Eq.(1)can be simpliﬁed to a quadratic equation set with only 6 variables:

⎧

⎪

⎨

⎪

⎩

f1=x21+y12+m21− e2=0, f2=x22+y22+m22− e2=0, f3=

(

x1+x2

)

2+

(

y1+y2

)

2+

(

m1+m2

)

2− e2=0, f4=− √ 3Ex1+Ey1+2hm1− l21+n=0, f5=−2Ey2+2hm2− l22+n=0, f6= √ 3E

(

x1+x2

)

+E

(

y1+y2

)

+2h

(

m1+m2

)

+l23− n=0. (3)

Notice that f4, f5and f6are linear equations, so they can be written in the matrix form:

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where,

μ

₌[ x1, x2, y2] T; T is a 3 ×3 matrix whose elements are the scale parameters and

υ

is a 3 ×1 vector of { y1, m1, m2}.

Therefore, we can obtain

μ

in terms of { y1, m1, m2} and substitute it into f1, f2 and f3 to achieve a quadratic equation set

about { y1, m1, m2}. It should be noted that through f1 and f4 it can be discovered that, after the substitution of

μ

, f1 does

not contain m2, and thus f1, f2and f3could be written as:

f1=a1y21+b1y1+c1=0, (5)

f2=a2m22+b2m2+c2=0, (6)

f3=a3m22+b3m2+c3=0, (7)

where, ai,biand ci are the coeﬃcients of corresponding variables, and these coeﬃcients are also functions of { y1, m1}. In

particular, a1, b1and c1only concern the variable m1. We then eliminate m2 from Eqs.(6)and ( 7):

(

a2c3− a3c2

)

2−

(

a2b3− a3b2

)(

b2c3− b3c2

)

=0. (8)

It can be shown from Eqs.(6)and ( 7) that aionly contains the scale parameters, and cionly contains quadratic variables.

Therefore, Eq.(8)is a quartic equation about { y1, m1} and can be rewritten as:

k1y41+k2y31+k3y21+k4y1+k5=0, (9)

where ki is a quartic function about m1. It is noted that now Eq. (9)and Eq.(5) are functions about y1. Therefore, the

Sylvester dialytic could be used here. We multiply Eq.(9)by y1, and multiply Eq.(5) by y1, y21 and y31 to obtain 4 extra

equations. We rewrite these 4 equations, Eq.(5)and Eq.(9)in matrix form:

⎡

⎢

⎣

0 k1 k2 k3 k4 k5 k1 k2 k3 k4 k5 0 a1 b1 c1 0 0 0 0 a1 b1 c1 0 0 0 0 a1 b1 c1 0 0 0 0 a1 b1 c1

⎤

⎥

⎦

⎛

⎜

⎝

y51 y4 1 y3 1 y2 1 y1 1

⎞

⎟

⎠

=0. (10) Because

{

y5 1,y41,y31,y21,y1,1

}

T

=0, the sufficient and necessary condition for this homogeneous function set to have nonzero solutions is that the determinant of the coefficient matric equals 0. By denoting the coefficient matrix as C, we achieve a high degree equation of m1:

det

(

C

)

=0. (11)

The equation above can be expanded and calculated in detail with the help of Matlab. It is proved to be an eight- degree equation of m1, and all its solutions could be solved by scientiﬁc computing software. Thus far, the motion constraint

equations are simpliﬁed to a single input–output equation.

After m1 is solved, each element of C can be determined. Then, y1can be uniquely determined by:

y1=

0 k1 k2 k3 −k5 k1 k2 k3 k4 0 a1 b1 c1 0 0 0 a1 b1 c1 0 0 0 a1 b1 0

0 k1 k2 k3 k4 k1 k2 k3 k4 k5 a1 b1 c1 0 0 0 a1 b1 c1 0 0 0 a1 b1 c1

. (12)

Then, a_i,b_iand c_ican all be calculated, and m2can be uniquely determined from Eqs.(6)and ( 7),

μ

=[ x1, x2, y2] T by Eq.

(4)and { x3, y3, z3} by G1. At this point, the motion constraint equations constructed by the coordinates of the hinge points

are solved completely. Therefore, it can be concluded that for a set of input parameters, the moving platform has at most 8 conﬁgurations.

