CS 545 Introduction to Robotics
Lecture XI
Newton-Euler Formulation
• Newton’s Equation
• Euler’s Equation
• Newton-Euler Formulation
http://robotics.usc.edu/~aatrash/cs545
Lagrangian Method
• Lagrangian (potential function): L=T-U
• Generalized Coordinates, Generalized Forces
• Kinetic Energy
� =1
2����
��� +1
2 �
��� �
• Potential Energy
� = �����(�)
• Solve entire system at once
Newton’s and Euler’s Equations
• Newton’s Equation
– Express the force acting at the center of mass for
accelerating body
� = �����
• Euler’s Equation
– Expresses the torque acting on a rigid body given
an angular velocity and angular acceleration
� = ��� + × ���
Newton-Euler Formulation
• Lagrange – Start with Lagrangian of entire system
• Idea of Newton-Euler– Goal: Force and torque balance at every link
– Examine each individual link
– Recursion through all links
• What do we need?– Angular velocities and accelerations of each link
– Linear velocities at each link
Approach Input:
Joint configurations,
Initial conditions on
velocities, accelerations
Approach Forward Recursion:
Determine linear acceleration,
angular velocity,
angular acceleration
Approach Backward Recursion:
Determine force and moments
Approach Final: Compute torque
ApproachHow do we do this?
Velocities and Accelerations
Forces and Moments
Forces and Moments
Velocities and Accelerations
Final Calculations
Final Calculations
Example – Two-Link Planar Arm
• Initial
Example – Two-Link Planar Arm
• Forward Recursion: link 1
Example – Two-Link Planar Arm
• Forward Recursion: link 2
Example – Two-Link Planar Arm
• Backward Recursion: link 2
Example – Two-Link Planar Arm• Backward Recursion: link 1
Lagrange vs. Newton-Euler
• Lagrange
– Systematic
– Analytic form. Good for control design
– Allows for more complex mechanics (deformation)
• Newton-Euler
– Recursive
– Computational efficient