Kinematics and optimization of the translating 3-CCR/3-RCC parallel mechanisms

KINEMATICS AND OPTIMIZATION OF THE TRANSLATING 3-CCR/3-RCC PARALLEL MECHANISMS

Massimo Callegari and Matteo-Claudio Palpacelli Polytechnic University of Marche, Department of Mechanics Ancona, Italy {m.callegari, m.palpacelli}@univpm.it

Abstract The paper presents the kinematics of the 3-RCC/3-CCR translating parallel mechanisms and several machines of such family are described in detail taking into account different possible kinds of actuations. They all share good kinematic properties as for instance simple closed-form relations and convex workspace, but differ for overall dimensions of the mobile platform and dynamic behaviour: therefore the concepts have been optimized and compared against common performance indices, to determine the best solutions for selected classes of applications. Based on such results, a prototype robot has been finally built.

Keywords: Parallel robots, Translating Parallel Machines, Kinematic analysis, Conceptual design, Optimization

1. Introduction The kinematics of parallel mechanisms with full spatial mobility is

usually very complex and sometimes cannot even be solved in closed form, as for example in the well-known case of the general Gough-Stewart platform (Liu and Fitzgerald, 2003). For this reason it is recently growing the interest of researchers and industry towards minor mobility mechanisms, able to perform elementary motions like pure rotations, pure translations or planar displacements: in this way the complexity of the analytical problem is greatly reduced while the advantages of closed-loop actuation are still preserved, eventually by having two parallel mechanisms mounted in series or cooperating in parallel for the accomplishment of a given task.

Many new translating parallel machines (TPMs) have been studied in the last years, including the 3-RPC mechanism by (Callegari and Tarantini, 2003) that showed good kinematic performances, e.g. simple equations, high stiffness, convex workspace, and so on. However the aforementioned concept presented also some clear weak points, such as a cumbersome moving platform, poor dynamic performances and an

overconstrained structure: therefore the study has been enlarged to the whole family of machines with 3-RCC/3-CCR kinematics (Callegari and Marzetti, 2003) to assess whether other mechanisms of this set show better features than the one previously identified.

The paper briefly presents three mechanisms selected among the most interesting elements of this class, develops their kinematics according to the possible different actuations and finally performs a kinematic optimization and compares the resulting performances.

2. Description of the family of mechanisms A whole family of mechanisms can be defined by the functional design

schematically represented in Fig. 1a: a mobile platform is connected to the fixed base by three identical limbs, that are composed by two members coupled by a cylindrical pair; the lower link of each leg is connected to the frame by a revolute joint while the upper one is connected to the platform by a cylindrical pair. Such kind of mechanisms are conventionally called 3-RCC to indicate the sequence of the joints in the three (identical) limbs, starting from the fixed frame and moving towards the platform; the 3-CCR architecture, shown in Fig. 1b, is simply obtained by kinematic inversion. It can be easily seen that the described architecture is characterized by 3 d.o.f.’s in space where in the general case spatial translations are coupled with changes of orientation of the platform. Nevertheless for particular geometrical configurations such mechanisms can provide motions of pure translation, i.e. iff (i) the axes of the revolute and outer cylindrical joint of ith limb are parallel to the same unit vector iu and (ii) ji uu ˆˆ ≠ for i ≠ j (i,j=1,2,3).

(a) (b) Figure 1. The 3-RCC (a) and 3-CCR (b) parallel mechanisms

It must be pointed out that in these cases the two links of each leg do not turn around the cylindrical pair, which could well be substituted by a prismatic joint, giving rise to the already mentioned 3-RPC (or 3-CPR) overconstrained mechanism.

3. Kinematics of some mechanisms In this section three different mechanisms are considered. The 3-RCC

architecture shown in Fig. 1a can be actuated either by driving the base revolute joints or by controlling legs’ extensions: the mobile platform of such concepts turns out to be rather bulky because of the (unavoidable) length of the stroke of the cylindrical pairs mounted on the platform itself. This drawback can be possibly avoided by inverting the kinematic structure of the mechanism, Fig. 1b: in this case the legs can be actuated by directly driving the sliders running along the fixed slideways. Unfortunately a mere inversion of the mechanism is not possible because the resulting kinematics shown in Fig. 1b is singular in its entire workspace; therefore two more concepts are presented, slightly different from first mechanism but both characterized by a 3-CCR architecture.

3.1 3-RCC with triangular configuration The configuration of the mechanism is symmetric, Fig. 1a, with the

axes of the three revolute pairs forming an equilateral triangle on the fixed base; in the same manner, another equilateral triangle is formed on the moving platform by the axes of the cylindrical pairs; moreover, the legs are perpendicular to the joints connecting them with the two bases.

