Industrial Robotics Theory Modelling and Control Part 5 ppt

8 A Complete Family of Kinematically-Simple Joint Layouts: Layout Models, Associated Displacement Problem Solutions and Applications Scott Nokleby and Ron Podhorodeski Podhorodeski an

5.1 Tensionability of DishBot

For the tensionability of DishBot, Theorem 1 given in Section 3 is not sufficient

The reason is that the spine force only affects the tension of drive cables and

has no influence on the passive ones As a result, it cannot leverage all the

ten-sions However, it should be noted that each passive cable has a pre-tensioning

spring to maintain the tension Therefore, the tensionability can be still proved

based on its definition in Section 2 For the proof of tensionability, we use the

idea of Theorem 1 (not the theorem itself), i.e tension can be generated in the

cables to any extent while the static equilibrium is satisfied

The free body diagram of the end-effector is shown in Fig 8 The passive

ca-bles are in parallel with the spine and their tensions are shown by a

super-script p while for the tension of drive cables, a supersuper-script d is used The static

equilibrium equations are found to be:

u u

u

¦ 1 ˆ1 2 ˆ2 3 ˆ3 1d 2d 3d ˆ

s d

d d

ττ

τ 's are passive cable forces, rj's

are the position vectors of the anchor points of the passive cables (j=1,2,3) and

wˆ

s

f is the spine force

Figure 8 Free body diagram of DishBot’s end-effector

A quick inspection of the above equations shows that Eq (35), which is the equilibrium of the moments, is a set of homogenous equations independent from the spine force which, in general, results in zero tension for passive ca-bles The tension of the drive cables are found from Eq (34) Note that the drive cables form a cone which contains the spine and hence, using Lemma 4, the drive cable tensions are positive as long as the equivalent spine force,

p p

p

s

f −τ1 −τ2 −τ3 , is positive (compressive) As a conclusion, tension in the drive cables can be generated to any desired level by choosing a large enough spine force but it does not affect the tension in the passive cables

In order for DishBot to be tensionable, we also need to show that the tension in

the passive cables can be increased to any desired level For this purpose, note

that, in Fig 8, if O′ is the geometrical center of the three anchor points of the passive cables (P1, P2 and P3) then Eq (35) has non-zero solutions for passive cable tensions In this case, the solution would be τ1p =τ2p =τ3p =τp where τp

is an arbitrary real value and thus can be positive It is known that such a

geometrical center coincides with the centroid of triangle ¨P1P2P3 As a result,

if O′ is the centroid of triangle ¨P1P2P3, positive equal tensions in passive bles are determined only by the pre-tensioning springs

ca-As a conclusion, DishBot is tensionable as long as the following conditions are

met:

1 O′ is the centroid of ¨P1P2P3,

2 The pretension of the pre-tensioning springs are equal (τ p) and positive (tensile),

3 The spine lies inside the cone of the drive cables

4 The spine force satisfies − 1p − 2p − 3p >0

s

s

f >3τ

6 Planar cable-based manipulators with translational motion

Planar manipulators with translational motion (in XY plane) are sufficient for

many industrial pick-and-place applications such as packaging and material handling Simplicity of these manipulators compared to spatial ones further facilitates their applications where a two axis motion is sufficient (Chan, 2005) Two new designs of planar cables-based manipulators with translational mo-tion are studied here that are tensionable everywhere in their workspace

Schematic diagrams of these manipulators are shown in Fig 9 The spine is connected to the base and end-effector by revolute joints The end-effector is constrained by three cables Two of the cables form a parallelogram which eliminates the rotation of the end-effector as long as the cables are taut As a

result, the end-effector can only move in X and Y directions

Figure 9 Planar cable-based manipulators with pure translational degrees of freedom

In the first design (Fig 9a), the parallelogram is maintained by two winches with a common shaft which makes them move simultaneously and hence,

keep the cable lengths equal Similar to BetaBot and DishBot, the workspace of

this manipulator is only limited by the minimum and maximum lengths of the

spine and hence it can theoretically span a half circle above the base In the second design (Fig 9b), a pair of synchronous rotating arms preserves the par-allelogram without changing the length of the cables and therefore, possess a smaller workspace The synchronization can be obtained by a pair of pulleys and a timing belt or a simple 4-bar parallelogram as seen in Fig 9b

The kinematics of these manipulators consist of a single planar cone and hence easy to formulate for both direct and inverse solutions In this paper, however, our main focus is on the their tensionability and rigidity which is presented in the following

6.1 Tensionability of Planar Manipulators

The planar manipulators of Fig 9 are both tensionable everywhere in their workspaces This can be proved using an approach similar to the one that was

used for BetaBot There are two geometrical conditions that should be met for

the tensionability of these two manipulators As depicted in Fig 10a, these two conditions are as follows:

