Kinematic Differential Geometry and Saddle Synthesis of Linkages

By Delun Wang and Wei Wang

With a pioneering methodology, the book covers the fundamental aspects of kinematic analysis and synthesis of linkage, and provides a theoretical foundation for engineers and researchers in mechanisms design.

• The first book to propose a complete curvature theory for planar, spherical and spatial motion
• Treatment of the synthesis of linkages with a novel approach
• Well-structured format with chapters introducing clearly distinguishable concepts following in a logical sequence dealing with planar, spherical and spatial motion
• Presents a pioneering methodology by a recognized expert in the field and brought up to date with the latest research and findings
• Fundamental theory and application examples are supplied fully illustrated throughout

• Language: English
• Publisher: Wiley
• Release date: May 8, 2015
• ISBN: 9781118255070

Book preview

Chapter 1

Planar Kinematic Differential Geometry

Kinematics, a branch of dynamics, deals with displacements, velocities, accelerations, jerks, etc. of a system of bodies, without consideration of the forces that cause them, while kinematic geometry deals with displacements or changes in position of a particle, a lamina, or a rigid body without consideration of time and the way that the displacements are achieved. As a combination of kinematic geometry and differential geometry both in content and approach, kinematic differential geometry describes and studies the geometrical properties of displacements.

There are a number of articles and books on kinematic geometry. Pioneers such as Euler (1765), Savary (1830), Burmester (1876), Ball (1871), Bobillier (1880), and Müller (1892) established the theoretical foundation and developed the classical geometrical and algebraic approaches for studying kinematic geometry in two dimensions some hundred years ago. The classical geometric and algebraic approaches are still in use today. Differential geometry is favored by many researchers studying the geometrical properties of positions of a planar object, changes in its positions, and their relationships. Invariants, independent of coordinate systems, are introduced to describe the geometric properties concisely. Thanks to the moving Frenet frame for describing infinitesimally small variations of successive positions, the positional geometry can be naturally and conveniently connected to the time-independent differential movement of a planar object.

This chapter deals with the kinematic characteristics of a two-dimensional object (a point, a line) in a plane without consideration of time by means of differential geometry. Though abstract, the explanation is judiciously presented step by step for ease of understanding and will be a necessary foundation for studying the kinematic characteristics of a three-dimensional object by means of differential geometry in later chapters.

1.1 Plane Curves

1.1.1 Vector Curve

A plane curve c01-math-0001 is represented in rectangular coordinates as

1.1 equation

where c01-math-0003 is a parameter. The above equation can be rewritten in the following way by eliminating the parameter c01-math-0004 :

1.2 equation

or in implicit form as

1.3 equation

In a fixed coordinate frame c01-math-0007 , the vector equation of curve c01-math-0008 can be written as

1.4 equation

or

1.5 equation

Obviously, both the magnitude and direction of c01-math-0011 in equation (1.5) vary.

To describe a curve in the vector form, a real vector function, represented by a unit vector c01-math-0012 with an azimuthal angle c01-math-0013 with respect to axis c01-math-0014 , measured counterclockwise, is defined as a vector function of a unit circle (see Fig. 1.1). A plane curve c01-math-0015 can be denoted by the following vector function:

1.6 equation

In the above equation, the magnitude and direction of vector c01-math-0017 depend on the scalar function c01-math-0018 and the vector function of a unit circle c01-math-0019 .

c01f001

Figure 1.1 Vector function of a unit circle

Another vector function of a unit circle c01-math-0020 can be obtained by rotating c01-math-0021 counterclockwise about c01-math-0022 by π/2 (in Chapters 1 and 2, c01-math-0023 is the unit vector normal to the paper and directed toward the reader).

The vector function of a unit circle has the following properties:

Expansion

1.7 equation

Orthogonality

For a unit orthogonal right-handed coordinate system c01-math-0025 consisting of c01-math-0026 , c01-math-0027 , and c01-math-0028 , we have the following identities:

1.8 equation

Transformation

1.9

equation

Differentiation

1.10 equation

The descriptive form of a curve depends on the chosen parameters and coordinates. A curve may have many descriptive forms, which differ in complexity if the parameters and reference coordinates are chosen differently. Below are three examples.

Example 1.1

A circle with radius c01-math-0032 and center point C is shown in Fig. 1.2. Write its equation in both vector and parameter forms.

c01f002

Figure 1.2 A circle

Solution

The parameter equation of a circle in rectangular coordinates c01-math-0033 can be written as

E1-1.1 equation

where c01-math-0035 are the coordinates of the center of the circle in the reference frame c01-math-0036 .

Alternatively, the same circle can be represented as a vector function of a unit circle:

E1-1.2 equation

Example 1.2

An involute is shown in Figs 1.3 and 1.4. Write its equation in both vector and parameter forms.

c01f003

Figure 1.3 An involute

c01f004

Figure 1.4 An involute with a unit circle vector function

Solution

The equation of an involute can be written in three different forms using polar coordinates, rectangular coordinates, and a vector function of a unit circle, where c01-math-0038 is the radius of the base circle.

Polar coordinates:

E1-2.1 equation

Rectangular coordinates:

E1-2.2 equation

Vector function of a unit circle:

E1-2.3 equation

Example 1.3

A planar four-bar linkage is shown in Fig. 1.5. Write the equation of the coupler curve in both parameter and vector forms.

c01f005

Figure 1.5 A planar four-bar linkage

Solution

As shown in Fig. 1.5, links c01-math-0042 and c01-math-0043 , in a planar four-bar linkage c01-math-0044 with link lengths c01-math-0045 , form an inclination angle c01-math-0046 and c01-math-0047 with respect to the fixed link. A moving rectangular coordinate system c01-math-0048 attached to link BC and a fixed coordinate system c01-math-0049 attached to the fixed link are established. Point c01-math-0050 in the coupler with polar coordinates c01-math-0051 can be represented in the coordinate system c01-math-0052 as

E1-3.1 equation

The parameter equation of coupler curves

A coupler curve traced by point c01-math-0054 can also be expressed in the fixed frame c01-math-0055 as

E1-3.2 equation

A sextic algebraic equation can be deduced for a coupler curve if parameters c01-math-0057 and c01-math-0058 are replaced by function c01-math-0059 in the displacement solution of a four-bar linkage.

