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Bicycle Dynamics III: Kinematics - Position Vectors

25 May 2014

Now that we have our coordinate systems, and our transformations between coordinate systems, the important position vectors can be written out. One we have those, we can take derivatives to get the important velocity vectors and acceleration vectors.

Before we get there, though, we need to define all of the different dimensions. The bike diagram is below but with the dimensions filled in. The convention for dimensions is that \(h\) dimensions are distances in the green, or \(2\), direction (h is for height) and \(l\) dimensions are distances in the red, or \(1\) direction (l is for length). These \(h\) and \(l\) dimensions are given in the coordinate frame closest to the inertial frame \(\hat e_i\), and the subscripts tell you between which two frames the distance describes (subscripts are right to left denoting farthest from inertial frame to closest to inertial frame). As an example, \(h_{ab}\) is the distance in the \(\hat a_2\) direction from the \(\hat b_i\) frame origin to the \(\hat a_i\) frame origin.

The exception to the above rule are the distances to the human \(h_m\) and \(l_m\). These are given in the \(\hat m_i\) frame. They have single subscripts to denote that they do not conform to the normal convention.

It is key to remember that these dimensions are not all in the inertial frame. For instance, the \(h_{bc}\) distance is described in the \(\hat b_i\) frame (the frame closest to inertial that is associated with).

So again, we have our bike diagram with coordinate systems but also with all dimensions and the vectors we want to define drawn in:

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Using the dimensions in their respective coordinate systems, the vectors can be written:

\[ \bar r_{raxle} = x\hat e_1 + r_r \hat a_2 \]

\[ \bar r_{hum} = l_{m} \hat m_1 + h_{m} \hat m_2 \]

\[ \bar r_{hand} = l_{ba} \hat a_1 + h_{ba} \hat a_2 \]

\[ \bar r_{faxle} = l_{bc} \hat c_1 + h_{bc} \hat c_2 \]

For convenience, I’ll sometimes use a shorthand notation that works nicely with the rotation matricies. The shorthand represents the components of a vector in a particular coordinate system. For instance, \(\bar r_{axle}\) is written:

\[ \bar r_{axle} = \begin{bmatrix} x \\ 0 \\ 0 \end{bmatrix}_{\hat e} + \begin{bmatrix} 0 \\ r_r \\ 0 \end{bmatrix}_{\hat a} \]

The subscript of the column vector denotes in what coordinate system the components are written. This is nice, because if you then want \(\bar r_{axle}\) written entirely in the \(\hat e\) frame you can do:

\[ \bar r_{axle} = \begin{bmatrix} x \\ 0 \\ 0 \end{bmatrix}_{\hat e} + \mathbf R_{el} \mathbf R_{la} \begin{bmatrix} 0 \\ r_r \\ 0 \end{bmatrix}_{\hat a} \]

And it is clear that the result of this is a component representation of \(\bar r_{axle}\) written entirely in the \(\hat e\) frame.

I realize all the notation is verbose and somewhat offensive at first glance, but with a complicated geometry problem like this it is really necessary. This way of doing things makes sure that there are no ambiguities.


We want to take time derivatives which is a bit tricky since almost everything in our system changes with time; some of the angles, some of the linear dimensions, and most of the coordinate systems.