Kinematic analysis of MacPherson strut …umts.iyte.edu.tr/wp-content/uploads/2015/06/66.pdfKinematic analysis of MacPherson strut suspension system Madhu Kodati K. Vikranth Reddyy

TrC-IFToMM Symposium on Theory of Machines and Mechanisms, Izmir, Turkey, June 14-17, 2015

Kinematic analysis of MacPherson strut suspension system

Madhu Kodati∗ K. Vikranth Reddy† Sandipan Bandyopadhyay‡

IIT Madras IIT Madras IIT MadrasChennai, India Chennai, India Chennai, India

Abstract—The MacPherson strut (MPS) suspension sys-tem is one of the most common independent automobile sus-pensions. This paper presents a kinematic analysis for thecomplete spatial model of the same. The presented solu-tion is built upon two key elements in the formulation andsolution stages, respectively: use of Rodrigue’s parametersto develop an algebraic set of equations representing thekinematics, and computation of Grobner basis as a methodof solving the resulting set of polynomial equations. It isfound that the final univariate equation representing all thekinematic solutions for a given pair of steering and roadprofile inputs is of 64 degree - which reflects the complex-ity observed in the kinematics of the mechanism. The realroots of this polynomial are extracted with high numericalprecision and are validated. Finally, they are used to findthe configuration of the mechanism for a particular set ofroad and steering inputs.

Keywords: MacPherson strut, Spatial kinematics, Polynomialequations, Grobner basis, Rodrigue’s parameters

I. Introduction

MacPherson strut suspensions are among the simplestand most commonly used suspension systems globally, inpassenger cars. American automotive engineer Earle S.MacPherson developed the MPS using a strut configura-tion [1]. Solid model of the suspension system is shownin Fig. 1. One end of the strut is attached to the knucklewith a prismatic joint and other end is fixed to the chassisby a spherical joint. The strut also contains the spring anddamping elements. The lower arm is connected to chassisby a revolute joint, and the other end is connected to theknuckle by a spherical joint. The steering link is attachedto the tie-rod using spherical joints and the tie-rod is con-nected to the rack with a universal joint. Though the MPShas less favourable kinematic characteristics compared to adouble wish-bone suspension [2], it is compact enough tobe compatible with the transversely mounted engines, andis thus used widely for front-wheel-drive cars.

Numerous techniques for the kinematic analysis of MPSsuspension exist in literature, a prominent one being [1].Modelling the MPS as a spatial mechanism, linear and non-linear position analysis was done based on vector algebra.

∗[email protected]†[email protected]‡[email protected]

King-pin axis

Lower A-arm

Tie-rod

Steering link

Knuckle

Rack

Strut

Fig. 1. Simplified solid model of the MacPherson strut

Later, H.A. Attia [3] presented a formulation in terms ofCartesian coordinates of some defined points in the linksand at the kinematic joints, whereas, D.A. Mantaras etal. [4] formed the constraint equations using Euler parame-ters.

The above methodologies implemented the Newton-Raphson method subsequently to solve the non-linear equa-tions for the unknown variables. However, it is well-knownthat this method has issues regarding not converging to asolution in the absence of a “good” initial guess solutionand its inability to capture all the possible solutions. Ad-dressing this, Y.A. Papegay et al. [5] tried to solve the MPSsuspension kinematics problem using symbolic computa-tion techniques. However, exact solution in symbolic formby employing Grobner basis on constraint equations wasreported to be unachievable.

The design of the MPS involves a relatively large numberof elements, and thus has a reasonably large design space.This affords better control over the kinematic response ofthe suspension. However, for the same reason, a significantdifficulty presents itself in any kinematic/dynamic study ofsuch systems: the complexity of their kinematics. As amechanism, the MPS possesses two degrees-of-freedom:one accounting for the steering input s(t), and the otheraccommodating jounce and rebound, the road profile in-put y(t)1.

1The steering input and road profile input are denoted henceforth by s

1


Finding the output or response of the MPS to these in-puts, e.g., determining the location and orientation of thekingpin axis (KPA) for a combination of s and y, whichis important for understanding the ride and handling char-acteristics, driving behaviour [2], is a formidable task.Once found, these kinematic solutions give greater accu-racy when used to study different design configurations,and optimization of the architecture parameters of the sus-pension mechanism with performance indices like camber,caster and toe as objectives.

