Our novel motion-creation algorithms build
a driverless car
that surpasses human’s driving skills


the car avoids accidents and earns trust from people


MotionLab, LLC, CEO: Yutaka Kanayama, Ph.D.
YouTube: https://www.youtube.com/user/motionlabLLC/videos
Science Robot, Math Mind: Mac motion simulator at App Store,

Version 3: November 2018
Version 2: October 2018
Version 1: August 2018


1. How do people perceive the latest driverless cars?

Let us review news on the driverless cars currently being built:

  • In January of this year, a self-driving car slammed into a fire truck on a freeway in Los Angeles.
  • In March, a pedestrian was killed by a self-driving car in Arizona, and another self-driving car hit a lane divider in Northern California and burst into flames, killing the driver.

These tragic accidents have put the self-driving car industry on notice. Further, recent polls tell stories about people’s perceptions of self-driving cars.

  • Reuters reported that an AAA survey in 2018 found that 63% of American drivers said they would be afraid to ride in a self-driving car.
  • According to an Axios/Survey Monkey poll, 64% of Americans are fearful of riding in a driverless car.
  • The Bay Area Council has found that a 2018 survey of 1,000 registered voters in the nine-county San Francisco Bay Area showed 46 percent of respondents say they would ride in a self-driving car, down from 52 percent in 2017.

In short, more than half of drivers are not at ease riding in the current driverless cars. They’re not even at ease with having them on the road! Thus, the driverless-car industry must produce cars that satisfy at least these two goals:

  • The first goal is to build safe and accident-free driverless cars.
  • The second goal is to earn trust from people.

These two goals are related, but their natures are different; the first one is purely technical and the second one is more related to people’s perceptionMotionLab possesses the best solution for driverless cars to attain these problems/goals. Furthermore, what we attained was building cars whose skills surpass human drivers’ ones.


2. Vehicle Motion is the only resource to earn trust from people

A driverless car appears to have no useful resources that might earn people’s trust. However, a driverless car has an output, its motion, which is the sole human-machine interface and is the only resource the vehicle can use. If people are to trust a car, the car must create motions in a way that the motion itself earns trust

Our goal is not only that a driverless car creates motion as skillful as a driver runs his or her automobile, but the car’s movement surpasses a human driver’s.

3. The motion-creation task is neither trivial nor easy
for both drivers and driverless cars

While driving a car, a driver executes the Motion-Creation Task. We call the rest of the entire task the Decision-Making Task. By definition, these tasks are independent and complementary (however, the driver may not be consciously aware of these tasks being independent because one brain performs both duties):

  • The Decision-Making Task, which includes evaluating the present situation, understanding traffic rules, recognizing traffic signals, recognizing and understanding the surroundings and other cars, deciding how to turn at an intersection, creating a motion specification, and sending it to the Motion-Creation task.
  • The Motion-Creation Task, which receives a motion specification from the Decision-Making Task and executes it by maneuvering the steering wheel to control the path curvature κ and the gas pedal/brake to control the velocity v. 

A 16-year-old boy who obtains a learner’s permit in California has few driving skills, especially when it comes to the second task. No one provides him with mathematical equations or a user manual about how to turn the wheel in a given situation. He can only watch what experienced drivers do in various cases. He cannot seamlessly translate what he recognizes into vehicle movement; his driving is, at first, less safe and less gentle. For him, there exists a steep learning curve before he masters the skill. 

Thus, we observe and conclude that:

  • The motion-creation task is neither trivial nor easy for human drivers.
  • Therefore, the task is neither trivial nor easy for driverless cars as well.

As the perfect solution for him, MotionLab invented exact equations/algorithms, Atomic Motions (Section 7), regarding how to steer wheels in a given situation.


The following Principles I, II, and III build entirely new driverless cars with Atomic Motions: 


4. Principle I

Motion-Creation Problem, as the major and independent problem

MotionLab extracted the motion-creation problem out of the entire car-control problem, as a major, independent, fully fledged problem. We did not treat the problem as a minor one under sensor-related problems. Both problems are independent, and the motion-creation problem is never trivial or straightforward; this website describes the whole spectrum of the issue. Namely, we divided the entire control system into the following two subsystems:

  • The Decision-Making Subsystem that executes the Decision-Making Task
  • The Motion-Creation Subsystem that executes the Motion-Creation Task

The following diagram shows the correspondence between the human driver’s tasks and the divided subsystems in a driverless car:

This system division is an application of a divide-and-conquer approach to solving a complicated problem into two or more simpler subproblems. 

