Derive the mobile kinematics, K, of a car-like robot with Ackermann steer- ing as given below, where the controls are forward speed (advance con- trol) v and heading direction turning speed (turn control) w, and 1.
Derive the mobile kinematics, K, of a car-like robot with Ackermann steer- ing as given below, where the controls are forward speed (advance con- trol) v and heading direction turning speed (turn control) w, and 1.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![### Deriving Mobile Kinematics for a Car-Like Robot with Ackermann Steering
#### Text Explanation:
The goal is to derive the mobile kinematics, \(K\), of a car-like robot utilizing Ackermann steering. The controls involved in this system are the forward speed (advance control) denoted as \(v\), and the heading direction turning speed (turn control) denoted as \(\omega\).
The kinematic model can be expressed as:
\[
\dot{q} = \begin{bmatrix} \dot{\phi} \\ \dot{x} \\ \dot{y} \\ \dot{\psi} \end{bmatrix} = K \begin{bmatrix} v \\ \omega \end{bmatrix}
\]
#### Diagram Explanation:
- The diagram depicts a car-like robot viewed from above.
- The axes labeled \(\hat{x}\) and \(\hat{y}\) represent the global coordinate frame.
- \(CoR\) refers to the Center of Rotation of the vehicle.
- \(r_\text{min}\) is the minimum turning radius of the vehicle's trajectory.
- The robot's body is shown as a rectangle with one pair of wheels aligned with the steering mechanism to illustrate Ackermann steering geometry.
- Parameters \(x\) and \(y\) indicate the position of the vehicle in the global coordinate system.
- \(\psi\) is the heading angle of the vehicle, and \(\phi\) is the steering angle of the wheels.
- \(\ell\) represents the wheelbase, the distance between the front and rear axles.
#### Conceptual Understanding:
The Ackermann steering geometry is employed to minimize the slip of tires during turning, effectively allowing each wheel to follow its unique circular path. By adjusting the steering angle \(\phi\), the robot can achieve a desired trajectory aligning with the kinematic model.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F712ebd5b-8d0a-4452-b86b-5a729421fa50%2F9b3be989-fba3-4487-a51a-7e56689d14d0%2F50e26w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Deriving Mobile Kinematics for a Car-Like Robot with Ackermann Steering
#### Text Explanation:
The goal is to derive the mobile kinematics, \(K\), of a car-like robot utilizing Ackermann steering. The controls involved in this system are the forward speed (advance control) denoted as \(v\), and the heading direction turning speed (turn control) denoted as \(\omega\).
The kinematic model can be expressed as:
\[
\dot{q} = \begin{bmatrix} \dot{\phi} \\ \dot{x} \\ \dot{y} \\ \dot{\psi} \end{bmatrix} = K \begin{bmatrix} v \\ \omega \end{bmatrix}
\]
#### Diagram Explanation:
- The diagram depicts a car-like robot viewed from above.
- The axes labeled \(\hat{x}\) and \(\hat{y}\) represent the global coordinate frame.
- \(CoR\) refers to the Center of Rotation of the vehicle.
- \(r_\text{min}\) is the minimum turning radius of the vehicle's trajectory.
- The robot's body is shown as a rectangle with one pair of wheels aligned with the steering mechanism to illustrate Ackermann steering geometry.
- Parameters \(x\) and \(y\) indicate the position of the vehicle in the global coordinate system.
- \(\psi\) is the heading angle of the vehicle, and \(\phi\) is the steering angle of the wheels.
- \(\ell\) represents the wheelbase, the distance between the front and rear axles.
#### Conceptual Understanding:
The Ackermann steering geometry is employed to minimize the slip of tires during turning, effectively allowing each wheel to follow its unique circular path. By adjusting the steering angle \(\phi\), the robot can achieve a desired trajectory aligning with the kinematic model.
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