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.,
)
Modern Robotics, Lynch and Park, Cambridge University Press 6
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.