The advantage of geometric dynamics analysis over most classical 3D analysis is that the equations are coordinate- neutral: as long as all quantities are expressed in the same coordinate frame, the equations are correct

Oort, G. van

Stramigioli, S.

English

van Oort, Gijs

Stramigioli, Stefano

University of Twente Research Information

Geometric dynamics analysis of humanoids — locked inertiaG. van OortImpact institute/University of Twenteg.vanoort@ewi.utwente.nlS. StramigioliImpact institute/University of Twentes.stramigioli@utwente.nl1 IntroductionAt the Control Engineering chair at the University ofTwente, research is done on geometric dynamics analysisof humanoids. We strive to find mathematics that representin an insightful, coordinate-free way the complex dynamicsof humanoid robots. The basis of the geometric dynamicsanalysis is screw theory [1].The advantage of geometric dynamics analysis over mostclassical 3D analysis is that the equations are coordinate-neutral: as long as all quantities are expressed in the samecoordinate frame, the equations are correct. Contrary tofor example euler angles, the dynamics equations are com-pletely singularity-free. Moreover, equations for transla-tions and rotations are combined, resulting in simple yetpowerful equations.2 The locked inertia tensor and locked inertia ellipsoidA rigid body is characterized by its inertia tensor, whichcontains all inertia properties (mass m and moments of in-ertia Ix, Iy and Iz). The total inertia of a system of rigidbodies that are not moving relatively to each other (they are‘locked’) —the locked inertia (also known as the compositerigid body inertia [2]) — can be found by simply summingthe inertia tensors of the individual bodies.The locked inertia enables us to regard the whole systemas one single entity (as long as the joints are locked). Thisis very useful in humanoid robots, where the most importantdynamics are the system as a whole pivoting around the con-tact point of the foot with the ground. Assuming that the in-ternal motion of the system is negligible (see also section 4),the dynamics equations simplify from full multi-body equa-tions of motion to those of an inverted pendulum.The locked inertia ellipsoid is a visualization of the lockedinertia tensor — its shape represents the inertial propertiesof the locked system (see figure 1). By analyzing it, wecan visually judge how much the locked dynamics are influ-enced by changes of the internal configuration of the robot.This can be used both as an analysis tool (how much does aninternal configuration disturbance influence the total dynam-ics) and as a control tool (deliberately changing the internalconfiguration in order to change the total dynamics).To visualize the locked inertia ellipsoid, its properties (e.g.the three radii) need to be extracted from the inertia tensor.Figure 1: A screen shot of the simulation showing the lockedinertia ellipsoid together with the system.The equations we have found for this can be easily imple-mented in a simulation environment such as 20-sim, pro-vided that SVD or eigenvalue decomposition algorithms areavailable.3 SimulationA system of interconnected rigid bodies was simulated in20-sim and the equations for the locked inertia ellipsoidwere implemented. The SVD algorithm was provided bycreating a DLL-file in C++ that uses a freeware math li-brary [3]. The simulations show that the locked inertia el-lipsoid indeed matches the inertia properties of the lockedsystem (figure 1).4 Future workThe notion of locked inertia as explained above only makessense when the joints in the system are locked, i.e. there isno relative motion between the bodies of the system. Weare working on extending the theory to systems that do havedynamic internal movement.References[1] R. M. Murray, Z. Li, and S. S. Sastry, A mathematical Intro-duction to Robotic manipulation. Boca Raton: CRC Press, 1994.[2] M. W. Walker and D. Orin, “Efficient dynamic computersimulation of robotic mechanisms,” ASME Journal of DynamicSystems, Measurement, and Control, vol. 104, pp. 205–211, Sept.1982.[3] Newmat C++ matrix library, http://www.robertnz.net/nm intro.htm.

Geometric dynamics analysis of humanoids - locked inertia

A mathematical Introduction to Robotic manipulation. Boca Raton:

Efﬁcient dynamic computer simulation of robotic mechanisms,”

https://ris.utwente.nl/ws/files/6135989/GeometricDynamicsAnalysisOfHumanoids-LockedInertia.pdf

Geometric dynamics analysis of humanoids - locked inertia

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