Figure 10 a) The configuration of the cables and spine in planar manipulators, b) The free body diagram of the end-effector

Condition 1. Cable 1 (Fid 10a) is always on the right and Cable 3 is

al-ways on the left side of the spine This is obtained if the spine is hinged to the base at a proper point between the two sets of cables

Condition 2. On the end-effector, the spine and Cable 3 are concurrent at

point E which is located somewhere between Cables 1 and 2

To prove the tensionability, we show that a compressive spine force can be

balanced by positive tensions in the cables The proof is quite similar to the one

of Theorem 2 and is briefly explained here

We first consider the force equilibrium of the end-effector subject to a

com-pressive spine force According to the free body diagram shown in Fig 10b, we

have:

w u

u 0

w u

uˆ1 3ˆ2 ˆ 1ˆ1 3ˆ2 ˆ

Due to Condition 1, the direction of the spine, wˆ is located between uˆ1and uˆ2

(cable directions) Therefore, the projection of − f swˆ on uˆ1and uˆ2 will be

posi-tive and hence σ1,τ3>0 Now, let:

1 2 1

2 2 1

E P P

It is clear that τ1+τ2 =σ1 Since σ1 >0 and due to the distribution given in Eq

(37), the moment of τ1ˆu1 about E cancels the one of τ2uˆ1 and hence, these two

forces can be replaced by σ1ˆu1 without violating the static equilibrium Finally,

since all three forces on the end-effector, σ1uˆ1, τ1ˆu1 and τ2uˆ1, are concurrent at

E, the equilibrium of the moments is also met which completes the proof

7 Conclusion

In this paper, several new cable-based manipulators with pure translational

motion were introduced and their rigidity where thoroughly studied The

sig-nificance of these new designs can be summarized in two major advantages

over the other cable-based manipulators:

1 Cables are utilized to provide kinematic constraints to eliminate rotational

motion of the end-effector In many industrial applications, reduced DoF

manipulators are sufficient to do the job at a lower cost (less number of

axes)

2 These manipulators can be rigidified everywhere in their workspace using

a sufficiently large pretension in the cables

In order to study the rigidity of these manipulators, the concept of

tensionabil-ity was used and a theorem was given to provide a sufficient condition for

ten-sionability Using this theorem, tensionability of each manipulator was proved

using line geometry and static equilibrium in vector form For each of these manipulators, it was shown that as long as certain conditions are met by the geometry of the manipulator, the tensionable workspace in which the manipu-lator can be rigidified, is identical to the geometrical workspace found from the kinematic analysis

BetaBot and the planar manipulators are tensionable everywhere and can be rigidified only by a sufficiently large spine force In DishBot, on top of the

geometrical conditions, a relation between the spine force and pre-tensioning springs of passive cables should be also satisfied to maintain the rigidity of the manipulator

8 References

Barrette G.; Gosselin C (2005), Determination of the dynamic workspace of ble- driven planar parallel mechanisms, Journal of Mechanical Design,

ca-Vol 127, No 3, pp 242-248

Behzadipour S (2005), High-speed Cable-based Robots with Translational Motion,

PhD Thesis, University of Waterloo, Waterloo, ON, Canada

Behzadipour S.; Khajepour A., (2006), Stiffness of Cable-based Parallel nipulators with Application to the Stability Analysis, ASME Journal of Mechanical Design, Vol 128, No 1, 303-310

Ma-Chan E (2005), Design and Implementation of a High Speed Cable-Based Planar Parallel Manipulator, MASc Thesis, University of Waterloo, Waterloo, ON,

Canada

Clavel R., (1991), Conception d'un robot parallele rapide a 4 degres deliberate, PhD

Thesis, EPFL, Lausanne

Dekker R & Khajepour A & Behzadipour S (2006), Design and Testing of an

Ultra High-Speed Cable Robot”, Journal of Robotics and Automation, Vol

21, No 1, pp 25-34

Ferraresi C.; Paoloni M.; Pastorelli S.; Pescarmona F, (2004), A new 6-DOF allel robotic structure actuated by wires: the WiRo-6.3, Journal of Robot- ics Systems, Vol 21, No 11, pp 581-595

par-Gallina P; Rosati G., (2002), Manipulability of a planar wire driven haptic vice, Mechanism and Machine Theory, Vol 37, pp 215-228

de-Gouttefarde M.; Gosselin C (2006), Analysis of the wrench-closure workspace

of planar parallel cable-driven mechanisms, IEEE Transactions on Robotics, Vol 22, No 3, pp 434-445

Kawamura S.; Choe W.; Tanak S Pandian S.R., (1995),Development of an

ul-trahigh speed robot FALCON usign wire driven systems, Proceedings of IEEE International Conference on Robotics and Automation, pp 215-220,

