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8 Parallel Wrists for Enhancing Grasping

Performance

Massimo Callegari, Luca Carbonari, Giacomo Palmieri and

Matteo-Claudio Palpacelli

Department of Industrial Engineering & Mathematical Sciences

Polytechnic University of Marche

Via Brecce Bianche – 60131 Ancona (AN) – Italy

Abstract Good grasping and effective manipulation heavily depend on the per-

formance of robotic wrists such as, e.g., the number of degrees of freedom, the

kind of motion that is generated, the dexterity of the operations, the stiffness and

the size of the mechanical structure; such characteristics heavily affect kinematic

and dynamic performance of the manipulation and can lead to a successful grasp

or to an unexpected failure, if not taken into consideration since the early design

steps. This chapter, after an introduction recalling the wrist structure of the indus-

trial manipulators, focuses on parallel kinematics wrists, a rather new kind of me-

chanical architecture that hasn’t found so far relevant industrial applications but

shows very promising features, such as mechanical stiffness, high accuracy,

lightweight construction, etc. After presenting a powerful kinematical tool for the

synthesis of parallel kinematics machines (SPM), which is based on Lie algebra,

the design of a novel spherical wrist is discussed in details. A prototype machine,

actuated by three brushless linear motors, has been built with the aim of obtaining

good static and dynamic performance.

8.1 Introduction

Since its origins, robotics has been inspired by the features of human beings, both

in terms of motor skills, related to the mechanical structure of human body, and in

terms of control architecture and mental processing.

Looking at the human body from outside, a serial anthropomorphic structure

can be recognized, typically adopted in industrial manipulators, which gives them

specific properties including large workspace and high manipulability. On the

contrary, by analyzing human joints in more detail, it is possible to identify paral-

lel kinematic chains which result from the complex pattern created by tendons and

bones.

Even the performance which are required to a grasping tool are usually com-

pared to those of the human hand, that is able to grab and manipulate several kinds

of objects, different in shape and dimension, and to keep them stable even in the

2

presence of disturbing forces. Industrial devices, however, imitate only partially

the hand behavior, preferring economy of scope and simplicity of realization to

more complex manipulation capabilities (see for example the common 2-finger

grippers which are only able to switch from the open to the close configuration).

On the other hand, industrial manipulators are usually equipped by a 2- or 3-axes

wrist that allow them to set the gripper in the best grasping conditions, overcom-

ing its motion limitations.

The human hand is able to rotate through the three rotational degrees of free-

dom provided by the wrist and the forearm: the former is responsible for the hand

radial/ulnar deviation and flexion/extension, while the latter for the hand prona-

tion/supination rotation; several research studies on wrists and grippers have been

proposed in the literature during past years (Rosheim 1989; Pennestrì 1991; Sara-

vanan et al. 2009; Charles and Hogan 2010), while a sample design of a humanoid

robot wrist is provided by Schäfer and Dillmann (2001).

The dexterity of a 6-axes serial manipulator is mainly due to its wrist, which

has to concentrate in a compact space three axes of rotation; the robotic arm, in-

stead, is mainly responsible for the positioning of the wrist. For a 6-dof fully par-

allel manipulator, such as the Gough-Stewart Platform (Fichter et al. 2009), rota-

tion and positioning are obtained by a proper coordination of robot actuators, since

the moving platform is driven by 6 actuated legs which concurrently manage its

six degrees of freedom: for this kind of spatial parallel manipulators end-effector

rotations and translations in space are highly coupled by the mechanical structure.

Sometimes it is preferable to decouple the motion referring to spherical parallel

wrists: in this case, a 3-axes platform rotates around a fixed point, which repre-

sents the wrist centre of rotation (Di Gregorio 2002).

8.2 Wrist kinematics

The most successful architectures of industrial robot wrists belong to the families

of Pitch-Yaw-Roll (PYR), Roll-Pitch-Roll (RPR) and 3-Roll wrists, according to

the orientation of joint axes.

The kinematics of PYR wrists closely resembles the human hand, with the

pitch and yaw axes mimicking the flexion/extension and the radial/ulnar deviation

and the roll axis performing the rotation of the forearm. The difficulty in laying

the three axes so that they intersect at a common point prevents the attainment of a

spherical motion: in this way singular configurations are avoided but the design is

more complex and highly coupled; as a matter of fact, nowadays PYR wrists are

not used anymore.

Figure 8.1 shows the most popular industrial wrists, based on the RPR and 3-

Roll serial kinematics. The joint axes of Roll-Pitch-Roll wrists always meet at a

common point, thus realizing a spherical motion: the possible occurrence of singu-

lar configurations at the center of their mobility range represents their major

3

drawback, but their simple and modular architecture paved the way to the success

of such design.

The 3-Roll solution, see Figure 8.1b, is characterized by three rotations around

consecutive roll axes: such architecture does not realize a spherical motion since

the three axes do not intersect at a common point, but the possibility to obtain

large rotations of the joints enhances the dexterity of the wrist. In Figure 8.1c two

incident axes of rotation, which follow the first roll axis, provide only one degree

of freedom because they are connected by a bevel gear transmission. Figure 8.1d

shows the Lemma wrist, which is similar to the PYR architecture and is still used

in some painting or coating applications.

(a) (b)

(c) (d)

Fig. 8.1. Typical industrial wrist architectures: Roll-Pitch-Roll (a), 3-Roll (b), modified 3-Roll

(c) and Lemma (d)

The weight and the dynamic loads of the hand, including the grasped object,

must be supported by the wrist in the most severe conditions: to this aim stiffness

and accuracy are required for the wrist in order to reduce possibilities of task fail-

ure and to achieve a more stable grasping operation.

A good design of robot grasping devices must aim at integrating actuators and

transmissions within its mechanical structure, maintaining a modularity where

each device is autonomous and responsible for carrying out its own functions: the

robot hand for the grasping, the wrist for the orientation of the hand and the robot-

ic arm for the positioning of the wrist.