In case the machine is driven by controlling legs’ lengths, IPK is characterized by a single configuration of the mechanism while DPK presents 8 different solutions that can be simply evaluated in closed-form (Callegari and Tarantini, 2003). When the unit vectors of the three limbs become linearly dependent, the manipulator gets stuck in a singular configuration: therefore the locus of all the singular points is given by a right cylinder, that however can be moved outside the workspace with a proper dimensioning of the machine. No constraint (rotation) singularities exist apart from the surface of the mentioned cylinder. This mechanism can be easily driven by means of rotary motors lumped at the end of the limbs and ball screws, see Fig. 2: such design yields very good static properties (e.g. a high thrust can be delivered since the legs are loaded only by normal forces) but the resulting dynamic behaviour is poor due to the relevant mass of the mobile platform and the high inertia of the spinning or tilting masses (Callegari et al., 2006).

Better dynamic performances can be obtained by directly powering the base revolute joints, even if the limbs are loaded by bending moments in this case. (Callegari and Marzetti, 2003) developed the complete kinematics of such mechanism: it is shown that a (very simple) unique solution exists for both direct and inverse position problem and no translation or rotation singularities are possible at all.

translatingplatform

fixed base

actuation limbs

driving motors

Figure 2. Sketch of the 3-RCC translational platform (actuation on legs’ length)

3.2 3-CCR with tetrahedral configuration The mere kinematic inversion of the 3-RCC mechanism just described

is shown in Fig. 1b: (Callegari and Marzetti, 2003) proved that such mechanism is useless since all the points of its workspace are singular. As a matter of fact, in order to give full mobility to the inverted mechanism it is necessary that the direction of the axes of the three pairs connecting the limbs to the ground are linearly independent.

(a) (b) Figure 3. Sketch of the tetrahedral 3-CCR architecture and loop-closure for ith limb

For instance, in the tetrahedral configuration shown in Fig. 3a such axes stem from the origin of the fixed frame ( )zyxO ˆ,ˆ,ˆ and are tilted by α radians with respect to the horizontal plane. If a frame ( )wvuP ˆ,ˆ,ˆ , parallel to ( )zyxO ˆ,ˆ,ˆ , is attached to the mobile platform at the intersection of the

three revolute joints, the vector OP=p can be taken to specify the position of the moving platform.

In order to simplify the study of the mechanism, other frames can be easily defined for each link as shown in Fig. 3b: e.g. the frame

)ˆ,ˆ,ˆ( iiii zyxA , attached at of ith limb, is obtained by starting at the global frame ( )zyxO ˆ,ˆ,ˆ , then it is rotated by iϕ around the (current) zi axis and then by α around the (current) yi axis, finally a translation ai along the direction of the (current) xi axis is performed, to allow for the variable sliding of the cylindrical pair. Moreover, the articular coordinates of the 3 joints are defined as follows: ai and iϑ for the first cylindrical pair, di and βi for the second one and γi for the revolute joint. It is also noted that the configuration of Fig. 3 is characterized by maximum symmetry, therefore

°=°=°= 240,120,0 321 ϕϕϕ and tti = for i=1,2,3. The loop-closure equation of ith limb is given by:

iii tdap ++= (1)

that can be expressed in the local iA frame as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

i

ii

ii

ii

i

i

i

z

y

x

ii

ii

ii

csd

ssscc

ta

ppp

csssccs

scscc

ϑϑ

βϑβϑ

β

ααϕαϕϕϕ

ααϕαϕ 0

000 (2)

The mobility analysis developed in the Appendix shows that unfortunately such architecture is liable of constraints singularities inside its entire workspace; therefore, in order to prevent it rotating, it is necessary to turn to an overconstrained architecture by replacing limbs’ internal cylindrical pairs with a prismatic pairs, resulting with a 3-CPR kinematics. The following analysis will be performed with reference to this case, where Eq. 2 still holds with βi=0.