IEEE, 1995

Khajepour A & Behzadipour S & Dekker R & Chan E (2003), Light Weight

Parallel Manipulators Using Active/Passive Cables, US patent provisional file No 10/615,595

Landsberger S.E.; Sheridan T.B., (1985), A new design for parallel link

manipu-lators, Proceedings of the International Conference on Cybernetics and Society,

Mechatron-Ming A.; Higuchi T (1994a), Study on multiple degree-of-freedom positioning

mechanism using wires (Part1), International Journal of Japan Society of Precision Engineering, Vol 28, No 2, pp 131-138

Ming A.; Higuchi T (1994b), Study on multiple degree-of-freedom positioning

mechanism using wires (Part2), International Journal of Japan Society of Precision Engineering, Vol 28, No 3, pp 235-242

Ogahara Y.; Kawato Y.; Takemura K.; Naeno T., (2003), A wire-driven ture five fingered robot hand using elastic elements as joints, Pro- ceedings of IEEE/RSJ International Conference on Intelligent Robots and Sys- tems, pp 2672-2677, Las Vegas, Nevada, 2003

minia-Oh s.; Makala K K.; Agrawal S., (2005a) Dynamic modeling and robust

con-troller design of a two-stage parallel cable robot, Multibody System namics, Vol 13, No 4, pp 385-399

Dy-Oh s.; Makala K K.; Agrawal S., Albus J S (2005b), A dual-stage planar cable robot: Dynamic modeling and design of a robust controller with posi-tive inputs, ASME Journal of Mechanical Design, Vol 127, No 4, pp 612-

620

Pusey j.; Fattah A.; Agrawal S.; Messina E., (2004), Design and workspace analysis of a 6-6 cable-suspended parallel robot, Mechanisms and Ma- chine Theory, Vol 39, No 7, pp 761-778

Robers G.R; Graham T.; Lippitt T (1998), On the inverse kinematics, statics, and fault tolerance of cable-suspended robots, Journal of Robotic Sys- tems,Vol 15, No 10, pp 581-597

Stump E.; Kumar V., (2006), Workspace of cable-actuated parallel tors, ASME Journal of Mechanical Design, Vol 128, No 1, pp 159-167

manipula-Tadokoro S.; Nishioka S.; Kimura T (1996), On fundamental design of wire configurations of wire-driven parallel manipulators with redundancy,

ASME Proceeding of Japan/USA Symposium on Flexible Automation, pp

151-158

Tsai L-W., (1996), Kinematics of A Three DOF Platform With Three Extensible

Limbs, In Recent Advances in Robot Kinematics, Lenarcic J and

Parenti-Castelli V., pp 401-410, Kluwer Academic, Netherlands

Verhoeven R.; Hiller M.; Tadokoro S (1998), Workspace, stiffness, singularities and classification of tendon-driven stewart platforms, In Advances in Robot Kinematics Analysis and Control, Lenarcic J and Husty L., pp 105-

114, Kluwer Academic, Netherlands

Verhoeven R.; Hiller M (2000), Estimating the controllable workspace of don- based Stewart platforms, In Advances in Robot Kinematics, Lenarcic

ten-J and Stanisic M., pp 277-284, Kluwer Academic, Netherlands

Yamaguchi F., (2002), A Totally Four-dimensional Approach / Computer-Aided Geometric Design, Springer-Verlag, Tokyo

8

A Complete Family of Kinematically-Simple Joint Layouts:

Layout Models, Associated Displacement

Problem Solutions and Applications

Scott Nokleby and Ron Podhorodeski

Podhorodeski and Pittens (1992, 1994) and Podhorodeski (1992) defined a nematically-simple (KS) layout as a manipulator layout that incorporates a spherical group of joints at the wrist with a main-arm comprised of success-fully parallel or perpendicular joints with no unnecessary offsets or link lengths between joints Having a spherical group of joints within the layouts ensures, as demonstrated by Pieper (1968), that a closed-form solution for the inverse displacement problem exists

ki-Using the notation of possible joint axes directions shown in Figure 1 and guments of kinematic equivalency and mobility of the layouts, Podhorodeski and Pittens (1992, 1994) showed that there are only five unique, revolute-only, main-arm joint layouts representative of all layouts belonging to the KS family These layouts have joint directions CBE, CAE, BCE, BEF, and AEF and are de-noted KS 1 to 5 in Figure 2

ar-Figure 1 Possible Joint Directions for the KS Family of Layouts

KS 1 - CBE KS 2 - CAE KS 3 - BCE

KS 4 - BEF KS 5 - AEF KS 6 - CCE

KS 7 - BBE KS 8 - CED KS 9 - ACE

KS 10 - ACF KS 11 - CFD KS 12 - BCF

KS 13 - CED

Figure 2 KS Family of Joint Layouts

Podhorodeski (1992) extended the work of Podhorodeski and Pittens (1992, 1994) to include prismatic joints in the layouts Podhorodeski (1992) con-cluded that there are 17 layouts belonging to the KS family: five layouts com-prised of three revolute joints; nine layouts comprised of two revolute joints and one prismatic joint; two layouts comprised of one revolute joint and two prismatic joints; and one layout comprised of three prismatic joints However, four of the layouts comprised of two revolute joints and one prismatic joint (layouts he denotes AAE, AAF, ABF, and BAE) are not kinematically simple,

by the definition set out in this chapter, due to an unnecessary offset existing between the second and third joints

