|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Roland Siegwart, Margarita Chli, Nick Lawrance
2021 | Mobile Robots Kinematics - add ons 1
Mobile Robot KinematicsAutonomous Mobile Robots
Spring 2021
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Manipulator arms versus mobile robots Robot arms are fixed to the ground and usually comprised of a single chain of actuated links The motion of mobile robots is defined through rolling and sliding constraints taking effect at
the wheel-ground contact points
2021 | Mobile Robots Kinematics - add ons 2
Mobile Robot Kinematics | Overview
C Willow GarageRide an ABB, https://www.youtube.com/watch?v=bxbjZiKAZP4
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Manipulator arms versus mobile robots Both are concerned with forward and inverse (backward) kinematics However, for mobile robots, encoder values don‘t map to unique robot poses And mobile robots can move unbound with respect to their environment There is no direct (=instantaneous) way to measure the robot’s position (end-effector) Robot positions must be integrated over time and depend on the path taken This leads to inaccuracies of the position (motion) estimate
Understanding mobile robot motion starts with understanding wheel constraints placed on the robot’s mobility
2021 | Mobile Robots Kinematics - add ons 3
Mobile Robot Kinematics | Overview
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Non-holonomic systems differential equations are not directly integrable to the final position. 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.
This is in strong contrast to actuator arms
2021 | Mobile Robots Kinematics - add ons 4
Non-Holonomic Systems
2121
212121
,,,
yyxxssssss LLRR
s1L s1R
s2L
s2R
yI
xI
x1, y1
x2, y2
s1
s2
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Forward kinematics: Transformation from joint to physical space
Inverse kinematics Transformation from physical to joint space Required for motion control
Due to non-holonomic constraints in mobile robotics, we deal with differential (inverse) kinematics Transformation between velocities instead of positions Such a differential kinematic model of a robot has the following form:
2021 | Mobile Robots Kinematics - add ons 5
Forward and Inverse Kinematics
Robot Model𝑥,𝑦,𝜃𝑣,𝜔
Control law
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Stability of a vehicle is be guaranteed with 3 wheels If center of gravity is within the triangle which is formed by the
ground contact point of the wheels Stability is improved by 4 and more wheel however, this arrangements are hyper static and require a flexible
suspension system. Selection of wheels depends on the application Bigger wheels allow to overcome higher obstacles but they require higher torque or reductions in the gear box.
Most arrangements are non-holonomic (see chapter 3) require high control effort
Combining actuation and steering on one wheel makes the design complex and adds additional errors for odometry.
2021 | Mobile Robots Kinematics - add ons 6
Mobile Robots with Wheels
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Two wheels
Three wheels
Four wheels
Six wheels2021 | Mobile Robots Kinematics - add ons 7
Different Arrangements of Wheels
COG below axle
Omnidirectional Drive Synchro Drive
Synchro Drive © J. Borenstein
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
2021 | Mobile Robots Kinematics - add ons 8
Kinematic Constraints | Fixed Standard Wheel
y.
x.
. v = r .
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
2021 | Mobile Robots Kinematics - add ons 9
3 - Mobile Robot Kinematics
l
Robot chassis
v = r .
x.
x sin.
x cos.
A
.
y
.y (cos
.
y sin
.
l).
(l)sin.
l)cos.
Rolling constraint →
Sliding constraint →
𝜉 𝑥 𝑦 𝜃
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
2021 | Mobile Robots Kinematics - add ons 10
Five Basic Types of Three-Wheel Configurations
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Purely friction based
2021 | Mobile Robots Kinematics - add ons 11
Wheeled Robot for Rough Terrain | Concepts for Object Climbing
Change of center of gravity(CoG)
Adapted suspension mechanism with
passive or active joints
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Passive adaptation to terrain 6 wheels with articulated suspensions two boogies on each side fixed wheel in the rear front wheel with spring
suspension Dimensions length: 60 cm height: 20 cm
Characteristics highly stable in rough terrain overcomes obstacles up to
2 times its wheel diameter
2021 | Mobile Robots Kinematics - add ons 12
Wheeled Robot for Rough Terrains | Articulated Suspensions
Shrimp (ASL EPFL/ETH)
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Vertigo – developed by students| the ultimate wall climberhttps://www.youtube.com/watch?v=KRYT2kYbgo4
Ascento| the balancing jumperhttps://youtu.be/1yvoZhRTX-U
2021 | Mobile Robots Kinematics - add ons 13
Wheeled Robots | beyond flat ground
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
The objective of a kinematic controller (not considering dynamics) is to follow a trajectory described by its position and/or velocity profiles as function of time.
Motion control is not straight forward because mobile robots are typically non-holonomic and MIMO systems.
