Kinematics Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Kinematics Semester 1, 2014-15 1 / 15
Introduction The kinematic quantities used are: position r, linear velocity ṙ, linear acceleration r, orientation angle α, angular velocity ω, angular acceleration ω, One must use ω for the angular velocity instead of α: in kinematic equations it is necessary to use the true angular velocity vector. If α is required, there are relations from ω to α and vice-versa. The motion equations are described by r(t) = g r (q(t),π,t) α(t) = g α (q(t),π,t) where π is a parameter vector that characterize the system from a geometrical, physical or structural point of view. B. Bona (DAUIN) Kinematics Semester 1, 2014-15 2 / 15
Kinematics: position equations If we use the pose vector p(t) T = [ r T (t) α T (t) ] we can write the direct position kinematics [ ] gr ( ) p(t) = g p (q(t),π,t) where g p ( ) = g α ( ) and the inverse position kinematics, given by the inverse nonlinear relation q(t) = g 1 p (p(t),π,t) This equation is in general much more difficult to express, since it requires the inversion of nonlinear trigonometric functions. B. Bona (DAUIN) Kinematics Semester 1, 2014-15 3 / 15
Kinematics: velocity equations One can express both the linear velocities ṙ(t) and the angular velocities α(t) of the rigid body as functions of the generalized velocities q(t), obtaining the direct linear velocity kinematic function and the direct angular velocity kinematic function ṙ(t) d dt g r(q(t),π,t) = J l (q(t),π,t) q(t)+ g r(q(t),π,t) t α(t) d dt g α(q(t),π,t) = J α (q(t),π) q(t)+ g α(q(t),π,t) t The derivative α is not the same as the angular velocity ω as we will detail later. B. Bona (DAUIN) Kinematics Semester 1, 2014-15 4 / 15
Kinematics: velocity equations The matrix J l is the linear Jacobian matrix [ ] [J l ]ij = gri (q(t),π,t) q j (t) The matrix J α is the angular Jacobian matrix [ ] [J α ] = gαi (q(t),π,t) ij q j (t) B. Bona (DAUIN) Kinematics Semester 1, 2014-15 5 / 15
Kinematics: velocity equations When the above functions do not explicitly depend on time, we obtain a simplified form ṙ(t) = J l (q(t),π) q(t) or simply ṙ = J l (q) q and α(t) = J α (q(t),π) q(t) or simply α = J α (q) q We observe that the two relations are linear in the velocities, since they are the product of the Jacobian matrices and the generalized velocities q i (t). We also observe that the Jacobian matrices are, in general, time varying, since they depend on the generalized coordinates q(t). B. Bona (DAUIN) Kinematics Semester 1, 2014-15 6 / 15
Kinematics: velocity equations Embedding ṙ and α in a single vector, we can write [ṙ(t) ] ṗ(t) = or equivalently ṗ(t) = α(t) [ ] v(t) α(t) The quantity ṗ takes the name of generalized velocity and is not a vector, since the time derivatives of the angular velocities are different from the components of the physical angular velocity vector ω. When we use the true geometrical angular velocity ω [ ] v(t) ṗ(t) = ω(t) is also called twist. B. Bona (DAUIN) Kinematics Semester 1, 2014-15 7 / 15
Kinematics: velocity equations We can now write in a compact form the kinematic function of the generalized velocities: ṗ(t) d dt g p(q(t),π,t) = J p (q p (t),π) dq(t) dt + g p(q(t),π,t) t where the Jacobian J p is a block matrix composed by J l and J α [ ] Jl (q(t),π) J p (q(t),π) = J α (q(t),π) If the kinematic position function g p does not explicitly depend on time, we can write. ṗ(t) = J p (q p (t),π) q(t) or simply ṗ = J p q B. Bona (DAUIN) Kinematics Semester 1, 2014-15 8 / 15
Kinematics: velocity equations This relation can be inverted only when the Jacobian is non-singular, i.e., detj p (q(t)) 0 In this case, if the kinematic equations do not depend on time, we have what we call the inverse velocity kinematic function q(t) = J p (q(t),π) 1 ṗ(t) or simply q = J 1 p ṗ The Jacobian depends on the generalized coordinates q i (t), and it can become singular for particular values of these ones; we say in this case that we have a singular configuration or a kinematic singularity. The coordinates q sing that produce the singularity are called singular configurations detj p (q sing ) = 0 The kinematic singularity problem is not treated in this course. B. Bona (DAUIN) Kinematics Semester 1, 2014-15 9 / 15
Angular velocity transformations If α(t) are the angular parameters (Euler, RPY, etc. angles), the analytical derivative α(t) is called analytical (angular) velocity. The analytical derivative α does not necessarily coincide with the physical angular velocity vector ω, and the second derivative α does not necessarily coincide with the physical angular acceleration vector ω. Let us assume that the orientation is described by the Euler angles α E = [ φ(t) θ(t) ψ(t) ] T ; the analytical angular velocity (Eulerian velocity) is then φ(t) α E (t) = θ(t) ψ(t) The Eulerian velocity α(t) is transformed into the geometrical (angular) velocity by ω(t) = b φ φ+bθ θ +bψ ψ, B. Bona (DAUIN) Kinematics Semester 1, 2014-15 10 / 15
Angular velocity transformations b φ = 0 0, b θ = cosφ(t) sin φ(t), b ψ = sinφ(t)sinθ(t) cosφ(t)sinθ(t) 1 0 cos θ(t) and we can define the transformation between α E (t) and ω(t) as φ ω(t) = M E (t) θ = M E (t) α E (t) ψ The transformation matrix 0 cosφ sinφsinθ M E (t) = 0 sinφ cosφsinθ 1 0 cos θ is not orthogonal and depends only on φ(t) and θ(t). B. Bona (DAUIN) Kinematics Semester 1, 2014-15 11 / 15
Angular velocity transformations When detm E (t) = sinθ = 0 the matrix is singular. The inverse is sinφcosθ cosφcosθ 1 sinθ sinθ M 1 E (t) = cosφ sinφ 0 sinφ cosφ 0 sinθ sinθ B. Bona (DAUIN) Kinematics Semester 1, 2014-15 12 / 15
Angular velocity transformations For the RPY angles α RPY = [ θ x θ y θ z ] T we have where ω(t) = M RPY (t) α RPY (t) cosθ z cosθ y sinθ z 0 M RPY (t) = sinθ z cosθ y cosθ z 0 sinθ y 0 1 For small angles we can approximate c i 1, s i 0 obtaining M RPY I; in this case ω(t) α RPY (t). B. Bona (DAUIN) Kinematics Semester 1, 2014-15 13 / 15
Angular velocity transformations When detm RPY (t) = cosθ y = 0 the matrix is singular. The inverse is M 1 RPY (t) = cosθ z sinθ z 0 cosθ y cosθ y sinθ z cosθ z 0 cosθ z sinθ y cosθ y sinθ z sinθ y 1 cosθ y B. Bona (DAUIN) Kinematics Semester 1, 2014-15 14 / 15
Analytical and geometrical Jacobians We have two types of angular Jacobians in, since we may write α = J α q (1) or ω = J ω q (2) J α is called the analytical jacobian. J ω is called the geometrical jacobian. The relation between the two is J ω = M(q)J α (3) where M = M E or M = M RPY B. Bona (DAUIN) Kinematics Semester 1, 2014-15 15 / 15