The vector equation of coupler curves

Link AB rotates about joint A of the fixed link AD, and link BC rotates about joint B of link AB. Since a circle can be expressed by a vector function of a unit circle, a coupler curve of a four-bar linkage can be written as

E1-3.3 equation

A point in link AB traces a circle vector c01-math-0061 in the fixed frame c01-math-0062 . A point in coupler link BC produces a circle vector c01-math-0063 in the reference frame of link AB. The subscripts inside the brackets are independent variables. Here, we deal with the coupler point relative to the coordinate system by the vector function of a unit circle.

Based on the above three examples, we observe that the description of a plane curve in terms of a vector function of a unit circle is simpler than the traditional algebraic equation. Moreover, since a vector function of a unit circle has intrinsic properties, its successive derivatives with respect to the chosen parameters can be conveniently obtained.

Invariants of a curve, independent of the coordinate system used, can be used to simplify the equation of the curve, which is considered a general rule in differential geometry. The arc length of a curve, which is also termed a natural parameter, is an invariant. Other invariants will be introduced in the later of this chapter and other chapters of the book. For equation (1.4), c01-math-0064 can be replaced by c01-math-0065 . The differential relationship between c01-math-0066 and c01-math-0067 can be written as

1.11

equation

Then, the vector equation of curve c01-math-0069 is expressed in terms of c01-math-0070 as

1.12 equation

It is recognized that c01-math-0072 . Using the Taylor expansion, curve c01-math-0073 can be expressed in the neighborhood c01-math-0074 of point c01-math-0075 by

1.13

equation

where c01-math-0077 .

1.1.2 Frenet Frame

In a fixed frame, a curve is traced by a point of a moving body. There exists a connection between the point path and the moving body. A frame that moves along the curve can be employed to study the intrinsic geometrical properties of the curve.

Assume that the unit tangent vector of a plane curve c01-math-0078 is always in the direction of increasing arc length. Adopting the right-handed rule, as in the case of equation (1.8), the unit normal vector of a curve may be defined as c01-math-0079 . A unit orthogonal right-handed coordinate system c01-math-0080 may be uniquely established for each point c01-math-0081 on the curve. This moving Cartesian reference frame is called the Frenet frame, or the moving frame of a plane curve (see Fig. 1.6). The Frenet frame for a plane curve may be defined as

1.14 equation

where c01-math-0083 , an invariant of the curve, is the curvature. Performing a dot product of both sides of the second equation in (1.14) with vector c01-math-0084 , we obtain

1.15

equation

If a vector equation with a general parameter c01-math-0086 is given, as in equation (1.4) for a plane curve c01-math-0087 , the unit tangent vector c01-math-0088 can be expressed as

1.16 equation

Utilizing the identity equation c01-math-0090 , the unit normal vector c01-math-0091 is obtained as

1.17 equation

According to equation (1.11), the relationship between c01-math-0093 and c01-math-0094 is c01-math-0095 . Hence, the curvature c01-math-0096 of a plane curve c01-math-0097 can be written in terms of c01-math-0098 as

1.18

equationc01f006

Figure 1.6 The Frenet frame of a plane curve

In order to explain the geometrical meaning of c01-math-0100 , c01-math-0101 and c01-math-0102 are projected onto each axis of the fixed coordinate system c01-math-0103 . We have c01-math-0104 and c01-math-0105 , where c01-math-0106 is the angle between c01-math-0107 and axis c01-math-0108 . The differentiation of c01-math-0109 with respect to c01-math-0110 can be written as

1.19 equation

From the second equation in (1.14), another expression for the curvature, c01-math-0112 , can be obtained. The geometrical meaning of c01-math-0113 is the rate of change in angle c01-math-0114 with respect to c01-math-0115 .

The curvature here may be positive or negative, depending on its direction around the curve, while the current curvature is always positive (used in the book for readers to identify the difference between them). As shown in Fig. 1.7, if c01-math-0116 points in the direction of increasing arc length, then c01-math-0117 always points toward the left-hand side of the curve. Hence, if c01-math-0118 and c01-math-0119 lie on opposite sides of the curve at a point, c01-math-0120 is positive, otherwise, c01-math-0121 is negative. In a special case, if c01-math-0122 at a point is zero, c01-math-0123 is coincidental with the curve at that point. The point is then called an inflection point.

c01f007

Figure 1.7 The curvature of a plane curve

With the definition of the curvature, both local and global properties of a plane curve can be determined. We have the following theorem:

Theorem 1.1

Given a continuously differentiable function c01-math-0124 in interval (sa, sb) and an initial point c01-math-0125 along with a unit tangent vector c01-math-0126 , there exists only one regular plane curve having the given curvature c01-math-0127 .

Although a plane curve may be expressed in different forms for different coordinate systems and parameters, the curvature determines the curve uniquely according to Theorem 1.1 because it is an invariant of the curve and independent of the coordinate system used. The curvature function is usually termed the natural equation of a curve in two dimensions.

As special cases, a circle has constant curvature while a straight line has zero curvature. The local shape of a plane curve is close to a circle in the neighborhood of point s if the curvature of the curve is constant at point c01-math-0128 or in its vicinity. To define closeness, a contact order between two curves is introduced. Two plane curves have two common points at two infinitesimally separated positions if they are tangent to each other. We define this case as first-order contact. Order-n contact is defined as the contact between two curves having n+1 common points at infinitesimally separated positions. According to this definition, a plane curve and a circle have first-order contact if the curve is tangent to the circle. Second-order contact implies that a curve and a circle have three common points at infinitesimally separated positions; the circle is then called the osculation circle (see Fig. 1.8), located by three infinitesimally separated points and whose radius is the radius of the curvature circle at the contact point. Third-order contact between a curve and a circle indicates that there are four common points at infinitesimally separated positions. In this case, the differential curvature of the curve with respect to the natural parameter at point c01-math-0129 must be zero, or c01-math-0130 . Similarly, if the contact order between a curve and a circle is n, the successive derivatives up to order n−2 of the curvature with respect to the natural parameter at position c01-math-0131 must be zero. The vector of center of the osculating circle, or the curvature center of a curve c01-math-0132 at the contact point c01-math-0133 , can be written as

1.20 equation

Each point on curve c01-math-0135 has a corresponding curvature center. The loci of all centers of the curvature circles of a curve is another curve, called the evolute of curve c01-math-0136 . In a special case, if curve c01-math-0137 is a circle, c01-math-0138 ., the evolute degenerates to a fixed point.

c01f008

Figure 1.8 The osculating circle of a plane curve

We can also define contact orders between a curve and a straight line. A plane curve and a straight line have first-order contact, which means that the line is tangent to the curve at the contact point. Second-order contact implies that a curve and a straight line have three common points at infinitesimally separated positions; the curvature of the curve at the contact point, or the inflection point of the curve, must be zero. Third-order contact requires that a curve and a line have four common points at infinitesimally separated positions, in which the curvature of the curve at the contact point is zero and whose differential with respect to the natural parameter at position c01-math-0139 is zero, or c01-math-0140 .

For convenience, in discussing the global geometrical properties of a curve in two dimensions, we introduce the following definitions.

Closed plane curve. A plane curve c01-math-0141 is a closed curve during interval c01-math-0142 if c01-math-0143 is satisfied.

Simple closed plane curve. A closed plane curve is a simple closed plane curve if it has no intersections with itself, or is without self-intersection points.

Simple convex closed plane curve. A simple closed plane curve is a simple convex closed plane curve if the tangent vector of the curve at every point is always located on one side of the curve.

As examples, the curve in Fig. 1.9(a) is a simple closed and non-convex plane curve. The curve in Fig. 1.9(b) is a simple closed and convex plane curve, often called an oval curve because of its similarity to a goose egg shape. The curve in Fig. 1.9(c) is a non-simple closed plane curve.

c01f009

Figure 1.9 (a) A simple closed and non-convex plane curve (b) An oval curve (c) A non-simple closed plane curve

From the definition of a simple convex closed plane curve and the geometrical meaning of curvature, we have:

Theorem 1.2

A simple closed plane curve is a simple convex closed plane curve if and only if the curvature remains greater than or equal to zero ( c01-math-0144 ) at every point for a properly chosen positive direction of increasing arc length.

In particular, a simple convex closed plane curve is an oval curve if the curvature is not equal to zero at any point of the curve. From Theorem 1.2, we have the following corollary:

Corollary 1.1

A simple closed plane curve must be an oval curve if the curvature remains sign-invariant at every point of the curve for a properly chosen positive direction of increasing arc length.

1.1.3 Adjoint Approach

A point c01-math-0145 moves along a plane curve c01-math-0146 in the fixed coordinate system c01-math-0147 . Another point c01-math-0148 , which does not belong to curve c01-math-0149 , traces a different curve c01-math-0150 in the same coordinate system c01-math-0151 . If each position of point c01-math-0152 at c01-math-0153 always corresponds to a position of point c01-math-0154 at c01-math-0155 , point c01-math-0156 is said to be adjoint to point c01-math-0157 , and curve c01-math-0158 is said to be adjoint to curve c01-math-0159 . c01-math-0160 is defined as the original curve, and c01-math-0161 is called the adjoint curve of c01-math-0162 (see Fig. 1.10).

c01f010

Figure 1.10 A plane curve adjoint to another plane curve

The Frenet frame c01-math-0163 is set up at the original curve c01-math-0164 , and the vector equation of the adjoint curve c01-math-0165 can be written as

1.21 equation

where c01-math-0167 are the coordinates of point c01-math-0168 in the Frenet frame c01-math-0169 of the original curve c01-math-0170 . Based on the Frenet formulas (1.14), the first derivative of the above equation with respect to the arc length c01-math-0171 of the original curve c01-math-0172 (not the arc length c01-math-0173 of the adjoint curve c01-math-0174 ) is given by

1.22 equation

where c01-math-0176 is the tangent vector of the plane curve c01-math-0177 . The absolute motion of point c01-math-0178 is examined in c01-math-0179 and expressed by the moving Frenet frame c01-math-0180 of the original curve c01-math-0181 . Hence, c01-math-0182 are the rates of change of the coordinates in the Frenet frame c01-math-0183 ; here, c01-math-0184 are the rates of change of the absolute motion of point c01-math-0185 in the fixed frame and expressed by the Frenet frame c01-math-0186 . In particular, c01-math-0187 is a fixed point in c01-math-0188 ; the absolute coordinates of point c01-math-0189 do not change with c01-math-0190 of the original curve c01-math-0191 , or c01-math-0192 are zero (i.e., c01-math-0193 ). The last two expressions of (1.22) can be written as

1.23 equation

The above equations are called the fixed point conditions of an adjoint plane curve, or Cesaro's fixed point conditions. That is, point c01-math-0195 in the Frenet frame c01-math-0196 at that instant has to meet the same conditions as if it remained absolutely still in the fixed coordinate system c01-math-0197 . The differential equation (1.23), in fact, implies the relationship between the motion of the Frenet frame c01-math-0198 with respect to the original curve c01-math-0199 and the motion of point c01-math-0200 relative to the fixed frame c01-math-0201 .

