Unless otherwise specified,
we use right-handed
frames in robotics
Why do we need to use so many frames?
Because we want to reason and express quantities relative to their local configuration.
For example: “grab the bottle behind the cereal bowl”
This lecture is about defining and representing frames of reference and reasoning about how to express quantities in one frame to quantities in the other.
Rigid-body motion
Motion that can be described by a rotation and translation.
All the parts making up the body move in unison, and there are no deformations.
Representing rotations, translations, and vectors in a given frame of reference is often a source of frustration and bugs in robot software because there are so many options.
The answer is meaningless unless I provide a definition of the coordinate frames
Color convention
for frames
Moving body (robot) frame
Fixed world frame
Always provide a frame diagram
Inertial frames of reference
G, the global frame of reference is fixed, i.e. with zero velocity in our previous example.
But, in general it can move as long as it has zero acceleration. Such a frame is called an “inertial” frame of reference.
Newton’s laws hold for inertial reference frames only. For reference frames with non-constant velocity we need the theory of General Relativity.
So, make sure that your global frame of reference is inertial, preferably fixed.
Today’s Agenda
Frames of reference
Ways to represent rotations
Simplified models of vehicles
Forward and inverse kinematics
Representing Rotations in 3D: Euler Angles
Specification ambiguities in Euler Angles
Need to specify the axes which each angle refers to.
There are 12 different valid combinations of fundamental rotations. Here are the possible axes:
z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y
x-y-z, y-z-x, z-y-x, x-z-y, z-y-x, y-x-z
E.g.: x-y-z rotation with Euler angles \((\theta, \phi, \psi)\) means the rotation can be expressed as a sequence of simple rotations \(R_x(\theta) R_y(\phi) R_z(\psi)\)
Specification ambiguities in Euler Angles
Simple rotations can be counter-clockwise or clockwise. This gives another 2 possibilities.
Rotation matrices are orthogonal matrices: their transpose is their inverse and they do not change the length of a vector, they just rotate it in space.
Suppose you have a point in the Body frame,\(^B\mathbf{p}\) which you want to transform/express in the World frame. Then you can do any of the two following options:
Composing rigid motions now becomes a series of matrix multiplications
Inverting a homogeneous transformation
Be careful:
\[
_B^W\mathbf{T}^{-1} \neq _A^W\mathbf{T}^t
\]
as was the case with rotation matrices.
Physical models of how systems move
Today’s Agenda
Frames of reference
Ways to represent rotations
Simplified models of vehicles
Forward and inverse kinematics
Why simplified?
“All models are wrong, but some are useful” – George Box (statistician)
Model: a function that describes a physical phenomenon or a system, i.e. how a set of input variables cause a set of output variables.
Models are useful if they can predict reality up to some degree .
Mismatch between model prediction and reality = error / noise
Noise
Anything that we do not bother modelling with our model
Example 1: “assume frictionless surface”
Example 2: Taylor series expansion (only first few terms are dominant)
With models, can be thought of as approximation error.
Simplified physical models of robotic vehicles
Omnidirectional motion
Dubins car
Differential drive steering
Ackerman steering
Unicycle
Cartpole
Quadcopter
Omnidirectional Robots
Omnidirectional Robots
The state of an omnidirectional robot
State := Configuration := \(\mathbf{X}\) := vector of physical quantities of interest about the system
\[
\mathbf{X} = [^Gp_x, ^Gp_y, ^G\theta]
\]
State = [Position, Orientation]
Position of the robot’s frame of reference C with respect to a fixed frame of reference G, expressed in coordinates of frame G. Angle is the orientation of frame C with respect to frame G.
Control of an omnidirectional robot
Control := \(\mathbf{u}\) := a vector of input commands that can modify the state of the system
\[
\mathbf{u} = [^Cv_x, ^Cv_y, ^Cw_z]
\]
Control = [Linear velocity, Angular velocity]
Linear and angular velocity of the robot’s frame of reference C with respect to a fixed frame of reference G, expressed in coordinates of frame C.
