		39-245 Rapid Design through Virtual and Physical Prototyping Carnegie Mellon University

Introduction to Mechanisms

Yi Zhang
with
Susan Finger
Stephannie Behrens

4 Basic Kinematics of Constrained Rigid Bodies

4.1 Degrees of Freedom of a Rigid Body

4.1.1 Degrees of Freedom of a Rigid Body in a Plane

The degrees of freedom (DOF) of a rigid body is defined as the number of independent movements it has. Figure 4-1 shows a rigid body in a plane. To determine the DOF of this body we must consider how many distinct ways the bar can be moved. In a two dimensional plane such as this computer screen, there are 3 DOF. The bar can be translated along the x axis, translated along the y axis, and rotated about its centroid.

Figure 4-1 Degrees of freedom of a rigid body in a plane

4.1.2 Degrees of Freedom of a Rigid Body in Space

An unrestrained rigid body in space has six degrees of freedom: three translating motions along the x, y and z axes and three rotary motions around the x, y and z axes respectively.

Figure 4-2 Degrees of freedom of a rigid body in space

4.2 Kinematic Constraints

Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigid body system.

The term kinematic pairs actually refers to kinematic constraints between rigid bodies. The kinematic pairs are divided into lower pairs and higher pairs, depending on how the two bodies are in contact.

4.2.1 Lower Pairs in Planar Mechanisms

There are two kinds of lower pairs in planar mechanisms: revolute pairs and prismatic pairs.

A rigid body in a plane has only three independent motions -- two translational and one rotary -- so introducing either a revolute pair or a prismatic pair between two rigid bodies removes two degrees of freedom.

Figure 4-3 A planar revolute pair (R-pair)

Figure 4-4 A planar prismatic pair (P-pair)

4.2.2 Lower Pairs in Spatial Mechanisms

There are six kinds of lower pairs under the category of spatial mechanisms. The types are: spherical pair, plane pair, cylindrical pair, revolute pair, prismatic pair, and screw pair.

Figure 4-5 A spherical pair (S-pair)

A spherical pair keeps two spherical centers together. Two rigid bodies connected by this constraint will be able to rotate relatively around x, y and z axes, but there will be no relative translation along any of these axes. Therefore, a spherical pair removes three degrees of freedom in spatial mechanism. DOF = 3.

Figure 4-6 A planar pair (E-pair)

A plane pair keeps the surfaces of two rigid bodies together. To visualize this, imagine a book lying on a table where is can move in any direction except off the table. Two rigid bodies connected by this kind of pair will have two independent translational motions in the plane, and a rotary motion around the axis that is perpendicular to the plane. Therefore, a plane pair removes three degrees of freedom in spatial mechanism. In our example, the book would not be able to raise off the table or to rotate into the table. DOF = 3.

Figure 4-7 A cylindrical pair (C-pair)

A cylindrical pair keeps two axes of two rigid bodies aligned. Two rigid bodies that are part of this kind of system will have an independent translational motion along the axis and a relative rotary motion around the axis. Therefore, a cylindrical pair removes four degrees of freedom from spatial mechanism. DOF = 2.

Figure 4-8 A revolute pair (R-pair)

A revolute pair keeps the axes of two rigid bodies together. Two rigid bodies constrained by a revolute pair have an independent rotary motion around their common axis. Therefore, a revolute pair removes five degrees of freedom in spatial mechanism. DOF = 1.

Figure 4-9 A prismatic pair (P-pair)

A prismatic pair keeps two axes of two rigid bodies align and allow no relative rotation. Two rigid bodies constrained by this kind of constraint will be able to have an independent translational motion along the axis. Therefore, a prismatic pair removes five degrees of freedom in spatial mechanism. DOF = 1.

Figure 4-10 A screw pair (H-pair)

The screw pair keeps two axes of two rigid bodies aligned and allows a relative screw motion. Two rigid bodies constrained by a screw pair a motion which is a composition of a translational motion along the axis and a corresponding rotary motion around the axis. Therefore, a screw pair removes five degrees of freedom in spatial mechanism.

4.3 Constrained Rigid Bodies

Rigid bodies and kinematic constraints are the basic components of mechanisms. A constrained rigid body system can be a kinematic chain, a mechanism, a structure, or none of these. The influence of kinematic constraints in the motion of rigid bodies has two intrinsic aspects, which are the geometrical and physical aspects. In other words, we can analyze the motion of the constrained rigid bodies from their geometrical relationships or using Newton's Second Law.

A mechanism is a constrained rigid body system in which one of the bodies is the frame. The degrees of freedom are important when considering a constrained rigid body system that is a mechanism. It is less crucial when the system is a structure or when it does not have definite motion.

Calculating the degrees of freedom of a rigid body system is straight forward. Any unconstrained rigid body has six degrees of freedom in space and three degrees of freedom in a plane. Adding kinematic constraints between rigid bodies will correspondingly decrease the degrees of freedom of the rigid body system. We will discuss more on this topic for planar mechanisms in the next section.

