Our three principles produce new driverless cars
that possess outstanding driving skills.
these cars are significantly safer and earn people’s trust.
Skills of these cars surpass those of
human drivers or current driverless cars.
MotionLab, LLC, CEO: Yutaka Kanayama, Ph.D.
Science Robot, Math Mind: Mac motion simulator at App Store,
Version 3: January 2019
Version 2: October 2018
Version 1: August 2018
How do people perceive current driverless cars?
Let us review news on current driverless cars:
- 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 and 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.
- Since Waymo started testing its driverless vans in Chandler, Arizona in 2017, people have attacked them nearly two dozen times, the New York Times reported on 12/31/2018. People slashed tires and pelted rocks, a Jeep had tried to run vans off the road, a woman screamed at a van, telling it to get out of her suburban neighborhood, and a man waved a .22‐caliber revolver at a vehicle and the backup driver at the wheel. Local residents are not even at ease with having them on the road.
Now, in current driverless cars, we identify at least two problems to be solved:
The driverless-car industry must solve at least two problems:
P1: How to build safer cars?
P2: How to build cars that earn people’s trust?
We have observed that the current driverless-car research and development principally take inductive approaches. However, driverless-car science extends over remarkably deep and wide areas, and only inductive approaches have reached only limited areas. MotionLab took a deductive approach with three principles as described below, and have solved the problems completely. No other ideas will solve these essential problems.
As the first step, let us first see how a human driver executes tasks. While driving a car, a driver performs a Motion-Creation Task and a Decision-Making Task, which we define as follows:
- The Motion-Creation Task receives a motion specification from the Decision-Making Task and executes it by maneuvering the steering wheel to control path curvature κ and the gas pedal and brake to control velocity v.
- The Decision-Making Task is the rest of the entire task after subtracting the Motion-Creation Task, including 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.
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.
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 an unambiguous instruction manual about how to turn the wheel in a given situation. He can only watch what experienced drivers do in various cases. His driving is, at first, less safe and less gentle. There exists a steep learning curve before he masters the skills. Thus, we observe and conclude that:
The Motion-Creation Task is neither trivial nor easy for drivers. Therefore, …..
the Motion-Creation Task is neither trivial nor easy for driverless cars as well.
Thus, MotionLab has taken the motion-creation task as a major and independent one, not as a minor one under sensor-related tasks. This entire website is devoted to this task.
Motion creation as a major and independent task
— A divide-and-conquer approach —
Principle I stands for dividing the entire control system of a driverless car into the following two subsystems. This method is a divide-and-conquer approach, which is commonly adopted when the problem to be solved is hard; dividing the original problem into two or more simpler subproblems:
- The Decision-Making Subsystem that executes the Decision-Making Task
- The Motion-Creation Subsystem that executes the Motion-Creation Task
Motion plays a significant role in a driverless car because the car’s only output is motion, and motion is the only interface between human and the car. The following diagram shows the correspondence between the human driver’s tasks and the divided subsystems in a driverless car.
2-DOF vehicle-motion control and the odometry capacity
This Principle II consists of two parts:
- We control vehicle motion using a two-degrees-of-freedom (2-DOF): (translation velocity, path curvature) = (v, κ) at every motion-control cycle.
- The second part is for the vehicle to evaluate and report the best estimate of the present 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 relative to the X-axis of the coordinate system. We call this vehicle capacity odometry.
This Principle II requires hardware installation.
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 novel Geometry produced all the algorithms invented at MotionLab including Atomic Motions. (Click here to read more about Symmetric Geometry)
These principles produced
which create vehicle motions.
MotionLab obtained a mathematical model of the human drivers’ motion-creation task. We first disassembled motions into minimal ones, which cannot be broken down into smaller movements. Second, we focused on what information creates movements:
For instance, a car (1) follows a lane, (2) follows a curb or a side wall, (3) follows another car, (4) avoids a car, pedestrian, or an object, (5) parks along a curb or in a parking spot using forward or backward movement, or (6) stops at a traffic signal or a toll gate. In these situations, the car keeps a geometrical relationship with objects nearby or afar.
