While rotational motion of rigid objects in 3D space seems quite complicated, the computation of angular velocity of the rigid object is in fact, remarkable simple. In this post, we show the superposition principle of angular velocities — the resulting angular velocity of the rigid object is simply the linear summation of each angular velocity. The superposition principle is related to the fact that angular velocity does not require a reference point to be defined, i.e., it is a free vector. Hence, one can simply superimpose angular velocity vectors by freely shifting around the vectors. We show a constructive proof for that.
Consider a rigid object with \(n\) sequence of rotations. The orientation of the rigid object can be expressed as a chain of rotation matrices: \[ {}^{0}\mathbf{R}_{n}(t) = {}^{0}\mathbf{R}_{1}(t) {}^{1}\mathbf{R}_{2}(t) \cdots {}^{n-1}\mathbf{R}_{n}(t) \] It is well known that the total angular velocity of the object, \({}^{0}\boldsymbol{\omega}\) is: \[ [{}^{0}\boldsymbol{\omega}] = {}^{0}\mathbf{\dot{R}}_{n}(t) \big\{ {}^{0}\mathbf{R}_{n}(t) \big\}^{\mathbf{T}} \] This shows that \[ [{}^{0}\boldsymbol{\omega}] = {}^{0}\mathbf{\dot{R}}_{1}(t) \big\{ {}^{0}\mathbf{R}_{1}(t) \big\}^{\mathbf{T}} + {}^{0}\mathbf{R}_{1}(t) {}^{1}\mathbf{\dot{R}}_{2}(t) \big\{ {}^{1}\mathbf{R}_{2}(t) \big\}^{\mathbf{T}} {}^{0}\mathbf{R}_{1}(t) \]
Consider the following system