Texts: A generic revolute joint of a robot should move through the sequence of four knots specified by the angular values q1 = 70, q2 = -45, q3 = 10, q4 = 100. Define an interpolating path in the form of a cubic spline q = q(s) C2, for s [0,1], such that at s1 = 0, s2 = 0.25, s3 = 0.5, and s4 = 1, we have q(0) = q1, q(0.25) = q2, q(0.5) = q3, and q(1) = q4.
The path tangent q'(s) at the initial and final knots is given by (q2 - q1)/(s2 - s1) and (q4 - q3)/(s4 - s3), respectively.
Provide the parametric expression and the associated numerical values of the coefficients for the three cubic polynomials in the spline solution.
Assuming that the path is executed at a constant speed s = 0.4/s:
- What is the total traveling time T from q1 to q4?
- What is the value of the joint velocity at the initial and final time, q(0) and q(T)?
- What is the maximum absolute value of the acceleration q(t) reached during the time interval [0, T], and which is the time instant t = ta at which this maximum occurs?