Angular constraints on the Inverted Double Pendulum - 'Acrobot'

In summary, the conversation discusses an attempt to recreate Sutton's work on the 'Acrobot' and develop a model for a gymnast swinging on a high bar. The implementation of the physics is similar, but with additional constraints to prevent unwanted movements. The angles of the two joints are measured independently, with positive and negative velocities resulting in different angle calculations. The challenge lies in preventing the inner angle from reaching zero in certain scenarios, and various solutions have been attempted. The conversation also addresses the issue of adding constraints on the opening angle, with suggestions such as a physical barrier or using angles as primary variables.
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I am attempting to recreate Sutton's work on the 'Acrobot' and have modeled a good solution to the following:
http://webdocs.cs.uAlberta.ca/~sutton/book/ebook/node110.html

The physics is implemented in exactly the same way, however my particular Java implementation requires some constraint satisfaction so that particular movements cannot occur. As it stands, the Acrobot will gain momentum and then begin the second pendulum will rotate 360°. The purpose is to model a gymnast swinging on a high bar.

WP_001320.jpg


From the case figures, the two angles of the two joints are measured independently of one another. Suppose the angular velocity of the first joint is positive and starts from 0, as it revolves, the accumulated angle will eventually reach 2∏ and be reset to zero. Vice versa, a negative velocity will accumulate a negative angle until it reaches -2∏ and be reset to 0. Another point of interest is in Figure 1, θ1 is true if the ω1 >0, however if ω1 < 0 then the θ (which is already known, so we don't need this calculation, but in the context of the Figure) would be -(2π - θ1).

If you imagine the second joint as the 'hips' of an acrobat, the inner angle between the two cannot become 0. On the other hand, suppose the acrobot is vertical and the second joint has a positive velocity, it is fine for the second pendulum to continue around, but here the new angle between must not reach zero.

So, what is required are constraints that stop the inner angle becoming zero in particular scenarios. I have attempted a faulty solution that you can see below. I have also attempted to think about measuring the angle between the two, but cannot figure out a method.

I would greatly appreciate some help, as this should be simple but perhaps I am over complicating it.

Attempt:
I created multiple cases, such that if ω1 > 0 and ω2 < 0, and θ1 + θ2 > ∏ then θ2 = -(∏ - θ1). Others include using just the angles, splitting the area into spaces and solely the velocities. All of them show bugs if the acrobot suddenly changes angular velocity or goes overhead.
 
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  • #2
I have also attempted to think about measuring the angle between the two, but cannot figure out a method.
I would use those angles as primary variables to describe the state of the acrobot.
How do you want to add the constraint on the opening angle? As physical barrier? Add a potential which is large for small opening angles, and negligible elsewhere.

Why did you open a second thread? It is easier if you post everything in one thread.
 

Related to Angular constraints on the Inverted Double Pendulum - 'Acrobot'

1. What is an inverted double pendulum?

An inverted double pendulum is a mechanical system that consists of two pendulums connected by a hinge. The lower pendulum is attached to a motor or actuator that can move it back and forth, while the upper pendulum is free to move in any direction.

2. How does the Acrobot work?

The Acrobot is an inverted double pendulum that uses angular constraints to control its movement. The lower pendulum is actuated by a motor to move it back and forth, while the upper pendulum is constrained by the angle between the two pendulums. By controlling the angle and velocity of the lower pendulum, the Acrobot can perform a variety of movements, including balancing in an upright position.

3. What are the applications of the Acrobot?

The Acrobot has various applications in robotics and control theory. It can be used as a model for studying the dynamics of complex systems, and its control algorithms can be applied to other mechanical systems. The Acrobot is also used in educational settings to demonstrate principles of control theory and robotics.

4. What are the challenges of controlling an inverted double pendulum?

The main challenge in controlling an inverted double pendulum is its inherent instability. The system is highly sensitive to small changes in its initial conditions, making it difficult to control. Additionally, the Acrobot's movement is affected by external factors such as friction and air resistance, which further complicate the control process.

5. How can angular constraints be used to stabilize the Acrobot?

Angular constraints play a crucial role in stabilizing the Acrobot. By constraining the angle between the two pendulums, the system's movement is limited, making it easier to control. Additionally, feedback control algorithms can be used to continuously adjust the angle and velocity of the lower pendulum, allowing the Acrobot to maintain its upright position and resist external disturbances.

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