Inspired by the generalized motor program (GMP) theory, this study presents a symbolic motion structure representation (SMSR) algorithm that identifies a basic spatial-temporal structure of a human motion. The algorithm resolves each joint angle-time trajectory of a multi-joint motion into a sequence of elemental motion segments and labels each motion segment with a symbol representing its shape ("U": monotonically increasing; "D": monotonically decreasing; "S": stationary). By concatenating symbols according to their order in time, the spatial-temporal structure of a joint angle-time trajectory is represented as a symbolic string. The structure of a multi-joint motion is then represented as a set of symbolic strings. A sample motion, whose structure is identified by the SMSR algorithm, can be generalized to produce an infinite number of similar motion variants. To generate a variant of a sample motion, segment boundary points of the sample motion are first relocated to new locations in the angle-time space, and then individual motion segments of the original joint angle trajectories are shifted and proportionally rescaled to fit the new segment boundary points. This motion generalization method provides a basis for developing GMP-based motion simulation models, and exploring ideas and hypotheses related to the GMP theory through simulation. As an application of the motion generalization method, a motion modification (MoM) algorithm is presented, which adapts existing reach motions for new target locations. Some examples generated by the MoM algorithm are illustrated.