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50 CHAPTER 6. CONSTRAINED RIGID BODY DYNAMICS
We will more or less follow this road map and generalize the concepts for rigid bodies. We will
however not define or derive the concepts once again from the ground up but utilize the fact that
rigid bodies can be treated as a special kind of particles moving in R 6 , implicitly transferring the
definitions, concepts and also results to rigid bodies.
6.1.1 Notation
We will use the following notation:
• generalized position r of a body
A block vector r consisting of two blocks storing the position x ∈ R 3 of the rigid body’s
center of mass and orientation θ ∈ R 3 by Euler angles, r = (x, θ) ∈ R 6 . Generalized
position serves the same purpose like the vector r of particles and corresponds to our view
of rigid bodies as particles in R 6 .
The representation of orientation by Euler angles θ stresses that rigid bodies have 3 rotational
degrees of freedom and allows to define the generalized velocity as a time derivative
of the generalized position (since angular velocity ω is formally defined as ˙ θ, see (5.3)).
Note that this is a formalism only. We still maintain the orientation of rigid bodies by
rotation matrices R or quaternions q and Euler angles θ can always be extracted from
R or q. Euler angles θ are used to define the generalized position r which is in turn
used to define position-level constraints for rigid bodies. Since it is often hard to express
rotational constraints in terms of θ on the position level, unlike of ω on the velocity level,
and constraints will finally be enforced on the acceleration level anyway, one might forget
about θ and formulate rotational constraints directly on the velocity or acceleration level,
in terms of ω or α. This is the reason why we do not study θ further.
• generalized velocity v of a body
A block vector v consisting of two blocks storing the rigid body’s linear velocity ˙ x ∈ R 3 and
angular velocity ω ∈ R 3 , v = ( ˙ x, ω) ∈ R 6 . Generalized velocity serves the same purpose
like vector ˙ r of particles and in fact v = ˙ r.
To prevent clash with the generalized velocity v we will denote the linear velocity by ˙ x.
• generalized acceleration a of a body
A block vector a consisting of two blocks storing the rigid body’s linear acceleration ¨ x ∈ R 3
and angular acceleration α ∈ R 3 , a = ( ¨ x, α) ∈ R 6 . Generalized acceleration serves the
same purpose like vector ¨ r of particles and in fact a = ¨ r.
To prevent clash with the generalized acceleration a we will denote the linear acceleration
by ¨ x.
• generalized force F exerted on a body
A block vector F consisting of two blocks storing the force f ∈ R 3 (imagined as if acting
at the body’s center of mass) and torque τ ∈ R 3 due to f, F = ( f, τ) ∈ R 6 . Generalized
force serves the same purpose like force F exerted on a particle and in fact F = M ·a for a
generalized acceleration a due to F , where M is the rigid body’s mass matrix (see (5.23)).
If F refers to the generalized total force exerted on the body, coriolis force is assumed to
be included into F .