The robot wrist is therefore concerned with the hand rotation, undergoing the

bending moment due to the load applied to the gripper: in order to provide the ro-

4

bot with the ability to manipulate different kinds of objects, in terms of weight and

dimension, a parallel kinematics may be conveniently considered as an interesting

alternative to a serial architecture for the wrist design, waiving part of the robot

mobility in favor of compactness and a high power-to-mass ratio with a greater

load capacity and stiffness.

8.3 Parallel kinematics wrists

Parallel Kinematics Machines (PKM’s) offer several features that make them at-

tractive for grasping operations: high mechanical stiffness, high accuracy, light-

weight construction, compact size (with reference to payload capacity); on the

contrary they generally have a limited workspace and a complex direct kinematics.

Therefore they can be profitably used in applications where the mentioned charac-

teristics are required, for instance in biomedicine and surgery (Bürger et al. 2001).

A great number of parallel kinematics machines have been proposed as haptic de-

vices in the field of human wrist or ankle rehabilitation, see e.g. Takaiwa and Nor-

itsugu (2005); Arata et al. (2011) proposed a redundantly actuated 3-dof parallel

machine to be used as haptic device, while Unal and Patoglu (2008) designed two

exoskeletons based on parallel kinematics and Saltaren et al. (2007) investigated

the performance of 3-dof spherical parallel manipulators in humanoid robotics.

Spherical Parallel Machines (SPM’s) are the class of parallel machines which

are able to perform motions of pure rotations: such research field was first intro-

duced by the work of Asada and Granito (1985) but the most relevant results have

been obtained by Gosselin and Angeles (1989), that conceived the Agile Eye, the

most known mechanism of this family upon which many prototype machines have

been designed since then. The Agile Eye is based on the 3-RRR overconstrained

kinematics, with three identical legs composed by two links and characterised by a

sequence of three revolute joints, as shown in Figure 8.2a. Few other studies on

the subject are available during the 90’s (Lee and Chang 1992; Innocenti and

Parenti-Castelli 1993; Alizade et al. 1994), while a renewed interest characterized

the beginning of the new century and produced interesting results.

The use of synthesis methods based on screw theory, for instance, has been ex-

ploited by Kong and Gosselin (2004a and 2004b) that provide comprehensive list-

ings of both overconstrained and nonoverconstrained SPM’s; Hervé and Karouia,

on the other hand, use the theory of Lie group of displacements to devise novel ar-

chitectures, as the four main families in (Karouia & Hervè 2002) or the 3-UPU, 3-

RCC, 3-CCR, 3-CRC kinematics specifically treated in (Karouia & Hervè 2000,

2002, 2006); Fang and Tsai (2004) use the theory of reciprocal screws to present a

systematic methodology for the structural synthesis of a class of 3-DOF rotational

parallel manipulators (e.g. 3-RRS, 3-CRU, 3-CRC, 3-UPC). Further interesting

non-overconstrained architectures are the 3-URC, 3-RUU, 3-RRS and 3-RSR,

studied by Di Gregorio (2001a, 2001b and 2004a, 2004b), and the 3-URU by Hu-

da and Takeda (2007).

5

Parallel kinematics becomes a forced option for small size devices, where the

scale factor requires the use of passive flexible joints and conjugate pairs are only

exploited for the actuated ground joints. Specific results have been reached in this

field by Yi et al. (2003), Lusk & Howell (2008) and Callegari et al. (2009).

Sometimes the wrist only needs 2 d.o.f.’s to perform the grasping, in which

case different design solutions have been identified: Fontana et al. (2006) present-

ed a novel robotic wrist with high angulation capability, Carricato and Parenti-

Castelli (2004) proposed a fully decoupled 2-dof parallel wrist. An example of a

2-dof spherical device is shown in Figure 8.2b, where a simplified version of the

Agile Eye is sketched: in this case the moving platform is not able to spin around

its own axis.

(a) (b)

Fig. 8.2. Kinematics of the Agile Eye: original 3-RRR kinematics (a) and simplified 2-dof ver-

sion (b)

8.4 Kinematic synthesis of spherical parallel machines

A useful aid for the kinematic synthesis of SPM’s is provided by the contribution

of Karouia and Hervé (2000), whose work has been then reformulated by Kong

and Gosselin (2004). By following such approach, Callegari et al. (2004) studied a

new kinematics based on the 3-CPU structure, upon which a spherical wrist has

been designed and eventually prototyped: the main synthesis steps of such ma-

chine are outlined in the following paragraphs.

It is anticipated that the synthesis process has been driven by some heuristic

rules. First of all, it was decided to focalize the study only on non-overconstrained

mechanisms: this choice avoids practical problems such as the strict dimensional

and geometric tolerances needed by overconstrained machines during manufactur-

ing and assembly phases. Moreover, the use of passive spherical pairs directly

joining the platform to the base has been avoided and for economic reasons only

modular solutions composed of three identical legs have been studied.

6

Fig. 8.3. Limb of connectivity 5 able to generate a spherical motion of the platform

A simple mobility analysis shows that, in order to design a moving platform

connected to the ground by identical legs and characterized by 3 d.o.f.’s, it must

be composed by 3 limbs of connectivity 5. In the general case, each leg is com-

posed by 4 links connected to each other (and to mobile platform and fixed ground

at the distal ends) by 5 lower joints (revolute R or prismatic P). To allow the end

effector have 3 d.o.f.’s of pure rotation, 3 out of the 5 pairs in each limb must be

revolute joints with concurrent axes: if such points of intersection are the same for

all the three legs, the platform will only rotate around this common point O. In this

way, each limb generates a 5-dimensional manifold that must contain the 3-

dimensional group of spherical motions around the point O. The well-known wrist

Agile Eye by Gosselin and Angeles is realized exactly in this way; nevertheless in

this case 2 revolute pairs in each limb are locked and the mechanism results over-

constrained, as shown in Fig. 8.3.