Yang et al (2001) used the concepts developed by Podhorodeski and Pittens (1992, 1994) to attempt to generate all unique KS layouts comprised of two revolute joints and one prismatic joint The authors identified eight layouts

Of these eight layouts, five layouts (the layouts they denote CAE, CAF, CBF, CFE, and CCE) are not kinematically simple, as defined in this chapter, in that they incorporate unnecessary offsets and one layout (the layout they denote CBE) is not capable of spatial motion

The purpose of this chapter is to clarify which joint layouts comprised of a combination of revolute and/or prismatic joints belong to the KS family The chapter first identifies all layouts belonging to the KS family Zero-displacement diagrams and Denavit and Hartenberg (D&H) parameters (1955) used to model the layouts are presented The complete forward and inverse displacement solutions for the KS family of layouts are shown The applica-tion of the KS family of joint layouts and the application of the presented for-ward and inverse displacement solutions to both serial and parallel manipula-tors is discussed

2 The Kinematically-Simple Family of Joint Layouts

The possible layouts can be divided into four groups: layouts with three lute joints; layouts with two revolute joints and one prismatic joint; layouts with one revolute joint and two prismatic joints; and layouts with three pris-matic joints

revo-2.1 Layouts with Three Revolute Joints

Using arguments of kinematic equivalency and motion capability, ski and Pittens (1992, 1994) identified five unique KS layouts representative of all layouts comprised of three revolute joints Referring to Figure 1, the joint directions for these layouts can be represented by the axes directions CBE, CAE, BCE, BEF, and AEF, and are illustrated as KS 1 to 5 in Figure 2, respec-tively

Podhorode-Fundamentally degenerate layouts occur when either the three axes of the main arm intersect to form a spherical group (see Figure 3a) or when the axis

of the final revolute joint intersects the spherical group at the wrist (see Figure 3b), i.e., the axis of the third joint is in the D direction of Figure 1 Note that for any KS layout, if the third joint is a revolute joint, the axis of the joint cannot intersect the spherical group at the wrist or the layout will be incapable of fully spatial motion

(a) Layout CBF (b) Layout CBD

Figure 3 Examples of the Two Types of Degenerate Revolute-Revolute-Revolute outs

Lay-2.2 Layouts with Two Revolute Joints and One Prismatic Joint

Layouts consisting of two revolute joints and one prismatic joint can take on three forms: prismatic-revolute-revolute; revolute-revolute-prismatic; and revolute-prismatic-revolute

2.2.1 Prismatic-Revolute-Revolute Layouts

For a prismatic-revolute-revolute layout to belong to the KS family, either the two revolute joints will be perpendicular to one another or the two revolute joints will be parallel to one another If the two revolute joints are perpendicu-lar to one another, then the two axes must intersect to form a pointer, other-wise an unnecessary offset would exist between the two joints and the layout would not be kinematically simple The prismatic-pointer layout can be repre-sented by the axes directions CCE and is illustrated as KS 6 in Figure 2

For the case where the two revolute joints are parallel to one another, in order

to achieve full spatial motion, the axes of the revolute joints must also be lel to the axis of the prismatic joint If the axes of the revolute joints were per-pendicular to the axis of the prismatic joint, the main-arm's ability to move the centre of the spherical group would be restricted to motion in a plane, i.e., fundamentally degenerate In addition, a necessary link length must exist be-tween the two revolute joints The axes for this layout can be represented with the directions BBE and the layout is illustrated as KS 7 in Figure 2

paral-2.2.2 Revolute-Revolute-Prismatic Layouts

For a revolute-revolute-prismatic layout to belong to the KS family, either the two revolute joints will be perpendicular to one another or the two revolute joints will be parallel to one another If the two revolute joints are perpendicu-lar to one another, then the two axes must intersect to form a pointer, other-wise an unnecessary offset would exist between the two joints and the layout would not be kinematically simple The pointer-prismatic layout can be repre-sented by the axes directions CED and is illustrated as KS 8 in Figure 2