Cesaro once used a metaphor for this: the original curve c01-math-0202 is like a winding river, whereas the Frenet frame c01-math-0203 is like a boat flowing and going with the water in the river (see Fig. 1.11). The tangent vector c01-math-0204 goes ahead downstream with the second axis, or the normal vector c01-math-0205 , perpendicular to the boat and toward the left. The boatman can see the breathtaking scenery of banks and mountains all from the viewpoint of the boat as the coordinate system, and hence he knows everything about the curves of the river and detailed information about the banks.

c01f011

Figure 1.11 Metaphor for adjoint movement

A point c01-math-0206 moves along a plane curve c01-math-0207 in c01-math-0208 . A straight line L passing through point c01-math-0209 , which does not belong to curve c01-math-0210 , traces a set of lines c01-math-0211 in c01-math-0212 , but each position of line L with point c01-math-0213 always corresponds to a position of point c01-math-0214 at c01-math-0215 , or line L with point c01-math-0216 is adjoint to point c01-math-0217 . Hence, c01-math-0218 is adjoint to curve c01-math-0219 . We designate c01-math-0220 as an original curve and c01-math-0221 as the set of adjoint lines of c01-math-0222 (see Fig. 1.12).

c01f012

Figure 1.12 A line adjoint to a plane curve

The Frenet frame c01-math-0223 is set up at the original curve c01-math-0224 , and the vector equation of the set of adjoint lines c01-math-0225 can be written as

1.24

equation

where c01-math-0227 is a parameter for the straight line, c01-math-0228 is a unit direction vector of the straight line described in the Frenet frame c01-math-0229 , which is a function of the arc length c01-math-0230 of the original curve c01-math-0231 . Based on the Frenet formulas in equation (1.14), the first derivative of the above equation with respect to c01-math-0232 of c01-math-0233 is given by

1.25 equation

If all points on the straight line L are fixed points in c01-math-0235 and do not change with c01-math-0236 of c01-math-0237 , the line is a fixed line. We define it as an absolute fixed line. Hence, the last four expressions of equation (1.25) are equal to zero in this case, which leads to the conditions of an absolute fixed line in the fixed frame c01-math-0238 :

1.26

equation

If line L is always collinear with a fixed line in c01-math-0240 , but slides along the fixed line, the line L satisfies the following conditions:

1.27 equation

Substituting equation (1.24) into the above equations, we introduce the quasi-fixed line conditions

1.28 equation

Example 1.4

Represent a coupler curve by an adjoint curve for a planar linkage.

Solution

As shown in Example 1.3, a coupler curve of a four-bar linkage is concisely expressed in terms of a vector function of a unit circle. In fact, the coupler point is always adjoint to joint B when it traces a coupler curve. Path c01-math-0243 of joint B in the fixed frame c01-math-0244 is a circle, and is consequently taken as the original curve, while a coupler curve is viewed as an adjoint curve of c01-math-0245 . Hence, the Frenet frame c01-math-0246 of the original curve c01-math-0247 is established as in Fig. 1.13. A coupler curve c01-math-0248 , or the adjoint curve of c01-math-0249 , can be expressed by the Frenet frame c01-math-0250 as

E1-4.1 equation

where c01-math-0252 , the coordinates of a coupler point c01-math-0253 in the Frenet frame c01-math-0254 , are

E1-4.2 equation

Here, angles c01-math-0256 are identical to those in Example 1.3. Both angles are measured in the counterclockwise sense.

c01f013

Figure 1.13 A coupler curve adjoint to a circle

1.2 Planar Differential Kinematics

1.2.1 Displacement

1.2.1.1 A General Description of Plane Displacement

To describe the displacement of a moving body c01-math-0257 relative to a fixed body c01-math-0258 from one position to another, different reference frames are established. A moving Cartesian reference frame c01-math-0259 is set up and attached to the moving body c01-math-0260 and a fixed Cartesian reference frame c01-math-0261 is established in the fixed body c01-math-0262 (see Fig. 1.14). For planar motion, a body has three-degrees-of-freedom motion, moving along both c01-math-0263 and c01-math-0264 and rotating about c01-math-0265 , or two linear displacements and one angular displacement. A point c01-math-0266 of c01-math-0267 is taken as a reference point, whose linear displacements c01-math-0268 present a given movement of c01-math-0269 in c01-math-0270 , and the angular c01-math-0271 of c01-math-0272 denotes the angular displacement around c01-math-0273 , completely describing a planar motion of c01-math-0274 relative to c01-math-0275 .

c01f014

Figure 1.14 A moving body c01-math-0276 relative to a fixed body c01-math-0277

An arbitrary point c01-math-0278 in c01-math-0279 corresponds to a position c01-math-0280 or a displacement in c01-math-0281 , which are related by

1.29 equation

We can analyze and determine the displacements of a point on c01-math-0283 if any two linear displacements c01-math-0284 and an angular displacement c01-math-0285 of c01-math-0286 relative to c01-math-0287 are given. In equation (1.29), the given movement of c01-math-0288 may be continuous, or discrete. For the former, its content belongs to the continuous kinematic geometry, or the kinematic geometry at infinitesimally separated positions, but the authors prefer the kinematic differential geometry since the continuous movement is achieved by the differential of the moving frame. The latter belongs to the discrete kinematic geometry (the authors appreciate this term, although its contents are not mature so far) or the kinematic geometry at finite separated positions corresponding to the classical curvature theory or the classical Burmester theory. All of these form a theoretical basis for kinematic synthesis of linkage. The kinematic geometry, either with continuous or discrete movement, has similar geometrical properties.

