Chapter 10: Kinematic Synthesis

J. Michael McCarthy

August 22, 2000

1. Shape Grammar Theory - G. Stiny, MIT

2. Engineering Shape Grammars - J. Cagan, CMU

3. Stochastic Optimization in MEMS Design - E. K. Antonsson, Caltech

4. Topology optimization - P. Papalambros, U. Michigan

5. Comparing Invention to Discovery, J. Cagan, K. Kotovksy, H. Simon, CMU

6. Evolutionary Synthesis - M. Jakeila, Washington U. of St. Louis

7. AI-based synthesis - T. Stahovich, CMU

8. Mechanical Design Compilers - A. Ward, Ward Design

9. Function-based Synthesis - K. Wood, U. Texas, Austin

10. Kinematic/Mechanism Synthesis - J. M. McCarthy, U.C. Irvine

11. Formal Theory of Design - T. Tomiyama, U. Tokyo

12. Synthesis of Structures - S. Fenves, CMU

13. Architectural Design Synthesis - W. Mitchell, MIT

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1 Introduction

This chapter surveys the kinematic synthesis of machines and its recent appli-cations in robotics, design, and manufacturing. A machine is often describedas a mechanical system that transforms a power source into a desired appli-cation of forces and movement. The earliest machines were wedges, levers,and wheels that amplified human and animal effort. Later wind and waterwere used to drive gearing that rotated millstones and pumps (Dimarogonas1993). Today chemical, nuclear, hydro-electric and solar energy drive ma-chines that build, manufacture, transport, and process items that affect allaspects of our lives.

Kinematic synthesis determines the configuration and size of the mechan-ical elements that shape this power flow in a machine. In this chapter weshow how researchers in machine design and robotics use the concepts ofworkspace and mechanical advantage as the criteria for the kinematic syn-thesis of a broad range of machines.

The workspace of a machine part is the set of positions and orientationsthat it can reach. We will see that there are many representations of thisworkspace, however the best place to start is with the kinematics equationsof the chain of bodies that connect the part to the base frame. The veloc-ity of the part is then obtained by computing Jacobian of these kinematicsequations. The principle of virtual work links this Jacobian directly to themechanical advantage of the system.

We begin by showing how virtual work relates the input-output speedratios of a system to its mechanical advantage. This is developed for generalserial and parallel chains of parts within a machine. Then we explore thecurrent design theory for serial and parallel robots and find that workspaceand mechanical advantage, clothed in various ways, are the primary con-siderations. We then develop in detail the current results in the kinematicsynthesis of constrained movement which focusses on satisfying workspaceconstraints formulated as algebraic equations. Finally, we outline the config-uration space analysis of machines that shows the versatility of these conceptsfor representing assembly, tolerances and fixtures.

We conclude by noting that computer-aided kinematic synthesis softwareexists for specialized systems in research laboratories, but little is availablein engineering practice. Also, current systems leave untouched many of themachine topologies available for invention. A software environment that en-ables designer/inventors to specify a workspace and a desired distribution of

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mechanical advantage combined with visualization and simulation of candi-date designs to evaluate performance can provide remarkable opportunitiesfor new devices to serve our needs.

2 Kinematics and Kinetics

The term kinematics refers to geometric properties of movement such asposition, velocity, and acceleration of components of a machine, while kineticsrefers properties of forces acting on and exerted by the part. Newton’s secondlaw of motion relates kinetics and kinematics by defining the acceleration ofa particle as proportional to the difference between the force applied to theparticle and the force it exerts,

Fin − Fout = mx. (1)

If these forces are equal then the particle moves at a constant velocity.Newton’s law can summed over all particles in the machine and integrated

as the system moves along a trajectory x(t) to define the change in energyof the system as the difference in the input and output work. It is a law ofmechanics that this equality of work and energy remains unchanged for smallvariations δx of the trajectory. Thus, the variations in work and energy mustcancel for all virtual displacements, that is

δWin − δWout = δE. (2)

A machine is designed to minimize energy losses, often due to friction andfatigue, so δE = 0, which means the input and output variations in workmust cancel. This is known as the principle of virtual work (Greenwood1977, Moon 1998).

At a specific instant of time, we can introduce the virtual displacementδx = vδt, where v is the velocity and δt is a virtual time increment. Thisallows us to define the virtual work δWin = Pinδt and δWout = Poutδt whereP = F·v is the instantaneous power. The result is that Equation (2) becomes

(Pin − Pout)δt = 0. (3)

Thus, the usual assumption is that the machine does not dissipate power andin every configuration the input and output instantaneous power are equal.Because power is force times velocity, we have that the mechanical advantageof a machine is the inverse of its speed ratio. In the next section we examinethis in more detail.

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Figure 1: The part M is connected to the base frame F by a series of links(vertices) and joints (edges).

2.1 Machine Topology

A machine is constructed from a variety of elements such as gears, cams, link-ages, ratchets, brakes, and clutches. Each of these elements can be reducedto a set of links connected together by joints. Perhaps the simplest jointto construct, though difficult to analyze, is the cam-and-follower formed byone link pushing against a second follower link. In this case, the movementof the output link depends on the shape of the contacting surfaces. In con-trast, pure rotary and sliding joints, and joints constructed from them, areconsidered simple joints because they are easy to analyze, though difficultto construct. These two classes of joints are often termed higher and lowerpairs, respectively, (Reuleaux 1875, Waldron and Kinzel 1998).

Each component M of a machine is connected by a series of links andjoints to the base frame F of the device, and the topology of the systemis presented as a graph with the links as vertices and joints as edges (Kota1993). For example, the typical robot arm has the graph shown in Figure 1.The mathematical relation that defines the position of each machine part Min the frame F is called its kinematics equations.

