Thursday, January 13, 2011

Rotational Kinematics

Rotational kinematics


Kinematics: The study and description of motion, without regard to its causes, for example, we can calculate the end point of a robot arm from the angles of all its joints. Alternatively, given the end point of the robot arm, we could calculate the angles and settings of all its joints required to put it there (inverse kinematics - IK). Kinematics can be studied without regard to mass or physical quantities that depend on mass.

Kinematics involves position, velocity and acceleration (and their rotational equivalents).
  • Position is the point in space that an object occupies, this needs to be defined in some coordinate system.
  • Velocity is the rate of change of position with respect to time.
  • Acceleration is the rate of change of velocity with respect to time.

 Rotational Kinematics

If you can do projectile motion problems, which involve straight-line motion, then you should be able to do rotational motion problems, because a circle is just a straight line rolled up. To solve rotational kinematics problems, a set of four equations is used; these are essentially the 1-D projectile motion equations in disguise.
If you spin a wheel, and look at how fast a point on the wheel is spinning, the answer depends on how far away the point is from the center. Velocity, then, isn't the most convenient thing to use when you're dealing with rotation, and for the same reason neither is displacement, or acceleration; it is often more convenient to use their rotational equivalents. The equivalent variables for rotation are angular displacement (angle, for short); angular velocity , and angular acceleration . All the angular variables are related to the straight-line variables by a factor of r, the distance from the center of rotation to the point you're interested in.

Although points at different distances from the center of a rotating wheel have different velocities, they all have the same angular velocity, so they all go around the same number of revolutions per minute, and the same number of radians per second. Angles (angular displacements, that is) are generally measured in radians, which is the most convenient unit to work with. A radian is an odd unit in physics, however, because it is treated as being unitless, and is often put in or taken out whenever it's convenient to do so.
It is helpful to recognize the parallel between straight-line motion and rotational motion. Writing down the four rotational kinematics equations reinforces that. Any equation dealing with rotation can be found from its straight-line motion equivalent by substituting the corresponding rotational variables.
The straight-line motion kinematics equations apply for constant acceleration, so it follows that the rotational kinematics equations apply when the angular acceleration is constant.

  Rotation of a Rigid body about a fixed axis  
Rotational kinematic parameters (q,w,a) and their relationships with linear kinematic parameters (x,v,a).

Table 1: Linear quantities and their rotational counterparts

Linear Rotational Relationship
displacement Dx (m; vector) Dq (rad;scalar) x = rq
velocity v (m/s; vector) w (rad/s;vector) v = rw
acceleration a (m/s2; vector) a (rad/s2; vector) a = ra

Table 2: Motion with constant acceleration

velocity, time v = vo + at w = wo + at
displacement. time x = xo + vot + (1/2)at2 q = qo + wot + (1/2)at2
velocity, displacement v2 = vo2 + 2a(x - xo) w2 = wo2 + 2a(q - qo)

A rotational examples

Consider an example of a spinning object to see how the rotational kinematics equations are applied. Imagine a ferris wheel that is rotating at the rate of 1 revolution every 8 seconds. The operator of the wheel decides to bring it to a stop, and puts on the brake; the brake produces a constant deceleration of 0.11 radians/s2.


(a) If your seat on the ferris wheel is 4.2 m from the center of the wheel, what is your speed when the wheel is turning at a constant rate, before the brake is applied?
(b) How long does it take before the ferris wheel comes to a stop?
(c) How many revolutions does the wheel make while it is coming to a stop?
(d) How far do you travel while the wheel is slowing down?


(a) The wheel is rotating at a rate of 1 revolution every 8 seconds, or 0.125 rev/s. This is the initial angular velocity. It is often most convenient to work with angular velocity in units of radians/s; doing the conversion gives:

Your speed is simply this angular velocity multiplied by your distance from the center of the wheel:

(b) We've calculated the initial angular velocity, the final angular velocity is zero, and the angular acceleration is -0.11 rad/s2. This allows the stopping time to be found:

(c) To find the number of revolutions the wheel undergoes in this 7.14 seconds, one way to do it is to use the equation:

This can be converted to revolutions:

(d) To figure out the distance you traveled while the wheel was slowing down. the angular displacement (in radians) can be converted to a displacement by multiplying by r: