Lagrangian dynamic formulation
Lagrangian formulation is an "energy-based" approach to dynamics.
Ch.4 Manipulator Dynamics
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Ch.4 Manipulator Dynamics
There are two problems related to the dynamics of a manipulator that we wish to solve.
0 gx0
D
1 2
c(
x&1
x&0 )2
W Fx1 Fx0
F
x0
F
M1
x1
k
c
M0
图4.1 一般物体的动能与位能
4.1 Dynamics of a Rigid Body
8
4.1.1 Kinetic and Potential Energy of a Rigid Body
x0 0, x1 is a generalized coordinate
机器人学基础
第四章 机器人动力学
Fundamentals of Robotics
Ch.4 Manipulator Dynamics
中南大学 蔡自兴，谢 斌 zxcai, xiebin@
2010
Fundamentals of Robotics
1
Contents
Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics
4.1 Dynamics of a Rigid Body
7
4.1 Dynamics of a Rigid Body
4.1.1 Kinetic and Potential Energy of a Rigid Body
K
1 2
M1 x&12
1 2
M0 x&02
P
1 2
k( x1
x0 )2
M1 gx1
M
Ch.4 Manipulator Dynamics
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4.1 Dynamics of a Rigid Body 刚体动力学
Langrangiຫໍສະໝຸດ n Function L is defined as:
LKP
(4.1)
Kinetic Energy
Potential Energy
Dynamic Equation of the system (Langrangian
Ch.4 Manipulator Dynamics
2
Ch.4 Manipulator Dynamics
Introduction
Manipulator Dynamics considers the forces required to cause desired motion. Considering the equations of motion arises from torques applied by the actuators, or from external forces applied to the manipulator.
Equation):
Fi
d dt
L qi
L qi
,i
1,2,
n
(4.2)
where qi is the generalized coordinates, qi represent corresponding velocity, Fi stand for corresponding torque or force on the ith coordinate.
Forward Dynamics: given a torque vector, Τ, calculate the resulting motion of the manipulator, , &, and && . This is useful for simulating the manipulator.
F
x0
F
d dt
K x1
K x1
D x1
P x1
W x1
M1
x1
k
c
① ②③ ④⑤
M0
① Kinetic Energy due to (angular) velocity ② Kinetic Energy due to position (or angle) ③ Dissipation Energy due to (angular) velocity ④ Potential Energy due to position ⑤ External Force or Torque
Inverse Dynamics: given a trajectory point, , &, and &&,
find the required vector of joint torques, Τ. This formulation of dynamics is useful for the problem of controlling the manipulator.
Ch.4 Manipulator Dynamics
5
Contents
Introduction to Dynamics Rigid Body Dynamics Lagrangian Formulation Newton-Euler Formulation Articulated Multi-Body Dynamics
Ch.4 Manipulator Dynamics
3
Ch.4 Manipulator Dynamics
Two methods for formulating dynamics model：
Newton-Euler dynamic formulation
Newton's equation along with its rotational analog, Euler's equation, describe how forces, inertias, and accelerations relate for rigid bodies, is a "force balance" approach to dynamics.