Our novel algorithms in a driverless car
create the best motions mathematically.

As a result, its motions surpass human driver’s ones.

The car reduces accidents and earns trust from people.
It becomes a friend of 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: December 2018
Version 2: October 2018
Version 1: August 2018


How do people perceive the latest driverless cars?

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

  • In January of 2018, a self-driving car slammed into a fire truck on a freeway in Los Angeles.
  • In March of 2018, 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 only 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 perceptions. To date, established theories and technologies including the ones presented in an article Towards Fully Autonomous Driving: Systems and Algorithms from CMU have led the direction of driverless-car research and development. However, these reports imply that the driverless-car industry might require novel ideas and principles to reduce accidents and change people’s perception.


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

The motion-creation task is neither trivial nor easy
for either drivers or 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 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 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 first step, we took the following Principles I, II, and III as ground rules:


Principle I

Motion creation as a major and independent problem.

MotionLab extracted the motion-creation problem out of the entire car-control problem, as a significant, independent, and fully fledged problems, not as a minor one under sensor-related problems. This problem is neither trivial nor straightforward because this entire website describes the whole spectrum of the issues regarding this problem.

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. 


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.

This Principle II has been working in the Swan robot and Science Robot; the same Principle also can work in any automobile or driverless car. The same kinematic analysis and motion creation algorithms work in all of these vehicles notwithstanding the size difference.


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 a 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 1990s, 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)

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 MotionsAll of the Atomic Motions are categorized into the following five types: 

  1. Direction-Tracking type
  2. Line-Tracking type
  3. Circle-Tracking type
  4. Curvature-Defined type
  5. Park type

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 recognizes that the part is linear, the Subsystem orders the Motion-Creation Subsystem (MCS) to use a Line-Tracking AM with an oriented line L = ((xy), θ) and a smoothness σ as input parameters.
  • If the DMS recognizes that the part is curved, the Subsystem orders the MCS to use a Circle-Tracking AM with an oriented circle C = ((xy), r) and a smoothness σ as input parameters.

To date, no one could give the 16-year-old boy unambiguous instructions on how to steer the car. However, we now have a perfect answer for him; just give him AMs.


AM-created motions surpass those created by human drivers.

Advantages, 1 to 6, are beneficial for driverless cars.
Benefits, 1 to 7, are beneficial for driverless-car makers.

MotionLab’s goal is NOT to build a car that is as skillful as a human driver; an imperfect human driver cannot be our role model. Instead, our goal is to build a car that creates perfect motions in every aspect, using the Principles and computers, which surpass human brains.

AMs create motions, which possess outstanding features and properties as follows:

  • AM-created motions demonstrate Advantages 1 to 6, which benefits driverless cars. These advantages tell us that AM-created motions surpass those produced by human drivers.
  • AM-created motions demonstrate Benefits 1 to 7, which benefits driverless carmakers. 


Advantage 1

AMs interact with geometric objects 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 accurately 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 AMs.

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.

Benefit 1

AMs 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.


Advantage 2

AMs’ extreme accuracy surpasses that of human drivers’.

Atomic-Motion algorithms consist of geometrical equations. Therefore, they create extremely accurate motions, which surpass the ones created by human drivers who have no 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.

Advantage 3

AMs respond more quickly than human drivers;
this swiftness reduces accidents and improves motion resolution

AMs execute their algorithms quickly because they consist only of geometrical equations. The sensor-motion sampling-control loop contains no AI-related computations and we obtain theoretically fastest response. The quick response in the Swan robot video below demonstrates quick steering surpassing that of human drivers’.

Quick responses significantly reduce accidents caused by driverless cars. If a vehicle detects any danger, an AM immediately responds to avoid it by steering or by changing the velocity. AMs takes minimum time lags in any situation.

Quick motions also make motion resolution higher. Because of quick responses, a vehicle responds to the object’s complex envelopes precisely to create high-resolution moves, as shown below. Such high-resolution motions let people understand the relationships between the geometrical layout and the corresponding movement. In this way, the car gives assurance and satisfaction and creates trust in 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.


Advantage 4

Optimal responsiveness of feedback-control type AMs
makes people at ease.

Atomic Motions have three tracking types 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 sharp the motion spatially converges.

  • 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 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.
          For detailed discussions, Click here to read more about Responsiveness of Tracking-Type Atomic Motions

The following video demonstration Squash game shows an example of optimally responding Atomic Motions. Each optimal turn, neither too quick nor too slow, makes people comfortable.

