The Kutzbach criterion is also called the mobility formula , because it computes the number of parameters that define the configuration of a linkage from the number of links and joints and the degree of freedom at each joint. Interesting and useful linkages have been designed that violate the mobility formula by using special geometric features and dimensions to provide more mobility than predicted by this formula. These devices are called overconstrained mechanisms. The mobility formula counts the number of parameters that define the positions of a set of rigid bodies and then reduces this number by the constraints that are imposed by joints connecting these bodies. A system of n rigid bodies moving in space has 6 n degrees of freedom measured relative to a fixed frame. This frame is included in the count of bodies, so that mobility is independent of the choice of the link that will form the fixed frame.
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Figure 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 Degrees of freedom of a rigid body in a plane 4. Figure Degrees of freedom of a rigid body in space 4. 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.
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. The types are: spherical pair , plane pair , cylindrical pair , revolute pair , prismatic pair , and screw pair.
Figure 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. Figure 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. Figure 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. Figure 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. Figure 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.
Figure 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. 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. For example, Figure shows several cases of a rigid body constrained by different kinds of pairs. Figure Rigid bodies constrained by different kinds of planar pairs In Figure a, 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 b, 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 c, 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 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 a,b , the degrees of freedom are reduced to 2. Similarly, if we create a higher pair Figure c , the degrees of freedom are reduced to 1. Example 1 Look at the transom above the door in Figure a.
The opening and closing mechanism is shown in Figure b. Let's calculate its degree of freedom. Example 2 Calculate the degrees of freedom of the mechanisms shown in Figure b. Figure a is an application 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. 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 a 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 a also has 1 degree of freedom. If an independent input is applied to link 1 e.
We can describe this motion with a rotation operator R 12 : where 4. We can describe this motion with a translation operator T 12 : where 4. We can represent these two steps by and We can concatenate these motions to get where D 12 is the planar general displacement operator : 4. 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.
For example, the general planar transformation for the three points A, B, C on a rigid body can be represented by 4. Suppose the rotational angle of the point about u is , the rotation operator will be expressed by where u x , u y , u z are the othographical projection of the unit axis u on x , y , and z axes, respectively.
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 and the rotation operator in Equation 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 x 1 y 1 z 1 and x 2 y 2 z 2. Let point P be attached to body 2 at location x 2 , y 2 , z 2 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 x 2 y 2 z 2 can be obtained from x 1 y 1 z 1 by combining translation L x1 along the x axis and rotation z about z axis.
We can derive the transformation matrix as follows: 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. 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 We can still write the transformation matrix in the same form as Equation Figure 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 x 2 y 2 z 2 with respect to coordinate system x 1 y 1 z 1.
However, the rotation z is still a variable. Therefore, kinematic constraints specify the transformation matrix to some extent. It can be used to represent the transformation matrix between links as shown in the Figure Figure 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-1 y i-1 z 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 The above transformation matrix can be denoted as T a i , i , i , d i for convenience.
Like a mechanism, a linkage should have a frame.
Figure 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.