Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 13 pps

Keywords: Conceptual design, complexity, kinematic chains, displacement sub-groups We propose here a formulation capable of measuring the complexity of the kinematic chains of robotic a

W.A Khan, S Caro, D Pasini, J Angeles

architecture

D.V Lee, S.A Velinsky

Robust three-dimensional non-contacting angular motion sensor

K Brunnthaler, H.-P Schr¨ ocker, M Husty

Synthesis of spherical four-bar mechanisms using spherical

kinematic mapping

R Vertechy, V Parenti-Castelli

Synthesis of 2-DOF spherical fully parallel mechanisms

G.S Soh, J.M McCarthy

Constraint synthesis for planar n-R robots

T Bruckmann, A Pott, M Hiller

Calculating force distributions for redundantly actuated

tendon- Stewart platforms

Complexity analysis for the conceptual design of robotic

817, Sherbrooke St West, Montreal, QC, Canada, H3A 2K6

{wakhan, caro}@cim.mcgill.ca, damiano.pasini@mcgill.ca, angeles@cim.mcgill.ca

Abstract We propose a formulation capable of measuring the complexity of

kine-matic chains at the conceptual stage in robot design As an example, two realizations of the Sch¨ onflies displacement subgroup are compared.

Keywords: Conceptual design, complexity, kinematic chains, displacement

sub-groups

We propose here a formulation capable of measuring the complexity

of the kinematic chains of robotic architectures at the conceptual-designstage The motivation lies in providing an aid to the robot designerwhen selecting the best design alternative among various candidates atthe early stages of the design process, when a parametric design is notyet available

In this paper, the complexity of three lower kinematic pairs (LKPs),the revolute, the prismatic and the cylindrical pairs, is first obtained.Then, a formulation to measure the complexity of kinematic bonds(Herv´e, 1978; Herv´e, 1999) is introduced Based on this formulation,the complexity of five displacement subgroups—the helical pair is leftout in this paper—is established Finally, as an application, two realiza-tions of the Sch¨onflies displacement subgroup (Angeles, 2004; Company

et al., 2001) are compared

and Kinematic Chain

A kinematic bond is defined as a set of displacements stemming fromthe product of displacement subgroups (Herv´e, 1978; Angeles, 2004),the bond itself not necessarily being a subgroup We denote a kinematicbond byL(i, n), where i and n stand for the integer numbers associated

© 2006 Springer Printed in the Netherlands.

359

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 359–368.

with the two end links of the bond There are six basic displacementsubgroupsR(A), P(e), H(A, p), C(A), F(u, v) and S(O) (Herv´e, 1978;

Herv´e, 1999; Angeles, 2004), each associated with a lower kinematic pair(LKP) In this notation, A stands for the axis of the kinematic pair in

question; e, u and v are unit vectors, O is a point denoting the center

of the spherical pair; and p is the pitch of the helical pair

A kinematic bond is realized by a kinematic chain A kinematic chain

is the result of the coupling of rigid bodies, called links, via kinematicpairs When the coupling takes place in such a way that the two linksshare a common surface, a lower kinematic pair results; when the cou-pling takes place along a common line or a common point, a higherkinematic pair is obtained Examples of higher kinematic pairs includegears and cams

There are six lower kinematic pairs, namely, revolute R, prismatic

P, helical H, cylindrical C, planar F, and spherical S These pairs can

be regarded as the generators of the six displacement subgroups listedabove Although the displacement subgroups can be realized by theircorresponding LKPs, it is possible to realize some of their displacementsubgroups by appropriate kinematic chains A common example is that

of the C(A) which, besides the C pair, can be realized by a suitableconcatenation of a P and a R pair

In this section, we propose a measure of the complexity of a givensurface We base this measure on the concept of loss of regularity LOR,defined as

LOR of the Surface of the R Pair. Typically, the surface ciated with the revolute pair is assumed to be a cylinder However, inorder to realize theR(A) subgroup, the translation in the axial direction

asso-of the cylindrical surface must be constrained This calls for additionalsurfaces, which must then be blended smoothly with the cylindrical sur-face in order to avoid curvature discontinuities

The above discussion reveals that the surface associated with a olute pair has to be a surface of revolution but cannot be an extrudedsurface; the cylindrical surface is both We should thus look for a gen-eratrixG other than a straight line, but with G2-continuity everywhere.

rev-The latter would allow a shaft of appropriate diameter to be blendedsmoothly on both ends The simplest realization ofG is a 2-4-6 polyno-mial, namely, P (x) = −x6+ 3x4− 3x2+ 1.

