#### Euler method

For most systems, the integration must be performed numerically. A system simulator based on numerical integration can be constructed by breaking into smaller intervals and iterating classical methods for computing numerical solutions to differential equations. The Euler method is the simplest of these methods. Let denote a small time interval over which the approximation will be made. This can be considered as an internal parameter of the system simulator. In practice, this is usually much smaller than the used in the discrete-time model of Section 14.2.2. Suppose that and are given and the task is to estimate .

By performing integration over time, the state transition equation can be used to determine the state after some fixed amount of time has passed. For example, if is given and is known over the interval , then the state at time can be determined as (14.14)

The integral cannot be evaluated directly because appears in the integrand and is unknown for time .

Using the fact that (14.15)

solving for yields the classic Euler integration method (14.16)

The approximation error depends on how quickly changes over time and on the length of the interval . If the planning algorithm applies a motion primitive , it gives as the time input, and the system simulator may subdivide the time interval to maintain higher accuracy. This allows the developer of the planning algorithm to ignore numerical accuracy issues.

Steven M LaValle 2020-08-14