This paper presents a kinematic model and analysis of theMPS suspension which follows a similar treatment of thedouble-wishbone suspension system presented in [6]. Thekinematic equations of the MPS are formulated in an al-gebraic manner by representing the orientation of the KPAin terms of Rodrigue’s parameters c = (c1, c2, c3)

> (seee.g., [7]). Retaining the symbolic form of these equationsand eliminating the unknowns which occur as linear terms,leads to a system of three quartic equations in the threeRodrigue’s parameters. Attempts to reduce this system ofequations in symbolic form to an univariate of degree 64 inone of the three parameters (c1, c2, c3), were unsuccessful.The equations are therefore solved by substituting the nu-merical values for the architecture parameters of the MPSalong with the inputs s and y. The solution technique in-volves finding the Grobner basis of the ideal generated bythe three equations and extracting the roots of final univari-ate with a high numerical precision. The real roots of thepolynomials are used to completely determine the config-uration of the suspension. The obtained results are vali-dated numerically to ascertain the correctness of the pro-posed method.

The rest of the paper is organised as follows: In Sec-tion II, formulation of the loop-closure equations, elimina-tion of the variables for two possible cases and methodol-ogy for finding solutions are presented. Finally, the con-figurations of the suspension for real solutions of the loop-closure equations are depicted graphically. In Section III,the conclusions are presented.

II. Kinematic analysis of MacPherson strut suspension

The eventual success in solving any position kinematicsproblem relies upon an appropriate kinematic modelling,efficient formulation, as well as effective solution tech-niques. The details of these, as applied to the present prob-lem, are presented in this section under the following mod-elling assumptions:• The cushioning effect of the bushes at the joints are notconsidered, i.e., only ideal joints are considered in the for-mulation.• The flexibility of the links are not considered, only rigidlinks are used in the formulation.

and y respectively, by dropping the explicit time dependence.

• The chassis is considered to be the fixed or the groundlink in the mechanism.• All the geometric parameters of the suspension mecha-nism are exactly known.• The universal joint between the tie-rod and the rack isreplaced with a kinematically equivalent spherical joint.

A. Geometry of the MPS

The schematic of the MPS suspension is shown in Fig. 2.Kinematically, it is a combination of a spatial four-bar loopo0p1o1o0 (which is an inverted slider mechanism that haschassis as the ground link and king-pin as the coupler), anda spatial five-bar loop o0p1p3p4p8o0. These two loops arecoupled at the king-pin (link 2© in Fig. 2).

Two coordinate systems are used to describe the config-uration of the MPS. The global frame of reference {0} isattached to o0, and its Z0-axis is aligned to the axis of thehinges of the lower A-arm. The lower A-arm is equiva-lently represented as a single link (link 1© in Fig. 2) whichmoves in the X0Y0-plane. The prismatic pair between p1

and o1 is considered to be a single link of variable length l2.A body-fixed frame of reference, {1}, is attached to p1.The X-axis of {1} (denoted by X1) is along the link vectorl2 = o1 − p1, and X1Y1-plane contains link 3©.

Fig. 2. Schematic of the MacPherson strut suspension mechanism

Any vector in the body-fixed frame of reference canbe transformed to the global frame of reference by pre-multiplying with an appropriate rotation matrix R ∈SO(3) (see, e.g., [8]). The rotation matrix 0

1R relates theorientation of {1} w.r.t. {0} i.e., the orientation of KPAw.r.t. the global frame of reference. In the presented for-

2


mulation, it is expressed in terms of the Rodrigue’s param-eters, c = (c1, c2, c3)

> ∈ IR3. This is a key step in thekinematic modelling, as this parametrisation leads to an al-gebraic description of the orientation of the KPA, whichin turn, helps in formulating the loop-closure equations inalgebraic terms, as explained in the next subsection.

B. Formulation of the loop-closure equations

For brevity, all vectors expressed in the global frame ofreference {0} are written henceforth without the leadingsuper-script ‘0’; e.g., as p, l instead of 0p, 0l respectively.The rack/steering input, s, is given at the point p8, and theroad profile input, y, is applied to the point p7 (see Fig. 2).The end points of link 2© can be expressed as:

p1 =o0 +RZ0(θ)[l1, 0, 0]>, and (1)

o1 =[o1x, o1y, o1z]>, (2)

respectively. The vector l2 can be expressed as:

l2 =o1 − p1, or (3)

l2 =01R[l2, 0, 0]

>. (4)

From Eqs. (1-4), one can write:

o1 − o0 −RZ0(θ)[l1, 0, 0]

> − 01R[l2, 0, 0]

> = 0. (5)

Eq. (5) models the kinematics of the four-bar loopo0p1o1o0. The expression on the left hand side of Eq. (5)is of the form:

η1 = (η1x, η1y, η1z)>, (6)

where η1x, η1y and η1z are given by:

η1x =− l2(c21 − c22 − c23 + 1

)− c∆l1 cos θ + c∆o1x,

(7)

η1y =− 2l2(c1c2 + c3)− c∆l1 sin θ + c∆o1y, (8)η1z =− 2l2(c1c3 − c2) + c∆o1z, where (9)

c∆ =1 + c21 + c22 + c23. (10)

The end points of link 4© can be expressed as:

p3 =p1 +01R

1l3, and (11)

p4 =[p4x + sx, p4y + sy, p4z + sz]>, (12)

where 1l3 = (l3x, l3y, l3z)> is the vector from p1 to 1p3

(expressed in {1}) and s = (sx, sy, sz)> is the rack input

vector in {0}, given by:

s =02R[s, 0, 0]>, where

02R =RZ(β1)RY (β2)RX(β3).