Fig. 8.4. Limb with subgroup RRR generating the subgroup of planar displacements

The spherical motion can be obtained also by using 5 revolute pairs R1-R5

where the axes of the joints R1, R3 and R5 still intersect at a common point while

the axes of pairs R2 and R4 are parallel to the direction of R3. In such a way, the 3

joints R2, R3 and R4 will generate the 3-dimensional subgroup of planar displace-

7

ments G(П), i.e. the set of translations lying in П and rotations around axes per-

pendicular to П. The same subgroup G(П) is generated also in case the axis of

revolute joint R3 is still perpendicular to plane П but does not cross the rotation

center O, as shown in Fig. 8.4; therefore also with this limb kinematics a spherical

wrist can be obtained.

On the other hand, by following the same line of reasoning, the same subgroup

of planar displacements G(П) can be generated by substituting one or two revolute

joints among the R2, R3, R4 set with prismatic pairs whose axes lie in the plane П,

thus obtaining limbs whose central joints are characterized by one of the sequenc-

es PRR, RPR, PPR, PRP, RRP, RPP. Of course, two adjacent joints in limbs kin-

ematics can be merged to yield simpler architectures with fewer links: for instance

two revolute joints with orthogonal axes can be superimposed to give a universal

(U) joint, while the set of one revolute joint and one prismatic pair with the same

axes are equivalent to a cylindrical (C) joint, as shown in Fig. 8.5.

(a) (b)

Fig. 8.5. Merge of two adjacent joints able to yield universal (a) or cylindrical (b) pairs

The kinematic chains described above prevent the ith limb’s end from translat-

ing in the direction normal to the plane Пi, i=1,2,3; therefore, if three such chains

are used for the limbs and the three normals to the planes Пi are linearly independ-

ent, all the possible translations in space are locked and the mobile platform, at-

tached to the three limbs, can only rotate around a fixed point.

In this way, seven alternative design concepts have been considered, which are:

3-URU, 3-CRU, 3-URC, 3-UPU, 3-CPU, 3-UPC, 3-CRC. Figures 8.6-8.10 show

the synthesis steps leading to the specific limb topology (a) and sketch a possible

arrangement of the introduced joints (b). If the three limbs of each machine are ar-

ranged in the simplest possible setting (at least from a merely visual point of

view), they will all lie within vertical planes: unfortunately in this case the 3 nor-

mals to limbs’ planes are all parallel to the horizontal plane and therefore result

linearly dependent, allowing the platforms to translate along the vertical direction.

Of course there are many different settings of these normal axes in space that grant

them to be linearly independent: Fig. 8.6b-8.10b show that limbs’ planes have

been tilted in such a way that they become mutually orthogonal in the initial con-

figuration (or “home” position) of the wrist. In this manner the kinematic relations

of the related machines are greatly simplified and above all, even if this arrange-

ment changes during operation of the machine, this configuration is the most far

from the singular setting previously outlined, therefore granting a better kinematic

manipulability of the wrists.

8

(a) (b)

Fig. 8.6. Synthesis of URU limbs (a) and sketch of the 3-URU mechanism (b)

(a) (b)

Fig. 8.7. Synthesis of CRU and URC limbs (a) and sketch of the 3-CRU mechanism (b)

(a) (b)

Fig. 8.8. Synthesis of UPU limb (a) and sketch of the 3-UPU mechanism (b)

9

(a) (b)

Fig. 8.9. Synthesis of CPU and UPC limbs (a) and sketch of the 3-CPU mechanism (b)

(a) (b)

Fig. 8.10. Synthesis of CRC limbs (a) and sketch of the 3-CRC mechanism (b)

The kinematics of all these concepts has been investigated and in view of the

design of a physical prototype, the 3-CPU architecture of Fig. 8.9 has been con-

sidered for further development; in making this choice it has been considered that:

if cylindrical pairs must be arranged on the mobile platform, its design will

be cumbersome due to the stroke of the linear motion of the joints, unless a

very limited workspace is accepted;

if the intermediate pair is revolute, machine’s kinematics is usually more

complex than the case it is prismatic: see for instance the kinematics of the

3-CRU (Callegari et al. 2009) in comparison with the present 3-CPU case;

the 3-UPU mechanism has been studied already by other researchers and,

in case it is actuated by motors placed on the ground, it suffers from singu-

larities in its home configuration, as shown for instance in (Zlatanov et al.

2002; Bonev and Zlatanov, 2001).

As a matter of fact, next sections will show many benefits of the 3-CPU archi-

tecture, such as the simplicity of the kinematic relations, the compactness of the

mechanical design, the easiness of actuation and, finally, the novelty of the con-

cept. Before entering the study of the 3-CPU SPM, it is marginally noted that the

same limbs’ topology, with different joints arrangements, is able to provide sever-

al kinds of motions (Carbonari 2012), included the pure translation (Callegari and

10

Palpacelli 2008); moreover, the 3-CRU mechanism of Fig. 8.7, even if character-

ized by a far more complex kinematics, is extensively studied in (Callegari et al.

2009) in view of the realization of a SPM for miniaturized assembly tasks.

8.5 Geometry and reference frames

Making reference to Fig. 8.11, the axes of cylindrical joints Ai, i=1,2,3 intersect at

point O (center of the motion) and are aligned to the axes x, y, z respectively of a

(fixed) Cartesian frame centered in O. The first member of each link (1) is perpen-

dicular to Ai and has a variable length bi due to the presence of the prismatic joint

Di: the second link (2) of the leg is set parallel the base cylindrical pair. The uni-

versal joint Bi is composed by two revolute pairs with orthogonal axes: one is per-

pendicular to leg’s plane while the other intersects at a common point P with the

corresponding joints of the other limbs; such directions, for the legs i=1,2,3 order-

ly, are aligned to the axes u, v, w respectively of a (mobile) Cartesian frame, lo-

cated in P and attached to the rotating platform. For a successful operation of the

mechanism, such manufacturing conditions must be accompanied by a proper

mounting condition: assembly should be performed in such a way that the two

frames O(x,y,z) and P(u,v,w) come to coincide. If these conditions are verified,

the points P and O remain coincident during the motion and the moving platform

performs a spherical motion. In the initial configuration the linear displacements ai

of the cylindrical joints are equal to the length c of the second link and the dis-

placements bi of the prismatic joints are equal to the constant length d for all the

limbs. It is also evident that, for practical design considerations, SPM’s based on

the 3-CPU concept are efficiently actuated by driving the linear displacements of

the cylindrical pairs coupling the limbs with the frame: therefore in the following

kinematic analysis it will be made reference to this case (i.e. joint variables ai,

i=1,2,3 will be considered the actuation parameters).

