Mobile robots kinematics

Question 1

What is kinematics and why we study it?

Answer 1

Definition
Is the most basic study of how mechanical systems behave.

Why?
In mobile robotics, we need to understand the mechanical behavior of the robot both to design appropriate mobile robots for tasks and to understand how to create control software for an istance of mobile robot hardware.

Question 2

How can we derive the model for the whole robot’s motion?

Answer 2

Bottom-up process: each wheel contributes and at the same time imposes constraints to the robot’s motion

Question 3

What is the pose of a robot?

Answer 3

Is a vector ξw = [x y Θ]

where x, y represent the position while Θ represents the orientation in the global ref frame

Question 4

Mobile robots wrt manipulator arms

Answer 4

• encoder values don’t map to unique robot poses
• they can move unbound with respect to their environment:
  - no direct way to measure the robot’s position
  - position must be integrated over time
  - leads to inaccuracies of the position estimation

Question 5

Main property of non-holonomic systems when dealing with kinematics

Answer 5

The measure of the traveled distance of each wheel is not sufficient to calculate the final position of the robot. One has also to know how this movement was executed as a function of time.

Question 6

Forward kinematic vs inverse kinematics

Answer 6

• forward kinematics: is giving the transformation from the joint space to the world space.
• inverse kinematics: is giving the transformation from the world space to the joint space (we want to reach a certain position: what are the commands?).

Question 7

Model the state of a mobile robot

Answer 7

Robot speed ξ. as a function of:
- wheel speed (ω.)
- steering angle (βi)
- steering speed (βi.)
- geometrical parameters of the chassis

• Forward kin: we want to find ξ. as a function of the above parameters
• Inverse kin: we want to find the motion parameters as a function of the robot speed ξ.

Question 8

Forward kinematic model

Answer 8

It would predict the robot’s overall speed in the global ref frame:

ξẇ = [ẋ ẏ Θ.] = R(Θ)^-1 * ξr = f(l, r, Θ, ω₁, ω₂)

where each wheel is at distance l from P, r is their radius, Θ is the angular difference between the local and the global ref frames, ωi are the spinning speed of the wheels.

Question 9

Forward kinematic model of a differential-drive chassis

Answer 9

We need to compute ξẇ = R(Θ)^-1 * ξr.

draw the sketch

Consider the motion of the point P between the two wheels and compute ξr. as a sum of the contributions of each wheel to the instantaneous motion.

ẋ = rω₁ / 2 + rω₂ / 2
ẏ = 0 (no lateral sliding)
Θ. = rω₁ / 2l - rω₂ / 2l

R(Θ)^-1 =
[ cos -sin 0 ]
[ sin cos 0 ]
[ 0 0 1 ]

Question 10

Assumptions for wheel kinematic

Answer 10

• horizontal plane
• massless robot
• single point of contact between wheels and floor
• not deformable wheels
• no sliding, no skidding, pure rolling
• no friction of the contact point when wheels are rotating
• steering axes of the wheels are parallel to the floor
• wheels are connected to a rigid frame

Question 11

Pure rolling constraint

Answer 11

All motion along the direction of the wheel plane must be accompained by the appropriate amount of wheel spin (so we have pure rolling at the contact point).

Question 12

No sliding constraint

Answer 12

No motion trasversal to the direction of the wheel plane.

Question 13

Wheels constraints

Answer 13

• each wheel imposes either zero o more constraints on the robot motion
• only fixed and stearable wheels imposes constraints (omnidirectional wheels no)

Question 14

How the non-lateral sliding constraint can be visualized?

Answer 14

Through the zero motion line.

The ZML represents the direction along which the wheels motion is zero at any istance in time.

draw the sketch of a car and of a bike

Question 15

What the robot mobility depends on?

Answer 15

On the contraints the wheels create (not exactly on the number of wheels).

example with bad wheels configuration (no ICR)

Question 16

Ackermann steering geometry

Answer 16

Is a geometric arrangement of linkages in the steering of a car or other vehicle designed to solve the problem of wheels on the inside and outside of a turn needing to trace out circles of different radius.

It was invented by the German carriage builder Georg Lankensperger in Munich in 1817, then patented by his agent in England, Rudolph Ackermann (1764–1834) in 1818 for horse-drawn carriages.

draw the sketch

Question 17

What is maneuverability for a mobile robot?

Answer 17

Is its ability to directly move in the environment.

The overall maneuverability of a robot is a combination of the kinematic sliding (δm) constraints of the standard wheels + additional freedom contributed by steering and spinning (δs) the steerable standard wheels.

δM = δm + δs (actual DoFs of the robot)

δM <= DoF

Question 18

How the degree of mobility can be computed?

Answer 18

How many actions can be done by just controlling the wheels speed?

Ranges from 0 to 3.

Examples:
* bicycle, car, tricycle: δm = 1
* differential drive robot with 1 spherical wheel: δm = 2
* robots composed only of omnidirectional wheels as swedish or spherical: δm = 3

Question 19

How the degree of steerability can be computed? (cases)

Answer 19

Ranges from 0 to 2.

• δs = 0 if no steerable wheels
• δs = 1 if 1 steerable wheel
• δs = 2 if the robot has no fixed standard wheels and has 2 steerable wheels

Question 20

Workspace parameters of a mobile robot

Answer 20

• POSITION: the location of the robot in the space (example: for mobile robot on a plane x,y)
• POSE: the values of the state variables of the robot in the space of its degrees of freedom (example: for a mobile robot on a plane x,y, ⍬)
• PATH: the sequences of poses in the space of its degrees of freedom
• TRAJECTORY: the path + the time, as an additional
  dimension.

Question 21

Omnidirectional robot vs two-steer robot

Answer 21

Both have δM = 3.

If we consider infinite accelerations and so any speed can be achieved istantaneusly, two-steer takes 5 phases to achieve the same path instead of 3 phases like omnidirectional.

draw the two cases (sketch + graph)

Question 22

What is the DDoF (differential degree of freedom) and how it is related to the standard DoF?

Answer 22

Is the number of indipendently achievable velocities. It governs the robot’s ability to achieve various paths.

DDoF = δm

Examples:
* omnibot: DDoF = 3 becasue it can reach all the 3 axes simultaneously
* bicycle: DDoF < 3

DDoF <= δM <= DoF

Question 23

Kinematic controller

Answer 23

Its objective is to follow a trajectory described by its position or velocity profile as a function of time.

• open-loop motion control: divides the path in motion segments of clearly defined shapes (3 disadvantages)
• feedback control is more reliable

Question 24

Feedback motion control

Answer 24

Control the motion by:
1. finding a control matrix K 2x3 where kij = k(t,e)
2. such that the control of v(t) and ω(t): 2x1 matrix = K*e where e = [x, y, Θ] is the pose error vector
3. brings lim e(t) -> 0

Question 25

Dynamics control vs kinematics control

Answer 25

Dynamic control takes into account the physical properties of the system and its main subject of study is how these properties change during the motion.

Kinematics control focus only on the motion of the system.

Question 26

Total number of distinctive event sequences N for a walking machine with k legs

Answer 26

N = (2k-1)!