This Principle I is the core of all three principles. To the best of our knowledge, this is unknown to the driverless-car community. For instance, the extensive article Towards Fully Autonomous Driving: Systems and Algorithms from CMU describes a broad spectrum of theories and technologies adopted in the community but does not mention anything close to the motion-creation problem as a major and independent problem.


5. Principle II

2-DOF vehicle-motion control and the odometry capacity

We need this Principle II to embody Principle I. Principle II requires hardware and software enhancements and consists of two parts:

The first part is to control a two-dimensional rigid-body vehicle on a horizontal plane by a restricted two-degrees-of-freedom (2-DOF): (translation velocity, path curvature) = (v, κ) at every motion-control cycle. 

The second part of this principle is to evaluate and report the best estimate of the vehicle frame q = ((x, y), θ) at every motion-control cycle, where (x, y) is the vehicle’s position in the global coordinate system and θ the direction of the vehicle heading relative to the X-axis of the coordinate system. We call this vehicle capacity odometry.

The 2-DOF vehicle kinematics commonly work in the Swan robot and automobiles. The kinematic analysis and motion creation algorithms worked in the Swan also work in cars.


6. Principle III

Symmetric Geometrya deductive approach 

In classical sciences, such as physics, researchers have made advancements through both inductive and deductive approaches.  At the turn of the century, led by Albert Einstein and other physicists, modern physics has thus accomplished revolutionary progress through both approaches.

On the other hand, in the driverless-car field, inductive approaches have dominated. Grand Challenges held by DARPA in 2004, 2005, and 2007 inspired us with the current standard testing method of cruising cars in deserts, highways, and streets. However, driverless science is vast, unexplored territory, and using only the inductive approaches may reveal only limited aspects of the area.

As we worked on the Swan robot in the 1990’s, it became apparent that we needed a new mathematical vocabulary to describe and solve vehicle-motion-related geometrical problems. This realization led to the formulation of a groundbreaking mathematical theory, which we call Symmetric Geometry. This Geometry is the first attempt to contribute to the rock-solid mathematical foundation of the driverless-car industry. Without a doubt, Symmetric Geometry will hold its value over time. (Click here to read more about Symmetric Geometry)


Principles I, II, and III create vehicle motions superior to the ones that human drivers do. The following Property 1 to Property 14 demonstrate the superiorities in numerous aspects.


7. Property 1
Atomic Motions as the answer for both
human drivers and autonomous vehicles

To implement Principle I, we invented the contents of the Motion-Creation Subsystem. It is a set of motion-creation algorithms, which we call Atomic Motions. (Click here to read more about Atomic Motions)
           All of the Atomic Motions are categorized into the following five types: (1) Direction-Tracking, (2) Line-Tracking, Circle-Tracking, (3) Curvature-Defined, (5) Park Atomic Motions.

The interface from the Decision-Making Subsystem to the Motion-Creation Subsystem is the set of geometric objects. Each Atomic Motion, given a geometrical object as input, creates vehicle motion (vκ) = (translation velocity, path curvature), as the output.

For instance, while a driverless car drives in a lane on a highway, the Decision-Making Subsystem senses the near-front part of the lane and decides whether the part is linear or curved:

  • If the DMS decides that the part is linear, the Subsystem sends the decision to the Motion-Creation Subsystem (MCS) with geometric elements of an oriented line L = ((xy), θ) and a smoothness σThen, the MCS executes a Line-Tracking Atomic Motions.
  • If the DMS decides that the part is curved, the Subsystem sends the decision to the MCS with geometric elements of an oriented circle C = ((xy), r) and a smoothness σThen, the MCS executes a Circle-Tracking Atomic Motions.

To date, no one could give the 16-years-old boy
unambiguous instructions on how to steer the car.
However, we now have an ideal answ
er for him; just give him Atomic Motions.