(a) (b)

Fig. 8.11. Reference frames in home configuration (a) and geometry of a single limb (b)

11

Two auxiliary frames, O*(x*,y*,z*) and P*(u*,v*,w*), are introduced, as

shown in Fig. 8.12a, in order to better define the spatial configuration of the plat-

form and to be able to easily visualize and understand the assigned tasks and the

obtained results. The frame P*(u*,v*,w*) is attached to the mobile platform and is

defined as follows:

the origin P* is located at the center of the equilateral triangle formed by the

points B1, B2 and B3;

the w* axis is normal to the platform (outward direction);

the u* axis has the direction (P* - B1);

the v* axis is placed according to the right-hand rule.

The fixed frame O*(x*,y*,z*) is coincident with the mobile frame

P*(u*,v*,w*) when the platform is in its initial configuration. Of course, since the

frames are not placed at the center of the spherical motion, the two origins O* and

P* will be coincident only in the home configuration, see Fig. 8.12b.

(a) (b)

Fig. 8.12. Placement of auxiliary frames O*(x*,y*,z*) and P*(u*,v*,w*): home pose (a) and per-

turbed setting (b)

The orientation of the platform can be easily described by assigning the relative

rotation between the mobile frame attached to the platform and the fixed frame

solid with the ground. In the development of robot’s kinematics, the two frames

O(x,y,z) and P(u,v,w) will be used for the purpose and the rotation matrix

will

be expressed by means of the following {

,

,

} Cardan angles set:

cccssscsscsc

csccssssccss

ssccc

O

Pzyx ,, RRRR

(8.1)

Path planning and wrist’s control, on the other hand, shall be developed by as-

signing the orientation of P*(u*,v*,w*) frame with respect to O*(x*,y*,z*) frame,

12

and the corresponding set {

x,

y,

z} of Cardan angles is defined in the same

manner by:

zzyyxxzyx

O

P

***

*

* ,, RRRR (8.2)

Since the attitude of the frame O* with respect to the frame O is provided by

the constant rotation matrix

∗

, the orientation of the mobile platform, if known

in the O(x,y,z) space, can be easily mapped in the O*(x*,y*,z*) space by:

RRRR O

O

O

P

TO

O

O

P**

*

* (8.3)

where the obvious identity

∗

∗

has been used.

8.6 Analysis of mobility

It is now shown that, if the mentioned assembly and mounting conditions are satis-

fied, the mobile platform is characterized by motions of pure rotation, as anticipat-

ed in previous section; with reference to Fig. 8.11, such conditions may be geo-

metrically expressed by:

1.

and

incident in P;

2.

perpendicular to the plane 〈

,

〉 (i.e.

∙

0 and

∙

0);

3.

lying in plane 〈

,

〉 (by taking into account (2), it leads to:

);

4.

not parallel to

, therefore

(for simplicity, the condition

∙

0 has been taken).

If the point P is considered belonging to the ith leg (i=1,2,3), its velocity can be

written as follows:

riii PPP 2 (8.4)

where

is the velocity of point P if considered attached to the 2nd link:

iiiiii dBP 4222 ˆ

wωBωBP (8.5)

and

is the velocity of point P relative to a frame attached to link 2 and centered

in point Bi:

iiiiiiri dBP 433 33 ˆ

ˆˆ wwwP

(8.6)

13

In (8.5), is the angular velocity of link 2:

iii 11 2 ˆ

wω

(8.7)

In the same way, with obvious meaning of the symbols, the vector describing

the velocity of point B can be expressed as:

riii BBB 1 (8.8)

where:

iiiii

iiiiiii

iiiiii

da

daa

ABa

4111

41111

111

ˆ

ˆˆ

ˆ

ˆˆˆ

ˆ

www

wwww

ωwB

(8.9)

iiri b2

ˆ

wB

(8.10)

If (8.5)-(8.10) are substituted back in (8.4), it is found:

iiiiiiii dab 433 12 ˆ

ˆˆˆ wwwwP

(8.11)

By dot-multiplying (8.11) by

and by taking into account the conditions (1)-

(4), it is finally obtained:

0

ˆ3 ii Pw (8.12)

that can be differentiated to yield:

0

ˆˆ 33 iiii PwPw

(8.13)

Equations (8.12) and (8.13), written for the 3 legs, build up a system of 6 linear

algebraic equations in 6 unknowns, the scalar components of

and

. Such a

system can be written in matrix form as follows:

0

P

P

L

i

i

(8.14)

The 6x6 matrix L collects the coefficients of the linear system and can be parti-

tioned as:

14

HH

0H

L (8.15)

with:

T

T

T

33

32

31

ˆ

ˆ

ˆ

w

w

w

H (8.16)

and 0 being the null 3x3 matrix.

If the matrix L is not singular, the system (8.14) only admits the trivial null so-

lution:

0PP (8.17)

which means that the point P does not move in space, i.e. the moving platform on-

ly rotates around P. On the other hand, it is possible to identify the singular con-

figurations of the wrist by posing:

0detdet 2 HL (8.18)

that leads to:

0

ˆˆˆ

det 333231

wwwH (8.19)

Equation (19) is satisfied only when the three unit vectors

,

and

are linearly dependent; therefore the platform will incur in a singularity configura-

tion if and only if the three vectors are linearly dependent. It might happen when:

the planes containing the three legs are simultaneously perpendicular to the

base plane;

such planes are coincident with the base plane (configuration not reachable);

at least two out of the three aforementioned planes are characterized by normal

unit vectors which are parallel.

