Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Manipulator Dynamics 2/2
Step 2 - Write the Newton and Euler equations for each link.
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Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Step 1 - Calculate the link velocities and accelerations iteratively from the
robot’s base to the end effector
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Iterative Newton-Euler Equations - Solution Procedure
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Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Step 3 - Use the forces and torques generated by interacting with the
environment (that is, tools, work stations, parts etc.) in calculating the joint
torques from the end effector to the robot’s base.
Iterative Newton-Euler Equations - Solution Procedure
Gravity Effect - The effect of gravity can be included by setting v&0 = g . This
is the equivalent to saying that the base of the robot is accelerating upward at 1
g. The result of this accelerating is the same as accelerating all the links
individually as gravity does.
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Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
0
Error Checking - Check the units of each term in the resulting equations
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Iterative Newton-Euler Equations - Solution Procedure
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Iterative Newton-Euler Equations - 2R Robot Example
G (θ )
( )
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
joint velocities) terms - nx1 Vector
- gravitational terms - nx1 Vector.
M (θ ) - Mass matrix (includes inertia terms) - nxn Matrix
V θ ,θ& - Centrifugal (square of joint velocity) and Coriolis (product of two different
τ = M (θ )θ&& + V (θ ,θ& ) + G (θ )
It is often convenient to express the dynamic equations of a manipulator in a
single equation
where
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Dynamic Equations - State Space Equation
•
[ ]
in a different form, we can
- Centrifugal coefficients(square of joint velocity)
- Coriolis coefficients (product of two different joint velocities)
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
This form can be useful for applications using force control. Each of the
matrices is a function of manipulator configuration only (that is, joint position)
and can be updated at a rate depending on the magnitude of joint changes.
B (θ )
C (θ )
(θ ,θ& )
τ = M (θ )θ&& + B (θ ) θ& θ& + C (θ ) θ& 2 + G (θ )
[ ]
By rewriting the velocity dependent term V
write the dynamic equations as
where
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Dynamic Equations - Configuration Space Equation
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&x& = J& (θ )θ& + J (θ )θ&&
F = J −T (θ )τ
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
By differentiation, we find
x& = J (θ )θ&
τ = J T (θ )F
we can substitute the joint moments using our definition of the Jacobian matrix:
•
τ = M (θ )θ&& + V (θ ,θ& ) + G (θ )
It can sometimes be desirable to have a relationship between the end effector’s
Cartesian accelerations and the joint torques. Beginning from the Configuration
Space equation
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Dynamic Equations - Cartesian State Space Equation
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( )
( )
( )
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
This equation relates the forces and moments at the end effector to the
Cartesian accelerations of the end effector and the manipulator joint positions
and velocities.
Gx (θ ) = J −T G (θ )
Vx θ ,θ& = J −T M (θ )J −1 J&θ& + J −T V θ ,θ&
M x (θ ) = J −T M (θ )J −1
F = M x (θ )&x& + Vx θ ,θ& + Gx (θ )
( )
F = J −T τ = J −T M (θ )J −1&x& − J −T M (θ )J −1 J&θ& + J −T V θ ,θ& + J −T G (θ )
Substitution yields
θ&& = J −1 &x& − J −1 J&θ&
Solving for joint acceleration gives
Where
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Dynamic Equations - Cartesian State Space Equation
Define a set of generalized forces (and moments) Qi for i=1,2,3…N
The generalized forces must satisfy
3.
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
where δqi is a small change in the generalized coordinate and δW is the work
done corresponding to that small change.
Qiδqi = δW
Define a set of generalized velocities q&i for i=1,2,3…N
2.
θ i if revolute joint
qi 
d i if prismatic joint
1. Define a set of generalized coordinates for i=1,2,3…N.
These coordinates can be chosen arbitrarily as long as they provide a set of
independent variables that map the system in a 1-to-1 manner. The usual
variable set for serial manipulators is:
Langrangian Formulation of Manipulator Dynamics 1/
1
1
T
mi vc i vc i + iω i c i I i iω i
2
2
P = f (qi , t )
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
L=K-P
Let L denote the Lagrangian given by:
1
P = ∑ mghi + kx 2
2
Let P denote the expression describing the potential energy. where
ki =
Let K denote the expression describing the kinetic energy. where K = f (q , q& , t )
i
i
4. Write the equations describing the kinetic and potential energies as functions of
the generalized coordinates as well as the resulting Lagrangian.
Langrangian Formulation of Manipulator Dynamics 2/
d  ∂K  ∂K ∂P
 −
+
Qi = 
dt  ∂q&i  ∂qi ∂qi
d  ∂L  ∂L

