Kinematics
Basilio Bona
DAUIN – Politecnico di Torino
Semester 1, 2014-15
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Introduction
The kinematic quantities used are: position r, linear velocity ṙ, linear
acceleration r̈, orientation angle α, angular velocity ω, angular
acceleration ω̇,
One must use ω for the angular velocity instead of α̇: in kinematic
equations it is necessary to use the true angular velocity vector.
If α̇ is required, there are relations from ω to α̇ and vice-versa.
The motion equations are described by
r(t) = gr (q(t), π, t)
α(t) = gα (q(t), π, t)
where π is a parameter vector that characterize the system from a
geometrical, physical or structural point of view.
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Kinematics: position equations
If we use the pose vector p(t)T = rT (t) αT (t) we can write the direct
position kinematics
gr (·)
p(t) = gp (q(t), π, t) where gp (·) =
gα (·)
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.
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Kinematics: velocity equations
One can express both the linear velocities ṙ(t) and the angular velocities
α̇(t) of the rigid body as functions of the generalized velocities q̇(t),
obtaining the direct linear velocity kinematic function and the direct
angular velocity kinematic function
ṙ(t) ≡
d
∂g (q(t), π, t)
gr (q(t), π, t) = Jℓ (q(t), π, t)q̇(t) + r
dt
∂t
α̇(t) ≡
d
∂g (q(t), π, t)
gα (q(t), π, t) = Jα (q(t), π)q̇(t) + α
dt
∂t
The derivative α̇ is not the same as the angular velocity ω as we will
detail later.
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Kinematics: velocity equations
The matrix Jℓ
∂gri (q(t), π, t)
[Jℓ ]ij =
∂qj (t)
is the linear Jacobian matrix
The matrix Jα
∂gαi (q(t), π, t)
[Jα ]ij =
∂qj (t)
is the angular Jacobian matrix
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Kinematics: velocity equations
When the above functions do not explicitly depend on time, we obtain a
simplified form
ṙ(t) = Jℓ (q(t), π) q̇(t) or simply ṙ = Jℓ (q)q̇
and
α̇(t) = Jα (q(t), π) q̇(t) or simply α̇ = Jα (q)q̇
We observe that the two relations are linear in the velocities, since they are
the product of the Jacobian matrices and the generalized velocities q̇i (t).
We also observe that the Jacobian matrices are, in general, time varying,
since they depend on the generalized coordinates q(t).
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Kinematics: velocity equations
Embedding ṙ and α̇ in a single “vector”, we can write
ṙ(t)
v(t)
ṗ(t) =
or equivalently ṗ(t) =
α̇(t)
α̇(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.
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Kinematics
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Kinematics: velocity equations
We can now write in a compact form the kinematic function of the
generalized velocities:
ṗ(t) ≡
d
dq(t) ∂gp (q(t), π, t)
gp (q(t), π, t) = Jp (qp (t), π)
+
dt
dt
∂t
where the Jacobian Jp is a block matrix composed by Jℓ and Jα
"
#
Jℓ (q(t), π)
Jp (q(t), π) =
Jα (q(t), π)
If the kinematic position function gp does not explicitly depend on time,
we can write
ṗ(t) = Jp (qp (t), π) q̇(t) or simply
ṗ = Jp q̇
.
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Kinematics: velocity equations
This relation can be inverted only when the Jacobian is non-singular, i.e.,
det Jp (q(t)) 6= 0
In this case, if the kinematic equations do not depend on time, we have
what we call the inverse velocity kinematic function
q̇(t) = Jp (q(t), π)−1 ṗ(t) or simply q̇ = J−1
p ṗ
The Jacobian depends on the generalized coordinates qi (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 qsing that produce the singularity are called singular
configurations
det Jp (qsing ) = 0
The kinematic singularity problem is not treated in this course.
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Angular velocity transformations
If α(t) are the angular parameters (Euler, RPY, etc. angles), the
analytical derivative α̇(t) is called analytical (angular) velocity.
The analytical derivative α̇ does not necessarily coincide with the physical
angular velocity vector ω, and the second derivative α̈ does not necessarily
coincide with the physical angular acceleration vector ω̇.
Let us assume that the orientation is described by the Euler angles
T
αE = φ(t) θ(t) ψ(t) ; the analytical angular velocity (Eulerian
velocity) is then


φ̇(t)
α̇E (t) =  θ̇(t) 
ψ̇(t)
The Eulerian velocity α̇(t) is transformed into the geometrical (angular)
velocity by
ω(t) = bφ φ̇ + bθ θ̇ + bψ ψ̇,
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Angular velocity transformations
 
0
bφ = 0 ,
1


cos φ(t)
bθ =  sin φ(t)  ,
0


sin φ(t) sin θ(t)
bψ = − cos φ(t) sin θ(t)
cos θ(t)
and we can define the transformation between α̇E (t) and ω(t) as
 
φ̇
ω(t) = ME (t)  θ̇  = ME (t)α̇E (t)
ψ̇
The transformation matrix


0 cos φ sin φ sin θ
ME (t) = 0 sin φ − cos φ sin θ
1
0
cos θ
is not orthogonal and depends only on φ(t) and θ(t).
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Angular velocity transformations
When det ME (t) = − sin θ = 0 the matrix is singular.
The inverse is
 sin φ cos θ
−

sin θ

−1

ME (t) =  cos φ

sin φ
sin θ
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cos φ cos θ
sin θ
sin φ
−
cos φ
sin θ
1



0


0
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Angular velocity transformations
For the RPY angles αRPY = θx
θy
θz
T
we have
ω (t) = MRPY (t)α̇RPY (t)
where

cos θz cos θy

MRPY (t) =  sin θz cos θy
− sin θy
− sin θz
cos θz
0
0


0
1
For small angles we can approximate ci ≃ 1, si ≃ 0 obtaining MRPY ≃ I;
in this case ω(t) ≃ α̇RPY (t).
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Angular velocity transformations
When det MRPY (t) = cos θy = 0 the matrix is singular.
The inverse is

cos θz
cos θy
− sin θz



M−1
(t)
=

RPY

 cos θz sin θy
cos θy
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sin θz
cos θy
cos θz
−
sin θz sin θy
cos θy
0



0



1
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Analytical and geometrical Jacobians
We have two types of angular Jacobians in, since we may write
α̇ = Jα q̇
(1)
ω = Jω q̇
(2)
or
Jα is called the analytical jacobian.
Jω is called the geometrical jacobian.
The relation between the two is
Jω = M(q)Jα
(3)
where
M = ME
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or
M = MRPY
Kinematics
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