This justifies the previous choice of having the legs laid on mutual orthogonal

planes, since such configuration results the most far from translation singularities.

15

8.7 Orientation kinematics

Orientation kinematics is based on the definition of the relative rotation between

mobile frame P(u,v,w) and fixed frame O(x,y,z), with P≡O; it can be solved by

writing one loop-closure equation for each leg as follows:

0

iiiiii BPDBADPA (8.20)

Equation (8.20) is conveniently expressed in a local frame Oi(xi, yi, zi), i=1,2,3

which is defined for each leg as follows: the xi axis is aligned with cylindrical

joint’s axis and the yi axis is chosen parallel to limb’s first link, when it is laid in

the initial configuration. It is obtained:

0RR

i

P

O

P

i

O

ii

ii

i

BP

c

sb

cb

a

0

0

0

0

0

1

1

(8.21)

In inverse kinematics the values of

Cardan angles (or equivalently the

elements rij of the rotation matrix

) are known and the joint variables ai must be

found; in this case loop closure equations (8.21) for i=1,2,3 represent three decou-

pled systems of non-linear algebraic equations in the unknowns ai,

1i and bi, that

can be solved to find the solutions:

13

21

3

13

11

3

112113

313

12

13

2

12

33

2

331312

232

11

32

1

11

22

1

223211

121

or

,atan2

or

,atan2

or

,atan2

s

rd

b

c

rd

b

rr

rdca

s

rd

b

c

rd

b

rr

rdca

s

rd

b

c

rd

b

rr

rdca

(8.22)

Equation (8.22) accounts for 8 different solutions of the inverse kinematics

problem, i.e. they are as many as the number of possible configurations of the 3

legs (2 possible configurations per leg). It is noted, however, that only one value is

found for the displacement of each actuated joint when the orientation of the plat-

form is given.

The direct kinematic problem, on the other hand, assumes the knowledge of

joint variables ai, i=1,2,3 and aims at finding all the admissible attitudes of the

platform in the space. The analysis is performed by means of simple trigonometric

manipulations; by substituting in (8.22) the expression of rij given in (8.1), it is ob-

tained:

16

d

ac

sscscr

d

ac

csr

d

ac

scr

3

31

2

23

1

12

(8.23)

Direct kinematics is solved when the full rotation matrix

has been comput-

ed, starting from the knowledge of the three components r12, r23, r31 given by

(8.23). By taking into consideration that the columns of

represent the unit vec-

tors of a Cartesian reference frame, the 3 relations of unitary modulus and the 3 re-

lations of mutual orthogonality could be exploited in order to find the 6 unknown

components, but the 6 arising non-linear equations would be pretty difficult to

solve in closed form. In the present case it is simpler to parameterize the rotation

matrix by means of the Cardan angles set given by (8.1) and to find the values of

by solving the non-linear system (8.23). It is noted that the parameteriza-

tion (but not wrist kinematics) degenerates when β=±π/2.

By considering separately the singular cases, from the first two equations in

(8.23) it is obtained:

s

r

r

s

12

23

(8.24)

and by considering the first and the last equations in (8.23) it results:

1

2

22

31

222

31

2

2

12

cc

ssrssr

s

r (8.25)

Equation (8.25) can be elaborated so as to obtain:

22222

31

222

31

222222

12

1

21

sssss

ssrssrsssssr

(8.26)

and taking into consideration (8.24):

0 24

ss (8.27)

with:

17

0

01

012

2

12

2

12

2

23

2

31

2

23

2

12

2

23

12

2331

r

rrr

r

r

r

r

rr

(8.28)

It can be observed that, if the discriminant of (8.27) is positive, the signs of

and

guarantee that the two solutions of (8.27) are positive real and there-

fore 4 real values for sin

and 8 values for the angle

can be found. The values of

the three parameters

and

which cause negative values of the discriminant,

correspond to reflections matrices (i.e. the determinant is equal to –1): since this

kind of matrices cannot be found using the parameterization of the Cardan angles,

in this case the solutions of (8.27) become complex. Once γ is known, the value of

cosβ can be obtained from the first equation in (8.23):

s

r

c12

(8.29)

It is noted that, for the chosen parameterization, angle

varies in the range

[-π/2, π/2]: since four of the eight previously obtained values of

are certainly in

quadrants I or IV whereas the other four are in quadrants II or III, there are at the

same time 4 positive values of cosβ (hence acceptable) and 4 negative values,

which cannot be considered any further. Hence, up to now, 8 acceptable solutions

for the pair of angles (γ, β) have been found.

Finally, the last two equations in (8.23) are used to work out the values of cosα

and sinα:

ccs

srcr

c

c

r

s

2331

23

(8.30)

Equation (8.30) provides just one value of

for each pair

) previously de-

termined. Hence the problem, and also the direct kinematics of the 3-CPU mecha-

nism, admits at most eight solutions (Innocenti and Parenti-Castelli 1993). Of

course the problem is comprehensively solved only when all the singular cases are

separately treated.

18

8.8 Differential kinematics

By direct differentiation of the equations (8.22), the analytic Jacobian matrix JA is

obtained:

A

csssccccsccss

sscc

ccss

d

a

a

a

J0

0

3

2

1

(8.31)

The vector of angular velocity ω of the platform is related to the derivatives of

the Cardan angles by the relation:

0

0

01

T

ccs

csc

s

z

y

x

(8.32)

Using equations (8.31) and (8.32) it is possible to work out the expression of

the geometric Jacobian JG of the mobile-platform:

z

y

x

G

z

y

x

AA

a

a

a

JTJJ 1

3

2

1

(8.33)

with:

0

0

0

ccsccss

scc

ssscccsssc

d

G

J (8.34)

It is noted that the geometric Jacobian JG is not a function of geometric pa-

rameters, therefore machine’s manipulability cannot be optimized by a proper se-

lection of functional dimensions.

