Dynamics

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Dynamics
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Why are we studying inertial dynamics and control?

• Kinematic vs dynamic models
• What were really doing is modeling the
  manipulator
• Kinematic models
• Simple control schemes
• Good approximation for manipulators at low
  velocities and accelerations when inertial
  coupling between links is small
• Not so good at higher velocities or accelerations
• Dynamic models
• More complex controllers
• More accurate

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Calculating Equations of Motion using the
Lagrangian
Kinetic energy
Potential energy
Generalized coordinate (joint angle)
Generalized force (torque)
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Example pendulum
Earlier, we derived the equation of motion of the
inverted pendulum by summing the torques
You can also do it using the Lagrangian
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Example pendulum
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Example pendulum
w/ torque and friction
Torque applied at motor
Coefficient of friction
If you want to think about a motor actuating the
joint
motor gear reduction
motor
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Moment of inertia
The moment of inertia of a single point mass
about a single axis is
Point mass
axis
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The Inertia Matrix

• If the link is not a point mass, then kinetic
  energy is calculated by integrating the kinetic
  energy of all points in the body.
• The moment of inertia encodes the distribution of
  mass for the purposes of calculating rotational
  kinetic energy

Center of gravity
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The Inertia Matrix
Calculate the kinetic energy of a body
Center of gravity
Integrate kinetic energy of all points in the
body
Express each point w.r.t. CG
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Time out for skew symmetric matrices!
Cross product
Where
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The Inertia Matrix
Translational energy
Rotational energy
Inertia matrix
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The Inertia Matrix

• Diagonal elements moments of inertia
• Off diagonals products of inertia
• Notice that the inertia matrix is symmetric

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The Inertia Matrix
Rotational kinetic energy
This equation holds for an arbitrary coordinate
frame

• In the above equation, it is assumed that angular
  velocity and the inertia matrix are expressed in
  the same reference frame.
• However, the inertia matrix is only constant when
  expressed in the reference frame of the body
• Therefore, we must convert the inertia matrix
  expressed in the body frame to an inertia in the
  base frame

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The Inertia Matrix
Link frame
Therefore
Where
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The Inertia Matrix Example 1
Calculate the moment of inertia of a cuboid about
its centroid
Since the object is symmetrical about the CG, all
cross products of inertia are zero
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Moment of Inertia Parallel Axis Theorem

• For a given axis, the moment of inertia about the
  CG is the minimum that axis.
• You can calculate the moment of inertia about any
  parallel axis by adding a term

new axis

• Where m is the mass of the object and d is the
  orthogonal distance from the CG
• The moment of inertia about a parallel axis is
  the CG inertia the moment of inertia of the
  body taken to be a point mass at the CG.

CG
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The parallel axis theorem and the inertia matrix
How does each term of the inertia matrix change?
new axis
CG
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Parallel Axis Theorem Example 1
Calculate the moment of inertia about a point at
one end of the cuboid
From before
For the new axis, it is
All the products of inertia remain zero.
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Kinetic Energy General Case
Velocity of CG of link
Where
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Kinetic Energy General Case
Note we need to use a Jacobian that calculates
the twist (velocity) of the CG of the relevant
link.
Suppose were calculating
the rest of the manipulator is irrelevant to
Important this Jacobian describes the motion of
the CG of the ith link, NOT the motion of the end
of that link.
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Kinetic Energy Jacobian Example
Suppose were calculating
The first link has length
We measure the velocity of the CG of the second
link a distance of
The third link doesnt matter to
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Kinetic Energy General Case

• Manipulator inertia matrix
• Notice that the inertia matrix is symmetric
• How do you know?

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Potential Energy General Case
Gravity vector
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Equation of Motion General Case
chain rule
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Equation of Motion General Case
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Equation of Motion General Case
This is essentially the manipulation eqn of
motion
Were going to modify the coriolis/centrifugal
term
Christoffel symbols
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Equation of Motion General Case
We will use Christoffel symbols to write the
coriolis/centrifugal term
where
where
Note that
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Equation of Motion General Case
Moment of inertia
Gravity
Centrifugal, coriolis
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Equation of Motion General Case
Moment of inertia
Gravity
Centrifugal, coriolis
You can add friction and external forces to the
eqn
Viscous friction
External forces
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Example two-link manipulator, inertia matrix
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Example two-link manipulator, inertia matrix
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Example two-link manipulator, inertia matrix
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Example two-link manipulator, inertia matrix
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Example two-link manip, coriolis centifugal
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Example two-link manip, coriolis centifugal
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Example two-link manip, gravity term
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Example two-link manip
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Inverse Dynamics

• The equation of motion allows you to compute the
  inverse dynamics
• Given a joint trajectory,
  , the inverse dynamics computes the
  sequence of torques that generate the motion.

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Direct (forward) Dynamics
Given a sequence of joint torques, what
trajectory does the manipulator follow?
Given acceleration, you integrate joint
velocities and positions to generate the full
trajectory.