Inverse position kinematics relations provide the actuated variables as functions of platform's position p:

iziyixi tspcspccpa +++= ααϕαϕ (3)

The same Eq. 3, written for i=1,2,3, is used to solve the direct position kinematics and find the values of zyx ppp ,, as functions of ia . The direct derivation of Eq. 1 is the base for the velocity analysis:

iiiiiip da dωdav ×+⋅+⋅= ˆˆ && (4)

By dot-multiplying both sides of Eq. 4 by the unit vector ia and collecting the 3 relations for i=1,2,3 in a single expression in matrix form,

av &=pJ , the expressions of the Jacobian matrix is obtained:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=ααϕαϕααϕαϕααϕαϕ

scsccscsccscscc

JT

T

T

33

22

11

3

2

1

ˆˆˆ

aaa

(5)

The constant Jacobian shows that this mechanisms belongs to the class of the so-called Cartesian parallel robots (Kim and Tsai, 2002). It is noted that the maximum value of ( ) ααscJ 2det = , granting an “optimal” manipulability, is obtained for °≈ 26.35α which corresponds to aligning the ground cylindrical pairs along a Cartesian frame.

3.3 3-CCR with Cartesian configuration The setting of the joints shown in Fig. 4 gives rise to a full Cartesian

kinematics; in fact inverse and direct position kinematics are solved by:

⎪⎩

⎪⎨

⎧

=++=

+=

z

yx

x

patspcpa

tpa

3

22

11

αα

⎪⎪⎩

⎪⎪⎨

⎧

=

−⋅++⋅−=

−=

3

2121

11

aps

ttcaacp

tap

z

y

x

ααα (6 a-b)

and velocity kinematics is expressed by the following Jacobian matrix:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=1000001

ˆˆˆ

2

2

1

αα scJ

T

T

T

T

aaa

(7)

showing that J is the identity matrix when the three ground cylindrical pairs are orthogonal one to the others (α=90°). Also this architecture is affected by rotation singularities inside the workspace therefore, just as before, the equivalent 3-CPR mechanism should be used instead.

Figure 4. Sketch of the Cartesian 3-CCR architecture

4. Optimization of the families The kinematic performances of the described machines strongly

depend on the specific geometric parameters of the four architectures: therefore according to what has been proposed by (Carretero et al., 2000), the machines have been individually optimized and then the resulting performances have been compared. The following non-dimensional performance index has been used for the optimization:

332211 FFFPI ⋅+⋅+⋅= βββ (8)

where F1 measures the volume of the workspace, F2 is proportional to mobile platform's overall dimensions, F3 quantifies the dexterity of the mechanism and βi, i=1,2,3 are the weights of the 3 functions; since the functions Fi are chosen such that 0≤Fi≤1 and βi are subject to the condition: β1+β2+β3=1, the performance index results bounded by 0≤PI≤1.

The measure of workspace volume F1 is chosen with reference to the Monte Carlo method proposed by (Stamper, 1997): robot's workspace is inscribed inside a cube of side 2dmax and is discretized in ntot points; then a numerical procedure determines the number of nin points belonging to robot's workspace and it is finally defined:

tot

inn

nF =1 (9)

Overall dimensions of mobile platform are evaluated as: ( )

max

max2 t

ttF −= (10)

where tmax is the maximum dimension of its height and is constrained to the maximum stroke of the ground sliders. Machine's dexterity is assessed by computing the condition number c of the matrix JTJ, with J Jacobian matrix, as suggested by (Gosselin and Angeles, 1991); by using the already defined discretization, it is therefore defined:

in

n

i i

tot

in

tot

n

i i

W

W

nc

nnd

ncd

dW

dWc

F

in

in

∑∑

∫

∫=

=

=≈= 12

3max

12

3max2

3

1

8

1

81

(11)

The variable parameters of the optimization are the geometric dimensions of both platforms, limbs' lengths and the incidence angle α between joint axes (or the offset distance e for the fully Cartesian machine); in order to deal with dimensionless parameters all variables are divided by maximum limbs’ lenght or by the stroke of ground sliders, i.e. dmax=1 or amax=1 is imposed according to the specific architecture.

If the three functions Fi are equally weighted, i.e. β1=β2=β3, the optimization process yields the results collected in Tab. 1: it is clear that

the two 3-CPR concepts yield the best performances also when the two 3-RCC architectures present a relatively high PI in spite of a virtually null workspace. In fact the optimization process drove geometry variations of the 3-RCC concepts towards very small values of mobile platforms sides, therefore constraining the displacements of its cylindrical pairs.

Table 1. Comparison of robots performances (optimization with equal weights)

RCC R(PR)C CPR (Tetrahedral)

CPR (Cartesian)

a 0.5 0.8 1 1 t 0.1 0.1 0 0 α — — 35.26° 90° PI 0.42 0.45 0.70 0.70 W 1.1% ≈0% 12.5% 12.5% e — — — 0

However, also in case the only workspace is maximized for the 4

mechanisms (β1=1, β2=β3=0) the two 3-CPR architectures still show a workspace comparable with the other concepts (see Fig. 5), that are in this case characterized by big overall dimensions and limited dexterity.