Turning to the analysis of singular poses, it is evident that limbs’ structure does

not allow for inverse kinematics singularities; on the other hand, direct kinematics

singularities can be found by letting the determinant of the geometric Jacobian JG

vanish:

2

23

det

sssccsd

GJ (8.35)

The zeroes of this equation all lie on closed surfaces of the three dimensional

space α, β, γ; their intersections with the coordinate planes generate straight lines,

as given by:

19

2

0

2

,

2

0

2

0

(8.36)

(a) (b)

Fig. 8.13. Singularity surfaces of ,, (a) and

∗,

,

(b)

(a) (b)

Fig. 8.14. Color plot of the determinant of ,, (left) and

∗,

,

(right) on several

planes (black regions represent almost null values)

Figures 8.13a and 8.13b represent the singularity surfaces in the α, β, γ and in

the ,

,

spaces respectively.

Due to the complexity of the geometries represented in Fig. 8.13, it can be use-

ful to plot the value of the determinant on several layers of the workspace, as

shown in Fig. 8.14 for the two Cardan angles sets used in this work.

20

In Fig. 8.15a, moreover, the workspace volumes whose determinant assumes

values in the range [–0,05, +0,05] have been taken out of the representation, while

the color map still represents the local determinant value: it is now more apprecia-

ble the extent of singularity-free regions inside the workspace, Fig. 8.15b, where

the planning of a motion could be performed: e.g. for the mechanism under design

a sphere with a radius of about 50° can be internally inscribed.

Finally, the translation singularities found in (8.19) can be easily expressed as

a function of articular coordinates

1i by:

131211131211

cccsss

(8.37)

and taking into consideration inverse kinematics (8.22) it is also obtained:

33 22 11 21 13 32

rrrrrr

(8.38)

Equation (8.38) is a useful expression of translation singularities in task space,

where the elements of the rotation matrix are used; the definition of the rotation

matrix (8.1) and few trigonometric manipulations yield an alternative expression

in function of Cardan angles

:

0

2

2

sssccs

(8.39)

It is noted that (8.39) is satisfied for the same values of

that make (8.35)

vanish: therefore translation singularities coincide with direct kinematics singu-

larities, i.e. no additional singular surfaces are present inside workspace.

(a) (b)

Fig. 8.15. Singularity surface in the space: color map representing local determinant values

(a) and close-up view of a connected singularity-free region (b)

21

8.9 Dynamics

8.9.1 Analysis of static loads

The static analysis is useful in the first phases of machine design for the selec-

tion of machine’s motors and for a first design of the links, with the related con-

necting bearings. The base relation is provided as usual by the well-known duality

between kinematics and statics, which allows a straightforward assessment of the

actuation efforts needed to balance a moment npl applied at the mobile platform:

plz

ply

plx

T

G

n

n

n

J

3

2

1

(8.40)

It must be noted that the application of a force fpl at the center of the spherical

motion does not require balancing forces by the actuators but it is entirely born by

frame bearings: the internal reactions at the bearings caused by the application of

the mentioned external wrench have been evaluated as well and used during struc-

tural design.

8.9.2 Inverse dynamics model

Depending on the kind of tasks the wrist will be required to perform, it may not be

sufficient to grant assigned static features, but wrist’s dynamics should be ac-

counted for as well. In this section an inverse dynamics model of the 3-CPU

mechanism is worked out by using the virtual work principle: it is assumed that

frictional forces at the joints are negligible, therefore the work produced by the

constraint forces at the joints is zero and only active forces (including the gravita-

tional effects) must be accounted in the developments.

In the derivation of the model, the notation is based on Fig. 8.11b and the se-

cond subscript i (i=1,2,3) indicates the ith limb while the first subscript j (j=1,2)

refers to the first or second link respectively. Namely, mji and Iji are the mass and

(central) inertia tensor of the jth member of the ith limb;

ji is its angular velocity

and vji is the linear velocity of its centre of mass; mpl, Ipl,

pl, vpl are the same

quantities referred to the mobile platform. The total wrench of active and inertial

effects acting on the centre of mass of jth member of the ith limb is written as:

jijijijiji

jijiji

ji

ji

ji

mm

ωIωωI

vg

n

f

F

(8.41)

22

In the same manner, the total wrench acting on the centre of mass of the mobile

platform is:

eplplplplpl

eplpl

pl

pl

pl

m

nωIωωI

fvg

n

f

F

(8.42)

where fe and ne are the external force and moment applied to its centre of mass;

it is accidentally noted that the centre of mass of the platform does not coincide

with the fixed point O. If is the vector of the actuation forces and q are the corre-

sponding displacements, the principle of virtual work can be written for the pre-

sent case:

0

3

1

2

1

ij

ji

T

jipl

T

pl

Tδδδ FxFxτq (8.43)

where the vector xji gathers the position of the centre of mass of jth member of

the ith limb and the orientation of the same link and xpl expresses the position of

the centre of mass of the mobile platform and its orientation. It is noted that all the

infinitesimal rotations appearing in (8.43) must be expressed as functions of the

angular velocity of the respective link, e.g. for the platform:

tωωωvvv T

plzplyplxplzplyplxpl

x (8.44)

Since all the virtual displacements in (8.44) must be compatible with the con-

straints, they are not independent but can rather be expressed as functions of an

independent set of Lagrangian coordinates; if the Cardan angles φ=[

]T of

the mobile platform are chosen for this purpose, the following relations hold be-

tween the introduced virtual displacements:

φJq δ δ φJx δ δjiji

φJx δ δplpl

(8.45)

where J, Jji and Jpl are proper Jacobian matrices that can be found through the

usual velocity analysis of the mechanism. Equation (8.43) can be written again as:

0

3

1

2

1

ij

ji

T

ji

pl

T

pl

TT FJFJτJφ

(8.46)

Since (8.46) is valid for any virtual displacement

φ of the platform, in non-

singular configurations it is:

23

3

1

2

1ij

ji

T

ji

pl

T

pl

TFJFJJτ (8.47)

Equation (8.47) completely describes manipulator’s dynamics; all the elements

in it have been worked out and the resulting model has been proofed by compari-

son with commercial packages’ output.