2.2 Kinematics Equations

If we model the local geometry of higher-pair joints using rotary and slidingjoints, then the position of every component of a machine can be obtainedfrom sequence of lines representing the axes Sj of equivalent revolute orprismatic joints. Between successive joint axes, we have the common normallines Aij which together with the joint axes forms a serial chain, Figure 2.This construction allows the specification of the location of the part relativeto the base of the machine by the matrix equation

[D] = [Z(θ1, ρ1)][X(α12, a12)][Z(θ2, ρ2)]...[X(αm−1,m, am−1,m)][Z(θm, ρm)],(4)

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S1

a12

A12

B1x

zα12

S2

S3

A23a23

α23

B2z

θ1

d1

θ2

d2

x

Figure 2: A machine part is located in space by a sequence of frames con-sisting of axes Sj and their common normals Aij.

known as the kinematics equations of the chain (Paul 1981, Craig 1989).The set of all positions [D] obtained as the joint parameters vary over theirrange of movement defines the workspace of the component, also called itsconfiguration space (Greenwood 1977, Arnold 1978).

The matrices [Z(θj, ρj)] and [X(αij, aij)] are 4 × 4 matrices that definescrew displacements around and along the joint axes Sj and Aij, respectively(Bottema and Roth 1979). The parameters αij, aij define the dimensions ofthe links in the chain. The parameter θj is the joint variable for revolutejoints and ρj is the variable for prismatic joints. The trajectory P(t) ofa point p in any part of a machine is obtained from the joint trajectory,�θ(t) = (θ1(t), . . . , θm(t))T , so we have

P(t) = [D(�θ(t))]p. (5)

A single part is often connected to the base frame by more than one serialchain, Figure 3. In this case we have a set of kinematics equations for eachchain,

[D] = [Gj][D(�θj)][Hj], j = 1, . . . , n, (6)

where [Gj] locates the base of the jth chain and [Hj] defines the positionof its attachment to the part. The set of positions [D] that simultaneouslysatisfy all of these equations is the workspace of the part. This imposesconstraints on the joint variables that must be determined to completelydefine its workspace (McCarthy 1990, Tsai 1999).

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Figure 3: A part M is often connected to the base frame F by parallel seriesof links and joints.

2.3 Speed Ratios

The speed ratio for any component of a machine relates the velocity P of a

point P to the joint rates �θ = (θ1, . . . , θm)T . The velocity of this point isgiven by

P = v + �ω × (P − d), (7)

where d and v are the position and velocity a reference point and �ω is theangular velocity of the part.

The vectors v and �ω depend on the joint rates θj by the formula

{v�ω

}=

∂v

∂θ1

∂v∂θ2

· · · ∂v∂θm

∂�ω∂θ1

∂�ω∂θ2

· · · ∂�ω∂θm

θ1...

θm

, (8)

or

V = [J ]�θ. (9)

The coefficient matrix [J ] in this equation is called the Jacobian and is amatrix of speed ratios relating the velocity of the part to the input jointrotation rates (Craig 1989, Tsai 1999).

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2.4 Mechanical Advantage

If the machine component exerts a force F at the point P, then the poweroutput is

Pout = F · P =m∑

j=1

F ·(

∂v

∂θj

+∂�ω

∂θj

× (P − d)

)θj. (10)

Each term in this sum is the portion of the output power that can be asso-ciated with an actuator at the joint Sj, if one exists.

The power input at joint Sj is the product τj θj of the torque τj and jointangular velocity θj. Using the principle of virtual work for each joint we cancompute

τj = F · ∂v∂θj

+ (P − d) × F · ∂�ω∂θj

, j = 1, . . . ,m. (11)

We have arranged this equation to introduce the six-vector F = (F, (P−d)×F)T which is the resultant force and moment at the reference point d.

The equations (11) can be assembled into the matrix equation

�τ = [JT ]F, (12)

where [J ] is the Jacobian defined above in (8). For a chain with six jointsthis equation can be solved for the output force-torque vector F,

F = [JT ]−1�τ . (13)

Thus, we see that the matrix that defines the mechanical advantage for thissystem is the inverse of the matrix of speed ratios. This is a more generalversion of the statement that to increase mechanical advantage we mustdecrease the speed ratio.

3 Simple Machines

The design of a lever and wedge illustrates the use of the kinematics equationsand virtual work in kinematic synthesis. The kinematics equations definemovement of the output in terms of input variables and the principle ofvirtual work relate the mechanical advantage and speed ratio. The designparameters are selected to achieve a desired performance.

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Figure 4: A lever is driven by a force at A to lift a load at B.

3.1 The Lever

A lever is a solid bar that rotates about a fixed hinge O called its fulcrum.This is the simplest serial chain and it has the kinematics equations

[D] = [Z(θ)], (14)

which defines a pure rotation about the fulcrum.Let A be the point of application of an input force and B the point where

the output force is exerted. See Figure 4. The velocities of these points arevin = aθ and vout = bθ, where a and b are the radial distances from thefulcrum O to A and B, respectively. The speed ratio between these twopoints is

voutvin

=bθ

aθ=

b

a= R. (15)

The velocities of A and B are directed parallel to the forces Fin and Fout.So over a virtual time period with no energy losses we have

(Finvin − Foutvout)δt = 0. (16)

Thus, the mechanical advantage of the lever as

FoutFin

=vinvout

=1

R. (17)

The law of the lever results from the fact that the speed ratio is also theratio of the radial distances R = b/a.

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Figure 5: A wedge is driven in the x direction to slide a load along the ydirection.

The kinematic synthesis of a lever determines the lengths a and b thatprovide the desired speed ratio and therefore the desired mechanical advan-tage.