Atomic Motions (AMs) are minimal motion-creation algorithms, which are controlled by geometrical entities. We categorized all of the AMs into the following five types (Click here to read more about Atomic Motions):
- Direction-Tracking type AM
- Line-Tracking type AM
- Circle-Tracking type AM
- Curvature-Defined type AM
- Park type AM
The interface between the Decision-Making Subsystem and the Motion-Creation Subsystem is the set of geometric objects. Each Atomic Motion, given a geometrical object as input, creates vehicle motion (v, κ) = (translation velocity, path curvature), as the output at every motion-control sampling time. These AM algorithms are the contents of the Motion-Creation Subsystem.
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 = ((x, y), θ) as an input parameter.
- 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 = ((x, y), r) as an input parameter.
Then, the car tracks the lane accurately as expected.
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 Atomic Motions (AMs). MotionLab has installed these AMs in the following two vehicles:
- The Swan robot
- The Science Robot simulator: Science Robot’s mathematically created motions are the reference of the AM algorithms.
- Because AMs are hardware/sensor independent, we can install them in any driverless car.
The Atomic Motions demonstrate
outstanding driving skills: Skills 1 to 5
Each skill stands for a single feature of driving skills.
AMs create high-resolution motions,
keeping a geometrical relationship
between objects and the car.
While cruising in streets and highways, The Decision-Making Subsystem (DMS) in a driverless car recognizes lanes, curbs, pedestrian crossings, other vehicles, bikes, people, and parking spaces.
Driverless cars will also share common spaces with people; in the parking lot of a grocery store, in garages, at gas stations, and so forth. In these situations, DMS recognizes nearby objects and people. The cars may have to avoid/follow obstacles or a person.
In either case, the car creates a motion so that the relationship between the car and the object are faithfully kept.
For instance, this sample motion “Hug wall” demonstrates a Swan motion that faithfully (accurately) maintains the distance between the car and the left walls constant. The Swan robot dynamically extracts linear features of left walls using sonars and a least-squares-fit algorithm and tracks the lines 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.
AMs create the most accurate motions mathematically.
An Atomic-Motion algorithm consists of geometrical equations, that keep a geometrical relationship between objects and the car. Therefore, the created motions not only surpass ones created by human drivers (who have no equations) but are the most accurate mathematically.
This motion Parallel Parking is an example of extremely accurate motions using Park Atomic Motions:
Swan’s sonars report
the geometrical situations to the car,
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.
AMs make the quickest responses theoretically.
They are much quicker than human drivers.
reduce accidents and improve 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, theoretically, we obtain the fastest response.
The quick response in a Swan robot video below demonstrates rapid and exact steering surpassing that of human drivers.
The Swan robot hugs
a “sparse can formation.”
Line-Tracking Atomic Motions execute these successive obstacle-avoiding motions. Dynamic changes of the relationships with obstacles faithfully reflect quick steering.
Swan responds to this complex object
with a super high-resolution motion.
If a car selects a correct AM type in a given situation, …
People see the motion as natural in the situation.
People understand a car’s intention and
anticipate what the car will do next.
People no longer regard the car’s algorithm as a black box.
As a result, the car earns people’s trust,
they regard the car as their “friend.”
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 driverless car selects the sole perfectly fit AM in a given situation as Hypothesis K2 claims, the following takes place:
- People see the motion as natural in the situation.
- People precisely recognize the direct relationship between the geometry of the surroundings and the created motion.
- People immediately and intuitively understand the motion’s intention.
- People anticipate what the car will do next.
- Therefore, people no longer regard the car’s algorithm as a mysterious black box, but as a transparent box.
- As a result, the driverless car earns people’s trust. They regard 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 these video shootings:
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!
Optimally responsive turnings in tracking-type AMs
earn people’s trust.
We have three tracking-type Atomic Motions:
- Direction-Tracking-type AM
- Line-Tracking-type AM
- Circle-Tracking-type AM
Each of these tracking-type AMs turns (steers) a car so that the forward motion exponentially converges into the given target (a direction, a line, or a circle) using a negative-feedback control rule in a critical-damping condition. We can control the turning speed (responsiveness) of these AMs.
The following videos show two versions of a Polygon Medley motion, which uses Line-Tracking AMs:
- This video demonstrates a version with sharper (smaller) turns.
- This video demonstrates a version with less sharp (larger) turns.
We can directly control the turning-convergence speed (responsiveness) using an input parameter σ, which we call smoothness.
- A smaller smoothness σ makes the convergence sharper and the transition part smaller. If the smoothness σ is too small, the car’s too-quick a turn may frighten people.
