ELEC 5530/6530 Mobile Robots Fall 2011 Dr. Roppel
In-Class Assignment #1 Friday Sept. 2, 2011
This is a “virtual” in-class assignment. Submit it on Blackboard anytime before the start of class n Wed.
9/7/11. Also bring hardcopy to class with you on Wed. and be prepared to submit it after discussion.
1. Consider Fig. 3.4 and Eq. 3.12 on p. 64 of the text.
Show that Eq. 3.12 can be written out in the form below:
sin(   ) xR  cos(   ) yR   l cos   R  r
where
xR is the instantaneous velocity of the robot* along its local x axis, y R is the
instantaneous velocity of the robot along its local y axis, and  R is the instantaneous rate of
rotation of the robot around point P (CCW = positive viewed from above).
Discussion: r is the radius of the wheel, and  is the rate of turning of the wheel (units of
r . The
wheel axle will move in the direction indicated by the vector v in the diagram, when  is
radians per second). Consequently, the linear speed of travel for the wheel’s axle is
increasing in the direction shown.
*The robot is considered to be a rigid body, so that “velocity of the robot” is equivalent to “velocity of
point P.”
2. Draw a simplified version of Fig. 3.4 for several cases, and confirm that they satisfy Eq. 3.12. One
case is given below by way of example.
Case 1: α = 0, β = 0. See drawing 2a below. Discussion: First, consider translational motion (that is,
motion parallel to the XR and YR axes). At the instant shown, the only possible motion of the robot
consistent with rolling of the wheel is along the YR axis. This is consistent with Eqn 3.12, since
sin(   ) =0 rendering the x-axis motion zero, and cos(   )
= 1 requiring all translational
motion to be along the yR axis. Note that motion of the robot in the positive YR axis requires the wheel to
roll in the opposite direction of the vector v. This fact is consistent with the minus sign in front of the cos
term in Eq. 3.12. Next, consider pure rotation. In order for the robot to rotate around point P at the
instant shown, the wheel must roll in the –v direction (up the page in the drawing). It must do so at such
a rate that, at the instant shown,
l  r . This can be seen from the auxiliary drawing 2b below.
Now it’s your turn. Following the example above, consider the following cases. For each case, make a
drawing and show that the rolling constraint equation is consistent with the physical situation.
Case2: α = 0, β = 900
Case3: α = 900, β = 0
3. Consider Fig. 3.17 in the text. This shows a robot planning a path made up of straight lines and
circular arcs. Layout a path like this for a room you are familiar with and have access to, such as the
main room in your apartment, the Green room in Broun Hall, etc. Make your drawing on
engineering paper or graph paper, or use a computer tool. Make it to scale with reasonable
accuracy. Start your path at an entrance and end at the farthest corner, or at another entrance.
What is the largest diameter of a round robot that can navigate your room without running into
anything?