When and when not to use the Inertia tensor in Kinetic energy computing? - Robotics Stack Exchange most recent 30 from robotics.stackexchange.com 2019-08-18T06:35:10Z https://robotics.stackexchange.com/feeds/question/15803 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://robotics.stackexchange.com/q/15803 3 When and when not to use the Inertia tensor in Kinetic energy computing? Ma ya https://robotics.stackexchange.com/users/20512 2018-06-02T17:20:15Z 2018-06-07T13:04:30Z <p>I'm trying to obtain the dynamic model of a 3D robot (academic problem), I have obtained the D-H table, the transformation matrix for each pair of links and the total transformation matrix. Now I'm struggling with computing the total Kinetic energy (for using it in the Lagrangian approach). </p> <p>In some texts, only the linear velocity is used in computing the kinetic energy, even if there are revolute links, that is:</p> <p>$K=\frac{1}{2}m~v^2$</p> <p>But in some others, both the linear and the angular velocities are considered :</p> <p>$K=\frac{1}{2}m~v^2 + \frac{1}{2} I~\omega^2$</p> <p>I'm a little bit confused with this, when and when not to use the angular contribution to the Kinetic energy?</p> https://robotics.stackexchange.com/questions/15803/-/15805#15805 1 Answer by N. Staub for When and when not to use the Inertia tensor in Kinetic energy computing? N. Staub https://robotics.stackexchange.com/users/18235 2018-06-04T09:50:57Z 2018-06-04T09:58:10Z <p>It all depends on your robot design, the joint category (linear/revolute) can have an impact but what you need to consider is the final motion of the parts for which you compute the Kinetic energy.</p> <p>The total Kinetic energy is the sum of all links and motors individual Kinetic energy. This is used because it makes the computation way easier as usually one consider a center of mass (CoM) for each motor and link.</p> <p>So you should consider both linear and angular components of Kinetic energies. Sometimes, based on your robot design, there are parts with no angular or linear velocity, which results in no contribution to the total Kinetic energy.</p> <p>Think for example about gantry crane which only have linear motion.</p> https://robotics.stackexchange.com/questions/15803/-/15814#15814 1 Answer by Long Smith for When and when not to use the Inertia tensor in Kinetic energy computing? Long Smith https://robotics.stackexchange.com/users/15392 2018-06-07T13:04:30Z 2018-06-07T13:04:30Z <p>In general you can represent a motion of a rigid body by two parts: rotation about some axis and linear motion of that axis.</p> <p>In robotic arm previous link movement affects the following ones by moving their rotation axes. It means that you have to use both parts of KE in your formula.</p> <p>You can compute linear and angular velocities that are used in formula by using a Jacobian:</p> <p>$\dot x = J^{-1}\dot q$. </p> <p>You can compute Jacobian geometrically for each COM frame and easily find angular and linear velocities of link's COM. Also note that $J$ is not always invertible so take a look at <em>Damped least-squares inverse</em> which is a standard way of approximating matrix inverse in case of it being non-square or singular.</p> <p>Quite a good example of computing KE you can find <a href="https://studywolf.wordpress.com/2013/09/07/robot-control-3-accounting-for-mass-and-gravity/" rel="nofollow noreferrer">here</a>.</p>