1.2.1.2 Descriptions of Plane Displacement by an Adjoint Approach

The point P of c01-math-0289 is located by the Cartesian coordinates c01-math-0290 or the polar coordinates c01-math-0291 in c01-math-0292 of c01-math-0293 , whose vector equation is

1.30 equation

The plane displacements of a moving body c01-math-0295 are examined in a fixed Cartesian reference frame c01-math-0296 of c01-math-0297 . The relationship between the coordinate axes of c01-math-0298 and those of c01-math-0299 is

1.31 equation

where c01-math-0301 is the angle between c01-math-0302 and c01-math-0303 . Hence, the plane displacements of c01-math-0304 can be represented by two linear displacements c01-math-0305 of point c01-math-0306 and the rotational displacement c01-math-0307 about c01-math-0308 . Point c01-math-0309 moves along curve c01-math-0310 , which is given as a regular plane curve with higher-order continuity.

A given curve c01-math-0311 in the fixed frame c01-math-0312 is designated as the curve traced by the origin point c01-math-0313 of the moving frame c01-math-0314 of c01-math-0315 (see Fig. 1.15); its vector equation can be written as

1.32 equation

Point c01-math-0317 of c01-math-0318 traces a path c01-math-0319 in the fixed frame c01-math-0320 ; its vector equation is

1.33

equation

Taking the derivative of equation (1.33) with respect to time c01-math-0322 , the absolute velocity of point c01-math-0323 can be written as

1.34

equation

The absolute velocity of the point c01-math-0325 can be viewed as the superposition of two velocities, the following velocity of the moving frame c01-math-0326 and the relative velocity of point c01-math-0327 with reference to the moving frame c01-math-0328 . The position of the origin c01-math-0329 and the orientation of c01-math-0330 in c01-math-0331 not only affect the property and complexity of the following velocity in equation (1.34), c01-math-0332 , but also change the magnitude of the relative velocity with reference to the moving frame.

c01f015

Figure 1.15 Adjoint descriptions of plane displacements

In general, point c01-math-0333 is a fixed point in the moving body c01-math-0334 , and the moving frame c01-math-0335 is also fixed in c01-math-0336 . There is no relative motion between c01-math-0337 and the moving frame c01-math-0338 , or the components of the relative velocity in equation (1.34), c01-math-0339 and c01-math-0340 , are zero. The following velocity of c01-math-0341 can be determined by equations (1.31) and (1.32).

As mentioned above, both the given curve c01-math-0342 and the relative angular displacement c01-math-0343 of c01-math-0344 completely define the kinematic properties of a moving body c01-math-0345 . A point c01-math-0346 of c01-math-0347 traces a path c01-math-0348 in c01-math-0349 , while the origin point c01-math-0350 of c01-math-0351 moves along the given curve c01-math-0352 in c01-math-0353 simultaneously. Each position of c01-math-0354 at c01-math-0355 always corresponds to a position of c01-math-0356 at c01-math-0357 . In other words, P is adjoint to c01-math-0358 . Hence, any point of c01-math-0359 traces a path, which can be viewed as an adjoint curve of the given curve c01-math-0360 , or the displacements of c01-math-0361 in plane motion can be expressed by the adjoint approach of curve c01-math-0362 if the given curve c01-math-0363 is a regular curve.

Utilizing the analytic properties of a given curve c01-math-0364 , the natural arc length c01-math-0365 is chosen as the parameter of the given curve c01-math-0366 . We can set up the Frenet frame c01-math-0367 of the given curve c01-math-0368 as

1.35a equation

or

1.35b

equation

The Frenet formulas are

1.36 equation

where invariant c01-math-0372 is the curvature of the given curve c01-math-0373 .

The given curve c01-math-0374 is taken as the original curve. Path c01-math-0375 traced by point c01-math-0376 of c01-math-0377 is an adjoint curve of c01-math-0378 . Hence, the adjoint curve c01-math-0379 in c01-math-0380 can be expressed in the Frenet frame c01-math-0381 of c01-math-0382 , or

1.37 equation

where c01-math-0384 are the coordinates of point c01-math-0385 in c01-math-0386 and the function of c01-math-0387 . Unit vectors c01-math-0388 can be derived from equation (1.35). We do not forget that point c01-math-0389 of c01-math-0390 has the Cartesian coordinates c01-math-0391 or the polar coordinates c01-math-0392 in c01-math-0393 .

Let c01-math-0394 be the angle between the unit vector c01-math-0395 and the axis c01-math-0396 of c01-math-0397 , which can be obtained from equations (1.31) and (1.35) (see Fig. 1.15). We obtain

1.38 equation

Substituting equations (1.33) and (1.38) into (1.37), we have

1.39 equation

Note that the path c01-math-0400 is examined in c01-math-0401 but expressed by equations (1.29), (1.33), and (1.37) in the different frames respectively, such as the moving frame c01-math-0402 of c01-math-0403 and the Frenet frame c01-math-0404 of c01-math-0405 . Equation (1.37) takes advantage of the analytic properties of the given curve by the adjoint approach.

Differentiating equation (1.37) with respect to c01-math-0406 of the original curve c01-math-0407 , and making use of the Frenet formulas in equation (1.14), we obtain

1.40 equation

where an overdot represents the derivative with respect to c01-math-0409 and c01-math-0410 is the curvature of the original curve c01-math-0411 and a function of c01-math-0412 .

