B.5.MPC:4.3 - Construct an interpretable mathematical question and expression

Choose mathematical objects and operations that retain the distinctions needed by the physical question. State what their important elements mean. For instance, let a vertex denote one particular gear, an edge denote a specified mesh, and a colour denote the sign of rotation when viewed from one common side. A colour number has meaning through that interpretation.

Construct the relation that connects the physical rules to the mathematical constraints. Derive a distance per motor increment from wheel circumference and transmission ratio; derive an opposite-colour constraint from the external-contact rule; derive a count invariant from a fixed stock and its permitted transfers. C.29:4.1 supplies the general correspondence-and-return method. When an operation or a compressed representation must preserve a result, use C.29.1 to compare performing the source operation and then transferring its result with transferring the inputs and then performing the receiving operation.

Work the distinction that could defeat the representation. A wheel orientation repeats after one turn; accumulated travel can continue to increase. A graph of opposite-direction contacts answers a different question from a graph in which an edge merely means “these parts are connected.” If two physical cases receive one mathematical representation but require different answers, retain their distinguishing information, restrict the cases or seek a weaker consequence.

Keep the relation available when the question changes. Identify which quantities are now given and which must be obtained, then derive the computational direction that serves that question. Under a model d = v*T and v = k*u, where u is a motor command setting, positive k, u and T permit T = d/(k*u) for a duration question or u = d/(k*T) for a command question. Obtain k and the range in which the speed model applies from the physical account. The device’s word “power” needs its interface meaning; u is not assumed to be physical power in watts. A resulting command outside the supported range returns a realizability question.

For a conditional example, let u be dimensionless and let k = 0.2 m/s. A distance d = 1 m at u = 0.5 takes T = 10 s. Changing the requested duration to T = 5 s requires u = 1. If the available range is 0 < u <= 0.8, the shortest duration under this model is 1 / (0.2 * 0.8) = 6.25 s. Returning that bound lets the requester change the deadline or seek a different realization.

An equality constrains the quantities in this model. A program assignment changes a stored value according to its execution rules. Construct the needed assignments or solver from the relation after selecting the givens and unknowns; C.29.2 supplies that formulation Method.

Make an expression that supports the next operation. Use A.6.3.RT:4.1 to express the givens and constraints under an available scheme and compare the result with its source. Put units and participant names where their absence permits the wrong operation. Keep a shared quantity recognizable across expressions, such as the same wheel revolution in the transmission ratio and circumference relation.

Notation can contribute to obtaining the result. A table can expose mutually exclusive card states; a graph can make a closed contact path traceable; a labelled equation can reveal the missing subtraction of an initial position. The subject Method supplies the construction or inference performed with those expressions. If the available scheme cannot express the needed distinction, change the scheme or obtain notation-design work before treating its expressions as adequate.