3.2 The Wedge

A wedge is a right triangle with apex angle α that slides horizontally along aflat surface and lifts a load vertically by sliding it along its inclined face andagainst a vertical wall, Figure 5. This system consists of two parallel chainsthat connect the base frame to the load. The first consists of the prismaticjoints formed by the base and face of the wedge, and the second is simply theprismatic joint of the load sliding against the wall. The kinematics equationsof these two chains are

[D] = [G1][Z(0, x)][X(α, 0)][Z(0, a)][H1]

and [D] = [G2][Z(0, y)][H2], (18)

where [Gi] and [Hi] locate the base and moving frames for the two chains.Often the relationship between the joint parameters in parallel chains can

be obtained from the geometry of the system. In this case we see that x andy are related by the equation of a line with slope tanα, that is

y = x tanα + k, (19)

where k is a constant. Thus, the vertical velocity y is related to the horizontalvelocity x by the speed ratio

y

x= tanα. (20)

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Assuming no energy losses the principle of virtual work yields

(Finx− Fouty)δt = 0, (21)

and we obtain the mechanical advantage

FoutFin

=1

tanα. (22)

The kinematic synthesis of this machine involves determining the physicalparameter α that provides a desired speed ratio.

The fundamental principles required to design the lever and wedge appearin the kinematic synthesis of serial and parallel robot arms. The primaryconcern is the workspace of the system and its mechanical advantage.

4 Serial Robots

A serial chain robot is a sequence links and joints that begins at a base andends with an gripper. See Figure 6. The position of the gripper is definedby the kinematics equations of the robot, which generally have the form

[D] = [Z(θ1, ρ1)][X(α12, a12)][Z(θ2, ρ2)]...[X(α56, a56)][Z(θ6, ρ6)], (23)

because the robot has six joints. The set of positions [D] reachable by therobot is called its workspace.

The links and joints of a robot are usually configured to provide separatetranslation and orientation structures. Usually, the first three joints areused to position a reference point in space and the last three form the wristwhich orients the gripper around this point (Vijaykumar et al. 1987, Gupta1987). This reference point is called the wrist center. The volume of space inwhich the wrist center can be placed is called the reachable workspace of therobot. The rotations available at each of these points is called the dexterousworkspace.

The design of a robot is often based on the symmetry of its reachableworkspace. From this point of view there are three basic shapes rectangular,cylindrical and spherical (Craig, 1989). A rectangular workspace is providedby three mutually perpendicular sliding, or prismatic, joints which form aPPPS chain called a Cartesian robot; S denotes the wrist which allows all

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Figure 6: A serial robot is defined a set of joint axes Si and the commonnormals Aij between them.

rotations about its center point. A rotary base joint combined with a ver-tical and horizontal prismatic joints forms a CPS chain with a cylindricalworkspace; C denotes a rotary and sliding joint with the same axis. TheP-joint can be replaced by a revolute joint that acts as an elbow in order toprovide the same radial movement. Finally, two rotary joints at right anglesform T-joint at the base of the robot that supports rotations about a verticaland horizontal axes. Radial movement is provided either by a P-joint or byan R-joint configured as an elbow. The result is a TPS or TRS chain with aspherical workspace.

It is rare that the workspace is completely symmetrical because joint axesare often offset to avoid link collision and there are limits to joint travel whichcombine to distort the shape of the workspace.

4.1 Design Optimization

Another approach to robot design uses a direct specification of the workspaceas a set of positions for the end-effector of a robotic system (Chen and Burdick1995, Chedmail and Ramstein 1996, Chedmail 1998, Leger and Bares 1998)which we call the taskspace. A general serial robot arm has two design

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parameters, link offset and twist, for each of five links combined with fourparameters each that locate the base of the robot and the workpiece in itsgripper, for a total of 18 design variables. The link parameters are oftenspecified so the chain has a spherical wrist and specific workspace shape.The design goal is often set the volume of the workspace and locate the baseand workpiece frames so the workspace of the system includes the specifiedtaskspace.

The taskspace is defined by a set 4× 4 transformations [Ti], i = 1, . . . , k.The problem is solved iteratively by selecting a design and using the associ-ated kinematics equations [D(�θ)] to compute the minimum relative displace-

ments [TiD−1(�θi)]. The invariants of each of these relative displacements are

used to construct an objective function

f(r) =k∑

i=1

‖[TiD−1(�θi)]‖. (24)

Parameter optimization yields the design parameter vector r that minimizesthis objective function.

Clearly, this optimization relies on the way the invariants are used todefine of a distance measure between the positions reached by the gripper andthe desired workspace. Park (1995), Martinez and Duffy (1995), Zefran etal. (1996), Lin and Burdick (2000) and others have shown that there is no suchdistance metric that is coordinate frame invariant. This means that unlessthis objective function can be forced to zero so the workspace completelycontains the taskspace, the resulting design will not be “geometric” in thesense that the same design is obtained for any choice of coordinates.

If the goal is a design that best approximates the taskspace, then wecannot allow both the location for the base frame of the robot and locationof the workpiece in the gripper to be design variables. For example, in theabove example, if the location of the base of the robot is given then we canfind a coordinate invariant solution for the location of the workpiece in thegripper. On the other hand, if the position of the workpiece is specifiedthe entire formulation may be inverted to allow a coordinate frame invariantsolution for the location of the base frame (Bobrow and Park 1995).

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4.2 4 × 4 Transforms as 4 × 4 Rotations

Etzel and McCarthy (1996) and Ahlers and McCarthy (2000) embed 4 × 4homogeneous transformations in the set of 4×4 rotation matrices in order toprovide the designer control over the error associated with coordinate framevariation within a specific volume of the world that contains the taskspace.

Consider the 4 × 4 screw displacement about the Z axis defined by

[Z(θ, ρ)] =

cos θ − sin θ 0 0sin θ cos θ 0 0

0 0 1 ρ0 0 0 1

. (25)

This defines a rigid displacement in the three-dimensional hyperplane x4 = 1of four dimensional Euclidean space, E4. The same displacement can bedefined in parallel hyperplanes, x4 = R, simply by dividing the translationcomponent by R, that is

[Z(θ, ρ)] =

cos θ − sin θ 0 0sin θ cos θ 0 0

0 0 1 ρR

0 0 0 1

. (26)

This matrix can be viewed as derived from the 4 × 4 rotation:

[Z(θ, γ)] =

cos θ − sin θ 0 0sin θ cos θ 0 0

0 0 cos γ sin γ0 0 − sin γ cos γ

. (27)

where tan γ = ρ/R. This formalism views spatial translations as rotations ofsmall angular values. The error ε associated with this approximation is lessthan

√L/R where L is the maximum dimension of the world space.

