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Rigid body dynamics describes the motion of objects whose shape remains fixed while they translate and rotate through space.
This is at least approximately true for aircraft and spacecraft. Unlike particle dynamics, where an object is represented by a single point mass,
rigid body dynamics accounts for the distribution of mass throughout the body and the resulting rotational behavior, adding degrees of freedom to the dynamical system.
While a point mass generally possesses three degrees of freedom, encoded by its 3D position vector, a rigid body can have up to six. Three degrees of freedom describe the position of a reference point, usually the center of mass,
while three additional degrees of freedom describe the body’s orientation, see Mozzi-Chasles’s theorem.
As with point masses, the translational motion of the center of mass of the rotating object is governed by Newton’s second law,
where
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To describe the orientation of a rigid body, it is useful to introduce two coordinate systems. The first is an inertial frame, denoted by axes
The body-fixed frame moves together with the object. Consequently, the coordinates of any point within the body remain constant when expressed in the body frame. For example, if a spacecraft carries a sensor located one meter along its body
in the body frame, regardless of how the spacecraft rotates in space.
This property greatly simplifies calculations because the mass distribution of the body remains constant in body-fixed coordinates.
The orientation of the body relative to the inertial frame can be represented by a rotation matrix,
The rotation matrix contains nine elements, but only three independent parameters are required to specify orientation. One common choice for these parameters is the set of Euler angles.
Euler angles describe orientation through a sequence of three rotations. Several conventions exist. One of the most common in aerospace engineering is the
The first rotation is a yaw angle
The resulting rotation matrix is
Explicitly,
and
Euler angles provide an intuitive description of orientation. Pilots, for example, naturally think in terms of roll, pitch, and yaw.
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Euler angles suffer from a mathematical singularity known as gimbal lock. For the
The rotational motion of a rigid body is characterized by its angular velocity vector
For a body-fixed frame,
where
Angular velocity determines how rapidly the body rotates. Any point in the body experiences a velocity
where
The relationship between Euler angle rates and angular velocity is, unfortunately, not trivial. In particular,
Instead, the relationship depends on the chosen Euler angle sequence. For the
This equation connects the kinematics of orientation with the rotational dynamics discussed next.
The rotational equivalent of mass is the moment of inertia. While mass measures resistance to linear acceleration, moment of inertia measures resistance to angular acceleration.
For a collection of mass elements
where
A body whose mass is concentrated far from the rotation axis has a larger moment of inertia and is more difficult to spin.
In three dimensions, rotational inertia is represented by the inertia tensor,
The diagonal terms are called moments of inertia,
The off-diagonal terms are called products of inertia,
and similarly for
For bodies possessing symmetry, it is particularly simple to find a coordinate system so that the products of inertia vanish. The resulting axes are called principal axes, and the inertia tensor becomes diagonal,
This simplification is extremely useful and is commonly employed.
The rotational analogue of linear momentum is angular momentum
Angular momentum is conserved when no external moments act on the body. This principle explains why to first order spacecraft can maintain attitude stability through rotation.
Analogous to
where
Because the body-fixed frame rotates with the body, the derivative of angular momentum must account for frame rotation. The result is Euler’s equation,
This equation is the rotational counterpart of Newton’s second law. Note that those hold only when
When the body axes coincide with the principal axes, Euler’s equations become
These three coupled differential equations describe the rotational motion of any rigid body.
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An important feature is the appearance of products of angular velocity. Unlike translational dynamics, rotational dynamics are inherently nonlinear. Even when no external moments are present,
the angular velocity components can still change because of the coupling terms. This phenomenon is responsible for effects such as torque-free precession and the famous tennis-racket instability.
Rigid body dynamics extends Newtonian mechanics to rotating objects. A body-fixed frame provides a convenient coordinate system that rotates with the object and keeps its mass distribution constant. Euler angles describe orientation through a sequence of rotations, while angular velocity characterizes the rate of rotation. The resistance of a body to rotational acceleration is quantified by the inertia tensor, whose principal moments depend on the mass distribution. Combining these concepts leads to the angular momentum equation and ultimately Euler’s equations,
Transforming between inertial and body-fixed frames is non-trivial and is most often achieved through the use of quaternions.