# A System Overview¶

Figure 1: Two wheeled differential drive robots with symbolic dimensions

You will model two-wheeled robots similar to the PCBbot shown in Fig. 1. The robot has two wheels of diameter d = 40 mm, separated by a distance w ≈ 110 mm. Each wheel is directly driven from a continuous rotation servo. They drag a tail for stability, that contact the ground at a distance l ≈ 100 mm behind their back edge. The position of each of these robots in the environment is defined relative to their centerpoints O. Keep all dimensions symbolic as defined here for the following sections.

We consider two laser range sensors and an IMU for extrinsic position sensing. The output of these sensors will be a function of the positional state of the robot relative to its environment.

## Actuation Model¶

Each wheel is powered independently by a continuous rotation servo —part number FS90R— with the angular velocity of the wheel controlled by a PWM signal from the microcontroller. The control input to the robot will be in (-1.0 to 1.0) for a total of 2 input variables and then it turned to PWM values when controlling hardware. This allows the robot to drive forwards or backwards at variable speed, or turn with any turning radius.

## Sensor Model¶

Consider adding two laser range sensors —part number GYVL53L0X— and an inertial measure- ment unit (IMU) —part ICM20948 IMU— onto your robot. The output of these sensors will be a function of the state of the robot within its environment.

The laser range sensors are mounted on the robot such that they measure

- the distance to a wall in a straight line in front of the robot
- the distance to a wall in a straight line to the right of the robot.

The IMU will return

- a measurement of the in-plane rotational speed from a angular rate (gyro) sensor
- the components of the measured magnetic field along each of the 2 in-plane coordinate axes, which can be used as a compass for absolute orientation relative to Earth’s magnetic field.

We will ignore the out-of-plane gyro and magnetometer axes, as well as the accelerometer on the IMU. Thus this robot will produce 5 output values.

## Environment¶

PCBbot runs within a circular bound with a radius of \(R_{env}=0.4 m\).

## Mathematical Formulation¶

The state of your robot will satisfy the Markov property, capturing the complete history of actuator inputs to the robot hardware, allowing for computation of the dynamics update as well as all of the sensor measurements. Define this state, and then write out the analytic mathematical models for the system dynamics and measurement processes, starting with an ideal theoretical model based on fundamental principles.

Be sure to clearly define and describe all variables and equations, and produce illustrative diagrams as necessary.

In the later lab sections we will consider uncertainty of the system.

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