8.9.3 Dynamic analysis in the task space

The dynamic expression (8.47) is usefully re-worked in order to explicit the

dependency on a proper set of Lagrangian coordinates and its derivatives. In the

case of parallel kinematics machines, the dynamic model results quite naturally

written in the task space, due to the (usually) difficult expression of DKP; there-

fore in the present case, after some cumbersome manipulation, it is obtained:

φGφφφCφφMhJτ

,

T

pl (8.48)

with: τJτT

, moments acting at the end-effector and corresponding to actual

forces at actuated joints;

φMφ, Cartesian mass matrix of the manipulator;

φφC

,

φ, vector of centrifugal and Coriolis terms;

φGφ, vector of gravity mo-

ments; h, vector of external forces and moments acting at the centre of mass of the

mobile platform.

Fig. 8.16. Values of mass matrix’ elements for null roll angle, i.e.

=0 (note the different scales

of the plots)

24

Fig. 8.17. Plots of mass matrix’ elements, normalised by determinant value, for null roll angle,

i.e.

=0 (note that all the scales of the graphs are multiplied by 10-6 but M(1,2) and M(2,3) which

are multiplied by 10-7).

In view of the realisation of possible control schemes based on the inversion of

manipulator’s dynamics, it is useful to study the variability of mass matrix

throughout the workspace. In fact, a major simplification of the model would be

yielded by neglecting the 6 non-diagonal terms of the mass matrix, whether actu-

ally allowed by their comparative magnitude; otherwise, all the elements in M

and C

could be considered constant. First simulation results show that in this case

both simplifying assumptions could be taken into consideration, even if the validi-

ty of the reduced models weakens when the operating trajectories get closer to

singularity surfaces, as expected.

Figure 8.16, for instance, shows the values of mass matrix’ elements in differ-

ent workspace configurations characterised by null roll angle, i.e.

=0: for robot’s

parameters it has been made reference to the virtual prototype, whose mass prop-

erties, presented in the following Table 8.1, are very similar to physical prototype

(h is the total length of the lower part of the three limbs). In Fig. 8.17 the same

plots have been normalised by dividing the matrix element by the (local) value of

matrix determinant, to allow a relative comparison among elements that have very

different magnitudes. It can be seen that near the isotropic point (

) the

diagonal elements are dominant and matrix variability is limited, while off-

diagonal elements show a stronger influence when getting closer to workspace

boundaries; moreover, element M(3,1) is generally an order of magnitude greater

than M(2,1) and M(2,3). Such behaviour gets even more evident if one moves

away from the plane

=0. The plots have been traced for pitch and yaw angles

varying between -50° and +50° because the sphere of 50° radius in the Cardan an-

gles space is completely free of singularities, as shown already in Fig. 8.15.

Other kinds of tests have been performed, aiming at identifying the relative

contribute of various dynamic terms: for instance it seems that, even for high dy-

namics manoeuvres, the contribute of gravity is never negligible, while Coriolis

25

and centrifugal forces account for 10%-16% maximum; on the other hand, the

mass and inertia of the mobile platform affect very slightly the overall dynamic

behaviour of the machine, possibly allowing for a major simplification of system’s

model.

Table 8.1. Main data used by the numerical model

Geometrical data Mass data

c 210 mm slider 7,15 kg

d 490 mm link 1 1,90 kg

h 280 mm link 2 2,21 kg

ai min 319 mm

p

latform 11,73 kg

ai ma

x

661 mm

bi min 130 mm

bi ma

x

210 mm

8.9.4 Dynamic manipulability

Machines’ manipulability is usually studied only from a kinematic point of

view, but high speed tasks may require to study also the dynamic behaviour of the

device when required to attain quick manipulation operations; on the other hand,

this is particularly important when the device to be studied or designed is a ma-

nipulator’s wrist, since the capability to develop high accelerations is usually one

of the most important features.

The this aim, the dynamic manipulability ellipsoids introduced by Yoshikawa

(1985, 2000) are a useful means to study the dynamic properties of a mechanism:

they express graphically the capability of a given device to yield accelerations in

all the directions stemming from one attitude of its workspace. As a matter of fact,

many other measures of manipulability have been proposed by different research-

ers since that pioneering work, see e.g. (Di Gregorio and Parenti-Castelli 2002),

but very few applications dealt with orienting devices.

Let us consider all the actuation forces with unit norm:

1ττT (8.49)

By manipulating (8.48) in order to work out , it is obtained:

biasφ

TφφMJτ (8.50)

having defined:

26

hJGφCMφT

pl

φφbias 1

(8.51)

A meaningful formulation of dynamic manipulability must be expressed as a

direct function of the angular acceleration ω

, therefore the mapping between the

rate of change of the Cardan angles φ

and the angular velocity must be made

explicit:

φφEω

css

csc

s

z

y

x

0

0

01

(8.52)

φφ,ωφφEω 1

(8.53)

If φ

is taken out of (8.53) and substituted in (8.50) it is then obtained:

biasφ

TωωEMJτ 1 (8.54)

having defined:

hJGφCMJEMωω T

pl

φφ

T

bias 11

1

(8.55)

The constraint expressed by (8.49) can be finally written in the following quad-

ratic form:

1 ΩφΓΩ T (8.56)

with obvious meaning of the introduced terms:

hJGφCMJEMωωωωΩ T

pl

φφ

T

bias

11

1

(8.57)

11

EMJJMEφΓ φ

TTT

(8.58)

The inspection of (8.57)-(8.58) shows that gravity merely induces a translation

of the dynamic manipulability ellipsoid while in general velocity has a complex,

non-negligible effect on manipulability. Making reference to the remarkable case

of a fixed platform ( 0φ

) with no external or gravity action applied (h=Gφ=0),

(8.56) provides:

1 ωφΓω T (8.59)

27

(a)

(b)

(c)

(d)

Fig. 8.18. Dynamic manipulability ellipsoids at different poses (

): (0°,0°,0°) (a),

(20°,20°,-5°) (b), (40°,40°,10°) (c), (54°,53°,10°) (d)

The quadratic form (8.59) represents an ellipsoid in the Cartesian space of the

angular accelerations: its eigenvalues express the square root of the maximum and

minimum accelerations that can be developed with unit actuator forces while the

eigenvectors represent the associated directions in the orientation space. Figure

8.18 represents graphically some dynamic manipulability ellipsoids of the robot in

the poses sketched aside.

