Energy-Based Control of Under-Actuated

Mechanical Systems: Remotely Driven Acrobot

By Xin Xin, and Taiga Yamasaki

Presentation by Mostafa Shokrian Zeini

Important Questions:

- What is an under-actuated mechanical system (UMS)? And what are its applications?

- What is the energy-based control?

- How to provide a global motion analysis of the UMS under the designed controller?

- What are the conditions on control parameters for achieving a successful swing-up control?

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Under-Actuated Mechanical Systems

Under-actuated Mechanical Systems (UMSs) possess fewer actuators than degrees of freedom.

The design of such mechanisms allows to reduce cost and weight by using fewer actuators and permits to increase fault tolerance to actuator failure.

Controlling these systems is challenging due to inherently complex nonlinear dynamics.

So

However

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Under-Actuated Mechanical Systems

Cart-Pendulu

m System

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Under-Actuated Mechanical Systems

Translational

Oscillators with

Rotating Actuator (TORA)

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Under-Actuated Mechanical Systems

Gyroscope

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Under-Actuated Mechanical Systems

Pendubot&

Acrobot

This is a two-link planar robot with a single actuator and with a first joint being attached to a passive joint.

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Acrobots

Acrobots

The Directly Driven Acrobot (DDA)

The second link is directly driven.

The Remotely Driven Acrobot

(RDA)

The second link is remotely driven by

an actuator mounted at a fixed base through a belt.

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Acrobots

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Acrobot

The motion equations

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Directly Driven Acrobot (DDA)

The motion equations

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The motion equations

Directly Driven Acrobot (DDA)

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Remotely Driven Acrobot (RDA)

The motion equations

For our RDA, the angle of link 2 with respect to the –axis, i.e. , is actuated via the timing belt:

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Remotely Driven Acrobot (RDA)

The motion equations

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Remotely Driven Acrobot (RDA)

The motion

equations

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Remotely Driven Acrobot (RDA)

The motion

equations

Regarding the RDA motion equations with respect to , and the DDA described with respect to , although their control input transformation matrices are the same, the other coefficient matrices are different. Thus, by no means the RDA motion equations are like the DDA dynamics.

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Energy-Based Swing-Up Controller

where is the potential energy of the RDA expressed as:

𝑃 (𝜃 )=𝛽1 (cos𝜃1 −1 )+𝛽2 (cos𝜃2−1 )

The total mechanical energy of the RDA is given by:

The control objective is to find a controller under which the RDA can be swung up:from any initial state to any small neighborhood of the Upright Equilibrium Point

(UEP)

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where the constant is a given reference of and satisfies

𝐸𝑟 ≥ min𝐸=−2(𝛽1+𝛽2)

In what follows, our goal is to design a controller such that:

lim𝑡→ ∞

𝐸=𝐸𝑟 , lim𝑡 → ∞

𝜃2=0 , lim𝑡→ ∞

�̇�2=0

Energy-Based Swing-Up Controller

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Lemma 1

Suppose constant control parameters , , and satisfy:

, and , respectively, then

contains no singular points. Moreover, as the closed-loop solution converges to the largest invariant set at which and where and are constants; and the following equations hold

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Energy-Based Swing-Up Controller

Energy-Based Swing-Up Controller

If can be chosen such that where :

Taking the time derivative of , together with the property yields

Consider the following Lyapunov function candidate:

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

Energy-Based Swing-Up Controller

Under the condition and using

Putting it into gives:

We obtain :

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

Energy-Based Swing-Up Controller

yields every approaches as .

Thus, and are constant in .

Let be the largest invariant set in . Using:

Yields:

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

It proves that the controller contains no singular points.

Energy-Based Swing-Up Controller

If , then is a constant denoted as . Putting and into the above equation yields:

Putting and into :

From , is also constant in : This proves

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

This completes the proof of lemma 1.

Energy-Based Swing-Up Controller

Therefore, since the (C2) holds for all , the following relation must hold:

where is constant. Since , is bounded: So the energy and the angular velocity are bounded.

Integrating it with respect to time yields:

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

Energy-Based Swing-Up Controller

where is constant. Similarly, . Thus:

is a constant noted as .

Integrating the above equation with respect to time yields:

Consequently:

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

Energy-Based Swing-Up Controller

Now, Let’s bring the Lemma 2.

