Generalized coordinates and constraints Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 1 / 25
Coordinates Each rigid body is defined by 6 coordinates (called d.o.f. or dof), 3 for the position χ , 3 for the orientation α of a body frame R B attached to it; the position and orientation stacked together are called the pose p of the body. p 1 ( t ) x 1 ( t ) p 2 ( t ) x 2 ( t ) � x ( t ) � p 3 ( t ) x 3 ( t ) p ( t ) def = = = p 4 ( t ) α 1 ( t ) α ( t ) p 5 ( t ) α 2 ( t ) p 6 ( t ) α 3 ( t ) A multibody system is a composition of several rigid bodies connected to each other and satisfying some physical or geometrical constraints. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 2 / 25
Coordinates A discrete (or “atomic”) rigid body B is composed by a finite set of N geometrical points P i , each one defined in the 3D space by its position vector with reference to some reference frame (usually the body frame) � x i 1 ( t ) � x i 2 ( t ) x i ( t ) = i = 1 ,..., N x i 3 ( t ) The body B is globally characterized by M = 3 N quantities. x 11 ( t ) x 12 ( t ) x 13 ( t ) χ 1 ( t ) . x 1 ( t ) . χ 2 ( t ) . . . . . x k 1 ( t ) . . ∈ X ⊆ R M χ ( t ) = x k ( t ) = x k 2 ( t ) = X = configuration space χ j ( t ) . x k 3 ( t ) . . . . . . . x N ( t ) . χ M ( t ) x N 1 ( t ) x N 2 ( t ) x N 3 ( t ) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 3 / 25
Coordinates We can express the χ i coordinates in many different ways; between the two representations we can define a transformation χ ′ = f ( χ ). The transformation f ( · ) must be non singular almost everywhere, i.e., the � ∂ f i � 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, 2016-17 4 / 25
Coordinates Example � T � x 1 Consider a point P described by cartesian coordinates x = x 2 x 3 or by polar/spherical coordinates x ′ = � T , where � x ′ x ′ x ′ 1 2 3 x 1 = x x 2 = y x 3 = z x ′ x ′ x ′ 1 = ρ 2 = θ 3 = φ In this case the transformations between x ′ and x are defined as follows � x 1 = x ′ 1 sin x ′ 2 cos x ′ x ′ x 2 1 + x 2 2 + x 2 1 = 3 3 f − 1 : x 2 = x ′ 1 sin x ′ 2 sin x ′ f : � x ′ ( x 2 1 + x 2 3 2 = arctan( 2 ) / x 3 ) x 3 = x ′ 1 cos x ′ x ′ 3 = arctan( x 2 / x 1 ) 2 B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 5 / 25
Coordinates The Jacobian J x of the transformation f is sin x ′ 2 cos x ′ ρ cos x ′ 2 cos x ′ − ρ sin x ′ 2 sin x ′ 3 3 3 sin x ′ 2 sin x ′ ρ cos x ′ 2 sin x ′ ρ sin x ′ 2 cos x ′ J x = 3 3 3 cos x ′ − ρ sin x ′ 0 2 2 with the determinant det J x = ρ 2 sin x ′ 2 If ρ � = 0 the determinant goes to zero only for θ = x ′ 2 = 0 ± 2 k π ; this configuration is called a singular configuration . For example, the spherical coordinates are useful to model a satellite orbit motion around Earth. B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 6 / 25
Constraints In general, kinematic constraints active on the body elements are defined by implicit function of the M = 3 N coordinates, and possibly also of time t , as: ψ ( χ 1 ,..., χ 3 N , t ) = 0 If the constraint are n c , a system of n c equalities arises ψ 1 ( χ 1 ,..., χ 3 N , t ) = 0 ψ 2 ( χ 1 ,..., χ 3 N , t ) = 0 . . . ψ n c ( χ 1 ,..., χ 3 N , t ) = 0 that is equivalent to the following matrix equation Ψ ( χ ( 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, 2016-17 7 / 25
Constraints It is also possible to write constraints involving the time derivative ˙ χ ( t ) Ψ ′ ( χ ( t ) , ˙ χ ( t ) , t ) = 0 . and, in general, some constraints can be expressed as inequalities Ψ ′′ ( χ ( t ) , ˙ χ ( 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 ). The various types of constraints can be written as: a) Ψ ( χ ( t ) , t ) = 0 Ψ ′ ( χ ( t ) , ˙ b) χ ( t ) , t ) = 0 Ψ ′′ ( χ ( t ) , ˙ c) χ ( t ) , t ) ≤ 0 The constraints that directly depend on time are called rheonomic , while the time-independent ones are called sclerononomic . B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 8 / 25
Constraints - An example Example The rigid system is composed by N = 4 point masses, with 0 � T 0 � T 0 � T � 0 0 � 1 0 � 0 1 � 0 0 1 � x 1 = x 2 = x 3 = x 4 = B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 9 / 25
Constraints - An example The rigid constraints are expressed as ( x 1 − x 2 ) T ( x 1 − x 2 ) − d 2 ( x 1 − x 3 ) T ( x 1 − x 3 ) − d 2 12 = 0 13 = 0 ( x 1 − x 4 ) T ( x 1 − x 4 ) − d 2 ( x 2 − x 3 ) T ( x 2 − x 3 ) − d 2 14 = 0 23 = 0 ( x 2 − x 4 ) T ( x 2 − x 4 ) − d 2 ( x 3 − x 4 ) T ( x 3 − x 4 ) − d 2 24 = 0 34 = 0 where d ij is the distance between the point masses. There are 3 N = 12 configuration variables and N ( N − 1) / 2 = 6 constraint equations, all independent. The three oriented segments x i − x 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 3 N − 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, 2016-17 10 / 25
Generalized Coordinates From now on, 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 . q ( t ) = . 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, 2016-17 11 / 25
Generalized Coordinates The set of independent generalized coordinates is not unique: many other sets of coordinates may represent the system motion. 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, 2016-17 12 / 25
Generalized Coordinates It is always possible to express each position vector x i , with i = 1 ,..., M , as a function of the n generalized coordinates x 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 χ , we can set the following transformation between q and χ : χ = g ( q 1 , q 2 ,..., q n , t ) = g ( q ( t ) , t ) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 13 / 25
Generalized velocities The generalized velocities are defined as � ˙ q ( t ) = d q ( t ) q n ( t ) � T ˙ = q 1 ( t ) ... ˙ d t The configuration velocities ˙ χ are defined as � ˙ χ ( t ) = d χ ( t ) χ M ( t ) � T ˙ χ 1 ( t ) ... ˙ = d t with M = 3 N . B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 14 / 25
Generalized velocities and accelerations The relation between ˙ χ and ˙ q is χ ( 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 ) [ b ] i = ∂ f i ( t ) ∂ q j ( t ) ∂ t J is called the transformation Jacobian ; b is non zero only if χ directly depends from time. The generalized accelerations are q ( t )+ ˙ q ( t )+ ˙ ¨ χ ( t ) = J ( t )¨ J ( t )˙ b ( t ) B. Bona (DAUIN) Generalized coordinates and constraints Semester 1, 2016-17 15 / 25
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