Generalized coordinates and constraints

Transcription

1 Generalized coordinates and constraints Basilio Bona DAUIN Politecnico di Torino Semester 1, B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

2 Coordinates A rigid body B is a set of point masses with geometrical constraints. Each rigid body is defined by 6 coordinates (called d.o.f. or dof); 3 for position x, 3 for orientation α, together called the pose p of the body. p 1 (t) x 1 (t) p [ ] 2 (t) x 2 (t) p(t) def x(t) p 3 (t) x 3 (t) = = = α(t) p 4 (t) α 1 (t) p 5 (t) α 2 (t) p 6 (t) α 3 (t) A set of points with constraints among them defines a rigid body. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

3 Coordinates A discrete rigid body B is composed by a set of N geometrical points P i, each one defined in the 3D space by its position vector [ ] ri1 (t) r i (t) = r i2 (t) i = 1,...,N r i3 (t) The body B is globally characterized by M = 3N quantities. r 11 (t) r 12 (t) r 13 (t) r 1 (t). χ 1 (t). χ. r k1 (t) 2 (t) x(t) = r k (t) = r k2 (t). =. r k3 (t) χ j (t) X R M X = configuration space. r N (t).. χ r N1 (t) M (t) r N2 (t) r N3 (t) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

4 Coordinates We can express the r ij coordinates in many different ways, for instance using cartesian or polar coordinates; between the two representations we can define a transformation r = f(r). The transformation f( ) [ must] be non singular almost everywhere, i.e., the fi transformation jacobian must be full rank χ j, with a possible χ j exception of a countable set of configurations. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

5 Coordinates Example Consider a point P described by cartesian coordinates r = [ r 1 r 2 r 3 ] T or by polar/spherical coordinates r = [ r 1 r 2 r 3] T, where r 1 = x r 2 = y r 3 = z r 1 = ρ r 2 = θ r 3 = φ In this case the transformations between r and r are defined as follows r 1 = r 1 sinr 2 cosr 3 r 1 r = 1 2 +r2 2 +r2 3 f : r 2 = r 1 sinr 2 sinr 3 f 1 : r r 3 = r 1 2 (r cosr 2 = arctan( 1 2 +r2 2 )/r 3) r 3 = arctan(r 2/r 1 ) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

6 Coordinates The jacobian J r of the transformation f is sinr 2 cosr 3 ρcosr 2 cosr 3 ρsinr 2 sinr 3 J r = sinr 2 sinr 3 ρcosr 2 sinr 3 ρsinr 2 cosr 3 cosr 2 ρsinr 2 0 with the determinant detj r = ρ 2 sinr 2 If ρ 0 the determinant goes to zero only for θ = r 2 = 0±2kπ; this configuration is called a singular configuration. The transformation can be useful to model a satellite orbit motion around Earth. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

7 Constraints In general, kinematic constraints are defined by implicit function of the M = 3N coordinates, and possibly also of time t, as: ψ(χ 1,...,χ 3N,t) = 0 If the constraint are n c, a system of n c equalities arises ψ 1 (χ 1,...,χ 3N,t) = 0 ψ 2 (χ 1,...,χ 3N,t) = 0. ψ nc (χ 1,...,χ 3N,t) = 0 that is equivalent to the following matrix equation Ψ(x(t),t) = 0. where Ψ is a n c 1 matrix containing the nonlinear functions of the coordinates. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

8 Constraints It is possible to write constraints involving the velocity ẋ(t) Ψ (x(t),ẋ(t),t) = 0. and, in general, some constraints can be expressed as inequalities Ψ (x(t),ẋ(t),t) 0. A direct time dependency is present when some constraints are varying according to an external time law, otherwise the constraints depend from time only through the coordinates χ i (t) can be written as: Ψ(x(t)) = 0 Ψ (x(t),ẋ(t)) = 0 Ψ (x(t),ẋ(t)) 0 The constraints that directly depend on time are called rheonomic constraints, while the time-independent ones are called sclerononomic constraints. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

9 Constraints - An example Example The rigid system is composed by N = 4 point masses, with r 1 = [ ] T r 2 = [ ] T r 3 = [ ] T r 4 = [ ] B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

10 Constraints - An example The rigid constraints are expressed as (r 1 r 2 ) T (r 1 r 2 ) d 2 12 = 0 (r 1 r 3 ) T (r 1 r 3 ) d 2 13 = 0 (r 1 r 4 ) T (r 1 r 4 ) d 2 14 = 0 (r 2 r 3 ) T (r 2 r 3 ) d 2 23 = 0 (r 2 r 4 ) T (r 2 r 4 ) d 2 24 = 0 (r 3 r 4 ) T (r 3 r 4 ) d 2 34 = 0 where d ij is the distance between the point masses. There are 3N = 12 configuration variables and N(N 1)/2 = 6 constraint equations, all independent. The three oriented segments r i r 1 form a basis of mutually orthogonal vectors, and are the ideal representation of a cartesian reference frame, the most simple example of a rigid body. The system has therefore only 3N n v = 12 6 = 6 free parameter; this number is the maximum number of dof of a rigid body in space. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

11 Generalized Coordinates We assume that all n c constraints are independent. The implicit function theorem guarantees that it is always possible to express n c variables as functions of the n = M n c remaining ones. We can therefore identify n = M n c independent variables q 1, q 2,..., q n. These variables are called generalized coordinates q 1 (t) q(t) =. Q q n (t) They univocally represent the motion of a multibody system, implicitly taking into account the kinematic constraints acting on the system. All the other n c configuration variables can be computed from them, using the constraint equations. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