Putting and into (C1) gives:

Putting and into the above equation yields , thus:

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𝑴 𝜽 (𝜽 ) �̈�+𝑯 𝜽 (𝜽 , �̇� )+𝑮𝜽 (𝜽 , �̇� )=𝑩𝑨𝝉

Lemma 2

Suppose constant control parameters , , and satisfy:

, and , respectively, and

If the convergent value of energy satisfies , then the closed-loop solution converges to an equilibrium point satisfying

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Global Motion Analysis of RDA

Let’s consider two situations:i- ii-

From (C1) : .

Thus by using :

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i-

Global Motion Analysis of RDA

We recall the result of Lemma 2:

We want to find conditions such that could be the unique solution of the above equation.

Clearly, is a solution of the above equation.But if , …

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ii-

Global Motion Analysis of RDA

ii-

Thus, is the unique solution of , if and only if:

Putting into yields , where:

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Global Motion Analysis of RDA

The closed-loop solution of system converges either to the UEP or the down-up EP.

Let’s Introduce the Theorem 1.

Under the above condition, is the unique solution of And according to the result of Lemma 2 ()

where

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Global Motion Analysis of RDA

Theorem 1Consider the closed-loop system with positive parameters , , and . Suppose that If , and control parameter satisfies

then under the designed controller, for any initial condition of RDA, either of the following two statements hold:i) , and the closed loop solution converges to ii) , and the closed loop solution converges either to the UEP or the down-up EP .

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Global Motion Analysis of RDA

Theorem 2

Consider the closed-loop system with positive parameters , , and . Suppose that . Suppose that . The following statements hold:i) The Jacobian matrix of evaluated at down-up EP has 2 and 2 eigenvalues in the open left- and right-half planes, respectively; such a down-up EP is unstable.ii) The Jacobian matrix of evaluated at UEP has 3 and 1 eigenvalues in the open left- and right-half planes, respectively; such an UEP is an unstable.

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Global Motion Analysis of RDA

The closed-loop solution converges to:

The closed-loop solution converges to the down-up EP which is unstable.

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ii)

Global Motion Analysis of RDA

With the quantities described in Theorem 1, if :

i)

Cosidering , starting from almost all initial state:

the closed-loop solution approaches as .

Since the down-up EP is unstable:the RDA under the designed controller cannot maintain at the this point

in practice.

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Global Motion Analysis of RDA

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Global Motion Analysis of RDA

The linearized model of the RDA around the UEP is:

where and:

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Global Motion Analysis of RDA

is controllable and one can obtain a locally stabilizing controller: , where is the state-feedback gain.

Defining the following small neighborhood of the UEP:

where is a given positive weighting matrix, and is a given small positive number.

When the RDA is swung up into the neighborhood of the UEP, we can switch the swing-up controller to the locally stabilizing controller.

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Global Motion Analysis of RDA

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Simulation Results

The switch was taken about at which the RDA was driven into the neighborhood of the UEP.

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Simulation Results

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Simulation Results

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Simulation Results

References

1. X. Xin, T. Yamasaki, “Energy-Based Swing-Up Control for a Remotely Driven Acrobot - Theoretical and Experimental Resultsˮ, 2012, IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, 20 (4), pp. 1048-1056.

2. X. Xin, S. Tanaka, J. She, T. Yamasaki, “New Analytical Results of Energy-Based Swing-up Control for the Pendubotˮ, 2013, International Journal of Non-Linear Mechanics, 52, pp. 110-118.

3. N. Adhikary, C. Mahanta, “Integral Backstepping Sliding Mode Control for Underactuated Systems - Swing-up and Stabilization of the Cart–Pendulum Systemˮ, 2013, ISA Transactions, 52, pp. 870-880.

4. B. Gao, H. Song, J. Zhao, C. Gong, “Dynamics and Energy-based Control of TORA System on a Slopeˮ, 2013, Proceeding of the IEEE Int. Conf. on Cyber Tech. in Automation, Control and Intelligent Systems, Nanjing, China, pp. 373-378.

5. M. Pourmahmoud Aghababa, H. Pourmahmoud Aghababa, “Chaos Synchronization of Gyroscopes using an Adaptive Robust Finite-time Controllerˮ, 2013, Journal of Mechanical Science and Technology, 27 (3), pp. 909-916. 43