Geometric Pseudospectral Method on SE(3) for Rigid-Body Dynamics with Application to Aircraft

Table 3

The top eight orders reduced truncated Magnus series expansion.

Order	𝑞	𝑝	Reduced Magnus series expansion
2	1	1	𝑢^[2](𝑡)=𝐼1(𝑡)
4	2	3	𝑢^[4](𝑡)=𝐼1(𝑡)+12𝐼2(𝑡)
6	3	5	𝑢^[6](𝑡)=𝐼1(𝑡)+12𝐼2(𝑡)+14𝐼3(𝑡)+1𝐼12^4(𝑡)+1𝐼24^5(𝑡)+1𝐼24^6(𝑡)+18𝐼8(𝑡)
8	4	7	𝑢^[8](𝑡)=𝐼1(𝑡)+12𝐼2(𝑡)+14𝐼3(𝑡)+1𝐼12^4(𝑡)+1𝐼24^5(𝑡)+1𝐼24^6(𝑡)+18𝐼8(𝑡)+1𝐼16^10(𝑡)+1𝐼48^11(𝑡)+1𝐼48^13(𝑡)+1𝐼48^15(𝑡)+1𝐼48^19(𝑡)+1𝐼144^20(𝑡)−1𝐼720^23(𝑡)+1𝐼24^7(𝑡)+1𝐼48^15(𝑡)+1𝐼144^16(𝑡)+1𝐼48^17(𝑡)