I want to find the equations of motion of an RRRR robot.I have studied about it a bit but I am having some confusion.
Here, in one of the lectures I found online, it describes an Inertia matrix of a link as $\bf{I}_i$ which is computed by $\tilde{\bf{I}}_i$ also described below?
In conclusion, the kinetic energy of a manipulator can be determined when, for each link, the following quantities are known:
- the link mass $m_i$;
- the inertia matrix $\bf{I}_i$, computed with respect to a frame $\mathcal{F}_i$ fixed to the center of mass in which it has a constant expression $\tilde{\bf{I}}_i$;
- the linear velocity $\bf{v}_{Ci}$ of the center of mass, and the rotational velocity $\omega_i$ of the link (both expressed in $\mathcal{F}_0$);
- the rotation matrix $\bf{R}_i$ between the frame fixed to the link and $\mathcal{F}_0$.
The kinetic energy $K_i$ of the i-th link has the form:
$$ K_i = \frac{1}{2}m_i\bf{v}_{Ci}^T\bf{v}_{Ci} + \frac{1}{2}\omega_i^T\bf{R}_i\tilde{\bf{I}}_i\bf{R}_i^T\omega_i \\ $$ It is now necessary to compute the linear and rotational velocities ($\bf{v}_{Ci}$ and $\omega_i$) as functions of the Lagrangian coordinates (i.e. the joint variables $\bf{q}$).
So $\tilde{\bf{I}}_i$ is computed wrt to fixed frame attached to the centre of mass.
However in another example below from another source there is no rotation matrix multiplication with ${I}_{C_1}$ and $I_{C_2}$ as shown above. Am I missing something?
$\underline{\mbox{Matrix M}}$
$$ M = m_1 J_{v_1}^TJ_{v_1} + J_{\omega_1}^TI_{C_1}J_{\omega_1} + m_2 J_{v_2}^TJ_{v_2} + J_{\omega_2}^TI_{C_2}J_{\omega_2} \\ $$
What is the significance of multiplying Rotation matrix with $I_{C_1}$ or $\tilde{\bf{I}}_i$?
I am using former approach and getting fairly large mass matrix. Is it normal to have such long terms inside a Mass matrix? I still need to know though which method is correct.
(A series of images showing Mathematica output of a 4x4 matrix with some very, very long terms - A, B, C)
The equation I used for the mass matrix is:
$$ \begin{array}{lcl} K & = & \displaystyle{\frac{1}{2}} \displaystyle{\sum_{i=1}^{n}} m_i\bf{v}_{Ci}^T \bf{v}_{Ci} + \displaystyle{\frac{1}{2}} \displaystyle{\sum_{i=1}^{n}} \omega_i^T\bf{R}_i\tilde{\bf{I}}_i\bf{R}_i^T\omega_i \\ & = & \boxed{ \frac{1}{2} \dot{\bf{q}}^T \sum_{i=1}^{n}\left[ m_i {\bf{J(\bf{q})}_{v}^{i}}^T {\bf{J(\bf{q})}_{v}^i} + {\bf{J(\bf{q})}_{\omega}^i}^T\bf{R}_i\tilde{\bf{I}}_i\bf{R}_i^T\bf{J(\bf{q})}_{\omega}^i \right] \dot{\bf{q}} } \\ & = & \displaystyle{\frac{1}{2}} \dot{\bf{q}}^T\bf{M(q)}\dot{\bf{q}} \\ & = & \displaystyle{\frac{1}{2}} \displaystyle{\sum_{i=1}^{n}} \displaystyle{\sum_{j=1}^{n}} M_{ij}(\bf{q})\dot{q}_i \dot{q}_j \\ \end{array} $$