The vector l4 can be expressed as:

l4 = p4 − p3.

The loop-closure constraints of the five-bar loop o0p1p3p4p8o0

can be reduced to the following equation:

(p4 − p3) · (p4 − p3)− l24 = 0. (13)

The expression on the left hand side of Eq. (13) is de-noted by η2. From the road input to the lower A-arm loop,o0p1p5p7o0, one can write:

[p7x + x, p7y + y, p7z + z]> − 01R

1l5

−01R[r2, 0, 0]

> −RZ0(θ)[l1, 0, 0]

> − o0 = 0, (14)

where x, z are lateral and longitudinal displacements of p7

w.r.t global frame of reference, which vary upon changing sand y; 1l5 = (l5x, l5y, l5z)

> is the vector from 1p5 to 1p7

(expressed in {1}). The parameter r2 is the distance be-tween p1 and p5. The expression on the left hand side ofEq. (14) is of the form:

η3 = (η3x, η3y, η3z)>, (15)

where η3x, η3y and η3z are given by:

η3x =− l5x(c21 − c22 − c23 + 1

)− 2l5y(c1c2 − c3)

− 2l5z(c1c3 + c2)− r2

(c21 − c22 − c23 + 1

)− c∆l1 cos θ

+ c∆p7x + c∆x, (16)

η3y =− 2l5x(c1c2 + c3)− l5y(−c21 + c22 − c23 + 1

)− 2l5z(c2c3 − c1)− 2r2(c1c2 + c3)− c∆l1 sin θ+ c∆p7y + c∆y, (17)

η3z =− l5z(−c21 − c22 + c23 + 1

)− 2l5x(c1c3 − c2)

− 2l5y(c1 + c2c3)− 2r2(c1c3 − c2) + c∆p7z + c∆z.(18)

Eqs. (7-9), Eq. (13) and Eqs. (16-18) together define thekinematics of the MPS suspension system.

TABLE I. Structure of the loop-closure constraints of the MPS suspensionmechanism

Equation LHS Variables Size (KB)Eq. (7) η1x c1, c2, c3, l2, cos θ 1.031Eq. (8) η1y c1, c2, c3, l2, sin θ 0.703Eq. (9) η1z c2, c1, c3, l2 0.687Eq. (13) η2 c1, c2, c3, cos θ, sin θ 15.937Eq. (16) η3x c1, c2, c3, x, cos θ 3.148Eq. (17) η3y c1, c2, c3, sin θ 2.820Eq. (18) η3z c1, c2, c3, z 2.805

C. Elimination of variables

The variables included in, and the sizes2 of the equationsobtained in the Section II-B are summarised in the Table I.

2All the symbolic computations have been performed using the commer-cial computer algebra software, Mathematica [9]. The “size”, in this con-text, refers to the amount of computer memory needed to represent/storean expression in Mathematica’s internal format.

3


The sequence of elimination of variables needs to be cho-sen so as to reduce the complexity of the resulting equa-tions. To begin with, it is to be noted that the loop-closureequations are linear in terms of sine and cosines of the an-gle θ and in l2. Discussed below are the two possible casesof kinematic solutions.

C.1 Case-1: (c2 − c1c3) 6= 0

The above-mentioned variables are eliminated in a se-quential manner, as shown schematically in Fig. 3.

Fig. 3. Sequence of elimination of cos θ, sin θ and l2 from Eq. (7-9),Eq. (13) & Eq. (17) for Case-1

First, l2 is computed from Eq. (9) and substituted inEq. (7) and Eq. (8). The functions η1x and η1y are lin-ear in terms of cos θ and sin θ. Hence Eq. (7) and Eq. (8)are solved simultaneously to obtain cos θ and sin θ, respec-tively. The eliminated variables, l2, cos θ and sin θ arefound to be:

l2 =−c∆o1z

2 (c2 − c1c3), (19)

cos θ =−2c3c1o1x + 2c2o1x + c21o1z − c22o1z − c23o1z + o1z

2 (c2 − c1c3) l1,

(20)

sin θ =c2o1y − c1c3o1y + c1c2o1z + c3o1z

(c2 − c1c3) l1. (21)