For the case where the two revolute joints are parallel to one another, the axes

of the revolute joints must also be parallel to the axis of the prismatic joint In addition, a necessary link length must exist between the two revolute joints The axes for this layout can be represented with the directions ADD Note that for this configuration, the layout is fundamentally degenerate, unless an addi-tional link length is added between joints two and three, since without the ad-ditional link length, the axis of the second revolute joint would always pass through the centre of the spherical joint group (see Figure 4a) Figure 4b illus-trates the non-degenerate KS layout with an additional link length between the second revolute joint and the prismatic joint However, the layout of Figure 4b

is kinematically equivalent to KS 7 and therefore is not counted as a unique KS layout

(a) Layout ADD (b) Layout ADD with Offset

Figure 4 Revolute-Revolute-Prismatic Layouts: a) Degenerate; b) Non-Degenerate

2.2.3 Revolute-Prismatic-Revolute Layouts

For a revolute-prismatic-revolute layout, in order to achieve spatial motion and belong to the KS class, the axes of the two revolute joints must be or-thogonal to one another The resulting KS layouts of axes can be represented

by the axes directions ACE and ACF and are illustrated as KS 9 and KS 10 in Figure 2, respectively

2.3 Layouts with One Revolute Joint and Two Prismatic Joints

Layouts consisting of one revolute joint and two prismatic joints can take on three forms: prismatic-revolute-prismatic; prismatic-prismatic-revolute; and revolute-prismatic-prismatic

2.3.1 Prismatic-Revolute-Prismatic Layouts

For a prismatic-revolute-prismatic layout, the two prismatic joints must be perpendicular to each other In order to achieve spatial motion and be kine-

matically simple, the axis of the revolute joint must be parallel to the axis of one of the prismatic joints The feasible layout of joint directions can be repre-sented by the axes directions CFD and is illustrated as KS 11 in Figure 2.

2.3.2 Prismatic-Prismatic-Revolute Layouts

For a prismatic-prismatic-revolute layout, the two prismatic joints must be perpendicular to each other In order to achieve spatial motion and be kine-matically simple, the axis of the revolute joint must be parallel to one of the prismatic joints The feasible layout of joint directions can be represented by the axes directions BCF and is illustrated as KS 12 in Figure 2

2.3.3 Revolute-Prismatic-Prismatic Layouts

For a revolute-prismatic-prismatic layout, the two prismatic joints must be perpendicular to each other In order to achieve spatial motion and be kine-matically simple, the axis of the revolute joint must be parallel to the axis of one of the prismatic joints The feasible layout of joint directions can be repre-sented by the axes directions CCD Note that this layout is kinematically equivalent to the prismatic-revolute-prismatic KS 11 Therefore, the revolute-prismatic-prismatic layout is not kinematically unique For a further discus-sion on collinear revolute-prismatic axes please see Section 2.5

2.4 Layouts with Three Prismatic Joints

To achieve spatial motion with three prismatic joints and belong to the KS class, the joint directions must be mutually orthogonal A representative lay-out of joint directions is CED This layout is illustrated as KS 13 in Figure 2

2.5 Additional Kinematically-Simple Layouts

The layouts above represent the 13 layouts with unique kinematics belonging

to the KS family However, additional layouts that have unique joint tures can provide motion that is kinematically equivalent to one of the KS lay-outs For branches where the axes of a prismatic and revolute joint are collin-ear, there are two possible layouts to achieve the same motion Four layouts,

struc-KS 6, 7, 11, and 12, have a prismatic joint followed by a collinear revolute joint The order of these joints could be reversed, i.e., the revolute joint could come first followed by the prismatic joint The order of the joints has no bearing on the kinematics of the layout, but would be very relevant in the physical design

of a manipulator Note that the d j and lj j elements of the corresponding rows

in the D&H tables (see Section 3.2) would need to be interchanged along with

an appropriate change in subscripts The presented forward and inverse placement solutions in Sections 4.1 and 4.2 would remain unchanged except for a change in the relevant subscripts

dis-In addition to the above four layouts, as discussed in Section 2.2.2, the layout shown in Figure 4b is kinematically equivalent to KS 7 Therefore, there are five additional kinematically-simple layouts that can be considered part of the

KS family

3 Zero-Displacement Diagrams and D&H Parameters

3.1 Zero-Displacement Diagrams

The zero-displacement diagrams (lj i = 0, for all revolute joints i ) for the KS

family of layouts for Craig's (1989) convention of frame assignment are sented in Figures 5 to 7 Note that the KS layouts in Figure 2 are not necessar-ily shown in zero-displacement The rotations necessary to put each of the KS Layouts from zero-displacement configuration into the configuration illus-trated in Figure 2 are outlined in Table 1

1

2 2

3.2 D&H Parameters

Table 2 shows the D&H parameters for the kinematically-simple family of joint

layouts The D&H parameters follow Craig's frame assignment convention

(Craig, 1989) and correspond to the link transformations:

00

coscos

sincossin

sin

sinsin

coscos

cos

sin

0sin

cos

1 1

1 1

1 1

1 1

1

ˆ ˆ

1 ˆ 1

ˆ

1

1 1

j j j

j j

j j

j j j

j j

j j

j j

j

j j

j j

j

j

d d a

Rot d Trans a

Trans Rot

j j

j j

αα

αθ

αθ

αα

αθ

αθ

θθ

where j−1jTis a homogeneous transformation describing the location and

orien-tation of link-frame F j with respect to link-frame F j−1, Rotxˆj−1( )αj− 1 denotes a

rotation about the xˆj−1 axis by αj−1, Transˆ −1( )a j− 1

0

0

1

1 1

j j

j

j

j

p R

where j−1jR is a 3x3 orthogonal rotation matrix describing the orientation of

frame F j with respect to frame F j−1 and j−1po j−1→o j is a vector from the origin of

frame F j−1 to the origin of frame F j

KS 11 KS 12

KS 13

Figure 7 Zero-Displacement Diagrams for Layouts with One Revolute Joint and Two

Primatic Joints (KS 11 and 12) or Three Prismatic Joints (KS 13)

4 Forward and Inverse Displacement Solutions

4.1 Forward Displacement Solutions for the KS Family of Layouts

The position and orientation of the spherical wrist frame F sph with respect to

the base frame F0 is found from:

0

0 0

3 2

3 1

p R T

T T

T

where the homogeneous transformation j−1jT is defined in equation (1) The

transformation 0T is the solution to the forward displacement problem: 0R

is the change in orientation due to the first three joints and 0 o0→o sph

p is the

loca-tion of the spherical wrist centre The homogeneous transformaloca-tions T0

sph for the KS family of layouts can be found in Tables 3 and 4 Note that in Tables 3 and 4, c i and s i denote cos( )θi and sin( )θi , respectively

KS F j− 1 αj−1 a j− 1 d j θj F j KS F j−1 αj−1 a j− 1 d j θj F j

1

F π/2 0 0 θ2 F2 F1 π/2 0 0 θ2 F2 2

sph

sph F

23

23 2 1 23 1 1 23 1

23 2 1 23 1 1 23 1

g c f s c s

g s f c s s s c c s

g s f c c s c s c c

−

−+

++

+

−

−

10

00

3 2 2 3

2 2

3 2

3 1 3 2 1 2 1 3 1 3 2 1 2 1 3 1 3 2 1

3 1 3 2 1 2 1 3 1 3 2 1 2 1 3 1 3 2 1

g s s f s s

s c

c s

g c c s c s f c s c c s c s s s s c c c s

g c s s c c f c c c s s c c s c s s c c c

−

−

−+

++

+

−

−

10

00

3 2 3

2 2

3 2

3 1 3 2 1 1 3 1 3 2 1 2 1 3 1 3 2 1

3 1 3 2 1 1 3 1 3 2 1 2 1 3 1 3 2 1

g s s s

s c

c s

g c c s c s f c c c s c s s s s c c c s

g c s s c c f s c s s c c s c s s c c c

00

3

3 12 1 3 12 12

3 12

3 12 1 3 12 12 3 12

g c c

s

g s s f s s s c

c s

g s c f c s c s c c

−

−+

++

+

−

−

10

00

3 2 3

2 2

3 2

3 1 3 2 1 1 3 1 3 2 1 2 1 3 1 3 2 1

3 1 3 2 1 1 3 1 3 2 1 2 1 3 1 3 2 1

g s s s

s c

c s

g c c s c s f s c c s c s s s s c c c s

g c s s c c f c c s s c c s c s s c c c

Table 3 Forward Displacement Solutions for KS 1 to 5

4.2 Inverse Displacement Solutions for the KS Family of Layouts

For the inverse displacement solution, the location of the spherical wrist centre

with respect to the base, 0 o0→o sph

Paul (1981) presented a methodology to solve the inverse displacement

prob-lem of 6-joint manipulators with a spherical wrist To demonstrate the

appli-cation of this methodology to the inverse displacement problem for the KS

family, KS 1 will be used as an example

3

3 2 3 2 2 3

2

3 2 3 2 2 3

2

d g c c s

g s s s s c c

s

g s c s c s c

00

01000

1

23 2 23 23

23 2 23 23

d

g c f s c s

g s f c s c

8

( ) ( ) ( )

00

2

3 2 1 2 1 1 2

1

3 2 1 2 1 1 2

1

g d c c

s

g d s s s s c c

s

g d s c s c s c

c

9

( ) ( )

00

3

2 3 1 3 1 1 3 1

2 3 1 3 1 1 3 1

g s s

c

d g c c c c s s c

d g c s c s c s

00

3

2 1 3 1 3 1 1 3

1

2 1 3 1 3 1 1 3

1

g c c

s

d c g s s s s c c

s

d s g s c s c s c

c

11

( ) ( )