Figure 1(a) is a 3D rendering of the surfaceSR obtained by revolvingthe generatrixG about the x-axis, so as to blend with a cylinder of unitradius

(a)

10 15 20 25 30 35

r LOR

where y = P + r and r is the radius of the cylindrical shaft The r.m.s

of the two principal curvatures, κµ and κπ, can now be obtained, i.e.,

κrms =

1

The LOR of SR can now be evaluated by eq (1),

Fig 1(b) Notice that LORR is not monotonic in r Further, LORR

reaches a minimum of 10.2999 at r = 0.1132 We thus assign LORR =10.2999

Conceptual Design of Robotic Architectures 361

and depicted in

LOR of the Surface of the P Pair. The most common crosssection of a P pair is a dovetail, but we might as well use an ellipse,

a square or a rectangle A family of smooth curves that continuouslyleads from a circle to a rectangle is known as Lam´e curves (Gardner,1965) In their simplest form, these curves are given by xm+ ym = 1,where m > 0 is an even integer When m = 2, the corresponding curve

is a circle of unit radius, with its center at the origin of the x-y plane

As m increases, the curve becomes flatter and flatter at its intersectionswith the coordinate axes, becoming more like a square For m → ∞,the curve is a square of sides equal to two units of length and centered

at the origin A Fourier analysis based on the curvature of these curvesconfirms the intuitively accepted notion that the spectral richness, ordiversity, of the curvature increases with m (Khan, Caro, Pasini andAngeles, 2006)

The LOR of the surface of the prismatic pair obtained by extruding

a square or a rectangle is expected to have a very high value A Lam´ecurve L with m = 4 is plausibly the best candidate for the cross section

of the prismatic pair This curve is shown in Fig 2(a) Figure 2(b) is a3D rendering of the surfaceSPobtained by extruding L along the z-axis

–1 –0.5

0.5 1

–1 –0.5 0.5 1

y

x s

The r.m.s of the two principal curvatures, κµ and κπ thus reduces to

κrms = κµ The length parameter s and the homogenizing length l are,correspondingly, the distance traveled along SP, depicted in Fig 2(a),and the total length l of the Lam´e curve, whence σ ≡ s/l

The loss of regularity LORP of SP, the surface associated with the Ppair, is thus LORP= 19.6802

.

LOR of the Surface of the F Pair. The F pair is a generator ofthe planar subgroup F and requires two parallel planes, separated by

an arbitrary distance In order to avoid corners and edges, a suitable

‘blending option’ is the use of the quartic Lam´e curve The concept

is shown in Fig 3(a) Notice that the female element of the pair is anextruded surfaceSFf while the male element is a solid of revolution SFm

–0.5

0.5 0.5 x, y

d D>>d

GFm

GFf

ξ η

(a)

44 46 48 50 52 54

0 2 4 6 8 10

LOR

d

(b)

Figure 3 (a) Cross section of the simplest realization of the planar pair; (b) LOR

vs diameter d of the male element

The LOR of the planar pair, LORF, is defined by both the male andthe female elements Further, the contribution of a flat surface to theLOR is zero, a plane being a sphere of infinite radius We thus obtain

LORplane= lim

κ→0

||κ
rms||2

κµ= ξ

η
− η

ξ
(ξ
2+ η
2)3/2, κπ = 1

ξ

where, from Fig 3(a), η = y and ξ = x + d/2, and d/2 is the tance between the y and the η axes The length parameter s and thehomogenizing length l are, correspondingly, the distance traveled alongthe generatrix GFm depicted in Fig 3(a) and its total length l, whence

dis-σ ≡ s/l

Figure 3(b) is a graph between the LOR of SFm, LORFm, and thediameter d Notice that LORFm grows monotonically with d Further,LORFm reaches a limit of approximately 56.0399, whence LORFm =56.0399 Finally, the LORFis defined as LORF ≡ (LOFFf+LOFFm)/2 =37.8601

Conceptual Design of Robotic Architectures 363

.