Dynamics of an omnidirectional robot
Dynamical System : = Dynamics := a function that describes the time evolution of the state in response to a control signal
Note: reference frames have been removed for readability.
Kinematics vs Dynamics
Kinematics considers models of locomotion independently of external forces and control.
For example, it describes how the speed of a car affects the state without considering what the required control commands required to generate those speeds are.
Dynamics considers models of locomotion as functions of their control inputs and state.
Special case of simple car: Dubins car
Can only go forward
Constant speed
\[
^Cv_x = \text{const} > 0
\]
You only control the angular velocity
Special case of simple car: Dubins car
Can only go forward
Constant speed
\[
^Cv_x = \text{const} > 0
\]
You only control the angular velocity
Dubins car: motion primitives
The path of the car can be decomposed to L(eft), R(ight), S(traight) segments.
RSR path
Instantaneous Center of Rotation
IC = Instantaneous Center of Rotation
The center of the circle circumscribed by the turning path.
Undefined for straight path segments.
Dubins car \(\color{blue}{\Rightarrow}\) Dubins boat
Why do we care about a car that can only go forward?
Because we can also model idealized airplanes and boats
Dubins boat = Dubins car
Dubins car \(\color{blue}{\Rightarrow}\) Dubins airplane in 3D
Pitch angle \(\phi\) and forward velocity determine descent rate
Yaw angle \(\theta\) and forward velocity determine turning rate
Position of the unicycle’s frame of reference U with respect to a fixed frame of reference G, expressed in coordinates of frame G. Angle is the orientation of frame U with respect to frame G.
Q: Would you put the radius of the unicycle to be part of the state?
A: Most likely not, because it is a constant quantity that we can measure beforehand. But, if we couldn’t measure it, we need to make it part of the state in order to estimate it.
Controls of a unicycle
\[
\mathbf{u} = [^Uw_z, ^Uw_y]
\]
Controls = [Yaw rate, and pedaling rate]
Yaw and pedaling rates describe the angular velocities of the respective axes of the unicycle’s frame of reference U with respect to a fixed frame of reference G, expressed in coordinates of U.
Position of the vehicle’s frame of reference D with respect to a fixed frame of reference G, expressed in coordinates of frame G. Angle is the orientation of frame D with respect to frame G.
Controls of a differential drive vehicle
ICR = Instantaneous Center of Rotation
\[
\mathbf{u} = [u_l, u_r]
\]
Controls = [Left wheel and right wheel turning rates]
Wheel turning rates determine the linear velocities of the respective wheels of the vehicle’s frame of reference D with respect to a fixed frame of reference G, expressed in coordinates of D.
State = [Position and velocity of cart, orientation and angular velocity of pole]
Controls of a single-link cartpole
\[
\mathbf{u} = [f]
\]
Controls = [Horizontal force applied to cart]
Balancing a triple-link pendulum on a cart
Extreme Balancing
The state of a double integrator
\[
\mathbf{x} = [^Gp_x]
\]
State = [Position along x-axis]
Controls of a double integrator
\[
\mathbf{x} = [^Gu_x]
\]
Controls = [Force along x-axis]
Dynamics of a double integrator
\[
\ddot x = F
\] This corresponds to applying force to a brick of mass 1 to move on frictionless ice. Where is the brick going to end up? Similar to curling.
A system is controllable if there exist control sequences that can bring the system from any state to any other state, in finite time.
For example, even though cars are subject to non-holonomic constraints (can’t move sideways directly), they are controllable, They can reach sideways states by parallel parking.
Passive Dynamics
Dynamics of systems that operate without drawing (a lot of) energy from a power supply.
Interesting because biological locomotion systems are more efficient than current robotic systems.
Passive Dynamics
Dynamics of systems that operate without drawing (a lot of) energy from a power supply.
Usually propelled by their own weight.
Interesting because biological locomotion systems are more efficient than current robotic systems.