2.2. Attitudeanglesofamovingplatform

We use superscript m and b to distinguish the coordinates in { m} and { b} for the same point. According to the Euler Transformation,

Ab

i=bRmAmi +ob, (13)

where b_R

mis the rotation matrix for transformation from { m} to { b}:

Rm=

⎡

⎣

C

β

C

γ

S

α

S

β

S

γ

− C

α

S

γ

C

α

S

β

C

γ

+S

α

S

γ

C

β

S

γ

S

α

S

β

S

γ

+C

α

C

γ

C

α

S

β

S

γ

− S

α

C

γ

−S

β

S

α

C

β

C

α

C

β

⎤

⎦

=[u

v

w]b, (14)

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Table 1

Results of numerical example.

No. x 1 (mm) y1 (mm) z1 (mm) x2 (mm) y2 (mm) z2 (mm) α( °) β( °) γ( °) 1 −119.71 39.36 620.31 119.19 80.09 679.27 −174.6488 37.0024 127.1249 2 −112.66 −73.36 638.41 −5.08 143.70 697.71 176.6758 22.5038 −179.2502 3 −71.18 −120.37 655.64 140.23 11.91 659.51 −13.3415 23.4394 −90.5340 4 80.43 115.49 659.51 −139.83 −1.46 655.64 −14.9656 22.4724 84.7648 5 121.91 −76.25 697.71 −119.86 −60.89 638.41 −158.7722 8.3361 −59.8465 6 128.95 63.18 679.27 −025.76 −123.35 620.31 −148.6721 21.4338 0.7424

7 4.62 + 164.02i −222.00 + 3.31i 689.40 + 40.69i −189.95–84.87i 115.01–140.39i 689.40–40.69i – – – 8 4.62–164.02i −222.00–3.31i 689.40–40.69i −189.95 + 84.87i 115.01 + 140.39i 689.40 + 40.69i – – –

where S represents sin and C represents cos. In addition, b_R

mis an orthogonal matrix, and therefore:

w=u×

v

. (15)

By substituting Ab

1, Ab2 as solved in Section2.1, and the known parameters A1m, Am2 and ob into Eq.(13)and combining

the result with Eqs.(14)and ( 15), b_R

m can be obtained. According to Eq.(14), the attitude angles of the moving platform

can be determined by

⎧

⎨

⎩

β

=Atan2

(

−u3,

u2 1+u22

)

,

α

=Atan2

(

v

3/C

β

,w3/C

β

)

,

γ

=Atan2

(

u2/C

β

,u1/C

β

)

, (16)

where A tan 2 is an arctan-function determined by two parameters. Eq.(16)limits

β

to the range of [ ₋₉₀°, 90 °] to ensure that the Euler angles describing the poses of the moving platform have a one-to-one correspondence to the actual poses.

2.3.Veriﬁcationofanumericalexample

In this section, a speciﬁc numerical example is used to verify the model proposed above. The scale parameters of the manipulator are given as follows: e₌144 mm, E₌200 mm and h₌690 mm. The input parameters are set as l1=700 mm,

l2= 700 mm and l3= 790 mm. According to the methods described above, the solutions and results are obtained and listed in

Table1.

Table1shows all eight forward kinematic solutions, of which two are plural and have no physical meaning. Therefore, only six effective conﬁgurations exist. We use each of these six groups of Euler angles and the given scale parameters to analyze the inverse kinematics. The results show that all of the obtained input parameters coincide with the setting parameters, which proves the forward kinematics method proposed above to be correct.

3. Singularityandconﬁgurationanalysis

3.1. Singularlociofposeparameters

In mechanism theory, a simple and effective way to identify a singularity is the Jacobian matrix, which is the generalized transmission ratio of input and output velocities. In singular positions, the transmission performance deteriorates sharply due to rank reduction in the Jacobian matrix.