LOR of the Surface of the C and S Pairs. The r.m.s of theprincipal curvatures of the cylindrical and the spherical surfaces is con-stant Hence, the loss of regularity is zero for the two surfaces, i.e.,LORC = LORS= 0.

We introduce here the geometric complexity of the LKPs based on theLOR introduced earlier: the geometric complexity KG|x of a pair x is

KG|x≡ LORx

where LORx is the loss of regularity of the surface associated with thepair x and LORmax ≡ max{LORR, LORC, LORP, LORF, LORS} Thegeometric complexity of the five LKPs of interest is, in the foregoingorder: 0.2721; 0; 0.5198; 1.0; and 0

In this section we lay the foundations for the evaluation of the plexity of any kinematic bond We first restrict our study to kinematicbonds that are realizable using LKPs; the study of bonds includinghigher kinematic pairs is as yet to be reported Next, we define thecomplexity K ∈ [0, 1] of a kinematic chain as a convex combination(Boyd, 2004) of its various complexities:

com-K = wJKJ+ wNKN + wLKL+ wBKB (8)where KJ ∈ [0, 1] is the joint-type complexity, KN ∈ [0, 1] the joint-number complexity, KL∈ [0, 1] the loop-complexity, and KB∈ [0, 1] thebond-realization complexity, with wJ, wN, wL, and wB denoting theircorresponding weights, such that wJ+ wN + wL+ wB = 1

Joint-type complexity is that associated with the type of LKPs used in

a kinematic chain We define a preliminary joint-type complexity KJ|x

as the geometric complexity KG|xof the x pair, the joint-type complexity

KJ of a kinematic bond L being defined as

KJ|L= 1

n(nRKJ|R+ nPKJ|P+ nCKJ|C+ nFKJ|F+ nSKJ|S) (9)where nR, nP, nC, nF and nS are the number of revolute, prismatic,cylindrical, planar and spherical joints, respectively, while n is the totalnumber of pairs

is the minimum number of LKPs required to produce a displacement ofbondL, and qN is the resolution parameter, to be adjusted according tothe resolution required Note that KN|L∈ [0, 1].

The loop-complexity KL|L of a kinematic bond is that associated withthe number of independent loops of the kinematic chain connecting thetwo links, i and n, of a kinematic bond L, with respect to the mini-mum required to produce the prescribed displacement set The loop-complexity can be evaluated by means of the formula:

KL|L= 1− exp(−qLL); L = l − lm (11)where l is the number of kinematic loops, lm the minimum number ofloops required to realize such a bond and qLthe resolution paramter

floating-of the geometric constraints, which is reported in (Khan, Caro, Pasiniand Angeles, 2006) A summary of the results of this analysis is dis-played in Table 1

The bond-realization complexity based on the geometric constraints

of its realization can now be defined as

Table 1 Realization cost of some geometric constraints

Intersection of two lines (e1× e2 )· q21 = 0 5A + 9M 9 Angle of intersection e1· e2 = cos α 2A + 3M 3 Parallelism b/w two lines e1× e2= 03 3A + 6M 6 Length of common normal ||q21− (q21· e1) e1 || 2 = d2 7A + 9M 9 Intersection of three lines det(C) = 0 30A + 36M 36

e1, e2 and e3 span 3D space det([ e1 e2 e3 ])
= 0 5A + 9M 9

Definition of the resolution parameters. Three resolution meters, namely qN, qLand qBwere introduced above These parametersprovide an appropriate resolution for the complexity at hand Since theforegoing formulation is intended to compare the complexities of two ormore kinematic chains, it is reasonable to assign a complexity of 0.9 tothe chain with maximum complexity and hence evaluate the normalizingconstant, i.e., for J = B, L, N ,

kine-Hence, using the geometric complexity of the LKPs as the ing joint-type complexities is not justified In order to solve this problem

correspond-we must look at the complexity of the displacement subgroups generated

by the five LKPs studied here

The basic displacement subgroups can be realized either by their responding pairs or by a kinematic bond The complexity of the dis-placement subgroups is defined as the complexity of the realization thatexhibits the minimum kinematic bond complexity

cor-The complexity of the five displacement subgroups generated by theLKPs considered here can now be evaluated In this vein, we apply theformulation introduced in the previous section to the different realiza-tions of the displacement subgroups under study Table 2 displays some

.

