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

Refer to the MAPLE worksheet fallcat3.mws for common notation, and the correct equation of motion.

• Rewrite the equation of motion in the form (da/dt) = M(q1(t),q2(t)) (dq1/dt) + N(q1(t),q2(t)) (dq2/dt) – i.e. solve for (da/dt).
• Explain why this is a useful way of writing the equation: Think of the practical meaning of the variables (dq1/dt), (dq2/dt) and (da/dt) in terms of the "WORKING MODEL" implementation supplied by Greg Mayer.
Explain why you might have expected that the coefficients M and N depend on
q1(t) and q2(t), but not on a(t).
• To better connect this project to the textbook, replace q1(t), q2(t), and a(t) by x(t), y(t), and z(t), respectively. In MAPLE use the "subs" command. Also specialize to the case where all densities are equal to 1 (e.g. kg/m) and the length of each link is equal to 2 (i.e. r0 = r1 = r2= 1). Again use the "subs" command.(Feel free to also explore different sizes and densities.)
• Calculate the "scalar curl" h(x,y) = (dN/dx)-(dM/dy) (partial derivatives) ( or ) of the "vector field" F(x,y)=M(x,y)i+N(x,y)j. Simplify the expression.
• Explain why it is so exciting that the vector field F(x,y) is not conservative --- i.e. if h(x,y) was identically equal to zero, explain what would be the net change
for any closed curve C in the "shape space" (xy-plane or q1q2-plane)?

Aside: Being not conservative is essentially the same as being controllable in the technical sense of geometric control theory. In this case controllable means that even though one has direct control of only q1(t) and q2(t), indirectly one can independently control all three states q1(t), q2(t), and a(t) by a judicious choice of the controls u1=(dq1/dt) and u2=(dq2/dt) – think again of the motor-speeds in the "WORKING MODEL".--- Also, look in the article by Brockett in the NRC publication for a discussion of control systems of the form (dx/dt)=u1, (dy/dt)=u2, (dz/dt)=M(x,y)u1+N(x,y)u2 (you can't miss it, it is easy to find!)

New restriction: Assume that each joint allows only angles between (-3p/2) and (3p/2), i.e no full rotation (or twist of the cat).

• Plot the graph of h(x,y) (i.e. h(q1,q2)), and use this to identify closed paths in the xy-plane (or q1q2-plane) which result in comparatively large changes Dz (or Da) in overall attitude a. (Use Green's theorem - ch.18 - and interpret double integrals - ch.15 - as signed volume under the graph, i.e. positive when above the xy-plane, and negative when below!)
• Find and describe a path in the xy-plane (or q1q2-plane) which starts and ends at (0,0) (what does this mean practically?), and that results in the change Da = p (what does this mean practically?). Note, there are infinitely many solutions. You should strive for a solution that clearly demonstrates the use of Green's theorem (compare previous item), whose simplicity is appealing and that appears to be a reasonable choice for engineering implementations. Also keep in mind that a path may cross itself (even several times) and it may e.g. wind around the same circle several times. Your description of the path may be any combination of formulas, list of vertices (e.g. for polygonal paths), and verbal descriptions, as long as it is concise and technically correct.
• For BONUS CREDIT: Devise an algorithm that will solve the previous problem for any given desired value Da = C, say, between -p and p. One possible solution might be a look-up table (possibly combined with formulas) for n and r where n is the number of times the path winds around a circle of radius r – but new ingenious solutions are much desired!
• For BONUS CREDIT: Now suppose that the objective is to keep a fixed (i.e. Da = 0), while changing the shape from an initial value (q11,q21) to a final value (q12,q22). E.g think of a path connecting (q1,q2)=(0,0) to (q1,q2)=(0,p) that holds Da = 0 (what does this mean practically?).
Finally, you may want to consider the complete problem of providing an algorithm that will provide a path in the shape space (i..e. in the
q1q2-plane) that connects any pair of points (q11,q21,a1) and (q12,q22,a2). A rather informal discussion is welcome, as long as it is systematic and clearly demonstrates the general methodology.