00

01000

1

3 2 2 2

3 2 2 2

d

d g c c s

d g s s c

00

0

010

0

1 3 3 3

2

3 3

3

d g c c s

d

g s s

001

010

100

1 2 3

d d

d g

Table 4 Forward Displacement Solutions for KS 6 to 13

From the forward displacement solution presented in Table 3 for KS 1, the

fol-lowing relation exists:

p p p

Pre-1 1 0

1 1

y x

p

p c p s

p s p c g

From the second row of equation (6), a solution for θ1 can be found as:

( , )or atan2( , )

where atan2 denotes a quadrant corrected arctangent function (Paul, 1981)

Squaring and adding the first three rows of equation (6) allows an expression

for s3 to be found thus yielding a solution for θ3 of:

atan2

2 2 2 2 2

3 2

3 3

3

fg

g f p p p s s

3

2

c = − ands23 =s2c3+c2s3, expressions for s2 and c2 can be found thus

yielding a solution for θ2 of:

A similar procedure can be followed for the other KS layouts Inverse

dis-placement solutions for all 13 of the KS layouts are summarized in Tables 5

and 6

KS Inverse Displacement Solutions

1 θ1 =atan2(p y,p x)oratan2(−p y,− p x)

atan2

2 2 2 2 2

3 2

3 3

3

fg

g f p p p s s

3 1

1 3 2

3 2 2

3 1

1 3 2

fg s g f

gp c p s p c g s f c fg

s g f

p g s f p s p c g c

++

−+

+

=+

+

+++

=2

atan2

2 2 2 2 2

3 2

3 3

3

fg

g f p p p s s

3 2

2 2 2

2

g s f

p s

2 2

3 2 3 1

1 1

y x

y x

y x

x y

p p

gp c p c g s f c p

p

gp c p c g s f s c

s

+

−+

=+

++

3 3

2 3 3

fg

g f p p p c c

3 2 2

2 2

2

g s

p s s

2 1 2

2

3 2 3

1 1

1

y x

y x

y x

y x

p p

p g c f gp s c c p

p

gp s c p g c f s c

++

2 2

2 2 2

fg s

p p g s f c c

3 2 3

2 1

2 2

3 2 3

2 1 1

1 1

&

where,,atan2

y x

y x

y x

y x

p p

gp s s p g s c f c

p p

p g s c f gp s s s c

2 2 2 2 2

3 2

fg

g f p p p s

f p s p c c g s

p s c

3

1 1 2 3

2 2

KS Inverse Displacement Solutions

p c

g s

p s

c

3 2 3

2 2

3 2

3 3

3

fg

g f p p s s

3 3

2 2 2 3 2 3

3 3

2 2

2 2

g c g s f

gp c p g s f c g c g s f

p g s f gp c s c

++

−+

=+

+

++

d =± x2+ y2+ z2 −

3

( )

g d

p c

g d

p s p c s c

+

=+

1 1 2 2

3 atan2 1 , , where

θ

2 2 3 2 2

d =± x+ y−

2 3

1 2

2 2 3

1 1

1

y x

y x

y x

x y

p p

p d gp s c p

p

p d gp s s c

d =± x2+ y2 −

3

( )

g d

p c

g d

p s

=

=

3 2 3

2 2

d1=

Table 6 Inverse Displacement Solutions for KS 6 to 13

Referring to Tables 5 and 6, for KS 1 to 6, 8, 9, and 10, up to four possible tions exist to the inverse displacement problem For KS 7, 11, and 12, up to two possible solutions exist for the inverse displacement problem For KS 13 there is only one solution to the inverse displacement problem.

solu-For the inverse displacement solutions presented, undefined configurations occur when the spherical wrist centre of the arm intersects either the first or second joint axes, provided the axes are for a revolute joint In such a configu-ration, the inverse solution becomes undefined, i.e., an infinity of possible so-lutions exist Looking at KS 3 of Figure 5 as an example, if s3 =0 as illustrated, the spherical wrist centre intersects the second joint axis and the solution for

2

θ becomes arbitrary Similarly, ifp x = p y =0, the spherical wrist centre sects the first joint axis and the solution for θ1 becomes arbitrary

inter-Table 7 reports all of the undefined configurations for the KS family of layouts

If an undefined configuration was encountered, a value would be assigned to the arbitrary joint displacement