Table 2 Complexity of five displacement subgroups

pertinent realizations The minimum complexity values found forR(A),

P(e), C(A), F(u, v) and S(O) are, correspondingly, 0.0907, 0.1733, 0,

0.5412 and 0.6907 Normalizing the above results so that the mum is given a complexity of 1, we obtain the complexities of the fivedisplacement subgroups as

maxi-KJ|R = 0.1313, KJ|P = 0.2509, KJ|C = 0, KJ|F = 0.7836 (13)

Notice that, although these are not the joint-type complexity defined inSection 5, which are rather based on form than on function, the abovevalues can be used to evaluate the joint-type complexity in eq.(9)

We apply our proposed formulation to compute the complexity of two

includes three independent translations and one rotation about an axis

of fixed orientation Figure 4(b) shows the joint and loop graphs of theMcGill SMG (Angeles, 2005) and the H4 robot (Company et al., 2001).Table 3 displays the different complexity values associated with thetopology of the two robots Here, we note that the overall complexity

of the McGill SMG is lower than that of the H4 robot

motion capability of this subgroupSch¨onflies-motion generators The

.

R R R R R R R R

R R R R R R R R

R R R R R frame

S S S S

S S S S

S S S frame

tool

(b)

Figure 4 Joint and loop graphs of: (a) the McGill SMG; and (b) the H4 robot

The complexity analysis of kinematic chains at the conceptual stage

in robot design was proposed in this paper To do this, the complexity

of five lower kinematic pairs and a formulation of the complexity of matic bonds were introduced The complexity values of two realizations

kine-of the Sch¨onflies displacement subgroup were computed

References

Angeles, J (2005) The degree of freedom of parallel robots: a group-theoretic proach Proc IEEE Int Conf Robotics and Automation, Barcelona, 1017–1024 Angeles, J (2004) The qualitative synthesis of parallel manipulators ASME Journal

ap-of Mechanical Design 126, 617–624.

Boyd, S and Vandenberghe, L (2004) Convex Optimization Cambridge University Press, Cambridge.

Parallel Robot European Patent EP1084802, March 21.

Gardner, M (1965) The superellipse: a curve that lies between the ellipse and the rectangle Scientific American, 213, 222–234.

Herv´ e, J (1978) Analyse structurelle des m´ ecanismes par groupes de d´ eplacements Mech Mach Theory 13, 437–450.

Herv´ e, J (1999) The Lie group of rigid body displacements, a fundamental tool for mechanical design Mech Mach Theory 34, 719–730.

Khan, W.A., Caro, S., Pasini, D and Angeles, J (2006) The Geometric Complexity

of Kinematic Chains Department of Mechanical Engineering and Centre for telligent Machines Technical Report CIM-TR 0601, McGill University, Montreal Oprea, J (2004) Differential Geometry and its Applications Pearson Prentice Hall, New Jersey.

In-Taguchi, G (1993) Taguchi on Robust Technology Development Bringing Quality Engineering Upstream ASME Press, New York.

ROBUST THREE-DIMENSIONAL

NON-CONTACTING ANGULAR MOTION SENSOR

Danny V Lee

Department of Mechanical and Aeronautical Engineering

University of California Davis

dvxlee@ucdavis.edu

Steven A Velinsky

Department of Mechanical and Aeronautical Engineering

University of California Davis

savelinsky@ucdavis.edu

Abstract

Keywords:

1 Introduction

In the literature, there are a variety of devices based on a sphere

2003, there is little work on spherical encoders or other means of dimensional, orientation feedback without mechanical coupling

three-In this paper, a non-contacting, angular velocity sensor based on magnetometry is presented The primary application for this sensor is a ball wheel mechanism, which will serve as the drive train in a robust omnidirectional mobile platform Designed for operation in unstructured environments, the spherical tire will be subject to contamination and rotating in a cradle These include: spherical motors as in Chirikjian and

For devices based on a sphere rotating in a cradle, the axis of rotation of the sphere is arbitrary and can change instantaneously Consequently, a non- contact means of velocity sensing is desirable For the ball wheel mechanism, which serves as the drive train for a class of omnidirectional mobile robots, most existing methods are not feasible, such as optical techniques based on surface-pattern distinction Thus, in this paper, a robust, three-dimensional angular velocity sensor based on magnetometry is presented that tracks the orientation of ferromagnet embedded in the sphere An algorithm based on vector orthogonality is used to approximate the angular velocity vector of the sphere from the sampled orientation data

Stein 1999, Dehez et al., 2005, spherical continuously variable trans- missions as in Ostrowski 2000, Gillespie et al., 2002, and omni-directional vehicles based on the ball wheel mechanism as in West andAsada 1997, Ferriere et al., 2001 However, according to Stein et al.,

Motion-tracking, sensor, spherical motion

© 2006 Springer Printed in the Netherlands.

