mobile robots (wheeled) · 2016-11-30 · • mobile robot dynamics - much simpler than legged...

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Page 1: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Mobile Robots (Wheeled)

(Take class notes)

Page 2: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Wheeled mobile robots

• Wheeled mobile platform controlled by a computer is called mobile robot in a broader sense

• Wheeled robots have a large scope of types and applications- Autonomous car- Autonomous wheelchair- Roomba vacuum cleaning robot- Mars rover- Unmanned aerial vehicle (UAV) - a special case- Automated helicopter (drone) – a special case

Page 4: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Wheeled mobile robots technical issues• Mobile robot dynamics

- Much simpler than legged mobile robots

• Mobile robot kinematics- Focus on path planning – how to move from one point to another

efficiently, avoiding obstacles while moving- How to navigate a mobile robot such as an autonomous vehicle

on highways

• Sensing for understanding the environments – necessary for autonomous vehicles- Visual sensors- Radar sensors- Laser sensors- Ultrasonic sensors

Page 5: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Wheeled robot dynamics• Simple Newton-Euler equations are sufficient to describe

the dynamics of a mobile robot

Driving wheels

Steering (support) wheels

Mobile platform

F1

F2

• The two driving wheels provide forces (torques) for moving (rotating) the robot

D1

D2

Page 6: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Dynamics equations

𝐹𝐹 = 𝑚𝑚𝑚𝑚

𝑁𝑁 = 𝐼𝐼�̇�𝑤 +𝑤𝑤 × 𝐼𝐼𝑤𝑤

• Newton’s equation

• Euler’s equation m: mass of the robotI: moment of inertia

• A common practice - Use PWM and PID controller to provide the force to the wheels

𝝉𝝉 = 𝐹𝐹1 × 𝐷𝐷1 + 𝐹𝐹2 × 𝐷𝐷2

𝐹𝐹 = 𝐹𝐹1 − 𝐹𝐹2

Page 7: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Recall the DC motor model

R L

Vb

i

τm

Bm Jm

�̇�𝜃 ,�̈�𝜃𝑚𝑚𝜃𝜃𝑚𝑚, �̈�𝜃�̇�𝜃𝑚𝑚, 𝜃𝜃 ,

τLnτLmτf

PWM supply++

--

Wheel

Page 8: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Forces by and on the wheels

Driving wheel

Torque τL

Driving force F

Supporting (passive) wheel

Shaft static and damping friction

Ground static and damping friction

Page 9: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Consider autonomous wheelchair as an example- SJTU Wheelchair

Joystick

IR Sensors

DC Motor

Camera

Bumper

Sonic Sensors

Page 10: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Wheelchair control block diagram

Wheel controller

PWM amplifier - left PWM amplifier - right

DC motor for the left wheel DC motor for the right wheel

DSP – TMS320LF2407A (TI)

Joystick IR Sensors CameraBumper Sonic Sensors

Page 11: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Navigation

• A great deal of research is on the navigation of autonomous robots- Indoor- Outdoor

• Ground autonomous vehicles will have a great scope of applications

• Aerial vehicles could use tele-operation as well – “drone” A pilotless aircraft operated by

remote control

Page 12: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Indoor robot navigation• Planning a path according to the map of a space

• Obstacle avoidance is more a challenge since there are many stationary and moving objects (people) to avoid

• For stationary objects one approach is to enlarge the size of the object and then plan the path as if the robot is a point

• Consider the following:

• Move the autonomous wheelchair from point A to point B in the space as shown

A

B

Page 13: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Path planning

Wheelchair

Wheelchair

• Enlarge the size of obstacles by the dimension of wheelchair• Consider the wheelchair as a single point – dimensionless• Program the shortest distance from the origin to the destination

Page 14: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Navigation and collision avoidance in depth• The robot detects an object. How to avoid it?• Many approaches have been developed• A well cited paper is by Oussama Khatib (Stanford

University)- It is called “artificial potential field” method- “Real-Time Obstacle Avoidance for Manipulators and Mobile

Robots,” Khatib, O., 1985

• Basic principle:1. For reaching the designation point, establish an Attraction

Potential2. For avoiding collision with obstacles, establish a Repulsive

Potential

Page 15: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

The mathematics

𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 𝑞𝑞 =

12𝜀𝜀𝑑𝑑2 , 𝑑𝑑 ≤ 𝑑𝑑∗𝑔𝑔𝑔𝑔𝑎𝑎𝑔𝑔

𝜀𝜀𝑑𝑑∗𝑔𝑔𝑔𝑔𝑎𝑎𝑔𝑔𝑑𝑑 −12

(𝑑𝑑∗𝑔𝑔𝑔𝑔𝑎𝑎𝑔𝑔)2,𝑑𝑑 > 𝑑𝑑∗𝑔𝑔𝑔𝑔𝑎𝑎𝑔𝑔

1. The attractive potential:

2. The repulsive potential:

𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟 𝑞𝑞 = �𝜖𝜖(

1𝐷𝐷 𝑞𝑞 −

1𝑄𝑄∗)2, 𝐷𝐷(𝑞𝑞) ≤ 𝑄𝑄∗

0, 𝐷𝐷(𝑞𝑞) > 𝑄𝑄∗

𝑑𝑑 = 𝑑𝑑 (𝑞𝑞, 𝑞𝑞𝑔𝑔𝑔𝑔𝑎𝑎𝑔𝑔 )

𝑞𝑞 = 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝𝑟𝑟𝑝𝑝, 𝑞𝑞𝑔𝑔𝑔𝑔𝑎𝑎𝑔𝑔= 𝑔𝑔𝑟𝑟𝑚𝑚𝑔𝑔 𝑝𝑝𝑟𝑟𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝𝑟𝑟𝑝𝑝

