TW x 2A0 x sint kx A0 k x2 cost kx

Matlab was used to solve the necessary equations, using the built in spline functions. The algorithm implemented was to minimize:

where 0i is the end slope of the spline in question and the design variable for the optimisation problem. For each unique value of 0i, there is a unique solution to the placement of the pivot point, since the end slope thereby defined; this was implemented in Matlab by searching for the zero of S = lt — ls where li is undeformed length of the spline in question and ls is as defined in equation 1. For this optimisation problem, xpi is the design variable. Once the spline endpoints are determined, the location of the caudal peduncle (xp5,yp5) is determined from geometry, followed by the location of the tip of the caudal fin (xpa,ypa), using:

4.3 SMA strains

Once the endpoints of the rigid Delrin blocks are determined (same as the locations of the endpoints of the splines), the dynamics locations of the ends of the SMA wires can be determined from simple geometry. The actual length l of the wires at all time steps is easily determined (from a knowledge of the location and orientation of the Delrin blocks), and compared to the undeformed length of the wire L. When the tail is straight, there is a pre-strain ep in each wire at length L'. Eq. (5) therefore describes the instantaneous strain in the wire and it is independent of L since there is no way to calculate it from the tail parameters, whereas L' can be computed from the geometry of the tail:

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