The Atomic Motions have already run the Swan robot and Science Robot, as shown in their numerous video demonstrations. They already have proven their usefulness and effectiveness.


8.  Property 2
Atomic Motions are object-driven with high resolution

An Atomic Motion, in general, moves to maintain a relationship with the geometry of an object. Namely, given a geometrical element, an Atomic Motion creates a movement that faithfully reflects the geometrical element. For instance, this sample motion “Hug wall” demonstrates that the vehicle produces a motion that faithfully reflects the geometrical disposition of the left walls.

The Swan robot dynamically extracts a linear feature of the left-side wall using sonars and a least-squares-fit algorithm and tracks the linear feature with Line-Tracking Atomic Motions.

Here is a sonar-based, custom-made autonomous vehicle Swan.

The Swan robot hugs a wall
with a left-handed rule.

Notice the vehicle’s object-driven
and high-resolution motion.

9.  Property 3

Atomic Motions are
vehicle-hardware independent and sensor-input independent

The output of Atomic Motions is (v, κ), which is passed to vehicle hardware to command its motion. Because these two real numbers control any hardware, this output of Atomic Motions is vehicle-hardware independent.
          On the other hand, the input to Atomic Motions is a geometric entity. Whatever sensor produces the input, the input interface is a geometrical entity and is sensor independent.
          Therefore, the Atomic Motion algorithms are also independent of hardware vehicles and sensors.

Let us give you a motion example, where identical Atomic Motions run in different hardware vehicles. A Circle-Train motion controls three different hardware vehicles:

1. Science Robot executes its instance at [3.1] Circle Medley on Circle-Tracking Atomic Motion page.
2. The Swan robot runs another instance of the same motion at [3.2] on the same page.
3. The Yamabico-11 robot also plays the same motion on About Us page.
These identical motion executions prove the hardware independence of Atomic Motions.


10. Property 4
Motion Accuracy that surpasses human drivers’ 

Atomic-Motion algorithms consist of geometrical equations. Therefore, they create extremely accurate motions, which surpass the ones created by human drivers because humans’ driving skills cannot match Atomic Motions’ geometrical equations

This motion Parallel Parking. is an example of accurate motions created by Park Atomic Motions:

Swan’s sonars report
the geometrical situations to the car,
which produces
backward and forward Park Atomic Motions
to create this “Parallel Parking” motion.

Notice that this motion is accurately responding
to the geometrical situations.

This exact movement is reproducible.

11. Property 5
AM algorithms respond quickly,
much faster than human drivers do;
this property reduces driverless cars’ 

Atomic Motions execute their algorithms quickly because they consist only of geometrical equations. The sensor-motion sampling-control loop contains no AI-related computations; we should not include any of those time-consuming processing because we want the theoretically fastest response. The quick response in the Swan robot video below demonstrates quick steering surpassing human drivers’ skills.

Quick responses drastically reduce accidents caused by driverless cars. If a vehicle detects any danger, the vehicle can immediately avoid obstacles or stop/reduce the velocity. Atomic Motions can handle any situation with the minimum time lags.

Quick motions also make motion resolution higher. Because of quick responses, a vehicle finely responds to the object’s complex envelopes precisely to create high-resolution moves, as shown below. Such high-resolution motions let people understand the relationship between the geometrical object layout and the corresponding movement. Then, the car gives assurance, satisfaction, and trustworthiness about its algorithms for people.

The Swan robot hugs a “can formation.”
Line-Tracking Atomic Motions execute these successive obstacle-avoiding motions. Dynamic changes of obstacles are translated into dynamic changes of target lines.

Therefore, Swan responds to this complex object with a super high-resolution motion.


12.  Property 6

Optimum/gentle responsiveness of feedback-control motions
makes people at ease

These are three tracking-type Atomic Motions with negative-feedback-control rules:

  1. Direction-Tracking Atomic Motions
  2. Line-Tracking Atomic Motions
  3. Circle-Tracking Atomic Motions

Each of these algorithms works under a feedback control rule in a critical-damping condition, demonstrating exponential convergence to a given target, which is direction, line, or circle.

