Explanation for Rigid Body Dynamics Formula: Understanding the Role and Composition of Matrices $M$ and $C(ω)$

Question

I'm working on understanding the dynamics of a rigid body, and I've come across a formula that I'm trying to make sense of. The formula is:

Where M is:

and C(w) is

The Parameters used in these formulas above are defined as

What I Think the Formula Represents:

The formula seems to represent the equations of motion for a rigid body under external forces and torques. The left side includes the effects of inertia (mass and moment of inertia) and possibly some Coriolis and centrifugal forces, while the right side represents external forces and torques applied to the body.

My Questions:

1. Is this formula correct in its current form? Should there be an additional term like to properly account for the Coriolis and centrifugal forces? This would make the equation of the dynamics of a rigid body be

2. What is the correct interpretation of each term in this formula? Specifically, how do the matrices M and C(ω) relate to the physical properties of the rigid body (like mass, center of mass, and inertia tensor)?

3. Is there a more standard form for expressing the rigid body dynamics equations that includes these terms, and how does this relate to my current understanding?

I would appreciate any insights or explanations, especially if they can clarify whether this formula is accurate or if there is a common version that I'm missing.

Thank you!

Answer 1

$\def \b {\mathbf}$

The equations of motion at point A

the position vector, velocity and acceleration at point A , given in inertial system \begin{align*} &\b P_A=\b P_C+\b R\,\b c\quad,\text{the velocity}\\ &\b{\dot{P}}_A=\b{\dot{P}}_C+\b{\dot{R}}\,\b c \quad,\text{the acceleration}\\ &\b{\ddot{P}}_A=\b{\ddot{P}}_C+\b{\ddot{R}}\,\b c \end{align*} with \begin{align*} & \b{\dot{R}}=\b R\,\b\omega^\times\quad, \b{\ddot{R}}=\b{\dot{R}}\,\b\omega^\times+\b R\,\b{\dot{\omega}}^\times\\ & \b{\ddot{R}}=\b R\,\b\omega^\times\,\b\omega^\times+\b R\,\b{\dot{\omega}}^\times\\ \end{align*} thus the acceleration \begin{align*} &\b{\ddot{P}}_A=\b{\ddot{P}}_C+\b R\left(\b\omega^\times\,\b\omega^\times+ \,\b{\dot{\omega}}^\times\right)\,\b c\quad,\text{or}\\ &\b R^T\,\b{\ddot{P}}_A=\underbrace{\b R^T\,\b{\ddot{P}}_C}_{\b{\dot{v}}}+ \left(\b\omega^\times\,\b\omega^\times+ \,\b{\dot{\omega}}^\times\right)\,\b c \end{align*}

the EOM's translation \begin{align*} &m\,\b{\ddot{P}}_A=\b F\quad,m\,\b R^T\,\b{\ddot{P}}_A=\b R^T\,\b F\quad\Rightarrow\\ &m\,\b{\dot{v}}+ \left(\b\omega^\times\,\b\omega^\times+ \,\b{\dot{\omega}}^\times\right)\,\b c= \b R^T\,\b F \end{align*} The EOM's rotation \begin{align*} &\b J_A\,\b{\dot{\omega}}+\b\omega^\times\left(\b J_A\,\b\omega\right)=\b\tau\quad,\text{where}\\ &\b J_A=\b J_C+m\,\b c^\times\,\b c^\times \end{align*} Translation and rotation \begin{align*} &\begin{bmatrix} m\,I & -\b c^\times \\ \b 0 & \b J_A \\ \end{bmatrix} \begin{bmatrix} \b{\dot{v}} \\ \b{\dot{\omega}} \\ \end{bmatrix}+\begin{bmatrix} \b\omega^\times\,\b\omega^\times\,\b c \\ \b\omega^\times\left(\b J_A\,\b\omega\right) \\ \end{bmatrix}= \begin{bmatrix} \b R^T\,\b F \\ \b \tau \\ \end{bmatrix}\quad,\text{with}\\ & \b\omega^\times= \left[ \begin {array}{ccc} 0&-\omega_{{3}}&\omega_{{2}} \\ \omega_{{3}}&0&-\omega_{{1}}\\ -\omega_{{2}}&\omega_{{1}}&0\end {array} \right] \quad, \b c^\times= \left[ \begin {array}{ccc} 0&-c_{{3}}&c_{{2}}\\ c_{ {3}}&0&-c_{{1}}\\ -c_{{2}}&c_{{1}}&0\end {array} \right] \end{align*}

• $~\b R~$ Rotation Matrix from body system to inertial system
• $~\b F~$ external force , given in inertial system
• $~\b\tau~$ external torque , given in body system
• $~\b J_C~$ Inertia tensor an the center of mass
• $~\b J_A~$ Inertia tensor, at point A , the coordinate system is parallel to the center of mass coordinate system

Answer 2

The context in which I see this notation used most often is for robotic manipulators. Although in that case, the system is made up of a number of rigid bodies with 1 DOF joints. $\bf M$ is called the "mass matrix" and depends ONLY on position variables. It is what multiplies the acceleration terms in the force/torque balance. $\bf C$ (also called $\bf V$) is a vector (not matrix) of coriolis and centrifugal forces. In robotics, they also include a (vector) $\bf G$ term to account for gravity effects and a $\bf \tau$ vector of applied forces and torques. A practical engineering approach with clear explanation (i.e. no fancy math jargon) can be found in "Introduction to Robotics" by John Craig.