The velocity constraint of the hinge points on the moving platform is:

(

vo+

ω

×−→oAi

)

· −−→ BiAi

−−→BiAi

=_l˙_i_,_i₌₁_,₂_,₃_, ₍₁₇₎

where, vo is the linear velocity of o. Because the height of the moving platform is determined by the initial adjustments

and remains unchanged during the working process, vo=0. Therefore, the Jacobian matrix derived would be dimensionally

homogeneous. The angular velocity of the moving platform (i.e., the output velocity) is

ω

, and the telescopic velocity of the driving leg (i.e., the input velocity) is l˙ i. Therefore, the Jacobian matrix of the manipulator is:

J=

⎡

⎣

oA→1× → B1A1

B1→A1

→ oA2× → B2A2

B2→A2

→ oA3× → B3A3

B3→A3

⎤

⎦

T 3×3 , (18)

where the coordinates of o and Biare only concerned with the scale parameters, and the coordinates of Aiare determined

by Eq.(13). Therefore, the pose parameters q(

α

,

β

,

γ

) can completely determine J. The singular condition is:

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Fig. 2. Singular loci of pose parameters.

Fig. 3. Singular loci of small attitude angles. The singular loci of the pose parameters described in Eq.(19)are visualized in Fig.2.

Fig.2shows that the singular loci of the pose parameters are extremely complicated, especially when the attitude angles are large (i.e., when near the singular loci), and the singular surfaces have a prominent stacking phenomenon. This means that when the moving platform simulates motions that have large attitude angles, there is a high possibility for the system to fall into singular space.

However, it should be noted that Fig.2(b) also shows that there are no stacking phenomena of the singular surfaces in the area around the center. For a better observation, a portion of the singular loci, which concern the small attitude angles, is given in Fig.3.

Fig. 3shows that when the attitude angles fall into the range of [ ₋₅₀°, 50 °], the singular loci degenerate to a single surface. In particular, when

α

and

β

fall into the range of [ −25 °, 25 °], the corresponding singular locus for

γ

is near 0 °. This means that in the motion design of the manipulator, if the required attitude angles are small (which is a very common requirement in practice), it is feasible to avoid the singularity if the attitude angles are designed according to

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Fig. 4. Distribution of conﬁguration numbers.

Fig.3. Furthermore, if the required

α

and

β

fall into the range of [ −25°, 25 °], and

γ

has no values near 0 °, then the design will be singularity-free.

3.2.Singularityanalysisbasedoninputparameters

Based on the forward kinematics proposed above, this section aims to obtain the positional relationships among the different conﬁguration spaces and to further analyze how they affect the singularities. To access the purpose and make the results intuitional and conducive for analysis, we hold the length of one driving leg constant while allowing the other two to vary in their operational ranges. Without loss of universality, we select the driving leg A1B1 to have a constant length.

According to the motion range that the electric cylinder can reach in practice, l2and l3vary in the range [650 mm, 800 mm],

and l1 is chosen to have a constant intermediate value of 700 mm. After solving all of the solutions in this 2-D net area of

input parameters according to the forward kinematics proposed above, the distribution of the conﬁguration numbers is given in Fig.4.

Fig.4shows that while l2and l3 are both small (i.e., <725 mm), the manipulator could not be assembled, and in the rest

area, at least two configurations exist. For the largest range of input parameters, only two configurations exist. If we define the set of the adjacent configurations that correspond with the adjacent input parameters as the configuration space, then these two configurations, which have an extensive correspondence to the input parameter spaces, could be defined as the basic configuration spaces. In the small zonal area of the input parameter space, there are some complicated configuration spaces, which are therefore defined as the special configuration spaces. The superposition of the basic and special configuration spaces results in the distribution shown in Fig.4. It should be noted that all of the configuration numbers are even, which means that in the mapping from the input parameter spaces to the configuration spaces, the latter always appear in pairs.

3.2.1. Basicconﬁgurationspaces

Fig.5shows the distributions of

γ

and

α

in the basic conﬁguration spaces.

Fig.5shows that in the 2 basic conﬁguration spaces, the values of

α

are basically equal to each other while the values of

γ

are almost the opposite of each other. The values of

β

are also basically equal to each other, though they are not shown in Fig.5. This means that the 2 basic conﬁguration spaces have a similar pattern of assembly conﬁguration in which only the directions of the rotation around the intermediate leg are opposite. In addition, the values of

α

and

β

fall into the range of [ −40 °, 40 °], and therefore most conﬁgurations in this space can be realized in practice. Moreover, in Fig.5, it also can be determined that there is an intersection of the conﬁguration surfaces when l2and l3 are both small and that there is a

trend of intersection when l2 and l3are both large. In other words, in the 2 boundaries of the input parameter spaces, there

is an overlap of the conﬁgurations or at least a trend to overlap.