𝑄𝑄∗ = 𝑟𝑟𝑡𝑟𝑟𝑡𝑡𝑝𝑝𝑡𝑟𝑟𝑔𝑔𝑑𝑑

3. The motion is realized by following the negative gradient of the sum of the attractive/repulsive potential (energy)

Page 16: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Total potential energy and its minimization

𝑈𝑈 𝑞𝑞 = 𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 𝑞𝑞 + 𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟 𝑞𝑞

Total potential energy:

The gradient of the potential:

𝛻𝛻𝑈𝑈 𝑞𝑞 = 𝛻𝛻𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 𝑞𝑞 + 𝛻𝛻𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟 𝑞𝑞

Gradient is a derivative of a function in several dimensions. If f(q1, ..., qn) is differentiable scalar-valued function in Cartesian coordinate, its gradient is the vector whose components are the n partial derivative of f, which is a vector.

Green represents U, and the arrows represent gradient

𝛻𝛻𝑈𝑈 𝑞𝑞 = [𝜕𝜕𝑈𝑈(𝑞𝑞)𝜕𝜕𝑞𝑞1

, 𝜕𝜕𝑈𝑈(𝑞𝑞)𝜕𝜕𝑞𝑞2

, … 𝜕𝜕𝑈𝑈(𝑞𝑞)𝜕𝜕𝑞𝑞𝑛𝑛

]𝑇𝑇

Page 17: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

We can use - 𝛻𝛻𝑈𝑈 𝑞𝑞 (negative gradient) to assign a velocity to the mobile robot:

+

Starting point

Designation

Page 18: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Outdoor navigation

• Outdoor navigation- Need a map- Use GPS for global path planning on the map- Use visual, sonar, radar, etc. sensors for local

maneuver – avoid obstacles, and find a path- Time constant is an issue

- How fast sensor updates its reading- How accurate the sensor readings are- How fast robot should be moving

Page 19: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Global Positioning System (GPS)• GPS is a space-based satellite navigation system that provides

location and time information

• In all weather conditions, anywhere on or near the earth where there is an unobstructed line of sight to four or more GPS satellites

• GPS uses trilateration to calculate the position of the receiver

• Trilateration is a mathematical technique used to calculate the position of a point from three surrounding points by using the intersection of circles in 2D and spheres in 3D

• Three satellites are for the trilateration, while the fourth satellite is for calibrating the clock

• The distance from each satellite is calculated as

𝑑𝑑𝑝𝑝𝑝𝑝𝑟𝑟𝑚𝑚𝑝𝑝𝑑𝑑𝑡𝑡 = 𝑑𝑑 � (𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑠𝑠)

c: speed of light; tr: time of received; ts: time message sent

Page 20: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Trilateration uses four satellite

• If only three satellites are visible, one can use a so-called the Pseudo-Satellite – setup on the ground

Page 21: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Local maneuver needs multiple sensors • Computer vision is a good approach to assess the environments

• Computer vision is difficult to obtain accurate 3D information of the surrounding environment

• Use radar to obtain the range information of individual objects

- Automobile radar is a hot topic in recent years (at 76 GHz)

- Radar vision integration (sensor fusion) is a topic of study

Vision detection of vehicles Radar detection of vehicles in the image

Page 22: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Radar depth association with vision detected vehicle

• Use the pinhole model to establish the relationship between the size and depth using computer vision

𝑝𝑝𝑝𝑝𝑠𝑠𝑡𝑡𝑟𝑟𝑟𝑟𝑔𝑔𝑝𝑝𝑟𝑟𝑝𝑝𝑎𝑎𝑟𝑟𝑝𝑝= 𝑔𝑔𝑟𝑟𝑟𝑟𝑔𝑔𝑝𝑝𝑟𝑟𝑝𝑝𝑎𝑎𝑟𝑟𝑝𝑝 × 𝑡𝑟𝑟𝑟𝑟𝑔𝑔𝑝𝑝𝑟𝑟𝑝𝑝𝑎𝑎𝑟𝑟𝑝𝑝

=𝛼𝛼𝛼𝛼 × 𝑔𝑔𝑡

𝑠𝑠2

where α and β are constant related to the focal length of the camera length

l h

• Size of the object in the camera can reveal the depth information if we know the size of the real objects exactly

Page 23: Mobile Robots (Wheeled) · 2016-11-30 · • Mobile robot dynamics - Much simpler than legged mobile robots • Mobile robot kinematics - Focus on path planning – how to move from

Approach

• Use camera to obtain rough depth for every vehicle in the image – assuming vehicles to have the same average size

• Use radar to obtain exact depth of every vehicle

• Use a so-called Hungarian algorithm to perform association

r(i) is the range data by radar; v(i) is the depth data by vision; m is the number of cars seen

𝑚𝑚𝑟𝑟𝑔𝑔 𝑚𝑚𝑝𝑝𝑝𝑝∑𝑖𝑖=1𝑚𝑚 ||𝑟𝑟 (𝑝𝑝) − 𝑣𝑣(𝑝𝑝)||2

based on the constraint

if 𝑟𝑟𝑚𝑚𝑝𝑝𝑟𝑟 𝑟𝑟 𝑝𝑝 < 𝑟𝑟𝑚𝑚𝑝𝑝𝑟𝑟 𝑟𝑟 𝑗𝑗 ,then 𝑑𝑑𝑝𝑝𝑝𝑝𝑟𝑟𝑚𝑚𝑝𝑝𝑑𝑑𝑡𝑡(𝑟𝑟 𝑝𝑝 ) < 𝑑𝑑𝑝𝑝𝑝𝑝𝑟𝑟𝑚𝑚𝑝𝑝𝑑𝑑𝑡𝑡 𝑟𝑟 𝑗𝑗