MAT 272 Calculus and Analytic Geometry III Fall 1997 Project 3: Tasks and deliverables for week 1

Final objectives and goals: At the end of the project you shall have completely understood the dynamics of the planar assembly of three linked rigid bodies with two actuated joints. This means, that when given an initial shape and initial orientation, and a final shape and a final orientation you can relatively quickly devise a maneuver (i.e. find a path in the parameter plane that encodes the shapes) that gets the assembly from one shape/orientation to the other. You should also be able to explain in simple terms how differential equations and vector calculus are used together to solve the problem.

Rough outline of the time-line:

Week one: Find the equations of motions for the model (physics and multi-variable calculus).

Understand the parameterization of the shapes (pictures).

Peek into the existing literature (surf the WWW)

Week two: Understand the constraints encoded in the equations of motion: Play with loops in shape space and observe.
Rewrite the equations of motion in various different ways (differential form, system of DEs , vector fields).
Calculate the "derivative of the constraint", investigate its graph, and relate its features to prior observations.

Week three: Formally connect differential equations and vector calculus, and use this to find superior solutions.
Develop an algorithmic solution that can be used to quickly solve any reorientation maneuver.

The time line may change a little, depending on progress and student feedback!

Specific tasks and deliverables for the first week: (30 points total – but this is also a prerequisite for all later work!).

1. Physics and some multi-variable calculus (curves, and velocity vectors):

Find the equations of motion for the three-link assembly.

Step-by-step hints (there are many different ways for doing this; the following is naïve, but uses lots of calculus!)
Suggestion: The formulae get quite messy; so it is advised to use MAPLE throughout. Due to idiosyncrasies of MAPLE, however, it is suggested to work with lists like r=[cos(alpha(t)),r1*sin(theta1(t))], rather than vectors. Also, calculate the cross products by hand, e.g. like u[1]*v[2]-u[2]*v[1] for the product of two lists (vectors) u and v.

• In a physics or mechanics book look up the definition of and formulae for angular momentum, as well as "conservation of angular momentum". Include a copy (Xerox, or hand-written) of these in your project folder.
• Draw a carefully labeled diagram of the assembly (can you do better than the old drawing which is on the WWW?)
• Consider each link as a line segment of zero-thickness with constant linear mass-density d=mi/(2ri) (like 1 kg/meter).
Parameterize each line segment – use s instead of the usual t. For each link you should get a formula for a vector r(s) from the center of the central link to the points along each line segment, e.g. s ranging from 0 to 2ri or from -ri to ri .
The position vectors r(s) depend on the constants r0, r1, and r2, as well as the angles
a(t), q1(t), and q2(t), which are functions of time.
• Calculate the velocity vectors v(s) for each point (parameterized by s) by differentiating r(s) with respect to t.
• Calculate the angular momentum of each link about the center of the central link by integrating e.g.
• Add the results, and equate the total angular momentum to zero. Solve for (da/dt).

(At the end of the first week , the class will compare the solutions found, discuss the problems encountered, and will share the correct solution as the basis for all future work.)

Deliverables: A labeled drawing of the assembly, derivation of the equns of motion.

2. Understand the parameterization of the shapes:

Consider a simple loop in the shape space, (straight lines from (0,0) to (p/2,0), to (p/2,p/2), to (0,p/2), and back to (0,0).

Sketch the loop in q1-q2 space (label axes, show tickmarks). Sketch the corresponding shapes of the three-body assembly (at least 5 frames), and briefly describe the motion in words. (Ignore the full equation of motion, and the resulting changes of orientation, concentrate on the shape only.). As a second curve consider the straight line segment from (0,0) to (p/2,p/2).In words explain the corresponding shape changes. Sketch at least one intermediate shape.

Deliverables: Two verbal descriptions, sketch of the first curve, sketches of the shapes along each curve.

3. Literature:

• Find the recent National Research Council publication related to this project (follow the links from the project description).
In this publication find the illustration of the "planar skater" (page 9), and illustrations of "geometric phase" – a spiral that "sits" above a loop in a plane (or surface) below (pages 7 and 33) – sketch copies of these illustrations (or print them out)..
• Search the WWW for "falling cat" (e.g. AltaVista comes up with a few dozen addresses) and visit at least three sites.

Deliverables: Write one paragraph either summarizing what you read, or reporting your overall impressions of the coverage of this problem in the literature. Suggestions: How old are the references (300 years?) What kind of people care about cats, and why do they do? Print out at least one illustration that you found, or sketch by hand the illustration w/ the spiral (with labels).
Bonus: Explain in words why an assembly consisting of two linked bodies is boring, and why the three body assembly is exciting. (You may keep this very informal, no technical details required here.