It is to be noted that the denominators in the expressionsfor l2, cos θ and sin θ do not vanish as per the stated as-sumption. The variable θ is eliminated using the identitysin2 θ + cos2 θ − 1 = 0, which gives rise to the equa-tion f1(c1, c2, c3) = 0. Note that f1 is free of any otherunknowns than those mentioned explicitly. Similarly, theexpressions for sin θ and cos θ are substituted in Eq. (13),which leads to the equation f2(c1, c2, c3) = 0. Finally, Theexpression for sin θ is substituted in Eq. (17), to obtain theequation f3(c1, c2, c3) = 0.

The functions f1, f2, f3 form a system of three quarticequations in the unknowns c1, c2, c3. They may be written

compactly as:

f1(c1, c2, c3) =

4∑i=0

4∑j=0

4∑k=0

uijkci1c

j2c

k3 , (22)

f2(c1, c2, c3) =

4∑i=0

4∑j=0

4∑k=0

vijkci1c

j2c

k3 , (23)

f3(c1, c2, c3) =

4∑i=0

4∑j=0

4∑k=0

wijkci1c

j2c

k3 ; (24)

i+ j + k ≤ 4.

The coefficients uijk, vijk, and wijk in Eqs. (22-24) arefunctions of the architecture parameters of the MPS. Theseare obtained symbolically using Mathematica3, andsimplified to their monomial-based canonical forms [10].The sizes of the polynomials after simplification are5.961 KB, 32.672 KB, and 5.844 KB respectively.

C.2 Case-2: (c2 − c1c3) = 0

In this case, Eq. (9) becomes

o1z = 0, as c∆ 6= 0, (25)

which is physically admissible. The variables l2, sin θand cos θ are eliminated in a different manner in this case,which is shown schematically in Fig. (4).

Fig. 4. Sequence of elimination of cos θ, sin θ and l2 from Eq. (7, 8),Eq. (13) & Eq. (17) for Case-2

The functions g1, g2 form a system of equations in the

3All the computations in this paper were done in Mathematica version-9.0.0 for 64-bit Linux O.S. on a system with 6-core Intel Core i7-4930KCPU @ 3.40 GHz clock speed and 64 GB of RAM.

4


unknowns c1, c3 and are written compactly as:

g1(c1, c3) =

4∑i=0

6∑j=0

mijci1c

j3, (26)

g2(c1, c3) =

2∑i=0

4∑j=0

nijci1c

j3. (27)

The coefficients mij , nij in Eqs. (26, 27) are again func-tions of the architecture parameters of the MPS. These co-efficients are obtained symbolically using Mathematica,and are simplified to their monomial-based canonicalforms. The sizes of the polynomials after simplification are24.305 KB and 19.305 KB respectively.

D. Solutions for system of quartic equations

D.1 Case-1: (c2 − c1c3) 6= 0

It would be ideal indeed to solve Eqs. (22-24) for themost general case, i.e., keeping all of the coefficients intheir symbolic forms. However, attempts to eliminate twoof the three remaining variables to obtain a univariate inthe third variable in terms of symbolic coefficients did notsucceed. Hence these equations are solved for given nu-merical instances of design parameters (given in Table II),and inputs (s, y). The solution procedure used, however,

TABLE II. Architecture parameter values of the MPS (base data is takenfrom [1] and transformed to match the given reference frame)

Parameter Symbol Value4

Length of lower A-arm l1 0.322Length of tie rod l4 0.361Link-3 vector w.r.t. {1} (l3x, l3y , l3z) (0.106, 0.108, 0)Position of origin of {0} o0 (0, 0, 0)Position of strut mounting o1 (0.136, 0.497, 0.011)Initial position of p4 (p4x, p4y , p4z) (-0.078, 0.105, -0.108)Euler angles relating {2} to {0} (β1, β2, β3) (-1.534, 0.019, 0.072)Initial position of p7 (p7x, p7y , p7z) (0.434, -0.333, -0.019)Link-5 vector w.r.t. {1} (l5x, l5y , l5z) (-0.505, 0.009, 0.054)Distance between p1 and p2 r1 0.106Distance between p1 and p5 r2 0.061

is semi-analytical in nature, and involves the computationof the Grobner basis (see, e.g., [11]) generated by theideal F = 〈f1, f2, f3〉. This prevents the degree-explosion,which happens otherwise, if pair-wise resultants are com-puted. The details of this final stage of elimination and so-lution are presented in the rest of this section. The sampleinputs considered for all numerical computations to explainthe methodology are s = 50mm and y = 50mm.

