Non-holonomic constraint example A unicycle

Transcription

1 Non-holonomic constraint example A unicycle A unicycle (in gray) moves on a plane; its motion is given by three coordinates: position x, y and orientation θ. The instantaneous velocity v = [ ẋ ẏ ] is along the wheel orientation. The wheel can rotate around its contact point with velocity θ. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

2 Non-holonomic constraints Assuming we express the constraint as x = q 1 y = q 2 θ = q 3 ẏ ẋ = tanθ i.e., q 1 q 2 = tanq 3 = sinq 3 cosq 3 i.e., q 1 cosq 3 q 2 sinq 3 = 0 In Pfaffian form: a 1 dq 1 +a 2 dq 2 +a 3 dq 3 = 0, where a 1 = cosq 3 a 2 = sinq 3 a 3 = 0 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

3 The differential form is exact? Check the conditions a 1 q 2 = a 2 q 1, a 1 q 3 = a 3 q 1, a 2 q 3 = a 3 q 2, If the coefficients a i s satisfy all the above relations, the differential form is integrable and the constraint is holonomic. Otherwise the form is not exactly integrable and the constraint is non-holonomic. It follows that 0 = 0 OK, sinq 3 = 0 NO, cosq 3 = 0 NO Since the conditions are not met, the constraint is NON-HOLONOMIC. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

4 Non-holonomic constraint example A wheel on a plane In Figure 1 we have represented a rigid wheel rolling on a plane without slipping; the wheel has a radius ρ, and negligible depth; the contact point O c (t) between the plane π c and the wheel moves as the wheel moves. A reference frame R 0 is defined on the plane π c for convenience. The wheel has its reference frame R r whose origin is located at the wheel center and unit vector j r orthogonal to the wheel plane π r ; this plane forms an angle α(t) with respect to the plane π c normal, that for convenience is considered horizontal. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

5 Figure: Example of non-holonomic constraint: a non slipping wheel rolling on a plane. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

6 The plane π r forms also an angle β(t) with a default direction chosen on the plane π c ; for example, the direction of the unit axis i 0. This angle can change during the motion and is called the steering angle. The plane motion constraint implies that the wheel contact point O c must always belong to the plane π c. Moreover, the non slipping condition implies: the impossibility to have a local component of the wheel motion orthogonal to the local velocity vector, given by the unit vector j r on π c, that in Figure 1 coincides with the direction of the unit vector j c. the wheel motion along the direction given by i c must guarantee the absence of slipping, i.e., it must be always verified that the space travelled on the plane by the contact point is equal to the circle arc subtended by the wheel angle θ(t). Otherwise we would experience the typical motion of the wheel with no displacement of the contact point, or a motion of the contact point without the related rotation of the wheel. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

7 The two conditions imply that the instantaneous velocity of the contact point O c (or of O r ) must be aligned with the unit vector i, that, by construction, is always aligned with i c. On the contrary, a rotation around the axis k r is permitted, since this will produce an ideal rotation around the contact point, that does not imply any slipping. In conclusions the wheel can proceed locally along the direction identified by i = i c and rotate around the instantaneous axis given by k r. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

8 Let us build the homogeneous matrix T 0 r that link the fixed frame R 0 to the wheel mobile frame R r. from R 0 to R c : T 0 c = Transl(d c(t)) Rot(k, β(t)), where d c = [ x c y c 0 ] T represents Oc in R 0 : from R c to R r : T c r = Rot(i,α(t)) Transl(ρ) Rot(j,θ(t)), where ρ = [ 0 0 ρ ] T B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

9 The quantities necessary for the motion definition are, with no constraints, the three components of d c (t) and the three angles α(t), β(t), and θ(t); therefore we have the generalized coordinate and the virtual displacements: q 1 = x c q 2 = y c q 3 = z c q 4 = α q 5 = β q 6 = θ δq 1 = δx c δq 2 = δy c δq 3 = δz c δq 4 = δα δq 5 = δβ δq 6 = δθ It seems plausible that a geometrical constraints will reduce the degrees of freedom from six to five due to the equation: d c k 0 = [ ] x c y c z c 0 0 = z c = q 3 = 0 δq 3 = 0 1 that constraints the contact point to belong to the plane π c. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

10 There is also the non slipping constraint, whose nature is not purely geometrical, but physical: it sets an identity between the wheel contact point linear velocity on the plane and the wheel angular velocity, according to the following relation, where we assume that all vector are represented in R 0 : ḋ c (t) = ρ θ(t)i c ḋc(t) ρ θ(t)i c = 0 Looking in detail, we see that with the conventions assumed in Figure 1, we can write d x c y c ρ θ(t) c β s β = 0 dt 0 0 i.e., dx c = ρ dθ c β dt dy c = ρ dθ s β dt B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

11 Dividing both terms of the two equations we have dx c = 1 dy c tanβ tanβ dx c +dy c = 0 (1) Replacing the infinitesimal motion with the virtual displacements we obtain tanβ δx c +δy c = 0 or tanq 5 δq 1 +δq 2 = 0 (2) Eqn. (1) represents a Pfaffian form constraint a 1 δq 1 +a 2 δq 2 +a 3 δq 3 +a 4 δq 4 +a 5 δq 5 +a 6 δq 6 = 0 with a 1 = tanβ and a 2 = 1; the other coefficients are zero: a 3 = a 4 = a 5 = a 6 = 0. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

12 Since a 1 q 5 = 1 cos 2 q 5 a 5 q 1 = 0 the conditions in (??) are not verified and therefore the constraint will be non-holonomic. We now ask how many degrees of freedom the wheel actually has: the relation (2) introduces a constraints that links the virtual displacement δx c to δy c, therefore, recalling that the degrees of freedom are defined as the dimension of the independent and complete set of virtual displacements, we have only four of these virtual displacements, namely δx c, δα, δβ and δθ, while δy c is linked to δx c by the relation (2), and δz c = 0 due to the planar motion constraint. It follows that for this type of wheel, the degrees of freedom are n = 4. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13

13 At last we note that despite the velocity constraint given by (2), all the points on the plane are reachable, but not through planar motion that require a sliding motion; Figure below, showing the wheel from above, presents a possible path that brings the wheel from point A to point B, arbitrarily near along a forbidden sliding direction. From A to B the wheel can complete a continuous path C D E F that does not violate any constraint. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 13