It is desired to move a single joint from position ${}_0$ to ${}_f,$ starting and ending at rest, in time $t_f.$ The values of ${}_0$ and ${}_f$ are given, but we wish to calculate $t_f$ so that the following constraints hold: $|{}^{}(t)|<msubsup><mrow><mi></mi></mrow><mrow><mtext>m</mtext><mtext>a</mtext><mtext>x</mtext></mrow><mrow><mi></mi></mrow></msubsup>$ and $|vec()(t)|<msubsup><mrow><mi></mi></mrow><mrow><mtext>m</mtext><mtext>a</mtext><mtext>x</mtext></mrow><mrow><mi></mi></mrow></msubsup>$ for all $t,$ where ${}_{\text{m}\text{a}\text{x}}^{}$ are ${}_{\text{m}\text{a}\text{x}}^{}$ are given positive constants.

a$)$ Using a single cubic polynomial, find eqfifations for the cubic's coefficients $c_i$ in terms of ${}_0,{}_f,$ and $t_f.$

b$)$ Using the velocity constraint, derive a condition on $t_f$ in terms of ${}_0,{}_f,$ and ${}_{\text{m}\text{a}\text{x}}^{}-$

c$)$ Using the acceleration constraint, derive a condition on $t_f$ in terms of ${}_0,{}_f,$ and ${}_{\text{m}\text{a}\text{x}}^{}$