After substitution of the numerical values as per the Ta-ble II, the coefficients uijk, vijk, and wijk in Eqs. (22-24) are found in their numerical forms with defaultMachinePrecision in Mathematica. The ideal Fis defined in SINGULAR [12] under a ring of real coeffi-cients with a numerical precision of 200 and lexicographi-cal monomial ordering on c1, c2, c3, i.e., c1 � c2 � c3. The

4Positions, lengths are given in metres and angles are given in radians.

Grobner basis of F consists of three polynomials: h1(c3),h2(c2, c3), and h3(c1, c3). Among these, h1 is a univariatepolynomial in c3 of degree 64:

h1 =a0c643 + a1c

633 + · · ·+ a64 = 0. (28)

Each of ais in Eq. (28) are floating-point numbers with 200significant digits. The polynomials h2, h3 are linear in c2and c1 respectively, and of degree 63 in c3.

Extraction of the real roots of the polynomial h1 usingthe SINGULAR routine laguerre_solve (for which,the internal computation precision as well as the output pre-cision is again set to 200) yields c3 = 1.436, 1.556. Thesereal solutions for c3 are substituted back in h2 and h3 toobtain the corresponding real solutions for c2 and c1: (c1,c2)= (-1.678, -2.599) and (0.879, 1.289).

The solutions of c1, c2 and c3 obtained above are back-substituted in Eqs. (7-9) and Eqs. (16-18) to compute thecorresponding solutions for θ, l2, x and z which are givenin Table III.

TABLE III. Values of θ, x and z for s = 50mm, y = 50mm for Case-1(c1, c2, c3) θ(rad) x(mm) z(mm) l2(mm)

(-1.678, -2.599, 1.436) 0.451 -6.517 -30.094 367.407(0.879, 1.289, 1.556) 0.332 94.594 26.519 403.550

The configurations of the mechanism for real solutionsof c1, c2 and c3 for various inputs are depicted in Figs. 5 -8, and the residual values of initial loop-closure equationsare listed in the Table IV, which establish the numericalaccuracy of the solutions obtained.

D.2 Case-2: (c2 − c1c3) = 0

Eq. (26) and Eq. (27) are solved following the same pro-cedure i.e., by defining the ideal G = 〈g1, g2〉 and findingthe Grobner basis of G, which consists of two polynomi-als: t1(c3) and t2(c1, c3). Among these, t1 is a univariatepolynomial in c3 of degree 28:

t1 =b0c283 + b1c

273 + · · ·+ b28 = 0. (29)

Each of bis in Eq. (29) are floating-point numbers with 200significant digits. The polynomial t2 is linear in c1 and ofdegree 27 in c3.

TABLE IV. Residuals of Eq. (5), Eq. (13) and Eq. (14) for Case - 1S.No s(mm) y(mm) (c1, c2, c3) Residuals×10−11

‖η1‖ |η2| ‖η3‖1 0 -100 (-0.958, -1.417, 1.411) 4.360 330.553 0.017

(0.491, 0.663, 1.430) 2.933 121.879 0.1312 50 50 (-1.678, -2.599, 1.436) 1.694 116.129 0.716

(0.879, 1.289, 1.556) 2.676 202.719 0.9693 0 60 (-1.044, -1.632, 1.459) 2.063 152.210 0.978

(0.595, 0.858, 1.548) 1.894 101.232 0.7674 50 -100 (-1.615, -2.358, 1.385) 3.477 245.735 0.205

(0.772, 1.057, 1.434) 4.563 314.927 0.460

5


(a) c = (−1.678,−2.599, 1.436)>

(b) c = (0.879, 1.289, 1.556)>

Fig. 5. Configurations corresponding to the real values of c1, c2, c3 fors = 50mm, y = 50mm for Case-1

Extraction of the real roots of the polynomial t1in SINGULAR yields c3 = 1.518, 1.520. These real solu-tions for c3 are substituted back in t2 to obtain correspond-ing real solutions for c1 as 0.861,−1.774. Finally, the cor-responding values for c2 are obtained using c2 = c1c3 as1.307,−2.696 respectively.

The values of c1, c2 and c3 are back-substituted inEqs. (7-8) and Eqs. (16-18) to compute the correspondingsolutions for θ, l2, x and z which are given in Table V.

TABLE V. Values of θ, x and z for s = 50mm, y = 50mm for Case-2(c1, c2, c3) θ(rad) x(mm) z(mm) l2(mm)

(0.861, 1.307, 1.518) 0.330 94.924 37.558 427.958(-1.774, -2.696, 1.520) 0.455 -6.553 -16.923 387.864

The configurations of the mechanism for real solutionsof c1, c2 and c3 for various inputs are depicted in Fig. 9-12,and the residual values of initial loop-closure equations arelisted in the Table VI.