The result is an optimization strategy that computes both the base andworkpiece frames yielding essentially the same design for all coordinate changeswithin the given volume of space. This approach also provides the opportu-nity to tailor the number of joints in the chain and set the values of internallink parameters to fit a desired task.

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4.3 Speed Ratios

A six axis robot has a 6 × 6 Jacobian [J ] obtained from (8) that is an arrayof speed ratios relating the components of the velocity v of the wrist centerand the angular velocity �ω of the gripper to each of the joint velocities. Theprinciple of virtual work yields the relationship

F = [JT ]−1�τ , (28)

which defines the force-torque F exerted at the wrist center in terms thetorque applied by each of the actuators. The link parameters of the robotcan be selected to provide a Jacobian [J ] with specific properties.

The sum of the squares of the actuator torques of robot is often used asa measure of “effort” (Gosselin 1998, Albro et al. 2000). From (28) we have

�τT�τ = FT [J ][JT ]F. (29)

The matrix [J ][JT ] is square and positive definite. Therefore, it can beviewed as defining an hyper-ellipsoid in six-dimensional space (Shilov 1974).The lengths of the semi-diameters of this ellipsoid are the inverse of the ab-solute value of the eigenvalues of the Jacobian [J ]. These eigenvalues may beviewed as “modal” speed ratios that define the amplification associated witheach joint velocity. Their reciprocals are the associated “modal” mechanicaladvantage, so the shape of this ellipsoid illustrates the force amplificationproperties of the robot.

The ratio of the largest of these eigenvalues to the smallest, called thecondition number, gives a measure of the anisotropy or “out-of-roundness”of the ellipsoid. A six-sphere has a condition number of one and is termedisotropic. When the gripper of a robot is in a position with an isotropic Ja-cobian there is no amplification of the speed ratios or mechanical advantage.This is considered to provide high-fidelity coupling between the input andoutput because errors are not amplified (Salisbury and Craig 1982, Angelesand Lopez-Cajun 1992). Thus, the condition number is used as a criterionin a robot design (Angeles and Chablat 2000).

In this case, it is assumed that the basic design of the robot providesa workspace that includes the taskspace. Parameter optimization finds theinternal link parameters that yield the desired properties for the Jacobian. Asin minimizing the distance to a desired workspace, optimization based on theJacobian depends on a careful formulation to avoid coordinate dependency.

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Figure 7: A parallel robot can have as many as six serial chains that connecta platform to the base frame.

5 Parallel Robots

A robotic system in which two or more serial chain robots support an end-effector is called a parallel robot. Each of the serial chains must have sixdegrees-of-freedom, however, in general only a total of six joints in the entiresystem are actuated. A good example is the Stewart platform formed fromsix TPS robots in which usually on the P joint in each chain is actuated,Figure 7 (Fichter 1987, Tsai 1999, Merlet 1999).

The kinematics equations of the TPS legs are

[D] = [Gj][D(�θj)][Hj], j = 1, . . . , 6, (30)

where [Gj] locates the base of the leg and [Hj] defines the position of itsattachment to the end-effector. The set of positions [D] that simultaneouslysatisfy all of these equations is the workspace of the parallel robot.

Often the workspace of an individual chain of a parallel robot can bedefined by geometric constraints. For example, a position [D] is in theworkspace of the jth supporting TPS chain if it satisfies the constraint equa-tion

([D]qj − Pj) · ([D]qj − Pj) = ρ2j . (31)

This equation defines the distance between the base joint Pj and the point ofattachment Qj = [D]qj to the platform as the length ρj is controlled by the

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actuated prismatic joint. In this case the workspace is the set of positions[D] that satisfy all six equations, one for each leg.

5.1 Workspace

The workspace of a parallel robot is the intersection of the workspaces of theindividual supporting chains. However, it is not the intersection of the reach-able and dexterous workspaces separately. These workspaces are intimatelycombined in parallel robots. The dexterous workspace is usually largest nearthe center of the reachable workspace and shrinks as the reference pointmoves toward the edge. A focus on the symmetry of movement allowed bysupporting leg designs has been an important design tool resulting in manynovel parallel designs (Herve 1977, 1999). Simulation of the system is usedto evaluate its workspace in terms of design parameters.

Another approach is to specify directly the positions and orientationsthat are to lie in the workspace and solve the algebraic equations that definethe leg constraints to determine the design parameters (Murray et al. 1997,Murray and Hanchak 2000). This yields parallel robots that are asymmetricbut have a specified reachable and dexterous workspace.

5.2 Mechanical Advantage

The force amplification properties of a parallel robot are obtained by consid-ering the Jacobians of the individual supporting chains. Let the linear andangular velocity of the platform be defined by the six-vector V = (v, �ω)T ,then from the kinematics equations of each of the support legs we have

V = [J1]�ρ1 = [J2]�ρ2 = · · · = [J6]�ρ6. (32)

Here we assume that the platform is supported by six chains, but it can beless. This occurs when the fingers of a mechanical hand grasp an object(Mason and Salisbury 1985).