Comparing equation (1.40) with equation (1.23), we apply Cesaro's fixed point conditions and have a point of c01-math-0413 with coordinates c01-math-0414 at an instant, which is

1.41 equation

Using the chain rule of differentiation, we have c01-math-0416 . This condition is the kinematic explanation of the Cesaro fixed point conditions, which is an instantaneous center of velocity for c01-math-0417 relative to c01-math-0418 , or the velocity center for short. From equation (1.41), the coordinates c01-math-0419 in the Frenet frame c01-math-0420 of c01-math-0421 for the velocity center at an instant of c01-math-0422 can be represented by

1.42 equation

From the above equations and the Frenet frame c01-math-0424 , the velocity center is located at the normal of the original curve c01-math-0425 with a distance c01-math-0426 . If the coordinates c01-math-0427 of the velocity center are transformed into the moving frame c01-math-0428 from the Frenet frame c01-math-0429 of the original curve c01-math-0430 , the moving centrode c01-math-0431 of the moving body c01-math-0432 can be obtained. If the coordinates c01-math-0433 are transformed into the fixed frame c01-math-0434 , the fixed centrode c01-math-0435 of the fixed body c01-math-0436 is formed. Detailed discussions are given in later sections.

1.2.2 Centrodes

1.2.2.1 Moving Centrode πm and its Curvature km

The velocity center of a moving body at an instant is a point in or at the extension part of c01-math-0437 , whose velocity is zero relative to the fixed body. For all instants, the velocity centers trace a curve c01-math-0438 in c01-math-0439 , called the moving centrode c01-math-0440 . Of course, the moving centrode c01-math-0441 is a curve of c01-math-0442 and described in c01-math-0443 of c01-math-0444 . By means of equations (1.37) and (1.42), we have the following vector expression of c01-math-0445 :

1.43 equation

where vector c01-math-0447 belongs to the Frenet frame c01-math-0448 of c01-math-0449 , whose direction changes with the arc length c01-math-0450 of c01-math-0451 . For c01-math-0452 , c01-math-0453 are similar to a set of orthogonal vectors, whose origin c01-math-0454 is coincident with the origin c01-math-0455 of c01-math-0456 , shown in Fig. 1.16. Vector c01-math-0457 makes angle c01-math-0458 with respect to c01-math-0459 . Vectors c01-math-0460 can be viewed as vector functions of a unit circle c01-math-0461 , with c01-math-0462 as the parameter at the origin point c01-math-0463 . Here, the Frenet frame c01-math-0464 of c01-math-0465 has no significance in c01-math-0466 . Based on the properties of a vector function of a unit circle, the derivatives of c01-math-0467 with respect to c01-math-0468 are obtained as c01-math-0469 , and the tangent vector of c01-math-0470 can be written as

1.44 equation

where c01-math-0472 is a general parameter representing the arc length of c01-math-0473 , not the natural parameter of c01-math-0474 . If the arc length of c01-math-0475 is denoted as c01-math-0476 , from the above equation, we may obtain the following differential relationship between c01-math-0477 and c01-math-0478 :

1.45

equationc01f016

Figure 1.16 The moving centrode c01-math-0480 adjoint to the given curve

The Frenet frame c01-math-0481 of c01-math-0482 can be established as

1.46

equation

This can be expressed in the moving frame c01-math-0484 of c01-math-0485 as

1.47

equation

The curvature of c01-math-0487 , deduced from the Frenet formulas (1.14), is written as

1.48

equation

1.2.2.2 Fixed Centrode πf and its Curvature kf

The velocity centers trace a curve c01-math-0489 in the fixed body c01-math-0490 for all instants, called the fixed centrode c01-math-0491 , which is attached to the fixed body c01-math-0492 and described in c01-math-0493 of c01-math-0494 . Substituting equation (1.42) into equation (1.37), we have the vector expression of c01-math-0495 :

1.49 equation

From the above equation, the fixed centrode c01-math-0497 is expressed as an adjoint curve of c01-math-0498 . The tangent vector of c01-math-0499 may be obtained by differentiating equation (1.49) with respect to c01-math-0500 :

1.50 equation

Based on the above equation, the differential relationship between c01-math-0502 and the arc length c01-math-0503 of c01-math-0504 can be derived as

1.51

equation

Comparing equation (1.45) with equation (1.51), we find that the differential arc length of c01-math-0506 is equal to that of c01-math-0507 , c01-math-0508 , or c01-math-0509 for short. That is, the moving centrode c01-math-0510 rolls on the fixed centrode c01-math-0511 without sliding.

The Frenet frame c01-math-0512 of c01-math-0513 is established as

1.52

equation

The relationship between the Frenet frame c01-math-0515 of c01-math-0516 and c01-math-0517 of c01-math-0518 is

1.53

equation

Differentiating c01-math-0520 of the above equation with respect to c01-math-0521 , we have the following curvature expression for c01-math-0522 :

1.54

equation

For the fixed centrode c01-math-0524 and the moving centrode c01-math-0525 at an instant, comparing equation (1.46) with equation (1.52), we find that their Frenet frames coincide at the velocity center. From both equations (1.45) and (1.51), the differentials of their arc lengths are equal to each other, c01-math-0526 . We have the following conclusion for plane motion of a rigid body:

For plane motion of a body c01-math-0527 relative to a fixed body c01-math-0528 , there exist a moving centrode c01-math-0529 on the moving body and a fixed centrode c01-math-0530 on the fixed body for all instants, c01-math-0531 rolls on c01-math-0532 without sliding, and the two Frenet frames of c01-math-0533 and c01-math-0534 are coincident at that instant.