(a) c = (−0.958,−1.417, 1.411)>

(b) c = (0.491, 0.663, 1.430)>

Fig. 6. Configurations corresponding to the real values of c1, c2, c3 fors = 0mm, y = −100mm for Case-1

TABLE VI. Residuals of Eq. (5), Eq. (13) and Eq. (14) for Case - 2S.No s(mm) y(mm) (c1, c2, c3) Residuals×10−13

‖η1‖ |η2| ‖η3‖1 0 -100 (0.484, 0.684, 1.414) 1.172 157.230 0.568

(-0.981, -1.417, 1.444) 1.271 1345.500 0.5682 50 50 (0.861, 1.307, 1.518) 0.568 202.367 0.036

(-1.774, -2.696, 1.520) 0.636 339.277 0.5683 0 60 (-1.097, -1.661, 1.514) 0.568 1444.240 0.568

(0.577, 0.878, 1.521) 0.568 527.021 0.5684 50 -100 (0.767, 1.081, 1.410) 1.137 368.108 0.604

(-1.662, -2.394, 1.441) 1.271 402.446 0.568

III. Conclusions

This paper presents a complete study of the position kine-matics of the MacPherson suspension system with the for-mulation based on algebraic parametrisation of the orien-tation of the KPA. Two elimination schemes are presentedbased on the relationship between the orientation param-eters, following which, the initial set of five simultane-ous algebraic-trigonometric loop-closure equations reducesto system of three and two polynomial equations. Thesetwo sets of polynomials are solved separately by finding

6


(a) c = (−1.044,−1.632, 1.459)>

(b) c = (0.595, 0.858, 1.548)>


Grobner basis of the ideal formed by the respective set af-ter substituting numerical values for the architecture pa-rameters and inputs. The methodology developed is semi-analytical in nature, as the loop-closure constraints are re-duced to three polynomials retaining the geometric param-eters and inputs in their symbolic form. The final univariatepolynomial in one of the parameters of interest turns out tobe of degree 64 and of 28 for the respective cases. Realsolutions with high numerical precision are extracted andare validated by computing the residuals of the initial set ofequations. Possible configurations of the MPS for all realsolutions obtained for various combinations of inputs aredepicted graphically. Out of 64 and 28 possible number ofsolutions in the two possible cases, two solutions turn outto be real and only one of them corresponds to a physicallyfeasible configuration of the MPS. While this is observedconsistently in all the numerical examples reported here,the number of real solutions in the generic case is yet to bestudied.

The main contribution of the paper is to present amethodology for finding solution of the MPS suspension

(a) c = (−1.615,−2.358, 1.385)>

(b) c = (0.772, 1.057, 1.434)>


position analysis problem in its full complexity. The mo-tivation behind this paper is to support and develop a for-mulation technique for detailed analysis/optimisation of theMPS design, as well the study of its dynamics. It is tobe noted that the entire formulation, which contains sym-bolic computations and numerical solution finding tech-niques with high precision takes around 0.6 seconds5 to ar-rive at the possible configurations of the mechanism for agiven set of input values. Future work would focus on find-ing the final resultant/univariate of the loop-closure equa-tions purely in symbolic form.

IV. Appendices

A. Rodrigue’s parameters

Rodrigue’s parameters provide an algebraic parametrisa-tion of SO(3). The rotation matrix 0

1R in terms of the three

5On a system with 64-bit Linux O.S. and 6-core Intel Core i7-4930KCPU @ 3.40 GHz clock speed and 64 GB of RAM.

7


(a) c = (0.861, 1.307, 1.518)>

(b) c = (−1.774,−2.696, 1.520)>


Rodrigue’s parameters (c1, c2, c3) is given by:

01R =

1

c∆

1 + c21 − c22 − c23 2(c1c2 − c3) 2(c1c3 + c2)2(c1c2 + c2) 1− c21 + c22 − c23 2(c2c3 − c1)2(c1c3 − c2) 2(c2c3 + c2) 1− c21 − c22 + c23

where, c∆ = 1 + c21 + c22 + c23.

