Kinematics Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Kinematics Semester 1, 2016-17 1 / 15
Introduction The kinematic quantities used to represent a body frame are: position (of the origin) x , linear velocity ˙ x , linear acceleration ¨ x , orientation (of the frame) α , angular velocity ω , angular acceleration ˙ ω . With α one indicates the generic angular parameters associated to the body frame orientation, e.g., Euler or RPY angles. Other parametrizations are possible, but they are more complex to deal with. One must use ω for the angular velocity instead of the time derivative ˙ α of the orientation angles: in kinematic equations it is necessary to use the true angular velocity vector. Notice that ω is a physical vector while ˙ α has no physical meaning (what represents the sum ˙ α 1 + ˙ α 2 ?). If ˙ α is required, there are relations from ω to ˙ α and vice-versa. B. Bona (DAUIN) Kinematics Semester 1, 2016-17 2 / 15
Kinematics: position equations The motion equations are described by two vectorial equalities x ( t ) = g x ( q ( t ) , λ , t ) α ( t ) = g α ( q ( t ) , λ , t ) where λ is a vector of (usually constant) parameters that characterize the system from a geometrical, physical or structural point of view. If one uses the pose vector p ( t ) T = � x T ( t ) α T ( t ) � the direct position kinematic function is given by a nonlinear equation � � g x ( · ) DPKF: p ( t ) = g ( q ( t ) , λ , t ) where g ( · ) = g α ( · ) and the inverse position kinematic function , is given by the inverse nonlinear relation IPKF: q ( t ) = g − 1 ( p ( t ) , λ , t ) This equation is in general much more difficult to solve, since it requires the inversion of nonlinear trigonometric functions. B. Bona (DAUIN) Kinematics Semester 1, 2016-17 3 / 15
Kinematics: velocity equations One can express both the linear velocities ˙ x ( t ) and the angular velocities α ( t ) of the rigid body as functions of the generalized velocitie ˙ ˙ q ( t ), obtaining the direct linear velocity kinematic function q ( t )+ ∂ g x ( q ( t ) , λ , t ) x ( t ) ≡ d ˙ d t g x ( q ( t ) , λ , t ) = J L ( q ( t ) , λ , t )˙ ∂ t and the direct angular velocity kinematic function α ( t ) ≡ d q ( t )+ ∂ g α ( q ( t ) , λ , t ) ˙ d t g α ( q ( t ) , λ , t ) = J A ( q ( t ) , λ , t )˙ ∂ t The derivative ˙ α is in general not equal to the angular velocity ω , as we will see below. B. Bona (DAUIN) Kinematics Semester 1, 2016-17 4 / 15
Kinematics: velocity equations The matrix J L � ∂ g xi ( q ( t ) , λ , t ) � [ J L ] ij = ∂ q j ( t ) is called the linear Jacobian matrix The matrix J A � ∂ g α i ( q ( t ) , λ , t ) � [ J A ] ij = ∂ q j ( t ) is called the angular Jacobian matrix B. Bona (DAUIN) Kinematics Semester 1, 2016-17 5 / 15
Kinematics: velocity equations When the above functions do not explicitly depend on time t , we obtain a simplified form x ( t ) = J L ( q ( t ) , λ ) ˙ ˙ q ( t ) or simply x = J L ( q )˙ ˙ q and α ( t ) = J A ( q ( t ) , λ ) ˙ ˙ q ( t ) or simply α = J A ( q )˙ ˙ q We observe that the two relations are linear in the velocities, since they are the product between 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, 2016-17 6 / 15
Kinematics: velocity equations Embedding ˙ x and ˙ α in a single “vector”, we can write � � � � ˙ x ( t ) v ( t ) p α ( t ) = ˙ or equivalently p α ( t ) = ˙ ˙ ˙ α ( t ) α ( t ) The quantity ˙ p 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 , we write � � � � x ( t ) ˙ v ( t ) p ω ( t ) = ˙ or equivalently p ω ( t ) = ˙ ω ( t ) ω ( t ) ω ( t ) � T is also called a twist . � v ( t ) this vector B. Bona (DAUIN) Kinematics Semester 1, 2016-17 7 / 15