4.4 Degrees of Freedom of Planar Mechanisms

4.4.1 Gruebler's Equation

The definition of the degrees of freedom of a mechanism is the number of independent relative motions among the rigid bodies. For example, Figure 4-11 shows several cases of a rigid body constrained by different kinds of pairs.

Figure 4-11 Rigid bodies constrained by different kinds of planar pairs

In Figure 4-11a, a rigid body is constrained by a revolute pair which allows only rotational movement around an axis. It has one degree of freedom, turning around point A. The two lost degrees of freedom are translational movements along the x and y axes. The only way the rigid body can move is to rotate about the fixed point A.

In Figure 4-11b, a rigid body is constrained by a prismatic pair which allows only translational motion. In two dimensions, it has one degree of freedom, translating along the x axis. In this example, the body has lost the ability to rotate about any axis, and it cannot move along the y axis.

In Figure 4-11c, a rigid body is constrained by a higher pair. It has two degrees of freedom: translating along the curved surface and turning about the instantaneous contact point.

In general, a rigid body in a plane has three degrees of freedom. Kinematic pairs are constraints on rigid bodies that reduce the degrees of freedom of a mechanism. Figure 4-11 shows the three kinds of pairs in planar mechanisms. These pairs reduce the number of the degrees of freedom. If we create a lower pair (Figure 4-11a,b), the degrees of freedom are reduced to 2. Similarly, if we create a higher pair (Figure 4-11c), the degrees of freedom are reduced to 1.

Figure 4-12 Kinematic Pairs in Planar Mechanisms

Therefore, we can write the following equation:

(4-1)

Where

F = total degrees of freedom in the mechanism

n = number of links (including the frame)

l = number of lower pairs (one degree of freedom)

h = number of higher pairs (two degrees of freedom)

This equation is also known as Gruebler's equation.

Example 1

Look at the transom above the door in Figure 4-13a. The opening and closing mechanism is shown in Figure 4-13b. Let's calculate its degree of freedom.

Figure 4-13 Transom mechanism

n = 4 (link 1,3,3 and frame 4), l = 4 (at A, B, C, D), h = 0

(4-2)

Note: D and E function as a same prismatic pair, so they only count as one lower pair.

Example 2

Calculate the degrees of freedom of the mechanisms shown in Figure 4-14b. Figure 4-14a is an application of the mechanism.

Figure 4-14 Dump truck

n = 4, l = 4 (at A, B, C, D), h = 0

(4-3)

Example 3

Calculate the degrees of freedom of the mechanisms shown in Figure 4-15.

Figure 4-15 Degrees of freedom calculation

For the mechanism in Figure 4-15a

n = 6, l = 7, h = 0

(4-4)

For the mechanism in Figure 4-15b

n = 4, l = 3, h = 2

(4-5)

Note: The rotation of the roller does not influence the relationship of the input and output motion of the mechanism. Hence, the freedom of the roller will not be considered; It is called a passive or redundant degree of freedom. Imagine that the roller is welded to link 2 when counting the degrees of freedom for the mechanism.

4.4.2 Kutzbach Criterion

The number of degrees of freedom of a mechanism is also called the mobility of the device. The mobility is the number of input parameters (usually pair variables) that must be independently controlled to bring the device into a particular position. The Kutzbach criterion, which is similar to Gruebler's equation, calculates the mobility.

In order to control a mechanism, the number of independent input motions must equal the number of degrees of freedom of the mechanism. For example, the transom in Figure 4-13a has a single degree of freedom, so it needs one independent input motion to open or close the window. That is, you just push or pull rod 3 to operate the window.

To see another example, the mechanism in Figure 4-15a also has 1 degree of freedom. If an independent input is applied to link 1 (e.g., a motor is mounted on joint A to drive link 1), the mechanism will have the a prescribed motion.

4.5 Finite Transformation

Finite transformation is used to describe the motion of a point on rigid body and the motion of the rigid body itself.

4.5.1 Finite Planar Rotational Transformation

Figure 4-16 Point on a planar rigid body rotated through an angle

Suppose that a point P on a rigid body goes through a rotation describing a circular path from P₁ to P₂ around the origin of a coordinate system. We can describe this motion with a rotation operator R₁₂:

(4-6)

where

(4-7)

4.5.2 Finite Planar Translational Transformation

Figure 4-17 Point on a planar rigid body translated through a distance

Suppose that a point P on a rigid body goes through a translation describing a straight path from P₁ to P₂ with a change of coordinates of (x, y). We can describe this motion with a translation operator T₁₂:

(4-8)

where

(4-9)

4.5.3 Concatenation of Finite Planar Displacements

Figure 4-18 Concatenation of finite planar displacements in space

Suppose that a point P on a rigid body goes through a rotation describing a circular path from P₁ to P₂' around the origin of a coordinate system, then a translation describing a straight path from P₂' to P₂. We can represent these two steps by

(4-10)

and

(4-11)

We can concatenate these motions to get

(4-12)

where D₁₂ is the planar general displacement operator :

(4-13)

4.5.4 Planar Rigid-Body Transformation

We have discussed various transformations to describe the displacements of a point on rigid body. Can these operators be applied to the displacements of a system of points such as a rigid body?