The above conclusion may be expressed as

1.55

equation

As a result, the moving centrode c01-math-0536 contacts tangentially the fixed centrode c01-math-0537 at the velocity center. Consequently, the induced curvature of centrodes, c01-math-0538 – c01-math-0539 , appears in the curvature analysis of point paths, implies the geometrical and kinematical properties of the moving body at the instant, and is also an important invariant for the plane motion of a body, which can be derived by subtracting equation (1.48) from equation (1.54), or

1.56

equation

1.2.2.3 Centrodes of a Planar Four-bar Linkage

For a general plane motion of the coupler for a planar four-bar linkage, both the moving and fixed centrodes can be derived by means of the adjoint approach. There exist special points on the coupler, or joints between the coupler and the other binary links, which constrain the motion of the coupler. As in Example 1.4, point c01-math-0541 is the joint of link c01-math-0542 and coupler c01-math-0543 . It traces a circle in the fixed frame. At an instant, the coupler can only move along the path of point B and rotate about joint B. Thus, the path, a circle, traced by joint B is taken as the original curve c01-math-0544 ; its arc length and differential are c01-math-0545 and c01-math-0546 . The Frenet frame of the original curve c01-math-0547 is comprised of a unit tangent vector and a unit normal vector of the circle. The curvature of the original curve c01-math-0548 is a constant, which is the reciprocal of the radius of the circle or the length of link c01-math-0549 , c01-math-0550 . The rate of change of c01-math-0551 with respect to the arc length c01-math-0552 is zero, or c01-math-0553 .

As defined in equation (E1-4.2) and shown in Fig. 1.13, the inclination angle c01-math-0554 between axis c01-math-0555 of c01-math-0556 and c01-math-0557 of c01-math-0558 is c01-math-0559 , which leads to c01-math-0560 . Substituting c01-math-0561 into equation (1.42), we obtain the coordinates (0, c01-math-0562 ) of the velocity center in c01-math-0563 at an instant, and the polar coordinates in c01-math-0564 of the coupler plane are

1.57 equation

Then we have c01-math-0566 , where c01-math-0567 are the angular velocities of link c01-math-0568 and coupler c01-math-0569 , respectively. Substituting the curvature of the original curve c01-math-0570 , the rotation angle c01-math-0571 , and equation (1.57) into the vector equations of (1.43) and (1.49), we can locate the moving centrode c01-math-0572 in c01-math-0573 of the coupler and the fixed centrode c01-math-0574 in c01-math-0575 of the fixed link:

1.58

equation

From equation (E1-4.2), the relationship between c01-math-0577 of the coupler and c01-math-0578 of the fixed link can be written as

1.59 equation

Substituting c01-math-0580 and c01-math-0581 into equations (1.48) and (1.54), we obtain the two curvatures c01-math-0582 of the moving centrode c01-math-0583 and c01-math-0584 of the fixed centrode c01-math-0585 for a planar four-bar linkage, respectively:

1.60 equation

In addition, we have the induced curvature c01-math-0587 of c01-math-0588 and c01-math-0589 for a planar linkage from equation (1.56):

1.61 equation

Example 1.5

Find the moving centrode c01-math-0591 and the fixed centrode c01-math-0592 for a planar four-bar linkage.

Solution

A planar four-bar linkage c01-math-0593 with the moving Cartesian reference frame c01-math-0594 and a fixed link with the fixed Cartesian frame c01-math-0595 are shown in Fig. 1.17. The displacements of the coupler, both the path of point c01-math-0596 and the angle c01-math-0597 , have to be solved to describe the movement of the coupler, or the displacement analysis of a four-bar linkage. In general, we can use the following vector form of the loop-closure equation:

E1-5.1 equation

Projecting the loop-closure equation into the axes c01-math-0599 and c01-math-0600 , and simplifying, we have

E1-5.2

equation

The above nonlinear equation defines a relationship between c01-math-0602 and c01-math-0603 . Singularity may appear as the binary link c01-math-0604 takes on different positions for four-bar linkages with different dimensions. It is a convenient way to solve the displacement equations of a planar four-bar linkage [1], which is converted into one of the two basic types of linkage by changing its fixed frame: a crank rocker for a Grashof kinematic chain and a double rocker for a non-Grashof one.

c01f017

Figure 1.17 The velocity center of the coupler of a planar four-bar linkage

Differentiating the above equation with respect to the arc length of the original circle, we obtain

E1-5.3

equation

Crank-rocker linkage

A crank-rocker linkage has the following dimensions: c01-math-0606 , c01-math-0607 . By substituting c01-math-0608 into equation (E1-5.3) to get c01-math-0609 , the coordinates of the velocity center can be located in both c01-math-0610 and c01-math-0611 by equation (1.58). For all instants c01-math-0612 , the moving centrode c01-math-0613 and the fixed centrode c01-math-0614 are calculated, which have first-order contact, or contact in tangency at instant c01-math-0615 (as shown in Fig. 1.18). The velocity center c01-math-0616 locates at c01-math-0617 on the coupler plane. The curvatures c01-math-0618 of c01-math-0619 and c01-math-0620 of c01-math-0621 are calculated as −0.1271 and −0.0874, respectively, from equation (1.60), and the induced curvature c01-math-0622 is c01-math-0623 .

c01f018

Figure 1.18 The centrodes of a crank-rocker linkage

Double-crank linkage

A double-crank linkage, shown in Fig. 1.19, is obtained by fixing link c01-math-0624 of the four-bar linkage in Fig. 1.17. The path of joint C, a circle in the fixed frame c01-math-0625 , is taken as an original curve, and the rotation angle of coupler CD around joint C is c01-math-0626 . For c01-math-0627 ∈(0,2 c01-math-0628 ), the rotation angle of c01-math-0629 relative to c01-math-0630 , the moving centrode c01-math-0631 , and the fixed centrode c01-math-0632 can be calculated, as shown in Fig. 1.20.

c01f019

Figure 1.19 A double-crank linkage

c01f020

Figure 1.20 The centrodes of a double-crank linkage

At c01-math-0633 , the velocity center, in polar coordinates, locates at c01-math-0634 of the coupler. The curvature c01-math-0635 of c01-math-0636 is −0.7219, c01-math-0637 of c01-math-0638 is −1.0015, and the induced curvature c01-math-0639 is −0.2796.