Kinematics: velocity equations We can now write in a compact form the kinematic function of the generalized velocities: d t g ( q ( t ) , λ , t ) = J ( q ( t ) , λ ) d q ( t ) + ∂ g ( q ( t ) , λ , t ) p ( t ) ≡ d ˙ ∂ t d t where the Jacobian J is a block matrix composed by J L and J A � � J L ( q ( t ) , λ ) J ( q ( t ) , λ ) = J A ( q ( t ) , λ ) If the kinematic position function g does not explicitly depend on time, we can write p ( t ) = J ( q ( t ) , λ ) ˙ ˙ q ( t ) or simply p = J ˙ ˙ q B. Bona (DAUIN) Kinematics Semester 1, 2016-17 8 / 15
Kinematics: velocity equations As we will detail below, when the generalized velocity is expressed using the angles derivative ˙ p α , then p α = J α ˙ ˙ q and when the generalized velocity is expressed using the angular velocity p ω , then ˙ p ω = J ω ˙ ˙ q B. Bona (DAUIN) Kinematics Semester 1, 2016-17 9 / 15
This relation can be inverted only when the Jacobian is non-singular, i.e., det J ( 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 ( q ( t ) , λ ) − 1 ˙ q = J − 1 ˙ ˙ p ( t ) or simply ˙ p The Jacobian depends on the generalized coordinates q i ( t ), and it can become singular for particular values of these coordinates; 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 det J ( q sing ) = 0 The kinematic singularity problem is not treated in this course, but is very important in robotics. B. Bona (DAUIN) Kinematics Semester 1, 2016-17 10 / 15
Angular velocity transformations – Euler angles If α ( t ) are the angular parameters (Euler angles, RPY angles, etc.), 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 ψ ( t ) � T ; the analytical angular velocity (Eulerian � φ ( t ) α E = θ ( t ) 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 the following relation ω ( t ) = b φ ˙ φ + b θ ˙ θ + b ψ ˙ ψ , B. Bona (DAUIN) Kinematics Semester 1, 2016-17 10 / 15
Angular velocity transformations – Euler angles Where 0 cos φ ( t ) sin φ ( t )sin θ ( t ) b φ = 0 b θ = sin φ ( t ) b ψ = − cos φ ( t )sin θ ( t ) , , 1 0 cos θ ( t ) and we can define the transformation between ˙ α E ( t ) and ω ( t ) introducing a square matrix ˙ φ ˙ = M E ( t ) ˙ ω ( t ) = M E ( t ) α E ( t ) θ ψ ˙ The transformation matrix 0 cos φ sin φ sin θ M E ( t ) = 0 sin φ − cos φ sin θ 1 0 cos θ is NOT a rotation matrix and depends only on φ ( t ) and θ ( t ). B. Bona (DAUIN) Kinematics Semester 1, 2016-17 11 / 15
Angular velocity transformations – Euler angles When det M E ( t ) = − sin θ = 0 the matrix is singular. The inverse is − sin φ cos θ cos φ cos θ 1 sin θ sin θ M − 1 E ( t ) = cos φ sin φ 0 sin φ − cos φ 0 sin θ sin θ B. Bona (DAUIN) Kinematics Semester 1, 2016-17 12 / 15
Angular velocity transformations – RPY angles � T we have � θ x For the RPY angles α RPY = θ y θ z ω ( t ) = M RPY ( t ) ˙ α RPY ( t ) where cos θ z cos θ y − sin θ z 0 M RPY ( t ) = sin θ z cos θ y cos θ z 0 − sin θ 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, 2016-17 13 / 15
Angular velocity transformations – RPY angles When det M RPY ( t ) = cos θ y = 0 the matrix is singular. The inverse is cos θ z sin θ z 0 cos θ y cos θ y M − 1 − sin θ z cos θ z 0 RPY ( t ) = cos θ z sin θ y − sin θ z sin θ y 1 cos θ y cos θ y B. Bona (DAUIN) Kinematics Semester 1, 2016-17 14 / 15
Analytical and Geometrical Jacobians According to the type of angle related velocity, we have two types of angular Jacobians. If we use ˙ α , then ˙ α = J α ˙ q (1) If we use ω , then ω = J ω ˙ q (2) J α is called the Analytical Jacobian . J ω is called the Geometrical Jacobian . The relation between the two Jacobians is given by a linear transformation J ω = M ( q ) J α (3) where M = M E or M = M RPY B. Bona (DAUIN) Kinematics Semester 1, 2016-17 15 / 15
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