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mod02lec07 - A Generalized Wheel (Kinematic) Model
NPTEL-NOC IITM
Overview
This lecture introduces a generalized kinematic model for wheels on mobile robots, focusing on complex wheel types like mecanum wheels. It derives the relationship between the robot's body-fixed velocities (forward, sideways, rotational) and the angular velocities of individual wheels. The process involves analyzing a single wheel, considering both its primary drive velocity and the velocity generated by passive rollers, and then transforming these velocities into the robot's main body frame. The ultimate goal is to express the robot's overall motion (u, v, r) as a function of its individual wheel angular velocities, forming the basis for controlling robots with various wheel configurations.
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Chapters
- The lecture aims to derive a generalized kinematic model for wheels on mobile robots.
- This model is particularly useful for complex wheel designs like mecanum wheels, which have passive rollers.
- The focus is on kinematics, relating wheel angular velocities to the robot's body-fixed linear and angular velocities (u, v, r).
Understanding generalized wheel models is crucial for controlling robots with diverse and complex wheel mechanisms, enabling them to achieve specific movements.
The lecture uses the mecanum wheel, characterized by passive rollers inclined relative to the wheel's main axis, as a complex example to motivate the need for a generalized model.
- A single wheel is analyzed in isolation, considering its own coordinate frame (ci).
- The primary drive velocity (V_drive) is related to the wheel's angular velocity (omega_i) and radius (r_i) via pure rolling (V_drive = omega_i * r_i).
- Passive rollers on wheels like mecanum wheels introduce a secondary velocity component (V_slide) that can be oriented laterally.
- The V_slide is also assumed to be a pure rolling velocity, dependent on the roller's angular velocity (beta_dot_i) and radius (rho_i).
Breaking down the motion into components and analyzing each independently simplifies the complex interactions within a wheel, allowing for a systematic derivation.
The V_drive is the tangential velocity due to the wheel's rotation, while V_slide is the tangential velocity due to the passive roller's rotation, which can be oriented at an angle phi_i relative to the wheel's main axis.
- The V_drive and V_slide are decomposed into components along the wheel's local coordinate axes (x_ci, y_ci).
- The lateral velocity (y_ci_dot) is directly related to the V_slide component along the y_ci axis.
- The longitudinal velocity (x_ci_dot) is a combination of V_drive and the V_slide component along the x_ci axis.
- An equation is derived relating V_drive to x_ci_dot, y_ci_dot, and the angle phi_i: V_drive = x_ci_dot + y_ci_dot * tan(phi_i).
- This equation is then expressed in matrix form to relate the wheel's local velocities (x_ci_dot, y_ci_dot) to its angular velocity (omega_i).
This step transforms the velocities from the wheel's local frame to a more useful representation, allowing us to connect individual wheel motion to the robot's overall movement.
The relationship x_ci_dot = omega_i * r_i - rho_i * beta_dot_i * sin(phi_i) and y_ci_dot = rho_i * beta_dot_i * cos(phi_i) are derived, and then combined to express omega_i in terms of x_ci_dot and y_ci_dot.
- The velocities in the wheel's local frame (x_ci_dot, y_ci_dot) are transformed into the robot's main body frame (x_b, y_b).
- This transformation uses the angle theta_bi between the body frame's x-axis and the wheel's local x-axis.
- The resulting velocities in the body frame (along x_b and y_b) are expressed as linear combinations of x_ci_dot, y_ci_dot, and trigonometric functions of theta_bi.
- The robot's body-fixed velocities (u, v, r) are also related to the velocities at the wheel's center (ci), considering the robot's angular velocity (r) and the position of ci relative to the body frame (dx_i, dy_i).
Connecting wheel velocities to the robot's main body frame is essential for understanding how each wheel contributes to the robot's overall translation and rotation.
The velocity component along the robot's x-axis (xb) is given by u - r*dy_i, and along the y-axis (yb) by v + r*dx_i, where dx_i and dy_i are the offsets of the wheel's center from the robot's origin in the body frame.
- The goal is to express the robot's body velocities (u, v, r) as a function of the individual wheel angular velocities (omega_i).
- By substituting the transformed wheel velocities into the equations relating body velocities to wheel velocities, a matrix equation is formed.
- This matrix equation, often denoted as zeta = W * omega, relates the robot's velocity vector (zeta = [u, v, r]^T) to the vector of wheel angular velocities (omega = [omega_1, omega_2, ..., omega_n]^T) via the input configuration matrix (W).
- The inverse of the transformation matrix allows expressing wheel velocities in terms of body velocities, and further substitution yields the desired relationship between body velocities and wheel angular velocities.
This final model provides a direct link between the robot's desired motion and the commands needed for its wheels, forming the core of the robot's kinematic control system.
The final generalized model expresses omega_i as a function of u, v, r, wheel radius, roller inclination angle, and the robot's pose relative to the wheel, effectively capturing the wheel's kinematic behavior within the robot's frame.
- The derived generalized wheel model can be used to construct the input configuration matrix (W) for robots with different wheel types.
- The structure of the W matrix determines the robot's maneuverability and capabilities.
- The next lecture will explore examples of deriving the W matrix for various wheel configurations discussed previously.
- Understanding the W matrix is key to classifying robot types based on their motion capabilities.
By analyzing the input configuration matrix, we can predict and understand the movement capabilities of different mobile robot designs.
The lecture mentions that different wheel configurations (like those discussed in Lecture 4) will be used in the next lecture to derive specific W matrices, demonstrating how the generalized model applies to real-world robot designs.
Key takeaways
- A generalized wheel kinematic model relates a robot's overall body velocities (u, v, r) to the angular velocities of its individual wheels.
- Complex wheels like mecanum wheels require a generalized model due to their passive rollers, which introduce lateral slip velocities.
- The derivation involves analyzing a single wheel, decomposing its velocities, and transforming them into the robot's main body frame.
- The relationship between body velocities and wheel angular velocities is captured by the input configuration matrix (W).
- The structure of the W matrix dictates the robot's maneuverability and classification.
- Understanding this model is fundamental for controlling mobile robots with diverse wheel systems.
Key terms
Generalized Wheel ModelKinematic ModelMecanum WheelPassive RollerBody FrameWheel Frame (ci)V_driveV_slidePure RollingLateral SlipInput Configuration Matrix (W)
Test your understanding
- How does the presence of passive rollers in a mecanum wheel affect its kinematic model compared to a conventional wheel?
- What is the primary purpose of transforming wheel velocities from the wheel's local frame (ci) to the robot's body frame (B)?
- Explain the relationship between the robot's linear and angular velocities (u, v, r) and the individual wheel angular velocities (omega_i) as described by the generalized model.
- How can the input configuration matrix (W) be used to understand a robot's maneuverability?
- What are the key components that contribute to the longitudinal and lateral velocities of a wheel's center in the generalized model?