The force on the platform applied by each chain is obtained from theprinciple of virtual work as

Fj = [JTj ]−1�τj, j = 1, . . . , 6. (33)

There are only six actuated joints in the system so we assemble the associatedjoint torques into the vector �τ = (τ1, . . . , τ6)

T . If Fi is the force-torque

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obtained from (33) for τij = 1, then, the resultant force-torque W applied tothe platform is

W = [F1,F2, · · · ,F6]�τ , (34)

or

W = [Γ]�τ . (35)

The elements of the coefficient matrix [Γ] define the mechanical advantage foreach of the actuated joints. In the case of a Stewart platform the columns ofthis matrix are are the Plucker coordinates of the lines along each leg (Merlet1989).

The principle of virtual work yields the velocity of the platform in termsof the joints rates �ρ as

[Γ]T V = �ρ. (36)

Thus, the inverse of [Γ] defines the speed ratios between the actuated jointsand the end-effector. The same equation can be obtained by computing thederivative of the geometric constraint equations (31), and [Γ] is the Jacobianof the parallel robot system (Kumar 1992, Tsai 1999).

The Jacobian [Γ] is used in parameter optimization algorithms to de-sign parallel robots (Gosselin and Angeles 1988) with isotropic mechanicaladvantage. The square root of the determinant |[Γ][Γ]T | measures the six-dimensional volume spanned by the column vectors Fj. The distribution ofthe percentage of this volume compared to its maximum within the workspaceis also used as a measure of the overall performance (Lee et al. 1996, 1998). Asimilar performance measure normalizes this Jacobian by the maximum jointtorques available and the maximum component of force and torque desired,and then seeks an isotropic design (Salcudean and Stocco 2000).

6 Linkage Design Theory

An assembly links and joints, which is our general definition of a machine,can be called a linkage. However, this term is generally restricted to machineelements that have much less than the six-degrees-of-freedom typical of arobotic system. Often they are one degree-of-freedom single-input-single-output devices such as the four-bar linkage.

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The kinematic synthesis theory presented above for robots is actually ageneralization of an approach originally developed for linkages. Beginningwith a set of task positions, Burmester (1886) obtained an exact geometricsolution to the constraint equations of a planar RR chain which he thenassembled into a four-bar linkage. This has grown into a rich theory for theexact solution of the geometric constraint equations for RR and RP planarchains, RR spherical chains, and TS, CC and RR spatial chains. See Chenand Roth (1967) and Suh and Radcliffe (1978).

Figure 8: A spatial RR chain.

6.1 The Spatial RR Chain

The principles of kinematic synthesis of linkages can be seen in the directsolution of the constraint equations of an RR chain. Figure 8 shows a spatialwhich can be considered the simplest robot. It has 10 design parameters,four for the fixed axis G and the moving axis W and the offset ρ and twistangle α.

The kinematics equations of the spatial RR chain are given by

[D] = [G][Z(θ, 0)][X(α, ρ)][Z(φ, 0)][H], (37)

which define its workspace. Choose a reference position [D1] and right trans-

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late the entire workspace to obtain

[D1k] = [D][D1]−1 =

([G][Z(θ, 0)][X(α, ρ)][Z(φ, 0)][H])([G][Z(θ1, 0)][X(α, ρ)][Z(φ1, 0)][H])−1.(38)

This equation can be simplified to obtain

[D1k] = [T (∆θ,G)][T (∆φ,W)], (39)

where

[T (∆θ,G)] = [G][Z(∆θ, 0)][G]−1,

[T (∆φ,W)] = ([G][[Z(θ1, 0)][X(α, ρ)])[Z(∆φ, 0)]([G][[Z(θ1, 0)][X(α, ρ)])−1.(40)

This defines the workspace of the RR chain as the composition of a rotationabout the moving axis W in its reference position followed by a rotationabout the fixed axis G, (∆θ = θ − θ1 and ∆φ = φ− φ1 measure the rotationfrom the reference position).

To design a spatial RR chain we determine G, W, α and ρ such that(39) includes the desired taskspace. The general case requires the solutionof 10 algebraic equations in 10 unknowns. We can simplify the presentationsignificantly by restricting attention to planar RR chains for which the axesG and W are parallel, which means the twist angle α = 0.

6.2 The Planar RR Chain

If G and W are parallel, then they can be located in the plane perpendicularto their common direction by the coordinates P = (x, y)T and Q = (λ, µ)T .The kinematics equations of the chain become

[D1k] = [T (∆θ,P)][T (∆φ,Q)], (41)

which is the composition of rotations parallel to this plane about the pointsQ then P.

The workspace of this chain can also be defined as the set of displacements[D1k] that satisfy the algebraic equation

([D1k]Q − P)T ([D1k]Q − P) = ρ2. (42)

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This is the geometric constraint that the displaced moving pivot [D1k]Q mustremain at a constant distance ρ from the fixed pivot P.

We use Equation (42) to directly determine the five design parametersr = (x, y, λ, µ, ρ). This is done by evaluating this equation at five taskpositions [Ti], i = 1, . . . , 5, so we have

([D1i]Q − P)T ([D1i]Q − P) = ρ2, i = 1, . . . , 5. (43)

The result is a set of five equations in the five unknown design parameters.The distance ρ is easily eliminated by subtracting the first equation from theremaining four. This also cancels the squared terms u2, v2, λ2 and µ2. Theresulting four equations are bilinear in the variables (x, y) and (λ, µ), andcan be written as

A2(x, y) B2(x, y)A3(x, y) B3(x, y)A4(x, y) B4(x, y)A5(x, y) B5(x, y)

{λµ

}=

C2(x, y)C3(x, y)C4(x, y)C5(x, y)

. (44)

In order for this equation to have a solution, the four 3 × 3 minors of theaugmented coefficient matrix must all be identically zero. This yields fourcubic equations in the coordinates x and y of the fixed pivot. These cubicequations can be further manipulated to yield a quartic polynomial in x(McCarthy 2000). Each real root of this polynomial defines a planar RRchain that reaches the five specified task positions.