- A larger smoothness σ makes the convergence less sharp and the transition part larger. If the smoothness σ is too large, the car’s too-slow a turn may not finish the motion in time.
In the above pairs of Polygon Medleys, we used smaller σ’s in the first Medley, and larger σ’s in the second one.
If we select optimal responsiveness in a situation, the motion looks gentle but not too slow and earns people’s trust. They feel at ease and regard the car as friendly. For detailed discussions, Click here to read more about Responsiveness of Tracking-Type Atomic Motions.
The following video Squash game shows an example of optimally responding AMs. People feel comfortable when each turn is neither too quick nor too slow.
The Swan robot bounces back, making the reflexive angle the same as the incident angle at the racket. The founder’s grandsons use “rackets” to determine reflection angles.
The Swan finds the racket’s normal direction in real time using sonars and a least-fit algorithm.
Skills 1 to 5 have solved the problems defined earlier:
P1: “How to build safer cars?”
P2: “How to build cars that earn people’s trust?”
Skill 1 (high resolution), Skill 2 (accuracy), and Skill 3 (quickness) of a driverless car minimize the risk of causing accidents. In this way, a car with Atomic Motions is a safer car, and, therefore, Problem P1 as defined earlier has been solved.
All of the five skills contribute to earning people’s trust, but the most critical is Skill 4, as described above. Thus, a driverless car with Atomic Motions has solved Problem P2 defined earlier.
Skills of AMs surpass those of human drivers.
Skills of AMs surpass those of current driverless cars.
We have examined the above five Skills and have come to these conclusions:
- Skill n of AMs surpasses the corresponding skill of human drivers for each 1 ≤ n ≤ 5.
- Skill n of AMs surpasses the corresponding skill of current driverless cars for each 1 ≤ n ≤ 5.
Therefore, we further conclude as a whole:
- The Skills of AMs surpass those of human drivers.
- The Skills of AMs surpass those of current driverless cars.
Atomic Motions provide significant benefits for carmakers:
Advantages 1 to 10.
Besides the five Skills, Atomic Motions provide other kinds of benefits, Advantages 1 to 10. They are, in various ways, beneficial to carmakers rather than car users.
Atomic-Motion algorithms have been installed
in object-oriented languages.
Thus, AMs possess full programmability.
We installed Atomic Motions in two distinct object-oriented programming languages: in Java® in the Swan vehicle and Xcode® in the Science Robot app. The contents of the coding are mostly geometrical data structures and geometrical equations, based on Symmetric Geometry. Therefore, the code is easily ported into any driverless car algorithms, when the time comes.
Therefore, AMs possess powerful programmability, which can create car motions with any complexity the same way Lego blocks construct a vast castle. Here are sample motions with a higher complexity:
- On Circle-Tracking Atomic Motion page: [3.4] Necklace (2)
- On Park Atomic Motion page: [5.5] Sea Urchin
- On Spatial Understanding page:  Vacuuming-area teaching and vacuuming-execution
- On Spatial Understanding page:  Map-Based Navigation
Atomic Motions are optimized and standardized.
The set of Atomic Motions is
a standard motion-creation language
that works for thinking,
coding, and documentation,
in algorithm-development teams.
MotionLab has been using AMs to control the Swan robot since 1999 and the Science Robot since 2016. During the period, we simplified, refined, and optimized their algorithms. The set of AMs is now standardized. Therefore, a firm can use and maintain only one copy of the standardized AMs 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 more training.
The set of Atomic Motions is a standard motion-creation language. This language works for thinking, communication, algorithm planning/design, programming, and maintenance in vehicle-motion development teams. Coding done today will be unambiguously understood by colleagues 20 years hence.
AMs are vehicle-hardware independent
and sensor-input independent.
- The input to Atomic Motions are geometric entities; whatever sensor produces the input, the input interface of a geometrical entity is sensor independent.
- The output of AMs, (v, κ), is not related to any individual vehicle hardware. Namely, the output of AMs is hardware independent.
- Therefore, each AM is hardware/sensor independent including its function.
For instance, a motion, Circle Train, runs three different hardware vehicles (if necessary, we can install this Circle Train in any driverless car):
- Science Robot executes its instance at [3.1] Circle Medley on Circle-Tracking Atomic Motion page.
- The Swan robot runs another instance of the same motion at [3.2] on the same page.
- The Yamabico-11 robot also plays the same motion on the About Us page.