B. Quartic polynomials in their symbolic form

The three quartic polynomials f1(c1, c2, c3), f2(c1, c2, c3),f3(c1, c2, c3) in Eqs. (22-24) representing the kinematics ofMPS in symbolic form are given by:

f1(c1, c2, c3) = c23c21(−4l21 + 4o2

1x + 4o21y − 2o2

1z) +

c2c3c1(8l21 − 8o2

1x − 8o21y + 8o2

1z) + c22(−4l21 + 4o21x +

4o21y−2o2

1z)−4c3c31o1xo1z+4c2c21o1xo1z+4c33c1o1xo1z+

4c22c3c1o1xo1z−4c3c1o1xo1z−4c32o1xo1z−4c2c23o1xo1z+4c2o1xo1z−8c2c3c21o1yo1z+8c22c1o1yo1z−8c23c1o1yo1z+8c2c3o1yo1z+c

41o

21z+2c22c

21o

21z+2c21o

21z+c

42o

21z+c

43o

21z+

2c22c23o

21z + 2c23o

21z + o2

1z;

f2(c1, c2, c3) = o1z(l3x − p4x − sx)c41 − 2c2o1z(p4y +

(a) c = (0.484, 0.684, 1.414)>

(b) c = (−0.981,−1.417, 1.444)>


sy)c31 + c3(−l21 + l24 − l23x − l23y − l23z − p2

4x − p24y − p2

4z −s2x − s2

y − s2z + 2l3yo1y + 2l3zo1z + 2o1xp4x − 2l3yp4y +

2o1yp4y−2l3zp4z+2o1xsx−2p4xsx+2l3x(−o1x+p4x+sx) − 2l3ysy + 2o1ysy − 2p4ysy − 2l3zsz − 2p4zsz)c

31 +

2c22l3xo1zc21−2o1z(−l3x+p4x+sx)c

21+4c2c3(l3y(−o1x+

p4x+sx)+ l3x(−o1y+p4y+sy))c21−2c3(o1z(p4y+sy)+

2l3z(−o1y +p4y + sy)+2l3y(o1z−p4z− sz))c21 + c2(l21−l24 + l23x + l23y + l23z + p2

4x + p24y + p2

4z + s2x + s2

y + s2z −

2l3yo1y−2l3zo1z−2o1xp4x+2l3yp4y−2o1yp4y+2l3zp4z+2l3x(o1x−p4x−sx)−2o1xsx+2p4xsx+2l3ysy−2o1ysy+2p4ysy+2l3zsz +2p4zsz)c

21−2c23(2l3z(o1x−p4x−sx)+

l3x(o1z − 2(p4z + sz)))c21 + 4c22(l3y(o1x − p4x − sx) +

l3x(o1y−p4y−sy))c1−2c32o1z(p4y+sy)c1+c23(4l3y(o1x−

p4x− sx)+4l3x(−o1y +p4y + sy))c1 + c2c23(4l3z(−o1y +

p4y + sy)− 2(o1z(p4y + sy) + 2l3y(o1z − p4z − sz)))c1−8c2c3l3x(−o1z +p4z +sz)c1 + c

22c3(−l21 + l24− l23x− l23y−

l23z−p24x−p2

4y−p24z− s2

x− s2y− s2

z−2l3yo1y +2l3zo1z +2o1xp4x+2l3yp4y+2o1yp4y−2l3zp4z+2o1xsx−2p4xsx−2l3x(−o1x + p4x + sx) + 2l3ysy + 2o1ysy − 2p4ysy −2l3zsz−2p4zsz)c1 + c

33(−l21 + l24− l23x− l23y− l23z−p2

4x−

8


(a) c = (−1.097,−1.661, 1.514)>

(b) c = (0.577, 0.878, 1.521)>


p24y − p2

4z − s2x− s2

y − s2z +2l3xo1x +2l3yo1y − 2l3zo1z +

2o1xp4x−2l3yp4y+2o1yp4y+2l3zp4z+2o1xsx−2p4xsx−2l3x(p4x + sx) − 2l3ysy + 2o1ysy − 2p4ysy + 2l3zsz −2p4zsz)c1 + c3(−l21 + l24 − l23x − l23y − l23z − p2

4x − p24y −

p24z−s2

x−s2y−s2

z−2l3yo1y−2l3zo1z+2o1xp4x+2l3yp4y+2o1yp4y+2l3zp4z+2o1xsx−2p4xsx+2l3x(−o1x+p4x+sx) + 2l3ysy + 2o1ysy − 2p4ysy + 2l3zsz − 2p4zsz)c1 −2c2(2l3z(o1y − p4y − sy) + o1z(p4y + sy) + 2l3y(−o1z +p4z+sz))c1+2c23l3xo1z+o1z(l3x−p4x−sx)+c42o1z(l3x+p4x + sx)+ c43o1z(l3x + p4x + sx)+ 2c22c

