Stanford OnlineStanford AA274A Principles of Robotic Autonomy | Autumn 2019 | Overview, Mobile Robot Kinematics
Stanford Online
Course aims 0:05
AA274A teaches you how to give a robot the ability to reason about its surroundings and act on that reasoning. The course is cross listed with electrical engineering and computer science. It aims to give you the math behind autonomy, the coding skills to implement it, and practice deploying those algorithms on a real robot using Robot Operating System, or ROS, a robot-focused operating system.
See think act 2:00
Robot autonomy is set against older automation, where every situation is known in advance and motions are preprogrammed. An autonomous robot must cope with surprises, like a self-driving car reacting to a child running into the road. The class uses a see think act cycle: sense the world, understand what is there, localize and map, decide what to do, then turn that choice into a trajectory and control it.
Course logistics 18:31
The lecture notes a few changes in the class. There is one more lecture each quarter, some material has moved to winter, and new topics include multi-sensor perception and open source software for autonomous systems. If you are in the 200-level version of the class, you should attend the Friday lectures. The midterms are in class by default, though another time can be arranged if needed.
Robot motion constraints 20:30
The discussion then turns to mobile robot kinematics. To plan or control a robot, you need a model of how it moves and what limits its motion. A robot is described by generalized coordinates, the smallest set of values that fixes its pose. For a simple wheel on a plane, that means the wheel center position and its heading angle.
Holonomic and nonholonomic 28:00
Some constraints depend only on position. These holonomic constraints reduce the set of poses the robot can reach, as with two rigid bodies joined by a revolute joint. Other constraints depend on position and velocity. These kinematic constraints limit what velocity you can have at a given instant, like a car that cannot move sideways. Not every kinematic constraint comes from a position constraint, and those that do not are called nonholonomic.
Wheel without slipping 36:03
A rolling wheel gives the clearest example. If the wheel rolls without slipping, the contact point has no sideways velocity. The heading angle sets the direction of motion, so the velocity vector and the sideways normal are perpendicular. The lecture then starts to write this condition in coordinates using a unit vector along the motion and the vector orthogonal to it.
Rolling constraint 40:02
The rolling disk has a no-side-slip constraint. Its velocity cannot point in the direction orthogonal to the wheel plane, so x dot sin theta minus y dot cos theta equals zero. This is a nonholonomic constraint because it is linear in the velocity derivatives but nonlinear in theta.
Wheel types 44:01
Standard wheels can spin around their own axle. Fixed standard wheels cannot turn around a vertical axis, while steerable standard wheels can. Caster wheels have the axle offset from the attachment point, so they line up with sideways motion. Active wheels are controlled by an actuator, while passive ones turn on their own.
Constraint matrices 48:00
When a robot has several kinematic constraints, you stack them in a matrix A. The admissible velocities must lie in the null space of A transpose. For wheel systems this form is common, so the class uses it to build robot models from the constraint equations.
Unicycle model 52:32
The null space gives a basis, and the robot velocity can be written as a linear combination of those basis vectors. For the unicycle, a convenient basis is cos theta, sin theta, 0 and 0, 0, 1. That leads to x dot, y dot, and theta dot expressed with two inputs: v for forward speed and omega for turning rate. This model is later bounded by maximum speed and maximum angular speed.
Differential drive robot 57:02
The unicycle model is useful because a differential drive robot has nearly the same kinematics. It has two active back wheels and a passive front wheel, often a caster. Driving the rear wheels at different speeds makes the robot rotate. The next step is to write its kinematic constraints and see how the wheel equations interact.
Wheel model mappings 59:31
The rear wheels of the differential drive give the same kinematic constraints as a unicycle, since the front caster wheel adds no side-slip constraint. The midpoint speed is the average of the left and right wheel speeds, while the robot’s turn rate is the difference between them divided by the wheelbase. That lets you plan with a unicycle model and then translate back to wheel speeds.
Simple car models 1:09:09
A more detailed car model adds two steerable front wheels, so the state includes position, heading, and steering angle. The front and rear no-side-slip constraints are both active, and the input space is two dimensional. Depending on the speed and steering bounds, this becomes a simple car, a Reeds-Shepp car, or a Dubins model.
From kinematics to dynamics 1:12:31
Kinematic models are only part of a fuller dynamic model. You can use them for planning, then send the planned speed and turn commands through a lower-level module that handles acceleration and other real inputs. You can also add state variables like velocity or steering angle and give them their own rate equations, but richer models cost more to optimize.
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