Most controllers (including the one presented here) are not considering the dynamics of the system
2021 | Mobile Robots Kinematics - add ons 14
Wheeled Mobile Robot Motion Control | Overview
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Trajectory (path) divided in motion segments of clearly defined shape: straight lines and segments of a circle Dubins car, and Reeds-Shepp car
Control problem: pre-compute a smooth trajectory
based on lines, circles (and clothoids) segments Disadvantages: It is not at all an easy task to pre-compute a feasible trajectory limitations and constraints of the robot’s velocities and accelerations does not adapt or correct the trajectory if dynamical changes
of the environment occur. The resulting trajectories are usually not smooth (in acceleration, jerk, etc.)
2021 | Mobile Robots Kinematics - add ons 15
Motion Control | Open Loop Control
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Find a control matrix 𝐾, if exists
𝐾 𝑘 𝑘 𝑘𝑘 𝑘 𝑘
with 𝑘 𝑘 𝑡, 𝑒such that the control of v(t) and (t)
𝑣 𝑡𝜔 𝑡 𝐾 · 𝑒 𝐾 ·
𝑥𝑦𝜃
lim→
𝑒 𝑡 =0
drives the error 𝑒 to zero→ MIMO state feedback control
2021 | Mobile Robots Kinematics - add ons 16
Motion Control | Closed Loop Feedback Control
Robot Model𝑥,𝑦,𝜃𝑣,𝜔
Control law
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Motion Control | Kinematic Position Control
The kinematics of a differential drive mobile robot described in the inertial frame 𝑥 ,𝑦 ,𝜃is given by,
where 𝑥 and 𝑦 are the linear velocities in the direction of the 𝑥 and 𝑦 of the inertial frame.
Let 𝛼 denote the angle between the 𝑥 axis of the robot reference frame and the vector connecting the center of the axle of the wheels with the final position.
2021 | Mobile Robots Kinematics - add ons 17
𝑥𝑦𝜃
𝑐𝑜𝑠𝜃 0𝑠𝑖𝑛𝜃 0
0 1
𝑣𝜔
∆𝑦
∆𝑥
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Kinematic Position Control | Coordinates Transformation
𝑦
𝑥
Coordinate transformation into polar coordinates with its origin at goal position:𝜌 ∆𝑥 ∆𝑦
𝛼 𝜃 atan 2 ∆𝑦,∆𝑥
𝛽 𝜃 𝛼 System description, in the new polar coordinates
𝜌𝛼𝛽
cos𝛼 0𝟏
0
𝑣𝜔 and
𝜌𝛼𝛽
cos𝛼 0𝟏
0
𝑣𝜔
for 𝛼 ∈ 𝐼 , for 𝛼 ∈ 𝐼 𝜋, ∪ ,𝜋 → 𝑣 𝑣
2021 | Mobile Robots Kinematics - add ons 18
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Kinematic Position Control | Coordinates Transformation
∆𝑦
∆𝑥
The coordinates transformation is not defined at 𝑥 𝑦 0→ Stop controller very close to the goal
For 𝛼 ∈ 𝐼 the forward direction of the robot points toward the goal, for 𝛼 ∈ 𝐼 it is the backward direction.
with 𝛼 ∈ 𝐼 , and 𝛼 ∈ 𝐼 𝜋, ∪ ,𝜋
By properly defining the forward direction of the robot at its initial configuration, it is always possible to have 𝛼 ∈ 𝐼 at 𝑡 0. However, this does not mean that 𝛼 remains in 𝐼 for all time 𝑡.
2021 | Mobile Robots Kinematics - add ons 19
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
Kinematic Position Control | The Control Law
𝑦
𝑥
It can be shown, that with 𝑣 𝑘 𝜌 and 𝜔 𝑘 𝛼 𝑘 𝛽
the feedback-controlled system 𝜌𝛼𝛽
𝑘 𝜌 cos𝛼𝑘 sin𝛼 𝑘 𝛼 𝑘 𝛽
𝑘 sin𝛼
will drive the robot to 𝜌,𝛼,𝛽 0,0,0 The control signal 𝑣 has thereby always constant sign: the direction of movement is kept positive or negative during movement parking maneuver is performed always in the most natural way and without ever inverting its
motion direction.
2021 | Mobile Robots Kinematics - add ons 20
|Autonomous Mobile RobotsRoland Siegwart, Margarita Chli, Nick Lawrance
ASLAutonomous Systems Lab
The goal is in the center and the initial position on the circle.
2021 | Mobile Robots Kinematics - add ons 21
Kinematic Position Control | Resulting Path
𝑘 𝑘 ,𝑘 ,𝑘 3, 8, 1.5
𝜶 ∈ 𝑰 /
𝐼𝜋2 ,𝜋2
→ forward pointing to goal
𝐼 𝜋, ∪ ,𝜋 → backward pointing to goal
𝑦
𝑥