(a) (b)

(c) (d) Figure 5 Optimized workspace of the 4 machines: 3-RCC with rotary (a) or

linear (b) actuation, 3-CPR with tetrahedral (c) or Cartesian (d) configuration

5. Conclusions The paper has presented the kinematic properties of the translating

platforms characterized by the 3-RCC architecture: it has been shown that also its 3-CCR kinematic inversion can be considered but, due to rotation singularities, the rotation of the inner cylindrical joint must be avoided by turning to the 3-CPR overconstrained architecture. After having optimized the mechanisms, it has been shown that the two 3-CPR concepts present quite better performances than the others: as a matter of fact they are both Cartesian parallel machines, being slightly different only due to the disposition of joint axes in the space and present a (comparably) wide workspace characterized by very high dexterity.

Moreover by still keeping the same geometry of the 3-CCR tetrahedron design of Fig. 3, but with a different disposition of the pairs, i.e. the 3-CPU architecture shown in Fig. 6a, it is possible to get rid of all singularities and obtain a non-overconstrained machine that still retains the good feature of the original concept (as a matter of fact the same kinematics equations at all). A prototype robot has been built with this structure, as described in (Callegari et al., 2005), and is shown in Fig. 6b: the first experimentations are still under way.

(a) (b)

Figure 6. Concept (a) and prototype (b) of the 3-CPU translation robot

References Callegari, M., Marzetti, P. (2003), Kinematics of a Family of Parallel Translating

Mechanisms, Proc. Intl. Workshop on Robotics in Alpe-Adria-Danube Region, Cassino (Italy), May 7-10.

Callegari, M., Palpacelli, M.-C., Principi, M. (2006), Dynamics Modelling and Control of the 3-RCC Translational Platform, submitted to Mechatronics

Callegari, M., Palpacelli, M.-C., Scarponi, M. (2005), Kinematics of the 3-CPU parallel manipulator assembled for motions of pure translation, Proc. Intl. Conf. Robotics and Automation, Barcelona, April 18-22, pp.4031-4036.

Callegari, M., Tarantini, M. (2003), Kinematic Analysis of a Novel Translational Platform, ASME Journal of Mechanical Design, vol. 125, pp. 308-315.

Carretero, J.A., Podhorodesky, R.P., Nahon, M.A., Gosselin, C.M. (2000), A Global Performance Index for the Kinematic Optimization of Robotic Manipulators, ASME Journal of Mechanical Design, vol. 122, pp. 17-24.

Gosselin, C., Angeles, J. (1991), A Global Performance Index for the Kinematic Optimization of Robotic Manipulators, ASME Journal of Mechanical Design, vol. 113, pp. 220-226.

Kim, H.S., Tsai, L-W. (2002), Evaluation of a Cartesian parallel manipulator. In Advances in Robot Kinematics: Theory and Applications, Lenarcic and Thomas Eds., Springer, pp.21-28.

Liu, K., Fitzgerald, J.M. (2003), Kinematic Analysis of a Stewart Platform Manipulator, IEEE Trans. Industrial Electronics, vol. 40, pp. 282-293.

Stamper, R.E. (1997), A Three Degree of Freedom Parallel Manipulator with Only Translational Degrees of Freedom, Ph.D. Thesis, University of Maryland, College Park, Maryland.

Appendix. Mobility analysis of the tetrahedral 3-CCR The angular velocity ω of mobile platform can be expressed in function

of ith limb’s articular coordinates as follows, see Fig. 3b:

iiiiii tdaω ˆˆˆ ⋅+⋅+⋅= γβϑ &&& (12)

By dot-multiplying Eq. 12 by id , it is obtained:

iTi β&=⋅ωd (13)

Taking into account Eq. 13, the derivation of the first line of Eq. 2 provides:

( )ω⋅+=⋅+⋅+⋅ Tiiiizyixi IIzstapspcspcc ˆβααϕαϕ &&&& (14)

If platform’s motors are held still ( 0=ia& ), due to the absence of translation singularities is also: 0=== zyx ppp &&& and therefore from Eq. 14:

( ) 0ˆ =⋅ωdTiii st β (15)

Initial mounting conditions require 0=iβ , therefore it is evident that a finite angular velocity ω can be initiated: however, since in case of pure translations the legs do not rotate around their own axes, the problem can be overcome by the introduction of prismatic pairs in place of the inner cylindrical joints (3-CPR architecture).