Video 7.
1
Vijay Kumar
Property of University of Pennsylvania, Vijay Kumar 1
Control of Affine Systems
State
Input
State equations
Output
Property of University of Pennsylvania, Vijay Kumar 2
Lie Derivative
Function
Vector Field X
Lie derivative of f along X
Property of University of Pennsylvania, Vijay Kumar 3
Lie Derivative
Lie derivative of f along X
Example: n=2
q m
Property of University of Pennsylvania, Vijay Kumar 4
Example: Controlling a Single Output
Output
Want
or
Need derivative of the output function
Lie Derivatives
Property of University of Pennsylvania, Vijay Kumar 5
Single Input, Single Output, First Order Dynamics
State equations
Output
Rate of change of output
Control law
Closed loop system behavior Error exponentially
converges to zero
Property of University of Pennsylvania, Vijay Kumar 6
Input Output Linearization
v u y
nonlinear original
feedback system
new system
Nonlinear feedback transforms the original nonlinear system to a new linear system
Linearization is exact (distinct from linear approximations to nonlinear systems)
Property of University of Pennsylvania, Vijay Kumar 7
Affine, Single Input Single Output
State x
Input u
State equations
Output Rate of change of output
Control law
(rate of change of output is independent of u)
Explore higher order derivatives of output nonzero?
Property of University of Pennsylvania, Vijay Kumar 8
Affine, Single Input Single Output
State x
Input u
State equations
Output Rate of change of output
Control law
(rate of change of output is independent of u)
Property of University of Pennsylvania, Vijay Kumar 9
Affine, Single Input Single Output
State x
Input u
State equations
Output Rate of change of output
Control law
Closed loop system behavior
Error exponentially converges to zero
Property of University of Pennsylvania, Vijay Kumar 10
Affine, Single Input Single Output
State x
Input u
State equations
Output …
Relative degree, r The index of the first nonzero term in the sequence
r=k+1
11
Property of University of Pennsylvania, Vijay Kumar 11
Video 7.2
Vijay Kumar
Property of University of Pennsylvania, Vijay Kumar 12
Example 1. Single degree of freedom arm
m
r=2
Property of University of Pennsylvania, Vijay Kumar 13
Affine, SISO
Linear control,
model independent
r=1
feed forward
feedback
r=2
r=3
General form of control law
Property of University of Pennsylvania, Vijay Kumar 14
Single degree of freedom arm
Property of University of Pennsylvania, Vijay Kumar 15
Input Output Linearization
Single Input, Single Output, Relative degree r
v u y
nonlinear original
feedback system
new system
Nonlinear feedback transforms the original nonlinear system to a new linear system
Linearization is exact (distinct from linear approximations to nonlinear systems)
Property of University of Pennsylvania, Vijay Kumar 16
Multiple Input Multiple Output Systems
State x
Input u
Output
Assume each output has relative degree r
Nonlinear feedback law
leads to the equivalent system
Property of University of Pennsylvania, Vijay Kumar 17
Fully-actuated robot arm
(n joints, n actuators)
Dynamic model
‣ M is the positive definite, n by n inertia matrix
‣ is the n-dimensional vector of Coriolis and centripetal
forces
‣ N is the n-dimensional vector of gravitational forces
‣ t is the n-dimensional vector of actuator forces and torques
Property of University of Pennsylvania, Vijay Kumar 18
Fully-actuated robot arm (continued)
Property of University of Pennsylvania, Vijay Kumar 19
Fully-actuated robot arm (continued)
Relative degree is 2
Control law
Method of computed torque Inverse dynamics approach to
(Paul, 1972) control (Spong et al, 1972)
Property of University of Pennsylvania, Vijay Kumar 20
Under Actuated Systems
The number of inputs is smaller than the number
of degrees of freedom!
Property of University of Pennsylvania, Vijay Kumar 21
Kinematic planar cart
State equations, inputs
2 inputs, 3 degrees of
freedom
Outputs
Relative degree is 1
22
Property of University of Pennsylvania, Vijay Kumar 22
Planar Quadrotor
2 inputs, 3 degrees of
freedom
Property of University of Pennsylvania, Vijay Kumar 23
Three-Dimensional Quadrotor
4 inputs, 6 degrees of
F3
freedom
M3
F2
F4 w3 M2
M4
w2
F1
w4
M1
w1
Property of University of Pennsylvania, Vijay Kumar 24