We can control the spatial converging speed of these Atomic Motions. They have another input parameter σ, which we call smoothness. This parameter controls how sharper or less sharp the motion geometrically converges to the target:

  • A smaller smoothness σ makes the convergence sharper and the transition part smaller.
  • A larger smoothness σ makes the convergence less sharp and the transition part larger.

If the smoothness σ is too small, the car’s too-quick response may frighten people; if the smoothness σ is too large, the car’s too-slow response may irritate people. If an optimal smoothness is selected, the motion looks gentle but not too slow; viewers would feel at ease.
          Selecting an optimal smoothness σ in each given situation is essential to earn trust from people. A car with optimal feedback-control responsiveness earns people’s trust; they regard the car as gentle, friendly and responsive enough.
          For detailed discussions, Click here to read more about Responsiveness of Tracking-Type Atomic Motions

The following motion Squash game shows an example of optimized-response Atomic Motions. Each optimized turn makes people comfortable, neither too quick nor too slow.

The Swan robot executes a similar motion to the above [1.2] Squash motion, while the founder’s grandsons use “rackets” to determine reflection angles.

The Swan finds the racket’s normal direction in real time using the sonars and the least-fit algorithm.

13. Property 7

Hypotheses K1 and K2
Completeness of the Atomic-Motions in its entirety

[13.1] Hypothesis K1: Necessity and Sufficiency 

MotionLab has experienced that Atomic Motions create any vehicle motion we need; the set of Atomic Motions are sufficient to create any vehicle motion for any purpose in any situation. Further, the set of all the Atomic-Motion types is necessary to create motions for various purposes in various situations. Thus, we claim this Hypothesis K1:

Hypothesis K1:
The set of Atomic Motions is necessary and sufficient to create any vehicle motion.

[13.2] Hypothesis K2: Perfection

Moreover, we found that there is no situation where two Atomic Motions respond with the same perfection. That is, there exists only one Atomic Motion that perfectly responds to a given situation. This fact leads us to the second Hypothesis K2:

Hypothesis K2:
One and only one Atomic Motion exists that perfectly responds to a given situation.

Hypothesis K2 is a stronger one than the first, K1. To date, we have seen no counterexamples of K1 or K2. Hypothesis K2 also means that each type of Atomic Motions has a unique role in motion creation; two Atomic Motion types are not interchangeable. These Hypotheses prove that the set of Atomic Motion is not an arbitrarily chosen set of minimal motions, but it is a perfectly designed complete set of minimal motions. (Click here to read more about Characterizing Atomic Motions)


14. Property 8

People understand vehicle motions’ intention and algorithms;
then, they trust the vehicle

Human drivers do not usually frighten people because people, in general, trust drivers’ skills and their goodwill toward others. On the other hand, people know nothing about driverless cars’ algorithms. People regard a driverless car as a black box; they never trust a black box, especially in light of widely publicized accidents.

However, when the car selects the sole perfect Atomic Motion in a situation, as Hypothesis K2 claims, we experience the following:

  1. The motion looks natural in the given situation. 
  2. People precisely recognize the direct relationship between the geometry of the surroundings and the created motion.
  3. People immediately and intuitively understand the motion’s intention.
  4. People anticipate what the car will do next.
  5. Therefore, people no longer regard the car’s algorithm as a mysterious black box, but as a white box
  6. As a result, the driverless car earns people’s trust.

The following video demonstrates that how Andrew Harding trusts the Swan robot:

The Swan robot faithfully responds
to the object’s envelope
with an optimal smoothness σ here.
Andrew Harding intuitively understands
Swan’s intention and anticipates
what it will do next.

The Swan’s algorithm
is not a black box anymore;
Andrew comes to trust the Swan vehicle.
The Swan is now his friend!

15. Property 9

We standardized Atomic Motions

All of the MotionLab’s Atomic Motions use the simplest, the most rigorous, and the quickest algorithms, based on Symmetric Geometry. They were refined, optimized, and standardized during decades of testing and application practices in the Swan vehicle and Science Robot.
          Therefore, a firm can use and maintain only one copy of the standardized set of Atomic Motions for all of its driverless cars built by the firm. 

