Structure and Synthesis of Robot Motion Dynamics Subramanian Ramamoorthy School of Informatics 2 February, 2009

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) 02/02/2009Structure and Synthesis of Robot Motion2

Newton’s Laws Three laws: 1.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 2.The mass m, acceleration a and applied force f are related by f = ma 3.The interaction forces of two bodies are of equal magnitude and in opposite directions 02/02/2009Structure and Synthesis of Robot Motion3

Motion of a Particle 02/02/2009Structure and Synthesis of Robot Motion4

Motion of a Lunar Lander 02/02/2009Structure and Synthesis of Robot Motion5

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 02/02/2009Structure and Synthesis of Robot Motion6

Describing a Free-Floating Rigid Body 02/02/2009Structure and Synthesis of Robot Motion7

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 02/02/2009Structure and Synthesis of Robot Motion8

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: 02/02/2009Structure and Synthesis of Robot Motion9 Optimization is over possible small perturbations in functional form

Variational Description of Dynamics 02/02/2009Structure and Synthesis of Robot Motion10

Deriving Equations of Motion from Lagrangians 02/02/2009Structure and Synthesis of Robot Motion11 Potential energy term (function of position) Kinetic energy term (function of velocity)

A Planar Robot 02/02/2009Structure and Synthesis of Robot Motion12 q1q1 q2q2 q3q3 agag External forces balance accel. terms

RP Manipulator 02/02/2009Structure and Synthesis of Robot Motion13

RP Manipulator Equations 02/02/2009Structure and Synthesis of Robot Motion14

Canonical Structure of Dynamics Equations 02/02/2009Structure and Synthesis of Robot Motion15

2-link Manipulator 02/02/2009Structure and Synthesis of Robot Motion16 CoM of A 1

2-Link Manipulator: M, C and g 02/02/2009Structure and Synthesis of Robot Motion17

2-Link Manipulator: Equations of Motion 02/02/2009Structure and Synthesis of Robot Motion18

Additional Structure ‘Velocity product’ terms can be derived from Inertia matrix 02/02/2009Structure and Synthesis of Robot Motion19

Canonical Equations of Motion, again 02/02/2009Structure and Synthesis of Robot Motion20

Many other Things to Keep in Mind e.g., velocity constraints 02/02/2009Structure and Synthesis of Robot Motion21

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 02/02/2009Structure and Synthesis of Robot Motion22