369

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 369–376.

wear As a result, optical encoder techniques that require surface

tracking in demanding environments, magnetic sensing is commonly

invasive gastrointestinal transit monitoring

Information on the configuration of a magnetic source provides a means of determining the configuration of the body to which the sources are attached In this case, the goal is to determine the axis of rotation and angular speed of the sphere given the absolute position of a point on the sphere, which the magnetometry scheme provides To solve this

1999 for applications in limb motion tracking in biomechanics

contrast or surface patterning, as proposed by Garner et al., 2001, Stein

et al., 2003, are not feasible for this application For non-contact motion employed Jacobs and Nelson 2001 use magnetic sensors to track use magnetic sensing for vehicle guidance; and Weitschies et al., 1994,

Figure 1 Schematic of field lines from magnetic dipole

abdominal cavity deflection in crash test dummies; Donecker et al., 2003

loyed This approach is described in Panjabi 1979, Halvorsen et al., 1999 inverse problem, a method based on vector orthogonality will be emp- Prakash and Spelman 1997 use magnetic marker tracking for non-

Consider the planar case as shown in Fig 1 The magnet is located at

originO mand the sensor is located at point Unit vector defines the

magnet axis,

S

ris the position vector fromO mtoO S, andT is the relative

orientation betweenrand The magnetic flux density vectorpˆ B is

decomposed into radial and tangential components and , and are,

TSP

sin 4

cos 2 3 0 3 0

r

M B r

M B

t

r

, (1)

whereP0is the permeability and M is the dipole moment The

relationship between the field components and the configuration

variables can be found in most texts on electromagnetic fields, such as

Shadowitz 1975 For the three-dimensional case, the expressions in Eq

1 can be used in the plane defined by vectors pˆand r It remains, then, to

find the relationship between the radial and tangential field components

and the the three-dimensional, measured field components A diagram of

the configuration is shown in Fig 2

eˆ eˆ t eˆ n

The magnetometer is positioned along the x-axis of the inertial reference

frame This significantly simplifies the geometry of the problem

{Bx,By,Bz} are the orthogonal field components from the magnetometer

Robust Three-Dimensional Non-Contacting Angular Motion Sensor 371

Figure 2 Sensor Diagram

and {l,m,n} are the direction cosines used to parameterize the magnet

axispˆ Next, orthogonal triad eˆr,eˆn,eˆt}is positioned atO Sand defined as,

u

n r t r

r n

p e

p e e r

r

eˆ , ˆ ˆˆ ˆˆ , ˆ ˆ ˆ (2) Unit vectore ris directed along the radial vector,e nis a unit vector normal

to the plane defined by and

pˆ r, and tis the tangential unit vector in the

n-plane, orthogonal to r Moreover, the trigonometric functions in Eq

1 can be expressed as a function of the direction cosines of ; as such,

l

T

T

(3)

For an arbitrary orientation of , the theoretical magnetic flux density

vector can be expressed as,

pˆ

t t r r

Eq 2 and Eq 3 provide the proper sign conventions through the

transformations Substituting Eq 1-3 into Eq 4 results in the following

n r M

m r M

l r M

B B

B

B

M z y x M

3 0 3 0 3 0

242

Eq 5 states that the direction cosines of are linearly proportional to the

measured field components In other words, this scheme directly

measures the motion of vector, fixed relative to the sphere, under

spherical motion It remians to solve the inverse kinematics problem of

extracting the angular velocity vector of the sphere given this data

pˆ

3 Inverse Kinematics of Spherical Motion

Calculating the angular displacement given the axis of rotation and

the trajectory of a point on the body, is a straightforward matter; several

orientation of the rotation axis and the angular displacement, given only

the trajectory of a point, is not well established A method for estimating

displacements are used to locate the instantaneous axis of rotation for

techniques are shown in Murray et al., 1994 However, determining the

these values can be found in Halvorsen et al., 1999 In this work, two