1. Lecture 22 The Spherical Bicycle 1

2. 2 Some relative dimensions with the wheel radius and mass as unity sphere radius 2, mass 50 fork length 4, radius 1/8, mass 1 treat the inertia terms as simple We will eventually need some numbers  = π/9 (20°)

3. 3 What do we have to do? Figure out equations of motion Find equilibria Assess stability (linear) Design stabilizing control (linear or nonlinear) Design tracking control (may well be beyond us at the moment that’s an autonomous bicycle) This is about as far as we are likely to get today

4. 4 We need to explore the nonsimple orientation constraints that we talked about before equation development Think about holonomic constraints There are two simple ones: that the rear wheel and the frame share a common K axis There are nine nonsimple connectivity constraints I elected to apply only the simple ones, giving me 22 generalized coordinates

5. 5 I want the K 1 axis to be horizontal I 1 is red J 1 is yellow K 1 is black the blue vector denotes k equation development

6. 6 I want the K 2 axis to slant back color codes are the same equation development

7. 7 I want the K 3 axis to point in the same general direction as K 1 Note that the J 3 axis points up equation development

8. 8 There are two simple nonholonomic constraints aligning K 4 and K 1, which we agreed to apply above, giving me 22 generalized coordinates There are nine connectivity constraints, defining the center of mass of links 2, 3 and 4 in terms of that of link 1 There are six rolling constraints, and they are independent I have a total of 9 + 4 + 6 = 19 constraints There are four nonsimple orientation constraints They are independent, leaving me with three components of a u vector equation development

9. 9 The elements of u are The first two of these will be zero at equilibrium and the third will be constant, which constant I will denote by  0 equation development

10. 10 The C matrix is 19 x 22 and the S matrix is 22 x 3 The system inertia matrix is 22 x 22 There will be 22 velocity equations (evolution of q) 22 Hamilton’s equations and 3 reduced Hamilton’s equations (evolution of u) The numerical simulation will have to deal with 25 first order equations with very complicated coefficients The state space for stability will have 25 dimensions and will need to be addressed very carefully equation development

11. The velocity equations are Hamilton’s equations are We need to reduce them, and we need to convert them to equations for u Start with the latter task 11 equation development

12. convert the rate of change of p, the left hand side of Hamilton’s equations 12 combine these equation development

13. 13 The right hand side and so we can combine the whole thing to give me Hamilton’s equations where I have changed a number of dummy subscripts to make them neater equation development

14. 14 Finally we need to reduce these using S equation development

15. 15 is of the form equation development Ideally we would construct a state space where

16. 16 The first step would be to solve for, but this is beyond the capability of Mathematica without some simplification We will have to go forward in pieces equation development

17. 17 To understand stability we need to identify an equilibrium and perturb this AND THE VELOCITY EQUATION The reduced Hamilton’s equations reduce to at equilibrium, and this has little content, because both terms are zero at lowest order equilibrium and stability

18. 18 We have an infinite set of equilibria, as they do not depend on  1 Here they are for  1 = 0 (we’re actually going to use π/4) equilibrium and stability

19. 19 Stability stability here is a little tricky, but we can follow the idea of stability by expanding around the equilibrium equilibrium and stability The velocity equations

20. 20 Finally get the equations for q’ equilibrium and stability

21. 21 For the u’ equations we can start with the symbolic version for convenience drop the generalized force for studying equilibrium equilibrium and stability

22. 22 the linear part where I use symmetry to combine the two Z 0 parts and again we need to find the perturbation parts equilibrium and stability

23. 23 equilibrium and stability

24. 24 equilibrium and stability We have a pair of linear vector equations that can be manipulated

25. 25 Now we can think about a state space picture of this

26. 26 The block matrix elements have dimensions The whole thing is a 25 x 25 matrix, and its eigenvalues determine the system stability It’s time to look at Mathematica — still incomplete