**
Kinematics of the Planar Continuum Snake Robot**

**New Explorations in Science, Technology + Math,
Manhattan**

**Summer Research Program for Science Teachers**

**August 2009**

Time: 45 Minutes

Level: Students who had a year of Physics, have taken, or are taking, Pre-Calculus/Calculus

**Objective****:**

Students should be able to write down the kinematic equations of the Planar Continuum Snake Robot

**Do Now:**

Calculate the length of an arc that subtends an angle θ at the center of a circle of radius R.

**Motivation:**

Throat surgery is unusually challenging because of the shape, length, and complexity of the organ. Snake robots have the dexterity to be used in such and in many others minimally invasive surgical procedures.

Materials:

Students will be given snake robots. In addition there will be one drawn on the blackboard as shown below in Figure 1.

**Figure 1:** Diagram of a
snake robot. ρ and ρ_{i} are the radius of curvature of the primary and
secondary backbones.

**Development of
Lesson**: First identify for the students the
primary backbone and the secondary backbone. By pulling/pushing on the secondary
backbones, the teacher will bend the robot alternatively in one direction and
then in the opposite direction. In groups of two students will work to obtain
the kinematics equations of the robot. They goal will be to derive an equation
that relates the length of the secondary backbone to that of the primary
backbone in terms of θ_{L }the angle with respect to the horizontal made
by the tangent to the primary backbone.

The teacher will walk around the class speaking to individual groups, encouraging them but not giving them the answer. Ask them to carefully observe the backbones and find out which backbone does not change in length as the robot bends. Once they have answered this question they are well on their way in obtaining the final equation which is,

Each group must present their work at the end of the class, check for units and explain why they think they are right. In a class of this nature they may be groups that may quickly arrive at the result. Those groups will start programming their equations using MATHLAB.

**Summary and
Conclusions**: I will summarize the main steps
in the derivation. I will ask the students to examine again the snake robot and
write down the differences they expect between a two dimension and three
dimension snake robot that is used in surgery. In this way students will get a
better appreciation of the dexterity of a snake robot and their potential in
minimal invasive surgery.

**
New York****
State**

**
STANDARD 1 - **
Analysis, Inquiry, and Design

Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seeks answers, and develop solutions.

**STANDARD 6** -
Interconnectedness: Common Themes

Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning. Students will understand that models are simplified representations of objects, or systems used in analysis, explanation, interpretation or design.

**STANDARD 7** -
Interdisciplinary Problem Solving

Students will apply the knowledge and thinking skill of mathematics, science and technology to address real-life problems and make informed decisions.