For the basic conﬁguration spaces, their singular status, which is measured by the reciprocal of the condition number of the Jacobian matrix (i.e., cond( J) −1), is shown in Fig.6.

When a mechanism has a singular status, the condition number of its Jacobian matrix will be infinite, and therefore the reciprocal of the condition will intuitively reflect its singular status. The smaller this value is, the worse the transmission performance will be. When the mechanism reaches a singular position, this value will be infinitely close to 0. Fig.6shows that the 2 basic configuration spaces have similar singular statuses. The manipulator has a stable transmission performance for most positions in these 2 spaces. The singularity appears in the same places where the overlap of configurations or their trend occurs in Fig.5. This proves that all of the singularities in the basic configuration spaces are forward singularities, which are caused by the overlaps produced by the forward kinematics.

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Fig. 5. Basic conﬁguration spaces.

3.2.2. Specialconﬁgurationspaces

Fig.7 shows the distribution of

α

in the special conﬁguration spaces. It is represented with a scatterplot because the curved conﬁguration surfaces are too concentrated and complicated, and the detailed analysis is unnecessary.

From Fig.7, it can be determined that in the special conﬁguration spaces, both the input and output parameters have highly limited variation ranges, which means that the manipulator will have many kinematical boundaries and result in a substantially reduced motion ability. It also should be noted that in particular, all the values of

α

fall outside the range of [ −100 °, 100 °], which means that the conﬁgurations in these spaces would be impossible to achieve in practice due to the interference of the limbs or the motion limits of the spherical hinges.

Fig.8shows the singular status of the special conﬁguration spaces.

Fig. 8 directly shows that the special configuration spaces consist of intensively distributed surfaces of configuration. However, all of the values of cond( J) −1 in these surfaces are smaller than 0.8, which has a significant distance from the values in the basic singular spaces. This means that the manipulator has a much worse transmission performance in the special configuration spaces. This result coincides with Fig.2in that when the manipulator simulates the motions that have large attitude angles, it has a high potential to fall into singularity and be damaged.

In summation, in practice we only need to concern ourselves with the basic conﬁguration spaces, where the singular status is substantially simpliﬁed. This would make singularity avoidance more practical and effective.

4. Singularconditionsbasedoninputparameters

Fig. 6shows that all of the singularities in the basic configuration spaces are forward singularities, which are caused by the overlaps in the forward kinematics solutions. To analyze whether this applies to all of the configuration spaces, the singular conditions of the manipulator based on Gosselin’s classification [2]are discussed below.

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Fig. 6. Singular status of basic conﬁguration spaces.

Fig. 7. Distribution of αin special conﬁguration spaces.

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The motion constraint equation of the manipulator is

Ab i− Bbi

T

Ab i− Bbi

− li2=0,i=1,2,3. (20)

Eq.(20)is also the constraint relationship of the output parameters q(

α

,

β

,

γ

) and the input parameters p( l1, l2, l3). We

can rewrite Eq.(20)as:

F

(

p,q

)

=

(

g1,g2,g3

)

T=0, (21)

where gi is the motion constraint equation of the corresponding leg. Differentiating Eq.(21)with respect to time obtains

the following:

Jpp˙=Jq ·

q_, (22)

where p· and q· are the input and output velocities, respectively, and

Jp=

∂

F

∂

p, Jq=

∂

F

∂

q. (23)

According to Gosselin’s classiﬁcation, the manipulator would have a singularity when either Jpor Jq becomes singular. It could be determined that for the manipulator considered in this paper:

Jp=

⎡

⎣

−2l1 0 0 0 −2l2 0 0 0 −2l3

⎤

⎦

. (24)

Therefore, the manipulator does not have either inverse singularities or composite singularities. The only possibility is a forward singularity, which is noted to be caused by the occurrence of a repeated solution in the forward position analysis [2].

