Lecture 23 The Spherical Bicycle II

1. Lecture 23 The Spherical Bicycle II (This is something of a guide to the Mathematica we will look at at the end which picks up where we left off last time.) The story so far: We’ve put together a model applied simple holonomic constraints and made a Lagrangian We found the constraint matrix and the null space matrix We “cheated” a little bit in assessing the rank of the constraint matrix but I have some confidence that it is actually full rank as found

2. Here’s the picture to remind us

3. A note on scaling I chose the radius of the wheels to be my length scale, the mass of the wheel to be my mass scale, and I can choose a time scale such that scaled g = 1 I can do all that without loss of generality, and it is possible to unscale for any real bicycle The scaled dimensions for the bicycle we have are then sphere radius and mass are 2 and 40, respectively the wheel radius and mass are, of course, both unity The fork is 4 units long, ¼ unit in diameter and has a mass of unity

4. Next we need equations of motion — velocity and momentum The velocity equations are the usual q has 22 components and u has three components The components of S (22 x 3) are complicated functions of q We identify the physical meaning of the components of u by examining the velocity equations

5. The components of q are The components of u are

6. The momentum equations are the usual reduced Hamilton equations which we have from last time — I’ll spare the details, which are posted which is of the form where

7. We have a total of 25 quite complicated equations and several tasks Questions: Is this system (infinitesimally, linearly) stable? Is this system (infinitesimally, linearly) stable? Can it be controlled? Build a simulation: Generalized forces Numerical issues

8. Is this system (infinitesimally, linearly) stable? The stability question requires: an equilibrium position linearized equations with no generalized forces I seek an equilibrium for which the bicycle is erect and moving in a straight line at a constant speed

9. What does this means for q0? suffices to satisfy the reduced Hamilton’s equations for We can deduce the rest of the equilibrium variables from the velocity equations

10. There are five variables that the differential equations don’t care about They are determined by the equilibrium, but do not enter the equations The equilibria contain two constants — the direction of the straightline motion and its speed

11. equilibrium and stability condensed from last time We have an infinite set of equilibria When we get to numbers we’ll be looking at

12. equilibrium and stability Stability stability here is a little tricky, but we can follow the idea of stability by expanding around the equilibrium The velocity equations

13. equilibrium and stability For the u’ equations we can start with the symbolic homogeneous version After considerable manipulation (see last lecture) we arrive at

14. equilibrium and stability We have a pair of linear vector equations that can be manipulated

15. Now we can think about a state space picture of this

16. The block matrix elements have dimensions The whole thing is a 25 x 25 matrix, and its eigenvalues determine the system stability If everything I have done is correct, the characteristic polynomial is of the form There are 21 zero roots, and four nonzero roots

17. The nonzero roots depend on the value of w0, but not f10 For the parameters of the present model there is: a pair of complex conjugate roots with negative real parts which grow rapidly with w0 one real negative root one real positive root the system is unstable!

18. The latter two roots vs. w0 unreal numerical glitch

19. Now we have an interesting question that we need to think about What do the 21 zero roots represent? This immediately leads us to another question We have 22 generalized coordinates and nineteen constraints — a three DOF system How many exponents do we expect for a three DOF system? SIX How can we reconcile this? Will that help with the first question?

20. If we look at the perturbation equations for q, we find that do not vary in the linear limit If we look at the linear equations for the evolution of q we find that they depend only on q4, q16, u1 and u2 The linear equations for the evolution of udepend only on q4, q5, q16, q17,u1 and u2 There are only six variables that do anything interesting

21. So we can define a reduced state and write the state equations in the usual linear form

22. We’ll need to look at the generalized forces to fill in what is happening Let me defer that a bit to see how stability analysis works for this problem We drop the generalized forces and look at the reduced matrix

23. This matrix is a 6 x 6. It has two zero eigenvalues. Its nonzero eigenvalues are the same as the nonzero eigenvalues of the 25 x 25 matrix The other 19 zero eigenvalues from the 25 x 25 problem are irrelevant for the physics of the problem, which is a good thing It’s worth noting that this simplification does not appear to extend to the full problem I suspect that that is because I have not given it enough thought there must be a comparable reduction

24. We now should be confident that we can restrict our linear analysis including possible control to the reduce 6 x 6 system The system is unstable — we’d like a control to stabilize it This means that the time has come to work out the generalized forces There are two torques, one applied to the fork and one applied to the rear wheel Each has a reaction torque

25. I can write the rate of work in terms of the vector torques and rotations I take the positive sense to be action on the receiving link I can calculate the generalized force in the usual way These need to be reduced, because they contribute to the reduced Hamilton’s equations

26. This is quite monstrous expression in the general nonlinear situation but it pretty simple in the linear situation, which is what I need for control These go into the reduced Hamilton equations, but the equations we are dealing with have been solved for So we need to do a few more things before we can find B and look at controllability

27. Substitute our equilibrium (we can do that at any time) Multiply by the inverse of the matrix multiplying Drop the last term, because u3 is not in the reduced state After this we find that the forcing depends only on t2

28. We have

29. The linear problem turns out not to be controllable the rank of W is five, not six We can explore the consequences of this better in Mathematica, so . . .