2-1 Chapter 2 Kinematics 2.1 Introduction 2.2 Degree of Freedom 2.2.1 DOF of a rigid body In order to control and guide the mechanisms to move as we desired, we need to set proper constraints. In order to set proper constraints, we need to study degree of freedom (DOF). In mechanics, it means how many independent motions a mechanism can possibly achieve. For a particle point, its DOF is 2 in 2D plane or 3 in 3D space without any constraint. In other words, a point can freely move at X and Y directions in a plane and X, Y and Z in a space. For a rigid body, since it comes with dimension (size and shape), it can rotate about certain axes. In planar motion, there is only one direction it can rotate. That is Z direction, or the direction that is perpendicular to the plane. Therefore, an unconstrained rigid body can have 3DOF in planar motion. When we put the rigid body in space, it will not only be able to move out of the plane, i.e., gain the translation at the Z direction, it will be able to rotate out from the plane, or possible to rotate about X and Y axes. Please note that since we did not specify any coordinate system at this point, these X, Y and Z axes are not necessary perpendicular to each other. As long as they are independent at the local point where the rigid body is residing, we can always observe 3 or 6 independent motions of a rigid body depending whether it is 2D or 3D. O X Y Z 2D Plane X Y Z R x R z Ry 3D space
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Transcript
2-1
Chapter 2 Kinematics
2.1 Introduction
2.2 Degree of Freedom
2.2.1 DOF of a rigid body
In order to control and guide the mechanisms to move as we desired, we need to set proper
constraints. In order to set proper constraints, we need to study degree of freedom (DOF). In
mechanics, it means how many independent motions a mechanism can possibly achieve. For
a particle point, its DOF is 2 in 2D plane or 3 in 3D space without any constraint. In other
words, a point can freely move at X and Y directions in a plane and X, Y and Z in a space. For a
rigid body, since it comes with dimension (size and shape), it can rotate about certain axes.
In planar motion, there is only one direction it can rotate. That is Z direction, or the direction
that is perpendicular to the plane. Therefore, an unconstrained rigid body can have 3DOF in
planar motion. When we put the rigid body in space, it will not only be able to move out of
the plane, i.e., gain the translation at the Z direction, it will be able to rotate out from the
plane, or possible to rotate about X and Y axes. Please note that since we did not specify any
coordinate system at this point, these X, Y and Z axes are not necessary perpendicular to
each other. As long as they are independent at the local point where the rigid body is
residing, we can always observe 3 or 6 independent motions of a rigid body depending
whether it is 2D or 3D.
O X
Y
Z
2D Plane
X
Y
Z
Rx
Rz
Ry
3D space
2-2
Figure 2-1: (Left) An unrestricted planar body can move at X and Y directions and rotate at Z
direction in a plane. The degree of freedom is 3. (Right) An unrestricted special body can move and
rotate at X, Y and Z directions. The degree of freedom is 6. You can get a more intuitive observation of
the degree of freedom from this interactive animation (add a link here). Select among 2D & 3D,
rotation, translation, or both.
For a rigid body in planar motion, since there are three possible independent motions, we
need to use three independent variables to identify them. We then define the two linear
translations and the one rotation as ̇ ̇ ̇ , where we implied a Cartesian coordinate
system here. As the motions are simply time derivative of the positions, their corresponding
position variables are . However, we note that there is some confusion with the
coordinate variables.
Let us look at the bicycle wheel of Figure 2-2. For the rotation, when the wheel is turning,
every single point will be turning at the same angular velocity. So we do not need to
differentiate any specific point. However, that is not the case for the linear velocities. As we
know from the Dynamics, when a wheel is rotating about its axis, even at a constant angular
velocity, the velocities of any two points in the rigid body will be different. For example, two
points A and C on the rim will have same speeds or magnitudes since they have same radii
about the rotating center O. Meanwhile, points A and B will have same direction since they
are along the same radial line. However, no two points will have same magnitudes and same
directions. Therefore, we need to specify which point we are representing using subscripts
on the linear velocity. Generally, we choose either the center of mass or the rotating center in
mechanics, say point O. Then we can describe the motion of the rigid body as
where a subscript is used to denote the point. For any other point, we can easily obtain its
linear velocity as long as we can find its relative position.
2-3
Figure 2-2: A rotating wheel.
Likewise, if we want to specify the pose (position and orientation) of a rigid body, we just
need to specify the coordinate of one point and a direction. The direction can be obtained by
measuring the angular displacement of a line that is fixed to the rigid body or simply by
getting the coordinates of another point. Unsurprisingly, these two methods are
essentially equivalent with some help of mathematics.
For example, if we know the coordinates of two points and . Then we can
choose any one as our base point, say A, and calculate the angle as:
(2-1)
However, there is a small problem with this equation. For example, look at Figure 2-3, point
A and point B are in opposite quadrant, but their tangent values of the corresponding angles
are the same. That is
(2-2)
VA
C
A
B
VB
VC
O
2-4
Figure 2-3: Points A and B will have same tangent values.
Therefore, the direct inverse tangent will not yield a
unique answer between 0 and . Further, when ,
Eq. (2-1) will not yield any result. Therefore, we will
introduce the four-quadrant inverse tangent, , such
as
{
2.2.2 DOF of linked rigid bodies
For a free floating rigid body with planar motion, its degree of freedom is 3. If we have
multiple, say N, independent free floating rigid bodies, the total DOF will be 3N since the
movements of them are completely independent from each other. This scenario might be
true in the computer arcade games. However, it is rarely true in the engineering. Indeed, we
can consider these independent rigid bodies as the parts of a mechanism. One purpose of
mechanical synthesis or mechanical design is to find ways to connect these N free floating
machine parts in a proper way so we will strip certain DOFs away with constraints (joints)
and force the parts to move with right motions at right timings.