`
`
`
`MOTHERSON
`MOTHERSON
`EXHIBIT 1012
`EXHIBIT 1012
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`
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`3/30/2020
`
`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
`
`39-245
`
`Rapid Design through Virtual and Physical
`Prototyping
`
`Carnegie Mellon University
`Introduction to Mechanisms
`
`Yi Zhang
`with
`Susan Finger
`Stephannie Behrens
`
`Table of Contents
`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
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
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`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)
`
`4.2.2 Lower Pairs in Spatial Mechanisms
`
`Figure 4-4 A planar prismatic pair (P-pair)
`
`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.
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
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`Figure 4-7 A cylindrical pair (C-pair)
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`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.
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
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`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)
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
`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.
`
`n = 4 (link 1,3,3 and frame 4), l = 4 (at A, B, C, D), h = 0
`
`Figure 4-13 Transom mechanism
`
`(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)
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`Example 3
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`Calculate the degrees of freedom of the mechanisms shown in Figure 4-15.
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
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`Figure 4-15 Degrees of freedom calculation
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`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
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
`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
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
`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
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
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`(4-15)
`
`where
`
`ux, uy, uz 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 Lx1 along the x
`axis and rotation z about z axis. We can derive the transformation matrix as follows:
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
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`(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 Lx1 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
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`In this figure,
`zi-1 and zi are the axes of two revolute pairs;
`i is the included angle of axes xi-1 and xi;
`di is the distance between the origin of the coordinate system xi-1yi-1zi-1 and the foot of the common perpendicular;
`ai is the distance between two feet of the common perpendicular;
`i is the included angle of axes zi-1 and zi;
`The transformation matrix will be T(i-1)i
`
`(4-19)
`The above transformation matrix can be denoted as T(ai,
`
`i,
`
`i, di) 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)
`
`Table of Contents
`
` 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
`4.2.2 Lower Pairs in Spatial Mechanisms
`
`4.3 Constrained Rigid Bodies
`4.4 Degrees of Freedom of Planar Mechanisms
`
`4.4.1 Gruebler's Equation
`4.2.2 4.4.2 Kutzbach Criterion
`
`4.5 4.5 Finite Transformation
`
`4.5.1 Finite Planar Rotational Transformation
`4.5.2 Finite Planar Translational Transformation
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`Chapter 4. Basic Kinematics of Constrained Rigid Bodies
`4.5.3 Concatenation of Finite Planar Displacements
`4.5.4 Planar Rigid-Body Transformation
`4.5.5 Spatial Rotational Transformation
`4.5.6 Spatial Translational Transformation
`4.5.7 Spatial Translation and Rotation Matrix for Axis Through the Origin
`
`4.6 Transformation Matrix Between Rigid Bodies
`
`4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies
`4.6.2 Kinematic Constraints Between Two Rigid Bodies
`4.6.3 Denavit-Hartenberg Notation
`4.6.4 Application of Transformation Matrices to Linkages
`
`5 Planar Linkages
`6 Cams
`7 Gears
`8 Other Mechanisms
`Index
`References
`
`sfinger@ri.cmu.edu
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