We used a 3 x 1 homogeneous column matrix to describe a vector representing a single point. A beneficial feature of the planar 3 x 3 translational, rotational, and general displacement matrix operators is that they can easily be programmed on a computer to manipulate a 3 x n matrix of n column vectors representing n points of a rigid body. Since the distance of each particle of a rigid body from every other point of the rigid body is constant, the vectors locating each point of a rigid body must undergo the same transformation when the rigid body moves and the proper axis, angle, and/or translation is specified to represent its motion. (Sandor & Erdman 84). For example, the general planar transformation for the three points A, B, C on a rigid body can be represented by

(4-14)

4.5.5 Spatial Rotational Transformation

We can describe a spatial rotation operator for the rotational transformation of a point about an unit axis u passing through the origin of the coordinate system. Suppose the rotational angle of the point about u is , the rotation operator will be expressed by

(4-15)

where

u_x, u_y, u_z are the othographical projection of the unit axis u on x, y, and z axes, respectively.

s = sin

c = cos

v = 1 - cos

4.5.6 Spatial Translational Transformation

Suppose that a point P on a rigid body goes through a translation describing a straight path from P₁ to P₂ with a change of coordinates of (x, y, z), we can describe this motion with a translation operator T:

(4-16)

4.5.7 Spatial Translation and Rotation Matrix for Axis Through the Origin

Suppose a point P on a rigid body rotates with an angular displacement about an unit axis u passing through the origin of the coordinate system at first, and then followed by a translation Du along u. This composition of this rotational transformation and this translational transformation is a screw motion. Its corresponding matrix operator, the screw operator, is a concatenation of the translation operator in Equation 4-7 and the rotation operator in Equation 4-9.

(4-17)

4.6 Transformation Matrix Between Rigid Bodies

4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies

For a system of rigid bodies, we can establish a local Cartesian coordinate system for each rigid body. Transformation matrices are used to describe the relative motion between rigid bodies.

For example, two rigid bodies in a space each have local coordinate systems x₁y₁z₁ and x₂y₂z₂. Let point P be attached to body 2 at location (x₂, y₂, z₂) in body 2's local coordinate system. To find the location of P with respect to body 1's local coordinate system, we know that that the point x₂y₂z₂ can be obtained from x₁y₁z₁ by combining translation L_x1 along the x axis and rotation z about z axis. We can derive the transformation matrix as follows:

(4-18)

If rigid body 1 is fixed as a frame, a global coordinate system can be created on this body. Therefore, the above transformation can be used to map the local coordinates of a point into the global coordinates.

4.6.2 Kinematic Constraints Between Two Rigid Bodies

The transformation matrix above is a specific example for two unconstrained rigid bodies. The transformation matrix depends on the relative position of the two rigid bodies. If we connect two rigid bodies with a kinematic constraint, their degrees of freedom will be decreased. In other words, their relative motion will be specified in some extent.

Suppose we constrain the two rigid bodies above with a revolute pair as shown in Figure 4-19. We can still write the transformation matrix in the same form as Equation 4-18.

Figure 4-19 Relative position of points on constrained bodies

The difference is that the L_x1 is a constant now, because the revolute pair fixes the origin of coordinate system x₂y₂z₂ with respect to coordinate system x₁y₁z₁. However, the rotation z is still a variable. Therefore, kinematic constraints specify the transformation matrix to some extent.

4.6.3 Denavit-Hartenberg Notation

Denavit-Hartenberg notation (Denavit & Hartenberg 55) is widely used in the transformation of coordinate systems of linkages and robot mechanisms. It can be used to represent the transformation matrix between links as shown in the Figure 4-20.

Figure 4-20 Denavit-Hartenberg Notation

In this figure,

z_i-1 and z_i are the axes of two revolute pairs;

_i is the included angle of axes x_i-1 and x_i;

d_i is the distance between the origin of the coordinate system x_i-1y_i-1z_i-1 and the foot of the common perpendicular;

a_i is the distance between two feet of the common perpendicular;

_i is the included angle of axes z_i-1 and z_i;

The transformation matrix will be T_(i-1)i

(4-19)

The above transformation matrix can be denoted as T(a_i, _i, _i, d_i) for convenience.

4.6.4 Application of Transformation Matrices to Linkages

A linkage is composed of several constrained rigid bodies. Like a mechanism, a linkage should have a frame. The matrix method can be used to derive the kinematic equations of the linkage. If all the links form a closed loop, the concatenation of all of the transformation matrices will be an identity matrix. If the mechanism has n links, we will have:

T₁₂T₂₃...T_(n-1)n = I

(4-20)

Complete Table of Contents

1 Introduction to Mechanisms

2 Mechanisms and Simple Machines

3 More on Machines and Mechanisms

4 Basic Kinematics of Constrained Rigid Bodies

4.1 Degrees of Freedom of a Rigid Body

4.1.1Degrees of Freedom of a Rigid Body in a Plane

4.1.2 Degrees of Freedom of a Rigid Body in Space

4.2 Kinematic Constraints

4.2.1 Lower Pairs in Planar Mechanisms