6.3 Design Software

Kaufman (1978) was the first to transform this mathematical result into aninteractive graphics program for linkage design called KINSYN. He used amodified game controller to provide the designer the ability to input a setof task positions. An important feature of this software was the decisionto only allow the designer to specify four, not five, positions. Rather thanobtain a finite number of RR chains, his software determined the cubic curveof solutions known as the center-point curve. This curve is obtained by settingthe minor obtained from the first three equations in (44) to zero. Kaufman’ssoftware would ask the designer to select two points on this curve in orderto define two RR chains that it assembled into the one degree-of-freedom 4Rlinkage, or four-bar linkage. Analysis routines evaluate the performance ofthe design and provide a simulation of its movement.

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Erdman and Gustafson (1977) introduced LINCAGES which, like KIN-SYN, focused on four task positions for the design of a 4R planar linkage.This software introduced a “guide map” which displayed the characteristicsof every four-bar linkage that could be constructed from points on the center-point curve. This software was extended by Chase et al. (1981) to design anadditional 3R chain to form a six-bar linkage.

Waldron and Song (1981) introduced the design software RECSYN whichagain sought 4R closed chains that guide a body through three or four taskpositions. Their innovation was an analytical formulation that ensured thelinkage would not “jam” as it moved between the design positions. In linkagedesign a jam is equivalent to hitting a singular configuration in a robot whichoccurs when the determinant of the Jacobian becomes zero. An importantfeature of this software was the growing reliance on graphical communica-tion of geometric information regarding the characteristics of available set ofdesigns.

Figure 9: A spherical RR chain.

Larochelle et al. (1993) introduced the “Sphinx” software for the designof spherical 4R linkages, which can be viewed as planar 4R linkages that arebent onto the surface of a sphere. A spherical RR chain is obtained whenthe link offset is ρ = 0, Figure 9. The fixed and moving axes of spherical RRchains are defined by the unit vectors G = (x, y, z)T and W = (λ, µ, ν)T .The workspace of relative rotations [A1k] reachable by this chain is defined

21

Figure 10: The desktop of the spherical linkage design software SphinxPC.

by the algebraic equation

GT [A1k]W = cosα. (45)

This equation is evaluated at five specified task orientations to obtain equa-tions that are essentially identical to (44) and solved in the same way (Mc-Carthy 2000).

Following the pattern established by KINSYN and LINCAGES, Sphinxasks the designer to specify four task orientations, and then generates thecenter-axis cone, which is the spherical equivalent of the center-point curve.The software also computes a “type map” which classifies via color coding themovement of every 4R chain that can be constructed from pairs of axes on thiscone. The typemap also included filters that eliminated designs with knowndefects. A later version of this software called SphinxPC also included planar4R linkage design (Ruth and McCarthy 1997). The display and typemapwindows of SphinxPC are shown in Figure 10.

The three dimensional nature of the interaction needed for spherical 4Rlinkage design presents severe visualization challenges. The designer findsthat specifying a task as a set of spatial orientations is an unfamiliar expe-rience. Furlong et al. (1998) used immersive virtual reality in their softwareIRIS to enhance this interaction.

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Fixed Cylindric Joint Moving Cylindric Joint

Figure 11: The CC open chain robot.

Larochelle (1998) introduced the SPADES software which provided thefirst interactive design of a truly spatial linkage system, the 4C linkage. ACC chain is the generalized robot link that allows both rotation about andsliding along each axis. Let G = (G,P × G)T and W = (G,Q × W)T bethe Plucker coordinates locating the fixed and moving axes in space. Theworkspace of this chain can be defined as the displacements [D1k] that satisfythe pair of geometric constraints,

GT [A1k]W = cosα,

(P × G)T [A1k]W + GT [D1k](Q × W) = −ρ sinα. (46)

These equations constrain the link parameters α and ρ to be constant for ev-ery position of the moving frame. There are 10 design parameters consistingof four parameters for each of the two axes, the link offset ρ and twist angleα.

By evaluating the two constraint equations at each of five task positions,we obtain 10 equations in 10 unknowns. Five of these are identical to thoseused for the synthesis of spherical RR chains and can be solved to determinethe directions G and W. The remaining five equations are linear in thecomponents of P × G and Q × W and are easily solved.

SPADES generates the center-axis congruence which is the set of spatialCC chains that reach four spatial task positions. It then assembles pairs ofthese chains into two degree-of-freedom 4C linkage. This software demon-

23

strates the significant visualization challenge that exists in the specificationof spatial task frames and evaluation of candidate designs.

These linkage design algorithms ensure that the workspace of each linkageincludes the specified taskspace. However, in each case designer is expected tosearch performance measures and examine simulations for many candidatesdesigns in order to verify the quality of movement between the individualtask positions.

Gimbal joint

Spherical joint

Figure 12: The TS open chain robot.

6.4 The TS Chain

Another example of the challenge inherent in the kinematic synthesis of spa-tial chains is found the design of the TS chain, Figure 12. This chain has theworkspace defined by the algebraic equation,

([D1k]Q − P)T ([D1k]Q − P) = ρ2. (47)

There are seven design parameters, the six coordinates of P = (x, y, z)T andQ = (λ, µ, ν)T that define the centers of the T and S joints, respectively, andthe length ρ. Therefore, we can evaluate this equation at seven spatial posi-tions. The result, however, is that we can compute as many as 20 TS chains(Innocenti 1995). If these are assembled into the single-degree-of-freedom5TS linkage, Figure 13, then over 15,000 designs must be analyzed whichis an extreme computational burden (Liao and McCarthy 1999). Prototype

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software shows that it is remarkably difficult for a designer to specify sevenspatial positions and obtain a useful design.

Recent research has focussed on developing higher resolution tasks andproviding methods for fitting a low dimensional workspace to a generaltaskspace.

P1

P2P3

P4

P5

Q4Q2

Q5

Q3

Q1

ρ1

Figure 13: The 5-TS platform linkage.