Example 1.6

A slider-crank linkage is shown in Fig. 1.21. Write the vector equation of both the moving centrode c01-math-0640 and the fixed centrode c01-math-0641 .

c01f021

Figure 1.21 A slider-crank linkage

Solution

For a slider-crank linkage, the loop-closure equation may be written as

E1-6.1 equation

Projecting the above equation onto axis c01-math-0643 of the fixed frame, we obtain

E1-6.2 equation

The above equation determines the relationship between the rotation angle c01-math-0645 of link c01-math-0646 and the inclination angle c01-math-0647 . Differentiating the above equation with respect to the arc length of the original curve, a circle in the fixed frame, we have

E1-6.3 equation

The velocity center point, both in c01-math-0649 of the coupler and c01-math-0650 of the base link, can be readily located by means of equation (1.58) with c01-math-0651 , c01-math-0652 , c01-math-0653 , and c01-math-0654 . For the linkage with c01-math-0655 , the moving centrode c01-math-0656 and the fixed centrode c01-math-0657 are calculated, and at the instant c01-math-0658 , the velocity center c01-math-0659 locates at c01-math-0660 and c01-math-0661 , c01-math-0662 , as shown in Fig. 1.22.

c01f022

Figure 1.22 The centrodes of a slider-crank linkage

1.2.3 Euler–Savary Equation

The planar motions of a moving body c01-math-0663 relative to the fixed body c01-math-0664 with three degrees of freedom, two translational displacements and a rotational one, are expressed as three equations (1.29), (1.33), and (1.37). Unfortunately, the intrinsic connection between the kinematics and the geometry of the movement of c01-math-0665 is not intuitively revealed in the three equations. In fact, the movement of c01-math-0666 is equivalent to that of the moving centrode c01-math-0667 , or the moving centrode c01-math-0668 rolls on the fixed centrode c01-math-0669 of the fixed body c01-math-0670 without sliding, which can be viewed as the differential movement of the Frenet frame c01-math-0671 along the fixed centrode c01-math-0672 . For both the moving and fixed centrodes, the curvature theory in instantaneous plane kinematics (in particular, the Euler–Savary equation) can be revealed in a natural, intuitive, and analytical way.

In Section 1.2.2, a velocity center locates at both the moving and the fixed centrodes at an instant, or generates a fixed centrode in the fixed frame and a moving centrode in the moving frame for all instants. Hence, point c01-math-0673 or c01-math-0674 in c01-math-0675 of c01-math-0676 traces a point path c01-math-0677 in c01-math-0678 of c01-math-0679 , which can be designated as an adjoint curve of the fixed centrode c01-math-0680 , or the fixed centrode c01-math-0681 is taken as an original curve. Referring to Fig. 1.23, we have

1.62 equation

where c01-math-0683 , the coordinates of point c01-math-0684 of c01-math-0685 in the Frenet frame c01-math-0686 of c01-math-0687 , are functions of the arc length c01-math-0688 , represented by the polar coordinates c01-math-0689 in the Frenet frame c01-math-0690 as follows:

1.63 equation

Differentiating equation (1.62) with respect to the arc length c01-math-0692 , and simplifying the so-obtained equation with the Frenet formulas in equation (1.14), we have

1.64

equationc01f023

Figure 1.23 The point path adjoint to the centrodes of a moving body

The point c01-math-0694 , a fixed point attached to the moving body c01-math-0695 , can be represented in the Frenet frame c01-math-0696 of c01-math-0697 ; its position does not change in the moving frame c01-math-0698 as c01-math-0699 moves, which can be expressed as

1.65 equation

where c01-math-0701 are the coordinates of point c01-math-0702 in c01-math-0703 of c01-math-0704 , and functions of the arc length c01-math-0705 . In particular, vector c01-math-0706 in equation (1.65) points from the origin c01-math-0707 of c01-math-0708 to the point c01-math-0709 via the velocity center c01-math-0710 in c01-math-0711 at the instant. While c01-math-0712 moves along c01-math-0713 , the coordinates c01-math-0714 change their values, although the positions of the starting point c01-math-0715 and the end point c01-math-0716 do not change. Hence, the point c01-math-0717 of c01-math-0718 is examined in the moving frame c01-math-0719 to be a path, which can be viewed as an adjoint curve of c01-math-0720 , or c01-math-0721 is an original curve. Differentiating equation (1.65) with respect to c01-math-0722 , and utilizing the Frenet formulas, we have

1.66

equation

The position of point c01-math-0724 in c01-math-0725 does not change since it is a fixed point in c01-math-0726 , or c01-math-0727 , which is Cesaro's fixed point conditions:

1.67 equation

or

1.68 equation

Note that c01-math-0730 moves along c01-math-0731 , and the position change of point c01-math-0732 is described in the Frenet frame c01-math-0733 as c01-math-0734 (just like the breath-taking scenery of the banks examined by the boatman, which implies the geometrical properties of the river). Hence, c01-math-0735 express the movement of point c01-math-0736 of c01-math-0737 relative to the Frenet frame c01-math-0738 of c01-math-0739 , and imply the geometrical properties of the moving centrode c01-math-0740 .

As mentioned in Section 1.2.2, the Frenet frame c01-math-0741 of the fixed centrode c01-math-0742 is coincident with the Frenet frame c01-math-0743 of the moving centrode c01-math-0744 at an instant. A point c01-math-0745 of c01-math-0746 has the same coordinates and changing