8.4 - Speed and Acceleration¶

Speed and Acceleration¶

Speed is defined as distance divided by time. For cars we use miles/hour; for a person running, perhaps meters/second. Distance, time, and speed are interrelated using these basic formulas:

speed = distance / time;
distance = speed * time;
time = distance / speed;

Since physical objects can’t be stationary one moment and traveling at 60 miles per hour the next, we should take into account acceleration and deceleration. Acceleration is the change in speed over time. An accelerating object is increasing in speed. A decelerating object is decreasing in speed. Any object that starts to move must accelerate to a given speed, travel at that speed for a certain amount of time, and then decelerate until the object is stationary. This is visualized in the diagram to the right.

We often make simplifying assumptions about motion when we create virtual worlds. Let’s consider three scenarios:

• Ignore acceleration entirely. (For simple animations this is sometimes acceptable.)
• Model acceleration as a simple linear change in speed.
• Model acceleration accurately, starting from zero acceleration – to increasing acceleration – to decreasing acceleration – to zero acceleration when a constant speed has been reached.

Let’s examine each of these scenarios separately.

Designing Motion¶

Given that speed = distance / time, the design of an animation requires that you specify two of the three values. Therefore, you have three design scenarios:

1. You decide the distance to travel and the time of travel. (speed is calculated)
2. You decide the speed of movement and the time of travel. (distance is calculated)
3. You decide the distance of travel and the speed. (time is calculated)

Option 1 is perhaps the most common, so our remaining discussion will assume that an animator will set starting and ending locations to specify a distance and the number of frames to specify time.

For an animation you can measure time in seconds or in frames. Distance can be measured using any units that are consistent with your virtual scene. Remember that WebGL is unit-less – but you must use consistent units for all measurements.

Ignore Acceleration¶

For simple motion it is sometimes acceptable to ignore acceleration and immediately start moving at a constant speed. For the parametric parameter t, if the differences between t_i and t_i+1 are equal, the change in values will be a constant, which means the object will be moving at a constant speed. To calculate the “change in t” (i.e, dt) for such a motion, simply calculate the number of discrete positions an object will have during the motion. Then divide this into 1.0, since t varies between 0.0 and 1.0. Note that given n frames, there are n-1 transitions between frames.

number_frames = ending_frame - starting_frame + 1;
dt = 1.0 / (number_frames-1);

Let’s walk through a specific example. In frame 100 an object is stationary and has a location of p1 = (10, 12,-4). You want the object to move to location p2 = (20, 8, 5) by frame 110. In frames 101 through 109 the object will be in a location calculated by the parametric equation. Using the above equations, the value for dt is 0.1 and the values for t for each frame will be:

The selection of the starting and ending frames determines the time for the animation. You can convert the number of frames into seconds by dividing by the frame rate. For example:

seconds = (number_frames-1) / (30 frames / 1 sec);

The selection of the starting and ending points determines the distance of the motion. You could calculate the exact distance using the distance formula:

distance = Math.sqrt( (p2x-p1x)*(p2x-p1x) + (p2y-p1y)*(p2y-p1y) + (p2z-p1z)*(p2z-p1z));

If we needed the exact speed of the motion we could calculate it:

speed = distance / seconds;

The example WebGL program from lesson 8.1 demonstrated constant speed with no acceleration and is repeated here if you would like to experiment.

Linear Acceleration¶

For more realistic motion you need to include the acceleration of an object as it starts to move and the deceleration as it slows down to stop. This adds complexity to your animation design and to your calculations.

To simplify our calculations as much as possible we model the acceleration and deceleration as linear functions, as shown in the diagram to the right. There are many scenarios we could consider, but let’s look at a straightforward scenario where an animator decides the time it takes to accelerate and to decelerate. The equations will be developed using the simple example we used above:

• In frame 100 an object is stationary and has a location of p1 = (10, 12,-4);
• You want the object to move to location p2 = (20, 8, 5) by frame 110;
• Acceleration takes 3 frames at the beginning of the motion; and
• Deceleration takes 2 frames at the end of the motion.

The table below shows the scenario with the values we need to calculate. Please note that we need to calculate the “change in t” between frames in order to calculate the the actual values of t. We will end up with 11 values for t, but 10 “change in t” values.

Let’s call the “change in t” dt. If dt is constant, then we have constant speed. If dt is increasing we have acceleration; if dt is decreasing we have deceleration. We use the slope of the acceleration and deceleration lines to approximate the dt value at the start and end of the motion. We don’t know the value for dt but we do know that the sum of all the values must be 1.0. For the specific example in the diagram, we can write the equation below. In the diagram the frames are marked with vertical red lines. Notice that we take the value for dt in the middle of each frame.

// Adding the dt values for acceleration, constant speed, and deceleration
1.0 = 1/6*dt + 3/6*dt + 5/6*dt + dt + dt + dt + dt + dt + 3/4*dt + 1/4*dt;

Therefore 7.5*dt === 1.0 and dt must be 1/7.5, or 0.133. Substituting this value for dt back into each term gives us the specific “change in t” for each frame. These values are show in the following table.

Let’s make some observations about the calculated values:

• For this problem the dt value is 0.133, while for the original “constant speed” problem above the dt value was 0.1. It makes sense that the speed during constant motion must be greater to make up for the time lost during acceleration and deceleration.
• The dt values are increasing during acceleration and decreasing during deceleration. Hopefully this makes sense from a physical perspective.
• The sum of the dt values for acceleration and deceleration can be written as the sum of odd integers times the slope of the line. In this example, the acceleration terms are (1/6)*(1 + 3 + 5) and the deceleration terms are (1/4)*(3 + 1). The sum of n odd integers is known to be n².

With a little work we can generalize this algorithm for any set of inputs. The function calculateAnimation in the demo code below will calculate and return an array of t values based on a starting frame, an ending frame, the number of frames to accelerate, and the number of frames to decelerate. Please study the code carefully. Then experiment with this WebGL program.

After you have experimented with the WebGL program above, consider its following deficiencies:

• The calculations are straightforward, but not elegant.
• There are limited visual clues about where acceleration stops and deceleration starts. Specifying the number of frames for these stages of motion does not facilitate an animator’s “visual design” of the motion.
• The motion is defined by varying the “change in time” for each frame during the acceleration and deceleration phases. This is very problematic, especially if multiple objects are moving at the same time, but at different rates. There is no easy way to sync the actual time each object is moving.
• The linear acceleration and deceleration is not visually consistent with what a person expects from the motion.

Non-Linear Acceleration¶

To extend the techniques just described to non-linear acceleration is mathematically too complex to pursue in this textbook.

Conclusion¶

Varying the “change in time” between frames to accomplish acceleration and deceleration is not a good idea. We need a better way to control “changes over time” that keeps the time steps between frames constant and consistent. There is a better way! We can use more complex basis functions for our parametric equations – which we will discuss next.

Glossary¶

speed

The rate of change in an object’s location and/or orientation, measured in terms of distance per change in time.

acceleration

The change in speed over a time interval. (Acceleration is often assumed to be a positive increase in speed.)

deceleration

A decrease in speed over a time interval.

Self Assessment¶