12 Generalized Coordinates The set of generalized coordinates is not unique: many other sets of coordinates may represent the system motion in all its parts. The set must be independent (no generalized coordinates q i shall exist that can be obtained as linear combinations of other generalized coordinates) complete (the motion of the constrained set is completely determined by the generalized coordinates included in the set) If the set is complete and independent, it is also minimal. The number n defines the dimension of the generalized coordinate space Q. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

13 Generalized Coordinates It is always possible to express each position vector r i, with i = 1,...,M, as a function of the generalized coordinates r i = h i (q 1,q 2,...,q n,t) = h i (q(t),t) where h i is a generic nonlinear vector function, whose derivatives with respect to its arguments exist up at least to the second order. Similarly, if we consider the configuration variables x, we can set the following transformation between q and x: x = g(q 1,q 2,...,q n,t) = g(q(t),t) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

14 Generalized velocities and accelerations The generalized velocities are defined as q(t) = dq(t) dt = [ q 1 (t)... q n (t) ] T The configuration velocities ẋ are defined as with M = 3N. ẋ(t) = dx(t) dt = [ ẋ 1 (t)... ẋ M (t) ] T B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

15 Generalized velocities and accelerations The relation between ẋ and q is ẋ(t) = J(t) q(t)+b(t) where J R M n and b R M 1 are defined as [J] ij = f i(t) q j (t) [b] i = f i(t) t J is called the transformation Jacobian b is non zero only if x directly depends from time. The generalized accelerations are ẍ(t) = J(t) q(t)+ J(t) q(t)+ḃ(t) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

16 Generalized constraints The constraints obtained considering the generalized coordinates and velocities are called generalized constraints If the constraints depend directly on time Φ(q(t), q(t),t) = 0 or Φ(q(t), q(t),t) 0 If they do not depend directly on time Φ(q(t), q(t)) = 0 or Φ(q(t), q(t)) 0 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

17 Virtual Displacements and Constraints Virtual displacements or admissible variations δr are a small (i.e., virtual, not real) displacements of body points, allowed by the kinematic constraints Virtual displacements can take place independently from time, i.e., are not subject to the law of physics. We assume δt 0. For every generalized coordinate q i there is a virtual displacement δq i. The number n dof of independent and complete virtual displacements δq i defines the degrees-of-freedom of the multibody system. Usually the number n of independent and complete generalized coordinates q i is equal to the degrees-of-freedom n = n dof. However, this is not always the case, and depends on the type of constraints; when the constraints are non-holonomic n dof < n. The term holonomic come from the Greek oλoς and means integer, integrable. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

18 Non-holonomic constraints Let us consider only the equality constraints, and, in particular, those that depend only on the positions Φ(q(t),t) = 0 The equality constraints that depend only on the positions are always holonomic constraints; non-holonomic constraints belong to two classes of constraints: Inequality constraints, described by: Φ (q(t), q(t),t) 0 Equality constraints that depend also on velocities but are not exactly integrable. Φ (q(t), q(t),t) = 0 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

19 Non-holonomic constraints For simplicity and without loss of generality, we restrict our attention to the constraints that depend only on the generalized velocities, Φ ( q(t),t) = 0 When these differential equations do not provide an exact integral, they represent non-holonomic constraints. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

20 Non-holonomic constraints Take a generic i-th holonomic constraint φ i (q(t),t) = 0, and derive it with respect to time to obtain the corresponding constraint, expressed as a function of the velocities where φ i ( q(t),t) = 0 a(q)t q+b(q) = 0 φ i (t) q 1 (t) a(q) =. φ i (t) q n (t) b(q) = φ i(t) t The two constraints φ i (q(t),t) = 0 and φ i ( q(t),t) = 0 are equivalent, since φ i can be obtained integrating φ i. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

21 Non-holonomic constraints The constraint written in differential form: a(q) T dq+b(q)dt = 0 represents the so-called Pfaffian form. If we replace the differential quantities with the corresponding virtual displacements, recalling that δt 0, we obtain: a(q) T δq = 0 If the Pfaffian form is integrable, i.e., if it represent an exact differential, it can substituted by its integral: in this case the constraint is holonomic. If on the contrary the Pfaffian form is not an exact differential, it cannot be integrated and the corresponding constraint is non-holonomic. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

22 Exact differentials Given the differential form dφ = a(q) T dq it is an exact form in R n if dφ does not depend on the integration path. This is true when dφ = ( φ) T dq where ( φ) T = (grad φ) T = [ φ q 1 The coefficients a(q) must satisfy the relation a i (q) = φ q i, i = 1,...,n ] φ q n B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

23 Exact differentials These identities between the second partial derivatives hold and this implies 2 φ = 2 φ, i,j = 1,...,n q j q i q i q j a i (q) q j = a j(q) q i, i, j = 1,...,n 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. From a physical point of view, the classical examples are those of a wheel rolling on a plane without slippage between wheel and plane at the contact point, or certain kind of sliding object, such as the ice skates. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

24 Virtual work Virtual displacements are important for the definition of the virtual work δw. Given a system consisting of N point masses, each defined by a position vector r i, on which acts a system of N forces f i, applied on the system and having their application point in r i, the virtual work δw is defined as: δw = N f i δr i i=1 N f T i δr i i=1 The system is said to be in static/dynamic equilibrium if the virtual work of the static/dynamic forces is zero, i.e., if δw = N f i δr i = 0 i=1 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25

25 Virtual work If both N f linear forces and N τ rotational moments act on the system, it is necessary to distinguish the relative contributions, as follows N f N τ δw = f i δr i + τ i δα i = 0 i=1 where now we have introduced the moments τ i and the virtual angular displacements δα i. i=1 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, / 25