Acrobat Robot Control and Stability

This assignment focuses on controlling and stabilizing a two-joint under-actuated robot known as the 'acrobat'. Students design controllers using Linear Quadratic Regulator (LQR), Linear Quadratic Gaussian (LQG), and H∞ methods to stabilize the system around its vertical equilibrium. The assignment involves linearizing the system, simulating control strategies with various initial conditions, and comparing the performance of different controller designs. It also touches upon concepts like domain of attraction and robustness in control systems.

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
UNIVERSITY OF AUCKLAND
ELECTENG 704 ADVANCED CONTROL SYSTEMS
ASSIGNMENT I
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ASSIGNMENT OVERVIEW
The acrobat is a two-joint under-actuated robot. It is roughly parallel to a funambulist swinging
on a high bar. The first link cannot apply torque but the second one can. The system has four
unremitting state variables: two joint positions and two joint speeds. The equations of the motion
are given as shown below,
The equations of motion are,
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The parameters in the equation of motion are,
SOLUTION
Part a
(i) Verify that the vertical position is an equilibrium.
(ii) Linearize the system about the equilibrium.
(iii) Verify that the equilibrium is unstable.
The under-actuated mechanical systems tend to have rarer control inputs as compared to the
degrees of freedom. They seem to arise in applications such as gymnastic automatons. These
systems cannot be controlled to trail an arbitrary trajectory
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The q1 is the angle of the shoulder link, and q2 is the angle of the elbow link. The dynamic
equation of the acrobat is as shown below,
Where m1 and m2 are the masses of link 1 and link 2 respectively.
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Part. 2
Design a LQR controller to stabilize the system about the equilibrium.
Check your results in a nonlinear simulation with several initial conditions slightly off the
vertical equilibrium. For this stabilizing controller, estimate the domain of attraction. Hint: Use
different initial conditions.
The equation of motion is derived using the Lagrangian dynamics. The derivation of the equation
is as shown in part 1. The dynamical system is a deterministic double pendulum. The actuation is
applied to the second joint solely. The undercurrents are in the form of
Where the are the two link angles while the u denotes the scalar control signal. It is the
torque that acts on the second link. The torque equations are represented as matrix yields where
F denotes the constant of friction,
Matlab computes the dynamics using the code snippet below,
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Linearization of the acrobat around the unstable upright point, one can obtain,
The linear dynamics follow directly from the equation of motion and the manipulator form of the
acrobat definitions or equations.
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The simulation on Matlab is as shown in the illustration below,
Part 3
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Design a LQG controller to stabilize the system about the equilibrium.
A nonlinear simulation with several initial conditions slightly off the vertical equilibrium.
For the stabilizing controller, a domain of attraction is obtained. Hint: Use different initial
conditions
The system is controllable when it has the following form of state variables,
Where x has dimension n. One can manage to construct an unimpeded control signal that seeks
to assign an initial state to any final state in a finite interval of time. When each state, within the
time interval is governable, the system is termed completely state governable. Another way to
ensure the stability is to test for non-repeating Eigen values.
The LQR design returns the solution S that is associated with Riccati equation. The relationship
with the Riccati equation is,
The closed loop Eigen values can, therefore, be obtained as,
Part 4
Design a LTR controller to stabilize the system about the equilibrium.
Domain of attraction
Compare with your LQG design
The regulator is achieved using set point trackers by constructing the LQ optimal gain as well as
the state estimator. The LQG design is formulated by connecting the LQ optimal gain and the
Kalman filter.
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The plant state and the measurement equations are of the form,
The inverted pendulum is a single input multiple output system. It has components as shown
where y is the vector that contains two angular position measurements q1 and q2,
Jd =
(2*k*r)/(a + b*k) - (b*(r*k^2 + q))/(a + b*k)^2
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Part 5
The H∞ control design does not seek the optimal controller as it seeks to produce the complete
minimal value. The design is finished using an iterative technique that seeks for reducing the
norm while still looking at other performance measures.
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The LQR produces a very robust controller while with the LQG robustness is not warranted.
There are other factors such as the stability radius and spectral values, that one needs to look at
the closed loop design. The H∞ design is sought as it provides robustness at the cost of a
pessimistic control law. The design assumes that the worst possible perturbation is acting on the
system at all times. The key limitation of the LQR is the requirement of full-state feedback. Only
a few of the states are measured.
APPENDICES
The entire Matlab script to solve the system requirements is as shown below,
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