 −
dt  ∂q&i  ∂qi
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
or, more practically, by
Qi =
5. The equations of motion are given by
Langrangian Formulation of Manipulator Dynamics 3/
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Langrangian Formulation - 2R Robot Example
and
q& 2 = θ&2
For i=1
•
k1 =
ki =
1
1
2 2
2
m1 Lg1 θ&1 + I1θ&1
2
2
1
1
T
mi vc i vc i + iω i c i I i iω i
2
2
Qi = τ i
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Kinetic Energy:
•
Step 4:
Step 3: Let external forces/torques
Step 2: Let q&1 = θ&1
Step 1: Let q1 = θ1 and q2 = θ 2
Langrangian Formulation - 2R Robot Example
•
•
•
(
[
(
)
(
)]
(
)
)
2
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
2
1
1
2 2
2 &
2
&
k 2 = m2 L1 θ1 + Lg 2 θ1 + θ 2 + 2 L1 Lg 2 c 2 θ&1 + θ&1θ&2 + I 2 θ&1 + θ&2
2
2
For i=2
)
2
2 2
2
2
T
vc i vc i = L1 θ1 + Lg 2 θ&1 + θ&2 + 2 L1 Lg 2 c 2 θ&1 + θ&1θ&2
(
 L1c1 + Lg 2 c12
Pg 2 = 

L
s
L
s
1
12
+
g
1
2


The derivative squared gives
0
To find the velocity of the center of mass of link 2, first consider its position given
by
Langrangian Formulation - 2R Robot Example
For i=1
For i=2
Lagrangian:
•
•
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L = k1 + k 2 − p1 − p2
p2 = m2 g (L1s1 + Lg 2 s12 )
p1 = m1 gLg1s1
p = ∑ mghi
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Potential Energy:
•
Langrangian Formulation - 2R Robot Example
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d  ∂K  ∂K ∂P

 −
+
&
dt  ∂qi  ∂qi ∂qi
Qi =
τ1 =
d  ∂K  ∂K ∂P

 −
+
&
dt  ∂θ1  ∂θ1 ∂θ1
d  ∂K  ∂K ∂P
τ 2 =  &  −
+
dt  ∂θ 2  ∂θ 2 ∂θ 2
d  ∂L  ∂L

 −
dt  ∂q&i  ∂qi
Qi =
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
Step 5: Solving
Langrangian Formulation - 2R Robot Example
g2
1
g2 2
2
2
g2
2
2
Instructor: Jacob Rosen Ph.D.
Models of Robot Manipulation - EE 543 - Department of Electrical Engineering - University of Washington
+ m2 gLg 2 c12
2
+ m2 L1 Lg 2 s2θ&1
2
τ 2 = [m2 ( Lg 2 2 + L1 Lg 2 c2 ) + I 2 ]θ&&1
+ [m L + I ]θ&&
+ m1 gLg1c1 + m2 g ( L1c1 + Lg 2 c12 )
2
− m2 L1 Lg 2 s2 (2θ&1θ&2 + θ&2 )
2
τ 1 = [m1 Lg1 + I1 + m2 ( L12 + Lg 2 2 + 2 L1 Lg 2 c2 ) + I 2 ]θ&&1
2
+ [m ( L + L L c + I ]θ&&
Langrangian Formulation - 2R Robot Example