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.

Benefit 2

Hypothesis K1: Necessary and Sufficient
Hypothesis K2: Uniqueness

The following two hypotheses prove that the set of five Atomic Motion types is not an arbitrarily chosen set of minimal motions; it is a carefully designed complete set of minimal motions. (Click here to read more about Characterizing Atomic Motions)

[1] Hypothesis K1: Necessity and Sufficiency 

MotionLab has observed all of the five AM types are necessary to create various motions in various situations. We also have observed that the set of five AM types is sufficient to create any vehicle motion for any purpose in any situation. Thus, we claim this Hypothesis K1:

Hypothesis K1:
The set of all of the five AM types is necessary and sufficient to create any vehicle motion.

[II] Hypothesis K2: Uniqueness

Moreover, we found that there is no situation where two AM types respond with the same perfection. That is, there exists only one AM type that perfectly responds to a given situation. Therefore, we have the second Hypothesis K2:

Hypothesis K2:
One and only one AM type exists that perfectly responds in a given situation.

Hypothesis K2 is stronger than the first, K1. To date, we have seen no counterexamples of K1 or K2. Hypothesis K2 also means that each AM type has a unique role in the motion-creation task; two AM types are never interchangeable. 


Advantage 5

People understand a car’s intentions and algorithms.
Therefore, they trust the car.
Now, the car and people are friends.

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 perfectly fit AM in a given situation as Hypothesis K2 claims, we experience the following:

  1. The motion looks natural in the 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. They take the car as their friend.

The following video demonstrates how Andrew Harding trusts the Swan. The Swan robot establishes a good rapport with the founder’s grandsons in numerous test runs:

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!

Benefit 3

We standardized Atomic Motions.

All of the Atomic Motions use the simplest, the most rigorous, and the quickest algorithms based on Symmetric Geometry. They have been refined, optimized, and standardized through decades of testing/running with the Swan vehicle and Science Robot.
          Therefore, a firm can use and maintain only one copy of the standardized Atomic-Motion version for all of its driverless cars built in the firm.

A novice driver requires hours of behind-the-wheel training 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 replaces many hours of behind-the-wheel classes.


Benefit 4

AMs possess full-fledged programmability.

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 produce none of these.


Benefit 5

Reducing hardware costs.

The Decision-Making Subsystem (DMS) may adopt AI algorithms, such as deep learning. However, we already have standardized the Motion-Creation Subsystem (MCS), and any AI-related hardware and computation time are not necessary, can even be harmful.

The removal of AI in the MCS significantly reduces the overall hardware cost of a driverless car.


Benefit 6

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

Each Atomic Motion requires only limited 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 set of driverless-car algorithms; the MCS lends significant influence in the functionality and system architecture of the DCS.

Advantage 6

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:

Benefit 7

Mathematics lasts, forever.

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 hence.



MotionLab possesses the following unique assets:

  • Principle I: Motion-Creation as the major, independent, and full-fledged problem to be solved
  • Principle II: 2-DOF vehicle-motion control and the odometry capacity
  • Principle III: Symmetric Geometry

These Principles birth Atomic Motion algorithms (AMs), which create motions exhibiting the following Advantages:

Advantage 1: object driven, high resolution
Advantage 2: highest accuracy
Advantage 3: quickest motion
Advantage 4: optimal responsiveness
Advantage 5: people discern a vehicle’s algorithms;
Advantage 6: motion abstraction

Each of these Advantages demonstrates that the AM-created motions surpass human-driver-created motions. No human-created motions do not show these advantages. Thus, AMs’ driving skills are much better than humans’.

AM-created motions also present the following Benefits that are valuable to driverless-car makers:

Benefit 1: hardware and sensor independence
Benefit 2: hypotheses K1 and K2
Benefit 3: standardization of AMs
Benefit 4: programmability
Benefit 5: hardware cost reduction
Benefit 6: the Motion-Creation Subsystem improves the Decision-Making Subsystem
Benefit 7: mathematics lasts

MotionLab already has installed AMs in the Swan robot and Science Robot. These robots already have demonstrated driving skills that surpass humans’. The dimension difference between an automobile and the Swan robot does not matter because they function primarily under the same kinematics theory.

MotionLab concludes that, as well as sensors, AI, and big data, our motion algorithm is one of the essential elements for driverless cars.