6.5 Task Specification

The workspace of a linkage is a subset of the group of all spatial displace-ments, denoted SE(3), reachable by a workpiece, or end-effector, of the sys-tem. The taskspace is a discrete set of points in SE(3) that located key-frames, or precision positions, that the linkage must reach. Research in mo-tion interpolation has provided techniques to generate smooth trajectoriesthrough a set of key-frames using Bezier-style methods.

Bezier interpolation is used in computer drawing systems to generatecurves through specified points (Farin 1997). Shoemake (1985) shows thatthis technique can be used to interpolate rotation key frames specified byquaternion coordinates (Hamilton 1860). Ge and Ravani (1994) generalizedShoemake’s results using dual quaternions to interpolate spatial displace-ments, and Ge and Kang (1995) refined this approach to ensure smoothtransitions at each key frame. The result is an efficient method to specify

25

Figure 14: Spatial tasks can be specified using Bezier motion interpolation.

task trajectories in the manifold SE(3) (Figure 14); see McCarthy (1990) fora discussion of quaternions and dual quaternions as Clifford Algebras whichrepresent the groups of rotations SO(3) and spatial displacements SE(3),respectively.

Ahlers and McCarthy (2000) show that the Clifford Algebra of 4 × 4rotations provides an efficient method for specifying spatial trajectories usingdouble quaternion equations. In McCarthy and Ahlers (2000) they designspatial CC chains for discrete subsets of a double quaternion trajectory, andselect the design that best fits the overall trajectory. This procedure combinesan analytical solution for the fixed and moving axes with an optimizationprocedure that bounds the error associated with coordinate transformationsin both the fixed and moving frames. This strategy was applied to the designof spatial RR robots in Perez and McCarthy (2000) Figure 15.

6.6 Mechanism Synthesis

While our focus has been systems constructed using lower-pair joints, thesame ideas apply to the design of cam-and-follower systems, see Norton(1993). Here the concepts of workspace and mechanical advantage appearas displacement diagrams and pressure angle. Three-dimensional cam-and-follower systems remain the focus of research (Gonzalez-Palacios and An-geles 1993). The design of gear trains has benefitted from graph-theoretic

26

Figure 15: The workspace of a spatial RR robot is a manifold can be fit toa desired trajectory by adjusting its design parameters.

representations of the machine topology (Tsai 1993). Computer enumerationcombined with heuristics for identifying isomorphic graphs has lead to newdesigns. These techniques have been extended to the design of robot wristsand cable drives for robotic applications (Tsai 1999).

In the following section we examine a representation of a workspace called“configuration space,” which has found important applications in the analysisof tolerances, assembly, and fixtures for mechanical systems.

7 Configuration Space

Configuration space is the set of values available to parameters that define theconfiguration of a mechanical system. It is a fundamental tool in robot pathplanning for obstacle avoidance (Lozano-Perez 1983). Though any link in thechain forming a robot may hit an obstacle, it is the gripper that is intendedto approach and move around obstacles such as the table supporting therobot and the fixtures for parts it is to pick up. Obstacles define forbiddenpositions and orientations in the workspace which map back to forbiddenjoint angles in the configuration space of the robot. Robot path plannersseek trajectories to a goal position through the free space around these jointspace obstacles (Latombe 1991).

27

Figure 16: A planar RR robot moves its end-effector between two walls.

The map of joint space obstacles provides a convenient illustration of themovement available to mechanical parts that are in close proximity (Joskow-icz and Sacks 1999). For this reason it has found applications far from robotpath planning, such as the modeling tolerances, assembly and fixtures.

7.1 An Example Planar RR Chain

An example map of joint space obstacles can be generated using a planarRR chain with a rectangular end-effector moving between two walls. SeeFigure 16. The position and orientation of the rectangle is defined by thejoint variables θ and φ, therefore its configuration space is two-dimensional.We show only the quadrant 0 ≤ θ ≤ 180◦ and 0 ≤ φ ≤ 180◦ in Figure 17with the free space in white.

The boundary curves of a joint space obstacle are defined by the modesof contact of the end-effector and a wall. For obstacles and links that arepolygons and circles, the robot-obstacle system forms a planar linkage thatcan be analyzed to determine this boundary (Ge and McCarthy 1989). Forexample, when a vertex of the rectangular end-effector moves along the leftwall from position 1 to 2, as shown, the system forms a slider-crank linkagethat is easily analyzed to determine φ(θ). Similar calculations can be donefor spatial polyhedra and spheres in contact in order to compute obstacleboundaries for spatial systems (Ge and McCarthy 1990).

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Figure 17: The joint angle values available to the robot are restricted by thepresents of the two walls.

7.1.1 Assembly

The map of joint space obstacles for our RR chain shows that a path existsfrom the (0, 0) configuration to a configuration between the two walls repre-sented by the channel to the right in Figure 17. Now suppose that the rightwall is moved closer to the left wall. It is easy to imagine the “pass” boundedby configurations 1 and 4 will pinch closed to form two isolated regions offree space. In one the end-effector is between the two walls and in the otherit is outside the two walls. This illustrates an assembly sequencing operation(Joskowicz and Sacks 1991, Blind et al. 2000, Xiao and Xuerang 2000). Ifthe end-effector is to be held between the two walls, clearly it must moveinto position before the right wall is moved into its final position.

7.1.2 Tolerances

Assume now that the end-effector is between the two walls, in which case itsfree to move in the free space rectangle with vertices 1, 2, 3, 4. When theright wall moves to a position near the left wall, the volume of this rectangle

29

shrinks. Its size shows the freedom of movement available to the end-effectorin its assembled configuration (Brost 1989, Ge and McCarthy 1991). In thisway the configuration space illustrates the assembly tolerance in the system(Sacks and Joskowicz 1998).

7.1.3 Fixtures

If the goal is to design a fixture that eliminates movement available to theend-effector, then the volume of configuration space surrounding its positionmust be reduced to zero (Rimon and Blake 1996, Brost and Goldberg 1996,Rimon and Burdick 1998). In this case the shape and size of the obstaclesare the focus of the design process.