It is noted that in Section 2.1, the analytic model of the forward kinematics was converted into a univariate-form polynomial equation (i.e., Eq.(11)). This provides the ability to mathematically express the singular condition of the input parameters. The singular condition can therefore be equivalent to when the input–output Eq.(11) has a repeated root in the real number ﬁeld. By denoting Eq.(11)as

λ

( m1), the singular condition of the manipulator can be written as:

λ

(

m1

)

=

λ

8m81+

λ

7m17+...+

λ

1m1+

λ

0=0,

λ

₍

_m₁

₎

₌₈

_λ8

_m7

1+7

λ7

m61+...+

λ1

=0,

(25) where

λ

i only contains the scale parameters and the input parameters. Similar to the elimination method in the forward

kinematics, the Sylvester dialytic method is applied to Eq. (25): By multiplying Eq.(25)by m6

1, m51, m41, m31, m21, and m1,

and then multiplying

λ

( m1) by m71, 13 extra equations are obtained. These extra equations combined with Eq.(25)can be

written as:

⎡

⎢

⎣

λ8

λ7

· · ·

λ2

λ1

λ0

._. . ... ... ... ...

λ

8

λ

7 · · ·

λ

0 8

λ

8 7

λ

7 · · ·

λ

1 ._. . ... ... 8

λ8

7

λ7

· · ·

λ1

⎤

⎥

⎦

15×15

⎛

⎜

⎝

m14 1 m13 1 . . . m2 1 m1 1

⎞

⎟

⎠

=0. (26)

Similarly, the sufficient and necessary condition for Eq. (26)to have nonzero solutions is that the determinant of the coefficient matrix is equals to 0. By denoting the coefficient matrix as D, then the singular condition equation for the input parameters is expressed as:

det

(

D

)

=0. (27)

By detecting and controlling the input parameters according to Eq.(27)in the actual operation phase, the manipulator could avoid falling into singularities in a more direct and effective way. In motion planning from the input parameters, Eq.(27)can be applied to singularity-free path design.

Eq.(27)would be overly complex after expansion because it contains the singularities of all of the configuration spaces. However, in practice, the manipulator usually only operates in the basic configuration spaces, and therefore, the singular conditions of the input parameters in the basic configuration spaces are more meaningful. Fig.3 shows the singular loci of the attitude angles in the interval of [ −50 °, 50 °], which almost includes all of the general attitude simulation angles. Therefore, it is feasible to obtain the singular loci of the input parameters in the basic configuration spaces by the inverse kinematics of each point in Fig.3. The results are shown in Fig.9.

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Fig. 9. Singular loci of input parameters in basic conﬁguration spaces.

Compared with Eq.(27), the singular loci of the input parameters given in Fig.9are much simpler and more effective in practice.

5. Conclusion

This paper focused on the forward kinematics, the configuration spaces, and the singularities of a 3SPS +1PS parallel hip joint simulator. This work is constructive as a new method for singularity avoidance in parallel manipulators based on the input parameters. First, the forward kinematic model is built with a single input–output equation of a higher degree. The singular loci of the pose parameters are obtained according to the rank reduction of the Jacobian matrix. It is determined that when the attitude angles are larger than certain values, the singular loci are extremely complex. As the attitude angles become smaller, the singular loci degenerate into a single surface. Then, the positional relationships among the different configuration spaces and how they will affect the singularities are analyzed. The configurations are divided into two types of spaces according to their distributions, and it is determined that in practice we only need to concern ourselves with the basic configuration spaces. Finally, the singular condition of the manipulator is proved to be equivalent to when the univariate-form input–output equation contains a repeated root in the real number field. Therefore, the singular condition equation of the input parameters and the singular loci of the input parameters in the basic configuration spaces are obtained. Overall, the method proposed is applicable to other types of mechanisms if the mechanism could be proved to only possess forward singularities and if the analytic model of the forward kinematics could be converted into a univariate-form polynomial equation. This work provides a new method for singularity avoidance in a parallel manipulator, especially for the singularity-free design in motion planning from the input parameters.

Acknowledgment

Financial support for this work, provided by the National Natural Science Foundation of China (Grant no. 51275512), the Foundation Research Project of Jiangsu Province (Natural Science Foundation) (BK20141128), the Priority Academic Pro- gram Development of Jiangsu Higher Education Institutions and the Fundamental Research Funds for the Central Universities (Grant no. 2013RC07), are gratefully acknowledged.

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