Where we are:

Chap 2 Configuration Space

Chap 3 Rigid-Body Motions

Chap 4 Forward Kinematics

Chap 5 Velocity Kinematics and Statics

Chap 6 Inverse Kinematics

Chap 8 Dynamics of Open Chains

8.1 Lagrangian Formulation

Chap 9 Trajectory Generation

Chap 11 Robot Control

Chap 13 Wheeled Mobile Robots

Modern Robotics, Lynch and Park, Cambridge University Press 1

Important concepts, symbols, and equations

• forward dynamics (for simulation): !, !̇, τ → !̈

• inverse dynamics (for control): !, !̇, !̈ → τ

• two equivalent approaches to computing the dynamics:

• Lagrangian (variational, based on energy)• Newton-Euler (“f = ma” for the rigid bodies)

Modern Robotics, Lynch and Park, Cambridge University Press 2

Modern Robotics, Lynch and Park, Cambridge University Press 3

Important concepts, symbols, and equations (cont.)

Lagrangian approach:

Lagrangian

eqs of motion

kinetic minus potential energy

∈ ℝn

in components

Modern Robotics, Lynch and Park, Cambridge University Press 4

Important concepts, symbols, and equations (cont.)

Standard forms of dynamic equations:

+

M(!) n×n symmetric positive definite mass matrixg(!) gravity (potential) terms c(!,!̇) velocity-product termsG(!) n×n×n tensor of Christoffel symbols (due to nonzero #M/#!)C(!,!̇) Coriolis matrix

Modern Robotics, Lynch and Park, Cambridge University Press 5

Important concepts, symbols, and equations (cont.)

Velocity-product terms (c(!,!̇) , C(!,!̇)!̇ , !̇TG(!)!̇) consist of

• centripetal terms, e.g.,

• Coriolis terms, e.g.,

)

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RP in gravity

= + ...

= L1 + ...product rule: !" # = !#" + !"#

chain rule: &' ( )&) = *'

*(&(&)

Modern Robotics, Lynch and Park, Cambridge University Press 7

L1 =

Contribution to !2:

RP in gravity

Contribution to !1:

Modern Robotics, Lynch and Park, Cambridge University Press 8

RP in gravity

Explain the velocity-product terms.