TennisOne Lessons

The Stick Balancer

by Jim McLennan

The Gravity Turn provides a sudden start when volleying or moving to hit a groundstroke. It is seemingly paradoxical: The initial movement (the foot drop) is away from the ball. However, the movement has a solid basis in physics, as we will show.

Engineering curriculum includes introduction to the physics of dynamical systems. One such system is the cart carrying an inverted pendulum. This model is used to introduce students to the task of developing mathematical models of such unstable systems. The ultimate task is to devise an automatic control system capable of balancing the stick and regulating the cart position in response to commands. In 1965 this task was considered PhD level research. Today, it is an undergraduate activity, and many students have contact with this and similar control-oriented subjects. The stick balancer actually has some engineering relevance in that it bears some similarity to stabilizing a rocket in flight.

In more detail, the system consists of a cart free to translate left and right. The cart carries a stick 2 meters in length, pinned to the cart by a rotational joint. This joint is totally free from restraint or friction of any kind. When released from vertical, the stick will fall, unless the cart executes the proper motion to maintain the stick upright. The cart is motorized, so it can accelerate and try to catch the stick.

The overall motion of the system can be set forth mathematically by appealing to Newton 's Laws of Motion. Suffice it to say, the governing equations are nonlinear differential equations not amenable to solution in closed form. The system can, however be simulated using a computer and appropriate numerical methods. Today, this is a routine task. The simulation output consists of an animation, which looks like a cartoon. In this case, though, the motion is generated by accurately applying the rules of physics.

The strategy to balance the stick comes from the field of control theory. There it is known that the “optimal” strategy consists of a feedback control in which the force exerted upon the cart comes from a weighted combination of the cart position, velocity, stick angle, and rotation rate. In an implementation, a suite of sensors would measure the cart position, the stick angle, etc. and this information would be processed by a computer mounted on the cart. The computer would then send a signal to the motor, telling it how much force to exert on the cart. This process would then repeat hundreds of times each second. We emphasize that there is no force exerted on the stick directly. It will fall unless it receives the correct pushes and pulls through the hinge connecting it to the cart.

When the controller is turned on, it stabilizes the stick. This is being done by pushing on the cart only. In the animation, the cart starts one meter to the left of the origin. At the initial instant, the cart is commanded to move two meters to the right, while balancing the stick. In response to the command, the cart's initial move is in the WRONG direction! Notice that the sudden movement to the left causes the stick to tip and fall to the right. The cart then accelerates to the right and catches the stick. The cart then decelerates and comes to rest at the commanded position. The cycle repeats and the cart moves back to the left, after first causing the stick to tip left by a quick move to the right.

The Cart Has Performed a Gravity Start!

The analogy to the tennis move is quite striking. By losing balance and initiating a fall, the system can respond more quickly to commands.

When viewing the animation it is intuitively obvious that the system can move more quickly by initiating a fall, and then recovering balance as the target position is reached. You can try the same experiment with a broomstick balanced on the palm of your hand. Without tipping the broomstick you can only move your hand slowly. If you want to reposition your hand rapidly, you first jerk your hand in the opposite direction. The stick falls in the direction you wish to move, and you reposition your hand while recovering balance of the stick. Try it!

References

The mathematics, physics and control of the stick balancer are discussed in the textbook:

“Dynamics of Physical Systems,” by Robert H. Cannon. Professor Cannon was for many years the Department Chairman of the Aero-Astro Department of Stanford University. This excellent book is widely used in courses on modeling and control system design and is readily available.

One of Professor Cannon's early students produced a PhD thesis whose subject was stick balancing. The thesis title is “On Bounded Control of Some Unstable Systems”, by John Schaefer, 1965. A copy of this thesis can be found in the Stanford University Engineering Library located in Terman Hall. Dr. Schaefer actually fabricated a cart and stick system, which succeeded in balancing the stick. He did this using transistor circuitry he soldered himself. (No mean feat in 1965!)

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