1 p x = p y =0Ÿθ1isarbitrary

arbitraryis

+g s fg

f

2 p x = p y =0Ÿθ1isarbitrary

arbitraryis

+ g s f

3 p x = p y =0Ÿθ1isarbitrary

arbitraryis

5 Discussion

5.1 Application of the KS Layouts

The KS family of layouts can be used as main-arms for serial manipulators or

as branches of parallel manipulators For example, KS 1 is a common arm layout for numerous industrial serial manipulators KS 4 is the branch configuration used in the RSI Research 6-DOF Master Controller parallel joy-stick (Podhorodeski, 1991) KS 8 is a very common layout used in many paral-lel manipulators including the Stewart-Gough platform (Stewart, 1965-66) KS

main-13 is the layout used in Cartesian manipulators

The choice of which KS layout to use for a manipulator would depend on tors such as the shape of the desired workspace, the ease of manufacture of the manipulator, the task required, etc For example, layout KS 1 provides a large spherical workspace Having the second and third joints parallel in KS 1 al-lows for the motors of the main-arm to be mounted close to the base and a simple drive-train can used to move the third joint

fac-5.2 Reconfigurable Manipulators

KS layouts are also very useful for reconfigurable manipulators ski and Nokleby (2000) presented a Reconfigurable Main-Arm (RMA) manipu-lator capable of configuring into all five KS layouts comprised of revolute only joints (KS 1 to 5) Depending on the task required, one of the five possible lay-outs can be selected

Podhorode-Yang, et al (2001) showed how KS branches are useful for modular figurable parallel manipulators

recon-5.3 Application of the Presented Displacement Solutions

5.3.1 Serial Manipulators

If a KS layout is to be used as a main-arm of a serial manipulator, the spherical wrist needs to be actuated Figure 8 shows the zero-displacement configura-tion and Table 8 the D&H parameters for the common roll-pitch-roll spherical-wrist layout The wrist shown in Figure 8 can be attached to any of the KS layouts

Figure 8 Zero-Displacement Diagram for the Roll-Pitch-Roll Spherical Wrist

1

−

j

F αj−1 a j− 1 d j θj F j sph

4

F −π/2 0 0 θ5 F5 5

Table 8 D&H Parameters for the Roll-Pitch-Roll Spherical Wrist

For the KS family of layouts with a spherical wrist, the forward displacement

sph

where ee6T is the homogeneous transformation describing the end-effector

frame F ee with respect to frame F6 and would be dependent on the type of

0

000

5 6

5 6

5

5 4 6 4 6 5 4 6 4 6 5

4

5 4 6 4 6 5 4 6 4 6 5

4

6

c s

s c

s

s s c c s c s s c c

c

s

s c c s s c c s s c

c

c

sph

For a 6-joint serial manipulator, Pieper (1968) demonstrated that for a

manipu-lator with three axes intersecting, a closed-form solution to the inverse

dis-placement problem can be found As demonstrated by Paul (1981), for a

6-joint manipulator with a spherical wrist, the solutions for the main-arm and

wrist displacements can be solved separately Therefore, the presented inverse

displacement solutions for the KS family of layouts (see Section 4.2) can be

used to solve for the main-arm joint displacements for serial manipulators that

use KS layouts as their main-arm and have a spherical wrist

For the inverse displacement solution of the main-arm joints, the location

(0 0 6

o

o→

p ) and orientation ( R0

6 ) of frame F6 with respect to the base frame in

terms of the known value T0

ee can be found from:

6 0

0 0

6 6

0 1 -

ee ee

ee

p R

T T T

o o o o o

o

o

p p p

0

0 0

0

0

p p

p

where p x, p y, and p z are found from equation (12) The inverse displacement

solutions for the KS family of layouts discussed in Section 4.2 can now be used

to solve for the main-arm joint displacements

For the inverse displacement solution of the spherical wrist joints, in terms of

the known value T0

ee , the orientation of F6 with respect to the base frame, R0

23 22 21

13 12 11 0

6 3 0 0

r r r

r r r T

R R R

R

Since the main arm joint displacements were solved above, the elements of

matrix R3

0 are known values and thus the right-hand-side of equation (14) is

known, i.e., r ij , i = 1 to 3 and j = 1 to 3, are known values

Substituting the elements of the rotation matrix 36R= sph3Rsph6R into equation

23 22 21

13 12 11

5 6

5 6

5

5 4 6 4 6 5 4 6 4 6 5 4

5 4 6 4 6 5 4 6 4 6 5 4 3 6

3

3

6

r r r

r r r

r r r

c s

s c

s

s s c c s c s s c c c s

s c c s s c c s s c c c

sph sph

where sph3R is dependent on the D&H parameter α3for the manipulator, i.e.:

2 if

, 0 1

0

1 0

0

0 0

1

2 if , 0 1

0

1 0

0

0 0

1

0 if , 1 0

0

0 1

0

0 0

1

3 3

3 3

3 3

πα

παα

Equation (15) can be used to derive expressions for the wrist joint

displace-ments θ4, θ5, and θ6 For example, if α3 =π 2, equation (15) becomes:

23 22 21

13 12 11

5 4 6 4 6 5 4 6

5 6

5

5 4 6 4 6 5 4 6

r r r

r r r

s s c c s c s s

s c

s

s c c s s c c s

33 4 4

4

4

s

r c s

r s c