AMs can be installed into any driverless car. Therefore, all of the motions installed in the Swan robot or Science Robot are easily ported into any driverless car with various dimensions with various hardware using the same kinematics. Dynamic control rules for individual hardware vehicles can be added as necessary.
Hypothesis K1: The five AM types are
necessary and sufficient
to create any vehicle motion.
Hypothesis K2: One and only one AM type exists
that correctly responds in a given situation.
These two hypotheses imply that the set of five Atomic Motion types is not arbitrarily chosen but is carefully designed as a complete set of minimal motions. (Click here to read more about Characterizing Atomic Motions)
[4.1] 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 found that the set of five AM types is sufficient to create any vehicle motion for any purpose in any situation. Thus, we claim the above Hypothesis K1.
[4.2] 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 correctly responds to a given situation. Thus, we claim the second Hypothesis K2.
Hypothesis K2 is stronger than the first, K1. Hypothesis K2 also means that each AM type has a unique role in the motion-creation task; two AM types are never interchangeable.
To date, we have seen no counterexamples of K1 or K2.
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) without AI capacity; any AI-related hardware for the MCS is not necessary, can even be harmful because of extra processing time.
The removal of AI in the MCS significantly reduces the overall hardware cost of a driverless car.
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 AMs in 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 standardized MCS works as a stable, fixed-point anchor in the entire set of driverless-car algorithms; the MCS affects significant influences in the functionality and system architecture of the DCS.
MotionLab is interested in discovering vehicle-motion abstractions rather than discussing individual car motions in a particular car. A foremost mission of Mathematics is building an abstract model of things, and Symmetric Geometry is no exception. It has given birth to Atomic Motions, the first example of motion abstraction in driverless-car science. Furthermore, the new geometry produced five more instances of Motion Abstractions including the following three:
More motions for the future
MotionLab has invented other motions, which may be useful for future driverless cars.
[8.1] Human-taught motions and their reproduction
These motion-creation methods work exceptionally well for leading a car in a complicated maze-like space or parking a car in a small space. A person feels comfortable being a master in these human-car interactions; the car never frightens people.
Further, these human-created motions can be reproduced.
These four motions are presented on Motion Reproduction page as well. We can implement all of the above motions simply using AMs.
[8.2] Omega-mode motions
We can control driverless electric cars or the Swan robot in the omega-mode, which is a motion-control method using two variables, (v, ω) = (translation velocity, rotation velocity). An internal-combustion-engine driverless car is not able to work in this motion mode.
Examples of omega-mode Swan motions are as follows:
- Kori’s hand controls the robot; the motion is reproduced backward.
- Andrew’s foot and hand pull the robot out from a garage and pushes back into it.
- In this maze-solving demonstration, the Swan uses a special U-turn at dead ends in the omega mode.
A significant advantage of omega-mode motions is that they run in a tight space with high resolution at a small or zero translational velocity. Automobiles cannot do it.
Mathematics lasts forever.
Based on the rock-solid Symmetric Geometry, Atomic Motions may not need to be modified for decades, possibly forever.
Full user satisfaction
Users of AM-equipped driverless cars will feel fully satisfied with their cars, knowing that they are significantly safe and trustworthy.
The set of Principles I, II, III is
the essential, productive, and ever-lasting foundation
in driverless-car science.
The foundation has produced the following revolutionary advancement in driverless-car science:
- Atomic Motions with five types
- Solved Problems P1 and P2
- Hypotheses K1 and K2
- At least five motion abstractions
- At least two kinds of future motions
The foundation has produced the outstanding advancements in driverless-car science as shown above. They were built not coincidentally, but as a natural, logical outcome of the foundation, forming a robust ecosystem. The foundation will also produce even more valuable fruits in the future. This development has proven the authenticity and productivity of the foundation.
The founder’s experiences of more than 40 thousand hours working with Atomic Motions and Symmetric Geometry in our robots have led us to the conclusion that only this foundation can build a driverless car, which people accept as a trustworthy friend, not as an untouchable cruising machine.
The founder personally holds all of the intellectual properties described in this letter and our website. MotionLab is an independent, self-funded firm and has the liberty to offer our technology to any carmakers.
Atomic Motions run the Swan robot and the Science robot with a vehicle kinematics model, which works for any driverless cars uniformly. However, when it comes to installing AMs in a driverless car in the real world, we may have to deal with its dynamic model for each hardware vehicle.