Kinematics

Kinematics of Wheeled Mobile Robots (WMRs)

Robot Pose and Velocity Definitions

• Robot pose in the inertial frame: ( $I \eta = [x, y, \theta]^T$ )

• Velocity in the robot frame: ( $R \dot{\eta} = [v_x, v_y, \omega]^T$)

• Parameters:

• ( $v_x$ ): Forward velocity

• ( $v_y$ ): Lateral velocity (often constrained)

• ( $\omega$ ): Angular velocity

Wheel Types and Constraints

1. Fixed Standard Wheel

• Fixed orientation ( $\beta$) relative to chassis at location ( $\alpha$), ( l ) from center.

• Rolling constraint: ( $(I_R R I \dot{\eta}) \cdot c_r = r \dot{\phi}$ )

• Slip constraint: ( $(I_R R I \dot{\eta}) \cdot c_s = 0$)

1. Steered Standard Wheel

• Similar to fixed, but steering angle ( $\beta$ ) is actively controlled as ( $\beta(t)$ ).

• Constraints are the same as Fixed Standard Wheel, but ( $\beta$ ) varies over time.

1. Caster Wheel

• Free steering joint with offset ( d ) between the steering axis and wheel contact point.

• Rolling constraint: ( $(I_R R I \dot{\eta}) \cdot c_r = r \dot{\phi}$))

• Slipping constraint (allows steering rotation): $( (I_R R I \dot{\eta}) \cdot c_s = -d \dot{\beta} )$

1. Omnidirectional Wheel

• Permits movement in any direction via rollers aligned obliquely to the main wheel.

• No slip constraint; unique rolling dynamics.

Differential Drive Model

• Two coaxial wheels independently driven with angular velocities $( \dot{\phi}_1 ), ( \dot{\phi}_2 )$.

• Distance between wheels: ( $2L$), wheel radius: ( $r$ ).

Velocity calculations:

• ( $v_x = \frac{r}{2} (\dot{\phi}_1 + \dot{\phi}_2)$ )

• ( $v_y = 0$ )

• ( $\omega = \frac{r}{2L} (\dot{\phi}_1 - \dot{\phi}_2)$)

Simple Car Model (Bicycle Model)

Parameters:

• ( $L$ ): Wheelbase distance

• ( $v$ ): Forward velocity (determined by drivetrain)

• ( $\phi$ ): Steering angle

Velocity calculations

• ( $\dot{x} = v \cos \theta$ )

• ( $\dot{x} = v \cos \theta$ )

• ( $\dot{x} = v \cos \theta$ )

Holonomic vs. Nonholonomic Systems

1. Holonomic

• Constraints integrable into positional form; all degrees of freedom controllable.

• ( $\delta_m = \text{DOF} = 3$ )

1. Nonholonomic

• Velocity constraints non-integrable; limits movement flexibility.

• ( $\delta_m < \text{DOF}$ )

Manipulator Kinematics & D-H Parameters

Link parameters:

• Link length: ( $a_i$)

• Link twist angle: ( $\alpha_i$ )

• Link offset: ( $d_i$ )

• Joint angle: ( $\theta_i$ )

Applications of D-H Parameters

1. Forward Kinematics

• Computes end-effector position from joint parameters.

1. Inverse Kinematics

• Calculates joint parameters to achieve a desired end-effector pose.

1. Robot Control

• Establishes a common reference for link movements.

1. Trajectory Generation

• Determines smooth paths by defining joint angle sequences