7.2 The Choice of Parameters

Configuration space analysis provides new capabilities for automation of thedesign and manufacture of mechanical systems. However, unlike robot sys-tems where the joint variables are easy to identify, a mechanical assemblydoes not have a natural set of configuration parameters.

One approach is to construct configuration space using the positions andorientations for all parts in the system (Joskowicz and Sacks 1999). This isan extension of the concept of workspace to include all bodies not just theend-effector. In this formulation the free space boundary for each part isobtained by analyzing the contact available with every other part. Here, asin the use of workspace for design optimization, care must be taken to ensurethat the coordinate dependency of measurement in the workspace does notaffect the results.

The set of positions and orientations of a body can be parameterized inmany useful ways. Quaternions have been found to provide a convenient rep-resentation for spatial rotations (Bottema and Roth 1979). Clifford Algebrashave been used to parameterize planar and spatial displacements in orderto obtain algebraic surfaces that represent geometric constraints (McCarthy1990, Collins and McCarthy 1998, Ge et al. 1998).

8 State of the Art

The kinematic synthesis of machines ranging from the lever to the robot hasprovided remarkable capabilities for redirecting power to our ends. Further-

30

more, it seems clear that the future holds more opportunities for mechanicalsystems that are tailored to enhance and augment our individual capabilitieswhether at the human or micro-scale.

Research in the kinematic synthesis of mechanisms and robots has demon-strated the central importance of workspace and mechanical advantage as de-sign criteria. The workspace, as defined the by kinematics equations, can beused for design, tolerancing, and assembly planning. The Jacobian of theseequations yields the speed ratios of the system, which, via the principle ofvirtual work, define its mechanical advantage. And, optimization techniqueshave proven effective for fitting the workspace and mechanical advantage ofserial and parallel chain systems to design specifications.

Unfortunately, computer tools for kinematic synthesis seem to exist onlyas special purpose algorithms developed by individual researchers. Com-mercial systems, such as LINCAGES 2000 (Univ. of Minnesota), SyMech(www.symech.com), and Watt (heron-technologies.com), focus solely on pla-nar four- and six-bar linkages. Kinematic synthesis in engineering applica-tions seems to consist of iterated analysis, in which little or no attempt ismade to use workspace and mechanical advantage criteria to generate de-signs.

A systematic procedure for the design of both serial and parallel linkagesand robots that can match workspace and mechanical advantage is needed,especially one that allows comparison of different machine topologies. Con-sider the following enumeration of spatial open chains of various degrees-of-freedom,

• 2 dof chains: 2R, RP, C, T.

• 3 dof chains: 3R, RRP, PPR, CR, CP, TR, RP, S.

• 4 dof chains: 4R, 3PR, 3RP, CRR, CPR, CC, TRR, TPR, TC, TT,RS, PS.

• 5 dof chains: 5R, 3PRR, RRS, PPS, CPC, CRC, CS, TRC, TPC, 2TR,2TP, TS.

Only a few of these topologies have been explored for use in the design ofspatial linkage systems. Clearly computer automation of the synthesis ofthese chains and their assembly into parallel systems can open the door toa wealth of new devices. This is particularly true when asymmetry exists inthe task, or the device must work around obstacles and people.

31

The generalization of Bezier curves to spatial motion provides a conve-nient way to specify continuous taskspaces. However, the extension of thistechnique to surfaces, solids, hypersolids, or hypersurfaces in SE(3) does notexist. This means that we cannot directly specify the workspace of a roboticsystem. Currently, we rely on general symmetry requirements.

Methodologies for the design of cam systems, gear trains, and linkageshave yet to be integrated into computer aided engineering tools. Whilesculptured surfaces are central to the operation of human and animal joints,they have yet to be exploited for robotics applications (Lenarcic et al. 2000,Parenti-Castelli and DiGregorio 2000).

Micro-electro-mechanical systems provide unique opportunities for multi-ple small systems constructed using layered manufacturing technologies typ-ical of electronic devices. The small size of these systems challenges the basicassumptions of virtual work and their design has benefitted from the synthe-sis theory for compliant linkages (Ananthasuresh and Kota 1995). The designof micro-robotic systems requires advances in the design and construction ofjoints and actuators (Will 2000).

A speculative direction for kinematic synthesis research involves the anal-ysis of proteins and rational drug design (Wang et al. 1998, LaValle etal. 2000). A protein can be viewed a serial chain of links, called aminoacids, each of which are similar except for molecular radicals that extend tothe side. A protein chain may contain 1000 amino acids which folds intoa complex spatial configuration. This configuration can shift between twopositions in order to perform a task, exactly like the machines that we havebeen considering in this chapter.

9 Conclusion

This chapter surveys the theoretical foundation and current implementa-tions of kinematic synthesis for the design of machines. The presentation isorganized to illustrate the importance of the concepts of workspace and me-chanical advantage in the synthesis process. While the workspace of a robotarm is a familiar entity, it is not generally recognized that concepts rangingfrom the geometric constraints used in linkage design, through configura-tion spaces used to define tolerances are simply different representations ofworkspace. Similarly, while mechanical advantage is clearly important to thedesign a lever and wedge, it is not as obvious that the Jacobian conditions

32

used in the design of robotic systems are, in fact, specifications on mechanicaladvantage.

The result of this study are the conclusions that (i) workspace and me-chanical advantage are effective specifications for the synthesis of broad rangeof mechanical devices; (ii) there already exist a large number of specialized al-gorithms that demonstrate the effectiveness of kinematic synthesis; and (iii)there are many opportunities for the application of new devices availablethrough kinematic synthesis. What is needed is a systematic developmentof computer tools for kinematic synthesis which integrates the efforts of acommunity of researchers from mechanical design, robotics, and computerscience.

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