Two dynamic figures can be defined in order to sum up some relevant aspects

of the dynamic manipulability analysis. The measure of the dynamic manipulabil-

ity, w, defined as:

φΓdetw (8.60)

results proportional to the volume of the manipulation ellipsoid and therefore

yields an overall information on the global manipulation capabilities, but fails to

capture the closeness to singular configurations or even the anisotropy of local dy-

namics. On the other hand, the index of dynamic manipulability, i, is independent

from the volume of the ellipsoid and vanishes close to singular configurations:

max

min

i

(8.61)

with

min,

max minimum and maximum eigenvalues of the matrix

φΓ

.

Usually a dynamic optimization of the design aims at trying to maximize the

global dynamic manipulability of the wrist while still guaranteeing a minimum

threshold of the local features. Figure 8.19, for instance, shows sample plots of the

index of dynamic manipulability as a function of actuators strokes ai on the three

coordinate planes for the final design of the wrist. For instance, in the four config-

urations shown in Fig. 8.18 the indexes assume the values: 0.7755, 0.1374,

28

0.3571, 0.0341 respectively, while it has been obtained a mean value of iave=0.502

over the central ± 30° span of the workspace.

(a) (b) (c)

Fig. 8.19. Plots of the index of dynamic manipulability as a function of actuators strokes on the

three coordinate planes a1=0 (a), a2=0 (b), a3=0 (c)

8.10 Prototype design

The design of a first prototype has been developed, with the aim of obtaining high

dynamic performance; as reference figures, the following requirements have been

posed:

orientation range (elevation and azimuth): 150°

maximum angular velocity: 500 °/s

maximum angular acceleration: 5 000 °/s2

spatial resolution: 0.01 °

overall dimensions of the machine: maximum volume of 1 m3.

(a) (b)

Fig. 8.20. CAD model of spherical wrist (a) and picture of a first laboratory prototype (b)

The particular form of the Jacobian matrix (8.34) does not allow for a mechani-

cal design based on the optimization of kinematic properties, since JG is not func-

tion of robot’s geometry, therefore heuristic considerations and several computer

simulations allowed to refine the design.

Figure 8.20a sketches the design of a prototype meeting the posed require-

ments; in Fig. 8.20b a picture of the machine is presented. The limbs are made of

29

avional (an aluminum-copper alloy) in order to join good mechanical properties

with a lightweight construction. The mobile platform is made of bronze, therefore

allowing the precise machining in a single placement of the 3 journal bearings that

have to meet orthogonally in a single point: in this way it has been obtained a high

stiffness together with precise geometrical alignments. It must be noted that such

revolute joints are idle, since no rotation occurs at all if all the manufacturing and

mounting conditions are correctly satisfied.

The actuation is based on 3 brushless linear motors Phase WVS 20.6.3, able to

provide a maximum thrust of 184 N at the speed of 6 m/s, with a maximum accel-

eration of 14.3 g and is controlled by Nation Instrument hardware (Flex-

motion/PXI architecture). Table 8.2 collects the most relevant data of prototype

actuation.

Table 8.2. Main actuation data

Characteristics of the three linear motors

Ms 2,95 kg Stator mass

Kt 58

NA

⁄Torque constant

In 3 A

N

ominal supply curren

t

T

n

184

N

N

ominal thrus

t

vn 6

ms

⁄

N

ominal spee

d

8.11 Conclusions

The chapter has presented the mechanical features of robotic wrists, especially ad-

dressing the kinematics of mechatronic devices to be used in grasping and ma-

nipulation tasks. A powerful kinematical tool for the synthesis of such kind of ma-

chines, based on Lie algebra, has been presented and an innovative spherical

parallel wrist developed at the Polytechnic University of Marche in Ancona has

been described, revisiting all the main design steps, from kinematic synthesis up to

physical prototyping.

The mechanical architecture of the machine is based on the 3-CPU kinematics,

that allows simple actuation and is characterized by easy direct and inverse rela-

tions; the 3-CRU variant has been discarded due to a much more complex kine-

matics but it could be taken into consideration for the design of mini- or micro-

machines.

Position and rate kinematics have been worked out in closed form and all the

singularity surfaces have been analyzed: it has been pointed out that the mecha-

nism does not possess inverse kinematics singularities, while direct kinematics

singularities and translation singularities lie on the same closed surface. The inner

space, where motion paths can be safely planned, has been identified and unfortu-

30

nately it cannot be enlarged by kinematics optimization because machine’s Jaco-

bian does not depend on geometrical parameters.

The kinematic concept of the mechanism has been developed up to the stage of

designing a spherical wrist to be used in cooperation with a translation machine

for assembly tasks. The physical prototyping of the machine allowed to validate

the good properties envisaged during the design phase, e.g. the high dynamics

mainly provided by the three direct drive linear motors. On the other hand, few

disadvantages of the concept itself have showed up: they are mainly due to the

scarce accessibility of the center of the spherical motion, which is common to

most parallel wrists, and to the difficult assembly, which requires a precise align-

ment of joints axes: this problem has been partially overcome by the manufactur-

ing of specific fixtures that are characterized by very high accuracy and are used

while assembling the machine.

In view of the strict cooperation of this wrist with a parallel translating machine

for grasping and assembly tasks, specific position and interaction control schemes

have been developed but the related results have been presented already elsewhere

by Carbonari et al. (2011) and Bruzzone and Callegari (2010).

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