A novice driver requires hours of behind-wheel training sessions to master motion-creation skills. However, our Standard Motion-Creation Subsystem is already a final product, which does not need any learning or training. The subsystem is just a finite static set of object-oriented classes.


16. Property 10

Atomic Motions’ Programmability
 systematically creates a variety of motions with immense complexity

Atomic Motions create car-motions with any complexity the same way Lego blocks construct a vast castle. The Swan robot and the Science Robot app present all sorts of motions powered by the programmability of Atomic Motions.

Atomic Motions are installed in the Swan vehicle using the Java® language and are installed in the Science Robot app using the Xcode®; thus, we installed Atomic Motions in two distinct object-oriented programming languages.

Here are examples of motions with a higher complexity. Human drivers can never create these motions of a higher complexity:

Human drivers can do none of these.


17. Property 11
Reducing hardware costs and computation costs

The Decision-Making Subsystem (DMS) may adopt AI algorithms, such as deep learning. However, the Motion-Creation Subsystem (MCS) is already standardized by geometrical computations, and any AI-related computation and hardware are not necessary for it; they are even harmful.

The removal of the use of AI in the Motion-Creation Subsystem significantly reduces the overall computational costs and hardware costs of a driverless car.


18. Property 12
The motion-creation subsystem influences and better organizes
the entire control system architecture

Each Atomic Motion requires only a couple of geometric data input delivered from the Decision-Making Subsystem (DMS). Sensors in the Subsystem may obtain millions of bytes in a millisecond, and the Subsystem may hold big data, but the geometrical data needed for the Motion-Creation Subsystem (MCS) is only a couple of numbers per motion-control sampling cycle. This undemanding output requirement makes the DMS simpler and better organized.

Therefore, even though the MCS is a smaller software system compared with the DCS, the standard MCS works as a stable, fixed-point anchor in the entire driverless-car algorithms; MCS gives significant influences in the functionality and system architecture of DCS.

19. Property 13
Motion Abstractions

MotionLab is interested in discovering vehicle-motion abstractions rather than individual car motions in a particular car. Mathematics is powerful in building an abstract model of things and Symmetric Geometry is no exception. It gave birth to Atomic Motions, the first example of motion abstraction.

Further, the new geometry produced five more instances of Motion Abstractions. The following three abstractions convey more significant concepts:

20. Property 14

Mathematics Lasts

Because the Atomic Motions are based on the rock-solid Symmetric Geometry, we will not need to modify them for decades, possibly forever. Coding done today might be unambiguously understood by colleagues 20 years later.


21. Conclusion

  • Principle I: Extracting Motion-Creation Problem as a major, independent, fully fledged problem
  • Principle II: 2-DOF vehicle-motion control and the odometry capacity
  • Principle III: Symmetric Geometry

These Principles I, II, and III create motions, which have Properties 1 to 14 (or Advantages, or Benefits, or Merits). We have seen any of these properties neither in the present driverless cars nor in human-driven vehicles.

Furthermore, every property demonstrates that the motions created by the Principles surpass human-created motions. Every property benefits us, but specifically, the following two properties solve the goals/problems mentioned in Section 1:

  • Property 5 creates quick and high-resolution motions that surpass human drivers and reduces accidents.
  • Property 8 makes viewers understand the motion’s intention and algorithms so that the vehicle does not frighten viewers anymore.

The following properties also help people trust our driverless cars:

  • Property 2 presents an object-driven property with high resolution.
  • Property 4 creates accurate motions that surpass human drivers’.
  • Property 10 systematically creates immensely complex motions.

MotionLab not only talk about these properties but we already have built, seen, and confirmed these properties in the Swan robot and Science Robot. The dimension difference between an automobile and the Swan robot does not matter because they function basically in the same kinematics theory.

We attained the 14 beneficial properties, but it was NOT a coincidence. That proves the competence of our rock-solid foundation, Symmetric GeometryWe may find even more useful properties in the future.

MotionLab concluded that an element the present driverless-car industry lacks might not be only in sensors, AI, or big data, but also crucial motion-creation algorithms.

We also have a conjecture:
MotionLab’s technology is not only one of the useful technologies but is the best one we can possess; it will never be taken over by a “better” technology in the future.