Structure and Synthesis of Robot Motion Dynamics

1. Structure and Synthesis of Robot MotionDynamics Subramanian Ramamoorthy School of Informatics 2 February, 2009

2. Describing Dynamics – Newton-Euler • Points in the world, W, are expressed using an inertial coordinate frame • One tries to make sensible choices (e.g., if playing squash, don’t put the frame on the ball – instead room may be better) • Some properties: • Laws of motion appear the same in any inertial frame • Any frame moving without rotation and at constant speed w.r.t. another inertial frame is also inertial • Then, Newton-Euler Mechanics happens in such a frame where all participants are explicitly modelled (closed system) Structure and Synthesis of Robot Motion

3. Newton’s Laws Three laws: • An object at rest tends to stay at rest, and an object in motion tends to stay in motion at fixed speed, unless a nonzero resultant force acts upon it • The mass m, acceleration a and applied force f are related by f = ma • The interaction forces of two bodies are of equal magnitude and in opposite directions Structure and Synthesis of Robot Motion

4. Motion of a Particle Structure and Synthesis of Robot Motion

5. Motion of a Lunar Lander Structure and Synthesis of Robot Motion

6. How to Move Beyond Point-Masses? • Clearly, we want to be able to model complex robots as they appear in practice • Within the Newton-Euler formalism, one can try to derive further conservation laws, e.g., momentum • This yields additional equations that act as differential constraints on the state space Structure and Synthesis of Robot Motion

7. Describing a Free-Floating Rigid Body Structure and Synthesis of Robot Motion

8. Some Limitations of Newton-Euler Method • Everything has to be described in terms of an orthogonal inertial frame • For complex robots, we also need to keep track of translational/rotational inertia, linear/angular momenta, etc. explicitly in order to do computations Structure and Synthesis of Robot Motion

9. Lagrangian Mechanics • Based on calculus of variations – optimisation over the space of paths • Motion is described in terms of the minimisation of an action functional: Optimization is over possible small perturbations in functional form Structure and Synthesis of Robot Motion

10. Variational Description of Dynamics Structure and Synthesis of Robot Motion

11. Deriving Equations of Motion from Lagrangians Potential energy term (function of position) Kinetic energy term (function of velocity) Structure and Synthesis of Robot Motion

12. A Planar Robot q2 q3 ag q1 External forces balance accel. terms Structure and Synthesis of Robot Motion

13. RP Manipulator Structure and Synthesis of Robot Motion

14. RP Manipulator Equations Structure and Synthesis of Robot Motion

15. Canonical Structure of Dynamics Equations Structure and Synthesis of Robot Motion

16. 2-link Manipulator CoM of A1 Structure and Synthesis of Robot Motion

17. 2-Link Manipulator: M, C and g Structure and Synthesis of Robot Motion

18. 2-Link Manipulator: Equations of Motion Structure and Synthesis of Robot Motion

19. Additional Structure ‘Velocity product’ terms can be derived from Inertia matrix Structure and Synthesis of Robot Motion

20. Canonical Equations of Motion, again Structure and Synthesis of Robot Motion

21. Many other Things to Keep in Mind e.g., velocity constraints Structure and Synthesis of Robot Motion

22. Summary • Three major ways to describe (rigid) robot dynamics • Newton-Euler: Direct description of forces and effects • Lagrangian: Variational approach • Hamiltonian (more advanced, did not discuss today) • Latter two are more general – can describe motion, e.g., submanifolds of c-space (where you can still do optimisation!) • While all of them yield similar equations in the end - for a given system, insights gained can vary significantly • One of the harder things to model is constraints • Typically, we can just begin with unconstrained equations and add constraints later when computing ‘actual’ motions • More sophisticated methods exist for specialized scenarios Structure and Synthesis of Robot Motion