23o1z(l3x + p4x +

sx)+c2c3(4l3y(−o1x+p4x+sx)+4l3x(o1y−p4y−sy))−2c33o1z(p4y+sy)−2c3o1z(p4y+sy)+c2(l

21−l24+l23x+l23y+

l23z +p24x+p

24y +p

24z + s

2x+ s

2y + s

2z +2l3yo1y +2l3zo1z−

2o1xp4x−2l3yp4y−2o1yp4y−2l3zp4z+2l3x(o1x−p4x−sx)−2o1xsx+2p4xsx−2l3ysy−2o1ysy+2p4ysy−2l3zsz+2p4zsz)+c2c

23(l

21− l24 + l23x+ l23y+ l23z+p2

4x+p24y+p

24z+

s2x + s2

y + s2z − 2l3yo1y + 2l3zo1z − 2o1xp4x + 2l3yp4y −

2o1yp4y−2l3zp4z−2o1xsx+2p4xsx+2l3x(−o1x+p4x+sx)+2l3ysy−2o1ysy+2p4ysy−2l3zsz+2p4zsz)+c

32(l

21−

l24+l23x+l

23y+l

23z+p

24x+p

24y+p

24z+s

2x+s

2y+s

2z+2l3yo1y−

(a) c = (0.767, 1.081, 1.410)>

(b) c = (−1.662,−2.394, 1.441)>


2l3zo1z−2o1xp4x−2l3yp4y−2o1yp4y+2l3zp4z−2o1xsx+2p4xsx+2l3x(−o1x+p4x+sx)−2l3ysy−2o1ysy+2p4ysy+2l3zsz+2p4zsz)−2c22c3(o1z(p4y+sy)+2l3z(−o1y+p4y+sy)+2l3y(−o1z + p4z + sz))+ c22(4l3z(o1x− p4x− sx)−2l3x(o1z − 2(p4z + sz)));

f3(c1, c2, c3) = c3c31(−l5y + o1y − p7y − y) + c2c

21(l5y −

o1y+p7y+y)+c33c1(−l5y+o1y−p7y−y)+c22c3c1(l5y+

o1y−p7y−y)+c3c1(l5y+o1y−p7y−y)+c32(−l5y−o1y+p7y+y)+c2(−l5y−o1y+p7y+y)+c2c

23(l5y−o1y+p7y+

y)+ c3c21(−2l5z−o1z)+ c2c

23c1(2l5z−o1z)+ c2c1(2l5z−

o1z)+c22c3(−2l5z−o1z)+c2c3c

21(2l5x+2r2)+c

23c1(2l5x+

2r2)+c22c1(−2l5x−2r2)+c2c3(−2l5x−2r2)−c2c31o1z−

c32c1o1z − c33o1z − c3o1z .

References[1] ARIKERE, A., SARAVANA KUMAR, G., AND BANDYOPADHYAY,

S. Optimisation of double wishbone suspension system using multi-objective genetic algorithm. In SEAL, vol. 6457 of Lecture Notes inComputer Science. Springer Berlin Heidelberg, 2010, pp. 445–454.

[2] ATTIA, H. Numerical kinematic analysis of the standard macpher-son motor-vehicle suspension system. KSME International Journal

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17, 12 (2003), 1961–1968.[3] BANDYOPADHYAY, S., AND GHOSAL, A. Geometric characteriza-

tion and parametric representation of the singularity manifold of a6-6 stewart platform manipulator. Mechanism and Machine Theory41, 11 (2006), 1377–1400.

[4] COX, D., LITTLE, J., AND O’SHEA, D. Ideals, Varieties, and Algo-rithms: An Introduction to Computational Algebraic Geometry andCommutative Algebra, 3rd. ed. Springer-Verlag, New York, 2007.

[5] CRONIN, D. Macpherson strut kinematics. Mechanism and MachineTheory 16, 6 (1981), 631–644.

[6] GHOSAL, A. Robotics: Fundamental Concepts and Analysis,1st. ed. Oxford University Press, New Delhi, 2006.

[7] GREUEL, G.-M., AND PFISTER, G. A Singular introduction to com-mutative algebra. With contributions by Olaf Bachmann, ChristophLossen and Hans Schonemann, 2nd. extended ed. Berlin: Springer,2007.

[8] JORNSEN REIMPELL, H. S., AND BETZLER, J. W. The AutomotiveChassis: Engineering Principles. SAE International, Warrendale,PA, 2001.

[9] MANTARAS, D. A., LUQUE, P., AND VERA, C. Development andvalidation of a three-dimensional kinematic model for the mcpher-son steering and suspension mechanisms. Mechanism and MachineTheory 39, 6 (2004), 603–619.

[10] PAPEGAY, Y. A., MERLET, J.-P., AND DANEY, D. Exact kinemat-ics analysis of cars suspension mechanisms using symbolic compu-tation and interval analysis. Mechanism and Machine Theory 40, 4(2005), 395–413.

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